Diffusion Processes During Drying of Solids

11

SERIES IN THEORETICAL AND APPLIED MECHANICS Edited by RKTHsieh

SERIES IN THEORETICAL AND APPLIED MECHANICS Editor: R. K. T. Hsieh Published Volume 1: Nonlinear Electromechanical Effects and Applications by G. A. Maugin Volume 2: Lattice Dynamical Foundations of Continuum Theories by A. Askar Volume 3: Heat and Mass Transfer in MHD Flows by E. Blums, Yu. Mikhailov, and R. Ozols Volume 5: Inelastic Mesomechanics by V. Kafka Volume 9: Aspects of Non-Equilibrium Thermodynamics by W. Muschik Forthcoming Volume 4: Mechanics of Continuous Media by L. Sedov Volume 6: Design Technology of Fusion Reactors edited by M. Akiyama Volume 8: Mechanics of Porous and Fractured Media by V. N. Nikolaevskij Volume 10: Fragments of the Theory of Anisotropic Shells by S. A. Ambartsumian Volume 12: Inhomogeneous Waves in Solids and Fluids by G. Caviglia and A. Morro

Diffusion Processes During Drying of Solids

K. PL Shukla

World Scientific Singapore • New Jersey • Hong Kong

Author K. N. Shukla Vikram Sarabhi Space Centre Trivandrum 695 022, India Series Editor-in-Chief R. K. T. Hsieh Department of Mechanics, Royal Institute of Technology S-10044 Stockholm, Sweden

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: 687 HaitweU Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging-in-Publication Data is available. DIFFUSION PROCESSES DURING DRYING OF SOLIDS Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without permission from the Publisher. ISSN 0218-0235 ISBN 981-02-0278-4

Printed in Singapore by JBW Printers & Binders Pte. Ltd.

V

Preface Modelling heat and mass transfer in porous media is an area of great

importance.

transfer

in

moisture

transfer

the

moisture

transfer. the

temperature

body.

The

itself

and

gradient

mechanism

is

drives

the

complex

is

a coupled

process of

that

heat

Also the coefficients of heat and mass diffusion and

moisture

content

making

the

moisture

because

a l t e r s the temperature gradient

hence it

temperature

nonlinear. of

The

the

the

drives

and

mass

v a r i e s with

process

highly

An attempt has been made to develop the basic equations

heat and

moisture transfer

in porous body with reference to the

drying of m a t e r i a l . The monograph begins with a brief comment on the laws of

the mutually

boundary

value

obtained

in

connected

problems

Chapter

analytical e x p r e s s i o n s phase

and

chemical

for

transfer

phenomena. Solutions of

axisymmetric third

and

chapter

spherical

2.

The

for

heat and moisture transfer

transformations

fundamental

in

is

spherical

the

cases

are

to

the

devoted

in presence of

body.

Chapter

k

considers the intensive drying of an infinite p l a t e . Besides molecular transfer,

the

conjugate

process

of

filtration

problem of interacting

reference

to

freeze

drying

is

also

included.

Finally

a

porous solid with a fluid stream in

is

analysed

in

Chapter

5.

A

short

description of the integral transforms is provided in the a p p e n d i x . The whole analysis i s presented in the dimensionless form with the

help

Integral

of

dimensionless

transform

variables

technique

is

the

and basic

the tool

similarity for

numbers.

the solutions of

the boundary value problems. The monograph scientists

of

is designed

applied

for the graduate students,

mathematics

and

engineering

research

sciences and

the

practising engineers in m a t e r i a l s , energy and s p a c e . The

material

from the a u t h o r s '

of

the

present

monograph

is

developed

r e s e a r c h e s on Heat and Mass Diffusion

mainly

c a r r i e d out

vi in Banaras Hindu University Professor

R.N.Pandey,

University I

thank

for

for

calculations Institute

of

reading

suggestions, in Chapter

my 5.

Technology,

suggestions

on

the

R.S.Pandey

for

typing

final

and

draft

Director,

Dr. V.Swaminathan, carefully

constructive

of

I am indebted

Technology,

Banaras

introducing me the subject and guiding the

Dr.S.C.Gupta,

publication, VSSC

during mid s e v e n t i e s .

Institute

Head, the

the first

World

and of

the

the

draft,

Scientific

Richard reviewer

text. Shri

I

for

some Royal

for

also Co.

offering

Hsieh,

valuable

thank

T.Thankappan

Publishing

its

Group,

and

M.J.Chacko

Professor

for

Dynamics

manuscript Mr.

research.

permission

Aero-Flight

colleague

Sweden

kind

entire

I thank

presentation

the

VSSC for

to

Hindu

Pvt.

Shri

Nair

for

Ltd.

for

publishing the book. The utmost care has been taken in checking the calculations

but it is

quite

possible that

some of them might

have

gone unnoticed. I e x p r e s s my gratitude to the r e a d e r s in advance for all suggestions for further improvement of the monograph.

K.N.Shukla

I

vii

Contents

Chapter

Chapter

1

Phenomenological

Laws of D i f f u s i o n

1.1

Phenomenological

Laws

1.2

Transfer

1.3

I n i t i a l and Boundary

I.*

Dimensionless Quantities

18

References

25

2

Integral

1 I

of Heat and M o i s t u r e i n Porous B o d i e s

Equation

Conditions

Approach

17

to Heat and

Mass

Transfer

Problems

28

2.1

The I n f i n i t e

Circular

2.2

Solution for

a Sphere

Cylinder

31 37

References Chapter

45

3

Heat and Mass T r a n s f e r

3.1

Statement

with

Chemical Transformations

of t h e P r o b l e m

S o l u t i o n of t h e P r o b l e m

51

3.3

A n a l y s i s of

66

Analysis

t h e Result

of t h e S o l u t i o n

75

References

Chapter

Appendix

46 47

3.2

3.U

Chapter

6

90

i = 1

=

4 | UAi =I

u

(1.2.1)

u. is defined in terms of the porosity

of the

body 1

"l

1

where

P. is the density

factor,

defined as the pore volume per unit body

a factor

related

to the

of the bound substance, IT is the

volume concentration

per unit

small,

vapour

the

and dry

specific

air

is

mass of the

bound substance varying in the process of mass transfer. masses of

porosity

volume and b

Since the

in the pores of the capillaries are

mass content

of

the bound matter

is equal to

the sum of the mass content of the ice and moisture i.e. 4

u

y i=1

ux . = u20 + u,3 .

Conservation of mass and energy: volume in the system. mass in any

Let us consider a small control

The differential

equation of the transfer of

phase in presence of sources or sinks may be written

by the continuity equation as 3(YQ u ) 3t

where

j

is

divj.

the density

of

+

Q.

,

the flux

(1.2.2)

of

l-phase

the strength of the source or sink of the i-th Eq.

(1.2.2) with respect to i ( i

3(y 0 u) £r— d t

i

since the sum of all

i.e.

I i

Q.

and Q

is

1,2,3,4), we obtain

4 I

div

matter

component. Summing

j

,

(1.2.3)

,

the mass sources or sinks is equal to zero,

0.

To obtain the differential equation of the heat transfer, we

8 consider

the transfer

rate

change

of

of

of e n t h a l p y . enthalpy

concentration

divergence of the enthalpy flux; £

(h0TQ

*

l

h

pressure,

'h'

is

the

equal

to

local the

thus, u.) =

i Y o

div (4 + l h

L

y ,

(1.2.*)

1

I

where the heat flux q* is defined q>

At constant

by the Fourier heat equation

AVT.

Let us denote the specific heat at constant p r e s s u r e by c . , dh„

JU an I -j=—

C

,

U -p=—

c

/ 1 ->

15

Equation (1.2.36)

is the usual heat conduction equation with a heat

E

source

PYn 3 u / 3 t due to moisture evaporation in the pores of the

porous body. For an intensive temperature

above

evaporation transport

there

of the moisture. phenomenon.

the

processes.

a pressure

This

of

presence

pressure

pressure

of

porous body at a

gradient

pressure gradient the

below 373K. Therefore,

influence

The

is

However,

occur at temperatures consider

heating of the c a p i l l a r y

373K,

influences

gradient it

gradient

a pressure

due to the the

may

also

is a p p r o p r i a t e on

gradient

the

to

transport

inside a

porous

body causes hydrodynamical motion (filtration) of vapour and liquid which a r e d e s c r i b e d by the Dercy law: I

=

X VP P

P where

X

is the coefficient

(1.2.37)

of filtration conductivity analogous to

X . q

The

system

of

differential

equations

describing

heat

and

mass

- ^

I c J ^ T

(,.2.38)

Vp)

(1.2.39)

transfer thus becomes fl=

div ( a q

v^T)

P3t

div (a

|£

d > v ( a

V~u m

+

+

a

c

6VT are

used

and the c o e f f i c i e n t s

L., II

a

for

heat,

L . ik

and

the symbols

pressure,

9.,

respectively

are

+ — a o L,_, c m ' 1 2 q

q

potentials,

moisture

, L-, ' 2 1

c

a

° m

q

a L „ 22

a

L

a

, L,, m ' U

e p — - 5 Kp , c ' P

L.,,. 23

a 33

p

a 6

c

y

a 6p m K

p

'

v

P

L

31

e a_ 6

c

'

L

32

c

P

P

These equations have been d e r i v e d by L u i k o v and M i k h a i l o v 26 and Narang have further modified these equations to

Kumar include

the

Inspite

of

hydrodynamical mass

hydrodynamical velocity

v\

diffusion,

motion

the E q .

~

div

DT Dt

dlV

of

if the

(1.2.35)

(am vu

effect

+

in

on the

moisture

and ( 1 . 2 . 3 6 )

the

transfer

phenomenon.

capillary

porous

occurs

some

at

body, average

become

a m 6V~T)

(1.2.42)

and ,

U

-t\

VT)

q

+e

P "c-

3u IT '

, , , ,,-> '-"3)

( K

where the s y m b o l ■=— stands for the s u b s t a n t i a l

m In Eqs.

sT the

(1.2.38)

derivative,

+ v v

'

same

i.e.

(1.2.W) way,

(1.2.40)

the

diffusion

can be m o d i f i e d

equations as

with

filtration

33

17

§1=

div (a

7"T)

|£.

+

|f

I

q

^ ut

- div (a

Vu + a m

cx J V T

(..2.45)

7p)

(1.2.46)

l

VT + a m

P

and

5? = div 1.3

(a v P ^

e

(K2 47)

f If

-

Initial and Boundary Condition In order

to make the differential

equations

for

the transfer of heat and mass physically sound, we need some laws which may d e s c r i b e the interaction between the surface of the body and the surrounding: (a) the

Initial conditions; system

potential

at

of

start

the

Initial

of the

system

conditions state the

process.

is

At t h i s

supposed

to

be

instant,

behaviour of the

arbitrary

transfer

and

is

a

function of the space coordinates only. Thus

jTI

f (I?)

