Study on the effect of flow maldistribution on the performance of the hollow fiber modules used in membrane distillation

Journal of Membrane Science 215 (2003) 11–23

Study on the effect of flow maldistribution on the performance of the hollow fiber modules used in membrane distillation Ding Zhongwei, Liu Liying, Ma Runyu∗ College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Received 11 June 2002; received in revised form 11 November 2002; accepted 13 November 2002

Abstract This paper investigates the effects of flow maldistribution on the performance of hollow fiber module used in membrane distillation (MD). As a base of this work, the mathematical model for ideal hollow fiber module was established. To get the flow maldistribution mathematically, sub-division of the module was employed to divide the fiber bundle into a number of sub-regions with different characteristics. The regional behavior was obtained from the analysis of ideal module, and the overall performance of a practical module was obtained by the combination of its sub-regions. At lumen side, the flow maldistribution is caused by the polydispersity of fiber inner diameter, and a Gauss distribution was introduced to divide the fiber bundle. At the shell side, the flow maldistribution is caused by the non-uniformity of fiber packing, and the Voronoi tessellation technique was adopted to determine the distribution of polygonal cell area. The effects of flow maldistribution at the two sides of the membrane were assessed individually. The result shows that the polydispersity of fiber inner diameter has some negative effect on the module permeate flux at the conditions considered, and the negative influence of the non-uniformity of fiber packing on the module performance seemed to be more important. More studies were focused on the shell side, the module flux was found increasing with the increase of packing fraction and flowrates at the two sides of the membrane. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Membrane distillation; Hollow fiber module; Flow maldistribution

1. Introduction The hollow fiber membrane module is a bundle of porous hollow fibers packed into a shell similar in configuration to a tube and shell heat exchanger. This device can be used as contractor that achieves gas/liquid and liquid/liquid mass transfer. Because of their very high rate of mass transfer, hollow fiber modules have been used in many practical applications, such as liquid/liquid extraction, artificial kidney, desalination ∗ Corresponding author. E-mail addresses: [email protected] (D. Zhongwei), [email protected] (L. Liying), [email protected] (M. Runyu).

and waste water treatment. Mass transfer and fluid dynamics in hollow fiber module have been extensively studied [1]. For the ease of calculations and the enormous amount of useful results in the available literatures of heat transfer studies, some assumptions are often adopted in many researches on hollow fiber module. For example, it is widely assumed that all the fibers in one module have the same inner diameter, and that they are arranged in an ordered ways, or evenly distributed on the shell side of the module. These assumptions may lead to the standpoint that the flow is uniformly distributed at both sides of the module. In practice, however, these modules often exhibit unexpected performance, and flow maldistribution is considered to be one of the most possible

0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0376-7388(02)00557-4

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Nomenclature

Sc Sh t T V W Y

membrane area (m2 ) cross-sectional area for flow (m2 ) heat capacity (J/kg ◦ C) membrane distillation coefficient (kg/m2 s Pa) fiber diameter (m) diffusion coefficient of water vapor in air (m2 /s) friction factor mass flowrate (kg/s) film heat transfer coefficient (W/m2 ◦ C) heat of vaporization (J/kg) thermal conductivity (W/m ◦ C) number of regions number of fibers mass flux (kg/m2 s) pressure (Pa) probability wetted perimeter (m) heat flux (W/m2 ) heat transfer rate (W) gas-law constant (J/mol K) Reynolds number number of nearest fibers around a polygonal cell Schmidt number Sherwood number celsius temperature (◦ C) kelvin temperature (K) volumetric flowrate (m3 /s) fractional flow mole fraction of air in membrane pores

Greek δ φ η σ

letters thickness of membrane (m) packing fraction viscosity (Pa s) standard deviation

A AC cp C d D f F h H k m n N p P Pw q Q R Re s

Subscripts e effective value for fiber bundle f fiber or in the bulk of the feed side fm the membrane surface of the feed side i ith region K Knudsen diffusion

ln m M p pm P s t v w

logarithmic mean average or membrane molecular diffusion the bulk of the permeate side the membrane surface of the permeated side Poiseuille flow shell side lumen side vapor water

