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Theoretical estimation of shell-side mass transfer coefficient in randomly packed hollow fiber modules with polydisperse hollow fiber outer radii
Journal of Membrane Science 284 (2006) 95–101
Theoretical estimation of shell-side mass transfer coefficient in randomly packed hollow fiber modules with polydisperse hollow fiber outer radii Weiping Ding a , Dayong Gao a,b,c,∗ , Zhen Wang c , Liqun He c a
Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA c Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei, Anhui 230027, PR China b
Received 10 March 2006; received in revised form 29 June 2006; accepted 5 July 2006 Available online 14 July 2006
Abstract The expression of average shell-side mass transfer coefficient (MTC) in hollow fiber modules is theoretically deduced in this paper. This expression considers the effects of both the polydispersity of hollow fiber outer radii and the randomicity of hollow fiber distribution on the shell-side mass transfer performance for the first time. In the expression, Gaussian function is used to model the polydisperse hollow fiber outer radii, Voronoi tessellation method is used to model the random hollow fiber distribution, and the two distributions are assumed to be independent. Then, the effect of polydisperse hollow fiber outer radii on average shell-side MTC is studied in detail. © 2006 Elsevier B.V. All rights reserved. Keywords: Mass transfer coefficient; Random distribution; Polydispersity; Packing density; Hollow fiber
1. Introduction Shell-side mass transfer of hollow fiber modules with shell and tube configuration has been studied widely in recent years. The shell-side mass transfer performance is affected by many factors [1,2]. Two of the main factors are the randomicity of hollow fiber distribution and the polydispersity of hollow fiber outer radii. Almost all previous works about the effect of hollow fiber distribution on shell-side mass transfer have based on the assumption of identical hollow fiber outer radii. Under such assumption, commonly there are two methods to deal with the distribution of hollow fibers. One is that the hollow fiber bundle is modeled as an infinite, spatially periodic medium, i.e. the hollow fibers are assumed to be arranged in regular arrays (e.g. square or triangular) [2–6] or random in a unit cell [5,6]. The other is that Voronoi tessellation method is used to model the random distribution of hollow fibers [2,7–10]. In practice, neither inner radii nor outer radii of hollow fibers are possible to be identical because of the imperfect manufactur-
∗
Corresponding author. Tel.: +1 206 543 1411; fax: +1 206 685 8047. E-mail address: [email protected] (D. Gao).
0376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2006.07.012
ing process. So, the polydisperse inner and outer radii absolutely will affect lumen-side and shell-side mass transfer performance, respectively. However, in the literature, only the effect of polydisperse inner radii was investigated in detail [10–13], the effect of polydisperse outer radii seldom was focused on. So, in this paper, the effect of polydisperse fiber radii (hollow fiber outer radii) on shell-side mass transfer is attempted to investigate theoretically in randomly packed hollow fiber modules, i.e. both the randomicity of hollow fiber distribution and the polydispersity of fiber radii are taken into account together. A theoretical average shell-side MTC expression is deduced and then the effect of polydisperse fiber radii is studied emphatically. In the literature, a system of removing oxygen from water often was chosen to study the shell-side mass transfer performance because of its availability and simplicity [2,4,11,14,15]. Here, we make the same choice: water containing oxygen flows outside of hollow fibers, and nitrogen flows inside of hollow fibers counter-currently. 2. Theory According to the literature [2,8–10], the following assumptions are used in this paper: (1) hollow fibers are rigid, and aligned axially but randomly arranged, i.e. the flow distribution
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W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101
For a certain module containing N hollow fibers, the average polygonal cell cross-section area can be acquired easily: a0 =
πR2M N
(4)
where RM is the module inner radius. Then, the packing density can be defined as φ=
Fig. 1. Schematic representation of the module cross-section subdivision with Voronoi tessellation method.
is consistent along the module length; (2) the effect of module wall on shell-side hydraulics is ignored; (3) the fluid flow profile is fully developed and laminar; (4) the effect of shell-side inlet and outlet is ignored. 2.1. Distribution of hollow fibers Voronoi tessellation is a mathematical method to describe the subdivision of space between randomly packed objects by drawing straight boundaries equidistant between neighboring objects, forming polygonal cells. It provides a good way to calculate the geometric characteristics of random spacing that can be used to calculate overall properties [7,16]. In this paper, Voronoi tessellation method is used to evaluate the shell-side flow distribution in a randomly packed hollow fiber module. Considering N hollow fibers are randomly placed in the module. Using Voronoi tessellation method, the cross-section area is then subdivided into polygonal cells, and each polygonal cell is associated with one hollow fiber (Fig. 1). The probability density distribution function of polygonal cell area can be written as below according to the literatures [2,7,11,17]: f (ϕ) =
ss
ϕs−1
ϕs (s − 1)!
ϕ = a − af
e−(sϕ)/ϕ
(1) (2)
where a is the polygonal cell cross-section area (including the fiber), af the hollow fiber cross-section area, s the number of nearest neighbor fibers, and the sign “” denotes statistical average in mathematics. At lower packing density, s = 4 appears to fit the simulation better; at higher packing density, the calculated distribution agrees with s = 6 better, but the effect of s on results is not so evident [2,7]. So in this paper only s = 6 is considered for the calculation of average shell-side MTC. Assuming that the hollow fiber distribution is independent of the hollow fiber outer radius distribution, the following equation can be obtained: ϕ = a0 − af
(3)
where a0 is the average polygonal cell cross-section area (including the fiber).
