Models and analyses of membrane gas permeators

Journal of Membrane Science, 73 (1992) l-23 Elsevier Science Publishers B.V., Amsterdam

Models and analyses of membrane gas permeators” A.S. Kowali, S. Vemury, K.R. Krovvidi and A.A. Khan Chemical Engineering Division, Indian Institute of Chemical Technology, Hyderabad 500 007 (India) (Received January

10,199l; accepted in revised form June 5,1992)

Abstract Transport models and parametric studies of membrane gas permeators for idealized flow patterns, e.g. plug and mixed, are reviewed here. Modeling aspects embracing binary and multicomponent mixtures with and without pressure drop effects are elucidated. Existing analytical and approximate solutions for different models along with their ranges of applicability are presented. The conditions under which nonidealities in flow patterns like longitudinal mixing, boundary layer and radial mixing effects exist are identified and their effects on the validity of the models are discussed. Possible future research needs are also presented. Keywords:

review; gas permeators;

models; pressure

1. Scope of the review There are excellent reviews on the fundamental mechanisms of gas permeation through polymers [l-5], on general and engineering features of membrane gas separations [ 6-101, and on modeling of gas separation in permeators [g-11]. Modeling of permeators forms an essential and integral aspect of gas separation processes in understanding the effects of various process parameters and flow patterns on the permeator performance. However, a detailed review of the modeling and simulation aspects of the transport equations in the permeators seems to be lacking. This review presents the salient features of various models and also attempts to Correspondence to: K.R. Krovvidi, Chemical Engineering Division, Indian Institute of Chemical Technology, Hyderabad, 500 007, India. *IICT Communication No. 2692.

0376-7388/92/$05.00

0 1992 Elsevier Science Publishers

drop effects

study the effects of process and membrane parameters for a successful design of gas separation units. The present review is classified into studies that were carried out without considering the effects of pressure drop along the flow channel and those which incorporated them. Further subclassification is based on the modeling aspects of binary and multicomponent mixture separations. Available analytical solutions, with their simplifying assumptions and specific conditions, are appropriately mentioned. The approximate solutions proposed for various situations are dealt with in detail. Multiple membrane units are gaining importance because of the advantages associated with them. These are also briefly discussed in this review. As recycle operations and cascade arrangements do not directly contribute to the understanding of modeling and analysis of gas

B.V. All rights reserved.

2

permeators, these have been omitted from the scope of the present discussion. The majority of the studies that appeared in the literature considered mainly idealized flow conditions: mixed, plug, cross and cross-plug. In some conditions, deviations from these idealized cases occur and they cannot be ignored in arriving at an overall picture of membrane gas separation. For this reason, studies dealing with longitudinal mixing, boundary layer and radial mixing effects are also discussed.

A.S. Kovvali et al./J. Membrane Sci. 73 (1992) 1-23

Permeation process Driving force for membrane gas separations is provided by the pressure difference across the membrane. Graham [ 161 deduced that the permeation process is a sequence of dissolution in, diffusion through, and desorption from the polymer of the permeating gas. This hypothesis was experimentally confirmed by Barrer [ 171. The rate of permeation of a gas through a thin nonporous polymeric film per unit area can be written, based on t,he Fick’s law of diffusion, as (1)

2. Introduction Separation of gases using polymeric membranes is gaining wide importance as an effective unit operation. These processes are becoming economically viable and competitive compared to other separation processes [ 121. This is due to synthesis of new polymers with improved selectivities, developments in membrane preparation, new designs and integration of modules etc. [ 13,141. In addition to the selection of a suitable polymer for a specific application, practical considerations like configuration, flow pattern, and operating conditions dominate the module performance. On the other hand, pressure drop effects, channel dimensions and membrane characteristics are some of the other important parameters that influence the performance of an individual permeator. Membrane modules can be designed as plate and frame, spiral wound or hollow fiber units. The latter two types are preferred over the plate and frame modules as they provide a higher area/volume ratio. Hollow fiber units provide a higher area/ volume ratio than spiral wound units but pressure drop in the fibers can be large. Hence, optimization of fiber dimensions becomes vital in balancing the economics and separation performances of modules [ 151.

where permeability coefficient, Qi, is the product of diffusivity and solubility coefficients of the permeating gas in the membrane, Ph and PI are the partial pressures of the gas on the high and low pressure sides of the membrane respectively, and d is the effective thickness of the membrane for permeation. Separation in membranes or separation index Selectivity or ideal separation factor (cy ) of the membrane to any two gases is defined by Qij = QJQj

(2)

where i and j are the components in the gas mixture. This definition of Crijis applicable for near vacuum conditions on the low pressure side and in the absence of interactions between the diffusing species, as well as between the species and the membrane, other than physical dissolution. Though permeability coefficients are generally functions of pressure and temperature, selectivity of the membrane may reasonably be assumed constant in engineering practice [9]. The actual separation in a binary system, on the other hand, is (3)

A.S. Kovvali et al/J. Membrane Sci. 73 (1992) l-23

where y and x are the mole fractions in the permeate and feed side streams respectively; (Yu and aactualare identical for P, << Ph, as in vacuum operation. A different separation index, the extent of separation was proposed by Rony [ 181. This index varies between 0 and 1, and is more convenient to apply in binary mixtures. It can be used as an objective function directly to optimize the separation process with respect to process design.

PERMEATE Ve * “a A

I

Ye,PI

t-

(a)

MEMBRANE

Le *re

S5.T

PERMEATE Ye .Ve

t ,y’.P,

I

REJECT

FEED Lt ,Xf

As mentioned earlier, when fluid streams pass through the narrow channels of permeators, they experience a pressure drop along the path due to frictional losses [ 191. However, the early attempts in modeling of gas separation neglected this phenomenon.

L,,% SrsT

s= 0 b)

REJECT

FEED

L, #XC

Lf ,Xf

3.1. Binary mixtures

NON POROUS

REJECT

FEED Lf#Xf

dV:dL

3. Separation studies with no pressure drop effects

I

SZST

s=o 03

Perfect mixing The first analytical study in binary separations with a non-porous membrane permeator was done by Weller and Steiner [ZO]. The model is based on the assumption that complete or perfect mixing of gas streams exists on both sides of the membrane (Fig. la). Due to this assumption, the transport equations can be written as: VJ’, =

&IPd be - D’e)/d

(4)

EP OR PURGE

PERME

Vt I Vt

VC,YC

REJECT

FEED

L, 2.2

Lf,Xf S: ST

szo (d)

Fig. 1. Flow patterns in membrane gas permeators: (a) mixed, (b) cross, (c) cocurrent, and (d) countercurrent.