Superscripts i ith region in module inlet out module outlet

causes. So there must be exist some considerable differences between these supposed situations mentioned above and the practical ones in the hollow fiber modules. Some of the possible factors include the polydisperisty of fiber inner diameter, the uniformity of fiber packing, fiber movement during operation, and module inlet and outlet effects [2], and the first two factors are thought to be the main causes of flow maldistribution in the lumen and shell, respectively. A few studies have focused on the effect of flow maldistribution on the performances of hollow fiber modules. Noda et al. [3] were among the first studying the shell-side flow maldistribution in dialysis process. They proposed a model incorporating flow maldistribution by assuming that the shell side of a hollow fiber module can be divided into two categories: fluid well distributed across the fiber bundle, and that bypass the bundle, and the later does not contribute to the overall mass transfer. Tompkins et al. [4] showed a study of solvent extraction process using axial flow hollow fiber module. In their study, the mass transfer performance were around 10% of those predicted, and this was attributed to fluid maldistribution on the shell side reducing the proportion of active membrane area. Lemanski and Lipscomb [5] studied the effect of shell-side residence time distribution (RTD) on mass transfer performance. They pointed out that plug flow would be obtained in an ideal hollow fiber

D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

module, but in real shell-side flow the distribution of fluid across the fiber bundle tended to broaden the RTD. Another way to study the effect of shell-side flow is to establish empirical equations to predict overall mass transfer coefficient [6–8]. The general form of these equations is: Sh = f (φ, Re, Sc), where φ is the packing fraction of the hollow fiber module. The existence of φ in these equations may indicate the influence of shell-side flow maldistribution. Recently, Voronoi tessellation was used to decide the distribution of fibers and shell-side flow in randomly packed fiber bundles assuming flow parallel to the fiber [2,9,10]. By calculating the theoretically exponential distribution of areas associated with each sub-division of the fiber bundles and estimating a local friction factor, the distribution of the shell-side flow and mass transfer coefficient were determined. The flow distribution in the lumen side is also often non-uniform. Using high-speed photography and dye tracer studies, Park and Chang [11] showed that the distribution depends on the inlet manifold type and size, tube length, fiber inner diameter, shell diameter, fiber packing fraction and Reynolds number. Wickramasinghe et al. [12], and Crowder and Cussler [13] showed that for polydisperse fiber inner diameter, the average mass transfer coefficient could be reduced by the polydispersity. Membrane distillation (MD), which is a relatively new membrane separation technique, is being developed for desalination and concentration of thermally labile substances. As a thermally driven process, MD can be significantly influenced by temperature polarization [14–16]. Among various types of membrane modules, hollow fiber module shows the least temperature polarization [17], so it must have a great future in this field. Although some researches on MD have employed hollow fiber membrane module [18–22], few studies focused on the analysis of the effect of flow maldistribution on the module performance. This paper investigates the effects of flow maldistribution on the performance of MD hollow fiber module by numerical analysis. As mentioned above, flow maldistribution may exist in both lumen and shell side of a module. In this study, it is assumed that the maldistribution at the lumen side is introduced by the polydispersity of fiber inner diameter, and that at the

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shell side is the inevitable result of non-uniform fiber packing. These two aspects are studied, respectively, in this study.