af Nr 2 = a0 R2M
(5)
2.2. Polydispersity of fiber radii According to the literature [11,12,18], the distribution of fiber radii g(r) is assumed to be a Gaussian function: e−(r−R0 ) /(2R0 ε0 ) √ 2πR0 ε0 2
g(r) =
2 2
(6)
where R0 is the average outer radius of hollow fibers, (R0 ε0 )2 the variance of this distribution, and ε0 denotes the polydispersity of fiber radii. Obviously, ∞ R0 = r = rg(r) dr (7) 0
Assuming the effective length of hollow fibers is identical and equals to L, the average hollow fiber outer surface area A is given by ∞ A = 2πL rg(r) dr = 2πrL (8) 0
2.3. Flow outside of hollow fibers in polygonal cells Using the assumptions that the entrance and exit losses are small and that flow areas are parallel and of the same length L [2,7,19,20], the shell-side pressure drop P for laminar flow through each of the separate flow area is the same as the whole bundle and is calculated by P =
4ρv2 L dh
(9)
where ρ is the density of water, dh = 2ϕ/(πr) the hydraulic diameter of each polygonal cell (4 × flow area/hollow fiber circumstance), and v is the mean flow velocity in each polygonal cell. Rearrangement of Eq. (9) gives the following equation: Pϕ v= (10) 2πrρL Then, the average shell-side flow rate Q is given by ∞ ∞ Q = ϕvf (ϕ)g(r) dϕ dr 0
0
(11)
W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101 Table 1 Parameters used in calculation
2.4. Local mass transfer outside of hollow fibers in polygonal cells In the literature, the available universal local MTC expression outside of one hollow fiber in each polygonal cell cannot be found. Wu and Chen used L´evˆeque equation to calculate this kind of local MTC, but L´evˆeque equation is valid only for high Gz number [2,21]. Miyatake and Iwashita theoretically studied the heat transfer coefficient expression of shell and tube heat exchangers with tubes arrayed regularly, laminar flow outside of tubes and constant temperature on the surface of tubes [22]. In this paper, the local MTC expression is acquired on the analogy of their heat transfer coefficient expression under the condition of the tubes arranged in regular triangular array:
k=
D 9 f 2 + g2 Gz2/3 2r 4
1/2 (12)
where D is the oxygen diffusion coefficient in water, Gz = ϕv/(LD) (Gz: 1–106 ) [22], the expressions of f and g are the following: f =
8.92(1 + 2.82γ) 1 + 6.86γ 5/3
g=
Nomenclature
Value
Module inner radius (m) Effective hollow fiber length (m) Density of water (300 K) (kg/m3 ) Oxygen diffusive coefficient (300 K) (m2 /s)
RM L ρ D
0.02 0.20 9.96 × 102 2.29 × 10−9
Combining Eqs. (17) and (18), we obtain ∞ ∞ −2krL/(ϕv) e ϕvf (ϕ)g(r) dϕ dr c = 0 0 ∞ ∞ c0 0 0 ϕvf (ϕ)g(r) dϕ dr
(19)
According to the above analysis, the average shell-side MTC can be described by ∞∞ ϕvf (ϕ)g(r) dϕ dr k = 0 0 ∞ 2L 0 rg(r) dr ∞∞ 0 ϕvf (ϕ)g(r) dϕ dr (20) × ln ∞ ∞0 −2krL/(ϕv) ϕvf (ϕ)g(r) dϕ dr 0 0 e
2.6. Parameters setting
2.34(1 + 24γ) 1/3
(14)
(15)
In this paper, the referred parameters are listed in Table 1. Mathematic 5.0 is used to calculate all expressions. 3. Results and discussions 3.1. Average shell-side MTC with the polydispersity close to 0
2.5. Average shell-side MTC Mass transfer of oxygen from water in shell-side to nitrogen in lumen side contains three steps: transfer of oxygen in water, diffusion of oxygen through the membrane pores that fully filled with gas, and diffusion of oxygen in nitrogen stream. Assuming that the phase interface between gas and liquid is on the hollow fiber outer surface and the oxygen concentration on the interface is 0. In this case, the mass transfer efficiency in the liquid phase dominates the overall mass transfer performance, i.e. the overall MTC could be approximated by the MTC in shell-side. According to the literature, the average shell-side MTC could be described by the following expression [2,12,23]: Q c0 ln A c
Parameter
(13)
(1 + 36.5γ 5/4 )(ϕ/r 2 ) √ 3(ϕ/r 2 + ) γ= −1 6
k =
97
One might try to follow the step similar to this paper and directly deduce the average shell-side MTC expression based on the assumption that all fiber radii are identical, but this is not necessary at all. The expression in this paper includes such ideal situation, i.e. the polydispersity is close to 0.
(16)
where c0 is the oxygen concentration at inlet, and c is the average oxygen concentration at outlet. The average oxygen concentration at outlet is given by ∞∞ cϕvf (ϕ)g(r) dϕ dr c = 0 ∞ 0 ∞ (17) 0 0 ϕvf (ϕ)g(r) dϕ dr The concentration in each polygonal cell area is given by [11] c = c0 e−2krL/(ϕv)
(18)
Fig. 2. Effects of hollow fiber outer radius and average shell-side flow velocity on average shell-side MTC.