From eqn. (4), the total membrane area in its dimensionless form is given by

(5) where y is the ratio of pressures on permeate and feed sides. Subscript e refers to the exit condition. A is the area for permeation. Equations (4) and (5) give

L!.kL=a l--Y,

%-?Ye

l-xc?-Y(l--y,)

(6)

(7) Since q
xe,min =

xf[l+

(a--1)Y(1-%)1 a(l-x,)+x,

(8)

4

A.S. Kouvali et al./J. Membrane Sci. 73 (1992) l-23

Equation (8) implies that feed stream composition can not be reduced below Xe,min, however large the membrane area may be. This model which assumes complete mixing on both sides of the membrane, does not include the effects of cell geometry and hence ignores other possibilities of flow patterns such as cocurrent, countercurrent and crossflow (Fig. lb,c,d). Yet, this study heralded modeling of gas separations in permeators. Further, the model helps to predict the minimum separation achievable and the maximum area required in a given situation. In this way, one can determine the bounds on area and separation efficiency. The model is also useful in experimental investigations using a permeation cell with a small membrane area, where perfectly mixed conditions can be safely assumed and pressure drop effects can be ignored. In a study on helium recovery by permeation through glassy membranes at low feed concentrations ( cl%), Stern et al. [21] found that with a good separation factor of the membrane, even a mixed flow pattern can give significant separation at low cut fractions (ratio of permeate flow to feed flow,
-dV*/dS

= - [cx(X-yy’)+1--x--y(1-y’)] L*&/dS=

--a!(~-yy’)+xdL*/ds

(9) (10)

Due to the crossflow condition, local permeate composition (y’ ) will be determined by the ratio of local fluxes of the components emanating from the membrane as

_A_-= (1-Y’)

a(=-7-Y’ 1 [l-x--Ytl-Y')l

(11)

Weller and Steiner [20] employed the ideal separation factor ((x) whereas Naylor and Backer [ 221 used the actual (in this case pointwise) separation factor (crY,,tual)in obtaining closed form solutions relating the mole fraction in the high pressure stream to that in the permeate stream. Weller and Steiner also suggested a shortcut approximation to the total dimensionless area as

s~=~[YatX-YY;)+YkltXe-~~)l/2

(12)

where 6 is the cut fraction. Saltonstall et al. [ 231 also presented expressions for membrane area, flow rates, and compositions for the crossflow model in terms of the feed and residue compositions, pressures and feed flow rate. For the condition where l/ xf is greater than a! and l/y, equations in the Weller and Steiner model can be approximated to [24]: &l_

(X/Xf)[“/(‘-Y)(~--l)-ll--l

(13)

S,=0l(l-y)

(14)

y/xr = [ 1- (x/X~):f)a’(l-y)(~-l)] /e

(15)

The crossflow model may closely represent the actual mass transfer conditions that exist in gas permeators employing asymmetric (and perhaps composite) membranes. Validity of this model in practical situations is discussed in Sections 3.3 and 6.3. Cocurrentlcountercurrent flow Blaisdell and Kammermeyer [ 251 carried out experiments using silicone rubber tube bundles in cocurrent and countercurrent flow patterns. They found that the data could not be ex-

A.S. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

5

plained by the above models [ 20,221 and so developed a model to take the other flow patterns into consideration (Fig. lc,d). By defining dimensionless flow rates, they could avoid using a trial and error procedure in solving the transport equations. Good agreement was found between the data and model predictions. The differential form of the transport equation of this model is given by L*

d.L*

-= dx

x+ (l-cw)+

(16)

&)l(x-W)

The overall material balance, for the countercurrent case, is y= (X,-L*X)/(l-L*),

X#X,

X#Xf

-d(L*x:)/dS=a!(x-yy)

= [1-x-x,,-Yc-Y-Yn,)l

I/dS

(20)

where x_, and y,, are mole fractions of the nonpermeating components in the feed and permeate respectively. Pan and Habgood [ 241 did not present any results showing the specific effects of the non-permeating component on permeator performance, but it is understandable that there will be an increase and a decrease in driving forces of other components across the membrane due to the presence of the non-permeating component in the low and high pressure streams respectively.

Parametric

(18)

where L* = L/Lf. Walawender and Stern [26] also presented similar equations for the same cases wherein they used the relationship between local permeate and feed stream compositions in simplifying the equations. Both the approaches [25,26] give identical predictions [27]. However, solving eqns. (16) and (17) involves a trial and error procedure if the cut fraction or product recovery rate is specified and integration is direct and less cumbersome if product purity is specified. Pan and Habgood [24] presented a unified mathematical development and calculation procedure for cocurrent and countercurrent flow patterns with two permeable components and a third non-permeating component in the feed as well as in the permeate. Assuming that plug flow exists on both sides of the membrane, the dimensionless forms of the transport equations are given as d(V*y)/dS=

-d[L*(l-x)

(17)

where L*= L/L,. For the cocurrent case, y= (X,--L*X)/(l-L*),

d[ V+(l-y)]/dS=

(19)

effects

Tranchino et al. [28] carried out experiments with binary gas mixtures of CHI and COz in a composite hollow fiber unit to study the effects of pressure, cut fraction, feed composition and flow regime. Similar studies were done earlier by Blaisdell and Kammermeyer [ 251 on the separation of OJN, mixtures with silicone capillaries while Pan and Habgood [ 241 studied the parametric effects numerically. Figure 2 shows the schematic of variation in permeate composition of the more permeable component with cut fraction for different pressure ratios. As 8+0, permeate stream composition in all the flow patterns reaches an asymptotic value. As y-+0, all the flow patterns (cocurrent, countercurrent and crossflow) perform identically. In general, the countercurrent flow pattern is the best, followed by the crossflow pattern with respect to membrane area requirement and recovery. Further, if l/y< l/xr, the ratio of feed to permeate pressures will be the limiting factor for the enrichment ratio (y/ xf) because the partial pressure of the component in the low pressure stream can not be greater than that in the high pressure stream regardless of the membrane selectivity [ 241. In

6

AS. Kovvali et al./J. Membrane Sci. 73 (1992) l-23 1

oc

Lx

requirement. The sacrifice to be made with purge is the dilution of permeate. When Q z+ l/ y and feed composition is low, permeate purging will be more effective in reducing the area requirement with not too significant dilution of the product stream. In a recent study, Sidhoum et al. [ 291 used the sweep gas technique to enhance the driving force across asymmetric hollow fibers. The results obtained were used to verify the hypothesis that the driving force across an asymmetric membrane is that as in the crossflow mode [ 30-321. This aspect is further discussed in detail in Section 6.3. 3.2. Approximate solutions

l.OC

Cut fraction

,

e

Fig. 2. Schematic of the effects of pressure ratio and cut fraction on permeate composition in binary gas separation. (---_) countercurrent; (-_) crossflow; (- - -) cocurrent.

fact, y/z, is little affected by membrane selectivity cy, for cy>> l/y or a!<< l/y. Pan and Habgood [24] showed that y/xf can never be larger than the smallest of c~, l/y or l/xf. For given feed and residue compositions and constant permeability of the fast permeating component, increase in permeability of the slower gas results in a decrease of the enrichment ratio and an increase in membrane area requirement. At a fixed permeability of the slower component, an increase in the permeability of the faster component decreases the membrane area needed with an increase in permeate composition of the fast component. Pan and Habgood [24] considered the concept of using sweep (or purge) gas in the permeate side (in cocurrent and countercurrent cases) to increase the driving force for permeation. Even a small flow rate of the purge, such as a small bleed from the feed stream or a nonpermeating gas introduced at a reduced pressure, can effectively lower the membrane area

