Ind. Eng. Chem. Res. 2005, 44, 7835-7843
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SEPARATIONS Corrections to Correlations for Shell-Side Mass-Transfer Coefficients in the Hollow-Fiber Membrane (HFM) Modules Tsun-teng Liang and Richard L. Long, Jr.* Department of Chemical Engineering, New Mexico State University, Las Cruces, New Mexico 88003
Mass-transfer correlations for the shell-side mass-transfer coefficients in hollow-fiber membrane (HFM) modules have been published by several authors. The uneven distribution of shell-side flow is due to the random packing of fibers in the module. The results of different authors give apparently conflicting results. This work provides an approach to estimating empirical correlations for HFM shell-side mass transfer. This approach is then applied to explain several literature observations and to offer corrections to earlier published correlations. The dependence of the Sherwood number on both the Reynolds number and the packing fraction are considered. Further work on the packing fraction dependence is required. 1. Introduction For hollow-fiber membrane (HFM) modules, masstransfer data have been correlated by empirical equations for the shell side by many researchers. Four representative correlations include those from Yang and Cussler in 1986,1 Dahuron and Cussler2 and Prasad and Sirkar in 1988,3 and Costello et al.4 in 1993. In 1999, Bao et al.5 developed a theoretical analysis for shellside mass transfer in the cases of constant wall flux and constant wall concentration, using a numerical approximation for the velocity gradient on the fiber surface. The maldistribution of the shell-side flow is due to random packing of fibers in the module. The effect of random packing of fibers in the HFM modules has been to cause inconsistent correlations in the literature. Yang and Cussler1 suspected that there was major channelling through the closely packed fibers (packing fraction of φ g 0.4) on the shell side. This problem is not noted in the work of Costello et al.4 in the middle and high packing-fraction range (φ ) 0.32-0.76). Bao et al.5 determined that the experimental measurements of shell-side mass-transfer coefficients are in poor agreement with theoretical values for either regular or random fiber packings, and they attributed these differences to deviations from the assumed axial flow pattern. With careful selection of the range of Reynolds number (Re), close agreement for the Sherwood number (Sh) versus Re between the research of Yang and Cussler1 and Prasad and Sirkar3 has been observed. However, significant differences in correlation results in the literature have also been observed in the work of Bao et al.,5 especially in the relation of the module-averaged Sherwood number (Shlm) to φ. Some research has taken the effect of module length on correlation into consideration, but some does not. The equations used to represent correlations in the literature * Corresponding author. Fax: (505) 646-7706; E-mail:
[email protected].
are different because the different authors choose different operational parameters in their experiments. Their correlation equations are developed from dimensional analysis. (See Tables 1-4 for comparisons of some shell-side mass-transfer data from the various researchers.) This note provides an approach to estimating the empirical correlations for HFM shell-side mass transfer. First, formulas and detailed schematic diagrams for estimation of the overall mass-transfer coefficients from experimental data based on the shell-side fluid (aqueous or organic phase) are provided and include cases of both counter-current flow and co-current flow operation with two different systems: (i) the gas/membrane/liquid system and (ii) the aqueous phase/membrane/organic phase system. Second, this work shows some errors in the literature for correlation of the HFM shell-side mass-transfer coefficient and corrects these inaccuracies. 