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Mass transfer into a laminar fluid stream from the moving interface of two immiscible fluids between parallel plates
The Chemical Engineering
Journal, 13 (1977)
179-l 83
@Elsevier Sequoia S.A., Lausanne. Printed in the Netherlands
Mass Transfer into a Laminar Fluid Stream from the Moving Interface of Two Immiscible Fluids between Parallel Plates H. HIKITA and K. ISHIMI Department
of Chemical Engineering,
University of Osaka Prefecture,
Sakai, Osaka (Japan)
(Received 10 August 1976)
Abstract
2
~=Ut2(3-21l+3(2-U) The effect of a moving interface on the mass transfer rate in one of two immiscible fluids in laminar cocurrent flow between parallel flat plates is studied theoretically. An analytical solution for the average Sherwood number is obtained in terms of the confluent hypergeometnk functions as a function of the Graetz number and the dimensionless interfacialfluid velocity.
INTRODUCTION
Various simple models have been suggested for describing mass transfer between immiscible fluid phases. However, in most of these models the influence of the fluid mechanics of one phase on that of the other phase has been neglected and the effect of the fluid velocity at the fluid-fluid interface on the mass transfer rate has been ignored. Recently, the effect of a moving interface was considered theoretically by Beek and Bakker’ and both theoretically and experimentally by Byers and King In the present work, the effect of a moving interface on the mass transfer in one of two immiscible fluids in laminar cocurrent flow between parallel flat plates is investigated theoretically.
a.
(1)
with U=
Ui/U,
(2)
where u is the fluid velocity, u, is the average fluid velocity, Ui is the interfacial fluid velocity, y is the distance from the interface, b is the thickness of the fluid layer and U is the dimensionless interfacial fluid velocity. In cocurrent flow of the two fluids, the value of the dimensionless interfacial fluid velocity U is always, positive. The flow pattern of the fluid depends on the value of U and Fig. 2 shows the velocity profiles for six different values. For U= 0 the flow of the fluid under consideration corresponds to single phase flow between two parallel plates. For U = 1.5 the flow of the fluid corresponds to thin film flow with a free surface where there is no shear stress. With U = 3 the velocity gradient of the fluid at the wall of the plate is equal to zero. Further, for U > 3 flow reversal occurs at some distance from the interface and circulation takes place in the fluid, i.e. the fluid near the interface flows in a positive direction along the z axis (in the flow direction), while the fluid near the plate wall flows in a negative direction
VELOCITY PROFILE
Consider two fluids flowing cocurrently between parallel flat plates. It is assumed that both fluid streams are laminar and that the flow is fully developed within a short distance from the point at which the two streams are brought into contact. The velocity profiles for this situation are shown in Fig. 1. In the present work, only one of the fluid phases (the righthand fluid phase in Fig. 1) is considered. The velocity profile for the fluid under consideration can be expressed as
0 f
Fig. 1. Flow model and coordinate system.
179
180
H. HIKITA,
u=o
a)
bl
Fig. 2. Velocity BASIC
U-O.5
profiles
EQUATIONS
c)
d)
U=l
~~1.5
of the fluid under consideration
FOR
MASS TRANSFER
e)
u=2
f)
K. ISHIMI
us
for various values of Ii.
ANALYSIS
Consider mass transfer from the interface to the fluid under consideration. It is assumed that there is no flow reversal in the fluid, i.e. U < 3, and that mass transfer takes place by molecular diffusion only in the direction perpendicular to the interface. Then the convective diffusion equation describing the diffusion of the solute can be written
ac
uz=D~
a*c ay
with the boundary
(3) conditions
z=O,OGy
c=c,
(4) (5)
z >o,
y=o
C= Ci
t>O,
y =b
acpy
=0
(6)
+, (6 - 3iY)t A,
where z is the distance in the flow direction, D is the diffusivity of the solute, C is the concentration of the solute, Ci is the interfacial solute concentration and C, is the average inlet solute concentration.
