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Dynamic simulation of multicomponent gas separation by hollow-fiber membrane module: Nonideal mixing flows in permeate and residue sides using the tanks-in-series model
Separation and Purification Technology 76 (2011) 362–372
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Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur
Dynamic simulation of multicomponent gas separation by hollow-fiber membrane module: Nonideal mixing flows in permeate and residue sides using the tanks-in-series model Takashi Katoh a , Masahiro Tokumura a , Hidemi Yoshikawa b , Yoshinori Kawase a,∗ a b
Research Center for Biochemical and Environmental Engineering, Department of Applied Chemistry, Toyo University, Kawagoe, Saitama 350-8585, Japan Education Support Center of Experiment and Training for Students, Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan
a r t i c l e
i n f o
Article history: Received 4 June 2010 Received in revised form 3 November 2010 Accepted 5 November 2010 Keywords: Membrane gas separation Dynamic performance Nonideal mixing Tanks-in-series model Relaxation simulation scheme
a b s t r a c t A new simulation model for the dynamic performance of gas separation membrane modules is presented. In order to take account of nonideal mixing flows in permeate and residue sides a tanks-in-series model is utilized. As a stable computational scheme, the relaxation method is applied to solve the governing ordinary differential equations for transport across the membrane, mass balance and pressure distributions in a hollow-fiber membrane module. The proposed simulation model and scheme are validated using the experimental data and simulation results hydrogen gas separation and air separation in the literature. Using the proposed simulation model and scheme the dynamic performance of membrane gas separation processes, hydrogen recovery process and two-stage methane separation process with residue recycle, is examined by varying the operating conditions, i.e., the bulk mixing degree (perfect mixing, plug flow and intermediate mixing), pressure drop and recycle ratio. The computational results indicate that effect of mixing degree in the feed side is more significant as compared with that in the permeate side and less mixing in the feed side results in higher performance. The retentate recycle is found to improve methane recovery efficiency. The proposed simulation model considering nonideal mixing in the membrane module provides more reliable examination of unsteady-state behaviors of hollow-fiber membrane gas separation modules. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Energy efficient membrane-based technology has become one of important separation processes over recent years (e.g., [1–5]). Since membrane separation processes offer a number of advantages over conventional separation technologies for certain gas separation, they have been successfully used in the industrial separation of gases [1–3]. Membrane separation has been also found in environmental application such as the removal of volatile organic compounds (VOCs). Commercial membrane separation units commonly use modules such as spiral-wound, hollow-fiber and capillary-fiber since a large membrane area can be packaged in a small volume [1,2,6,7]. The hollow-fiber membrane module finding wide applications consists of a large number of membrane fibers housed in a module shell. In other words, the module is constructed similar to the shell-and-tube type heat exchanger. Feed can be introduced on either the fiber or the shell side. Typically the high-pressure feed gas enters the shell side of the module at one end
∗ Corresponding author. Tel.: +81 49 239 1377; fax: +81 49 231 1031. E-mail address: [email protected] (Y. Kawase). 1383-5866/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2010.11.006
and leaves at the other end. The permeate which is enriched with the more permeable components is withdrawn from the inside of the fibers through the openings on the fiber tubesheet. The modeling of hollow-fiber membrane gas separation has been studied by a number of investigators [1,2,6,7]. Although the flow of the bulk fluid in the commercial membrane modules is unlikely to be perfectly mixed or plug flow, in the modeling of membrane modules the idealized mixing, i.e., the perfect mixing or the plug flow, has been widely assumed to describe mixing in the hollow-fiber and the shell sides (e.g., [6–12]). Since analytical solutions are available, highly simplified approximations such as perfectly mixed permeate and residue have been often applied. The gas flows in permeate and residue sides are also assumed to be in plug flow. Certainly, these approximate models have been useful for quick design calculations. While in the hollow-fiber side the axial dispersion is probably negligible and the plug flow assumption may be reasonable, in the shell side the flow pattern is close to be perfectly mixed and non-ideal mixing must be sometimes considered. Furthermore, the presence of baffles in the shell side significantly affects the mixing in the shell side and as a result the performance of gas separation [13]. More detailed mixing simulation models are necessary for more accu-
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Nomenclature A Am Cn D Ez F J L P Q Sp Sr t U u V x y Z z
membrane surface area, m2 membrane surface area in a tank for residue = A/Sr , m2 the number of gas components inside diameter of hollow fiber, m axial dispersion coefficient, m2 s−1 total gas flow rate, Nm3 h−1 permeate flux through the membrane, Nm3 h−1 fiber length, m total pressure, MPa permeance in the membrane, Nm3 m−2 h−1 MPa−1 the number of tanks for permeate side the number of tanks for residue side time, s gas flow rate in the fiber, m h−1 superficial gas velocity, m s−1 volume, m3 mole fraction in residue mole fraction in permeate dimensionless axial position = z/L axial distance, m
Greek letters t time interval, s residence time, s gas density, kg m−3 dimensionless time = t/ Subscripts e sealed end i component i in inlet j j-th tank in residue n n-th tank in permeate out outlet p permeate r residue
rate simulation studies of gas separation by follow-fiber membrane modules. The axial dispersion model and tanks-in-series models have been widely used to represent nonideal mixing [14]. Marriott and Sorensen [15] presented the description of the axial dispersion model for membrane modules. The axial dispersion model, which has been frequently applied to represent in mixing, can describe satisfactorily only mixing deviating not largely from the plug flow. Furthermore, a set of differential equations with initial and boundary conditions obtained for the axial dispersion model has to be solved by rather complicated numerical techniques [16,17]. A tanks-in-series model is more realistic and advantageous as compared with the axial dispersion model. Consequently, we have developed a simulation scheme for nonideal mixing hollow-fiber membrane separation on the basis of the tanks-in-series model. It should be noted that the number of tanks (Sp or Sr ) representing the mixing degree in the tanks-in-series is approximately related with the axial dispersion coefficient (Ez ) representing the mixing degree in the axial dispersion model by Sp (or Sr ) ≈ uL/(2Ez ) [18]. Dindore et al. [19] measured shell-side dispersion coefficients in a rectangular cross-flow hollow fiber membrane module and the gas phase axial dispersion coefficient Ez was 2.5 × 10−4 to 1.5 × 10−3 m2 s−1 . This range corresponds to the Sr range of 40–7. Their results indicate that as described below the mixing in the shell side may be non-
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ideal. Thundyil and Koros [20] proposed the simulation algorithm for hollow-fiber membrane modules based on the succession of states method. In their algorithm the module is divided into a number of elements. They found that 125,000 elements are required to obtain the accurate solution. Coker et al. [21] presented a staged model approach for countercurrent multicomponent permeation. They found that the number of well-mixed stage is usually 100 to obtain good agreement with the analytical solution. It should be emphasized that they assumed plug flow in both shell and fiber sides and therefore the number of stage does not represent the mixing degree in the membrane module. The number of the elements or stages in the above models [20,21] is related to the convergence of the trial-and-error calculation in their computational schemes instead of the mixing degree in the module. In other words, the above two models are not a tanks-in-series model describing nonideal mixing flows in fiber and shell sides of a hollow-fiber membrane module. Bouton and Luyben [22] examined the optimum economic design and control of a gas permeation membrane coupled with the hydroalkylation process using a cell model. Although they selected the number of cells being 10, they did not relate it to mixing degree in the membrane module. In their study the pressure drop of the permeate stream was neglected and the number of cells in the residue side was fixed to be equal to that in the permeate side. Computational fluid dynamics (CFD) provides a more rigorous technique. Due to the complexity of the problem there have been few attempts at modeling hollow-fiber membrane modules [e.g., 23, 24] and complete descriptions of the modules have not been presented. The approximate approaches described above are still very effective. Although a number of mathematical models do exist, they usually rely on a number of fixed assumptions. One of them is steady-state condition assumption. Most of available models including the above models [20,21] for gas membrane separation have been developed for steady-state. An understanding of dynamic performance of gas membrane separation processes is essential for rational design and operation of industrial scale gas separation membrane modules. A dynamic model taking account of nonideal mixing can provide practical information for startup and shutdown operations, the transition between two stationary states, optimal design and operability studies besides a steady-state performance [16,17]. To the best of our knowledge, on the basis of the tanks-in-series concept the unsteady-state phenomena of the membrane separation have not yet been modeled in the open literature. The aim of this study is to model dynamic performance of nonideal mixing hollow-fiber membrane modules for gas separation on the basis of the tanks-in-series model. A rational modeling and a numerical scheme, which is not time-consuming, is required for reliable and useful simulation gaining a great understanding of the membrane separation processes. We apply the relaxation method in solving simultaneously a series of differential equations for gas membrane separation based on the tanks-in-series model. The relaxation method making use of the unsteady-state equations for the determination of the steady-state solution has been effectively applied to multicomponent separation calculations [25–28]. Due to the unknown permeate pressure profile inside the fiber a trial-and-error shooting method is generally required to obtain the solution [e.g., 20, 21]. However, the relaxation method can provide a straightforward iteration scheme to obtain the dynamic and steady-state solutions. A robust dynamic simulation based on a nonideal mixing model is required for the optimal design and operation of multicomponent gas separation by hollow-fiber membrane modules. The experimental data and computational results for steady-state gas separation by hollow-fiber membrane modules in the literature are used for the validation of the proposed modeling and simulation scheme. The applicability of the modeling
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Fig. 1. Schematic of the tanks-in-series model for hollow-fiber membrane module (the case for 2Sp = Sr ): feed is introduced on the shell side and permeate is withdrawn from the inside of the fibers in a countercurrent manner. For simplicity, only one fiber is shown.
