Basic Equations of the Mass Transport Through a Membrane Layer

Basic Equations of the Mass Transport through a Membrane Layer Basic Equations of the Mass Transport through a Membran...

Author: Endre Nagy

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Basic Equations of the Mass Transport through a Membrane Layer


Endre Nagy Research Institute of Chemical and Process Engineering University of Pannonia, Veszpre´m, Hungary

AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO '

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Elsevier 32 Jamestown Road, London NW1 7BY 225 Wyman Street, Waltham, MA 02451, USA First edition 2012 Copyright r 2012 Elsevier Inc. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-416025-5 For information on all Elsevier publications visit our website at elsevierdirect.com This book has been manufactured using Print On Demand technology. Each copy is produced to order and is limited to black ink. The online version of this book will show color figures where appropriate.

Dedication

To my beloved wife, E´va, and my children, Anita and Andra´s.

Preface

My introduction to the membrane process, namely pervaporation, was as a newly graduated chemical engineer in 1969 in the Research Institute of Chemical and Process Engineering of the Hungarian Academy of Sciences. At that time, membrane separation technology made its first steps toward becoming a real industrial technology. My research group was the first in my home country that started with investigation of the pervaporation process for separation of alcohol (with low carbon number)/water binary mixture applying cellulose-hydrate, cellulose-acetate foils. Since then, pervaporation has become one of the most important separation processes for both aqueous/organic and organic/organic mixtures. During these early experiments, we learned that the mass transfer process cannot be described correctly by Fick’s diffusion, because the pretreatment of the membrane and/or the feed concentration of the alcohol strongly affected the separation efficiency. This experience encouraged me to study the mass transport phenomena during different membrane processes. The mass transport through membranes is affected by several factors originating mainly in the membrane/component interaction, membrane structure, and the operating conditions. All differential balance equations used for description of the transport (mass, momentum, energy) can be obtained by simplification of the Navier Stokes flow models. These equations are mostly adapted for membrane transport as well as flow models for the fluid phases flowing on the feed and/or permeate sides of an active or supporting membrane layer. The existing texts primarily focus on the theoretical description of the mass transport processes through a separating and/or reacting membrane layer. It defines the membrane inlet/outlet mass transfer rates depending on the constant or variable mass transfer parameters (e.g., diffusivity, convective flow, external mass transfer resistances), membrane curvature (plane, capillary), and on chemical/biochemical reactions (analytical solution for first- and zero-order reactions, analytical approach for the second-order reaction, or Michaelis Menten kinetics). Many equations have been developed by the author and published here for the first time. Several of them should then be adapted to real situations before their application. The effect of constant and variable transport parameters is illustrated in several figures in this book. The first two chapters give a brief survey of the transport equations through membranes considering the diffusive and the convective transport applying the solution-diffusion model, Maxwell Stefan, Flory Huggins approaches (Chapter 1), and equations to predict the diffusion coefficient in the fluid and membrane phases (Chapter 2). In Chapters 3 6, we give the most important mass

xii

Preface

transfer rate and concentration distribution expressions, mostly developed by the author. Diffusion with constant and variable diffusion or solubility coefficients is modeled for plane interface (rectangular coordinate, Chapter 3), and for capillary/ tube membrane (Chapter 4), as well as mass transfer accompanied by chemical reaction in plane sheet membranes (Chapter 5) and cylindrical membranes (Chapter 6). These chapters also discuss the mass transport through asymmetric membranes due to its importance in the membrane separation processes. The flow models of the fluid phases in capillary or plane membranes are summarized in Chapter 7. Starting from the general Navier Stokes equations, we give simplified variations of these equations and some special cases considering the convective flows discussed in this section. The application of the mass transfer expression previously developed and balance equation listed is discussed in Chapters 8 12 in the cases of membrane technologies as membrane reactor (Chapter 8), membrane bioreactor (Chapter 9), nanofiltration (Chapter 10), pervaporation (Chapter 11), and membrane contactors (Chapter 12). The radial flow rate is expressed under different operation conditions in enzyme membrane reactors (Chapter 9). Unified models have been developed and discussed which take into account the simultaneous effect of the membrane and the polarization layers in Chapters 10 and 11, respectively. Chapter 12 gives a brief overview of mass transfer equations for the membrane contactors, applying them to absorption, extraction, or distillation. Mass transfer equations defining in explicit, closed forms are important to predict easily the effectiveness of a separation and/or chemical reaction process or for planning devices or technologies. This book intends to make easier the work of the engineers and technologists who are working, researching, and learning the membrane separation processes.

1 On Mass Transport Through a Membrane Layer

1.1

General Remarks

1.1.1

Transport of Dilute Solution

A membrane may be defined as a permselective barrier between two homogeneous phases. Two main potential differences are important in membrane processes, namely the chemical potential difference (Δμ) and the electrical potential difference (ΔF) (the electrochemical potential is the sum of the chemical potential and the electrical potential). Other possible forces such as magnetic fields, centrifugal fields, and gravity will not be considered here. In passive transport, components or particles are transferred from high potential to low potential. In the case of multicomponent mixtures, fluxes often cannot be described by simple phenomenological equations because the driving forces and fluxes are coupled. In practice, that means that the individual components do not permeate independently from each other. For example, a pressure difference across the membrane not only results in a solvent flux, but also leads to mass flux and the development of a solute concentration gradient. On the other hand, a concentration gradient not only results in diffusive mass transfer, but also leads to a building up of hydrostatic pressure. Potential difference arises as a result of differences in pressure, concentration, temperature, or electrical potential. Only charged molecules or ions are affected by the electrical field. The chemical potential in the presence of electrical field can be given as μi 5 μ0i 1RT ln ai 1Vi p1Fzi E

ð1:1Þ

where R is the gas constant (0.082 atm m /kmol K); T is the temperature (K); μ is the chemical potential (atm m3/kmol); ai is the activity of component i; Vi is the molar volume of component i (m3/kmol); p is the pressure (atm); F is the Faraday constant (96.5 kC/mol); and E is the electrical potential (V). The electrical field forces on the charged species only. Most of the transport processes take place because of a difference in chemical potential Δμ. Under isothermal conditions (constant T), pressure and concentration contribute to the chemical potential of a component according to 3

μi 5 μ0i 1 RT ln ai 1Vi p Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00001-6 © 2012 Elsevier Inc. All rights reserved.

ð1:2Þ

2


The concentration or vapor composition is given in terms of activities ai in order to express nonideality: ai 5 γ i x i

ð1:3Þ

where xi is the mole fraction of component i and γ i is the activity coefficient of component i (kmol/kmol). The chemical potential change dμi 5 RT d lnðγ i xi Þ 1 Vi dp

ð1:4Þ

Taking into account that the driving force is the potential gradient, and there is no coupling of flows, the mass transfer rate, Ji (Flux, kg/m2s, kmol/m2s, or in cgs unit gmol/cm2 s) perpendicular to the membrane interface (y is the distance of diffusion in the membrane in meters) can be given as (Wijmans and Baker, 1995; Baker, 2004): Ji 52 Li

dμi dy

ð1:5Þ

where dμi /dy is the chemical potential gradient of component i and Li is a coefficient of proportionality linking this chemical potential driving force to flux. Driving forces such as gradients in concentration, pressure, temperature, and electrical potential can be expressed as chemical potential gradients, and their effect on flux is expressed by Eq. (1.5). If the pressure gradient inside the membrane is negligible (e.g., mass transport through dense membrane), one can get Ji 5 2RTLi

d ln ai RTLi dai 2 dy ai dy

ð1:6Þ

For ideal solutions, the activity coefficient γ i-1, and the activity ai becomes equal to the mole fraction xi. Introducing the more practical concentration, φi (kg/m3) is defined as φi 5 Mi ρxi

ð1:7Þ

where Mi denotes the molecular weight of i (kg/kmol) and ρ is the molar density of the membrane (kmol/m3). From the value of φi defined by Eq. (1.7), the concentration in weight fraction ( 5 φi/ρ) or in volume fraction [5 φiVi/Mi, where Vi is molar volume (m3/kmol) and Mi is molar weight (kg/kmol)] can easily be gotten. Thus, from Eq. (1.6), the Fick’s law in a stationary fluid (i.e., there has been no net movement of the bulk fluid or convection flow of the entire phase) can be given as Ji 5 2

RTLi dφi dφ 2Di i φi dy dy

ð1:8Þ

On Mass Transport Through a Membrane Layer

3

with Di 5

RTLi φi

ð1:9aÞ

where Di is called the diffusion coefficient (m2/s) and is a measure of the mobility of the component i. For single-component permeation where the activity coefficient is not constant, the mass transfer rate can be derived from Eqs (1.5) and (1.6) as Ji 5 2 Li RT

@ ln ai @φi dφ 2Di ðφi Þ i @φi @y dy

ð1:9bÞ

That means that the diffusion coefficient can be expressed as Di ðφi Þ 5 2Li RT

@ ln ai @φi

ð1:10Þ

The function of ai 5 f(φi) can be given by applying the FloryHuggins thermodynamics, as will be shown later. It should also be expressed as the link between the thermodynamic diffusion coefficient DTi and Di. According to Lonsdale (1982), the thermodynamic diffusion coefficient can be expressed as Ji 5 2

DTi dμi φ RT i dy

ð1:11Þ

Now, taking into account Eqs (1.5), (1.10), and (1.11), one can get (Meuleman et al., 1999): Di ðφi Þ 5 DTi φi

@ ln ai @ ln ai DTi @φi @ ln φi

. DTi 5 Di

@ ln φi @ ln ai

ð1:12Þ

During separation of binary mixtures, often coupled transport may occur, which means that the flux of a component of a mixture may change not only by the presence of the other component, but also by its movement. Both the solubility and the diffusivity of component i can depend on the concentration of both components. According to Eq. (1.5), the following equation system can be given for a binary system with components i and j: Ji 5 2Lii

dμj dμi 2 Lij dy dy

ð1:13Þ

Jj 5 2Lji

dμj dμi 2 Ljj dy dy

ð1:14Þ

4


The first term on the right-hand side of Eq. (1.13) describes the flux of component i due to its own gradient, and the second term of this equation describes the flux of component i due to the gradient of component j. This second term represents the coupling effect. Taking into account Eqs (1.9a), (1.9b), and (1.10), that the permeation rate of component i can be given as (Mulder, 1981; Meuleman et al., 1999): ! ! @ ln ai @φi @ ln ai @φj @ ln aj @φi @ ln aj @φj 1 1 2 Lij RT Ji 5 2Lii RT @φi @y @φj @y @φi @y @φj @y ð1:15Þ Rewriting Eq. (1.13), one can get ! ) @ ln ai @ ln aj @φi @ ln ai @ ln aj @φj 1 Lii Lii 1Lij 1Lij @φi @φi @y @φj @φj @y

( Ji 5 2RT

ð1:16Þ thus, dφj dφi 1 Aij Ji 5 2 Aii dy dy

ð1:17Þ

dφj dφ Jj 5 2 Aji i 1 Ajj dy dy

ð1:18Þ

@ ln ai @ ln aj Aii 5 RT Lii 1 Lij @φi @φi

ð1:19Þ

and

with

@ ln ai @ ln aj Aij 5 RT Lii 1 Lij @φj @φj

1.1.2

! ð1:20Þ

Transfer Rate of Concentrated Feed Solution

For more concentrated feed solutions, bulk convection terms must be included. When describing the transport process, it is necessary to specify the frame of reference of the transport process (Bird et al., 1960). In the case of fixed frame of reference transport, the membrane can be used as the frame of reference because


5

the membrane is stationary at steady state. For this static reference frame, the transport process is a sum of diffusional and convective flux. Generally, the following simplified equations can be used for binary (components i and j) mixture, assuming no coupling of the transport process: Ji 5 2Di

dφi φ 1 i ðJi 1Jj Þ dy ρ

ð1:21aÞ

Jj 5 2Dj

dφj φj 1 ðJi 1Jj Þ dy ρ

ð1:21bÞ

and

as well as from Eq. (1.21a), Ji can be expressed as 1 dφi φi Jj Ji 5 2Di 1 dy ρ Ji 1 2 φj =ρ

ð1:21cÞ

where φi(Ji 1 Jj)/ρ is the bulk flow term (convective mass transfer); Di and Dj are the effective binary diffusion coefficients of components; and J is the total mass flux (diffusive 1 convective flux) (kmol/m2 s) of a component with respect to a fixed frame of reference. (Note that here the diffusive flux and the total mass flux is not distinguished in this chapter as it is generally made in the literature where the total flux is mostly denoted by N). In the case of binary mixture the mass flux of the membrane material also has to be given. The polymer mass flux will be zero, while the diffusional mass flux of membrane is equal in value and opposite in the direction to the flow with respect to its bulk mass flux (not given here) (Kamaruddin and Koros, 1997). Note that the fluxes in Eq. (1.21a) or (1.21b) are the sum of the diffusive mass transfer rate and the convective one, accordingly the values of J in Eqs (1.8), (1.21a), and (1.21b) should be different in presence of convective velocity. The convective velocity can be expressed from Eq. (1.21a) or (1.21b) as (Bird et al., 1960; Geankoplis, 2003): υ5

Ji 1 Jj ρ

ð1:22Þ

The υ convective velocity in Eq. (1.22) is a product of diffusion of components in the membrane matrix. Accordingly, the total flux can be given as Ji 5 2Di

dφi 1 φi υ dy

ð1:23Þ

The convective velocity can exist also due to the pressure difference on the two sides of a porous membrane which has to be involved in the υ value. Diffusive mass

6


transport takes place in a dense polymer layer with no visible pores. The transport can be described well by the solution-diffusion model that will be discussed briefly in Section 1.2. This is the case for reverse osmosis, pervaporation, and polymeric gas separation (Baker, 2004). The spaces between the polymer chains in these membranes are less about 0.5 nm in diameter, and these spaces are caused by the thermal motion of the polymer chains. Ultrafiltration, microfiltration, and microporous Knudsen-flow gas separation membranes are all clearly microporous, and transport occurs by convective flow through the pores (see Section 1.3). The pore size can be stated to be larger than 1 nm for these separation processes. The mass transport equations of the diffusive plus convective mass transport will be separately discussed in Chapter 3 and 5, respectively. The extended NernstPlank equation, for the membrane in electrical field, is as (Geraldes and Brites, 2008): Ji 2 Di;p

dφ zi φ Di;p dψ F 1 Ki;c Jv 2 i RT dy dy

ð1:24Þ

where Ji is solute flux of component i (kg/m2 s); Jv is permeate volume flux (m3/ m2 s); Di,p is hindered diffusion coefficient (m2/s); Ki,c is hindrance coefficient for convection; zi is charge number; F is Faraday constant (9.64867 3 104 C/eq); R is ideal gas constant (8.314 J/kmol); ψ is electric potential (V); and Di,N is the diffusivity of the i species in water at infinite dilution (m2/s).

1.2

Transport Through Dense Membrane: Solution-Diffusion Theory

We can broadly classify transport in solids into two types of diffusion: diffusion that can be considered to follow Fick’s law and does not depend primarily on the actual structure of the solid membrane (e.g., dense polymer membrane); and diffusion in porous solids (e.g., zeolite membrane), where the actual structure and channels are important. Both types will be briefly considered. The separation process is illustrated in Figure 1.1. The solution-diffusion is the generally accepted mechanism of mass transport through nonporous membranes (Lonsdale, 1982). It is held that the component permeation through a homogeneous membrane consists of five fundamental processes: (1) the solute molecules must first be transported or diffused through the liquid (or gas; in this latter case the mass transfer resistance of the boundary layer is negligible, consequently this step can also be omitted) film (Figure 1.1B) of the feed phase on the feed side of the solid membrane; (2) the solution of the solute molecule in the upstream surface of the membrane matrix; (3) diffusion of the dissolved species across the membrane matrix; (4) desorption of the solute molecules in the downstream side (permeate side) of the membrane; and (5) diffusion through the boundary layer of the permeate phase. The first and the fifth steps can be omitted when there are no mass transfer resistances between the


(A)

7

(B)

δ

φ*

Co C* Ji

Ji φ*δ

C*δ

o

Cδ

Figure 1.1 (A) Transport through dense membrane. (B) Concentration profile and the important notations.

continuous and membrane phases. This can often be the case, especially for gas permeation or liquid permeation with high flow rates in the fluid phases on the two sides of membrane. The thermodynamic equilibrium between the continuous phase and the membrane interface assumes that the chemical potentials are equal to each other, therefore: μfi 5 μmi

ð1:25Þ

where subscripts f and m denote fluid and membrane, respectively. According to Eq. (1.25) for liquid phase in contact with membrane phase (Wijmans and Baker, 1995; Baker, 2004; Wijmans, 2004): sat RTd lnðγ if xif Þ 1 Vif ðp 2 psat i Þ 5 RTd lnðγ im xim Þ 1 Vim ðp 2 pi Þ

ð1:26Þ

Assuming that the molar volume difference is negligible, it can be understood from Eq. (1.26) as γ if xif 5 γ im xim

ð1:27Þ

Taking into account Eq. (1.7) and applying it for the membrane phase, we can obtain φi 5

γ if ρm ci H i ci γ im ρf

ð1:28Þ

8


with Hi 5

γ if ρm γ im ρf

ð1:29Þ

where ρ is the density (kg/m3). The value of H solubility coefficient depends on the separation process to be applied according the chemical potential. Baker (2004) and (in part) Wijmans (2004) define this solubility coefficient for different membrane processes as dialysis, reverse osmosis, pervaporation, gas separation, and hyperfiltration. Accordingly, let us define the partition coefficient as φi =ci 5 Hi

ð1:30Þ

where c and φ represent the concentration of species in the membrane surface and the feed, respectively, and H is the equilibrium distribution coefficient (similar to Henry’s law coefficient for gas and liquid; note that the inverse of H is often defined in the literature as equilibrium coefficient), which is a characteristic parameter dependent upon interaction of the species with the membrane. Taking into account the H partition coefficient, and integrating Fick’s first law, given by Eq. (1.8) with boundary conditions illustrated in Figure 1.1B, one can obtain Ji 5 Di

φi 2 φδi Di Hi o Pi ðci 2 coδi Þ ðcoi 2 coδi Þ δ δ δ

ð1:31Þ

where Pi is the permeability coefficient (m2/s) and ðcoi 2 coδi Þ is the driving force. The Di and Hi and thus Pi are concentration dependent for many systems. Thus, these values should be considered as averaged over the membrane thickness in Eq. (1.31).

1.3

Convective Transport Through a Porous Membrane Layer

Pressure-driven convective flow, the basis of the pore flow model, is most commonly used to describe flow in a capillary or a porous medium. Look first at the flow through cylindrical capillary tubes. Let us assume that there are cylindrical capillaries in the membrane matrix perpendicular to the interface. The laminar fluid flow through these capillaries can be described by means of the differential momentum balance equation, Newton’s law of viscosity as (Bird et al., 1960; Geankoplis, 2003): τ ry 5 2μ

dυy dr

ð1:32Þ


9

where τ ry is shear stress or force per unit area [((kg m/s)/m2 s) N/m2]; μ is the viscosity of the flowing fluid (Pa s or kg/m s), r denotes cylindrical coordinate in the capillary tube (Figure 1.2; note, assuming that the capillary pores is perpendicular to the membrane interface, the r space coordinate is parallel to the interface, while y is perpendicular to it). On the other hand, the τ ry shear stress can be expressed as τ ry 5

p 2 pδ r 2δ

ð1:33Þ

where p 2 pδ is the pressure drop for a capillary length δ (N/m2 or Pa); δ is the thickness of the membrane layer; p is the feed side, while pδ is the permeate side pressure (Pa). Substituting Eq. (1.32) into Eq. (1.33), we obtain the following differential equation for the convective velocity: dυy p 2 pδ 52 r dr 2μδ

ð1:34Þ

After integrating Eq. (1.34) using suitable boundary conditions (Bird et al., 1960), we get the well-known HagenPoiseuille equation for the average convective velocity through the porous membrane: υ5

Δprp2 8μδ

ð1:35Þ

where Δp is the pressure difference between the two sides of the membrane (Pa) and rp is the pore radius (m). Figure 1.2 Transport through a porous membrane.

10


The specific volume flow rate can also be calculated by means of Eq. (1.36), namely (Baker, 2004): υo 5

Δpεrp2 8μδ

ð1:36Þ

where υo is the flow rate related to the membrane total area (m3/m2 s) and ε is the porosity of the membrane (ε 5 Nπrp2 ; where N is the number of capillaries related to the membrane surface). For porous medium, Darcy’s law is a basic equation to describe the rate of fluid flow [Eq. (1.37)] and the mass transport through it [Eq. (1.38)]. It can be written as υ5

K 0 dp μ dy

ð1:37Þ

where υ is the convective velocity (m/s); μ is the fluid dynamic viscosity (Pa s); K0 is an empirical constant (m2); p is the pressure (Pa); and y is the space coordinate perpendicular to the porous layer’s interface. Ji 5

K 0 dp φ μ i dy

ð1:38Þ

where dp/dy is the pressure gradient existing in the porous membrane; φi is the concentration of component i in the membrane; K0 is a coefficient reflecting the nature of the membrane; and Ji is the mass transfer rate of component i (kg/m2 s). This equation defines the mass transfer rate through porous membrane as it is given in Eq. (1.8) for diffusive mass transport through dense membrane. The transition between the two models depends on the size of free-volume elements (pores). According to Baker (2004), this transition regime can exist in the pore size range of 0.51 nm (1 nm 5 1 3 1029 m). For the sake of completeness, let us give the following often-used form of the convective velocity, in its one-dimensional form, namely: υ52

Bo dp μ dy

ð1:39Þ

where the permeability Bo is characteristic of the membrane structure and has to be determined experimentally, along with the porositytortuosity factor (ε/τ); μ is the viscosity of the fluid mixture (Pa s). The permeability coefficient Bo can be calculated for some typical structures. For a cylindrical pore, the permeability is calculated from the Poiseuille flow relationship [see Eq. (1.35)]: Bo 5

rp2 8

ð1:40Þ


11

For a suspension of spheres of diameter do, the RichardsonZaki correlation gives Bo 5

d2o 2:7 ε 18

ð1:41Þ

where ε is the porosity of the suspension. The CarmanKozeny equation also should be mentioned for cases in which if the membrane consists of a packed bed of particles (Silva et al., 2008): Bo 5

d2o ε2 180 ð1 2 εÞ2

ð1:42Þ

Thus, υ52

d2p

ε3 dp 180ð1 2 εÞ η τ dy 2

ð1:43Þ

where dp is the particle size (m); ε is the porosity (); τ is the tortuosity factor (); and υ is total volumetric flux (m/s). For a corresponding bed of fibers (Krishna and Wesselingh, 1997): Bo 5

do2 ε2 80 ð1 2 εÞ2

ð1:44Þ

The viscous flow contribution is important for transport in membrane with open structures and it is relatively unimportant for transport in membranes with dense structures, such as those present in gas permeation and pervaporation membranes.

1.4

Component Transport Through a Porous membrane

The mass transport through a porous layer is extensively studied. Its type depends on the pore size comparing it with the mean path of the transported molecules. In the case of gas transport, three regions are distinguished regarding the transport (Seidel-Morgenstern, 2010): 1. Molecular and/or viscous flow, when э , dp 2. Knudsen diffusion, when ds , dp , э 3. Molecular sieving transport (configurational diffusion), dsDdp

12


where dp is the pore size; ds is the size of transported species; and э is the free path of molecule. The value of the mean free path of the gas molecule i, эi, can be calculated as (Lawson and Lloyd, 1997): kB T 3i 5 pffiffiffi 2 2πpds;i

ð1:45Þ

where kB is the Boltzman constant (kB 5 1.34 3 10223 J/K); p is the mean pressure within the pore (Pa); and T is the absolute temperature (K). Molecular diffusion and/or viscous flow are dominated by the interaction between the transported molecules when the pore diameter is considerably larger than the mean free path. The real diffusivity is, however, reduced by the ratio between porosity, ε, and tortuosity, τ. Thus, we can get the effective diffusion coefficients De, as ε De;i 5 Di ð1:46Þ τ By application of kinetic gas theory to a single, straight, and cylindrical pore, the coefficient of Knudsen diffusion can be derived as (Mason and Malinauskas, 1983): rffiffiffiffiffiffiffiffiffi dp ε 8RT ð1:47Þ DK;i 5 3 τ πMi where R is the gas constant (R 5 8.314 J/mol K); Mi is the molecular weight of species i (kg/kmol); and dp is the pore size (m). By analog to the molecular diffusion, the Knudsen diffusion flux can be calculated to Ji 5 2DK;i

dci DK;i dpi 2 dy RT dy

ð1:48Þ

where Ji the mass transfer rate (kmol/m2 s). The gradient of the partial pressure, dpi/dy, is the driving force of the transport. Note the diffusion is regarded as one dimensional. The viscous flow can exist when the pore size is larger than the mean free path. For a single capillary, the viscous flow can be calculated according to HagenPoiseuille [see Eq. (1.35)], thus: Ji 5 υ

pi pi ε dp2 dp 52 RT RT τ 32 dy

ð1:49Þ

where p is the total pressure and pi is the partial pressure. Note that the viscous flow does not contribute to the separation of different species. [But as discussed by Eqs (10.3)(10.5), in the case of liquid phase, the fluid viscosity will be increased in the nanopores, and the convective velocity of the molecules will also be hindered depending on the ratio of ds (solute molecule size) to dp.]


13

When both the molecular and the viscous flows exist, the total mass transfer rate is expressed by the sum of these two rates, as ! 1 ε dp2 dp dpi pi 1 DK;i ð1:50Þ Ji 5 2 dy RT τ 32 dy The total (p) and the partial pressure of species i (pi) can vary partly independently. This change can be given as dp 1 dpi dxi 2p 5 dy dy xi dy

ð1:51Þ

where xi is the mol fraction of species i. Assuming that the mol fraction of the diffusing component does not change during the transport (there is no source or ^ sink term), from Eq. (1.51) it can be obtained that ðci 5 ct xi ; ci 5 pi =RTÞ: dp 1 dpi ct RT dpi 5 dy xi dy pi dy

ð1:52Þ

Replacing Eq. (1.52) into Eq. (1.50), one can obtain 1 Ji 5 2 RT

! ε dp2 dpi ct RT 1 DK;i dy τ 32

ð1:53Þ

After integration of Eq. (1.53) with y 5 0 pi 5 pe,i and with y 5 δ pi 5 pt,i (the thickness of the catalyst sponge layer is neglected here; the thickness of the active layer is generally a few micrometers, while that of the distributor a few hundred micrometers), it can be obtained for the mass transfer rate: ( ) 1 ε dp2 DK;i ct 1 ðpe;i 2 pt;i Þ Ji 5 RT δ τ 32

ð1:54Þ

where subscript e and t denote the shell side and the tube side and ct is the total concentration (kmol/m3). The thickness of a porous gas distributor in a membrane reactor [e.g., in case of partial oxidation in an oxygen permeable membrane reactor (Hoang et al., 2005)] can be relatively large, a few hundred micrometers. In the case of low feed side radius, the cylindrical effect should also be taken into account. This effect can be predicted applying the following equation: 1d dJi 50 r dr r dr

ð1:55Þ

14


After integration of Eq. (1.55), applying the boundary conditions at r 5 ro then pi 5 pt,I and r 5 ro 1 δ then pi 5 pe,I, one sees ( ) 1 ε dp2 DK;i ct 1 ðpe;i 2 pt;i Þ Ji 5 RT ro lnð1 1 δ=ro Þ τ 32

ð1:56Þ

where pt and pe are total pressure on the tube and shell side (Pa). Equations (1.54) and (1.56) do not involve the molecular diffusion in the membrane pores. The molecular diffusion coefficient often is combined with the Knudsen diffusion coefficient. Thus, the total diffusion flux will be the sum of molecular and Knudsen diffusion. The overall diffusion coefficient, Di,eff, is given by Bosanquet equation (Seidel-Morgenstern, 2010, p. 41): Di;eff 5

1 ð1=Dei;j Þ 1 ð1=DK;i Þ

ð1:57Þ

where Dei;j is the effective molecular diffusion coefficient of component i (its value involves the ε/τ factor) (m2/s) and DK,i is the effective Knudsen diffusion coefficient [Eq. (1.47), m2/s]. The value of Dei;j should then be substituted into Eq. (1.54) or (1.56), replacing DK,i with it. If the pore diameters approach the range of the sizes of the molecules, the configurational diffusion regime is reached. The corresponding flux density can be described by (Thomas et al., 2001): Ji 5 2

1 ε c dpi D RT τ i dz

ð1:58Þ

with Dci

rffiffiffiffiffiffiffiffiffi 8RT 2ðEic =RTÞ 5 ρ g dp e πMi

ð1:59Þ

where Dci is the diffusion factor (m2/s) and Eic is activation energy of diffusion (kJ/ kmol). There are no well-accepted formulas to predict the Dci diffusivity (Thomas et al., 2001). In the region of very small pores, the effect of the surface diffusion might be significant. The surface diffusion flux can also be important in the mass transport through pores in the Knudsen regime due to the adsorption, consequently this flux will be added to the above-discussed molecular and Knudsen diffusion and the viscous flow (Tuchlenski et al., 1998; Thomas et al., 2001). Physically adsorbed molecules are relatively mobile and, although the surface mobility is in general several orders of magnitude lower compared to the mobility in fluid phase, surface fluxes might be relatively high due to larger molecular densities


15

on the pore surface. The most general approach for multicomponent surface diffusion was proposed by Krishna and Wesselingh (1997). Based on the physical picture of molecules moving on the surface, the generalized MaxwellStefan equations were applied to describe interactions mechanistically (Tuchlenski et al., 2001). This model is applied for bulk diffusion through a membrane layer in the next section. For adaptation of these equations, see Krishna and Wesselingh (1997) or Krishna (1990).

1.5

Application of the MaxwellStefan Equations

The generalized MaxwellStefan equations are based on the assumption that movement of species is caused by a driving force which is balanced by the friction that the moving species experience from each other and their surroundings. The generalized form of this equation applying it for multicomponent fluid mixtures is given as (Krishna and Wesselingh, 1997; Wesselingh and Krishna, 2000; Amundsen et al., 2003): 2

j5n X rμi xj ðui 2 uj Þ 5 ; RT Dij j51

i 5 1; 2; . . . ; n

ð1:60Þ

j 6¼ i

where μi is the chemical potential of species i (kJ/kmol); R is the gas constant (8.314 J/mol K); T is the absolute temperature (K); xj is the mol fraction of species j; ui, uj are molar-averaged mixture velocity of species i or j (m/s); Dij is the MaxwellStefan diffusivity (m2/s) and represents inverse friction factor between molecules i and j. Accordingly the force balance on the species 1. e.g. in a ternary fluid mixture takes the form for one-dimensional diffusion as 2

dμ1 x2 ðu1 2 u2 Þ x3 ðu1 2 u3 Þ 5 1 RTdy D12 D13

ð1:61Þ

where y is the space variable (m); ternary system is a membrane with two-component separation, thus this system can be originated by the above equation. The molar flux Ji, with respect to a laboratory-fixed coordinate reference frame, can be given by the following equation (Krishna and Wesselingh, 1997): J i 5 c i ui c t x i ui

ð1:62Þ

where ct is the total molar concentration of the fluid mixture (kmol/m3); ci is the molar concentration of species i (kmol/m3), n is the number of diffusing component. Because motion of molecules must be balanced, i.e., movement of one

16


molecule must be balanced by the motion of another molecule, thus it can be given as i5n X

i5n X

Ji 5 0;

i51

xi 5 1

ð1:63Þ

i51

Thus, any one of the differential equations can be replaced by one of the two conditions above. Taking into account Eq. (1.62), we can obtain for a multicomponent system as 2

j5n X rμi xj Ji 2 xi Jj 5 RT ct Dij j51

ð1:64Þ

j 6¼ i

Accordingly, for component 1 in a ternary system can be given as (Amundsen et al., 2003): dx1 x2 x3 x1 x1 52 J1 1 1 J2 1 J3 ct dy D12 D13 D12 D13

ð1:65Þ

The thermodynamic potential gradient rμi can be expressed in terms of the mole fraction gradients (Krishna and Wesselingh, 1997) or in terms of the gradients of the surface occupancies (Kapteijn et al., 1995; Krishna and Baur, 2003) by the matrix of thermodynamic factors. For a fluid mixture, it will be j5 n 21 X xi rμi 5 Γij rxi ; RT j51

Γij 5 δij 1 xi

d ln γ i dxj

ð1:66Þ

6 j) and γ is the activity where δij is the Kronecker delta (δij 5 1 for i 5 j, δij 5 0 for i ¼ coefficient of species i. Combining Eqs (1.64) and (1.66), it can be obtained (Krishna and Wesselingh, 1997): ðJÞ 5 2ct ½B 21 ½ΓðrxÞ

ð1:67Þ

with Bii 5

k5 n 21 X xi xk 1 ; Din D ik k51

Bijði 6¼ jÞ 2 xi

1 1 2 Dij Din

k 6¼ i

where (J) represents the column vector (n 21) diffusion fluxes (kmol/m2 s); the element of the matrix [B] can be derived by the above equation. The application


17

of the above equation for zeolitic membrane is discussed in Section 5.2. It is common to define a matrix of Fick diffusivities [D] by using (n 21) 3 (n 21) matrix notation: ½D 5 ½B 21 ½Γ

ð1:68Þ

Applying the above basic equation given for a fluid system, the generalized MaxwellStefan equation can be reformulated for membrane separation systems. Application of the above equation will be discussed briefly in the next sections.

1.5.1

The MaxwellStefan Approach to Mass Transfer in a Polymeric, Dense Membrane

The various species transport through a polymer matrix or inorganic ceramic layer involves different transport mechanisms, different diffusion mechanism as bulk gas diffusion, bulk liquid diffusion, Knudsen diffusion inside pores, interface diffusion, and so forth. The Fick’s law of diffusion is a basic design procedure and postulates a linear dependence of flux on its composition gradient as it is given by Eq. (1.8). The deviation from the linearity is taken into account by variable diffusion coefficient. After the literature data, the MaxwellStefan formulation provides the most general approach for describing mass transport for nonideal systems, thus, for the mass transport during membrane separation (Krishna and Wesselingh, 1997; Wesselingh and Krishna, 2000). This approach can be applied for mass transport through both the dense membrane [Eqs (1.17)(1.20)] and inorganic membrane [Eqs (1.13)(1.16); Krishna and Baur, 2003]. The following equation can be obtained for multicomponent mass transfer through solid membrane layer, applying the MaxwellStefan approach regarding the membrane as a motionless (n 11)th component and (Krishna and Wesselingh, 1997): 2

n X φj Ji 2 φi Jj φi dμi φ Ji 5 1 m φt RT dy φ D φ ij t t Dim j51

ð1:69Þ

j 6¼ i

with j 5X n11

φj 5 φt ð ρÞ

j51

where φi is the concentration of ith species in the membrane (kg/m3 or kmol/m3); Ji is the mass transfer rate of species i (kg/m2 s or kmol/m2 s); Dij is the MaxwellStefan counterexchange diffusivity of components i, j, and it represents the inverse friction factor between molecules (m2/s); Dim is the diffusivity of i in the membrane (m2/s).

18


For diffusion of binary mixture in a membrane (μi 5 μoi 1 RT ln xi ; φ1 1 φ2 1 φm 5 ρ): 2ρ

dμi φ J1 2 φ1 J2 φm J1 5 2 2 dy D12 D1m

ð1:70Þ

The driving force for diffusion is the chemical potential gradient (dμi/dy) on the left-hand side of Eq. (1.70). Friction is the result of interaction between transported (adsorbed) molecules (first term on the right-hand side) and interaction between a solute molecule and the membrane matrix (second term on the right-hand side). For details, see Krishna and Wesselingh (1997) or Wesselingh and Krishna (2000). For binary mixture (i 5 1, j 5 2): 2ρ

dμ1 φ J1 2 φ 1 J2 φ J1 5 2 1 m dy D12 D1m

ð1:71Þ

2ρ

dφ2 φ J2 2 φ2 J1 φ J2 5 1 1 m dy D21 D2m

ð1:72Þ

and

The mass transfer rate of the individual components can be expressed by means of Eqs (1.71) and (1.72). Assuming that D12 5 D21 and dμi 5 d ln φi, one can obtain (Heintz and Stephan, 1994a,b; Iza´k et al., 2003; Nagy, 2004): dφ dφ J1 5 2 A11 1 1 A12 2 dy dy

ð1:73Þ

with

D12 1 φ1 D2m A11 5 D1m ρ D12m 1 φ1 D2m 1 φ2 D1m A12 5 D1m ρ

φ1 D2m D12 1 φ1 D2m 1 φ2 D1m

ð1:74Þ

ð1:75Þ

as well as

dφ1 dφ2 1 A22 J2 5 2 A21 dy dy

ð1:76Þ

with A21 5 D2m ρ

φ2 D1m D12 1 φ1 D2m 1 φ2 D1m

ð1:77Þ


and

D12 1 φ2 D1m A22 5 D2m ρ D12 1 φ1 D2m 1 φ2 D1m

19

ð1:78Þ

The separation of ethanolwater binary mixture by pervaporation, using poly (vinyl)alcohol membrane is illustrated briefly in Section 3.2.5.2. How can we express the link between the MaxwellStefan diffusion coefficient and the Fickian diffusion coefficient? This can be made by the so-called Darken correction factor, namely: D 5 DΓ

ð1:79Þ

The Γ thermodynamic factor can be given of the difference of chemical potential change in real and ideal system. In a real system, the mass transfer rate is (D 5 RTLi/φ): Ji 5 2

RTLi d ln ai dφi dφ Di i φi d ln φi dy dy

ð1:80Þ

Accordingly, Γ5

d ln ai φi dai d ln φi ai dy

ð1:81Þ

Note, if we assume that ai 5 γ ixi [Eq. (1.3)], it easily can be understood that (Wesselingh and Krishna, 2000): Γ 5 1 1 xi

dðln xi Þ dxi

ð1:82Þ

Note that Eq. (1.80) defines the mass transfer rate for constant H partition (solubility) coefficient. The solubility coefficient is determined by the fact that the chemical potential of the continuous fluid and the membrane phases have to be equal in equilibrium [Eq. (1.30)]. If we want to estimate the solubility coefficient, the value of chemical potential as a function of the concentration both in the fluid and membrane phases have to be predicted. For the latter, several methods can be used. The most-often recommended approach for it is the FloryHuggins theory briefly shown in Section 1.6.

1.5.2

The MaxwellStefan Approach to Mass Transfer in a Ceramic (Zeolite) Membrane

Aluminosilicate (e.g., zeolites, silicalite) and other inorganic membrane layers have high chemical and thermal stability. They operate as molecular sieves and

20


have very promising industrial potential as material for separation (Gardner et al., 2002). They separate molecules based on differences in adsorptive and diffusive properties of the membrane. Mostly the Langmuir adsorption isotherms and the MaxwellStefan diffusion model are applied to describe the transport through an aluminosilicate membrane (Kapteijn et al., 1995; van de Graaf et al., 1999; Krishna and Baur, 2003). The diffusion in ceramic (mostly zeolite) membrane takes place in narrow channels both in the bulk channel phase and on the channel surface. The general form of the generalized MaxwellStefan equation applied to surface diffusion is given as (Wesselingh and Krishna, 1990; Krishna et al., 1999; van de Graaf et al., 1999): 2

n X θi dμi θ j Ji 2 θ i Jj Ji 5 1 ; RT dy ρθ D ρθ max ij max Di j51

i 5 1; 2; . . . ; n

ð1:83Þ

j 6¼ i

where θi is the molecular loading within the zeolite, expressed in molecules per unit cell; θmax is the maximum molecular loading; θmax 5 θA 1 θB for twocomponent diffusion; ρ represents here the number of unit cells per m3 of membrane; D1 is the diffusivity of species i in the membrane (m2/s) (note that D1 D1m Þ: The often-used form of the MaxwellStefan equation is given as (Kapteijn et al., 2000): 2ρ

n θi dμi X qj J i 2 qi J j Ji 5 1 ; RT dy q q D q i;sat Di j 5 1 i;sat j;sat ij

i 5 1; 2; . . . ; n

ð1:84Þ

j 6¼ i

where θi is the fractional occupancy defined as θi 5 qi/qi,sat; qi is adsorbed concentration in zeolite (cell or kmol/kg), qi,sat is saturation loading (cell or kmol/kg), ρ is the density of membrane, number of unit cell per m3 (kg/m3). The gradient of the thermodynamic potential can be expressed in terms of thermodynamic factors as it was shown at beginning of this section (Krishna, 1993; Kapteijn et al., 1995; Martinek et al., 2006) as n X θi rμi 5 Γij rθj ; RT j51

Γij 5

θi @pi ; pi @θj

i; j 5 1; 2; . . . ; n

ð1:85Þ

where pi is the partial pressure of species i (Pa). The combination of Eqs (1.84) and (1.85) gives the surface flux of a component through the membrane in multicomponent system. Thus, the mass transfer rates can be given in matrix form as ðJÞ 5 2ρ½qsat ½B 21 ½ΓðrθÞ

ð1:86Þ


21

The value of [B]21 matrix can be obtained by inversion of the [B] matrix as j5n X 1 θj 1 ; D Di ij j51

Bii 5

Bij 5 2

θi ; i 6¼ j Dij

j 6¼ i

The surface occupancy, θi, is related to the partial pressure by the adsorption isotherm. The choice of the adsorption model determines the mathematical form of the thermodynamic factor obtaining according to sense from Eq. (1.79). Applying the extended single-site Langmuir equation, the fractional occupation is as θi 5

11

bp Pin i

j51

bj pj

;

i; j 5 1; 2; . . . ; n

ð1:87Þ

Note that the dual-site (Krishna et al., 1999) or triple-site (Krishna et al., 2007) Langmuir models take into account the preferential adsorption or location of molecules at certain sites in the membrane structure. The sorption characteristics of the different sites can differ from each other (Krishna et al., 1999). The Langmuir isotherm can be given for single-component adsorption as θ1 5

b1 p 1 1 b1 p

ð1:88Þ

Applying Eq. (1.85), one can get Γ5

1 1 2 θ1

ð1:89Þ

Thus, the flux through the membrane for single-component adsorption is given by (Kapteijn et al., 1995; van de Graaf et al., 1999): Ji 5 qi;sat ρ

Di dθi 1 2 θi dy

ð1:90Þ

For a single-site, extended Langmuir isotherm, the thermodynamic correction factors for a binary mixture are (i 5 1; j 5 2): Γ11 5

1 2 θ2 ; 1 2 θ1 2 θ2

Γ11 5

θ1 1 2 θ1 2 θ2

Γ21 5

θ2 ; 1 2 θ1 2 θ2

Γ22 5

1 2 θ1 1 2 θ1 2 θ2

ð1:91Þ

22


The Γ factor gives much more complex equation, e.g., for dual-site adsorption isotherm (Krishna et al., 1999). Combining Eqs (1.84)(1.90) and assuming D12 5 D21 and solving the fluxes yield (Kapteijn et al., 2000; Martinek et al., 2006): J1 5 q1;sat ρD1

Γ11 1 θ1 ðD2 =D12 ÞðΓ11 1 Γ21 Þ ðdθ1 =dyÞ 1 Γ12 1 θ1 ðD2 =D12 ÞðΓ12 1 Γ22 Þ ðdθ2 =dyÞ Ξ

ð1:92Þ

J2 5 q2;sat ρD2

Γ22 1 θ2 ðD1 =D12 ÞðΓ22 1 Γ12 Þ ðdθ2 =dyÞ 1 Γ21 1 θ2 ðD1 =D12 ÞðΓ21 1 Γ11 Þ ðdθ1 =dyÞ Ξ

ð1:93Þ

where Ξ 5 1 1 θ2

D1 D2 1 θ1 D12 D12

Substituting Γ values from Eq. (1.47) into Eq. (1.48), relatively simple expression of the fluxes, J1 and J2, will be obtained (Kapteijn et al., 1995; van de Graaf et al., 1999; Nagy, 2004), taking also into account the role of the porosity of the support layer, ε, as: dθ1 dθ2 1 A12 ð1:94Þ J1 5 2 A11 dy dy with D2 Q; A11 5 q1;sat ερD1 1 2 θ2 1 θ1 D12 and

D2 A12 5 q1;sat ερD1 θ1 1 θ1 Q D12

dθ1 dθ2 1 A21 J2 5 2 A22 dy dy

ð1:95Þ

with D1 Q; A21 5 q2;sat ερD2 θ2 1 θ2 D12

D1 A22 5 q2;sat ερD2 1 2 θ1 1 θ2 Q D12

where Q5

1 ð1 2 θ1 2 θ2 Þ 1 1 θ1 ðD2 =D12 Þ 1 θ2 ðD1 =D12 Þ

ð1:96Þ


23

An important simplification is that there is no interaction between the components, that is, the terms with the D12 vanish (numerically D12 -NÞ: The flux expression for component 1 becomes J1 2 ρεD1 ½Γ11 rθ1 1 Γ12 θ1 5 2 ρD1

θ1 @p1 θ2 @p1 rθ1 1 rθ2 p1 @θ1 p1 @θ2

ð1:97Þ

For application of the above equation in order to predict the binary gas separation by zeolite membrane, see Section 3.2.5.1. A single-component adsorption also can be described by dual-site Langmuir model (Zhu et al., 1998; Krishna et al., 1999; Kapteijn et al., 2000) as qi 5

θiA biA pi θiB biB pi biA pi biB pi 1 qiA;sat 1 qiB;sat 1 1 biA pi 1 1 biB pi 1 1 biA pi 1 1 biB;sat pi

qi;sat 5 qiA;sat 1 qiB;sat

ð1:98Þ ð1:99Þ

where the different sites A and B may be represented by channel interiors (e.g., site A) and the intersections (site B). For a binary mixture (components 1 and 2), the dual-site Langmuir isotherm can be given as (Krishna et al., 1999): θ1 5

ðθ1A b1A 1 θ1B b1B Þp1 1ðθ1A 1 θ1B Þb1A b1B p21 1 1ðb1A 1 b1B Þp1 1 b2A b2B p21 1ðb2A 1 b2B Þp2 1 b2A b2B p22

ð1:100Þ

The Γ thermodynamic factor can be obtained by means of Eq. (1.44). For it, the pi 5 f(θi) function should be expressed, then it is relatively easy to get @p/@θ differential quotient. For details regarding the Γ values, see Krishna et al. (1999). According to paper of Krishna and Baur (2003), the thermodynamic correction factor for the dual-site Langmuir model is Γ5

Θ1A Θ1

1 1 2ðΘ1A =Θ1A;sat Þ 1

Θ1B Θ1

1 2ðΘ1B =Θ1B;sat Þ

ð1:101Þ

with θ1

Θ1 q1 5 Θ1;sat q1;sat

The interchange coefficient, D12 ; is not measurable. Its value can be predicted by means of measured data, as it was done for polymeric mass transport by Heintz and Stephan (1994a,b); Iza´k et al. (2003); and Nagy (2006); or it can be calculated as a logarithmic average of the single-component MaxwellStefan diffusivities, as

24


it is recommended by Krishna and Baur (2003) for mass transport through a zeolite membrane layer, as D12 5 D1 θ1 =ðθ1 1 θ2 Þ D2 θ1 =ðθ1 1 θ2 Þ

ð1:102Þ

When the saturation loading of the two components differs, and the singlecomponent MaxwellStefan diffusivity is occupancy dependent, Skoulidas et al. (2003) suggest a modified formula in which D12 6¼ D21 (this is not shown here). Equation (1.102) is based on the empirical approach of Vignes (1966) for describing the concentration dependence of the diffusivity in a binary liquid mixture, using as inputs the infinite dilution diffusivity values. Note that Eq. (1.102) is similar to the modified Vignes equation published by Bitter (1991), which defines the real diffusion coefficient of components in a (polymeric) membrane, but they must not be confused [see Eq. (3.111)]. Recent experiments show that the MaxwellStefan diffusion coefficient (Di ; note that Di Dim Þ of gases as CO2, CO, O2, CH4, and so forth in zeolites that consist of cages separated by narrow windows, such as e.g. CHA chabasite), DDR (decadodecasil R), LTA (Linde Type A), ERI (erionite), MFI mordenite framework inverted), can be strongly loading dependent, especially at high pressure, i.e. at high loading (Krishna et al., 2007; Li et al., 2007a,b). To quantify the loading dependence, the model developed by Reed and Ehrlich (1981) is the mostoften applied one. This model gives the following expression: Di 5 Di ð0Þ

ð1 1 εi Þz 21 ð1 1 εi =φi Þ2

ð1:103Þ

where z is the coordination number, representing the maximum number of neighbors within a cage; φi is the ReedEhrlich parameter, dimensionless; Di ð0Þ is the zero-loading MaxwellStefan diffusivity of species i (m2/s); the other parameters are defined as [see Krishna et al. (2007) for more details]: εi 5

ðβ i 21 1 2θi Þφi ; 2ð1 2 θi Þ

βi 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 4θi ð1 2 θi Þð1 2 1=φi Þ;

θi 5

qi qi;sat

Results obtained by the above model can be found in the papers of Li et al. (2007a,b) and Krishna et al. (2007).

