Residence Time Distribution Analysis of a Hollow-Fiber Contactor

Article pubs.acs.org/IECR

Residence Time Distribution Analysis of a Hollow-Fiber Contactor for Membrane Gas Absorption and Vibration-Induced Mass Transfer Intensification Weidong Zhang,* Zisu Hao, Jiang Li, Junteng Liu, Zihao Wang, and Zhongqi Ren* Beijing Key Laboratory of Membrane Science and Technology, State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, Beijing 100029, PR China ABSTRACT: The relationship between the flow conditions and the mass transfer in a hollow-fiber membrane contactor was quantitatively correlated by coupling a residence time distribution (RTD) method with a multicontinuous stirred tank reactor model (mCSTR). A vibration technique was applied to manipulate and intensify the mass transfer performance of the hollowfiber contactor (HFC). Absorption behaviors of CO2 in deionized water were investigated as a model system. The effects of liquid feeding velocity, vibration direction and frequency, operating modes, and packing density on the flow conditions and mass transfer performance were investigated and quantitatively examined by the RTD analysis. The results indicated that the flow conditions as well as the mass transfer performance can be greatly improved at a low liquid-phase velocity or a high packing density. Similarly, the mass transfer intensification was more remarkable, when the absorbent flowed through the shell side of HFC, or the vibration and flow directions were vertical. The mass transfer coefficients obtained from the RTD-mCSTR model showed good agreement with those obtained from the experiments. It appeared that the vibration technique was a powerful method to improve the flow conditions and greatly enhanced the mass transfer performance.

1. INTRODUCTION The hollow-fiber contactor (HFC) has been proven to be a promising separation unit for chemical processing, environmental engineering, and bioengineering applications.1−6 Among other significant applications, CO2 capture with HFC has attracted tremendous attention in recent decades.7−11 Owing to the large mass transfer areas, the mass transfer efficiency of HFC is approximately 1 order of magnitude greater than that of conventional separation units, such as distillation and extraction column.12,13 To further improve the mass transfer performance, various types of HFC were invented.14−18 Among them, regular packing HFCs showed efficient mass transfer performance,17,18 which were attributed to the improvement of the flow conditions.19−23 To optimize the structure of HFC and screen suitable process intensification techniques, it is necessary to better understand the relationship between flow conditions and mass transfer process. A possible way is to couple a proper mass transfer model with a method by which the flow conditions can be quantitatively illustrated. The residence time distribution (RTD) analysis was used to evaluate the flow conditions on the shell side of HFC.21,22,24,25 As an early work for RTD analysis for HFC, Noda et al.24 investigated the effect of flow maldistribution on hollow fiber dialysis. It was found that the performance of real hollow fiber dialyzers deviated from the ideal flow condition, which was attributed to the muldistribution of solution flow caused by the random packing structure. Lemanski and Lipscomb25 analyzed the fluid flow within the shell of a hollow fiber bundle on the basis of the volume averaging of the relevant conservation of mass and momentum equations. They predicted residence time distribution and performance accounting for fiber packing fraction and module aspect ratio. Wang et al.21 investigated the © 2014 American Chemical Society

residence time distribution and performance accounting for HFCs. They found that the random packing of fibers leads to the nonideal flow and significantly deteriorate the mass transfer performance. Recently, Ren et al.26 investigated the RTD of a hollow fiber renewal liquid membrane technique and proposed a mass transfer correlation in which the effect of flow condition, summarized as the Peclet number, was considered. However, for most of the previous studies, RTD curves were only used to qualitatively show the flow conditions on the shell side of HFC, whereas the mass transfer performance was estimated by using empirical correlations.21,26 It appears to us that a quantitative relationship between the flow conditions and the mass transfer is important for the design of HFC as well as evaluating the performance of process-intensification methods. In our recent research, Zhang et al. applied the third-phase particles (such as solid particles27 and bubbles28) on the shell side of the HFC to improve the mass transfer performance. The results revealed that the improvement of the flow conditions was responsible for the mass transfer intensification.27 However, the quantitative relationship between the flow conditions and the mass transfer process was not evaluated. Vibration is an effective and energy-saving method to change the flow conditions in the target reactor/contactor. It has been widely used to improve the mass transfer performance in various biological and chemical engineering processes.29−34 Although vibration may not be favorable for the industrial application, it may serve as an effective method for changing the Received: Revised: Accepted: Published: 8640