and the capability of the proposed simulation scheme are demonstrated with the numerical examples of hydrogen separation and two-stage methane separation with recycle stream. 2. Modeling for dynamic simulation of hollow-fiber membrane module 2.1. Tanks-in-series model for nonideal mixing in membrane module The tanks-in-series model consisting of a series of equal sized, well-mixed tanks is applied to represent dynamic performance of hollow-fiber membrane separators under a wide range of operating conditions. The tanks-in-series model can describe the axial changes in the gas compositions and flow rates of both the feed and permeate streams along the permeator due to transport through the membrane under nonideal mixing conditions. The number of the hypothetical well-mixed tanks represents the degree of bulk flow mixing. When the number of the hypothetical tanks is unity, the tanks-in-series model describes perfect mixing. We confirmed that the plug flow can be satisfactorily approximated by the tanksin-series model with the 20 hypothetical tanks. Fig. 1 illustrates a schematic of the tanks-in-series model for a hollow-fiber module operated in countercurrent flow pattern. For simplicity only one hollow fiber is shown. This configuration corresponds to the hydrogen recover process discussed below. The tanks in the permeate and residue sides are numbered from the feed end to the residue end. The hollow fiber bundle is sealed on both ends by epoxy tubesheets. The hollow fibers are closed at one end of the tube bundles. Feed gas is introduced on the bore of the hollow fibers or on the shell side of the module. The permeate gas inside the fibers flows countercurrent to the shell-side flow and is collected in a chamber where the open ends of the fibers terminate. The permeate flows exit through an epoxy tubesheet. There is no permeation through epoxy tubesheets but they contribute to the fiber pressure drop. In general the mixing in the shell side is more well-mixed as compared with that in the hollow-fiber side, i.e., Sr ≤ Sp for the flow pattern in Fig. 1 (feed gas is introduced on the shell). Furthermore, the number of tanks in the fiber side must be equal to that in the shell side or even number times of that [17]. In the model of Coker et al. [21], incidentally, the number of stages in the fiber side must be equal to that in the shell side. It should be noted that the tanksin-series model for the countercurrent flow is readily extended to other flow patterns, i.e., radial crossflow and cocurrent flow.
A mathematical model should include the relationship governing transport across the membrane, material balance equations, pressure drop relations for both the permeate/feed sides and boundary and initial conditions that reflect the configuration and operation of the hollow-fiber membrane module. The tanks-inseries model can predict changes in gas compositions and flow rates in both the feed and permeate streams along the membrane module due to transport through the membrane. The principle assumptions of the mathematical model are as follows: 1. 2. 3. 4.
The gas behavior is ideal. The operation is isothermal. The deformation of the hollow fiber under pressure is negligible. Membrane permeabilities are independent of pressure and concentration and pure component permeabilities are the same as the mixed gas permeabilities. 5. The pressure change in the hollow fiber is given by the Hagen–Poiseuille equation and the shell side has negligible pressure drop. 6. There is no concentration polarization. These assumptions have been frequently applied in modeling hollow-fiber membrane modules [e.g., 8,12,20,21]. With the above assumptions, the governing unsteady-state equations for multicomponent gas mixture permeation in hollow-fiber membrane module operated countercurrent (feed is introduced on the shell side and permeate is withdrawn from the inside of the fibers) are as follows: 1. The rate-of-permeation equations for component i (i = 1, . . ., Cn ) from j-th tank in residue (j = 1, . . ., Sr ) to n-th tank in permeate (n = 1, . . ., Sp ): Ji,n = Qi Am (Pr j xi,j − Pp n yi,n )
(1)
Here j is the largest integer in n/(Sp /Sr ). 2. The rate of change in concentration of component i (i = 1, . . ., Cn ) for n-th tank (n = 1, . . ., Sp−1 ) in permeate: dyi,n = dt
Fp n+1 yi,n+1 + Ji,n − Fp n yi,n Vp /Sp
(2)
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
For the outlet epoxy sealed end (n = 0) where there is no permeation, we have
dyi,0 = dt
Fp 1 yi,1 − Fp out yi,0 Ve
(3)
For the closed epoxy sealed end of permeate (n = Sp ), Eq. (2) reduces to
dyi,Sp
=
dt
Ji,Sp − Fp Sp yi,Sp
(4)
Vp /Sp
3. The rate of change in concentration of component i (i = 1, . . ., Cn ) for j-th tank (j = 2, . . ., Sr ) in residue: dxi,j dt
=
Fr j−1 xi,j−1 − Ji,j − Fr j xi,j
Vr /Sr
(5)
where Sp /Sr ·j
Ji,j =
Ji,n
(6)
n=1+Sp /Sr ·(j−1)
For the 1st hypothetical tank (j = 1) in residue, Eq. (5) yields to dxi,1 = dt
Fr in xi,in − Ji,1 − Fr 1 xi,1 Vr /Sr
(7)
4. The changes in gas flow rate between the adjacent tanks in permeate (n = 1, . . ., Sp ) and residue (j = 1, . . ., Sr ) sides: Fp,n = Fp,n+1 +
Cn
Ji,n
(8)
i=1
and Fr,j = Fr,j−1 −
Cn
(9)
Ji,j
i=1
respectively. 5. The pressure change between n-th tank and (n − 1)-th tank in permeate or tube side may be written as: Pn = Pp n−1 − Pp n
(10)
The Hagen–Poiseuille equation is used to evaluate the pressure drop in the fiber and tubesheet by considering the flow under laminar flow regime [8,12,29]. Pn = 4
16 DUn /
L/S U 2 p n D
2
(11)
The pressure on the shell side of the fibers (Prj , j = 1, . . ., Sr ) is generally taken to be constant and equal to the feed pressure (Pr in ). 2.2. The relaxation method as simulation scheme The resulting set of 2{i(Sp + Sr ) + 1} first-order ordinary differential equations with the initial values was solved using the fourth order Runge-Kutta integration method. In order to obtain the solution of the ordinary differential equations, a straightforward relaxation method including no trial-and-error scheme is utilized in this study. The relaxation method has been widely applied to find the steady-state solution through the change in the separation process with time [25–28]. The specified boundary conditions are the flow rate (Fr in ), compositions (xi,in ) and pressure (Pr in ) of the feed and the permeate outlet pressure (Pp 0 ). Of course the configuration of the membrane module such as the outer and inner diameters
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of hollow-fiber, the length of hollow-fiber, the number of hollowfiber, fiber pot length, active membrane area and the volume of permeator must be specified as input variables. The calculation procedure is explained for the membrane module with shell side feed and a countercurrent permeate stream illustrated in Fig. 1 as follows. Firstly the initial values for all hypothetical tanks in both the residue and permeate sides are specified. The initial condition is the steady-state which is based on the uniform profiles of concentrations and pressures and stagnant fluids. It is assumed that initially gas having extremely small permeance occupies the module, all compositions in the membrane module are zero (xi,j = yi,n = 0 at t = 0) and gas flow rate is zero (Fp n = Fr j = 0 at t = 0). The gas having very low permeability and occupying initially both the hollow fiber and shell sides is assumed to be unable to permeate through the membrane. The initial pressure profiles in the residue and permeate sides are assumed to be uniform at the feed pressure Pr in and the specified permeate outlet pressure Pp 0 , respectively. It should be noted that the temperature in the module is assumed to be constant. By introducing the feed on the residue side the initial unsteady-state starts to change. At the same time the calculation also starts to simulate really the course of changes in the module with time. The gas fed into the residue side sweeps the low permeability gas and permeates through the follow-fiber membrane. For the iterative calculation of unsteady-state operation the time increment t is specified. The changes in compositions, flow rates and pressures of the tanks in both the residue and permeate sides with time are pursued. Therefore the time increment must be properly chosen. If it is too large, the dynamic change in gas separation process will be incorrectly described. On the other hand, if it is too small, extremely long computation time will be required. From the simulation examples presented below we found the appropriate dimensionless time increment (=t/) is 0.03. In every time period the calculation starts at the feed end of the module. The small changes from the initial steady-state started by the introduction of feed during the small time increment t are propagated step by step from 1st tank in the residue (j = 1) to the residue outlet (j = Sr ). In order to determine the transport through the membrane, the partial pressures on both sides need to be known and consequently the compositions in both sides needs to be known. Therefore the computation for the residue side proceeds from the feed inlet to the outlet and simultaneously that for the permeate side is carried out from the permeate outlet to the closed end. Using Eq. (1) the local permeate flux through the membrane Ji,1 is calculated using the feed flow rates, compositions and pressures and the initial conditions for 1st tank in the residue (j = 1) and 1st tank in the permeate (n = 1). If Sp = 2Sr (as described above, Sp > Sr ), the calculation for n = 2 must be carried out at the same time. By the permeation through the membrane the gas flow rate and composition for j = 1 in the residue side change. At the same time those for n = 1 and n = 2 in the permeate side as well as for n = 0 (the epoxy tubesheet) change. Since the inlet pressure of the feed in the residue side (Pr in ) and the exit pressure of the permeate in the permeate side (Pp 0 ) are specified, the pressures for j = 1 and n = 1 and 2 (Pr 1 , Pp 1 and Pp 2 ) can be calculated by considering the pressure drops based on the newly determined gas flow rates in both the residue and permeate sides. As a result the gas flow rate, composition and pressure for n = 1 and 2 in the permeate side are simultaneously computed with those for j = 1 in the residue at the end of the time period. The calculations in both the residue and permeate sides are successively and straightforwardly carried out from the feed end (j = 1) to the residue outlet (j = Sr ) and from the permeate outlet (n = 0) to the fiber closed end (n = Sp ), respectively. At the beginning of the first time period, the compositions, flow rates and pressures at 1st tank (j = 1) in the residue and 1st and 2nd tanks (n = 1 and n = 2) in the permeate are firstly evaluated using those at the initial steady-state. Successively the compositions, flow rates and pressures at j-th tank
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
in the residue and n-th and (n + 1)-th tank in the permeate are evaluated using those at (j − 1)-th tank in the residue and (n − 1)-th tank in the permeate. When the calculations for the residue side from the feed end (j = 1) to the residue outlet (j = Sr ) are completed, the computation for the permeate side are simultaneously completed from the permeate outlet end (n = 0) to the closed end of the fiber (n = Sp ). After calculating the compositions, flow rates and pressures at each tank in both the residue and permeate sides at t = t, the calculation from the feed end to the residue outlet at t = t + t is successively repeated using known values at t = t. After a certain time, the gas flow rates, compositions and pressures in both the residue and permeate sides reaches the new steady-state. Since in the proposed simulation scheme the outlet gas flow rate and gas composition are not specified and the active membrane area is given, the socalled shooting method is not necessary. The simulation scheme for countercurrent flow described above can be easily extended to that for concurrent flow as described below.