2. Theory As the estimation of the effect of flow maldistribution is algebraically complicated, we begin with the ideal hollow fiber module, and then discuss the more practical one. 2.1. The model for ideal hollow fiber membrane module used in MD The configuration of a hollow fiber module is similar to a heat exchanger working in counter-current flow with fluid flowing on both sides. The ideal hollow fiber membrane module is a postulated one, which has the two features. (1) There is no polydispersity of fiber inner diameter, so the flow in the lumen side of the module is uniform. (2) The fibers are evenly and regularly distributed on the shell side of the module, so the flow in the shell side is uniform. The following assumptions are introduced to establish the heat balance equations for the module. (1) There is no heat lost from the module. (2) Axial diffusion is negligible. (3) The streams on both sides of the membrane are in hydrodynamically and thermally developed laminar flow. (4) There exist no entrance and exit effects in this module. In the following sections, the situation that two streams of pure water are feed into the lumen and shell duct of a hollow fiber membrane module, respectively, in countercurrent flow to perform MD is to be considered. In a hollow fiber module used in MD process, the fluids’ temperatures and transmembrane flux may vary axially alongside the module. Considering a differential element in the module, the following differential equations were obtained by the enthalpy conservation for the element to describe the temperatures of the

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feed and permeate stream alongside the module [N H + (km /δ)(tfm − tpm )]nf πdm dtf = dz Ff cpf

that the transmembrane flux is proportional to the vapor pressure difference across the membrane: (1)

with the boundary condition: tf = tfin

z = 0,

and [N H + (km /δ)(tfm − tpm )]nf πdm dtp = dz Fp cpp

(2)

(3)

with the boundary condition: z = L,

tp = tpin

(4)

where dm is the logarithmic mean diameter of the hollow fiber. Eqs. (1)–(4) form a system of ordinary differential equations with splitting boundary values, and a shooting method can be employed to solve this model. The permeate temperature at z = 0 (tpout ) is guessed and the boundary condition given in Eq. (2) is applied. The solution is marched forward using Eqs. (1) and (3) until z = L, and then the guessed value of tpout is modified by an implicit secant method. This procedure is repeated until the boundary condition shown in Eq. (4) is met. While the differential equations are being solved, the transmembrane mass flux at each differential element, N, must be known. The following paragraphs show how to calculate N value for each element according to the principles of MD. MD is a complicated physical process in which both heat and mass transfer are involved. For heat transfer in direct contact membrane distillation (DCMD) at steady state, the temperatures adjacent to the membrane for a given flux can be expressed, in terms of the bulk feed and permeate temperatures and the three heat transfer coefficients [20]: tfm =

(km /δ)(tp + tf hf / hp ) + hf tf − N H (km /δ) + hf (1 + (km / hp δ))

(5)

tpm =

(km /δ)(tf + tp hp / hf ) + hp tp + N H (km /δ) + hp (1 + (km / hf δ))

(6)

For mass transfer through the membrane, because of its complexity, most investigators adopted a more empirical approach to describe this process, assuming

N = C[p(tfm ) − p(tpm )]

(7)

where C is the membrane distillation coefficient (MDC), and can be determined from MD experiments. In a pervious work, we proposed a model named Knudsen molecular-Poiseuille transition (KMPT) to describe the mechanism of mass transfer through the membrane in MD, and this model provides us with a method to calculate MDC value:
1 C= (1/(CK (M/RTm )0.5 ))+(1/(CM (DM/Yln RTm )))   Mpm (8) +CP ηv RTm where the model parameters CK , CM and CP should be determined from MD experiment. Eqs. (5)–(7) form a system of non-linear algebraic equations, and an explicit secant method is suitable to solve it. 2.2. Modeling the lumen side flow distribution On the base established above, we now investigate the effect of polydispersity of fiber inner diameter. Here the size distribution of fiber inner diameter is considered, but it is assumed that fibers are packed evenly, so the flow distribution in the shell side of the module is completely uniform. It has been recognized that the fiber inner diameters are generally non-uniform, but follow a Gauss distribution [1,8,13]. So the probability density function for this random variable is 1 2 2 g(x) = √ e−(x−dim ) /2σ (9) 2σ and the probability that a fiber has an inner diameter between x1 and x2 is  x2 1 2 2 P {x1 < x ≤ x2 } = √ e−(x−dim ) /2σ dx (10) 2σ x1 where dim is the average fiber inner diameter, and σ is its standard deviation. Theoretically, the analysis should be carried out for each fiber in the module, but in practice a histogram describing the fiber diameter distribution is constructed consisting of M categories.

