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Effect of fiber variation on the performance of cross-flow hollow fiber gas separation modules
Journal of Membrane Science 153 (1999) 33±43
Effect of ®ber variation on the performance of cross-¯ow hollow ®ber gas separation modules J. Lemanskia, B. Liub, G.G. Lipscombb,* a
Chemical Engineering Department, University of Cincinnati, Cincinnati, OH 45221-0171, USA Chemical and Environmental Engineering Department, University of Toledo, Toledo, OH 43606-3390, USA
b
Received 3 April 1998; received in revised form 28 July 1998; accepted 29 July 1998
Abstract A quantitative analysis of the effects of variable ®ber properties on the performance of a cross-¯ow hollow ®ber gas separation module is presented. The effects of variations in size, permeance, and selectivity are considered. Fiber variability is detrimental to performance. The recovery and ¯ow rate of an enriched retentate stream decrease as variability increases. Some ®bers may actually stop producing product as purity increases. Additionally, performance is poorer if the permeate from all ®bers is not well mixed. The results of this work can be used to determine quality control guidelines for ®ber manufacture and evaluate process enhancements. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Fiber membranes; Gas separations; Modules; Theory
1. Introduction As membrane based gas separation processes mature, manufacturers push them to their performance limits. Economic success requires systems to operate as close as possible to their `theoretical' maximum. Unfortunately, deviations from this ideal behavior can be signi®cant. Such deviations may be attributable to a number of factors: poor ¯ow distribution, ®ber size variation, transport property variation, tube sheet leaks, temperature variations, pressure variations, concentration polarization, concentration dependent properties and others [1]. The literature discusses the qualitative and quantitative effects of many of these factors. However, a *Corresponding author. Tel.: +1-419-530-8088; fax: +1-419530-8086; e-mail: [email protected]
quantitative analysis of the effects of ®ber size and transport property variation is not available. The literature does suggest that the effects can be detrimental based on the reduction in liquid mass transfer coef®cientsthataccompaniesa®bersizevariation[2±5]. We present here a quantitative analysis of the effect of ®ber property (size, permeance, and separation factor) variation on the performance of a cross-¯ow, hollow ®ber, gas separation module for binary separations. From the analysis, one can determine the extent to which ®ber variability reduces performance. Furthermore, the results may be used to establish quality control guidelines for ®ber production. 2. Theoretical analysis We base the performance analysis on the following assumptions:
0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00245-2
34
1. 2. 3. 4. 5. 6. 7. 8. 9.
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Gas phases contain two components and are ideal. Shell pressure is constant. The retentate flows through the fiber lumens. Lumen pressure obeys the Hagen±Poiseuille equation for compressible flows. Lumen pressure drop is much smaller than the feed pressure. Concentration polarization and axial diffusion are negligible. Material properties are concentration independent. Isothermal operation. Properties of a single fiber do not vary along its length.
If ®ber properties vary over length scales much greater than the length of the ®ber bundle, ®ber properties are constant along the length of a given ®ber but may vary from ®ber to ®ber within the bundle. Let g( ) d be the fraction of ®bers for which the value of the material property lies in the interval ( , d ). may be the inner ®ber diameter, ID, the permeance (the ratio of the intrinsic, speci®c permeability to the effective membrane thickness based on the permeation area calculated using the ®ber outer diameter [1]), Q, or the separation factor, . If a property is Gaussian distributed, g is given by 2 =22 exp ÿ ÿ p ; g 2
Zmax
f g d ;
d 2 ÿN p ; dz
(2)
min
where f( ) is the ¯ow in ®bers for which the variable material property takes on a value of . If more than one material property is variable, then one must evaluate a multiple integral over each of the properties. This work, however, only considers the variation of a single property at a time. A momentum and mass balance about each ®ber yields the following dimensionless performance equations [1]:
(3) (4) (5)
where z is the dimensionless ®ber length (actual length/active ®ber length), x the mole fraction of the faster permeating species, the dimensionless retentate molar ¯ow (actual retentate ¯ow/feed ¯ow), and is the dimensionless retentate pressure (retentate pressure/feed pressure). J1 and J2 are the dimensionless molar permeation rates of the faster and slower permeating species, respectively, and are given by J1 N h x ÿ y; h
J2 N 1 ÿ x ÿ 1 ÿ y;
(6) (7)
where is the dimensionless permeate to retentate pressure ratio and y is the fast gas mole fraction in the permeate. If the retentate pressure drop is much smaller than the feed pressure, is approximately independent of z. The dimensionless groups Np and Nh are given by Np
256Rg TZF ; ID4 p2f
(8)
Nh
OD ZQ2 pf ; F
(9)
(1)
where is the average value and is the standard deviation of the distribution. One can calculate the average ¯ow per ®ber in a bundle of ®bers that possess a single variable material property from f
d x ÿJ1 ; dz d ÿ J1 J2 ; dz
where is the gas viscosity, Rg the ideal gas constant, T the temperature, Z the active ®ber length, ID the ®ber inner diameter, OD the ®ber outer diameter, F the feed molar ¯ow rate, pf the feed pressure, and the subscript 2 indicates the slower permeating component of the binary gas mixture. Np represents a dimensionless viscous resistance to ¯ow (viscosity) while Nh represents a dimensionless membrane area or, equivalently, a dimensionless permeance. The boundary conditions for Eqs. (3)±(5) are: x z 0 xf ;
(10)
z 0 1;
(11)
2 z 0 1:
(12)
To solve these equations one must also specify ®ber properties (OD, ID, Z, Q2, ), material properties (),
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
operating conditions (T, pf, ) and a molar feed rate for each ®ber (F). In practice, one does not specify the feed ¯ow directly but speci®es a pressure drop across the bundle; each ®ber has the same pressure drop imposed across it. The feed ¯ow adjusts to a value that permits the pressure to satisfy the following additional boundary condition: 2 z 1 21
(13)
Therefore, the feed ¯ow becomes an additional variable that is calculated simultaneously with the other unknowns. One must solve Eqs. (3)±(13) for each ®ber in the bundle to determine the overall performance. If the ®bers possess identical properties, only one set of these equations, three coupled ordinary differential equations, require simultaneous solution. If some material property varies, however, one must solve these three equations for all values of that material property. This large set of equations is coupled through the value of y, the permeate fast gas mole fraction, since all ®bers contribute to the permeate. An ef®cient numerical approximation to the solution of these equations can be developed by noting that we are interested in the performance of the entire bundle, not the performance of each individual ®ber in the bundle. For cases in which a high purity retentate is the desired product, the important performance parameters are retentate composition, retentate recovery, and retentate ¯ow. The fast gas composition in the product produced by combining the retentate ¯ows from each ®ber is given by R max R max min xRg d xFg d R min ; (14) xr R max max min Rg d min Fg d where R is the product retentate ¯ow in ®bers for which the varying material property equals . From Eq. (2), the denominator in Eq. (14) is equal to the total molar retentate ¯ow rate while the numerator is equal to the total fast gas ¯ow rate; the ratio is equal to the mixing cup average mole fraction. The retentate recovery is given by R max R max Fg d min Rg d R minmax : (15) Recovery R max min Fg d min Fg d
35
