Modeling of Microporous Hollow Fiber Membrane Modules Operated

784

Ind. Eng. Chem. Res. 1997, 36, 784-793

Modeling of Microporous Hollow Fiber Membrane Modules Operated under Partially Wetted Conditions A. Malek, K. Li,* and W. K. Teo Department of Chemical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Gas-liquid mass transfer has been studied theoretically in a microprous hollow fiber membrane module operated under partially wetted conditions in the laminar flow regime. Dissolved oxygen removal has been used as an example for the simulation study. The mathematical model developed consists of nonlinear partial differential equations and is solved using the orthogonal collocation technique. The effect of membrane wetting pressures on the overall mass transfer performances of the module has been examined. The results indicate that under partially wetted operating mode, a maximum overall mass transfer coefficient is attainable with respect to the water velocity, which is completely different from the results obtained under both wetted and nonwetted conditions where the overall mass transfer coefficient is generally increased with water velocity. The phenomena of partial wetting of the membrane may provide an explanation to the observation that the overall mass transfer coefficient is increased to a maximum value and then is decreased with water velocity in a gas-liquid contactor. Introduction Microporous hollow fiber membrane modules employed in gas absorption or stripping processes have attracted considerable attention in the past and were first studied by Zhang and Cussler (1985a,b) and Cooney and Jackson (1989). The hollow fiber membrane used by them acts as a fixed interface and keeps the gas and liquid phases separated while mass transfer of gases takes place through the membrane. Depending on the membrane material, the physicochemical properties of the liquid, and the operating pressures employed, the pores of the membrane can be filled with either gas or liquid, which will result in great differences in the mass transfer resistance of the membrane employed (Karoor and Sirkar, 1993). Compared with conventional absorption or stripping processes such as bubble columns and packed beds, there are several advantages of using microporous hollow fiber modules for gas absorptions or stripping. These include larger interfacial area per unit volume, independent control of gas and liquid flow rates without any flooding, loading, foaming, etc., and known gasliquid interfacial area. The advantages have led to a number of investigations on the use of the membrane modules for gas absorption and stripping (Yang and Cussler, 1986; Ahmed and Semmens, 1992; Kreulen et al., 1993a,b; Costello et al., 1993; Li et al., 1994). In most studies, however, emphasis has been focused on nonwetted operating mode whereby the membrane pores are filled with gases. The primary reason for such an operating mode is quite obvious, since the usual physical absorption processes used in industry are limited by the mass transfer rates in the liquid phase. The overall mass transfer coefficients in these cases are between 10-5 and 10-4 m/s (Danckwerts, 1970). Adding the mass transfer resistance of the nonwetted membrane, which is several orders of magnitude smaller than that of liquid, will result in a negligible effect on the overall mass transfer coefficient. Due to the considerably larger interfacial area, the overall mass transfer per unit volume, Kla for the membrane module, * To whom correspondence should be addressed. Tel: (65) 7726388. Fax: (65) 7791936. E-mail: [email protected]. S0888-5885(96)00529-5 CCC: $14.00

is therefore, much higher than that of conventional columns. Thus, the membrane-based gas absorption and stripping offer a promising alternative. It is pertinent to point out, however, that the membrane resistance in the module can be significantly higher if it is operated in the wetted mode (Kreulen et al., 1993c). For given hollow fiber membranes and liquid, the operation of the module under nonwetted, wetted, or partially wetted modes, shown in Figure 1, will essentially depend on the operating pressure. In order to prevent the membrane from getting wet, the liquid pressure should always be operated lower than the wetting pressure of the membrane employed (Kim and Harriott, 1987; Callahan, 1988; Sirkar, 1992). For a process where the liquid is fed into the fiber lumen of a membrane module, because of the small size of the fiber diameter, pressure drop of the liquid in the fiber lumen will inevitably be increased with the liquid velocity and length of the fiber used according to the HagenPoiseuille law. In this case, the membrane may be partially wetted due to the pressure gradient along the fiber lumen (Figure 1c). Such a phenomenon has been observed by Tai (1995), who found that water droplets are formed at the outer surface of the hollow fiber membrane, especially near the inlet of the membrane module. The overall mass transfer coefficients obtained by Tai (1995) at different liquid velocities for the O2H2O system exhibit a maximum value which cannot be explained by existing mass transfer models (Yang and Cussler, 1986). Conventionally, for the membrane modules operated under nonwetted mode, increase of the water flow will generally increase the overall mass transfer coefficients for the O2-H2O system. The existence of a maximum value of the overall mass transfer coefficient may suggest that the membrane is partially wetted from the inlet of the module due to high liquid pressure in the fiber lumen. The partially wetted membrane may result in a reduction of the overall mass transfer coefficient compared with the nonwetted mode, hence giving a maximum value of the overall mass transfer coefficient as a function of the water velocity. In this paper, a theoretical analysis is carried out for microporous hollow fiber membrane modules employed for gas absorption or stripping processes. Removal of © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 785