Transport equations in the models assuming plug flow of feed and/or permeate streams are coupled, nonlinear ordinary differential equations (o.d.e.‘s). Their exact solutions are nonexistent and approximate solutions can be obtained only under certain simplifying assumptions. In order to reduce computational efforts, especially in cascade analyses, and for quick estimates of compositions, product recoveries and/or membrane area requirements, approximate solutions of the models are extremely useful. The first attempt in this direction was done by Boucif et al. [ 331 who proposed a series solution approach to obtain approximate solutions to the transport equations for both cocurrent and countercurrent cases. Compositions of the streams at any cross-section of the permeator were expressed as x=a, +a,S+azS2+a3S3

(21)

y=bo+blS+b2S”+b3S3

(22)

where a0,a 1,a2,ab,b0,b1,b2,b3 are coefficients and S is dimensionless membrane area. Origin of the coordinate (S) is taken at the feed entry and residue ends in the cocurrent and countercurrent patterns respectively. Equations (21) and (22) upon substitution in the equations for

A.S. Kovvali et al./J. Membrane

Sci. 73 (1992) l-23

d_r/dS and dy/dS obtained from eqns. (19) and (20) yield eight algebraic equations containing the unknown coefficients which can be solved sequentially. The series solutions obtained in the above manner showed some divergence from the exact results as eqns. (21) and (22) contain a limited number of terms in S. It was fund that the solutions are effective at low cut fractions (near the inlet of the permeator) or when permeabilities are low. They are helpful in predicting the permeator performance as long as selectivity is not too large. Disadvantages with the series solutions are that the ranges in which they are effective are not known a priori [34] and the method is not easily amenable to multicomponent situations, because of the above disadvantages. Rautenbach and Dahm [ 341 obtained solutions for the binary countercurrent case only under different limiting conditions: - Selectivity is very high (l/a M0); - Permeate partial pressure is zero (y=O); and - Constant partial pressure of each component in the permeate ( w z constant ) . The above conditions, considered separately, yield analytical expressions which predicted better than the series solutions under the same conditions. But, the first two conditions do not represent the real situations, whereas the third condition ceases to be valid for the cocurrent and crossflow conditions. Further, this approach can not be extended for multicomponent mixtures. Basaran and Auvil [35] performed an asymptotic analysis of the cross plug-flow model choosing a+ 1 and cy+co as the asymptotic limits. This approach provides quick estimate of the membrane area. The analysis showed that when 3cf<< y, c~does not have much effect on the membrane area, and when a! is large and X, k y, practically little or no area is required. However, the analysis can not be extended for other flow patterns and multicomponent systems. It

7

is not effective in the intermediate range of selectivities, as are normally encountered in practical situations. Recently, two approaches for obtaining approximate solutions in the cocurrent and countercurrent flow patterns were proposed by Krovvidi et al. [36,37]. One method (Operating Line Method) assumes a linear relationship between the feed and permeate stream compositions: y=mx+b

(23)

The slope (m) and intercept (b) in the y vs. x relation can be obtained from known boundary conditions (at feed entry or residue end) and the expression for dy/dx. The second method (Driving Force Method) is based on the variation in driving force of each of the components along the permeator length. This variation is assumed to be a quadratic function of the membrane area as given by X-yy=u~

+a,S+a,S2

(24)

where c&=x--

(25a)

a, = dx/dS - ydy/dS

(25b)

u2= d2x/dS2- yd2y/dS2

(25c)

The coefficients in eqn. (25) are obtained based on the initial conditions of x and y. The operating line method (eqn. 23) yields an expression for the dimensionless membrane area as a function of the feed and residue compositions of the fast permeating gas while the driving force method (eqn. 24) gives expressions for X, y and flow rates in terms of the dimensionless membrane area. It is possible to neglect the third term in eqn. (24) and assume a linear variation in S [ 361, but the quadratic variation will certainly improve the predictions. Figure 3 shows the predictions by the above two methods compared with the exact and series solutions. As can be seen, the above two

8

AS. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

methods predict better than the series solutions and do not suffer from divergence problems. In another study [ 381, Kameswara Rao et al. presented an analysis based on the combination of series and perturbation methods to study specifically the effect of y on the compositions. This approach involved setting y=O initially and seeking solutions in the form of eqns. (21) and (22). Subsequently, the effect of pressure ratio can be estimated to the second order:

(26)

(27)

Feed

stream

The numerical analysis showed that the correction term (dy/dy),,o is necessarily negative, but it may be a monotonically decreasing function or a function with a maximum in S.

I 0.4

I

0.3 composition.x

X lo2

3.3. Multicomponent mixtures

0’

Feed

stream

0.5

0.4

0.3

02

composition,

xX

lo2

Fig. 3. y versus x profiles in binary gas permeators for different selectivities (x,=0.005; ~~0.01). (---) Exact solutions; ( - - - ) operating line method; (- -) driving force approximation method; (- - -) series solutions). (a) Cocurrent, (b) countercurrent pattern (from Ref. [37] ).

In practice, separation of binary mixtures is rarely encountered. In contrast, separations involving multicomponent mixtures like hydrogen recovery from ammonia synthesis, helium recovery from natural gas, acid gas separation etc. are more common. Brubaker and Kammermeyer [39] studied separation of ternary and quaternary mixtures based on the complete mixing model. They studied the effects of pressure variation on separation and observed that separation can increase, decrease or remain constant when the pressures on both sides of the membrane are varied independently. They further observed that in multicomponent gas separations, compositions of intermediate components (that are other than the fastest and slowest gases) in the permeated product can pass through a maximum value at a certain cut fraction. Stern et al. [ 211 presented an alternative formulation for

A.S. Kovvali et al./J. Membrane

Sci. 73 (1992) l-23

ternary mixture separation based on the complete mixing model. Stern and Leone [40] studied experimentally the separation of krypton and xenon from Kr-Xe-02--Nz mixtures of nuclear reactor atmospheres by permeation through silicone rubber capillaries operating in countercurrent mode. Without considering any pressure drop effects within the capillaries, they included in the model in an empirical way, the effect of pressure difference across the membrane on permeability coefficients. At high pressure differentials, due to collapse of the capillaries, the observed separations agreed well with the model predictions based on the perfect mixing conditions. Theoretical studies on multicomponent separation have been very few. McCandless [41] suggested a method of solving the equations of countercurrent and recycle permeator models with a modified Box method iteratively by trial and error procedures. Shindo et al. [ 421 and Li et al. [ 431 presented formulations and suitable calculation procedures for five flow patterns: cocurrent, countercurrent, crossflow, perfect mixing, and mixing on permeate side of the permeator. Simulations were performed on the effects of stage cut, pressure ratio and feed flow on module performance and membrane area requirements. The results showed that countercurrent flow pattern in general is the most effective among the flow patterns considered. Without considering the pressure drop effects, Pan and Habgood [30] presented the transport equations based on a significant assumption that the crossflow driving force (xi - pi ) adequately describes the permeation for any flow pattern in permeators using an asymmetric membrane. This was based on the presumption that the porous support layer of a membrane prevents mixing of local permeate streams emanating from the surface of the skin as well as negligible mixing with the permeate bulk within the porous layer. Mathematically this is given by, for a differential increment,