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Fig. 2 shows the effects of hollow fiber outer radius R0 and average shell-side flow velocity V on average shell-side MTC when the polydispersity is close to 0. Under the given packing density, the decrease of hollow fiber outer radius consequentially leads to the increase of hollow fiber number, which causes the better hollow fiber packing/distribution, and then enhances the average shell-side MTC. (When the number of hollow fibers increases, the distribution of hollow fibers runs to regular array. The average shell-side MTC with regular fiber array is bigger than that with random fiber array [2,5].) Certainly, the increase of hollow fiber number also results in the more dead zones (regions of low velocity where hollow fibers are close or touch), which reduce the average shell-side MTC [2,4,5], but this reduction is not significant comparing to the enhancement at lower packing density. When the packing density increases, the effect of dead zones becomes more and more prominent and disadvantageous. So with increasing the packing density, the average shell-side MTC curve with the smaller hollow fiber outer radius firstly has decreasing tendency, i.e. the optimal packing density corresponding to the maximal average shell-side MTC is smaller. Practically, another important method to improve the average shell-side MTC is to increase the average shell-side flow velocity. The higher the average flow velocity, the bigger the average MTC. Moreover, Fig. 2 also clearly shows the fact that with increasing the average flow velocity, the optimal packing density increases, which has not been shown in the literature. From Fig. 2, one also could deduce that in the available range of packing density the average shell-side MTC will increase monotonously with increasing the packing density when the average shell-side flow velocity is very high (high Gz number), which is consistent with the theoretical result in the literature qualitatively [5]. Reliable experimental data for the validation of this study about the effect of polydisperse fiber radii on average shell-side MTC are not found in the literature. However, when the polydispersity is close to 0, the average shell-side MTC has been investigated widely in previous work [2,4,5,15]. Fig. 3 shows the comparison of average shell-side MTC between the litera-
ture and this paper under such ideal condition. In the literature, the conclusion about the effect of packing density on the average shell-side MTC is inconsistent. One representative standpoint is that with increasing the packing density, the average shellside MTC first increases and then decreases (low Gz number) [4], while the other is contrary [2]. These discrepancies within the literature itself could be mainly caused by their different experimental conditions, such as the hollow fiber number, the hollow fiber length, the hollow fiber radius, and the inlet and outlet design. The simulative result in this paper supports the first standpoint, but the departure between the experimental results of the literature [4] and the theoretical values of this paper is still somewhat large. One important reason is that the assumed axial laminar flow pattern in this study somewhat deviates from the actual flow. Firstly, the current shell inlet and outlet design is not perfect, which causes cross/transverse flows at the shell entry/exit regions. Secondly, the hollow fibers maybe are not strictly parallel, and axial flows could terminate or split to produce additional cross-flows. Thirdly, some imperfect designs could produce unexpected secondary flows [5]. Cross-flows or secondary flows could promote mass transfer, especially at lower packing density. However, our model cannot calculate the average shell-side MTC under such flow conditions. So, our theoretical values are far less than the experimental values. Another possible reason which causes the smaller theoretical values of this paper is that the local shell-side MTC expression used in this study is not ideal for the actual mass transfer outside of hollow fibers in polygonal cells. After all, this expression is under the condition of regular triangular array. So, the universal local shell-side mass transfer correlation needs to be established. Additionally, this model also fails to accurately represent dead zones and channeling flows (flows mainly pass through larger channels between the fiber bundle and the external case or within the fiber bundle) in experiments, which also causes the theoretical values of this paper imperfect. 3.2. Average shell-side MTC with the polydisperse fiber radii When the polydispersity is not close to 0, can the effect of polydisperse fiber radii on average shell-side MTC be ignored? Wickramasinghe et al. in their work incidentally gave a theoretical expression to estimate the average shell-side MTC under such general situation [11]. Their result implied that the polydisperse effect cannot be neglected unadvisedly, but their expression does not refer to the effect of random fiber packing/distribution. In order to conveniently discuss the effect of polydisperse fiber radii, the relative departure η = 1 − k/ lim k is defined in this paper, which shows the difference ε0 →0
Fig. 3. Comparison of average shell-side MTC between the literature and this paper with the polydispersity close to 0.
between the real value with the polydisperse fiber radii and the ideal one with the identical fiber radii. Fig. 4 shows the comparison of relative departure between the literature and this paper. Qualitatively, the two estimated results are consistent from the point of view of variation tendency. However, the average shellside MTC calculated by the literature is far greater than the one shown by this paper. Although the polydispersity of fiber radii
W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101
99
absolutely results in the decrease of average shell-side MTC, we think, the expression in the literature to some extent magnifies the polydisperse effect. Why does the polydispersity affect the shell-side flow field and further affect the shell-side mass transfer? Polydisperse fiber radii easily cause the emergence of dead zones and channeling flows. For example, if in a polygonal cell the outer radius of the hollow fiber increases, the probability of the hollow fiber to touch other hollow fibers is higher, dead zones could be shaped easily, and the flow could bypass the hollow fiber. Such fact is equal to the sharp decrease of local flow velocity. If the outer radius of the hollow fiber decreases, although dead zones could not easily occur, channeling flows potentially would be shaped (dead zones and channeling flows are relative). Then, how does the polydispersity affect the average shellside MTC under different conditions, such as different average shell-side flow velocities, different shell-side pressure drops and different total shell-side flow rates? Fig. 5 shows the effect of average shell-side flow velocity V on relative departure. For the given packing density and polydis-
persity, the increasing average shell-side flow velocity not only increases the average shell-side MTC, but also to some extent improves the flow conditions of the hollow fibers around which dead zones and channeling flows are shaped, so the relative departure decreases when the average shell-side flow velocity increases. For the given average shell-side flow velocity and packing density, the increasing polydispersity causes the effects of dead zones and channeling flows increasingly more serious and then results in the decreasing average MTC, so the relative departure increases with increasing the polydispersity. For the given average shell-side flow velocity and polydispersity, when the packing density increases, dead zones become more and more, so the relative departure increases with increasing the packing density. In Fig. 5, when the average shell-side flow velocity is 0.05 m/s, the polydispersity is 0.15, and the packing density is about 75%, the decrease of average shell-side MTC actually reaches above 10%, so the effect of polydisperse fiber radii on shell-side mass transfer cannot be neglected optionally. Additionally, although the polydispersity affects and decreases the value of average shell-side MTC, it cannot change the variation tendency of average shell-side MTC curve. The effect of shell-side pressure drop P on relative departure almost is the same as the one of average shell-side flow velocity. According to the analysis mentioned in above paragraphs, one could easily understand the likeness. Here, the most important thing needed to be emphasized is the difference between the two situations about the effect of packing density on shell-side flow field. When the shell-side pressure drop is given, the increasing packing density increases the probability of dead zones, and at the same time causes the quickly decreasing average shell-side flow velocity, which also aggravates the polydisperse effect (the figure under this situation is omitted because in variation tendency it almost is the same as Fig. 5). Fig. 6 shows the effect of total shell-side flow rate Q0 on relative departure. The comparison between Figs. 6 and 5 shows that the variation tendency of relative departure versus packing density under the given polydispersity and total shell-side flow rate, i.e. the relative departure firstly increases and then decreases with increasing the packing density, remarkably differs from the
Fig. 5. Effect of average shell-side flow velocity on relative departure.