developed for the membrane module operated under nonwetted, wetted, and partially wetted conditions. Nonwetted Hollow Fiber. Using the above assumptions, the steady state material balance equations for oxygen transfer across the membrane are given as follows:

(

)

∂Ca ∂2Ca 1 ∂Ca ) Da + ∂z r ∂r ∂r2

Uz

( ( ))

2Um 1 -

r R

2

(

)

∂Ca ∂2Ca 1 ∂Ca ) Da + ∂z r ∂r ∂r2

(1)

with boundary conditions

Ca|z)0 ) Caf

Figure 1. Microporous hollow fiber membrane operated under (a) nonwetted mode; (b) wetted mode; and (c) partially wetted mode.

dissolved oxygen from water is used as an example to illustrate the module performances operated under the nonwetted, wetted, or partially wetted conditions. It is realized that unless the membranes used for absorption or stripping processes are extremely hydrophobic, partial wetting of the membranes due to the increased pressure gratient in the fiber lumen will become inevitable. Therefore, analysis of this problem is useful in studying the effects of operating conditions such as feed flow rate and module length on the overall performance of the microporous fiber membrane modules. Formulation of the Model Theoretical models are developed to analyze microporous hollow fiber membrane modules operated under the wetted, nonwetted, and partially wetted modes. The stripping of dissolved oxygen from water is used as an example to describe the module performance. Water saturated with dissolved oxygen is fed in the fiber lumen, while purified nitrogen, used as sweep gas, is fed into the shell side. The following assumptions are adopted for the model development: (1) Oxygen concentration in the liquid phase is small so that the density and the flow velocity in the liquid stream are constant along the fiber length. (2) There is fully developed laminar flow in the tube side. (3) There is negligible axial dispersion in the tube side. (4) Equilibrium is attained instantaneously between the bulk gas flow and the liquid interface. (5) Shell side pressure is constant. (6) The system is operated isothermally and in steady state. Assumption 1 approximates the system realistically and simplifies the mass balance equations considerably. Assumption 2 is valid for a Reynolds number not exceeding 2100. The entrance effect for the flow profile in the fiber lumen is ignored because it is not significant compared to the length of fiber used. Assumption 3 is admissible since the flow velocity in the tube side can generally be expected to be greater than axial diffusion. On the basis of the above assumptions, the general conservation equations for the stripping process can be

(0 e r < R)

(2a)

)0

(0 e z e L)

(2b)

Ca|r)R ) Cai

(0 e z e L)

(2c)

|

∂Ca ∂r

r)0

The shell side material balance can be coupled to the liquid side mass transfer as follows. Taking a differential segment of the membrane module, the number of moles of oxygen entering the shell side is equal to that diffusing across the liquid boundary. Hence

∆nae ) -2πRn∆zDa

|

∂Ca ∂r

r)R

(3)

The porosity of the membrane used has not been taken into the consideration as it has been shown that the porosity of the membrane gives no significant effect on the mass transfer (Kreulen et al., 1993). Taking limits

|

∂Ca dnae ) -2πRnDa dz ∂r

r)R

(4)

Writing nae in terms of the shell side mole fraction (nae ) yaNg) gives

(

| )

∂Ca dya 2πRn ) -Da dz Ng ∂r

r)R

(5)

where Ng is total molar flow rate in the shell side. Since the bulk gas is in equilibrium with the liquid interface, Henry’s law holds

ya )

CaiHa Cfps

(6)

where Cf is the total liquid molar concentration. Substituting eq 6 into eq 5, we have

(

| )

∂Ca dCai 2πRnpsCf ) -Da dz NgHa ∂r

r)R

(7)

The boundary condition in this case is

Cai|z)0 ) 0

(8)

