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GENERAL RESEARCH Analytical Study of Multipass Laminar Counterflow Mass Exchangers through a Parallel-Plate Channel C. D. Ho* and J. W. Tu Department of Chemical Engineering, Tamkang University, Tamsui, Taipei, Taiwan 251, Republic of China
A multipass mass exchanger through a parallel-plate channel by inserting three permeable barriers with uniform wall concentration was designed and the mathematical formulation was investigated for solving such a conjugated Graetz problem by using an orthogonal expansion technique. Considerable improvement in the mass-transfer efficiency is obtainable by such multipass operations with suitable adjustment of the permeable-barrier locations. Comparisons of analytical results have shown that the multipass device can enhance the mass-transfer efficiency compared with those of the single-pass (without permeable barrier inserted) and doublepass operations of the same size in the previous work. The ratio of the improvement of transfer efficiency to the increment of power consumption has been determined to make good economic sense. 1. Introduction Laminar forced convection of heat transfer in a bounded conduit at steady state with negligible axial conduction is well-known as the Graetz problem.1-4 Axial conduction in liquid metals with a small axial Peclet number can have a significant effect through extension of the influences in the so-called extended Graetz problems.5-11 The larger the Peclet number, the smaller the heat transfer by conduction in the flow direction. The application of recycle effect is widely used in many separation processes and chemical reactors, such as distillation,12 extraction,13 adsorption,14 mass diffusion,15 thermal diffusion,16 loop reactors,17 air-lift reactor,18 and draft-tube bubble column.19 In these processes, the effects of recycle play an important role in the heat- and mass-transfer mechanism and in turn enhance improvement in device performance. However, the interaction between streams or phases of multistream or multiphase problems, as in conjugated Graetz problems,20-25 presents additional difficulties in their solution. A mathematical formulation has been developed to investigate the transfer efficiency of multipass effects on separation processes26,27 as well as on heat and mass exchangers.28-30 Catalytic reactors,31 filters,32 gas scrubbers,33 and gas absorbers34 are designed and widely used in industrial processes. The previous work29,30 was designed to be a doublepass mass exchanger while the present paper is extended to four-pass devices. The analytical solution of the multipass operation follows the double-pass operation very closely except for the additional two governing equations and one extra subchannel thickness ratio. The purpose of this work was to study the improvement in * To whom correspondence should be addressed. Fax: 8862-26209887. E-mail:
[email protected].
mass-transfer efficiency and develop a mathematical formulation for a multipass mass exchanger by inserting three permeable barriers. The device performance with barrier position and mass Graetz number is also discussed. The same procedure certainly occurs in dealing with many further boundary conditions, especially for mutual conditions at the boundaries in two or more contiguous streams and phases of multistream or multiphase problems. This is the value of the present method, easy application in solving other conjugated Graetz problems of heat- and mass-transfer devices. 