- Email: [email protected]
An exact solution of the diffusion equation with shrinking
The Chemical
Engineering
Journal,
55 (1994)
135
135-138
An exact solution of the diffusion equation with shrinking Pascual E. Viollaz and Constantino Suarez Depatiamento (Argentina]
de Industrius,
Facultad
& Ciencias
Exactas
Ciudad
y Natmales,
Universitaria,
1428 Buenos Aires
(Received June 21, 1993; in final form November 6, 1993)
Abstract An exact solution for diffusion from a shrinking slab of finite thickness, assuming a constant mutual diffusion coefficient, has been obtained. This has been accomplished by first casting the appropriate differential equation into the form of an equation describing diffusion in a semi-infinite slab with a fixed domain of integration but with a quadratic concentration dependence of the diffusivity and then making use of Fyjita’s analytical solution for that equation. The present results are of potential interest in the fields of drying, dyeing, ion exchange and solid-liquid extraction.
1. Introduction
diffusant across
In the experimental investigation of diffusion of low molecular weight substances into high polymeric solids the diffusion coefficients are generally very strongly dependent upon the concentration of the diffusing substances [l-4]. In order to obtain solutions of the diiusion equations, Crank and Henry [ 5 1 have developed numerical methods. Later, F’ujita [6 ] was able to obtain analytical solutions of the diffusion equation when the diffusion coefficient varies with concentration in linear or quadratic form. It is the purpose of this work to obtain an exact solution of the diffusion equation following Fyjita’s method for the problem of diffusion from a shrinking slab of 6nite thickness. It is shown in the present work that if an appropriate reference frame is, adopted for the diffusion coefficient, an apparent quadratic functionality of the diffusion coefficients with concentration appears, even with a constant mutual diffusion coefficient.
F,=D;
2. The diffusion
is obtained where 0: is the diffusion coefficient for a reference frame fixed to the inert solid B. This coefficient is related to the mutual diffusion coefficient D by the equation [ 71
equation
for a shrinking
slab
In order to obtain the differential equation of a difhrsantA in a finite slab of solid B which undergoes shrinking during the process, we will follow the procedure developed by Crank [7]. The hypothesis of constant specific volume and volume additivity will also be used. With this hypothesis the flux of
5
a tied
section
of B is (1)
B
Here Cz is the concentration of component A per unit volume of B, which is related to the diffusant content on a dry basis U, by the expression c,” =up, where ps is the mass density of fully dried solid, and .& is the volume of dry solid per unit transversal area, defined as d&= @ dx PS
being the apparent density of solid B and x the coordinate along the diiusion path measured from the centre of the slab. By means of a differential mass balance the equation
pi
DE=D
0
2
f?
PS
It was demonstrated in a previous work [8] that eqn. (2) can be transformed as
0923-0467/94/$07.00 0 1994 Elsevier Science S.A. All rights reserved SSDI 0923-0467(93)02836-L
136
P.E. VioUaz, C. Suarez
/ Exact solution
a -_ pgau ax ps”az
au -=-
( 1
aF,
R.cJ.2 1
u=u 0
z=o
u=u. 1
z=l
au -= a.2
8
equation
with shrinJcing
( psui1
(12)
1+
PW
This equation together with the boundary conditions F,=O
of [email protected]
(4) (5)
0
R, and Ri being the initial and equilibrium halfthicknesses of the slab respectively. By means of eqns. (7)-(12), eqn. (3) can be written in the form
au*
-=_at
Da
1
R: ax
&&*
(~-uu*)~
a,2
where
has been used in a previous work [8] to model the drying process in a system that undergoes a change in volume. It is the purpose of this work to demonstrate that eqn. (3) and the corresponding boundary conditions can be transformed into the form of a diffusion equation for a finite slab without shrinkage but with a quadratic concentration dependence of the diffusivity. A solution of the transformed equation can be obtained by adopting the analytical solution developed by Fujita [6] for the unidimensional diffusion equation with a concentration-dependent diffusion coefficient.