100
90
H2 purity (%)
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Coker et al. (1998) Pfin=76.9 bar 60 Present result Pfin=42.4 bar Present result Pfin=76.9 bar 50 0
20
40
60
80
100
H2recovery (%) (a) Hydrogen gas separation simulations (Coker et al. [21])
3.1. Validation of the proposed model and simulation scheme The proposed model and simulation scheme are verified using experimental and simulation results for steady-state multicomponent membrane gas separation processes reported in the literature.
H2purity in permeate (%)
100
95
90 Present model 85
Experimental data (Pan, 1986) Model predictions (Marriott and Sorensen, 2003)
80 35
40
45
50
55
60
65
Stage cut (%) (b) Hydrogen recovery from purge gas of ammonia plant (Pan [8]) 100
90
N2purity ( %)
3.1.2. Hydrogen recovery from purge gas of ammonia plant (Pan [8]) Pan [8] reported the experimental results for hydrogen recovery from simulated purge gas of ammonia plant which consists of 51.78% H2 , 24.69% N2 , 19.57% CH4 and 3.96% Ar. For the separation a shell-side-feed parallel-flow hollow-fiber module was used. Marriott and Sorensen [15] used the experimental data for H2 purity in permeate reported by Pan [8] to verify their approach based on the plug flow model. In Fig. 2b the predictions of the present simulation model represented by a solid line are compared with the expermental data of Pan [8] and the simulation results of Marriott and Sorensen [15]. The H2 purity decreases with increasing stage cut defined as the fraction of feed removed as permeate. Excellent agreement is found between the proposed model and the experimental data over the whole operation range. Marriott and Sorensen [11,15] proposed the detailed model consisting of three sub-models and the resulting sets of partial differential and algebraic equations are solved by the use of both finite difference and orthogonal collection on finite element methods. In their two-dimensional flow
70 Coker et al.(1998) Pfin=42.4 bar
3. Simulation of membrane gas separation process
3.1.1. Separation of hydrogen in gas stream from hydrotreater (Coker et al. [21]) The membrane hydrogen purification for a hydrotreater simulated by Coker et al. [21] is examined to validate the present simulation scheme. The simulation parameters such as composition and conditions of the feedstream and pemeance are given in their paper. The feed gas consisting of 65.0%H2 , 21.0%CH4 , 8.0%C2 H6 , 3.5%C3 H8 and 2.5%C2 H4 is directed to flow through the shell side of the module. The pressure difference between the feed and permeate stream is mantained constant at 34.7 bar. The number of fibers and active membrane area are 500,000 and 377 m2 , respectively. In Fig. 2a, H2 purity/recovery relationships obtaned by the staged model of Coker et al. [21] with the number of stages of 100 are compared with the predictions of the proposed simulation model. At higher ratio of feed to permeate pressure, higher hydrogen recovery is found. As can be seen, the simulation results by the proposed model, in which the degrees of mixing in both the fiber and shell sides are assumed to be represented by 10 hypothetical tanks, agree reasonably well with the numerical solutions of Coker et al. [21] for the feed pressure (Pr,in ) of 42.2 and 76.9 bar.
80
80
Present result Pfin=10 bar Present result Pfin=5 bar
70
Coker et al.(1998) Pfin=10 bar Coker et al.(1998) Pfin=5 bar 60
0
20
40
60
80
100
Overall residue recovery (%) (c) Air separation (Coker et al. [21]) Fig. 2. Validation of the proposed simulation scheme using the experimental data and simulation results in the literature [8,21].
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372 Table 1 Operating and feed conditions assumed in process simulation. Feed Feed flow rate (Fr in ) Feed composition (xi in ) Feed pressure (Pr in ) Permeate pressure (Pp 0 ) Feed temperature Fiber OD/ID Fiber length (L) Number of fibers ID of membrane module
Shell side 2000 Nm3 h−1 0.70H2 (i = 1), 0.15CH4 (i = 2), 0.10C2 H6 (i = 3), 0.05C3 H8 (i = 4) 3 MPa 0.5 MPa 80 ◦ C 400/200 m 2m 100,000 0.2 m
submodel, the dispersion coefficients in the axial and radial directions are taken into account. The present simulation results almost coincide with those of Marriott and Sorensen [15]. This comparison indicates the validity of the proposed simulation scheme based on the tanks-in-series model. 3.1.3. Air separation or nitrogen production from air (Coker et al. [21]) The air separation for the generation of purified N2 was simulated by Coker et al. [21]. The air fed to the countercurrent membrane module was modeled as a four component mixture of 78.41% N2 , 20.84%O2 , 0.03% CO2 and 0.72% H2 O. The feed gas is introduced on the bore side. In Fig. 2c, the simulation results obtaned by Coker et al. [21] at feed pressures of 5 or 10 bar for no purge stream from the residue product are compared with the predictions of the proposed simulation scheme in which the hypothetical numbers of tanks in both the permeate and residue sides are assumed to be 10. The figure shows the N2 purity as a function of the overall residue recovery. As feed flow rate is increased, residue recovery increases and nitrogen purity decreases. Nitrogen purity increases with increasing feed pressure. Reasonable agreement in Fig. 2c suggests the validity of the proposed simulation model.
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tion. First the calculation of concentration, pressure and flow rate in the residue is carried out from 1st tank (feed inlet end) to Sr -th tank (residue end). Simultaneously, the calculation in the permeate is succeeded from 0-th tank (outlet epoxy sealed end) to Sp -th tank (closed end of hollow-fiber). The time increment t plays an important role in the accuracy of the calculation and we used the dimensionless time increment of 0.03 in all calculations. In the computer simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively. Typical results for dynamic changes in gas flow rates and hydrogen concentration in permeate and residue sides are presented in Fig. 3a–d. In these figures, the relative position is the normalized axial coordinate in the module by the active fiber length, L and the dimensionless time is the normalized time by the residence time based on the feed flow rate in the module. The relative positions of 0 and 1 correspond the permeate and residue ends, respectively. The feed inlet in residue is at the relative position of 0. Of course at the closed epoxy sealed end of permeate the gas flow rate is always zero and the gas flow rate at the residue inlet is constantly 2000 Nm3 h−1 . As shown in Fig. 3a and b, as expected, the gas flow rate in the permeate increases rapidly from the permeate end to the outlet and on the contrary that in the residue decreases from the feed inlet to the residue end. The gas flow rates near the permeate outlet and residue inlet change more quickly as compared with those near the permeate sealed end and residue outlet, respec-
3.2. Dynamic simulation of hydrogen recovery from refinery gases The proposed scheme for dynamic simulation of hollow-fiber membrane modules is examined using examples for hydrogen recovery from refinery gases. The use of membranes for hydrogen separation has been applied commercially in recent years [1]. The high-pressure feed gas is introduced on the shell side and permeate being enriched with hydrogen gas is withdrawn from the inside of the fibers. The operating and feed conditions assumed in process simulation are given in Table 1. The length of epoxy tubesheet is 0.1 m and therefore the permeating length of the hollow fibers in the module is 1.9 m. The kinetic viscosity and density of the permeate gas were assumed to be 2 × 10−5 m2 s−1 and 0.46 kg m−3 , respectively. The permeances of the membrane to gases used in the simulation are presented in Table 2. As described above, the calculations for concentration and flow rate profiles in the membrane module starts with the initial conditions; xi,j = 0, yi,n = 0, Fr j = Fp n = 0 and Pr j = Pp n = Pr in (i = 1, . . ., Cn , j = 1, . . ., Sr and n = 0, . . ., Sp ). The calculation is straightforward and a trial-and-error scheme is not required to obtain the soluTable 2 Permeability used in process simulation. Component
Permeance (Qi ) GPU
H2 (i = 1) CH4 (i = 2) C2 H6 (i = 3) C3 H8 (i = 4)
300 3 2.4 1.92
1GPD = 7.501 × 10−12 Nm3 m−2 s−1 Pa−1 .