Fig. 6. Effect of total shell-side flow rate on relative departure.
Fig. 4. Comparison of relative departure between the literature and this paper with the polydisperse fiber radii.
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W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101
one under the given polydispersity and average shell-side flow velocity. In fact, when the total shell-side flow rate is given, because of conservation of mass, the increasing packing density consequentially leads to the increase of average shell-side flow velocity. Whereas, the higher average shell-side flow velocity is advantageous to alleviate the polydisperse effect. So, with increasing the packing density, the advantage becomes more and more predominant and the relative departure curve appears the decreasing tendency. 4. Conclusions In this paper, the expression of average shell-side MTC in hollow fiber modules with the polydisperse fiber radii is theoretically deduced. To the best of our knowledge, for the first time the effects of both the randomicity of hollow fiber distribution and the polydispersity of hollow fiber outer radii on shell-side mass transfer are considered together. Then, under the conditions of different average shell-side flow velocities, different shell-side pressure drops and different total shell-side flow rates, the effect of polydisperse fiber radii on average shell-side MTC is studied emphatically. Our results show that the polydispersity of hollow fiber outer radii results in the decrease of average shell-side MTC. For the given average flow velocity/pressure drop/total flow rate and packing density, the relative departure between the real value and the ideal one increases with increasing the polydispersity. For the given packing density and polydispersity, the higher the average flow velocity/pressure drop/total flow rate, the smaller the relative departure. For the given average flow velocity/pressure drop and polydispersity, the relative departure increases with increasing the packing density; however, for the given total flow rate and polydispersity, the relative departure firstly increases and then decreases with increasing the packing density.
Nomenclature a af a0 A c c0 dh D k L N P Q Q0 r RM
polygonal cell cross-section area (including the fiber) (m2 ) hollow fiber cross-section area (m2 ) average polygonal cell cross-section area (including the fiber) (m2 ) hollow fiber outer surface area (m2 ) oxygen concentration at outlet (ppm) oxygen concentration at inlet (ppm) hydraulic diameter of each polygonal cell (m) oxygen diffusion coefficient in water (m2 /s) shell-side mass transfer coefficient (m/s) effective hollow fiber length (m) number of hollow fibers pressure (Pa) shell-side flow rate (m3 /s) total shell-side flow rate (m3 /s) variable module inner radius (m)
R0 s v V
average outer radius of hollow fibers (m) number of nearest neighbor fibers mean flow velocity in each polygonal cell (m/s) average shell-side flow velocity (m/s)
Greek symbols ε0 polydispersity of fiber radii η relative departure ρ density of water (kg/m3 ) φ packing density ϕ variable
References [1] A. Gabelman, S.-T. Hwang, Hollow fiber membrane contactors, J. Membr. Sci. 159 (1999) 61–106. [2] J. Wu, V. Chen, Shell-side mass transfer performance of randomly packed hollow fiber modules, J. Membr. Sci. 172 (2000) 59–74. [3] I. Noda, C.C. Gryte, Mass transfer in regular arrays of hollow fibers in countercurrent dialysis, AIChE J. 25 (1979) 113–122. [4] M.J. Costello, A.G. Fane, P.A. Hogan, R.W. Schofield, The effect of shell side hydrodynamics on the performance of axial flow hollow fibre modules, J. Membr. Sci. 80 (1993) 1–11. [5] L. Bao, B. Liu, G.G. Lipscomb, Entry mass transfer in axial flows through randomly packed fiber bundles, AIChE J. 45 (1999) 2346– 2356. [6] L. Bao, G.G. Lipscomb, Mass transfer in axial flows through randomly packed fiber bundles with constant wall concentration, J. Membr. Sci. 204 (2002) 207–220. [7] V. Chen, M. Hlavacek, Application of Voronoi tessellation for modeling randomly packed hollow fiber bundles, AIChE J. 40 (1994) 606– 612. [8] J.D. Rogers, R. Long, Modeling hollow fiber membrane contactors using film theory, Voronoi tessellations, and facilitation factors for systems with interface reactions, J. Membr. Sci. 134 (1997) 1–17. [9] Y.J. Wang, F. Chen, Y. Wang, G.S. Luo, Y.Y. Dai, Effect of random packing on shell-side flow and mass transfer in hollow fiber module described by normal distribution function, J. Membr. Sci. 216 (2003) 81– 93. [10] Z.W. Ding, L.Y. Liu, R.Y. Ma, Study on the effect of flow maldistribution on the performance of the hollow fiber modules used in membrane distillation, J. Membr. Sci. 215 (2003) 11–23. [11] S.R. Wickramasinghe, M.J. Semmens, E.L. Cussler, Mass transfer in various hollow fiber geometries, J. Membr. Sci. 69 (1992) 235– 250. [12] 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. [13] R.O. Crowder, E.L. Cussler, Mass transfer in hollow-fiber modules with non-uniform hollow fibers, J. Membr. Sci. 134 (1997) 235– 244. [14] M.C. Yang, E.L. Cussler, Designing hollow fiber contactors, AIChE J. 32 (1986) 1910–1916. [15] R. Prasad, K.K. Sirkar, Dispersion-free extraction with microporous hollow fibre modules, AIChE J. 34 (1988) 177–188. [16] D.Y.C. Chan, B.D. Hughes, L. Paterson, Fluid capacity distribution of random porous media, Transport Porous Media 3 (1988) 81–94. [17] J.M. Zheng, Y.Y. Xu, Z.K. Xu, Flow distribution in a randomly packed hollow fiber membrane module, J. Membr. Sci. 211 (2003) 263– 269. [18] A.R. Goerke, J. Leung, S.R. Wickramasinghe, Mass and momentum transfer in blood oxygenators, Chem. Eng. Sci. 57 (2002) 2035–2046.