D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

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Each category represents a size interval of the fiber inner diameter, and the probability of a category can be calculated by Eq. (10). Thus, we have divided a hollow fiber module into M regions, and each one was regarded as an ideal hollow fiber module. These regions have the same packing fraction, but the fiber numbers and fiber inner diameters in one region are different from another, so there exist a flow distribution in the lumen side of the module. As the flowrate in a duct is proportional to the fourth power of its hydraulic diameter at laminar flow, the flowrate in the lumen side for kth region can be determined by the following equation: 4 nk dim,k Ft,k = M Ft 4 j =1 nj dim,j

(11)

where nk is the number of fibers in kth region, dim,k the fiber inner diameter of kth region, Ft the total flowrate at the lumen side, and Ft,k is the flowrate at lumen side for kth region. With of the assumption that the shell is uniformly packed, the flowrate on the shell side for kth region is proportional to its number of fibers: nk Fs,k = M (12) Fs j =1 nj where Fs is the total flowrate on the shell side, and Fs,k is the flowrate at shell side for kth region. As each region is ideal, Eqs. (1)–(8) can be used to describe its performance. Solving these equations is then conducted over each of these regions individually, using the calculated feed and permeate flowrates from Eqs. (11) and (12), and associated membrane area. The total flux for the module is the sum of the production rates of all regions divided by the total membrane area:  Ni Ai N=  (13) Ai The outlet temperatures of the feed and permeate should be calculated from the following equations, weighted to allow for the flowrates as follows:  i,out i  i,out i tp Fp tf Ff out out tf = , tp = (14) Ff Fp 2.3. Modeling the shell-side flow distribution Theoretically, the fibers in a bundle can be spaced regularly across the shell of a module, and the array

Fig. 1. The random distribution of hollow fiber on the cross section of the module.

of fiber can be configured in either square pitch or triangular pitch, as in tube and shell heat exchanger. In most industrial modules, however, the distribution of fibers is far more arbitrary, the fibers are randomly packed in the shell. This leads to a range of duct sizes and shapes in the shell, or the module shows a certain extent variation of the local packing fraction, as illustrated in Fig. 1. Because the distribution of flow across the bundle lies on the hydraulic resistance of each duct for fluid to pass through, the randomness of fibers’ arrangement can result in flow maldistribution on the shell side. The following sections describes how to determine the influence of flow maldistribution on MD. The basis of this work is to divide all the fibers of a randomly packed module into a number of groups or regions of different packing fraction, and it is assumed that all fibers in a group are evenly distributed. The MD performance in each of these regions can be obtained independently by solving Eqs. (1)–(8), and then these results are combined by Eqs. (13) and (14) to evaluate the overall performance of a module. 2.3.1. The local packing fraction distribution In order to characterize local variation in fiber packing in the hollow fiber module, sub-division of the modules into areas associated with each fiber is necessary. Voronoi tessellation is a mathematical modeling technique to describe the sub-division of space between randomly packed points, and has been proved to have a good agreement with the ball packing

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2.3.2. Flow distribution A crucial variable for us to determine the flow distribution for the overall fiber bundles is the product of the friction factor versus Reynolds number, f Re. By solving the momentum equation for the field of longitudinal laminar flow between cylinders arranged in regular array, Sparrow developed an effective method to calculate the f Re for this situation [24]:   
 r04 dp θ0 2θ0 − + θ − f Re = 0 V ηw dz 1−ε (1 − ε)2 (18) Fig. 2. Illustration of Voronoi tessellation of the module cross section.

experiments. This attractive technique has been applied in some researches [2,9,10] to divide hollow fiber bundle into a number of polygonal cells, as shown in Fig. 2. In these works, fiber cross section was described using probability function to determine cell size distribution. The probability density function, f, obtained by Chan et al. [23], is an exponential one, and was applied to approximate the probability that there is no other fiber within a polygonal area surrounded by s nearest fibers: f (ψi ) =

s s ψis−1 −sψi /ψ¯ e ψ¯ s (s − 1)!

