중공사형 막모듈에서 순수 기체 투과의 섭동 해석해

장대준, 민준호, 오세헌, 문기호

현대중공업 산업기술연구소 환경연구실

Perturbation Solution of Hollow-Fiber Membrane Module

for Pure Gas Permeation

D. Chang, J. Min, S. Oh, and K. Moon

Environment Research Department, Hyundai Industrial Research Institute,

Hyundai Heavy Industries

1. Introduction

One of the most important objectives in experimental study of hollow-fiber module is to precisely estimate the permeability of pure gas molecules  through hollow-fiber membrane. Though transport of a component is often affected by presence of other penetrants due to plasticization, hydrostatic compression, or competitive sorption effects, the study on permeation of the pure component provides a baseline for further investigations on permeation of multi-component mixtures. Usually, permeability is estimated from the permeate flow rate for given pressures applied to tube and shell sides. While the pressure or velocity of shell-side stream are relatively easy to measure, those of tube-side streams are not due to the small radial dimension of hollow fibers. The pressure of tube side is often deduced from either values at fiber ends.

Some analytic approaches have been suggested by several authors for some limiting cases. For plug flow cases are available analytic solutions neglecting the pressure drop in feed and permeate streams. Boucif et al. proposed for a binary mixture approximate solutions by expressing each state variable as a truncated series in terms of dimensionless membrane area. Rautenbach and Dahm developed analytic solutions for three extremes, each of which assumes the infinite selectivity, negligible permeate-side pressure, and constant permeate pressure, respectively. The approach is refined by Smith et al. assuming constant pressure in the feed and permeate streams  and perfect-mixing in the permeate stream. Basaran and Auvil carried out an asymptotic analysis for the case of a binary mixture in a cross-flow module. They formulated perturbation expansions, identifying a parameter closely related with pressure ratio and selectivity.

In this study an analytic approach of perturbation expansion is suggested for gas permeation of a pure gas in a countercurrent hollow-fiber module. The variation of pressure, as well as that of velocity, is allowed  in the feed and permeate streams. Perturbation methods have found its successful application to analysis of systems whose behavior is characterized by small parameters. A hollow-fiber module  will be described by equations for pressure drop and mass conservation. When diffusion and elastic effect are negligible, the equations will be characterized by several parameters. Provided the parameter to characterize pressure drop is small, perturbation method can be applied to give birth to functional dependence of pressure and velocity on the module length as well as on the operating and designing values for the module.

2. Theory

Under the condition of negligible diffusion, end effects, and physical deformation of membrane, the governing equations of hollow-fiber module  are described as following:

[eqn: {d P^t}over{dZ} &=& {8 mu}over{r_i^2} U^t , ~~ {d P^s}over{dZ} &=& -{8 mu}over{r_s^2} U^s , ~~ a_s {d C^s U^s}over{dZ} &=& -pi r_o J(r_o ) , ~~ a_t {d C^t U^t}over{dZ} &=& -pi r_i J(r_i ),]

where the flux is given [eqn: J(r)= {k (P^s -P^t )} over{r \ln(r_o /r_i )}].  Here, [eqn: P, ~U,~ C,~ mu,~ Z,~ r,~ a, {rm and} k] are pressure, velocity, mole concentration, viscosity, axial position, radius, area, and permeability, respectively. Subscripts 'i' and 'o' for radius denote the inner and outer radii of a fiber, and superscripts 's' and 't' represent the shell and tube variables. These equations are expressed in dimensionless form,

[eqn: {d p^s}over{dz} = -delta_1 u^s , ~ {d p^t}over{dz} = delta_2 u^t , ~ {d p^s u^s}over{dz} =- delta_3 (p^s -p^t ) , ~ {d p^t u^t}over{dz} =- delta_4 (p^s -p^t ),]

where the parameters are given in terms of the characteristic  scales and additional  design specifications.

[eqn: delta_1 &=& {8 mu U_c L_c }over{r_s^2 P_c},~ delta_2 &=& delta_1 eta_r ,~ delta_3 &=& {\pi k RT L_c}over{a_s U_c \ln(r_o /r_i )} ,~ delta_4 &=& delta_3 eta_a] Here, [eqn: eta_r~ and~ eta_a] are defined by [eqn: r_s^2 /r_i^2~ and ~a_s /a_t].

Usually, for gas permeation membranes [eqn: delta_1] is far less than [eqn: delta_3] with [eqn: eta_r ~and~eta_a] being of [eqn: O(1)]. In the case, regular perturbation method is capable of providing a precise analytic solution.

3. Results and Discussion

The analytic solution by perturbation method is obtained as follows

[eqn: & ~~~~~~~epsilon^0 & ~~~~~~~~~epsilon^1 ## p^s~~~~~~&~~~~~~~~1 &delta_3 (1-p_r ) (z^2 -1)/2 -(z-1)## p^t~~~~~~&~~~~~~~~p_r &delta_4 eta_r (1-p_r ) (z-z^2 /2)/p_r## u^s~~~~~~&-delta_3 (1-p_r ) z+1 &-delta_3 phi_1 (z) - u_0^s (z) [p_1^s -p_1^s (z) ]## p^s~~~~~~&delta_4 (1-p_r ) (1-z) /p_r~~~~~~~~~~~&delta_4 [phi_1 (1) -phi_1 (z) ]/p_r -p_1^t (z) u_0^t (z) /p_r]

where [eqn: epsilon] is equal to [eqn: delta_1].

Figure 1 shows the results of perturbation solution and numerical simulation with [eqn: epsilon=delta_1 =0.005,~eta_r =3.5, delta_3 =0.2, delta_4 =0.3,~ and~ p_r =0.2]. The dotted line corresponds to the numerical results, and dashed and dash-dot-dot lines to the [eqn: O(epsilon^0 )] and [eqn: O(epsilon)] solutions, respectively. Since \orderzero solutions of pressure are constant [eqn: (p_0^s (z)=1~ and ~p_0^t =p_r )], they cannot predict the actual pressure drops. The results do not mean that the velocity solution is more precise than that of pressure. The relative deviation of [eqn: p_0^s (z)] from numerical solution is also of [eqn: O(epsilon)].

Note that the pressure variation of tube side is considerable while that of shell side is not. In a laboratory experiment the permeability of a membrane is determined from flow rates and pressure difference. The result indicates that the permeability will be incorrectly estimated if the outlet pressure of permeate side is taken for the average pressure of permeate stream.

References

1. N. Yi-Yan, R. Felder, and W. Koros, J. Appl. Polym. Sci., 25, 1755 (1980).

2. E.S. Sanders and W.J. Koros, J. Polym. Sci., Polym. Phys. Ed., 24, 175 (1986).

3. S.-T. Hwang and K. Kammermeyer, Membranes In Separations, Robert E. Krieger Publishing Company, Malabar, 1984.

4. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods For Scientists and Engineers, McGraw-Hill Book Company, New York, 1978.

Figure 1. Variation of pressure and velocity along with [eqn: epsilon=delta_1 =0.005,~eta_r =3.5, delta_3 =0.2, delta_4 =0.3,~ and~ p_r =0.2]. Dotted, dash-dot, dash-dot-dot curves represent the results of numerical simulation, [eqn: O(epsilon^0 )] perturbation solution, and [eqn: O(epsilon)]-perturbation solution, respectively.