786 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

In dimensionless form, eqs 1-8 above become

(

)

∂xa ∂2xa 1 ∂xa ψa ) + ∂Z (1 - Γ2) ∂Γ2 Γ ∂Γ

|

∂xa dxai ) -φa dZ ∂Γ

Γ)1

(9)

(10)

with boundary conditions

xa|Z)0 ) xaf

(0 e Γ < 1)

xai|Z)0 ) 0

|

∂xa ∂Γ

(11a) (11b)

)0

(0 e Z e 1)

(11c)

xa|Γ)1 ) xai

(0 e Z e 1)

(11d)

Γ)0

A few points are worth mentioning here. In these equations, the liquid phase concentrations are nondimensionalized on the basis of the feed oxygen concentration, Caf. Using Caf to nondimensionalize the liquid concentrations gives a better normalization of this variable. In addition, eq 11d above implies that, for the nonwetted operation, the inside wall concentration is equal to the interface concentration. It should also be pointed out that the boundary conditions can be easily modified to simulate either a constant wall concentration (including zero) or a constant wall flux process. For a constant wall concentration model, the equilibrium interface expression, eq 10, can be dropped. Instead, xai is set equal to the wall concentration throughout the length of the fiber. Similarly, for a constant wall flux process, the equilibrium interface expression can be dropped. In its place, the following boundary condition can be used

|

∂xa ∂Γ

Γ)1

) constant

(0 e Z e 1)

Wetted Hollow Fiber. When the microporous hollow fiber membrane is totally wetted, the general conservation equations remain the same, except that the interface concentration and the fiber wall concentration are generally not the same. These two concentrations are now related as follows (Karoor and Sirkar, 1993)

Da

|

∂Ca ∂r

r)R

) -Da

1 (C | - Cai) τ (ln Ro - ln R)R a r)R (12)

|

Γ)1

) -ζ(xa|Γ)1 - xai)

through the hollow fibers, i.e. wetting of the fiber, however, becomes possible if the hydraulic pressure drop across the fiber wall exceeds the wetting pressure. Therefore, the wetting phenomena of a hollow fiber module can be related to the hollow fiber wetting pressure, Pw, described by the Laplace equation (Sirkar, 1992). The fiber wall is considered to be wetted for portions of the fiber length where the hydraulic pressure drop across the fiber wall exceeds the wetting pressure (Figure 1c). The shell side pressure can generally be taken as atmospheric pressure, while the tube side gauge pressure can be calculated using the HagenPoiseuille equation as follows

pt )

32Umµ(L - z) di2

(14)

where z is the length of fiber measured from the inlet. On the basis of eq 14, the pressure profile in the fiber lumen can be calculated and is shown in Figure 2. As can be seen, the pressure in the fiber lumen is dependent on both the liquid velocity and the length of the fiber. Hence, the partially wetted model solves for mass transfer using the wetted hollow fiber model where the wall pressure drop is larger than the wetting pressure, pw, and switches to the nonwetted model otherwise. Evaluation of Overall Mass Transfer Coefficient. The overall mass transfer coefficient, Kl, for a particular operating condition can be determined on the basis of the following expression derived by Sirkar (1992) and Tai et al. (1994)

Kl )

-Ql πdiLn(1 + λ)

ln[(1 + λ)xae - λ]

(15)

The “mixing cup” outlet concentration, xae, required to calculate the overall mass transfer coefficient is based on the following formula (Skelland, 1974a)

or in dimensionless form

∂xa ∂Γ

Figure 2. Pressure profile in the fiber lumen for module 1.

(13)

where ζ represents the mass transfer resistance of the stagnant liquid in the fiber wall. Here again, xai is in equilibrium with the bulk gas phase concentration. Partially Wetted Hollow Fiber. In actual hydrophobic hollow fiber systems, it is likely that the membrane is partially wetted. The extent of wetting of the hollow fibers is as yet an unknown function of the operating conditions. Water may not penetrate through the hydrophobic hollow fibers due to the presence of a finite wetting pressure acting along the direction opposite to that of penetration. The penetration of water

∫0R2πrUxa dr xae ) ∫0R2πrU dr ∫01Γ(1 - Γ2)xa dΓ ) ∫01Γ(1 - Γ2) dΓ

(16)