.

I u J where r is the position (b)

Boundary

transferred

a

f 2 (?) vector.

conditions:

At

the

surfaces,

the

moisture

is

under the influence of potential gradient of moisture and

heat. Applying t h e mass balance at the surface, we have X

The

m

(V

"*u)s

quantity

utilized

partly

of

+ X

heat

m

6

(

^ s

transferred

* V(t)

to

the

(, 3 2

= °-

surface

- - >

of

the

body

is

in the evaporation of the l i q u i d . Applying the heat

balance at the surface, we have X (v"Hs + q (t)

( l - e ) p qm(t)

= 0

(1.3.3)

18 In and

the

the

case of

system,

the

convective law,

the

interaction

exchange

of

between

heat

and

the

gaseous

mass

takes

medium

place

by

i.e. q

a

(Tc

Ts)

and q ^m w h e r e the s u b s c r i p t transfer heat

potentials.

and

mass

6Y n (U '0 s

U ) , c '

s stands f o r

the surface and c f o r t h e

The c o e f f i c i e n t s

transfer,

ambient

a and 6 are the c o e f f i c i e n t s

respectively,

thus

Eqs.

(1.3.2

of

1.3.3)

become,

X

m(Vs

+

X

m

6

+

= °

'3-*>

and Xq(VT)s + Equations

a (Tc -Ts)

(1.3.4

+ BY0(US

1.3.5)

UC)

can also

0

(1.3.5)

be e x p r e s s e d

in

general

form as

-»

—»

(Vu)s + a 2 ( V T ) s

+ B

U

2

s

*2

( t )

°

(1.3.6)

and (VT)

where

+ a,T + B|U

a.,

a2>

thermophysicai

8|

82

coefficients

and

(1.3.7)

tne

4> ( t )

aggregates are

of

the

fluxes

a

process

the of

known

heat

and

reflect

the

experiments.

Dimensionless Quantities Differential

physical equation the

by

0;

are

and

moisture to be determined 1. k

#t(t)

picture is

change

amount

equations of

the

a consequence in

energy

liberated

from

of

dealing

process. of

the

energy

t h e system the

with For

system.

example, equation

as the Thus

the

diffusion

which

describes

equivalent a

f o r m of

differential

the

equation

19 occurring

in t h e

formulation

of a problem

describes

the

physical

laws which govern the system. The c h a r a c t e r i s t i c v a r i a b l e s the relation to

between the s e p a r a t e terms of the equations. We have

establish

similarity

define

such

relationship

theory

gives

a

among

method

the

to

different

transform

terms.

the

The

expressions

having differential operator into the simplest algebraic form. Now medium;

consider

the

the transfer

interaction

of

solid

with

the

phenomenon in t h i s case is governed

gaseous by

the

convective law X (-ajj) 3X s where T

<MT s

T ), a

(1.0.1)

is the ambient t e m p e r a t u r e . Let us further

form of an infinite

assume that the solid

body is taken in the

plate of finite thickness R and the temperature

drop over the t h i c k n e s s R is p r e s c r i b e d to T then for ( 3 T / 3 x )

a

constant, we have 3J 6T * 3x ~ R If the temperature difference T §1

_

AT

h being t h e heat transfer The r i g h t

(1.0.2)

-

T

is denoted by AT, then

*S

(1 4,3)

X

Ki.t.JJ

coefficient.

hand s i d e of Eq.

(1.5.1)

becomes

dimensioniess

and for t h i s we call 3iot number ' B i ' a t t r i b u t e d to the name of the s c i e n t i s t who has made a significant contribution to the development of t h i s

field. In

defining

the

variables,

a ,R

into

introducing

this

number

Biot

one is

number,

variable that

once

we

Bi. we

The fix

have

reduced

other a,R

benefit as

the

two in basic

20 quantities,

we

obtain

an

infinite

number

of

other

sets

quantities and thus an infinite number of phenomena are Now consider t h e basic equation of heat 2

of

these

determined.

diffusion

a

The term

3T/ 3t r e p r e s e n t s the rate of change of temperature

with

respect

to time and can be replaced by 6T / t . Similarly the term 2 2 hand s i d e 3 T/3x is the square r a t e of change of T 2 r e s p e c t to x and it can be replaced by 6I* R /x , where t h e

of the r i g h t with suffixes

t

and

temperature T,

R denote

the

time r a t e

and

space

rate

change

in

therefore 6 T

6T

t

- ^

=

R

a —|

,

(1.0.5)

x

For a plate of t h i c k n e s s R, the Eq. ( 1 . 5 . 5 )

!Ii

at

6T

R2

R

becomes

'

the r i g h t hand s i d e of t h i s equation being dimensionless This

represents

a

generalised

variable

which

we

quantity. call

Fourier

number ^-4 R Thus

the

Fourier

Fo.

number

is

(1.0.6) defined

as

the

ratio

of

time

rate

change in temperature with the space r a t e change in t e m p e r a t u r e . It has also t h e importance that it first reduces the t h r e e v a r i a b l e s a, t,

R into one and second once a set of a,

obtain

an infinite

different

set

of other

t,

combinations

R is e s t a b l i s h e d , each

characterising

we a

process. In

this

way,

we

see

that

the

introduction

of

these

dimensionless v a r i a b l e s not only reduces the number of v a r i a b l e but it generalises the r e s u l t s and c r e a t e s a firm scientific base for

the

21 mutually connected transfer analysis

has

transport

phenomena. A list

wide

of

phenomena. This is why the dimensional

acceptance

the

various

in

tackling

the

dimensioniess

problems

variables

of

the

which

are

often used in the transfer of energy and mass are given below. Biot number, Bi

= T— q

Eckert number, E Fedorov number, Fe

2 V 0 —T^F c AT q

EKO Pn

'

c q

Fourier number, Fo

a t ~~j

Kirpichev number, Ki

T^AT

R q

v ■u u v Kossovich number, Ko

pAu = -—TJ q

Luikov number, Lu '

a — a q

Peclet number, Pe Prandtl number, Pr Predvoditelev number, Po

V0R a

q — V

¥ q 2.~ OR q

Posnov number, Pn

Reynolds number, Re

6AT —g^j jd ——e

22 The multitude

growing

importance

applications

transmitting

such

energy,

in

supersonic

generated

tremendous

processes

in

the

as

of

interest

porous

the

heat

and

in

porous pipes,

cryogenic

hypersonic

modelling

media.

materials

Also

wind

heat

and

is

lack

there

for

lines

for

tunnels mass of

a

has

transfer

a

unified

theory

to d e s c r i b e the t o t a l process i n v o l v e d i n the v a r i e t y of t h e 29-32 porous b o d i e s . L u i k o v developed the equations f o r i n t e r n a l and

external

heat

solutions

of

presented

and

mass

the

heat

transfer and

for

mass

porous

transfer

bodies.

problems

Analytical have

been

under

Luikov

and

types

of

different boundary c o n d i t i o n s i n a t r e a t i s e of 33 Mikhailov . M i k h a i l o v and O z i s i k i} (2.1.10)

and t h e constant c o e f f i c i e n t s a r e g i v e n by

L]

(-l)J ^ V ;

(-,) J ^H-

L.J

v, - v 2 , r * L ,

(-1) v u

, ■

J

2

Pn fn,

L L

v2 - v ,

] 2

2 . v -(-l) —! V W 2 v2

M

I -v V

i

E KoPn •

2

v,

2 EKO

(-I)J — L _

1/Lu

J

2 .

° '

v2 - V | ,

2

eK

V

2

2

Pn L U V .

i-

.

,

MJ

.

(-I)J

v

' 2

v

x 1

E Ko Lu v . , J

33 2

?

]l

(

"')J

.

v

J

Lu

2

2

Pn

M

'

Applying for

the

inversion

temperature

(

2 J

v2 -v,

expressions

2

i /i ?

l/Lu-v ~2 2 ^ v2 - v ,

"I)J

formulae

and

Lu

(2.1.5)

moisture

Pn

and

distributions

-

(2.1.6), are

the

obtained

as

e^x.Fo) -

l

[L;k

P;k 0 j k

] ,

(2...,,)

) , K- I

where

P..

2[

1 f x f, ( x ) d x 0

°° 3Q(pnx) ; —= exp(-Lu n=l J Q ( p n )

+

/ i

x f, ( x ) k

p

v

Fo)

Jn(p x ) d x ] u *n

(2.1.12)

and

Fo Qjk

jk

2[

j

Xk(u)du k

0

+

°° I

3Q(pnx) 1 (o ) —

n=l

W

Fo 2 2 f Xk(u)exp(-p v Lu F o - u ) | , 0

k

n

J

(2.1.13) provided

the c o n v o l u t i o n The

problem.

expressions

In o r d e r

determine

exists.

x .(Fo)

(2.1.11)

to s o l v e from

the

the

are

the

present

original

solutions

problem, boundary

it

to is

the

auxiliary

necessary

conditions

(V)

(VI). S u b s t i t u t i n g the

36 values of = 1 and 6 . at x

( 2 . 1 . 1 ) and ( 2 . 1 . 1 1 ) , we obtain

1 from E q s .

to and

34

X.(Fo)

2 I

+

i

, (A

.>k=,

M I

Fo Mf ) [ / X k (u)du

+ B i

JK

JK

0

00

+

r J n ( p x ) F o . — -— / " 30(pn) 0

7

L

9

exp(-p z v / n ■ J

Lu Fo-u) j ( . ( u )

= V F O ) J , (A, Lk + B. L V

du]

p Jk

(2 , , )

--"

j , k- I

and A

00 +

^o^Pn y ^ l

nil

*2(Fo)

B

1

The values

2 I j,k=1

x,(Fo) + X7(Fo) + 2 B Z l * '

of

l

integral

the

X

Fo

'

exp(

I

1 j,k =l

equations

functions

Xi(Fo)

"pn

A*

Jk

2

P

J

FO ( / Xk(u)du k 0

2 j

Lu Fo-u) X k ( u )

(2.1.15)

by

and

(2.1.15)

substituting

X. (u) in ( 2 . 1 . 1 3 ) , we obtain the e x p r e s s i o n s for Q. expressions

for

the

temperature

du)

Jk

(2.1.14) and

v

M*

and

moisture

determine the

values

the of

and hence the

distributions

are

determined^ In order to simplify

the calculations, we s h a l l determine an

approximate solution of Xi(Fo) for small values of generalised Under

the

Laplace

transformation,

the

integral

time.

equations

(2.1.14) and (2.1.15) take the form

2

A X(s) 1

+

2

I J,k=l

1

(A M '

7

1 ° °

♦ B, M k ) ( l '

t

^(p

x n

)

I

° i L _

n=l

0

K

n

35

*

)

4 s+Lu v . J

P

Xk(s)

= g* (s)

(2.1.16)

n

and A

2 _ J Mk J j,k=l

A

X2(s)

+

2

(s)

X|

2B2

+

}

4 ~ s+Lu v p J "

Xk(s)

. (I

°° I n=l

+

Jn(p x) ° ^ _ 0 Kn

= gj(s),

x

(2.1.17)

where

g^s)

e-sFo

|

p,(Fo)

.|=](A,

Ljk+B,

LJk)PjkjdFo

(2.1.18) and

g^s)

s F

/e-

°

^2(Fo)

L2,

B 2 [ k __,

PjkldFo, -*

where

it

is

assumed

that

<J>.(Fo)

e x p r e s s i b l e i n the e x p o n e n t i a l l y

Equations

(2.1.16)

converges u l t i m a t e l y .

in

Xi(s)

ar|

function

(2.1.17)

are

two

ordinary

A

d

)(,(s)

and contain a s e r i e s

which

Now a p p r o x i m a t i n g the term

2 s + Lu v .

