Theory of Calorimetry (Hot Topics in Thermal Analysis and Calorimetry)

THEORY OF CALORIMETRY Hot Topics in Thermal Analysis and Calorimetry Volume 2 Series Editor: Judit Simon, Budapes...

Author: W. Zielenkiewicz | E. Margas

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THEORY OF CALORIMETRY

Hot Topics in Thermal Analysis and Calorimetry

Volume 2

Series Editor:

Judit Simon, Budapest University of Technology and Economics, Hungary

The titles published in this series are listed at the end of this volume.

Theory of Calorimetry

by

Wojciech Zielenkiewicz Institute of Physical Chemistry,

Polish Academy of Sciences, Warsaw, Poland

and

Eugeniusz Margas Institute of Physical Chemistry,

Polish Academy of Sciences, Warsaw, Poland

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-48418-8 1-4020-0797-3

©2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

Contents

Preface

ix

Chapter 1 The calorimeter as an object with a heat source 1.1. The Fourier law and the Fourier-Kirchhoff equation 1.2. Heat transfer. Conduction, convection and radiation 1.3. General integral of the Fourier equation. Cooling and heating processes 1.4. Heat balance equation of a simple body. The Newton law of cooling

1.5. The heat balance equations for a rod and sphere 1.6. General heat balance equation of a calorimetric system

1

2

10

14

20 26

33

Chapter 2 Calorimeters as dynamic objects 2.1. Types of dynamic objects 2.2. Laplace transformation 2.3. Dynamic time-resolved characteristics 2.4. Pulse response 2.5. Frequential characteristics 2.6. Calculations of spectrum transmittance 2.7. Methods of determination of dynamic parameters 2.7.1. Determination of time constant 2.7.2. Least squares method 2.7.3. Modulating functions method 2.7.4. Rational function method of transmittance approximation Determination of parameters of spectrum transmittance 2.7.5.

37

39

41

47

55

58

61

66

66

74

76

79

81

CONTENTS

vi

Chapter 3 Classification of calorimeters. of heat effects

Methods

of determination

3.1. Classification of calorimeters 3.2. Methods of determination of heat effects General description of methods of determination of heat ef3.2.1. fects 3.2.2.

Comparative method of measurements 3.2.3. Adiabatic method and its application in adiabatic and scan

ning adiabatic calorimetry 3.2.4. Multidomains method 3.2.5. Finite elements method 3.2.6. Dynamic method 3.2.7. Flux method 3.2.8. Modulating method 3.2.9. Steady-state method 3.2.10. Method of corrected temperature rise 3.2.11. Numerical and analog methods of determination

of thermokinetics 3.2.11.1 Harmonic analysis method 3.2.11.2. Method of dynamic optimization 3.2.11.3. Thermal curve interpretation method 3.2.11.4. Method of state variables 3.2.11.5. Method of transmittance decomposition 3.2.11.6. Inverse filter method 3.2.11.7. Evaluation of methods of determination of total heat effects

and thermokinetics 3.3. Linearity and principle of superposition

85 85 97

97

101

103

104

109

111

114

114

116

119

123

123

124

125

127

128

129

131

136

Chapter 4 Dynamic properties of calorimeters 4.1. Equations of dynamics 4.2. Dynamic properties of two and three-domain calorimeters with

cascading structure 4.2.1. Equations of dynamics. System of two domains in series 4.2.2. Equations of dynamics. Three domains in series 4.2.3. Applications of equations of dynamics of cascading systems

139

139

143

143

148

151

CONTENTS

4.3. Dynamic properties of calorimeters with concentric configuration 4.3.1. Dependence of dynamic properties of two-domain calorimeter with concentric configuration on location of heat sources and temperature sensors 4.3.2. Dependence between temperature and heat effect as a function of location of heat source and temperature sensor Apparent heat capacity 4.3.3. 4.3.4. Energy equivalent of calorimetric system

vii 154 155 165 168 171

Final remarks

177

References

179

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Preface

Calorimetry is one of the oldest areas of physical chemistry. The date on which calorimetry came into being may be taken as 13 June 1783, the day on which Lavoisier and Laplace presented a contribution entitled ,,Memoire de la Chaleur“ at a session of the Academie Française. Throughout the existence of calorimetry, many new methods have been developed and the measuring techniques have been improved. At pre sent, numerous laboratories worldwide continue to focus attention on the development and applications of calorimetry, and a number of compa nies specialize in the production of calorimeters. The calorimeter is an instrument that allows heat effects in it to be determined by directly measurement of temperature. Accordingly, to determine a heat effect, it is necessary to establish the relationship be tween the heat effect generated and the quantity measured in the calo rimeter. It is this relationship that unambiguously determines the mathematical model of the calorimeter. Depending on the type of calo rimeter applied, the accuracy required, and the conditions of heat and mass transfer that prevail in the device, the relationship between the measured and generated quantities can assume different mathematical forms. Various methods are used to construct the mathematical model of a calorimeter. The theory of calorimetry presented below is based on the assumption of the calorimeter as an object with a heat source, and as a dynamic object with well-defined parameters. A consequence of this assumption is that the calorimeter is described in terms of the relation ships and notions applied in heat transfer theory and control theory. With the aim of a description and analysis of the courses of heat effects, the method of analogy is applied, so as to interrelate the thermal and the

x

PREFACE

dynamic properties of the calorimeter. As the basis on which the thermal properties of calorimeters will be considered, the general heat balance equations are formulated and the calorimeter is taken as a system of linear first-order inertial objects. The dynamic properties of calorimeters are defined as those corre sponding to proportional, integrating and inertial objects. Attention is concentrated on calorimeters as inertial objects. In view of the fact that the general mathematical equations describing the properties of inertial objects contain both integrating and proportional terms, a calorimeter with only proportional or integrating properties is treated as a particular case of an inertial object. The thermal and dynamic properties that are distinguished are used as a basis for the classification of calorimeters. The methods applied to determine the total heat effects and thermokinetics are presented. For analysis of the courses of heat effects, the equation of dynamics is for mulated. This equation is demonstrated to be of value for an analysis of various thermal transformations occurring in calorimeters. The considerations presented can prove to be of great use in studies intended to enhance the accuracy and reproducibility of calorimetric measurements, and in connection with methods utilized to observe heat effects in thermal analysis.

Chapter 1

The calorimeter as an object with a heat source A calorimeter can be treated as a physical object with active heat sources inside it. An analysis of the thermal processes occurring inside the calorimeter, and those between the calorimeter and its environment, requires utilization of the laws and relations defined by heat transfer theory [1–5]. The relations arising from heat transfer theory are applied to design the mathematical models of calorimeters, which express the dependence of the change in temperature measured directly as a function of the heat effect produced. There is an understandable tendency to attempt to ex press these models in the simplest way. In practice, this is achieved by applying simplifications to the original formulas. To make use of them wisely, one has to understand precisely the assumptions made. This chapter will present a detailed consideration of this topic. Selected problems from heat transfer theory are also presented. Spe cial attention is paid to a discussion of the processes occurring in a nonstationary heat transfer state. An understanding of these processes is of importance for a proper interpretation of calorimetric measurements. The general heat balance equation is introduced into the considerations. Particular forms of this equation will be applied to consider problems that form the subject of this book.

2

CHAPTER 1

1.1 The Fourier law and the Fourier-Kirchhoff

equation

Heat transfer by conduction in a homogenous, isotropic body is mathematically described by the Fourier law [1]:

which assumes proportionality between the heat flux q and the tempera is a vector determin ture gradient grad T. The heat flux q, in ing the rate of heat Q transferred through unit surface at point P per unit time dt (Fig. 1.1):

or

Quantity T in Eq. (1.1) is a function of the coordinates x, y, z and time t. For any time t the value of T determines the scalar temperature field. At every point upon it, the temperature at this instant is the same. Such a surface is called the isothermal surface for temperature T. The gradient of temperature, grad T, is a vector:

At points of an isothermic surface, the absolute gradient values are equal to

where denotes differentiation along an outward-drawn line normal to the surface. The value of is an experimentally determined coeffi cient called the thermal conductivity, expressed in

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

3

defines the amount of heat conducted through unit area in unit time if a unit of temperature gradient exists across the plane in which the area is measured. The value of for isotropic materials is a scalar, its value depending on pressure and temperature changes. If the range of tem perature is limited, the variation in may not be large and as a reason able approximation it can be assumed to be constant. The reciprocal of the thermal conductivity of a material is called its thermal resistivity. The relation between the heat energy, expressed by the heat flux q, and its intensity, expressed by temperature T, is the essence of the Fou rier law, the general character of which is the basis for analysis of vari ous phenomena of heat considerations. The analysis is performed by use of the heat conduction equation of Fourier-Kirchhoff. Let us derive this equation. To do this, we will consider the process of heat flow by con duction from a solid body of any shape and volume V located in an environment of temperature [5,6]. When heat is generated in the body, two processes can occur: the ac cumulation of heat and heat transfer between the body and its environ ment. Thus, the amount of heat generated, dQ, corresponds to the sum of the amount of heat accumulated, and the amount of heat ex changed,

