Effect of Fiber Property Variation on Hollow Fiber Membrane Module

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Ind. Eng. Chem. Res. 2010, 49, 12074–12083

Effect of Fiber Property Variation on Hollow Fiber Membrane Module Performance in the Production of a Permeate Product Santosh A. Sonalkar, Pingjiao Hao, and G. Glenn Lipscomb* Department of Chemical and EnVironmental Engineering, UniVersity of Toledo, Toledo, Ohio 43606-3390, United States

The literature demonstrates the detrimental effects of variability in the inner diameter, permeance, and selectivity of hollow fiber membranes for the production of a retentate product, for example, nitrogen purification from air. This past work is extended to the production of a permeate product. Additionally, simultaneous variation of multiple fiber properties is considered. Fiber variability is detrimental to permeate production; the ratio of product to feed flow rate decreases with an increase in variability. However, the permeate product flow rate remains unchanged in contrast to the large changes observed for a retentate product. Simultaneous, independent variation of multiple fiber properties makes performance worse. However, correlations between the variations in fiber properties can improve performance. Theoretical performance predictions compare favorably with experimental measurements for the production of oxygen from air if either simultaneous variation of fiber size with permeance occurs or significant deviations from countercurrent flow exist. The results may be used to develop manufacturing guidelines for the production of hollow fiber membranes. Introduction Hollow fiber membranes are used for a wide range of gas separations. These membranes are gathered together to form a hollow fiber bundle that is placed in a case to create a single stage or module. The case possesses ports that allow the introduction of a feed and removal of a high-pressure, retentate product and a low-pressure, permeate product as illustrated in Figure 1. Note that the feed may be introduced into either (i) the fiber lumens through ports on the end of the module or (ii) the shell, the space external to the fibers, through ports on the periphery of the case. For many gas separations, the desired product is the retentate that leaves the module. These separations include the production of nitrogen from air and the drying of compressed air; the permeate often is released to the environment and not used. However, the permeate that leaves the module is the desired product for other applications such as the production of enriched oxygen and recovery of hydrogen from industrial process streams. The performance of real modules, in terms of product to feed flow rate ratio and absolute product flow rate, often is poorer than that predicted by ideal performance models. This deviation may be due to a number of different factors1-4 including variability in fiber properties (such as permeance and size), poor lumen-side flow distribution due to lumen header design, poor shell-side flow distribution due to shell header design, shellflow maldistribution due to fiber packing nonuniformity, and the presence of significant cross-flow regions in the module due to poor module design. The literature suggests variability in fiber properties is one of the most significant, especially for production of a high-purity product. Hollow fiber membranes are made industrially by polymer solution spinning. In wet spinning processes, commonly 10-40 fibers are spun simultaneously and pass through different process zones as a group or tow. In dry spinning processes, up to 100 fibers may be spun simultaneously with a collection of spinnerets. These zones may include a gas-filled gap after the * Corresponding author: Phone: (419) 530-8088. Fax (419) 5308086. E-mail: [email protected].

spinneret and one or more liquid baths where the final fiber properties are set through a combination of momentum, heat, and mass transfer. Maintenance of uniform conditions around each fiber is confounded by the number of fibers moving together (some fibers are on the outside of the tow while others are inside) and the rate at which the fibers move (25-150 m/min is common). Consequently, one expects some inherent variability in fiber properties.

Figure 1. Schematic diagram of typical hollow fiber membrane modules: (a) countercurrent and (b) cross-flow. Shell-side flows are indicated by the dotted lines and lumen flows by the solid lines. The shell-side region is indicated by darker shading.

