Ind. Eng. Chem. Res. 2001, 40, 5839-5846
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Mass-Transfer Enhancement in Double-Pass Mass Exchangers with External Refluxes C. D. Ho,* H. M. Yeh, and S. C. Chiang Department of Chemical Engineering, Tamkang University, Tamsui, Taipei, Taiwan 251, R.O.C.
The enhancement of mass transfer in double-pass countercurrent-flow parallel-plate mass exchangers with uniform wall concentration and external refluxes at the ends has been developed by either inserting an impermeable sheet or preferably inserting a permeable barrier. The analytical treatment was carried out by the orthogonal expansion technique with the assumptions of fully developed laminar flow and constant fluid properties. Considerable improvement in mass transfer is obtainable by employing double-pass devices with a permeable barrier instead of using double-pass devices with an impermeable sheet and single-pass devices. Analytical results also show that the reflux ratio and permeable-barrier (or impermeable-sheet) position are two important manipulated parameters resulting in substantial improvement of the mass-transfer efficiency compared with that in single-pass operations. 1. Introduction The Graetz problem is a well-known process which describes heat or mass transfer in a confined conduit with a fully developed velocity distribution.1-4 Because the early studies of the Graetz problem were all carried out with axial conduction or diffusion neglected, which are not always valid, particularly for low Prandtl number fluids such as liquid metals, the modified Graetz problems of considering axial conduction or diffusion for laminar flow systems, called extended Graetz problems, were later solved.5-12 Beyond treating the single-stream problem, the conjugated Graetz problems13-19 are referred to to deal with heat- and mass-transfer processes between two or more contiguous phases coupled with mutual conditions at the boundaries. Application of the recycle-effect concept in absorption, fermentation, and polymerization with the use of internal or external refluxes, such as loop reactors,20,21 airlift reactors22,23 and draft-tube bubble columns, at both ends is broadly used and economically feasible.24,25 Some analytical solutions were obtained for these conjugated Graetz problems by use of orthogonal expansion techniques.26-34 It is the purpose of this work in designing a new device of inserting an impermeable sheet or a permeable barrier to divide an open duct into two channels and to investigate the improvement of the double-pass mass exchanger with external refluxes at the outlet, which are the devices that were designed to change the composition of flowing streams, such as catalytic reactors,35 filters,36 gas scrubbers,37 and gas absorbers.38 Analytical results also show that the reflux ratio and permeable-barrier (or impermeable-sheet) position are two important manupulated parameters resulting in substantial improvement of the mass-transfer efficiency compared with that in single-pass operations. The two conflicting influences of the operation performance, the desirable effect of increasing the fluid velocity for decreasing mass-transfer resistance and the undesirable effect of reducing the mass-transfer driving force due * Corresponding author. Fax: 886-2-26209887. E-mail: cdho@ mail.tku.edu.tw.
to a decrease in the concentration difference, produced by double-pass operation are also discussed. 2. Mathematical Statement 2.1. Concentration Distribution in the DoublePass Device. Consider a parallel-plate conduit of thickness W, length L, and width B (.W) which is divided into two subchannels, subchannel a and subchannel b, with thickness ∆W and (1 - ∆)W, respectively, by inserting a permeable barrier (or impermeable sheet) with negligible thickness d (,W), as shown in Figure 1. Before entering the lower subchannel, subchannel a for a double-pass operation, the fluid entering with volumetric flow rate V and concentration Ci will premix the fluid exiting from the upper subchannel, subchannel b, with reflux flow rate RV and outlet concentration CF which is regulated with the aid of a conventional pump situated at the beginning of subchannel a. In obtaining the theoretical formulation, the following assumptions are made: (a) constant physical properties and wall concentration; (b) negligible axial diffusion as well as entrance length and end effects; (c) pure fully developed laminar flow in each channel; (d) mass transfer by diffusion only through the permeable barrier. After the following dimensionless variables are introduced:
ηa )
xa xb z , ηb ) , ξ) , Wa Wb L V(Wa + Wb) VW GZ,m ) ) , Wa ) ∆W, DBL DBL Wb ) (1 - ∆)W (1)
the velocity distributions in dimensionless form may be written as
va(ηa) ) va(6ηa - 6ηa2), 0 e ηa e 1
(2)
vb(ηb) ) vb(6ηb - 6ηb2), 0 e ηb e 1
(3)
in which va ) [(R + 1)V/WaB] and vb ) -[(R + 1)V/WbB].