Analytical solutions for the gas diffusion through nonwetted hollow fiber can be derived using the methods given by Leveque and Greatz (Skelland, 1974b,c). The Graetz solution is an infinite series solution which


is obtained by solving eq 9 using the method of separation of variables. This series solution is given as

Sh ) 0.5Gzθ where

∞

1-

∑ j)1

θ) ∞

1+

∑ j)1

( )

-4Bj dφj βj2

dΓ

( )

-4Bj dφj βj2

dΓ

[ ] ( ) [ ] -βj

exp

exp

()

1 ∂

2L di

(17a)

( )

Γ ∂Γ

Γ

∂xa ∂Γ

(j) )

2L

dxa

di

dZ

Re Sc

(17b)

βj ) (-1)j-1 × 2.84606βj-2/3

(17c)

(j) )

ψa (1 - Γ(j)2)

∑i B(j,i) xa(i)

(21)

Note that the ordinary differential equations are solved for j ) 1, 2, ..., M since at j ) M + 1 (i.e. the wall concentration) xa(M+1) ) xai from boundary condition 11d. Furthermore, the expression for the equilibrium interface concentration, xai , i.e. eq 10, is converted to

dxai

Γ)1

(20)

where j ) 1, 2, ..., M and i ) 1, 2, ..., M + 1.

βj ) 4(j - 1) + 8/3; j ) 1, 2, 3, 4, ...

( )

∑i B(j,i) xa(i)

(19)

This converts eq 9 above into

and

Bj dφj 2 dΓ

∑i A(j,i) xa(i)

where j ) 1, 2, ..., M + 1 and i ) 1, 2, ..., M + 1.

Re Sc

-βj2

Γ)1

∂xa (j) ) ∂Γ

(17)

2

Γ)1

internal points was chosen. The first and Laplacian derivatives are given by the A and B matrices as follows

dZ

) -φa

∑i A(M+1,i) xa(i)

(22)

where i ) 1, 2, ..., M + 1. ) 1.01276βj-1/3

(17d)

The Leveque solution, on the other hand, is more restricted since it is derived on the basis of the assumption that concentration gradients in the liquid are limited to a thin layer near the fiber wall. This assumption also means that ∂xai/∂Z can be taken as constant along the fiber length. One important consequence of this assumption is that the model is only applicable for Gz numbers exceeding 400, that is, for short fibers or large liquid flow rates. The Leveque equation is given as

Sh ) 1.615Gz1/3

The solution algorithm for the nonwetted operating mode is given in Figure 3. Wetted Mode. Similarly, the ODES relating the radial concentrations, i.e. eq 21, are solved for j ) 1, 2, ..., M. In contrast to the nonwetted mode, the wall concentration xa(M+1) must be solved from boundary condition 13 as follows

∑A(M+1,i) xa(i) + A(M+1,M+1) xa(M+1) )

-ζ(xa(M+1) - xai) (23)

where i in the summation range from 1 to M. Rearranging, we get

ζxai +

(18) xa(M+1) )

It is clear from the derivation of the analytical solutions of Graetz and Leveque that these two models are only applicable to nonwetted mode of operation.

Method of Solution The system of eqs 9-13 includes several partial differential equations of first and second order. Solution of these equations can be attempted in a number of different ways, for example, the well-known implicit (Lapidus, 1962) and explicit (Liu, 1969) finite difference techniques, the quasilinearization techniques developed by Lee (1966), etc. However, Finlayson (1971) has shown that orthogonal collocation techniques yield more accurate results with less computation time compared to the above mentioned methods. We therefore use this technique for all solutions in this paper, and the solution details are given below. Nonwetted Mode. Using symmetrical orthogonal collocation for cylindrical coordinates, a solution for 11

∑i A(M+1,i) xa(i)

(24)

A(M+1,M+1) + ζ

The equilibrium interface concentration, xai, which is required in determining the wall concentration is as given previously (eq 22). Therefore, eqs 21, 22, and 24 can be simultaneously solved for the wetted operating mode. The solution algorithm is given in Figure 4. Partially Wetted Mode. The solution for partially wetted mode is straightforward. At points along the fiber length where pt < pw, the nonwetted model is used for calculation. When pt > pw, the model for wetted mode must be employed. The detailed algorithm is given in Figure 5. Experimental Section The hollow fibers employed were specially prepared for this study. The fibers were made of polyethersulfone (Goodfellow Company of Cambridge, England) by the phase inversion technique in a dry-wet spinning process (Zhang, 1994). A polyethersulfone/NMP/ethylene glycol ternary spinning dope was spun through the orifice spinneret into a water bath using water as both the internal and external coagulant. The nascent