2. Mathematical Statements 2.1. Concentration Distribution in a Multipass Device. Consider the mass transfer in four subchannels with thickness Wa, Wb, Wc, and Wd, respectively, and the ratio of channel thickness is defined as β ) Wa/Wb ) Wd/Wc, which is obtained by inserting three permeable barriers with negligible thickness δ (,W) into a parallel conduit of thickness W, length L, and width B (.W), as shown in Figure 1. The inlet fluid with volume flow rate V and the inlet concentration CI enters two outer subchannels (flow pattern A) and then exits from the inner subchannles for a four-pass operation accomplished with the aid of a conventional pump situated at the end of the channel, as shown in Figure 1a. The inlet fluid may flow through two inner subchannels first and then turn back to flow through the outer subchannles (flow pattern B), as shown in Figure 1b. In each flow pattern, the fluid is assumed to be well-mixed at the inlet and the outlet of each subchannel. The following analysis assumptions were made to obtain the theoretical formulation: (a) constant physical properties and wall concentrations of the two outer channels; (b) steady state and purely fully developed laminar flow in each channel, (c) negligible entrance length and end effects as well as the concentration
10.1021/ie0207461 CCC: $25.00 © 2003 American Chemical Society Published on Web 06/10/2003
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3471
polarization effect on the permeable barrier; (d) only diffuse mass transfer through the permeable barrier and negligible axial diffusion. After the following dimensionless variables are introduced:
xi , Wi
z ξ) , L 2V(Wa + Wb + Wc + Wd) 2VW Gzm ) ) , DBL DBL Ci - Cs Ci - C I W , ψi ) , θ ) 1 - ψi ) , γ) δ CI - Cs i Cs - CI i ) a, b, c, d (1)
ηi )
the velocity distributions and equations of mass transfer in dimensionless form may be written as follows:
∂2ψi(ηi,ξ) ∂ηi2
[
]
Wi2vi(ηi) ∂ψi(ηi,ξ) ) i ) a,b,c,d LD ∂ξ
(2)
The analytical solutions to both flow patterns may be obtained by the use of an orthogonal expansion technique with the eigenfunction expanding in terms of an extended power series. Separation of variables in the form ∞
ψi(ηi,ξ) )
∑ Si,mFi,m(ηi)Gm(ξ)
applied to eqs 2 and 3 lead to
Gm(ξ) ) e-λm(1-ξ) F ′′i,m(η) -
[
]
λmWi2vi(ηi) Fi,m(η) ) 0 i ) a,b,c,d LD
Fa,m(0) ) 0
vj a )
V -V -V , vj b ) , vj c ) , BWa BWb BWc V and vj d ) for flow pattern A (4) BWd -V V V , vj ) , vj ) , BWa b BWb c BWc -V and vj d ) for flow pattern B (5) BWd
Sa,mF ′a,m(1) ) -
Sa,mF ′a,m(1) ) -
(6)
∂ψa(1,ξ) Wa ∂ψb(1,ξ) )∂ηa Wb ∂ηb
(7)
γWa ∂ψa(1,ξ) [ψa(1,ξ) - ψb(1,ξ)] )∂ηa W
(8)
Wb ∂ψc(0,ξ) ∂ψb(0,ξ) )∂ηb Wc ∂ηc
(9)
γWb ∂ψb(0,ξ) )[ψb(0,ξ) - ψc(0,ξ)] ∂ηb W
(10)
Wc ∂ψd(1,ξ) ∂ψc(1,ξ) )∂ηc Wd ∂ηd
(11)
γWc ∂ψc(1,ξ) [ψ (1,ξ) - ψd(1,ξ)] )∂ηc W d
(12)
ψd(0,ξ) ) 0
(13)
Sb,mF ′b,m(0) ) -
θF ) (θF,ab + θF,cd)/2 ) 1 - (ψF,ab + ψF,cd)/2 ) (CF,ab + CF,cd)/2 - CI (14) Cs - CI
Wa S F ′ (1) Wb b,m b,m
Wb S F ′ (0) Wc c,m c,m
γWb [S F (0) - Sc,mFc,m(0)] W b,m b,m
Sc,mF ′c,m(1) ) -
Sc,mF ′c,m(1) ) -
and the dimensionless outlet concentration is
(17)
(18) (19)
γWa [S F (1) - Sb,mFb,m(1)] (20) W a,m a,m
Sb,mF ′b,m(0) ) -
The boundary conditions for solving eq 2 are
ψa(0,ξ) ) 0
(16)
and also the boundary conditions in eqs 6-13 can be rewritten as
vi(ηi) ) vj i(6ηi - 6ηi2) 0 e ηi e 1 i ) a,b,c,d (3) vj a )
i ) a,b,c,d (15)
m)0
Wc S F ′ (1) Wd d,m d,m
γWc [S F (1) - Sd,mFd,m(1)] W c,m c,m Fd,m(0) ) 0
(21)
(22)
(23)
(24) (25)
where the primes on Fa,m(ηa), Fb,m(ηb), Fc,m(ηc), and Fd,m(ηd) denote the differentiations with respect to ηa, ηb, ηc, and ηd, respectively. Combinations of eqs 19 and 20, eqs 21 and 22, and eqs 23 and 24 give eqs 26, 27, and 28, respectively,
γWaFb,m(1) Sa,m WaF ′b,m(1) ) )Sb,m γWaFa,m(1) + WF ′a,m(1) WbF ′a,m(1)
(26)
γWbFc,m(0) WbF ′c,m(0) Sb,m ) )Sc,m γWbFb,m(0) + WF ′b,m(0) WcF ′b,m(0)
(27)
γWcFd,m(1) WcF ′d,m(1) Sc,m ) )Sd,m γWcFc,m(1) + WF ′c,m(1) WdF ′c,m(1)
(28)
in which the eigenfunctions Fa,m(ηa), Fb,m(ηb), Fc,m(ηc), and Fd,m(ηd) were assumed to be polynomials to avoid the loss of generality. With the use of eqs 18 and 25,
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Figure 1. Multipass parallel-plate mass exchanger.