(S-$*)1
SOLUTION
x
(8)
Equation (3) and eqn. (1) with the boundary conditions (4)-(6) have been solved numerically on a computer by Byers and King’ for several values of Cr. In this paper, an analytical solution valid for all values of U is presented. The problem can be solved by the method of separation of the variables, and the solution for the concentration profile in the fluid is obtained as
The expansion
coefficients A, are defined by
(9) ~=~A,,F,,(f,&,)
exp(-Sk:) The eigenvalues A, are the roots of the equation (7) . ,
where F, are the eigenfunctions, X, are the eigenvalues and A, are the expansion coefficients. The eigenfunctions F,, can be expressed in terms of the confluent hypergeometric functions3 M(a, /3,x) as
wniaY),=b = 0
(10)
The average outlet solute concentration Ca , i.e. the cup-mixing concentration at the outlet z = 2 of the two parallel plates, can be obtained by multiplying both
MASS TRANSFER
BETWEEN IMMISCIBLE FLUIDS
181
sidesof eqn. (7) by (u/n,b)dv, by integrating from y = 0 toy = b and by letting z = Z. The resulting equation is
Sh = 2(UGz/n)t
(11) where B, are the average expansion
coefficients
given
by
(12)
y=o
and Gz is the Graetz number defined by Gz = u,b2/DZ
(13)
If the average mass transfer coefficient entire length of the plate is defined by u,b
C2 -- - C, (AC),,
kc = ?-
k, over the
(20)
If U is equal to zero, the flow under consideration corresponds to single phase laminar flow between two parallel plates. Therefore, when the value of Gz is very large the physical situation approaches the L&eque model for mass transfer from a stagnant surface into a fluid with a linear velocity profile, and the solution in terms of the average Sherwood number reduces to Sh = 1.47 Gzf
(21)
On the other hand, when the value of Gz is very small only the first term in the summation in eqn. (11) is significant and thus the average Sherwood number approaches the asymptotic value given by Sh = A:
(22)
(14) RESULTS AND DISCUSSION
with c2 Whm
=
In I(Ci
-
Cl
Ci )/CC
-
(15) C2)l
the average Sherwood number Sh can be expressed as
(16) and the substitution
of eqn. (11) into eqn. (16) gives
Sh = -Gz In 2 B, exp (-Xz/Gz) (-?I=1
(17) 1
When the value of Gz is very large, the concentration profile of the solute in the fluid is confined to a region near the interface, if U is not equal to zero. Therefore, in this case, eqn. (3) can be written as
aczDa2c
uiaz
(18)
ay2
with the boundary
conditions
z=O,O
C=C,
(4)
z>o,y
Cc Ci
(5)
z>o,y+=
=0
C=C1
(19)
Condition (19) indicates that the fluid can be considered to be infinitely deep. This situation corresponds to the Higbie penetration model and the solution of eqn. (18) in terms of the average Sherwood number is given by
The eigenvahres h,, the expansion coefficients A,, and the average expansion coefficients B, were calculated from eqns. (lo), (9) and (12), respectively, using eqn. (8) for six different values of U. The first six sets of values of X,,, A,, and B, for each value of U are given in Table 1. For U = 0, Butler and Plewes4 have obtained a series solution and have calculated the first two sets of values of X, and B, . Their values are in good agreement with those obtained in the present work. For U = 1 S, Olbrich and Wild’ have presented a series solution consisting of ten terms. Again, the values of A, and B, obtained by Olbrich and Wild are in excellent agreement with those obtained in the present work. Further, for U = 2 and U = 3, Galiullin and Semenov6 have presented two exact analytical solutions in terms of Bessel functions. For these special cases, the present analytical solution, eqn. (7) with eqn. (8), reduces to the exact solutions derived by Galiullin and Semenov. Figure 3 shows the computed results for the fractional degree of saturation (C, - Cr)/(Cr - Cr) as a function of the Graetz number Gz for the six values of U. The numerical solution obtained by Byers and King’ almost exactly matches the present solution for U=O.S, 1, 1.5 and 2. As can be seenin the figure, at the same value of Gz the fractional degree of saturation increases as the value of U increases. This indicates that, for a given system, if the average fluid velocity U, is kept constant the mass transfer rate increases with increasing interfacial fluid velocity Ui.