9

d(k) --'y; dL

(28)

where y{ is the local permeate composition of component i. On the other hand, for a symmetric or dense membrane the driving force will be (Xi-yYi) where yi is the bulk composition of component i in the permeate side. Pan [31] computed permeate stream compositions for different operating pressures for both symmetric and asymmetric membranes and found that the model based on a symmetric fiber showed a significant effect of the flow pattern which was not substantiated by the experimental results. On the other hand, the model for an asymmetric fiber predicted virtually identical performances for both cocurrent and countercurrent flow patterns and were in reasonable agreement with the experimental results. However, recent experimental findings do not fully confirm this view (see section 6.3). Saltonstall [44] presented a simplified approach for a multicomponent crossflow model neglecting pressure drop effects. This approach can reasonably predict the area for a required amount of reduction in composition from the feed stream. As pointed out by Saltonstall, the model is valid only for negligible pressure drops. It is not suitable for either cocurrent or countercurrent situations. Further, this model is restricted by the condition that the local permeate composition does not vary significantly over a differential increment of area. 4. Studies considering the pressure drop effects The assumption of constant feed and permeate pressures may be reasonable for flat sheet and tubular membrane modules at low permeabilities and small cut fractions. But, in modules employing hollow fiber or spiral wound membranes, significant pressure drops can develop due to flow in narrow channels. Pressure

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AS. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

variation can also occur when there is significant permeation across the membrane. Based on the analysis of Berman [ 191 for porous walled tubes, for very low mass transfer rates across the wall as normally encountered in membrane separations, one can use the HagenPoiseuille equation for pressure drop estimation. For permeate flow inside a circular channel, the pressure gradient is given by

@I -= d.2

- 128/&T V lrP,df

(29)

where P, is pressure inside the tube, V is molar flow rate of the permeate stream, and p is viscosity of the gas mixture. 4.1. Binary mixtures Thorman et al. [45] incorporated the effect of pressure drop in a study on the separation of binary mixtures employing silicone rubber capillaries. The calculated pressure profiles using the Poiseuille equation matched with the experimental results for gases whose permeabilities are relatively low, but large deviations were observed for gases with high permeabilities. In a hollow fiber module, for feed flowing outside the fibers, pressure build-up inside fibers (of permeate) can significantly alter the ratio of permeate to feed pressures, influencing the module performance along the area. Generally, this mode of operation is used where small cut fractions are required. On the other hand, feed flow inside the tube may minimize the effect of pressure drop on the module performance. The latter mode of flow pattern is used in cases of large cut fractions [46]. Boundary conditions for the transport equations are incompletely known at either end when feed flows outside fibers because of the unknown permeate pressure profile inside fibers. This requires a trial and error approach to solve the equations in either countercurrent or cocurrent flow patterns. Calculations are

straight forward for feed flowing inside fibers and permeate in cocurrent flow as it is an initial value problem. It requires an iterative procedure (though still an initial value problem) when permeate flows in the countercurrent mode. When permeate is flowing on the shell side, pressure loss or buildup may not be significant enough to influence the permeator performance, even though feed inside will experience normal pressure losses. Based on this reasoning, Antonson et al. [47] found that the feedinside of fibers mode is better than the feedoutside type of operation, In all cases, the Hagen-Poiseuille equation is assumed to be applicable for estimating the pressure profiles. With shell side feed, most of the pressure drop occurs near the feed inlet, while in the tube feed mode, the pressure differential across the membrane is almost a linear function of the distance from the feed end. Among the six flow patterns that were examined, tube side feed flow with countercurrent permeate flow gave the maximum enrichment ratio (ratio of fast gas composition in the permeate to its composition in the feed) and fast gas recovery. They observed that tube side feed flow gave a better performance than shell side feed flow for all flow patterns because of the larger driving forces in the former case. For the tube side feed countercurrent mode, at fixed feed flow, there exists a maximum enrichment ratio under the limiting conditions of zero recovery, i.e. near the inlet and zero pressure drop [ 471. An increase in feed flow rate leads to higher permeate flow rates but lowers the recovery of the fast permeating gas. The study also presented the effects of design parameters (fiber length, fiber O.D. to I.D. ) , operating variables (feed pressure, feed flow, feed composition and pressure ratio), physical properties, fast gas permeability and selectivity on the module performance. Chern et al. [151 considered the case of shell side feed and countercurrent pattern in a par-

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A.S. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

ametric study on the effect of process and design variables. Permeability variations due to pressure and competitive sorption were analyzed by the generalized dual sorption model. Fiber dimensions were found to play a significant role, even with the shell side feed, in affecting the permeate flow rate and purity. Smaller fiber O.D. gives higher purity and greater packing density of fibers with an increase in the production capacity per unit volume of the permeator. Pressure drop obviously increases with decreasing fiber I.D. and has deleterious effect on the permeation driving force, especially in the lean feed and high recovery situations. 4.2. Multicomponent mixtures The effect of pressure drop in hollow fibers on separation of multicomponent mixtures was considered by Pan [ 321. With the feed on the shell side, the model equations based on the driving force for crossflow permeation are: d(L*x,) =ai(xi -wi) dS d(L*xi) -=yi dL*

i=l,....,N,

i= l,....,N,

(30) (31)

where c,

256RTpul(W2 x2N 2d0d:P3 (QJd)

(33)

4.3. Approximate solutions

and

ain=(Qild)l(QN,ld)

shooting method is generally needed to obtain the solutions. However, for a given membrane module, with specified feed pressure, composition, mole fraction of one component in the residue or permeate, and permeate exit pressure, a direct iterative procedure can be used to obtain the solution. To facilitate this, Pan [31] expressed composition variations as o.d.e.‘s with the mole fraction of the component whose exit composition is specified as the independent variable. The developed model was verified with experimental data on hydrogen recovery from ammonia plant purge gas and on separation of acid gases from natural gas streams. At high concentrations of acid gases (C02, H,S), permeabilities of the gases in the mixtures were significantly different from those of the pure components, perhaps due to accompanying plasticization [32]. Plasticization increases permeation rates and reduces selectivities; hence variations in selectivities with pressure also need to be taken into account. Even in the absence of a plasticizing effect, deviations from pure gas permeabilities can occur due to competitive sorption [ 48,491. Competitive sorption alone was found to reduce the product recovery with a concomitant increase in the product purity [ 151. Thus these two factors act in opposing ways, but plasticization is usually significant at high pressures or in the presence of plasticizers or good solute-membrane interaction, while competitive sorption effect is present in almost all situations.