1.5.3

The MaxwellStefan Approach for Mass Transfer in Porous Media

There are several excellent reviews available analyzing the mass transfer in a porous layer (Mason and Malinauskas, 1983; Wesselingh and Krishna, 1997). It is widely accepted (Thomas et al., 2001) that the mass transfer in a large range of pore sizes can be quantified using the dusty gas model (Mason and Malinauskas, 1983). This model takes into account molecular and Knudsen diffusion, as well as the contributions of viscous flux. However, in the region


25

of micropores, other effects began to dominate and different models have been suggested. One of the most advanced concepts is based on the assumption of activated configuration diffusion in the small pores [see Eq. (1.58)]. Accordingly, the molar flux densities Ji for all components are as (Thomas et al., 2001; Pedernera et al. 2002): 2

n X p xi Beo xj Ji 2 xi Jj Ji p rp 5 rxi 2 1 11 e ^ ^ DK;i μ Dij DK;i RT RT j51

ð1:104Þ

j 6¼ i

where p is the total pressure (Pa); Beo is effective permeability constant (m2); μ is the viscosity (Pa s); Ji is molar flux density (kmol/m2 s); DK,i and Deij are as sffiffiffiffiffiffiffiffiffi ^ 8RT ; DK;i 5 Ko πMi

Deij 5

ε Dij τ

ð1:105aÞ

with [for the values of Ko and Beo , see Eqs (1.47) and (1.40)]: Ko 5

2 ε rp ; 3 τ

Beo 5

1 2ε r 8 pτ

ð1:105bÞ

Generally, three gas phase transport mechanisms should be considered by the above model (see 1.4 for details): 1. Molecular diffusion 2. Knudsen diffusion 3. Viscous flux

For a single gas permeation through a membrane, the following flux can be obtained from Eq. (1.104) (Thomas et al., 2001): 0

1 sffiffiffiffiffiffiffiffiffi e ^ 1 @ 8RT Bo A p rp 1 Ji 5 2 Ko ^ μ πMi RT

ð1:106Þ

Eq. (1.106) is in harmony with Eq. (1.50).

1.6

FloryHuggins Theory for Prediction of the Activity

The FloryHuggins equation deals with molecules that are similar chemically, but differ greatly in length. The model is based on the idea that the chain elements arrange themselves randomly on a three-dimensional structure. The resulting equation for the activity of the solvent is a simple proportional function of the volume fraction of the solvent. Note that the volume fraction is

26


denoted here by ε to distinguish it from the φ concentration [εi 5 φiVi/Mi, where Mi is molar weight (kg/kmol); Vi is molar volume of i (kmol/m3); and εi is the volume fraction (m3/m3)]. The activity of a component in the membrane can be described according to FloryHuggins thermodynamics (Flory, 1963; Lue et al., 2004) by Vi 1 χim ð1 2 εi Þ2 ln ai 5 ln εi 1ð1 2 εi Þ 1 2 Vm

ð1:107Þ

where χim is an interaction parameter between the component i and the membrane called the FloryHuggins interaction parameter; Vi and Vm are molar volumes of solvent and membrane, respectively (kmol/m3). The χ interaction parameter is a dimensionless quantity characterizing the difference in interaction energy of a solvent molecule immersed in pure polymer compared with one in pure solvent. Its value can be positive or negative. If χ . 0, then the solvent and polymer “dislike” each other; if χ 5 0, then the solvent and polymer are similar; and if χ , 0, the solvent and polymer attract each other (Wesselingh and Krishna, 2000). After differentiating Eq. (1.107), it can be shown as d ln ai 1 Vi 22χim ð1 2 εi Þ 5 1 12 dεi Vm εi

ð1:108Þ

The mass transfer rate can be given according to Eqs (1.9a) and (1.9b) as [ε Pi 5 φiρ/ρi, that is, dφi 5 ρidεi/ρ, where ρ is the membrane’s density, ρ 5 nj5 1 ρi φi (kg/m3)]: Ji 5 DTi φi

d ln ai dφi ρi 1 Vi ρ dφi T 2 2χim 1 2 φi 5 Di φi 1 12 dφi dy ρ φi Vm dy ρi ð1:109Þ

where Ji is here the mass transfer rate (kg/m2 s). Integration of Eq. (1.109) over the membrane layer (with φ 5 φ and φ 5 φδ at y=0 and y=δ, respectively), the mass transfer rate can be: DT Ji 5 i δ

(

! !) 2 3 3 φi 2 φ2 ρi Vi ρ φi 2 φδ;i δ;i ðφ 2φδ;i Þ 1 1 2 22χim 1 2χim ρ i Vm 2 3 ρi ð1:110Þ

In the case of polymeric membrane and a binary liquid mixture, a ternary system (solvent components and membrane matrix), the activity ai and aj of liquid


27

components i and j in the polymeric membrane are given by (Mulder and Smolders, 1984): Vi Vi 2 εm 1ðχij εj 1 χim εm Þðεj 1 εm Þ Vj Vm @χij Vi 2χjm εj ε 1 u1 u2 ε1 Vj @uj

ln ai 5 ln εi 1ð1 2 εi Þ 2 εj

ð1:111Þ

Vj Vj Vj ln aj 5 ln εj 1ð1 2 εj Þ 2 εi 2 εm 1 χij εi 1 χjm εm ðεi 1 εm Þ Vi Vm Vi Vj Vj 2 @χij 2χim εi εm 1 u1 εj Vi Vi @u2 ð1:112Þ with ui 5

εi ; εi 1 εj

uj 5

εj εi 1 εj

where subscripts i, j, and m denote solvent components and membrane, respectively; χim and χjm are interaction parameters between components and membrane. Substituting Eq. (1.111) into Eqs (1.9a) and (1.9b), one can get the mass transfer rate for component i as

Ji 52 φi DTi ðφi ;φj Þ

9 > > > =

8 Vi Vi > ln εi 1ð12 εi Þ 2 εj 2 εm > > < V V

j m d dφi @χ V > > i ij dy dφi > 1ðχ ε 1 χ ε Þðε 1 ε Þ2 χ > εj εm 2 ui uj εj > > j m ij j im m jm : V @u ; j

j

ð1:113Þ

For the sake of completeness, the differential quotient of Eqs (1.111) and (1.112) will here be given for separation of binary mixture. Note that ε1 1 ε2 1 εm 5 1 for this system (Mulder, 1984; Meuleman et al., 1999): @ ln a1 1 V1 V1 5 21 1 2 χ12 ε2 2 χ1m εm 1χ2m ε2 @ε1 Vm V2 ε1 @χ12 ðu2 Þ @χ @2 χ12 1 u22 ðu1 2 u2 Þ 12 1 u1 u32 1ðε2 1 εm Þ 2χ1m 1 ε2 @ε1 @u2 @u22 ð1:114Þ

28


@ ln a1 V1 V1 @χ 52 1 1 χ12 ð12 ε1 Þ 1χ1m ðε1 21Þ1ð12 ε1 Þε2 12 @ε2 V2 Vm @ε2 V1 @χ @2 χ12 1 χ2m ðε1 1 2ε2 21Þ2ðu21 u2 2u1 u22 1 u1 u2 Þ 12 2 u21 u22 V2 @u2 @u22 ð1:115Þ

@ ln a2 V2 V2 V2 V2 @χ12 52 1 1 χij ð1 2ε2 Þ 2χ12 ð1 2ε2 Þ 1ð12 ε2 Þε1 @ε1 V1 Vm V1 V1 @ε1 2 V2 V2 2 @χ12 2 2 @ χ12 2 χ1m ð12 2ε1 2 ε2 Þ 1 22u1 u2 2 u1 u 2 V1 V1 @u2 @u22 ð1:116Þ

@ ln a2 1 V2 V2 V2 @χ12 5 21 1 2 χ12 ε1 2χ2m ð12 ε1 2 ε2 Þ 1ð1 2 ε2 Þε1 @ε2 Vm V1 V1 @ε2 ε2 V2 2 V2 @χ12 2ð1 2ε2 Þχ2m 1ε1 χ1m 22u2 u1 V1 V1 @u2 2 @ χ12 V2 2 @χ12 1 u1 u32 1 u1 V1 @u2 @u22 ð1:117Þ

1.6.1

MaxwellStefan Equation with the FloryHuggins Theory

Let us start with the general expression of the MaxwellStefan equation that can be applied for a multicomponent mixture, e.g. in the dusty gas model (Krishna and Wesselingh, 1997; Schaetzel et al., 2001), namely:

2

n X 1 dμi cj Ji Jj 5 2 ; c D ci cj RT dy j 5 1 t ij j 6¼ i

where

ct 5

j5n X

cj

ð1:118Þ

j51

where n is the number of species in the mixture; Ji is the average diffusive molar flux of species i (kmol/m2 s); ci is the molarity of species i (kmol/m3); Dij are generalizations of MaxwellStefan diffusivities (m2/s); μi is the chemical potential of species i (J/mol or kJ/kmol). Adapting Eq. (1.118) to a polymer (m) and single solvent (i) system, the MaxwellStefan equation (1.118)


29

becomes (Schaetzel et al., 2001) as (Jm 5 0; ct 5 cm 1 ci; Mmcm 5 φm; Mici 5 φi; ct M 5 ρÞ: d ln ai cm Ji φm M Ji 5 dy ct Dim ci Mm ρDim φi

ð1:119Þ

where Mm is the molar weight of membrane (kg/kmol); M is average molar weight of the system (e.g., M 5 ðcm Mm 1 ci Mi Þ=ρÞ; ρ is the density of the membrane/component system (it is assumed that its value is constant, kg/m3); φi is the concentration of diffusing species (kg/m3); Ji is the mass transfer rate (kg/m2 s). The relationship between φi and φm is Vi Vm φi 1 φ 51 Mi Mm m

ð1:120Þ

Inserting Eq. (1.108) and Eq. (1.120) into Eq. (1.109) and taking into account that d ln ai /dy 5 (d ln ai /dφi)dφi /dy, the following differential expression can be obtained: Ji 5 Dim

ρ φi ρi 1 Vi ρ dφi 22χim 1 2 φi 1 12 Vm dy ρi M ð1 2 Vi φi =Mi Þ ρ φi ð1:121Þ

Let us integrate Eq. (1.121) term by term in order to check the solution. Thus, one can get 0 1 0 1 0 2 31 9φδ;i 8 > > > > ρ M V V M M V i i i i i i > i > > ln@12 φi A 1 @1 2 22χim A @ 2φi 1 ln41 2 φi 5A > > > > > = < ρ V M V V V M i i m i i i Dim ρ 0 1 Ji 52 * + > δ M> 2 > > ρ Mi Mi @ Vi φ > > > > > > 2φi 1 ln 1 2 φi A2 i 12χim > > ; : Vi Mi 2 ρi Vi φi

ð1:122Þ

Rearranging Eq. (1.122), it can be obtained as 9 8 = 1 2ðVi =Mi Þφi Dim ρ < ρ M i 2 1 Bðφi 2 φδ;i Þ 2 χim Ji 5 ðφ2 2 φ Þ A ln δ;i ; ρ i Vi i δM : 1 2ðVi =Mi Þφ δ;i

ð1:123Þ where

2 ρi Mi Vi ρ Mi A5 1 12 22χim 1 2 ρ Vi Vm ρi Vi2

ð1:124Þ

30


Vi Mi ρ B52 12 22χim 12 Vm Vi ρi

ð1:125Þ

It was assumed during the solution of Eq. (1.119) or (1.121) that the value of ct ðct 5 MρÞ should be constant and consequently, the values of M; ρ should also be constant. In reality, according to Eq. (1.120): Vi Mm φm 5 1 2 φi Mi Vm

ð1:126Þ

Accordingly, Eq. (1.121) can be rewritten as D E 1 2 VMi φi i Mm =Vm 1 φi φi d ln a i

Ji 5 2 Dim V i φi dy 1 2 Mi Mm =Vm

ð1:127Þ

Inserting Eq. (1.108) into Eq. (1.127), the mass transfer rate can be given by the following differential equation: Ji 52Dim

ρi 1 φi φ2i φ3i dφi 1A 1B 2C ρ 12ðVi φi =Mi Þ 12ðVi φi =Mi Þ 12ðVi φi =Mi Þ 12ðVi φi =Mi Þ dy ð1:128Þ

where A512

Vi Vi Vm ρ i 22χim 2 2 Mi Mi Mm ρ

ρ Vi Vm Vi 12 B 5 2χim 2 2 22χim Mi Mm Mi ρi C 5 2χim

ρ ρi

Solving Eq. (1.128), one can get a rather complex expression, namely: Dim Ji 5 δ

( *

) 2 + Mi ρi Mi Mi 12ðVi =Mi Þφi 2 A1B2C 1 A 1 B 2C ln 1T ðφi 2 φδ;i Þ 1 Vi ρ Vi Vi 12ðVi =Mi Þφδ;i

ð1:129Þ


31

where ! ! 2 3 2 2 φi 2 φ3 Mi Mi φi 2 φδ;i Mi δ;i T 52 B2C 1C Vi Vi 2 Vi 3

1.7

UNIQUAC Model

According to the well-known, solution-diffusion model (Paul, 1976), sorption is usually considered one of the key steps for mass transfer through dense polymeric membranes. Often recommended model is the UNIQUAC (short of UNIversal QUAsiChemical) model, originally proposed by Abrams and Prausnitz (1975), for the prediction of the nonideal solubilities of mixtures in a dense active layer (Heintz and Stephan, 1994a,b; Janquie´res et al., 2000). UNIQUAC is the initial approach for predicting the activity of the solvent components in the membrane from which several other models have been developed as UNIQUAC-HB (Janquie´res et al., 2000). The detailed discussion of these models is not a topic of this material, only the UNIQUAC approach, originally proposed by Abrams and Prausnitz (1975), will be shown briefly. This model accounts for the different sizes and shapes of the molecules, as well as for the different intermolecular interactions between the mixture components, including polymeric components. The UNIQUAC model requires only binary interaction parameters for the description of multicomponent mixtures. The thermodynamic activity ai of a component i dissolved in a polymer membrane material, in presence of n solvent components, is given by (Heintz and Stephan, 1994a,b; Janquie´res et al., 2000): n X Z θi ri Z qm 12 εj lj 2 ri εm 21 ln ai 5 ln εi 1 qi ln 1 li 2 εi rj rm 2 2 j51 ð1:130Þ

j 6¼ m

1 qi 2 qi ln

n X j51

!

θj χij

2 qi

n X j51

θj χij k 5 1 θk χkj

Pn

with li 5

Z ðri 2 qi Þ 2ðri 21Þ 2

ð1:131Þ

where εi is volume fraction of i, θi is the surface fraction of component i, χij, τ ji are binary interaction parameters; Z is the coordination number; ri and qi are dimensionless parameters for the relative molecular size and surface of component i related to the size and surface of a CH2 segment in polymer membrane,

32


respectively; n is number of components including the membrane as well. Altogether, n(n 21) χ parameters are needed for describing an n-component mixture with the UNIQUAC model. The volume fraction can be calculated by φ =ρ εi 5 Pn i i j 5 1 φj =ρj

ð1:132Þ

The UNIQUAC model gives excellent results in the case of polar liquid mixtures and hydrophilic membranes (Heintz and Stephan, 1994a,b).

References Abrams, D.S., and Prausnitz, J.M. (1975) Statistical thermodynamics of liquid mixtures: a new conception for the excess gibbs energy of partly or completely miscible mixtures. Am. Inst. Chem. 21, 116128. Amundsen, N.R., Pan, T.-W., and Paulsen, V.I. (2003) Diffusing with Stefan and Maxwell. AIChE J. 49, 813830. Baker, R.W. (2004) Membrane Technology and Application, 2nd ed. John Wiley and Sons, Chichester. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena. John Wiley and Sons, New York. Bitter, J.G.A. (1991) Transport Mechanisms in Membrane Separation Processes. Shell-Laboratorium, Amsterdam. Flory, P. (1963) Principles of Polymer Chemistry. Cornell University Press, New York. Gardner, T.Q., Falconer, J.L., and Noble, R.D. (2002) Adsorption and diffusion properties of zeolite membranes by transient permeation. Desalination 149, 435440. Geankoplis, C.J. (2003) Transport Processes and Separation Process Principles, 4th ed. Prentice Hall, New Jersey. Geraldes, V., and Brites, A.M. (2008) Computer program for simulation of mass transport in nanofiltration membranes. J. Membr. Sci. 321, 172182. Heintz, A., and Stephan, W. (1994a) A generalized solution-diffusion model of the pervaporation process through composite membrane, Part I. Prediction of mixture solubilities in the dense active layer using the UNIQUAC model. J. Membr. Sci. 89, 143151. Heintz, A., and Stephan, W. (1994b) A generalized solution-diffusion model of the pervaporation process through composite membrane, Part II. Concentration polarization, coupled diffusion and the influence of the porous layer. J. Membr. Sci. 89, 153169. Hoang, D.L., Chan, S.H., and Ding, O.L. (2005) Kinetic modeling of partial oxidation of methane in an oxygen permeable membrane reactor. Chem. Eng. Res. Design 82, 177186. Iza´k, P., Bartovska´, L., Friess, K., Sipek, M., and Uchytil, P. (2003) Description of binary liquid mixtures transport through non-porous membrane by modified MaxwellStefan equations. J. Membr. Sci. 214, 293309. Janquie´res, A., Perrin, L., Arnold, S., Cle´ment, R., and Lochon, P. (2000) From binary to ternary systems: general behavior and modeling of membrane sorption in purely organic systems strongly deviating from ideality by UNIQUAC and related models. J. Membr. Sci. 174, 255275.


33

Kamaruddin, H.D., and Koros, W.J. (1997) Some observation about the application of Fick’s first law for membrane separation of multicomponent mixtures. J. Membr. Sci. 135, 147159. Kapteijn, F., Bakker, W.J.W., Zheng, G., and Poppe, J. (1995) Permeation and separation of light hydrocarbons through a silicalite-1 membrane. Application of the generalized MaxwellStefan equations. Chem. Eng. J. 57, 145153. Kapteijn, F., Moulijn, J.A., and Krishna, R. (2000) The generalized MaxwellStefan model for diffusion in zeolites: sorbate molecules with different saturation loadings. Chem. Eng. Sci. 55, 29232930. Krishna, R. (1990) Multicomponent surface diffusion of adsorbed species: a description based on the generalized MaxwellStefan equations. Chem. Eng. Sci. 45, 17791791. Krishna, R. (1993) A unified approach to the modeling of intraparticle diffusion in adsorption processes. Gas. Sep. Purif. 7, 91. Krishna, R., and Baur, R. (2003) Modeling issues in zeolite based separation processes. Sep. Purif. Technol. 33, 213254. Krishna, R., and Wesselingh, J.A. (1997) The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 52, 861911. Krishna, R., Vlugt, T.J., and Smit, B. (1999) Influence of isotherm inflection on diffusion in silicalite. Chem. Eng. Sci. 54, 17511757. Krishna, R., van Baten, J.M., Garcia-Perez, E., and Calero, S. (2007) Incorporating the loading dependence of the MaxwellStefan diffusivity in the modeling of CH4 and CO2 permeation across zeolite membranes. Ind. Eng. Chem. Res. 46, 29742986. Lawson, K.W., and Lloyd, D.R. (1997) Membrane distillation: review. J. Membr. Sci. 124, 125. Lonsdale, H.K. (1982) The growth of membrane and technology. J. Membr. Sci. 10, 81181. Li, S., Falconer, J.I., Noble, R.D., and Krishna, R. (2007a) Modeling permeation of CO2/CH4, CO2/N2 and N2/CH4 mixtures across SAPO-34 membrane with the MaxwellStefan equations. Ind. Eng. Chem. Res. 46, 39043911. Li, S., Falconer, J.I., Noble, R.D., and Krishna, R. (2007b) Interpreting unary, binary and ternary mixture permeation cross a SAPO-34 membrane with loading-dependent MaxwellStefan diffusivities. J. Phys. Chem. C 111, 50755082. Lue, S.J., Wang, F.J., and Hsiaw, S.-Y. (2004) Pervaporation of benzene/cyclohexane mixtures using ion-exchange membrane containing copper ions. J. Membr. Sci. 240, 149158. Martinek, J.G., Gardener, T.Q., Noble, R.D., and Falconer, J.L. (2006) Modelling transient permeation of binary mixtures through zeolite membrane. Ind. Eng. Chem. Res. 45, 60326043. Mason, E.A., and Malinauskas, A.P. (1983) Gas Transport in Porous Media: The Dusty Gas Model. Elsevier, Amsterdam. Meuleman, E.E.B., Bosch, B., Mulder, M.H.V., and Stratman, H. (1999) Modeling of liquid/ liquid separation by pervaporation: toluene from water. AIChE J. 45, 21532160. Mulder M. (1981) Pervaporation Separation of EthanolWater and Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Mulder, M.H.V., and Smolders, C.A. (1984) On the mechanism of separation of ethanol/ water mixtures by pervaporation I. Calculations of concentration profiles. J. Membr. Sci. 17, 289307. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical dense membrane. J. Membr. Sci. 274, 159168. Paul, D.R. (1976) The solution-diffusion model for swollen membranes. Sep. Purif. Methods 5, 35.

34


Pedernera, M., Alfonso, M.J., Meneńdez, M., and Santamaria, J. (2002) Simulation of a catalytic membrane reactor for the oxidative dehydrogenation of butane. Chem. Eng. Sci. 57, 25312544. Reed, D., and Ehrlich, G. (1981) Surface diffusion, atomic jump rates and thermodynamics. Surf. Sci. 102, 588609. Schaetzel, P., Bendjama, Z., Vauclair, C., and Nguyen, Q.T. (2001) Ideal and non-ideal diffusion through polymers: application to pervaporation. J. Membr. Sci. 191, 95102. Seidel-Morgenstern, A. (2010) Membrane Reactors. Wiley-VCH, Weinheim. Silva, P., Peeva, L.G., and Livingston, A.G. (2008) Nanofiltration in organic solvents, in Advanced Membrane Technology and Applications, Ed. by, N.N., Li, A.G., Fane, W.S.W., Ho, and T., Matsuura. John Wiley and Sons, New Jersey, pp. 451468. Skoulidas, A.I., Sholl, D.S., and Krishna, R. (2003) Correlation effects in diffusion of CH4/CF4 mixtures in MFI zeolite. A study linking MD simulation with the MaxwellStefan formulation. Langmuir 19, 79777988. Thomas, S., Schafer, R., Caro, J., and Seidel-Morgenstern, A. (2001) Investigation of mass transfer through inorganic membrane with several layers. Catalysis Today 67, 205216. Tuchlenski, A., Uchytil, P., and Seidel-Morgenstern, A. (2001) An experimental study of combined gas phase and surface diffusion in porous glass. J. Membr. Sci. 140, 165184. Van de Graaf, J.M., Kapteijn, F., and Moulijn, J.A. (1999) Modeling permeation of binary mixtures through zeolite membranes. AIChE J. 45, 497511. Vignes, A. (1966) Diffusion in binary solutions. Ind. Eng. Chem. Fund. 5, 189199. Wesselingh, J.A., and Krishna, R. (1990) Mass Transfer. Ellis Horwood, Chichester, U.K. Wesselingh, J.A., and Krishna, R. (2000) Mass Transfer in Multicomponent Mixture. Delft University Press, Delft. Wijmans, J.G. (2004) The role of permeant molar volume in the solution-diffusion model transport equations. J. Membr. Sci. 237, 3950. Wijmans, J.G., and Baker, R.W. (1995) The solution-diffusion model: a review. J. Membr. Sci. 107, 121. Zhu, W., Graaf, J.M., Broeke, L.J.P., Kapteijn, F., and Moulijn, J.A. (1998) TEOM: a unique technique for measuring adsorption properties. Light alkanes in silicalite-1. Ind. Eng. Chem. Res. 37, 19341942.

2 Molecular Diffusion 2.1

Introduction

Molecular diffusion of molecular transport can be defined as the transfer or movement of individual molecules through a fluid by means of the random, Brownian, or thermal motion of the molecules. Molecules move at high speeds but travel extremely short distances before colliding with other molecules. The migration of individual molecules, therefore, is slow except at quite low molecular densities. Because the molecules travel in random paths, mainly due to the frequent collision with other moving molecules, molecular diffusion is often called a random-walk process (Geankoplis, 2003). The diffusion coefficient is often defined as the ratio of flux density to the negative of the concentration gradient in direction of diffusion, then according to Fick’s law: Ji 5 2Di

dφ dy

ð2:1Þ

where J is the transfer rate (kg/m2 s); φ is the concentration in the membrane (kg/m3); and y is the space coordinate in the direction of diffusion (m). The quantity, Di, is the diffusion coefficient, dφ/dy is the concentration gradient, and it can be regarded as the driving force for diffusion. But the real force can be obtained by means of thermodynamic consideration. The mean energy per mole in a uniform system is the partial molar Gibbs function, Gi ; or chemical potential, μi. The mean energy per molecule is μ=N; where N is Avogadro’s number. If energy were dissipated in moving a molecule down the chemical potential gradient, the driving force, Fi, per molecule of species, i, would be (Bungay et al., 1983): Fi 5 2

1 dμ N dy

ð2:2Þ

Assuming a velocity-dependent frictional resistance, fi, then the mean velocity, υi, from one plane to the next will be 1 dμ ð2:3Þ υi 5 2 Nfi dy If the molar concentration of species i is φi, then the number of moles passing through unit area in unit time, Ji, will be Ji 5 υ i φ i Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00002-8 © 2012 Elsevier Inc. All rights reserved.

ð2:4Þ

36


For a nonideal medium, the chemical potential is as μi 5 μoi 1 RT ln ai

ð2:5Þ

Equation (2.5) involves the concentration effect; it does not contain the effect of temperature, pressure, or electrical field. Thus, Ji 5 2

RT d ln ai φi dy Nfi

ð2:6Þ

Ji 5 2

RT @ ln ai @φi RT @ ln ai @φi 2 φi @φ @y Nfi Nfi @ ln φi @y i

ð2:7Þ

or

Considering Fick’s law [Eq. (2.1)], one can get Di 5

RT @ ln ai RT φi @ai φi @φi Nfi Nfi ai @φi

ð2:8aÞ

Di 5

RT @ ln ai Nfi @ ln φi

ð2:8bÞ

or

This latter equation can be rewritten as (ai 5 γ iφi): RT φi @γ i 11 Di 5 γ i @φi Nfi

ð2:8cÞ

For ideal solution, Ji 5 2 Di 5

RT d ln φi RT dφi 2 φi dy Nfi Nfi dy

RT Nfi

ð2:9Þ ð2:10aÞ

In the ideal solution, the quantity RT=Nfi is sometimes called the thermodynamic diffusion coefficient, DT. Taking into account Eq. (2.8b), the link between the Di and DT is as, see Eq. (1.12): DT 5 Di

@ ln ai @ ln φi

ð2:10bÞ

Molecular Diffusion

2.2

37

Gas Diffusivities

Gas diffusivities can be estimated quite well from the kinetic theory of gases. In the simplest form of this theory, molecules are hard spheres. In a simplified treatment, it is assumed that there are no attractive or repulsive forces between molecules. The derivation uses the mean free path λ, which is the average distance that a molecule has traveled between collisions. The final equation is DAB 5

1 uλ 3

ð2:11Þ

where u is the average velocity of the molecules. A more accurate and rigorous treatment must consider the intermolecular forces, attraction and repulsion between molecules as well as different sizes of molecules A and B. Chapman and Enskog (Hirschfelder et al., 1954) solved the Boltzmann equation, which uses a distribution function instead of the mean free path, λ. For a pair of nonpolar molecules, a reasonable approximation to the forces is the LennardJones function. The final relation for predicting the diffusivity of a binary pair of A and B molecules is 1:8583 3 1027 T 3=2 1 1 1=2 1 ð2:12Þ DAB 5 MA MB Pσ2AB ΩD;AB where DAB is the diffusivity (m2/s); T is temperature (K); MA and MB are the molecular weights of components A and B (kg/kg mol); and P is absolute pressure (atm). The term σAB is an “average collision diameter”; ΩD,AB is a collision integral based on the LennardJones potential. The effect of concentration of components is not included in Eq. (2.12), but this effect can be neglected in most cases. Equation (2.12) is relatively complicated to use, and often some of the constants such as σAB are not available or are difficult to estimate. Hence, the semiempirical method of Fuller et al. (1966), which is much more convenient to use, often is preferred. The equation is 1:0 3 1027 T 1:75 1 1 1=2 ð2:13Þ DAB 5 P 1=3 2 MA 1 MB 1=3 P P υA 1 υB P where υA 5 sum of structural volume increments, DAB is in m2/s. This method can be used for mixtures of nonpolar gases or for a polarnonpolar mixture.

2.3

Prediction of Diffusivities in Liquids

The equations for predicting diffusivities of dilute solutes in liquids are by necessity semiempirical, as the theory for diffusion in liquids is not well established as yet.

38


The StokesEinstein equation, one of the first theories, was derived for a very large spherical molecule (A) diffusing in a liquid solvent (B) of small molecules. Stokes’ law was used to describe the drag on the moving solute molecule. Accordingly, the diffusion coefficient (DAB, m2/s) can be estimated by DAB 5

kT 6πrs η

ð2:14Þ

where k is Boltzmann’s constant; rs is the radius of solute (m); and η is the solute dynamic viscosity (cP). The equation is a good approximation for large solutes with radii greater than ˚ (Baker, 2004). But, as the solute becomes smaller, the approximation of 510 A the solvent as a continuous fluid becomes less valid. Thus, the above equation was modified by assuming that all molecules are alike and arranged in a cubic lattice, and by expressing the molecular radius in therms of molar volume (Geankoplis, 2003): DAB 5

9:96 3 10216 T

ð2:15Þ

1=3

ηVA

where T is temperature (K); η is viscosity of solution (Pa s); and VA is the solute molar volume at its normal boiling point (m3 kg mol). The modified WilkeChang equation is as DAB 5 1:173 3 10216 ðψMB Þ1=2

T ηVA0:6

ð2:16Þ

where MB is the molecular weight of solvent B; η is viscosity (Pa s); ψ is an “association parameter” of the solvent, where ψ is 2.6 for water, 1.9 for methanol, 1.5 for ethanol, etc. (Geankoplis, 2003, p. 432).

2.4

Diffusion of an Electrolyte Solution

Electrolytes in aqueous solution dissociate into anions and cations. Each ion diffuses at a different rate. If the solution is to remain electrically neutral at each point, the cations and anions diffuse effectively as one component, and the ions have the same net flux. The well-known NernstHaskell equation for a dilute, single-salt solution can be used to predict the overall diffusivity DAB of the salt A in the solvent B (Geankoplis, 2003, p. 434): DAB 5 8:928 3 1010 T

ð1=n1 Þ 1ð1=n2 Þ ð1=λ1 Þ 1ð1=λ2 Þ

ð2:17Þ

Molecular Diffusion

39

where DAB is in cm2/s; n1 is the valence of the cation; n2 is the valence of the anion; λ1 and λ2 are the limiting ionic conductances in very dilute solutions ((A/cm2)(V/cm) and (g eq/cm2), respectively). The diffusion coefficient of an individual ion i at 25 C can be calculated by Di 5 2:662 3 1027

λi ni

ð2:18Þ

Substituting Eq. (2.9) into Eqs (2.8a)(2.8c), one can obtain DAB 5

2.5

n1 1 n2 ðn2 =D1 Þ 1ðn1 =D2 Þ

ð2:19Þ

Diffusion in a Membrane

2.5.1

Diffusion in a Dense Membrane

One can broadly classify transport in solids into two types of diffusion: diffusion that can be considered to follow Fick’s law and does not depend primarily on the actual structure of the solid, and diffusion in porous solids where the actual structure and void channels are important. This latter process is discussed in the next section. If the diffusion coefficient is independent of penetrant concentration, it can be determined by time lag or by equilibrium sorption measurements (Vieth et al., 1976). In the case of time lag measurements, flux through a membrane is measured as a function of time when pressure is applied to one side of the membrane and vacuum is pulled at the other. Extrapolation of the linear region of a plot of the steady-state flux versus time provides an intercept with the time axis called the time lag, θ, from which D is obtained as follows (Barrer, 1939): D5

δ 6θ

ð2:20Þ

where δ is the membrane thickness (m) and D is diffusion coefficient (m2/s). In the case of gas or vapor sorption by a membrane, D can be obtained from the ratio of the mass sorbed gas at time t(Mt) to the equilibrium sorption mass (MN) by the following relationship (Crank and Parks, 1968): Mt 4 5 1=2 MN π

rffiffiffiffiffi Dt δ2

ð2:21Þ

The diffusion coefficient can be obtained from the initial gradient of a plot of qffiffiffiffiffiffiffiffiffi Mt/MN versus t=δ2 :

40


As concentration dependency of the diffusion coefficient, the most often suggested function is the exponential and linear concentration dependency of the diffusion coefficient (Mulder, 1981; Qin and Cabral, 1998). Equations for these cases are discussed in detail in Chapters 3 and 11. In the two- or multicomponent separation, the coupling of the diffusion of the components occurs in most cases. The MaxwellStefan approach for coupling is discussed in Chapter 3 in cases of pervaporation and gas separation by application of zeolite membrane (Nagy, 2004). Another often recommended mass transport theory is the so-called FloryHuggins approach (Meuleman et al., 1999). This theory is especially applicable for organophilic pervaporation of organic compounds in water. The FloryHuggins theory that gives the chemical potential as a function of the components’ concentration, and consequently to give the mass flux through the membrane, is also recommended to describe the binary separation by pervaporation (Smart et al., 1998; Meuleman et al., 1999; Nagy, 2006). For polymers with high molecular weight, the activity in a membrane, according to the FloryHuggins approach, in a case of single-component transport, can be given as [see Eq. (1.107)]: Vi 1 χim ð12εi Þ2 ln ai 5 ln εi 1ð12εi Þ 12 Vm

ð2:22Þ

Expressing the value of d ln a/dε, the concentration dependence of the diffusion coefficient can be expressed [Eq. (1.108)]. For detailed analysis of the mass transport for unary and binary systems, see Chapter 1.

2.5.2

Diffusion in a Porous Membrane

Mass transfer mechanisms depend strongly on the pore size during membrane distillation; accordingly, different flux equations can be obtained. Basically, the following transfer rate can be defined depending on the pore size (Soni et al., 2009).

2.5.2.1 Knudsen-Limited Diffusion In membranes with small pores (r{λ; λ is the mean free path of the diffusing molecules), the molecule-pore wall collisions are dominant, therefore the Knudsen equation can used to describe the transport; the mass transfer rate can be given by Eq. (2.23). The flux is directly proportional to the difference of partial pressures. Ji 5 2DK;i

dci DK;i dpi 2 dy RT dy

ð2:23Þ

Molecular Diffusion

41

2.5.2.2 Knudsen-Viscous Transition Diffusion The mean free path of diffusing chemical component, λ, is similar to the membrane pore diameter, consequently the convective flow also can exist during the transport. The Knudsen-viscous transition equation is (see Chapter 1): rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8RTavg pi;avg 1 Ko ΔP Δpi 1 Bo Ji 5 πMi μ RTavg δ

ð2:24Þ

with Bo 5

r2 ε ; 8τ

Ko 5

1 dp 3

ð2:25Þ

where Δpi is the partial pressure difference of transported component on the two sides of the membrane (Pa); ΔP is the pressure difference (Pa); μ is the viscosity (kg/ms); and δ is the membrane thickness (m).

2.5.2.3 Knudsen-Molecular Diffusion For membranes with smaller air-filled pores (about , 0.5 μm; Soni et al., 2009), even the molecular diffusion also can affect the transport rate, thus both mechanisms should be taken into account. Accordingly: Ji 5 2

2.6

1 1 1 21 dpi 1 1 21 dci 2 1 1 dy dy RT Do;i DK;i Do;i DK;i

ð2:26Þ

Transport with Convective Velocity Due to the Component Diffusion

Due to the diffusion, the movement of the whole phase also can be moved especially in the case of a high concentration of the diffusing component. The rate at which moles of A passed a fixed point (JAo kmol/m2 s) can be converted to a velocity of diffusion of A by o 5 υAd cA JAc

ð2:27Þ

where υAd is the diffusion velocity of A (m/s) and cA is the concentration of A (kmol/m3). Look at briefly the situation when the whole fluid is moving in bulk or convective velocity. The molar average velocity of the whole fluid relative to a stationary point is υm (m/s). The velocity of A relative to the stationary point

42


is the sum of the diffusion velocity and the average or convective velocity (Bird et al., 1960; Geankoplis, 2003): υA 5 υAd 1 υM

ð2:28Þ

Multiplying Eq. (2.28) by cA, it can be obtained as υA cA 5 υAd cA 1 υM cA

ð2:29Þ

The term υAcA can be represented by the flux JAo as follows: o o o JAo 5 JAd 1 υM cA JAd 1 JAc

ð2:30Þ

o o where JAd is the diffusion flux relative to the moving fluid (kmol/m2 s) and JAc is 2 the convective flow of component A (kmol/m s). Let Jo be the total convective flux of the whole binary stream relative to the stationary point, then:

J o 5 υM c 5 JAo 1 JBo

ð2:31Þ

JAo 1 JBo c

ð2:32Þ

or υM 5

where c is the total concentration of the fluid (kmol/m3). Substituting Eq. (2.32) into Eq. (2.28), one can obtain o 1 JAo 5 JAd

cA o ðJ 1 JBo Þ c A

ð2:33Þ

Applying the known diffusive flow, the general equation for diffusion plus convection can be given as JAo 5 2DAB

dcA cA 1 ðJAo 1 JBo Þ dy c

ð2:34Þ

Equation (2.34) is valid for mass transport in solid phase as well assuming the stationary coordinate is fixed to the laboratory device or to the membrane module. Thus, for the mass transport through a membrane layer [see also Eqs (1.21a) and (1.21b)]: JAo 5 2DAB

dφA cA 1 ðJAo 1 JBo Þ dy c

ð2:35Þ

Molecular Diffusion

2.7

43

Ion Transport and Hindrance Factors

The extended NerstPlank equation for the membrane is as (Geraldes and Brites, 2008): Ji 2 Di;p

zi φ Di;p dψ dφ F 1 Ki;c Jv 2 i RT dy dy

ð2:36Þ

where Ji is the solute flux of component i (kg/m2 s); Jv is the permeate volume flux (m3/m2 s); Di,p is the hindered diffusion coefficient (m2/s); Ki,c is the hindrance coefficient for convection; zi is the charge number; F is the Faraday constant (9.64867 3 104 C/eq); R is the ideal gas constant (8.314 J/kmol); ψ is the electric potential (V); and Di,N is the diffusivity of the i species in water at infinite dilution (m2/s): Di;p 5 Ki;d Di;N

ð2:37Þ

Hindrance factors Ki,d, Ki,c are functions of λi and are related to hydrodynamic coefficients such as the enhanced drag and lag coefficients. According to Dechadilok and Deen (2006), for λi # 0.95: A22:81903λ4i 1 0:270788λ5i 21:10115λ6i 20:435933λ7i ð12λi Þ2

ð2:38Þ

A 5 1:01ð9=8Þλi ln λi 21:56034λi 1 0:528155λ2i 11:91521λ3i

ð2:39Þ

Ki;d 5 with

For λi . 0.95, the results of Mavrovouniotis and Brenner (1988) can be applied: 12λi 5=2 ð2:40Þ Ki;d 5 0:984 λi For convection, the hindrance factor can be as: Ki;c 5

11 3:867λi 21:907λ2i 20:834λ3i 111:867λi 20:741λ2i

ð2:41Þ

or as given by Deen (1987): ^ i gð11 0:054λi 20:988λi 2 1 0:44λi 3 Þ Ki;c 5 f2 2 Φ where λi 5

rs;i rp

and

^ i 5 ½12λi 2 Φ

ð2:42Þ

44


References Baker, R.W. (2004) Membrane Technology and Application, 2nd ed. John Wiley and Sons, Chichester. Bird, B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena. John Wiley and Sons, New York. Bungay, P.M., Lonsdale, H.K., and de Pinho, M.N. (1983) Synthetic Membranes: Science, Engineering and Application, NATO ASI Series, D. Reidel Publishing Company, Vol 181. Barrer, R.M. (1939) Permeation, diffusion and solution of gases in organic polymer. Trans. Faraday Soc. 35, 629643. Crank, J., and Parks, G.S. (1968) Diffusion in Polymers. Academic Press, London. Dechadilok, P., and Deen, W. (2006) Hindrance factors for diffusion and convection in pores. Ind. Eng. Chem. Res. 45, 6953. Deen, W.M. (1987) Hindered transport of large molecules in liquid-filled pores. AIChE J. 33, 14091425. Fuller, E.N., Schettler, P.D., and Giddings, J.C. (1966) Ind. Eng. Chem. 58, 19. Geankoplis, Ch.J. (2003) Transport Process and Separation Process Principles, 4th ed. Prentice Hall, Upper Saddle River. Geraldes, V., and Brites, A.M. (2008) Computer program for simulation of mass transport in nanofiltration membranes. J. Membr. Sci. 321, 172182. Hirschfelder, J.Q., Curtiss, C.F., and Bird, R.B. (1954) Molecular Theory of Gases and Liquids. John Wiley and Sons, New York. Mavrovouniotis, G.M., and Brenner, H. (1988) Hindered sedimentation diffusion and dispersion coefficient for Brownian spheres in circular cylindrical pores. J. Colloid Interface Sci. 124, 269. Meuleman, E.E.B., Bosch, B., Mulder, M.H.V., and Strathmann, H. (1999) Modeling of liquid/liquid separation by pervaporation: toulene from water. AIChE J. 45, 21532160. Mulder, M. (1981). Pervaporation Separation of EthanolWater and Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Qin, Y., and Cabral, J.M.S. (1998) Lumen mass transfer in hollow fiber membrane processes with nonlinear boundary conditions. AIChE J. 41, 836848. Smart, J., Starov, V.M., Schucker, R.C., and Lloyd, D.R. (1998) Pervaporative extraction of volatile organic compounds from aqueous systems with use of a tubular transverse flow module. Part II. Experimental results. J. Membr. Sci. 143, 159179. Soni, V., Abildskov, J., Jonsson, G., and Gani, R. (2009) A general model for membrane-based separation process. Computers Chem. Eng. 33, 644659. Vieth, W.R., Howel, J.M., and Hsieh, J.H. (1976) Dual sorption theory. J. Membr. Sci. 1, 177220.