~

\_ s '

p

have

X,(s)

1

continuous

decreasing f o r m .

I

we

a piecewise

and

A

simultaneous equations

is

(2.1.19)

♦ 2

I

j,k=1

L(A

M

'

♦ B

Jk

M2 ) ( J +

'

JK

J

n=l

°

W

n

) 1 J

s

(s)J

k

36

g,(s)

(2.1.20)

and A

2

A

X2(s)

+

A2 X , ( s )

-,

2B2 I

+

[M

»

k(U

J

j ,k= 1

JQ(p

n=I

r

0

A

2

. ( 2B [g ( s ) l W - ^ l s '

A

X,fs) 1

- i M2

1

,

A

X?(s)

d

(2.1.21)

anQl

restric-

3

°°

JZ

j=1

ar,

(1+

.

n

g2(s)

Solving these two equations f o r X i ( s ) I t i n g to the terms of o r d e r — o n l y , we o b t a i n

x)

T T ^ l - ' s >= k ( s ) ]

1

x) n(p On n=l J 0 k F V

i A

I

J

2

f g2(s)

I

1:0 (l

+

(A, M ] 2 ♦ B jM 2 2 )

3 n ( p x) TT77T— > 0ypn'

I n=1

x

]/d*^)

(2.1.22)

s

and r

A

x2(s)

2B

2

[-g/sifv-^ + g 2 (s){l +

f

oo

J

( D

I

3°(n

(,t

n=l

n

,

*

^ MJ, (U

^

J

n^P

x

*

1

3^-)}

I (A, Mj |+B| M*,) x X )

)

0lFn

> \ J/ U + 2|) ,

(2.1.23)

s

'

where a

2 £

(A,

M.,

♦ B 2 M^ 2 - A 2 A ,

M^

» (1

+

A^,

J0(pn

i . 57(71 n=1

0

r

n

M^)

x)

>■

x

37 The

inversion

can

be

c a r r i e d by expanding (1+2^- ) and s 1 0 order s and s only. The inverted

considering the term of e x p r e s s i o n s of X i ( s ) are

X,(Fo)

g (Fo)

3 n ( p x)

°° ( , +

«>

3_(p

I n=1

Fo }

I 3 (p ) n=1 0 V F V

( , +

Fo J g . ( u ) d u + 2B 0

2a

?

M%

2

x

J

2 8)(u)du '

i 0

x)

.

1 (A,M j=l ' J

. +B

M< )

x

JZ

'

Fo }

3 (p ) CTpn;

2 V j=l

&2(u)du

/ 0

(2.1.2*)

z

and

X 2 (Fo)

A2 g,(Fo)

+ g2(Fo)

Fo

2

j g (u)du

+2a

2B

Fo

the

g.(u)du + 2

(I

+

functions 2.2

y,(Fo)

values

?

I

M< ( U

Fo j

g,(u)du

co

3n(p

J

(

u

x)

"

)

x

2

/ 0

co I n=l

Expressions functions

2 a A2

of

3.(p , | n W

(2.1.2*)

which, the

V j=l

after

function

(A

M

+ B. M j . )

J

x) )

and

Fo / 0

x

J

g?(u)du.

(2.1.25)

(2.1.25)

determine

substitution

in E q .

Q., JK

ultimately

which

the

(2.1.13),

auxiliary give

determine

out the

6.(x,Fo).

Solution for a Sphere We s h a l l

moisture

now determine the solutions for the temperature and

distributions

in

a

sphere

with

the

differential

equations

38 (III) and (V

(IV) for

VIII).

problem

For t h i s

defined

equations

f = 2 under the boundary and i n i t i a l

is

in

purpose we shall Eq.

solved

(2.1.1).

by

the

also consider

The

system

simultaneous

of

the

the

applications

conditions

of

auxiliary

differential finite

sine

transform and Laplace transform. The is defined

finite

sine transform

with

respect

to space

variable

as

~ 6(p,Fo)

I f x 6 (x,Fo) 0

1

where p

p

SIR

pX

dx,

P

(2.2.1)

are the roots of the c h a r a c t e r i s t i c equation tan p

p.

(2.2.2)

The inversion formula of ( 2 . 2 . 1 ) can be obtained by 00

Vx.Fo)

where

x

aQ .

I n= I

sin p x an — - - = - ,

defining (2.2.3)

a

(n = 0, I , 2 , . . . , °° ) are the constant coefficients. The n 2 coefficient a„ is determined by multiplying equation ( 2 . 2 . 3 ) by x and integrating

between the limits

x

0 and x

1 . Thus we find

that 1 , a. - 3 | x ' e,(x,Fo)dx 0 The

other

coefficients

equation ( 2 . 2 . 3 )

a.,

a.,,...

by x sin p

(2.2.f) are

determined

by

multiplying

x and integrating within limit 0 to 1,

which gives

1

2p

an

2~sin

p n

r

/ 0

x

^(x.Fo)

sln

P n x dx.

(2.2.5)

39 Thus, the inversion formula e^x.Fo)

°° + 2 I

3 e^o.Fo)

becomes

p

s in p ^-2

2

n=1

sin

x

_^ 6j(pn,Fo)

p

(2.2.6) The

simultaneous

application

of

( 2 . 1 . * ) to E q s . ( I l l ) and (IV) for T

the

(2.2.1)

A

Tjtp)

S1

- x,(s)

p

"

and

2 gives

A

sT,

transforms

^

p2 6 , ( p , s )

+e K o ( s 9 2 - F 2 ( p ) ) (2.2.7)

and

s^

p 2 e 2 (p,s)]

f"2(P) - Lu Lx 2 (s) 5in_E + Lu PnLx,(s)

S1

"

P

p2 ^ ( p . s )

(2.2.8)

A

These

two equations

are

solved

for

A

6.

and

6,.

After

the

application of the inversion formula ( 2 . 2 . 6 ) , we obtain

^Cp.Fo)

I

[LJk (Vjk)

MJ k ( W j k ) j ,

+

(2.2.9)

j, k - 1

where "~ V,

jk

_ jk

and

the

e x vp ( - v sin

symbols

2 j

Lu rp

p

Fo j

L ,

and

p

Jk

^

A

k

2

Fo) f. ( p ) , k

(u)

M,

Jk

e x p (

have

.v/

7

j

Lu

been

7 p^

Fo-u)du,

defined

in

the

preceding

section.

Applying

the

inversion

formula

(2.1.6),

the

expression

40 (2.2.10) i s reduced to 6l (x,Fo)

J

[L^ Vjk

= )

Mjk Wjk]

+

(2.2.10)

where 1 3 f x i. (x) dx + 2 I k 0

V.. jk '

9

exp(-p

1 /

7

v

n

Lu Fo) J

oo . n=i LI

p ^ . 2 sin Kp

sin

sin p

x — dx

EL_

x x

n

x f. (x) k

0

p x

(2.2.11)

%

and

W

J

Fo °o f X k (u)du ♦ 2 I 0 n=1

= 3

Fo 0

ex

P

x "

sin

XL-( U )

J

p - p

sin p r

x n

(" v -

2 J

Lu p

2

Fo-u)

n

du,

(2.2.12)

a r e , again, the roots of the c h a r a c t e r i s t i c equation tan p = p . In order

the function

to solve our main problem, u

X|( )

from

the original

we have to

determine

boundary conditions (V) and ( VI).

Now substituting the values of X(Fo)

+2

+

2 I

(A

o o s i n p x F o I x sin p I Xk n=1 ^n 0

$,(Fo)

I j , k- I

and

1_ and in Eqs.(V) and (VI), we get 3x . , F M + B M ) [3 J ° xk(u)du

(A, L j k

+

_ ex

P

B, l]k)

)

jk V

J, K- I To solve these equations, we shall again apply the Laplace transform

with

respect

to

variable

Fo.

The

integral

equations

(2.1.13) and ( 2 . 1 . H ) then become A X,(s) X|

+

2 . _ , ) ( A , M1.. + B. M Z . ) [^ t 2 j,k = 1 ' Jk ' Jk s

]

4—2 s+Lu v

i(s)

8*

o o s i n p x ) —x sin pn n A,

(s)

*

(2.2.15)

p

and A

A0

2

v,(s) A

? ) . /• .

A

+ Ax-,<s) + B_

2

2

j,k=1

1

sin p x V ; — 2 — *■ . x sin Kp

3

[-

+ 2

s

A

A

1 s+Lu v .

2

Bz. jk '

2~ ] xk(s)

n=1

x

n

,

g2(s),

(2.2.16)

p

where g?(s)

I

/

e-sF°

[ ♦ (Fo)

I

0

V

JdFo Jk

I

j

)

(A

k

=

, I

L1

Jk

♦ B

L2 ) *

I

Jk (2.2.17)

42 and 2

g*(s)

e"sFo

J

1

[* ( F o ) - B ,

0

2

Equations

(2.2.15)

j)k=,

and

(2.2.15)

and

(2.2.16)

in

(2.2.18)

Jk

are

two

ordinary

A

X.(s)

are

jk

(2.2.16)

A

simultaneous equations

L2.. V - . ] dFo.

I

2

and

X7(s).

convergent

and

The s e r i e s may

terms of

Eqs.

be a p p r o x i m a t e d

for

large value of s by !

\_

. 2 2 s+ Lu v . p J n as a r e s u l t of w h i c h E q s .