The numerator of Eq. (15) is the total retentate ¯ow exiting the ®ber bundle while the denominator is the total feed ¯ow to the ®ber bundle. For a given retentate composition, one wants the recovery to be as high as possible. To obtain a numerical approximation to Eqs. (14) and (15), we use quadrature formulas based on polynomial interpolating functions for the integrand. This converts the integrals to ®nite sums: Pn i1 xi i Fi gi wi xr P ; (16) n i1 i Fi gi wi Pn i1 i Fi gi wi ; (17) Recovery P n i1 Fi gi wi where n is the number of points used in the sum, the subscript i indicates the value evaluated at a speci®ed value of ( i referred to as base points), and the wi are the weighting factors. Values for the base points and weights for various values of n are tabulated in the literature [6]. Increasing n increases the accuracy of the simulation. To avoid the ambiguous assignment of speci®c values for max and min, we extend the integrals in Eqs. (16) and (17) to positive and negative in®nity [2,4]. For distributions that possess small standard deviations, one would expect this to be a good approximation. Additionally, this permits us to use Gauss± Hermite quadrature base points and weights developed speci®cally for integrals involving terms of the 2 form eÿx [6]. To evaluate retentate composition and recovery from Eqs. (16) and (17), one needs to know the values of x and only for values of equal to the base points. Therefore, one must solve 3n coupled ordinary differential equations to evaluate performance and one can determine the accuracy of the approximation by increasing n. For the results reported here, only distributions in ®ber ID, Q2 and are considered. These variations arise from a number of factors in the spinning process including: uniformity of spinneret holes in multi-hole spinnerets, differences in heat and mass transfer along the spinline, uniformity of ®ber tension, changes in raw materials with time, and many others. Moreover, manufacturers of asymmetric hollow ®ber membranes are driven to reduce skin thickness to increase productivity. A limited number of defects in the skin are
36
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
acceptable if the reduction in the effective separation factor is compensated for by a large enough increase in permeance. Poor ``control'' of these defects can exacerbate variations in permeance and are primarily responsible for observed selectivity variations. Of the three material properties considered here, one would expect the variation in to be least well represented by a Gaussian distribution. The distribution will have a lower bound of unity (no discriminating layer) and an upper bound equal to the maximum intrinsic selectivity (which may be a function of material history); one would expect the distribution to be skewed towards the upper bound. Although the distribution is taken as Gaussian, the procedure could be readily modi®ed to account for any distribution if the population statistics are known. If all other properties are held constant, one may calculate Np and Nh from F ; ID4 Q2 ; N h Ch F
N p Cp
(18) (19)
where Cp and Ch are given by 256Rg TZ==p2f and ( OD Zpf), respectively. Moreover, if one speci®es Nh then Np is given by Q2 1 N p Cp Ch : (20) Nh ID4 Substituting Eq. (20) into Eq. (5) gives a set of three differential equations, Eqs. (3)±(13) with boundary conditions (10)±(12), and one additional constraint, boundary condition (13) to solve for the four unknowns x, , , and Nh for each of the quadrature base points. We used two different expressions to calculate the permeate composition, y. If the permeate from ®bers with different properties is well mixed in the shell, the value of y is identical for all ®bers and is equal to R1 Pn J1 Fg d J1 i Fi gi wi Pn i1 ; y R 1 ÿ1 J J J Fg d 1 J2 i Fi gi wi 2 i1 ÿ1 1 (21)
where the numerator is equal to the molar permeation rate of the fast gas from all ®bers and the denominator is equal to the total molar permeation rate. Eqs. (6) and (7) give J1 and J2, respectively. If no mixing
occurs, y varies from ®ber to ®ber. The value for a given ®ber is equal to the ratio of the fast gas permeation rate to the total permeation rate for that ®ber: yi
J1 i i xi ÿ yi ; J1 J2 i i xi ÿ yi 1 ÿ xi ÿ 1 ÿ yi (22)
where the subscript i denotes the value evaluated for base point i. One substitutes a ®ber's material properties into Eq. (22) to calculate its value of y. The results indicate that Eqs. (21) and (22) give limiting values of y that place an upper and lower bound on performance. Both equations are quadratic equations for y that one solves simultaneously with the other performance equations. Numerical approximations to the differential equations governing performance were obtained using two different techniques: direct integration with a Runge± Kutta algorithm and a ®nite difference algorithm. A description of each technique follows. A combined fourth±®fth order Runge±Kutta algorithm was implemented in MATLAB using ODE45 [7]. Integration error was controlled to give four signi®cant ®gures in the approximation. One must specify a value for Nh for each base point before integrating the equations. Since these values are unknown, they were guessed initially and varied until the calculated exit retentate pressure differed by less than 0.1% from the speci®ed value. A modi®ed multivariate Newton±Raphson algorithm was used to improve the guesses in-between integrations. The (k1)th estimate for each Nh was obtained from " # Nih k ÿ Nih kÿ1 h k1 h k Ni Ni 1 ÿ ki ; ki ÿ kÿ1 i (23) where the superscripts k1, k, and kÿ1 denote the guess number and the subscript i denotes the value evaluated for base point i. One would expect Eq. (23) to be a good approximation to a full multi-variate Newton±Raphson technique if the performance of a given ®ber depends relatively weakly on the performance of other ®bers. We found that the values of Nh converged rapidly using Eq. (23) in all the cases. Finally, the number of base points used in the quadrature formulas was increased to test solution convergence. The results did not differ to four sig-
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
ni®cant ®gures upon increasing the number of base points from 3 to 5. The ®nite difference algorithm used second-order central differences to represent each derivative in the governing differential equations. First-order backward differences were used at z1, where a central difference approximation does not exist, and the boundary conditions, Eqs. (10)±(12), were speci®ed at z0 in lieu of the discretized differential equations. Specifying the pressure at z1, in addition to using the discretized differential equation for , provides the additional equation needed to calculate Nh for each base point. Solutions to the resulting non-linear algebraic equations for the value of x, , and at each node and the value of Nh for each base point were obtained using a full multi-variate Newton±Raphson algorithm. The number of nodes were varied from 20 to 100 to evaluate numerical error. The difference between 50 and 100 node points was found to be less than 1%. Both algorithms gave results for uniform ®bers that differed by less than 0.1% from the analytical solution [1]. Additionally, both algorithms gave results for nonuniform ®bers that differed by less than 0.1% for those cases in which both could be used; Section 3 discusses why both algorithms could not be used for all cases. From the results obtained with either algorithm, the retentate composition and recovery were calculated from Eqs. (16) and (17) using the converged values of Nh. Module performance for a range of retentate compositions was obtained by varying the ®nal pressure.
37
will use percent variation to indicate the value of the standard deviation, as a percentage of the average value, for a distribution; for example a 15% variation implies that the standard deviation is 15% of the average value. In the ®gures, retentate ¯ows are shown relative to the retentate ¯ow from a module with uniform ®bers. Fig. 1 compares the effects of similar variations in ID, Q2 and on performance for the well mixed case. Performance decreases slightly for 15% variation in either the permeance or separation factor; a much larger performance decline is observed for a 10% variation in inner radius. Therefore, ®ber variation increases the work (due to the decrease in recovery) and area (due to the decrease in retentate ¯ow) requirements to produce a product of speci®ed ¯ow rate and purity.
3. Results The results presented here are for base (average) 8 and 0:1 which correspond to values of values representative of commercial oxygen/nitrogen products; such products are used to produce high purity nitrogen for a broad range of applications. The results are independent of average values for the other material properties. In the discussion that follows, we will refer to the retentate recovery ± retentate fast gas (oxygen) mole fraction relationship and the retentate ¯ow ± retentate fast gas (oxygen) mole fraction relationship as performance or the performance curves. Additionally, we
Fig. 1. The effect of variations in ID, Q2, and on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; short dash±long dash, 15% variation; intermediate dash, 15% Q2 variation; short dash, 10% ID variation. Note the lines for and Q2 overlap in (a).