(15)

where ψ i is a polygonal area, and ψ¯ is the average polygonal area. From the above equation, we can obtain the equation for calculating the probability that a polygonal cell has a packing fraction between φ 1 and φ2:  λ2 λs ss P = e−sλ dλ (16) (s − 1)! λ1 where (1 − φi )φ λi = (1 − φ)φi

(i = 1, 2)

(17)

and φ is the packing fraction of the hollow fiber module. Thus, it has been shown that Voronoi tessellation technique can provide us with a way to determine the distribution of polygons areas and therefore the distribution of the local packing fraction.

Under the conditions: (1) identical pressure drop in all cells; (2) the total mass flowrate is equal to the sum of the individual cell mass flowrates, the effective f Re for the overall fiber bundle can be decided by the f Re values of all polygonal cells [9]     nf Pw 2 ACi 3 1 1 = (19) (f Re)e (f Re)i Pwi AC i=1

where Pwi and ACi are the wetted perimeter and the flow area of the ith polygonal cell, respectively, Pw and AC the total wetted perimeter and total cross-sectional area for flow in the overall bundle, respectively, and nf is the total number of fibers in the overall bundle. Similar to the analysis of the lumen side, we introduce a histogram consisting of m categories to describe the distribution of the cells’ area for flow in the module. The effective f Re is then given by     m 1 1 Pw 2 ACi 3 = (20) nfi (f Re)e (f Re)i Pwi AC i=1

where nfi is the number of fibers or cells in the ith category. Inspiringly, except nfi , all the other items at the right-hand side of Eq. (20) can be substituted by the local packing fraction of a category, φ i , and the module’s packing fraction, φ. By introducing Pi = nfi /nf , the following equation was obtained [25]:
 m 1 (1 − φi )φ 3 1 = Pi (21) (f Re)e (f Re)i (1 − φ)φi i=1

where Pi is the probability of the ith category occurring, and it can be determined from Eq. (16). Once (f Re)i and (f Re)e are determined, the flow distribution on the shell side of the module can be decided by the

D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

following equation, expressed by the fractional flow in the shell side for the ith category, Ws,i [25]:
Ws,i = Pi

(f Re)e (f Re)i




(1 − φi )φ (1 − φ)φi

3 (22)

Because the polydispersity of fiber inner diameter is not considered here, the lumen side flowrate in a category is proportional to its membrane area or number of fibers, so the fractional flow in the lumen side of the ith category, Wt,i , can be expressed as Wt,i = Pi

(23)

Once the flowrates in the both sides of the membrane for each category are obtained, the behavior of each category can be predicted by solving Eqs. (1)–(8), in which the two flowrates is decided from Eqs. (22) and (23). The total permeate flux of the module and the outlet temperatures of feed and permeate can also be obtained from Eqs. (13) and (14), respectively.

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3. The results and discussion 3.1. The effect of the polydispersity of fiber inner diameter To study the effect of polydisperisty of fiber inner diameter quantitatively, the calculation should aim at a specific hollow fiber module. The characteristics of the module considered here are given in Table 1. CK , CM and CP are the parameters of KMPT model, and they have been determined in the authors’ previous study [25]. Fig. 3 shows the effect of the polydispersity of fiber inner diameter on the module flux at various flowrates of feed and permeate. The standard deviation of fiber inner diameter, σ , is used as abscissa, and the correction factor is read as ordinate. The correction factor is defined as the module flux with a positive value of σ divided by that with the value of σ = 0. The latter is the ideal hollow fiber module mentioned above. It can be seen that the correction factor is always less than

Table 1 Characteristics of the module used in calculation Membrane area (m2 )

Packing fraction

Fiber inner diameter (mm)

Number of fibers

Fiber length (m)

CK (10−4 )

CM (103 m−1 )

CP (10−11 m)

1

0.6

0.30

3000

0.34

15.18

5.10

12.97

Fig. 3. Correction factor as a function of standard derivation of fiber inner diameter.