3 Diffusion Through a Plane Membrane Layer

3.1

Introduction

In this chapter, we consider various cases of one-dimensional diffusion, perpendicular to the membrane surface, into a solid layer with thickness of δ bounded by two parallel plane surfaces. For a description of this process, the solution-diffusion model is applied. That means that the transported component is adsorbing on the solid interface and then it is transported by molecular diffusion through the membrane layer to its external surface, and from here (through an external mass transfer resistance) to the bulk permeate phase. Note that the concentrations are in equilibrium at the interface, between the continuous phase and membrane layer, and that this sorption process is an instantaneous process. Most of the membrane modules used for separation are capillary membrane (hollow-fiber) modules. The question arises, under which conditions the mass transport through a cylindrical membrane layer can be considered as a plane layer. The diffusional mass transfer equation through a cylindrical membrane is defined in Chapter 6. For limiting cases, this condition can be obtained as if δ=ro -0

then

Di Di Δφi - Δφi ro lnð11δ=ro Þ δ

ð3:1Þ

where ro is the internal radius of fibers, δ is the membrane thickness, Di is the diffusion coefficient of component i (m2/s), and φi is the concentration in the membrane (kg/m3). The selective (or skin) layer of an asymmetric membrane is very thin; its thickness falls generally in the range of 0.12 μm, while the radius of a capillary is about 100150 μm. Thus, the value of δ/ro equals about 1022, and consequently, the cylindrical effect can generally be neglected in this case. The support or spongy layer is much thicker; it is about 100 μm; thus in this case, the cylindrical effect must not be neglected. Besides these special cases, the mass transport through a membrane layer through plate-and-frame, tubular, and spiral-wound modules should be regarded as that through a plane interface.

Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00003-X © 2012 Elsevier Inc. All rights reserved.

46


3.2

Steady-State Diffusion

Let us consider the case of diffusion through a plane membrane of thickness δ and diffusion coefficient D, whose surfaces, y 5 0 and y 5 δ, are maintained at constant concentrations, Hco 5 φ and Hcoδ 5 φδ ; respectively (Figure 3.1). When the value of c . Cδ ; then the diffusion process occurs from one side of the membrane to the other side due to the driving force existing in the membrane layer. For a longer time, a steady state is reached in which the concentration in the membrane remains constant. The one-dimensional, steady-state diffusion equation in one dimension then can be given, in dimensionless variables [Φ 5 φ/φ φ/(Hco), where c denotes the membrane concentration, co is the fluid phase’s concentration] with c H 5 φ , Y 5 y/δ as D

d2 φ 50 dY 2

ð3:2Þ

The membrane concentration, c, is given here in a unit of measure of kgmol/m3. This can be easily obtained by means of the usually applied g/g unit of measure with the equation of φ 5 wρ/M, where w is concentration (kg/kg); ρ is membrane density (kg/m3); and M is molar weight (kg/mol). Its dimensionless form can be given by Φ 5 φ/(coH), where H is the partition coefficient of the transported component in the membrane phase. After integration of Eq. (3.2) (its general integration is well known: Φ 5 TY 1 S) and using the boundary conditions without external mass transfer resistance (continuous line in the fluid phase boundary layer) in Figure 3.1, one can get the concentration as Y5

φ 2 φ φ 2 Hco Φ21 5 o o o φδ 2 φ Hcδ 2 Hc Cδ 2 1

ð3:3Þ

Figure 3.1 Illustration of the concentration distribution and the important variables through mass transport through a membrane layer.

Membrane βo βofδ

βof Co

C* Φ* Φ*δ ,

C*δ Cδo

δf

δfδ δ

Diffusion Through a Plane Membrane Layer

47

According to Eq. (3.3), the concentration inside the membrane layer changes linearly from φ to φδ through the layer that is from Hco to Hcoδ or from H to HCδo : The mass transfer rate can be given as (note that the value of J here, in this chapter, denotes the specific mass transfer rate without chemical reaction, kg/m2s or kmol/m2s) J5 2

DHco dΦ D D 5 ðφ 2 φδ Þ Hðco 2 coδ Þ β o Hco ð12Cδo Þ δ dY δ δ

ð3:4aÞ

Introducing the permeability constant P with P 5 DH, one can get from the mass transfer rate: J5

P o c ð12Cδo Þ δ

ð3:4bÞ

According to Eq. (3.4b), the permeability coefficient can be expressed here in units such as m2/s, because H is dimensionless due to its definition. Because different investigators can use different units, the permeability constant can have different units and even different definitions. Look at the situation when feed phase and also sweep fluid (permeate side; this is the case during dialysis or often during pervaporation) are flowing on the two sides of the membrane layer. Then a polarization layer can be formed in the fluid phases, which can essentially affect the mass transfer rate into and/or out of the membrane. This situation is illustrated by Figure 3.1 plotted by dotted line. The general solution of Eq. (3.1) is well known, φ 5 TY 1 S, where the constants T and S should be determined by means of the suitable boundary conditions. Taking into account the mass transfer resistance on both sides of the membrane, the boundary conditions can be given as at Y 5 0

then

β of ðco 2c Þ 5 2β o T

ð3:5Þ

at Y 5 1

then

β ofδ ðcδ 2coδ Þ 5 2β o T

ð3:6Þ

where β o 5 D/δ and β of 5 Df =δf as well as β ofδ 5 Dfδ =δfδ : After solution, the values of T and S can be expressed, for φ 5 TY1S, as T5

β oov o o ðc 2c Þ βo δ

ð3:7Þ

and S 5 Hcoδ 1

β oov o o o cδ 2 c βf

ð3:8Þ

where 1 1 1 1 o 5 o1 o1 o β ov β f Hβ β fδ

ð3:9Þ

48


The mass transfer rate can be given as: J 5 β oov ðco 2coδ Þ

ð3:10Þ

3.2.1 Concentration-Dependent Diffusion Coefficient It is clear that the classic solution-diffusion theory is only valid for governing permeation through essentially nonswollen membranes, for example, in the case of removal of dilute organics from water. When membranes are used for pervaporation dehydration, or organicorganic separation, appreciable membrane swelling usually occurs, and both the partition and diffusion coefficient become concentration dependent. Therefore, the classic solution-diffusion theory should be modified to adapt to the generally swollen pervaporation membranes. When a membrane is swollen or plasticized by transporting species, the interaction between polymer chains tends to be diminished, and the membrane matrix will therefore experience an increase in free volume. It is generally true that in a given membrane, increased free volumes correspond to increased diffusion coefficients of the penetrants. When a membrane is plasticized by more than one species, the diffusion coefficient of a species is facilitated by all the plasticizants. Many membranologists have found that the diffusion coefficient of species i in a ternary system of membrane/species i/species j could generally be expressed as (Schaetzel et al., 2004): Di 5 Dio expðα~ i φi 1 β~ j φj Þ

ð3:11aÞ

or in dimensionless form as: Di 5 Dio expðαi Φi 1β j Φj Þ

ð3:11bÞ

where Dio represents the diffusion coefficients of species i at infinite solution; φi and φj represent the local concentrations of the species of i and j in the membrane, respectively (kg/m3); and α~ and β~ are usually interpreted as the plasticization coefficients of the two species for the membrane (m3/kg). The plasticization coefficient of the less-permeable species can be neglected during dehydration processes, because dehydration membranes generally show overwhelming affinity for water, and the concentration of the less-permeable species in the membrane is negligibly small. The diffusion coefficients of both the species in the membrane are thus dependent on the concentration of water in the membrane phase alone. Thus, the diffusion coefficient of species i can be written as (Mulder, 1984): Di 5 Dio expðα~ i φi Þ

ð3:12Þ


49

Some other relationships were also found to be adequate for depicting the concentration-dependent diffusion coefficient as Di 5 Dio ð11 α~ i φi 1 β~ j φj Þ Both concentration dependencies are discussed briefly in this section. If the D is concentration dependent, Eq. (3.1) is to be replaced by d dφ D 50 dy dy

ð3:13Þ

Integrating between φ and φδ ; the two surface concentrations (Figure 3.1), one can get the mass transfer rate as 1 J 52 δ

φðδ

D dφ 5 φ

DH o ðc 2 coδ Þ δ

ð3:14Þ

where D5

1 ðφ 2 φδ Þ

φðδ

D dφ φ

ð3:15Þ

where D is the average value of the diffusion coefficient. It is important to note that the concentration depends on the concentration dependency of D, thus its value is not known, and thus, the real concentration distribution can be calculated by iteration, supposing a concentration distribution at beginning. Starting from this value, we should then calculate the value of Dm and then the concentration distribution. About three to four calculation steps are enough to get the correct concentration distribution by means of the real diffusion coefficients.

3.2.1.1 Exponential Concentration Dependency, ~ Do expðαΦÞ D 5 Do expðαφÞ Integrating Eq. (3.13) and applying Eq. (3.12), the mass transfer rate can be given ~ Þ: as ðα 5 αφ J 52

Do co H αΦ dΦ e δ dY

ð3:16Þ

After integration of Eq. (3.16), one can get as Φ5

1 αδ ln ðS2JYÞ α Do

ð3:17Þ

50


where S and J parameters should be determined by means of the boundary conditions as at Y50, Φ51 and y51, Φ 5 Φδ : Accordingly, one can get for the concentration distribution as Φ5

1 α ln e ð12 YÞ 1 YeαΦδ α

ð3:18Þ

The concentration distribution is plotted in Figure 3.2 as a function of the α exponent. The mass transfer rate can be given, applying the measurable concentra~ o Þ: ~ αHc tions, as ðα 5 αφ J5

Do Hco α o ðe 2 eαCδ Þ αδ

ð3:19Þ

If one takes into account the mass transfer resistances on both sides of the membrane, ~ o : it can be calculated applying the following equations ½Φ 5 φ=ðHco Þ; α 5 αHc J 5 β of ðco 2 c Þ

ð3:20Þ

J 5 β ofδ ðcδ 2 coδ Þ

ð3:21Þ

and J5

Do Hco αΦ αΦδ ðe 2e Þ αδ

ð3:22Þ

Taking into account the equality of the above equations, one can get as: J5

o o o Do Hco n αð12J=½βof co Þ o e 2 eαðJ=½βfδ c 1Cδ Þ αδ

ð3:23Þ

Figure 3.2 Concentration distribution with variable diffusion coefficient, D 5 Do eαΦ, without external mass transfer resistance.

Concentration distribution

1.0 α=10 0.8

5 3

0.6 1 0

0.4 0.2 0.0 0.0

0.2

0.4

0.6

Membrane layer Y

0.8

1.0


51

By Eq. (3.23), the J value can be obtained by simple iterative manner. Knowing the value of J, the S value can be calculated (in order to predict the concentration distribution) by one of the boundary conditions, for example replacing the limiting case of Eq. (3.17), namely if Y 5 0, into Eq. (3.20), one can get as: J

5 β of co

1 αδ 12 ln S α Do

ð3:24Þ

and from that becomes: S5

Do αð12J=½β of co Þ e δα

ð3:25Þ

Thus, the concentration distribution can be calculated in presence of mass transfer resistance in the feed and permeate phases by means of Eqs (3.17), (3.23), and (3.25). The average value of the diffusion coefficient can be calculated by Eq. (3.15) and thus, the mass transfer rate can be calculated by Eq. (3.14). Using this equation, the external mass transfer resistance can easily be taken into account by the following equation: J5

co ð1 2 Cδo Þ 1 δ 1 1 1 β of HD β ofδ

ð3:26Þ

The effect of the external mass coefficient is illustrated in Figure 3.3 on the mass transfer enhancement. It can be seen that the relatively large value of the external mass transfer resistance can strongly diminish the effect of the exponentially increasing diffusion coefficient.

Figure 3.3 The effect of the external mass transfer resistances on the mass transfer rate (β o 5 Do/δ; Jo denotes here the overall mass transfer rate with α 5 0 and J with α . 0, and β with α . 0; β of 5 β ofδ Þ:

3.0 Enhancement (J/J*)

o o βf /β →∞

2.5

10

5

2.0 1

1.5 1.0 0.1

0.1 1.0 Exponent (α)

10.0

52


~ Do ð1 1 αΦÞ 3.2.1.2 Linear Concentration Dependency, D 5 Do ð11 αφÞ

The mass transfer rate can be given, in this case, as follows (Φ 5 φ/φ ): J 52

Do dΦ φ ð11αΦÞ dY δ

ð3:27Þ

After solution of Eq. (3.13) with suitable boundary conditions, as were used for the case of exponential concentration dependence, the concentration distribution for the membrane layer can be obtained as (φ 5 Hco): φ 1 Φ 52 1 φ α

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2δ S2JY 1 2 α Do φ α

ð3:28Þ

with S5

Do o α Hc 11 δ 2

ð3:29Þ

J5

o αCδ Do o α c H 11 11 2Cδo δ 2 2

ð3:30Þ

and

Note that S and J values in Eqs (3.28)(3.30) have a dimension of kg/(m2 s), and α is dimensionless. Knowing the values of S and J, the concentration distribution can be calculated by means of Eq. (3.28). In order to take into account the external mass transfer resistances, the following algebraic equations system should be solved [Φ 5 φ /(Hco) where due to the external mass transfer resistance φ , Hco]: Do o αΦ αΦδ c H Φ 11 2 Φδ 11 J5 δ 2 2

ð3:31Þ

with C 5 1 2

J β of co

Cδ 5 Cδo 1

J β ofδ co

ð3:32Þ

ð3:33Þ


53

Replacing the C and Cδ (C 5 Φ ; Cδ 5 Φδ Þ from Eqs (3.32) and (3.33), respectively into Eq. (3.31), one gets a second-order algebraic equation to be solved. This becomes: J5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2B6 B2 24AE 2A

ð3:34Þ

with ! 1 1 α A5 2 o2 o2 βf β fδ 2co2 1 1 1 Cδo 1 1 1 α 1 1 1 B5 β ofδ β of β ofδ β of β o H co

ð3:35Þ ð3:36Þ

as well as E5

α ð12Cδo2 Þ 2 ð12Cδo Þ 2

ð3:37Þ

The effect of the external mass transfer resistances can be predicted by means of Eq. (3.34).

3.2.1.3 Optional Concentration Dependency of the Diffusion Coefficient The concentration dependency of the diffusion coefficient can be essentially different from the above-discussed function. Thus, a general solution of the problem will be shown in this section. The solution methodology will be discussed in this chapter. The membrane is divided into N sublayers with ΔY thickness, and the diffusion coefficient Di was assumed to be constant for every sublayer. See e.g. section A2 for the solution’s methodology. The concentration distribution for ith sublayer can be given as Φ 5 Ti Y 1 Si

at

Yi21 # Y # Yi

ð3:38Þ

where Ti 5 2

ð12 Cδo Þ H PN Di j 5 1 ðΔY=Dj Þ

ð3:39Þ

and Si 5 ΔY

i 21 X j51

Tj 2ði21ÞΔYTi 11

ð3:40Þ

54


The mass transfer rate can be given, for example, as co ð12Cδo Þ J 5 H PN δ j 5 1 ðΔX=Dj Þ

ð3:41Þ

As mentioned, the Di value depends on the concentration, thus the real concentration distribution and the mass transfer rate can be obtained by iterative method. The external mass transfer resistance can easily be taken into account in the mass transfer rate as J 5H

co ð12Cδo Þ PN o ðH=β f Þ1δ j51 ðΔY=Dj Þ1ðH=β of Þ

ð3:42Þ

3.2.2 Concentration-Dependent Solubility Coefficient, H 5 H(c) The second key factor determining permeability in a membrane is the sorption coefficient. The sorption coefficient can also vary as a function of the concentration, but this dependency is much lower than that of the diffusion coefficient (Chandak et al., 1998). The main types of this dependency are discussed briefly here, namely linear and the Langmuir-type absorption isotherms. Accordingly, assuming linear concentration dependency, one can get as ~ φ 5 Ho ð11αcÞc

ð3:43Þ

Thus in dimensionless form, it will be as (C 5 c/co): φ Φ 5 ð11αCÞC Ho co

ð3:44Þ

or in case of Langmuir-type sorption property (φsat is saturation sorption concentration at which all excess free volume sites are filled, kg/m3): φ 5 φsat

~ αc ~ 11 αc

ð3:45aÞ

~ o ; Φ 5 φ=φsat Þ : or ðα 5 αc Φ5

αC 11αC

ð3:45bÞ

According to the dual-sorption model, gas sorption in polymer occurs in two types of sites (Baker, 2004; Follain et al., 2010; Koros, 1980; Vieth, 1988). The total sorption can be written as: φ 5 Ho c 1

~ φsat αc ~ 1 1 αc

ð3:46Þ


55

According to the ENSIC (engaged species induced clustering) model (Favre et al. 1996; Shah et al., 2007), the activity of the solvent in the polymer is given by 1 A2B a 5 γc 5 log 11 φρ ð3:47Þ A2B B From Eq. (3.47), the φ(c) function can be expressed as φ5

B ðexp½γcðA2BÞ21Þ A2B

ð3:48Þ

where γ is the activity coefficient; ρ is the density of the swollen polymer membrane (kg/m3); and A and B are the parameters of the model. The ENSIC model is essentially an empirical model, but it can fit the sorption data for most polymer/solvent systems very well (Favre et al., 1996). Alternatively, a more thermodynamically rigorous model like the extended FloryHuggins model can also be used. According to the extended FloryHuggins model for a single component, the activity can be given as (Mulder, 1984) log a 5 logðφρÞ1ð12φρÞ1ðχ0 1χ1 φρÞφρ

ð3:49Þ

It can be obtained for φ(c) as φ 5 γc expð11½χ0 21φ1χ1 φ2 Þ

ð3:50Þ

where χ0 and χ1 are interaction parameters of the model. The role of the concentration dependency is especially important when the external mass transfer resistance cannot be neglected. In this case, the membrane’s interface concentration depends on the external mass transfer coefficients; thus, the H value changes according to the interface concentrations, namely to φ and φδ : Look briefly at how the concentration distribution can be obtained in the presence of limiting values of β of and β ofδ (see Figure 3.1, dotted line).

3.2.2.1 Linear Solubility Dependency The mass transfer rate, when the solubility coefficient changes according to Eq. (3.43), can be expressed as follows, without external mass transfer resistance, β of 5 β ofδ -N: J 5 β o co Ho ð11α 2 ½11αCδo Cδo Þ where βo 5

D δ

ð3:51Þ

56


A more important case is when the mass transfer resistances are not negligible. For that case, the mass transfer rate can be given by the following expression: J 5 β o Hco ð1 1 αC ÞC 2 1 1 αCδ Cδ ð3:52aÞ Replacing e.g. Eq. (3.32) and (3.33) into Eq. (3.52a) one can obtain a second order algebraic equation for J to be solved, namely: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2B6 B2 24AE ð3:52bÞ J5 2A with 1 1 A5 2 o2 o2 βf β fδ

!

α co2

1 112α 1 1 2αCδo 1 1 1 β ofδ co β of βo H E 5 1 1 α 2 1 1 αCδo Cδo B5 2

ð3:53Þ ð3:54Þ ð3:55Þ

3.2.2.2 Langmuir-Type Dependency ~ o; The mass transfer rate, without external mass transfer resistance, is as (α 5 αc 3 o o o kg/m ; Cδ 5 cδ =c Þ : ~ oδ ~ o αc α αCδo αc o sat β ð3:56Þ 2 2 φ J 5 β o φsat ~ o 1 1 αc ~ oδ 11 αc 11α 11αCδo The situation is rather complicated in this case; thus, the mass transfer rate is only defined here taking into account the external mass transfer resistance of the fluid phases as Cδo 1J=½β ofδ co J 12J=½β of co 2 5 β o φsat α 11αð12J=½β of co Þ 11αðCδo 1J=½β ofδ co Þ

ð3:57Þ

By means of Eq. (3.57), the J value can be easily obtained, for example, by iterative method. The other possibility is that by rearranging this equation, one can get a third-order algebraic equation that has an analytical solution.

3.2.2.3 Dual-Sorption Model The linear concentration distribution is not changed due to the sorption isotherm, thus the mass transfer rate can be given as ~ oδ ~ o αc αc 2 J 5 β o ðφ 2 φδ Þ 5 β o ðco 2 coδ ÞHo 1 φsat ~ o 1 1 αc ~ oδ 1 1 αc

ð3:58Þ


57

The mass transfer rate in presence of external mass transfer resistances can be given similarly as it was made previously: J 5 J1 1 J2

ð3:59Þ

J1 5 β oov ðco 2 coδ Þ

ð3:60Þ

with

J2 5 β o φsat α

12J=½β of co Cδo 1J=½β ofδ co 2 11αð12J=½β of co Þ 1 1 αðCδo 1J=½β ofδ co Þ

ð3:61Þ

with 1 1 1 1 o 5 o 1 o 1 o β ov β f Ho β β fδ where J1 and J2 denote the mass transfer rate according to the Henry’s law and to the Langmuir’s equation, respectively [kg/(m2 s)]. It has been demonstrated that the sorption isotherms of small molecule penetrant gases such as carbon dioxide, methane, argon, etc. in glassy polymers are generally concave to the pressure axis (Koros, 1980; Vieth, 1988). The dual mode sorption model is widely used to describe such behavior. The adsorbed amount will be the sum of the Henry’s and the Langmuir’s isotherm, namely: 0

φHbp φ 5 φD 1φH 5 kD p1 11bp

ð3:62Þ

where φ is the solubility (cm3 (STP)/cm3 polymer); kD is the Henry’s law dissolution constant (cm3 (STP)/cm3 polymer); b is the hole, microvoid, affinity constant 0 (1/Pa); φH is the hole saturation constant (cm3 (STP)/cm3 polymer); and p is the pressure (Pa). The following equation can be obtained from Eq. (3.61) (Koros et al., 1981; Vieth, 1988): φ 5 φD 1

KφD 11αφD

ð3:63Þ

with 0

φ b K5 H ; kD

p5

φD ; kD

α5

b kD

58


Dual mobility transport of a gas in a glassy polymer is described by Fick’s law, assuming that the two modes of sorption occur simultaneously and the diffusion can occur in both modes, though the diffusion coefficients are different in the two modes, as follows (Paul and Koros, 1976): J 5 JD 1JH 5 2DD

dφD dφ 2 DH H dy dy

ð3:64Þ

where φD and φH are given in Eq. (3.62); DD and DH are the respective diffusion coefficients of the two sorbed populations; J is the total diffusional flux; and JD and JH are the respective fluxes of the two populations. Another approach for penetrant transport is to define a phenomenological diffusion coefficient as the “effective diffusivity,” which expresses the flux in terms of the total concentration, φ: J 5 2Deff

dφ dy

ð3:65Þ

The gradient of φH can be expressed as a function of φD, according to Eq. (3.62): dφH K dφD 5 2 dy ð11αφD Þ dy

ð3:66Þ

Accordingly, the total mass transfer rate can be expressed as DH K=DD dφD J 5 2DD 11 ð11αφD Þ2 dy

ð3:67Þ

The question arises how the J mass transfer rate can be expressed as a function of the φ total concentration. Differentiating Eq. (3.63), one can get as: dφ K dφD 5 11 2 dy dy ð11αφD Þ

ð3:68Þ

Replacing Eq. (3.68) into Eq. (3.67), one can get the value of the Deff given in Eq. (3.69) as (Vieth, 1988; Follain et al., 2010): Deff 5 DD

DH K=DD 1 11 K 2 11 ð11αφD Þ ð1 1 αφ

ð3:69Þ DÞ

2


59

Deff diffusion constant contains the value of φD. For its elimination, its value should be expressed as a function of φ concentration. Applying Eq. (3.63), one can get as:

φD 5

2ð11K2 αφÞ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11K2αφÞ2 14αφ 2α

ð3:70Þ

Replacing Eq. (3.70) into Eq. (3.68), now the mass transfer rate can be estimated as a function of the total concentration. To develop Deff as a function of position in the membrane layer, after integration of Eq. (3.65), one can obtain 2Jy 1 S 5 DD φD 1DH φH

ð3:71Þ

The values of J and T can be determined by the boundary conditions as (Vieth, 1988): y50

then

S 5 DD φD 1DH φH

ð3:72aÞ

y5δ

then

2Jδ1T 5 0

ð3:72bÞ

Thus, J5

DD φD 1DH φH δ

ð3:73Þ

as well as y DD φD 1DH φH 5 ðDD φD 1DH φH Þ 12 δ

ð3:74Þ

where φD and φH are the values at the boundary. Expressing φH in terms of φD [Eqs. (3.60) and (3.61)], φD can be expressed as a function of y, and the effective diffusivity can subsequently be expressed as a function of y, namely: φD 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T1 T 2 1 4αξð12y=δÞ 2α

where T 5 11

DH y K2αξ 1 2 DD δ

ð3:75Þ

60


and ξ5

DD φD 1DH φH DD

Substituting Eq. (3.75) into Eq. (3.69), φD can be expressed as a function of y, and the effective diffusivity can then be expressed also as a function of position in the membrane layer.

3.2.2.4 ENSIC Model The membrane surface concentration can be obtained by this model as φ5

B ðeγcðA2BÞ 21Þ A2B

ð3:76Þ

Thus, the mass transfer rate without external mass transfer resistance can be expressed as J 5 βo

B o o ðeγðA2BÞc 2eγðA2BÞcδ Þ A2B

ð3:77Þ

The mass transfer rate with external mass transfer resistances will be as J 5 βo

B ðeγðA2BÞc 2eγðA2BÞcδ Þ A2B

ð3:78Þ

with c 5 co 2

J β of

ð3:79Þ

cδ 5 coδ 1

J β ofδ

ð3:80Þ

and

Applying Eqs (3.78)(3.80), the J mass transfer rate can be determined by trial-and-error method.

3.2.2.5 FloryHuggins Model According to this model, the φ(c) function is

1 2 φ exp 2 1 1 χ0 21 φ 1 χ1 φ 5c γ

ð3:50aÞ


61

In the presence of the external mass transfer resistance, the values of the membrane interface concentrations have to be determined, because J 5 β o ðφ 2φδ Þ: Taking into account Eqs (3.50a), (3.79), and (3.80), the surface membrane concentrations can be expressed in implicit equations. Thus: φδ 5 φ 2

β of φ exp ψ βo 1 fo co o β γ β

ð3:81aÞ

φ 5 φδ 1

β ofδ φδ exp ψδ β ofδ o 2 o cδ βo β γ

ð3:81bÞ

ψ 5 11ðχ0 2 1Þφ 1 χ1 φ2 ψδ 5 11ðχ0 2 1Þφδ 1 χ1 φ2 δ The values of φ and φδ can be determined by iteration. e.g. replacing φδ in Eq. (3.81a) into Eq. (3.81b), the value of φ (then that of φδ ) can be obtained. By means of Eq. (3.81), the value of φδ can be calculated by trial and error.

3.2.3

Mass Transfer Through a Composite Membrane

Two essential conditions should be taken into account: the thickness of the sublayers can be different and the solubility coefficient can also be different. The diffusion coefficient is also different, but its value is constant inside of every sublayer. It is assumed that the solubility coefficient for the sublayers can be defined as follows: Hi ci 5 Hi11 ci11

for i51 to N

ð3:82Þ

For the external phases: Hoco5H1c1 and HN cN 5 Hδ coδ (or in the case of external mass transfer resistances, according to Figure 3.1, Hoc 5H1φ and HN φδ 5 Hδ cδ Þ: The values of the parameters in Eq. (3.38) can be given as (C 5 c/co): Ti 5 2

1 ðHo 2 Hδ Cδo Þ N Di P ðHj ΔYj Þ j51

ð3:83Þ

Dj

and Si 5

i21 X Ho Hj 1 Tj ΔYj 2 Ti Yi21 Hi Hi j51

ð3:84Þ

Accordingly, the mass transfer rate is J 52

D1 T1 5 β o co ðHo 2 Hδ Cδo Þ δΔY1

ð3:85Þ

62


Taking into account the external mass transfer resistances on the both sides of the membrane, the mass transfer rate can be given as follows: J 5 co

3.2.4

Ho =H1 C o 2 Hδ =HN Cδo Ho =H1 ko 1 1=β o 1 Hδ =HN kδo

ð3:87Þ

Binary, Coupled Component Diffusion Transport

In addition, coupling phenomena could occur in most cases during the separation of liquid mixtures. A strong coupling of diffusion of components to be separated takes place in the MaxwellStefan approach (Wesselingh and Krishna, 2000). This approach could be applied to describe mass transport, with strongly concentrationdependent diffusion, during pervaporation of binary wateralcohol mixture, with low carbon number (Heintz and Stephan, 1994), as well as during membrane extraction of organic components (Kubaczka et al., 1998). Separation by means of zeolite, or, generally, of inorganic membranes is another important group of membrane separation processes where the MaxwellStefan approach to mass transfer is recommended (Krishna and Wesselingh, 1997; van den Graaf et al., 1999; van den Broeke et al., 1999). Another often recommended mass transport theory is the so-called FloryHuggins approach (Meuleman et al., 1999; Ghoreyshi et al., 2008). This theory is especially applicable for organophilic pervaporation of organic compounds in water. For the estimation of the countersorption diffusivity, Krishna and Wesselingh (1997) and Bitter (1991) proposed to use a generalized form of Vignes equation, which also means strong concentrationdependent diffusion coefficients. The differential mass balance equations, for binary, diffusional mass transport processes, can generally be given by Eqs (3.1) and (3.2) for all theories discussed earlier. The solution of these equations can only be done by numerical methods, which makes their applications difficult. Recently, Nagy (2006) gave an analytical approach for the solution, but the equations developed are very complex. The aim of this work was to develop a much simpler mathematical model that defines the concentration distribution of components in the cases of single component as well as that of binary, coupled diffusion processes in the membrane layer and the mass transfer rates of the diffusing components at the membrane interface, by means of explicit mathematical equations. This approach should generally be used independently of what function the diffusion coefficient of components could depend on the concentrations. As a practical example, the binary mass transport through zeolite membrane will be shown by applying the well-known MaxwellStefan approach. Obviously, the model can be used for multilayered membranes having space-dependent mass transport parameters.


63

3.2.4.1 Modeling of the Coupled Diffusion Some important assumptions of the mass transport for modeling (see also Chapter 1): (1) the binary mass transport takes place by diffusion only; (2) concentrations at the interfaces are in equilibrium with the gas (or liquid) phase; (3) the membrane can be plane or can have a cylindrical interface and diffusion in direction of the radius is taken into account; (4) mass transport is a steady-state process; and (5) diffusion coefficients are concentration/space dependent and the diffusion of the components is coupled to each other. The differential mass balance equations for binary, coupled diffusion through a plane membrane layer can generally be given as follows: d dφ d dφ LA A 1 LB B 5 0 dy dy dy dy

ð3:88Þ

d d dφB dφA 1 50 L LB dy dy A dy dy

ð3:89Þ

The membrane concentration is given here in a unit of measure of kgmol/m3. This can easily be obtained by means of the usually applied kg/kg unit of measure with the equation of φ 5 wρ/M, where w is the concentration (kg/kg); ρ denotes the membrane density (kg/m3); M is the molar weight (kg/mol). For a binary mixture, we have four effective diffusion coefficients, LA, LB, LA ; LB (Wesselingh and Krishna, 2009) that could strongly depend on both concentrations and on the real Fickian diffusion coefficients. Their values can be given, as we can see it later, by the given function of the concentrations in the membrane matrix or in membrane pores, φA, φB, and the diffusivities, DA, DB, DAB, depending on the approaches used for describing the mass transport. Basically, there are two main theories for the coupling diffusion process, namely the Maxwell Stefan theory (Heintz and Stephan, 1994; van den Graaf et al., 1999), and the FloryHuggins theory (Meuleman et al., 1999). It also should be mentioned that the Vignes and the modified Vignes equations can be used (Bitter, 1991) for the concentration dependency of the real diffusion coefficients given by Eq. (3.32). All theories mentioned above involve the above effective diffusion coefficients, and the component transport can be described by the differential equations (3.88) and (3.89). The specific mass transfer rate in dimension of m/s, at the membrane interface, can be given as follows: dφ dφ JA 5 2 LA A 1 LB B dy dy dφ dφ JB 5 2 LA A 1 LB B dy dy

at y 5 0

ð3:90Þ

at y 5 0

ð3:91Þ

64


In order to receive the mass transfer rates of component A and B, the differential equation system, given by Eqs (3.88) and (3.89), is to be solved. The schematic diagram of the physical model applied in order to get an analytical approach of the solution is illustrated in Figure 3.4. The membrane is divided into N sublayers with thickness of ΔY perpendicular to the membrane interface (ΔY 5 δ/N), with constant diffusion coefficient in all these sublayers. Consequently, the differential mass balance equations of the diffusing components for the nth sublayer becomes (Yn 21 , Y , Yn): LAn

2 d2 φAn d φBn 1L 5 0; Bn dY 2 dY 2

n51; . . . ; N

ð3:92Þ

LAn

d2 φAn d2 φBn 1LBn 5 0; 2 dY dY 2

n51; . . . ; N

ð3:93Þ

As general solutions of Eqs (3.92) and (3.93), the concentration distribution for components A and B can be given as follows: φAn 5 Tn Y 1Qn

ð3:94Þ

φBn 5 Sn Y 1En

ð3:95Þ

and

Figure 3.4 Schematic figure of concentration distribution of membrane sublayers and notations (for flat membrane, its value will be infinite).

δ

ο

CA

φ*

φ1

φn

φN ο

β fδ ο

βf

L1

Ln

L1

∗

Ln

Ln

ΔYn

ΔYn

ΔYN

∗

L N φδ* ∗

ο

Cδ A


65

The boundary conditions for the interface of the sublayers, at Y 5 Yn, are as LAn

LAn

dφAn dφ dφ dφ 1LBn Bn 5 LAn11 An11 1LBn11 Bn11 dY dY dY dY at Y5Yn ; n51; . . . ; N21

ð3:96Þ

dφAn dφ dφ dφ X1LBn Bn 5 LAn11 An11 1LBn11 Bn11 dY dY dY dY at Y5Yn ; n51; . . . ; N21

ð3:97Þ

as well as φAn 5 φAn11

at Y5Yn ;

n51; . . . ; N21

ð3:98Þ

φBn 5 φBn11

at Y5Yn ;

n51; . . . ; N21

ð3:99Þ

and

The boundary conditions at the external interfaces of the membrane layer can be distinguished in two cases, namely the external mass transfer resistances could be neglected or not. Both cases involve large parts of membrane processes depending on the hydrodynamic conditions of the streaming phases, as well as the transfer rate inside of the membrane. During gas mixture separation, it could often be neglected, while during liquid separation process often not. In this text, the mass transfer rate without external mass transfer resistance will be shown. This case can be used for mass transport of gas components through a membrane. The external boundary conditions will be: cA HA 5 φA ;

cB HB 5 φB

at Y 5 0

ð3:100Þ

and cAδ HAδ 5 φAδ ;

cBδ HBδ 5 φBδ

at X 51

ð3:101Þ

Through some mathematical manipulation, it can be obtained that the two differential equations [Eqs (3.77) and (3.78)] can be solved separately. Their solution is well known, and they are given in Eqs (3.79) and (3.80). Assuming N sublayers in the membrane, one can get N algebraic equations for the concentration distribution of both components. Each component has 2N parameters to be determined (Nagy, 2006). These parameters can be calculated by means of the 4N boundary conditions defined by Eqs (3.96)(3.99). The algebraic equation system containing 2N equations, obtained using Eqs (3.96)(3.99), was solved by means of the well-known Cramer rules using properties of determinants. Only the end results of this procedure will be

66


given here. The mass transfer rate of the transporting components can be given as follows at Y 5 0: JA 5 2ðLA1 P1 1LB1 S1 Þ

ð3:102aÞ

JB 5 2ðLA1 P1 1LB1 S1 Þ

ð3:102bÞ

where P1 5

MN ðφAδ 2φA Þ2FN ðφBδ 2φB Þ LA1 γ 1 ðVN ZN 2WN YN Þ

ð3:103Þ

with N N X X 1 βi ; 1ΔY 2 MN 5 ΔY LBi γ i L i51 Bi i52

FN 5 ΔY

N X βi L γ i52 Ai i

as well as VN 5

N X ΔY 1 αi ; 1ΔY 1 LA1 LAi LAi γ i i52

WN 5 2

αi 5

N X ΔY 1 1ΔY ; LA1 γ 1 L γ i52 Ai i

LA1 LAi 1 ; LA1 LAi

βi 5 2

ZN 5

N X ΔY αi 2ΔY γ L LB1 i52 Bi i

UN 5 2ΔY

LB1 LBi 1 ; LB1 LAi

γi 5

N X αi L γ i52 Bi i

LBi LAi 2 ; LBi LAi

i 5 22N

The value of S1 for the component B is as follows: S1 5

VN ðφAδ 2φA Þ 2 YN ðφBδ 2φB Þ LB1 ðVN ZN 2WN YN Þγ 1

ð3:104Þ

By means of Eqs (3.102)(3.104), the mass transfer rate, at the membrane interface, can be predicted for both components. If one wants to calculate the concentration distribution through the membrane layer, it can be done by means of the following equations according to the internal boundary conditions for components A and B, respectively: Pi11 5

JAi LBi11 2 JBi LBi11 ; LAi11 LBi11 2 LAi11 LBi11

2 # i # N21

ð3:105Þ


Si11 5

JBi LAi11 2 JAi LAi11 ; LAi11 LBi11 2 LAi11 LBi11

67

2 # i # N21

ð3:106Þ

as well as (Yi 5 iΔY) Qi11 5 φAi 2 Pi11 Yi

ð3:107Þ

Ei11 5 φBi 2 Si11 Yi

ð3:108Þ

and

3.2.5

Case Studies

We will show two examples for mass transfer through membrane using the MaxwellStefan approach: (1) separation of binary liquid mixture by pervaporation (Heintz and Stephan, 1994; Iza´k et al., 2003; Nagy, 2004, 2008) and (2) binary gas separation by zeolite membrane (van den Graaf et al., 1999; Wesselingh and Krishna, 2000).

3.2.5.1 Binary Gas Separation by Zeolite Membrane The mass transfer through solid membrane and, thus through zeolite layer, was extensively investigated in the literature (van den Broeke et al., 1999; van den Graaf et al., 1999). The diffusion coefficients for coupled diffusion, taking also into account the friction between the diffusing molecules (this effect is represented by the value of DAB), on the pore interface can be given as follows (Nagy, 2004): DB Qn ; LAn 5 qsat ερDA 12θBn 1θAn DABn DA Qn ; LAn 5 qsat ερDB θBn 1θBn DABn

DB Qn LBn 5 qsat ερDA θAn 1θAn DABn ð3:109aÞ

DA Qn LBn 5 qsat ερDA 12θAn 1θBn DABn ð3:109bÞ

and Qn 5

1 B A ð12θAn 2θBn Þ 11θAn DDABn 1θBn DDABn

ð3:110Þ

where θ is fractional loading (5 q/qsat); DA and DB are the diffusion coefficients of components in zeolite layer (m2/s); DABn is the MaxwellStefan interaction

68


parameter (m2/s); ε is porosity; qsat is the saturated concentration (kmol/kg); and ρ is the density (kg/m3). The values of DA and DB are assumed to be constant while the value of DAB is concentration dependent. The value of DABn can be given by the modified Vignes equation (Bitter, 1991): ðθ =ðθAn 1θBn ÞÞ

DABn 5 DA An

ðθ =ðθAn 1θBn ÞÞ

ð3:111Þ

DB Bn

The concentration distribution and the mass transfer rate of methane (component A) and ethane (component B) mixture were calculated and illustrated in two figures (Nagy, 2004). Both the concentrations and mass transfer rates were calculated by means of the predicted value of DAB, using the modified Vignes equation given by Eq. (3.96) (Figure 3.5; dotted lines), and for the case when DAB -N (Figure 3.5; continuous lines; this is the so-called single-file diffusion). The data used for calculation are given in Table 3.1 (van den Graaf et al., 1999). Values of KA and KB are parameters of the Langmuir isotherm, θA 5 qA/qsat. The overall pressure was kept to be 1 kPa in this side. Both the concentration and mass transfer rates (Figure 3.5) were calculated by the predicted value of DABn using the above-mentioned Vignes equation (dotted lines) and those without interaction, namely in the case when DABn -N (continuous lines). As can be seen, the mass transfer rate of methane is strongly affected by the friction between the molecules (Figure 3.3); consequently, the separation factor is also strongly altered as a function of the membrane concentration. 40.0 - - - - - - Coupled _____ Noncoupled

Flux (mmol m–2 s–1)

30.0

Methane

20.0

Ethane

10.0

0.0

1

25

P ethane (kPa)

75

99

Figure 3.5 The mass transfer rates of methane and ethane as a function of ethane partial pressure with adsorbateadsorbate interactions (dotted lines) and without that interaction (continuous lines; single-file diffusion, DAB-N).


69

Table 3.1 Parameters for the Zeolite Layer Applied for Calculation DA 5 21.4 3 10210 m2/s KA 5 3.1 3 1026 Pa21 θsat 5 1.85 mmol/g Psum 5 100 kPa Pδ 5 1 kPa

DB 5 5 3 10210 m2/s KB 5 57 3 1026 Pa21 ρ 5 1.8 3 106 g/m3 ε 5 0.2 δ 5 10 μm

It also is interesting to compare the change of the diffusion coefficients of the adsorbed components, in zeolite layer (not shown here). All L values, LA, LB, LA ; and LB ; change strongly in the membrane, and consequently, with the concentration. The concentrations of the components alter drastically the diffusion rate of the other components. The effect of the adsorbateadsorbate interaction is also essential.

3.2.5.2 Binary Transport for Pervaporation The separation of ethanol/water by poly(vinyl alcohol) membrane was discussed in detail by Heintz and Stephan (1994). The effective diffusion coefficients for the nth section of the membrane, applying the MaxwellStefan approach, can be given, for components A and B, as follows (Heintz and Stephan, 1994): LAn 5 DAn ρ LBn 5 DAn ρ LAn

5 DBn ρ

D12 1φAn DBn D12 1φAn DBn 1φBn DAn φAn DBn D12 1φAn DBn 1φBn DAn φBn DAn D12 1φAn DBn 1φBn DAn

ð3:112Þ ð3:113Þ ð3:114Þ

and LBn 5 DBn ρ

D12 1φBn DAn D12 1φ1 DBn 1φBn DAn

ð3:115Þ

where DAn and DBn are the diffusion coefficients of components in the membrane given for the concentration in the nth sublayer (m2/s); DAB is the MaxwellStefan interaction parameter (m2/s); and ρ is density of swollen membrane (kg/m3). Note that here DAn and DBn diffusion coefficients are not average values, their values should be fitted to the real concentration in the membrane. Thus, the calculation of the concentration distribution or the mass transfer rate needs a few iteration steps in order to obtain the real diffusion coefficients to the real concentration values. Equations (3.111)(3.113) can easily be obtained by the MaxwellStefan theory.