38
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Why do the performance curves for the ID variation case end at a fast gas mole fraction of 0.045? The calculations were performed until the combined product fast gas mole fraction dropped below 0.0025 or the recovery at any of the quadrature points dropped below 5%. The latter criterion stops the calculation when ®ber performance at a quadrature point drops too low and the ®ber appears to stop producing product. The quadrature point corresponding to the smallest ID ®bers ID=ID 0:7 met this criteria when the overall retentate composition was 0.045. Flow rates decrease as size decreases but permeation rates do not change since permeance and outer diameter are held constant. Thus, the feed to ®bers smaller than some critical ®ber size appears to permeate completely. This minimum size is less than the smallest value of ID at a quadrature point for product compositions >0.045 but is greater than the smallest value for compositions <0.045. Fig. 1 does not contain results for a 15% variation in ID since the quadrature point corresponding to the smallest ID ®bers ID=ID 0:55 had a recovery of less than 5% over the entire retentate composition range. These results clearly indicate the signi®cance of ID variation on performance. ID variations have a greater impact on performance because the relationship between pressure drop and ¯ow rate depends on ID4 as indicated by Eqs. (5) and (8); a 5% change in ID results in 20% change in ID4 and the associated pressure drop ± ¯ow rate relationship. Do the smallest ID ®bers produce any product retentate or do they completely stop producing gas? The analytical solution for the performance of cross¯ow modules with uniform ®bers indicates that complete permeation occurs when Nh equals Nh
xf 1 ÿ xf ; 1 ÿ
(24)
as demonstrated in Appendix A; Appendix A also discusses the physical signi®cance of this result in more detail. This suggests that ®bers can completely stop producing gas. If some ®bers do not produce any retentate, what happens at the product end of these ®bers? The feed that enters these ®bers permeates completely before the end of the ®ber is reached. The pressure at this
point of zero retentate ¯ow will drop below that at the retentate exit to induce a ¯ow into the ®ber from the product end. All of the retentate that enters the ®ber from the product end will permeate as well. One can modify the solution algorithms described previously to determine the location and pressure at the point of zero retentate ¯ow. To use the direct integration method, one must integrate the performance equations from both the feed and product ends; the composition and pressure at both ends are known or can be calculated. The entering gas ¯ow rates at both ends are then varied until both retentate ¯ows vanish and the pressures are equal at a common point. To use the ®nite difference method, one simply replaces the differential equation for x at the product end, z1, with the boundary condition that the retentate composition is equal to the composition of the product produced by the other ®bers, xxr as given by Eq. (16). Since the latter modi®cation was signi®cantly easier than the former, only the ®nite difference algorithm was used to determine performance when ®bers stopped producing product. With this modi®cation, performance calculations were possible for product retentate compositions down to 0.0025 for a 10% ID variation as shown in Fig. 2. Results for a 5% variation are also shown. Clearly, as the percent variation increases the drop in performance increases. For example, for a fast gas mole fraction of 0.005 retentate recovery decreases from 30% to 28±23% and relative retentate ¯ow decreases from 1 to 0.9±0.65 as the ID variation increases from 0% to 5±10%. Such changes would signi®cantly have an impact on the design of a system for the production of a 99.5% nitrogen stream. Fig. 2 also illustrates the effects of permeate mixing. Performance for the no mixing case is signi®cantly poorer than that for the well-mixed case. At the highest retentate purities, the decrease in recovery or relative retentate ¯ow more than doubles in the absence of mixing. One would expect actual performance to lie somewhere in between these two limiting cases. These results suggest that module designs which promote permeate mixing can help mitigate the effects of ®ber variation. The patent literature describes numerous design features that attempt to improve shell mixing on a macroscopic, module length scale [1]. Additionally, if modules are wound from multiple
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Fig. 2. The effect of variations in ID on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; short dash±long dash, 5% variation/well mixed; intermediate dash, 5% variation/no mixing; short dash, 10% variation/well mixed; dot, 10% variation/no mixing.
®ber lots with varying ®ber properties, mixing ®ber tows together would help promote mixing on a microscopic, ®ber length scale. Fig. 3 shows typical pro®les of composition, retentate ¯ow rate, and pressure along the module. These pro®les are for a module with a 10% ID variation and no permeate mixing producing a product with a fast gas composition of 0.0029. The corresponding retentate recovery and relative ¯ow are 0.066 and 0.20, respectively. The pro®les are for the ®ve quadrature points used in the simulation, ID=ID 0:71, 0.86, 1.0, 1.14, and 1.29. Fig. 3(a) shows that the fast gas composition drops rapidly initially but the rate of change decreases with axial distance. At z0.62 the feed to the smallest ®ber shown, ID=ID 0:71, completely permeates while the feed to the next smallest ®ber completely perme-
39
ates at z0.81. This is dif®cult to see due to the scale of Fig. 3(a) but is clearly shown in Fig. 3(b) with a change of scale. Prior to complete permeation, composition increases with increasing ®ber ID at a ®xed position; larger ®bers have higher feed gas ¯ow rates so permeation reduces the composition less quickly. In contrast, past the point of complete permeation, composition increases with decreasing ®ber ID at a ®xed position. This behavior is due to higher ¯ow rates into the smaller ®bers of the product retentate from the product end; more gas must enter the smaller ®bers since permeation occurs over a greater distance from the product end. One might expect smaller ®bers to possess lower ¯ow rates but the pressure at the complete permeation point is much smaller in the smaller ®bers than the large ®bers to drive the ¯ow, see Fig. 3(d). Fig. 3(c) illustrates how retentate recovery changes with axial distance. Positive values indicate a ¯ow from the feed end to the product end while negative values indicate a ¯ow in the opposite direction. Negative values are scaled with the same feed ¯ow rate, though, as positive values. The complete permeation point corresponds to the point where the recovery equals zero. Clearly, only those ®bers with ID=ID > 1 contribute to the product produced at this operating point since the recovery is negative for ID=ID < 1 at z1. The relative pressure pro®les in Fig. 3(d) also shows the complete permeation point as the point at which the pressure is a minimum. The minimum value decreases dramatically as ®ber size decreases. These differences are responsible for higher ¯ow rates of product back into smaller ®bers than larger ®bers as discussed previously. One can directly compare the no mixing results with the analytical solution since each ®ber performs as if it were in a module of uniform ®bers. In all cases complete permeation ®rst occurred when Nh was equal to 0.907 (to within less than 1%), consistent with the analytical prediction. Moreover, the calculated values of x and on either side of the point where 0 and the location of this point were in excellent agreement with the analytical solution. This suggests that the numerical method was robust enough to capture the zero permeation point and permit accurate performance calculations in the presence of ®bers that did not produce the product.
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J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Fig. 3. Simulation results for a module with 10% ID variation and no permeate mixing producing a product with xr0.0029: (a) retentate fast gas composition, (b) fast gas composition near the product end of module, (c) retentate recovery, and (d) retentate pressure. The lines in all figures are: dot, ID=ID 0:71; intermediate dash, ID=ID 0:86; solid line, ID=ID 1:0; short dash±long dash, ID=ID 1:14; short dash, ID=ID 1:29.
Fig. 4 illustrates the effects of percent variation in Q2 and permeate mixing on module performance. As observed for the ID variation cases, ®bers with the highest permeation rate to feed rate ratio stop producing product and start consuming retentate as the retentate purity increases. Similar to the ID variation results, performance decreases as variation increases and performance is much worse in the absence of permeate mixing. For example, recovery and retentate ¯ow for a fast gas mole fraction of 0.01 decreases by 40% with a 30% permeance variation and no permeate mixing. This implies one must increase the area and compressor horsepower by a factor of 1.6 to provide the same performance as a module with no ®ber variability! Fig. 5 illustrates the effect of percent variation in and permeate mixing on module performance. As observed for size and permeance variations, perfor-
mance decreases as the percent variation increases. The magnitude of the change is comparable to the permeance variation case. However, ®bers did not stop producing gas in any of the simulations with selectivity variations. In contrast to the size and permeance variation cases, permeate mixing had no effect on the performance. The performance changes due to mixing are highly dependent on how permeate composition varies with ®ber properties for the speci®ed operating conditions. In general, the performance of some ®bers improves (due to a decrease in the permeate fast gas composition) while that of the others decline (due to an increase in the permeate fast gas composition) and the net effect can be either an overall performance increase or decrease. The results indicate that the net effect is a performance improvement with mixing for size and
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Fig. 4. The effect of variations in Q2 on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; short dash±long dash, 15% variation/well mixed; intermediate dash, 15% variation/no mixing; short dash, 30% variation/well mixed; dot, 30% variation/no mixing.
permeance variations but no change occurs for selectivity variations. These observations may have an impact on the operation of ®ber manufacturing processes. If process changes affect both selectivity and permeance, reducing permeance variation will have a greater impact on module performance than reducing selectivity variation since the effects of permeate mixing are minimized. 4. Conclusions We present an analysis of the effects of ®ber material property variations on the performance of a cross-¯ow hollow ®ber gas separation module. The effects of variation in ®ber size, permeance, and separation factor are explicitly considered.