Fig. 3. Dynamic performance of hydrogen recovery. Countercurrent operation in which feed is introduced on the shell side and permeate is withdrawn from the inside of the fiber. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively.
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100
600 H2 Recovery [%] (PD) H2 Recovery [%] (No PD) total membrane area [m2] PD: pressure drop in hollow fiber
500
400
80 300 70 200 60
Total membrane area, A [m2]
H2recovery[%]
90
100
50 0
200
400
600
800
1000
0 1200
Inner diameter of hollow fiber, D [μm] Fig. 4. Effect of pressure drop in the permeate (hollow-fiber) side on hydrogen recovery. Countercurrent operation in which feed is introduced on the shell side and permeate is withdrawn from the inside of the fiber. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively.
Fig. 3. (Continued).
tively. The permeate outlet flow rate rapidly increases and reaches the steady-state flow rate of 1283.18 Nm3 h−1 at about = 30. The increase in residue outlet flow rate is slow and the steady-state flow rate of 716.81 Nm3 h−1 cannot be accomplished for < 30. It is seen in Fig. 3c that the change of H2 concentrationin with time in the permeate near the outlet of permeate is more rapid as compared with that near the permeate end. The change in H2 concentration at the closed end of hollow-fiber (y1,0 ) starts at after = 10. The transient of H2 concentration profile in the residue side from the initial steady-state to the new steady-state is seen in Fig. 3d. The H2 concentration in the residue outlet (x1,Sr ) gradually approches the steady-state concentration of 0.2068. The inner diameter of hollow-fiber is changed to examine the effect of pressure drop in the hollow-fiber. In the simulation, the mixing degrees in the shell and fiber sides are represented by 10 and 20 (plug flow) hypothetical tanks, respectively. Fig. 4 depicts the effect of pressure drop in the hollow-fiber (permeate side) on hydrogen recovery. Since the number and thickness of membrane are fixed, the active membrane area decreases with increasing the inner diameter of follow-fiber as reperented by a broken line. The flux of permeate through the membrane is constant with constant membrane thichness. As a result, the total permeation flux through the hollow-fibers decreases as the active membrane area decreases. H2 recover decreases with increasing inner diameter of
hollow-fiber decreasing active membrane area. When the pressure drop for flow in the hollow-fibers is not considered, H2 recovery monotonously increases with decreasing hollow-fiber diameter. The pressure drop in the hollow-fiber causes the depression of H2 recovery and its effect is signuficant for D < 400 m and the maximum H2 recovery is obtained at around D = 200 m. At smaller fibers gas flow rate decreases due to the increase in pressure drop and as a result the H2 recovery decreases. Fig. 5 illustrates effects of mixing degree in hollow-fiber membrane modules. The results for Sr :Sp = 1:1, Sr :Sp = 1:20 and Sr :Sp = 10:20 corresponds to those for the perfect mixing in both the residue and permeate sides, the perfect mixing in the residue side and plug flow in the permeate side and the intermediate mixing degree in the residue side and plug flow in the permeate side, respectively. According to the simulated results in Fig. 5a and b, the number of hypothetical tanks in the permeate side insignificantly affects the dynamic changes in the outlet H2 concentration in the permeate side or the product and H2 recovery. The increase in the number of tanks or decrease in the mixing in the residue side causes quick changes in the outlet H2 concentration in the permeate side or the product and H2 recovery. Effects of mixing degree in the residue side is more significant as compared with that in the permeate side. This is because of the gas transfer from the residue side to the permeate side. The concentration variation in the residue or feed side dominates the driving force for gas transfer and as a result permeate flux through the membrane rather than that in the permeate side. Fig. 5a shows that the decrease of mixing degree in the residue side improves the hydrogen purity. As depicted in the insert of Fig. 5a, overshoots of y1,0 compared with their final steady-state values are found. The value of y1,0 increases, passes through the maximum and then finally approaches to its steady-state values. The overshoot of H2 concentration in the product stream is more considerable at higher mixing degree and for the perfect mixing in both the residue and permeate sides the overshoot is the highest. The steady-state values for Sr :Sp = 1:1, Sr :Sp = 1:20 and Sr :Sp = 10:20 are 96.53, 96.60 and 97.24%, respectively. The overshoots for the systems in which the mixing in the residue side is perfect mixing are more distinguishable as compared with that for the system with imperfect mixing in the residue side. The H2 recovery for imperfect mixing in both the permeate and residue sides deviates rather considerably from that for perfect mixing in both the permeate and residue sides. The transient to the steady-state condition slows down with
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Table 3 Operating and feed conditions assumed in the simulation of methane separation process consisting of two-stage cascade with recycle stream. Feed (Module 1 and 2) Feed flow rate (Fr in ) (Module 1) Fresh feed composition (xi in ) (Module 1)
Tube side 2300 Nm3 h−1 0.796CH4 (i = 1), 0.193CO2 (i = 2), 0.021H2 (i = 3) 0.75 MPa 0.13 MPa 0.89 MPa 0.13 MPa 40 ◦ C 175/110 m 0.95 m 290,000 0.15 m
Feed pressure (Pr in ) (Module 1) Permeate pressure (Pp out ) (Module 1) Feed pressure (Pr in ) (Module 2) Permeate pressure (Pp out ) (Module 2) Feed temperature (Module 1 and 2) Fiber OD/ID (Module 1 and 2) Fiber length (L) (Module 1 and 2) Number of fibers (Module 1 and 2) ID of membrane modules (Module 1 and 2)
gases are presented in Tables 3 and 4, respectively. It can be seen in Table 4 that the membrane used is more permeable to CO2 and H2 than to CH4 . The production of this membrane separation process is a high purity less permeable gas, CH4 . The configurations of two membrane modules connected in series are same. The modules are operated with the feed entering on the tube side. Unlike the hydrogen recovery process discussed above, the high-pressure feed gas is fed into the fiber and the permeate is withdrawn from the shell side of the module. Since as noted above the mixing in the shell side is more well-mixed than that in the fiber side, we have the relationship of Sr ≥ Sp . The recovery of CH4 and CO2 concentration in the product are specified to be larger than 95% and less than 3%, respectively. Some of the governing equations for the hollow-fiber membrane module operated countercurrent in which the feed is introduced on the fiber side and the permeate is withdrawn from the shell side are different from a set of the equations presented above, Eqs. (1)–(11). The governing equations for this configuration are as follows: Fig. 5. Effects of mixing degree in hollow-fiber membrane modules on dynamic performance of hydrogen recovery. Countercurrent operation in which feed is introduced on the shell side and permeate is withdrawn from the inside of the fiber (, Sr :Sp = 1:1; complete mixing in both residue and permeate sides, , Sr :Sp = 1:20; complete mixing in the residue side and the plug-flow in the permeate side, 䊉, Sr :Sp = 10:20; intermediate mixing in the residue side and plug flow in the permeate side).
decreasing the mixing degree in the residue side. The steady-state H2 recoveries for Sr :Sp = 1:1, Sr :Sp = 1:20 and Sr :Sp = 10:20 are 79.79, 74.33 and 88.68%, respectively. Fig. 5b illustrates the reduction of mixing degree in the residue improves the H2 recovery but that in the permeate side reduces the H2 recovery. 3.3. Dynamic simulation of two-stage membrane separation with recycle stream for methane recovery Recycling allows an increase in the performance of separation. The effect of recycle streams on the dynamic performance of membrane separation processes is examined using separation of CH4 from ternary CH4 /CO2 /H2 mixture containing 78.6 mol% CH4 , 19.3 mol% CO2 and 2.1 mol% H2 . The processing cost is dominated by the cost of CH4 lost in the permeate. Some of the lost CH4 can be recovered by utilizing two or more membrane modules connected in series and provided with recycle streams. The configuration examined in this study is a two-stage cascade with residue recycle (Fig. 6). In this cascade arrangement, the permeate from the 1st module after compression is sent as feed to the 2nd module while the residue from the 2nd module is recycled and mixed the feed to the 1st module. The residue from the 1st module is then the product stream. The operating and feed conditions assumed in the simulation and permeances of the membrane to
1. The rate-of-permeation equations for component i (i = 1, . . ., Cn ) from j-th tank in residue (j = 1, . . ., Sr ) to n-th tank in permeate (n = 1, . . ., Sp ): Ji,j = Qi Am (Pr j xi,j − Pp n yi,n )
(12)
Here n is the smallest integer in j(Sr /Sp ). 2. The rate of change in concentration of component i (i = 1, . . ., Cn ) for j-th tank (j = 1, . . ., Sr ) in residue can be written by Eq. (5). For the 0-th tank (the inlet epoxy tubesheet) and (Sr + 1)-th tank (the outlet epoxy tubesheet) in residue, Eq. (5) yields dxi,0 = dt
and dxi,Sr +1 = dt
Fr in xi,in − Fr 0 xi,0 Ve
(13)
Fr Sr xi,Sr − Fr Sr +1 xi,Sr +1 Ve
(14)
respectively. 3. The rate of change in concentration of component i (i = 1, . . ., Cn ) for n-th tank (n = 1, . . ., Sp−1 ) in permeate is given by Eq. (2). The Table 4 Permeability used in the simulation of methane separation process consisting of two-stage cascade with recycle stream. Component
Permeance (Qi ) GPU
CH4 (i = 1) CO2 (i = 2) H2 (i = 3)
5.8 107 540
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Fig. 6. Two membrane stages in series with recycle stream for methane recovery.