H. HIKITA, K. ISHIMI
182 TABLE 1 Values of h,, A,, and B, for various values of (i ______
CJ=O
1 2 3 4 5 6
1.5590 4.8570 8.1340 11.406 14.675 17.944
2.17654 1.47272 1.19360 1.06378 0.97689 0.91293
0.895561 0.060500 0.018041 0.008177 0.0045 36 0.002835
u= 0.5
1 2 3 4 5 6
1.6411 4.9436 8.2016 11.446 14.683 17.916
2.30596 1.76533 1.63357 1.57344 1.53957 1.51825
0.856263 0.072234 0.024285 0.012011 0.007141 0.004730
I 2 3 4 5 6
1.7363 5.0345 8.2876 11.529 14.766 18.001
2.47361 2.15319 2.09946 2.08049 2.07166 2.06684
0.8205 36 0.084950 0.030567 0.015652 0.009501 0.006379
II= 1.5
1 2 3 4 5 6
1.8478 5.1420 8.4163 11.686 14.955 18.223
2.69640 2.57148 2.55661 2.55201 2.55000 2.54892
0.789703 0.097255 0.036094 0.018686 0.011402 0.007676
u = ‘2.0
1 2 3 4 5 6
1.9796 5.2904 8.6171 11.947 15.278 18.609
3.00000 3.00000 3.00000 3.00000 3.00000 3.00000
0.765565 0.107187 0.040402 0.021019 0.012853 0.008663
u= 3.0
1 2 3 4 5 6
2.3167 5.9156 9.5355 13.160 16.785 20.411
4.00000 4.00000 4.00000 4.00000 4.00000 4.00000
0.745298 0.114304 0.043992 0.023098 0.014198 0.009601
II= 1.0
2 0.6 1
2
L
610
2
L
6
100
2
Fig. 3. Fractional degree of saturation as a function of GZ for various values of U.
-c
-- .-~
1
6 II "'0.6 1
I 2
Eq (221
-~--
Eq
(201
--~
Eq
121)
IiII L610
I 2
l/II 66100
2
Gt Fig. 4. Average Sherwood number as a function of Gz for various values of U.
Figure 4 shows the computed values of the average Sherwood number Sh as a function of the Graetz number Cz for the six values of U shown in Fig. 3. The solid lines represent the analytical solution given by eqn. (17) and the dotted lines show the asymptotic solution given by eqn. (22) for very small values of Gz. Further, the broken lines and the chaindotted line represent the asymptotic solutions given by eqns. (20) and (21) respectively, for very large values of Gz for U f 0 and for U = 0. It can be seen that the value of Sh increases with an increase in both Gz and U.
NOMENCLATURE
expansion coefficients in eqn. (9) average expansion coefficients in eqn. (12) b thickness of the fluid layer concentration of the solute C interfacial solute concentration ci average inlet solute concentration Cl average outlet solute concentration logarithmic concentration driving force &hm diffusivity of solute D eigenfunctions defined by eqn. (8) Frl Graetz number u, b2/QZ GZ average mass transfer coefficient kc M(Q,0, x) confluent hypergeometric function of argument x, parameters 01and 0 index of the series n Sh average Sherwood number k,b/D dimensionless interfacial fluid velocity Ui/U, CT
AtI 4
183
MASS TRANSFER BETWEEN IMMISCIBLE FLUIDS U
ui um Y
Z Z
fluid velocity interfacial fluid velocity average fluid velocity distance perpendicular to the interlace length of the plate distance in the flow direction
Greek symbols parameters a, P eigenvalues bl
REFERENCES 1 W. J. Beek and C. A. P. Bakker. Appl. . . Sci. Rex, Sect. A, 10 (1961) 241. 2 C. H. Bvers and C. J. Kine.A.Z.Ch.E. J.. 13 (19671 628. 3 M. Abramowitz and I. A.%tegun, HandbookofM&hematial Functions, National Bureau of Standards, Washington D.C., 1964. 4 R. M. Butler and A. C. Plewes, Chem. Eng. Prog. Symp. Ser. No. 10 (1954) 121. 5 W. E. Olbrich and J. D. Wild, Chem. Eng. Sci., 24 (1969) 25. 6 M. F. Galiullin and P. A. Semenov, Theor. Found. Chem. Eng. (USSR), 2 (1968) 143.