41
Fig. 5. The effect of variations in on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; intermediate dash, 15% variation/no mixing or well mixed; dot, 30% variation/no mixing or well mixed. Note that the differences between the well mixed and no mixing results are not distinguishable.
Increasing the variation decreased the module performance. For comparable levels of variability, the effects are greatest for a size distribution. Permeance and separation factor variations have similar, less pronounced effects. As product retentate purity increases, some ®bers may stop producing product and actually consume part of the product produced by other ®bers. This phenomenon was observed for size and permeance variations but not selectivity variations. Additionally, for size and permeance variations, performance is highest if the permeate from all ®bers is well mixed. Permeate mixing had little effect for selectivity variations. This is due to how permeate mixing changes the performance of the distribution of ®bers.
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J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
The results presented here can help guide the development of ®ber manufacturing processes. One can use the analysis to determine how closely ®ber properties must be controlled to achieve desired performance and evaluate the relative effects of process changes on performance.
Acknowledgements
5. List of symbols
The analytical solution for the performance of a cross-¯ow module with uniform ®ber properties is given implicitly by [8,9]:
Ch Cp f F g ID J Nh Np OD pf Q R Rg T w x y z Z
constant defined in Eq. (19) (m2 Pa) constant defined in Eq. (18) (m4 s molÿ1) flow rate (mol sÿ1) feed rate (mol sÿ1) distribution function fiber inner diameter (m) dimensionless molar permeation rate defined by Eqs. (6) and (7) dimensionless group defined by Eq. (9) dimensionless group defined by Eq. (8) fiber outer diameter (m) feed pressure (Pa) permeance (mol mÿ2 Paÿ1 sÿ1) product retentate flow (mol sÿ1) gas constant (Pa m3 molÿ1 Kÿ1) temperature (K) quadrature weighting factors mole fraction of faster permeating gas component in retentate mole fraction of faster permeating gas component in permeate dimensionless active fiber length active fiber length (m) separation factor arbitrary material property ratio of permeate to feed pressure viscosity (Pa s) dimensionless retentate pressure standard deviation dimensionless molar retentate flow rate
Subscripts and superscripts f value for feed r value for product retentate max maximum value min minimum value 1 value for faster permeating gas component 2 value for slower permeating gas component average value Å
The authors acknowledge partial support of this work by the National Science Foundation through grant CTS-9408414. Appendix A
N h 1 ÿ
xf ÿ xr r 1 ÿ xf ÿ 1 ÿ xr r ; (A.1)
1 ÿ1 = 1ÿ ÿ1 yr r yf 1 ÿ yr ÿ ÿ1 = 1ÿ ÿ1 1 ÿ yf ÿ ÿ 1yr ; ÿ ÿ 1yf
(A.2)
where the subscripts r and f indicated the value evaluated for the product retentate and feed, respectively, and y is the local permeate composition given by Eq. (22). Given a value for Nh, the membrane area, one solves Eqs. (A.1), (A.2) and (22) simultaneously for the values of xr, yr, and r. Alternatively, one can specify a value for xr and solve Eqs. (A.1), (A.2) and (22) for Nh, yr, and r. These equations predict that r becomes identically zero at a ®nite value of Nh. Eq. (A.2) indicates that yr must be identically zero if r is identically zero, and Eq. (22) indicates that xr must be identically zero if yr is identically zero. Therefore, Eq. (A.1) gives the value of Nh at which r becomes identically zero as Nh
xf 1 ÿ xf : 1 ÿ
(A.3)
Eq. (A.3) indicates that as approaches 1 this value becomes unbounded; an in®nite area is required for the feed to completely permeate as intuition suggests. As approaches 1, this value becomes 1/(1ÿ ) which is also consistent with intuition. The membrane is equivalent to a porous frit and permeation across this frit is equivalent to ¯ow. The permeate composition is the same as the feed composition and if the feed pressure is maintained higher than the permeate pres-
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
sure all of the feed will permeate, or ¯ow, across the membrane in a ®nite area. For values of >1 and >1, however, Eq. (A.3) predicts that the feed completely permeates across the membrane for a ®nite value of Nh or membrane area which might appear inconsistent with intuition. For example, given the values considered here of xf0.21, 8, and 0.1, this value of Nh is 0.907 while, for !1, this value is (1ÿxf)/(1ÿ ). How is this possible? First consider the in®nite selectivity result. As the slow gas permeance becomes arbitrarily small, !1 and Nh approaches the ®nite ®xed value given above. The de®nition of Nh, Eq. (9), indicates that this is equivalent to a ®xed value for the product of membrane area and slow gas permeance. Therefore, as the slow gas permeance becomes arbitrarily small the area required for complete permeation becomes arbitrarily large consistent with intuition. The ®nite value of Nh predicted for the conditions considered here is not consistent with intuition engendered from mass transfer in other membrane processes in which one or more solutes are removed by diffusion across the membrane, such as pure dialysis. In such processes an in®nite area is required to completely remove a solute. However, a fundamental difference exists between pure dialysis and gas separation: in gas separation all components of the feed permeate while in pure dialysis the solvent does not. In cross-¯ow gas separation processes, slow gas permeation increases the fast gas concentration in the retentate above the value that would be observed in the absence of slow
43
gas permeation. Eliminating slow gas permeation by letting its permeance become arbitrarily small makes the slow gas equivalent to the solvent in dialysis and, as discussed above, the selectivity and required membrane area for complete permeation become in®nite. References [1] G.G. Lipscomb, Design of hollow fiber contactors for membrane gas separations, in: The 1996 Membrane Technology Reviews, Business Communication Co., Inc., Norwalk, CT, 1996, pp. 23±102. [2] S.R. Wickramasinghe, M.J. Semmens, E.L. Cussler, Mass transfer in various hollow fiber geometries, J. Membr. Sci. 69 (1985) 235±250. [3] H. Kruelen, C.A. Smolders, G.F. Verteeg, W.P.M. van Swaaij, Microporous hollow fibre membrane modules as gas±liquid contactors. Part 1. Physical mass transfer process, J. Membr. Sci. 78 (1993) 197±216. [4] S. Elmore, G.G. Lipscomb, Analytical approximations of the effect of a fiber size distribution on the performance of hollow fiber membrane separation devices, J. Membr. Sci. 98 (1995) 49±56. [5] R.O. Crowder, E.L. Cussler, Mass transfer in hollow-fiber modules with non-uniform hollow fibers, J. Membr. Sci. 134 (1997) 235±244. [6] B. Carnahan, H.A. Luther, J.O. Wilkes, Applied Numerical Methods, Krieger, Malabar, FL, 1990. [7] MATLABTM, The MathWorks Inc., Natick, MA. [8] C.Y. Pan, H.W. Habgood, Gas separation by permeation. Part I: Calculation methods and parametric analysis, Can. J. Chem. Eng. 56 (1978) 197±209. [9] C.Y. Pan, Gas separation by high-flux, asymmetric hollowfiber membrane, AIChE J. 32 (1986) 2020±2027.