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D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

Fig. 4. Cumulative flow of the lumen side as a function of cumulative membrane area.

1.0, this means that the polydispersity of fiber inner diameter always has a negative effect on the module flux. The explanation is due to the flow maldistribution in the lumen side. Eqs. (11) and (12) show that there is some difference in flow distribution between lumen and shell side, and this is shown visually in Fig. 4. This figure gives the cumulative fraction of flow in lumen side compared to the cumulative fraction of membrane area (or of fibers). The curves’ deflexion from diagonal indicates a flow maldistribution at the lumen side. Because of the polydispersity of fiber inner diameter, the fiber bundle may be regarded as a combination of many regions of different fiber inner diameter. In some regions, the flowrates of the lumen side are larger than that of the shell side, and in the other regions the situation is opposite. In the regions that the flowrate in one side is higher than that of the other side, the temperature in the latter would be close to that in the former, and this means a reduction of the driving force on the MD process. 3.2. The effect of non-uniform packing 3.2.1. MD performance in a non-uniformly packed module To study the effect of maldistribution caused by the non-uniform packing of fibers, the fibers bundle is di-

vided into a number of categories of different packing fraction. The calculation for each categories is performed individually, and then the results are combined to give the overall performance of the module. Table 2 summarizes the regional results for a module with overall packing fraction of 0.5, and the other characteristics of the module are shown in Table 1. For the purpose of comparison, the simulation results for the ideal hollow fiber module are also listed in the bottom line of Table 2. Some discussions are needed to recognize the performance of non-uniformly packed module. 1. The vast majority of the MD process occurs in the regions with the local packing fraction, φ i , between 0.3 and 0.6. Production rate (93%) of the module is from these regions, and they occupy only 75% of the overall membrane area of the module. In the regions with φ i larger than 0.6, the permeate flowrates are too much less than that of the feed, so their temperatures are very close to that of the feed. This means that more than 20% of the feed stream goes through the module almost without any driving force for MD process, so the associated membrane area, more than 20% of the total, is ineffective. 2. Even in the regions with φ i between 0.3 and 0.6, the permeate fluxes are much less than the ideal

D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

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Table 2 Summarized regional performance of a non-uniformly packed module with φ = 0.5 Local packing fraction, φ i 0.909 0.769 0.667 0.588 0.526 0.476 0.435 0.400 0.370 0.345 0.323 0.303 0.286 0.270 0.256 0.256 Non-uniformly packed module Ideal hollow fiber module

Membrane area fraction (%) 0.00 7.00 14.20 17.60 16.90 13.90 10.40 7.20 4.70 3.00 1.80 1.10 0.60 0.30 0.20 0.10

Ff,i (%)

Fp,i (%)

0.00 7.00 14.20 17.60 16.90 13.90 10.40 7.20 4.70 3.00 1.80 1.10 0.60 0.30 0.20 0.10

0.00 0.10 0.60 2.20 4.80 8.00 10.70 12.30 12.70 11.90 10.40 8.50 6.70 5.00 3.60 2.50

tfout (◦ C)

tpout (◦ C)

Flux (kg m−2 h)

Flux × area (kg h−1 )

70.00 69.33 68.01 64.07 56.35 41.51 28.12 26.06 25.57 25.42 25.42 25.32 25.31 25.30 25.32 25.33

70.00 70.00 70.00 69.99 69.99 69.85 61.78 48.73 40.30 35.51 32.25 30.43 28.77 27.53 27.35 26.69

0.00 0.12 0.35 1.02 2.27 4.27 5.83 6.15 6.20 6.20 6.20 6.21 6.22 6.24 6.26 6.28

0.0000 0.0084 0.0497 0.1795 0.3836 0.5935 0.6063 0.4428 0.2914 0.1860 0.1116 0.0683 0.0373 0.0187 0.0125 0.0063

