1471
Ind. Eng. Chem. Res. 1990,29, 1471-1476
Mass Transfer in Single Oil-Containing Microporous Hollow Fiber Contactors Hiroshi Takeuchi,* K a t s u r o k u T a k a h a s h i , and Makoto N a k a n o Department of Chemical Engineering, Nagoya University, Chikusaku, Nagoya, 464-01 Japan
Studies were made of mass transfer in a single hydrophobic microporous hollow fiber (HF) contactor in two operating modes: bound membrane and supported liquid membrane. Two aqueous film mass-transfer coefficients, k, and k,, on the shell and lumen sides were obtained with an extraction system of waterliodineln-heptane. The values of k, were in reasonable agreement with previous correlations, whereas the k, values were given by two empirical correlations for laminar and turbulent flows. These transfer coefficients for the oil-containing hydrophobic HFs were smaller than the literature values for solid tube flows by a factor of 10-15%. The membrane-transfer coefficient had a tendency to decrease with an increase in the inner-to-outer diameter ratio of the HF. Microporous hollow fibers (MHFs) have been widely used in reverse osmosis (RO) modules and gas permeators or separators, because of their large surface area per volume advantage. Most recently, the applicability of MHF contactors to equilibrium separation of aqueous solutions or gaseous mixtures is being explored in two operating modes-supported liquid membrane (SLM) and bound membrane (BM). In the contactors, two fluid flows are almost completely independent; as a result, there is no constraint due to flooding, loading, or channeling. There are, however, some problems to be solved for the practical application of SLMs: the membrane stability and its regeneration. However, the bound membranes are free of such constraints and therefore offer an efficient liquidmembrane extractor or absorber. For contactor design, aqueous film diffusion and membrane diffusion are crucial. Prasad and Sirkar (1987,1988) and Dahuron and Cussler (1988) have reported the correlations of two liquid-phase mass-transfer coefficients, k, and k,, on the tube (lumen) and shell sides in multiple MHF modules. However, it is difficult to make a multiple HF module having a calming section at the entrance and exit on the shell side, while a single MHF contactor can be designed to eliminate any end effects. Although having no commercial use because of a small surface area per volume ratio, such single HF devices might be suitable for investigating interphase mass transfer. In the present study, systematic measurements were made to obtain more accurate correlations of k, and k, in single hydrophobic MHF contactors using an extraction system of waterliodineln-heptane. The membrane mass-transfer coefficient, k,, was also studied. The present extraction system is characterized by use of I2as a solute: (1)the distribution coefficient, m, can be changed only by adjusting the concentration of potassium iodide in an aqueous solution without a significant change in the mass-transfer parameters, and (2) the resistance in I, transport through aqueous boundary film either on the lumen side or on the shell side in an MHF module can be eliminated by utilizing a reaction with thiosulfate. Therefore, 12, determination becomes simpler and more accurate. Theoretical Basis for Experimental Measurements In this work, we have adopted three contacting modes shown in Figure 1 for determining k, k, and k,. In all cases, hydrophobic membrane pores are assumed to be filled with an organic liquid (n-heptane). The aqueous feed was an iodine solution with KI, which contains the iodine species 12,I-, and 1,- (Ramette and Sandford, 1965):
Iz + I-
I
700 deviate remarkably from each line. This suggests that a transition from the laminar to the turbulent flow arose at about 700-800 in the Re range for the aqueous flow inside the oil-containing MHF, unlike fluid flow inside solid tube. Thus, a comparison was made between the present results and the Chilton and Colburn (1934) equation, as shown by dashed line in Figure 3. Figure 4 and 5 show the effects of length and inner diameter of the MHF on Sh, respectively, giving a relation of Sh a (di/LY3. Here, assuming the 2/3 power of Sc from previous theories on mass transfer and heat transfer, we find the following correlation in the Graetz number range ( C r = (di/L)ReSc) of 50-1000 ktdi/D = 1.4(di/L)1/3(diut/u)1/3(v/D)1/3 (10) This equation is lower by about 12% than a result for gas absorption in a BM contactor by Yang and Cussler (1986) as well as that for a nonporous tube by Leveque (1928). In addition, Prasad and Sirkar (1988) have reported that the aqueous tube-side film transfer coefficient in a hydrophobic MHF is smaller than that in a hydrophilic membrane. Such a difference may be due to the interfacial motion of a stagnant oil sublayer adhering on the surface of a hydrophobic MHF. This suggests that a nonslip condition for fluid flow at the interface is not valid for aqueous streamline flows over the oil-containing MHF.