(34)

Equations (30)- (34) are applicable for both cocurrent and countercurrent patterns by taking the appropriate sign for permeate flow direction. Integration of these equations is always a boundary value type due to the unknown permeate pressure profile and a trial and error

Recently, Boucif et al. [50] extended their series solution approach to obtain expressions for feed and permeate compositions, and pressure inside fibers as functions of the dimensionless membrane area. However, the resulting expressions for the coefficients in the series contain total membrane area and permeate

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AS. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

pressure at the closed end of the fiber as parameters, which are not known initially. The procedure, hence, involves solving simultaneous, nonlinear algebraic equations which can be numerically demanding. The driving force method developed by Krowidi et al. [37] can be applied directly to treat multicomponent separations with or without the pressure drop effect for any flow pattern [ 511. The procedure for studying the pressure drop effect is briefly presented here. As discussed earlier (section 3.2, eqn. 24), the driving force for each of the components across the membrane can be assumed to be in the form xi-_yy:=~,i+U,,iS+U2,iS2

(35)

with

ao,i= (xi-m) dxi

dy:

$j-yds-Yia

d’xi a2,i=

$35-733

(36)

Is=0 ,dY

dy dyl

$j-Yis

(37)

s=.

, d2y s=.

(38)

subject to the conditions Cxi = Cyi = 1. Calculations can begin from either end of the permeator. Equations (35 )- (38) were obtained assuming that only permeate pressure varies and pressure of the feed stream remains constant along the permeator. Equations (37) and (38) contain terms dy/dS and d2y/ds2 which take into account the pressure drop effect in the permeator. In the absence of a pressure drop effect, these terms become zero. The Driving Force Method simplifies integration of the governing equations and the ensuing computational efforts. It unifies the procedures for all situations and needs very few iterations to be performed as opposed to other methods in the literature [ 15,30,32,41,42]. Recently, a generalized model was developed considering pressure drops in both feed and permeate streams [52,53]. The equations are

unified to represent all possible situations, i.e. feed flowing outside or inside the fibers, and feed and permeate flow in cocurrent or countercurrent mode. 5. Separations using multiple membranes Separation of gas mixtures is usually carried out in permeators having a single type of membrane. However, the extent of separation and recovery of the desired component(s) can be enhanced by using two or more membranes of different selectivities in the same unit as shown in Fig. 4. Often, more than two product streams are obtained in a dual membrane module at lower energy and capital costs. Membranes with differing selectivities can also be housed in separate units, which can be arranged either in series or parallel. The first attempts employing two types of membranes in a single unit were made by Kimura et al. [ 541 and Ohno et al. [ 551 for binary mixture separations. They modeled binary mixture separation using two membranes assuming complete mixing to prevail on both sides of the membrane and for zero reject flow rate. Stern et al. [56] showed by numerical analysis that higher concentration of product is obtained when both the membranes of different selectivities are housed in the same unit. The analysis showed that concentration of component 1 in the permeate from membrane I selective to component 1 increases as the stage cut PERMEATE MEMBRANE

I

“1*

REJECT

FEED

L*

L1.Q

MEMBRANE

’ Y1*

II ’

I XI

PERMEATE “2. s=o

5 Y2*

s = 5.7

Fig. 4. Dual membrane (or asymmetric) configuration.

permeator

13

AS. Kovvali et al/J. Membrane Sci. 73 (1992) 1-23

from membrane II (selective to component 2) increases. For certain stage cuts (19,< Or,), composition of component 1 in the reject can be higher than its composition in the entering feed. By contrast, this never happens in single membrane permeators arranged in series, for any flow pattern, where the residue composition is always lower than the entering feed composition. Further, in single membrane permeators arranged in series with 0, < Bri,separation performance decreases in the order: countercurrent > crossflow > cocurrent > completely mixed flow patterns. In the dual membrane unit, the above order may not be followed (Fig. 5). Because of this, a combination of cocurrent and countercurrent flow patterns in the unit is recommended, as perfectly mixed flow conditions can not be easily implemented in practice [ 561. Perrin and Stern [ 571 modeled binary gas separations in a dual membrane unit for mixed, cocurrent, and countercurrent flow patterns and found that countercurrent flow pattern is

the most effective of all the patterns. For a dual membrane unit, an overall separation factor can be defined [ 54,551 as as1

=

(%/%

aI and cyIIare the selectivities defined in the usual way for membranes I and II respectively for component 1. In general, cysl is a function of individual cut fractions (0, and On) and total membrane area [58]. Perrin and Stern [57] plotted cz,i as a function of 0, and Onand found that large as1 is achieved by high crI and low cyII,i.e. when a maximum amount is recovered through I and a minimum through II. Figure 6 shows the schematic of cy,i variation with Oi/ e,, for different flow patterns. In general, cocurrent and mixed flow patterns give higher separation than countercurrent pattern. This trend is reversed for the ratio of cut fractions

Ratio

”

Cut

fraction

from

membrane

I,

81

Fig. 5. Optimal flow pattern for dual membrane permeator performance. (Adapted from Ref. [57] ).

(39)

11

of cut

fractions,

81

I8tt

Fig. 6. Schematic of variation of overall separation factor, (Y,~with the ratio of cut fractions t&/O,,for different flow patterns. (Adapted from Ref. [57]).

14

(&/&) > 1. The overall separation decreases as 13,increases and it increases as & increases. The latter effect is due to the fact that as en increases, more of component 2 passes through II thereby increasing the driving force for 1 through I. Perrin and Stern [59] found a weak dependence of permeate compositions on the flow pattern because of the low cut fractions ( < 0.2) and a small membrane area chosen in the study. Sirkar [ 601, in an earlier study, extended the model of Ohno et al. [ 551 to binary and ternary mixture separation in a dual membrane unit for finite reject flow rate. He considered the cases of mixed and crossflow situations and explored, in general, the conditions under which the asymmetric or multiple membrane permeator is advantageous. Sengupta and Sirkar [61] simulated models of dual membrane units separating ternary mixtures for cross, parallel, countercurrent, and perfectly mixed cases neglecting the axial pressure drop effect. It was found that the countercurrent flow pattern yields the best performance and that one twomembrane permeator performs better than two single-membrane permeators arranged in series. Experiments on ternary mixtures also supported the above studies [ 62,631. As an extension of the dual-membrane concept, an internally staged permeator (ISP) was conceptualized by Sidhoum et al. [ 641 using a single type of membrane but having two stages in the same unit and carried out experiments on the separation of 0,-N, and C02-N2 mixtures. The ISP works on the principle of splitting the overall driving force into a two-step process using the two stages. The permeate from one membrane acts as a feed to the second membrane at no extra cost of pressurization. It was found that in an ISP, cocurrent pattern works better than the countercurrent one. The ISP performs better than a conventional countercurrent permeator at low stage cuts only ( ~0.3) and poorly at higher cut fractions.