10.1021/ie100649q  2010 American Chemical Society Published on Web 11/24/2010

Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010

The effect of these variations on module or stage performance for the production of a retentate product is well-documented in the literature.5-8 Fiber property variation is detrimental to module performance. Variability reduces the fraction of the feed that can be recovered as the retentate product. Additionally, variability reduces the product flow rate and thereby increases the membrane area required to produce a desired product flow rate. Variability in fiber diameter has a much greater impact on performance than variability in permeance and selectivity. The detrimental effects of variability can be mitigated by module staging. Moreover, staging configurations that do not improve performance in the absence of fiber variability can dramatically improve performance in the presence of fiber variability. However, the effect on performance for permeate production has not been discussed. The theoretical analysis of module or stage performance in the presence of fiber variations developed for retentate production is used here for permeate production. Additionally, the analysis is modified to account for simultaneous variation in fiber inner diameter, permeance, and selectivity. Theoretical predictions of performance are compared to experimental measurements for a commercial air separation module. Module performance is poorer than predicted. To determine why performance is poorer, the fiber size distribution of the fiber bundle is measured. Predictions of performance that account for the measured size distribution are closer to experiment, but experimental performance still is poorer. The existence of significant cross-flow regions in the module or simultaneous variation in permeance with fiber size can account for the differences. This work is significant in its focus on the permeate product. Results are reported for low to high permeate recovery, which leads to the unexpected finding that permeate flow rate does not change significantly with fiber variability. Additionally, results are reported for the first time for simultaneous variation in fiber properties. The best performance that can be obtained by a module in the absence of shell-flow maldistribution is the focus of this work. The results establish a performance limit that cannot be exceeded and thereby determine the maximum variation in fiber properties that is allowable in manufacture. Deviations from this limit due to shell-flow maldistribution would have to be addressed separately and do not affect this limit. One may be able to compensate for shell-flow maldistribution by reducing fiber property variation, but the analysis of shell-flow distribution and its impact on module performance is the subject of ongoing research9 that is beyond the scope of the present work.

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(7) Changes in membrane and gas material properties with gas composition are negligible. The validity of these assumptions has been discussed extensively in the literature,3,4 and past work supports their use in the permeate purification processes considered here. The variation in ID, Q2, and R is assumed to be given by a multivariate distribution function g(ID, Q2, R), where g(ID, Q2, R) dID dQ2 dR is the fraction of fibers that possess values of inner diameter, permeance, and selectivity that fall in a region bounded by the intervals [ID, ID + dID], [Q2, Q2 + dQ2], and [R, R + dR]. In this work only Gaussian distributions of each variable will be considered. The Gaussian distribution for a single variable is given by g(φ) )

[

1 (φ - φ¯ )2 exp 2σ2 σ√2π

]

(1)

j is where φ is the value of the fiber property (ID, Q2, or R), φ the mean value, and σ is the standard deviation. Although only Gaussian distributions are considered, any distribution can be accommodated by the theoretical approach. For a bundle of fibers, the average flow per fiber is given by jf )

∫

IDmax

IDmin

∫

Q2,max

Q2,min

∫

Rmax

Rmin

fg dID dQ2 dR

(2)

where f is the flow in fibers that possess an inner diameter, permeance, and selectivity of ID, Q2, and R, respectively. The overbar indicates an average value, and the subscripts max and min indicate the maximum and the minimum values of the material property, respectively. Figure 1 illustrates a typical hollow fiber membrane module and the flows within it. For a lumen-fed module, the feed gas enters a feed header from which it is distributed into the fiber lumens. Similarly, the product retentate exits the lumens and is collected in the retentate header. The gas that permeates from inside the fibers to the shell (the space outside the fibers) may be forced to flow in a countercurrent (opposite) direction to the retentate (Figure 1a) and removed from a port on the periphery of the case holding the fiber bundle; if multiple ports are present, only the ports located near the feed end of the module are used. Alternatively (Figure 1b), the permeate may flow in a crossflow direction to the feed. In cross-flow operation, both ports shown on the module may be open for removal of the permeate. An overall mass balance and a mass balance on the faster permeating component for the retentate and permeate flows give the following dimensionless equations for flow rate and composition for each fiber in the bundle:8

Theory The theoretical analysis of oxygen (permeate) purification from air follows directly from the analysis used for nitrogen (retentate) purification7,8 and is based on the following assumptions: (1) The properties of a single fiber are constant along its length, but the properties may vary from fiber to fiber. (2) Fiber inner diameter (ID), slow gas permeance (Q2), and selectivity (R) may vary simultaneously, or a single property may vary while the others remain constant. (3) Air is assumed to be a binary mixture of ideal gases with an oxygen mole fraction of 0.21 and the balance nitrogen. (4) Module operation is steady-state and isothermal. (5) Negligible pressure drop occurs in the shell, and the lumen pressure drop is governed by the Hagen-Poiseuille equation. (6) Concentration polarization and axial diffusion are negligible.