70


Diffusion coefficient (10–10 m2/s)

20.0

15.0 B: Ethanol

L

A: Water

A

10.0

LB

L* B

5.0

0.0 0.0

L* A

0.2

0.3

0.4

0.6

0.8

0.9

1.0

Membrane thickness

Figure 3.6 Change of effective diffusion coefficients obtained by Eqs (3.113)(3.115) in the membrane top layer. Coupled diffusion coefficient: D12 5 25 3 10215 m2 =s; noncoupled diffusion: D12 -N; DB 5 30 3 10215 m2 =s; DA φwater-0 5 10 3 10215 m2 =s (the water diffusivity is increased with its concentration; see Heintz and Stephan, 1994).

As illustration, we show typical concentration distributions of water and ethanol for two cases, namely with coupled (D12 5 25 3 10 211 cm2/s) and uncoupled ðif D12 -NÞ diffusion. The diffusion coefficient of water and ethanol in the membrane used for our calculations, as a function of concentrations, were measured by Hauser et al. (1989). The value of D12 was also predicted by Heintz and Stephan (1994). The interface concentrations of the upstream side (at y 5 0) were also used measured values. Concentrations were chosen to be zero for both components at the permeate side (at y5δ). The external mass transfer resistance and that for the porous support layer were neglected. As seen in Figure 3.2, the coupling of diffusion decreases the concentration change of components and, thus changes the mass transfer rates. The effective different diffusion coefficients, according to Eqs (3.112)(3.115), are plotted in Figure 3.6 as a function of the space coordinate in the membrane. [Measured diffusion coefficients of water and ethanol in the membrane, DA and DB and the calculated MaxwellStefan interaction parameter, DAB, used by us, are given by Heintz and Stephan (1994) and their related articles.] Their strong changes can be observed as a function of x, thus, as a function of the concentrations.

3.3

Nonsteady-State Diffusion

Let us consider a membrane layer with constant, uniform concentration. Its value can be equal to that existing on the membrane side with lower concentration; thus,


71

at t 5 0 the membrane concentration is kept at a concentration of φoδ 5 Hcoδ (Figure 3.1). Keeping the concentration on the other side at higher value, φ 5 Hc , the diffusion process will be started through the membrane toward the lower concentration side. To calculate the amount of species transferred into the membrane, it is first necessary to determine the concentration distribution of the transferred species within the membrane layer as a function of position and time. The differential mass balance equation can be the well-known Fick II equation, for the membrane layer under unsteady conditions as @φ @2 φ 5D 2 @t @y

ð3:116Þ

in which the diffusion coefficient is considered constant. The initial and the boundary conditions are at at at

t50 y50 y5δ

φ 5 Hcoδ φ 5 Hco φδ 5 Hcoδ

for all y t.0 t.0

ð3:117Þ

The concentration difference on the two sides of the membrane is kept constant; thus, the mass transport through the membrane is permanent, and after a certain time it will be constant, that is, steady state. This transitional state is described by Eq. (3.116), while the steady-state process is described by Eq. (3.1). Let us introduce the following dimensionless quantities: C o 2C Co 2Cδo

Φ5

c ; Hco

Y5

y δ

ð3:119Þ

τ5

Dt δ2

ð3:120Þ

Φ5

ð3:118Þ

Thus, one can obtain a dimensionless form of Eq. (3.121) as @Φ @2 Φ 5 2 @τ @Y at at at

τ 50 Y 50 Y 51

ð3:121Þ Φ 5 Cδo Φ51 Φ 5 Cδo

ð3:122Þ

72


Note that the boundary condition at Y 5 1 is not homogeneous and, as a result, the method of separation of variables cannot be applied. To overcome this problem, a solution is sought in the following form: Φðτ; YÞ 5 ΦN ðYÞ 2 Φt ðτ; YÞ

ð3:123Þ

in which ΦN(Y) is the steady-state solution, i.e.: d2 Φ N 50 dY 2

ð3:124Þ

with the following conditions: at Y 5 0 ΦN 5 1 at Y 5 1 ΦN 5 Cδo

ð3:125Þ

The solution will be as ΦN 5 11ðCδo 21ÞY

ð3:126Þ

However, the transient contribution of Φt(τ,Y) satisfies Eq. (3.121): @Φt @ 2 Φt 5 @τ @Y 2

ð3:127Þ

From Eqs (3.122) and (3.126), Φt 5 Y2Φ, thus the initial and boundary conditions become: at at at

τ 50 Y 50 Y 51

Φt 5 Cδo 2ΦN Φt 5 0 Φt 5 0

ð3:128Þ

Now the differential equation, Eq. (3.127), can be solved by the method of separation of variables that assumes that the solution can be represented as a product of two functions of the form Φt ðτ; YÞ 5 f ðτÞgðYÞ

ð3:129Þ

Substitution of Eq. (3.129) into Eq. (3.127) and rearranging it, one can get 1 df 1 d2 g 5 f dτ g dY 2

ð3:130Þ


73

The left side of Eq. (3.130) is a function of τ only, the right side depends on Y only. This is possible only if both sides of Eq. (3.130) are equal to a constant, 2λ2: 1 df 1 d2 g 5 2λ2 5 f dτ g dY 2

ð3:131Þ

The choice of a negative constant is due to the fact that the solution will decay to zero as time increases. Equation (3.131) results in two ordinary differential equations. The equation for f is given by df 1λ2 f 5 0 dτ

ð3:132Þ

The solution of Eq. (3.132) is f ðτÞ 5 e2λ

2

τ

ð3:133Þ

However, the equation g is d2 g 1λ2 g 5 0 dY 2

ð3:134Þ

with the boundary conditions at at

Y 50 Y 51

g50 g50

ð3:135Þ

The solution of Eq. (3.134) is gðYÞ 5 T sinðλYÞ 1 S cosðλYÞ

ð3:136Þ

where T and S parameters should be determined by means of boundary conditions given by Eq. (3.136). The boundary conditions given at Y51 in Eq. (3.135) give that S 5 0. The use of the boundary condition defined at Y51 in Eq. (3.135) results in T sin λ 5 0

ð3:137Þ

For nontrivial solution, the eigenvalues are given by λn 5 nπ;

n 5 1; 2; 3; . . .

ð3:138Þ

74


The most general solution of the Φt value is obtained by adding the solutions for all integrals from n 51 to n 5N; thus, the transient solution is Φt 5

N X

An e2n

π τ

2 2

ð3:139Þ

sinðnπYÞ

n51

The unknown coefficient An can be determined by using the initial condition in Eq. (3.128) at τ 5 0. The result is Cδo 2ΦN 5

N X

ð3:140Þ

An sinðnπYÞ

n51

Because the eigenfunctions are simply orthogonal, multiplication of Eq. (3.140) by sin(mπY) and integration from Y50 to Y51 gives ð1 ðCδo 2ΦNÞsinðmπYÞdY

5

N X

ð1 An sinðnπYÞsinðmπYÞdY

n51

0

ð3:141Þ

0

Note that the integral on the right side of Eq. (3.141) is zero when n6¼ m and nonzero when n 5 m. Therefore, when n 5 m, the summation drops out and Eq. (3.141) reduces to the form ð1

ð1 ðCδo 2ΦN ÞsinðnπYÞdY

5 An sin2 ðnπYÞdY

0

ð3:142Þ

0

Eigenvalues of the integral give An 5

2 ð12Cδo ÞcosðnπÞ; nπ

n 5 61; 62; 63; . . . ; 6N

ð3:143Þ

The transient solution takes the form Φt 5 2

N 2X 12Cδo 2 2 cosðnπÞe2n π τ sinðnπYÞ π n51 n

ð3:144aÞ

In limiting case, if Cδo -0, Eq. (3.144a) will be as Φt 5 2

N 2X 1 2n2 π2 τ e sinðnπYÞ π n51 n

ð3:144bÞ


75

Substitution of the steady-state and the transient solutions, Eqs (3.126) and (3.144) into Eq. (3.104) and taking into account Eqs (3.118) and (3.119) gives the solution as (Carslaw and Jaeger, 1959; Crank, 1975; Slattery, 1999): N 2 Cδo 21 2X 2 2 cosðnπÞe2n π Dm t=δm sinðnπYÞ Φ 5 1 2 1 2 Cδo Y 1 π n51 n

ð3:145Þ

The specific mass transfer rate at the membrane interface, at Y 5 0, as a function of time, can be given as: J 52

3.3.1

N X 2 DHco 2 2 ð12Cδo Þ e2ðn π D=δ Þt δ n51

ð3:146Þ

Mass Transport with External Mass Transfer Resistance on the Feed Side

In this case, the permeate side mass transfer resistance is neglected, β oδ -N: Thus, the initial and boundary conditions to be applied for the solution of Eq. (3.116) are as follows: at

t50

at

y50

at

y5δ

φ 5 φ @φ 2D 5 β of ðco 2c Þ @y φ50

for all y

ð3:147Þ

Applying dimensionless variables, given in Eqs (3.118) and (3.119) with φ 5 0 at y 5 δ, the above conditions can be expressed as follows: τ 5 0;

Φ 5 1 for all Y

ð3:148aÞ

Y 5 0;

Φ50

ð3:148bÞ

Y 5 0;

@Φ β of 5 o Φ 5 BiΦ β @Y

ð3:148cÞ

with Bi 5

β of βo

With solution by the separation of variables, the general solution can be given, similar to that discussed in Eqs (3.107)(3.134), as Θ 5 e2λ τ fT sinðλYÞ1S cosðλYÞg 2

ð3:149Þ

76


According to boundary conditions given by Eqs (3.148b) and (3.148c), one can get as T sin λ1S cos λ 5 0

ð3:150Þ

2Tλ cos 01Sλ cos 0 5 BiðT sin 01S cos 0Þ

ð3:151Þ

From Eqs (3.150) and (3.151), the following expression between the parameters T and S can be obtained: S 5 2T tan λ

ð3:152Þ

The eigenvalues of λ can be obtained by the following expression: Bi 5

λn tan λn

ð3:153Þ

Therefore, the transient solution after a few manipulations is Θ5

N X

An e2λn τ

n51

2

sin½λn ðY 21Þ cos λn

ð3:154Þ

Applying the initial boundary condition, Eq. (3.148a), the unknown An coefficient can be obtained by the following equation: 15

N X n51

An

sin½λn ðY 21Þ cos λn

ð3:155Þ

Applying the methodology used by Eqs (3.141) and (3.142), the An can be given as λn 1 sinð2λn Þ 2 An 5 cos λn ðcos λn 21Þ 2 4λn

ð3:156Þ

Thus, the transient solution takes the form N X co 2c λn 1 sinð2λn Þ 2λ2n τ sin½λn ðY 21Þ 5 e ϕ o c 20 n 5 1 cos λn ðcos λn 21Þ 2 4λn cos λn with λn given by Eq. (3.153).

ð3:157Þ


3.3.2

77

Solution of Fickian Diffusion by Boltzmann’s Transformation

Solution of Eq. (3.116) for short time can be made by transformation. At small values of time, the component does not penetrate very far into the membrane layer. Under these circumstances, it is possible to consider the slab as a semiinfinite medium in y-direction. The initial and boundary conditions become: at

t50

φ 5 φo

ð3:158aÞ

at

z50

φ 5 φ

ð3:158bÞ

at

z 5 N φ 5 φo

ð3:158cÞ

Introduction of the dimensionless concentration: φ2φo φ 2φo

Φ5

ð3:159Þ

Equation (3.116) reduces to @Φ @2 Φ 5D 2 @t @y

ð3:160Þ

at

t50

ð3:161aÞ

at

Y 50

at

Y 5N Φ50

Φ50 Φ51

ð3:161bÞ ð3:161cÞ

Introducing the variable η as pffiffiffiffiffi η y=ð2 DtÞ

ð3:162Þ

The chain rule of differentiation gives @Φ @f dη 1 η df 5 52 @t @η dt 2 t dη

ð3:163Þ

as well as @2 Φ d2 f @η 2 df @2 η 1 d2 f 5 1 5 2 2 2 @y dη @y dη @y 4Dt dη2

ð3:164Þ

78


Substitution of Eqs (3.163) and (3.164) into Eq. (3.160) gives d2 f df 12η 50 2 dη dη

ð3:165Þ

The boundary conditions, from Eqs (3.161b) and (3.161c), will be as at

η50

f 51

at

η5N

ð3:166aÞ

f 50

ð3:166bÞ 2

The integrating factor for Eq. (3.165) is exp(η ). Multiplication of Eq. (3.163) by the integrating factor yields d η2 df e 50 dη dη

ð3:167Þ

which gives that df 2 5 C1 e2η dη

ð3:168Þ

After integration of Eq. (3.168), one can get as f 5 C1

ðη

e2u du1C2 2

ð3:169Þ

0

where u is a dummy variable of integration. Application of the boundary condition at η50 gives C251. However, application of the boundary condition at η 51 gives pffiffiffi C1 5 22= π: Therefore, the solution becomes ðη 2 2 f 5 12 pffiffiffi e2u du 5 12erfðηÞ π

ð3:170Þ

Φ2Φo y p ffiffiffiffiffiffiffi ffi 5 12erf Φ 2Φo 4Dt

ð3:171Þ

0

or

The specific mass transfer rate will be as (Tosun, 2002): Φ 2Φo @φ 5 pffiffiffiffiffiffiffiffi J 5 2D @y y 5 0 Dπt

ð3:172Þ


3.3.3

79

Solution with a Variable Diffusion Coefficient

There is a variable diffusion coefficient, and its value can be changed as a function of concentration and/or space coordinate. The Fick’s law to be solved is as @φ @ @φ 5 D @t @y @y

ð3:173Þ

where D is a function of φ, e.g., D 5 D0 eαφ. In certain cases, the Boltzmann transformation may be employed to convert this to an ordinary differential equation as it is done in Eqs (3.162) and (3.164). The solution is discussed by Follain et al. (2010). For a numerical solution, see Section A.1.

References Baker, R.W. (2006) Membrane Technology and Applications, 2nd ed. Wiley, Chichester. Bitter, J.G.A. (1991) Transport Mechanisms in Membrane Separation Processes. ShellLaboratorium, Amsterdam. Chandak, M.V., Lin, Y.S., Ji, W., and Higgins, R.J. (1998) Sorption and diffusion of volatile organic compounds in polydimethylsiloxane membranes. J. Appl. Polym. Sci. 67, 165175. Carslaw, H.S., and Jaeger, J.C. (1959) Conduction of Heat in Solids. Clarendon Press, Oxford. Crank, J. (1975) Mathematics of Diffusion. Clarendon Press, Oxford. Favre, F., Nguyen, Q.T., Cle´ment, R., and Neél, J. (1996) The engaged species induced clustering (ENSIC) model: a unified mechanistic approach of sorption phenomena in polymers. J. Membr. Sci. 117, 227236. Follain, N., Valleton, J.-M., Lebrun, L., Alexandre, B., Schaetzel, P., and Metayer, M., et al., (2010) Simulation of kinetic curves in mass transport phenomena for a concentrationdependent diffusion coefficient in polymer membranes. J. Membr. Sci. 349, 195207. Ghoreyshi, A.A., Jahanshahi, M., and Peyvandi, K. (2008) Modeling of volatile organic compounds removal from water by pervaporation process. Desalination 222, 410418. Hauser, J., Heintz, A., Schmitttecker, B., and Lichtenthaler, R.N. (1989) Sorption equilibria and diffusion in polymeric membranes. Fluid Phase Equilib. 51, 369381. Heintz, A., and Stephan, W. (1994) A generalized solution-diffusion model of the pervaporation process through composite membrane. J. Membr. Sci. 89, 153169. Iza´k, P., Bartovska´, L., Friess, K., Sipek, M., and Uchytil, P. (2003) Description of binary liquid mixtures transport through nonporous membrane by modified MaxwellStefan equation. J. Membr. Sci. 214, 293309. Koros, W.J. (1980) Model for sorption of mixed gases in glassy polymers. J. Polym. Sci. Phys. Ed. 18, 981992. Koros, W.J., Smith, G.N., and Stanett, V.T. (1981) High-pressure sorption of carbon dioxide in solvent-cast poly(methyl methacrylate) and poly(ethyl methacrylate) films. J. Appl. Polym. Sci. 26, 159170.

80


Krishna, R., and Wesselingh, J.A. (1997) The MaxwellStefan approach to mass transfer. Chem. Eng. Sci. 52, 862906. Kubaczka, A., Burghardt, A., and Mokrosz, T. (1998) Membrane-based solvent extraction in multicomponent systems. Chem. Eng. Sci. 53, 899917. Meuleman, E., Bosch, B.B., Mulder, M.H.V., and Strathmann, H. (1999) Modeling of liquid/ liquid separation by pervaporation: toulene from water. AIChE J. 45, 21532160. Mulder, M.H.V. (1984) Pervaporation Separation of EthanolWater and of Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Nagy, E. (2008) Mass transport with varying diffusion and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Des. 86, 723730. Paul, D.R., and Koros, W.J. (1976) Effect of partially immobilizing sorption on permeability and the diffusion time lag. J. Polym. Sci. Phys. Ed. 14, 675685. Schaetzel, P., Vauclair, C., Nguyen, Q.T., and Bouzerar, R. (2004) A simplified solutiondiffusion theory in pervaporation: the total solvent volume fraction model. J. Membr. Sci. 244, 117127. Shah, M.R., Noble, R.D., and Clough, D.E. (2007) Measurement of sorption and diffusion in nonporous membranes by transient permeation experiments. J. Membr. Sci. 287, 111118. Slattery, J.C. (1999) Advanced Transport Phenomena. Cambridge University Press, Cambridge. Tosun, Í. (2002) Modelling in Transport Phenomena. Elsevier, New York. van den Broeke, L.J.P., Bakker, W.J.W., Kapteijn, F., and Moulijn, J.A. (1999) Binary permeation through a silicalite-1 membrane. AIChE J. 45, 977985. van den Graaf, J.M., Kapteijn, F., and Moulijn, J.A. (1999) Modeling permeation of binary mixtures through zeolite membranes. AIChE J. 45, 497. Vieth, W.R. (1988) Membrane Systems: Analysis and Design. Hanser Publisher, Munich. Wesselingh, J.A., and Krishna, R. (2000) Mass Transport in Multicomponent Mixtures. Delft University Press, The Netherlands.

4 Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

4.1

Introduction

The catalytic membrane reactor as a promising novel technology is widely recommended for carrying out heterogeneous reactions. A number of reactions have been investigated by means of this process, such as dehydrogenation of alkanes to alkenes, partial oxidation reactions using inorganic or organic peroxides, as well as partial hydrogenations and hydration. As catalytic membrane reactors for these reactions, intrinsically catalytic membranes can be used (e.g., zeolite or metallic membranes), or membranes that have been made catalytic by dispersion or impregnation of catalytically active particles such as metallic complexes, metallic clusters or activated carbon, and zeolite particles throughout dense polymeric or inorganic membrane layers (Markano and Tsotsis, 2002). In the majority of the above experiments, the reactants are separated from each other by the catalytic membrane layer. In this case, the reactants are absorbed into the catalytic membrane matrix and then transported by diffusion (and in special cases by convection) from the membrane interface into catalyst particles where they react. Mass transport limitation can be experienced with this method, which can also reduce selectivity. The application of a sweep gas on the permeate side dilutes the permeating component, thus increasing the chemical reaction gradient and the driving force for permeation (Westermann and Melin, 2009). For their description, two types of membrane reactors should generally be distinguished, namely intrinsically catalytic membranes and membrane layers with dispersed catalyst particle, either nanometer-sized or micrometer-sized catalyst particles. Basically, in order to describe the mass transfer rate, a heterogeneous model can be used for larger particles and/or a pseudohomogeneous one for very fine catalyst particles (Nagy, 2007). Both approaches, namely the heterogeneous model for larger catalyst particles and the homogeneous one for submicron particles, will be applied for mass transfer through a catalytic membrane layer.

Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00005-3 © 2012 Elsevier Inc. All rights reserved.

82


4.2

Steady-State Condition

The differential mass balance equation can generally be given by the following equation for the catalytic membrane layer with various geometries, perpendicular to the membrane interface (Ferreira et al., 2001; Julbe et al., 2001): d dφ D 2Q50 dy dy

ð4:1Þ

The membrane concentration, φ, is given here in a unit of measure of kmol/m3 or kg/m3. This can easily be obtained by means of the usually applied g/g unit of measure with the equation of φ 5 wρ/M, where w is the concentration (kg/kg); ρ is the membrane density (kg/m3); and M is the molar weight (kg/mol). The boundary conditions can depend on the external mass transfer resistance as discussed here. As mentioned, the catalytic membrane can be intrinsically catalytic or the membrane matrix can be made catalytic by dispersed catalytic particles. For a membrane with dispersed catalyst particles, the Q source term should involve the mass transport in the membrane matrix to the catalyst particle and the simultaneous internal transport, as well as the internal chemical reaction. Accordingly, the source term can be strongly different for a membrane reactor with dispersed catalyst particles or that for an intrinsically catalytic membrane layer. The mathematical description of the mass transport through these membrane layers can be different depending on the size of the catalytic particle. Thus, presentation of the mass transport equations is divided into two parts, namely: 1. Mass transport through intrinsically catalytic or nanometer-sized catalytic particles are dispersed in the membrane layer; in this case, it can be assumed that the mass transport inside the catalytic particles or the mass transport to the catalytic interface is instantaneous and catalytic particles can be located in every differential volume element of the membrane; accordingly, the membrane can be regarded as a continuous catalytic layer. 2. The dispersed catalytic particles fall into the micrometer-sized regime, the internal mass transport mechanism, inside of catalyst particles, must be taken into account. In this case, the so-called heterogeneous model should be used, which takes into account the internal mass transport as well.

4.2.1

Mass Transport with an Intrinsically Catalytic Layer or a Membrane with Fine (Nanometer-Sized) Catalyst Particles (Pseudohomogeneous Model)

In both cases, the membrane matrix is regarded as a continuous phase for the mass transport. Assumptions made for expression of the differential mass balance equation to the catalytic membrane layer are: G

G

Reaction occurs at every position within the catalyst layer. Mass transport through the catalyst layer occurs by diffusion.

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

Figure 4.1 Illustration of a catalytic membrane layer with concentration distribution and with important notations [layer with dispersed fine catalysts (points in the membrane layer) or intrinsically catalytic membrane].

Membrane

βfo

βo

83

βofδ φδ∗

J Cο

φ∗

C*δ o

Cδ

C∗ Jδ

x

G

G

G

G

y

The partitioning of the components (substrate, product) is taken into account (thus, c H 5 φ , where φ denotes membrane concentration on the feed interface; see Figure 4.1). The mass transport parameters (diffusion coefficient, partitioning coefficient) are constant. The effect of the external mass transfer resistance should also be taken into account. The mass transport is steady state and one dimensional.

In the case of dispersed catalyst particles, they are uniformly distributed and they are very fine particles with a size of less than 1 μm, i.e., they are nanometer-sized particles. It is assumed that catalyst particles are placed in every differential volume element of the membrane reactor. The reactant first enters in the membrane layer and from that, it enters into the catalyst particles where the reaction of particles is porous as active carbon, zeolite (Vital et al., 2001), occurs or enters onto the particle interface and reacts [the particle is nonporous as a metal cluster (Vancelecom and Jacobs, 2000)]. Consequently, the mass transfer rate into the catalyst particles has to be defined first. In this case, the whole amount of the reactant transported in or on the catalyst particle will be reacted. Then this term should be placed into the mass balance equation of the catalytic membrane layer as a source term. Thus, the differential mass balance equation for intrinsically catalytic membranes and membranes with dispersed nanometer-sized particles differ only by their source term. The cylindrical effect can be significant only when the thickness of a capillary membrane can be compared to the internal radius of the capillary tube as shown by Nagy (2006). On the other hand, the application of a cylindrical coordinate hinders the analytical solution for first- or zero-order reactions as well. Thus, the basic equations will be shown here for plane interface and in the section 6.3.2, an analytical approach will be presented for a cylindrical tube as well.

4.2.1.1 Reaction Terms for a First-Order Reaction Intrinsically catalytic membrane: The reaction term can be expressed by the following equation, where k1 is the reaction rate constant: Q 5 k1 φ

ð4:2Þ

84


where φ is the reactant concentration in membrane (kg/m3) and k1 is the first-order reaction rate constant (1/s). Catalyst with dispersed particles; reaction takes place inside of the porous particles: The differential mass balance equation for the catalytic membrane can be given as Dp

! dφ2p 2 dφp 2 k1 φp 5 0 1 dr 2 r dr

ð4:3Þ

with boundary conditions at r 5 Rp

φp 5 φp

ð4:4Þ

at r 5 0

dφp 50 dr

ð4:5Þ

where φp is the particle concentration (kg/m3); subscript “p” denotes the catalyst particles; r is the radial coordinate (m); Rp 5 r/rp; rp is the particle radius (m); and Dp is the diffusion coefficient in the catalytic particles (m2/s). It is assumed in Eq. (4.3) that the reaction takes place inside of the catalyst particles. The reactant diffuses in the catalyst particle and it reacts. This case, when the reaction occurs at the catalyst surface, is also briefly discussed later. The membrane layer with nanometer-sized catalyst particles is illustrated in Figure 4.1 where the very fine particles, illustrated by points, are uniformly distributed in the membrane matrix. For a catalytic membrane with dispersed nanometer-sized particles, the mass transfer rate into the spherical catalyst particle must be defined. The internal specific mass transfer rate in spherical catalyst particles, for steady-state conditions and when the mass transport is accompanied by a first-order chemical reaction, can be given as follows (Nagy and Moser, 1995): j 5 β p φp

ð4:6Þ

where βp 5 and ϑp 5

Dp ϑp 21 rp tanh ðϑp Þ

ð4:7Þ

sffiffiffiffiffiffiffiffi k1 rp2 Dp

where j is the mass transfer rate into catalytic particles (kg/m2 s) and φp is the concentration on the catalyst surface (kg/m3).


φ*

φ

j

βpo

δp

j

j

Figure 4.2 Mass transfer into the nanometer-sized catalyst particles in the membrane layer and the important notations.

dp

rp

βp

85

j h

The external mass transfer resistance through the catalyst particle depends on the diffusion boundary layer thickness, δp. According to Figure 4.2, the value of δp can be estimated from the distance of particles from each other, h [e.g., h 5 0.806 dp /ε1/3, where ε denotes the holdup of spherical catalyst in the membrane layer; the h value is discussed by Nagy (2007), in the case of cubic particles]. Namely, its value is limited by the neighboring particles; thus, the value of β op will be slightly higher than that which follows from the well-known equation of 2 5 β op dp =D; where the value of δp is supposed to be infinite. Thus, one can obtain (Nagy et al., 1989): β op 5

2D D 1 dp δ p

ð4:8Þ

h 2dp 2

ð4:9Þ

where δp 5

where D is the diffusion coefficient in the membrane around particles (m2/s); δp is the thickness of the boundary layer around particle (m); and h is the distance of particles modeled as cubic ones (m). From Eqs (4.6) and (4.8), one can obtain for the mass transfer rate with the overall mass transfer resistance j 5 β p tot φ 5

φ ð1=β op Þ1ð1=Hp β p Þ

ð4:10Þ

Accordingly, the k1 value in Eq. (4.2) can be expressed as follows (Nagy et al., 1989; Nagy, 2007): k1 5

ω β 1 2 ε p tot

ð4:11Þ

86


The ω value denotes the specific particle interface (ω 5 6ε/dp) in the membrane, (1 2 ε) means the portion of the membrane layer for mass transport, and ε denotes the catalyst particle holdup. Reaction occurs on the interface of the catalytic particles (Nagy, 2007): It often might occur that the chemical reaction takes place on the interface of the particles, for example, in cases of metallic clusters; the diffusion inside the dense particles is negligible. Assuming the Henry’s sorption isotherm of the reacting component onto the spherical catalytic surface, Hs (CHs 5 qs), applying D dφ/dr 5 ks1Hsφp boundary condition at the catalyst’s interface, at r 5 rp, the k1 reaction rate constant can be given according to Eq. (4.11) with the following β tot value: β p tot 5

1 ð1=β op Þ1ð1=ks1 Hs Þ

ð4:12Þ

where ks1 is the interface reaction rate constant (m3/m2 s). The above model is obviously a simplified one. Figure 4.3A illustrates the effect of catalyst phase holdup on the inlet mass transfer rate into the catalytic membrane layer, applying the pseudohomogeneous model (fine particles are dispersed in the membrane structure). The k1 value was calculated by Eq. (4.11). Both the reaction modulus and the catalyst holdup can strongly affect the inlet mass transfer rate of the membrane layer.

4.2.1.2 Mass Transfer Accompanied by First-Order Reaction Herewith, first the reaction source term will be defined in the case of intrinsically catalytic membrane and the solution of the differential mass balance equation under different boundary conditions. The differential mass balance equation for the catalytic membrane layer becomes D

d2 φ 2Q50 dy2

ð4:13Þ

The differential mass balance equation for the reactant entering the catalytic membrane layer is in dimensionless form of the space coordinate (Y 5 y/δ): d2 φ 2 ϑ2 φ 5 0 dY 2

ð4:14Þ

with sffiffiffiffiffiffiffiffiffi k1 δ 2 ϑ5 D

ð4:15Þ


J

31

87

(A) ϑP = 5

J0

1

21

0.5

11

0.15 1 0.0

0.1

0.2

0.3

0.4

Catalyst phase holdup, ε (B) 9.0 Enrichment (J/J o)

ο

ο

β f = β fδ → ∞ 7.0 5.0

ο

ο

β f = β fδ = 10 1

3.0

0.1

1.0 0.1

1.0

10.0

Reaction modulus

Figure 4.3 (A) The mass transfer rate as a function of the catalyst phase holdup obtained by the pseudohomogeneous model (D 5 1 3 10210 m2/s; Cδo 5 0; β of 5 β of;δ -N; dp 5 2 μm; δ 5 30 μm; δ is the thickness of the membrane layer). (B) The change of the relative value of the mass transfer rate as a function of the reaction modulus at different values of the external mass transfer coefficient coδ 5 0; β of 5 β ofδ :

Solution of Eq. (4.14) is well known: φ 5 TeϑY 1Se2ϑY

ð4:16Þ

For the sake of generalization, in the boundary conditions we should take into account the external mass transfer resistance on both sides of the membrane, though it should be noted that the role of the β oδ will be gradually diminished with the increase of the reaction rate. At the end of this subsection, the limiting cases will also be given briefly. Thus (both sides of boundary conditions are divided by co) it becomes

88


Y 5 0;

Y 5 1;

β of ðco

2 c Þ 5 2β dY

o dφ

β ofδ ðcδ 2 coδ Þ 5 2β o

ð4:17aÞ Y50

dφ dY Y51

ð4:17bÞ

Parameters of Eq. (4.16), applying Eqs. (4.17a) and (4.17b), will be as (they can easily be obtained by means of the known Cramer rules): 1 β ofδ coδ β of β of co β ofδ 1 o 11 o eϑ 12 o S5 βo β Hϑ β β Hϑ N

ð4:18Þ

with N 52

β of 1 β ofδ β of β ofδ 2sinh ϑ cosh ϑ 1 ϑ 1 βoH ðβ o H Þ2 ϑ

o 1 β ofδ coδ β of β of co β fδ 2ϑ T5 2 o 21 e 11 o βo β Hϑ β β o Hϑ N

ð4:19Þ

The mass transfer rate on the upstream side of the membrane can be given as follows (Nagy, 2007): Jov 5 βco ð1 2 FCδo Þ

ð4:20Þ

with

β 5 βo

β o ϑH 1 1 o tanh ϑ β ! fδ o2 1 β ϑH 1 1 1 o o tanh ϑ 1 β o o 1 o Hϑ β f β fδ βf β fδ

ð4:21Þ

and F5

1 β o ϑH tanh ϑ cosh ϑ 1 1 β ofδ

ð4:22Þ

with β of 5

Df ; δf

βo 5

D δ

The strong influence of the external mass transfer resistances is illustrated on the inlet enhancement in Figure 4.3B.


89

Similarly, the mass transfer rate for the downstream side of the membrane at Y51: β o ϑH o Cδo ð4:23Þ Jδ 5 β δ Hc 1 2 cosh ϑ tanh ϑ 1 β of with βδ 5

βo ϑ cosh ϑ

1 ðβ o ϑÞ2 2 1 1 Hβ o ϑ tanh ϑ 1 1 o o H 1 1 β f β fδ β of β ofδ

ð4:24Þ

Limiting cases: The transfer rate without external mass transfer resistances, namely when β of -N and β ofδ -N; can easily be obtained from Eq. (4.22) as a limiting case: β o ϑHco Cδo 12 ð4:25aÞ J5 tanh ϑ cosh ϑ Similarly, the outlet mass transfer rate without external mass transfer resistance will be as: Jδ 5

β o ϑHco

1 2 Cδo cosh ϑ sinh ϑ

ð4:25bÞ

Equation (4.25) is a well known mass transfer equation for liquid mass transfer accompanied by first-order reaction. Note that the overall mass transfer rate can also be obtained by means of resistance-in-series model. For that the J inlet mass transfer rates for the boundary layer of the feed side [J 5 β of ðco 2 c Þ] and that for the membrane layer Eq. (4.25a) as well as for the Jδ values for the outlet membrane layer

Eq. (4.25b) and for the boundary layer on the downstream side [Jδ 5 β ofδ cδ 2 coδ ] should be applied. The mass transfer rate can similarly be obtained for the case when the outlet concentration.

ð4:26aÞ J 5 β tot co 2 coδ where β tot 5

1 tanh ϑ 1 o 1 o Hϑβ βf

ð4:26bÞ

Avoiding the outlet flow of reactant is an important requirement for the membrane reactors. For this, the operating conditions should be chosen correctly. The concentration distribution is illustrated in Figure 4.4 without external mass transfer resistances, that is, β ofδ -N and β of -N:

4.2.1.3 Mass Transfer in an Ultrafiltration Operating Mode, i.e., dφ/dy 5 0 at y 5 δ For the sake of completeness, this mode will be shown here briefly. The solution of the mass balance equation is as given in Eq. (4.16). Boundary conditions neglecting the external mass transfer resistances are

90



1.0 0.8 ϑ=0

0.6

1

0.4

3

0.2 0.0 0.0

5

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 4.4 Concentration distribution p at ffiffiffiffiffiffiffiffiffiffiffiffi different ffi values of reaction modulus, without external mass transfer resistances ϑ 5 k1 δ=D; Cδo 5 0:5:

φ 5 φ

at y 5 0

ð4:27aÞ

dφ 50 dy

at y 5 δ

ð4:27bÞ

The concentration distribution will be as φ 5 φ

cosh½ϑð1 2 YÞ cosh ϑ

ð4:28Þ

where sffiffiffiffiffiffiffiffiffi k1 δ 2 ; ϑ5 D

Y 5 y=δ

The inlet mass transfer rate is as J 5 β o φ ϑ tanh ϑ

ð4:29Þ

Figure 4.5 illustrates the effect of the reaction rate on the concentration at zero outlet mass transfer rate. Figures 4.4 and 4.5 clearly demonstrate the significant effect of the operation modes on the concentration distribution and, consequently on the inlet mass transfer rate.

4.2.2

Mass Transfer Accompanied by Zero-Order Reaction

4.2.2.1 Source Term for Zero-Order Reaction The zero-order reaction is important primarily for biocatalytic reactions. That is why it will be briefly discussed here for the sake of completeness.


91


1.0 0.8 ϑ=0 0.6 0.4 3 0.2 5 0.0

0.0

0.2

0.4

0.6

0.8

1.0


Figure 4.5 Typical concentration distribution for ultrafiltration mode at different values of the reaction modulus dφ/dy50 at y5δ.

For intrinsically catalytic membranes, the source term is equal to k0 (Q 5 k0), where it is the reaction rate constant with a dimension of mol/m3/s. In the case of dispersed fine catalyst particles, the specific mass transfer rate should be given in order to use it as a reaction term in the mass balance equation given for the membrane phase. The differential mass balance equation for a spherical catalyst particle is as 2 dφ 2 dφ 1 2 k0 5 0 D dr 2 r dr

ð4:30Þ

with boundary conditions at r 5 Rp

φ 5 φ

ð4:30aÞ

at r 5 0

dφ 50 dr

ð4:30bÞ

Solving Eq. (4.28) with the boundary conditions (4.29) and (4.30), the concentration distribution can be given as φp 5 φp 2

k0 2 ðr 2 r 2 Þ Dp p

ð4:31Þ

The specific mass transfer rate, j, related to the particle interface can be expressed as j 5 2k0 rp

ð4:32Þ

92


Taking into account that the catalyst-specific interface in the membrane layer is equal to 3ε/rp, the reaction term for the mass balance equation of the membrane, Eq. (4.1), can be given as Q5j

ω 6k0 ε 5 12ε 12ε

ð4:33Þ

When the zero-order reaction takes place on the spherical catalyst surface, the reacted amount related to the surface can be given as j 5 ks0, thus the mass transfer rate due to the reaction can be given as (the dimension of ks0 is mol/m2/s): Q5

ω ks0 12ε

ð4:34Þ

The external mass transfer resistance can also affect the reacted amount. As applied, the mass transfer rate through the boundary layer around the catalyst particles is j 5 β op ðφ 2 φp Þ: Accordingly, the maximum value of the reaction term is determined by the external transfer rate, jmax 5 β op φ; thus Q # jmax.

4.2.2.2 Mass Transfer for Zero-Order Reaction In this case, the reaction rate is independent of the concentration of reactant in the membrane layer. The differential mass balance equation can be given as d2 Φ 5 ϑ2 dY 2

ð4:35Þ

The value of ϑ can be given for an intrinsically catalytic membrane as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6k0 δ2 ε ϑ5 DHco ð12 εÞ

ð4:36Þ

The general solution of Eq. (4.35) gives Φ 5 TY 1 S 1

ϑ2 Y 2

ð4:37Þ

Applying the known boundary conditions without external mass transfer resistance (at Y50, Φ5Φ 51 and at Y51, Φ 5 Φδ Cδo Þ; one can obtain for the concentration distribution in the membrane as φ 5 ðφδ 2 φ ÞY 1 φ 1 φ ðY 2 1ÞY

ϑ2 2

ð4:38aÞ


93

or in dimensionless concentrations: Φ 5 ðΦδ 2 1ÞY 1 1 1

ϑ2 YðY 2 1Þ 2

ð4:38bÞ

The mass transfer rate entering the catalytic membrane (at Y 5 0) is as J5

D ϑ2 φ ð1 2 Φδ Þ 1 2 δ

ð4:39aÞ

The outlet mass transfer rate can be given as D ϑ2 Jδ 5 φ ð1 2 Φδ Þ 2 2 δ

ð4:39bÞ

Let us look at the mass transfer with external mass transfer resistances. For the solution of Eq. (4.37), let us use the following boundary conditions: D ϑ2 Φ 2 Φδ 1 2 δ

at Y 5 0

β of ð1 2 C Þ 5 J

at Y 5 1

β ofδ ðCδ 2 Cδo Þ 5 Jδ

ð4:40aÞ ð4:40bÞ

as well as [from Eqs (4.39a) and (4.39b)]: Jδ 5 J 2

D 2 ϑ δ

ð4:40cÞ

From the above equations, it can be shown: J

5 β oov Hco

1 2 Cδo

β o ϑ2 1 o β fδ

ð4:41Þ

where 1 1 1 1 5 o1 1 o β oov βf Hβ o β fδ In order to determine the T and S parameters, Eqs (4.40a)(4.40c) can be rewritten, taking into account Eq. (4.37), as follows: 12

S βo 5 2T o βf H

ð4:42aÞ

94


1 ϑ2 βo 2 Cδo 5 2 o ðT 1 ϑ2 Þ T 1S2 2 β fδ H

ð4:42bÞ

Then, from Eqs (4.42a) and (4.42b), one can obtain for T and S: β oov βo 2 o T 5 o Cδ 2 12 o ϑ β β fδ

ð4:43aÞ

and S5

4.2.3

βoH 1H β of

ð4:43bÞ

Mass Transfer Accompanied by Second-Order Reaction

It is assumed that the reagents (components A and B) are fed separately on the feed (component A) and on the shell sides of the membrane reactor (component B), and that they are diffusing through the membrane layer concurrently (Figure 4.6) or in the countercurrent. The reaction term can be given for an intrinsically catalytic membrane as follows: Q 5 k2 φ A φ B

ð4:44Þ

Substituting the reaction term into Eq. (4.1) for both reactants and the plane interface, as well as steady-state condition (DA, DB are the diffusion coefficients for components A and B, respectively; their values can be constant or variable), one can get d dφA 2 k2 φA φB 5 0 DA dy dy

ð4:45aÞ

d dφB 2 k2 φA φB 5 0 DB dy dy

ð4:45bÞ

Figure 4.6 Illustration of the concentrations for a second-order reaction.

δ

o

CA

o

φBi

CB φAi

o CBδ

i

o

CAδ


Figure 4.7 Division of the catalytic membrane layer assuming constant parameters in these sublayers including the external mass transfer coefficients as well (Nagy, 2006, 2010).

δ

φ*

ο

CA

φ1 φi

D1

Di

95

φN

DN φδ* ο

CδA

ΔYl

ΔYi

ΔYN

This equation can be solved either by numerical method or by developing an analytical approach. The physical model to get an analytical approach is illustrated in Figure 4.7. Essential to this method is that the membrane layer is divided into N very thin sublayers (N $ 100) and parameters ϑAi ϑBi are assumed to be constant in every sublayer. Thus, one can get a second-order differential equation with a linear source term that can be solved analytically. In dimensionless space coordinate, the mass balance equation, for the ith sublayer, will be as d2 φ A 2 ϑ2Ai φA 5 0 for Yi 2 1 # Y # Yi dY 2

ð4:46aÞ

where ϑAi

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 δ2 φBi 5 DA;i

d2 φ B 2 ϑ2Bi φB 5 0 dY 2

for Yi 2 1 # Y # Yi

ð4:46bÞ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 δ2 φAi ϑBi 5 DB;i where φBi and φAi denote the average concentration of components in the ith sublayer (Figure 4.6). The general solution of Eq. (4.46) is well known [Eq. (4.16)]. The solution for every sublayer has two parameters that can be determined by

96


the suitable boundary conditions. The boundary conditions will be for model A as follows: at Y 5 0

φA 5 φA ;

at Yi 2 1 # Y # Yi at Yi 2 1 # Y # Yi at Y 5 1

φ 5 φBδ

ð4:47Þ

Di dφA Di 1 1 dφA 5 with i 5 1; 2; . . . ; N δ dY Yi2 δ dY Yi1 φA Yi2 5 φA Yi1 with i 5 1; 2; . . . ; N

φ 5 φAδ ;

φB 5 φB

ð4:48Þ ð4:49Þ ð4:50Þ

The internal boundary conditions for component B will be the same as for component A, adapting them for component B [Eqs (4.48) and (4.49)]. The N algebraic equations obtained can be solved using the well-known Cramer rules. After solution of the N differential equation with 2N parameters to be determined, the T1 and S1 parameters for the first sublayer can be obtained as (ΔY is the thickness of the sublayers): ! 1 φAδ T ξN φA 2 N T1 5 2 O 2ξN coshðϑA1 ΔYÞ Li 5 2 coshðϑAi ΔYÞ

ð4:51Þ

! 1 φAδ S ξN φA 2 N S1 5 O 2ξN coshðϑA1 ΔYÞ Li 5 2 coshðϑAi ΔYÞ

ð4:52Þ

and

For details of the general solution, see Section A.2. Note that in order to determine the T1 and S1 parameters, the average concentration of the B component should be known. For this, its correct value should be used. At the starting calculation, we should assume values for φBi ; it can also be zero. After differentiating Eq. (4.16) and applying it for the first sublayer, the mass transfer rate of component A can be expressed as 0 J5

1

B DA1 ϑA1 ξSN 2 ξTN B BφA 2 @ δ 2ξO coshðϑ ΔYÞ A1 N

C C C N A S T ðξN 2 ξ N Þ L coshðϑAj ΔYÞ φAδ

ð4:53Þ

j52

where ξ ij 5 ξ ij2 1 1 κij2 1

tanhðϑAi ΔYÞ zi 2 1

for i 5 2; 3; . . . ; N

and j 5 S; T; O

ð4:54Þ


97

and κij 5 ξ ij2 1 tanhðϑAi ΔYÞ 1

κij2 1 zi 2 1

for i 5 2; 3; . . . ; N 2 1 and j 5 S; T; O ð4:55Þ

The starting values of ξ1j and κ1j are as follows: ξT1 5 e 2 ϑA1 ΔY ;

ξS1 5 eϑA1 ΔY ;

ξO 1 5 tanhðϑA1 ΔYÞ

and κT1 5 2 e 2 ϑA1 ΔY ;

κS1 5 eϑA1 ΔY ;

κO 1 51

as well as zi 2 1 5

DAi ϑAi DAi 2 1 ϑAi 2 1

ð4:56Þ

Obviously, in order to get the inlet mass transfer rate of component A, the exact concentration distribution of component B is needed. Thus, for prediction of the J value, the concentration of component B must be known. It is easy to learn that a trial-and-error method should be used to get the component concentrations alternately. Three to four calculation steps are enough to get the correct value of one of the components. The concentration of a component gradually and automatically approaches its correct value belonging to a given concentration of the other component. Steps of calculation of concentration of both components can be as follows: 1. A starting concentration distribution for component B should be given and one calculates the concentration distribution of component A applying Eqs. (4.16) and (4.51)(4.57d) in three to four calculation steps, using the previously obtained concentration for every new step. 2. The indices of sublayers for component A have to be changed and adjusted to that of B starting from the permeate side of membrane, i.e., at Y 5 Yi, thus, subscript i of A, namely Ai, should be replaced by Y 5 YN 2 i. 3. Now applying the previously calculated averaged value of concentration of A ðΦAi Þ; one can predict the concentration distribution of component B, using Eqs. (4.51)(4.57d), adapting them to component B; ϑB,i parameters can significantly differ from the value of ϑA,i depending on the DB and DA values. 4. These three steps should be repeated three to four times until concentrations of both components do not change anymore.