Effect of ®ber variation on the performance of cross-¯ow hollow ®ber gas separation modules J. Lemanskia, B. Liub, G.G. Lipscombb,* a
Chemical Engineering Department, University of Cincinnati, Cincinnati, OH 45221-0171, USA Chemical and Environmental Engineering Department, University of Toledo, Toledo, OH 43606-3390, USA
b
Received 3 April 1998; received in revised form 28 July 1998; accepted 29 July 1998
Abstract A quantitative analysis of the effects of variable ®ber properties on the performance of a cross-¯ow hollow ®ber gas separation module is presented. The effects of variations in size, permeance, and selectivity are considered. Fiber variability is detrimental to performance. The recovery and ¯ow rate of an enriched retentate stream decrease as variability increases. Some ®bers may actually stop producing product as purity increases. Additionally, performance is poorer if the permeate from all ®bers is not well mixed. The results of this work can be used to determine quality control guidelines for ®ber manufacture and evaluate process enhancements. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Fiber membranes; Gas separations; Modules; Theory
1. Introduction As membrane based gas separation processes mature, manufacturers push them to their performance limits. Economic success requires systems to operate as close as possible to their `theoretical' maximum. Unfortunately, deviations from this ideal behavior can be signi®cant. Such deviations may be attributable to a number of factors: poor ¯ow distribution, ®ber size variation, transport property variation, tube sheet leaks, temperature variations, pressure variations, concentration polarization, concentration dependent properties and others [1]. The literature discusses the qualitative and quantitative effects of many of these factors. However, a *Corresponding author. Tel.: +1-419-530-8088; fax: +1-419530-8086; e-mail: [email protected]
quantitative analysis of the effects of ®ber size and transport property variation is not available. The literature does suggest that the effects can be detrimental based on the reduction in liquid mass transfer coef®cientsthataccompaniesa®bersizevariation[2±5]. We present here a quantitative analysis of the effect of ®ber property (size, permeance, and separation factor) variation on the performance of a cross-¯ow, hollow ®ber, gas separation module for binary separations. From the analysis, one can determine the extent to which ®ber variability reduces performance. Furthermore, the results may be used to establish quality control guidelines for ®ber production. 2. Theoretical analysis We base the performance analysis on the following assumptions:
0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00245-2
34
1. 2. 3. 4. 5. 6. 7. 8. 9.
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Gas phases contain two components and are ideal. Shell pressure is constant. The retentate flows through the fiber lumens. Lumen pressure obeys the Hagen±Poiseuille equation for compressible flows. Lumen pressure drop is much smaller than the feed pressure. Concentration polarization and axial diffusion are negligible. Material properties are concentration independent. Isothermal operation. Properties of a single fiber do not vary along its length.
If ®ber properties vary over length scales much greater than the length of the ®ber bundle, ®ber properties are constant along the length of a given ®ber but may vary from ®ber to ®ber within the bundle. Let g( ) d be the fraction of ®bers for which the value of the material property lies in the interval ( , d ). may be the inner ®ber diameter, ID, the permeance (the ratio of the intrinsic, speci®c permeability to the effective membrane thickness based on the permeation area calculated using the ®ber outer diameter [1]), Q, or the separation factor, . If a property is Gaussian distributed, g is given by 2 =22 exp ÿ ÿ p ; g 2
Zmax
f g d ;
d 2 ÿN p ; dz
(2)
min
where f( ) is the ¯ow in ®bers for which the variable material property takes on a value of . If more than one material property is variable, then one must evaluate a multiple integral over each of the properties. This work, however, only considers the variation of a single property at a time. A momentum and mass balance about each ®ber yields the following dimensionless performance equations [1]:
(3) (4) (5)
where z is the dimensionless ®ber length (actual length/active ®ber length), x the mole fraction of the faster permeating species, the dimensionless retentate molar ¯ow (actual retentate ¯ow/feed ¯ow), and is the dimensionless retentate pressure (retentate pressure/feed pressure). J1 and J2 are the dimensionless molar permeation rates of the faster and slower permeating species, respectively, and are given by J1 N h x ÿ y; h
J2 N 1 ÿ x ÿ 1 ÿ y;
(6) (7)
where is the dimensionless permeate to retentate pressure ratio and y is the fast gas mole fraction in the permeate. If the retentate pressure drop is much smaller than the feed pressure, is approximately independent of z. The dimensionless groups Np and Nh are given by Np
256Rg TZF ; ID4 p2f
(8)
Nh
OD ZQ2 pf ; F
(9)
(1)
where is the average value and is the standard deviation of the distribution. One can calculate the average ¯ow per ®ber in a bundle of ®bers that possess a single variable material property from f
d x ÿJ1 ; dz d ÿ J1 J2 ; dz
where is the gas viscosity, Rg the ideal gas constant, T the temperature, Z the active ®ber length, ID the ®ber inner diameter, OD the ®ber outer diameter, F the feed molar ¯ow rate, pf the feed pressure, and the subscript 2 indicates the slower permeating component of the binary gas mixture. Np represents a dimensionless viscous resistance to ¯ow (viscosity) while Nh represents a dimensionless membrane area or, equivalently, a dimensionless permeance. The boundary conditions for Eqs. (3)±(5) are: x z 0 xf ;
(10)
z 0 1;
(11)
2 z 0 1:
(12)
To solve these equations one must also specify ®ber properties (OD, ID, Z, Q2, ), material properties (),
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
operating conditions (T, pf, ) and a molar feed rate for each ®ber (F). In practice, one does not specify the feed ¯ow directly but speci®es a pressure drop across the bundle; each ®ber has the same pressure drop imposed across it. The feed ¯ow adjusts to a value that permits the pressure to satisfy the following additional boundary condition: 2 z 1 21
(13)
Therefore, the feed ¯ow becomes an additional variable that is calculated simultaneously with the other unknowns. One must solve Eqs. (3)±(13) for each ®ber in the bundle to determine the overall performance. If the ®bers possess identical properties, only one set of these equations, three coupled ordinary differential equations, require simultaneous solution. If some material property varies, however, one must solve these three equations for all values of that material property. This large set of equations is coupled through the value of y, the permeate fast gas mole fraction, since all ®bers contribute to the permeate. An ef®cient numerical approximation to the solution of these equations can be developed by noting that we are interested in the performance of the entire bundle, not the performance of each individual ®ber in the bundle. For cases in which a high purity retentate is the desired product, the important performance parameters are retentate composition, retentate recovery, and retentate ¯ow. The fast gas composition in the product produced by combining the retentate ¯ows from each ®ber is given by R max R max min xRg d xFg d R min ; (14) xr R max max min Rg d min Fg d where R is the product retentate ¯ow in ®bers for which the varying material property equals . From Eq. (2), the denominator in Eq. (14) is equal to the total molar retentate ¯ow rate while the numerator is equal to the total fast gas ¯ow rate; the ratio is equal to the mixing cup average mole fraction. The retentate recovery is given by R max R max Fg d min Rg d R minmax : (15) Recovery R max min Fg d min Fg d
35