W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101 [19] E.M. Sparrow, A.L. Loeffler, Longitudinal laminar flow between cylinders arranged in regular array, AIChE J. 5 (1959) 325–330. [20] R.K. Shah, A.L. London, Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data, Academic Press, New York, 1978 (Chapters 15 and 17). [21] J.A. L´evˆeque, Les lois de la tranmission de chaleur par convection, Ann. Mines 13 (1928), pp. 201, 305, 381.
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[22] O. Miyatake, H. Iwashita, Laminar-flow heat transfer to a fluid flowing axially between cylinders with a uniform surface temperature, Int. J. Heat Mass Transfer 33 (1990) 417–425. [23] H. Kreulen, C.A. Smolders, G.F. Versteeg, B.P.M. van Swaaij, Microporous hollow fibre membrane modules as gas–liquid contactors. Part I: physical mass transfer process, J. Membr. Sci. 78 (1993) 197–216.
Theoretical estimation of shell-side mass transfer coefficient in randomly packed hollow fiber modules with polydisperse hollow fiber outer radii Weiping Ding a , Dayong Gao a,b,c,∗ , Zhen Wang c , Liqun He c a
Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA c Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei, Anhui 230027, PR China b
Received 10 March 2006; received in revised form 29 June 2006; accepted 5 July 2006 Available online 14 July 2006
Abstract The expression of average shell-side mass transfer coefficient (MTC) in hollow fiber modules is theoretically deduced in this paper. This expression considers the effects of both the polydispersity of hollow fiber outer radii and the randomicity of hollow fiber distribution on the shell-side mass transfer performance for the first time. In the expression, Gaussian function is used to model the polydisperse hollow fiber outer radii, Voronoi tessellation method is used to model the random hollow fiber distribution, and the two distributions are assumed to be independent. Then, the effect of polydisperse hollow fiber outer radii on average shell-side MTC is studied in detail. © 2006 Elsevier B.V. All rights reserved. Keywords: Mass transfer coefficient; Random distribution; Polydispersity; Packing density; Hollow fiber
1. Introduction Shell-side mass transfer of hollow fiber modules with shell and tube configuration has been studied widely in recent years. The shell-side mass transfer performance is affected by many factors [1,2]. Two of the main factors are the randomicity of hollow fiber distribution and the polydispersity of hollow fiber outer radii. Almost all previous works about the effect of hollow fiber distribution on shell-side mass transfer have based on the assumption of identical hollow fiber outer radii. Under such assumption, commonly there are two methods to deal with the distribution of hollow fibers. One is that the hollow fiber bundle is modeled as an infinite, spatially periodic medium, i.e. the hollow fibers are assumed to be arranged in regular arrays (e.g. square or triangular) [2–6] or random in a unit cell [5,6]. The other is that Voronoi tessellation method is used to model the random distribution of hollow fibers [2,7–10]. In practice, neither inner radii nor outer radii of hollow fibers are possible to be identical because of the imperfect manufactur-
∗
Corresponding author. Tel.: +1 206 543 1411; fax: +1 206 685 8047. E-mail address: [email protected] (D. Gao).
0376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2006.07.012
ing process. So, the polydisperse inner and outer radii absolutely will affect lumen-side and shell-side mass transfer performance, respectively. However, in the literature, only the effect of polydisperse inner radii was investigated in detail [10–13], the effect of polydisperse outer radii seldom was focused on. So, in this paper, the effect of polydisperse fiber radii (hollow fiber outer radii) on shell-side mass transfer is attempted to investigate theoretically in randomly packed hollow fiber modules, i.e. both the randomicity of hollow fiber distribution and the polydispersity of fiber radii are taken into account together. A theoretical average shell-side MTC expression is deduced and then the effect of polydisperse fiber radii is studied emphatically. In the literature, a system of removing oxygen from water often was chosen to study the shell-side mass transfer performance because of its availability and simplicity [2,4,11,14,15]. Here, we make the same choice: water containing oxygen flows outside of hollow fibers, and nitrogen flows inside of hollow fibers counter-currently. 2. Theory According to the literature [2,8–10], the following assumptions are used in this paper: (1) hollow fibers are rigid, and aligned axially but randomly arranged, i.e. the flow distribution
96
W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101
For a certain module containing N hollow fibers, the average polygonal cell cross-section area can be acquired easily: a0 =
πR2M N
(4)
where RM is the module inner radius. Then, the packing density can be defined as φ=
Fig. 1. Schematic representation of the module cross-section subdivision with Voronoi tessellation method.