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Figure 1. Multicontinuous stirred tank reactor model (mCSTR) of a hollow fiber membrane contactor.

flow conditions in a HFC, which is expected to impact the mass transfer performance of the absorption unit. Krantz et al.34 explored a method using axial vibrations of a membrane tube bundle to increase oxygen transfer to the intralumenal liquid flow. According to the experiments, the axial membrane vibrations increased the mass transfer coefficient by at least a factor of 2.65. In this study, we established the relationship between the flow conditions and the mass transfer on the shell side of a HFC by using the RTD-multicontinuous stirred tank reactor (RTD-mCSTR) model. As a process intensification method, vibration was applied in a bench-scale gas absorption system to evaluate the relationship among the flow condition, RTD curves, and mass transfer. In this system, CO2-deionzied water was used, and the effects of the frequency and direction of vibration, liquid velocity, and fiber packing density on the mass transfer performance were investigated. RTD analysis was conducted to evaluate the effects of flow conditions on the mass transfer behavior. Based on the parameters obtained in the RTD analysis, the theoretical mass transfer coefficients, which were in good agreement with the experimental mass transfer coefficients, were obtained by using the RTD-mCSTR model.

where θ is the dimensionless time, defined by eq 6. θ=

σθ 2 =

σ2 tm 2

N=

1 σθ 2

tN =

tm N

∞

(1)

kiL = 2

D πtc

(10)

where tc is the exposure time of the liquid and is equal to tN. The overall mass transfer flux based on the RTD analysis in the HFC can then be determined by using eq 11. N

∞

Q RTD =

∞

(2) 2

∫0 (t − tm) C(t ) dt ∞

∫0

C(t ) dt

=

∞ 2 t C(t ) 0 ∞

∫

∫0

dt

C(t ) dt

− tm 2 (3)

Q exp = A m kLΔC L

(12)

where Qexp is the absorption rate of the CO2 based on experimental results, Am is the mass transfer area, ΔCL and ΔCLi are the overall and the ith mass transfer driving forces, respectively.

(4)

∞

E(θ ) dθ = 1

(11)

where QRTD is the absorption rate of the CO2 based on the RTD analysis, and Ami is the mass transfer area of the ith CSTR cell. For the gas absorption experiments, the overall mass transfer flux is

Then, the dimensionless RTD, E(θ), is expressed by eqs 4 and 5: E(θ ) = tmE(t )

∑ A mikiLΔCiL i=1

∫0 C(t ) dt ∞

∫0

(9)

Assuming that the liquid-side mass transfer in each CSTR is governed by the penetration theory,37 the mass transfer coefficient of the ith CSTR is expressed by eq 10.

∫0 tC(t ) dt

σ2 =

(8)

The mean residence time in each CSTR cell, tN, is expressed by eq 9

where E(t) is a RTD function and C(t) is the outlet tracer concentration of the liquid phase at time t. Based on the RTD curves obtained, the first-order moment and the second-order moment of the RTD function, i.e., tm, the mean residence time, and, σ2, the variance about the mean residence time, can be determined by using eqs 2 and 3.35 tm =

(7)

2.2. Multicontinuous Stirred Tank Reactor (mCSTR) Model. The mass transfer behavior of the gas absorption process in a HFC can be evaluated by using a mCSTR model.36 A hollow fiber is divided into N ideal mCSTR cells, as shown in Figure 1. CL and CG are the concentrations of liquid and gas phases, respectively. In this study, pure CO2 and deionized water were used and flowed respectively through the lumen and shell sides of the HFC counter-currently. Based on the RTD analysis, the total number of the CSTR cells, N, can be derived by using eq 7

C(t )

∫0 C(t ) dt

(6)

Further, the dimensionless variance is expressed by eq 7,

2. THEORY 2.1. Introduction of RTD Analysis. In this study, RTD was used to derive the parameters of the model for the HFC. Pulse experiments were conducted with RTD curves obtained through the following eq 1. E (t ) =

t tm

(5) 8641

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The log-mean concentration difference, ΔCL, is expressed as eq 13 ΔC L =