10.1021/ie0009334 CCC: $20.00 © 2001 American Chemical Society Published on Web 11/06/2001
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Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001
Then
θF ) 1 - ψF )
1 GZ,m
[
∞
∑ m)0
1 - e-λa,m Sa,mF′a,m(0) + λa,m∆ 1 - e-λb,m
∞
∑
m)0λb,m(1
Figure 1. Double-pass parallel-plate mass exchangers with external refluxes at both ends.
2.1.1. Impermeable Sheet Inserted. For the doublepass device, the permeable barrier in Figure 1 is exchanged by inserting an impermeable sheet. The equations of mass transfer in dimensionless form may be obtained as 2
∂ ψa(ηa,ξ) 2
)
∂ηa
∂2ψb(ηb,ξ) ∂ηb2
( ) ( ) 2
)
Wa va ∂ψa(ηa,ξ) LD ∂ξ
(4)
After the eigenvalues and associated expansion coefficients are obtained, the dimensionless outlet concentrations (ψF and θF ) 1 - ψF) are readily calculated by eq 12. Also, the mixed inlet concentration of the lower channel is calculated by
ψa(ηa,0) R [R+1 )
∫01 vbWbBψb(ηb,0) dηb] + V
)
Wb2vb ∂ψb(ηb,ξ) LD ∂ξ
(5)
1
)
1
(7)
ψb(0,ξ) ) 0
(8)
∂ψa | )0 ∂ηa ηa)1
(9)
∂ψb | )0 ∂ηb ηb)1
(10)
By following a mathematical treatment similar to that performed in our previous works,33,34 the analytical solution to this type of problem will be obtained by use of an orthogonal expansion technique. The dimensionless outlet concentrations, which are referred to as the average concentration, are obtained by making the overall mass balances on both subchannels. It was in terms of the mass Graetz number (GZ,m), eigenvalues (λa,m and λb,m), expansion coefficients (Sa,m and Sb,m), location of the impermeable sheet (∆), and eigenfunctions (Fa,m(ηa) and Fb,m(ηb)). The result is as follows:
V(1 - ψF) ) DBL ∂ψa(0,ξ)
∫01 W
[
∂ηa
a
DBL Wa
dξ +
∞
∑
Sa,mF′a,m(0)
m)0
DBL Wb
∞
∑ m)0
DBL ∂ψb(0,ξ)
∫01 W
V(R + 1) WbB
∫
1 -λa,m(1-ξ) e 0
∫
1 -λb,m(1-ξ) e dξ 0
]
∑
V R + 1m)0
1-
WbB
( )
R
LD
-λb,m
∞
e
∑ Sb,m λ 2 m)0
V R+1 W b
)
1
[
R+1
1-
R
1
∞
∑
GZ,m(1 - ∆) R + 1m)0
(
)
e-λb,mSb,m λb,m
]
{F′b,m(1) b,m
F′b,m(0)}
]
]
{F′b,m(1) - F′b,m(0)} (13)
2.1.2. Permeable Barrier Inserted. Similarly, a permeable barrier was inserted, as shown in Figure 1, for the double-pass device and the equations of mass transfer in dimensionless form may also be obtained, with the notation of t being included and with the boundary conditions of eqs 9 and 10 being changed as follows:
-
∂ψa,t(1,ξ) Wa ∂ψb,t(1,ξ) ) ∂ηa Wb ∂ηb
(14)
∂ψa(1,ξ) Wa [ψb(1,ξ) - ψa(1,ξ)] ) ∂ηa δ
(15)
The mathematical treatment is similar to that in the previous section, except the orthogonality conditions of conjugated Graetz problems. The dimensionless inlet and outlet concentrations [θ(ηa,0) and θF,t] for doublepass devices by inserting a permeable barrier are