2. Estimation of the Overall Mass-Transfer Coefficients, Based on the Shell-Side Fluid (Liquid Phase) in the Gas/Membrane/Liquid System 2.1. Counter-current Flow Operation. There is a once-through mode with counter-current flow operation, which is shown in Figure 1. There seems to be an error in eq 4 in ref 4 for calculating the shell-side masstransfer coefficients. The schematic in Figure 1 is used to match their experimental design, such as solute A penetrating from the liquid phase at the shell side to gas phase at the lumen side of the filaments. In addition, the pores of the membrane are filled with gas. For the notation used here, please refer to the notes of Figure 1. Before deriving the overall mass-transfer coefficients based on the shell-side fluid (liquid phase), several assumptions are made: (i) a dilute system (that is, QL|in = QL|out ) QL and QG|in = QG|out ) QG); (ii) steady-state conditions; (iii) no chemical reaction; (iv) uniform shell-side flow distribution; (v) the thickness
10.1021/ie040265c CCC: $30.25 © 2005 American Chemical Society Published on Web 08/30/2005
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Table 1. Comparison of the Overall Mass-Transfer Coefficients Based on Shell-Side Fluid (Liquid Phase) between the Counter-current Flow and Co-current Flow Operations for Stripping and Absorption in the Gas/Membrane/Liquid System counter-current flow operation
[
][
]
QL CLAbin - (1/mi)(CGAbout) 1 K hL ) ln A 1 - (1/mi)(QL/QG) CLAbout - (1/mi)CGAbin
stripping
special case (a): at miQG . QL and CGAbin ) 0: absorption
( ) [
]
QL CLAbin - (1/mi)CGAbout ln A CLAbout QL CLAbin - (1/mi)(CGAbout) 1 K hL ) ln A 1 - (1/mi)(QL/QG) CLAbout - (1/mi)CGAbin K hL )
[
][
[
][
]
-(1/mi)(CGAbout) QL 1 ln A 1 - (1/mi)(QL/QG) CLAbout - (1/mi)CGAbin QL CLAbout - (1/mi)(CGAbin) special case (b): hL ) ln at miQG . QL and CLAbin ) 0: K A -(1/mi)CGAbout special case (a): at CLAbin ) 0:
K hL )
[
]
]
co-current flow operation
{ [ ( )( ) ( ) { [ ( )( ) { [ [ ]
-(QL/A) CLAbin - CGAbout K hL ) ln 1 + h′ h′ (1/mi)CGAbin - CLAbin 1 QL where h′ ) 1 + mi Q G CLAbin QL ln K hL ) A CGAbout -(QL/A) CLAbin - CLAbout ln 1 + h′ h′ (1/mi)CGAbin - CLAbin 1 QL where h′ ) 1 + mi Q G -CLAbout (QL/A) ln 1 + h′ K hL ) h′ (1/mi)CGAbin - CLAbin QL CLAbout ln 1 K hL ) A (1/mi)CGAbin K hL )
]} ]}
]}
Table 2. Comparison of Correlationsa for Shell-Side Mass Transfer authors
correlations
conditions used
Sh ) dhks/DAB; Shlm,c ) 2Rks/DAB Yang and Cussler1
Prasad and Sirkar3
Costello et al.4
Bao et al.5
Re ) dhus/v; Re2 ) 2Rus/v Re ) deus/v 0.93 0.33 Sh ) 1.25(Redh/L) Sc φ ) 0.03, 0.26; Re = N/A; Sc = 477 Restriction 1: The correlation is based on Sh vs [usdh2/(vL)] at fixed φ (where φ ) 0.03 or 0.26) and fixed L (where L ) 6.4, 11.1, 21.3, or 10.2 cm). Restriction 2: The experiment is performed under liquid f gas phase, and both membrane and gas-phase resistance are usually ignored. Sh ) 5.85(1 - φ)(dh/L)Re0.66Sc0.33 φ ) 0.05, 0.06, 0.12, 0.22; Re ) 0-500; Sc ) 300-1000 Restriction 1: The obtained correlation is based on Sh vs us at fixed φ (where φ ) 0.05, 0.06, 0.12, or 0.22) and fixed L, to estimate the exponent of Re and the coefficient of (1 - φ)(dh/L)Re0.66Sc0.33. Restriction 2: The experiment is performed under liquid f liquid phase. Comments: Additional studies will have to be done with different modules. Having different lengths and packing fractions before the above observations can be assumed to be generally valid for MHF extraction.3 Sh ) (0.52 - 0.58φ)Re10.53Sc0.33 φ ) 0.32-0.76; Re1 ) 24-350; Sc = 300-1000 Restriction 1: The contribution of both membrane and gas-phase resistances to mass transfer can usually be ignored. Restriction 2. The experiment is performed under liquid f gas phase, and both membrane and gas-phase resistances are usually ignored. Shlmic ) 1.38(-0.07 + 2.35φ) φ g 0.03; Re2 ) N/A; Sc ) N/A Restriction 1: Where (1 - φ)/φ = dh/do under [π(Di/4)dh]/nπ(do2/4) , dh/do and φ g0.03. Restriction 2: Derived through momentum balance and mass balance in the case of the constant wall concentrations and a high Graetz number region (i.e., module length (L f 0)).
a Note: In this table, the parameters used to make the correlations are defined as follows: d ) 4 × (cross-sectional area of flow)/ h (wetted perimeter); de ) 4 × (flow area)/(total fiber circumference); do is the outer diameter of the filament; Di is the inner diameter of the module; R is the outer radius of the filament; and n is the number of filaments.