L * − C inL) − (C in* − Cout (Cout )

ln

* − C inL Cout L C in* − Cout

C* L C * −Cout

(14)

(Ci*− 1 − CiL) − (Ci* − CiL− 1) ln

Ci*− 1 − CiL Ci* − CiL‐ 1

=

CiL− 1 − CiL ln

(1 < i < N )

C * −CiL C * −CiL− 1

(15)

where i = 1 and i = N denote the inlet and outlet, respectively. Combining eqs 4, 6, and 7, the mass transfer coefficient based on the liquid phase can be expressed as eq 16. N

∑i = 1 kiL·

N

kL =

∑i = 1 A mi kiL·ΔCiL A m ·ΔC L

=

CiL‐ 1 − CiL ln

C* L C * −Cout

(16)

In fact, because the feed gas was pure CO2, the fractional mass transfer resistances for the gas and membrane phases were negligible.27,38 Based on the resistance-in-series model,39 the overall mass transfer coefficient and the mass transfer coefficient for the liquid phase should approximately be the same. 2.3. Determination of the Overall Mass Transfer Driving Force, ΔCLi . Because a pure CO2-deionized water system, i.e., a slow mass transfer system/weak chemical reaction condition,38 was applied, certain specific assumptions were made for the analysis in the current work: (1) The system was assumed under the steady-state condition. (2) Fiber bundle was treated as one hollow fiber, which has the equivalent mass transfer area of radius rd, to simplify the calculation. This assumption is accurate and reasonable for this model, because the effect of nonideal flow condition on the shell side of the hollow fiber module has been considered in eq 10. (3) The axial diffusion was neglected. (4) The liquid flow velocity did not vary axially and radially. Based on these assumptions, the mass balance differential equation for a hollow fiber membrane gas absorption system can be described by eq 17. uz

⎡ 1 ∂ ⎛ ∂C ⎞⎤ ∂C ⎜r ⎟⎥ = D⎢ ⎣ r ∂r ⎝ ∂r ⎠⎦ ∂z

L C = Cout

(18)

Table 1. Characteristics of Modules

C * −CiL C * −CiL‐ 1

L Cout

ln

∂C =0 ∂x

3. EXPERIMENTAL SECTION 3.1. Chemicals and Apparatus. Pure CO2 (Beijing Praxair Industrial Gas Co., Ltd., China) of >99.9% purity was used as the feed gas, whereas the absorbent was deionized water. The HFCs used in this study were made of hollow cylindrical glass with an inner diameter of 10 mm. Polypropylene hollow fibers with an inner diameter of 350 μm, a membrane thickness of 50 μm, and an outer diameter of 450 μm were supplied by Hangzhou Qiushi Membrane Technology Co., Ltd. (Hangzhou, China). The detailed information on the modules is listed in Table 1.

For the ith CSTR, eq 13 can be expressed as ΔCiL =

r = rD

where rD is the inner radius of the shell side of the module. The set of equations along with the boundary conditions can be solved using the finite volume method,38 thus obtaining the concentration profile. ΔCLi can be calculated by using the concentration profile of the ith CSTR cell. Then, the mass transfer coefficient based on the liquid phase can be estimated by substituting the solute concentration of the inlet and outlet of the mCSTR cells in eq 16.

L Cout

ln

C = C*

z=l

(13)

where Cin* and Cout * are the equilibrium concentrations at the inlet and outlet, respectively. Because a pure gas was used in this study, C*in = C*out = C*. Similarly, because the absorbent was fresh, CLin = 0; then, eq 13 becomes ΔC L =

r = rd

no.

fiber length, l (m)

no. of fibers, n (−)

mass transfer area, Am (cm2)

packing fraction, φ (%)