dθR dθP ) -(J1 + J2) ) dz dz

(3)

d(xθR) d(yθP) ) -J1 ) dz dz

(4)

where θ is the dimensionless molar flow rate (ratio of retentate or permeate flow to feed flow) and z is the dimensionless fiber length (ratio of distance from the feed end to fiber length). J1 and J2 are the dimensionless molar permeation rates of the faster and slower permeating components, respectively, and are given by J1 ) RNh(x - γy)

(5)

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J2 ) Nh[(1 - x) - γ(1 - y)]

(6)

where x is the retentate mole fraction of the fast gas, y is the permeate mole fraction of the fast gas, γ is the pressure ratio (ratio of permeate pressure to retentate pressure), R is the ideal separation factor (ratio of fast gas permeance Q1 to slow gas permeance Q2), and Nh is the dimensionless mass transfer coefficient given by π(OD)LQ2pf F

Nh )

(7)

where OD is the fiber outside diameter, L is the fiber length, pf is the feed pressure, and F is the molar feed rate. Permeance (Q) is the ratio of the intrinsic, specific permeability to the effective membrane thickness based on the permeation area calculated from the fiber OD. The lumen pressure drop is calculated from dΠ2 ) -NpθR dz

(8)

where Π is the ratio of the retentate pressure (pR) to the feed pressure and the dimensionless group Np is given by 256µRgTLF

Np )

π(ID)4pf2

(9)

where Rg is the universal gas law constant, T is the ambient temperature, and pf is the feed pressure. Equation 8 is the Hagen-Poiseuille law and is expected to be valid when permeation rates are small relative to the volumetric flow rate. At the feed end of the module (z ) 0), the retentate flow rate is equal to the feed flow rate and the retentate composition is equal to the feed composition: θR ) 1

(10)

x ) xf

(11)

Additionally, the retentate pressure is equal to the feed pressure, Π ) 1. For countercurrent flow, at the retentate outlet (z ) 1), the permeate flow rate is zero and the permeate composition is equal to the composition produced under cross-flow conditions: P )0 F

(12)

J1 J1 + J 2

(13)

θP ) y)

For cross-flow, eq 13 is used to calculate the permeate composition along the length of the module. Equations 10-13 provide the four boundary conditions required to solve eqs 3 and 4. Moreover, mass balances around the retentate product end of the module provide the following j ) and composition expressions for the average permeate flow (P j (yj) in terms of the average retentate flow (R) and composition (xj): j (z) ) R j (z) - R j (z ) 1) ) P

∫

IDmax

IDmin

∫

∫

Q2,max

∫

Rmax

Q2,min Rmin IDmax Q2,max

IDmin

∫

Q2,min

j ) ) yP(z) ) xR(z) - xR(z ) 1) ) (yj)(P IDmax Q2,max Rmax xθRFg dID dQ2 dR ID Q R

∫

∫

min

∫

IDmax

IDmin

∫

2,min

∫

Q2,max

min

∫

Rmax

∫

Rmax

Rmin

θR z)1Fg dID dQ2 dR (14)

(15)

where the overbar indicates the average flow per fiber in the bundle; that is, the total flow is obtained by multiplying by the number of fibers. One must solve eqs 3, 4, and 8 for each fiber in the bundle. This requires specifying each fiber’s material properties (R, Q2, ID, OD, and l) and operating conditions (pf and F) as well as the gas viscosity. Additionally, the parameters of the distribution function for the material properties, g, must be specified. For a lumen-fed module, the values of F for each fiber are not independent. One must specify a set of values that leads to the calculation of the same pressure drop (∆p) for each fiber. The pressure drop must be the same because the fibers share a common inlet and outlet header in which pressure changes are negligible; see Figure 1a. To calculate a set of feed flow rates that satisfy this requirement, a common pressure drop is specified and the feed flow rate to each fiber is varied to produce this common value. The mass balances for each fiber are linked by the value of the permeate composition, y. All fibers contribute to the permeate and therefore the composition depends on the extent of mixing of the permeate from adjacent fibers. Past studies have considered two mixing limits: (1) no mixing and (2) complete mixing. In the nomixing case, permeate mixing between fibers does not occur; the permeate from each fiber remains adjacent to the fiber. In the complete-mixing case, permeate from all fibers is well-mixed, so for a given axial position the permeate composition is the same for all fibers. One expects the mixing that occurs in real modules to fall between these two limits. Equations 3 and 4 and boundary conditions (eqs 10-13) apply directly to the no-mixing case. For the well-mixed case, one must replace boundary condition 13 with

jy(z ) 1) )