The numerator of Eq. (15) is the total retentate ¯ow exiting the ®ber bundle while the denominator is the total feed ¯ow to the ®ber bundle. For a given retentate composition, one wants the recovery to be as high as possible. To obtain a numerical approximation to Eqs. (14) and (15), we use quadrature formulas based on polynomial interpolating functions for the integrand. This converts the integrals to ®nite sums: Pn i1 xi i Fi gi wi xr P ; (16) n i1 i Fi gi wi Pn i1 i Fi gi wi ; (17) Recovery P n i1 Fi gi wi where n is the number of points used in the sum, the subscript i indicates the value evaluated at a speci®ed value of ( i referred to as base points), and the wi are the weighting factors. Values for the base points and weights for various values of n are tabulated in the literature [6]. Increasing n increases the accuracy of the simulation. To avoid the ambiguous assignment of speci®c values for max and min, we extend the integrals in Eqs. (16) and (17) to positive and negative in®nity [2,4]. For distributions that possess small standard deviations, one would expect this to be a good approximation. Additionally, this permits us to use Gauss± Hermite quadrature base points and weights developed speci®cally for integrals involving terms of the 2 form eÿx [6]. To evaluate retentate composition and recovery from Eqs. (16) and (17), one needs to know the values of x and only for values of equal to the base points. Therefore, one must solve 3n coupled ordinary differential equations to evaluate performance and one can determine the accuracy of the approximation by increasing n. For the results reported here, only distributions in ®ber ID, Q2 and are considered. These variations arise from a number of factors in the spinning process including: uniformity of spinneret holes in multi-hole spinnerets, differences in heat and mass transfer along the spinline, uniformity of ®ber tension, changes in raw materials with time, and many others. Moreover, manufacturers of asymmetric hollow ®ber membranes are driven to reduce skin thickness to increase productivity. A limited number of defects in the skin are
36
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
acceptable if the reduction in the effective separation factor is compensated for by a large enough increase in permeance. Poor ``control'' of these defects can exacerbate variations in permeance and are primarily responsible for observed selectivity variations. Of the three material properties considered here, one would expect the variation in to be least well represented by a Gaussian distribution. The distribution will have a lower bound of unity (no discriminating layer) and an upper bound equal to the maximum intrinsic selectivity (which may be a function of material history); one would expect the distribution to be skewed towards the upper bound. Although the distribution is taken as Gaussian, the procedure could be readily modi®ed to account for any distribution if the population statistics are known. If all other properties are held constant, one may calculate Np and Nh from F ; ID4 Q2 ; N h Ch F
N p Cp
(18) (19)
where Cp and Ch are given by 256Rg TZ==p2f and ( OD Zpf), respectively. Moreover, if one speci®es Nh then Np is given by Q2 1 N p Cp Ch : (20) Nh ID4 Substituting Eq. (20) into Eq. (5) gives a set of three differential equations, Eqs. (3)±(13) with boundary conditions (10)±(12), and one additional constraint, boundary condition (13) to solve for the four unknowns x, , , and Nh for each of the quadrature base points. We used two different expressions to calculate the permeate composition, y. If the permeate from ®bers with different properties is well mixed in the shell, the value of y is identical for all ®bers and is equal to R1 Pn J1 Fg d J1 i Fi gi wi Pn i1 ; y R 1 ÿ1 J J J Fg d 1 J2 i Fi gi wi 2 i1 ÿ1 1 (21)
where the numerator is equal to the molar permeation rate of the fast gas from all ®bers and the denominator is equal to the total molar permeation rate. Eqs. (6) and (7) give J1 and J2, respectively. If no mixing
occurs, y varies from ®ber to ®ber. The value for a given ®ber is equal to the ratio of the fast gas permeation rate to the total permeation rate for that ®ber: yi
J1 i i xi ÿ yi ; J1 J2 i i xi ÿ yi 1 ÿ xi ÿ 1 ÿ yi (22)
where the subscript i denotes the value evaluated for base point i. One substitutes a ®ber's material properties into Eq. (22) to calculate its value of y. The results indicate that Eqs. (21) and (22) give limiting values of y that place an upper and lower bound on performance. Both equations are quadratic equations for y that one solves simultaneously with the other performance equations. Numerical approximations to the differential equations governing performance were obtained using two different techniques: direct integration with a Runge± Kutta algorithm and a ®nite difference algorithm. A description of each technique follows. A combined fourth±®fth order Runge±Kutta algorithm was implemented in MATLAB using ODE45 [7]. Integration error was controlled to give four signi®cant ®gures in the approximation. One must specify a value for Nh for each base point before integrating the equations. Since these values are unknown, they were guessed initially and varied until the calculated exit retentate pressure differed by less than 0.1% from the speci®ed value. A modi®ed multivariate Newton±Raphson algorithm was used to improve the guesses in-between integrations. The (k1)th estimate for each Nh was obtained from " # Nih k ÿ Nih kÿ1 h k1 h k Ni Ni 1 ÿ ki ; ki ÿ kÿ1 i (23) where the superscripts k1, k, and kÿ1 denote the guess number and the subscript i denotes the value evaluated for base point i. One would expect Eq. (23) to be a good approximation to a full multi-variate Newton±Raphson technique if the performance of a given ®ber depends relatively weakly on the performance of other ®bers. We found that the values of Nh converged rapidly using Eq. (23) in all the cases. Finally, the number of base points used in the quadrature formulas was increased to test solution convergence. The results did not differ to four sig-
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
ni®cant ®gures upon increasing the number of base points from 3 to 5. The ®nite difference algorithm used second-order central differences to represent each derivative in the governing differential equations. First-order backward differences were used at z1, where a central difference approximation does not exist, and the boundary conditions, Eqs. (10)±(12), were speci®ed at z0 in lieu of the discretized differential equations. Specifying the pressure at z1, in addition to using the discretized differential equation for , provides the additional equation needed to calculate Nh for each base point. Solutions to the resulting non-linear algebraic equations for the value of x, , and at each node and the value of Nh for each base point were obtained using a full multi-variate Newton±Raphson algorithm. The number of nodes were varied from 20 to 100 to evaluate numerical error. The difference between 50 and 100 node points was found to be less than 1%. Both algorithms gave results for uniform ®bers that differed by less than 0.1% from the analytical solution [1]. Additionally, both algorithms gave results for nonuniform ®bers that differed by less than 0.1% for those cases in which both could be used; Section 3 discusses why both algorithms could not be used for all cases. From the results obtained with either algorithm, the retentate composition and recovery were calculated from Eqs. (16) and (17) using the converged values of Nh. Module performance for a range of retentate compositions was obtained by varying the ®nal pressure.
37
will use percent variation to indicate the value of the standard deviation, as a percentage of the average value, for a distribution; for example a 15% variation implies that the standard deviation is 15% of the average value. In the ®gures, retentate ¯ows are shown relative to the retentate ¯ow from a module with uniform ®bers. Fig. 1 compares the effects of similar variations in ID, Q2 and on performance for the well mixed case. Performance decreases slightly for 15% variation in either the permeance or separation factor; a much larger performance decline is observed for a 10% variation in inner radius. Therefore, ®ber variation increases the work (due to the decrease in recovery) and area (due to the decrease in retentate ¯ow) requirements to produce a product of speci®ed ¯ow rate and purity.
3. Results The results presented here are for base (average) 8 and 0:1 which correspond to values of values representative of commercial oxygen/nitrogen products; such products are used to produce high purity nitrogen for a broad range of applications. The results are independent of average values for the other material properties. In the discussion that follows, we will refer to the retentate recovery ± retentate fast gas (oxygen) mole fraction relationship and the retentate ¯ow ± retentate fast gas (oxygen) mole fraction relationship as performance or the performance curves. Additionally, we
Fig. 1. The effect of variations in ID, Q2, and on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; short dash±long dash, 15% variation; intermediate dash, 15% Q2 variation; short dash, 10% ID variation. Note the lines for and Q2 overlap in (a).