4

CHAPTER 1

The heat flux through a surface element dS in time dt, in conformity with Eq. (1.5), corresponds to

where dS = ndS is a normal vector to a surface element in the external direction, so that the amount of heat transferred through the whole surface S in time dt is equal to

On application of the Gauss-Ostrogrodsky theorem, which states that the surface integral of a vector is equal to the volume integral of the divergence of the vector, Eq. (1.8) becomes

The amount of heat generated in the body by the inner heat sources of density g (the amount of heat developed by unit volume in unit time) in element dV of volume V in time dt is equal to

Thus, the heat generated in the total volume V of the body in time dt is equal to

The amount of heat accumulated, to

according to Eq. (1.6), is equal

When Eqs (1.9) and (1.11) are taken into account, Eq. (1.12) can be written in the form


5

According to the First Law of Thermodynamics, the amount of heat can be described for each volume element of the accumulated, body as

where dh is the increase in specific enthalpy (due to unit volume), and p is pressure. The increase in specific enthalpy is proportional to the increase dT in temperature T:

where

is the specific heat capacity at constant pressure, in

and is the density, expressed in the body of volume V is equal to

The increase in enthalpy in

The second term on the right-hand side of Eq. (1.14) may be written as

With regard to Eqs (1.16) and (1.17), Eq. (1.12) becomes

Comparison of both sides of Eqs (1.13) and (1.18) gives

Equation (1.19) is valid in any element of the body if

Division of both sides of Eq. (1.20) by dt gives

6

CHAPTER 1

On introduction of Eqs (1.15) and (1.1), Eq. (1.21) can be written in the form

For a solid body with a distribution of temperature T at time t given by

we have

Thus, the substantial derivative of temperature takes the form

Introduction of the velocity vector w

into Eq. (1.25) lends to

In a similar way, pressure changes as a function of (x, y, z, t) can be expressed by


7

Substitution of Eqs (1.27) and (1.28) into Eq. (1.22) yields

Equation (1.29) is called the Fourier-Kirchhoff equation. When the process takes place under isobaric conditions, i.e.

the Fourier-Kirchhoff equation (Eq. (1.29)) takes the form

When the velocity vector is equal to zero, i.e.

Eq. (1.31) becomes

Equation (1.33) is called the Fourier equation or the equation of con duction of heat. Let us use the Fourier equation to consider the following processes: a) Stationary (steady-state) heat transfer, which occurs when the changes in temperature T are not time-dependent:

and the distribution of temperature is a function only of the Cartesian coordinates (x,y, z). Relation (1.34) is fulfilled, when g = const, b) A non-stationary (non-steady-state) heat transfer process, which takes place when the changes in temperature T are time-dependent:

8

CHAPTER 1

When g = 0, the temperature changes depend only on the initial dis tribution of temperature. They characterize the heating or cooling proc esses occurring in the thermal by passive body. The investigation of heat processes for which

is the subject of great numbers of calorimetric determinations. and are constants independent of both pressure and When temperature, parameter a is often applied:

This is called the thermal diffusivity coefficient, expressed in Introduction of coefficient a into Eq. (1.33) gives

where is a Laplace operator: The Fourier-Kirchhoff differential equation and the equation of con duction of heat describe the transfer of heat in general form. In order to obtain the particular solutions of these equations, it is necessary to de termine the initial and boundary conditions. The initial conditions are be to understood in that the temperature throughout the body is given arbitrarily at the instant taken as the origin of the time coordinate t. It is usually assumed that the temperature at the beginning of the process is constant. For steady-state processes, the course of the temperature changes does not depend on the initial condi tions. The boundary conditions prescribe:


9

a) Dimensionless parameters which characterize the shape and dimen sions of the body. An example is a homogenous ball with radius R, where

In the spherical system,

is expressed by

In the case of a cylinder, it is convenient to operate with cylindrical coordinates r, x in which the Laplacian has the form

where

b) Physical parameters such as specific heat density and thermal

conductivity coefficient which characterize the properties of the body

and the environment. It is determined that they are constant or a function

of temperature. In all the problems discussed in this book, it is considered that and are constants.

c) Surface conditions. These conditions are as follows:

1. The prescribed temperature distribution on the surface S of the body:

2. The prescribed distribution of the heat flux or temperature gradient on the surface of S of the body:

3. The defined relation between the temperature and the heat flux on the surface S of the body:

10

CHAPTER 1

where is the surface heat transfer coefficient (see § 1.2). Mixed boundary conditions, which are subsequent to the assumption that particular parts of surface S are characterized by various types of boundary conditions, can also be used. d) The form of function g, which describes the inner heat sources. When the initial and boundary conditions are known, the physical problem of heat conduction is to find adequate solutions of the FourierKirchhoff equation or the Fourier equation.

1.2 Heat transfer.

Conduction, convection and radiation

“When different parts of a body are at different temperatures, heat flows from the hotter to the colder parts. The transfer of heat can take place in three distinct ways: conduction, in which the heat passes through the substance of the body itself; convection, in which heat is transferred by relative motion of portions of the heated body; and radia tion, in which heat is transferred directly between distant portions of the body by electromagnetic radiation” [1]. Heat conduction is a type of transfer of heat in solids and liquids, in terpreted as the imparting of kinetic energy resulting from collisions between disorderly moving molecules. The process occurs without any macroscopic motions in the body. The conductivity of diamond without traces of isotope is the highest. The conductivity of a metals is also high. The lowest conductivity is that of a gas. Heat transfer by conduction is defined by the Fourier equation (1.1). The application of Eq. (1.1) in calculations encounters difficulties be cause the temperature gradient of the wall must be defined, as well as its increments around the whole surface S of the body. Accordingly, for practical reasons the Newton equation is usually applied:


11

where coefficient is called the surface conductance or the coefficient of surface heat transfer. It defines the amount of heat transferred in unit time through unit surface when the existing temperature gradient is 1 deg. For using of the Newton equation, there is no need to introduce any simplifying assumptions. All the complicated character of the heattransfer phenomenon is enclosed in the value of coefficient which depends on many parameters. When the heat transfer conditions are described, appropriate attention must be paid to the proper choice of and the definition of the influence of different factors on its value. For a description of the heat flow phenomenon on the border of the body, a differential equation is used:

Equation (1.47) results from the Fourier law [Eq. (1.1)] and the New ton equation [Eq. (1.46)], expressed by Eqs (1.48) and (1.49), respec tively

by equating the right-hand sides. When the geometry of the considered system is simple, an exact solution of Eq. (1.47) can easily be deduced. The value of surface heat transfer coefficient is strongly affected by the presence of heat bridges and thermal resistances. Imperfect con tact between touching surfaces which are at different temperatures causes a lack of thermal equilibrium inside the free space between them. The magnitude of the thermal resistance depends on the surface condi tions, the number and shape of surface irregularities and the conditions of heat conduction through the gas present in the space between the contact points. Temperature variations appear even though the thickness of the gas layer may be close to the size of the free distance in the gas molecule. It has been found [6] that even surface irregularities within the range of tens of microns influence the value of the surface heat transfer

12

CHAPTER 1

coefficient. The variations in thermal resistance between two solid bod ies are the cause of errors in the heat determination. also de The coefficient of surface heat transfer by conduction pends on the existence of heat bridges in the system. If a layer of sub stance of low heat conductivity isolates two objects, then the heat ex change is not intensive. If a third object with heat conductivity higher than that of an insulating substance joins those two objects, then through this object (called a “heat bridge”) heat will flow, intensively enhancing the total heat exchange. Unfortunately, it is often very difficult in calorimetric practice to eliminate the existence of unwanted heat bridges and thermal resistances. It is crucial to decrease their contribution to the value of Heat transfer by convection occurs in liquids and gases where there is a velocity field caused by extorted fluid motion or by natural fluid motion caused by a difference in density. The former case involves forced convection, and the latter case free convection. Combined con vection occurs when both forced and free convection are present. The defining the heat ex convection coefficient of surface heat transfer, change in the contact boundary layer between fluid and solid, is deter mined. Coefficient is often expressed by equations containing criteria numbers, such as those of Nusselt (Nu), Prandtl (Pr), Reynolds (Re) and Grashof(Gr):