we have ∞
Fa,m(ηa) )
pm,nηna , ∑ n)0
pm,0 ) 0, pm,1 ) 1 (selected) (29)
the associated eigenfunctions (Fa,m, Fb,m, Fc,m, and Fd,m) are found, the dimensionless outlet concentration ψF,ab or ψF,cd, referred to as the bulk concentration for flow pattern A (or flow pattern B), may be calculated by
∞
Fb,m(ηb) )
∑
qm,nηnb ,
qm,0 ) 1 (selected) (30)
n)0
ψF,ab )
Fc,m(ηc) )
∑
rm,0 ) 1 (selected)
e-λmSb,m {F ′b,m(1) - F ′b,m(0)} (33) WbGzmm)0 λm -2W
(31)
n)0 ∞
Fd,m(ηd) )
tm,nηnd, ∑ n)0
tm,0 ) 0, tm,1 ) 1 (selected)
∞
∑
or
(32)
Substituting eqs 29-32 into eq 17, all the coefficients pm,n, qm,n, rm,n, and tm,n may be expressed in terms of eigenvalues λm after using eqs 18 and 25, as referred to in the Appendix. Once all of the eigenvalues (λm), the expansion coefficients, (Sa,m, Sb,m, Sc,m, and Sd,m) and
)
V
∞
rm,nηnc ,
∫01 vbWbBψb(ηb,0) dηb
∫01 vcWcBψc(ηc,0) dηc
ψF,cd )
)
V e-λmSc,m {F ′c,m(1) - F ′c,m(0)} (34) WcGzmm)0 λm -2W
∞
∑
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3473
Figure 2. Single- and double-pass parallel-plate mass exchanger.
or may be examined by eq 37, which is readily obtained from the following overall mass balance on the outer walls,
V(1 - ψF,ab) + V(1 - ψF,cd) )
∂ψa(0,ξ) dξ + ∂ηa 1 DBL ∂ψd(0,ξ) dξ (35) 0 W ∂ηd d
∫01 DBL Wa
∫ thus,
(1 - ψF,ab) + (1 - ψF,cd) ) 2
[∑
Gzm
(1 - e-λm)W
∞
Sa,mF ′a,m(0)
m)0
]
∞
(1 - e-λm)W
m)0
λmWd
∑ Sd,mF ′d,m(0)
or
+
λmWa
(36)
ψF,ab + ψF,cd )
[∑
(1 - e-λm)W Sa,mF′a,m(0) + 2Gzm m)0 λmWa 2
∞
]
(1 - e-λm)W Sd,mF′d,m(0) (37) λmWd m)0 ∞
∑
2.2. Concentration Distributions in Single- and Double-Pass Devices. Single- and double-pass devices with the same working dimension as that of the multipass devices are presented in parts (a) and (b), respectively, of Figure 2. The calculation procedure for a single- and double-pass device is rather simpler than that for multipass devices. The outlet concentrations for double-pass (θF) and single-pass devices (θ0,F) were obtained in terms of the mass-transfer Graetz number (Gzm), eigenvalues (λm and λ0,m), expansion coefficients (Sa,m, Sb,m, and S0,m), and eigenfunctions (Fa,m(ηa), Fb,m(ηb), and F0,m(η0)). The results are
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θF ) 1 - ψF )
[∑
(1 - e-λm) W
∞
1 Gzm
λm
m)0 ∞
∑
(1 - e-λm) W
m)0
θ0,F ) 1 - ψ0,F )
1
Sa,mF ′a,m(0) + Wa
∞
∑
Gzmm)0
[
λm
]
Sb,mF ′b,m(0) (38) Wb
(1 - e-λ0,m) S0,mF ′0,m(0) λ0,m
]
(1 - e-λ0,m) S0,mF ′0,m(1) (39) λ0,m
3. Improvement in Mass-Transfer Efficiency By following the same definition in the previous work,29 the average Sherwood number for double- and multipass devices by inserting three permeable barriers may be obtained as follows:
kmW 2VW ) (1 - ψF) ) 0.5Gzm(1 - ψF) ) D 2DBL 0.5GzmθF (40)
Sh )
Similarly, for the single-pass device,
km,0W 2VW ) (1 - ψ0,F) ) 0.5Gzm(1 - ψ0,F) ) Sh0 ) D 2DBL 0.5Gzmθ0,F (41) The improvement in performance by employing double- and multipass devices are best illustrated by calculating the percentage increase in mass-transfer rate, based on the mass transfer of a single-pass operation without recycle of the same mass-transfer Graetz number as