A.S. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

Based on model calculations, Sidhoum et al. [ 641 recommended recycling of the shell (intermediate stream) reject stream and combining with the feed to improve the performance of an ISP at higher cut fractions. Simulations showed the existence of pressure ratios across the stages at which maximum permeate composition can be obtained. Li et al. [ 651 modeled separation of air in the cocurrent and countercurrent flow patterns in an ISP but carried out simulations for the mixed flow in the stages as a baseline case. It was found that the performance (permeate composition) goes through a maximum at approximately equal pressure ratios in the stages for a given overall stage cut (0) and the optimum is relatively independent of 8. There is also an optimum ratio of stage cuts for an overall 8, and the performance decreased with increasing individual stage cuts (>0.2) as observed by Sidhoum et al. [ 641. As regards the effect of flow pattern, the cocurrent pattern (in each stage) again showed a better performance. 6. Deviations from the ideal flow patterns The models discussed so far describe the idealized versions of the actual flow conditions that exist in permeators. Separations obtained in practice are usually less than the maximum attainable under idealized conditions due to: incomplete use of the available area, channeling of streams, mixing in feed and permeate, concentration polarization etc. For practical estimates, “effective” separation factors in the model formulation can instead be used to predict performance of the permeator [ 661. These factors can be obtained by fitting the model(s) to data obtained from operating the same or similar separation system under the actual conditions. The idealized flow patterns: mixed, plug and cross-plug encompass many of the real situa-

A.S. Kovvali et al./J. Membrane

Sci. 73 (1992) l-23

15

tions, thereby giving valuable information on the practical limits of separations. At the same time, it is desirable to know under what conditions a rigorous model will reduce to the ideal cases. 6.1. Partial or longitudinal mixing As the fast permeating components are depleted from the feed stream along the flow direction, resulting concentration gradients in the longitudinal direction induce mixing by molecular diffusion. This mixing causes driving forces to decrease at the upstream side and their increase at the downstream side, and an overall reduction in efficiency in general since most of the separation takes place at the upstream side. Breuer and Kammermeyer [ 671 modeled incomplete mixing in feed and permeate streams by introducing dispersion due to molecular diffusion. Equations incorporating the partial mixing effect for the parallel flow mode are [ 91: 1 dx d(V*y) -pd&*x) dS +P,a=dS---

1 dy PPm dS (40)

=ck!(x-yy) -

d[L*(l-x)] dS

1 d3c --P,dS

=W*U-y)l+ dS

1 dy PPm dS

= Cl-Y)---Y(l--Y)

(41)

where the dimensionless mixing parameter P, is defined by P,=

(longitudinal convective diffusion)* (longitudinal molecular diffusion) (permeation rate)

= h%WmJA,ChQZPh

(42)

P, applies to the high pressure stream only. For P, >> 1, flow tends to be plug flow and for P, = 0, complete mixing exists. Similarly for the

permeate stream, /lPmis the measure of its degree of mixing, where P=ohDhCh/(aJ)~G)

(43)

a, D and C are channel cross-sectional area, molecular diffusivity, and total molar concentration respectively. Subscripts h and 1 refer to high and low pressure streams respectively. Boundary conditions for the cocurrent pattern with a closed fiber-end, are L*=l

and V*=OatS=O

xf=x-l/P,(&/dS);

(44)

dy/dS=O at S=O (45)

dx/dS=dy/dS=O

at S=ST

(46)

These are the well known Danckwert’s conditions. In the countercurrent pattern, V * = 0 at S=O in eqn. (44) is replaced by V*=O at S=&. Breuer and Kammermeyer [ 671 analyzed the cross plug flow case (permeate in the crossflow mode) for dense homogeneous membranes. Figure 7 shows that plug flow is a reasonable assumption for a P, value of 50 or above. But, there are no studies on the parallel flow pattern incorporating partial mixing on both sides of the membrane and hence, it is not known if and when the crossflow model can replace the parallel flow model. In general, partial mixing is ignored to keep the mathematics tractable. But, in practice, one should be aware of its presence and its effect in reducing performance of the permeator [ 91. The importance of the partial mixing effect was brought out by studies on the continuous membrane column (CMC) . Modeling of CMC [68] began with the assumption that the streams are in plug flow. As already noted, any presence of mixing in the flow direction has the tendency to lower the performance. The plug flow model equations become singular in behavior at the bottom of the stripper if there is

16

A.S. Kovvali et al/J. Membrane Sci. 73 (1992) l-23

Fig. 7. Schematic of the influence of mixing parameter, P,,,, on permeator performance versus cut fraction. (Adapted from Ref. [9] ).

zero flow rate (without recycle) [ 24,691. With axial dispersion terms not included, the equations were found to be very sensitive to small changes in the feed composition when the internal flow rate is low. To avoid this, Kao et al. [69,70] modified the CMC model by taking into account the axial diffusion effect. With this modification, the problem becomes a boundary valued one, requiring specification of conditions at both the ends and computations can begin from either end. It was shown that the axial diffusion effect is indeed important for low permeate flow rates (low cut fractions) and is very critical for the stripper in general. It is not inappropriate to conclude that this effect needs to be taken into account in permeators (even in dual membrane and recycle units) with low cut fractions, and especially near the entrance of the permeator. 6.2. Boundary layer effect Due to permeation, there will be either

buildup or depletion of components within the thin boundary layer adjacent to the membrane surface on the high pressure side. This imposes an additional resistance to the permeation process and decreases permeator performance. In general, this resistance is neglected in the case of gas separations due to the much higher diffusion coefficients in the bulk compared to those in the membrane. But, occasionally, it becomes important when diffusion through the membrane is significant (as in porous, and rubbery membranes, due to plasticization effects) and/or when concentrations in the bulk are very high [ 71-731. The effective thickness of the boundary layer close to the membrane surface depends on the flow conditions (laminar or turbulent) under which the process is run. These in turn depend on the flow rate, channel spacing, fiber diameter, pressure, temperature, permeation rate etc. For rigorous treatment, the Stefan-Maxwell equations have to ‘be solved for diffusion of gases in the boundary layer coupled with the permeation rate equations through the membrane surface. Narinsky [ 711 treated the boundary layer phenomenon in a general manner for a binary system and obtained an upper limit for the concentration gradient perpendicular to the membrane surface. The concentration profile inside the boundary layer on the high pressure side can be given by w

&+

2 a2

w

&izD.a2xi

“an

‘an2

(47)

IV, and IV, are the magnitudes of local feed flow and permeate flux respectively. The coordinates parallel and perpendicular to the membrane surface are z and n respectively. At the membrane surface, W,=L;

Wn=-J;Xi]n=O=Xi~

(Wnxi -DidxJdn)

(46)

1n=O = - Ji

(The reader is referred to Ref. [71] for de-

17

A.S. Kovvaliet al/J. Membrane Sci. 73 (1992) l-23

tails.) The upper limit on the concentration difference between the bulk and the membrane surface is obtained as [ 711 (x1 -&w)

(49)

< (Y; -Y1 Mua

for the fast permeating component 1 in a binary mixture, where y; is the local mole fraction on the permeate side andy, is its bulk concentration in the permeate. The boundary layer thickness is 8,. Di/S, is the diffusional mixing rate in the boundary layer and JS,/Di is the diffusion Peclet number. Thus plug flow on the high pressure side is ensured only when J<< (D/6,) and/or when turbulence is present to increase mixing in the direction normal to membrane. Haraya et al. [72], on the other hand, defined a polarization modulus as M, = (Y-X)l(Y-%v)

(50)

in a study using porous membranes. Based on the implicit assumption that y, the permeate bulk composition, is not a function of .It,Haraya et al. [ 721 obtained the expression M,=exp(-JV/k)

(51)

where J, is the volumetric flux of permeate and Kis the mass transfer coefficient. As can be seen from eqn. (51) , the magnitude of Mp (polarization) decreases with increasing flow rate of the feed stream and increases with increasing permeation rate. 6.3. Radial mixing in permeate

stream

On the low pressure (permeate) side, polarization may occur due to the singular behavior of asymmetric membranes. The porous support layer can cause incomplete (radial) mixing of the local permeate emanating from the membrane with the bulk stream. Membranes used in the early gas separation studies were dense in structure having homogeneous properties, but with low fluxes. Accordingly, the permeation models were based on driving forces caused