Journal, 13 (1977)
179-l 83
@Elsevier Sequoia S.A., Lausanne. Printed in the Netherlands
Mass Transfer into a Laminar Fluid Stream from the Moving Interface of Two Immiscible Fluids between Parallel Plates H. HIKITA and K. ISHIMI Department
of Chemical Engineering,
University of Osaka Prefecture,
Sakai, Osaka (Japan)
(Received 10 August 1976)
Abstract
2
~=Ut2(3-21l+3(2-U) The effect of a moving interface on the mass transfer rate in one of two immiscible fluids in laminar cocurrent flow between parallel flat plates is studied theoretically. An analytical solution for the average Sherwood number is obtained in terms of the confluent hypergeometnk functions as a function of the Graetz number and the dimensionless interfacialfluid velocity.
INTRODUCTION
Various simple models have been suggested for describing mass transfer between immiscible fluid phases. However, in most of these models the influence of the fluid mechanics of one phase on that of the other phase has been neglected and the effect of the fluid velocity at the fluid-fluid interface on the mass transfer rate has been ignored. Recently, the effect of a moving interface was considered theoretically by Beek and Bakker’ and both theoretically and experimentally by Byers and King In the present work, the effect of a moving interface on the mass transfer in one of two immiscible fluids in laminar cocurrent flow between parallel flat plates is investigated theoretically.
a.
(1)
with U=
Ui/U,
(2)
where u is the fluid velocity, u, is the average fluid velocity, Ui is the interfacial fluid velocity, y is the distance from the interface, b is the thickness of the fluid layer and U is the dimensionless interfacial fluid velocity. In cocurrent flow of the two fluids, the value of the dimensionless interfacial fluid velocity U is always, positive. The flow pattern of the fluid depends on the value of U and Fig. 2 shows the velocity profiles for six different values. For U= 0 the flow of the fluid under consideration corresponds to single phase flow between two parallel plates. For U = 1.5 the flow of the fluid corresponds to thin film flow with a free surface where there is no shear stress. With U = 3 the velocity gradient of the fluid at the wall of the plate is equal to zero. Further, for U > 3 flow reversal occurs at some distance from the interface and circulation takes place in the fluid, i.e. the fluid near the interface flows in a positive direction along the z axis (in the flow direction), while the fluid near the plate wall flows in a negative direction
VELOCITY PROFILE
Consider two fluids flowing cocurrently between parallel flat plates. It is assumed that both fluid streams are laminar and that the flow is fully developed within a short distance from the point at which the two streams are brought into contact. The velocity profiles for this situation are shown in Fig. 1. In the present work, only one of the fluid phases (the righthand fluid phase in Fig. 1) is considered. The velocity profile for the fluid under consideration can be expressed as
0 f
Fig. 1. Flow model and coordinate system.
179
180
H. HIKITA,
u=o
a)
bl
Fig. 2. Velocity BASIC
U-O.5
profiles
EQUATIONS
c)
d)
U=l
~~1.5
of the fluid under consideration
FOR
MASS TRANSFER
e)
u=2
f)
K. ISHIMI
us
for various values of Ii.
ANALYSIS
Consider mass transfer from the interface to the fluid under consideration. It is assumed that there is no flow reversal in the fluid, i.e. U < 3, and that mass transfer takes place by molecular diffusion only in the direction perpendicular to the interface. Then the convective diffusion equation describing the diffusion of the solute can be written
ac
uz=D~
a*c ay
with the boundary
(3) conditions
z=O,OGy
c=c,
(4) (5)
z >o,
y=o
C= Ci
t>O,
y =b
acpy
=0
(6)
+, (6 - 3iY)t A,
where z is the distance in the flow direction, D is the diffusivity of the solute, C is the concentration of the solute, Ci is the interfacial solute concentration and C, is the average inlet solute concentration.