Table 3. Comparison of the Hydraulic Diameter Defined in Costello et al.4 with That Defined in This Worka hydraulic diameter (mm) filament
from this work
outer diameter, do (mm)
inner diameter, di (mm)
inner diameter of module, di (mm)
number of filaments, n
from ref 4
de
dh
0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
15.4 15.4 15.4 15.4 15.4 15.4 15.4 15.4 15.4 15.4 15.4 15.4 15.4
399 373 346 320 292 280 252 240 224 213 168 320 320
0.21 0.28 0.31 0.43 0.54 0.59 0.73 0.80 0.91 0.99 1.43 0.43 0.43
0.22 0.28 0.35 0.44 0.54 0.59 0.73 0.80 0.91 0.99 1.44 0.44 0.44
0.205 0.265 0.331 0.407 0.503 0.549 0.673 0.735 0.826 0.895 1.264 0.407 0.407
a Hydraulic diameter is defined from ref 4 to be equal to 4 × (flow area)/(total fiber circumference). The parameters d and d are h e defined as follows for these calculations: dh ) 4 x {π(Di2/4) - [nπ(do2/4)]}/(πDi + nπdo) and de ) 4 x {π(Di2/4) - [nπ(do2/4)]}/(nπdo). HFM characteristics were taken from Table 1 of Costello et al.4
of the interfacial fluid film is small, so that concentration at any location in section x shown in Figure 1b can
be assumed to be the same as the bulk concentration in the fluid at x.
Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7837 Table 4. Comparison of the Shell-Side Void Fraction Used in Table 2 of Prasad and Sirkar3 and That Obtained after Correction of the Hydraulic Diameter filament module shell-side void fraction, (1 - φ)a
inner diameter membrane
module number
outer diameter, do (µm)
di (µm)
Di (mm)
number of filaments, N
length, L (cm)
before correction
after correction
Celgard X-20 Celgard X-20 Celgard X-20 Cuprophan Cuprophan
1 2 3 1 2
290 290 290 200 200
240 240 240 140 140
12.5 6.25 6.25 6.25 6.25
114 54 102 50 50
15.8 16.0 16.0 18 6.0
0.96 0.803 0.6 0.913 0.913
0.939 0.884 0.780 0.949 0.949
a φ is the fiber packing fraction. The shell-side void fraction is given as 1 - φ ) 1 - (total filament outer cross-sectional area)/(inner cross-sectional area of the module) ) 1 - {nπ(do2/4)/[π(Di2/4)]}.
Figure 1. Schematics of (a) concentration profiles at a filament and (b) a hollow-fiber membrane (HFM) module for once-through countercurrent flow operation. Notes: (i) The solute mass transfer is from the liquid phase (at the shell side) through the gas phase (at the lumen side), and it is a once-through counter-current operation (i.e., no recycling action). (ii) Assume that it is a dilute system, so that the flow rates of the liquid and gas phases are fixed and expressed as QL and QG, respectively. (See Nomenclature section for a description of the terms shown in this figure.)
Consider the overall mass balance of solute A (see Figure 1b) to be as follows:
QL(CLAbin - CLAbout) ) QG(CGAbout - CGAbin)
(1)
where the symbol b in subscripts LAbin, LAbout, GAbout, GAbin, and so on represents the bulk concentration. Consider the local mass balance of solute A in the system from x ) 0 to x to be as follows (see Figure 1b):
QL(CLAbin - CLAbx) ) QG(CGAbout - CGAbx)
(2)
where the symbol x in subscripts LAbx and GAbx
represents the concentration at location x. Consider the local mass balance of solute A in the system from x ) x to x ) ∆x to be as follows (see Figure 1):
QL(CLAbx - CLAbx + ∆x) ) / ) (3) K h Lnπdo(dx)(CLAbx - CLAbx
where K h L is a module-averaged overall mass-transfer coefficient based on the liquid phase (i.e., independent of the module length), n the number of filaments in the / module, and CLAbx a hypothetical liquid-phase concentration in equilibrium with the bulk gas-phase concen-
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Figure 2. Schematic of a HFM module for once-through co-current flow operation.