Figure 3. Solution algorithm for calculation of the overall mass transfer coefficient of the module operated under the nonwetted condition.

hollow fibers formed were gelated and kept in water bath for 3-4 days. They were then dried under ambient conditions (25 °C) for 2 days. After pore size, porosity, and tortuosity of the fibers were tested using methods given by Mulder (1991), the fibers were then mounted in a stainless steel tube acting as the module shell. The hollow fibers through the two openings at the ends of the module were glued with epoxy resin and then cured for 2 days. The geometrical characteristics of the module prepared are shown as module 3 in Table 1. The apparatus for the removal of dissolved oxygen from water using the membrane module is shown in Figure 6. Deionized water, which was saturated with purified air at ambient temperature, was used in all the experiments. The feed water containing the saturated dissolved oxygen was fed into the hollow fiber lumen, while purified nitrogen used as a purge gas was introduced into the shell side. The inlet flow rate of water and nitrogen were controlled by a rotameter and a mass flow controller (Brooks 5850), respectively. DO concentrations in both inlet and product streams were continuously monitored using a DO meter (Martek at a range of 0-20 ppm with an accuracy of (0.1 ppm). It usually took about 30 min to stabilize DO concentration in the product stream after each time that the H2O or N2 flow was changed. The reproducibility of the DO data are generally good. The measured DO values was

used for obtaining xae and the overall mass transfer coefficient, Kl, according to eq 15. Results and Discussion The dissolved oxygen removal has been used as an example for the analysis of the microporous hollow fiber membrane modules operated under the nonwetted, wetted, and partially wetted modes. Simulations were carried out using all the three modules shown in Table 1, while experimental results were only obtained from module 3. Analysis of the theoretical and experimental results obtained are presented below. Mass Transfer under Nonwetted, Wetted, and Partially Wetted Conditions. Figures 7-9 study the mass transfer of the microporous hollow fiber membrane modules operated under the nonwetted, wetted, and partially wetted conditions. The simulation results have been plotted in terms of Sherwood number, Sh, versus Graetz number, Gz. The definition of the Sh and Gz are given in the notation of the paper. The mass transfer coefficient, Kl, used for calculation of the Sherwood number is the overall one obtained from eq 15. In Figure 7, the effect of operating conditions on the module performance is illustrated. As could be expected, an increase in Graetz number due to the increase in feed flow rate results in an increase in mass transfer coefficient for the nonwetted mode. The ana-


Figure 4. Solution algorithm for calculation of the overall mass transfer coefficient of the module operated under the wetted condition.

lytical solutions of Leveque and Graetz (Skelland, 1974b,c) have also been plotted and show almost identical results at higher Graetz number compared with the theoretical model derived here. At low Graetz number, the Leveque solution predicts a lower Sherwood number. This is expected due to the limiting assumptions used in the model. Operation under wetted mode is also demonstrated in Figure 7. In this case, the pores of membrane are filled with liquid and the mass transfer resistances are thus increased. As a result, the overall mass transfer coefficient is dramatically reduced, which is clearly reflected by the plot (curve 4) in Figure 7. It is interesting to note that under the wetted mode of operation, the increase in the Sherwood number with the Graetz number is much less compared to that for nonwetted mode. This indicates that the membrane resistance may also become important in the overall mass transfer process. For the oxygen-water system, it is generally believed that resistance in the liquid film controls oxygen mass transfer. As a result, increase of water flow will increase the overall mass transfer coefficient. The

existing theoretical models, however, cannot explain well the situation where the overall mass transfer coefficient reaches a maximum value and then decreases with water flow rate. As has been illustrated in the preceeding section, it is likely that the membrane modules are operated under the partially wetted conditions if the liquid pressure at a point in the fiber lumen is higher than the wetting pressure. For a given liquid and membrane material, i.e., given wetting pressure, the overall mass transfer coefficients of the membrane modules may be functions of both the liquid flow rate, i.e. Graetz number, and the wetting pressure. Figure 8 shows the simulation results for both modules 1 and 2 at different wetting pressures. It can be seen from the figure that as the Graetz number is increased, the Sherwood number for the modules operated at either wetted or nonwetted modes is increased. When the wetting pressure is greater than 1.5 bars, a maximum value of the mass transfer coefficient is observed. This maximum value is dependent on both the Graetz number and the wetting pressure. The figure also indicates that the maximum Sherwood number occurs