Fig. 8. Schematic of the structure of an asymmetric membrane. (Adapted from Refs. [71,76] ).

by differences in partial pressures at the bulk stream conditions and negligible attention was paid to the actual structure of the membrane (a black box approach). However, the development of asymmetric membranes by phase inversion processes resulted in higher fluxes due to reduced thickness of the separating layer. Electron microscopic characterization studies on asymmetric RO membranes [ 74,751 showed the presence of a thin dense skin on top of a layer with somewhat tight porosity, followed by a layer of looser structure with greater porosity and much larger thickness than the intermediate layer (Fig. 8). However, it is customary to assign an effective thickness to the dense skin and intermediate layer together since in practice it is difficult to ascertain the individual thicknesses. This has prompted later researchers to model the membrane with its nonhomogeneous properties in a more realistic way. Some modeling studies in this direction are presented below. (i) Two-layer graded porosity model

Sirkar [76], in one of the early attempts, modeled the structure of the separating medium to consist of an ‘effective’ thin dense layer followed by a single graded porous substrate layer, neglecting the intermediate layer. The model was based on: flux through the top layer is governed by Fick’s law of diffusion whereas Knudsen diffusion is present in the porous layer. The overall flux is then given by

18

A.S. Kovvali et d/J.

(Pih -pil)

Ji = si

, Jmd-4)

Qim’

a0

(See Fig. 8 for the explanation of variables). The constant ~o=uoed,, depends on R, T etc., dpi is the pore diameter and Eiis the porosity at distance 6i from the top respectively. Qimis the permeability through skin layer, & is the ‘effective’ skin thickness, and d is the total membrane thickness. fii dependence occurs because of Knudsen diffusion through the pores. The average pore diameter (Eli) was varied empirically with distance from the skin surface obtaining an expression for selectivity ((x1,) between the components of a binary system. The expression for aI2 reduced to QIJQnm as in dense homogeneous membrane or r M2 M, as in the case of a microporous membrane with Knudsen diffusion for PI << Ph. The model assuming an exponential variation in &i with distance from the top was found to give the least deviation (upto < 30% ) from the experimental results compared to either quadratic or no variation in Eli. This model needs one empirical parameter to be obtained from experiments. Perhaps the above discrepancy could be reduced, as pointed out by Sirkar, by incorporating a more exact pore size distribution, and giving allowance for viscous flow [ 761 within the pores and skin surface due to ‘leaks’. On the whole, the model performed reasonably well.

Sci. 73 (1992) 1-23

structure of the porous sub-layer and its contribution to the permeation process. The governing equations are given earlier (eqns. 3034). Equation (30) in effect assumes the crossplug flow model and y: is the local composition of permeate. There is no ambiguity in y since pressure in the porous backing will be the same as in the permeate bulk [ 771. (iii) Three-layer cross-permeation model Chekalov et al. [78] considered the membrane to consist of a three-layer composite as shown in Fig. 8. They assumed diffusion to take place within the skin and diffusion-convection through capillaries in the porous sub-layer and backing layer. In the porous layers, the conservation equation is given by [ 781 (53) where t is an ‘equivalent’ porosity (no gradation was assumed) and 6 is the total length of the porous layers (see Fig. 8). Di is the diffusion coefficient of component i. r( = n/6) is the dimensionless distance (measured from the center of the channel) and defined with respect to S. Chekalov et al. [ 781 applied eqn. (53) from the center of the channel (r=O) to the interface between skin and porous sublayer. They set the interface as the <=l position which is an error according to the definition of <. Further, applicability of eqn. (53) to flow within the permeate channel is questionable since they used the initial condition as Y:(O)

(ii) Single-layer cross-permeation model Pan [ 301 proposed a model based on the assumption that permeation proceeds by the solution-diffusion mechanism through the skin, the driving force for this being the same as in crossflow. The local permeate stream issuing from the membrane mixes with the bulk permeate stream outside the porous layer and not vice versa. This picture completely ignored the

Membrane

(54)

=yi

where yi is the bulk permeate composition. Integration of eqn. (53) subject to eqn. (54) gives an expression for the local permeate composition as

(-iJ$t>

yX5) =yi exp

+$[1 -exi( -$$)I

(55)

A.S. Kovvali et al/J. Membrane

19

Sci. 73 (1992) 1-23

When the factor exp ( -JS/eDi) ~0, this corresponds to the crossflow model; when it is one this is equivalent to the plug flow model. For practical estimates, 6 can be approximated to the thickness of the membrane and an average c can perhaps be substituted, though it varies along the membrane thickness. Diffusivity coefficients inside the capillaries can be obtained from

(56) I& is the diffusivity of i in the bulk stream. Tortuosity (r), can be estimated by the expression of Meares [ 791: 2

(57) where &, is the polymer volume fraction in the porous layer. The calculated compositions based on the above model were found to lie between the observed cocurrent and countercurrent values for air separation with PVTMS membranes. Giglia et al. [BO] recently found (for binary mixtures of 0,-N, and He-N2) that radial mixing has only a very small effect on the calculated flow rate and pressure profiles of the permeate stream. As regards compositions, there were significant differences between the predictions incorporating the radial mixing effect and the observed values, more so at higher stage cuts where the predictions were much larger than the observed values. The observed data in the countercurrent pattern were found to lie between the predicted values of the parallel-flow model (complete radial mixing) and the crossflow model (complete polarization). They attributed this to partial radial mixing within the porous layer and/or flow imperfections on the feed side due to flow maldistribution, and perhaps due to axial flow within the porous layer. But they concluded that complete radial mixing takes place, based on agreement between

the model predictions and the data obtained in the cocurrent mode. Sidhoum et al. [ 291 studied the effect of concentration polarization (i.e. radial mixing effect) on the permeate side by employing a purge gas to increase the flux across asymmetric membranes in order to create favorable conditions for the effect to be observed unambiguously. According to them, presence of polarization to any significant extent would have validated the assumption of a crossflow type driving force across the skin as expounded by Pan and coworkers [30-32,461. But, on the contrary, the experimental observations showed [ 291, rather inconclusively, that the asymmetric membrane can still be treated as a dense membrane as far as the driving force and modeling aspects are concerned. Based on the above studies, it is still inconclusive whether gas permeators employing asymmetric membranes behave in the parallel flow or in the crossflow mode as far as their separation characteristics are concerned. 7. Concluding remarks The advent of composite and hollow fiber membranes has changed the scenario of gas separations. Newer possibilities in terms of novel applications and additional challenges in optimizations are being brought to the fore. Such challenges include influences of fiber dimensions, effective pressure drops through the module, and optimum flow arrangements. Modeling and analysis of membrane permeators constitute essential and integral parts of an overall successful design of gas separation operation. Apart from identifying and developing polymers suitable for selective separations, study of individual permeators is the first step which precedes optimization of the process to be carried out. Such an exercise will help identify the parameters that have direct influence and others that have a secondary, indirect

20

A.S. Kovvali et al./J. Membrane Sci. 73 (1992) l-23

or little role to play. An attempt was made here to review the models and analytical studies reported in the literature. Controversy still exists about the actual driving force across asymmetric membranes; high flux hollow fiber devices are being used to study this aspect closely. Data on permeability coefficients of mixtures, whether due to competitive sorption or plasticization or both, should be known for rigorous analysis [ 811. There is a need to study time dependent effects [ 82,831, multicomponent separations with pressure drop effects, and nonideal flow patterns such as partial mixing, boundary layer, flow maldistribution and polarization. Development of analytical or approximate solutions will be useful for fundamental understanding of permeator behavior, to find the effects of process and design parameters, for quick estimates in the comparison of various permeators and for process optimization studies. Though recycle permeators and cascade units have been omitted from the present review, additional material balances, boundary conditions, and possible variations in stream conditions associated with them, no doubt, will have an influence on the final separation. There is a need to study these aspects in depth for a better design and integration into overall processes.