[∑
1 - e-λm,t t t ′(0) + Sa,m Fa,m m)0 λm,t∆ ∞
1 GZ,m
∞
(11)
∫01 vb(ηb)Fb,m(ηb) dηb
Sb,me-λb,m
θF,t ) 1 - ψF,t )
dξ +
∞
R
∂ηb
b
Sb,mF′b,m(0)
dξ )
1-
[
R+1
The boundary conditions for solving eqs 4 and 5 are
ψa(0,ξ) ) 0
[
R+1
in which
Ca - Cs Cb - Cs , ψb ) , θa ) 1 - ψa, ψa ) Ci - Cs Ci - Cs θb ) 1 - ψb (6)
]
Sb,mF′b,m(0) (12) - ∆)
1 - e-λm,t
∑ m)0λ
]
t t Sb,m Fb,m ′(0) (16) m,t(1 - ∆)
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5841
and
θ0,F ) 1 - ψ0,F )
ψa,t(ηa,0) R [R+1 )
∫0
)
)
1
1-
WbB
R
∞
∑
V R + 1m)0
1-
WbB
R
∫
t vb(ηb) Fb,m
]
( )
-λm
∞
LD
e
t Sb,m ∑ 2 λ m)0
V R+1 W b
)
[
R+1
1-
∞
R
1
∑
GZ,m(1 - ∆) R + 1m)0
m
( ) t e-λmSb,m
λm
]
]
t t {Fb,m ′(1) - Fb,m ′(0)} (17)
2.2. Concentration Distribution in a Single-Pass Device without Reflux. For the single-pass device of the same size without recycle, R ) 0, the permeable barrier in Figure 1 is removed and, thus, ∆ ) 1, Wa ) W, and ηa ) η0. The velocity distribution and equation of mass transfer in dimensionless form may also be obtained with the notation of 0 being included, and the velocity distribution and the boundary conditions are
v0(η0) )
[
1 - e-λ0,m S0,mF′0,m(0) λ0,m
]
1 - e-λ0,m S0,mF′0,m(1) (22) λ0,m
V (6η - 6η02), 0 e η0 e 1 WB 0
By following the same definition in the previous work,39 the Sherwood number for double-pass devices by inserting an impermeable sheet and by inserting a permeable barrier may be obtained as follows:
kmW VW ) (1 - ψF) ) 0.5GZ,m(1 - ψF) ) D 2DBL 0.5GZ,mθF (23)
Sh )
t {Fb,m ′(1) -
t ′(0)} Fb,m
1
GZ,m
∞
∑ m)0
3. Improvement of Mass-Transfer Efficiency
1 t Sb,m e-λm 0
(ηb) dηb
[
R+1
vbWbBψb,t(ηb,0) dηb] + V V(R + 1)
[
R+1
1
1
1
for inserting an impermeanble sheet and
Sht )
for inserting a permeable barrier. Similarly, for a singlepass device without recycle
Sh0 )
k0,mW VW ) (1 - ψ0,F) ) 0.5GZ,mθ0,F D 2DBL
(25)
The improvements of mass-transfer efficiencies, Im and Im,t, for double-pass devices by inserting an impermeable sheet and by inserting a permeable barrier, respectively, are best illustrated by calculating the percentage increase in the mass-transfer rate, based on the mass transfer of a single-pass device of the same mass Graetz number without recycle as
(18)
and
kmW VW ) (1 - ψF,t) ) 0.5GZ,m(1 - ψF,t) ) D 2DBL 0.5GZ,mθF,t (24)
Im )
Sh - Sh0 ψ0,F - ψF θF - θ0,F ) ) Sh0 1 - ψ0,F θ0,F
(26)
and
ψ0(0,ξ) ) 0
(19)
ψ0(1,ξ) ) 0
(20)
∫0 v0WBψ0(η0,0) dη0 1
)
)
V WB
∞
e-λ ∑ V m)0
1
∞
∑
GZm)0
(
S0,m
0,m
λ0,m
∫01v0(η0) F0,m(η0) dη0
)
e-λ0,mS0,m
Sht - Sh0 ψ0,F - ψF,t θF,t - θ0,F ) ) (27) Sh0 1 - ψ0,F θ0,F
The results are shown in Tables 1 and 2.
Also, the dimensionless inlet concentration would be the constraint for the whole system.