100

100

100

49.516

43.821

2.995

2.995

100

100

100

28.103

62.508

5.700

5.700

With the operation conditions: Ff = Fp = 2 l min−1 , tfin = 70 ◦ C, tpin = 25 ◦ C, the permeate is in the shell side.

hollow fiber module. This indicates a not very active associated membrane area, about 35% of the total. In the two regions with the φi = 0.526 and 0.588, the permeate flowrates are still much less than that of feed side, and the temperature differences between the two sides alongside the module are still very small. In these two regions, the outlet permeate temperatures are almost equal to that of the feed inlet, this means that part of associated membrane area is ineffective. 3. The region with φi = 0.435 shows the closest performance to ideal hollow fiber module. The explanation is due to its Ff /Fp value closest to 1.0. Unfortunately, this region occupies only 10.40% of the total membrane area, so its contribution to the module is not so much. 4. In the regions with φ i between 0.25 and 0.4, the flow imbalance is in the opposite direction, with about 70% of the total permeate stream in contact with only 19% of the total feed stream or membrane area. This imbalance indicates excess driving forces in these regions. Consequently, the permeate fluxes in them are even higher than that in the ideal hollow fiber module. But these fluxes occur in a very small

portion of the membrane area, so they contribute very little to the overall performance of the module. From Table 2 and the above discussion, it can be concluded that the flow imbalance or flow maldistribution on the shell side may largely reduce the driving force of MD process. This reduction will make only a very small part of total membrane area (30% in Table 2) provide the approximate flux of ideal hollow fiber module, and even a portion of the membrane area is completely ineffective. 3.2.2. The effect of packing fraction on the performance of non-uniformly packed module Fig. 5 shows the effect of packing fraction on the permeate flux of a non-uniformly packed module. The ordinate is also a correction factor, which is defined as the flux of a non-uniformly packed module divided by the flux of an ideal module. The explanation of the inflexion on the curve is due to an assumption adopted during the calculation. In case of φ being less than 0.6, the number of the nearest neighbors for a fiber is considered to be four, i.e. s = 4; when φ is larger than 0.6, however, the number is assumed to be six, i.e. s =

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Fig. 5. The effect of packing fraction on the flux of a non-uniformly packed module: Ff = Fp = 2 l min−1 , tfin = 70 ◦ C, tpin = 25 ◦ C, A = 1 m2 .

6. This way to decide the s value in Eq. (15) is proved to be effective by Chen and Hlavacek’s experiment [9]. From Fig. 5, we may conclude that packing fraction has a significant effect on the performance of a non-uniformly packed module. There are two possible mechanisms for the flux enhancement by increasing

packing fraction. It is well known that heat transfer at the shell side can be improved by the increment of packing fraction in a tube and shell heat exchanger. This is also true for a hollow fiber module, as shown in Fig. 6. However, Fig. 6 also shows that packing fraction almost has no influence on the flux of an ideal

Fig. 6. The effect of packing fraction on the heat transfer coefficient and the flux of ideal hollow fiber module with constant membrane area: Ff = Fp = 2 l min−1 , tfin = 70 ◦ C, tpin = 25 ◦ C, A = 1 m2 .

D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

module. This implies that the limiting step of MD process in hollow fiber is no longer the heat transfer in the thermal boundary layer at the two sides of the membrane. In fact, because of the very small size of flow ducts, the heat transfer coefficients at both sides of the membrane are very high, so temperature polarization can almost be neglected. Another possible mechanism is that the increment of packing fraction may mitigate the flow maldistribution on the shell side, so the regional temperature differences between the two sides of the membrane alongside the module increase. Fig. 7 shows the shell-side flow distribution at various packing fractions, expressed by cumulative flow fraction of the shell side compared to the cumulative fraction of membrane area. The deflexions of these curves from diagonal are just the evidence of flow maldistribution in the shell side of the module. With the increase of packing fraction, the curve is more and more closer to the diagonal. This means that the shell side experiences a more uniform flow at a higher packing fraction, and larger temperature differences between the two streams can be obtained. Some literatures presented the results that for MD module with very low packing fraction, e.g. φ = 0.1, the value of the permeate flux were significantly higher than those with some higher packing fraction, e.g. φ = 0.4. It seems that these results are opposite to that presented here. However, it should be noticed