The criteria numbers are calculated by use of material constants such as – thermal conductivity coefficient; a – thermal diffusity coefficient; and v – kinematic viscosity. In the expressions in (1.50), l is a distinctive dimension of the body; w is the distinctive velocity; g is the acceleration is the difference in temperature, and is the due to gravity; thermal expansivity coefficient. Coefficient usually depends on the difference in temperature be tween the body and its environment. In the microcalorimeter described by Czarnota et al. [7], free convection occurs in the spherical space is defined by around the calorimeter, and as a result the coefficient the equation


13

A frequent goal in calorimetry is to obtain a constant value of irre spective of existing temperature gradients. A more common trend is to decrease or eliminate the heat transfer by convection. A very effective way to achieve this is the creation of a vacuum. The contribution of the heat transfer by convection can also be reduced by increasing the share of heat conduction in the total heat transfer. Radiation is the transfer of thermal energy in the form of electro magnetic waves; it occurs in processes of emission, reflectivity and ab radiated through a transparent gas sorptivity. The quantity of heat layer from surface at temperature to surface at temperature completely enclosing the first surface, is

where is a substitute emission coefficient depending on the emission from a surface, the geometry and the reflectivity, and is a radiation Introduction of the relation express constant ing the total radiant energy leaving unit surface area:

yields

or, on analogy with Eq. (1.47):

where

is called the radiation heat transfer coefficient. When the temperature difference

14

CHAPTER 1

is much lower than the temperatures

and

can be expressed by

where higher–power exponents of ratio are neglected. If the value of this fraction is very small, then a simplified formula is often valid:

The heat transfer in calorimeters is described in terms of the effective heat transfer coefficient, involving the heat transfer of convection, con duction and radiation. In calorimetry, it is more common to use coeffi cient G, called the heat loss coefficient, which is equivalent to the effec tive heat transfer coefficient calculated for the whole surface S.

1.3. General integral of the Fourier equation.

Cooling and heating processes

Let us consider the temperature changes in a body, due to the initial temperature difference between the body surface S and the environment [1, 6, 8]. The initial condition is defined as

Additionally, it is assumed that

In the examined case, the solution of the Fourier equation will refer to the heating or cooling processes of a thermally passive body. Any assumptions made will not impose any restrictions on the solution. When heat is generated in the body and the expression of g (x, y, z, t) is known, the particular solution of the Fourier equation, can be found, on the assumption that the initial conditions are zero, T(x, y, z) = 0. Accord


15

ing to the superposition rule, the solution will be the sum of the sepa rately determined solutions. Let the heat transfer between the body and its environment proceed through the surface S according to the boundary condition of third kind, i.e.

If it is assumed that the physical parameters are constants and the Fourier equation (Eq. (1.38)) becomes

and can be written in the form

To solve the differential Eq. (1.63), the Fourier method of separated variables will be used. According to this function, T(x, y, z, t) can be described as

Then

and

Substituting the above equations into Eq. (1.63) gives

or, after transformation

16

CHAPTER 1

The right–hand side of Eq. (1.69) is a function only of coordinates (x, y, z), whereas the left–hand side is a function only of the independent variable t. To satisfy Eq. (1.69), both sides must be equivalent to the same constant value. This means that this value should be negative. Thus, when the initial temperature of the body is higher than the envi ronment temperature, a cooling process occurs and

whereas, when the initial temperature of the body is lower than the envi ronment temperature, the heating process is characterized by

Thus:

In order to fulfil this condition, let us denote by the value of the sides of Eq. (1.69). Thus, instead of one differential equation, we obtain the two following differential equations:

Equation (1.73) may be written as


17

and its solution is

which can easily be checked.

Equation (1.74) may be written in the form

Solving this equation gives a set of values

equivalent to the set of solutions of the function

whereas the compatible function

is equal to

The magnitudes are called the eigenvalues of the differential Eq. (1.65). The functions in Eq. (1.80) corresponding to the particular eigenvalues are called eigenfunctions. Since Eq. (1.62) is a linear differ ential equation, its solution is also the sum of eigenfunctions:

or, with regard to Eq. (1.80), it can be written in the form

On introduction of the notation

Equation (1.82) can be written as

18

CHAPTER 1

or

We see that the course of the temperature T (x, y, z, t) changes at any point of the body is the sum of infinite number of exponential functions. Whereas the sequence of eigenvalues is an increasing sequence, the monotonously decreases [9]. As a sequence of constants higher accuracy of determination of temperature T (x, y, z, t) is needed, a greater number of exponential terms must be used for a proper descrip tion of the changes in temperature of the body within time. In the limiting case, it is possible to neglect all the exponential terms, excluding the one having in the exponent the smallest value of the constant Equa tion (1.84) then transforms to an equation equivalent to the mathemati cal expression of Newton’s law. This type of description is mostly used in calorimetry and thermal analysis. The expression given by Eq. (1.85) was used for the first time in microcalorimetry to describe short–duration heat processes investigated in a Calvet microcalorimeter [10–13]. The temperature T rise caused by heat effect was expressed by the following equation:

where the adopted relations made possible the determination of T for an exponential temperature course of second or third order. The excellent monograph by Camia [14] is one of the most important works on this field. A number of methods are currently used to determine the heat effects and thermokinetics of short–duration processes, based on the assumption of a multiexponential course of temperature changes. These methods are disscused in Chapter 3. The question of the determination of total heat effects for a multiex ponential temperature course in time was investigated by Hattori et al. [15] and Tanaka and Amaya [16].


19

Hattori et al. [15] consider the calorimeter in terms of a onedimensional model of distributed parameters. The objects distinguished are the calorimetric vessel A, the heat conductor B and the medium C, at constant temperature, that surrounds the calorimeter (Fig. 1.2). For the solution of the Fourier equation, the following assumptions were made: heat power is generated in the calorimetric vessel at homogenous tem perature and with constant heat capacity; the heat conductor along which the heat flows has a well-insulated lateral surface. Its ends are defined as x = 0 and x = L; there exists a heat bridge of the calorimetric vessel with the conductor; the environment is kept at constant temperature. The heat transfer takes place only through the cross-section for x = 0. Tanaka and Amaya [16] consider the calorimeter as a concentric and (Fig. 1.3), which are solids. model of three domains, in which heat q heat is generated or adsorbed, is surrounded Domain by domain Domain is surrounded by domain at constant tem perature. These three domains represent the calorimetric vessel, and the heat conductor between the vessel and the shield. In both of the above papers, it was shown that, independently of the

20

CHAPTER 1

number of exponential terms that describe the temperature course in time, it is possible to calculate the total heat effect determining the sur face area between the registrated temperature course in time and the time axis below the temperature course in time if the measurement starts from equilibrium conditions and the initial and final temperatures are equal.

1.4. Heat balance equation of a simple body.

The Newton law of cooling

Linear differential equation of first order called the heat balance equation of a simple body, has found wide application in calorimetry and thermal analysis as mathematical models used to elaborate various methods for the determination of heat effects. It is important to define the conditions for correct use of this equation, indicating all simplifica tions and limitations. They can easily be recognized from the assump tion made to transform the Fourier-Kirchhoff equation into the heat bal ance equation of a simple body. Let us consider [1, 6, 8, 17] that the heat transfer process takes place under isobaric conditions, without mass exchange and that the thermal parameters of the body are constant. The Fourier-Kirchhoff equation can then be written as

Integration of both sides of Eq. (1.87) with respect to the volume V of the body of extremal surface S gives


21

The left–hand side of Eq. (1.88) can be transformed as follows:

where

is the average temperature of the body of volume V. The first term on the right–hand side of Eq. (1.88) can be trans formed by using the Gauss-Ostrogradsky theorem for the vector gradi ent:

where dS is an oriented element of the surface S of the body. Applica tion of the average value theory gives

where is the average flux of heat across the surface S of the body. Application of the boundary condition of the third kind leads to

where is the average temperature of the external surface S of the body, is the heat transfer coefficient, and is the temperature of the environment. The integral of the second term on the right–hand side of Eq. (1.88):

corresponds to the change in the heat power dQ/dt within time t. Thus, the second term on the left–hand side of Eq. (1.88) can be written in the form

22

CHAPTER 1

From Eqs (1.89) – (1.95) and Eq. (1.88), and putting

where C is the total heat capacity of the body, we have

Equation (1.97) is accurate, but does not give an explicit solution, and the temperature on the because the relation between temperature surface is not defined.

If it is assumed that

Equation (1.97) becomes

or

Equations (1.99) and (1.100) are commonly known as the heat bal ance equations of a simple body. From the above considerations, it is clear that these heat balance equations and the Fourier-Kirchhoff equa tion [Eq. (1.87)] are equivalent to each other when: 1. the temperature in the total volume of the body is homogenous and only a function of time; 2. the temperature on the whole surface is homogenous and only a function of time; 3. the above temperatures are identically equal to one another at any moment of time; 4. the heat capacity C and the heat loss coefficient G are constant and not functions of time and temperature.