ψ0(η0,0) ) 1 )
Im,t )
{F′0,m(1) - F′0,m(0)} (21)
The calculation procedure for a single-flow device is rather simpler than that for a double-pass device. The outlet concentration for single-pass devices (θ0,F) is
4. Results and Discussion 4.1. Double-Pass Devices with External Refluxes. After the eigenvalues and associated expansion coefficients are obtained, the dimensionless mixed inlet concentrations [θa,t(ηa,0) and θa(ηa,0)] and the dimensionless outlet concentrations (θF,t and θF) are readily calculated respectively from eqs 12, 13, 16, and 17. Figures 2 and 3 show respectively the dimensionless inlet concentrations θa(ηa,0) and θa,t(ηa,0) of the fluid after mixing and dimensionless outlet concentrations θF and θF,t with R and (W/δ) as parameters for ∆ ) 0.5. Figure 2 is the graphical representation of θa,t(ηa,0) and θa(ηa,0) versus GZ,m with R and (W/δ) as parameters. It is seen from this figure that, as expected, the dimensionless mixed inlet concentrations θa,t(ηa,0) and θa(ηa,0) increase with an increase of the reflux ratio but with a decrease of the value of (W/δ). It is shown in Figure 3 that the dimensionless inlet concentration of
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Table 1. Improvement of the Transfer Efficiency with Reflux Ratio and Sheet Position as Parameters of Double-Pass Operations by Inserting an Impermeable Sheet R ) 1.0
R ) 2.0
R ) 5.0
Im (%)
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
GZ,m ) 1 10 100 1000
-2.07 13.06 63.91 75.61
-3.90 -1.83 29.39 32.10
-2.07 13.06 63.91 75.61
-1.23 16.31 65.10 75.79
-1.29 2.19 29.89 32.20
-1.23 16.31 65.10 75.79
-0.03 23.49 67.09 75.99
-0.33 6.29 30.40 32.31
-0.03 23.49 67.09 75.99
Table 2. Improvement of the Transfer Efficiency with Reflux Ratio and Barrier Position as Parameters of Double-Pass Operations by Inserting a Permeable Barrier [E(W/δ) ) 5] R ) 1.0
R ) 2.0
R ) 5.0
Im,t (%)
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
GZ,m ) 1 10 100 1000
-11.68 50.89 371.68 572.32
-13.47 31.29 193.54 256.06
-10.14 36.58 215.73 288.15
-8.16 54.08 375.25 573.05
-9.83 33.91 195.13 256.30
-7.56 38.53 217.02 288.35
-4.64 57.21 378.85 573.79
-6.38 36.53 196.72 256.54
-5.21 40.47 218.32 288.54
Figure 2. Dimensionless average inlet concentration of fluid after mixing. Reflux ratio as a parameter: ∆ ) 0.5, 0 e ηa e 1.
fluid after mixing increases with the reflux ratio but decreases with the mass Graetz number. It was also found in Figure 3 and Tables 1 and 2 that, for a fixed reflux ratio, the dimensionless average outlet concentration of double-pass operations decreases with an increase in the mass Gratez number owing to the short residence time of fluid, but the dimensionless average outlet concentration of double-pass operations increases with a decrease in (W/δ). For small mass Graetz number, say GZ,m < 6, no improvement in the transfer efficiency can be achieved in the double-pass device with a permeable barrier inserted. Under this region, one may prefer to employ the double-pass device where insertion of an impermeable sheet is preferred rather than using the double-pass one with a permeable barrier inserted or single-pass devices operating at such conditions. Because the dimensionless outlet concentrations θb,t(ηb,0) (or θF,t) and θb(ηb,0) (or θF) increase as the residence time of the fluid increases (or as V decreases or as L increases), the dimensionless outlet concentra-
Figure 3. Dimensionless average outlet concentration of three devices vs GZ,m with the reflux ratio as a parameter: ∆ ) 0.5.
tions θF,t and θF increase as the mass Graetz number decreases but increases with the reflux ratio, as shown in Figure 3, for ∆ ) 0.5 and with R and (W/δ) as parameters. Figure 4 is another graphical expression for θF,t and θF vs GZ,m but with ∆ and R as parameters for (W/δ) ) 5. It is shown that the permeable-barrier position has much influence on the outlet concentration θF,t and θF, θF,t, and θF increase as the ratio of the thickness ∆ diverges from 0.5, especially for ∆ < 0.5 by inserting a permeable barrier. However, the two curves in Figure 4 of the double-pass device for inserting an impermeable sheet with ∆ ) 0.25 and 0.75 are overlapped. Sherwood numbers and hence the improvements of mass-transfer efficiencies are proportional to θF (θF,t or θ0,F), as shown in eqs 23-27, so the higher improvement of the performance is really obtained by employing a double-pass device with a permeable barrier inserted, instead of using double-pass devices with an imperme-
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5843
Figure 4. Dimensionless average outlet concentration of both double-pass devices vs GZ,m with ∆ as a parameter: R ) 5.