(2.2.15)

~ s

and ( 2 . 2 . 1 6 )

a 6 ™ ( —y * 7 T7 ' m m m -.2 r or 3r

+

/ ^ i ->i (3.1.2)

and 3u

d

, ^Ud

a

TT

{

d

3u

2

+

T^ F

d ,

) + a

"17

6

.

d d

(

dr 3u - g - p , 0 < r < R,

where

T

T(r,t)

and

t

u

distributions respectively.

chemical

the

thermal

reaction

2

F

3T ,

3? >

> 0;

(3.1.3)

u(r,t)

are

the

heat

and

moisture

The thermophysical p r o p e r t i e s are assumed

constant in writing the E q s . (3.1.1 In

, 32T TT + or

3.1.3).

decomposition

depends

upon

the

of

the

body,

concentrations

of

the

rate

of

the

reacting

components and the products of decomposition. The r a t e of reaction to a

first

approximation

is

a

function

of

the

concentration

of

the

reactants and thus 3u

s

~gf

kf< (U s ) ,

where f (u ) is a given The transformed

system

of

(3....)

function. differential

by using dimentionless

equations variables.

(3.1.1)

(3.1.**)

is

49 a t

_ £

v

and

R '

"

u

T

q

rFo„ _

_L

fl

2 ' °1 R

x0

Q

'

°2 "

the

field

T

_y_

0 ' u

B

D

s

3

0 u s

u. 64 = -Sg- , U

d

and s i m i l a r i t y (i)

numbers:

the

Luikov

number

of

p r o d u c t s of decomposition i n r e l a t i o n to

Lu

a m = — and m a q

(ii) p r o d u c t s of

the

Lu .

Posnov

d

m

temperature

matter

and

the

field

number

for

bound

matter

and

the

gaseous

decomposition

=

(iii)

6.T0 and

— Q u

the

Pn

d

— 0 ud

Kossovich

number

gaseous p r o d u c t s of

„ Ko

bound

a d — a q

6mT° Pn

of

for

the

bound

matter

and

the

(3.1.*)

now

decomposition

0

m

p u . „ = — — « - and Ko . d T0 c T q

(iv)

t h e Hess number

Q, u° d

c

s ^

T0

q T

and

-

Ge

kR2

Ia — d

The becomes

system

of

,

(u

0\n'-1

sS )

differential

equations

(3.1.1)

50

32(x9,)

3(x9,) -, r-

3 Fo

3(x0 ac

3(x62)

,

+ e Ko

. 2 3x ) Lu

3Fo

m

3(x64) ',, r3 Fo

Lu, d

m

m

32(x6^) =; _ 2

3x

K o . -5—p

3 Fo

32(x9,) r—— + Lu ., 2 3x

3(x63)

-, c

,

d 3 Fo

(3.1.5)

32(x9 ) =—— ,

Pn

m

.

+ Lu, Pn, d d „

3x

(3.1.6)

2

32(x6]) ^ 2

3x

, 31x6^ rrr- - rW„ 3 Fo

(3.1.7)

Fo > 0.

(3.1.8)

0

and

3 6, r-p^

Lu d Ge f ( 6 3 ) ,

The

boundary

equations ( 3 . 1 . 5 )

8.

O.Fo)

T

0 < x < 1

conditions

for

and

the

system

of

differentiai

( 3 . 1 . 8 ) can be described as

A, 9 , ( 1 , F o )

+ B, 6 2 ( 1 , F o )

92

x

(l,Fo)

+ A2 9,

x(1,Fo)

94

x(1,Fo)

+ A3 9,

x

v(0,Fo)

0 ,

(',Fo)

*,(Fo),

(3.1.9)

f B2 9 2 ( 1 , F o )

$2 )

l

/sx)

2

y

sinh(x-£ ) v . / s

1

sinh(x-g)v2/s}

y

v

' "

S

) C^ e x p ( - v 2

d£ + j

F(x,s)],

(3.2.8)

where F(x,s)

-xf,(x)+ I

The conditions

Ko

constants and

the

m

C.

xf_,(x) 2

L u . Ge K o . x f ( 0 - > ) . d d 3

are

to

be

determined

condition

of

symmetry.

by

The

the

latter,

boundary on

the

a p p l i c a t i o n of Laplace t r a n s f o r m , now becomes

f,(0,s)

The c o n d i t i o n s two

(3.2.9)

(3.2.9)

reduce t h e four a r b i t r a r y

constants

to

i.e. C.

and

0

thus

the

- - C 7 and C ,

expressions

s i m p l i f c a t i o n reduce to

in

Eq.(3.2.6)

-C.

and

Eq.

(3.2.8),

after

54 A

|

iMx.s)

C sinh v. / s x

|-7-smh(x-C)v/s V - r

O

^ D sinh v ? / s x

+

—K— s i n h ( x - £ ) v , / s ]

1

V —V O

X

r

- — (V| - v 2 )s

/•R(Cs) i

d£

(3.2.10)

£

and * —p zKo

A

¥_(x,s) I

[(l-v, m

-> )C s i n h 1 .

X +

(v,

' - v 2 )s

1 -v? -|

where

2

I — V

? R(c,s) { — ^ £ 1

sinh

o ( l - v . )D s i n h v , / s x 2 I

v,/sx + 1

(x-C)v2/s}

sinh

tx-C>v/s

. d? + J F ( x , s )

,

C and D a r e new constants t o be d e t e r m i n e d

boundary

(3.2.11)

by t h e f i r s t

two

conditions. The

expressions

in

Eq.

(3.2.10)

and E q .

(3.2.11)

can be

w r i t t e n as

A r 6. ( x , s ) = — s i n h

/ D v . / s x + — sinh

{ —J-r- s i n h ( x - £ ) v . / s

1 v./sx +

=

~

(

R{£ , s )

( v , - v 2 ) s x >0 7 - sinh(x-£) v /s } d £ (3.2.12)

and

0_(x,s)

—JT— [ ( 1 - v . m 1 +

—~2 2— ( v | "v2 )SX

) — sinh

x

/ 0

v . / s x + ( l - v ? ) — sinh

. 2 '"vl R(C,s) { — ^ '

sinh(x-c)v]/s

v? ^ix

55

'v

V

2

2

J

s i n h ( x - C ) v 2 / s } d £ + -± F ( x , s ) ]

(3.2.13)

The boundary conditions ( 3 . 1 . 9 ) and (3.1.10) are transformed as

e,

x(i,s)+A,e!(t,s)+B,e2(i,s)

62

x(1,s)

*,(s)

(3.2.1^)

and + A2e)

x(1,s)+B282(1,s)

*2(s).

Substituting the vaiues of A6 . ( x , s )

and i t s first d e r i v a t i v e at

x

1 in Eq. ( 3 . 2 . H ) and Eq. ( 3 . 2 . 1 5 ) , we obtain

C

v JVP7Q Q 1 2 1 v 27

[

7( v~. ^ - v 2~)s 2

(Q

2

R

Q

1

2R2

(3.2.15)

P

2S1

+

P

2S2)

a m

Q 2 B, F ( 1 , s )

P2B2 F ( 1 , s ) ) ]

and

°' w ^ [z^i

(P S| PS2 Q|Ri

'

'

•

w

-(P,J 2 (.|-Q,{,(s)). - g J - j (P, i H i ^ i - Q . B . F d . s ) + P,B2 F ( 1 , s ) ) ] , where

56

P. J

B 2 1 (-1+A.+(1-v ) —r;—)sinh ' J eKom

1-v.2 Q. - ( A . + - p — ! — ) v . / s j 2 Kom j

R J

v . / s + v . / s cosh v . / s , j j j

cosh v . / s + ( ( B - - 1 ) j 2

B 7 i i (_1+A,+(1-V; ) - ^ - ) - V 1 J eKom v./s

' /

R(C,s)sinh(1-5)v./s j

0

1 / R(C,s)cosh(1-C)v / s J 0

+

(1-v.2) —p—J -A.,) s i n h v . / s , e Kom 2 j

d?

d£

and

S.

2

((B2-!)(1-v

) ^ ~

2

1-v + ( g ^

1

. | 0

+A2)

m

Thus,

the

-A2) — ^ 7 - /

R(£,s)sinh(1-Ov ./sd£

R ( C , s ) c o s h ( l - C ) v VsdC '

solutions

for

transfer

•

potentials

for

heat

and

matter under t h e t r a n s f o r m can be w r i t t e n as A

8.(x,s)

= t ( Q i sinh

2

/sx-?2

-(P.

sinh

v _ / s x - Q_ sinh

sinh

(Q,

sinh

A

$.(s)

(s)}/x(Q.P2-P

sinh v 2 / s x - P 2 sinh

v 2 / s x - Q 2 sinh

V)/sx)}

v./sx)-(R.-R_)x

/ { (v| -v

)(QIP2-P,Q2)xs}

2

2 2

* -

Q )

v

{(5.-S2)(P|

♦

v./sx) *

v./sx)

2 * 2 , ( v . - v - )sx

I 0

RCS.sH^Wsinhtx-Uv/s 1

57

—y— s i n h ( x - 5 ) v _ / s ) d ^

- P 2 sinh

+ ( P . sinh v . / s x

v , / s x ) ( 3 F 8 ( ^ s ) + B2

+ B. F ( l , s ) ( Q 2 sinh

(eKo m (Q ] P 2

v./sx

F(l,s))/(EKom(Q|P2-P|Q2)xs)

Q. s i n h

v?/sx)/

P 1 Q2) x s )

(3.2.16)

and ^ 2(x,s)

[ ( Q . ( 1 - v _ 2) s i n h

v V s x - Q ( 1 - v 2) s i n h

A

v . / s x ) $ (s)

6 (P)(l-v2 )sinh

v2/sx

P2(l-V) )sinh

2

2

x f Q ^-

m

( e Ko + [(S,-S2)

v ( /sx) $2(s) ] /

P,Q2))

{ Pj(l-v2 )sinh

v2/sx

P2( 1 - v ] ) s i n h

2 2)

- (Rj-R /((v,2

(Q](1-v2 )

2

sinh

v2/sx

Q2( 1 - V j 2 ) s i n h

Vj/sx}

2

v22)(Q,P2

P,Q2) eKom xs)

1 f

v,/sx}

'

( v .1 - v2.JTZ )eKom x s —2

I0

v

' " R(?

1 ' S ) * V^

|

2

sinhfx-Ov/s

1 -v 2

r— s i n h ( x - C ) v . / s } d £ + ( (21 - v _

v_ *s

2

2

) P , sinh v . / s x

1

2

]

58

P2(l-V|2)sinh

(e2Ko2

v,/sx)

(Q,P2

xs

* B2 F ( ! , s ) ) /

+ B. F( 1 , s ) (Q 2 ( 1 - v , 2 ) s i n h

e2Ko2

v2/sx)/(

F(x,s)

eKo

dF { ,S) dl

P]Q2)xs)

Q](1-v22)sinh

+ -p-J

(

v,/sx

(Q|P2-P]Q2)xs)

.