is consistent along the module length; (2) the effect of module wall on shell-side hydraulics is ignored; (3) the fluid flow profile is fully developed and laminar; (4) the effect of shell-side inlet and outlet is ignored. 2.1. Distribution of hollow fibers Voronoi tessellation is a mathematical method to describe the subdivision of space between randomly packed objects by drawing straight boundaries equidistant between neighboring objects, forming polygonal cells. It provides a good way to calculate the geometric characteristics of random spacing that can be used to calculate overall properties [7,16]. In this paper, Voronoi tessellation method is used to evaluate the shell-side flow distribution in a randomly packed hollow fiber module. Considering N hollow fibers are randomly placed in the module. Using Voronoi tessellation method, the cross-section area is then subdivided into polygonal cells, and each polygonal cell is associated with one hollow fiber (Fig. 1). The probability density distribution function of polygonal cell area can be written as below according to the literatures [2,7,11,17]: f (ϕ) =
ss
ϕs−1
ϕs (s − 1)!
ϕ = a − af
e−(sϕ)/ϕ
(1) (2)
where a is the polygonal cell cross-section area (including the fiber), af the hollow fiber cross-section area, s the number of nearest neighbor fibers, and the sign “” denotes statistical average in mathematics. At lower packing density, s = 4 appears to fit the simulation better; at higher packing density, the calculated distribution agrees with s = 6 better, but the effect of s on results is not so evident [2,7]. So in this paper only s = 6 is considered for the calculation of average shell-side MTC. Assuming that the hollow fiber distribution is independent of the hollow fiber outer radius distribution, the following equation can be obtained: ϕ = a0 − af
(3)
where a0 is the average polygonal cell cross-section area (including the fiber).
af Nr 2 = a0 R2M
(5)
2.2. Polydispersity of fiber radii According to the literature [11,12,18], the distribution of fiber radii g(r) is assumed to be a Gaussian function: e−(r−R0 ) /(2R0 ε0 ) √ 2πR0 ε0 2
g(r) =
2 2
(6)
where R0 is the average outer radius of hollow fibers, (R0 ε0 )2 the variance of this distribution, and ε0 denotes the polydispersity of fiber radii. Obviously, ∞ R0 = r = rg(r) dr (7) 0
Assuming the effective length of hollow fibers is identical and equals to L, the average hollow fiber outer surface area A is given by ∞ A = 2πL rg(r) dr = 2πrL (8) 0
2.3. Flow outside of hollow fibers in polygonal cells Using the assumptions that the entrance and exit losses are small and that flow areas are parallel and of the same length L [2,7,19,20], the shell-side pressure drop P for laminar flow through each of the separate flow area is the same as the whole bundle and is calculated by P =
4ρv2 L dh
(9)
where ρ is the density of water, dh = 2ϕ/(πr) the hydraulic diameter of each polygonal cell (4 × flow area/hollow fiber circumstance), and v is the mean flow velocity in each polygonal cell. Rearrangement of Eq. (9) gives the following equation: Pϕ v= (10) 2πrρL Then, the average shell-side flow rate Q is given by ∞ ∞ Q = ϕvf (ϕ)g(r) dϕ dr 0
0
(11)
W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101 Table 1 Parameters used in calculation
2.4. Local mass transfer outside of hollow fibers in polygonal cells In the literature, the available universal local MTC expression outside of one hollow fiber in each polygonal cell cannot be found. Wu and Chen used L´evˆeque equation to calculate this kind of local MTC, but L´evˆeque equation is valid only for high Gz number [2,21]. Miyatake and Iwashita theoretically studied the heat transfer coefficient expression of shell and tube heat exchangers with tubes arrayed regularly, laminar flow outside of tubes and constant temperature on the surface of tubes [22]. In this paper, the local MTC expression is acquired on the analogy of their heat transfer coefficient expression under the condition of the tubes arranged in regular triangular array:
k=
D 9 f 2 + g2 Gz2/3 2r 4
1/2 (12)
where D is the oxygen diffusion coefficient in water, Gz = ϕv/(LD) (Gz: 1–106 ) [22], the expressions of f and g are the following: f =
8.92(1 + 2.82γ) 1 + 6.86γ 5/3
g=
Nomenclature
Value
Module inner radius (m) Effective hollow fiber length (m) Density of water (300 K) (kg/m3 ) Oxygen diffusive coefficient (300 K) (m2 /s)
RM L ρ D
0.02 0.20 9.96 × 102 2.29 × 10−9
Combining Eqs. (17) and (18), we obtain ∞ ∞ −2krL/(ϕv) e ϕvf (ϕ)g(r) dϕ dr c = 0 0 ∞ ∞ c0 0 0 ϕvf (ϕ)g(r) dϕ dr
(19)
According to the above analysis, the average shell-side MTC can be described by ∞∞ ϕvf (ϕ)g(r) dϕ dr k = 0 0 ∞ 2L 0 rg(r) dr ∞∞ 0 ϕvf (ϕ)g(r) dϕ dr (20) × ln ∞ ∞0 −2krL/(ϕv) ϕvf (ϕ)g(r) dϕ dr 0 0 e
2.6. Parameters setting
2.34(1 + 24γ) 1/3
(14)
(15)
In this paper, the referred parameters are listed in Table 1. Mathematic 5.0 is used to calculate all expressions. 3. Results and discussions 3.1. Average shell-side MTC with the polydispersity close to 0
2.5. Average shell-side MTC Mass transfer of oxygen from water in shell-side to nitrogen in lumen side contains three steps: transfer of oxygen in water, diffusion of oxygen through the membrane pores that fully filled with gas, and diffusion of oxygen in nitrogen stream. Assuming that the phase interface between gas and liquid is on the hollow fiber outer surface and the oxygen concentration on the interface is 0. In this case, the mass transfer efficiency in the liquid phase dominates the overall mass transfer performance, i.e. the overall MTC could be approximated by the MTC in shell-side. According to the literature, the average shell-side MTC could be described by the following expression [2,12,23]: Q c0 ln A c
Parameter
(13)
(1 + 36.5γ 5/4 )(ϕ/r 2 ) √ 3(ϕ/r 2 + ) γ= −1 6
k =
97
One might try to follow the step similar to this paper and directly deduce the average shell-side MTC expression based on the assumption that all fiber radii are identical, but this is not necessary at all. The expression in this paper includes such ideal situation, i.e. the polydispersity is close to 0.