Shell-Side Liquid Film Mass-Transfer Coefficient, k,. The film mass-transfer coefficient for the aqueous flow on the shell side was obtained in the contacting mode shown in Figure lb, where the direction of the solute transfer was from the annular fluid onto the external surface of the oil-containing MHF. The results were also
ds/do
[-I
Figure 8. Generalized shell-side Sherwood numbers for single MHF modules.
examined in general dimensionless correlations. Here, we used a hydraulic equivalent diameter, de, defined as d, do, where d, is the inner diameter of the shell tube, as the characteristic length in the annular space of the single MHF module. The values of k, are plotted against the shell-side fluid velocity in the dimensionless form in Figure 6, wherein solid lines represent eq 11 in the lower Re region and eq 12 in the higher Re region. For laminar flow on the shell side, the 1/3 power of Re is found to be the same as that on the tube side, while for turbulent flow we assumed the exponent 0.8 from the Chilton and Colburn analogy on account of the small experimental range of Re covered in this work. For laminar flow in the annular space, the effect of the length in the flow direction, L, on k, is illustrated as a plot of Sh vs L in Figure 7. The data points located on each straight line are the constant Reynolds number. From the assumption that the Sh varies with the power of de/L, the values of ShRe-1/3(de/L)-1/4are plotted against d,/do in Figure 8. With the exponent of 1/3 on Sc, the following equation can be derived for laminar flow in the annular space: Sh = 0.85(d,/d,)0~45(de/L)1/4Re1/3SC'/3 (11) For the turbulent flow, on the other hand, the effect of L on k,, may be ignored in the same way as that for the usual flows inside a tube, though they are not clear in the
1474 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 6 v 4
I
I
I
I
100
1
Monrad- R l t o n
,'
,,
m
I u
2
4
6
d,/do
[
810
s'u IO 2 v,
20
-1
Figure 9. Deviation of shell-side mass-transfer coefficient for annular flow from the Chilton and Colburn equation for tube-side flow. The theoretical line represents the result for heat transfer in annular space by Monrad and Pelton.
I IO'
present data. The dependence of Sh on the diameter ratio of the shell tube to the MHF, d,/d,, was examined in Figure 9 on the basis of Shcx from the Chilton and Colburn equation for tube-side flows. The present results deviate from their equation with an increase in d,/d,, and then the film transfer coefficient can be expressed by changing the constant, 0.023, into 0.017(d,/d,)0~57:
Sh = 0.017(d,/d,)0~57Re08SC'i3
(12)
Equation 12, shown by the solid line in Figure 9, is also compared with the result of heat transfer in annular tubes (dotted line) by Monrad and Pelton (1942). A dash-dotted line in the figure represents the result of their theoretical computation for heat transfer through a viscous flow in annular space, which is based on the ratio of the velocity profile at the internal surface of the shell tube to that at the external surface of the inner tube. Reasonable agreement is found between the theoretical line and the present data. However, considering the existence of a stagnant oil layer on the outer surface of the MHF, as mentioned above, we can recommend eq 12 for evaluating the aqueous film mass-transfer coefficient on the shell side. In addition, we attempted a comparison between the results obtained for the single MHF modules and previous correlations for multiple MHF extractors. Though being significantly different from that in the single module, the shell-side flow in multiple MHF modules should approach annular flow under a limiting condition. Prasad and Sirkar (1987) have provided a correlation of the form
Sh = 5.8[de(l - 4)/L]Re0%c0.33
(13)
where 4 is the packing fraction of the MHF in the module. Also, Dahuron and Cussler (1988) have developed a correlation for the shell-side flow in multiple MHF extractors:
Sh = 8.8(d,/L)Rel.OS~'/~
(14)
Figure 10 shows the comparison of the shell-side mass-transfer coefficients in the single and multiple MHF modules, in terms of two parameters, d,/L and d,/d,, where Cp in eq 13 corresponds to (d,/dJ2 for the single HF module. Multiple MHFs have more significant effects of d,/L and Re on k,, as compared with single HFs. In their structure, their multiple HF modules had no calming section on the shell side; hence, it is considered that the velocity profile of the shell-side fluid was not fully developed in the devices having comparatively short length. This indicates the possibility of an d,/L overdependence of k,. In the single MHF module, it is noticed that eq 11 tends to converge to the Leveque solution as both values of d,/L and d,/d, become smaller. Such a behavior suggests that there is no substantial difference in the aqueous
102
lo3
Re [ - I Figure 10. Comparison of shell-side mass-transfer coefficients in a single MHF with those for multiple MHFs.