Knowing the T1 and S1, the other parameters, namely Ti and Si (i 5 2, 3, . . . ,N), can be easily calculated by means of the internal boundary conditions given by

98


Eqs (4.48) and (4.49), starting from T2 and S2 up to TN and SN. Thus, one can get the following equations for prediction of the Ti and Si from Ti21 and Si21, for the component A: TAi eϑAi YAi 1 SAi e 2 ϑAi Yi 5 ΓAi 2 1

ð4:57aÞ

DAi ϑAi ðTAi eϑAi Yi 2 SAi e 2 ϑAi Yi Þ 5 ΞAi 2 1

ð4:57bÞ

ΓAi 2 1 5 TAi 2 1 eϑAi 2 1 Yi 1 SAi 2 1 e 2 ϑAi 2 1 Yi

ð4:57cÞ

ΞAi 2 1 5 DAi 2 1 ϑAi 2 1 ðTAi 2 1 eϑAi 2 1 Yi 2 SAi 2 1 e 2 ϑAi 2 1 Yi Þ

ð4:57dÞ

with

Now knowing the Ti and Si (with i 5 1, 2, . . . , N) parameters, the concentration distribution can be calculated easily through the membrane, its value at Y5Yi with i51 N: φA 5 TAi eϑAi Yi 1 SAi e 2 ϑAi Yi ;

Yi iΔY

ð4:58Þ

Now, knowing the value of TN and SN, the outlet mass transfer rate can be predicted by the following equation: JAδ 5 2

DAN ϑAN ðTN eϑAN 2 SN e 2 ϑAN Þ δ

ð4:59Þ

During the calculation, the concentration distribution of the component B will also be known. From that, its mass transfer rate can easily be calculated by a similar manner as for the component A. The concentration distribution is illustrated in Figure 4.8 with the sweep phase on the permeate side (model A, i.e., dφ/dy . 0). The reaction modulus was chosen five times higher for component B, thus its value of component A was relatively low; it varied between 0.2 and 1.2. As can be seen, the concentration of component A varies only in lesser extent.

4.2.3.1 The Concentration Gradient Is Zero on the Outlet Surface, dφB/dY 5 0 at Y 5 0 It can often occur during membrane processes that one of the reactants cannot enter the flowing (sweep) phase on the other side of the catalytic membrane, when it does not solve in that phase. Accordingly, its concentration gradient will be zero at Y51 (model B). Therefore, it seems important to analyze this mass transport process. The starting equations are the same as in the previous case. Only one of


99

1.0


0.8

ϑBi = 1 2

4

0.6 6 0.4 0.2 ϑAi = ϑBi / 5 0.0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 4.8 Concentration distribution for a second-order chemical reaction ðφAδ =φA 5 φBδ =φB 5 0:2Þ (continuous lines: component B; dotted lines: component A).

the external boundary conditions, namely that at Y 5 1, is different from the previous case, namely: at Y 5 1

then

dφB 50 dY

ð4:60Þ

The solution methodology remains the same as used for the previous case or for the second-order reaction. The values of T1 and S1 obtained will be as T1 5 2

κTN φB 2κO N coshðϑB1 ΔYÞ

ð4:61Þ

and S1 5

κSN φB 2κO N coshðϑB1 ΔYÞ

ð4:62Þ

j The value of κTN ; κSN ; and κO N should be calculated by Eq. (4.55), as well as ξ i j and κi (j5T, S, O; j512N21) and transforming it for component B [Eqs (4.61) and (4.62) are valid for component B, only while for the component A Eqs (4.51)(4.57) remain valid due to its unchanged boundary conditions]. The concentration distribution is plotted in Figure 4.9 at different values of reaction modulus for the case when the B reactant cannot leave the catalytic membrane

100



1.0

ϑBi = 1 2

0.8

4

0.6

6

0.4 0.2 0.0

ϑAi = ϑBi / 5 0.0

0.2

0.4

0.6

0.8

1.0


Figure 4.9 Concentration distribution of a second-order chemical reaction with no outlet flow of component B, dφB/dy50 at y50 (continuous lines: component B; dotted lines: component A).

layer. Strong effects of the reaction rate can be observed for component B. Accordingly, the inlet mass transfer rate of component B significantly increases with increasing values of the reaction modulus. On the other hand, applying the above expressions, the concentration level of the B reactant can be adjusted to the technological requirements. The concentration change of the component A is rather low.

4.2.4

Mass Transfer Accompanied by MichaelisMenten (or Monod) Kinetics

This example of reaction kinetics is especially important in the case of biochemical reactions, which is why this case will be discussed briefly.

4.2.4.1 With Sweep Phase on the Permeate Side, dφ/dY . 0 at Y51 This assumption is important regarding the boundary condition on the permeate side. Namely, it often occurs during catalytic or biocatalytic membrane reactor that there are feeding phases on both sides of the membrane reactor. Considering the membrane bioreactors, in order to increase the membrane transport rate, phases can be circulated on both sides of membrane. Thus, the external boundary conditions given by Eqs (4.47) and (4.50) should be used, assuming negligible external mass transfer resistances. The role of the general MichaelisMenten kinetics (or the Monod kinetics, in the case of living cells) is crucially important (Moser,


101

1988; Nagy, 2009) for biochemical reactions. Therefore, the analytical approach of this reaction will also be briefly discussed here, applying the approach used in the case of second-order reaction with constant concentration B. The reaction rate of the substrate in the catalytic membrane layer can be given as Q5

rmax φ KM 1 φ

ð4:63Þ

The source term can be rewritten for the ith sublayer as follows: Qi 5

rmax φ D ki φ KM 1 φi

ð4:64Þ

ki 5

rmax KM 1 φi

ð4:65Þ

with

Note that the value of φi denotes the average concentration of the substrate in the ith sublayer. It can be applied for its algebraic average value. In dimensionless form, one can get the following equation (Y5y/δ): d2 φ 2 ϑ2i φ 5 0 dY 2

for

Yi # φ # Yi 1 1

ð4:66Þ

with sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ 2 ki δ2 vmax 5 ϑi 5 Di Di KM 1 φi The solution of Eq. (4.66) is well known [Eq. (4.16)]. For calculation of the φi value, we should use its previously predicted value. This calculation should be repeated until the concentration will be constant in every sublayer. The prediction will automatically tend to its correct value, as this problem is a self-adjusted one. The concentration distributions are illustrated in Figure 4.10 at the case of firstorder, zero-order reactions, and an intermediate reactor rate. The starting value of ϑ was chosen to be constant for every case. Both parameters, namely vmax and KM, were chosen in such a way that the starting value of ϑ remains the same. During the process, its value changed according to the actual substrate concentration. Figure 4.10 clearly shows that there is an essential difference between results obtained by the limiting cases. On the other hand, the limiting cases, namely the first-order or zero-order reaction, might be used in special cases, only if someone wants to obtain real results.

102



1.0

0.8

0.6

0.4 First order 0.2 KM / φ*= 0.01

Zeroorder 0.0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 4.10 Concentration change applying the MichaelisMenten kinetics (the starting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of reaction modulus: ϑ 5 ðvmax =φ Þ=ðKM =φ 1 1Þ 5 2Þ:

4.2.4.2 The Concentration Gradient Is Zero on the Outlet Surface, dφ/dY 5 0 If the liquid phases, flowing on the two sides of the membrane, are immiscible and the reactant does not dissolve in the liquid on the permeate side, then its concentration gradient will be zero at Y 5 1. The same is true when there is no sweep phase. This is often the case for biocatalytic processes, e.g., hydrolysis of organic esters. Therefore, it seems important to analyze this mass transport process. The starting equations are the same as in the previous case. Only the external boundary condition is different from the previous case, namely: at Y 5 1 then

dφ 50 dY

ð4:67Þ

The solution methodology remains the same as used for the previous case or for the second-order reaction. The values of T1 and S1 obtained will be as T1 5 2

2κO N

κTN φ coshðϑ1 ΔYÞ

ð4:68Þ

and S1 5

2κO N

κSN φ coshðϑ1 ΔYÞ

ð4:69Þ


103

The values of κTN ; κSN ; κO N ; and those values necessary for the prediction of these three values are given by Eqs (4.54)(4.56). The inlet mass transfer rate then will be as J5 2

D1 κS 2 κTN φ ϑ1 O N δ 2κN coshðϑ1 ΔYÞ

ð4:70Þ

The concentration distribution can be calculated applying the internal boundary conditions as done in the previous case.

4.2.5

Mass Transfer Through an Asymmetric Catalytic Membrane

The mass transport through an asymmetric membrane is an important process in membrane separation. Most often the skin layer is noncatalytic; it serves as a layer retaining the catalytic particles in the sponge side of the membrane, while the sponge layer contains the catalytic particles, e.g., enzymes and microorganisms. Thus, chemical or biochemical reactions occur in the sponge layer only. In principle, feeding of the reactant substrate can be carried out on both sides of the membrane (Figure 4.11). Because the diffusion coefficient can be essentially different between the two membrane layers (mostly the diffusivity of the skin layer is one to two orders of magnitude less than that in the sponge layer), the inlet mass transfer rates can be different on the two sides of membrane.

4.2.5.1 Reactant Is Fed on the Sponge Side (Figure 4.11A) The differential mass balance equations for the sponge and the skin layers, respectively for first-order chemical reaction in the sponge layer, will be as D1

d2 φ 2 k1 φ 5 0; dy2

D2

d2 φ 5 0; dy2

Catalytic layer (1) C

J φ*1

C C C

(A)

ð4:71Þ

δ1 # y # δ

ð4:72Þ

Figure 4.11 Illustration of the mass transport through an asymmetric, catalytic membrane layer (for the internal interface: e.g., H1 φ1;δ 5 H2 φ2 Þ:

Noncatalytic layer (skin)

C

J φ*2

C C

C C C

0

0 # y # δ1

δ1 δ

C C C C

Jδ ,φ*2,δ (B)

C C C

C C C

0 δ2

δ

Jδ ,φ*1,δ

104


The general solution of the above equation system, with dimensionless space coordinate (Y 5 y/δ with δ 5 δ1 1δ2, where the subscripts 1, 2 denote the sponge and skin layers, respectively): φ 5 Aeðϑ

YÞ

1 Be 2 ðϑ

ð4:73Þ

YÞ

and φ 5 EY 1 F

ð4:74Þ

Boundary conditions for defining the parameters are as at Y 5 0

then φ1 5 A 1 B

ð4:75Þ

at Y 5

δ1 δ

δ1 then H1 ðAeϑ 1 Be 2 ϑ Þ 5 H2 E 1 F δ

ð4:76Þ

at Y 5

δ1 δ

then

D1 ϑ D2 ϑ ðAe 2 Be 2 ϑ Þ 5 E δ δ

ð4:77Þ

at Y 5 1

then φ2;δ 5 E 1 F

ð4:78Þ

with sffiffiffiffiffiffiffiffiffi k1 δ 2 δ ϑ ; ϑ 5 D1 δ1

sffiffiffiffiffiffiffiffiffi k1 δ21 ϑ5 D1

where H1 and H2 are the solubility constants, D1 and D2 are the diffusivities for the sponge and the skin layers, respectively, δ1 is the thickness of the sponge layer, δ denotes the overall thickness of the layers, and φ and φδ are the membrane surface concentrations as they are given in Figure 4.11. After solution, one can get the following parameter values: H2 δ2 2 ϑ D2 D2 2 ϑ ϑ e φ1 2 φ2;δ 2 e φ1 DA 5 H1 δ1 D1 D1

DB 5

H2 δ2 D2 D2 ϑ ϑ eϑ φ1 2 φ2;δ 1 e φ1 H1 δ1 D1 D1

δ DE 5 ϑ δ1

2 φ1

H2 1 φ cosh ϑ H1 2;δ

ð4:79Þ

ð4:80Þ

ð4:81Þ


DF 5 D5

D2 H2 δ φ2;δ sinh ϑ 2 ϑφ2;δ cosh ϑ 1 ϑ φ1 δ1 D1 H1

D2 H2 δ 2 sinh ϑ 1 ϑ cosh ϑ D1 H1 δ 1

105

ð4:82Þ ð4:83Þ

with I5

Di ; D

where I 5 A; B; E; F

ð4:84Þ

Let us give the inlet, J, and the outlet mass transfer rate, Jδ, which can be used in the differential mass balance equation given for the lumen and the shell sides: β o2 H1 1 β o2 1 1 φ1 2 o φ2;δ tanh ϑ 1 o β H β ϑ ϑ cosh ϑ 1 2 1 J 5 β o1 ϑ ð4:85Þ β o2 H1 1 11 o tanh ϑ β 1 H2 ϑ The outlet mass transfer rate, that at y5δ, is as H1 φ1 2 φ2;δ o H2 cosh ϑ Jδ 5 β 2 β o H1 1 tanh ϑ 1 1 2o β 1 H2 ϑ

ð4:86Þ

with sffiffiffiffiffiffiffiffiffi k1 δ21 ϕ5 ; D1

β o1 5

D1 ; δ1

β o2 5

D2 ; δ2

δ2 5 δ 2 δ1

where subscripts 1 and 2 represent the sponge (catalytic layer) and the skin (noncatalytic) layers, respectively. The overall mass transfer rate J given by Eq. (4.85) can also be obtained by the mass transfer rates given for the resistance-in-series model, as well. For it, the inlet and outlet mass transfer rates of the sponge layer, J1 and J1,δ, as well as the inlet mass transfer rate of the skin layer, J2, should be given, namely [these equations are partly known at gasliquid systems (Danckwerts, 1970; Nagy et al., 1982)]: J J1 5 β o1 J1;δ 5 β o1

φ1;δ ϑ φ1 2 cosh ϑ tanh ϑ

ϑ ðφ 2 cosh ϑφ1;δ Þ sinh ϑ 1

J2 5 β o2 ðφ2 2 φ2;δ Þ

ð4:87aÞ ð4:87bÞ ð4:87cÞ

106


Taking into account that J2 5 J1,δ, and H1 φ1;δ 5 H2 φ2 ; the value of φ1;δ can be expressed and it can be replaced into Eq. (4.87a). Thus, one can get Eq. (4.85).

4.2.5.2 Reactant Is Fed on the Skin Side Figure 4.11B illustrates this situation. Equations obtained for this case serve as a good opportunity to see the difference between the two feeding modes regarding the mass transfer rates. The differential balance equations and boundary conditions remain the same. The absolute values of the boundary conditions should be changed to get the desired transport direction. The concentration distribution in the skin layer was obtained to be as φ1;δ H2 φ2 2 δ H1 cosh ϑ y 1 φ2 φ2 5 2 ϑ D2 δ1 H2 δ2 δ ϑ 1 tanh ϑ H1 δ1 D1

with

0 # y # δ2

ð4:88Þ

That for the sponge layer is as

φ1 5 Aeϑ Y 1 Be 2 ϑ with

Y

1 D2 H2 2 ϑδ=δ1 D2 H2 δ2 2 ϑδ2 =δ1 e 2 e φ2 1 1 ϑ φ1;δ D1 H1 D1 H1 δ 1 G 1 D2 H2 ϑδ=δ1 D2 H2 δ2 ϑδ2 =δ1 B5 e e φ2 2 2 ϑ φ1;δ D1 H1 δ1 G D1 H1 A5

ð4:89Þ

ð4:90Þ ð4:91Þ

and

D2 H2 G52 sinh ϑ 1 ϑδ2 =δ1 cosh ϑ D1 H1

ð4:92Þ

The mass transfer rates on the feed side is as H1 φ1;δ H1 φ1;δ φ2 2 H2 cosh ϑ H2 cosh ϑ J 5 β o2 1 1 H1 1 β o2 H1 1 tanh ϑ 1 o tanh ϑ 11 o βo β 1 H2 ϑ β 1 H2 ϑ

φ2 2

ð4:93aÞ

The second form of Eq. (4.93a) is a well-known expression in the gasliquid systems where the first-order chemical reaction takes place in the liquid phase (Westerterp et al., 1984). The ratio of the mass transfer rates given by Eqs (4.85) and (4.93a) is plotted in Figure 9.9. This figure illustrates very well the difference between the two operation modes.


107

The outlet mass transfer rate is as o β o2 1 1 β 2 H1 1 φ2 2 1 tanh ϑ φ1;δ o β o1 H2 ϑ o β 1 ϑ cosh ϑ Jδ 5 β 1 ϑ β o H1 1 1 1 2o tanh ϑ β 1 H2 ϑ

ð4:93bÞ

The values of ϑ, β o1 ; β o2 are the same as in Eq. (4.86). The subscripts denote the same layer as the previous one, namely 1 denotes the spongy layer, while 2 denotes the skin layer. As expected, the mass transfer rates are independent of the direction of the transport process. Equations (4.85) and (4.86) are, by suitable application, the “same” as it was obtained by means of the countercurrent diffusion expressed by Eqs (4.93a) and (4.93b).

4.2.6

Mass Transfer with Micrometer-Sized, Dispersed, Catalyst Particles: Applying the Heterogeneous Model

Mostly, depending also on the membrane thickness, when particles fall into the micrometer-sized regime, the internal mass transport mechanism, inside of catalyst particles, must be taken into account. A simple physical model could be applied for the description of the process in this case, as schematically illustrated in Figure 4.12. The gas (or liquid) reactant enters first the catalytic membrane layer and then diffuses to the first catalytic particle, perpendicular to the membrane interface. The chemical reaction, namely a first-order, irreversible chemical reaction, takes place in the catalyst particles only. It is assumed that the concentration of the organic reactant should be much higher in a hydrophobic polymer membrane than that of the reactant investigated, such as peroxides, oxygen, hydrogen, and so forth. Then the nonreacted reactant diffuses through the first catalytic particle to its other side and enters again the βof

βofδ

O2

Jo

CH H2

J

Co

or

J

Coδ

Product

Figure 4.12 Membrane reactor with dispersed catalyst particles (for heterogeneous model, the spherical particles are modeled as cubic ones).

108


polymer membrane matrix, and so on (the route of this mass transfer process is illustrated by an arrow denoted by J in Figure 4.12). This diffusion path exists only for the heterogeneous part of the membrane interface (which is the projection of the cubic catalyst particle onto the membrane interface). There can be a portion of membrane interface, that is the so-called homogeneous part of the interface, where the diffusing reactant does not cross any catalyst particle (this mass stream is denoted by Jo in Figure 4.12). This also affects the resultant mass transfer rate. The assumed cubic (Mehra, 1999; Yawalkar et al., 2001; Nagy, 2002) catalyst particles are supposed to be uniformly distributed in the polymer membrane matrix. For the description of this transport process, the catalyst membrane layer should be divided into 2N 11 sublayers, perpendicular to the membrane interface. Namely, N sublayers for catalyst particles located perpendicular to the membrane interface, N 11 sublayers for the polymer membrane matrix between particles (ΔY) and between the first particle (Y1) and the last particle ð1 2 YN Þ and the membrane interfaces (Figure 4.13B). In order to get a mathematical expression for the mass transfer rates, a differential mass balance equation should be given for each sublayer. Thus, one can obtain a differential equation system containing 2N 11 second-order differential equations. This equation system with suitable boundary conditions can be solved analytically, which is also demonstrated in this chapter. The number of particles, N, and the distance between them, ΔY, can be calculated from the particle size, dp and the catalyst phase holdup, ε (Nagy, 2007). The distance of the first particle from the membrane interface, Y1 (Y1 5 y1/δ), can be regulated by the preparation method of the catalytic membrane layer. The differential mass balance equations for the sections of the polymer membrane phase and for that of the catalyst particles can be given, in dimensionless form, as follows, respectively: D d2 Φ 5 0; δ2 dY 2

0 # Y # Y1 ;

d2 Φ p k1 δ2 2 Φp 5 0; dY 2 Dp

(A)

Co C*

Yi 1

d # Y # Yi 1 1 ; δ

Yi # Y # Yi 1

Yi # Y # 1

ð4:94Þ

d δ

ð4:95Þ

(B)

φ∗

1

Cδo

Y1 0Y

1

i

N

ΔY

d Y •1

Yi

Y •i

YN

Y •N1

Figure 4.13 Concentration distribution in the membrane reactor (A) and a particle line with notations (B).


109

The solutions of the above differential equations, for the ith sections, are well known, namely: Φ 5 Ti Y 1 S i ;

1#i#N 11

ð4:96Þ

Φp 5 Ei expðϑYÞ 1 Fi expð2 ϑYÞ;

1#i#N

ð4:97Þ

with sffiffiffiffiffiffiffiffiffi k1 δ 2 ϑ5 Dp Thus, one can obtain 2N11 algebraic equations with twice as many parameters, Ti, Si (i 5 1, 2, 3, . . ., N11), as well as Ei and Fi (i 5 1, 2, 3, . . ., N), which are to be determined. Their values can be determined by means of suitable boundary conditions at the external interfaces of the membrane, at Y 5 0 and Y 5 1, as well as at the internal interfaces of every segment in the membrane matrix, at Yi and Yi with i 5 1, 2, 3, . . ., N. The effect of the external mass transfer resistances should be taken into account: β of ðco

2c Þ

β of

at Y 5 0

then

at Y 5 1

then β ofδ ðcδ 2 coδ Þ β ofδ

T1 DH dΦ 52 2β o HT1 c 2 H δ dY o

TN 1 1 1 SN 1 1 DH dΦ 2 coδ 5 2 H δ dY

ð4:98Þ

ð4:99Þ

2β HTN 1 1 o

The boundary conditions for the internal interfaces of the sublayers are also well known (Figure 4.11B): at Y 5 Yi

ðTi Yi 1 Si ÞHp 5 Ei eϑYi 1 Fi e2ϑYi

ð4:100Þ

at Y 5 Yi

DTi 5 Dp ϑðEi eΦYi 2 Fi e2ΦYi Þ

ð4:101Þ

as well as for the other side of the catalyst particles, namely at Y 5 Yi : at Y 5 Yi 1

d 5 Yi δ

ðTi 1 1 Yi 1 Si 1 1 Þ Hp 5 Ei eϑYi 1 Fi e2ϑYi

ð4:102Þ

at Y 5 Yi 1

d 5 Yi δ

DTi 1 1 5 Dp ϑ Ei eϑYi 2 Fi e2ϑYi

ð4:103Þ

110


Equations (4.100) and (4.102) express that there is equilibrium on the sublayer interfaces, while Eqs (4.101) and (4.103) show that there is no accumulation or source at the internal interfaces. Thus, an algebraic equation system with 2(N 11) equations can be obtained that can be solved analytically with a traditional method using the Cramer rules. The solution is briefly discussed in Nagy’s papers (2007, 2008) and in Section A.2. As a result of this solution, the mass transfer rate on the upstream side of the membrane interface, related to its heterogeneous part (which is the projection of the cubic catalyst particle onto the membrane interface), can be given as follows:

J 5 βHco 1 2 ΨCδo

ð4:104Þ

where β 5 β o Hp

UNT 1 1 1 Hp βo

N

L

UNO 1 1 βofδH 1 Hp i 5 1

αTi αO i

ð4:105Þ

as well as Ψ5

β ofδ

T N UN 1 1 1 Hp L αTi cosh ϑp

ð4:106Þ

i51

with " UNj 1 1

5

Hp ð1 2 YN Þ 1

ξ jN

#

αjN

YN 5 Y1 1 ðN 2 1ÞΔY 1 N

with J 5 T; O

d δ

ð4:107Þ

ð4:108Þ

as well as Hp φ 5 φp ;

Hco 5 φ ;

Hcδ 5 φδ

where φ , φδ denote the membrane concentration at membrane interfaces, subscript p denotes the catalyst particles in the membrane layer; The values of Uij , αji , and ξ ji with j 5 T, O should be calculated from sub-layer to sub-layer, that is from 1 to N (αji , ξji ) or N+1 (Uij ) from equations given in Table 4.1. Also it may be important to know the portion of the reactant that reacts in the catalytic membrane layer during its diffusion, or that there is an unreacted portion of the diffused reactant that passes on the downstream side of the membrane into


111

Table 4.1 The List of Variables That Should Be Applied for the Calculation of the Mass Transfer Rates For i 5 2 to N and j 5 T, O

αji 5 1 1 κUij tanh ϑp ; ξ ji 5

tanhðϑp Þ 1Uij ; κ

the starting values of parameters:

U1T 5 1; αT1 5 κ tanh ϑp ; ξ T1 5 1;

O αO 1 5 1 1 κU1 tanh ϑp ; as well as qffiffiffiffiffiffi2ffi ϑp 5 kD1 dp ;

κ5

Dp D

qffiffiffiffi k1 Dp

ξO 1 5

tanhðϑp Þ κ

βo

i21

o U1O 5 Hp Y1 1 Hp H ββ o ; f

1 U1O ;

ΔY 5 Yi 2 Yi2 1 ;

β po ϑp ;

ξj

Uij 5 Hp ΔY 1 αi21 ; j

β op 5

Dp d

;

βo 5

D δ

the continuous phase. This outlet mass transfer rate, for the heterogeneous part of the membrane interface, at X 5 1, can be given as follows: 0

1

J δ 5 β δ @1 2

Hp Cδo D E A ξO β o BN 2 AN αNO βfo N

ð4:109Þ

with β δ 5 β Hc o

o

ξO

BN 2 AN αNO N

β of βo

cosh N ϑp

UNO 1 1 1 Hp H ββo

o

ð4:110Þ

fδ

with Ai 5 2

1 βO tanh ϑp βO i 21 i21 ; B i5 2; . .. ; N B 1 A 5 2 B 1 A i 21 i21 O i i21 i 21 O αi 2 1 κUiO αi 2 1 UiO

The starting values of A and B will be as: Hp Y1 A1 5 1 2 O ; U1

tanh ϑp Hp Y1 B1 5 21 κ U1O

O O The values of αO i , β i and Ui (i 5 1, . . . ,N) are given in Table 4.1. The physical mass transfer rate for the heterogeneous part of the interface is as follows: o 5 β otot Hco ð1 2 Cδo Þ Jtot

ð4:111Þ

112


The physical mass transfer coefficient, with external mass transfer resistances, for the portion of the membrane interface where there are particles in the diffusion path, taking into account the effect of the catalyst particles as well, can be given by the following equation: β otot 5

1 H β of 1 β ofδ

1

1 βo

1

Nd Dp

1 Hp

2

Dp D

ð4:112Þ

Depending on the value of the diffusion coefficient Dp, solubility coefficient H, (φp5Hpφ), as well as the number of particles perpendicular to the interface, N, the value of the physical mass transfer coefficient of the membrane with catalytic particles, β otot , might be completely different from that of the membrane layer without catalyst particles, β o (β o5D/δ). For example, in the case of a polymer membrane filled with zeolite particles as catalyst, the value of Dp can be lowered by about four orders of magnitude than that in the polymer matrix (Yawalkar et al., 2001). The specific mass transfer rate related to the total catalytic membrane interface (Nagy, 2007) can be given as Jave 5 KJε2=3 1 J o ð1 2 Kε2=3 Þ

ð4:113Þ

or that for the outlet mass transfer rate: Jave δ 5 KJδ ε2=3 1 J o ð1 2 Kε2=3 Þ

ð4:114Þ

The value of the mass transfer rate can be easily obtained for the homogeneous part of the interface, Jo, namely: J o 5 β o Hco ð1 2 Cδo Þ

ð4:115Þ

In order to calculate the enhancement during the mass transfer accompanied by chemical reaction, the physical mass transfer rate related to the total membrane interface also should be defined: o o 2=3 5 KJtot ε 1 J o ð1 2 Kε2=3 Þ Jave

ð4:116Þ

The value of the factor K can be obtained from the distribution of catalyst particles in the polymer membrane matrix. Its value, depending on the particles distribution in the membrane matrix, should be K51 or K51.8715 (for details, see Nagy, 2007). The effect of the catalyst particle size is illustrated in Figure 4.14 at different reaction rate, applying the heterogeneous model. The mass transfer rate is very sensitive to the particle size, namely its value strongly decreases with the increase of the size. With increasing size decreases the number of particles, because the distance between them increases. Accordingly, the particle size should be decreasing as


13

113

Limiting value

11

J 0ave

Jave

Had=10

3

9

1

7

0.5

5

0.2

3 1 0.01

0.1 0.10

1.00

10.00

Catalyst particle size, dp, (μm)

Figure 4.14 Effect of the catalyst particle size on the mass transfer rate related to the total membrane interface, as a function of the reaction modulus (H 5 Hp 5 1; D 5 1 3 10210 m2/s; o Cfδ 5 0; β of 5 β ofδ -N; y1 5 1 μm; δ 5 30 μm; ε 5 0.1).

low as possible. Detailed analysis of this model and its comparison to the homogeneous one is given in Nagy’s paper (2007). The effect of the catalyst phase holdup is illustrated in Figure 4.15 at different reaction rates by heterogeneous (continuous lines) and the homogeneous (dotted lines) models. The mass transfer rates obtained by the homogeneous and the heterogeneous models are plotted in Figure 4.15 as a function of the catalyst holdup, at different reaction modulus. The results are in good agreement up to about ϑp 5 Had 5 1, proving that the two models give practically the same results at low particles size (dp , about 1 μm) and reaction rate. But they can differ from each other in the fast reaction rate regime.

4.2.7

Approaching an Analytical Solution of the Mass Transport with Variable Parameters

It is assumed that both the diffusion coefficient and reaction rate constant can vary as a function of space coordinate or concentration. The differential mass balance equation for first-order chemical reaction can be given as d dφ D½φ; y 2 kðyÞφ 5 0 dy dy

ð4:117Þ

Let us consider negligible external mass transfer resistance for the fluid phases, thus the usual boundary conditions will be used: at y 5 0, φ 5 φ and at y5δ, φ 5 φδ : For the solution of Eq. (4.117), the catalytic membrane should be divided into N sublayer (Figure 4.7), in the direction of the mass transport, that is

114


31 ϑp = 5 1

J 0ave

Jave

21

0.5

11

0.15 1 0.0

0.1

0.2

0.3

0.4

Catalyst phase holdup, ε

Figure 4.15 Enhancement as a function of the catalyst phase holdup at different reaction rates for heterogeneous model (continuous lines) and the pseudohomogeneous one (dotted lines) (Dp 5 D 5 1 3 10210 m2/s, β of 5 β of;δ -N; coδ 5 0; ϑp 5 Had, x1 5 0.01 μm) (x1 denotes the distance of the first particle to the fluid/membrane interface), δ 5 30 μm (δ is the membrane thickness), dp 5 2 μm (dp is the catalytic particle size).

perpendicular to the membrane interface, with thickness of Δδ (Δδ5δ/N) and with constant transport parameters in every sublayer for details of a general solution (see Nagy, 2008, 2009). Thus, for the nth sublayer of the membrane layer, using dimensionless quantities, it can be obtained: Dn

d2 φn 2 kn φn 5 0; dy2

yn 2 1 , y , yn

ð4:118Þ

In dimensionless form, one can get the following equation: d2 φ n 2 ϑ2n φn 5 0 dY 2

ð4:119Þ

where ϑn 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 kn =Dn

The solution of Eq. (4.118) is well known [see Eq. (4.16)]: φ 5 Tn eðϑn YÞ 1 Sn eð 2 ϑn YÞ ;

Y n 2 1 , Y , Yn

ð4:120Þ

Parameters Tn and Pn of Eq. (4.120) can be determined by means of the boundary conditions for the nth sublayer (with 1 # n # N). The boundary conditions at the


115

internal interfaces of the sub-layers (1 # n # N 2 1; Yn 5 nΔY; ΔY 5 1/N) can be obtained from the following two equations [Eqs (4.121) and (4.122)]: dφn Dn 1 1 dφn 1 1 5 dY Dn dY

at Y 5 Yn

φn Y 5 Y 1 5 φn 1 1 Y 5 Y 2 n

n

ð4:121Þ

at Y 5 Yn

ð4:122Þ

After solution of the N differential equation with 2N parameters to be determined, the T1 and S1 parameters for the first sublayer can be obtained as (ΔY is the thickness of the sublayers): ! 1 φ δ φ ζ TN 2 N ð4:123Þ T1 5 2 O 2ζ N coshðϑ1 ΔYÞ Li 5 2 coshðϑi ΔYÞ and 1 φδ S1 5 O φ ζ SN 2 N 2ζ N coshðϑ1 ΔYÞ Li 5 2 coshðϑi ΔYÞ

! ð4:124Þ

For details of the general solution, see Section A.2. Knowing the T1 and S1, the other parameters, namely Tn and Sn (n 5 2, 3, . . ., N), can be easily calculated by means of the internal boundary conditions given by Eqs (4.121) and (4.122) starting from T2 and S2 up to TN and SN, as it is given by Eqs (4.57a)(4.57d) in subscript i parameter. After differentiating Eq. (4.120) and applying it for the first sublayer, the mass transfer rate of the reactant component can be expressed as: ! D1 ϑ1 ζ SN 2 ζ TN φδ φ 2 S ð4:125Þ J5 N δ 2ζ O ðζ N 2 ζ TN ÞLj 5 2 coshðϑj ΔYÞ N coshðϑ1 ΔYÞ where ζ ji 5 ζ ji 2 1 1 κji 2 1

tanhðϑi ΔYÞ zi 2 1

for i 5 2; 3; . . . ; N

and j 5 S; T; O

ð4:126Þ

κji 2 1 zi 2 1

for i 5 2; 3; . . . ; N

and j 5 S; T; O

ð4:127Þ

and κji 5 ζ ji 2 1 tanhðϑi ΔYÞ 1

The starting values of ξj1 and κj1 are as follows:

116


Figure 4.16 Typical curves for illustrating the effect of variable diffuion coefficient (δ 5 100 μm; k1 5 0.1 s21; D 5 5 3 1029 25 3 10210 m2/s) [e.g., when D 5 (5 24.5i/N) 3 1029, N 5 100 then ϑ varies between 0.6 and 1.9].


1.0 0.8 0.6 0.4 ϑ = 0.6 − 1.91

0.2 0.0

ϑ = 0.6 − 1.91 0.0

0.2

0.4

0.6

0.8

1.0


ζ T1 5 e 2 ϑ1 ΔY ;

ζ S1 5 eϑ1 ΔY ;

ζO 1 5 tanhðϑ1 ΔYÞ

and κT1 5 2 e 2 ϑ1 ΔY ;

κS1 5 eϑ1 ΔY ;

κO 1 51

as well as zi 2 1 5

Di ϑi Di 2 1 ϑi 2 1

ð4:128Þ

For illustration, the concentration distribution is plotted with variable diffusion coefficients and consequently with variable reaction rate modulus (Figure 4.16). The diffusion coefficient was varied about one order of magnitude in the two cases, namely with increasing or decreasing values. The curves do not show significant difference with varying diffusion coefficients.

4.2.7.1 Mass Transfer in Ultrafiltration Mode, dφ/dy 5 0 at y 5 δ The general solution is given by Eq. (4.97) for that case as well. After solution of the N differential equation with 2N parameters to be determined, the T1 and S1 parameters for the first sublayer can be obtained as (ΔY is the thickness of sublayers): T1 5 φ and

κTN O 2κN coshðϑi ΔYÞ

ð4:129Þ


Figure 4.17 Concentration variation obtained by increasing reaction modulus [D 5 (2.5 20.25 3 9 3 i/ N) 3 1029], at the average ϑ value (D 5 1.25 3 1029), as well as by decreasing reaction modulus D5 (0.25 10.25 3 9 3 i/N) 3 1029, δ 5 100 μm; k1 5 0.1 s21.


1.0 0.9 0.8

ϑ = 1.9 → 0.63

0.7

ϑ = 0.89

0.6 ϑ = 0.63 → 1.9 0.5

0.0

0.2

0.4

0.6

0.8

117

1.0


S1 5 φ

κSN 2κO N coshðϑi ΔYÞ

ð4:130Þ

where κjN 5 tanhðϑi ΔYÞ 1

κji 2 1 zi 2 1

with j 5 T; S; O

ð4:131Þ

The other values of κji (i 5 2 2 N 21) are the same as it given by Eq. (4.55). Accordingly, other parameters are also the same as in Eqs. (4.126)(4.128). Figure 4.17 illustrates the concentration distribution with increasing and decreasing reaction modulus due to the change of the diffusion coefficient in the membrane, as well as at the average value of ϑ. In this case, the difference between the curves is relatively large. This indicates that the transport essentially can be affected by the anisotropy of a membrane.

4.3

Unsteady-State Diffusion and Reaction

Here we consider a case where dilute A is absorbed at the surface of a solid, catalytic membrane layer and then unsteady-state diffusion and reaction occur in the membrane. The absorbed A reacts by a first-order reaction: A 1 B-C

ð4:132Þ

and the rate of generation is k1φ. The mass balance equation will be then as

118


@φ d2 φ 5 D 2 2 k1 φ 5 0 @t dy

ð4:133Þ

The initial and boundary conditions are t 5 0;

φ50

for y . 0

ð4:134aÞ

y 5 0;

φ 5 φ

for t . 0

ð4:134bÞ

y 5 δ;

φ 5 φδ

for t . 0

ð4:134cÞ

Let us solve Eq. (4.133) with the conditions (4.134a)(4.134c) by Laplace transformation. The Laplace-transformed form of Eq. (4.133) will be as ~ d2 Φ sδ2 ~ 2~ Φ 2 ϑ Φ 5 D dY 2

ð4:135Þ

~ is the Laplace transform where Φ is the dimensionless concentration, Φ 5 φ/φ , Φ of Φ(s) and Y 5 y/δ. After integration of Eq. (4.135), one can get

~ 5 TeΘY 1 Se 2 ΘY Φ

ð4:136Þ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sδ2 Θ 5 ϑ2 1 D The T and S parameters can be determined by the following boundary conditions: Y 5 0;

~ 5 1=s Φ

ð4:137aÞ

Y 5 1;

~ 5 φ δ Φδ Φ φ s s

ð4:137bÞ

~ Φ5 12

Φδ Φδ 2 eΘ ΘY e 1 e 2 ΘY 2s sinh Θ 2s sinh Θ

ð4:138Þ

The dimensionless concentration distribution can be obtained by the inverse transformation of Eq. (4.138) as


119

~ ΦðtÞ 5 L 2 1 ½ΦðsÞ The solution for fast reaction where the catalytic membrane can be considered as a semiinfinite medium (Danckwerts, 1950): pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi φ 1 y p ffiffiffiffiffi expð 2 y k 5 =D Þerfc t 2 k 1 1 φ 2 2 tD pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 y 1 expðy k1 =DÞerfc pffiffiffiffiffi 1 k1 t 2 2 tD

ð4:139Þ

References Danckwerts, P.V. (1950) Absorption by simultaneous diffusion and chemical reaction. Trans. Faraday Soc. 46, 300305. Danckwerts, P.V. (1970) Gas-Liquid Reactions, McGraw-Hill, New York. Ferreira, B.S., Fernandes, P., and Cabral, J.M.S. (2001) Design and modeling of immobilized biocatalytic reactors, in Multiphase bioreactor design. Ed. by J.M.S. Cabral, M. Mota, J. Tramper), Taylor & Francis, London, pp. 85180. Julbe, A., Farusseng, D., and Guizard, C. (2001) Porous ceramic membranes for catalytic reactors—overview and new ideas. J. Membr. Sci. 181, 320. Marcano, J.G.S., and Tsotsis, T.T. (2002) Catalytic Membranes and Membrane Reactions. Wiley-VCH, Weinheim. Mehra, A. (1999) Heterogeneous modeling of gas absorption in emulsion. Ind. Eng. Chem. Res. 38, 24602468. Moser, A. (1988) Bioprocess Technology. Springer, Wien. Nagy, E. (2002) Three-phase oxygen absorption and its effect on fermentation. Adv. Biochem. Eng. Biotechnol. 75, 5181. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Nagy, E. (2007) Mass transfer through a dense, polymeric, catalytic membrane layer with dispersed catalyst. Ind. Eng. Chem. Res. 46, 22952306. Nagy, E. (2008) Mass transport with varying diffusion- and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Design 86, 723730. Nagy, E. (2009) Mathematical Modeling of Biochemical Membrane Reactors, in Membrane Operations, Innovative Separations and Transformations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 309334. Nagy, E. (2010) Convective and diffusive mass transport through anisotropic, capillary membrane. Chem. Eng. Process. Process Intens. 49, 716721. Nagy, E., and Moser, A. (1995) Three-phase mass transfer: Improved pseudo-homogeneous model. AIChE J. 41, 2334. Nagy, E., Blickle, T., and Ujhidy, A. (1989) Spherical effect on mass transfer between fine solid particles and liquid accompanied by chemical reaction. Chem. Eng. Sci. 44, 198201.

120


Nagy, E., Blickle, T., Ujhidy, A., and Horva´th, K. (1982) Mass transfer accompanied by first order intermediate reaction rate in two phase cocurrent flow with axial dispersion. Chem. Eng. Sci. 37, 18171819. Vancelecom, I.F.J., and Jacobs, P.A. (2000) Dense organic catalytic membrane for fine chemical synthesis. Catalysis Today 56, 147157. Vital, J., Ramos, A.M., Silva, I.F., Valenete, H., and Castanheiro, J.E. (2001) Hydration of α-pinene over zeolites and activated carbons dispersed in polymeric membranes. Catalysis Today 67, 217223. Yawalkar, A.A., Pangarkar, V.G., and Baron, G.V. (2001) Alkene epoxidation with peroxide in a catalytic membrane reactor: a theoretical study. J. Membr. Sci. 182, 129213. Westermann, T., and Melin, T. (2009) Flow-through catalytic membrane reactors—principles and applications. Chem. Eng. Process. 48, 1728. Westerterp, K.R., van Swaaij, W.P.M, and Beenackers, A.A.C.M. (1984) Chemical Reactor Design and Operation. John Wiley and Sons, New York.

5 Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

5.1

Introduction

Convective mass transport can take place if the transmembrane pressure difference exists between the two membrane sides. There are several membrane processes when the diffusive driving force, for the mass transport, is enlarged by the pressure difference between the two sides of membrane, causing convection flow as well. The investigation of the simultaneous effect of the diffusive and convective flows is especially important when the measures of the two flows are comparable with each other. This can be the case by nanofiltration membrane bioreactor and so forth.

5.2

Mass Transport Without Chemical Reaction

The mass balance equation for a plane membrane layer with constant transport parameters to be solved as (D denotes here the effective diffusion coefficient in the membrane layer): D

d2 φ dφ 50 2υ 2 dy dy

ð5:1Þ

where φ is the concentration in the membrane layer (kg/m3); υ is the convective velocity (m/s); y denotes space coordinate, perpendicular to the membrane surface (m); D is the diffusion coefficient in the membrane (m2/s) (note that the membrane diffusion coefficient means the effective one in the membrane which should involve the effect of the membrane structure, the interaction between the transported molecules and membrane material, etc.; see Chapters 3 and 4). In dimensionless space, coordinate (Y 5 y/δ): d2 φ dφ 50 2 Pe 2 dY dY Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00006-5 © 2012 Elsevier Inc. All rights reserved.