1 2 3 4 5

0.30 0.30 0.30 0.30 0.30

20 50 100 150 180

0.008 48 0.021 21 0.042 41 0.063 62 0.076 34

4 10 20 30 36

3.2. Experimental Setup and Operation. A schematic diagram of the membrane absorption experimental setup is shown in Figure 2. The membrane module was fixed on a shaking table with a shaking diameter of 2.0 cm, and the maximum frequency was 300 rpm. Two operating modes with vibration direction controlled to be vertical or parallel to the liquid flow direction, were achieved by changing the position of the module on the shaking table, as shown in Figure 3. The gas was introduced from the gas cylinder continuously through the lumen side of the HFC. The inlet and outlet gas volume flow rates were measured by two soap-bubble flowmeters. The gas pressure was maintained constant. A peristaltic pump (Masterflex BT00-300T, Baoding Longer Precision Pump Co., Ltd. China) was used to pump the fresh liquid absorbent, through a liquid-flow meter, into the shell side of the HFC counter-currently. A U-pressure gauge was used to indicate the inlet and outlet pressures of the gas and liquid phases. To prevent the bubble formation, the liquid-phase pressure was slightly larger than that at the gas phase; however, the trans-membrane pressure drop was 120 rpm) was applied. According to the basic theory of RTD, the ideal plug flow condition is favorable for the mass transfer. Therefore, the mass 8646

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through the shell side of the HFC by applying a proper intensification method, and then the related hydrodynamic results can be used as a guidance to improve the mass transfer performance. As described in section 4.2, this problem can be solved by applying the vibration. In Figure 8, the application of vibration improved the flow conditions, and the enhancement factor significantly increased with increasing vibration frequency. When the vibration frequency was 300 rpm, the mass transfer coefficient was 6.2 × 10−6 m/s, which was nearly 2 times larger than that without applying vibration. This result indicated that the vibration-induced mass transfer intensification was favorable for most of the processes in the HFC. In the above analysis, the effect of vibration on the gas phase was not considered. In this study, pure CO2 was used in the experiments; therefore, the mass transfer resistance of the gas phase can be neglected. Thus, the degree of nonideal flow of the gas phase was insignificant. Although the application of a mixed gas in most of the membrane gas absorption processes may lead to a considerable mass transfer resistance, the enhancement caused by vibration for the gas phase is still negligible because the density of gas is much less than that of liquid. 4.4. Effect of the Direction of Vibration. The fluid flow on the shell side of the HFC was approximately unidirectional; therefore, the vibration direction, vertical (mode 1) or parallel (mode 2) to the flow direction, may have significant effect on the flow conditions. As shown in Figure 9, using RTD analysis, this effect was quantitatively determined at the vibration frequencies 200, 240, and 300 rpm. The RTD curve of mode 2 was very similar to that when no vibration was applied, even when the vibration frequency was increased. When the vibration direction was parallel to the liquid flow direction, although the mixing of fluid between the liquid boundary layer and liquid bulk had a positive effect on forming the plug flow condition, the backmixing of the fluid flow on the shell side of the HFC became severe, which played a negative effect. Consequently, the increase in the vibration frequency enhanced the positive and negative effects simultaneously; therefore, they offset each other. On the other hand, when the vibration direction was vertical to the flow direction (mode 1), the RTD curves were all closer to the dimensionless time of 1 than those of mode 2. The higher the vibration frequency, the closer was the RTD curve and the dimensionless time of 1. The improvement in the flow conditions was attributed to the mixing at the radial direction and convection between the liquid boundary layer and the liquid bulk. Moreover, the negative effect of the back-mixing of mode 2 was not significant, thus reducing the dead zone and channeling on the shell side of the HFC because of the vertical direction of the vibration and the fluid flow. With increasing vibration frequency, the flow conditions became close to the ideal plug flow at an increasing rate. When the vibration frequency was 300 rpm, the tailing of the RTD curve was decreased, and the θp was very close to the dimensionless time of 1. A series of mass transfer experiments were conducted to verify the efficiency of the RTD analysis, as shown in Figure 10. The mass transfer performance was insignificant at the vibration frequency 120 rpm for both the intensification modes, because a higher vibration frequency provided a larger energy

Figure 12. Effect of the fiber packing density on (a) mass transfer coefficient and (b) enhancement factor with varying vibration frequency (uL = 0.011 m·s−1, vibration and liquid flow directions vertical).