flux through the membrane Ji,n in Eq. (2) may be written as: Sr /Sp ·n
Ji,n =
Ji,j
(15)
j=1+Sr /Sp ·(n−1)
4. The changes in gas flow rate between the adjacent tanks in permeate (n = 1, . . ., Sp ) and residue (j = 1, . . ., Sr ) sides can be calculated by Eqs. (8) and (9). 5. The fiber-side pressure change can be written as: Pr,j = Pr,j−1 + Pj where Pj = 4
16 DUj /
(16)
L/Sr D
Uj2 2
(17)
Fig. 7 illustrates dynamic performance of methane recovery with the operating and feed conditions given in Table 4. The flow
rate ratio of the Residue 2 to Feed 0 is 0.46. The concentration profiles of CH4 and CO2 in the product stream or the residue side for Module 1 are presented in Figs. 7a and b, respectively. In the computer simulation, for both Module 1 and Module 2 the numbers of hypothetical tanks in the shell and fiber sides are 10 and 20, respectively. The mixing degree in the fiber side is assumed to be close to the plug-flow. It can be found in Fig. 7a that the concentration of slow permeating gas, CH4 , along the residue increases more quickly near the feed inlet as compared with that at the residue outlet. For < 40 the outlet CH4 concentration in the residue (x1,Sr+1 ) from Module 1 is almost zero. This indicates that the feed stream takes about 40 times of the average residence time (40) to reach the residue outlet. The CH4 concentration in the product stream (relative position of 1) rapidly increases after = 40 and approaches to its steady-state concentration of 98.04%. It is interesting that the maximum CH4 concentration in the central region of the fiber can be observed. However, the distribution of CH4 is rather uniform through the hollow-fiber. As shown in Fig. 7b, there is an initial sharp increase in the concentration of fast permeating gas, CO2 ,
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near the residue inlet from the initial value of 0 for short time but the increase in x2,j near the residue outlet is considerably slow. The CO2 concentration in the residue outlet of Module 1 slowly increases from 0 and after a certain time reaches the steady-state concentration of 0.0194 which is about 1/10 of the inlet concentration. A very significant distribution in CO2 concentration from the inlet of the feed to outlet can be found. The overshoots of the variations for CH4 and CO2 concentrations with time are seen near the residue inlet at around = 20. Fig. 7c depicts the dynamic increase of CH4 concentration profile in the residue of Module 2. The increase of x1,j in Module 2 is somewhat slower as compared with that in Module 1. As the time progress the CH4 concentration at the residue outlet of Module 2 (x1,Sr+1 ) approaches to the steady-state of 0.7710. It can be seen in Fig. 7d that CO2 concentration in the residue outlet of Module 2 (x2,Sr+1 ) is zero for < 40 and monotonously increases to the steady-state of 0.2248. Since CO2 concentration in the residue outlet of Module 1 is rather lower than that in the feed, as expected that of Module 2 is higher than that in the feed. In Module 2 the variations for CH4 and CO2 concentrations with time have not their overshoots. Effect of recycle ratio is examined by varying the pressure of the feed to the 2nd stage module or Permeate 1. The increase in the pressure of feed to the 2nd module increases the flow rate of Permeate 2 and as a result decreases that of Residue 2 or the recycle ratio. Fig. 8 shows the performance characteristics of a two-stage
Fig. 7. (Continued).
membrane module are plotted as a function of recycle ratio defined as the ratio of the recycled residue flow rate from the 2nd stage (Residue 2) to the flow rate of fresh feed (Feed 0). The numbers of hypothetical tanks in the shell and fiber sides are assumed to be
20 99.66 (CH4)
90 10 80
70 0.34 (CO2 )
60
50
■
CH4 recovery
● ▲
CH4 purity
50.0 (CH4 recovery)
0 0.1 (single stage) Fig. 7. Dynamic performance of methane recovery. Countercurrent operation in which feed is introduced to the inside of hollow-fibers and permeate is withdrawn from the shell side. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug-flow), respectively.
0
0.2
CO2 concentration
CO2 concentration in Residue 1(mol%)
CH4 recovery and CH4 purity in Residue 1 (%)
100
-10 0.3
0.4
0.5
0.6
Recycle ratio (-)
Fig. 8. Effects of recycle ratio on CH4 recovery and CH4 purity and CO2 concentrations in the residue from the 1st membrane module (Residue 1) or product of the membrane separation. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively.
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10 and 20, respectively. For the single membrane stage, as plotted on the left-hand y-axis in Fig. 8, the concentrations of CH4 and CO2 in the outlet residue are 99.66 and 0.34%. The CH4 recovery is 50%. According to the specifications the purity of CH4 is larger than 95% and the product must contain <3% CO2 . Certainly the specifications are satisfied by the single stage operation. However, the CH4 recovery of 50% is very low. With the recycle ratio the CH4 purity slightly decreases and the corresponding CO2 concentration increases but the CH4 recovery given by the solid line significantly is improved. For instance, at the recycle ratio of 0.46 the CH4 recovery is 95.78% with the CH4 purity of 98.04% and the CO2 concentration of 1.94%. It should be noted here that commercial multistage cascade arrangement may be expensive due to compression of permeate between stages. 4. Conclusions We developed a new simulation model available to predict the dynamic performance of hollow-fiber gas separation membrane modules. A tanks-in-series model was used to describe nonideal mixing flows in the membrane module. The relaxation method is applied to solve the governing simultaneous ordinary differential equations for transport across the membrane, mass balance and pressure distributions in a hollow-fiber gas separation module. It is a straightforward scheme and includes no trial-and-error procedures. In the relaxation method, the changes in compositions, flow rates and pressures of the tanks in both the residue and permeate sides with time are pursued from the initial steady-state based on the uniform profiles of concentrations and pressures and stagnant fluids. After certain time the gas flow rates, compositions and pressures in both the residue and permeate sides reaches the new steady-state. The validity of the proposed simulation model and scheme was proven using the experimental data and simulation results for hydrogen gas separation in the stream from hydrotreater, hydrogen recovery from purge gas of ammonia plant and air separation reported in the literature. Using the proposed simulation model the dynamic performance of membrane gas separation processes, hydrogen recovery process and two-stage methane separation process with residue recycle from the second stage to feed, is examined by varying the operating conditions, i.e., flow pattern (countercurrent and concurrent), the bulk mixing degree (perfect mixing, plug flow and intermediate mixing), pressure drop and recycle ratio. The computational results indicate that effects of mixing degree in the feed side is more significant as compared with that in the permeate side and less mixing in the feed side results in higher performance. It is possible that the perfect mixing and plug-flow assumptions result in underestimation and overestimation of the dynamic performance of hollow-fiber membrane module, respectively. For the two-stage CH4 separation with recycle stream for methane recovery, the residue recycle to feed is found to improve CH4 recovery efficiency. The proposed simulation model considering nonideal mixing in membrane modules and simulation scheme provide more reliable
examination of unsteady-state behaviors of hollow-fiber membrane gas separation modules and as a result enable their optimal design and operation. References [1] R.W. Baker, Membrane Technology and Applications, 2nd ed., John Wiley & Sons, Chichester, 2004. [2] E. Drioli, L. Giorno, Membrane Operations, Wiley–VCH, Germany, 2009. [3] P. Bernardo, E. Drioli, G. Golemme, Mambrane gas separation, Ind. Eng. Chem. Res. 48 (2009) 4638–4663. [4] S.-L. Wee, C.-T. Tye, S. Bhatia, Membrane separation process—pervaporation through zeolite membrane, Sep. Purif. Technol. 63 (2008) 500–516. [5] E. Favre, R. Bounaceur, D. Roizard, A hybrid process combining oxygen enriched air combustion and membrane separation for post-combustion carbon dioxide capture, Sep. Purif. Technol. 68 (2009) 30–36. [6] C.J. Geankoplis, Transport Processes and Separation Process Principles, Prentice-Hall, New Jersey, 2003. [7] P.C. Wankat, Separation Process Engineering, 2nd ed., Prentice-Hall, New Jersey, 2007. [8] C.Y. Pan, Gas separation by high-flux, asymmetric hollow-fiber membrane, AIChE J. 32 (1986) 2020–2027. [9] S. Sridhar, A.A. Khan, Simulation studies for the separation of propylene and propane by ethylcellulose membrane, J. Membr, Sci. 159 (1999) 209–219. [10] S. Tessendorf, R. Gani, M.L. Michelsen, Modeling, simulation and optimization of membrane-based gas separation systems, Chem. Eng. Sci. 54 (1999) 943–955. [11] J.I. Marriott, E. Sorensen, I.D.L. Bogle, Detailed mathematical modeling of membrane modules, Comp. Chem. Eng. 25 (2001) 693–700. [12] A.S. Kovvali, S. Vemury, W. Admassu, Modeling of multicomponent countercurrent gas permeators, Ind. Eng. Chem. Res. 33 (1994) 896–903. [13] K.L. Wang, E.L. Cussler, Baffled membrane modules made with hollow fiber fabric, J. Membr. Sci. 85 (1993) 265–278. [14] O. Levenspiel, Chemical Reaction Engineering, 3rd ed., John Wiley & Sons, New York, 1999. [15] J. Marriott, E. Sorensen, A general approach to modeling membrane modules, Chem. Eng. Sci. 58 (2003) 4975–4990. [16] T. Kanai, J. Ichikawa, H. Yoshikawa, Y. Kawase, Dynamic modeling and simulation of continuous airlift bioreactors, Bioprocess Eng. 23 (2000) 213–220. [17] M. Tokumura, M. Baba, Y. Kawase, Dynamic modeling and simulation of absorption of carbon dioxide into seawater, Chem. Eng. Sci. 62 (2007) 7305–7311. [18] C.Y. Wen, L.T. Fan, Models for Flow Systems and Chemical Reactors, Marcel Dekker, Inc., New York, 1975. [19] V.Y. Dindore, A.H.G. Cents, D.W.F. Brilman, G.F. Versteeg, Shell-side dispersion coefficients in a rectangular cross-flow hollow fibre membrane module, Chem. Eng. Res. Des. 83 (2005) 317–325. [20] M.J. Thundyil, W.J. Koros, Mathematical modeling of gas separation permeators for radial cross flow, countercurrent, and concurrent hollow fiber membrane modules, J. Membr. Sci. 125 (1997) 275–291. [21] D.T. Coker, B.D. Freeman, G.K. Fleming, Modeling multicomponent gas separation using hollow-fiber membrane contactors, AIChE J. 44 (1998) 1289–1302. [22] G.R. Bouton, W.L. Luyben, Optimum economic design and control of a gas permeation membrane coupled with the hydrodealkylation (HAD) process, Ind. Eng. Chem. Res. 47 (2008) 1221–1237. [23] D.E. Wiley, D.F. Fletcher, Techniques for computational fluid dynamics modelling of flow in membrane channels, J. Membr. Sci. 211 (2003) 127–137. [24] H. Takaba, S. Nakao, Computational fluid dynamics study on concentration polarization in H2 /CO separation membranes, J. Membr. Sci. 249 (2005) 83–88. [25] A. Rose, R.F. Sweeny, V.N. Schrodt, Continuous distillation calculations by relaxation method, Ind. Eng. Chem. 50 (1958) 737–740. [26] T. Vanek, V. Hlavacek, M. Kubicek, Calculation of separation columns by nonlinear block successive relaxation methods, Chem. Eng. Sci. 32 (1977) 839–844. [27] E. Kenig, K. Jakobsson, P. Banik, J. Aittamaa, A. Gorak, M. Koskinen, P. Wettmann, An integrated tool for synthesis and design of reactive distillation, Chem. Eng. Sci. 54 (1999) 1347–1352. [28] Z. Lei, C. Li, Y. Li, B. Chen, Separation of acetic acid and water by complex extractive distillation, Sep. Purif. Technol. 