∫

IDmax

IDmin

∫

IDmax

IDmin

∫

∫

Q2,max

Q2,max

Q2,min

∫

∫

Rmax

Rmin

Q2,min

Rmax

Rmin

J1Fg dID dQ2 dR

[J1 + J2]Fg dID dQ2 dR

|

z)1

(16) and use a common value for y in the permeation expressions 5 and 6. For retentate production, module performance is better for the well-mixed case than for the no-mixing case.8 This is because the performance of poorer-performing fibers increases more than the performance of better-performing fibers decreases. Module performance is characterized by the fraction of the faster permeating component in the feed that is recovered in the permeate, the permeate recovery, as a function of the permeate composition. The permeate recovery is given by yP(z ) 0) ) xfF

yP(z ) 0)

∫

IDmax

IDmin

∫

Q2,max

Q2,min

∫

Rmax

Rmin

xfFg dID dQ2 dR

(17) and the average product permeate composition is given by jy(z ) 0) )

θRFg dID dQ2 dR -

(xθR) z)1Fg dID dQ2 dR

Rmin

Q2,min

yP(z ) 0) j (z ) 0) P

(18)

j are given by eqs 14 and 15.These quantities where yP and P will be used to quantify module performance in the next section.

Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010 Table 1. Operating Conditions for Membrane System feed O2 composition feed N2 composition feed pressure permeate outlet pressure permeance O2 permeance N2 O2/N2 selectivity operating temperature

0.21 0.79 90 psig 0 psig 5.88 × 10-6 cm3 (STP)/(cm2 s cm of Hg) 1.00 × 10-6 cm3 (STP)/(cm2 s cm of Hg) 5.88 25 °C

A finite difference method is used to obtain an approximate numerical solution to the governing mass and momentum balances. Forward and backward differences are applied at the end nodes, and second-order central differences are used for

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internal nodes. A multivariate Newton-Raphson algorithm is used to solve the resulting algebraic equations for permeate recovery, permeate fast gas mole fraction, and pressure. Gauss-Hermite quadrature is used to evaluate the integrals in eqs 14-17.8 The quadrature replaces integrals with sums over a finite number of weight points. For example, eq 2 becomes jf ≈ 1 π3/2

n

n

n

∑ ∑ ∑ w w w f(ID, Q , R) i j k

2

(19)

i)1 j)1 k)1

where the weight points determine the values of ID, Q2, and R used in eq 19. This approximation requires extending the limits

Figure 2. Effect of variation in individual fiber properties on module performance: (s) no property variation; (- - -) 30% permeance variation; (- - -) 10% inner diameter variation; (- · -) 20% inner diameter variation.

Figure 3. Effect of simultaneous variation in fiber properties on module performance: (s) no property variation; (- - -) 10% inner diameter and 30% permeance variation positively correlated; (- - -) 10% inner diameter variation; (- · -) 10% inner diameter and 30% permeance variation uncorrelated; ( · · · ) 10% inner diameter and 30% permeance variation negatively correlated.

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Figure 4. Axial changes in (a) permeate recovery and (b) composition for a module producing a permeate product with 37.4% oxygen with 6.8% inner diameter variation.

of integration to ( ∞. We expect this to be a good assumption due to the narrowness of the distributions considered. Moreover, it avoids having to specify the maximum and minimum values of each material property, which can be difficult to determine. With the use of quadrature, one can predict the performance of the whole module by simulating only fibers that possess material properties associated with each weight point. All calculations were performed with MATLAB. Results The results presented here are for the operating conditions presented in Table 1. This work is an extension of work done for nitrogen purification, hence the same operating conditions and membrane modules were considered.8 For all the simulations, a five-point quadrature and 81 node points were considered. Reducing the number of quadrature

points to three or the number of nodes to 41 resulted in less than 1% change in the solution. In the discussion that follows, percent Variation refers to the ratio of the standard deviation of a material property to its mean value expressed as a percentage. The term permeate recoVery refers to the ratio of the oxygen flow rate in the permeate to the oxygen flow rate in the feed as defined by eq 17. RelatiVe permeate flow rate is the ratio of the permeate flow rate for a module with a variable fiber property to the flow from a module with no variation. Module performance refers to the value of recovery and relative flow for a given product purity. Better performance is associated with higher recovery or relative permeate flow for a given permeate purity. Figure 2 illustrates the effect of variation in indiVidual fiber properties on the performance of a lumen-fed, countercurrent, well-mixed shell module. Property variation is detrimental to