Figure 5. Average Sherwood number of three devices vs GZ,m with the reflux ratio as a parameter: ∆ ) 0.5.
able sheet and single-pass devices for large mass Graetz numbers. It is seen from Figures 3 and 4 that the difference θF - θF,t with any values of (W/δ) of outlet concentrations is positive for small GZ,m values but then turns to a minus sign with any values of R and ∆. Figure 5 shows the average Sherwood numbers with the reflux ratio as a parameter for ∆ ) 0.5, while Figure 6 and Tables 1 and 2 show the ratio of the thickness ∆ as a
Figure 6. Average Sherwood number both double-pass devices vs GZ,m with ∆ as a parameter: R ) 5.
parameter. It is concluded that the Sherwood number increases with increasing R but with ∆ diverging from 0.5, especially for ∆ < 0.5 by inserting a permeable barrier. 4.2. Improvement in Transfer Efficiencies. The Sherwood numbers as well as the mass-transfer coefficients can be calculated from eqs 23-25 respectively for a double-pass device with an impermeable sheet inserted and with a permeable barrier inserted and for a single-pass device without recycle. Figures 3 and 4 show the dimensionless outlet concentrations, θF,t, θF, and θ0,F. It is found in Figures 3 and 4 that all θF,t, θF, and θ0,F decrease with an increase in the mass Graetz number GZ,m owing to the short residence time of the fluid (either large V or small L), while θF,t and θF increase as the ratio of the thickness ∆ diverges from 0.5, especially for ∆ < 0.5 by inserting a permeable barrier. Figure 5 gives the graphical representations of Sht, Sh, and Sh0 versus GZ,m, with R as a parameter for (W/δ) ) 5 and ∆ ) 0.5. Figure 6 is the other representation for the effect of ∆ on Sht and Sh. It is seen in Figures 5 and 6 that all of Sht, Sh, and Sh0 increase with GZ,m while Sht and Sh increase also when ∆ diverges from 0.5, especially for ∆ < 0.5 by inserting a permeable barrier. On the other hand, as shown in Figure 5, Sht - Sh0 and Sh - Sh0 increase with GZ,m, but the increase with GZ,m is limited as GZ,m approaches infinity. The improvement of performance, Im and Im,t, can be obtained from eqs 26 and 27, respectively. Some results are given in Tables 1 and 2 with the mass Graetz number and the ratio of thickness ∆ as parameters. It is found from Tables 1 and 2 that the improvement in the mass-transfer coefficient of a double-pass device with recycle, based on that of a single-pass device of the same dimensions and fluid flow rate without recycle, increases with the mass Graetz number and reflux ratio as well as with the ratio of thickness ∆ diverging from 0.5, especially for ∆ < 0.5 by inserting a permeable
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barrier. It should be mentioned that the improvement turns to negative when the mass Graetz number is sufficiently small, as shown by the negative signs in Tables 1 and 2. This result is also indicated in Figure 3 for dimensionless outlet concentrations. In this case, a single-pass mass exchanger without recycle is recommended rather than using the double-pass one even with recycle. 4.3. Enlargement of Hydraulic Dissipated Power. The friction loss in conduits may be estimated by
lwf )
2fv2L De
(28)
where v and De denote the bulk velocities in the conduits and the equivalent diameters of the conduits, respectively, while f is the Fanning factor, which is a function of the Reynolds number, Re. Accordingly
v0 )
(R + 1)V (R + 1)V V , va ) , vb ) (29) BW B∆W B(1 - ∆)W
De,0 ) 2W, De,a ) 2∆W, De,b ) 2(1 - ∆)W (30) Thus, from eqs 29 and 30, one obtains
Re0 )
Rea Reb ) R+1 R+1
(31)
and then we have for laminar flow
f0 ) (R + 1)fa ) (R + 1)fb )
Re0 8Bµ ) 16 VF
(32)
The enlargement of power consumption, Ip, may be defined as
Ip ) )
)
PR - P0 P0 [(R + 1)VFlwf,a + (R + 1)VFlwf,b] - VFlwf,0 ) VFlwf,0 [(R + 1)(lwf,a + lwf,b)/lwf,0] - 1 (R + 1)2 ∆3
+
(R + 1)2 (1 - ∆)3
-1
(33)
while the hydraulic dissipated energy in one-pass device without recycle is
P0 ) VFlwf,0 )
8VµL 8µL2BD ) GZ,m W2 W3
(34)
Though the increment of power consumption does not depend on the mass Graetz number, it increases as ∆ diverges from 0.5. It is readily obtained from eq 33 that Ip increases with the reflux ratio as well as with ∆ diverging from 0.5, and the results of Ip as well as of P0 ) 6.6 × 10-11, 6.6 × 10-10, 6.6 × 10-9, and 6.6 × 10-8 hp, respectively, for GZ,m ) 1, 10, 100, and 1000 for the aqueous solution of water flowing through a parallel conduit made of benzoic acid (D ) 6.67 × 10-10 m2/s, µ ) 1.032 × 10-3 N‚s/m2) with B ) 0.2 m, L ) 0.6 m, and W ) 0.02 m are presented in Table 3.