ð5:2Þ

122


where (Pe denotes the Peclet number) Pe 5

υδ D

The concentration distribution, without chemical reaction, is very simple to obtain. The solution of the differential equation, Eq. (5.2), without source term is as follows: φ 5 TePeY 1 S

ð5:3Þ

where Y denotes the dimensionless space coordinate (Y 5 y/δ) and δ is the thickness of the membrane layer (m). Values of parameters T and S can be determined, without external diffusive mass transfer resistances, by the boundary conditions according to the value of concentration gradient on the downstream (permeate) side of membrane. There can be diffusive flux into the fluid phase on the permeate side, i.e. dφ/dy . 0 at y 5 0. The boundary conditions are as (see Fig. 11.1 for denotes and the possible concentration distribution): at Y 5 0

φ 5 φ

ð5:4aÞ

at Y 5 1

φ 5 φδ

ð5:4bÞ

The concentration distribution, applying Eqs (5.4a) and (5.4b) as boundary conditions, is as follows: ePeY=2 Pe PeY 2Pe=2 ½1 2 Y φ 1 e sinh sinh φδ φ5 sinhðPe=2Þ 2 2

ð5:5Þ

The concentration distribution is illustrated in the membrane layer as a function of the Peclet number in Figure 5.1. The outlet membrane concentration was chosen to be 0.1. The overall mass transfer rate, namely the sum of the convective and the diffusive flows, can be expressed as J 5 2D

dφ 1 υφ dy

ð5:6Þ

Thus, the physical mass transfer rate, through the membrane, can be obtained by means of Eqs (5.6) and (5.7) as follows: J o 5 β}o ðφ 2 e2Pe φδ Þ

ð5:7Þ

where β}o 5

D ePe Pe ePe=2 βo Pe Pe e 21 δ 2 sinhðPe=2Þ

ð5:8Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

Figure 5.1 Concentration distribution in a layer as a function of the Peclet number, without external mass transfer coefficient.

1.0 Concentration distribution

10 0.8

3 1

Pe = 0.01

0.6

123

5

2

0.1 0.4 0.2 0.0 0.0

φδ*/ φ* = 0.1 0.2

0.4

0.6

0.8

1.0

Membrane thickness

Note that the physical mass transfer coefficient in the case of the simultaneous effect of the diffusion and convection, β}o (it denotes the overall mass transfer coefficient in the presence of convective and diffusive flows), can be much higher than that of the diffusion mass transfer coefficient, β o 5 D/δ. The β}o value increases linearly if Pe . about 3. It should be noted also that the driving force depends on the Peclet number as it is given by Eq. (5.7). With the increase of the Pe number, the driving force also increases. In a limiting case, namely if Pe-~, the value of e2Pe φδ 5 0: The physical mass transfer rate can also be given by diffusive mass transfer resistances in the boundary layers of fluid phases (see Figure 5.3A). The mass transfer rate for the boundary layers can be expressed as follows: if Y 5 0

then J o 5 β of} ðco 2 e2Pef c Þ

ð5:9Þ

where, according to Eq. (5.8) (Pef 5 υδf/Df; f subscript denotes the external feed phase, while δ subscript denotes the permeate or shell phase): β of} 5 β of

Pef ePef =2 2 sinhðPef =2Þ

ð5:10Þ

as well as for the permeate side (with subscript fδ) boundary layer ðPefδ 5 υδfδ =Dfδ υ=β ofδ Þ: if

Y 5 1 then

J o 5 β ofδ} ðcδ 2 e2Pefδ coδ Þ

ð5:11Þ

The value of β ofδ} can be obtained with the same equation than β of} [Eq. (5.10)] with β ofδ 5 Df δ =δf δ :

124


Applying Eqs (5.7), (5.8), and (5.11), the mass transfer rate for the overall mass transfer resistances will be as J o 5 β oov co 2 e2ðPef 1 Pe 1 Pefδ Þ coδ

ð5:12Þ

1 1 e2Pef e2ðPef 1 PeÞ 1 o 5 o 1 o β ov} β f} β} H β ofδ}

ð5:13Þ

with

Concentration distribution with concentration boundary layer on the feed side with dφ/dy . 0 at Y 5 1: Equation (5.12) gives the mass transfer rate with mass transfer resistances on both sides of the membrane. Regarding its importance, the concentration distribution will be given here also in the presence of a concentration polarization layer on the feed phase only. The mass transfer resistance is neglected on the permeate phase. For it, the mass balance equation for both layers should be solved as a differential equation system. Equation (5.2) should also be adapted to the feed boundary layer as well. Its solution is the same as in Eq. (5.3). The external boundary conditions for the solution of the parameters are as usual: y 5 0, c 5 co and at y 5 δf 1 δ then c 5 coδ (β ofδ -N; φδ 5 Hcoδ ). The internal boundary conditions express that the total transfer rate and the concentration are equal to each other on the inner edge of the boundary layer and on the feed side of the membrane, that is, at y 5 δ. The parameters obtained will be as (Hc 5 φ). For the liquid (fluid) phase (Nagy and Borbe´ly, 2007): c 5 Tf ePef 1 Sf

ð5:14Þ

where Tf 5

co ð1 2 e2Pe 2 HÞ 1 Hcoδ e2Pe HePef 2 e2Pe 1 1 2 H

ð5:15aÞ

and Sf 5 H

co ePef 2 coδ e2Pe HePef 2 e2Pe 1 1 2 H

ð5:15bÞ

For the membrane phase with parameters T, S in Eq. (5.3): T 5H S 5 Sf

2co ePef 1 coδ ðHePef 1 1 2 HÞ Pe ePeð11δf =δÞ½He f 2ePe112H

ð5:16aÞ ð5:16bÞ


125


Pe = 10 0.8 Pef = 1

0.6

1

o β f = 1×10 –4 m/s

0.4

0.5

δf / δ = 1

0.2

2

0.1

0.0 0.0

0.4

0.8

1.2

1.6

2.0

Boundary layer and membrane thickness

Figure 5.2 Concentration distribution in the boundary layer and membrane layer at different Pe numbers.

The concentration distribution in both layers is illustrated in Figure 5.2. It is clearly shown that the value of the concentration on the membrane interface, namely at y 5 δ (dotted vertical line), strongly depends on the Pe value. The overall mass transfer rate in the presence of the feed side mass transfer resistance also will be increased. For two-layer diffusive plus convective mass transfer, the overall transfer rate will be as [see Eq. (5.12)] (Nagy, 2007a; Nagy and Kulcsa´r, 2009): J o 5 β oov ðco 2 e2ðPef1PeÞ coδ Þ

ð5:17Þ

with β oov 5

1 ð1=β of} Þ 1ðe2Pef =β}o Þ

where Pe and Pef are the Peclet numbers of the membrane and the polarization layers, respectively. Mass transfer without diffusive flux into the permeate side, i.e., dφ/dy 5 0 at Y 5 1. This situation can exist when there is no sweep phase on the downstream side or the transported component is immiscible in the sweep phase. This letter case can often occur, for example, at enzymatic bireactions, when the two reactants are in immiscible phases. Let us look first the case when the membrane is considered as a “black box,” and the boundary conditions of the boundary layer at memo o 5 JY51 Þ: brane interfaces are as ðJY50 2D

dc 9 1 υc9Y50 5 υcp dy Y50

ð5:18aÞ

126


and c 5 co

Y 50

ð5:18bÞ

Applying the general solution of the concentration, given by Eq. (5.3) in the membrane, for the boundary layer, one can get the concentration distribution of the boundary layer as: c 2 cp 5 ePef Y co 2 cp

ð5:18cÞ

From the above expression, the following well-known equation can be obtained at Y 5 1 (where c 5 c ): c 2 cp 5 ePef Y co 2 cp

ð5:18dÞ

Details about the origin of the above expression are given in Section 11.4. Equations (5.18c) and (5.18d) do not contain any information on the concentration distribution or its gradient inside the membrane (it can be larger than zero as it is used for description of pervaporation process; see Chapter 11). Assuming that dφ/dy 5 0 at Y 5 1 (here at y 5 δf 1 δ; see Figure 11.1), then the concentration gradient and, due to it, the diffusive flow will be zero along the whole membrane, accordingly the outlet mass transfer rate ðJδo Þ as well. Thus, Jδo 5 υφ

ð5:18eÞ

The mass transfer rate, given by Eq. (5.9) for the boundary (polarization) layer, and that for the membrane layer [Eq. (5.18e)] are equal to each other. Applying the resistance-in-series model, one can get for the overall mass transfer rate as: o 5 Jov }

υePef β of Pef ePef o c co o HυðePef 2 1Þ 1 1 Hβ f Pef ðePef 2 1Þ 1 1

ð5:18fÞ

It is easy to see when the solubility coefficient is equal to unit, i.e., H 5 1 (the transported component is not dissolved in the membrane matrix), then o 5 β of Pef co υco Jov }

ð5:18gÞ

Thus, there is diffusive flux neither in the membrane layer and nor in the boundary layer. Note that it was assumed that the convective velocity has the same value in both layers. But it can also occur that the υ value differs in the membrane and the boundary layers. Especially, when the diffusive and the convective flows have similar values. This can be the case of the nanofiltration which is discussed in detail in Chapter 10. This case is not discussed here.


5.3

127

Diffusive Plus Convective Mass Transport with an Intrinsic Catalytic Layer or with Fine Catalytic Particles

Recently, it was proved in the literature (Ilinitch et al., 2000; Nagy, 2010) that the presence of convective flow can improve the efficiency of the membrane reactor. Thus, the study of the mass transport in the presence of convective mass flow can be important in order to predict the reaction process. On the other hand, the use of convective flow is rather rare, because the aim is mostly to minimize the outlet rate of the reactant on the permeate side.

5.3.1

Mass Transport Accompanied by First-Order Reaction

The differential mass balance equation for the polymeric or macroporous ceramic catalytic membrane layer, for steady state, taking both diffusive and convective flow into account, can be given as D

d2 φ dφ 2υ 2 k1 φ 5 0 2 dy dy

ð5:19Þ

or in a dimensionless form of space coordinate d2 φ dφ 2 Pe 2 ϑ2 φ 5 0 dY 2 dY

ð5:20Þ

where υδ Pe 5 ; D

sffiffiffiffiffiffiffiffiffi k1 δ 2 ϑ5 D

where υ denotes the convective velocity; D is the diffusion coefficient of the membrane; and δ is the membrane thickness. In the case of dispersed catalyst particle in the membrane matrix, the reaction takes place in the particles or at their interface. In this case, the k1 “reaction rate” can be defined by the following equation, as discussed in detail in Chapter 4 (Nagy, 2007b): k1 5

ω β 1 2 ε ptot

ð5:21Þ

where β ptot is determined by the mass transfer rate into the catalyst particles [see Eq. (4.12) or Eq. (28) in Nagy, 2007b]; ω is the specific interface of catalyst particles in the membrane ( 5 6ε/dp, where dp is the particle size, m) (m2/m3); and ε is the catalyst phase holdup.

128


φ~ 5 φe2PeY=2

ð5:22Þ

Introducing a new variable, φ~ [Eq. (5.17)], the following differential equation is obtained from Eq. (5.20): d2 φ~ 2 Θ2 φ~ 5 0 dY 2

ð5:23Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 1 ϑ2 Θ5 4

ð5:24Þ

The general solution of Eq. (5.23) is well known, so the concentration distribution in the catalytic membrane layer can be given as follows: ~

φ 5 TeλY 1 SeλY

ð5:25Þ

with Pe λ~ 5 2 Θ; 2

λ5

Pe 1Θ 2

The inlet and the outlet mass transfer rate can easily be expressed by means of Eq. (5.25). The overall inlet mass transfer rate, namely the sum of the diffusive and convective mass transfer rates, is given by J 5 υφ9Y50 2

D dφ ~ 1 λSÞ 9 5 β o ðλT δ dY Y50

ð5:26Þ

with βo 5

D δ

The outlet mass transfer rate is obtained in a similar way as Eq. (5.26) for Y 5 1: ~ λ 1 λSeλÞ ~ Jδ 5 β o ðλTe

ð5:27Þ

The value of parameters T and S can be determined from the boundary conditions. For the sake of generality, two models, namely models A and B, will be distinguished according to Figure 5.3 (for details, see Nagy, 2010). The essential difference between the models is that, in the case of model A, there is a sweep


(A)

Figure 5.3 Illustration of the concentration distribution for mass transport with diffusion 1 convection for models A (there is a diffusive outlet flux, dφ/dy . 0 at y 5 δ) and B (there is no diffusive outlet flux dφ/dy 5 0 at y 5 δ).

Catalyst particles (B)

φ∗δ

Cο

Cο

ο

Cδ

129

Cδ

φ∗

βof βo βfoδ

phase that can remove the transported component from the downstream side, providing the low concentration of the reacted component in the outlet phase and due to it, a high diffusive mass transfer rate. There is no sweep phase on the permeate side, in the case of model B, thus the outlet phase is moving from the membrane due to the lower pressure on the permeate side, that is, due to convective flow, only because there is no diffusive outlet flow. Mass transport without external resistances, for dφ/dy . 0 at y 5 δ. An important limiting case should also be mentioned, namely the case when the external diffusive mass transfer resistances on both sides of membrane can be neglected, i.e., when β of -N and β ofδ -N: Accordingly, the boundary conditions will be: at Y 5 0, φ 5 φ (φ 5 Hco) and at Y 5 1 then φδ 5 Hcoδ : For that case, the concentration distribution and the inlet mass transfer rate can be expressed by Eqs (5.28) and (5.29), respectively. o ePeðY21Þ=2 n Pe=2 e sinh½Θð1 2 YÞφ 1 φδ sinhðΘYÞ sinh Θ Θ J 5 β φ 2 Pe=2 φδ e ½ðPe=2Þsinh Θ 1 Θ cosh Θ

φ5

ð5:28Þ ð5:29Þ

with β5

β o ð½Pe=2tanh Θ 1 ΘÞ tanh Θ

ð5:30Þ

In several cases, the value of the outlet mass transfer rate, Jδ, can be important during catalytic or biocatalytic reactions in a membrane reactor. Its value will be as Θ 2ððPe=2Þtanh ΘÞ cosh Θφδ Jδ 5 β δ φ 2 ΘePe=2

ð5:31Þ

130



Pe = 30 0.8 10

0.6 5 0.4 1 0

0.2 0.0

3

0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.4 Concentration distribution in a catalytic membrane by means of Eq. (5.28); ϑ 5 3.

with βδ 5

D ΘePe=2 δ sinh Θ

ð5:32Þ

The effect of the convective flow, namely the Peclet number, is illustrated in Figure 5.4. The outlet concentration was chosen to be zero, while the reaction modulus was 3 for this calculation (ϑ 5 3). This is a typical figure regarding the concentration distribution. By increasing the Peclet number, the trend of curves will gradually change from convex to concave ones. Looking at the concentration gradients close to the inlet interface, it can be seen clearly that the inlet diffusive flow strongly decreases with the increase of the Peclet number. It is also obvious from this figure that the outlet diffusive flow also strongly increases with the increase of the Peclet number, owing to the increasing portion of the unreacted amount of reactant in the catalytic membrane. This value can be predicted exactly by means of Eq. (5.32) for the plotted case, namely at φδ 5 0: The membrane reactor makes it possible to supply the reactant on both sides of the membrane layer. This can be useful when the reactant concentration should be maintained above a critical level, such as in case of biocatalytic processes, especially in the case of sparingly soluble oxygen supply for cell culture. This situation is illustrated in Figure 5.5 where the effect of the reaction modulus is plotted at given interface concentrations. Obviously, the mass transport takes place by diffusion, only (Pe 5 0) in countercurrent mode. The application of this mode can be useful, only if the two reactants are fed, separatedly, on the two sides of the catalytic membrane layer. How the Peclet number affects the enhancement as a function of the reaction rate is illustrated in


131


1.0 ϑ=0

0.8

1

0.6

3

0.4

5

0.2 0.0

10

Pe = 0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.5 The reactant is fed on both sides of the catalytic membrane layer.

Pe = 0.1

Enhancement (β/βo)

17.0

1 13.0

1 2 4

9.0 10

5.0 1.0 1.0

25 10.0

100.0

Reaction modulus

Figure 5.6 Effect of the reaction modulus and Pe number on the enhancement of the mass transfer rate.

Figure 5.6. As expected, the increase of the convective velocity can significantly decrease the effect of the chemical and biochemical reactions. The reacted amount of the reactant can also be given by the following expression: ð1 o ePeðY 21Þ=2 n Pe=2 Jreact 5 δ k1 φ e sinh½Θð1 2 YÞ 1 φδ sinhðΘYÞ dY ð5:33Þ sinh Θ 0 Thus, by integration of the above equation, one can obtain Jreact 5 β r ðφ 2 F2 φδ Þ

ð5:34Þ

132


with βr 5

D ΘððPe=2Þ 1 1Þ 2ðΘePe=2 =cosh ΘÞ δ tanh Θ

ð5:35Þ

F2 5

Θ½ðe2Pe1 =2 =cosh ΘÞ 2 1 1ðPe1 =2Þtanh Θ Θ½1 2ðePe1 =2 =cosh ΘÞ 1ðPe1 =2Þtanh Θ

ð5:36Þ

The same result can be obtained by the difference of J 2 Jδ, applying Eqs (5.29) and (5.31). Mass transfer resistance is at both sides of membrane: In this case, due to the effect of the sweeping phase, the external mass transfer resistance on both sides of the membrane should be taken into account in the boundary conditions, though the role of β oδ is gradually diminished as the catalytic reaction rate increases (model A). The concentration distribution in the catalytic membrane when applying a sweep phase on the two sides of the membrane is illustrated as well in Figure 5.3A. On the upper part of the catalytic membrane layer, in Figure 5.3A, the fine catalyst particles are illustrated with black dots. It is assumed that these particles are homogeneously distributed in the membrane matrix. Due to sweeping phase, the concentration of the bulk phase on the permeate side may be lower than that on the membrane interface. The boundary conditions can be given for that case as υc 1 β of ðco 2 c Þ 5 J

at Y 5 0

υcδ 1 β ofδ ðcδ 2 coδ Þ 5 Jδ

at Y 5 1

ð5:37Þ ð5:38Þ

where the values of J and Jδ are expressed in Eqs (5.26) and (5.27), respectively; c and cδ denote the fluid phase concentration on the feed side and permeate side of the membrane, respectively (kg/m3). Boundary conditions, given by Eqs (5.37) and (5.38), are only valid in the case of two external flowing phase flows. Where coδ denotes the concentration on the downstream side of the continuous phase, β of and β ofδ are the diffusive mass transfer coefficients in the continuous phases. The solution of the algebraic equations, applying Eq (5.25) as well as Eqs (5.37) and (5.38), can be obtained by means of known mathematical manipulations (not shown here). Thus, the values of T and S obtained for Eq. (5.25) in order to describe the concentration distribution are as fol lows (Hc 5 φ ; here β of 5 Df =δf ; β ofδ 5 Dfδ =δfδ Þ; β o 5 D/δ: T 52

β of ξ2 co 2 β ofδ ξ 4 coδ 1 ξ2 ξ3 2 ξ1 ξ4 β o

ð5:39Þ

and S5

β of ξ 1 co 1 β ofδ ξ3 coδ 1 ξ2 ξ3 2 ξ1 ξ4 β o

ð5:40Þ


133

1.5 Pe = 0.01

J/J o

1.4

Pe f =1

0.1

1.3 1

1.2 1.1

10

1.0 0.10

1.00

10.00

Reaction modulus

Figure 5.7 The ratio of J/Jo as a function of first-order reaction modulus, ϑ, at different values of Pe number and in the presence of external mass transfer resistances ðβ of 5 β o 5 1 3 1024 m=sÞ:

where Pe β oδ ~ 1 o 2 λ eλ ; ξ1 5 β H H ξ3 5

Pe βo ~ 2 o f 2 λ; β H H

ξ4 5

Pe β oδ ~ 1 o 2 λ eλ; ξ2 5 β H H Pe βo 2 o f 2 λ; β H H

Knowing the value of T and S from Eqs (5.39) and (5.40), the inlet and outlet mass transfer rates can be calculated by Eq. (5.26) and Eq. (5.27), respectively (not written here). The effect of the enhancement is illustrated at different Peclet numbers, in Figure 5.7 in presence of the external mass transfer resistance, according to Eqs (5.26), (5.39), and (5.40). Jo denotes the physical (mass transfer without chemical reaction) mass transfer rate. The curves tend to a limiting value as it is expected. This value is limited by the external mass transfer resistances. No sweep phase on the one side of membrane, thus dφ/dy 5 0 at Y 5 δ. This case of catalytic membrane reactor is operating in dead-end mode as in Figure 5.3B. In this case, the concentration of the permeate phase does not change during the mass transport from the membrane interface (there is only convective outlet flow). If there is no sweep phase on the downstream side, then the correct boundary conditions will be as υc 1 β of ðco 2 c Þ 5 J υφδ 5 Jδ

at Y 5 1

at Y 5 0

ð5:41Þ ð5:42Þ

134


After solution, one obtains T 52 S5

β of ξ2 co 1 ξ2 ξ3 2 ξ1 ξ4 β o

ð5:43Þ

β of ξ1 co 1 ξ2 ξ3 2 ξ1 ξ4 β o

where ξ1 5

Pe ~ λ 2λ e ; H

ð5:44Þ

ξ2 5

Pe ~ 2 λ eλ H

The values of ξ 3 and ξ 4 are the same as they are given after Eq. (5.40). Mass transport without mass transfer resistances. During the catalytic membrane processes, it can often occur that the transported component cannot enter the permeate side, accordingly dφ/dy 5 0 at y 5 δ. Assuming that there is no mass transfer resistance in the feed phase, thus, at y 5 0 then φ 5 φ , the solution of the differential equation given by Eq. (5.19) gives the following concentration distribution in the membrane and mass transfer rate: φ 5 φ ePeY=2 with

½ sinh½Θð1 2 YÞ Pe 2 2 Θ cosh Θð1 2 YÞ Pe 2 sinh Θ 2 Θ cosh Θ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 Θ5 1 ϑ2 ; 4

ϑi 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 k1 =D;

Pe 5

ð5:45Þ

υδ D

Note that Eq. (5.45) does not involve the external mass transfer resistances. The inlet mass transfer rate, namely the sum of the diffusive and the convective flows, can be given as J 5 βφ with

ð5:46Þ

2 2 Θ tanh Θ D Pe β5 δ 2 tanh Θ 2 Θ Pe2 4

The outlet mass transfer rate, Jδ, is the convective flow; thus, Jδ 5 υφδ, because the diffusive flow is zero in this case. The concentration distribution is plotted at different values of the Peclet number with zero outlet diffusive flows for mass transfer accompanied by a first-order reaction (Figure 5.8). The value of reaction modulus chosen was relatively high for better illustration the effect of the Peclet number. The concentration level of the


135


Pe = 30 0.8 10

0.6 5 0.4

1

3

0

0.2

ϑ=3 0.0

0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.8 Concentration distribution when there is no diffusive outlet flow on the permeate side (dφ/dY 5 0 at Y 5 1); ϑ 5 3.

reactant essentially can be affected by the Peclet number. Obviously, this level can strongly depend on the ϑ value. Let us compare the concentration change close to the inlet region of the catalytic membrane in Figures 5.4 and 5.8. At a given value of Pe, the concentration has practically the same values between Y 5 0 and 0.2 for the two operating modes, namely when the outlet diffusion flow can be larger than zero (Figure 5.4) or it is equal to zero (Figure 5.8). This fact should mean that the values of the inlet diffusive flows are practically the same in the two operating modes. Accordingly, the same mass transfer rate can be achieved with a higher concentration level by the operating mode when dφ/dY 5 0 at Y 5 1. The operating mode, when a reactant cannot diffuse into the permeate phase (e.g., because it is not dissolvable in it) might often exist. On the other hand, this is not discussed in detail in the literature. Figure 5.9 illustrates how the reaction modulus can affect the concentration distribution at a given Peclet number. In the slow reaction rate regime, ϑ , 0.30.5, the concentration remains above about 0.9. In the intermediate reaction regime, ϑ 5 0.33, the value of φδ will generally be higher than zero, and its value can strongly depend on the Peclet number as well. If ϑ . 3, then the concentration level can be maintained above zero by higher Peclet value only.

5.3.1.1 Mass transport with polarization layer Mass transport in the case of dφ/dy . 0 at y 5 δ. It is assumed the concentration polarization affects the mass transfer rate of the inlet reactant. In the boundary layer, there is no chemical reaction, thus the mass transfer rate through it can be given according to Eq. (5.7). Thus, the mass transfer rate in the liquid phase, at the membrane interface, will be as J o 5 β of} ðco 2 e2Pef c Þ

ð5:47Þ

136


1.0

0.3


0.5 0.8

1

0.6 ϑ=2

0.4

ϑ=3

0.2 0.0

0.0

0.2

0.4 0.6 Catalytic membrane layer

0.8

1.0

Figure 5.9 Effect of the reaction modulus on the concentration distribution (dφ/dY 5 0 at Y 5 1), Pe 5 1.

where β of} 5

Df ePef Pef ePef =2 β of Pef Pe f δf e 21 2 sinhðPef =2Þ

ð5:48Þ

The co denotes the bulk concentration in the fluid, c represents the liquid concen tration at the membrane interface (Hc 5 φ ). The J value defined by Eq. (5.29) and the Jo in Eq. (5.47) are equal to each other, thus the overall mass transfer rate can be given by means of the resistance-in-series principle. The overall inlet mass transfer rate is, for a first-order chemical reaction, as follows (note dφ/dY . 0 at Y 5 1 by this model): J 5 β ov} ðco 2 Ke2Pef coδ Þ

ð5:49Þ

where β}tot 5

1 1 e 2 Pef o 1 Hβ β f}

ð5:50Þ

and K5

Θ Θ 1 Θ cosh Θ

ePe=2 ½ðPe=2Þsinh

ð5:51Þ

The value of β is defined by Eq. (5.30). The parameter values in order to predict the concentration distribution are as follows: ePef e 2 Θ Peco 2 ePef 2 1 λH 1 Pe e2Pe=2 coδ 1 T 5 λδ =δ ð1 2 e2Pef ÞA 2 Pe 2e f cosh Θ H tanh Θ

ð5:52aÞ


137

with A5Θ1

Pe tanhΘ 2

~ 1 Pe e 2 Pe=2 co ePef eΘ Peco 2 ePef 2 1 λH 1 δ S5 ~ ð1 2 e 2 Pef ÞA 2 Pe 2eλδf =δ coshΘ H tanhΘ o 2Pe=2 o A 2 Pe cδ =coshΘ H tanhΘ c 2 Θe Tf 5 ð1 2 ePef ÞA 2 Pe tanhΘ H

ð5:52bÞ ð5:53aÞ

and S f 5 c o 2 Tf

ð5:53bÞ

with Pe λ~ 5 2 Θ; 2

λ5

Pe 1 Θ; 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe ϑ2 Θ5 1 1 4 2; Pe 2

sffiffiffiffiffiffiffiffiffi k1 δ2 ϑ5 D

δf denotes the thickness of the polarization layer, m; δ is the membrane thickness, m. Formulas given by Eqs (5.52a)(5.53b) are rather complex ones and it is rather difficult to handle them. That is why it is important to note that if one knows the mass transfer rate defined by Eq. (5.46), there is a much easier way to get the values of T, S, Tf and Sf parameters. Using, e.g, Eq. (5.46) and the boundary condition at Y 5 0, namely C 5 Tf 1 Sf, the values of Tf and Sf can simply be predicted. The same method can be used to get the values of T and S, namely applying Eq. (5.46) again and the boundary condition at Y 5 1 1 δf/δ [Eq. (5.4b) and Eq. ~ (5.25), that is HCoδ 5 Teλ 1 Seλ ], the T and S values can be obtained. Typical concentration distributions are shown in Figure 5.10A at different values of the reaction modulus. The ϑ can significantly lower the concentration level not only in the membrane layer, but also in the concentration boundary layer as well. At larger values of ϑ, the concentration curves have inflexion points, at Y 5 1, at the feed membrane interface, where the curves change their concave trends to convex. The relative value of the outlet concentration was chosen to be 0.1. As can be seen, the concentration can decrease below its outlet value, at a high value of the reaction modulus. That should mean that to maintain the given outlet concentration, additional feeding of the reactant is needed on the outlet side of the catalytic needed on the outlet side of the catalytic membrane. Mass transport in the case of dφ/dy 5 0 at y 5 δ. The mass transfer rate is given here. Applying Eqs. (5.46) and (5.47), one can easily get the overall mass transfer rate as: Jov} 5

1 β of}

co 2 Pe 1 e Hβ f

ð5:54Þ

138


1.0


ϑ = 0.01 0.8

1

Pef = Pe = 5

0.6

2

δf / δ = 1

5

0.4 βfo = βo = 1×10–4 m/s 0.2 Boundary layer 0.4

10 Membrane layer

0.0 0.0

0.1

0.8

1.2

1.6

2.0

Boundary layer + membrane

Figure 5.10A The effect of a first-order chemical reaction on the concentration distribution in the two layers at different values of reaction modulus ðcoδ =co 5 0:1; β of 5 β o 5 1 3 1024 m=sÞ:

with β [according to Eq. (5.46)] as

2 2 Θ tanh Θ D Pe β5 δ 2 tanh Θ 2 Θ Pe2 4

ð5:46Þ

where the value of β of} is given by Eq. (5.48).

5.3.1.2 Remarks for Application of the Fick’s Diffusive Transfer Plus Convective Flow The mass transfer resistance given by Fick’s law is often applied for diffusive mass transport through a boundary layer ( 5 β oΔc). The question arises whether this can also be used when there is convective flow in the boundary layer as well as it was done the mass transport is accompanied by chemical reaction in the membrane phase [see boundary conditions given by Eqs. (5.37) and (5.38)]. The Fick’s law is true in case of linear concentration distribution. This is not fulfilled when convective velocity also exists in the boundary layer (see for example Figure 5.2). The diffusive flow, JD, can be expressed, in the presence of convective flow as JD} 5 2β of Pef

Δc ePef 2 1

ð5:55aÞ


Ratio of mass tranfer rates,–

1.0

139

C* = 0.90 0.75

0.8

0.50

0.6

0.25

0.4

0.10

0.2 0.0 0.0

1.0

10.0

Peclet number, Pef – o o Figure 5.10B The value of JFi /J EX as a function of the Pef at different value of C interface concentration.

According to Eq. (5.55a), the diffusive mass transfer coefficient ðβ oD} Þ for diffusive plus convective flow can be expressed as (here Y 5 y/δf). At Y 5 Y, then β oD} 5 2β of

Pef ePef Y ePef 2 1

ð5:55bÞ

At Y 5 0, then β oD} 5 2β of

Pef 21

ð5:55cÞ

Pef ePef ePef 2 1

ð5:55dÞ

ePef

At Y 5 1, then β oD} 5 2β of

Accordingly, the diffusive mass transfer coefficient, β oD} ; continuously increases as a function of space coordinate in the case of so-called exact solution. Thus, the overall mass transfer rate will be unchanged with the decrease of the concentration. The value of β oD} # β of at Y 5 0 and β oD} $ β of at Y 5 1. Let us compare the mass transfer rates of the boundary layer, entering the membrane phase, namely at y 5 δf, obtained the exact solution and by usage of Fickian mass transfer coefficient

140


for expressing the mass transfer resistance. Look at the ratio of these two mass transfer rates, namely: o JFi β of ðco 2 c Þ 1 υc f1 1 ðPef 2 1ÞC gðePef 2 1Þ 5 o JEx β of} ðco 2 e2Pef c Þ Pef ePef ð1 2 e2Pef C Þ

ð5:55eÞ

where subscripts Fi and Ex denote the Fick’s and the exact solutions, C 5 c /co. It is obvious that in limiting case, namely if Pef tends to zero, the above ratio tends to unit. On the other hand, when the Pef value increases the difference between the o o =JEx ratio is decreasing, in the range of C two models also increases, i.e., the JFi between 0 and 1. Figure 5.10B illustrates the change of the ratio of mass transfer rates as a function of Pef at different values of C . It can be stated that the approach by the Fick’s law can only be used at very low values of Pef and relatively large values of C .

5.3.2

Mass Transport Accompanied by Zero-Order Reaction

The effect of the zero-order reaction will be discussed here for an intrinsically catalytic membrane layer only. This reaction has no important role in the case of a membrane reactor. The differential mass balance equation to be solved is as D

d2 φ dφ 2 k0 5 0 2υ 2 dy dy

ð5:56Þ

Similar to Eq. (5.2), the differential mass balance equation for the catalytic membrane can be given as d2 φ dφ 5 φ ϑ2 2 Pe dY 2 dY

ð5:57Þ

where sffiffiffiffiffiffiffiffiffi k0 δ 2 ϑ5 Dφ

ð5:58Þ

Mass transport with dφ/dy . 0 at Y 5 δ. First look at the solution with the following boundary conditions: Y 50

then φ 5 φ

ð5:59aÞ

Y 51

then φ 5 φδ

ð5:59bÞ


141

The general solution of Eq. (5.56) is as φ 5 TePeY 2φ

ϑ2 Y 1S Pe

ð5:60Þ

Applying the boundary conditions [Eqs (5.59a) and (5.59b)], one can get

Pe PeY ð1 2 YÞ 1 S φ 1 e2Pe=2 sinh φδ 2 2

φ5

ePeY=2 sinhðPe=2Þ

S5

ϑ2 2Pe=2 PeY Pe e 2Y Sinh sinh Pe 2 2

with

sinh

ð5:61Þ

ð5:62Þ

The mass transfer rate can be given as e2Pe φδ J 5 β φ 2 11T

ð5:63Þ

where β 5 β}o ð1 1 T Þ

ð5:64Þ

and T 52

ϑ2 e2Pe Pe 1 1 2 ePe 2 Pe

ð5:65Þ

and according to Eq. (5.48) β}o 5

D ePe Pe Pe e 21 δ

The outlet mass transfer rate should also be given as e2Pe φ Jδ 5 β} ζ φ 2 ζ δ o

ð5:66Þ

where ζ 512

ϑ2 2Pe=2 ϑ2 Pe Pe 2Pe=2 ½ Þ e ð1 2 e 1 2 Pe 1 2 Pe 22e sinh Pe2 Pe2 2 ð5:67Þ

142


Mass transfer rate with polarization layer. Now look at the solution with mass transfer resistance in the feed phase, applying the resistance-in-series model: υC 2 β of

dc 5J dY

ð5:68Þ

The physical mass transfer rate through the boundary layer can be expressed, in the case of convective and diffusive flows, as follows (Nagy and Kulcsa´r, 2009): J o 5 β of} ðco 2 e2Pef c Þ

ð5:69Þ

with β of} 5

Df ePef Pef Pe δf e f 21

ð5:70Þ

where Pef 5

υδf Df

The overall mass transfer rate can be given, applying Eqs. (5.63) and (5.69), which two mass transfer rate are equal to each other, as follows: J 5 β}ov ðco 2 Te2Pef coδ Þ

ð5:71Þ

with β}ov 5

1 ð1=β of} Þ 1ðe2Pef =HβÞ

ð5:72Þ

The values of β ofδ and β are defined in Eqs (5.70) and (5.64). Because the zero-order reaction rate is independent of the concentration, the effect of the chemical reaction, on the concentration level, can be much higher than in the case of the first-order reaction (Figure 5.11). The concentration lowers quickly down to zero when the ϑ value is larger than two to three. Comparing these results to that of the first-order reaction (Figure 5.10A),there can be a huge difference between them (Nagy and Kulcsa´r, 2009). There is no diffusive mass transport into the permeate side, dφ/dY 5 0 at Y 5 1. For example, the dimensionless concentration in the membrane layer is as Φ5

2 1 ϑ PeY ϑ2 Pe 2 Y ðe 2 1Þ 1 Pee Pe Pe Pe Pe e

ð5:73Þ


143

1.0


ϑ = 0.01 0.8

1

Pef = Pe = 5 0.6

δf / δ = 1

2

0.4 βfo = βo = 1×10–4 m/s

3

0.2 5 Boundary layer

10

0.0 0.0

0.4

0.8

Membrane layer 1.2

1.6

2.0

Boundary layer + membrane

Figure 5.11 The effect of the zero-order reaction on the two-layer concentration distribution ðcoδ =co 5 0:1; β of 5 β o 5 1 3 1024 m=sÞ:

with sffiffiffiffiffiffiffiffiffi k0 δ 2 ϑ5 ; Dφ

Pe 5

υδ D

The inlet mass transfer rate is as 1 J 5 β o φ Pe 2 ϑ2 Pe 2 1 e

ð5:74Þ

The above equation gives the mass transfer rate without external mass transfer resistances. The overall mass transfer rate can similarly be obtained as it was done to get Eq. (5.71).

5.3.3

Mass Transport with Variable Parameters

It is assumed that both the diffusion coefficient and reaction rate constant can vary as a function of space coordinate or concentration. The differential mass balance equation for a first-order chemical reaction can be given as d dφ dφ D½φ; y 2υ 2 kðyÞφ 5 0 ð5:75Þ dy dy dy

144


For analytical solution of Eq. (5.75), it should be linearized. The solution methodology of this type of differential equation was given by Nagy (2009a) for diffusional mass transport through a membrane reactor and by Nagy (2007b) for diffusive plus convective mass transport with variable parameters. Essentially, this solution methodology serves the mass transfer rate and the concentration distribution in a closed, explicit mathematical expression. For the solution of Eq. (5.75), the catalytic membrane should be divided by the N sublayer, in the direction of the mass transport, that is perpendicular to the membrane interface, with thickness of Δδ (Δδ 5 δ/N) and with constant transport parameters in every sublayer. Thus, for the ith sublayer of the membrane layer, using dimensionless quantities, it can be obtained as Di

d2 φ dφ 2 ki φ 5 0; 2υ 2 dy dy

yi21 , y , yi

ð5:76Þ

In dimensionless space coordinate d2 φ dφ 2 Pei 2 ϑ2 i φ 5 0 dY 2 dr

ð5:77Þ

where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϑi 5 δ2 ki =Di ;

Pei 5

υδ2 Di

~ Let us introduce the following variable φ: φ~ 5 φe2Pei Y=2

ð5:78Þ

After a few manipulations, one can get the following differential equation to be solved: d2 φ~ 2 Θ2i φ~ 5 0 dR2

ð5:79Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pei 1 ϑ2i Θi 5 4 The solution of Eq. (5.79) can be easily obtained by well-known mathematical methods as follows: ~

φ 5 Ti eλi Y 1 Si eλi Y

ð5:80Þ


145

with Pei λ~ i 5 1 Θi ; 2

λi 5

Pei 2 Θi 2

The solution of the parameters Ti and Si strongly depends on the operating mode and thus, on the external boundary conditions. Accordingly, see models A and B (Figure 5.3). This will be shown in the next solution and several figures illustrate the effect of variable parameters. Model Adφ/dY . 0 at Y 5 1. The Ti and Pi parameters of Eq. (5.80) can be determined by means of the external [Eqs (5.81a) and (5.81d)] and the internal boundary conditions [Eqs (5.81b) and (5.81c)] for the ith sublayer (with 1 # i # N 2 1; Yi 5 iΔY; ΔY 5 1/N). These can be given by the following equations for model A as φ 5 φ ;

Y 50

ð5:81aÞ

dφi dφ 1 Pei φi 5 2 i 1 1 1 Pei11 φi11 dY dY φi 1 5 φi11 2 at Y 5 Yi Y5Y Y5Y 2

i

at Y 5 Yi

ð5:81bÞ ð5:81cÞ

i

φ 5 φδ ;

Y 51

ð5:81dÞ

Taking into account the concentration distribution [Eq. (5.80)], the algebraic equation system to be solved will be as φ 5 T1 1 S1

ð5:82Þ

~

~

Ti eλi Yi 1 Si eλi Yi 5 Ti11 eλi11 Yi 1 Si11 eλi11 Yi

ð5:83Þ

~ ~ Di ðTi λi eλi Yi 1 Si λ~ i eλi Yi Þ 5 Di11 ðTi11 λi11 eλi11 Yi 1 Si11 λ~ i11 eλi11 Yi Þ

ð5:84Þ

~

φδ 5 TN eλN 1 SN eλN

ð5:85Þ

After solution of the algebraic equation system containing 2N equations, applying the well-known Cramer rules, the value of the integration parameters for the first sublayer can be expressed as ζT T1 5 NO ζN

φ 2

φδ

!

1 o N ζ TN eðPeN =2Þ Li52 coshðΘi ΔYÞ 2coshðΘ1 ΔYÞ

ð5:86Þ

146


The value of ξ i 21 (with i 5 1 2 N 2 1), ζ ji and ψji (with j 5 T, O and i 5 1 2 N) can be calculated by the following expressions: ξi21 5 ehPei2Pei21 iYi =2

ð5:87Þ

*

ζ ji

tanhðϑi ΔY Þ ψji21 Di 2 1 Pei 5 12 2 j 2 ϑi ζ i21 Di

!+

ζ ji21 ξi21

with j 5 T; O;

i 5 2; . . . ; N ð5:88Þ

* ψji

5 A i 2 Bi

ψji21 Di21 Pei 2 j D 2 ζ i21 i

!+

ζ ji21 ξi21

with j 5 T; O;

i 5 2; . . . ; N 2 1 ð5:89Þ

Ai 5

Pei tanhðϑi ΔYÞ 2 Θi ; 2

i 5 1; . . . ; N 2 1

ð5:90Þ

Bi 5

Pei 2 Θi tanhðϑi ΔYÞ; 2

i 5 2; . . . ; N 2 1

ð5:91Þ

The initial values of ζ ji and ψji ; namely ζ j1 and ψj1 (j 5 T, O), are as ζ T1 5 e2Θ1 ΔY ; ψT1 5 λ~ 1 e2Θ1 ΔY ;

ζO 1 5 2tanhðΘ1 ΔYÞ ψO 1 5 2A1

The value of S1 can be predicted by means of Eq. (5.82). For details of this solution, see the Appendix. The mass transfer rate at place of Y 5 0, as a sum of the diffusive and the convective flows, can be given as J5

D1 ðλ1 T1 1 λ~ 1 S1 Þ δ

ð5:92Þ

Knowing T1 and S1, the other parameters, namely Ti and Si (i 5 2,3,. . .,N), can be easily calculated by means of the internal boundary conditions given by Eqs (5.83) and (5.84), starting from T2 and S2 up to TN and SN. Thus, one can get the following equations for prediction of the Ti and Si from Ti 21 and Si 21, for the component A: ~

Ti eλi Yi 1 Si eλi Yi 5 Γi21 ~

Di ðTi λi eλi Yi 1 Si λ~ i eλi Yi Þ 5 Ξi21

ð5:93Þ ð5:94Þ


147

with ~

Γi21 5 Ti21 eλi21 Yi 1 Si21 eλi21 Yi

ð5:95Þ

~ ~ λi21 Yi Þ Ξi21 5 Di21 ðTi21 λi21 eλi21 Yi 1 Si21 λe

ð5:96Þ

Now knowing the Ti and Si (with i 5 1,2,. . .,N) parameters, the concentration distribution can be calculated easily through the membrane, i.e., its value for every sublayer. The outlet mass transfer rate can also be expressed as Jδ 5

DN ðλN TN eλ~ 1 λ~ N SN eλ Þ δ

ð5:97Þ

Some typical figures will be shown in order to illustrate the effect of the variable Peclet number. Figure 5.12 shows how the concentration distribution changes at different types of an anisotropic membrane where the Peclet number changes, due to the variable diffusion coefficient. The dotted lines give the concentration distribution at a low reaction rate, namely at ϑ 5 0.2, while the continuous lines show the fast reaction rate, namely at ϑ 5 3. The change of Pe as a function of the space coordinate can significantly affect the concentration distribution, as consequently the value of the diffusive inlet mass transfer rate. Decreasing the Pe number with high initial value provides a high inlet mass transfer rate, accordingly,

1.0


1 0.8 2

1 0.6

2 3

0.4 0.2 ............. 0.0

0.0

0.2

3

ϑ=3 ϑ = 0.2 0.4

0.6

0.8

1.0


Figure 5.12 Concentration distribution with variable Peclet number and at constant reaction modulus φδ 5 0:1: Curves: (1) Pe decreases from Pe 5 10 (at Y 5 0) down to 1 (at Y 5 1); (2) Pe 5 5.5; and (3) Pe increases from Pe 5 1 (at Y 5 0) to Pe 5 10 (at Y 5 1).