Im )
(Sh - Sh0) Sh0
× 100% ) (1 - ψF) - (1 - ψ0,F) (1 - ψ0,F)
× 100% (42)
4. Results and Discussion The calculation procedure is similar to those performed in the previous work;29,30 the results are discussed as follows. 4.1. Flow Pattern A. 4.1.1. Outlet Concentration and Mass-Transfer Efficiency in Multipass Devices. Table 1 shows calculation results for the first two eigenvalues with associated expansion coefficients and the dimensionless outlet concentration for βab ) βcd ) 1, γ ) W/δ ) 5 and Gzm ) 1, 10, 100, and 1000 for flow
Figure 3. Dimensionless outlet concentration vs Gzm with βab and βcd as parameters for γ ) 1 (flow pattern A).
pattern A. It is experiential that only the first negative eigenvalue is taken into account during the concentration distribution calculation due to the rapid convergence. Comparisons of the dimensionless outlet concentrations, θF and θ0,F, and average Sherwood numbers, Sh and Sh0, are shown in Figures 3-6. Sherwood numbers and the mass-transfer efficiency improvement are proportional to θF (θ0,F), as shown in eqs 40-42. Figures 3 and 4 show the relation of another more practical form of dimensionless outlet concentrations and average Sherwood numbers vs mass Graetz numbers with the subchannel thickness ratio as a parameter for γ ) 1 while Figures 5 and 6 with the design parameter γ as a parameter for βab ) βcd ) 1. It is found that in Figures 3 and 5 that, for a specified γ, the dimensionless average outlet concentration decreases with increasing mass-transfer Graetz number Gzm as well as channel thickness ratio βab and βcd, and with increasing γ for large mass Graetz numbers, say, Gzm > 50. 4.1.2. Improvement in Mass-Transfer Efficiency Based on Single-Pass Devices. Figures 4 and 6 give a graphical representation of average Sherwood numbers, Sh and Sh0, as well as show that all Sh and Sh0 values increase with mass-transfer Graetz numbers Gzm, owing to increment of convective transfer coef-
Table 1. Eigenvalues and Expansion Coefficients as Well as Dimensionless Outlet Concentrations in Multipass Devices for βab ) βcd ) 1 and γ ) 5; Gzmλ1 ) -27.284 and Gzmλ2 ) -27.552 (Flow Pattern A) Gzm 1 10 100 1000
m 0 1 0 1 0 1 0 1
λm -27.2839 -27.5517 -2.7284 -2.7552 -0.2728 -0.2755 -0.0273 -0.0276
Sa,m 10-12
2.1 × 4.5 × 10-28 0.0995 2.9 × 10-17 1.6830 4.7 × 10-17 2.4939 3.2v10-16
Sb,m 10-13
5.3 × 6.3 × 10-43 0.0251 7.6 × 10-32 0.4249 -2.4 × 10-31 0.6296 3.3 × 10-30
Sc,m 10-13
5.3 × -1.3 × 10-43 0.0251 -9.7 × 10-33 0.4249 1.3 × 10-31 0.6296 -1.1 × 10-30
Sd,m 10-12
2.1 × 1.8 × 10-28 0.0995 -7.4 × 10-18 1.6830 4.7 × 10-17 2.4939 3.2 × 10-16
ψF (λ1)
ψF (λ1,λ2)
0.5659
0.5659
0.5824
0.5824
0.8452
0.8452
0.9798
0.9798
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3475
Figure 4. Average Sherwood number vs Gzm with βab and βcd as parameters for γ ) 1 (flow pattern A).