Table 3. Increment of Power Consumption with Reflux Ratio and Barrier Position as Parameters Ip R
∆ ) 0.25
∆ ) 0.5
∆ ) 0.75
0.5 1.0 2.0 5.0
148.33 264.48 596.33 2388.33
17.00 31.00 71.00 287.00
148.33 264.48 596.33 2388.33
5. Conclusion The equations of mass transfer for liquid flowing through a double-pass parallel-plate channel with external refluxes as well as through a single-pass device have been studied analytically, with the eigenfunctions expanding in terms of an extended power series. Though the phenomenon of mass transfer in the present study could be mostly analogized from that of heat transfer in our previous works,33,34 the manners of transport across the interphase (from channel b to channel a) are somewhat different. In our previous works, we assumed that heat could transfer across the interphase without thermal resistance (i.e., the temperatures at both sides of the impermeable sheet are the same), while in the present study, some amount of solute in channel b is transferred by diffusion through the pores of a permeable barrier (with permeability ) and then into channel a. Accordingly, the boundary conditions at this point are different, as shown in eq 15, and the mathematical treatments and analytical solutions are not the same either. The application of recycle operation to mass exchangers actually has two conflicting effects: the desirable effect of increasing the fluid velocity, resulting in an increase of the mass-transfer coefficient, and the undesirable effect of lowering the concentration difference due to mixing of the inlet fluid with the recycling stream, resulting in a decrease of the driving force of mass transfer. Further, the strategies for improving the performances in mass-transfer devices are increasing the concentration difference to enhance the driving force of the mass transport, increasing the fluid velocity to decrease the mass-transfer resistance, and increasing the residence time. Accordingly, at small mass Graetz number, GZ,m (either small V or large L), the residence time is long; therefore, the increase of the fluid velocity created by applying the recycle with a reflux ratio R which is not large enough cannot compensate for the situation that the driving force of mass transfer in the mass-transfer device decreases. Thus, the mass transfer in this recycle system of small GZ,m with low R cannot be over that in a device of the same size without recycle, as shown in Figures 3 and 4 as well as in Tables 1 and 2. On the other hand, the introduction of a reflux still has positive effects on the mass transfer for larger mass Graetz number. This is due to the increase of the fluid velocity having more influence here than the effect of a decrease of the concentration difference, as is also shown in Figures 3 and 4 and in Tables 1 and 2. It is seen in Figures 3-6 that the dimensionless outlet concentration and the average Sherwood number change greatly with the mass Graetz number at high reflux ratio but change very little with the reflux ratio for large mass Graetz number. One may notice in Figures 5 and 6 that the values of Sht are higher than those of Sh at large mass Graetz numbers because the permeable barrier provides the mass transfer from subchannel b to subchannel a while the impermeable sheet does not, leading to
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5845
improved performance. The results in Figures 4 and 6 show that the permeable-barrier position has much influence on the mass-transfer behavior. It is presented in Tables 1 and 2 that the improvement of mass-transfer efficiency increases with the mass Graetz number and with the reflux ratio while the average Sherwood number increases as the ratio of the thickness ∆ diverges from 0.5, especially for ∆ < 0.5 by inserting a permeable barrier. The reason that decreasing ∆ from 0.5 is more effective than increasing ∆ from 0.5 may be due to the fact that the enhancement of mass transfer in subchannel a is more effective than that in subchannel b. Moreover, it was found that the suitable selection of the value of (W/δ) may also lead to improved device performance with a permeable barrier inserted. Roughly, the value of (W/δ) for the optimal mass-transfer efficiency is about 1.0, but not for all mass Graetz numbers. This means that the