(3.2.17)

'

m The product equation

expression

under

(3.2.3)

condition

f o r t h e transfer

t h e transform with

potential by s o l v i n g

t h e help of t h e m o d i f i e d

of t h e gaseous the differential

form of t h e boundary

( 3 . 1 . 1 1 ) , and t h u s :

s i n h v<s/Lu .)

A 6

i s obtained

4

( x

'

s )

=

x [ ^s/Lud)cosh

B.,

A

[Ms)

, ,

3

f i

*

^/Lud)+

(Bj-Dsinh

'

vfs/Lu d >]

9 8.

; / { £ f , ( 0 + Lu. Pn.(

is Lu ,)

Q

d

**

d

3 8.

^- + 2 ^2

?(6 ) } sinh /(s/Lu^ «-1)dt -A- f f 1 o *

Lu

(

d%

{ ^/L|U

2 A

^N

3 6.

3 6.

T ^+ 3x

2

_

3-x-'x=C s + f

f(6

o

+ —zri

x

3 6.

^

3) 1

yv

36.

: / (£f„U)+Lu, P n . ( — y 1 * 2 ~ )

x ^ Lud)

J

Q

it

d

d

3 x

2

~)

+ jJ- J d0

s i n h / ( s / L u d ) ( 5 - D - c o s h / ( s / L u d ) ( C - l ) } d£ ? A

.

8 x

3x x = ?

x= £

UfJO

59 Lu . Ge A + — | Cf03)) 0

sinh

4s/Lud) (C-x)dC

,

(3.2.18)

„2-

A

39,

are,

w h e r e t h e values of —5— and — = — are determined dx

» z. d X

by

(3.2.16).

A

The e x p r e s s i o n s .(s) and f ( 9 , ) ,

context.

the t r u e nature of w h i c h

Therefore,

to

e x p r e s s i o n s , we s h a l l (i)

the

determine

i s not

the

inversion

the

terms

yet defined

inverted

form

in

the

of

these

analysis,

where

apply: theorem

of

the

complex

the e x p r e s s i o n s contain a l l the w e l l defined (ii)

contain t h e

1

A

convolution

theorem

for

the

terms, terms

of

A

$.(s)

and

f(9j).

To a p p l y expression

in

the

the

inversion

denominator

is always greater

theorem, has only

we s h a l l simple

suppose

roots

that

and i t s

the

degree

than t h a t of the e x p r e s s i o n s of the numerator.

Now

to f i n d the r o o t s of denominator, we equate i t to zero and thus

fQ(s)

This

s(Q,

P2

P, Q 2 )

0.

gives

(i)

s = sn

(ii)

s - sn

where s

,

s a t i s f i e s the equation

VSn> or more

0 (a zero r o o t )

Q

1

P

2

P

,

Q

2

°

clearly

Q , P , v nl n2

P. nl

Q n 7 - 0. n2

(3.2.19)

60 where

the

hyperbolic

sines

cosine by s u b s t i t u t i n g s characteristic

equation

P .= u v . nj n j

= -y

and cosines 2 , u

(3.2.19).

cos y *n

are

changed

being

into

sine

and

root

of

the

and Q . are

given

the

The values of P

2 1 v. + (-l+A, + ( 1 - v . ) —r;—)sin y j 1 j eKom' >n

v. j

(3.2.20)

and i 2 . 2 1 -v . I-v. ( A , + —r;—^—)y v . cos y v . + ( ( B _ - 1 ) —r;—*— 2 eKo n J n j 2 eKo m ' ' m

Q . v n iJ

A.Jsiny v. 2 n J) (3.2.21)

For

determination

of

d e r i v a t i v e of t h e denominator



where

^

* %

v,

*n

Qn2 A n , +

the

r e s i d u e , we need the 2 at s = - y . This gives

value

,

of

the

(3-2.22)

v ^ B ^

v ^ A ^

v , ? ^ ,

(3.2.23)

The q u a n t i t i e s A . and B . are nj nj A . nj

B 2 1 ( A , + ( 1 - v . ) —7}—) 1 j cKo m

cos y n

v. j

y v . sin n j

y v. n j

(3.2.24)

and B 2 ) —Q j eKo

2

B . ni

(1-v

The

1-v.2 cos y v - ( A , + —Q—>—) y v . s i n n j 2 eKo n l

inverted

expressions

of

the

transfer

y v. n I

(3.2.25)

potentials

6 (x,Fo)

can be w r i t t e n as

e,(x,Fo)

-

oo p Fo I f / t(Qn, n=l n o

sin y n v 2

x

- ( P , siny v . x - P _ siny v,x)4>_(u)] nl n 2 nz n 1 2

Qn2 sin y v

e x rp ( - y

n

x)«

Fo-u)du

(u)

61

a 1

f

2

m

; 2 2 2, ^ i H T ( v . - v - ) x n=t K n n

-(RnrRn2)(Qn1

+

s i n

m

1

, exp(-y^

2

(P

pH V

n=1

n1

n n

Fo)](B2 F,(l,s)

B

s i n

Qn2

°° I.

[

nr5n1)(Pn1

x

V 2

B A7-BT " \

EKo-

[(S

+

sinp

V l

x )

Sin

V

nv2

]ex

x

-

P

P("p n

2X"Pn2

n2

s i n

Fo)

sin

v

l

x)

3F.(l,s) ^ )

°°

+ —77—[- -7— eKo A. m 1

— Ly — „ ; — (Q _, sinu v . x - vQ . siny v-.x) v x , y f n2 n I n1 n 2 n=l n n

.

Lu . Ge Ko . FO

e x p ( - i £ Fo)] F . ( l , s ) n '

+ —~ v,

=-* / v2 6

nv2

P

B.

H, (u^-v-^1 2

GO

+

x

I -TTT~ n=1 n n

Lu.

Ge Ko r f

+

v. - v 2

(P

nl

FO Jf 0

sinU

H

x

I (u)[- r -

+

n2

2 -

1

Q . s i n y v_ x )

FO

B

i0H 3 ( u , [ A T I7 1 2 exp(-yn

exp(-y

Fo-u)]

sin

^ n v | x > e x P ( - t^Fo^u)]

°° 1 I T p r (Qn2 siny v n=1 n n

x

Lu . Ge Ko , du + —-rjrm

»

x n=l 2 Kyn4n- ( P n l

sin

V 2 X"Pn2

sin

Lu . Ge Ko , FO . F o - u ) ] d u + —^ B, / H^(u)[- j - + m 0 I

Vlx)x 2

-

du

Vl

x )

62 00

I —y— (Q n 2 s i n u n v ( n=l n n

x

Qn] sinyn v2x) e x p ( - p n

Fo-u)]du, (3.2.26)

OO

9

2(x'Fo)

d_(u)] e x p ( - U Fo-u)du n2 1 n 1 2 n

+ riT— [ f . ( 1 ) - f , ( 0 ) + eKo m f , ( l ) £Ko

I

m

m

2

e Ko f,(G)- J m 2

tS-i 2

ff(l) I

a 2

m (v.

(Pn|(l-v22)sinynv2x-

- v _ )x

n=l

n n

Pp2( I - v, 2 ) s i n ^ v , x)-(R*n)-R"n2)

2 2 2 ( Q n | ( l - v 2 )sin u n v 2 x - Q n 2 ( l-Vj ) s i n y n VjX)] e x p ( - u n Fo) OO

+

I. I 7 V [ ( , - V 2 2 ) P n 1 n=l n n

~^2— e Ko x m -

SlnlJ

nV2X-('-Vl2)Pn2

3F ( 1 , s )

exp(-u n 2 F o ) [ B 2 F , ( , , s ) ,

'

d

2

2)

V

lx]

» -

I

- ^ n n

2 2 sin un v ) x - Q n ] ( 1 - v 2 ) s i n p n v 2 x ) e x p ( - u n F o ) F . ( 1 , s )

Lu . Ge Ko .

♦

n

2B.

]

e Ko x n=l m

2 ( O - V j )Q n

SlnW

'

( v . - v . , )eKo x 1 2 m

eKo x

»

I H,(Fo^,)[-B;=L.

Fo

2 I

0

2

n=l

,

^ ( P n n

n

,

63

2 ( 1 - v , )sinu v , x c n 2

2 2 P , ( l - v . )sinu v . x ) e x pr ( - u u)]du n2 1 n 1 n

2 L u . Ge Ko , 2 T (v, - v A K o x

♦

I

2

m

2

Qn2(1-v, Fo j

Fo °° 1 / H (Fo-u) I —^0 ' n=l V n

2 [Q , ( 1 - v - )siny v x ni z n z

*"Ud ^ e

2

) s i n u n v , x ] e x p ( - p n u)du

^°rl

- ^ ^ m

x Lu . Ge Ko , Fo f H s ( C , u ) ( x - ? ) d u d£ + -=S— ° J H (Fo-u)

0

0

E

Ko x m

eKo x [—^T

i

0

B

2

CO

2

P

I. U V ^ n= 1 n n

n1(,-V22)

n

V

iX

u)]du ♦

m

[Q 2 ^ ' " v i

+

£-|

j

d?

+(A2

< j * f P 3> I 1

x = ?

0

+£

«"(e3)

m

1-v.2 1 (Cf(63) j - J KQ J ) f m 0 m

-Cf"(63)-2f'(e3))cosh(l-C)v Vs d 5

and

h

J

(-, + A, + (i-v 23 ^L-)

(S)

'

^ J m

j

-2f'(e3))sinh(1-£)v Vs

dC

-2f'( 03))cosh(1-C)v./s

d£.

3.3

-xf.(x) 1

T eKo

m

Cf"(6

)

xf-,(x). 2

Analysis of the Result For a k - t h

order" chemical r e a c t i o n equation ( 3 . 1 . 8 )

ae,(x,Fo) ~ ^ The s o l u t i o n under

c f"(e 3 )

m

+ / ( j ^ - C £( 6 ) 0 m

J

F,(x,s) I

Uf(e 3 ) ^ 0

becomes

. - L u d Ge[ 6 3 ( x , F o ) ] K

expressed

by

the

Eq.

, k > 0

(3.2.28)

is

(3.3.1) thus

modified

as

67

93(x,Fo)

= f3(x)[1-(k-1)Lud

1

Ge Fo f ^ '

'(x)]-1

/ ( k _ l ]

,

k? 1

(3.3.2) and

6 3 (x,Fo)

f3(x)+exp(-Lud

Generally greater reaction

than of

2

it

order

common

interest

(3.2.27)

and

is

are

found

rare I

to

(3.2.31)

that

and,

takes

Ge F o ) , k

the

in

the

chemical

various

place

reduce

1

(3.3.3)

reactions

power

frequently.

complicated

of

plants,

Therefore

it

expressions

9(x,0)

fo.