23

When


If it is assumed that


Equation (1.104) is equivalent to Newton’s law of cooling. The coef ficients and of Eqs (1.97) and (1.104) can be characterized as fol lows [9]. Coefficient determines the amount of heat transfer by unit surface in unit time when the temperature difference is equal to 1 deg. Constant is called the cooling constant and characterizes the rate of body cooling. This quantity is the reciprocal of the body time constant It has the same value for any point of the body. Constant does not depend on the initial temperature field, but depends on the shape and dimensions of the body, the thermal parameters of the body (e.g. the thermal diffusivity coefficient) and the conditions of heat transfer. Coefficient is a quantity describing the measure of the ability of the given body to react to cooling or heating of the environment. The influence of this environment on the body is characterized by the heat transfer coef ficient and/or the heat loss coefficient G. The dependence between quantities G, C and given by Eq. (1.103) When these temperatures are is appropriate only when different, Eqs (1.97) and (1.105) can be supplemented by a new parame ter, the coefficient of heterogenity of the temperature field

24

CHAPTER 1

Thus, Eq.( 1.97) becomes

In order to solve this non–stationary differential equation, it would This condition is difficult to be necessary to know the function fulfil. To solve the theory of the problem of ordered state heat transfer [9], two periods of cooling or heating process of the body are specified.

The first period is characterized by a disordered course of temperature field changes in time. The second, the well-ordered heat transfer period, comes after a certain period of time (Fig. 1.4). It is assumed that, for the ordered state heat transfer, the relation between heat capacity C, heat loss coefficient G, cooling constant and coefficient is expressed as follows:


25

Coefficient is a dimensionless quantity; its value can vary between one and zero. The more differs from unity, the greater is the inequality in the field of temperature. This coefficient is a function of Biot’s num Coefficient decreases to zero as constant Bi increases ber to infinity. Let us assume that the same body is observed under various conditions of heat transfer from the body to the environment and that as a result there are various values of the surface heat transfer coefficient For the sequence of increasing values of there is a sequence of increasing values of the constants

According to the discussed theory of the ordered state heat transfer, together with the increase in the value the reaction of the body be comes weaker and the value of the cooling constant drifts towards the limit value (Fig. 1.5). Only in the case of certain small values of coefficient can it be assumed that a homogenous temperature distribu exists in the examined body. It also indicates that the range tion where the heat balance equation can be accurately used for determina tion of the heat power changes within time (the P(t) function called thermokinetics) is limited. However, this equation is also used as a mathematical model in determining the P(t) function in instruments with

26

CHAPTER 1

different values of coefficient It is applied as a mathematical model in conduction microcalorimeters, in which heat exchange between the body and environment is extensive and coefficient is significant. For these reasons, it is very important to verify the accepted model experi mentally. If a maker of a calorimetric system decides to apply the method of determining the heat effect resulting from Eq. (1.99), it would be most convenient to establish a set of parameters (e.g. such that the is generated fulfils the conditions calorimeter in which the heat needed to apply this equation. A probe to determine such a set of pa rameters was undertaken by Utzig and Zielenkiewicz [18] for a simple physical model as an approximation of a real calorimetric system. The the physical pa relation between the dimensionless parameter rameters of the system and was elaborated. In the parameter the value corresponds to the time interval after which the body temperature changes can be described by one exponential term, while is a time constant.

1.5. The heat balance equations

for a rod and sphere

A real calorimeter is composed of many parts made from materials with different heat conductivities. Between these parts one can expect the existence of heat resistances and heat bridges. To describe the heat transfer in such a system, a new mathematical model of the calorimeter was elaborated [8, 19] based on the assumption that constant tempera tures are ascribed to particular parts of the calorimeter. In the system discussed, temperature gradients can occur a priori. Before defining the general heat balance equation, let us consider the particular solutions of the equation of conduction of heat for a rod and sphere. For a body treated as a rod in which the process takes place under isobaric condi tions, without mass exchange, the Fourier-Kirchhoff equation may be written as


Consider the above equation for

27

(Fig. 1.6):

If we put

Eq. (1.109) becomes

If we expand the function T(x, t) as a Taylor series in the neighbor hood of the point and neglect terms of higher order than the sec ond, we have

Consideration of the function T(x, t) at points gives the following set of equations:

and

28

CHAPTER 1

If we put

this set of equations becomes

The solutions of the equation set defined by Eq. (1.115) are

Substitution of Eq. (1.116) in Eq. (1.111) gives


29

Let the volume of the n-th element be

where F is the cross–sectional area of the rod. Multiplication of both sides of Eq. (1.118) by gives

where

defined by

is the heat capacity of domain n, whereas

defined by

is the heat power generated in domain n. The quantities

are the heat transfer coefficients between domains n and n+l, and do mains n–1 and n, respectively. The quantities

denote the heat loss coefficients. Thus, Eq. (1.120) may finally be writ ten as

30

CHAPTER 1

where

denote the amounts of heat exchanged between domains n and n+1 and between domains n–1 and n in time interval dt, respectively. Equa tion (1.125) is the desired heat balance equation for a rod considered as a system of domains arranged in a row. The same procedure is applied to deduce the equation of heat conduction for a homogenous sphere of radius r, where an isobaric process without mass exchange takes place. The Fourier-Kirchhoff equation written with spherical coordinates be comes

for

Let us consider the above equation with

The expansion of the function T(r, t) into a Taylor series in the neglecting terms of higher order than neighborhood of the point the second, leads to

Substitution of

and

gives the set of equations


Let us put

and neglect derivatives of higher order than the second:

The solutions of the above set of equations are

31

CHAPTER 1

32

From Eqs (1.132), (1.134), (1.135) and (1.128) we have

Multiplication of both sides of Eq. (1.13 6) by

where

and putting

gives the following differential equation

The heat balance equations for the rod and sphere described as Eqs (1.125) and (1.138) are identical in form. They are derived on the basis of the same assumptions: in the examined bodies several elements (parts, domains) are distinguished; each is characterized by a constant heat capacity and homogenous temperature in the total volume; the heat exchange between these parts is characterized by heat loss coef ficient G. The first term on the left–hand side of these equations deter mines the amount of accumulated heat in the domain of the body of indicator n; the second and third terms are the amounts of heat ex changed between this part and the neighboring domains of indicators


33

n+1 and n–1. The heat exchange between the body and environment is characterized by the boundary condition of the third kind.

1.6. General heat balance equation

of a calorimetric system

Let us assume Eqs (1.126) and (1.139) as the basis on which to de rive the generalized heat balance equation of the calorimeter treated as a multidomain (elements, parts) system of various configurations [19-21]. Let us also assume that the heat transfer in the body can take place not only between the neighboring domains, but also between any domains characterized by heat capacities and temperatures Each of the separate domains has a uniform temperature throughout its entire volume; its temperature is a function only of time t, and the heat capacity of domain is constant. The domains are separated by centers characterized by loss coefficient and the heat exchange between the domains and between the domains and the environment of temperature takes place through these centers. Temperature gradients appear only in these centers and between the domains and the environment; their heat capacities are, by assumption, negligibly small. Furthermore, a heat source or temperature sensor may be positioned in any of the domains. The amount of heat exchanged between domains j and i in the time interval dt is proportional to the temperature difference of these domains; the heat loss coefficient is the propor tionality factor:

The condition is fulfilled. The amount of heat ex changed between domain j and the environment in time interval dt is equal to

Thus, the total amount of heat dQ”(t) exchanged between domain j and the remaining domains and the environment is equal to

34

CHAPTER 1

Taking into account Eqs (1.140) and (1.141), we have

The amount of heat dQ´(t) accumulated in domain j is equal to

The amount of heat generated in domain j in time interval dt is equal to the sum of the heat accumulated in this domain dQ´(t) and the heat exchanged between this domain and the remaining domains and the environment dQ´´(t). Thus:

From Eqs (1.143) and (1.144), we have

Dividing both sides of the above equation by dt gives

or

where


35

v–n,

is the heat power generated in domain j. In Eq. (1.148), the first term on the right–hand side determines the amount of heat accumulated in domain j, the second term the amount of heat exchange between domain j and the environment, and the last term the amount of heat exchange between domain j and remaining N–1 do mains. The differential equation Eq. (1.148) is called the general heat bal ance equation of domain j, and the set of these equations is the general heat balance equation of a calorimetric system. The heat balance equa tion Eq. (1.147) describes in general form the courses of the heat effects in a calorimeter of any configuration of domains and any localization of heat sources. The general heat balance equation corresponds to the formalization of the general calorimetric model by means of the set of equations with lumped parameters. To consider the thermal properties of a calorimeter, the detailed form of this equation has to be derived. It is necessary to define in it the number and configuration of the distinguished domains and the centers which separate these domains and where heat transfer takes place. The representation of the calorimeter by mathematical models de scribed by a set of heat balance equations has long traditions. In 1942 King and Grover [22] and then Jessup [23] and Churney et al. [24] used this method to explain the fact that the calculated heat capacity of a calo rimetric bomb as the sum of the heat capacities of particular parts of the calorimeter was not equal to the experimentally determined heat capac ity of the system. Since that time, many papers have been published on this field. For example, Zielenkiewicz et al. applied systems of heat balance equations for two and three distinguished domains [25-48] to analyze various phenomena occurring in calorimeters with a constant– temperature external shield; Socorro and de Rivera [49] studied microeffects on the continuous-injection TAM microcalorimeter, while Kumpinsky [50] developed a method or evaluating heat-transfer coeffi cients in a heat flow reaction calorimeter.