t
'@\
d /do Figure 11. Variations of membrane-transfer Coefficients through the oil-containing MHF with the diameter ratio.
film transfer coefficients on the two sides of the HF. Using eqs 11and 12, we can obtain a critical Reynolds number, Recrit,for the annular flow, giving the transition from the laminar to turbulent region: Recrit= 4.46
X
103(d,/L)0.54(d,/d,)~.26
(15)
Overall, eqs 11,12, and 15 well substantiate the experimental results for the mass transfer on the shell side in the single MHF contactor, as illustrated with the solid lines in Figure 6. Membrane-Transfer Coefficient, k,. Membranetransfer coefficients was examined for two liquid membranes: SLM and BM. In the SLM experiment, iodine was transferred from the shell-side flow into the tube-side fluid containing Na2S203;then, the value of k, from eq 8 was determined by using k, from eq 11or eq 12. In the case of the bound membrane, k, from eq 7a or 7c was evaluated by use of kat and 12, from eq 10 and eq 11or 12, respectively. The results of k, for the various MHFs are shown as a plot of k,/(De/b) against dildoin Figure 11, wherein Dt/6 is a theoretical membrane-transfer coefficient across a homogeneous oil layer of 6 thickness, and hence the ordinate represents the reciprocal of tortuosity of the microporous membrane, 117. It is evident that the value of k , depends not only on the geometry of the MHF but also the operating mode. If the MHFs used in this study had uniform pore structure, the value of k, or T obtained should be independent of dildoas well as the direction of the solute transfer through the liquid membrane layer.
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1475 Thus, it is to be expected that Gore Tex fiber is not uniform in the porosity profile through the wall thickness. According to Pons (1987), the fraction of surface free area, f , of microporous sheets is not identical with the membrane porosity. Also, one can easily observe that microporous poly(tetrafluoroethy1ene)membranes such as Gore Tex and Fluoropore are asymmetrical (Takeuchi et al., 1987). It is considered that the asymmetry of the pore structure in MHF is responsible for the behavior of k, in Figure 11. In the present MHF contactors, it is to be noted that the aqueous-organic interface is not always on the surface of the polymeric solid, as mentioned above. The thickness of the stagnant organic sublayer on the solid surface varies with the hydrophobicity of the polymeric support and also operating conditions such as the pressure difference between the tube side and shell side in an MHF module. This leads to a uncertainty in the path length of solute diffusion across the oil-containing membrane. Also, the pore entrance and exit effects might become more significant with an increase in the thickness of the oil sublayer, as was suggested by Malone and Anderson (1977). When flowing on the tube side or shell side, the oil could be partly interchanged with the membrane phase. This causes an apparent enhancement of k, for the bound membrane. From this discussion,a conclusion can be drawn that the application of the tortuosity factor to the oil-containing MHF membrane is limited. No further discussion of the membrane-transfer coefficient will be made here, since there is no reliable evidence that the aqueous-oil interface was just on the surface of the microporous solid support. Conclusions Studies were made for obtaining more accurate correlations of the mass-transfer coefficients on the tube side and shell side in a single oil-containing MHF contactor, as well as the membrane-transfer coefficient. The experiments were carried out in two operating modes: extraction by a bound-type membrane and permeation across an SLM. The tube-side mass-transfer coefficients are comparable to the Leveque solution and to previous correlations for laminar flow inside tubes. Also, for laminar flow in the annular space, the film mass-transfer coefficient approaches the Leveque solution with a decreasing ratio of the hydraulic diameter to the fiber length. The shell-side mass-transfer coefficients