36 (2004) 131–138. [29] R. Wang, S.L. Liu, T.T. Lin, T.S. Chung, Characterization of hollow fiber membranes in a permeator using binary gas mixtures, Chem. Eng. Sci. 57 (2002) 967–976.
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Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur
Dynamic simulation of multicomponent gas separation by hollow-fiber membrane module: Nonideal mixing flows in permeate and residue sides using the tanks-in-series model Takashi Katoh a , Masahiro Tokumura a , Hidemi Yoshikawa b , Yoshinori Kawase a,∗ a b
Research Center for Biochemical and Environmental Engineering, Department of Applied Chemistry, Toyo University, Kawagoe, Saitama 350-8585, Japan Education Support Center of Experiment and Training for Students, Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan
a r t i c l e
i n f o
Article history: Received 4 June 2010 Received in revised form 3 November 2010 Accepted 5 November 2010 Keywords: Membrane gas separation Dynamic performance Nonideal mixing Tanks-in-series model Relaxation simulation scheme
a b s t r a c t A new simulation model for the dynamic performance of gas separation membrane modules is presented. In order to take account of nonideal mixing flows in permeate and residue sides a tanks-in-series model is utilized. As a stable computational scheme, the relaxation method is applied to solve the governing ordinary differential equations for transport across the membrane, mass balance and pressure distributions in a hollow-fiber membrane module. The proposed simulation model and scheme are validated using the experimental data and simulation results hydrogen gas separation and air separation in the literature. Using the proposed simulation model and scheme the dynamic performance of membrane gas separation processes, hydrogen recovery process and two-stage methane separation process with residue recycle, is examined by varying the operating conditions, i.e., the bulk mixing degree (perfect mixing, plug flow and intermediate mixing), pressure drop and recycle ratio. The computational results indicate that effect of mixing degree in the feed side is more significant as compared with that in the permeate side and less mixing in the feed side results in higher performance. The retentate recycle is found to improve methane recovery efficiency. The proposed simulation model considering nonideal mixing in the membrane module provides more reliable examination of unsteady-state behaviors of hollow-fiber membrane gas separation modules. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Energy efficient membrane-based technology has become one of important separation processes over recent years (e.g., [1–5]). Since membrane separation processes offer a number of advantages over conventional separation technologies for certain gas separation, they have been successfully used in the industrial separation of gases [1–3]. Membrane separation has been also found in environmental application such as the removal of volatile organic compounds (VOCs). Commercial membrane separation units commonly use modules such as spiral-wound, hollow-fiber and capillary-fiber since a large membrane area can be packaged in a small volume [1,2,6,7]. The hollow-fiber membrane module finding wide applications consists of a large number of membrane fibers housed in a module shell. In other words, the module is constructed similar to the shell-and-tube type heat exchanger. Feed can be introduced on either the fiber or the shell side. Typically the high-pressure feed gas enters the shell side of the module at one end
∗ Corresponding author. Tel.: +81 49 239 1377; fax: +81 49 231 1031. E-mail address: [email protected] (Y. Kawase). 1383-5866/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2010.11.006
and leaves at the other end. The permeate which is enriched with the more permeable components is withdrawn from the inside of the fibers through the openings on the fiber tubesheet. The modeling of hollow-fiber membrane gas separation has been studied by a number of investigators [1,2,6,7]. Although the flow of the bulk fluid in the commercial membrane modules is unlikely to be perfectly mixed or plug flow, in the modeling of membrane modules the idealized mixing, i.e., the perfect mixing or the plug flow, has been widely assumed to describe mixing in the hollow-fiber and the shell sides (e.g., [6–12]). Since analytical solutions are available, highly simplified approximations such as perfectly mixed permeate and residue have been often applied. The gas flows in permeate and residue sides are also assumed to be in plug flow. Certainly, these approximate models have been useful for quick design calculations. While in the hollow-fiber side the axial dispersion is probably negligible and the plug flow assumption may be reasonable, in the shell side the flow pattern is close to be perfectly mixed and non-ideal mixing must be sometimes considered. Furthermore, the presence of baffles in the shell side significantly affects the mixing in the shell side and as a result the performance of gas separation [13]. More detailed mixing simulation models are necessary for more accu-
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Nomenclature A Am Cn D Ez F J L P Q Sp Sr t U u V x y Z z
membrane surface area, m2 membrane surface area in a tank for residue = A/Sr , m2 the number of gas components inside diameter of hollow fiber, m axial dispersion coefficient, m2 s−1 total gas flow rate, Nm3 h−1 permeate flux through the membrane, Nm3 h−1 fiber length, m total pressure, MPa permeance in the membrane, Nm3 m−2 h−1 MPa−1 the number of tanks for permeate side the number of tanks for residue side time, s gas flow rate in the fiber, m h−1 superficial gas velocity, m s−1 volume, m3 mole fraction in residue mole fraction in permeate dimensionless axial position = z/L axial distance, m
Greek letters t time interval, s residence time, s gas density, kg m−3 dimensionless time = t/ Subscripts e sealed end i component i in inlet j j-th tank in residue n n-th tank in permeate out outlet p permeate r residue
rate simulation studies of gas separation by follow-fiber membrane modules. The axial dispersion model and tanks-in-series models have been widely used to represent nonideal mixing [14]. Marriott and Sorensen [15] presented the description of the axial dispersion model for membrane modules. The axial dispersion model, which has been frequently applied to represent in mixing, can describe satisfactorily only mixing deviating not largely from the plug flow. Furthermore, a set of differential equations with initial and boundary conditions obtained for the axial dispersion model has to be solved by rather complicated numerical techniques [16,17]. A tanks-in-series model is more realistic and advantageous as compared with the axial dispersion model. Consequently, we have developed a simulation scheme for nonideal mixing hollow-fiber membrane separation on the basis of the tanks-in-series model. It should be noted that the number of tanks (Sp or Sr ) representing the mixing degree in the tanks-in-series is approximately related with the axial dispersion coefficient (Ez ) representing the mixing degree in the axial dispersion model by Sp (or Sr ) ≈ uL/(2Ez ) [18]. Dindore et al. [19] measured shell-side dispersion coefficients in a rectangular cross-flow hollow fiber membrane module and the gas phase axial dispersion coefficient Ez was 2.5 × 10−4 to 1.5 × 10−3 m2 s−1 . This range corresponds to the Sr range of 40–7. Their results indicate that as described below the mixing in the shell side may be non-
363