tration. The solute distribution coefficient mi is defined as follows (see Figure 1a):
mi
CGAbx / CLAbx
)
CGAw CLAw
(4)
Therefore,
(
)
CGAbx h Lnπdo dx CLAbx -QLdCLAbx ) K mi -
∫CC
-
∫CC
1
LAbout
LAbout
LAbin
LAbout
CGAbx (CLAbx) mi
dCLAbx )
nπdo QL
∫0LKh L dx
(6)
1 dCLAbx ) CLAbx - (CGAbx/mi) nπdoL A K hL ) K h (7) QL QL L
[
]
QL 1 × A 1 - (1/mi)(QL/QG) CLAbin - (1/mi)(CGAbout) (8) ln CLAbout - (1/mi)(CGAbin)
[
]
Case 1. If the influent concentration of solute A in the gas phase is zero (that is, CGAbin ) 0), then
K hL )
[
] [
QL 1 × A 1 - (1/mi)(QL/QG) CLAbin - (1/mi)CGAbout (9) ln CLAbout
]
Case 2. If the influent concentration of solute A in the gas phase is zero (that is, CGAbin ) 0) and miQG . QL, then
[
]
QL CLAbin - (1/mi)(CGAbout/CLAbout) K hL ) ln A CLAbout
do do 1 + + ) K hL midikG midtlmkm resistance: overall gas phase membrane do (11) d ok L liquid phase
(5)
where A ) nπdoL (that is, the interfacial area between both gas and liquid phases). Consider the integration of the aforementioned equation with the help of eqs 1 and 2. Then,
K hL )
gas/liquid microporous HFM system. As we know (see 1/K h w with organic in tube (see Table 2-1 on page 733 of the work by Ho and Sirkar6),
(10)
Case 3. Generally, mass-transfer resistance from the liquid phase is much greater than mass-transfer resistance from both the membrane and gas phase in the
where kG, km, kL are the individual mass-transfer coefficients of the gas, membrane, and liquid phase, respectively; do and di are the outer and inner diameter of a filament, and dtlm is the logarithmic mean diameter of hollow fiber. Then,
1 1 = K h L kL
(that is, K h L = kL)
(12)
Also, case 2 (miQG . QL) has been met at the same time. Then,
kL )
[
]
QL CLAbin - (1/m1)(CGAbout/CLAbout) ln A CLAbout
(13)
2.2. Co-current Flow Operation. Figures 2 and 1 are the same for the once-through mode, except that Figure 2 is under the co-current flow operation conditions and Figure 1 is under the counter-current flow operation conditions. All notation in Figure 2 refers to Figure 1. The same procedures, same conditions, and same assumptions as those made in the previous section are used to estimate the overall mass-transfer coefficients based on the shell-side fluid (i.e., liquid phase). First, consider the overall mass balance of solute A in Figure 2:
QL(CLAbin - CLAbout) ) QG(CGAbout - CGAbin)
(14)
Second, consider the local mass balance of solute A in the system from x ) 0 to x (see Figure 2):
QL(CLAbin - CLAbx) ) QG(CGAbx - CGAbin)
(15)
Third, consider the local mass balance of solute A in the system from x ) x to x + ∆x (see Figure 2): / QL(CLAbx - CLAbx + ∆x) ) K h Lnπdo(dx)(CLAbx - CLAbx ) (16)
Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7839
Figure 3. Estimation of the overall mass-transfer coefficients based on the shell-side fluid in the once-through mode (i.e., without recirculation) of HFM-based solvent extraction with counter-current flow operation.
The solute distribution coefficient mi is defined as follows (see Figure 1a):
mi )
CGAbx / CLAbx
)
CGAw CLAw
(17)
(
)
-
∫CC
LAbout
LAbin
nπdo 1 dCLAbx ) QL CLAbx - (CGAbx/mi)
K hL )
[
]
-QL 1 × A 1 + (1/mi)(QL/QG)
{
Then,
CGAbx h Lnπdo dx CLAbx -QL dCLAbx ) K mi
Consider the integration of eq 19 with application of eqs 14 and 15. Then,
ln 1 -
}
[1 + (1/mi)(QL/QG)(CLAbin - CLAbout)] CLAbin - (1/mi)CGAbin
(20)
(18)
∫0LKh L dx (19)
Finally, comparison of the overall mass-transfer coefficients based on the shell-side fluid (liquid phase) between the counter-current flow and co-current flow operation in the gas/membrane/liquid system is shown in Table 1.