36

CHAPTER 1

The calorimeter as a system of several domains has often been pre sented by using a method based on thermal-electric analogy [51–53]. This is based on the similarity of the equations describing heat conduc tion and electric conduction. Such a study generally involves both the use of circuit theory and the principle of dimensional similarity. In a corresponding network analog, the thermal resistances and capacities are simulated by electric resistors and capacitors, respectively. In construct ing such circuits, it is considered that there is an equivalence between the quantities: electric current and heat flow; electric voltage difference and temperature difference; electric resistance and thermal resistance; and electric capacity and heat capacity. The thermal–electric analogy method has been used to analyze and represent the operation mode of a number of calorimetric and thermal analysis devices. For example, Rouquerol and Boivinet [54] presented analogs of electric circuits for the following heat-flux and power com pensation devices: the conventional DTA instrument of Mauras [55], the differential scanning microcalorimeter of Arndt and Fujita [56], and the Tian-Calvet microcalorimeter [57]. Wilburn et al. [58] used both passive and active analogs to investigate the effect of the holder design on the shape of DTA peaks. The thermal-electric analogy was applied by Nico laus [59] to analyze the problem of reconstruction of the heat flux curve. Ozawa and Kanari [60] illustrate the discussed heat balance equations for constant heating rate DSC by an equivalent electrical circuit. The method of analogy is also used in the papers of Claudy et al. [61–63]. The thermal-electric analogy method is useful to represent the ac cepted structures, which distinguish the domains and the modes of their connection with themselves and the environment. It is also useful to formulate a suitable system of the balance equations for the bodies (do mains). The increase in the number of its applications is related to the development of analog computer calculation methods. It will be demonstrated below that the extended information range relating to the investigated calorimeter allows analysis of both its ther mal and dynamic properties. The use of the method based on the anal ogy of the thermal and dynamic properties of the investigated systems is profitable in this case.

Chapter 2

Calorimeters as dynamic objects

Calorimeters are physical objects that can be described in different ways. The steering theory treats a calorimeter as a dynamic object, in which the generated heat effects (input signals) are transformed to the quantity measured directly in the calorimeter, e.g. temperature (output signal). Let us describe the input signals by the functions and the output signals by the functions In fact, in calorimetry the output function never reproduces the input function directly. The calorimeter is a kind of transducer, which transforms the input functions into the output functions. Thus, to reproduce with the required accuracy on the basis of the relation between these two types of functions should be determined:

and the procedure should be selected so as to allow the inverse opera tion, i.e.

which permits determination of the input function on the basis of a given course of the output function of the calorimeter. When the unchanging relation between the input and output functions is determined, it fur nishes an explicit description of the dynamic properties of the calo rimeter. A calorimeter is usually regarded as an object that can be described by one differential equation or a system of linear differential equations with constant coefficients. These equations are treated as the mathemati cal models of calorimeters. If there are many output functions, then the dynamic object (calorimeter) is described by a system of n differential equations. Assuming linearity and applying the superposition rule, one

38

CHAPTER 2

may reduce two or n objects to one. On analysis of an object described by one input and one output function, we do not lose generality. A convenient method for solving linear equations is to apply the Laplace or Fourier transform. In the transfer function (transmittance), the dynamic properties are encoded. The function is defined as the quo tient of the Laplace transforms of the output and input functions. The spectrum transmittance is defined as the quotient of the Fourier trans forms of the output and input functions under zero initial conditions. The transmittance (transfer function) of the object has the form

where s is a complex variable; x(s) and y(s) are the Laplace transforms of the output signal x(t) and the input signal y(t), respectively. Equation (2.3) written in the form

can be visualized by a block diagram (Fig. 2.1). The objects of the

structure shown in Fig. 2.1 are called open objects in the steering theory.

Let us consider the quantities H(s), x(s) and y(s) of Eq. (2.4). Each of them can be obtained by using the two others: 1. The transmittance H(s) can be determined only on the basis of ex perimentally determined y(s) and x(s). The calorimeter is then treated as a “black box” and knowledge of their physical parame ters is neglected. It is assumed that the calorimeter can be treated as a linear, stationary and invariant object. 2. The transmittance H(s) of the calorimeter is based on a mathe matical model of the calorimeter with distinguished physical pa rameters and mutual relation between them. The experimental de

CALORIMETER AS DYNAMIC OBJECT

39

mination of the functions y(s) and x(s) is applied for evaluation of the values of the dynamic and static parameters of the transmit tance. This procedure is equivalent to the calibration of the calo rimeter. 3. The transform x(s) is determined in the knowledge of the form of the transmittance H(s) and the values of their parameters. Input function y(s) is determined experimentally. This procedure is convenient to verify the agreement between experimentally de termined and calculated functions x(s). 4. The transform y(s) is evaluated on the basis of previously known H(s) and experimentally determined function x(s). It corresponds to the determination in the domain of the complex variable of the course of the investigated heat effect. The form of the transmittance depends on the type of the dynamic object.

2.1. Types of dynamic objects Among the open systems, the following types of dynamic objects can be distinguished [64, 65]: Proportional type, when the input function y(t) is proportional to the output function x(t):

where k is the proportionality coefficient, and the transmittance H(s) has the form

while the spectrum transmittance

is expressed by Eq. (2.7):

Integrating type, when the input function is proportional to the derivative of the output function:

40

CHAPTER 2

while

First-order inertial type, when the input function is a linear combination of the output function and its derivative:

while

Let us compare the equations describing the dynamic properties of the distinguished types of objects with suitable heat transfer equations: 1. We compare Eq. (2.5), which describes the dynamic properies of a proportional object, with the Newton heat transfer equation [Eq. (1.46)]; 2. We compare Eq. (2.8), which describes the dynamic properties of an integrating object, with the first term of the left-hand term of Eq.(1.99); 3. We compare Eq. (2.11), which describes the dynamic properties of an inertial object, with the heat balance equation of a simple body [Eq. (1.99)]. It can readily be imagined that a calorimeter in which the course of heat power within time corresponds (with the accuracy of the factor) to the course of the changes in the output function (e.g temperature) has the properties of the proportional object. The properties of integrating


41

objects are those of adiabatic calorimeters, inside which the accumula tion of heat occurs. Calorimeters that are inertial objects comprise the most numerous group of calorimeters. Let us assume that a calorimeter has only the integrating or proportional properties of the object, as a certain idealization. Of course, this idealization is well-founded in many cases. The relation describing the dynamic properties of the object is equivalent to the mathematical model of the calorimeter. It is expressed as a function of time, frequency or a complex variable domain.