When the liquid phase flowed through the lumen side, owing to its regular structure, the flow conditions were similar to those of the ideal plug flow. The average mass transfer coefficient in this mode (9.0 × 10−6 m/s) was almost 2 times larger than that in another mode (liquid phase flowed through the shell side). In addition, the mass transfer coefficient did not change with varying vibration frequency, indicating that the enhancement factor was almost 1. This phenomenon demonstrated that the effect of the vibration was not significant when the liquid phase flowed through the lumen side because the ideal plug flow had fully developed in the lumen side of the capillary hollow fiber even when the vibration was not applied. The liquid phase flowing through the lumen side of HFC seems to be a good and convenient method at a glance; however, it cannot be considered as the optimal selection for industrial applications mainly because of certain intrinsic disadvantages. The pumping of the liquid phase into the hollow fibers consumed more energy and resulted in a large pressure drop between the inlet and outlet. When the pressure difference between the lumen and shell sides became higher than the breakthrough pressure, i.e., a membrane with a large pore size or a higher lumen side velocity, the mass transfer system irreversibly broke down. Therefore, another operating mode, i.e., the liquid phase flows through the shell side of the HFC, was applied by most of the researchers. However, in this mode, because of the irregular structure of the shell side, the flow condition was nonideal, resulting in the dead zone and channeling, as shown in the RTD curves in Figure 4a. Therefore, the mass transfer performance deteriorated. In fact, the operating mode of the liquid-phase flows through the shell side of the HFC widely exists in other membrane separation techniques, particularly in the liquid−liquid processes, such as membrane solvent extraction and hollow fiber liquid membrane techniques. It is worthwhile to analyze the RTD in which the liquid phase flows 8647

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As shown in Figure 12b, mass transfer intensification increased as the packing density increased from 4 to 36% for the vibration frequencies considered. The reason can also be directly obtained from the RTD analysis. Simultaneously, this study indicated that vibration could be an effective approach to decrease the mass transfer resistance by improving the flow conditions and benefits to the application of compact HFC with a high packing density. By analyzing the RTD curves, the dimensionless time and variance can be obtained and used to predict the mass transfer coefficient via a mCSTR model. The calculated and experimental mass transfer coefficients agreed well. Thus, this is an effective and simple approach to establish and correlate the relationship between the flow conditions and the mass transfer behavior. It also demonstrates the close relationship between the mass transfer and the flow conditions, i.e., the nonideal flow condition contributes to the deviation between the ideal and nonideal mass transfer behaviors. This work demonstrated that by using the RTD curves, this deviation could be quantitatively characterized. By applying vibration in the mass transfer system, this deviation can be greatly eliminated, as shown by the enhancement factor.

input per unit time. This result indicated that the mass transfer intensification caused by vibration would not be effective unless a sufficient frequency was applied. When the vibration frequency was >120 rpm, the mass transfer performance of mode 1 was significantly higher than that of mode 2, and the difference in the enhancement factors between modes 1 and 2 increased with increasing vibration frequency. When the vibration frequency was 300 rpm, the enhancement factor of mode 1 was 1.5 times higher than that of mode 2. In summary, the conclusion drawn in the experiments supports the evidence shown by the RTD curves, which can be used to establish the relationship between flow conditions and mass transfer. 4.5. Effect of Packing Density. One of the critical advantages of HFC is that the mass transfer surface area can be freely adjusted by changing the fiber number. To obtain a large surface area per unit volume, a high density of fibers is often required in a HFC, thus increasing the packing density. However, the increase in the packing density would lead to the severe nonideal flow condition on the shell side of the HFC, mainly because of the random packing of fibers.19,42−44 Therefore, the vibration was applied and proved as an effective alternative. In this section, the effect of the packing density with varying vibration frequency was investigated by the RTD analysis, as shown in Figure 11. When the packing density was relatively low and the vibration was not applied, as shown in Figure 11a,b, the RTD curves show a profile similar to that of the ideal plug flow, indicating that a loosely packed module resulted in few dead zones and negligible channeling because of the widely spaced hollow fibers.19,23,44 The increase in the packing density may result in maldistribution of the feed and bypass the liquid flow on the shell side of the HFC.21,44 Therefore, the nonideal flow condition caused by the dead zones and channeling would become severe. This explanation was proved by the RTD curves shown in Figure 11c−e. When no vibration was applied, the deviation between the real RTD curves and those of ideal plug flow gradually increased with increasing packing density. When the vibration method was applied to the mass transfer system with a high packing density, as shown in the RTD curves of Figure 11, the flow condition became close to that of ideal plug flow, particularly when a high vibration frequency was applied. The reason for this enhancement can be attributed to the increased radial mixing and reduction in the dead zones and channeling effect. Therefore, when the vibration frequency was 300 rpm, the θp values for the five packing densities were close to the dimensionless time of 1, indicating that vibration was an effective and flexible method to improve the flow conditions on the shell side of HFC. Under the conditions similar to those of the RTD analysis, the corresponding mass transfer experiments were conducted, and the trend of mass transfer coefficient and enhancement factor are shown in Figure 12. When the vibration was not applied, the mass transfer coefficient decreased with increasing packing density, which can be attributed to the loss of mass transfer area caused by the dead zone and the channeling effect. This explanation agrees with the conclusion obtained from the RTD analysis that the nonideal flow condition was responsible for the unfavorable mass transfer performance. When vibration was applied, the mass transfer coefficient increased with increasing vibration frequency for a certain packing density, indicating that the mass transfer can be enhanced by improving the flow conditions.