Therefore

the Eqs.

p o t e n t i a l s are uniform

is

Further,

initially:

(constant)

(3.3.

u u Vn n n 2 * , ( u ) ] s i n T L — x e x p ( - u n F O - U X T J — COS 7 5 — d d o

U

.

+ (B,-1)sin -rr^-V 3 /Lu. d

,

du + x

co LT

. m=l

FO

A I m i 0

[(P . D , - P ,D ,),j>,(u) ml m2 m2 m l * 2

v j -1 -(Q , D _-Q _,D , ) * , (u)]sin -TT^- X exp(-v Fo-u)(P ,Q ,-Q ,P , ) v K m1 m2 ^m2 ml 1 /Lu. m ml^m2 ^ml m2 d 00

du

|

-A,

fo (

I -irV[(B2Pn2-BlQn2)f02 n=l n n

, Qn2

exp^Fo)

fo

+

2 D m2- A 1

D

+

nrQn1Dn2)](7Cu7

I

cos

D

nl+ (QnlBrPnlB2)fo2

TTu^ + < V 1 ) s i n

^

n2

)

J -m_ [ ( B 2 P m 2 - B l Q m 2 ) f o 2 D m] + ( Q ^ B , - ? ^ ) m=1 v m

fo (

l Qm2DmrQmlDm2)] ^ m . ^ " ^ ! ^

eX

P("vm

2Lu . Ge Ko , A. °° , y n ~ 1 —3T- (Q |D ,-Q ,D ,)sin j y ^ - x x nl n2 n2 n / L u n=1 WJn ' d

exp(-u*Fo)

D

( ^ d

cos7T^-+(B3-l)s1n o

^ J o

Fo)

70

Lu , Ge Ko . A. x

sin ^ -

oo A l t -2 m=1 v m

, (

x exp(-^Fo)

Qm2

D

m l-Qmi

- \ ^ -

D

m2

)(P

(

P

)

ml 3m2-Qml m2 "

(Q°2 D'-Q' D^MP', Q^-QT, P* 2 )"'

d

sin /Ge x ( / G e cos / G e + ( B ^ - l ) s i n / G e ) "

e x p ( - L u . Ge Fo)

2

+ 2Pn

I

d

d

U FO

Y1 1 i< p n l ~ 2 ni

n=l *n 0

2

v?

sin y

nv2 n

v2^-1/Lud

x

P

v.

n2 ~~1—]

l

ni

v^-1/Lud

2 V

sin

v

U

x ) « (u)

(Qnl

2 V

2 siny v

2

x-Q

v2 - 1 / L u d

sin

1

v, - 1 / L u d

u v, x)

Lu . Ge Ko . A

, / , 2 ' L u . Ge-u d n

^

and E

n2

j f V tP nl (W m -B 2 to i> + «nl (A l f0 1 +,i 1 & 2- Ki q>- A 1 J T ^ T n n The

Q

nl ]

Lu d Ge-U n mean

value

of

the

transfer

potentials

in

sphere

are

obtained from the relation 1

< 6 (Fo) 1

>

3 / 0

x

2

e.(x,Fo)dx. *

(3.3.16)

74

Thus

the

expressions

for

mean

values

of

the

transfer

potentials are obtained from E q s . ( 3 . 3 . 1 2 ) , ( 3 . 3 . 1 3 ) and ( 3 . 3 . 1 5 ) as B , Ki - B , Ki

2

q

1

I

m +3

2

°°

2

y

y ^ ( ^ v

n=1 j = 1

E .

c

°*Vj

1

n j

-

- s i n g v.) e x p ( - u Fo)+3Ko , [ - ■=■ + A. { (^Lii. Ge) v_

Q 1 sin/(Lu , Ge) v ? ) — y -(/(Lu . Ge) v. v2

cosy(Lu . Ge) v 7

Q

"1

2 , 1 G e ) v . ) — j ] j——g— (Q 2

a coSf{Lud Gei Vj-sin/tLu d

V,

d

Pj-P^)

d

Ge F o ) ,

(3.3.17)

exp(-Lu

a Ki2



exp(-u

g-S 2

OD I n=

+

3

I

2 i=1

En .

Y -SI A

Fo) + 3 :

1-v v. .

(

2L)(unvj

'

'

'

cosw v - s i n u n v . )

) 1Kod =— (/j-u, Ge) v_ cos»{Lu, Ge) V ? d

-sin L u . Ge v_) —=- -(/(Lu , Ge) v, cos/iLu. G e ) v . d l l d I d V 2 Q

°2 . "' sin /(Lu d Ge) v , ) — j } (Q2P"-P*2Q* ) exp(-Lu V l

Ge Fo) (3.3.18)

and <eltU,Fo)>

Ki . -^~ 3

♦ 3

°= A Lu , J - n y - * [Ki m=l

v m

B fo

75

Lu , Ge B-.

v 7-JlTrj Wo(LuH G e - v " ) ^ d m 2

v cos -p. /Lu

[1-

WQ

/Lu

d

] * d

3B3(/Ge cos / G e - s i n / G e ) cos/Ge+(B3-l)sin/Ge)]

(

e x p ( - v m Fo)-

v sin -p.

eGe x p (Fo) -Lud 3.4

x

Ge(/Ge

(3.3.19)

Analysis of the Solution The

graphical

characteristic

roots

method.

this

For

u (n

1 , 2 , . . . , °° )are

purpose,

the

obtained

trigonometrical

by

equation

(3.2.19) i s written in the form

u

N

A,-l

(3 1 1) U.*.U

'

where M = Q n 2 usin uvj

Qn|usinyv2

and N

cosuv

Q j(uv2

Q

2

+

( u v , cos iiv,

2 C-v2 '

+(l-v,

2

e

B 1 K0

m

sinuv2)

B l ) -^p-

sinuv,). m

The M/N

and

a b c i s s a e of the

characteristic to determine 0.3,

Ko

the

points

straight

line

roots u

at different

the

= 1.2,

values of u e

m

slope of the line Y located in the f i r s t

0.5,

B,

of

interaction

Y = u /(A.-I) at 1.8,

give

of the

the curves Y values

of

the

A.. Figure 3.1 has been plotted different B,

A.

(Lu

10.0 and A,

0.3,

Fe

0.5).

The

i

u / ( A . - 1 ) is (Aj-1)~ . For A. > 1, the line is quadrant and for A. < 1, it is located in the

(a) 0.05 < A.

< 5.0

(b)

51.0

r o o t s of E q . ( 3 . 4 . 1 ) 6.0 < A,
0, i t that

the

reaction

is

of

endothermic

type

and

proceeds

the

quantity it

signifies with

the

absorption of heat and for Ko . < 0, it signifies that the reaction i s of exothermic type and proceeds with the evolution of h e a t . Now potentials

we

consider

applicable

for

approximate

solutions

small

of

values

for

these

the generalised

transfer time,

Fo.

For small values of Fo, the values of s is large and for large s we have v./s sinh v . / s

i?

cosh v

/ s *?

? e

]

Under t h e s e approximations and r e s t r i c t i n g to the terms of o r d e r s only, the E q s . (3.2.16) e,(x,Fo)-fo. I

I

(3.2.19)

give

= 77-[(N._ fo--A.N,_ f o . ) / F o i e r f c ( v . Mx

- ( N . . fo 2 -N 2 ]

1L

I

\

Aj f o , ) / F o

11

I

i erfc(v2

1

f^x/Fo)

Hoc/2/Fo)

3/2

81

w\ —

r-v (^

N O 0 ON iO

-3-^-1 ON m

JS

N

00

-3-

00

(N

OO

—

0 - 3 -

o r ^ i

O

CN

O

CN

O

CN

O

CN

—

CN

O ^ J -

N

O

CN

^3-00 O

—

ON

tf>

N N as

.

N

•

_

O

CN

OO

j - o

CN.

O

— N

u"\ N

0 0 0

O

I A C N

o

C

N

O

C

N

O

f

N

O

C

N

CN

N ON

o ^ N w\ -3- (N — ON C N N O ' J S O O v D v O — ON N ON .joo M o o —

m

> u_

o

■*

3

< > CN ID -I to

O

C

N

r^i\D N .3CN, ^

-3-30> OO

O \£ f\. N

\£> O N ON o o o

vD M u"\ oo

-3" (M oO

N

00 u^ vr\ _ — .3OS .3-

ON ON

ON

0 - 3 -

O

CN

O

CN

O

CN

O

CN

O

O

—

— CM

W\

O

r*N

O

ON

tr\

\C

c»N in

^ CN, CN -3- -3.3- N 0 0 " " \ \ 0 0 0 - 3 - C N . C N — A ON CN ON N ON 3-

ON

O

-3-

O

O

N

W

CN

-3" CN 00 TN ^ \ D N

O

-3"

N

O

C

N

CN.

0 - 3 -

O

vO ON O 0 ON

U"\ 0 0 S f ON

ON 00 — ""N

^ ON O O

O

fN

M

fv.CN «"\ — O

S

0 0 0

ON

CN \D

G

V

D

o — ^

N

00 CT>

O t N

O C N

O C N

O f N

O

v O — — CN i r \ —

c*N — 3- CN n O N

(■** \D —

0 0 0 0 0 — ON CN

0*^1 — OO oo —

O

N

N

O

O t N

CN,

N

H

O

N

CN

O

N

v£) — OO —

—

O O O O O O N

O f N

O C N

O

—

-3" — ^ W\

N ©o - a - j "A g^ ON o

CN. 3-

— CN r«-\

CN —

ON

OO

Of-vj

O —

O

—

0 - 3 -

O

CN.

O C N

tN

ON

o N — f«N

O C N

OV

N

°*N

ON

vo 3r«N ON

CN

•— ON 00 — sD OOtN ON ON CN 00 N CN \ 0 v O O OO.* O O N CN^O N 0 0 C N r * % . O N v O \ D v O 0 N u ~ \ N » . — ON\D ONtN O N O ON 00 O N N

Or^\

O C N

O C N

O t N

O —

O

—

m o o .3-r-K (*>\o oooo ON S *r\,oo r ^ . o T^.fn ■& a\ — ■& r ^ t n m o i o v m &■ & O N o o o N N O N O o o v o o o f ^ N v o r v m ON-* O N — O N O N O N t N O N O O N O O O N r ^

•— •

O

*

O — O C N O N 0 - 3 — ■—,

—

>

>>

CN

>

OC'N.