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Figure 5. Axial changes in (a) retentate recovery and (b) composition for a module producing a permeate product with 37.4% oxygen with 6.8% inner diameter variation. The lines correspond to results for individual fibers with material properties associated with each quadrature point: (- - -) ID/ID ) 1.19; (- - -) ID/ID ) 1.09; (s) ID/ID ) 1; (- · -) ID/ID ) 0.91; (- · · -) ID/ID ) 0.81.

performance. For a given permeate product purity, permeate recovery decreases in the presence of fiber variability. Inner diameter variability has a much larger impact than permeance or selectivity variability. Permeate recovery for a permeate composition of 0.45 decreases from 0.6 to 0.54 in the presence of 10% inner diameter variability, while 10% variability in permeance or selectivity leads to negligible changes and results are not shown in Figure 2 because the curves are indistinguishable from the no-variability result. The permeance variability must increase to 30% for the performance change to be comparable to that for 10% inner diameter variability as illustrated in Figure 2. The much stronger dependence on inner diameter variability arises from the dependence of flow rate on fiber size. For a fixed pressure drop, the Hagen-Poiseuille law predicts flow

rate is proportional to inner diameter raised to the fourth power. Thus, a 10% increase in diameter leads to ∼50% increase in flow rate. This large variation in flow rate leads to the dramatic performance changes. The magnitude of the decrease in recovery increases as the percent variation increases. For example, for an oxygen concentration of 0.45, the permeate recovery is 0.6 in the absence of diameter variation, but the recovery decreases to 0.54 for 10% variation and 0.42 for 20% variation. In stark contrast to the results for a retentate product reported in the literature,5-8 changes in permeate flow with fiber variability are negligible. The changes in permeate flow are less than 1% for the results illustrated in Figure 2. However, for high-purity retentate products, retentate flows can decrease by 80%.5-8

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Figure 6. Axial changes in pressure drop for a module producing a permeate product with 37.4% oxygen with 6.8% inner diameter variation. The lines correspond to results for individual fibers with material properties associated with each quadrature point: (- - -) ID/ID ) 1.19; (- - -) ID/ID ) 1.09; (s) ID/ID ) 1; (- · -) ID/ID ) 0.91; (- · · -) ID/ID ) 0.81.

Figure 3 illustrates the performance changes predicted for simultaneous variation in fiber inner diameter and permeance. Three results are shown for 10% inner diameter and 30% permeance variability: (1) uncorrelated, (2) positively correlated, and (3) negatively correlated. These results are compared to performance in the absence of fiber variability and performance with 10% inner diameter variation. The uncorrelated results correspond to independent, uncorrelated variation of inner diameter and permeance. Both properties are independently Gaussian-distributed about their respective average values. Performance decreases with the addition of permeance variability to inner diameter variability. The changes are not as large as the changes observed in Figure 2 when the no-variability case is compared to the case of 10% permeance variability. The positively correlated results correspond to a permeance variation that is positively correlated to the inner diameter variation: as inner diameter increases, permeance increases, and vice versa. The relationship between permeance and inner diameter for each fiber is given by ¯2 Q2 - Q ID - ID ) σID σQ2

(20)

Figure 3 indicates that for this positive correlation module performance is nearly identical to that in the absence of fiber variability. Increasing the permeation rate for larger fibers compensates for the higher flow rates through these fibers and leads to similar performance for all fibers. The negatively correlated results correspond to a permeance variation that is negatively correlated to the inner diameter variation: as inner diameter increases, permeance decreases, and vice versa. The relationship between permeance and inner diameter for each fiber is given by ¯ 2 - Q2 Q ID - ID ) σID σQ2

(21)