148


higher concentration level in the catalytic membrane. Increasing the Pe number with low initial value serves a much lower concentration level in the catalytic membrane. Line 2 gives the concentration distribution by constant, by the average Peclet number, namely Pe 5 5.5. In the case of a variable Peclet number, the Pe was varied between 1 and 10, with expression of Pe 5 1 19i/N or with Pe 5 10 29i/N, for its increasing and decreasing values, respectively. Mass transport with boundary condition of dφ/dY 5 0 at Y 5 1. The starting expressions are the same as that in φ 5 φ ;

Y 50

ð5:98aÞ

dφ dφ 1 Pei φ 5 Di11 2 1 Pei11 φ Di 2 dY dY φ 1 5 φ 2 Y5Y Y5Y

at Y 5 Yi

ð5:98bÞ ð5:98cÞ

at Y 5 Yi

i11

i

dφ 5 0; dY

Y 51

ð5:98dÞ

The solution of this problem differs from the model A, due to the difference in the last, namely Nth, sublayer regarding the two models. Accordingly, one can get the following solution for T1 and S1: *

A^ B^ 2 ΘN

ψTN21 DN21 PeN 2 T 2 ζ N21 DN

A^ B^ 2 ΘN

ψSN21 DN21 PeN 2 S 2 ζ N21 DN

T1 5 φ

*

S1 5 φ

!+

!+

ζ TN 2 1 1 1 ξN 2 1 Ω 2coshðΘ1 ΔY Þ

ð5:99Þ

ζ SN 2 1 1 1 ξN 2 1 Ω 2coshðΘ1 ΔY Þ

ð5:100Þ

with A^ Ω 5 B^ 2 ΘN

ψO PeN N21 DN 2 1 2 O D 2 ζ N21 N

!

ζO N21 ξN21

ð5:101Þ

where PeN tanhðΘN ΔYÞ 1 ΘN A^ 5 2

ð5:102Þ

PeN B^ 5 1 ΘN tanhðΘN ΔYÞ 2

ð5:103Þ


149

and ξi21 5 ehPei 2 Pei21 iYi =2

ð5:104Þ

* ζ ji 5

tanhðϑi ΔY Þ ψji21 Di21 Pei 12 2 2 ϑi ζ ji21 Di

!+

ζ ji21 ξ i21

with j 5 T; S; O; i 5 2 2 N 2 1 ð5:105Þ

* ψji 5

A i 2 Bi

ψji 2 1 Di21 Pei 2 2 ζ ji 2 1 Di

!+

ζ ji21 ξ i21

with j 5 T; S; O; i 5 2 2 N 2 1 ð5:106Þ

Ai 5

Pei tanhðϑi ΔYÞ 2 Θi ; i 5 2 2 N 21 2

ð5:107Þ

Bi 5

Pei 2 Θi tanhðϑi ΔYÞ; i 5 2 2 N 21 2

ð5:108Þ

The initial values of ζ ji and ψji ; namely ζ j1 and ψj1 (j 5 T, S, O), are as ζ T1 5 e2Θ1 ΔY ; ψT1 5 λ~ 1 e2Θ1 ΔY ;

ζ S1 5 eΘ1 ΔY ;

ζO 1 5 2tanhðΘ1 ΔYÞ

ψS1 5 λ1 eΘ1 ΔY ;

ψO 1 5 2A1

The concentration distribution is plotted in Figure 5.13 at different membrane properties, at ϑ 5 3. As can be seen, there is a significant difference in the concentration depending on the variable Peclet number. With a decreasing Peclet number, the concentration can be maintained at very high level even in the fast reaction rate regime as well. For comparison, the concentration distribution is plotted at a low value of the reaction rate, ϑ 5 0.2 (Figure 5.14). Surprisingly, the concentration of the reactant strongly increases far above unit, in the catalytic membrane layer. This should mean that the diffusive mass transfer flows in countercurrent direction to the convective one, decreasing the overall transport rate. At constant value of Pe, Pe 5 5.5, the concentration falls close to unit, its value is more than 0.99. In the case of an increasing Peclet number, the concentration decreases down to 0.2, even at low reaction rate. In reality the convective velocity, related to the total membrane interface, is generally not changed during a process. The υ value in Eq. (5.75), as can be seen in Chapter 10, should mean that the convective velocity is related to the total membrane interface. The change of the Peclet number can occur due to the change of the diffusion coefficient as a result of the membrane anisotropy. The D value can change depending on the membrane structure’s properties. Accordingly, the value of reaction

150


1.0


Pe = 10 → 1 0.8 0.6 Pe = 5.5 0.4 0.2 Pe = 1 → 10

ϑ=3 0.0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.13 Concentration distribution with variable Peclet number and with boundary condition of dφ/dY 5 0 at Y 5 1 (ϑ 5 3). Pe 5 10 29i/N for Pe 5 10-1 and Pe 5 1 19i/N for Pe 5 1-10.

10.0


ϑ = 0.2 8.0 Pe = 10 → 1 6.0

4.0

2.0

Pe = 5.5 Pe = 1 → 10

0.0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.14 Concentration distribution with variable Peclet number and with boundary condition of dφ/dY 5 0 at Y 5 1 at lower value of ϑ; (ϑ 5 0.2). Pe 5 10 29i/N for Pe 5 10-1 and Pe 5 1 19i/N for i 5 1-10.


151

3.2 ϑ = 0.087 − 0.27


2.8

Pe = 10 → 1

2.4 2.0 1.6

Pe = 2.5

1.2 0.8

ϑ = 0.13

Pe = 1 → 10

0.4 0.0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.15 Concentration distribution with variable Peclet number at low reaction rate (dφ/dY 5 0 at Y 5 1; Pe 5 10 29i/N for Pe 5 10-1 and Pe 5 1 19i/N for Pe 5 1-10); (D 5 1 3 10291 3 10210 m2/s; υ 5 1 3 1024 m/s; δ 5 10 3 1026 m; k1 5 0.075 s21).

modulus should also change at a constant reaction rate. Now, let us look at the concentration distribution when the diffusion coefficient changes both the values of Peclet number and ϑ reaction modulus. Figures 5.15 and 5.16 illustrate this situation at two different values of reaction modulus, namely with its average values of 0.13 (Figure 5.15) and 1.25 (Figure 5.16). The Peclet number was changed between 1 and 10 in an increasing or decreasing manner. The effect of the decreasing diffusion coefficient, that is, decreasing Pe and ϑ, is basically different from that of increasing the diffusion coefficient. These figures prove that the transport process essentially can be affected by the anisotropy structure of the membrane.

5.3.4

Mass Transport Through an Asymmetric Catalytic Membrane

A two-layer membrane, where the catalytic particles are distributed in one of these layers while the other layer is noncatalytic is assumed (Figure 5.17). This can serve as a support layer or a separating layer. In the membrane reactors, the catalytic layer is often a very thin layer, and the noncatalytic layer is a distributor and/or support layer. Or in the case of biocatalyzed membrane reactor, the sponge layer holds the biocatalytic particles, enzymes, or living cells, while the skin layer separates the catalyst particles from the permeate side. That is why the mass transport through an asymmetric membrane reactor will be analyzed. We consider here a general case, namely when diffusion plus convection can take place in both layers. From that, the limiting cases, in which there is only diffusion mass transport in the skin layer, can easily be obtained.

152


1.4 Pe = 10 → 1


1.2 1.0 0.8

Pe = 2.5

0.6

ϑ = 1.85

0.4 Pe = 1 → 10

0.2 ϑ = 1.2 − 3.8

0.0 0.0

0.2

0.4

0.6

0.8

1.0


Figure 5.16 Concentration change with varying Peclet number and accordingly with varying reaction modulus; Pe 5 110; ϑ 5 1.23.8 (D 5 1 3 10291 3 10210 m2/s; υ 5 1 3 1024 m/s; δ 5 10 3 1026 m; k1 5 15 s21).

Noncatalytic Figure 5.17 Schematic illustration of an asymmetric catalytic layer (skin) membrane layer.

Catalytic layer (1) C

J φ∗1

C C C

C C C

C C C

0

Jδ , φ∗2,δ

δ1 δ

5.3.4.1 Reactant Is Fed on the Catalyst Layer Look at a first-order reaction. If you do not want to know the concentration distribution of reactants in the membrane, the overall mass transfer rates can be expressed by means of those that are given for both the single layers. The starting expressions are the mass transfer rate of a symmetric membrane layer, as discussed previously. The inlet and the outlet rates of a single catalytic membrane layer are as follows (Nagy, 2009a, b): Inlet stream [see Eqs (5.28)(5.32)]: J 5 βðφ1 2 E1 φ1;δ Þ

ð5:109Þ


β5

D1 β o ð½Pe1 =2tanh Θ 1 ΘÞ δ1 tanh Θ

E1 5

ePe1 =2 ½ðPe

153

ð5:110Þ

Θ 1 =2Þsinh Θ 1 Θ cosh Θ

ð5:111Þ

Outlet stream [see Eqs (5.30) and (5.31)]: J1;δ 5 β δ ðφ1 2 E2 φ1;δ Þ

ð5:112Þ

βδ 5

D1 ΘePe1 =2 δ1 sinh Θ

ð5:113Þ

E2 5

Θ 2ðPe1 =2Þtanh Θ cosh Θ ΘePe1 =2

ð5:114Þ

and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe21 Θ5 1 ϑ2 ; 4

sffiffiffiffiffiffiffiffiffi k1 δ21 ϑ5 ; D1

Pe1 5

υδ1 ; D1

βo 5

D δ

The mass transfer rate for the skin layer, assuming that there is no chemical reaction in this layer, is as [see Eqs (5.69) and (5.70)]: J2o 5 β o2 ðφ2 2 e2Pe2 φ2;δ Þ

ð5:115Þ

where β o2} 5

D2 ePe2 Pe2 Pe δ2 e 2 21

ð5:116Þ

where the subscripts 1 and 2 denote the sponge and the skin layers, respectively; D1 and D2 represent the effective diffusivities of the layers (m2/s); and Pei 5 υiδi =Di with i 5 1, 2. The overall inlet mass transfer rate can be expressed by combinations of Eqs (5.110), (5.112), and (5.115), for example, by the following way: the φ1;δ value is expressed by Eqs (5.112) and (5.115) ðJδ 5 J2o ; H1 φ1;δ 5 H2 φ2 Þ and replaced into Eq. (5.110). Thus, one can obtain J 5 β ov ðφ1 2 Gφ2;δ Þ

ð5:117Þ

154


with β ov 5 β

β o2} ðH1 =H2 Þ 1 β δ ðE2 2 E1 Þ β o2} ðH1 =H2 Þ 1 β δ E2

ð5:118Þ

and G5

ββ o2} e2Pe2 o β 2} ðH1 =H2 Þ 1 β δ ðE2

2 E1 Þ

ð5:119Þ

The overall outlet mass transfer rate at δ 5 δ1 1 δ2 can be obtained by Eqs (5.112) and (5.115) with expression H1 φ1;δ 5 H2 φ2 (note J1;δ 5 J2o Jδ Þ as H2 2Pe2 Jδ 5 β δ;ov φ1 2 E2 e φ2;δ H1

ð5:120Þ

with β δ;ov 5

1 ð1=β δ Þ 1ðE2 H2 =H1 Þð1=β o2} Þ

ð5:121Þ

and Pe2 5

υδ2 D2

5.3.4.2 Reactant Is Fed on the Noncatalytic Layer For the sake of completeness, we give here the overall mass transfer rate when the feed of the reactant is carried out through the nonreactive skin layer. This can be done more easily than in the previous case, though the reacted amount is also needed to give the outlet rate in the spongy layer. This mass transfer is illustrated in Figure 5.18. Note that the subscripts 1 and 2 denote here also the skin and the catalytic sponge layers as it was made in section 5.3.4.1. Equations (5.115 )(5.117) and (5.120) were applied in order to express the mass transfer rates at y 5 0 and y 5 δ1 1 δ2 5 δ. The inlet mass transfer rate will be as H2 2Pe2 φ1;δ J 5 β ov φ2 2 E1 e H1

ð5:122Þ

with β ov 5

1 ð1=β o2} Þ 1ð1=βÞðH2 =H1 Þe2Pe2

ð5:123Þ


C

J φ∗2

C C C

Figure 5.18 Illustration of the mass transport in the case when the substrate is fed on the skin side of the asymmetric catalytic membrane.

C C C C

C C 0 δ2

155

Jδ , φ∗2,δ

δ

The value of β and β o2} are given in Eqs (5.110) and (5.116), respectively. The outlet mass transfer rate, namely that on the permeate side, at y 5 δ [taking into account that the value of overall mass transfer rate, J, in Eq. (5.122) is equal to J2o given by Eq. (5.115), the value of φ1 can be expressed; then it should be replaced into Eq. (5.112)] can be given as Jδ 5 β δ;ov ðφ2 2 E3 φ1;δ Þ

ð5:124Þ

where β H2 Pe2 e β δ;ov 5 β δ 1 2 ov o β 2} H1

ð5:125Þ

and E3 5

E2 2 E1 ððβ o2} =β ov Þ 21ÞðH2 =H1 ÞePe2

ð5:126Þ

The values of β ov, β δ, E1, and E2 are given by Eqs (5.123), (5.113), (5.111), and (5.114), respectively. The mass transfer rate through asymmetric membrane, when the transport is accompanied by zero-order reaction is given in Chapter 9 because this reaction often can be important in the case of biocatalytic reactions applying asymmetric membranes.

References Ilinitch, O.M., Cuperus, F.P., Nosova, L.V., and Gribov, E.N. (2000) Catalytic membrane in reduction of aqueous nitrates: operational principles and catalytic performance. Catalysis Today 56, 137145. Nagy, E. (2007a) The effect of the concentration polarization and a membrane layer mass transport on the membrane separation. J. Appl. Membr. Sci. 6, 18. Nagy, E. (2007b) Mass transport through a dense, polymeric, catalytic membrane layer with dispersed catalyst. Ind. Eng. Chem. Res. 46, 22952306.

156


Nagy, E. (2009a) Mathematical Modeling of Biochemical Membrane Reactors, in Membrane Operations, Innovative Separations and Transformations, Ed. by, E. Drioli, and L. Giorno. Wiley-VCH, Weinheim, pp. 309334. Nagy, E. (2009b) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270278. Nagy, E. (2010) Mass transport through a convection flow catalytic membrane layer with dispersed nanometer-sized catalyst. Ind. Eng. Chem. Res. 49, 10571062. Nagy, E., and Borbe´ly, G. (2007) The effect of the concentration polarization and the membrane layer mass transport on the membrane separation. J. Appl. Membr. Sci. Technol. 6, 916. Nagy, E., and Kulcsa´r, E. (2009) Mass transport through biocatalytic membrane reactors. Desalination 245, 422436.

6 Diffusion in a Cylindrical Membrane Layer

6.1

Introduction

The capillary, a hollow fiber membrane, is the most often-applied membrane module in the separation industry. This proves the importance of the mass transport through a cylindrical membrane layer. The schematic illustration of a capillary is given in Figure 6.1.

6.2

Steady-State Diffusion

We consider a circular cylinder, a capillary membrane in which diffusion everywhere is radial. Concentration is then a function of radius r, only under steady-state conditions. Thus, the mass balance equation will be as d dφ rD 50 dr dr

ð6:1Þ

Rearranging Eq. (6.1) in dimensionless form with R 5 r/ro and Φ 5 φ/φ 5 φ/(Hco), C=c/co (where co is the bulk fluid concentration, kg/m3 or kmol/m3; H is the solubility coefficient, -; φ is concentration in membrane, kg/m3 or kmol/m3, see Figure 3.1) and assuming that D is constant, it can be get as: d dΦ R 50 dR dR

ð6:2Þ

Assuming a hollow cylinder whose membrane thickness is δ and the inner radius is ro, as well as considering the D value as a constant, the general solution of Eq. (6.1) is Φ 5 T ln R 1 S

ð6:3Þ

where T and S are constants to be determined from the boundary conditions at R 5 1 and R 5 1 1 δ/ro. If the surface of the membrane at R 5 1 is kept at a Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00004-1 © 2012 Elsevier Inc. All rights reserved.

158


δ

J

r0

J

z

r

r0 A

δ

βof

βo βofδ

Figure 6.1 Schematic representation of the transfer conditions and important notations for a capillary membrane.

constant concentration φ , Φ 5 φ /(Hco) 5 1 and R 5 1 1 δ/ro at Φoδ ; then (note that there is no mass transfer resistance in the fluid boundary layer): φ5

φ ln½ðro 1 δÞ=r 1 φδ lnðr=ro Þ ln½ðro 1 δÞ=ro

ð6:4Þ

In dimensionless form (without external resistance Φ 5 1): Φ5

Φ ln½ð1 1 δ=ro Þ=R 1 Φδ ln R ln½ð1 1 δ=ro Þ=R 1 Cδo ln R ln½ð1 1 δ=ro Þ ln½ð1 1 δ=ro Þ

The mass transfer rate can be defined as dφ D Dco H o J 52D ðφ 2 φδ Þ ð1 2 Cδo Þ 5 dr r5ro ro lnð1 1 δ=ro Þ ro lnð1 1 δ=ro Þ

ð6:5Þ

ð6:6Þ

It is easy to see that ro ln(1 1 δ/ro)-δ if ro-N, i.e., the mass transfer rate tends to that of plane interface in limiting cases. Defining the β o mass transfer coefficient for cylindrical interface, the mass transfer rate can be written in a form obtained for plane interface, without external mass transfer resistance, as J o 5 β o ðφ 2 φδ Þ β o Hðco 2 coδ Þ

ð6:7Þ

with βo 5

D ro lnð1 1 δ=ro Þ

ð6:8Þ

Diffusion in a Cylindrical Membrane Layer

159

Note that, according to Eq. (6.2), the specific mass transfer rate depends on the value of the r space coordinate. Thus, the connection between the inlet, at R 5 1, and the outlet mass transfer rate, at R 5 1 1 δ/ro is as J o R51 5 ð1 1 δ=ro ÞJ o R511δ=ro ð6:9Þ Look at the mass transfer rate when there is external mass transfer resistance on both sides of a cylindrical (capillary) membrane. The mass transfer rate can be given for the fluid phase boundary layer as at r 5 ro

β of ðco 2 c Þ β of co ð1 2 C Þ 5 2D

at r 5 ro 1 δ

dφ Dφ dΦ 5 Jo ro dR dr r5ro

β ofδ ðcδ 2 coδ Þ β ofδ co ðCδ 2 Cδo Þ 5 J o

1 ð1 1δ=ro Þ

ð6:10Þ ð6:11Þ

The overall mass transfer rate and mass transfer coefficient, β oov ; taking into account Eqs (6.7) and (6.8), are easy to express, similarly to that for plane interface: J o 5 β oov ðco 2 coδ Þ β oov co ð1 2 Cδo Þ

ð6:12Þ

1 1 1 1 5 o1 1 o β oov βf Hβ o β fδ ð1 1 δ=ro Þ

ð6:13aÞ

1 1 1 1 5 o1 1 o β oov βf Hβ o ðβ fδ Þr5ro

ð6:13bÞ

with

or

where β of 5

Df ro lnðro =½ro 2 δf Þ

β ofδ 5

Dfδ ro ð1 1 δ=ro Þlnð½1 1 δ 1 δfδ =½ro 1 δÞ

ð6:14Þ

and ð6:15aÞ

where δf and δfδ are the thickness of the boundary layer for feed and permeate side, respectively (m), and Df and Dfδ are the diffusivity of the continuous phase (m2/s). Note that the β ofδ gives the mass transfer coefficient at r 5 ro 1 δ. If this coefficient is related to the place of r 5 ro, its value will be as ðβ ofδ Þr 5 ro 5

Dfδ ro lnð½1 1 δ 1 δfδ =½ro 1 δÞ

ð6:15bÞ

160


Note that the value of βof in Eq. (6.14) is also related here to the place of r 5 ro. The T and S parameters, for calculation of the concentration distribution through the membrane, can be expressed, in dimensionless form, for Φ 5 T ln R 1 S, with Φ 5 φ/(Hco), as follows: ro ð6:16Þ T 5 β oov ðCδo 21Þ D and

S 5 Hβ oov

1 1 Cδo 1 o 1 βf ð1 1 δ=ro Þβ ofδ Hβ o

ð6:17Þ

6.2.1 Concentration-Dependent Diffusion Coefficient Two important cases will be discussed in this section, namely the exponential and linear concentration dependency.

~ 6.2.1.1 Exponential Concentration Dependency, D 5 Doexp(αφ) Here also two cases will be discussed briefly, namely exponential and linear concentration dependency of the diffusion coefficient. Look at first an exponential ~ ~ : Replacing it or D 5 Doexp(αΦ) with α 5 αφ function, namely D 5 Doexp(αφ) into Eq. (6.1), one can get d ~ dφ αφ e 50 ð6:18Þ Do dr dr After integration of Eq. (6.18), we get ~ dφ T~ 5 reαφ dr

ð6:19aÞ

Separating the variables, we get T~ ~ dr 5 eαφ dφ r

ð6:19bÞ

Integrating Eq. (6.19a), one can get a general solution of this equation as ~ eαφ 5 T~ ln r 1 S~ α~

ð6:20Þ

In Eq. (6.20), parameters T~ and S~ have a dimension of kg/m3. Rewriting the membrane concentration and the space coordinate of Eq. (6.19b) into dimensionless form, it becomes eαΦ 5 T ln R 1 S α

ð6:21Þ


161

Regarding negligible mass transfer resistance in the fluid phase boundary layer, assuming that fluid phase concentration is kept constant, namely at r 5 roφ 5 φ and r 5 ro 1 δφ 5 φδ ; the concentration distribution can be expressed as (Φ 5 φ/φ ; here Φδ 5 Cδo Þ: 1 ln R ð6:22Þ 1 eα Φ 5 ln ðeαΦδ 2 eα Þ α lnð1 1 δ=ro Þ The mass transfer rate expressing with measurable concentrations, at R 5 1, can be expressed as Jo 5

β o Hco α o ðe 2 eαCδ Þ α

βo 5

Do ro lnð1 1 δ=ro Þ

ð6:23Þ

The mass transfer rate can be expressed, in the presence of external mass transfer resistances, considering Eqs (6.10) and (6.11), as Jo 5

β o Hco αA ðe 2 eαB Þ α

ð6:24Þ

with A512

Jo β of co

B 5 Cδo 1

Jo ½1 1 δ=ro β of δ co

Accordingly, the Jo value can easily be determined by trial-error method by means of Eq. (6.24).

6.2.1.2 Linear Concentration Dependency, D 5 Do(1 1 αΦ) The differential mass transfer equation to be solved becomes d dΦ Rð1 1 αΦÞ 50 dR dR

ð6:25Þ

From that, after integration Rð1 1 αΦÞ

dΦ 5T dR

ð6:26Þ

162


The general solution of Eq. (6.26) is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 T ln R 1 S 12 Φ52 1 α α2 α

ð6:27Þ

with (note Φ 5 1, Φδ 5 Cδo Þ: α S5 11 2 and T5

ð6:28Þ

n o 1 α ðCδo 21Þ 1 ðCδo2 21Þ lnð1 1 δ=ro Þ 2

ð6:29Þ

The mass transfer rate can be given, taking into account Eqs (6.26) and (6.29), as J o 5 2Do ð1 1 αΦÞ

n o Hco dΦ α 5 β o Hco ð1 2 Cδo Þ 1 ð1 2 Cδo2 Þ ro dR 2

ð6:30Þ

Look at the mass transfer rate with external mass transfer resistance in the fluid phases flowing in the cylindrical membrane. The mass transfer rates in the external boundary layer can be given as it is done in Eqs (6.10) and (6.11). The mass transfer rate through the membrane, taking into account that the interface concentration can change according to the membrane resistance, can be given as n o α ð6:31Þ J o 5 β o φ ðΦ 2 Φδ Þ 1 ðΦ2 2 Φ2 δ Þ 2 According to Eqs (6.10), (6.11), and (6.31), the mass transfer rate with external mass transfer resistance can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2B 6 B2 2 4AE ð6:32Þ Jo 5 2A with ( ) α 1 1 2 A5 2 ðβ of co Þ2 ð½β ofδ r5ro co Þ2 (

1 1 αCδo 11α 1 B5 2 1 o o 1 o ½β f δ r5ro βf β H E 5 1 2 Cδo 1

α

1 2 Cδo2 2

ð6:33Þ ) 1 co

ð6:34Þ ð6:35Þ

The values of β o, β of and (β ofd)r5ro are defined by Eqs. (6.8), (6.14) and (6.15b), respectively.


163

6.2.1.3 Optional Concentration and/or Local-Coordinate Dependency of the Diffusion Coefficient The concentration dependency of the diffusion coefficient can be essentially different from the above discussed function. On the other hand, preparation and application of anisotropic membrane become more important. The diffusion coefficient can vary from place to place. Note that in the case of D 5 D(R), the concentration distribution can be directly calculated, while in the case of D 5 D(Φ), a few iteration steps are needed to get the correct D and Φ values in the membrane. Thus, a general solution of the problem will be shown in this section. The membrane is divided into N sublayers with ΔR thickness and the diffusion coefficient Di is assumed to be constant for every sub-layer (see Fig.4.7). The concentration distribution for ith sublayer, without external mass transfer resistance on the boundary layer, can be given as Ri 5 1 1 iΔR and ΔR 5 δ/(roN) (Nagy, 2006): Φ 5 Ti ln R 1 Si

at

Ri21 # R # Ri

ð6:36Þ

where (Ri 5 ri/ro) Ti 5 2

1 ðΦ 2 Φδ Þ PN Dj j51 ð1=Dj ÞlnðRj =Rj21 Þ

1 ð1 2 Cδo Þ 2 PN Dj j51 ð1=Dj ÞlnðRj =Rj21 Þ

ð6:37Þ with i 5 1; . . . ; N

and (with S1 5 Φ 5 1) Si 5

i21 X

Tj ln Rj 2 Ti ln Ri 1 Φ 1 1

j51

i21 X

Tj ln Rj 2 Ti ln Ri with i 5 2; . . . ; N

j51

ð6:38Þ The mass transfer rate can be given as: D1 dΦ o 5 β o Hco ð1 2 Cδo Þ J 5 2 Hco ro dR R51 βo 5

1 1 P ro Nj51 ð1=Dj ÞlnðRj =Rj21 Þ

ð6:39Þ ð6:40Þ

As mentioned, when the Di values depend on the concentration, the real concentration distribution and the mass transfer rate can be obtained by iterative method. The external mass transfer resistance can easily be taken into account applying Eqs. (6.13b) and (6.40). Thus, the mass transfer rate can be obtained as: ð6:41Þ J o R51 5 β oov co ð1 2 Cδo Þ

164


with N 1 1 1 ro X 1 Rj 1 ln o 5 o 1 o H j51 Dj Rj21 β ov βf ðβ f δ Þr5ro

ð6:42Þ

For the value of ðβ ofδ Þr5ro ; see Eq. (6.15b).

6.2.2 Concentration-Dependent Solubility Coefficient, H Two cases are discussed, namely linear and Langmuir-type concentration dependency, as given in Sections 3.2.2.1 and 3.2.2.2.

6.2.2.1 Linear Concentration Dependency of the Sorption Coefficient, ~ with Ho ð1 1 αc ~ Þc 5 φ H 5 Ho ð1 1 αcÞ The sorption isotherm can be given, in dimensionless form {C 5 φ =ðHo co Þ Φ ; ~ o g; as α 5 αc Φ 5 ð1 1 αCÞC

ð6:43Þ

According to Eqs (6.5) and (6.43), the concentration distribution in the membrane, without external mass transfer resistance, can be given as Φ5

ð1 1 αÞln½ð1 1 δ=ro Þ=R 1 ð1 1 αCδo ÞCδo ln R ln½ð1 1 δ=ro Þ

ð6:44Þ

The mass transfer rate will be as J o 5 β o Ho co ð1 1 α 2 ð1 1 αCδo ÞCδo Þ

ð6:45Þ

The overall mass transfer rate with mass transfer resistance, applying Eqs (6.10) and (6.11) as well as (6.45) can be given, after some manipulation of the equation system, as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð6:46Þ 2B6 B2 2 4AE J5 2A with (

)

1 1 A5α 2 o o 2 o ðβ f c Þ ð½β f δ r5ro co Þ2 1 B5 2 o c

(

) 1 1 1 2αCδo 1 1 2α 1 1 ðβ ofδ Þr5ro β o Ho β of

ð6:47Þ

ð6:48Þ


165

E 5 1 1 α 2 Cδo ð1 1 αCδo Þ

ð6:49Þ

6.2.2.2 Langmuir-Type Sorption Isotherm φ 5 Ho

Φ5

~ αc ~ 1 1 αc

ð6:50Þ

φ αC Ho 1 1 αC

ð6:51Þ

Note that the Ho value has a dimension of kg/m3 here. The concentration distribution is not affected by the isotherm used, the interface concentration depends on the fluid concentration only. Thus, the concentration distribution can be given when there is no external mass transfer resistance, as Φ 5 T ln R 1 S

ð6:52Þ

where Cδo α 1 2 T5 lnð1 1 δ=ro Þ 1 1 αCδo 1 1 α

ð6:53Þ

and S5

α 11α

ð6:54Þ

The mass transfer rate can be expressed as

Cδo 1 2 J 5 β Ho c α 1 1 α 1 1 αCδo o

o

ð6:55Þ

o

Look at the concentration distribution regarding the external mass transfer resistance as well. The mass transfer rate through the fluid boundary layer can be given as it is done in Eqs (6.10) and (6.11). The mass transfer rate through the membrane will be according to Eq. (6.55):

C Cδ 2 J 5 β Ho c α 1 1 αC 1 1 αCδ o

o

o

ð6:56Þ

166


Expressing the value of C and Cδ from Eqs (6.10) and (6.11), respectively, and replacing them into Eq. (6.56), the following equation can be obtained: 1 2 J o =ðβ of co Þ Cδo 1 J o =hðβ ofδ Þr 5 ro co i Jo 5 2 β o Ho co α 1 1 αð1 2 J o =½β of co Þ 1 1 α½Cδo 1 J o =ð½β ofδ r 5 ro co Þ

ð6:57Þ

The Jo value can easily be obtained by an iteration method. It is also possible to rearrange Eq. (6.57) to get the Jo value in explicit form. Thus, one can get a 3rd order algebraic equation from that the Jo can be calculated by the well-known traditional method.

6.2.2.3 Dual-Sorption Isotherm According to the dual-sorption isotherm: φ 5 Ho c 1

~ φsat αc ~ 1 1 αc

ð6:58Þ

or in dimensionless concentration, it will be (Φsat 5 φsat/co) Φ5

φ Φsat αC 5 C 1 Hco 1 1 αC

ð6:59Þ

The mass transfer rate without external mass transfer resistance will be as apply~ o ; C 5 c/co), assuming that the diffusion coefficients are the ing Eq. (6.7), (α 5 αc same in the two models (see also section 3.2.2.3): φsat αCδo φsat α ð6:60Þ 2 Cδo 2 J o 5 β o H o co 1 1 1 1 αCδo 11α Taking into account the external mass transfer resistance, we have three equations in order to determine the mass transfer rate, namely [see Eqs (6.10) and (6.11); Φ 5 φ /(Hoco)]: Jo 11 co β of Φsat αC Φsat αCδo o o o o 2 Ho Cδ 2 J 5 β c Ho C 1 1 1 αC 1 1 αCδo C 5 2

Cδ 5 J o

1 1 Cδo ð1 1 δ=ro Þβ ofδ

ð6:61aÞ ð6:61bÞ ð6:61cÞ

Substituting Eqs (6.61a) and (6.61c) into Eq. (6.61b), the following implicit equation can be obtained for calculating the mass transfer rate at R 5 1: Jo Ho J o Φsat αð1 2 J o =β of Þ 2A 5 2 1 1 1 β o co β of 1 1 αð1 2 J o =β of Þ

ð6:62Þ


A5 2

Φsat αð1 2 J o =½β ofδ r5ro 1 Cδo Þ Ho J o 1 Cδo 1 o ½β f δ r5ro 1 1 αð1 2 J o =½β ofδ r5ro 1 Cδo Þ

167

ð6:63Þ

The Jo mass transfer rate can be calculated by trial-and-error method by means of Eqs (6.62) and (6.63).

6.2.2.4 Freundlich Sorption Isotherm The sorption isotherm of the species can be written in this case as φ 5 Hcn

or

Φ5

φ 5 Cn Hcon

ð6:64Þ

The dimension of H is (m3/kg)n 21 in the Freundlich equation. Consider the fluid phase concentration without external mass transfer resistance (see Figure 6.1) and Eq. (6.5), the concentration distribution inside the membrane is as Φ5

Φ ln½ð1 1 δ=ro Þ=R 1 Φδ ln R ln½ð1 1 δ=ro Þ=R 1 Cδon ln R ln½ð1 1 δ=ro Þ ln½ð1 1 δ=ro Þ

ð6:65Þ

Thus, the mass transfer rate is as J 5 β o Hcon ðΦ 2 Φδ Þ β o Hcon ð1 2 Cδon Þ

ð6:66Þ

The expression of β o is given in Eq. (6.8). The mass transfer rate can be similarly given in the presence of mass transfer resistance as it was made previously. Applying Eqs (6.10), (6.11), and (6.66), the mass transfer rate is as J o 5 β o Hcon fðC o 2 J o=½β of Hcon Þn 2 ðCδo 1 J o=½β ofδ h1 1 δ=ro iHcon Þn g

ð6:67Þ

The Jo value can be determined by e.g. iteration.

6.2.3

Mass Transfer Through a Composite Membrane

Two essential conditions should be taken into account: the thickness of the sublayers can be different and the solubility coefficient can also be different. The diffusion coefficient is also different but its value is constant inside of every single sublayer. It is assumed that the solubility coefficient for the sublayers can be defined as follows: Hmi φi 5 Hmi11 φi11

for i 5 1 to N 2 1

ð6:68Þ

168


For the external phases: Hoco 5 H1φ and HN φN 5 Hδ cofδ (or in the case of external mass transfer resistances, according to Figure 6.1, Hoc 5 H1φ and HN φN 5 Hδ cfδ Þ: After solution of the algebraic equation system obtained by the boundary conditions, the parameters have been determined. The values of the parameters in Eq. (6.36) can be given as: Ti 5 2

ðHo Co 2 Hδ Cδo Þ 1 PN Di j51 ðHj =Dj ÞlnðRj =Rj 21 Þ

with i 5 1; . . . ; N

ð6:69Þ

and (note that S1 5 Ho/H1) i21 X Ho Hj Rj 1 Tj ln 2 Ti ln Ri21 Hi H R i j 21 j51

Si 5

with i 5 2; . . . ; N

Accordingly, the mass transfer rate is D1 H1 co o J 52 T1 5 β oov H1 co ðHo 2 Hδ Cδo Þ ro R51

ð6:70Þ

ð6:71Þ

where βo 5

1 1 P ro Nj51 ðHj =Dj ÞlnðRj =Rj21 Þ

ð6:72Þ

Taking into account the external mass transfer resistances on the both sides of the membrane, the mass transfer rate can be given as follows: J5

Ho 2 Hδ Cδo D 1 co 1 Hδ ro Ho o 1 o 1 o βf ½β fδ r5r0 β

ð6:73Þ

The β o value is expressed in Eq. (6.72), while the β of and ½β ofδ r5r0 are given by Eqs. (6.14) and (6.15b), respectively.

6.3

Diffusion Accompanied by Chemical Reaction

6.3.1

Solution as a Bessel Function

Let us look at the solution of mass transfer accompanied by first-order chemical reaction in a cylindrical hollow fiber membrane. The mass balance equation for the catalytic membrane will be as d2 φ 1 dφ 1 2 ϑ2 φ 5 0 dR2 R dR

ð6:74Þ


169

with rffiffiffiffiffiffiffiffi k1 ro2 ϑ5 D where k1 is the reaction rate constant (1/s); ro is the capillary radius (m); and D is the diffusion coefficient in the capillary membrane (m2/s). The general solution of Eq. (6.74) will be as (O’Neil, 1987): φ 5 TIo ðϑRÞ 1 SKo ðϑRÞ

ð6:75Þ

where Io is a Bessel function of the first kind of order zero and Ko is a Bessel function of the second kind of order zero. The parameters T and S should be determined by the suitable boundary conditions as R 5 1;

φ 5 φ

R 5 1 1 δ=ro ;

ð6:76aÞ φ 5 φδ

ð6:76bÞ

After solution, we get T5

φ φ Io ðϑ½1 1 δ=ro Þ 2 φδ Io ðϑÞ Ko ðϑÞ 2 Io ðϑÞ Ko ðϑÞIo ðϑ½1 1 δ=ro Þ 2 Ko ðϑ½1 1 δ=ro ÞIo ðϑÞ Io ðϑÞ

ð6:77Þ

S5

φ Io ðϑ½1 1 δ=ro Þ 2 φδ Io ðϑÞ Ko ðϑÞIo ðϑ½1 1 δ=ro Þ 2 Ko ðϑ½1 1 δ=ro ÞIo ðϑÞ

ð6:78Þ

and

The Io and Ko are well-known functions and their values can be found in O’Neil (1987, p. 311).

6.3.2

Analytical Approach for Solution

The analytical solution of the Bessel function is a rather complex task. On the other hand, the parameters are often concentration- or local-coordinate dependent. Thus, an analytical approach for the solution can be very useful. Look at a general case here, namely when there is diffusion plus convection, as well as a chemical reaction. Let us look at a first-order reaction. The methodology of the solution is discussed in details in Sections 4.2.3 and A2. How this methodology can be extended to the solution of second-order or

170


Michaelis-Menten reactions is also discussed in Chapter 4. Accordingly, the ith sublayer of the membrane can be given in the following balance equation: Di

d2 φ 1 dφ 1 2 υ 2 ki φ 5 0; dr 2 ri dr

ri21 , r , ri ;

ð6:79Þ

where the value of ki, first-order reaction constant, can be regarded as constant or variable. Equation (6.79) is in dimensionless form as [R 5 r/ro, Φ 5 φ/φ , Φi 5 ðΦi21 1 Φi Þ=2: d2 Φ 1 dΦ 2 Pei 2 2 ϑ2 i Φ 5 0 dR2 Ri dR

ð6:80Þ

where Ri denotes the average value of Ri in the ith sublayer of the catalytic membrane layer [Ri 5 1 1ði 20:5ÞΔR; ΔR 5 δ/(Nro)], ro is inner radius of the cylindrical membrane (m). ϑi 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ro2 ki =Di ;

Pei 5

υ i ro Di

~ Let us introduce the following new variable Φ: ~ 5 Φe2Peoi R=2 Φ

ð6:81Þ

with Peoi 5 ðPei 2 1=Ri Þ After a few manipulations, one can get the following differential equation to be solved: ~ d2 Φ ~ 50 2 Θ2i Φ dR2

ð6:82Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Peo2 i 1 ϑ2i Θi 5 4 The solution of Eq. (6.82) can be easily obtained by well-known mathematical methods as it follows (Nagy, 2008, 2009; Nagy and Borbe´ly, 2009): ~

Φ 5 Ti eλi R 1 Si eλi R

ð6:83Þ


171

with Peo λ~ i 5 i 1 Θi ; 2

λi 5

Peoi 2 Θi 2

Parameters Ti and Pi of Eq. (6.83) can be determined by means of the boundary conditions for the ith sublayer (with 1 # i # N). The boundary conditions at the internal interfaces of the sublayers (1 # i # N 21; Ri 5 1 1 iΔR; ΔR 5 δ/[Nro]) can be obtained from the following two equations [Eqs (6.84a) and (6.84b)]: dΦi dΦi11 1 Pei Φi 1 Pei 1 1 Φi11 R5R1i at R 5 Ri 5 Di11 2 Di 2 R5R2i dR dR ð6:84aÞ Φi

R5R2 i

5 Φi11

R5R1 i

at R 5 Ri with i 5 1; . . . ; N 2 1

ð6:84bÞ

where Φi is the substrate concentration of the ith sublayer in the biocatalytic membrane at R 5 Ri. The external boundary conditions, namely that at R 5 1 and R 5 1 1 δ/ro can depend on the operating conditions. Two cases will be discussed here: (1) the concentrations on the both sides of the membrane are determined by the feed and sweep phases (there is diffusion transport into the sweep phase, thus, generally saying, dφ/dR . 0 at R 5 1 1 δ/ro); and (2) the permeate compound is immiscible in the sweep phase or there is no sweep phase, accordingly dφ/dR 5 0 at R 5 1 1 δ/ro. Mass transfer with sweep phase on the permeate side, dΦ/dR . 0 at R 5 1 1 δ/ro (Case A). Thus, the external boundary conditions will be as ~

at R 5 1 then Φ 5 1 T1 eλ1 1 S1 eλ1

ð6:84cÞ

~

ð6:84dÞ

at R 5 1 1 δ=ro then Φ 5 Φδ TN eλN ð11δ=ro Þ 1 SN eλN ð11δ=ro Þ

The solution methodology is discussed in Nagy’s papers (Nagy, 2008; Nagy and Borbe´ly, 2009). After solution of the algebraic equation system containing 2N equations, the value of the integration parameters for the first sublayer can be expressed as 0 1 T1 5

ΩTN B B Bφ 2 @ ΩO N

e2PeN ðδ=ro Þ=2 o

ΩTN

N

L coshðΘi ΔRÞ

~ C e2λ1 C φδ C A 2 coshðΘ1 ΔRÞ

ð6:85Þ

i52

0 S1 5

ΩSN B B Bφ 2 @ ΩO N

1 e2PeN ðδ=ro Þ=2 o

ΩSN

N

L coshðΘi ΔRÞ i52

C e2λ1 C φδ C A 2 coshðΘ1 ΔRÞ

ð6:86Þ

172


with *

ΩNj

j tanhðΘN ΔRÞ ψN21 DN21 PeoN 5 12 2 j 2 ΘN ΩN21 DN

!+

j ΩN21 ξ N21

with j 5 T; S; O ð6:87Þ

The value of ξ i21, Ai, and Bi (with i 5 1 2 N 21) as well as Ωij (with j 5 T, S, O and i 5 1 2 N 21), and ψij (with j 5 T, S, O and i 5 1 2 N) can be calculated by the expressions given below: ξ i21 5 ehPei2Pei21 iRi21 =2

with i 5 1 2 N

* Ωij 5

j tanhðΘi ΔRÞ ψi21 Di21 Peoi 12 2 j 2 Θi Ωi21 Di

ð6:88Þ

!+

j Ωi21 with j 5 T; S; O; i 5 2 2 N 21 ξi21

ð6:89Þ * j 5 ψi21

Bi21 2 Ai21

j ψi21 Di21 Peoi 2 j Di 2 Ωi21

!+

j Ωi21 with j 5 T; S; O; i 5 2 2 N 21 ξ i21

ð6:90Þ Ai 5

Peoi Peo tanhðΘi ΔRÞ 2 Θi ; Bi 5 i 2 Θi tanhðΘi ΔRÞ with i 5 1 2 N 21 2 2 ð6:91Þ

The initial values of Ωij and ψij ; namely Ω1j and ψ1j (j 5 T, S, O), are as ΩT1 5 e2Θ1 ΔR ; ψT1 5 λ~ 1 e2Θ1 ΔR ;

ΩS1 5 eΘ1 ΔR ;

ΩO 1 5 2tanhðΘ1 ΔRÞ

ψS1 5 λ1 eΘ1 ΔR ;

ψO 1 5 2A1

ð6:92Þ ð6:93Þ

It is important to note that the calculation of theΩij and ψij values for i 5 1 to N 21 (or to N) requires a very accurate process. All calculated variables should be given or calculated, even the value of ΔR ( 5 δ/(roN) with accuracy. Each step of the calculation should be carried out by a quick basic computer program with an accuracy of 14 decimals. This is the maximal accuracy of this program. To get the concentration distribution, the values of Ti and/or Si, with i 5 2 2 N, should be determined. [The N value should be chosen to be not less than 100 during calculation; note that in reality it is enough to predict either T1 or S1 because if you know one of these two parameters, the other one can be obtained from the boundary condition given by Eq. (6.84c).]