ideal. Thundyil and Koros [20] proposed the simulation algorithm for hollow-fiber membrane modules based on the succession of states method. In their algorithm the module is divided into a number of elements. They found that 125,000 elements are required to obtain the accurate solution. Coker et al. [21] presented a staged model approach for countercurrent multicomponent permeation. They found that the number of well-mixed stage is usually 100 to obtain good agreement with the analytical solution. It should be emphasized that they assumed plug flow in both shell and fiber sides and therefore the number of stage does not represent the mixing degree in the membrane module. The number of the elements or stages in the above models [20,21] is related to the convergence of the trial-and-error calculation in their computational schemes instead of the mixing degree in the module. In other words, the above two models are not a tanks-in-series model describing nonideal mixing flows in fiber and shell sides of a hollow-fiber membrane module. Bouton and Luyben [22] examined the optimum economic design and control of a gas permeation membrane coupled with the hydroalkylation process using a cell model. Although they selected the number of cells being 10, they did not relate it to mixing degree in the membrane module. In their study the pressure drop of the permeate stream was neglected and the number of cells in the residue side was fixed to be equal to that in the permeate side. Computational fluid dynamics (CFD) provides a more rigorous technique. Due to the complexity of the problem there have been few attempts at modeling hollow-fiber membrane modules [e.g., 23, 24] and complete descriptions of the modules have not been presented. The approximate approaches described above are still very effective. Although a number of mathematical models do exist, they usually rely on a number of fixed assumptions. One of them is steady-state condition assumption. Most of available models including the above models [20,21] for gas membrane separation have been developed for steady-state. An understanding of dynamic performance of gas membrane separation processes is essential for rational design and operation of industrial scale gas separation membrane modules. A dynamic model taking account of nonideal mixing can provide practical information for startup and shutdown operations, the transition between two stationary states, optimal design and operability studies besides a steady-state performance [16,17]. To the best of our knowledge, on the basis of the tanks-in-series concept the unsteady-state phenomena of the membrane separation have not yet been modeled in the open literature. The aim of this study is to model dynamic performance of nonideal mixing hollow-fiber membrane modules for gas separation on the basis of the tanks-in-series model. A rational modeling and a numerical scheme, which is not time-consuming, is required for reliable and useful simulation gaining a great understanding of the membrane separation processes. We apply the relaxation method in solving simultaneously a series of differential equations for gas membrane separation based on the tanks-in-series model. The relaxation method making use of the unsteady-state equations for the determination of the steady-state solution has been effectively applied to multicomponent separation calculations [25–28]. Due to the unknown permeate pressure profile inside the fiber a trial-and-error shooting method is generally required to obtain the solution [e.g., 20, 21]. However, the relaxation method can provide a straightforward iteration scheme to obtain the dynamic and steady-state solutions. A robust dynamic simulation based on a nonideal mixing model is required for the optimal design and operation of multicomponent gas separation by hollow-fiber membrane modules. The experimental data and computational results for steady-state gas separation by hollow-fiber membrane modules in the literature are used for the validation of the proposed modeling and simulation scheme. The applicability of the modeling
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Fig. 1. Schematic of the tanks-in-series model for hollow-fiber membrane module (the case for 2Sp = Sr ): feed is introduced on the shell side and permeate is withdrawn from the inside of the fibers in a countercurrent manner. For simplicity, only one fiber is shown.
and the capability of the proposed simulation scheme are demonstrated with the numerical examples of hydrogen separation and two-stage methane separation with recycle stream. 2. Modeling for dynamic simulation of hollow-fiber membrane module 2.1. Tanks-in-series model for nonideal mixing in membrane module The tanks-in-series model consisting of a series of equal sized, well-mixed tanks is applied to represent dynamic performance of hollow-fiber membrane separators under a wide range of operating conditions. The tanks-in-series model can describe the axial changes in the gas compositions and flow rates of both the feed and permeate streams along the permeator due to transport through the membrane under nonideal mixing conditions. The number of the hypothetical well-mixed tanks represents the degree of bulk flow mixing. When the number of the hypothetical tanks is unity, the tanks-in-series model describes perfect mixing. We confirmed that the plug flow can be satisfactorily approximated by the tanksin-series model with the 20 hypothetical tanks. Fig. 1 illustrates a schematic of the tanks-in-series model for a hollow-fiber module operated in countercurrent flow pattern. For simplicity only one hollow fiber is shown. This configuration corresponds to the hydrogen recover process discussed below. The tanks in the permeate and residue sides are numbered from the feed end to the residue end. The hollow fiber bundle is sealed on both ends by epoxy tubesheets. The hollow fibers are closed at one end of the tube bundles. Feed gas is introduced on the bore of the hollow fibers or on the shell side of the module. The permeate gas inside the fibers flows countercurrent to the shell-side flow and is collected in a chamber where the open ends of the fibers terminate. The permeate flows exit through an epoxy tubesheet. There is no permeation through epoxy tubesheets but they contribute to the fiber pressure drop. In general the mixing in the shell side is more well-mixed as compared with that in the hollow-fiber side, i.e., Sr ≤ Sp for the flow pattern in Fig. 1 (feed gas is introduced on the shell). Furthermore, the number of tanks in the fiber side must be equal to that in the shell side or even number times of that [17]. In the model of Coker et al. [21], incidentally, the number of stages in the fiber side must be equal to that in the shell side. It should be noted that the tanksin-series model for the countercurrent flow is readily extended to other flow patterns, i.e., radial crossflow and cocurrent flow.
A mathematical model should include the relationship governing transport across the membrane, material balance equations, pressure drop relations for both the permeate/feed sides and boundary and initial conditions that reflect the configuration and operation of the hollow-fiber membrane module. The tanks-inseries model can predict changes in gas compositions and flow rates in both the feed and permeate streams along the membrane module due to transport through the membrane. The principle assumptions of the mathematical model are as follows: 1. 2. 3. 4.
The gas behavior is ideal. The operation is isothermal. The deformation of the hollow fiber under pressure is negligible. Membrane permeabilities are independent of pressure and concentration and pure component permeabilities are the same as the mixed gas permeabilities. 5. The pressure change in the hollow fiber is given by the Hagen–Poiseuille equation and the shell side has negligible pressure drop. 6. There is no concentration polarization. These assumptions have been frequently applied in modeling hollow-fiber membrane modules [e.g., 8,12,20,21]. With the above assumptions, the governing unsteady-state equations for multicomponent gas mixture permeation in hollow-fiber membrane module operated countercurrent (feed is introduced on the shell side and permeate is withdrawn from the inside of the fibers) are as follows: 1. The rate-of-permeation equations for component i (i = 1, . . ., Cn ) from j-th tank in residue (j = 1, . . ., Sr ) to n-th tank in permeate (n = 1, . . ., Sp ): Ji,n = Qi Am (Pr j xi,j − Pp n yi,n )
(1)
Here j is the largest integer in n/(Sp /Sr ). 2. The rate of change in concentration of component i (i = 1, . . ., Cn ) for n-th tank (n = 1, . . ., Sp−1 ) in permeate: dyi,n = dt
Fp n+1 yi,n+1 + Ji,n − Fp n yi,n Vp /Sp
(2)
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
For the outlet epoxy sealed end (n = 0) where there is no permeation, we have
dyi,0 = dt
Fp 1 yi,1 − Fp out yi,0 Ve
(3)
For the closed epoxy sealed end of permeate (n = Sp ), Eq. (2) reduces to
dyi,Sp
=
dt
Ji,Sp − Fp Sp yi,Sp
(4)
Vp /Sp
3. The rate of change in concentration of component i (i = 1, . . ., Cn ) for j-th tank (j = 2, . . ., Sr ) in residue: dxi,j dt
=
Fr j−1 xi,j−1 − Ji,j − Fr j xi,j
Vr /Sr
(5)
where Sp /Sr ·j
Ji,j =
Ji,n
(6)
n=1+Sp /Sr ·(j−1)
For the 1st hypothetical tank (j = 1) in residue, Eq. (5) yields to dxi,1 = dt
Fr in xi,in − Ji,1 − Fr 1 xi,1 Vr /Sr
(7)
4. The changes in gas flow rate between the adjacent tanks in permeate (n = 1, . . ., Sp ) and residue (j = 1, . . ., Sr ) sides: Fp,n = Fp,n+1 +
Cn
Ji,n
(8)
i=1
and Fr,j = Fr,j−1 −
Cn
(9)
Ji,j
i=1
respectively. 5. The pressure change between n-th tank and (n − 1)-th tank in permeate or tube side may be written as: Pn = Pp n−1 − Pp n
(10)
The Hagen–Poiseuille equation is used to evaluate the pressure drop in the fiber and tubesheet by considering the flow under laminar flow regime [8,12,29]. Pn = 4
16 DUn /
L/S U 2 p n D
2
(11)
The pressure on the shell side of the fibers (Prj , j = 1, . . ., Sr ) is generally taken to be constant and equal to the feed pressure (Pr in ). 2.2. The relaxation method as simulation scheme The resulting set of 2{i(Sp + Sr ) + 1} first-order ordinary differential equations with the initial values was solved using the fourth order Runge-Kutta integration method. In order to obtain the solution of the ordinary differential equations, a straightforward relaxation method including no trial-and-error scheme is utilized in this study. The relaxation method has been widely applied to find the steady-state solution through the change in the separation process with time [25–28]. The specified boundary conditions are the flow rate (Fr in ), compositions (xi,in ) and pressure (Pr in ) of the feed and the permeate outlet pressure (Pp 0 ). Of course the configuration of the membrane module such as the outer and inner diameters
365