Figure 5. Solution algorithm for calculation of the overall mass transfer coefficient of the module operated under the partially wetted condition. Table 1. Characteristics of Fibers and Modules module number dimension

1

2

3

fiber o.d., mm fiber i.d., mm number of fibers length of fibers, mm

0.40 0.34 10 1000

0.40 0.34 10 500

0.85 0.56 10 500

at the higher Graetz number as the wetting pressure is increased. For module 1, for example, the maximum mass transfer coefficient can be obtained at a Graetz number of Gz ) 7 for a wetting pressure of pw ) 1.5 bars. When the wetting pressure is increased to pw ) 12 bars, the maximum mass transfer coefficient is not attainable unless the Graetz number is increased to Gz ) 200. At the low wetting pressure of pw ) 1.05 bars, the maximum value of the overall mass transfer coefficient cannot be observed and the Sherwood number plotted in Figure 8 generally decreases as the Graetz number is increased. It is apparent that as the wetting pressure is reduced, the maximum mass transfer coef-

Figure 6. Experimental apparatus.

ficient is further shifted to the lower values of Graetz number which are beyond the range of the Graetz number employed in this study. Figure 8 further revealed that the mass transfer coefficients for module 2 are comparatively higher than those for Module 1 at the same wetting pressure. This may be attributed to the shorter length of Module 2 where the percentage of


Figure 7. Sherwood number versus Graetz number for the modules operated under wetted and nonwetted modes.

Figure 10. Comparison between experimental results and theoretical calculations; filled circles are the experimental data.

Figure 8. Sherwood number versus Graetz number for the modules operated under nonwetted, wetted and partially wetted modes.

values of Graetz number, increase of the wetting pressure results in a sharp increase in the Sherwood number, indicating that a shift from wetted mode to nonwetted mode takes place with slight incremental increase in the wetting pressure. When the Graetz number becomes larger, the shift of the operating conditions from wetted mode to nonwetted mode becomes gradual and the partially wetted mode can result. Under the latter operating mode, a maximum in the overall mass transfer coefficient for dissolved oxygen transfer can be observed. Figure 9 also shows that at low wetting pressures, the overall mass transfer coefficient reaches the maximum value at an extremely low Graetz number, and thereafter, the overall mass transfer coefficient decreases as the Graetz number is increased. It may, therefore be concluded that for the gas absorption and stripping processes, the inherent wetting pressure for a given membrane and liquid must be higher than the pressure drop along the fiber lumen so that the module can be operated at the optimum condition, i.e. nonwetted operating mode. In reality where the partially wetted mode becomes inevitable, the optimum flow condition, i.e. the Graetz number at which maximum Sherwood number is obtained, must be operated in order to achieve the maximum mass transfer. Comparison with Experimental Data. Figure 10 compares the theoretical results with the experimental data obtained from module 3. It can be seen that the observed experimental data shows a maximum value of the mass transfer coefficient at a Graetz number around 350. The experimental data agrees exceptionally well with the theoretical results at lower Graetz number region, i.e. nonwetted mode. As the Graetz number increases, the partial wetting of the module becomes inevitable resulting in a decrease in the mass transfer coefficient. Overall, the experimental data is in fairly good agreement with the simulation results obtained at pw ) 1.5 bars. When the partial wetting of the fiber takes place in the module, the experimental results deviate from the theoretical results; however, the general trends are sufficiently clear to merit a meaningful comparison between the simulation and experimental results. It was also observed that after a week operation, the DO concentration in the product stream was slightly increased for the water flow range studied. Nevertheless, the maximum overall mass transfer coefficient calculated from the measured DO data was always obtained. Performance of the Module. The performance of the module for removal of the dissolved oxygen under nonwetted, wetted, and partially wetted conditions has

Figure 9. Sherwood number versus Graetz number for the membrane with different wetting pressures. (The simulated results are obtained for module 2.)

nonwetted mode operated is considerably higher than that in module 1. Figure 9 illustrates the effects of Graetz number and wetting pressures on Sherwood number plotted in threedimension. It can be seen from the figure that at low