(16)
where c0 is the oxygen concentration at inlet, and c is the average oxygen concentration at outlet. The average oxygen concentration at outlet is given by ∞∞ cϕvf (ϕ)g(r) dϕ dr c = 0 ∞ 0 ∞ (17) 0 0 ϕvf (ϕ)g(r) dϕ dr The concentration in each polygonal cell area is given by [11] c = c0 e−2krL/(ϕv)
(18)
Fig. 2. Effects of hollow fiber outer radius and average shell-side flow velocity on average shell-side MTC.
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Fig. 2 shows the effects of hollow fiber outer radius R0 and average shell-side flow velocity V on average shell-side MTC when the polydispersity is close to 0. Under the given packing density, the decrease of hollow fiber outer radius consequentially leads to the increase of hollow fiber number, which causes the better hollow fiber packing/distribution, and then enhances the average shell-side MTC. (When the number of hollow fibers increases, the distribution of hollow fibers runs to regular array. The average shell-side MTC with regular fiber array is bigger than that with random fiber array [2,5].) Certainly, the increase of hollow fiber number also results in the more dead zones (regions of low velocity where hollow fibers are close or touch), which reduce the average shell-side MTC [2,4,5], but this reduction is not significant comparing to the enhancement at lower packing density. When the packing density increases, the effect of dead zones becomes more and more prominent and disadvantageous. So with increasing the packing density, the average shell-side MTC curve with the smaller hollow fiber outer radius firstly has decreasing tendency, i.e. the optimal packing density corresponding to the maximal average shell-side MTC is smaller. Practically, another important method to improve the average shell-side MTC is to increase the average shell-side flow velocity. The higher the average flow velocity, the bigger the average MTC. Moreover, Fig. 2 also clearly shows the fact that with increasing the average flow velocity, the optimal packing density increases, which has not been shown in the literature. From Fig. 2, one also could deduce that in the available range of packing density the average shell-side MTC will increase monotonously with increasing the packing density when the average shell-side flow velocity is very high (high Gz number), which is consistent with the theoretical result in the literature qualitatively [5]. Reliable experimental data for the validation of this study about the effect of polydisperse fiber radii on average shell-side MTC are not found in the literature. However, when the polydispersity is close to 0, the average shell-side MTC has been investigated widely in previous work [2,4,5,15]. Fig. 3 shows the comparison of average shell-side MTC between the litera-
ture and this paper under such ideal condition. In the literature, the conclusion about the effect of packing density on the average shell-side MTC is inconsistent. One representative standpoint is that with increasing the packing density, the average shellside MTC first increases and then decreases (low Gz number) [4], while the other is contrary [2]. These discrepancies within the literature itself could be mainly caused by their different experimental conditions, such as the hollow fiber number, the hollow fiber length, the hollow fiber radius, and the inlet and outlet design. The simulative result in this paper supports the first standpoint, but the departure between the experimental results of the literature [4] and the theoretical values of this paper is still somewhat large. One important reason is that the assumed axial laminar flow pattern in this study somewhat deviates from the actual flow. Firstly, the current shell inlet and outlet design is not perfect, which causes cross/transverse flows at the shell entry/exit regions. Secondly, the hollow fibers maybe are not strictly parallel, and axial flows could terminate or split to produce additional cross-flows. Thirdly, some imperfect designs could produce unexpected secondary flows [5]. Cross-flows or secondary flows could promote mass transfer, especially at lower packing density. However, our model cannot calculate the average shell-side MTC under such flow conditions. So, our theoretical values are far less than the experimental values. Another possible reason which causes the smaller theoretical values of this paper is that the local shell-side MTC expression used in this study is not ideal for the actual mass transfer outside of hollow fibers in polygonal cells. After all, this expression is under the condition of regular triangular array. So, the universal local shell-side mass transfer correlation needs to be established. Additionally, this model also fails to accurately represent dead zones and channeling flows (flows mainly pass through larger channels between the fiber bundle and the external case or within the fiber bundle) in experiments, which also causes the theoretical values of this paper imperfect. 3.2. Average shell-side MTC with the polydisperse fiber radii When the polydispersity is not close to 0, can the effect of polydisperse fiber radii on average shell-side MTC be ignored? Wickramasinghe et al. in their work incidentally gave a theoretical expression to estimate the average shell-side MTC under such general situation [11]. Their result implied that the polydisperse effect cannot be neglected unadvisedly, but their expression does not refer to the effect of random fiber packing/distribution. In order to conveniently discuss the effect of polydisperse fiber radii, the relative departure η = 1 − k/ lim k is defined in this paper, which shows the difference ε0 →0
Fig. 3. Comparison of average shell-side MTC between the literature and this paper with the polydispersity close to 0.