38
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Why do the performance curves for the ID variation case end at a fast gas mole fraction of 0.045? The calculations were performed until the combined product fast gas mole fraction dropped below 0.0025 or the recovery at any of the quadrature points dropped below 5%. The latter criterion stops the calculation when ®ber performance at a quadrature point drops too low and the ®ber appears to stop producing product. The quadrature point corresponding to the smallest ID ®bers ID=ID 0:7 met this criteria when the overall retentate composition was 0.045. Flow rates decrease as size decreases but permeation rates do not change since permeance and outer diameter are held constant. Thus, the feed to ®bers smaller than some critical ®ber size appears to permeate completely. This minimum size is less than the smallest value of ID at a quadrature point for product compositions >0.045 but is greater than the smallest value for compositions <0.045. Fig. 1 does not contain results for a 15% variation in ID since the quadrature point corresponding to the smallest ID ®bers ID=ID 0:55 had a recovery of less than 5% over the entire retentate composition range. These results clearly indicate the signi®cance of ID variation on performance. ID variations have a greater impact on performance because the relationship between pressure drop and ¯ow rate depends on ID4 as indicated by Eqs. (5) and (8); a 5% change in ID results in 20% change in ID4 and the associated pressure drop ± ¯ow rate relationship. Do the smallest ID ®bers produce any product retentate or do they completely stop producing gas? The analytical solution for the performance of cross¯ow modules with uniform ®bers indicates that complete permeation occurs when Nh equals Nh
xf 1 ÿ xf ; 1 ÿ
(24)
as demonstrated in Appendix A; Appendix A also discusses the physical signi®cance of this result in more detail. This suggests that ®bers can completely stop producing gas. If some ®bers do not produce any retentate, what happens at the product end of these ®bers? The feed that enters these ®bers permeates completely before the end of the ®ber is reached. The pressure at this
point of zero retentate ¯ow will drop below that at the retentate exit to induce a ¯ow into the ®ber from the product end. All of the retentate that enters the ®ber from the product end will permeate as well. One can modify the solution algorithms described previously to determine the location and pressure at the point of zero retentate ¯ow. To use the direct integration method, one must integrate the performance equations from both the feed and product ends; the composition and pressure at both ends are known or can be calculated. The entering gas ¯ow rates at both ends are then varied until both retentate ¯ows vanish and the pressures are equal at a common point. To use the ®nite difference method, one simply replaces the differential equation for x at the product end, z1, with the boundary condition that the retentate composition is equal to the composition of the product produced by the other ®bers, xxr as given by Eq. (16). Since the latter modi®cation was signi®cantly easier than the former, only the ®nite difference algorithm was used to determine performance when ®bers stopped producing product. With this modi®cation, performance calculations were possible for product retentate compositions down to 0.0025 for a 10% ID variation as shown in Fig. 2. Results for a 5% variation are also shown. Clearly, as the percent variation increases the drop in performance increases. For example, for a fast gas mole fraction of 0.005 retentate recovery decreases from 30% to 28±23% and relative retentate ¯ow decreases from 1 to 0.9±0.65 as the ID variation increases from 0% to 5±10%. Such changes would signi®cantly have an impact on the design of a system for the production of a 99.5% nitrogen stream. Fig. 2 also illustrates the effects of permeate mixing. Performance for the no mixing case is signi®cantly poorer than that for the well-mixed case. At the highest retentate purities, the decrease in recovery or relative retentate ¯ow more than doubles in the absence of mixing. One would expect actual performance to lie somewhere in between these two limiting cases. These results suggest that module designs which promote permeate mixing can help mitigate the effects of ®ber variation. The patent literature describes numerous design features that attempt to improve shell mixing on a macroscopic, module length scale [1]. Additionally, if modules are wound from multiple
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Fig. 2. The effect of variations in ID on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; short dash±long dash, 5% variation/well mixed; intermediate dash, 5% variation/no mixing; short dash, 10% variation/well mixed; dot, 10% variation/no mixing.
®ber lots with varying ®ber properties, mixing ®ber tows together would help promote mixing on a microscopic, ®ber length scale. Fig. 3 shows typical pro®les of composition, retentate ¯ow rate, and pressure along the module. These pro®les are for a module with a 10% ID variation and no permeate mixing producing a product with a fast gas composition of 0.0029. The corresponding retentate recovery and relative ¯ow are 0.066 and 0.20, respectively. The pro®les are for the ®ve quadrature points used in the simulation, ID=ID 0:71, 0.86, 1.0, 1.14, and 1.29. Fig. 3(a) shows that the fast gas composition drops rapidly initially but the rate of change decreases with axial distance. At z0.62 the feed to the smallest ®ber shown, ID=ID 0:71, completely permeates while the feed to the next smallest ®ber completely perme-
39
ates at z0.81. This is dif®cult to see due to the scale of Fig. 3(a) but is clearly shown in Fig. 3(b) with a change of scale. Prior to complete permeation, composition increases with increasing ®ber ID at a ®xed position; larger ®bers have higher feed gas ¯ow rates so permeation reduces the composition less quickly. In contrast, past the point of complete permeation, composition increases with decreasing ®ber ID at a ®xed position. This behavior is due to higher ¯ow rates into the smaller ®bers of the product retentate from the product end; more gas must enter the smaller ®bers since permeation occurs over a greater distance from the product end. One might expect smaller ®bers to possess lower ¯ow rates but the pressure at the complete permeation point is much smaller in the smaller ®bers than the large ®bers to drive the ¯ow, see Fig. 3(d). Fig. 3(c) illustrates how retentate recovery changes with axial distance. Positive values indicate a ¯ow from the feed end to the product end while negative values indicate a ¯ow in the opposite direction. Negative values are scaled with the same feed ¯ow rate, though, as positive values. The complete permeation point corresponds to the point where the recovery equals zero. Clearly, only those ®bers with ID=ID > 1 contribute to the product produced at this operating point since the recovery is negative for ID=ID < 1 at z1. The relative pressure pro®les in Fig. 3(d) also shows the complete permeation point as the point at which the pressure is a minimum. The minimum value decreases dramatically as ®ber size decreases. These differences are responsible for higher ¯ow rates of product back into smaller ®bers than larger ®bers as discussed previously. One can directly compare the no mixing results with the analytical solution since each ®ber performs as if it were in a module of uniform ®bers. In all cases complete permeation ®rst occurred when Nh was equal to 0.907 (to within less than 1%), consistent with the analytical prediction. Moreover, the calculated values of x and on either side of the point where 0 and the location of this point were in excellent agreement with the analytical solution. This suggests that the numerical method was robust enough to capture the zero permeation point and permit accurate performance calculations in the presence of ®bers that did not produce the product.
40
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Fig. 3. Simulation results for a module with 10% ID variation and no permeate mixing producing a product with xr0.0029: (a) retentate fast gas composition, (b) fast gas composition near the product end of module, (c) retentate recovery, and (d) retentate pressure. The lines in all figures are: dot, ID=ID 0:71; intermediate dash, ID=ID 0:86; solid line, ID=ID 1:0; short dash±long dash, ID=ID 1:14; short dash, ID=ID 1:29.
Fig. 4 illustrates the effects of percent variation in Q2 and permeate mixing on module performance. As observed for the ID variation cases, ®bers with the highest permeation rate to feed rate ratio stop producing product and start consuming retentate as the retentate purity increases. Similar to the ID variation results, performance decreases as variation increases and performance is much worse in the absence of permeate mixing. For example, recovery and retentate ¯ow for a fast gas mole fraction of 0.01 decreases by 40% with a 30% permeance variation and no permeate mixing. This implies one must increase the area and compressor horsepower by a factor of 1.6 to provide the same performance as a module with no ®ber variability! Fig. 5 illustrates the effect of percent variation in and permeate mixing on module performance. As observed for size and permeance variations, perfor-
mance decreases as the percent variation increases. The magnitude of the change is comparable to the permeance variation case. However, ®bers did not stop producing gas in any of the simulations with selectivity variations. In contrast to the size and permeance variation cases, permeate mixing had no effect on the performance. The performance changes due to mixing are highly dependent on how permeate composition varies with ®ber properties for the speci®ed operating conditions. In general, the performance of some ®bers improves (due to a decrease in the permeate fast gas composition) while that of the others decline (due to an increase in the permeate fast gas composition) and the net effect can be either an overall performance increase or decrease. The results indicate that the net effect is a performance improvement with mixing for size and
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
Fig. 4. The effect of variations in Q2 on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; short dash±long dash, 15% variation/well mixed; intermediate dash, 15% variation/no mixing; short dash, 30% variation/well mixed; dot, 30% variation/no mixing.
permeance variations but no change occurs for selectivity variations. These observations may have an impact on the operation of ®ber manufacturing processes. If process changes affect both selectivity and permeance, reducing permeance variation will have a greater impact on module performance than reducing selectivity variation since the effects of permeate mixing are minimized. 4. Conclusions We present an analysis of the effects of ®ber material property variations on the performance of a cross-¯ow hollow ®ber gas separation module. The effects of variation in ®ber size, permeance, and separation factor are explicitly considered.