2.2. Laplace transformation The Laplace transformation [66] is the operation of changing one expression into another by integration. In this transformation, the function f(t) of the real variable is changed into the complex function F(s) of the complex variable s. The Laplace transform is defined by Eq. (2.14):

and abbreviated as

where while for the existence of the transform F(s) the condition must be fulfilled. The inverse Laplace transformation is defined by Eq. (2.16):

where C is a contour that outlines all extremes of the function in the integral formula. This operation is abbreviated as The Laplace transformation is very convenient to use. Its advantages include: 1) The Laplace transforms of simple functions can be deter

CHAPTER 2

42

mined by direct integration or integration by parts. In most cases, the simple function f(t) and the transform F(s) representing the function transform pairs are tabulated; 2) the Laplace transformation is a linear transformation for which superposition holds; 3) by application of the Laplace transformation, an ordinary differential equation is reduced to the algebraic equation of the transform, called the subsidiary equation of the differential equation. Let us apply the Laplace transform to the heat balance equation of a simple body (Eq. 1.99):

On dividing by G and putting

we obtain

Use of the Laplace transformation for Eq. (2.19):

gives the solution for Eq. (2.19) in the complex domain

After simple rearrangement, Eq. (2.21) becomes

Equation (2.22) is called the subsidiary equation of differential Eq. (1.99); T(s) is the response transform of the output function; F(s) and are driving forces; and is the characteristic function of the object. The first and third terms on the right-hand side of Eq. (2.22), the function of the initial conditions, are the transforms of the transient


43

solution. The second term, which is independent of the initial condi tions, represent the transform of the steady-state solution. The inverse Laplace transformation defines the function T(t) charac terizing the course of the temperature changes of the calorimeter. When the heat effects are not generated, and and thus T(t) depends only on the initial temperature difference T(0) between the calorimeter and its environment. Then:

In a similar way, the Laplace transform can be applied to obtain the solution of Eq. (2.20) for the other input functions. If it is assumed that two first-order inertial objects in series are dis tinguished in the calorimeter, while the output function of the first object is at the same time the input function of the second object (Fig. 2.2), then the calorimeter transmittance H(s) has the form

In the time domain, the system is described by the following differ ential equations:

Increase of the number of inertial objects causes significant changes in the course of the output function. Let us assume that the objects are arranged in series in such a way that the input function of the next iner

44

CHAPTER 2

tial first-order object is the output function of the previous object (Fig. 2.3).

A graphical presentation of the output functions for numbers of objects ranging from one to six, caused by input function forcing corresponding to the unit step function, is shown in Fig. 2.4.

The block diagrams presented in Figs 2.1 - 2.3 are characteristic for the open systems and differ from one another only in the number of inertial objects. The dynamic objects are not always arranged in series. In many cases, as a result of self-arrangement of the objects and their configurations, we must consider the set of differential equations presented by the block diagrams of closed-loop systems. For example, for the calorimeter described by the differential equation

the resulting block diagram is a in Fig. 2.5, while for the calorimeter described by the following set of differential equations:


45

where

the resulting block diagram is b in Fig. 2.5.

When a larger number of inertial objects are distinguished, the calorimeter transmittance will have a more complicated form. For N objects, the transmittance has the form

and we obtain the form of the calorimeter transmittance H(s):

In this case, in the time domain the calorimeter is described by the following set of differential equations:

where

CHAPTER 2

46

When the physical parameters of the system are neglected, Eq. (2.30) takes the form

On application of the Laplace transformation, Eq. (2.33) under zero initial conditions can be written in the form Eq. (2.4):

where

is the transmittance of the analyzed system; m < N; and N denotes the system rank. Determination of the transmittance expressed by Eq. (2.34) is eqivalent to calculation of the polynomials

and

The equation

is called the characteristic equation and its roots are the “eigenvalues” or “poles” of transmittance. If it is assumed that in Eq. (2.35) the polynomial has only single zero values, we can write

The roots

of Eq. (2.37)


47

are named the “zeros” of transmittance, and the polynomial in this equa tion is expressed by

On substitution of Eqs (2.36) and (2.38) into Eq. (2.34), the transmittance H(s) can be written in the form

The poles and zeros of transmittance express the inertial properties of the calorimetric system as a dynamic object. With the introduction of

transmittance H(s) becomes

where

is a constant called the static factor.

2.3. Dynamic time-resolved characteristics The relation that describes the output function changes in time caused by the action of the input function is given by the dynamic timeresolved characteristics. In calorimetry, the same input functions are used for their description as in control theory [64]. However, the termi nology used for this purpose is different. Thus,

48

CHAPTER 2 1. The input function described by a short-duration heat pulse of relatively high amplitude, called in control theory a unit pulse function (impulse function, Dirac function) (Fig. 2.6), is expressed by

The unit pulse whose surface area is equal to one has a Laplace transform y(s) equal to one. 2. The input function described by a constant heat effect in time corresponds to the unit step function (Fig. 2.7):


49

The unit step function corresponds to the integral of the unit pulse function with respect to time. The Laplace transform of the unit step function is 3. The input function described by a heat effect that is constant in time over a determined interval of time corresponds to the input step function of amplitude b and time interval a, called the rectangular pulse (Fig. 2.8): where u(t) is determined by

while u(t–a) is expressed by

This is the shape of the input function that is applied when the cali bration of the calorimeter consists in generation of a Joule effect that is constant in time for a defined duration. The exceptions to the rule are those instruments in the calibration of which the frequency characteris tics are used. The Laplace transform of the rectangular pulse is expressed by

CHAPTER 2

50

4. The input function described by a heat effect rising linearly in time is presented in Fig. 2.9 and expressed by Eq. (2.49): The generation of such a forcing function is used in steering theory as well as in adiabatic and scanning calorimetry. The ramp function has the following Laplace transform:

5. Generation of the heat effect of the first-order kinetic reaction is expressed by the exponential function (Fig. 2.10)

The Laplace transform of which is


51

6) To evaluate the dynamic properties of calorimeters and calibrate the instruments, period input functions have also been used (Fig. 2.11).

where A is the amplitude of oscillation, and

is the oscillation frequency,

The Laplace transform of the sinusoidal input function is

These periodic heat forcing functions are the basis for some calo rimetric methods, e.g. those used in modulated scanning calorimetry. Determination of both the transmittance of the investigated object and the Laplace transform of the input function y(t) furnishes the output function x(s) = y(s)· H(s). With the inverse transformation, we obtain y(t). Output functions x(t) of proportional, integrating and inertial ob jects for various input functions are collected in Table 2.1. The time-resolved dynamic characteristics presented in Table 2.1 show that the shapes of the output functions depend strongly on the type of the dynamic object. For proportional objects, the output and input functions have the same shape, while their values are equal to each other with the accuracy of the factor. This means that in a calorimeter with the dynamic properties of a proportional object the output function gives direct information on the course of the output function; in other words, the course of the experi mentally determined function T = T(t) corresponds to the course of the changes in heat power P in time t.

52

CHAPTER 2


53

54

CHAPTER 2

For integrating objects, the course of the output function corresponds to the accumulation process and to the operation of integration. The object responds to a generated unit pulse with an output signal, which is equivalent to the unit step function; the response of the object to the unit step function is a linearly rising function; production of the ramp forcing function stimulates the response of the object according to the relation For inertial objects, the course of the output function is induced by the inertial properties of the object (calorimeter). This results from the transmittance form H(s), which is expressed by the operator 1/(Cs+k) or the operator while the relation between the input function y(s), the output function x(s) and the transmittance H(s) is presented graphically by a block diagram (Fig. 2.12)

When the trasmittance is represented by the single symbol H(s), it is depicted by the block diagram presented in Fig. 2.1. The form of the transmittance H(s) indicates that the time constant is a decisive parameter for characterizing the inertial properties of the object (calorimeter). This also means that the value of the time constant determines the course of the output function, the character of which is approached more closely for either proportional or integrating objects. Simply, the values of control the inertial, damping properties of the object. Different values of the function x(t), depending on the values of are responses to the same heat forcing (Fig. 2.13). The courses of the output functions caused by the generation of the sinusoidal input function for proportional, integrating and inertial firstorder objects are also presented in Table 2. 1. It is seen that, for a propor tional object, only the value of the sinusoidal (harmonic) oscillation frequency changes. For an integral object, the sinusoidal input function is transformed by the object to a cosinusoidal function. The amplitude of the output function is then inversely proportional to the frequency of the


55

sinusoidal input function. The frequency phase lag is 90° relative to the input function. For inertial objects, the sinusoidal input function is trans formed by the object to an other sinusoidal function that has different phase and amplitude. In the following relation, expressing the output function x(t) for the steady state

where

the factor expresses in terms of f requency the ratio of the output and input function amplitudes. The values of the factor and the size of the phase shift characterize the dynamic properties of the iner tial object. For a frequency close to zero, when sinusoidal changes have low frequency, the course of the output function is similar to that of the input function y(t). The phase shift is then close to zero and the proper ties of inertial and proportional objects become very similar. The shift in the output function course relative to the input function results from the rise in frequency. For infinitely high frequency, the shift in the course of x(t) is expressed as radians or –90°. When the frequency is related to then the oscillation frequency is 0.707 the conversed time constant radians or 45°. This is the frequency related to and the phase shift is the transfer function of an inertial object, given by the operator or

2.4. Pulse response The pulse response function is the output function h(t) caused by the action of the input impulse function (Dirac function). It is applied for determination of the particular forms of the Laplace transmittance. It can be obtained by applying the Laplace inverse transformation to the transmittance Eq. (2.41):

CHAPTER 2

56

where coefficients

have the form

from the Cauchy theorem of residues. The pulse response function is a positive function of real argument t. It fulfils the condition given by Eq. (2.43):