5. CONCLUSION RTD analysis was used to establish and correlate the relationship between the flow conditions and the mass transfer in a HFC. The effects of vibration, operating modes, vibration direction, liquid velocity, and fiber packing density on the mass transfer performance were studied. The experimental results indicated that vibration improved the flow conditions in a HFC. When the vibration frequency was 300 rpm, the mass transfer coefficient could be enhanced approximately 2 times larger than when vibration was not applied. The RTD curves quantitatively showed the extent of nonideal flow by the deviation from the dimensionless time of 1. Under low liquid-phase velocity or high packing density conditions, the corresponding RTD curves showed a significant deviation in comparison with that of the ideal plug flow. Therefore, the mass transfer performance under these conditions can be significantly increased by applying the vibration. Similarly, compared to when the absorbent flowed through the lumen side of HFC or the vibration direction was parallel to the flow direction, the enhancement in the mass transfer process was more remarkable when the absorbent flowed through the shell side of HFC or the vibration and flow directions were vertical. The important parameters obtained by the RTD analysis were used to predict the mass transfer coefficients of the HFC via a mCSTR model. The experimental mass transfer coefficients were in good agreement with the predicted mass transfer coefficients. In summary, based on the RTD-mCSTR model, the relationship between flow conditions and mass transfer in the HFC was successfully established and correlated. In the future, HFC with various structures, such as hollow fiber fabric and helically wound HFC, would be investigated to better understand the relationship between flow conditions, RTD, and mass transfer.

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AUTHOR INFORMATION

Corresponding Authors

*W. Zhang: tel, +86-10-64423628; fax, +86-10-6443-6781; email, [email protected]. 8648

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*Z. Ren: tel, +86-10-64423628; fax, +86-10-6443-4925; e-mail, [email protected].