JV

List of symbols

Y

d

4 D

Dbi do 4 J

channel cross-sectional area membrane area total molar concentration; constant in eqn. (34) effective membrane thickness average pore diameter molecular diffusivity diffusivity of component i in the bulk stream outside diameter of the fiber inside diameter of the fiber flux across the membrane

k L L*

;Ir NC M

MP P

PIZI Q Qim R

s T V V*

W

W x

z

volumetric flux of permeate mass transfer coefficient feed side molar flow rate dimensionless feed flow; = L/Lf (cocurrent crossflow); = L/L, and (countercurrent) coordinate normal to the membrane number of fibers in the permeator number of components in the feed stream molecular weight polarization modulus as defined in eqn. (50) pressure dimensionless mixing parameter, defined in eqn. (42) permeability across the membrane permeability of the component i through skin layer gas constant dimensionless membrane area, = (Q2/ d)PhAILf temperature permeate side molar flow rate dimensionless permeate flow, = V/L, (cocurrent and crossflow); = V/L, (countercurrent ) membrane width, in eqn. (42 ) molar flow rate mole fraction of the component on the feed side mole fraction of the component in the permeate side length coordinate

Greek symbols a p””

Y 8, si

selectivity, (QJd)l (Q214 overall separation factor for component 1 partial mixing parameter ratio of permeate to feed side pressures, WPll boundary layer thickness effective thickness of skin

21

AS. Kovvali et al. jJ. Membrane Sci. 73 (1992) l-23

;

P Pl

r ;

porosity cut fraction, V./L, viscosity of the gas mixture viscosity of the component 1 dimensionless distance, n/S tortuosity volume fraction

Subscripts/superscripts

I h i,i 1 n np F W

z * I

exit condition feed entry condition high pressure side (feed side) components i,j low pressure side (permeate side) normal direction nonpermeable component polymer total wall direction dimensionless variable local condition

8

9 10

11

12

13 14

15

16 17 18

References P. Meares, The physical chemistry of transport and separation by membranes, in: Membrane Separation Processes, Elsevier, Amsterdam, 1976, pp. l-38. J. Crank and G.S. Park, in: Diffusion in Polymers, Academic Press, New York, NY, 1968. H.L. Frisch and S.A. Stern, Diffusion of small molecules in polymers, CRC Critical Reviews in Solid State and Materials Science, Vol. II, No. 2, p. 123, CRC Press, Boca Raton, FL, 1983. C.E. Rogers, Permeation of gases and vapors in polymers, in: J. Comyn (Ed.), Polymer Permeability, Elsevier, NY, 1985, Chapt. 2, pp. 11-73. V.S. Stannett, W.J. Koros, D.R. Paul, H.K. Lonsdale and R.W. Baker, Recent advances in membrane science and technology, Adv. Polym. Sci., 32 (1979) 69. W.J. Koros and R.T. Chern, Separation of gaseous mixtures using polymer membranes, in: R.W. Rousseau (Ed.), Handbook of Separation of Process Technology, Wiley & Sons, New York, NY, 1987, Chapt. 20, pp. 862-953. S.L. Matson, J. Lopez and J.A. Quinn, Separation of gases with synthetic membranes, Chem. Eng. Sci., 38 (1983) 503.

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R. Albrecht and R. Rautenbach, Membrane Processes, Wiley & Sons, Chichester, 1989, Chapt. 5, pp. 131-171, Chapt. 13, pp. 422-454. S.-T. Hwang and K. Kammermayer, Membranes in Separations, Wiley-Interscience, New York, NY, 1975. S.A. Stern, Gas permeation processes, in: R.E. Lacey and S. Loeb (Eds.), Industrial Processing with Membranes, Wiley-Interscience, New York, NY, 1972, Chapt. 13, pp. 279-339. S.A. Stern and W.A. Walawender, Analysis of membrane separation parameters, Sep. Sci. Technol., 4 (1969) 129. R.W. Spillman, M.G. Barret and T.E. Cooley, Gas membrane process optimization, AIChE National Meeting, New Orleans, LA, March 1988. R.E. Kesting, Synthetic Polymeric Membranes, 2nd edn., Wiley & Sons, New York, NY, 1985. D.R. Lloyd (Ed.), Materials Science of Synthetic Membranes, ACS Symp. Ser., No. 269, American Chemical Society, Washington, DC, 1985. R.T. Chern, W.J. Koros and P.S. Fedkiw, Simulation of a hollow-fiber gas separator: The effects of process and design variables, Ind. Eng. Chem., Proc. Des. Dev., 24 (1985) 1015. T. Graham, Phil. Mag., 32 (1866) 401; cited in Ref.

PI.

R.M. Barrer, Diffusion in and Through Solids, Cambridge Press, London, 1941. P.R. Rony, The extent of separation: On the unification of the field of chemical separations, AIChE Symp. Ser., Vol. 68, No. 120 (1972) 89. A.S. Berman, Laminar flow in channels with porous walls, J. Appl. Phys., 24 (1953) 1232. S. Weller and W.A. Steiner, Engineering aspects of separation of gases, Chem. Eng. Prog., 46 (1950) 585. S.A. Stern, T.F. Sinclair, P.J. Garies, N.P. Vahldieck and P.H. Mohr, Helium recovery by permeation, Ind. Eng. Chem., 57 (1965) 49. R.W. Naylor and P.O. Backer, Enrichment calculations in gaseous diffusion: Large separation factor, AIChE J., 1 (1955) 95. C.W. Saltonstall, R.W. Lawrence and D. Niu, Calculation of performance in membrane separation, AIChE Spring National Meeting, Houston, TX, 1983. C.Y. Pan and H.W. Habgood, An analysis of the single-stage gaseous permeation process, Ind. Eng. Chem. Fundam., 13 (1974) 323. C.T. Blaisdell and K. Kammermeyer, Counter-current and co-current gas separation, Chem. Eng. Sci., 28 (1973) 1249. W.P. Walawender and S.A. Stern, Analysis of membrane separation parameters. II. Countercurrent and cocurrent flow in a single permeation stage, Sep. Sci., 7 (1972) 553.

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