in the turbulent flow can be correlated with a modification of the constant in the Chilton and Colburn equation. It was found that the aqueous film mass-transfer coefficients for both sides of the oil-containing hydrophobic MHF are 1 6 1 5% smaller than those on solid tubes. An empirical equation was proposed to evaluate a critical Reynolds number for the annular flow in single oil-containing MHF modules. The membrane-transfer coefficient depended on the direction of the solute transfer as well as the operating mode, tending to lower with an increase in the inner-toouter diameter ratio of the MHF. Thus, the greatest care must be taken in the application of tortuosity factor for the interpretation of mass transport across the wall of oil-containing MHF itself. Nomenclature A = contact area, m* D = diffusion coefficient of iodine, m2/s di, do = inner and outer diameters of the hollow fiber, m d, = inner diameter of the shell tube, m
de = hydraulic diameter of the shell side of the single hollow
fiber module, cm Gr = Graetz number, (di/L)ReSc J = mass-transfer flux, mol/(m2-s) K,, K 2 = equilibrium constants for eqs 1 and 2 k,, k, = film mass-transfer coefficients on the shell and tube sides of HF module, m/s k , = membrane-transfer coefficient, m/s k,, = apparent mass-transfer coefficient for organic flow including hydrophobic membrane phase, m/s k, = organic-phase film mass-transfer coefficient on the shell or tube side of HF module, m/s k , = aqueous phase film mass-transfer coefficient on the shell or tube side of HF module, m/s L = length of the hollow fiber, m m = iodine distribution coefficient between organic and aqueous phases N = extraction rate of iodine, mol/s Q,, = organic flow rate, m3/s Re = Reynolds number, diu,/v or deu,/v Sc = Schmidt number, v / D u = fluid velocity, m/s Greek Letters 6 = membrane thickness, m c = porosity of membrane u = kinematic viscosity, mz/s E = fractional surface area 7 = tortuosity of membrane 4 = packing fraction of HF module Subscripts and Superscripts ex = external F = aqueous feed side i = interface j = feed 1 or 2 m = membrane phase o = oil phase R = recovery side s = shell side T = total t = tube (lumen) side w = aqueous phase lm = log mean 1 = feed 1 2 = feed 2 - = organic phase [ ] = concentration
Literature Cited Chilton, T. H.; Colburn, A. P. Mass Transfer (Absorption) Coefficients, Prediction from Data on Heat Transfer and Fluid Friction. Ind. Eng. Chem. 1934,26, 1183. Dahuron, L.; Cussler, E. L. Protein Extractions with Hollow Fibers. AIChE J. 1988,34, 130. Darrall, K. G.; Oldhan, G. The Diffusion Coefficients of the Triiodide Ion in Aqueous Solutions. J. Chem. SOC.A 1968, 2584. Leveque, M. A. Les Lois de la Transmission de Chaleur par Convection. Ann. Mines Rec. Mem. L’Explotas. Mines 1928,13,201. Malone, D. M.; Anderson, J. L. Diffusional Boundary-Layer Resistance for Membranes with Low Porosity. AIChE J. 1977,23, 177.
Monrad, C. C., Pelton, J. F. Heat Transfer by Convection in Annular Spaces. Trans. AIChE 1942, 38, 593. Pons, M. N. Monte-Carlo Model of Microporous Plane Membranes. Chem. Eng. J. 1987,35, 201. Prasad, R.; Sirkar, K. K. Solvent Extraction with Microporous Hydrophilic and Composite Membranes. AIChE J. 1987,33, 1057. Prasad, R.; Sirkar, K. K. Dispersion-Free Solvent Extraction with Microporous Hollow-Fiber Modules. AIChE J. 1988, 34, 177. Ramette, R. W.; Standford, R. W., Jr. Thermodynamics of Iodine Solubility and Triiodide Ion Formation in Water and in Deuterium Oxide. J. Chem. SOC.1965, 87, 5001.
Ind. Eng. Chem. Res. 1990,29, 1476-1485
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Takahashi, K.; Nakano, M.; Takeuchi, H. Mass Transfer Coefficients in a Flat Type Supported Liquid Membrane. Kagaku Kogaku Ronbunshu 1987,13, 256. Takeuchi, H.; Takahashi, K.; Goto, W. Some Observations on the Stability of Supported Liquid Membranes. J. Membrane Sci.
Yang, M. C.; Cussler, E. L. Designing Hollow-Fiber Contactors.
AIChE J. 1986,32, 1910. Received for review August 1, 1989 Revised manuscript received January 23, 1990 Accepted February 21, 1990
1987, 34, 19.