173

From the internal boundary conditions given by Eqs (6.84a) and (6.84b), the parameters Ti and Si (i 5 2 2 N) can be calculated by the following expressions: Ti 5

ðΓi21 λ~ i 2 Ξi21 Þ λi ½11ði 21ÞΔR e 2Θi ePei ½1 1ði 21ÞΔR

with i 5 2 2 N

ð6:94Þ

Si 5

Ξi21 2 λi Γi21 λ~ i ½11ði 21ÞΔR e 2Θi ePei ½1 1ði 21ÞΔR

with i 5 2 2 N

ð6:95Þ

where ~

Γi 5 Ti eλi ð11 iΔRÞ 1 Si eλi ð11iΔRÞ Ξi 5

ð6:96Þ

o Di n ~ Ti λi eλi ð11iΔRÞ 1 Si λ~ i eλi ð11iΔRÞ Di11

ð6:97Þ

The mass transfer rate at place of Y 5 0, as a sum of the diffusive and the convective flows, can be given as J5

D1 ðλ1 T1 eλ~ 1 λ~ 1 S1 eλ Þ ro

ð6:98Þ

Mass transfer without sweep phase on the permeate side, dΦ/dR 5 0 at R 5 1 1 δ/ro (Case B). ~

at R 5 1 then Φ 5 1 T1 eλ1 1 S1 eλ1

ð6:99aÞ

~ at R 5 1 1 δ=ro then Φ 5 Φδ TN λN eλN ð11δ=ro Þ 1 SN λ~ N eλN ð11δ=ro Þ

ð6:99bÞ

The internal ones with i 5 2, 3, . . ., N 21 are given by Eqs (6.84a) and (6.84b). After the solution, one can obtain for the values of T1 and S1 as T1 5 φ e2λ1

ΨTN 1 2 coshðΘ ΨO 1 ΔRÞ N

ð6:100Þ

S1 5 φ e2λ1

ΨSN 1 2 coshðΘ ΨO 1 ΔRÞ N

ð6:101Þ

~

where * ψjN

5 B N 2 AN

ψjN21 DN21 Peoi 2 2 ΩjN21 DN

!+

ΩjN21 ξN21

with j 5 T; S; O

ð6:102Þ

174


and AN 5

PeoN tanhðΘN ΔRÞ 1 ΘN ; 2

BN 5

PeoN 1 ΘN tanhðΘN ΔRÞ 2

ð6:103Þ

Similar to the previous case, the value of ξi 21, Ai, and Bi [with j 5 T, S, O; i 5 1 2 (N 21)] as well as Ωji and ψji [with j 5 T, S, O and i 5 1 2 (N 2 1)] can be calculated by expressions given by Eqs (6.87)(6.91). The difference between the two models is caused by the outlet boundary condition, thus, the expressions differ at expression related to place of r 5 ro 1 δ. Knowing T1 and S1, the other parameters, namely Ti and Si (i 5 2, 3, . . ., N), can easily be calculated by means of the internal boundary conditions given by Eqs (6.84a) and (6.84b), starting from T2 and S2 up to TN and SN. Thus, the values of Ti and Si can be predicted from Ti21 and Si21 using Eqs. (6.94)(6.97).

6.3.3

Diffusion Accompanied by Zero-Order Reaction

Mass transfer without sweep phase on the permeate side, dΦ/dR . 0 at R 5 1 1 δ/ro. The mass balance equation to be solved will be, for steady-state, as: 2 d φ 1 dφ 1 2 k0 5 0 D dr 2 r dr

ð6:104Þ

In dimensionless form d2 Φ 1 dΦ 2 ϑ2 5 0 1 dR2 R dR

ð6:105Þ

where

sffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi k0 ro2 k0 ro2 ϑ5 Dφ DHco

The solution of Eq. (6.105), with boundary conditions of Φ 5 1 at R 5 1 and Φ 5 Φδ at R 5 1 1 δ/ro, will be as Φ 5 T ln R 1 S 1 where

T5

Φδ

ϑ2 2 R 4

212

ϑ2 4

2δ ro

ln 1 1

1

δ ro

ð6:106Þ h i2 δ ro

ð6:107Þ


175

and S512

ϑ2 4

ð6:108Þ

The inlet mass transfer rate is as: D o dΦ J 5 2 Hc ro dR

8 9 ϑ2 δ δ < 1 2 Φ 1 2 1 δ ro 4 ro D o ϑ2 = 5 Hc 2 : 2; ro ln 1 1 rδo R51

ð6:109Þ

The outlet mass transfer rate, Jδ, i.e., at R 5 1 1 δ/ro is as: 8 9 ϑ2 δ δ < 1 2 Φ 1 2 1 δ ro 4 ro D ϑ2 δ = 2 11 Jδ 5 2 ro : 1 1 δ ln 1 1 δ ro ; ro ro

ð6:110Þ

Mass transfer without sweep phase on the permeate side, dΦ/dR 5 0 at R 5 1 1 δ/ro. After solution of Eq. (6.105) with this new boundary condition on the downstream side of the capillary membrane, the concentration distribution can be expressed as follows: Φ5

ϑ2 δ 2 ϑ2 11 ð1 2 ln RÞ 1 1 1 ðR2 2 1Þ 2 4 ro

ð6:111Þ

The inlet mass transfer rate will be as: D o 2δ δ 11 J 5 Hc ϑ ro ro 2ro

ð6:112Þ

References Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Nagy, E. (2008) Mass transport with varying diffusion- and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Design 86, 723730. Nagy, E. (2009) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270278. Nagy, E., and Borbe´ly, G. (2009) Mass transport through anisotropic membrane layer. Desalination 240, 5463. O’Neil, P.V. (1987) Advanced Engineering Mathematics. Wadsworth Inc., Belmont.

7 Transport of Fluid Phase in a Capillary Membrane

7.1

Introduction

A complete description of a membrane module requires that local transport equations that describe the flow and transport conditions should be simultaneously solved. There has been a substantial amount of published research on developing equations to characterize the transport properties (Kelsey et al., 1990; Mondor and Moresoli, 1999; Geraldes et al., 2001; Calabro´ et al., 2002; Long et al., 2003; Marriott and Sorensen, 2003; Richardson and Nassehi, 2003; Damak et al., 2004; Godongwana et al., 2007). There are different regions (lumen side, membrane matrix, and shell side), all of which should be modeled. A structure of the model system is illustrated in Figure 7.1 (Marriott and Sorensen, 2003). As it will be shown, the starting NavierStokes flow models with component and energy equations are very complex equation systems. Thus, significant simplification is needed to obtain more easily applicable model equations. Careful analysis of the separation equipment should be carried out regarding the flow and operating conditions. The pressure distribution in the membrane can be determined by application of the overall balance of linear momentum. The utilization of this conservation principle is complicated here by the fact that an external force must be applied to keep the polymer membrane stationary. Consequently, the balance of linear momentum must include the effects of this externally applied force in addition to inertia, viscous forces, the gravitational force, and pressure forces. Inertia effects are negligible in the membrane because velocities are small, and the effects of viscous forces are also negligible because deformation gradients are small. Consequently, the externally applied force is effectively used to balance pressure and gravitational forces. For example, in a capillary membrane module with permeable membrane wall (Figure 7.2), three regions of flow should be considered: flow in the lumen (0 # r # ro), flow within the membrane matrix (ro # r # ro 1δ), and flow in the extracapillary space (ro 1δ # r). The differential balance equation should be given for every region, thus, the continuity, momentum equations, as well as the component mass balance and/or energy balance equations. Thus, in order to describe behavior of a membrane module, three submodels are required: two that describe the flow or transport on either side of the membrane and a third model that characterizes the separation properties of the membrane and any porous support material (for details of the membrane transport, see Chapters 36). Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00007-7 © 2012 Elsevier Inc. All rights reserved.

178


Operating conditions

Transport parameters (Pe, Re, etc.)

Model equation

Physical properties

Simplified equations

Feed side Flow model

Qi

Transport models for membrane

Qi

Permeate side Flow model

Principal model

Figure 7.1 A model structure for modeling a membrane module.

7.2

Flow Models for Fluid Phases on Both Sides of Capillary Membrane Modules

The general model equations listed by Eqs (7.1)(7.8) should be adapted for fluids in the lumen and shell-side regions. Note that the balance equations given below do not contain any source terms. In given cases, the production or consumption of momentum, component, or heat should additionally be taken into account in their balance equations, as will be made in a few chapters. The equation of continuity (conservation of mass) in a cylindrical coordinate system (Bird et al., 1960) (note there is no source term in the balance equations for the fluid phase) is: @ρ 1 @ 1 @ @ 5 ðρrυr Þ1 ðρυθ Þ1 ðρυz Þ50 @t r @r r @θ @z

ð7:1Þ

where ρ is the mass density (kg/m3); υ is the convective velocity (m/s); t is the time (s); r is the radial coordinate (m); z is the axial coordinate (m); and subscripts r, θ, and z denote the different coordinate directions.

Transport of Fluid Phase in a Capillary Membrane

Membrane matrix

179

Figure 7.2 Schematic diagram of the flow in a capillary tube with a permeable wall.

Membrane lumen Shell

ro

δ

The equation of motion in cylindrical coordinates (in terms of Newtonian fluid with constant ρ and μ): r-component: ρ

@υr @υr υθ @υr υ2θ @υr 1υr 1 2 1υz @t @r r @θ r @r @p 1@ @υr υr 1 @2 υr 2 @υθ @2 υr 2 21 2 2 2 2 1 2 1ρgr 5 2 1μ r @r r @z @r r @r r @θ r @θ

ð7:2Þ

θ-component: ρ

@υθ @υθ υθ @υθ υr υθ @υθ 1υr 1 2 1υz @t @r r @θ r @r 1 @p @ 1@ 1 @2 υθ 2 @υr @2 υθ 1 2 1ρgθ 52 1μ ðrυθ Þ 1 2 2 2 2 @z r @θ @r r @r r @θ r @θ

ð7:3Þ

180


z-component: @υz @υz υθ @υz @υz @p 1@ @υz 1 @2 υz @2 υz ρ 1υr 1 1υz 52 1μ 1 2 2 1 2 1ρgz r @t @r r @θ @z @r @z @z r @r r @θ ð7:4Þ where p is the pressure (Pa); μ is the dynamic viscosity (Pa s 5 kg/ms); and g is the gravitational acceleration (9.80665 m/s2). Equation (7.1) defines the mass balance in the cylindrical system (tube, membrane). According to Eq. (7.1), the component balance equation can be defined as well. The convective transported components, inside the flowing phase, also can be transported by molecular diffusion. Let Ji define the overall transport rate of the component i, Eq. (7.1) can be rewritten for component i, defining the three-dimensional distribution of i, as (Labecki et al., 2004): @ci 1 @ðrJr;i Þ 1 @Jθ;i @Jz;i 5 1 1 @t @z r @r r @θ

ð7:5Þ

where ci is the local concentration of component i (kg/m3); Jr,i, Jθ,i, and Jz,i are component flux in r, θ, and z direction, respectively (kg/m2 s). Upon expansion of the component flux into diffusive and convective terms, the following equation governing the i solute transport is obtained: @ci 1@ @ci 1 @ Dθ;i @ci @ @ci 5 2rυr;i ci 1 2υθ;i ci 1 2υz;i ci rDr;i Dz;i @t @r @z r @r r @θ r @θ @z ð7:6Þ Equation (7.6) considers variable diffusion coefficient and convective flow rate. In the case of constant diffusivity and convective velocities, one can get the following, known equation (Bird et al., 1960, p. 559): @ci @ci @ci υθ @ci 1@ @ci 1 @2 c i @2 c i 1u 1υr 1 5 Di 1 2 21 r @t @z @r r @θ @r @z r @r r @θ

ð7:7Þ

The equations of energy in terms of the transport properties (with Newtonian fluids of constant ρ and k): @T @T @T υθ @T @ 1 @T 1 @2 T @2 T ^ 1u 1υr 1 5k 1 2 21 1ℑ ρC p r @θ @t @z @r @r r @r r @θ @z ð7:8aÞ where C^ p denotes the heat capacity at constant pressure, per unit mass (kJ/kg K m2/s2 K); k is the thermal conductivity (kW/m K kg m/s3 K); and


181

u is the axial convective velocity (m/s). Note that k 5 λρC^ p ; where λ is the thermal diffusivity (m2/s), ρ is the fluid density (kg/m3). In Eq. (7.8a), the terms associated with viscous dissipation also are given according to Bird et al. (1960, p. 319). These terms [Eq. (7.8b)] should be involved for systems with large velocity gradients only (for details, see Bird et al., 1960), thus this term can be neglected, but in special cases should be taken into account. In a capillary and even in a tubular membrane, these terms are generally not important. ( 2 2 ) @υr 2 1 @υθ @u 1υr 1 1 ℑ 5 2μ r @θ @z @r ( 2 ) @υθ 1 @u @u @υr 2 1 @υr @ υθ 2 1 1 1 1 1 1μ @z @z r @θ @r r @θ @r r

ð7:8bÞ

Let us consider a steady-state, laminar, incompressible, viscous, and isothermal flow with a permeable wall. The flow is axisymmetric, so only half of the tube (capillary) should be considered. Moreover, with a sufficiently long entrance region in the tube, the flow is fully developed at the permeable tube entrance. A schematic diagram of the physical model and coordinate system is given in Figure 7.2 for a vertically oriented tube. Thus, the above NavierStokes equations [Eqs (7.1)(7.4)] can be simplified for the tube side of a capillary as follows (Wiley and Flechter, 2003; Damak et al., 2004): 1@ @u ðrυÞ1 50 r @r @z

ð7:9Þ

@u @u @p @ 1 @ðruÞ @2 u ρ υ 1u 5 2 1μ 1 2 @r @z @z @r r @r @z

ð7:10Þ

@υ @υ @p @ 1 @ðrυÞ @2 υ ρ υ 1u 5 2 1μ 1 2 @r @z @z @r r @r @z

ð7:11Þ

and for solute concentration in the case of constant density u

@c @c @ @c 1 @ðDcÞ @ @c 1υ 5 D 1 1 D @z @r @r @r r @r @z @z

ð7:12Þ

as well as for the temperature distribution 2 @T @T @ T 1 @T @2 T ^ ρC p u 1 1υ 5k 1 @z @r @r 2 r @r @z

ð7:13Þ

182


where u and υ are the axial and radial flow rates, respectively; the diffusion coefficient can be varied as a function of the concentration. Equations (7.9)(7.13) can further be reduced depending on the transmembrane convective velocity, υ. Its value depends strongly on the transmembrane pressure [see Eqs 1.351.37]. In the case of ultrafiltration or nanofiltration, the permeation velocity falls between about 1024 and 1026 m/s, while the axial convective velocity may be between 1 3 1023 and 3 3 1021 m/s, depending strongly on the necessary operating conditions, that is, depending on the membrane process itself. As the reader can see, the axial velocity often can be much larger than that of the permeation rate. Look at the value of the axial (Peax 5 uL/Df, where L is the capillary length, and Df is diffusivity in fluid) and radial Peclet numbers (Perad 5 υro/Df, where ro is the internal radius of capillary); their values can help for the further simplification of Eqs (7.8)(7.10). The Peclet number is the ratio of the convective velocity to the diffusive mass transfer coefficient. Let the radius of the capillary be 150 μm, while the diffusion coefficient in the flowing liquid changes between 1028 and 1029 m2/s. Thus, we can see the axial Peclet number is between about 10 and 4.5 3 104, thus the axial diffusion term and the axial thermal conductivity term can be neglected. On the other side, the radial Peclet number can change between about 1025 and 1 thus, the radial concentration gradient is rather low, accordingly, momentum equation of the transverse velocity can be neglected. The question is that the momentum radial convective velocity term in Eqs (7.9) and (7.10) as well as Eqs (7.12) and (7.13) can be neglected when the radial velocity is relatively low. This can only be done when the axial volumetric flow rate does not change significantly during the flow trough of a capillary tube with a length of L. The ratio of the volumetric permeation rate to the volumetric axial rate, that is υro/uL, can orient us. If this ratio is much less then 1, the transverse term can be neglected. Accordingly, Eqs (7.8)(7.11) can be simplified as @u 1 @ðrυÞ 1 50 @z r @r

ð7:14Þ

In the most viscous flows, normal stress effects, @2u/@z2, are not as important as shear stresses (Godongwana et al., 2007) and thus, Eq. (7.10) becomes 052

@p 1@ @u 1μ r @z r @r @r

ð7:15Þ

The component and energy balance will be as

u

2 @ci @ ci 1 @ci 5D 1 @z @r 2 r @r

ð7:16Þ


183

and 2 @T @ T 1 @T ^ ρC p u 1 5k @z @r 2 r @r

ð7:17Þ

Similarly, the balance equations can be simplified for the membrane and for the shell side, as well. In the following sections, some special cases will be shown briefly.

7.3

Special Cases

7.3.1

For the Axial Flow of an Incompressible Fluid in a Circular Tube with an Impermeable Wall (Figure 7.3)

We consider a long tube and set υθ and υr equal to zero. The remaining velocity component υz (let it be denoted by u) will not be a function of θ because of cylindrical symmetry. The z-component of the equation of motion for constant ρ and μ may then be written as @u @p 1@ @u @2 u r 1 2 ρu 5 2 1μ @z @z r @r @r @z

ð7:18Þ

This equation may be further simplified by taking advantage of the equation of continuity, which reduces here to @u 50 @z

ð7:19Þ

In the most viscous flows, normal stress effects, @2u/@z2, are not as important as shear stresses (Godongwana et al., 2007), and thus Eq. (7.12) becomes @p 1@ @u r 0 5 2 1μ @z r @r @r

ð7:20Þ

Figure 7.3 Flow in a capillary tube with an impermeable wall. r z

z=0

z=L

184


Integration twice with respect to r and use of the boundary conditions u50 at r5R, and u is finite at r50 gives ðp0 2pδ Þro2 u5 4μδ

7.3.2

(

2 ) r 12 ro

ð7:21Þ

Flow Equations for Ultrafiltration in a Capillary Tube with a Permeable Wall (Song, 1998)

During ultrafiltration, the axial volumetric velocity will change in an axial direction, depending on the transverse volumetric flow rate. Consequently, both the radial and axial convective velocities have to be taken into account (Figure 7.4). The equations to be solved for isothermal ultrafiltration are as follows: Continuity equation: 1@ @u ðrυÞ1 50 r @r @z

ð7:22Þ

Momentum equation: @u @u @P @ 1 @ðruÞ ρ υ 1u 5 2 1μ @r @z @z @r r @r

ð7:23Þ

Mass balance equation: υ

@c @c @ 1 @ðrcÞ 1u 5 D @r @z @r r @r

ð7:24Þ

It is worth noting here that u and υ are the bulk flow velocities, that is, the convective transport of the fluid and particles is the same. These equations are subject to the following boundary conditions: At the inlet of the capillary lumen side, at z50: c5co ;

z50;

υ50;

u5uo

Figure 7.4 Transport in a capillary tube with a permeable wall.

ν u r z

z=0

for all z

z=L


185

At the membrane wall, r5ro: at r5ro

u50;

at r50

@υf 5 0; @r

υw 5 Δp=μRm ; @c 50 @r

uc2Df

@c @φ 5 υw φ2Dm @r @r

for all z

for all z

where subscripts w, f and m denote wall, fluid and membrane, respectively, R is membrane resistance, μ is viscosity. Parameters without subscripts are related to the flowing fluid phase.

7.3.3

Capillary Transport with Low Transverse Convective Velocity

Several membrane separation processes often involve low convective velocity in a radial direction such as nanofiltration, membrane reactors, gas separation, pervaporation in porous ceramic membrane where the momentum and continuity equations, and consequently, the component and energy equations can significantly be reduced or neglected, as briefly discussed by Eqs (7.14) and (7.15). In this case, the membrane is relatively impermeable, that is, the membrane is the main resistance to flow. Regarding its importance, a solution of this case will be shown here. The continuity and momentum equations to be solved are (Kelsey et al., 1990; Piret and Cooney, 1990; Mondor and Moresoli, 1999): @u 1 @ðrυÞ 1 50 @z r @r 052

ð7:14Þ

@p 1@ @u 1μ r @z r @r @r

ð7:15Þ

When the momentum equation [Eq. (7.15)] is integrated twice with respect to r, one can get u5

dp r 2 1A ln r1B dz 4

ð7:25Þ

The integration constants A and B can be determined by the following boundary conditions (Bird et al., 1960; Mondor and Moresoli, 1999): at r 5 ro

u50

at r 5 0

@u 5 0 for all r @z

for all r

ð7:26Þ ð7:27Þ

186


Consequently, it can be obtained as 2 ! ro2 r dpf 12 u52 4μ dz ro

ð7:28Þ

Replacing Eq. (7.28) into Eq. (7.15), one can get for the radial convective velocity in the lumen side (for distinction for the membrane’s velocity, subscript f denotes the fluid phase here in the lumen side) ( ) ro3 r 1 r 3 d2 pf 2 ð7:29Þ υf 5 8μ ro 2 ro dz2 From Eq. (7.29), the wall velocity, υw, can be given as (Brotheton and Chau, 1990): υw 5

ro3 @2 pf 16μ @z2

ð7:30Þ

Look at the radial convective velocity in the membrane itself. The flow through this membrane is proportional to the transmembrane pressure drop and inversely proportional to the fluid viscosity. The continuity equation for the membrane (ro #r#ro 1δ) is as follows: 1@ @υ r 50 ð7:31Þ r @r @r Let us apply Darcy’s law to obtain the radial convective in the membrane: υ52

K dp μ dr

ð7:32Þ

where K is the Darcy permeability (m2); μ denotes the viscosity (kg/ms); ε is the porosity of the membrane; υ is the radial convective velocity in the membrane (m/s); and p is the pressure (Pa kg/m s2). Replacing Eq. (7.32) into Eq. (7.31), one can obtain the following equation (the constant factor can be omitted): 1@ @p r 50 ð7:33Þ r @r @r When Eq. (7.33) is integrated twice with respect to r using the boundary conditions, and when r 5 ro then p 5 po and when r 5 ro 1 δ then p 5 pδ, the following membrane pressure is obtained: p5

ΔpðzÞ r ln 1po ðzÞ lnð11δ=ro Þ ro

ð7:34Þ


187

where Δp 5 po 2 pδ. Replacing Eq. (7.34), after derivation, into Eq. (7.32), the radial velocity can be obtained as υ5

K ro po ðzÞ2pδ ðzÞ εμ r lnð11δ=ro Þ

ð7:35Þ

Note that pressures are a function of the axial coordinate according to the change of the boundary conditions in axial direction, thus po(z), Δp(z), and pδ(z) should be considered. Equation (7.35) expresses the radial convective velocity distribution in the membrane at a given axial point. The wall velocity will be from this equation as υw 5

K po ðzÞ2pδ ðzÞ εμ lnð11δ=ro Þ

ð7:36Þ

In the following case, we assume that the pressure in the shell side does not change as an axial direction, that is, pδ 5 constant. From equality of Eqs (7.30) and (7.36), one can obtain a second-order differential equation as (the radial pressure gradient in the lumen is negligible) d2 p 2A2 p 5 2A2 pδ dz2

ð7:37Þ

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16K A5 3 ε ro lnð11δ=ro Þ

The following boundary conditions can be used (Brotheton and Chau, 1990): a. In an open system:

at z50 at z5L

p5pin p5pout

ð7:38Þ

b. In a closed system:

at z50 at z5L

p5pin dp 50 dz

ð7:39Þ

The main resistance is created by the capillary membrane in the closed system. In an open system, the outlet pressure is mostly zero. The solution of Eq. (7.37) is well known p 5 T eAz 1Se2Az 1pδ

ð7:40Þ

188


The pressure axial distribution will be as: a. For an open system:

p5

pin 2pout cosh ALð12z=LÞ 1pδ coshðALÞ

ð7:41Þ

b. For a closed system:

p5

pin 2pout sinh ALð12z=LÞ 1pδ sinhðALÞ

ð7:42Þ

With the knowledge of p(z), now one can determine the wall velocity in an axial direction applying Eqs (7.36) and (7.41) or (7.42), for the open and closed systems, respectively, as υw 5

K pin 2pout cosh ALð12z=LÞ εμ lnð11δ=ro Þ coshðALÞ

ð7:43Þ

υw 5

K pin 2pout sinh ALð12z=LÞ εμ lnð11δ=ro Þ sinhðALÞ

ð7:44Þ

or

The mass balance over a slice dz of the fiber can give du 2 5 2υw dz ro

ð7:45Þ

where u denotes the average axial velocity (for laminar flow u 5 umax =2; where umax is the maximal velocity of the parabolic flow profile). Note that the value of u is radial average one; it depends on the axial coordinate. By means of Eq. (7.45), the distribution of the average axial lumen velocity can be determined, namely ðu uo

du 5 2

2 ro

ðL

υw dz

ð7:46Þ

0

Accordingly, we can get for the open and closed systems, respectively. u 5 uo 2Ξ tanh½AL with

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ro KΔp Ξ5 2 4μ εL2 lnð11δ=ro Þ

ð7:47Þ

ð7:48Þ


189

and u 5 uo 2Ξ

½12coshðALÞ coshðALÞ

ð7:49Þ

The above methodology was used and discussed with varying axial pressure on the shell side of the capillary membrane (Brotheton and Chau, 1990; Kelsey et al., 1990; Mondor and Moresoli, 1999).

7.3.4

Mass Transport Through a Capillary Membrane: Component Model for the Feed Side

Most of the membrane separation processes involve the component transport as well. One hundred percent retain of a component, particle, colloids by a membrane layer, is very rare. However, the membrane enables the selective retain and permeation of solute components (e.g., gas separation, pervaporation, nanofiltration, dialysis, and membrane reactor) depending on the operating conditions, membrane structure, and so on. The description of such processes needs the solution of the differential balance equations, discussed above, for both the feed and the permeate phases. The connection between the two departments is determined by the transport through the membrane. Thus, the balance equation also should be solved for the membrane phase. These results are discussed in Chapters 36. General balance equations [e.g., Eqs (7.1)(7.8)] can be essentially simplified in most of the membrane processes. Let us assume isothermal process with zero radial convective velocity and steady-state conditions. This situation is illustrated schematically in Figure 7.5. In the case of a dilute solution, the differential component balance equation can be given as for the feed side (fiber lumen) of a capillary membrane, assuming a fully developed parabolic velocity profile (cf,i denotes the fluid concentration while the concentration in membrane will be given without subscript) as 2 r 2 @cf;i @ cf;i 1 @cf;i 2 5 Di ð7:50Þ 1 2uo ro 12 2 ro @z @r 2 r @r Let it be given in dimensionless form with R5r/ro; Z5z/L; Ci 5ci =coi 2 2uo ro2 @Cf;i @ Cf;i 1 @Cf;i ð12R2 Þ 5 1 Df;i L @Z @R2 R @R ν=0

Ji uo

uo

cio

r,y z

Ci,out Ji

Figure 7.5 Illustration of a component transport through a membrane.

ð7:51Þ

190


One should apply the common boundary conditions ( ) as follows: at Z 50 Ci 51 for all R at R 5 0

@Ci 5 0 for all Z @R

at R 5 1

Df;i

ð7:52aÞ ð7:52bÞ

@Cf;i @Φi 5 Di 2Ji @R @R

ð7:52cÞ

where Φ is the dimensionless concentration in the membrane Φ 5 φ/φ (Hc 5 φ where concentration with stars are interface concentrations); Df and D are diffusivity in the fluid phase and in the membrane layer (m2/s); and J denotes the specific mass transfer rate (kg/m2 s). Note the third boundary condition contains the connection between the fluid phase and the membrane layer. Equation (7.52c) expresses that there is no accumulation or consumption of component on the membrane interface. In Chapters 3 and 6, Ji values have been defined under different conditions. One of these equations should be substituted in Eq. (7.52c). Average concentration distribution was calculated by numerical solution of Eq. (7.51) (Nagy and Hadik, 2002). The Ji mass transfer rate is given by Eq (3.23) or (3.26). Figure 7.6 clearly demonstrates how strongly the membrane mass 1.0

1.0

0.8

0.8 α=0

0.6

0.6

Ci

0.3 0.5

0.4

0.4

1.0

0.2

0.0

0.2

0

0.2

0.4

0.6

0.8

1

0.0

Z

Figure 7.6 Average concentration as a function of axial direction when the diffusion coefficient exponentially varies with the membrane concentration ðDi 5Do;i eαCi ; H55, ro5100 μm, Do,i51 3 1029 m2/s, uo55 3 1022 m, Sh5βroH/Di50.5, β5Do,i/δ55 3 1026 m/s).


191

transport parameters can influence the axial average concentration distribution, and further, the Ji mass transfer rate as well.

7.3.5

Flow Models for Plane Membrane Modules (Rectangular Coordinates, x, y, z)

As mentioned previously, there are a lot of membrane modules where rectangular coordinates should be used. In this section, the basic balance equations for these cases are given. These general forms of equations should then be simplified according to the operating conditions. According to Eqs (7.1)(7.8), the balance equations for a rectangular coordinate system are as follows: For z-component (note the axial velocity is denoted by u, while that for perpendicular direction, y, is by υy, similar to the cylindrical coordinate; in x direction, the convective velocity is denoted by υx): 2 @u @u @u @u @p @ u @2 u @2 u 1ρgz 1υy 1υx 1u 5 2 1μ 1 1 ρ @t @y @x @z @z @x2 @y2 @z2

ð7:53Þ

For y-component: 2 @υy @υy @υy @υy @p @ υ y @2 υ y @2 υ y 1υy 1υx 1u 5 2 1μ 1 2 1 2 1ρgy ρ @t @y @x @z @x2 @y @z @y ð7:54Þ For x-component: 2 @υx @υx @υx @υx @p @ υ x @2 υ x @2 υ x 1υy 1υx 1u 5 2 1μ ρ 1 2 1 2 1ρgx @t @y @x @z @x2 @y @z @x ð7:55Þ For component balance: 2 @ci @ci @ci @ci @ ci @2 ci @2 ci 1υx 1υy 1u 5 Di 1 21 2 @t @x @y @z @x2 @y @z

ð7:56Þ

For energy balance: 2 @T @T @T @T @ T @ 2 T @2 T ^ ρC p 1 1 2 1ℑ 1υx 1υy 1u 5k @t @x @y @z @x2 @y2 @z

ð7:57Þ

192


( ) @υx 2 @υy 2 @u 2 ℑ 5 2μ 1 1 @x @y @z ( ) @υx @υy 2 @υx @u 2 @υy @u 2 1μ 1 1 1 1 1 @y @x @z @z @x @y

ð7:58Þ

The value of ℑ that is associated with viscous dissipation can be neglected, as it can be done usually (Bird et al., 1960, p. 319). These terms should be involved for systems with large velocity gradients (for details, see Bird et al., 1960).

References Bird, R.B., Stewart, E., and Lightfoot, E.N. (1960) Transport Phenomena. John Wiley and Sons, New York. Brotheton, J.D., and Chau, P.C. (1990) Modeling analysis of an intercalated-spiral alternate-dead-ended hollow-fiber bioreactor for mammalian cell culture. Biotechnol. Bioeng. 35, 375394. Calabro´, V., Curcio, S., and Iorio, G. (2002) A theoretical analysis of transport phenomena in a hollow fiber membrane bioreactor with immobilized biocatalyst. J. Membr. Sci. 206, 217241. Damak, K., Ayadi, A., Zeghmati, B., and Schmitz, P. (2004) A new NavierStokes and Darcy’s law combined model for fluid flow in crossflow filtration tubular membrane. Desalination 161, 6777. Geraldes, V., Semiato, V., and de Pinho, M.N. (2001) Flow and mass transfer modeling of nanofiltration. J. Membr. Sci. 191, 109128. Godongwana, B., Sheldon, M.S., and Solomons, D.M. (2007) Momentum transfer inside a vertically oriented capillary membrane bioreactor. J. Membr. Sci. 303, 8699. Kelsey, L.J., Pillarella, M.R., and Zydney, A.L. (1990) Theoretical analysis of convective flow profiles in a hollow-fiber membrane bioreactor. Chem. Eng. Sci. 45, 32113220. Labecki, M., Piret, J.M., and Bowen, B.D. (2004) Effects of free convection on three-dimensional protein transport in hollow-fiber bioreactor. AIChE J. 50, 19741990. Long, W.S., Bhatia, S., and Kamaruddin, A. (2003) Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester. J. Membr. Sci. 209, 6988. Marriott, J., and Sorensen, E. (2003) A general approach to modeling membrane modules. Chem. Eng. Sci. 58, 49754990. Mondor, M., and Moresoli, C. (1999) Theoretical analysis of the influence of the axial variation of the transmembrane pressure in cross-flow filtration of rigid spheres. J. Membr. Sci. 152, 7187. Nagy E., and Hadik P. (2002) Analysis of mass transfer in hollow-fiber membranes, Desalination. 145, 147152. Piret, J.M., and Cooney, C.L. (1990) Model of oxygen transport limitations in hollow fiber bioreactors. Biotechnol. Bioeng. 37, 8092. Richardson, C.J., and Nassehi, V. (2003) Finite element modeling of concentration profiles in flow domains with curved porous boundaries. Chem. Eng. Sci. 58, 24912503. Song, L. (1998) A new model for the calculation of the limiting flux in ultrafiltration. J. Membr. Sci. 144, 173185. Wiley, D.E., and Flechter, D.F. (2003) Techniques for computational fluid dynamics modeling of flow in membrane channels. J. Membr. Sci. 211, 127137.

8 Membrane Reactor 8.1

Introduction

The laboratory application and investigation of different types of membrane reactors as a promising unit operation started in the 1980s (Marcano and Tsotsis, 2002; Seidel-Morgenstern, 2010). It is well known that the selectivity in reaction networks toward a target compound can be increased by properly adjusting the local concentration of the reactants involved. A membrane separation unit can be applied especially for adjustment of the reactant’s and/or product’s concentration simply by coupling the membrane separation unit with a chemical/biochemical reactor. This coupling or integration often is made in the same unit. During the last three decades, this technical concept has attracted substantial worldwide research and process development efforts (Marcano and Tsotsis, 2002). There are several books and reviews published in the fields of both membrane reactors (Reij et al., 1998; Coronas and Santamarıá, 1999; Saracco et al., 1999; Julbe et al., 2001; Marcano and Tsotsis, 2002; Paturzo et al., 2002; Dittmeyer et al., 2004; Charcosset, 2006; Judd, 2006; McLearly et al., 2006; Ozdemir et al., 2006; Seidel-Morgenstern, 2010) and membrane bioreactors (Marcano and Tsotsis, 2002; Rios et al., 2004; Fenu et al., 2010; Santos et al., 2010). It is not the aim of this work to discuss in detail the properties, applications, or devices of this process. We look at this unit operation as deeply as needed for its mathematical modeling or description. On the other hand, considering the essential differences in the operation and behaviors between the membrane reactors and the membrane bioreactors, mainly due to the other kinetic models, especially for the living organisms, these two reactor types will be discussed separately. Basically, two configurations of the membrane reactor system can be applied: in the first case, the reactor and the membrane separation equipment are simply connecting in series, while in the second case, the real membrane reactor concept combines these two different processing units (namely, reactor and a membrane separator) into a single unit (Figure 8.1). The membrane can serve as a distributor of one of the reactants or as an active catalyst and permselective layer. The subject of this chapter is to analyze briefly the mass transport in this latter membrane reactor configuration.

8.2

Membrane Reactor Configurations

Membrane-based reactive separation processes are mostly applying thin permselective, porous, or dense layers prepared by means of materials that are organic, inorganic, metal, and so on. The choice of a porous or a dense film and the type of material Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00008-9 © 2012 Elsevier Inc. All rights reserved.

194


used for manufacturing depends on the desired separation process, operating temperature, and driving force used for separation; the choice of material depends on the desired permeance and selectivity, and on thermal and mechanical stability requirements (Marcano and Tsotsis, 2002). There are various membrane reactor configurations at laboratory scale that focus on the reactant/product distribution in order to improve selectivity-conversion performances. According to Saracco et al. (1999) and Seidel-Morgenstern (2010), the membrane reactor concept can be divided into six groups which are schematically illustrated in Figure 8.2. Products in sweep fluid

Sweep fluid Reactants

Reactants

Catalytic and permselective membrane

Figure 8.1 Integrated membrane reactor system. (I)

(II)

Reactants

Homogeneous catalyst Ji

A

C

B

C

(III) A

(IV)

A

B⫹C

Sweep

B

C

A

B

D

(V) A Sweep

Catalytic membrane layer (A ⫹ B C)

A D⫹B

B⫹C

E

C E

B

(VI) A⫹B D⫹B JD

D E

E

A

D

B

A⫹B D⫹B

D E

JB

Figure 8.2 Illustration of most often-applied membrane reactor concepts (IVI).

D

Membrane Reactor

195

According to Figure 8.2, the essentials of this membrane concept are: 1. The reaction product is separated continuously from the homogeneous catalyst, thus, this process can be operating in continuous mode. 2. The membrane serves as a catalyst layer; reactants can supply in regular manner which enables the avoidance of side reactions. 3. The selective removal of the product B enables the enhancement of the productivity to shift the reaction to production of compound B. 4. This is a realization of the selective transport of product B which participates in a second reaction. 5. This concept enables the removal of the product in order to avoid the undesirable consecutive reaction. 6. This is a controlled addition of a reactant through the membrane in order to achieve higher selectivities and yields.

All these reactor configurations serve for better selectivity and yield of the chemical reaction. Thus, it is particularly important to understand the relation between local concentrations, temperatures, and the selectivity-conversion behavior. In the following section, a few basic expressions of chemical reaction engineering that are important for understanding how membrane reactor should act to achieve higher reaction efficiency are discussed.

8.3

Reaction Rate

The reaction rates are the key information required to quantify chemical reactions and to describe the performance of chemical reactors. The specific rate of a single reaction in which N components are involved is defined; for details, see Levenspiel (1999) and Westerterp et al. (1984): Qi 5

1 dci ; ξ i dt

i 5 1; . . . ; N

ð8:1Þ

where Qi is the reaction rate of component i (kg/m3s, kmol/m3s); c is the concentration (kg/m3, kmol/m3); and ξ i is the stoichiometric coefficient (equal to zero for inerts or diluents). Equation (8.1) is applied for a homogeneous catalyst reaction. In heterogeneous catalysis, often the mass or surface area of the catalyst is used for relation, thus the reaction rate measures as kmol/kgs, kmol/m2s, and so forth. Obviously, the chosen scaling quantity should be used consistently for calculation of the Qi reaction rate. The reaction rates can depend on temperature and the molar concentration change of reactants. Conversion for constant volume can be defined as Xi 5

coi 2ci coi

ð8:2Þ

196


where coi is the initial or inlet concentration and Xi is the conversion. The reaction rate can be given for reaction of A 1 B2E 1 F reversible reaction as cA cB 2cE cF ð8:3Þ Q 5 k2 Keq where k2 is the reaction rate constant (m3/kmols); Keq is the equilibrium constant; and Q is the reaction rate (kmol/m3s). Selectivity of component E, σE, is the ratio between the amount of desired product E obtained and the amount of a key reactant, A, converted: σE 5

cP 2coPξ A coA 2cAξ P

ð8:4Þ

where coi is the initial concentration of components (kmol/m3) and ξi is the stoichiometric coefficient. Thiele modulus (ϑ), which is the ratio of characteristic time for radial diffusion to the characteristic time for reaction in the membrane, is described by the following equation: sffiffiffiffiffiffiffiffiffi δ2 Qi ð8:5Þ ϑ5 D i ci where δ is the membrane thickness; c-i is the average concentration of i (kmol/3); Qi is the reaction rate (kmol/m3s); and Di is the effective diffusion coefficient in the membrane layer (m2/s). For first-order chemical reaction, Q 5 k1c (k1 is the reaction rate constant (1/s)), thus the value of ϑ is as sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi k1 D k1 δ 2 5 ϑ5 D βo

ð8:6Þ

Equation (8.6) is the well-known Ha-number for fluid phase, Ha (Ha ϑ, with β o5D/δ). Let us express ϑ for cylindrical space. Taking into account Eq. (3.8), the Thiele modulus for cylindrical space can be expressed as follows: rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi k1 D k1 ro2 ϑ5 5 o D β lnð11δ=ro Þ

ð8:7Þ

with βo 5

D ro lnð11δ=ro Þ

ð8:8Þ

Membrane Reactor

197

The cylindrical mass transfer coefficient, β o, tends to the plane mass transfer coefficient, β o 5 D/δ, in limiting cases, namely when ro-N.

8.4

Modeling of Membrane Reactors

As seen in Figure 8.1, different membrane’s configurations can be obtained depending on the placement of a catalyst (tube and/or shell side or in/on membrane), and motion or not of catalyst particles (packed beds, fluidized beds). The mass balance equations should be given according to real operating conditions. The basis of the continuum models are the differential balance equation given by Eqs (7.1)(7.8) for the fluid zones, completing them by source or sink terms. Because fluid phases and catalyst and/or permselective membrane layers must be treated simultaneously, a multiphase approach is necessary (Seidel-Morgenstern, 2010). Accordingly, the balance equations should be given every three sections of a capillary membrane reactor, namely for the lumen and shell fluid phases as well as for the catalytic (or noncatalytic) membrane. Here we give the general equations that should be applied for every section, taking into account operating conditions of every section. The complete equations for conservation of momentum, total mass, component mass, and energy must be considered in every real fluid and membrane matrix. Thus, one needs for description of such a system continuity equation, momentum transport energy balance, and (n 21) species balance equations (n is number of components of the system). Accordingly, the starting equation system to be used is as it is given in Chapter 7. The equation of continuity (conservation of mass) in cylindrical coordinate system and with variable μ, ε, and ρ (Bird et al., 1960): @ðρεÞ 1 @ðrερυr Þ 1 @ðερυθ Þ @ðερυz Þ 1 1 1 50 @t r @r r @θ @z

ð8:9Þ

The equation of motion in cylindrical coordinates (in terms of Newtonian fluid with constant ρ and μ and porosity ε; these equations are assumed to be valid for membrane or packed-bed reactors as well; the fluid phase in packed membrane reactors are considered as quasihomogeneous flowing phase): r-component: @ðρευr Þ υr @ðrρευr Þ υθ @ðρευr Þ ρευ2θ @ðρευr Þ @ðεpÞ 1υz 1 1 2 1 r r @θ r @t @r @r @r 2 0 1 0 1 0 13 1 @ @ @ðευr ÞA 1 @ @ @ðευr ÞA 2 @ðευθ Þ @ @ @ðευr ÞA5 μr 1 2 μ 2 2μ 1 μ 24 r @r @r r @θ @θ r @θ @z @z 2ερgr 5 2εfr ð8:10Þ

198


θ-component: 0

1 @@ðρευθ Þ 1υr @ðρευθ Þ 1 υθ @ðρευθ Þ 2ρε υr υθ 1υz @ðρευθ ÞA1 1 @ðεpÞ r r @t @r @θ @r r @θ 2 0 0 1 0 13 1 @ 1 @ 1 @ @ðευθ ÞA 2 @ðευr Þ @ @ @ðευθ ÞA5 2 2μ 1 μ 24 @ μr ðrυθ ÞA1 2 @μ @r r @r r @θ @θ r @θ @z @z 2ερgθ 5 2εfθ ð8:11Þ z-component: 0

1 @ðρευ Þ υ @ðrρευ Þ υ @ðρευ Þ @ðρευ Þ z r z θ z z @ A1 @ðεpÞ 1 1 1 υz r r @θ @t @r @z @z 2 0 1 0 1 0 13 1 @ @ðευ Þ 1 @ @ðευ Þ @ @ðευ Þ z z z @μr A1 @μ A1 @μ A52 ερgz 52εfz 24 r @r @r r 2 @θ @θ @z @z ð8:12Þ where the source terms in the momentum conservation equations (SeidelMorgenstern, 2010): fj 5 f1 υj 1 f2 υj jυj with j 5 r; q; z

ð8:13Þ

The fr ; fz source terms in the momentum conservation equations can be neglected for lumen and shell without solid particles. This is not the case in porous membranes or in packed-bed lumens or shells. Equation (8.13) expresses that friction and inertial forces caused by flow through pores lead to an additional loss of momentum, accounted for by the source term f. The parameters f1 and f2, taking into account the pressure drop during transport through membrane layer, can be calculated using the coefficient determined by application of the dusty gas model (Wesselingh and Krishna, 2000) in the case of gasfluid phase: f1 5

μf ; 2 =τ32Þ1ðDμ =pÞ ðεdpore f

f2 5 0

ð8:14Þ

where μf is the dynamic viscosity of the gas phase (Pa s); dpore is the pore size; p is pressure (Pa), ε is porosity; τ denotes the tortuosity; and D is diffusivity of the key component (m2/s). The factor f1 takes into account the viscous slip at pore walls, the parameter f2 can be set to zero due to the laminar character of the flow (SeidelMorgenstern, 2010).

Membrane Reactor

199

The component balance equation will be as: 0

1

@@ðερxi Þ 1υz @ðερxi Þ 1υr @ðερxi Þ 1 υθ @ðερxi ÞA r @θ @t @z @r 8 0 1 0 1 0 19 =

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