(S-$*)1
SOLUTION
x
(8)
Equation (3) and eqn. (1) with the boundary conditions (4)-(6) have been solved numerically on a computer by Byers and King’ for several values of Cr. In this paper, an analytical solution valid for all values of U is presented. The problem can be solved by the method of separation of the variables, and the solution for the concentration profile in the fluid is obtained as
The expansion
coefficients A, are defined by
(9) ~=~A,,F,,(f,&,)
exp(-Sk:) The eigenvalues A, are the roots of the equation (7) . ,
where F, are the eigenfunctions, X, are the eigenvalues and A, are the expansion coefficients. The eigenfunctions F,, can be expressed in terms of the confluent hypergeometric functions3 M(a, /3,x) as
wniaY),=b = 0
(10)
The average outlet solute concentration Ca , i.e. the cup-mixing concentration at the outlet z = 2 of the two parallel plates, can be obtained by multiplying both
MASS TRANSFER
BETWEEN IMMISCIBLE FLUIDS
181
sidesof eqn. (7) by (u/n,b)dv, by integrating from y = 0 toy = b and by letting z = Z. The resulting equation is
Sh = 2(UGz/n)t
(11) where B, are the average expansion
coefficients
given
by
(12)
y=o
and Gz is the Graetz number defined by Gz = u,b2/DZ
(13)
If the average mass transfer coefficient entire length of the plate is defined by u,b
C2 -- - C, (AC),,
kc = ?-
k, over the
(20)
If U is equal to zero, the flow under consideration corresponds to single phase laminar flow between two parallel plates. Therefore, when the value of Gz is very large the physical situation approaches the L&eque model for mass transfer from a stagnant surface into a fluid with a linear velocity profile, and the solution in terms of the average Sherwood number reduces to Sh = 1.47 Gzf
(21)
On the other hand, when the value of Gz is very small only the first term in the summation in eqn. (11) is significant and thus the average Sherwood number approaches the asymptotic value given by Sh = A:
(22)
(14) RESULTS AND DISCUSSION
with c2 Whm
=
In I(Ci
-
Cl
Ci )/CC
-
(15) C2)l
the average Sherwood number Sh can be expressed as
(16) and the substitution
of eqn. (11) into eqn. (16) gives
Sh = -Gz In 2 B, exp (-Xz/Gz) (-?I=1
(17) 1
When the value of Gz is very large, the concentration profile of the solute in the fluid is confined to a region near the interface, if U is not equal to zero. Therefore, in this case, eqn. (3) can be written as
aczDa2c
uiaz
(18)
ay2
with the boundary
conditions
z=O,O
C=C,
(4)
z>o,y
Cc Ci
(5)
z>o,y+=
=0
C=C1
(19)
Condition (19) indicates that the fluid can be considered to be infinitely deep. This situation corresponds to the Higbie penetration model and the solution of eqn. (18) in terms of the average Sherwood number is given by
The eigenvahres h,, the expansion coefficients A,, and the average expansion coefficients B, were calculated from eqns. (lo), (9) and (12), respectively, using eqn. (8) for six different values of U. The first six sets of values of X,,, A,, and B, for each value of U are given in Table 1. For U = 0, Butler and Plewes4 have obtained a series solution and have calculated the first two sets of values of X, and B, . Their values are in good agreement with those obtained in the present work. For U = 1 S, Olbrich and Wild’ have presented a series solution consisting of ten terms. Again, the values of A, and B, obtained by Olbrich and Wild are in excellent agreement with those obtained in the present work. Further, for U = 2 and U = 3, Galiullin and Semenov6 have presented two exact analytical solutions in terms of Bessel functions. For these special cases, the present analytical solution, eqn. (7) with eqn. (8), reduces to the exact solutions derived by Galiullin and Semenov. Figure 3 shows the computed results for the fractional degree of saturation (C, - Cr)/(Cr - Cr) as a function of the Graetz number Gz for the six values of U. The numerical solution obtained by Byers and King’ almost exactly matches the present solution for U=O.S, 1, 1.5 and 2. As can be seenin the figure, at the same value of Gz the fractional degree of saturation increases as the value of U increases. This indicates that, for a given system, if the average fluid velocity U, is kept constant the mass transfer rate increases with increasing interfacial fluid velocity Ui.