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Figure 4. Estimation of the overall mass-transfer coefficients based on the shell-side fluid in the once-through mode (i.e., without recirculation) of HFM-based solvent extraction with co-current flow operation.
3. Estimation of the Overall Mass-Transfer Coefficients Based on the Shell-Side Fluid (Aqueous or Organic Phase) in the Aqueous Phase/ Membrane/Organic Phase System
current flow operation condition, and Figure 4 is the co-current flow operation condition.
The solution procedure used to derive mass-transfer coefficients based on the shell-side fluid in the aqueous phase/membrane/organic phase system is similar to the previous procedures. The illustrations for describing the mass-transfer situation and concentration profiles with the estimated overall mass-transfer coefficients in the once-through mode of the HFM-based solvent extraction are shown in Figures 3 and 4; Figure 3 is the counter-
We note several apparent errors that have been found in the literature. Tables 5 and 6 show a list of errors in the literature and corrections from this work. For the error of eq 15 in Bao et al.,5 the correction is shown in Appendix A. For Figure 12 of Costello et al.,4 they observed a significant difference between their experimental and theoretical results that they could not explain. We have found that the theoretical results
4. Apparent Errors Found in the Literature
Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7841 Table 5. Errors in the Literature (References 6 and 5) and Correction from This Work reference Ho and
Sirkar6
Bao et al.5
error d d dto 1 to to ) + + Kw midtikios midtlmkimo dtokiws which is an equation from Table 2-1 on page 733 of ref 6 N 2π ∂c dθ 0 ∂ξ w,i i)1 2Rk )2 Shloc ) N D 2π (cb - cw)i dθ
∑∫
()
correction d d dto 1 to to ) + + Kw midtikiot midtlmkimo dtokiws
N
Shloc )
∑∫ i)1
2Rk D
∑∫ )2
i)1
N
∑∫
0
i)1
2π
0
2π
0
() ∂c
∂ξ
dθ
w,i
(Cb - Cw)i dθ
which is eq 15 from ref 5 (page 2349)
where c ) (C - Cw)/(Cb - Cw), as defined in eq 5 of ref 5
(1) Figures 11 and 12, page 2354 of ref 6
Must eliminate the curves in Figures 11 and 12 of ref 6 representing Prasad and Sirkar,3 because of the fiber packing fraction used in Prasad and Sirkar3 (φ < 0.2). For verification, see the middle of the left-hand side, page 458 of ref 7. π(Di/4) π(Di/4) 1-φ 1 1 - φ dh 1 = + and at . ) dh φ do nπ(d 2/4) φ do do nπ(d 2/4)
(2) 100 × φ ) 4-40 for Prasad and Sirkar,3 Table 4, ref 5 Note below Table 4 of ref 5: Note that the factor (1 - φ)/φ is equal to the ratio of the hydraulic diameter to the fiber diameter
[
o
]
o
This will affect eqs 21, 22, 23, 25, and 26 in ref 5, as well as Table 4 of ref 5. Inappropriately use Re ) 450 in Figure 12 of ref 5 to represent data from Prasad and Sirkar.3
Eliminate the curve representing the Prasad and Sirkar3 data in Figure 12 of ref 5 because Re ) 2RVb/v ) 0-125 for Prasad and Sirkar3 with φ < 0.2 (see Table 4 of ref 5).
Inappropriately use Sc ) 1000 in Figures 11 and 12 of ref 5 for fitting ref 1 and ref 4.
Sc ≈ 500
Equation 23 of ref 5 and Re(1 - φ)/φ of Table 4 of ref 5 for fitting Costello et al.4
Make note of the difference between dh and de, where dh )
4 x (cross-sectional area) 4 x (cross-sectional area) and de ) wetted perimeter total fiber circumference
where wetted perimeter ) total fiber circumference + modular inner circumference. Thus, the wetted perimeter is greater than or equal to the total fiber circumference. Generally, for middle and high φ, dh = de. Verification by comparison of dh and de is given in Table 1 of ref 4. Table 6. Errors in the Literature (References 4, 3, and 1) and Correction from This Work reference Costello et al.4
error
correction
Theoretical results of Figure 12 of ref 4, packing densities used in Prasad and Sirkar,3 Table 3 of ref 4.
See Figure 5. Packing densities of