u=l-
5
(14)
D
Rljita [6] was able to obtain an exact solution for the unidimensional diffusion equation with a variable diffusion coefficient. In particular, for a semi-infinite medium initially free from diffusing substance and its surface maintained at a constant concentration throughout the process, the mathematical expression used by Fyiita [6] was (15) To solve eqn. (15), the following used [S]:
3. Exact
solution
of the differential
equation
In order to transform eqn. (3), we define the dimensionless diffusant content as U*=
U,--U ?A,-Ui
(7)
The hypothesis of volume write the relation
additivity
&%+&=I Pw P.s
allows us to
(8)
Since the diffusant content on a dry basis can be expressed as a function of concentration per unit volume, we have u=
PA
(9)
PB Combining
C(z, 0) = 0,
o
C(O,t>= cm
t>o
conditions were
03
D(0) and A are arbitrary constants used by Fyiita [6] to define a variable diffusion coefficient with the concentration of diffusant of the form D(C) =D(O)l(l - hC)2. According to Fyjita [S], this concentration dependence of the diffusivity is commonly encountered in many polymer-solvent systems. By means of a similarity transformation, eqn. (15) together with the corresponding boundary conditions can be expressed as
(16)
c=o,
q=CU
(17)
c= 1,
q=o
(18)
Equation (13) can also be transformed by means of the following change in variables:
eqns. (8) and (9) gives -1
PB -= Ps
(13)
( ) 1+p,u
(19)
PW
The hypothesis of unidirectional shrinkage and volume additivity allows us to write
R,=R.(l+ p.u.) \ f% /
(111
In this way eqn. (13) and the corresponding boundary conditions become
(20)
P.E. Vi&a.z, C. Sum-ez / Exact soltiiun of d@ibon
u*=o,
7’03
(21)
u*=1,
q=o
(22)
equation with shrinking
From the definition of the dimensionless diffusant content u* it follows that (30)
The solution of eqn. (20) with boundary conditions (21) and (22) can now be obtained by adopting Fujita’s solution, i.e.
u*= .f7@i-3 a[1 -a+f(e,
_f(&P) = ~9 q= +a
(23)
p)]
(24)
exp(P)[ 1 - --f(W) I
(1 -a+f(0,
P)0-exp[P2(1
-e2>l}
(25)
where erf is the error function. Following the mathematical derivation, it is easily shown that 8= 1 and 8= CQcorrespond to 17= 0 and 77= 00 respectively. Thus the intermediate variable 8 varies from unity at the interface to infinity. From this fact and boundary condition (22) it follows that f(I, P>=o
(31) Since there is no explicit relation between u* and 7, the derivative must be evaluated in the form
au* au*/ae -=aqbe %
Ri prln exp(P2) erfc(p) = 1 - z
(27)
0
In this way the solution of the problem, i.e. the values of u* VS. 7, can be obtained from eqns. (23) and (25), 13being an intermediate parameter.
(32)
These derivatives can be obtained from eqns. (23) and (25) respectively, leading to $flae -&u* -l-u -
ae =
arl -=
+$
ae
(33)
(l-u+j)a
a
(
(26)
Using this condition in eqn. (24), the corresponding value of the constant p can be obtained from the equation
137
l-u,+0$
+2p26J exp[j32(1-82)]
1 (34)
Performing the ratio indicated in eqn. (32), evaluating the derivatives at the interface (0= 1) and taking into consideration that f(l) = a results in
au* -=arl
2@(1- a)2
(35)
a
Then, according to eqns. (29~(31) and (35), the loss of difhrsant per unit dry solid is (36)
4. Variation
in the mean moisture
content
Thus the total loss of diffusant per unit dry solid can be obtained by integrating eqn. (36) to give
with time
From eqn. (1) the flux of diffusant A with reference to a frame fixed to the solid B at the surface is
_ u”-u=
D 2(u,-u@(l-a)t’/2 aD’B E
(37)
This equation can be tinally simplified by using the definition of a and rearranging as where 8” is the flux of diffusant per unit crosssectional area and time. Dividing this expression by the weight of dry solid per unit cross-sectional area, R,p,, the flux of diffusant per unit time and dry weight can be calculated. By normalizing the coordinate & with the dry half-thickness R, and making use of the relation P&i =K PB~=R Ps
Gw
the following equation can be obtained:
(29)
ii-T.4
uo--ui =
’-
w (R,
-Rip
lnt In
(33)
where p is the positive root of eqn. (27). 5. Results and discussion Equation (38) was used to simulate the drying process of a semi-infinite slab for three initial moisture contents: 0.15, 1.0 and 3.0 g water per 1 g dry solid (g/g). The results are given in Fig. 1 together with the numerical integration of eqn. (3) for the same drying conditions. It is observed that
P.E. Viollaz, C. Suarez
/ Exact solution of dmion
equation with shrinking
5 J. Crank and M.E. Henry, Trans. Faraday Sot., 45 (1949) 1119. 6 H. Fujita, Textile Res. J., 22 (1952) 757. 7 J. Crank, Mathematics of D@jksion, Clarendon, Oxford, 1956, p. 224. 8 P.E. Viollaz and C. Suarez, AZChB J., 31 (1985) 1566. 9 H. Fuji& Textile Res. J., 22 (1952) 823.