Figure 5. Dimensionless outlet concentration vs Gzm with γ as a parameter for βab ) βcd ) 1 (flow pattern A).
ficient for large V. They also increase with a decreasing channel thickness ratio βab and βcd. As shown in Figures 4 and 6, the difference (Sh - Sh0) of average Sherwood numbers is negative for small Gzm values and then turns positive with increasing Gzm, but the increment with Gzm is limited as Gzm approaches infinity with any values of γ, βab, and βcd. The mass-transfer improvement, Im, in the multipass operations are shown in Table
Figure 6. Average Sherwood number vs Gzm with γ as a parameter for βab ) βcd ) 1 (flow pattern A).
2 with the mass-transfer Graetz number and channel thickness ratio βab and βcd as parameters. The negative signs in Tables 2 and 3 indicate that the single-pass device without external refluxes is recommended rather than multipass devices with three permeable barriers under such circumstances. It is noted that the masstransfer improvement, Im, increases with increasing the mass-transfer Graetz number but with decreasing the channel thickness ratio βab and βcd. 4.2. Flow Pattern B. After following a calculation procedure similar to the one in the previous section of flow pattern A, the dimensionless average outlet concentration, average Sherwood numbers, and masstransfer improvement with mass-transfer Graetz number and channel thickness ratio as parameters were shown in Figures 7-10 and Table 3 for flow pattern B. Figures 7 and 9 show that the differences (θF - θ0,F) and (Sh - Sh0) of dimensionless outlet concentration and average Sherwood numbers, respectively, are negative for small mass-transfer Graetz numbers and then revert to a position number with any value of βab and βcd for Gzm > 8, and these two differences increase with decreasing channel thickness ratio. Therefore, the multipass devices with three permeable barriers with smaller βab and βcd values are recommended for large Gzm and γ. Some numerical values of the mass-transfer improvement Im were given in Table 3. The masstransfer improvement can be achieved with any masstransfer Graetz numbers and channel thickness ratio for a given design parameter γ ) 5. 4.3. Increment of Power Consumption. It is wellknown that in the design of a multipass mass exchanger, the criterion of device performance of interest is not the transfer efficiency improvement Im alone, but the ratio of mass-transfer efficiency improvement to the power consumption increment, which is presented here in the form of Im/Ip. The power consumption increment Ip due to the friction losses lwf,a, lwf,b, lwf,c, and lwf,d for
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Table 2. Improvement of the Mass-Transfer Efficiency with the Channel Thickness Ratio as a Parameter γ ) 5 (Flow Pattern A) Im (%)
βab ) βcd ) 1/3
βab ) βcd ) 2/3
βab ) βcd ) 1
βab ) βcd ) 3/2
βab ) βcd ) 3
βab ) 1/3, βcd ) 1
Gzm ) 1 Gzm ) 10 Gzm ) 100 Gzm ) 1000
-51.90 -12.81 214.72 363.87
-55.68 -20.85 133.61 208.53
-56.57 -23.72 100 151.57
-57.42 -26.55 75.3 113.1
-22.81 -10.62 28.08 37.3
-45.83 -4.07 186.28 287.15
Table 3. Improvement of the Mass-Transfer Efficiency with the Channel Thickness Ratio as a Parameter for γ ) 5 (Flow Pattern B) Im (%)
βab ) βcd ) 1/3
βab)βcd ) 2/3
βab ) βcd ) 1
βab ) βcd ) 3/2
βab ) βcd ) 3
βab ) 1/3, βcd ) 1
Gzm ) 1 Gzm ) 10 Gzm ) 100 Gzm ) 1000
0.05 79.94 327.07 383.34
0.05 73.86 202.33 218.46
0.05 67.42 150.57 158.38
0.05 60.59 114.58 118.14
0.05 50.47 77.25 77.63
0.05 72.78 252.42 297.64
Figure 7. Dimensionless outlet concentration vs Gzm with βab and βcd as parameters for γ ) 1 (flow pattern B).