41
Fig. 5. The effect of variations in on performance: (a) retentate recovery as a function of retentate fast gas mole fraction and (b) relative retentate flow rate as a function of retentate fast gas mole fraction. The lines in both figures are: solid line, no variation; intermediate dash, 15% variation/no mixing or well mixed; dot, 30% variation/no mixing or well mixed. Note that the differences between the well mixed and no mixing results are not distinguishable.
Increasing the variation decreased the module performance. For comparable levels of variability, the effects are greatest for a size distribution. Permeance and separation factor variations have similar, less pronounced effects. As product retentate purity increases, some ®bers may stop producing product and actually consume part of the product produced by other ®bers. This phenomenon was observed for size and permeance variations but not selectivity variations. Additionally, for size and permeance variations, performance is highest if the permeate from all ®bers is well mixed. Permeate mixing had little effect for selectivity variations. This is due to how permeate mixing changes the performance of the distribution of ®bers.
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J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
The results presented here can help guide the development of ®ber manufacturing processes. One can use the analysis to determine how closely ®ber properties must be controlled to achieve desired performance and evaluate the relative effects of process changes on performance.
Acknowledgements
5. List of symbols
The analytical solution for the performance of a cross-¯ow module with uniform ®ber properties is given implicitly by [8,9]:
Ch Cp f F g ID J Nh Np OD pf Q R Rg T w x y z Z
constant defined in Eq. (19) (m2 Pa) constant defined in Eq. (18) (m4 s molÿ1) flow rate (mol sÿ1) feed rate (mol sÿ1) distribution function fiber inner diameter (m) dimensionless molar permeation rate defined by Eqs. (6) and (7) dimensionless group defined by Eq. (9) dimensionless group defined by Eq. (8) fiber outer diameter (m) feed pressure (Pa) permeance (mol mÿ2 Paÿ1 sÿ1) product retentate flow (mol sÿ1) gas constant (Pa m3 molÿ1 Kÿ1) temperature (K) quadrature weighting factors mole fraction of faster permeating gas component in retentate mole fraction of faster permeating gas component in permeate dimensionless active fiber length active fiber length (m) separation factor arbitrary material property ratio of permeate to feed pressure viscosity (Pa s) dimensionless retentate pressure standard deviation dimensionless molar retentate flow rate
Subscripts and superscripts f value for feed r value for product retentate max maximum value min minimum value 1 value for faster permeating gas component 2 value for slower permeating gas component average value Å
The authors acknowledge partial support of this work by the National Science Foundation through grant CTS-9408414. Appendix A
N h 1 ÿ
xf ÿ xr r 1 ÿ xf ÿ 1 ÿ xr r ; (A.1)
1 ÿ1 = 1ÿ ÿ1 yr r yf 1 ÿ yr ÿ ÿ1 = 1ÿ ÿ1 1 ÿ yf ÿ ÿ 1yr ; ÿ ÿ 1yf
(A.2)
where the subscripts r and f indicated the value evaluated for the product retentate and feed, respectively, and y is the local permeate composition given by Eq. (22). Given a value for Nh, the membrane area, one solves Eqs. (A.1), (A.2) and (22) simultaneously for the values of xr, yr, and r. Alternatively, one can specify a value for xr and solve Eqs. (A.1), (A.2) and (22) for Nh, yr, and r. These equations predict that r becomes identically zero at a ®nite value of Nh. Eq. (A.2) indicates that yr must be identically zero if r is identically zero, and Eq. (22) indicates that xr must be identically zero if yr is identically zero. Therefore, Eq. (A.1) gives the value of Nh at which r becomes identically zero as Nh
xf 1 ÿ xf : 1 ÿ
(A.3)
Eq. (A.3) indicates that as approaches 1 this value becomes unbounded; an in®nite area is required for the feed to completely permeate as intuition suggests. As approaches 1, this value becomes 1/(1ÿ ) which is also consistent with intuition. The membrane is equivalent to a porous frit and permeation across this frit is equivalent to ¯ow. The permeate composition is the same as the feed composition and if the feed pressure is maintained higher than the permeate pres-
J. Lemanski et al. / Journal of Membrane Science 153 (1999) 33±43
sure all of the feed will permeate, or ¯ow, across the membrane in a ®nite area. For values of >1 and >1, however, Eq. (A.3) predicts that the feed completely permeates across the membrane for a ®nite value of Nh or membrane area which might appear inconsistent with intuition. For example, given the values considered here of xf0.21, 8, and 0.1, this value of Nh is 0.907 while, for !1, this value is (1ÿxf)/(1ÿ ). How is this possible? First consider the in®nite selectivity result. As the slow gas permeance becomes arbitrarily small, !1 and Nh approaches the ®nite ®xed value given above. The de®nition of Nh, Eq. (9), indicates that this is equivalent to a ®xed value for the product of membrane area and slow gas permeance. Therefore, as the slow gas permeance becomes arbitrarily small the area required for complete permeation becomes arbitrarily large consistent with intuition. The ®nite value of Nh predicted for the conditions considered here is not consistent with intuition engendered from mass transfer in other membrane processes in which one or more solutes are removed by diffusion across the membrane, such as pure dialysis. In such processes an in®nite area is required to completely remove a solute. However, a fundamental difference exists between pure dialysis and gas separation: in gas separation all components of the feed permeate while in pure dialysis the solvent does not. In cross-¯ow gas separation processes, slow gas permeation increases the fast gas concentration in the retentate above the value that would be observed in the absence of slow
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gas permeation. Eliminating slow gas permeation by letting its permeance become arbitrarily small makes the slow gas equivalent to the solvent in dialysis and, as discussed above, the selectivity and required membrane area for complete permeation become in®nite. References [1] G.G. Lipscomb, Design of hollow fiber contactors for membrane gas separations, in: The 1996 Membrane Technology Reviews, Business Communication Co., Inc., Norwalk, CT, 1996, pp. 23±102. [2] S.R. Wickramasinghe, M.J. Semmens, E.L. Cussler, Mass transfer in various hollow fiber geometries, J. Membr. Sci. 69 (1985) 235±250. [3] H. Kruelen, C.A. Smolders, G.F. Verteeg, W.P.M. van Swaaij, Microporous hollow fibre membrane modules as gas±liquid contactors. Part 1. Physical mass transfer process, J. Membr. Sci. 78 (1993) 197±216. [4] S. Elmore, G.G. Lipscomb, Analytical approximations of the effect of a fiber size distribution on the performance of hollow fiber membrane separation devices, J. Membr. Sci. 98 (1995) 49±56. [5] R.O. Crowder, E.L. Cussler, Mass transfer in hollow-fiber modules with non-uniform hollow fibers, J. Membr. Sci. 134 (1997) 235±244. [6] B. Carnahan, H.A. Luther, J.O. Wilkes, Applied Numerical Methods, Krieger, Malabar, FL, 1990. [7] MATLABTM, The MathWorks Inc., Natick, MA. [8] C.Y. Pan, H.W. Habgood, Gas separation by permeation. Part I: Calculation methods and parametric analysis, Can. J. Chem. Eng. 56 (1978) 197±209. [9] C.Y. Pan, Gas separation by high-flux, asymmetric hollowfiber membrane, AIChE J. 32 (1986) 2020±2027.