3
k
=r 0.6-F
Appendix
A: Nomenclature
6
constant defined in eqn. (14) diffusing component (liquid) concentration of diffuser in Frjita’s solution concentration of component A per unit volume of dry solid B (g cmP3) mutual diffusion coefficient parameter in Fqjita’s solution diffusion coefficient for frame fixed to inert solid B defined in eqn. (24) flux of A for frame fixed to inert solid B Dt/Rz, Fourier number based on dry halfthickness of slab flux of A at surface of slab half-thickness of slab with equilibrium moisture content initial half-thickness of slab half-thickness of totally dried slab time moisture content of solid on a dry basis surface equilibrium moisture content of slab on a dry basis initial moisture content of slab on a dry basis mean moisture content on a dry basis (u, - u)l(uO - ui>, dimensionless moisture content coordinate along diffusion path &JR,, dimensionless coordinate along diffusion path 1 -y, dimensionless coordinate measured from surface of slab
0.2 1
0
05
Fl/Z
1.0
s
Fig. 1. Dimensionless diffusant content vs. Fourier number based on dry ha.Whickness of slab. Curves 1, 3 and 5 obtained from eqn. (38) for adsorption. Curves 2, 4 and 6 obtained from numerical solution of eqn. (3). Curves 1 and 2: u,= 0.15 g/g, ui=O.O g/g. Curves 3 and 4: u,=l.O g/g, w=O.O g/g. Curves 5 and 6: u,=3.0 g/g, u+= 0.0 g/g. True density of dry solids in all cases: ps= 1.2 g cme3. Density of water: h= 1.0 g cm?.
for relatively low Fourier numbers both solutions agree. Such agreement indicates that during a certain interval of drying the finite slab behaves as a semiinfinite solid. Once the perturbation reaches the centre of the slab, the analytical and numerical solutions differ. However, given that the diffusion coefficients in solids are in general very low, the analytical solution can be used to describe the diffusive process during relatively large intervals of time. It should be noted that the solution found here assumes a constant diffusion coefficient and that the results agree with Fyjita’s equation [6] where a quadratic concentration dependence of the diffusion coefficient was assumed, a fact that is observed in many experimental situations [91. It appears, however, that such a concentration dependence is fictitious and can be eliminated using an appropriate frame of reference. References J. Crank and G.S. Park, Trans. Faraduy Sot., 45 (1948) 240. G.S. Park, Tras. Faraday Sot., 46 (1950) 684. S. Prager, J. Cha. Phys., I9 (1951) 537. S. Prager and F.A. Long, J. Am. Chem. Sot., 73 (1951) 4072.