trend of the masstransfer efficiency decreases as (W/δ) goes away from 1.0, but no regularity was found, especially for (W/δ) < 1.0. The values of parameters (W/δ) chosen for the numerical calculations provide illustrations for theoretical results; the appropriate values of those parameters (, W, and δ) may be selected for practical application. It is seen in Table 3 that the enlargement of power consumptions due to recycling operation though increases rapidly with the reflux ratio, especially for the case that ∆ goes far from 0.5. Table 3 shows that the power consumptions calculated by assuming only the friction losses to the walls were significant. With operation under these conditions, the theoretical predictions show that the friction losses estimated by eq 33 actually are still small even for ∆ ) 0.25 and 0.75 with R ) 5. As an illustration, for the aqueous solution of water flowing through a parallel conduit made of benzoic acid (D ) 6.67 × 10-10 m2/s and µ ) 1.032 × 10-3 N‚s/m2), P ) 1.6 × 10-7, 1.6 × 10-6, 1.6 × 10-5, and 1.6 × 10-4 hp respectively for GZ,m ) 1, 10, 100, and 1000 with B ) 0.2 m, L ) 0.6 m, and W ) 0.02 m. Therefore, the power consumptions of a double-pass operation with recycle may be ignored. The present paper is actually the extension of other recycle problems carried out in the previous work,39 except the type of reflux and two double-pass devices are included. To explain how the present device improves on the previous work, Figure 7 illustrates some results obtained in the previous work,39 with the same parameter values used in Figure 5 for comparison. With this comparison, the advantage of the present results is evident with any mass Graetz number. It is concluded that the recycle effect can enhance the mass transfer for the fluid flowing through a mass exchanger under double-pass operations by inserting an impermeable sheet for low mass Graetz number or by inserting a permeable barrier with any mass Graetz number, and the systematic approach derived in this work may also be applied to solve similar problems analytically. It is apparent that the mathematical formulation in conjugated Graetz systems derived in this study may also be applied to the heat exchanger with external refluxes at both ends, except with the boundary condition on the impermeable sheet instead of on the permeable barrier. Acknowledgment The authors thank the National Science Council of the Republic of China for financial support.
Figure 7. Dimensionless average outlet concentration of both double-pass devices in the present work and in the previous work39 vs GZ,m with ∆ as a parameter: R ) 5.
Nomenclature B ) conduit width, m C ) concentration in the stream, mol/m3 D ) ordinary diffusion coefficient in binary mixtures, m2/s De ) equivalent diameter of the conduit, m Fm ) eigenfunction associated with eigenvalue λm f ) friction factor GZ,m ) mass Graetz number VW/DBL Im ) improvement of the mass transfer, defined by eqs 26 and 27 Ip ) enlargement of power consumption, defined by eq 40 km ) average convection mass-transfer coefficient, m/s lwf ) friction loss in conduits, kJ/kg L ) conduit length, m P0 ) hydraulic dissipated energy in a single pass without recycle, kJ/s PR ) hydraulic dissipated energy in a double pass with reflux, kJ/s Sh ) Sherwood number Sm ) expansion coefficient associated with eigenvalue λm Re ) Reynolds number V ) input volumetric flow rate of the conduit, m3/s v ) velocity distribution of fluid, m/s v ) average velocity, m/s W ) thickness of the conduit, m x ) transversal coordinate, m z ) longitudinal coordinate, m ∆ ) ratio of the thickness between the forward flow channel and the conduit, Wa/W 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 ξ ) longitudinal coordinate, z/L ψ ) dimensionless concentration, (C - Cs)/(Ci - Cs)
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Superscripts and Subscripts a ) in the forward flow channel b ) in the backward flow channel F ) at the outlet of a double-pass device i ) at the inlet L ) at the outlet, ξ ) 1 0 ) in a single-pass device without recycle s ) at the wall surface t ) in a double-pass device with insertion of a permeable barrier
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Received for review October 31, 2000 Accepted September 13, 2001 IE0009334