Figure 3 indicates that, for this negative correlation, module performance decreases dramatically. Reducing permeation rates from larger fibers with higher flow rates exacerbates the effect of these fibers on performance. Clearly any correlation between the variability in fiber properties can affect performance. Figures 4-6 illustrate axial changes in composition and flow for a lumen-fed, countercurrent, well-mixed shell module with 6.8% inner diameter variation producing 37.4% oxygen permeate. Figure 4a illustrates how the permeate recovery changes with axial position. The permeate recovery at the feed end (z ) 0) is highest and decreases to zero at the retentate end (z ) 1) as required by the boundary condition (eq 12). Figure 4b illustrates the permeate fast gas (i.e., oxygen) composition as a function of position in the module. The composition decreases from the permeate outlet to the retentate outlet and reflects the decrease in retentate composition that occurs simultaneously; as the retentate composition decreases, the composition of the permeate it produces must decrease. Figure 5 panels a and b illustrate how the retentate recovery, given by θR (ratio of the retentate to feed flow rate), and retentate fast gas composition change with axial position, respectively. The magnitude of the retentate recovery is greater for larger fibers because of the dependence of flow rate on ID4 in the Hagen-Poiseuille law, eq 8. For a fixed pressure drop, flow rates per unit mass transfer area are higher through larger fibers, and higher flow rates lead to higher recoveries since the gas spends less time in the module and a smaller percentage of the feed can permeate. Figure 6 illustrates how retentate pressure varies with axial distance. The pressure decreases monotonically from feed to retentate end and the overall pressure drop is the same for all fibers. Experiment The experimental setup used to test the theoretical predictions is illustrated in Figure 7. The feed to the module (Permea, Inc., Model PPA-22AD) is laboratory compressed air at ∼90 psig.

Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010

Figure 7. Experimental system for study of oxygen enrichment. Table 2. Properties for Hollow Fiber Membrane Module, Model PPA-22AD membrane property

value

maximum operating pressure maximum operation temperature fiber outer diameter fiber length fiber number

100 psig 150 °F 400 µm 0.5 m 1000

Table 3. Experimental Permeance Values for Module Used to Obtain Data for Theoretical Comparisons O2 permeance N2 permeance

6.09 × 10-6 cm3(STP)/(cm2 s cm of Hg) 1.04 × 10-6 cm3(STP)/(cm2 s cm of Hg)

The air passes through a filter and a pressure regulator before entering the module. Properties are listed in Table 2. Permeate and retentate mass flow rates are measured (Sierra Instruments, Inc., Model 820 or a Precision wet test meter) along with oxygen compositions (Engineering Systems and Designs, Model 600 Oxan). The retentate flow rate is controlled by a needle valve. Mass balances for the experimental system closed to within 5% or less. To obtain a priori estimates of module performance, permeation rates were measured for pure oxygen and nitrogen. The permeances calculated from the measured permeation rates based on the module properties in Table 2 are provided in Table 3. These values differ from those reported previously8 because the previously reported values were not obtained by pure gas

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permeation experiments; the values were obtained by curvefitting theoretical predictions of retentate recovery to experimental measurements. The selectivity calculated from these permeance values is the same as that reported in Table 1 and was used for the performance simulations reported in the previous section. Figure 8 compares experimental and theoretical performance results for the operating conditions in Table 1 and the material and module properties in Tables 2 and 3; a lumen-fed, countercurrent, well-mixed shell module with no fiber variability is assumed. Experimental recovery values are significantly smaller than the predictions. The differences are greatest for permeate concentrations greater than 0.35. To determine the origin of this discrepancy, a section of the module tubesheet was removed and the size of 60 fibers was determined by optical microscopy; the results of the previous section suggest that fiber size variability will have the greatest effect on performance. The cumulative fiber size distribution, the fraction of fibers with a given inner diameter or less, is illustrated in Figure 9. The average fiber size for the 60 fiber sample is 210 µm and the standard deviation is 14.3 µm or ∼6.8% of the average. Predictions of module performance with a 6.8% inner diameter variation are illustrated in Figure 8. Although the fiber size distribution moves the predictions toward the experimental measurements, significant differences still exist. To test potential reasons for the performance differences, further simulations were performed and are illustrated in Figure 8. Predictions are in excellent agreement with experiment if a fiber size variation of 15% is used. However, such variability is not supported by the experimental size measurements. The addition of 30% variability in permeance, which is negatively correlated with the fiber inner diameter as described previously, also brings theoretical prediction into good agreement with experiment. A sufficiently large variability in selectivity has a similar impact. Assumption that the module operates in pure cross-flow, instead of countercurrent flow, with a well-mixed shell and a 6.8% inner diameter variability is

Figure 8. Comparison of experimental permeate recovery measurements with theoretical predictions: (2) experimental measurements; (s) prediction with no fiber variability; (- - -) prediction with 6.8% inner diameter variation; (- - -) prediction with 15% inner diameter variation; (- · -) prediction with 6.8% inner diameter variation and 30% permeance variation; ( · · · ) cross-flow prediction with 6.8% inner diameter variation.