between the real value with the polydisperse fiber radii and the ideal one with the identical fiber radii. Fig. 4 shows the comparison of relative departure between the literature and this paper. Qualitatively, the two estimated results are consistent from the point of view of variation tendency. However, the average shellside MTC calculated by the literature is far greater than the one shown by this paper. Although the polydispersity of fiber radii
W. Ding et al. / Journal of Membrane Science 284 (2006) 95–101
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absolutely results in the decrease of average shell-side MTC, we think, the expression in the literature to some extent magnifies the polydisperse effect. Why does the polydispersity affect the shell-side flow field and further affect the shell-side mass transfer? Polydisperse fiber radii easily cause the emergence of dead zones and channeling flows. For example, if in a polygonal cell the outer radius of the hollow fiber increases, the probability of the hollow fiber to touch other hollow fibers is higher, dead zones could be shaped easily, and the flow could bypass the hollow fiber. Such fact is equal to the sharp decrease of local flow velocity. If the outer radius of the hollow fiber decreases, although dead zones could not easily occur, channeling flows potentially would be shaped (dead zones and channeling flows are relative). Then, how does the polydispersity affect the average shellside MTC under different conditions, such as different average shell-side flow velocities, different shell-side pressure drops and different total shell-side flow rates? Fig. 5 shows the effect of average shell-side flow velocity V on relative departure. For the given packing density and polydis-
persity, the increasing average shell-side flow velocity not only increases the average shell-side MTC, but also to some extent improves the flow conditions of the hollow fibers around which dead zones and channeling flows are shaped, so the relative departure decreases when the average shell-side flow velocity increases. For the given average shell-side flow velocity and packing density, the increasing polydispersity causes the effects of dead zones and channeling flows increasingly more serious and then results in the decreasing average MTC, so the relative departure increases with increasing the polydispersity. For the given average shell-side flow velocity and polydispersity, when the packing density increases, dead zones become more and more, so the relative departure increases with increasing the packing density. In Fig. 5, when the average shell-side flow velocity is 0.05 m/s, the polydispersity is 0.15, and the packing density is about 75%, the decrease of average shell-side MTC actually reaches above 10%, so the effect of polydisperse fiber radii on shell-side mass transfer cannot be neglected optionally. Additionally, although the polydispersity affects and decreases the value of average shell-side MTC, it cannot change the variation tendency of average shell-side MTC curve. The effect of shell-side pressure drop P on relative departure almost is the same as the one of average shell-side flow velocity. According to the analysis mentioned in above paragraphs, one could easily understand the likeness. Here, the most important thing needed to be emphasized is the difference between the two situations about the effect of packing density on shell-side flow field. When the shell-side pressure drop is given, the increasing packing density increases the probability of dead zones, and at the same time causes the quickly decreasing average shell-side flow velocity, which also aggravates the polydisperse effect (the figure under this situation is omitted because in variation tendency it almost is the same as Fig. 5). Fig. 6 shows the effect of total shell-side flow rate Q0 on relative departure. The comparison between Figs. 6 and 5 shows that the variation tendency of relative departure versus packing density under the given polydispersity and total shell-side flow rate, i.e. the relative departure firstly increases and then decreases with increasing the packing density, remarkably differs from the
Fig. 5. Effect of average shell-side flow velocity on relative departure.
Fig. 6. Effect of total shell-side flow rate on relative departure.
Fig. 4. Comparison of relative departure between the literature and this paper with the polydisperse fiber radii.
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one under the given polydispersity and average shell-side flow velocity. In fact, when the total shell-side flow rate is given, because of conservation of mass, the increasing packing density consequentially leads to the increase of average shell-side flow velocity. Whereas, the higher average shell-side flow velocity is advantageous to alleviate the polydisperse effect. So, with increasing the packing density, the advantage becomes more and more predominant and the relative departure curve appears the decreasing tendency. 4. Conclusions In this paper, the expression of average shell-side MTC in hollow fiber modules with the polydisperse fiber radii is theoretically deduced. To the best of our knowledge, for the first time the effects of both the randomicity of hollow fiber distribution and the polydispersity of hollow fiber outer radii on shell-side mass transfer are considered together. Then, under the conditions of different average shell-side flow velocities, different shell-side pressure drops and different total shell-side flow rates, the effect of polydisperse fiber radii on average shell-side MTC is studied emphatically. Our results show that the polydispersity of hollow fiber outer radii results in the decrease of average shell-side MTC. For the given average flow velocity/pressure drop/total flow rate and packing density, the relative departure between the real value and the ideal one increases with increasing the polydispersity. For the given packing density and polydispersity, the higher the average flow velocity/pressure drop/total flow rate, the smaller the relative departure. For the given average flow velocity/pressure drop and polydispersity, the relative departure increases with increasing the packing density; however, for the given total flow rate and polydispersity, the relative departure firstly increases and then decreases with increasing the packing density.
Nomenclature a af a0 A c c0 dh D k L N P Q Q0 r RM
polygonal cell cross-section area (including the fiber) (m2 ) hollow fiber cross-section area (m2 ) average polygonal cell cross-section area (including the fiber) (m2 ) hollow fiber outer surface area (m2 ) oxygen concentration at outlet (ppm) oxygen concentration at inlet (ppm) hydraulic diameter of each polygonal cell (m) oxygen diffusion coefficient in water (m2 /s) shell-side mass transfer coefficient (m/s) effective hollow fiber length (m) number of hollow fibers pressure (Pa) shell-side flow rate (m3 /s) total shell-side flow rate (m3 /s) variable module inner radius (m)
R0 s v V
average outer radius of hollow fibers (m) number of nearest neighbor fibers mean flow velocity in each polygonal cell (m/s) average shell-side flow velocity (m/s)
Greek symbols ε0 polydispersity of fiber radii η relative departure ρ density of water (kg/m3 ) φ packing density ϕ variable
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