multipass operations while lwf,0 for single-pass operations with the assumption of laminar flow in the flow channels can be readily derived as29
Ip ) )
P - P0 (lwf,a + lwf,b + lwf,c + lwf,d) - (2lwf,0) ) P0 2lwf,0 (43)
( ) ( ) ( ) ( )
1 W 4 Wa
3
+
1 W 4 Wb
3
+
1 W 4 Wc
3
+
1 W 4 Wd
3
-1
(44)
where P ) VF[lwf,a + lwf,b + lwf,c + lwf,d] for flow patterns A and B. The power consumption of a singlepass device will be illustrated using working dimensions as follows: L ) 1.2 m, W ) 0.04 m, B ) 0.2 m, V ) 1 × 10-5 m3/s, µ ) 8.94 × 10-4 kg/m‚s, and F ) 997.08 kg/ m3. From these numerical values, the friction loss and hydraulic dissipated energy P0 of a single-pass device was calculated as follows:
P0 ) 2VF(lwf,0) ) 1.073 × 10-4 W ) 1.44 × 10-7 hp (45)
Figure 8. Average Sherwood number vs Gzm with βab and βcd as parameters for γ ) 1 (flow pattern B).
Some results for Ip are presented in Table 4 for flow patterns A and B. It is seen from this table that the power consumption increment does not depend on masstransfer Graetz numbers but increases as βab (or βcd) goes away from 1. Also, Figures 11 and 12 show the results of the ratio of mass-transfer efficiency improvement to the power consumption increment, Im/Ip, vs mass-transfer Graetz numbers with the design parameter, βab and βcd, as parameters. The methods for improving the performance in mass-transfer devices are either in the multipass operation with fluid first flowing through two inner subchannels or two outer subchannels. It is seen in Tables 2 and 3 that an enhancement of mass-transfer efficiency in flow pattern B, compared to flow pattern A, is evident with the same power consumption increment. One may notice in Tables 2 and 3 as well as Figures 11 and 12 that the values of Im and Im/Ip in flow pattern B are higher than those in flow pattern A since the outlet fluid of flow pattern B is concentrated twice in two inner and then outer subchannels. This is because the fluid in two outer sub-
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3477 Table 4. Power Consumption Increment with the Channel Thickness Ratio as a Parameter (Flow Patterns A and B) Ip βab ) βcd ) 1/3
βab ) βcd ) 2/3
βab ) βcd ) 1
βab ) βcd ) 3/2
βab ) βcd ) 3
βab ) 1/3, βcd ) 1
264.52
80.02
63
80.02
264.52
163.74
Figure 9. Dimensionless outlet concentration vs Gzm with γ as a parameter for βab ) βcd ) 1 (flow pattern B).
Figure 11. Values of Im/Ip vs Gzm with βab (or βcd) as a parameter for γ ) 5 (flow pattern A).
Figure 10. Average Sherwood number vs Gzm with γ as a parameter for βab ) βcd ) 1 (flow pattern B).
Figure 12. Values of Im/Ip vs Gzm with βab (or βcd) as a parameter for γ ) 5 (flow pattern B).
channels of flow pattern B is employed for mass transferring through the permeable barriers to two
inner subchannels while two outer subchannels in flow pattern A is concentrated only from the walls.
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5. Conclusion
pm,2 ) 0
Mass transfer through multipass parallel-plate operations with three inserted permeable barriers was investigated analytically with taking the design parameter (γ, βab, and βcd) and operating parameter (Gzm) into account. The basic mass-transfer mechanism with diffusion through the permeable barrier was employed to develop the mathematical formulation for mass transfer in multipass parallel-plate channels and the orthogonal expansion technique with the eigenfunctions expanding in terms of an extended power series was used to find the dimensionless outlet concentration, average Sherwood numbers, and mass-transfer improvement as well. The present paper dealing with four-pass operations is actually an extension of our previous works29,30 of double-pass devices, and the more complicated mathematical formulations for such multistream systems were developed. Figures 3-10 illustrate the results obtained in the previous work30 with the same design and operating parameter values for comparisons. The advantage of the present results is evident in four-pass operations. The multipass effect can enhance the mass transfer for a fluid flowing through a parallel-plate channel with three parallel permeable barriers inserted under large mass-transfer Graetz numbers. More discussion on the suitable selections based on Im/Ip with respect to the continuous variables (Gzm) and the system configuration (βab and βcd) has been presented in Figures 11 and 12 for flow patterns A and B, respectively. The larger the value of Im/Ip obtained, the higher the mass-transfer improvement achieved. It is seen from Figures 11 and 12 that the values of Im/Ip > 0 for the mass-transfer Graetz number are greater than some values, say Gzm > 20 for flow pattern A while the value of Im/Ip is positive for any mass-transfer Graetz number for flow pattern B. It is concluded from Figures 11 and 12 that there exists optimal performance of mass-transfer efficiency at βab ) βcd ) 2/3 for both flow patterns A and B and should be technically and economically feasible in the design of multipass mass exchangers.