Nomenclature

Figure 11. Performances of the membrane module for dissolved oxygen removal under nonwetted, wetted and partially wetted modes.

been simulated, and the results are presented in Figure 11. As could be expected, the nonwetted mode gives the best dissolved oxygen removal in the product stream, while the wetted mode of operation is least efficient. The performance of the module for partially wetted conditions approaches that for the nonwetted conditions as the wetting pressure of the membrane is increased or the feed flow rate is reduced. This is understandable as reduction of the feed flow rate will reduce the velocity of the liquid feed and hence the pressure drop in the fiber lumen. As long as the pressure drop in the fiber lumen is low, a nonwetted operating mode is expected to prevail, thus giving better performance in terms of dissolved oxygen removal. Conclusions A theoretical model for a microporous hollow fiber membrane module operated under partially wetted mode has been postulated. This model provides an explanation for a situation where the overall mass transfer coefficient is increased to a maximum value and then is decreased with the liquid velocity. Numerical techniques using the orthogonal collocation principle have been used for the solution of the resulting equations. The equations, of the partial differential, nonlinear type, are solved faster by the present method because the solutions are effected at only selected positions along the radius, namely the collocation points. Dissolved oxygen removal from water has been used as an example to demonstrate the present model. Numerical simulation of the dissolved oxygen mass transfer with perturbation of the wetting pressure of the membrane definitely reveals that the mass transfer performance of the membrane module is dependent on the water flow rate and nature of the membrane used. For a given membrane material, the optimum operating conditions of the membrane module for dissolved oxygen removal could be obtained at conditions that the flow rate of water is controlled at a level where the pressure drop across the membrane is less than the wetting pressure of the membrane. Acknowledgment The authors gratefully acknowledge the research funding (RP3930639) provided by the National University of Singapore. We would also thank Dr. D. Wang for preparation of the hollow fibers and the hollow fiber module and Dr. S. Farooq for the useful discussion on this article.

AR ) fiber cross-sectional area (m2) Ca ) concentration of dissolved oxygen (kmol/m3) Cae ) concentration of dissolved oxygen in the outlet stream (kmol/m3) Cai ) interface concentration of dissolved oxygen (kmol/ m 3) Cf ) liquid concentration (kmol/m3) Caf ) dissolved oxygen concentration in feed (kmol/m3) di ) fiber inner diameter (m) Da ) diffusivity of oxygen in water (m2/s) (Coulson and Richardson, 1980) Gz ) Umdi2/(zDa); Graetz number Ha ) Henry’s constant of oxygen (bar/mole fraction) (Perry and Chilton, 1973) Kl ) overall mass transfer coefficient in liquid phase (m/s) L ) effective fiber length (m) nae ) molar rate of oxygen diffusing from tube side to shell side (kmol/s) n ) number of fibers in permeator Ng ) gas (shell side) molar rate (kmol/s) ps ) shell side pressure, bar pt ) gauge pressure in fiber lumen at a given length of the fiber, bar pw ) wetting pressure of the membrane, bar Ql ) liquid flow rate (m3/s) r ) radial coordinate (m) R ) fiber inner radius (m) Ro ) fiber outer radius (m) Sh ) Kldi/Da; Sherwood number Um ) Ql/ARn; average velocity in fiber (m/s) xa ) dimensionless concentration of dissolved oxygen Ca/ Caf y ) gas phase mole fraction z ) axial coordinate (m) Z ) dimensionless axial coordinate, z/L Symbols ) membrane porosity ( ) 0.3 is used for eq 12) τ ) tortuosity of the membrane pores (τ ) 4.5 used for eq 12) Γ ) dimensionless radial coordinate, r/R φa ) (2πnLpsCfDa)/NgHa λ ) Ql/(QgHa) ψa ) (DaL)/(2UmR2) ζ ) (/τ)(1/((ln Ro - ln R)R)) µ ) liquid viscosity Subscripts a ) oxygen e ) module outlet f ) feed g ) gas phase l ) liquid phase

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Received for review August 23, 1996 Revised manuscript received November 20, 1996 Accepted November 26, 1996X IE960529Y

X Abstract published in Advance ACS Abstracts, January 15, 1997.

Modeling of Microporous Hollow Fiber Membrane Modules Operated

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