Greek
P
23 PA PEI Ps
Pw
letters
root of eqn. (24) for f(0, p) =a similarity variable in Fqjita’s solution volume of solid B per unit transversal area mass concentration of component A per unit volume mass concentration of component B per unit volume mass density of fully dried solid (skeletal density) mass density of diffusant
Engineering
Journal,
55 (1994)
135
135-138
An exact solution of the diffusion equation with shrinking Pascual E. Viollaz and Constantino Suarez Depatiamento (Argentina]
de Industrius,
Facultad
& Ciencias
Exactas
Ciudad
y Natmales,
Universitaria,
1428 Buenos Aires
(Received June 21, 1993; in final form November 6, 1993)
Abstract An exact solution for diffusion from a shrinking slab of finite thickness, assuming a constant mutual diffusion coefficient, has been obtained. This has been accomplished by first casting the appropriate differential equation into the form of an equation describing diffusion in a semi-infinite slab with a fixed domain of integration but with a quadratic concentration dependence of the diffusivity and then making use of Fyjita’s analytical solution for that equation. The present results are of potential interest in the fields of drying, dyeing, ion exchange and solid-liquid extraction.
1. Introduction
diffusant across
In the experimental investigation of diffusion of low molecular weight substances into high polymeric solids the diffusion coefficients are generally very strongly dependent upon the concentration of the diffusing substances [l-4]. In order to obtain solutions of the diiusion equations, Crank and Henry [ 5 1 have developed numerical methods. Later, F’ujita [6 ] was able to obtain analytical solutions of the diffusion equation when the diffusion coefficient varies with concentration in linear or quadratic form. It is the purpose of this work to obtain an exact solution of the diffusion equation following Fyjita’s method for the problem of diffusion from a shrinking slab of 6nite thickness. It is shown in the present work that if an appropriate reference frame is, adopted for the diffusion coefficient, an apparent quadratic functionality of the diffusion coefficients with concentration appears, even with a constant mutual diffusion coefficient.
F,=D;
2. The diffusion
is obtained where 0: is the diffusion coefficient for a reference frame fixed to the inert solid B. This coefficient is related to the mutual diffusion coefficient D by the equation [ 71
equation
for a shrinking
slab
In order to obtain the differential equation of a difhrsantA in a finite slab of solid B which undergoes shrinking during the process, we will follow the procedure developed by Crank [7]. The hypothesis of constant specific volume and volume additivity will also be used. With this hypothesis the flux of
5
a tied
section
of B is (1)
B
Here Cz is the concentration of component A per unit volume of B, which is related to the diffusant content on a dry basis U, by the expression c,” =up, where ps is the mass density of fully dried solid, and .& is the volume of dry solid per unit transversal area, defined as d&= @ dx PS
being the apparent density of solid B and x the coordinate along the diiusion path measured from the centre of the slab. By means of a differential mass balance the equation
pi
DE=D
0
2
f?
PS
It was demonstrated in a previous work [8] that eqn. (2) can be transformed as
0923-0467/94/$07.00 0 1994 Elsevier Science S.A. All rights reserved SSDI 0923-0467(93)02836-L
136
P.E. VioUaz, C. Suarez
/ Exact solution
a -_ pgau ax ps”az
au -=-
( 1
aF,
R.cJ.2 1
u=u 0
z=o
u=u. 1
z=l
au -= a.2
8
equation
with shrinJcing
( psui1
(12)
1+
PW
This equation together with the boundary conditions F,=O
of [email protected]
(4) (5)
0
R, and Ri being the initial and equilibrium halfthicknesses of the slab respectively. By means of eqns. (7)-(12), eqn. (3) can be written in the form
au*
-=_at
Da
1
R: ax
&&*
(~-uu*)~
a,2
where
has been used in a previous work [8] to model the drying process in a system that undergoes a change in volume. It is the purpose of this work to demonstrate that eqn. (3) and the corresponding boundary conditions can be transformed into the form of a diffusion equation for a finite slab without shrinkage but with a quadratic concentration dependence of the diffusivity. A solution of the transformed equation can be obtained by adopting the analytical solution developed by Fujita [6] for the unidimensional diffusion equation with a concentration-dependent diffusion coefficient.