O C N

O t N

O C N

O —

O O O O

O O N O — O 00 Ou'N

O O O \D O f N . O C N

O O O O O O O O

O N 0 " A O C N O O O

—

—

—

C N C N v O —

—

tN

>

(N

CN

—

>>

CN

CN

—

>>

fN

CN

—

CN

> >

—

O

O c - \ O O O O N O v O

—

—

—

—

tN

—

P U-

o 0

«

" O

—

\

O O

—

u

O

N C

O w N f N O

O

^ O C * N O

.

«

—

> >CN

> >

r -1

O

CN — o

CN oo

\ 0 \ D 00 OO u*N ON OS M

r>.vr\ O .3- vs. ON 0 N O N O 0 ON^JON

tN

3

vo (N

Y * N - 3 - C N . 3 -

— w\ 3- .=»• — OO — (N r ^ J 0 0 0 N O N O CN oo ON — -3-oo oo CN (N o ^D-=f o o < ^ r ^ . - 3 - > n O N - * \ o ( N i A — O O N O O ON "N. ON CN o\ *c o> m as — as o oo oo

"^




3

tr\ —

ON

^

ON

rt

4 : o

B^fo0+( A, f o . - B . f o . ) (I - v, 2 +eKo A.,))

(eKo

1

m

2

2

1

1

1

2

1

m 2

erfc v 2 /2vFo]+ K o d ( 1 - e x p ( - L u d Ge F o ) ) ,

¥°'F0)-f02

, 2 2 , , [ ( l - v 2 2 ) ( , - v , 2 + e K o m A2>Ki - K i J ( v 2 - v , )eKo m

erfc v . / 2 / F o 2

(1 -v , 2 ) ((1 -v 2 + eKo A,)Ki 1 2 m 2 q

erfc v , / 2 / F o +(B. fo, + (A, f o . - B , I

(l-v

2

2

(3.4.5)

2

2

1

I

I

Ki ) m

fo,)( 1 -v ,P_ +eKo 2

I Z

m

x

A_)) 2

x

) e r f c v - ^ v f o - t B , fo- + A. f o . - B . fo,) (1 -v , 2 + e Ko A,)) 2 2 2 1 1 1 2 2 m 2

( l - v , 2 ) erfc v , / 2 / F o ]

x

(3.4.6)

and 9(0,Fo)

= 2Lu d (Ki d

B ? fo^) erfc l / 2 / L u d F o .

(3.4.7)

86

The

Eqs.

(3.4.2

3.4.5)

functions which are sufficiently the

influence

temperature thermal almost

of

these e r r o r

over

the

destruction linearly

for

can be seen that proportional

contain

integral

of

the

error

small. From Eq. ( 3 . 4 . 2 ) , neglecting functions,

initial

distribution

(chemical

reaction)

small

the

values of

the formation

we find

that the excess of

in the of

time.

the

process body

From

is

and

due

it

to

varies

the Eq. ( 3 . 4 . 4 ) ,

of the gaseous products

to the square root of the generalised

time,

is

it

directly

Fo at

the

surface of the body. Figure 3.3 shows the relation between (8-.-fo-J/Ki and Fo for ° I 2 m the surface and centre of the s p h e r e (Ki Ki ) under the simple v

m

Y

q

boundary conditions of second kind (A, B. = B,, 0, E ' I l i = 1.2 and A- - P 0 . 5 ) . The transfer potentials for with

time. In small range of the generalised time, the matter

transferred

from

the

reaction

surface

speedily

for in

small

values

comparison

of

m is

unaffected is

chemical

matter

generalised transfer

the

0 . 5 , Ko

the

with

the

of moisture from the c e n t r e . Figures 3.4 and 3.5 show

the

distribution of moisture and i t s gradient inside the s p h e r e . From the c u r v e s , we observe that the moisture transfer

from the layer

nearer

to the surface occurs at a fast r a t e and the rate slows down as the layer goes farther

to the surface.

The moisture is transferred

from

the centre towards the surface. The e x p r e s s i o n s small

values

for

mean

values of

of Fo are obtained

6.(x,Fo),

by using the Eq.

i

(3.3.16).

we have < (e , ( x , F o ) - f o 1 ) > = ^

(Nn

fo 2

TT(V,,FO)

2

[ ( N 1 2 i°2~A\

N 2 ] A] f 0 ] ) n ( v 2 ,

(N„

N 2 2 fo,) 3i ( v ] ; F o )

Fo) + (N 3 ] K i m - N 2 ] Ki )

„ Ki ) 7i ( v . , F o ) ] 32 Ki m- N 22 q 1

1,2,3

for

Thus,

87

Figure 3.3 - ( 0- - f o _ ) / K i

versus Fo for sphere

88

Figure 3A

Distribution of moisture in a sphere (Ki m

Figure 3.5

Ki ) q

Distribution of moisture gradient in a sphere

89 + K o d ( l - e x p ( - L u d Ge Fo))

(3.4.7)

< ( 6 2 ( x , F o ) - f o 2 ) >= ^ [ ( N , 2 fo 2 -A, N 2 2 fo, )(I - v , 2 ) it ( v ( , F o )

-(Nn

+

(N

N 2) A, f 0 ) ) ( 1 - v 2 2 ) IT ( v 2 , F o )

fo 2

3I

Kl

N

m

Kiq)(.-v22)ir(v2,Fo)

2I

N 2 2 Ki ) ( ' - v , 2 ) T ( v ) f

(N 3 2 Ki m

Fo)]

(3.4.8)

and <e^(x,Fo)>

3(Ki d

B 3 f0i()TT ( L u d , F o ) ,

(3.4.9)

where 3/2 - ^ 3v. A J

Tt ( v . , F o ) = — ^ - ( 4 F o ) 3 / 2 i 3 erfc ( v . / 2 v F o ) - F o / v . J J J v. J J - ' ,2,3; Equations

(3.4.7

3.4.9)

,

l/Lud •

v3

contain the term

i

erfc

(v / 2 / F o )

which is very small for small values of Fo. Neglecting the influence of

this

depending process.

term, upon

we the

see

that

these

generalised

transfer

time

Fo

processes in

the

are

beginning

linearly of

the

90 REFERENCES 1.

Lebedev, P . D . , Int.3.Heat Mass Transfer

2.

Luikov,

A.V. and

Mikhailov,

Y.A.

Transfer" (Pergamon P r e s s , Oxford,

"Theory

Ralko, A.V., I n t . 3 . Heat Mass Transfer

k.

Shukla,

5.

1973).

Tripathi,

and

G.,

Shukla

and mass transfer Chemical

transformation

International (Belgrade, 6.

Tripathi, Transfer

K.N.

in an infinite under

Mass

I (1961) 273-279.

"Heat and Mass Diffusion",

Hindu University ( V a r a n a s i ,

of Energy and

1965).

3.

K.N.,

1 (1961) 294-301.

Pandey

P h . D . Thesis

R.N.,

Banaras

simultaneous

heat

plate in presence of phase and boundary

conditions,

Seminar on Recent Development on Heat

generalised

Exchangers,

1972). G.,

Shukla,

K.N. and Pandey,

18 (1975) 351-362.

R.N.,

Int. J.Heat

Mass

91

Chapter *f HEAT AND MASS TRANSFER DURING INTENSIVE DRYING

An analytical approach has been made to determine the temperature, moisture and pressure distributions in the drying of an infinite plate. The expressions for mean values of these distributions over the plate thickness have also been obtained. The variations in these distributions and their gradients with respect to space and time are presented graphically. Analytical result indicates that the process is intensified by the filtrational drying. In the process of drying, the moisture is transferred the

material

which

evaporates

from

the

surface

of

inside

material

to

surrounding medium. In general, the rate of drying depends upon the intensity

of moisture from within the material towards its surface.

3 7 a The experimental researches of Lebedev , Maximov proved that intensive

the

phenomena of the exchange of

drying

hydrodynamical

are

influenced

forces.

Luikov

by

the

and others have

heat and matter

action

has shown that

of for

the

in

various

non-isothermal

conditions the total flow of mass in this case is equal to the sum of the mass flow through the process of diffusion, thermodiffusion and filtration. The system of differential equations of the exchange of heat and moisture with

the molecular

and filtrational transfer of energy

and matter can be presented as: C

q Y0 | T =

div( X

q Srand

I i

c

m yQ | f =

div(X

T

>

+

Ci(qm

m Srad

u

+ X

£ P cm ^ 0 f f grad T)

m

&

Srad

(XII)

T

*

X

p 8 r a d P>

92 and

C

where

Y

p

3p 0 I t

the

first

the change i n denotes the

the

tionai)

denotes

on the

for

(XII)

Eqs.

moisture

porous

the

convective

convective a number

type

drying of

Tien

basic

Toei

In (XII

the XIV)

hand

due

content

and

Eq.

bodies

arbitrary

functions

of

like

V I

„ ,

the second

term

transformation

and

heat

by

are

currents

the

t e r m of r i g h t may

infinite

be

plate

conditions 2

have

intensive

chapter,

The of

denotes

of

differential (filtra-

hand s i d e

of

neglected

in

and

and

with

proposed

a

sphere uniform

new

porous b o d i e s .

the

solved transfer

space

system

under

of

the

potentials

coordinates

at

are the

for

the

in

body

laws.

differential most

of

initial

However,

h e a t i n g , the surface of

in

under

model

the c o n v e c t i v e t y p e of i n t e r a c t i o n

been

conditions.

(XII)

,

hydrodynamicai

(XIV)

mechanism of c a p i l l a r y

boundary

the

Eq.

heat,

phase of

( X I V )

i n v e s t i g a t e d the condensation process g Mikhailov has solved the system

Okazaki

present

of

XIV)

the t h i r d

boundary

have

of

to

3T 3T

BIT

^

.

of

processes

side

transfer

(XIII

and

does not a l w a y s f o l l o w

3u 3T

m ^0

terms.

insultation,

for

distributions.

and

£ C

diffusion

convective

and the l a s t term of

equations the

due to

the

comparison to t h e o t h e r

the

right

a zonal c a l c u l a t i o n ,

Ogniewiez

"

temperature

the

the

. P)

in

motion r e s p e c t i v e l y For

Eq.

, Srad

temperature

Similarly

equations

,, p

term

change

third

matter.