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Figure 9. Experimental cumulative fiber size distribution.

equally effective in bringing theoretical predictions into good agreement with experiment. Cross-flow predictions with no shell mixing differ by less than 5% from the predictions that assume a well-mixed shell (these results are not shown in Figure 8), which indicates shell mixing is not a significant factor. These calculations suggest that fiber size variability is not sufficient to predict module performance. A combination of variability in permeance and deviation from countercurrent flow due to module design can bring theory into agreement with experiment but will require significant additional module characterization to characterize. Conclusions A study of the effects of fiber property variation on the performance of a hollow fiber membrane module producing a permeate product is presented. The effects of variability in fiber inner diameter, slow gas permeance, and selectivity are studied. Variation is detrimental to performance: for a given permeate purity, permeate recovery decreases as variability increases. Variation in inner diameter has a greater effect than variation in permeance and selectivity. In comparison to the results for production of a retentate product, the observed decrease in product recovery is similar but the decrease in product flow is much smaller. This is due to the fact that, for a specified feed pressure, if retentate pressure drops are small, the total permeation rate does not change dramatically with composition. Simultaneous variation in multiple fiber properties leads to further decreases in performance if the variations are uncorrelated. Correlated variations in fiber properties may improve performance if the performance changes introduced by the property variations compensate for each other. The results of experimental measurements for oxygen production from air differ significantly from theoretical predictions that account for the fiber inner diameter variation. Inclusion of simultaneous variation in permeance and deviation from countercurrent flow can account for the observed differences. Fiber manufacturers might use the results presented here to set quality control limits. The theoretical analysis would allow determination of the maximum variability permissible to achieve

module performance goals. Additionally, the results may be applied directly to other separations that involve the production of a high-purity permeate, such as the purification of hydrogen or helium, and to the production of a high-purity retentate when simultaneous variation in fiber properties occurs. Nomenclature List of Symbols F ) feed flow rate (mol s-1) g ) fiber property distribution function ID ) fiber inner diameter (µm) J ) dimensionless component molar permeation rate n ) number of Gauss-Hermite quadrature points N ) number of fibers L ) length of fibers (m) Nh ) dimensionless group defined by eq 7 Np ) dimensionless group defined by eq 9 OD ) fiber outer diameter (µm) p ) pressure (Pa) P ) permeate flow rate (mol s-1) Q ) permeance, ratio of intrinsic permeability to effective membrane thickness [cm3(STP)/(cm2 · s · cm of Hg)] R ) retentate flow rate (mol s-1) Rg ) ideal gas law constant (Pa m3 mol-1 K-1) T ) temperature (K) wi ) Gauss-Hermite quadrature weight factor x ) mole fraction of the faster permeating species in the retentate y ) mole fraction of the faster permeating species in the permeate z ) dimensionless flow path length, ratio of actual to active length Greek Letters R ) permselectivity ratio of fast gas to slow gas permeance γ ) pressure ratio, permeate to retentate ∆ ) change in a particular property θ ) ratio of actual to feed flow rate φ ) arbitrary fiber property Π ) ratio of actual to feed pressure σ ) standard deviation µ ) viscosity (Pa s)

Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010 Subscript and Superscripts 0 ) initial value 1 ) faster permeating gas component 2 ) slower permeating gas component f ) feed max ) maximum value min ) minimum value P ) permeate R ) retentate

Literature Cited (1) Rautenbach, R.; Dahm, W. Simplified calculations of gas-permeation hollow fiber modules for the separation of binary mixtures. J. Membr. Sci. 1991, 62, 165. (2) Crowder, R. O.; Cussler, E. L. Mass transfer in hollow fiber modules with non uniform hollow fibers. J. Membr. Sci. 1997, 134, 235. (3) Lipscomb, G. G.; Sonalkar, S. A. Sources of non-ideal flow distribution and their effect on the performance of hollow fiber gas separation modules. Sep. Purif. ReV. 2004, 33, 41.

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ReceiVed for reView March 16, 2010 ReVised manuscript receiVed October 25, 2010 Accepted October 26, 2010 IE100649Q