A Simple Equation of State for Nonspherical and Associating Molecules J. Richard Elliott, Jr.* and S. Jayaraman Suresh Chemical Engineering Department, University of Akron, Akron, Ohio 44325-3906
Marc D. Donohue Chemical Engineering Department, The Johns Hopkins University, Baltimore, Maryland 21218
A simple semiempirical model is presented that gives an accurate representation of the thermodynamic properties for hydrogen bonding and nonspherical molecules. The model is shown to provide a good representation of available molecular simulation data for hard chains, hard spherocylinders, square-well spheres, Lennard-Jones chains, and dimerizing spheres. T h e simplicity of the model leads to an equation of state that can be applied easily and generally via corresponding states. The equation of state is cubic for nonassociating fluids and has only three real roots for associating fluids. Analysis of hydrogen-bonding systems shows that specifically accounting for molecular association clearly improves the equation’s ability to represent mixture phase equilibrium data.
Introduction A number of thermodynamic models can be used to make accurate predictions of the vapor-liquid equilibria for nonpolar and weakly polar compounds. Methods based on a corresponding states approach, such as the Soave (1972) equation or Peng-Robinson (1976) equation, have been remarkably successful for nonassociating mixtures. For example, Elliott and Daubert (1985) have reported an accurate representation of vapor-liquid equilibria with the Soave equation for a large number of binary systems containing hydrocarbons, hydrogen, nitrogen, hydrogen sulfide, carbon monoxide, and carbon dioxide. Elliott and Daubert (1987) also have shown that vapor-liquid critical properties can be predicted accurately for these systems. Michelsen (1986) has obtained a remarkably detailed representation of multiphase and tricritical phenomena. These results extend over a wide range of temperatures and pressures and treat multicomponent as well as binary mixtures. However, no simple generalized theory is available for associating systems as yet. Components like water and ethanol tend to self-associate, mutually associate, and multiply associate to form trimers, tetramers, etc. The complexity of these associations makes it difficult to characterize such mixtures even with sophisticated analytical chemistry. Furthermore, the types of phase behavior exhibited by these mixtures are difficult to describe quantitatively. For example, liquid-liquid immiscibilities are common in mixtures containing a hydrogen-bonding component. Such systems are more difficult to represent than systems that exhibit vapor-liquid equilibria. The desire to accurately predict such multiphase equilibria over wide ranges of temperature and pressure makes it necessary to develop models that take into account the physical and chemical interactions that make these mixtures distinctly different from nonassociating mixtures. Counter to the need for representing these complex molecular interactions, simplicity is essential when considering engineering applications. The heart of most engineering applications is the fugacity evaluation. This must be repeated over and over, and it must never fail. This OSS8-5885/90/2629-1476$02.50/0
necessity for simplicity is the reason many recent advances in molecular physics have not made their way into routine engineering calculations. The purpose of this paper is to partially resolve the difference between these two competing needs. Hence, we present expressions that accurately mimic the results of molecular physics and maintain a clear physical intepretation but are simple enough to allow repetitive fugacity evaluations. We concentrate on three molecular attributes that are important to the chemical industries. These are (1) the effect of nonsphericity on repulsive forces, (2) the effects of attractive dispersion forces, and (3) the effects of molecular association. Slight modifications of available theories are presented that yield simple forms and yet lead to a generalized cubic equation of state. The simple forms are compared to the results of molecular simulations and fundamental theories to show that they accurately mimic the relevant physics. The resulting final equation is compared to the Soave equation of state to show how a realistic physical model improves the ability to represent experimental data with little penalty in computational complexity.
Repulsive Forces Several approaches are available for representing the contributions of repulsive forces in the equation of state. We would like to present a method that is capable of representing the essential features of the various approaches while retaining the simplicity of popular engineering equations. There are two well-developed but distinctly different approaches to the treatment of repulsive forces for nonspherical molecules: convex-body models and chainmolecule models. In both cases, the equations reduce to the Carnahan-Starling equation (1969) for spherical molecules. The convex-body models are typified by the works of Nezbeda and Boublik (1978) and co-workers and are closely related to scaled particle theory. These models have the limitation that the molecules are treated as rigid. This tends to be a poor approximation for larger molecules. Studies of chain molecules relax the assumption of rigidity 0 1990 American Chemical Society