Taking into account Eq. (2.57) and integrating, we have

If we take advantage of the theorem of the initial value of the original, i.e. the function h(t), the initial value of the pulse response h(t) on the basis of Eq. (2.34) can be given by

The pulse response can be obtained experimentally as the response of the calorimetric system: a) to a rectangular heat pulse of short duration (“experimental” Dirac pulse) expressed as a sequence of discrete values

where and is the sampling period; or b) to the unit step function lasting a sufficiently long time to achieve the stationary state of heat transfer. In this case, the pulse response is expressed by the derivative of the calorimetric response


With the measured values of the response respectively

57

at times

and

numerical derivation has to be performed in order to obtain discrete values of the pulse response h(t). In the first case, the accuracy of obtaining the values of the unit pulse response depends on the degree to which the “experimental” pulse ap proximates to the “ideal” Dirac function and on the accuracy of the measurement of the calorimetric signal. In the second case, the proce dure of numerical derivation influences the accuracy of obtaining the pulse response. Since the experimentally obtained response of the sys tem does not fulfil the condition [Eq. (2.49)], it is necessary to calculate the integral from the course obtained for the calorimetric signal and divide all the values of the signal by the value of this integral. Thus, we obtain a new course that fulfils the condition needed. From Eqs (2.59) and (2.60), the dependences between the amplitudes and time constants can be obtained:

a) for the first-order system:

b) for the second-order inertial system:

c) for the second-order inertial system whose transmittance contains

one zero

It results from Eqs (2.65) and (2.66) that the values of coefficient and depend not only on the time constants, but also on the zeros of transmittance.

CHAPTER 2

58

In Fig. 2.13, the plots of pulse responses of calorimeters of various orders are shown. As the order of the system becomes higher, the pulse response is “flattened”, and its maximum value drifts more in time.

2.5. Frequential characteristics To analyze the dynamic properties of calorimeters, frequential char acteristics, similarly as time-resolved characteristics, are determined [8, 67]. To obtain the frequency characteristic, Fourier transforms are used. The Fourier transform which is a complex function of the real variable can be written as follows: or in the form If it is assumed that

where and the transmittance

are the real (even) and imaginary (odd) parts of respectively, we can write


59

Because

from Eq. (2.69) we have

Magnitude

is called the amplitude. Division of Eq. (2.69) leads

to

Thus Magnitude is called the phase and is equal to the argument of transmittance The phase describes the relative amounts of sine and cosine at a given frequency. The spectrum transmittance of an N-order inertial system has the form

Equation (2.75) is equivalent to Eq. (2.67), assuming that In the spectrum transmittance described by Eq. (2.75), let us distinguish the component transmittances and

If this becomes

and for

It results from Eqs (2.78) and (2.79) that the amplitude of the trans is equal to the product of the amplitudes of particular mittance

CHAPTER 2

60

transmittances. Thus, the phase of the transmittance is equal to the sum of the particular transmittances of the phases. is described by According to Eqs (2.75) and (2.79), the phase the equation

If the magnitude approaches infinity, then, according to Eq. (2.80), the value of the limit phase approaches

To analyze the dynamic properties of the object, the amplitude char acteristics, phase characteristics and amplitude-phase characteristics are used. The amplitude characteristic is a relation expressing the ratio of the amplitude of the output function to that of the input function. The dependence of a phase shift in frequency is called the phase characteris tic of the object. The amplitude-phase characteristic presents the ampli tude changes and the phases of the output function. Two types of plots are usually used to draw the amplitudes: one in the coordinates and the other in the coordinates and two types of plots for the phase: in the coordinates or The plot in gives the amplitude-phase characteristics. the coordinates Let us analyze the amplitude-phase characteristics for a few types of spectrum transmittance. is described According to Eqs (2.72) and (2.75), the amplitude by the function

For sufficiently large values of approximated by the equation

the function

can be


61

Taking logarithms of both sides of the above equation gives

It results from the above equation that the plot of the amplitude in the asymptotically approaches a straight line coordinate system with direction coefficient equal to –(N – m). In this way, the asymptotic plot permits an estimation of the difference between the number of poles and the number of zeros of the transmittance. The plot of helix shape which passess through the –(N – m) guater of the system in the coordi– nates will be obtained. The number of –(N – m) is related with the phase shift [Eq. (2.81)].

2.6. Calculations of spectrum transmittance The spectrum transmittance can be obtained as the Fourier transform of the pulse response or as the quotient of the Fourier transform of the system response to a known heat effect.

where is a heat pulse of constant heat power and time interval u (input step function)(Fig. 2.14a); is the temperature response to this heat effect (Fig. 2.14b). In order to determine the spectrum transmittance it is neces sary to calculate the integrals on the right-hand side of Eq. (2.85). Ac cording to the Euler formula:

where

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and are the real (even) and imaginary (odd) parts of the Fourier transform of the response function, respectively. The result of the calorimetric measurement is obtained as the number sequence temperature data [Eq. (2.66)]

at times respectively. Thus, the integrals of the calorimetric [Eqs (2.87) and (2.88)] are to be calculated numerically by signal applying a convenient approximation, e.g.

Then, Eqs (2.87) and (2.88) become


63

Integration and rearrangement of Eqs (2.90) and (2.91) gives

In this way we obtain the real and imaginary parts of the Fourier transform of the response function to a heat pulse. The Fourier transform of the input function should also be determined. Let us determine the transforms for the following input functions: a step input function, a pulse function, and a periodic (sinusoidal) func tion). These functions are often used for spectrum transmittance deter mination. The Fourier transform of the heat pulse can be written in the form

where

are the real and imaginary parts, respectively, of the Fourier transform of the input pulse. According to Eqs (2.85), (2.86) and (2.92) – (2.94), we have

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or

where

When the calorimeter is calibrated by a unit pulse of amplitude and sufficiently long duration, the spectrum transmittance can be expressed as

where is the calorimetric response to a unit pulse heat effect. Ap plication of the approximation of the calorimetric response given by Eq. (2.89) yields

and, after integration, the spectrum transmittance obtained can be writ ten in the form

where


65

On the basis of the above method, the algorithm of the calculation of the Fourier transforms expressed by Eqs (2.85) and (2.100) is calculated by using the Fast Fourier Transform Method [68]. The Fourier transform of the impulse response h(t) can be obtained in the case of a sinusoidal input function (Fig. 2.11):

The Fourier transform is equal to

Hence

becomes

where

It results from Eqs (2.96)–(2.99) that the accuracy of the determina depends on the accuracy of the tion of the spectrum transmittance determination of and Errors connected with their determi nation are due to the approximation of the thermograms or and also connected with the accuracy and precision of the measurements made.

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2.7. Methods of determination

of dynamic parameters

Selected methods will now be discussed that are used to determine the dynamic properties of calorimeters as inertial objects. Several of these methods are similar to those used in steering theory. Different methods are presented for determining the time constant of a calorimeter treated as an inertial object of first order. The methods used to determine the dynamic parameters of calorimeters that are inertial objects of order higher than one are also discussed. All these numerical methods, algorithms and listing programs are described in detail in [67]. To apply each of the methods presented below, a knowledge of the physical properties of the investigated object is not necessary. These methods have been qualified as useful in calorimetry to identify the dynamic parameters and to study thermokinetics. To obtain the most information about the properties of the calorime ter, it is recommended to determine the physical properties of the par ticular domains of the calorimeter and quantities characterizing the heat transfer between the domains themselves and between the domains and the environment. When we follow this procedure, the dynamic proper ties of a calorimeter can be determined by using the method of Ndomains based on the general heat balance equation [Eq. (1.147)]. This method will be presented in Chapter 3.2.4.

2.7.1. Determination of time constant The dynamic properties of a calorimeter treated as an inertial object of first order are characterized unambiguously by the time constant To evaluate the time constant on the basis of the heat balance equa tion of a simple body, different input functions are used. Consider the determination of by applying the input step function [Eq. (2.45)] under conditions where the initial temperatures of the calorimeter and isother mal shield are the same. Equation (2.19) can then be written in the form


67

where

is determined by Eqs (2.45) and (2.47).

The Laplace transform of the function y(t) corresponds to

while after the Laplace transformation Eq. (2.111) becomes

After simple rearrangement, Eqs (2.113) and (2.114) can be written in the form

Inverse Laplace transformation of Eq. (2.115) gives

The function T(t) expressed by Eq. (2.116) can be presented graphi cally by using the curves I, II and III shown in Fig. 2.15. The first term on the right-hand side of Eq. (2.116) is related to curve I, and the second term on the right-hand side of this equation to curve II. Curves I and II have identical shapes, but there is a shift in time between them, related to the duration of the heat impulse produced. For this period of time, the courses of the changes in the calorimeter temperature T in time t are represented by the interval 0K of curve I. When t > a, the course of the to the shield temperature is presented in curve III. function T(t) from This is the cooling curve of the calorimeter. The changes in temperature that occur here are only a result of the existing difference in tempera The interval KM of curve III graphically represents the differ tures ence between the values T(t) of the first and second terms on the righthand side of Eq. (2.116).