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that the range of packing fraction of the module used in this study is 0.4–0.8. The possible mechanism for the results obtained from the module with very low packing fraction, e.g. 0.1–0.4, is that the transverse flow or radial mixing in the shell side of the module become more serious with the decrease of packing fraction. This was proved by Wu and Chen’s [2] experiment, in which a minimum mass transfer coefficient at φ = 0.5 was obtained when they studied the effect of packing fraction on mass transfer within the range 0.1–0.8. Analysis of the measured pressure drop as a function of axial flow velocity by them showed that the shell-side pressure drop versus velocity component (n) was higher than 1.4 when φ was lower than 0.5, and n dropped to 1.15 at the φ = 0.5. This indicates that more significant turbulent component exists in the press drop of module with lower packing fraction. The transverse flow may exert a positive effect on MD module performance by making the temperature profile on the cross section of the shell more uniform. With the increase of packing fraction, transverse flow or radial mixing would become less important, and flow maldistribution would be the crucial factor in MD process. 3.2.3. The effect of flowrates Fig. 8 shows how the flowrates of feed and permeate affect the flux of a non-uniformly packed hollow fiber

Fig. 7. Cumulative shell-side flow fraction as a function of cumulative fraction of membrane area.

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D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

Fig. 8. The effect of flowrates on module flux: φ = 0.6, tfin = 70 ◦ C, tpin = 25 ◦ C.

module. Obvious improvement of flux with increasing flowrates is observed. In the above section, we have excluded the possibility of the flux enhancement by the improvement of heat transfer in the module. So although the heat transfer can be improved by increasing flowrates, it is not the reason that MD process gets improvement. When the permeate flowrate increases, its temperature will become less affected by heat transfer and vapor condensation from the feed side of the membrane, and so does the feed stream. This means that the increment of flowrates can enlarge the temperature difference between these two streams in the module, and in this way MD process is improved. Despite the positive effect of increasing flowrates, there exist a trade-off between the permeate flux and the heat recovery in MD system [19], i.e. the improvement of heat recovery is restricted by the demand of a certain flux level, and vice versa.

4. Conclusions Because of the polydispersity of fiber inner diameter, the fiber bundle of a module may be regarded as combination of many regions with different fiber inner diameter, and this is the reason that the flow maldistribution exists at the lumen side of a hollow fiber module. This flow maldistribution can exert a negative

effect on the module flux by weakening the driving force of MD process, and this effect would become more serious with the increase of the standard deviation of fiber inner diameter. In the worst case considered, the module flux can reduce by about 11% when the inner diameter has a standard deviation of 13.3%. At the shell side, the randomness of fiber packing leads to a certain extent variation of the local packing fraction on the cross section of the fiber bundle, so the flow maldistribution occurs. The simulation result shows that the packing fraction of a module is a crucial factor affecting flow distribution at the shell side. The shell-side flow tends to be more unevenly distributed in the module with a lower packing fraction. The maldistribution makes the local flowrates at the two sides of membrane quite different, so the temperature differences between the two streams is largely reduced, and even some regions experience almost no driving force for MD process. The module flux is largely influenced by the effect of maldistribution. The non-uniformly packed module exhibits flux from some 42% of ideal module to 88% with the variation of packing fraction from 0.4 to 0.8. The increment of module flux with packing fraction is due to the corresponding mitigation of flow maldistribution, not the improvement of temperature polarization in MD. Due to the transverse flow or radial mixing in the shell side, the reported results obtained from the module of very low packing

D. Zhongwei et al. / Journal of Membrane Science 215 (2003) 11–23

fraction was opposite to that obtained in this study. The increase of flowrates also has a positive effect on module flux, and this can be attributed to the corresponding enhancement of MD driving force.

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