H. HIKITA, K. ISHIMI
182 TABLE 1 Values of h,, A,, and B, for various values of (i ______
CJ=O
1 2 3 4 5 6
1.5590 4.8570 8.1340 11.406 14.675 17.944
2.17654 1.47272 1.19360 1.06378 0.97689 0.91293
0.895561 0.060500 0.018041 0.008177 0.0045 36 0.002835
u= 0.5
1 2 3 4 5 6
1.6411 4.9436 8.2016 11.446 14.683 17.916
2.30596 1.76533 1.63357 1.57344 1.53957 1.51825
0.856263 0.072234 0.024285 0.012011 0.007141 0.004730
I 2 3 4 5 6
1.7363 5.0345 8.2876 11.529 14.766 18.001
2.47361 2.15319 2.09946 2.08049 2.07166 2.06684
0.8205 36 0.084950 0.030567 0.015652 0.009501 0.006379
II= 1.5
1 2 3 4 5 6
1.8478 5.1420 8.4163 11.686 14.955 18.223
2.69640 2.57148 2.55661 2.55201 2.55000 2.54892
0.789703 0.097255 0.036094 0.018686 0.011402 0.007676
u = ‘2.0
1 2 3 4 5 6
1.9796 5.2904 8.6171 11.947 15.278 18.609
3.00000 3.00000 3.00000 3.00000 3.00000 3.00000
0.765565 0.107187 0.040402 0.021019 0.012853 0.008663
u= 3.0
1 2 3 4 5 6
2.3167 5.9156 9.5355 13.160 16.785 20.411
4.00000 4.00000 4.00000 4.00000 4.00000 4.00000
0.745298 0.114304 0.043992 0.023098 0.014198 0.009601
II= 1.0
2 0.6 1
2
L
610
2
L
6
100
2
Fig. 3. Fractional degree of saturation as a function of GZ for various values of U.
-c
-- .-~
1
6 II "'0.6 1
I 2
Eq (221
-~--
Eq
(201
--~
Eq
121)
IiII L610
I 2
l/II 66100
2
Gt Fig. 4. Average Sherwood number as a function of Gz for various values of U.
Figure 4 shows the computed values of the average Sherwood number Sh as a function of the Graetz number Cz for the six values of U shown in Fig. 3. The solid lines represent the analytical solution given by eqn. (17) and the dotted lines show the asymptotic solution given by eqn. (22) for very small values of Gz. Further, the broken lines and the chaindotted line represent the asymptotic solutions given by eqns. (20) and (21) respectively, for very large values of Gz for U f 0 and for U = 0. It can be seen that the value of Sh increases with an increase in both Gz and U.
NOMENCLATURE
expansion coefficients in eqn. (9) average expansion coefficients in eqn. (12) b thickness of the fluid layer concentration of the solute C interfacial solute concentration ci average inlet solute concentration Cl average outlet solute concentration logarithmic concentration driving force &hm diffusivity of solute D eigenfunctions defined by eqn. (8) Frl Graetz number u, b2/QZ GZ average mass transfer coefficient kc M(Q,0, x) confluent hypergeometric function of argument x, parameters 01and 0 index of the series n Sh average Sherwood number k,b/D dimensionless interfacial fluid velocity Ui/U, CT
AtI 4
183
MASS TRANSFER BETWEEN IMMISCIBLE FLUIDS U
ui um Y
Z Z
fluid velocity interfacial fluid velocity average fluid velocity distance perpendicular to the interlace length of the plate distance in the flow direction
Greek symbols parameters a, P eigenvalues bl
REFERENCES 1 W. J. Beek and C. A. P. Bakker. Appl. . . Sci. Rex, Sect. A, 10 (1961) 241. 2 C. H. Bvers and C. J. Kine.A.Z.Ch.E. J.. 13 (19671 628. 3 M. Abramowitz and I. A.%tegun, HandbookofM&hematial Functions, National Bureau of Standards, Washington D.C., 1964. 4 R. M. Butler and A. C. Plewes, Chem. Eng. Prog. Symp. Ser. No. 10 (1954) 121. 5 W. E. Olbrich and J. D. Wild, Chem. Eng. Sci., 24 (1969) 25. 6 M. F. Galiullin and P. A. Semenov, Theor. Found. Chem. Eng. (USSR), 2 (1968) 143.