,.

d l v U

=

equations

general

type

of

to

be

moment

of

supposed initial

time. 4.1

Statement of the Problem The system of

transfer 2R for

of energy

differential

and matter

equations

in an i n f i n i t e

with

molecular

p l a t e of f i n i t e 9 10 the zonal c a l c u l a t i o n s may be d e s c r i b e d as '

and

molar

thickness

93

3T

C

,

q Y0 "37 =

C

3 T

q ^7

Yn I T

*

m ' 0 3t

_

+

e P C

Y m

^ ^

3u

i,

A J ^ * J

m _ 2 3x

(

0 a"'

^ |

m . 2 3x

, ,\

*-K,)

(4.1.2)

P ^ 2 r dx

and c

3p , 3 p Yn j f = ^ — o p 0 3t P g 2

3u ecYn"5-m ' 0 3t

,, , , . (4. . 3)

where the thermophysical p r o p e r t i e s a r e assumed constant. For s i m p l i c i t y ,

we s h a l l

transform

equations

(4.1.1

in the dimensionless form by defining the non-dimensional x

r / R , Fo

a t / R 2 , 6!

T / T ° , &2 = u/u° and Q^

and Luikov number for the field to temperature

a — , a

Lu

q Kossovich

number pu° T

Posnov

number

Pn

—Q- ,

u and Bulygin number Bu

=

=

a —2- ; a q

c

c pPP -E-5c T q

p/p°

of matter and filtration in relation

field Lu

4.1.3)

variations:

,

94 where

the

characteristic

entity

denotes

the

respective

potential

drop. The E q s . (4.1.1 32 9 —r1 -, 2 3x

36 -SET3Fo

4.1.3)

become,

39 + eKo -s=^ , 3Fo

(4.1.4)

32 9 326 Lu Bu Lu — ^ T Lu Pn —=-^ + g

39 ~

-2 3x

dro

, 2 3x

Ko

329, — ^

(4.1.5)

,.2 dx

and 3 0, 3

3 2 6, 3

,

„ Ko

3 9, 2

,.

-

T~E~ - Lu —=— eH "5c- • 3 Fo p _ 2 Bu 3Fo The boundary conditions for equations ( 4 . 1 . 4 )

the

system

( 4 . 1 . 6 ) may be p r e s c r i b e d

9,

x(',Fo)

62

x(l,Fo)-A2

+ A, e i ( 1 , F o ) + B ] 6 2 ( l , F o )

6,

x(l,Fo)+B2

, ,>

(4.1.6) of

differential

as = *t(Fo),

92(l,Fo)+C|

9j

(4.1.7)

x(1,Fo)

$2(Fo)

(4.1.8)

and 93 ( l , F o ) where

A.,

B,

(1

1,2)

physicai coefficients

and

which may be determined In order that

the

therefore

system

and

C.

are

(4.1.9)

aggregate

of

known

,(Fo) are p r e s c r i b e d fluxes at the

thermosurface

by the experiment.

to simplify is

$ 3 (Fo) ;

the

present

problem,

symmetrical, thermally

and

we s h a l l

suppose

geometrically

and

95 8

(0,Fo)

0

(4.1.10)

1, x

For the

the complete

potential

statement

distributions

at

the

of the

problem

initial

moment of

cribed functions of the space v a r i a b l e S^x.O) 4.2

we shall

specify

time as

pres­

i.e.

f^x).

(4.1.11)

Solution of the Problem The solution

4 . 1 . 6 ) is obtained

of the system of differential

equations

by the application of Laplace transform.

the Laplace transform

to E q s .

(4.1.4

4.1.6)

— ~ - + eKo(s6 2 dx

f2(x)),

(4.1.4 Applying

and using the

initial

conditions ( 4 . 1 . 1 1 ) , we obtain sG,

f,(x)

7 A

s6_

f.(x)

d 8 Lu — ~ , L dx

£

f ( x )

L

Z

I

(4.2.1)

■) A

7 A

d^8 + Lu Pn — ~ , I dx

d^8 + Lu ^M —-J. n l\0 , L r dx

(4.2.2)

and s

2 •*• 3_£K^ dx

K

Eliminating help

of

Eq.

d

(4.2.1),

from

Eq.

we

find

A

differential equations in 8. and 8.

( s

-

h

(4.2.2) a

set

and of

Eq.

two

(4.2.3)

homogeneous

with

A

/sx)+L(x,s),

(4.2.4)

where 2 V

Ux s)

-

the

partial

8 , which gives

I C* e x p ( - v . y s x ) + I D. e x p ( v J J j=1 J j=1 J

1

u > 2 3 )

-—2—2T7-2—277^—27^( V ] - v 2 ) ( v 2 - v 3 ) ( v 3 - v ] )s

[

2

x

- V

f

VT7sJ-/R(x's) * '

o

96

v

sinh

vJs(x-x')dx'*

2 3

2

"vl

*-

2

2

v

+

-V

— v

x

I

v?ys(x-x ' )dx'

x /

-2— I R ( x ' , s ) Js J o

J

R(x',s)sinh

0

sinh

v, 7 s ( x - x ' ) d x ' ] . 3

and

R(x s)

uTT^f,(x)

'

+ {

ur

+

l f ) s fi' (x)

p -ffv(x) The determined for

g

2

by

3 I,

the

C

j

(

C , D a r e t h e a r b i t r a r y c o n s t a n t s to be J ) b o u n d a r y and s y m m e t r y c o n d i t i o n s . T h e e x p r e s s i o n

by s u b s t i t u t i n g t h e v a l u e s of into E q .

'-VJ2)

f2(x) +

EKO j ^ - £ * V ( x ) . U P

coefficients

(4.2.4)

d b

3

p

♦ eKo f * v ( x )

6L i s o b t a i n e d

from E q .

esf (x)

^

( 4 . 2 . 1 ) . This

eXp(

"Vj

'SX)

,

Ffe

+

d

3 I , ■>*(1-v j

,

f (x) E-KoT l

— s —

f

9. and

9./dx

gives

)exp(Vjysx)

,

MCo"

L ( x

'

s )

e-Ko^L"(x's)(4.2.5)

On ( s 67 4.2.5),

%

substituting

f?(x))

from

the

Eq.

value

(4.2.3)

of

d 6,/dx

and

making

from use

of

Eq.

(4.2.2)and

Eqs.

(4.2.4

we o b t a i n

r k

C

I,

{

+

3

j

°j e x p ( - v . ; s x )

( x )

1-e

s

* —

E

^

I 5 Bu

f

[L(x,s)- i

. iv, , ~ ^x>s>

] ,

1

( x )

s

+

_i-

^

D*0j

exp(v.ysx)

Lu Ko , " , v I ,", . , 5 ( f , ( x ) - —— f . ( x ) ) Bu 2 I e Ko

L"(x,s)]

e

-|^

[(I*

1 -=2

£KoPn)L"(x,s)

(4.2.6)

97 The s y m b o l s o . and v are defined as J J (l-e)(l-v.2)-Lu

o.

v.2(l-v.2)-eKoPnLu

v.2

(4.2.7)

and Vj

where

7(yj

y

* | « ) ;

j

1,2,3;

(4.2.8)

are the r o o t s of the c u b i c

equation

3 y

11

+

Tt,y + 1T2

1

a

'

a2 1 "

+ B

( ,

-e)

+

0 ;

a

'

2 3 27 °

*2

+

U7

+ £ Ko

L ^

1 o 3aB

+

Pn

Y

'

P 6

(I-E)

j^-t Lu

(I

t EKo Pn ♦

)

T^-

Lu

r^— Lu P

and 1 Lu Lu

Y

' P

Equations hyperbolic A

3

6.

I

ft, 2

(4.2.4

4.2.6)

can

also

be

expressed

in

the

form 3

C. cosh

- i eKo

J >,

sinh

v. y s x

+

C ( l - v 2 ) cosh j j

v. y s x

I

D. sinh

v. / s x j

L(x,s) ♦ -pj^—

v. / s x

+ -4eKo

L"(x,s) EKos

| >,

t L(x,s),

D.()-v.2) j )

f l(xJe~l t ' j . s

2 2 Bu s

e Ko

n

- L"(X,S)] s

- ^ ebus

I

[(UeKoPn)L"(x,s)

(it.2.11)

The c o n d i t i o n s of symmetry

under t h e t r a n s f o r m

become

3 6.(0,s)

3x which

reduce ' I

A

6.(x,s)

the

C

J

j=' (1 - e) }

-g- sinh

cosh

/

J

sinh

* {a 1 v 3 " ° 3 v i

v^sx

+

v^/sx

2 2 2 2 v_ - o 2 v 3 " ' v 2 ~ v 3 )

l°2vi

"°|v2 " " " £ ^

"^v3 ~v]

v

1

ys sinh

I —-j—

r t- L ' ( 0 , s ) L { o ,

v /sx

1 v

° >

sinh

v3/s

v

xj- -

/sx

v,/sx 2

~v2 M

2

L{a,

+(0,(0,-0.)

+{a.(o, 1 I I

2

( o 2 - o 3 )+6 , ( v 2 - v 3 ) } *■

1 7- sinh v^/s

|

Hl-e)}

-o,) 2

+ ^i'v3

? 2 ~vi M *

2 2 t 6,(v. -v, ) } * 1 1 2 '

x

99

\

7 sinh v . V s x ] v./s 3

+ (v, -v.

j

)

2

+ —=-[(v_ 2 2

2

2

7—sinh v - / s x

1

v, /s

\

v, ) 7—sinh 3 v . / s

+ ( v , -v., )

1

I

v./sx 1

7—sinh

v,/ s

z

i

* L(x,s),

82(x,s)

(f.2.12)

I C (1-v 2 ) j=1 ' '

~

cosh v V s x '

+ - ^

e

v

v

£

v

v

" ° 2 3 - C - ) ( 2 " 3 )J

y s sinh

-('-

H

3

+ {c^Vj - o , v 2 - 0 - e ) ( v .

[

i

+ { a,(o3

" ,

)}

-v2 )}

v

/s

s i n n

I-V32 - ^s

2 2 '~v2 - a j ) + B , ( v 3 - V j ) } - ys I-V32 ) } - /

+ Bt C v1 -v2

v2/sx

sinh

(«,(V 0 3 )+ 6 1 ( V 2 2 - V 3 2 ) J 7 7 s -

+ {a)(a]-o2)

Vj/sx

2

,_v |V 3

+ {o2vi

v2/sx

V

lkL'(°'s)[{a3V22-°2v32

-

sinh v^/sx

cBu s

sinh

/sx

v(/sx

s i n h

+ Bj(v2 -v 3 ) } - ys

°2 -j—gr

v

,,", > 2 (X)

(f

v,/sx] 3

1 TOS

, ", f

l

+ s

.,

( X ) ) +

U"(x,s)]

- ^ - [(1+EKoPn)

s

e DUS

1-e. EBG

L"(X,S)

t

Lii*!] s (t.2.1f)

where

101

a, = j J - tfj(0)

e Ko f^O)

B, = jyp [ eBu f j ( 0 )

L'"

+ L'"(0,s)]

(l-E)fj(O)

( L U ( 1 + E KoPn) + ( 1-e))