u=l-
5
(14)
D
Rljita [6] was able to obtain an exact solution for the unidimensional diffusion equation with a variable diffusion coefficient. In particular, for a semi-infinite medium initially free from diffusing substance and its surface maintained at a constant concentration throughout the process, the mathematical expression used by Fyiita [6] was (15) To solve eqn. (15), the following used [S]:
3. Exact
solution
of the differential
equation
In order to transform eqn. (3), we define the dimensionless diffusant content as U*=
U,--U ?A,-Ui
(7)
The hypothesis of volume write the relation
additivity
&%+&=I Pw P.s
allows us to
(8)
Since the diffusant content on a dry basis can be expressed as a function of concentration per unit volume, we have u=
PA
(9)
PB Combining
C(z, 0) = 0,
o
C(O,t>= cm
t>o
conditions were
03
D(0) and A are arbitrary constants used by Fyiita [6] to define a variable diffusion coefficient with the concentration of diffusant of the form D(C) =D(O)l(l - hC)2. According to Fyjita [S], this concentration dependence of the diffusivity is commonly encountered in many polymer-solvent systems. By means of a similarity transformation, eqn. (15) together with the corresponding boundary conditions can be expressed as
(16)
c=o,
q=CU
(17)
c= 1,
q=o
(18)
Equation (13) can also be transformed by means of the following change in variables:
eqns. (8) and (9) gives -1
PB -= Ps
(13)
( ) 1+p,u
(19)
PW
The hypothesis of unidirectional shrinkage and volume additivity allows us to write
R,=R.(l+ p.u.) \ f% /
(111
In this way eqn. (13) and the corresponding boundary conditions become
(20)
P.E. Vi&a.z, C. Sum-ez / Exact soltiiun of d@ibon
u*=o,
7’03
(21)
u*=1,
q=o
(22)
equation with shrinking
From the definition of the dimensionless diffusant content u* it follows that (30)
The solution of eqn. (20) with boundary conditions (21) and (22) can now be obtained by adopting Fujita’s solution, i.e.
u*= .f7@i-3 a[1 -a+f(e,
_f(&P) = ~9 q= +a
(23)
p)]
(24)
exp(P)[ 1 - --f(W) I
(1 -a+f(0,
P)0-exp[P2(1
-e2>l}
(25)
where erf is the error function. Following the mathematical derivation, it is easily shown that 8= 1 and 8= CQcorrespond to 17= 0 and 77= 00 respectively. Thus the intermediate variable 8 varies from unity at the interface to infinity. From this fact and boundary condition (22) it follows that f(I, P>=o
(31) Since there is no explicit relation between u* and 7, the derivative must be evaluated in the form
au* au*/ae -=aqbe %
Ri prln exp(P2) erfc(p) = 1 - z
(27)
0
In this way the solution of the problem, i.e. the values of u* VS. 7, can be obtained from eqns. (23) and (25), 13being an intermediate parameter.
(32)
These derivatives can be obtained from eqns. (23) and (25) respectively, leading to $flae -&u* -l-u -
ae =
arl -=
+$
ae
(33)
(l-u+j)a
a
(
(26)
Using this condition in eqn. (24), the corresponding value of the constant p can be obtained from the equation
137
l-u,+0$
+2p26J exp[j32(1-82)]
1 (34)
Performing the ratio indicated in eqn. (32), evaluating the derivatives at the interface (0= 1) and taking into consideration that f(l) = a results in
au* -=arl
2@(1- a)2
(35)
a
Then, according to eqns. (29~(31) and (35), the loss of difhrsant per unit dry solid is (36)
4. Variation
in the mean moisture
content
Thus the total loss of diffusant per unit dry solid can be obtained by integrating eqn. (36) to give
with time
From eqn. (1) the flux of diffusant A with reference to a frame fixed to the solid B at the surface is
_ u”-u=
D 2(u,-u@(l-a)t’/2 aD’B E
(37)
This equation can be tinally simplified by using the definition of a and rearranging as where 8” is the flux of diffusant per unit crosssectional area and time. Dividing this expression by the weight of dry solid per unit cross-sectional area, R,p,, the flux of diffusant per unit time and dry weight can be calculated. By normalizing the coordinate & with the dry half-thickness R, and making use of the relation P&i =K PB~=R Ps
Gw
the following equation can be obtained:
(29)
ii-T.4
uo--ui =
’-
w (R,
-Rip
lnt In
(33)
where p is the positive root of eqn. (27). 5. Results and discussion Equation (38) was used to simulate the drying process of a semi-infinite slab for three initial moisture contents: 0.15, 1.0 and 3.0 g water per 1 g dry solid (g/g). The results are given in Fig. 1 together with the numerical integration of eqn. (3) for the same drying conditions. It is observed that
P.E. Viollaz, C. Suarez
/ Exact solution of dmion
equation with shrinking
5 J. Crank and M.E. Henry, Trans. Faraday Sot., 45 (1949) 1119. 6 H. Fujita, Textile Res. J., 22 (1952) 757. 7 J. Crank, Mathematics of D@jksion, Clarendon, Oxford, 1956, p. 224. 8 P.E. Viollaz and C. Suarez, AZChB J., 31 (1985) 1566. 9 H. Fuji& Textile Res. J., 22 (1952) 823.