modified PVDF hollow fiber membrane contactor. J. Membr. Sci. 2013, 427, 270−275. (8) Rongwong, W.; Assabumrungrat, S.; Jiraratananon, R. Rate based modeling for CO2 absorption using monoethanolamine solution in a hollow fiber membrane contactor. J. Membr. Sci. 2013, 429, 396−408. (9) Wang, R.; Li, D.; Zhou, C.; Liu, M.; Liang, D. Impact of DEA solutions with and without CO2 loading on porous polypropylene membranes intended for use as contactors. J. Membr. Sci. 2004, 229, 147−157. (10) Qi, Z.; Cussler, E. L. Microporous hollow fibers for gas absorption: I. Mass transfer in the liquid. J. Membr. Sci. 1985, 23, 321− 332. (11) Qi, Z.; Cussler, E. L. Microporous hollow fibers for gas absorption: II. Mass transfer across the membrane. J. Membr. Sci. 1985, 23, 333−345. (12) Li, J.; Chen, B. Review of CO2 absorption using chemical solvents in hollow fiber membrane contactors. Sep. Purif. Technol. 2005, 41, 109−122. (13) deMontigny, D.; Tontiwachwuthikul, P.; Chakma, A. Comparing the absorption performance of packed columns and membrane contactors. Ind. Eng. Chem. Res. 2005, 44, 5726−5732. (14) Wang, K. L.; Cussler, E. L. Baffled membrane modules made with hollow fiber fabric. J. Membr. Sci. 1993, 85, 265−278. (15) Wickramasinghe, S.; Semmens, M. J.; Cussler, E. Mass transfer in various hollow fiber geometries. J. Membr. Sci. 1992, 69, 235−250. (16) Yang, M. C.; Cussler, E. L. Designing hollow-fiber contactors. AIChE J. 1986, 32, 1910−1916. (17) Bhaumik, D.; Majumdar, S.; Sirkar, K. K. Absorption of CO2 in a transverse flow hollow fiber membrane module having a few wraps of the fiber mat. J. Membr. Sci. 1998, 138, 77−82. (18) Cussler, E. L. Hollow fiber contactors. In Membrane Processes in Separation and Purification; Crespo, J. G., BoÈddeker, K. W., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994; pp 375−394. (19) Wu, J.; Chen, V. Shell-side mass transfer performance of randomly packed hollow fiber modules. J. Membr. Sci. 2000, 172, 59− 74. (20) Zheng, J.; Xu, Z.; Li, J.; Wang, S.; Xu, Y. Influence of random arrangement of hollow fiber membranes on shell-side mass transfer performance: a novel model prediction. J. Membr. Sci. 2004, 236, 145− 151. (21) Wang, Y.; Chen, F.; Wang, Y.; Luo, G.; Dai, Y. Effect of random packing on shell-side flow and mass transfer in hollow fiber module described by normal distribution function. J. Membr. Sci. 2003, 216, 81−93. (22) Zhang, W.; Hakim, A.; Gao, J. Fluid hydrodynamics and mass transfer characteristics in a multi-inlet hollow fiber membrane absorber. Journal of Beijing University of Chemical Technology (Natural Science Edition) 2006, 33, 1−4. (23) Zhang, W.; Li, Y.; Dai, Y. A sub-channels model on the mass transfer process in hollow fiber modules. Membr. Sci. Technol. 1996, 16, 56−61. (24) Noda, I.; Brown-West, D. G.; Gryte, C. C. Effect of flow maldistribution on hollow fiber dialysisexperimental studies. J. Membr. Sci. 1979, 5, 209−225. (25) Lemanski, J.; Lipscomb, G. G. Effect of shell-side flows on hollow-fiber membrane device performance. AIChE J. 1995, 41, 2322− 2326. (26) Ren, Z.; Yang, Y.; Zhang, W.; Liu, J.; Wang, H. Modeling study on the mass transfer of hollow fiber renewal liquid membrane: Effect of the hollow fiber module scale. J. Membr. Sci. 2013, 439, 28−35. (27) Zhang, W.; Chen, G.; Li, J.; Liu, J. Intensification of mass transfer in hollow fiber modules by adding solid particles. Ind. Eng. Chem. Res. 2009, 48, 8655−8662. (28) Zhang, W.; Zhu, S.; Luo, G.; Dai, Y.; Wang, J. Intensification of mass transfer performance in hollow fiber contained liquid membrane. Membr. Sci. Technol. 1998, 18, 53−57.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support from the National Natural Science Foundation of China (No. 21076012 and 21276011) and Natural Science Foundation of Beijing (No. 3121003) are gratefully acknowledged. In addition, we also acknowledge the research support from National Development and Reform Commission (No. SHGS-KJ-CO-2012-12).

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NOMENCLATURE Am contact area based on the fiber outside diameter (m2) C concentration of the solute (mol·m−3) ΔCm logarithmic mean concentration difference of CO2 (mol· m−3) Est enhancement factor caused by vibration (−) E(t) residence time distribution function (s−1) E(θ) dimensionless residence time distribution function (−) fst vibration frequency (rpm) KL mass transfer coefficient of liquid phase (m·s−1) n number of hollow fibers in the module (−) Q CO2 absorption rate (mol·s−1) t time (s) tm mean residence time (s) uL velocity of absorbent (m·s−1) Greek Letters

θ normalized time (−) θp the corresponding θ of the highest value of E(θ) on a RTD curve σ2 the variance about the mean of RTD (s2) φ hollow fiber packing density of the module (%) Superscripts and Subscripts

cal exp G in L out RTD

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calculation experimental gas phase inlet of liquid phase liquid phase outlet of liquid phase residence time distribution

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dx.doi.org/10.1021/ie500583v | Ind. Eng. Chem. Res. 2014, 53, 8640−8650