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When the heat generation period is long enough for a new state of thermal equilibrium to be reached, characterized by Eq. (2.19) can be written in the form

The solution of Eq. (2.117) when T(0) = 0 is

which corresponds to the first term on the right-hand side of Eq. (2.116). Equation (2.118) describes the heating process occurring in the calorimeter (curve 0I, Fig. 2.15). Let us use Eq. (2.118) to present several procedures for determining the time constant Procedure 1. We find the time derivative of the heating curve

From Eq. (2.119), we have that a tangent to the curve 0K crosses an (Fig. 2.15). asymptote of the curve at the point relating to


69

Procedure 2. At every point of the heating curve characterized by the interval 0K of the curve I (Fig. 2.15), the length of the subtangent determined by its point of intersection with the straight line is related to the time constant From Eqs. (2.117) and (2.118), we have

If we take into account the graphical presentation of the heating curve in (Fig. 2.15), we have

where

Thus, the length of the subtangent is equal to the time constant value Procedure 3. After time

T(t) is equal to

After time the temperature becomes and after we have Proceeding in such a way for the time known values of and T(t) of the heating process, we can obtain the value of the time constant or its multiplicity. Procedure 4. The value of the time constant can be determined analytically. The values of T(t) related to time moments and cor respond to

Hence, Eqs (2.124) – (2.126) become

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Thus

and after rearrangement we have

Procedure 5. To obtain the time constant can be used.

the integration method

Hence, the time constant is equal to

Integral

represents the surface area F1 between the

straight line and the heating curve (Fig. 2.16a). The changes in temperature T(t) in time, caused only by the initial temperature difference between the calorimeter and shield, are also used to determine the time constant In this case, it is assumed that y(t) = 0, and the cooling process is expressed by equation


71

The solution of Eq. (2.134) is

The course of T(t) is graphically presented by the cooling curve (curve IV) in Fig. 2.15. The procedures used to determine the time constant on the basis of the cooling curves are not much different from the procedures presented above. Procedure 1. The same as in the case of the heating curve; the length of the subtangent at every point of the cooling curve is equal to the time constant (see heating curve, Fig. 2.16, Procedure 2). Procedure 2. After time the decrease in T(t) is 36.8% com pared to the initial value This results from Eq. (2.135). In a similar way, the value of T(t) can be determined for the time constant multiplic ity. Knowing temperature and choosing a matching value of T(t), one can determine the value of the time constant. Procedure 3. The value of the time constant can be determined ana lytically if at least two temperature values of the cooling process defined as a function of time are known. When, after time of the cooling proc

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ess, the value of T(t) becomes then according to Eq. (2.135)

and, after time

it becomes

Division of both sides of Eq. (2.136) by Eq. (2.137) gives

Hence

Procedure 4. The time constant for the cooling process can be calculated by an integrating procedure as for the heating process (Fig. 2.16b). On calculation of the integral:

Hence

Procedure 5. Taking logarithms of both sides of Eq. (2.135) gives

Hence

Equation (2.143) is used for the graphical determination of the time constant When the experimentally determined T and the t data for the cooling process have been obtained, the plot in the coordinates (t, lnT)


73

can be drawn, as shown in Fig. 2.17a [8]. When the dynamic properties of the calorimeter are characterized by one time constant the relation lnT = f(t) is expressed by a straight line (Fig 2.17a), which forms an angle with the t axis, for which

This line cuts the axis at the point of the ordinate. Thus

When the dynamic properties of the calorimeter are characterized by more than one constant of time [Eq. (2.57)], at the begining of the cooling process the relation lnT = f(t) is not linear, as shown in Fig. 2.17b. In this case, the discussed method can be applied to evaluate the higher time constants. The following procedure is used. The straight line interval (Fig. 2.17b) is extended to the point of intersection with the lnT axis and the first time constant is determined. Next, a new function of the form

is created and a new plot in the coordinate system can be drawn. is determined. The iterative procedure is The second time constant repeated until a plot similar to the plot given in Fig. 2.17a is obtained.

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The number of time constants which can be distinguished depends on the properties of the calorimeter and the accuracy of the determination of the experimental data.

2.7.2. Least squares method The least squares method for determination of the transmittance pa rameters was proposed by Rodriguez et al. [69]. This method allows the and the zeros of the transmittance nominator to be time constants obtained by approximation of the pulse response of the calorimeter by the least squares nonlinear curve-fitting procedure described by Marquardt [70]. In order to use this method, it is necessary to assume or determine the order of the model number of poles and the number of transmittance zeros. According to these assumptions, the pulse response is a func tion of time t, poles and zeros

where m < N. Introducing vector and zeros:

the components of which are poles

Equation (2.147) can be written in the form As is a nonlinear function of the components, expansion of this function as a Taylor series in the neighborhood of point is applied:

neglecting the derivatives of higher order. As a criterion of fitting the and approximated pulse experimentally determined pulse response it is assumed that response


75

Equation (2.150) then becomes

where < 0, T > is the time interval in which the changes in pulse response are measured, and

In possession of the discrete values of the pulse response, instead of Eq. (2.152) the following fitting criterion is accepted:

where and is the sampling period. From the necessary condition of the minimum of the function expressed by Eq. (2.154):

we have the following set of equations:

Solving the set of Eqs. (2.156) with respect to leads to the poles and zeros of transmittance according to the equation

Since the linear approximation is applied to the nonlinear function, we can obtain the values of the parameters only with large error in one iteration. Thus, it is necessary to repeat the iteration, assuming the calculated values as initial values and repeat the iterative procedure until we obtain a suitable approximation of the pulse response.

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2.7.3.

Modulating functions method

The modulating functions method was proposed by Ortin et al. [71]. This method permits determination of the poles and zeros of transmit tance. The poles of transmittance are determined on the basis of an ac cepted model in the form of a differential equation. As the pulse re sponse h(t) is the finite sum of exponential functions [Eq. (2.157)]:

it can be assumed that it satisfies the differential equation with constant coefficients:

where is k times the derivative with respect to time t of the pulse Next, the time interval in response of the system, and which the changes in the pulse response are measured (observed) is assumed as the basis of the modulating functions which fulfils the conditions

for i = 1, 2, ... , N; k = 0, 1,..., N – 1 is also assumed. These functions have continuous derivatives of desired order. The order and the smooth ness are connected with the order of differential Eq. (2.159). Multiply ing both sides of Eq. (2.159) by and intergrating with respect to time t in the interval yields

Putting

leads with respect to the coefficients equations:

to the following set of algebraic


77

The above set of equations can be solved with respect to From the values of the coefficients from the characteristic equation

it is possible to determine the eigenvalues (poles) of the transmittance, taking into account that the time constants The expressions in Eq. (2.162) contain the derivatives of the pulse response function h(t). To avoid the calculations of these derivatives, which can result in the introduction of large errors, Eq. (2.162) (after integrating by parts) can be presented in the form

taking into account the conditions given by Eq. (2.160). When we have determined the poles of transmittance and know the value of the pulse response h(t) in Eq. (2.158), only the amplitudes are unknown. These coefficients can be determined by applying the modulating function procedure given above to the pulse response:

On putting

we obtain the following set of equations:

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The coefficients of the pulse response h(t) can be determined by solving the above set of algebraic equations after previous calculations of the coefficients From the determined time constants and coefficients the Laplace transform to the pulse response gives

where

is the denominator of the transmittance, and

is the m–degree polynomial m < N with respect to s. By solving the equation the zeros of the transmittance H(s) are obtained. The general form of the equation expressing the relation between the generated heat power P(t) and the temperature changes T(t) of the calorimetric system has the form

Multiplying both sides of the above differential equation by the modulating function and integrating with respect to time t in the interval leads to the following set of algebraic equations with respect to coefficients and

where


79

The obtained set of Eq. (2.175) has a similar form as that of the set of Eq. (2.162). In the particular case when the calorimeter is calibrated by a constant heat effect of heat power and duration u smaller than the relation for coefficients given by Eq. (2.177) (taking into account the conditions Eq. (2.159) and integrating by parts), can be simplified to the following form:

The flow diagram of the modulating functions method, the program algorithm and the listing program are given in [67].

2.7.4. Rational function method

of transmittance approximation

In this method, the transmittance H(s) is approximated by a rational function [67, 72, 73]:

where D(s) and L(s) are polynomials of degree m and N(m

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