pm,3 ) 0
Acknowledgment The authors wish to thank the National Science Council of the Republic of China for its financial support. Appendix Equation 17 can be rewritten as
Wa 1 F ′′a,m(ηa) - λmGzm (6ηa - 6ηa2)Fa,m(ηa) ) 0 2 W
(A1)
Wb 1 F ′′b,m(ηb) + λmGzm (6ηb - 6ηb2)Fb,m(ηb) ) 0 2 W
(A2)
Wc 1 F ′′c,m(ηc) + λmGzm (6ηc - 6ηc2)Fc,m(ηc) ) 0 2 W
(A3)
Wd 1 F ′′d,m(ηd) - λmGzm (6ηd - 6ηd2)Fd,m(ηd) ) 0 2 W
(A4)
Combining eqs A1-A4 and 18 and 25 yields
Wa 1 pm,4 ) λmGzm 4 W l pm,n )
3λmGzm(pm,n-3 - pm,n-4) Wa W n(n - 1)
(A5)
qm,2 ) 0 Wb 1 qm,3 ) - λmGzm qm,0 2 W l qm,n ) -
3λmGzm(qm,n-3 - qm,n-4) Wb W n(n - 1)
(A6)
rm,2 ) 0 Wc 1 rm,3 ) - λmGzm rm,0 2 W l rm,n ) -
3λmGzm(rm,n-3 - rm,n-4) Wc W n(n - 1)
(A7)
tm,2 ) 0 tm,3 ) 0 Wd 1 tm,4 ) λmGzm 4 W l tm,n )
3λmGzm(tm,n-3 - tm,n-4) Wd W n(n - 1)
(A8)
Nomenclature B ) conduit width, m C ) concentration in the stream, mol/m3 D ) ordinary diffusion coefficient in binary mixture, m2/s De ) equivalent diameter of the conduit, m Fm ) eigenfunction associated with eigenvalue λm f ) friction factor Gzm ) mass-transfer Graetz number, 2VW/DBL Gm ) function defined during the use of the orthogonal expansion method Im ) improvement of mass transfer, defined by eq 42 Ip ) increment of power consumption, defined by eq 43 km ) average convection mass-transfer coefficient, m/s L ) conduit length, m lwf ) friction loss in conduit, N‚m/kg pmn ) coefficient in the eigenfunction Fa,m P ) power consumption, N‚m/s qmn ) coefficient in the eigenfunction Fb,m rmn ) coefficient in the eigenfunction Fc,m Re ) Reynolds number Sm ) expansion coefficient associated with eigenvalue λm Sh ) average Sherwood number tmn ) coefficient in the eigenfunction Fd,m V ) input volume flow rate of conduit, m3/s v ) velocity distribution of fluid, m/s vj ) average velocity of fluid, m/s W ) distance between two parallel plates, m
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3479 x ) transversal coordinate, m z ) longitudinal coordinate, m β ) ratio of channel thickness, Wa/Wb ) Wd/Wc γ ) the parameter of the permeable barrier, γ ) (W/δ) δ ) thickness of the permeable barrier, m ) permeability of the permeable barrier η ) transversal coordinate, x/W θ ) dimensionless concentration, (C - CI)/(CS - CI) λm ) eigenvalue µ ) viscosity of the fluid, kg/m‚s ξ ) longitudinal coordinate, z/L F ) density of the fluid, kg/m3 ψ ) dimensionless concentration, (C - CS)/(CI - CS) Superscripts and Subscripts a ) the channel a b ) the channel b c ) the channel c d ) the channel d F ) at the outlet I ) at the inlet L ) at the end of the channel 0 ) in a single-pass device without recycle s ) at the wall surface
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Received for review September 24, 2002 Revised manuscript received May 2, 2003 Accepted May 8, 2003 IE0207461