3
k
=r 0.6-F
Appendix
A: Nomenclature
6
constant defined in eqn. (14) diffusing component (liquid) concentration of diffuser in Frjita’s solution concentration of component A per unit volume of dry solid B (g cmP3) mutual diffusion coefficient parameter in Fqjita’s solution diffusion coefficient for frame fixed to inert solid B defined in eqn. (24) flux of A for frame fixed to inert solid B Dt/Rz, Fourier number based on dry halfthickness of slab flux of A at surface of slab half-thickness of slab with equilibrium moisture content initial half-thickness of slab half-thickness of totally dried slab time moisture content of solid on a dry basis surface equilibrium moisture content of slab on a dry basis initial moisture content of slab on a dry basis mean moisture content on a dry basis (u, - u)l(uO - ui>, dimensionless moisture content coordinate along diffusion path &JR,, dimensionless coordinate along diffusion path 1 -y, dimensionless coordinate measured from surface of slab
0.2 1
0
05
Fl/Z
1.0
s
Fig. 1. Dimensionless diffusant content vs. Fourier number based on dry ha.Whickness of slab. Curves 1, 3 and 5 obtained from eqn. (38) for adsorption. Curves 2, 4 and 6 obtained from numerical solution of eqn. (3). Curves 1 and 2: u,= 0.15 g/g, ui=O.O g/g. Curves 3 and 4: u,=l.O g/g, w=O.O g/g. Curves 5 and 6: u,=3.0 g/g, u+= 0.0 g/g. True density of dry solids in all cases: ps= 1.2 g cme3. Density of water: h= 1.0 g cm?.
for relatively low Fourier numbers both solutions agree. Such agreement indicates that during a certain interval of drying the finite slab behaves as a semiinfinite solid. Once the perturbation reaches the centre of the slab, the analytical and numerical solutions differ. However, given that the diffusion coefficients in solids are in general very low, the analytical solution can be used to describe the diffusive process during relatively large intervals of time. It should be noted that the solution found here assumes a constant diffusion coefficient and that the results agree with Fyjita’s equation [6] where a quadratic concentration dependence of the diffusion coefficient was assumed, a fact that is observed in many experimental situations [91. It appears, however, that such a concentration dependence is fictitious and can be eliminated using an appropriate frame of reference. References J. Crank and G.S. Park, Trans. Faraduy Sot., 45 (1948) 240. G.S. Park, Tras. Faraday Sot., 46 (1950) 684. S. Prager, J. Cha. Phys., I9 (1951) 537. S. Prager and F.A. Long, J. Am. Chem. Sot., 73 (1951) 4072.
Greek
P
23 PA PEI Ps
Pw
letters
root of eqn. (24) for f(0, p) =a similarity variable in Fqjita’s solution volume of solid B per unit transversal area mass concentration of component A per unit volume mass concentration of component B per unit volume mass density of fully dried solid (skeletal density) mass density of diffusant