of hollow-fiber, the length of hollow-fiber, the number of hollowfiber, fiber pot length, active membrane area and the volume of permeator must be specified as input variables. The calculation procedure is explained for the membrane module with shell side feed and a countercurrent permeate stream illustrated in Fig. 1 as follows. Firstly the initial values for all hypothetical tanks in both the residue and permeate sides are specified. The initial condition is the steady-state which is based on the uniform profiles of concentrations and pressures and stagnant fluids. It is assumed that initially gas having extremely small permeance occupies the module, all compositions in the membrane module are zero (xi,j = yi,n = 0 at t = 0) and gas flow rate is zero (Fp n = Fr j = 0 at t = 0). The gas having very low permeability and occupying initially both the hollow fiber and shell sides is assumed to be unable to permeate through the membrane. The initial pressure profiles in the residue and permeate sides are assumed to be uniform at the feed pressure Pr in and the specified permeate outlet pressure Pp 0 , respectively. It should be noted that the temperature in the module is assumed to be constant. By introducing the feed on the residue side the initial unsteady-state starts to change. At the same time the calculation also starts to simulate really the course of changes in the module with time. The gas fed into the residue side sweeps the low permeability gas and permeates through the follow-fiber membrane. For the iterative calculation of unsteady-state operation the time increment t is specified. The changes in compositions, flow rates and pressures of the tanks in both the residue and permeate sides with time are pursued. Therefore the time increment must be properly chosen. If it is too large, the dynamic change in gas separation process will be incorrectly described. On the other hand, if it is too small, extremely long computation time will be required. From the simulation examples presented below we found the appropriate dimensionless time increment (=t/) is 0.03. In every time period the calculation starts at the feed end of the module. The small changes from the initial steady-state started by the introduction of feed during the small time increment t are propagated step by step from 1st tank in the residue (j = 1) to the residue outlet (j = Sr ). In order to determine the transport through the membrane, the partial pressures on both sides need to be known and consequently the compositions in both sides needs to be known. Therefore the computation for the residue side proceeds from the feed inlet to the outlet and simultaneously that for the permeate side is carried out from the permeate outlet to the closed end. Using Eq. (1) the local permeate flux through the membrane Ji,1 is calculated using the feed flow rates, compositions and pressures and the initial conditions for 1st tank in the residue (j = 1) and 1st tank in the permeate (n = 1). If Sp = 2Sr (as described above, Sp > Sr ), the calculation for n = 2 must be carried out at the same time. By the permeation through the membrane the gas flow rate and composition for j = 1 in the residue side change. At the same time those for n = 1 and n = 2 in the permeate side as well as for n = 0 (the epoxy tubesheet) change. Since the inlet pressure of the feed in the residue side (Pr in ) and the exit pressure of the permeate in the permeate side (Pp 0 ) are specified, the pressures for j = 1 and n = 1 and 2 (Pr 1 , Pp 1 and Pp 2 ) can be calculated by considering the pressure drops based on the newly determined gas flow rates in both the residue and permeate sides. As a result the gas flow rate, composition and pressure for n = 1 and 2 in the permeate side are simultaneously computed with those for j = 1 in the residue at the end of the time period. The calculations in both the residue and permeate sides are successively and straightforwardly carried out from the feed end (j = 1) to the residue outlet (j = Sr ) and from the permeate outlet (n = 0) to the fiber closed end (n = Sp ), respectively. At the beginning of the first time period, the compositions, flow rates and pressures at 1st tank (j = 1) in the residue and 1st and 2nd tanks (n = 1 and n = 2) in the permeate are firstly evaluated using those at the initial steady-state. Successively the compositions, flow rates and pressures at j-th tank
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
in the residue and n-th and (n + 1)-th tank in the permeate are evaluated using those at (j − 1)-th tank in the residue and (n − 1)-th tank in the permeate. When the calculations for the residue side from the feed end (j = 1) to the residue outlet (j = Sr ) are completed, the computation for the permeate side are simultaneously completed from the permeate outlet end (n = 0) to the closed end of the fiber (n = Sp ). After calculating the compositions, flow rates and pressures at each tank in both the residue and permeate sides at t = t, the calculation from the feed end to the residue outlet at t = t + t is successively repeated using known values at t = t. After a certain time, the gas flow rates, compositions and pressures in both the residue and permeate sides reaches the new steady-state. Since in the proposed simulation scheme the outlet gas flow rate and gas composition are not specified and the active membrane area is given, the socalled shooting method is not necessary. The simulation scheme for countercurrent flow described above can be easily extended to that for concurrent flow as described below.
100
90
H2 purity (%)
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Coker et al. (1998) Pfin=76.9 bar 60 Present result Pfin=42.4 bar Present result Pfin=76.9 bar 50 0
20
40
60
80
100
H2recovery (%) (a) Hydrogen gas separation simulations (Coker et al. [21])
3.1. Validation of the proposed model and simulation scheme The proposed model and simulation scheme are verified using experimental and simulation results for steady-state multicomponent membrane gas separation processes reported in the literature.
H2purity in permeate (%)
100
95
90 Present model 85
Experimental data (Pan, 1986) Model predictions (Marriott and Sorensen, 2003)
80 35
40
45
50
55
60
65
Stage cut (%) (b) Hydrogen recovery from purge gas of ammonia plant (Pan [8]) 100
90
N2purity ( %)
3.1.2. Hydrogen recovery from purge gas of ammonia plant (Pan [8]) Pan [8] reported the experimental results for hydrogen recovery from simulated purge gas of ammonia plant which consists of 51.78% H2 , 24.69% N2 , 19.57% CH4 and 3.96% Ar. For the separation a shell-side-feed parallel-flow hollow-fiber module was used. Marriott and Sorensen [15] used the experimental data for H2 purity in permeate reported by Pan [8] to verify their approach based on the plug flow model. In Fig. 2b the predictions of the present simulation model represented by a solid line are compared with the expermental data of Pan [8] and the simulation results of Marriott and Sorensen [15]. The H2 purity decreases with increasing stage cut defined as the fraction of feed removed as permeate. Excellent agreement is found between the proposed model and the experimental data over the whole operation range. Marriott and Sorensen [11,15] proposed the detailed model consisting of three sub-models and the resulting sets of partial differential and algebraic equations are solved by the use of both finite difference and orthogonal collection on finite element methods. In their two-dimensional flow
70 Coker et al.(1998) Pfin=42.4 bar
3. Simulation of membrane gas separation process
3.1.1. Separation of hydrogen in gas stream from hydrotreater (Coker et al. [21]) The membrane hydrogen purification for a hydrotreater simulated by Coker et al. [21] is examined to validate the present simulation scheme. The simulation parameters such as composition and conditions of the feedstream and pemeance are given in their paper. The feed gas consisting of 65.0%H2 , 21.0%CH4 , 8.0%C2 H6 , 3.5%C3 H8 and 2.5%C2 H4 is directed to flow through the shell side of the module. The pressure difference between the feed and permeate stream is mantained constant at 34.7 bar. The number of fibers and active membrane area are 500,000 and 377 m2 , respectively. In Fig. 2a, H2 purity/recovery relationships obtaned by the staged model of Coker et al. [21] with the number of stages of 100 are compared with the predictions of the proposed simulation model. At higher ratio of feed to permeate pressure, higher hydrogen recovery is found. As can be seen, the simulation results by the proposed model, in which the degrees of mixing in both the fiber and shell sides are assumed to be represented by 10 hypothetical tanks, agree reasonably well with the numerical solutions of Coker et al. [21] for the feed pressure (Pr,in ) of 42.2 and 76.9 bar.
80
80
Present result Pfin=10 bar Present result Pfin=5 bar
70
Coker et al.(1998) Pfin=10 bar Coker et al.(1998) Pfin=5 bar 60
0
20
40
60
80
100
Overall residue recovery (%) (c) Air separation (Coker et al. [21]) Fig. 2. Validation of the proposed simulation scheme using the experimental data and simulation results in the literature [8,21].
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372 Table 1 Operating and feed conditions assumed in process simulation. Feed Feed flow rate (Fr in ) Feed composition (xi in ) Feed pressure (Pr in ) Permeate pressure (Pp 0 ) Feed temperature Fiber OD/ID Fiber length (L) Number of fibers ID of membrane module
Shell side 2000 Nm3 h−1 0.70H2 (i = 1), 0.15CH4 (i = 2), 0.10C2 H6 (i = 3), 0.05C3 H8 (i = 4) 3 MPa 0.5 MPa 80 ◦ C 400/200 m 2m 100,000 0.2 m
submodel, the dispersion coefficients in the axial and radial directions are taken into account. The present simulation results almost coincide with those of Marriott and Sorensen [15]. This comparison indicates the validity of the proposed simulation scheme based on the tanks-in-series model. 3.1.3. Air separation or nitrogen production from air (Coker et al. [21]) The air separation for the generation of purified N2 was simulated by Coker et al. [21]. The air fed to the countercurrent membrane module was modeled as a four component mixture of 78.41% N2 , 20.84%O2 , 0.03% CO2 and 0.72% H2 O. The feed gas is introduced on the bore side. In Fig. 2c, the simulation results obtaned by Coker et al. [21] at feed pressures of 5 or 10 bar for no purge stream from the residue product are compared with the predictions of the proposed simulation scheme in which the hypothetical numbers of tanks in both the permeate and residue sides are assumed to be 10. The figure shows the N2 purity as a function of the overall residue recovery. As feed flow rate is increased, residue recovery increases and nitrogen purity decreases. Nitrogen purity increases with increasing feed pressure. Reasonable agreement in Fig. 2c suggests the validity of the proposed simulation model.
367
tion. First the calculation of concentration, pressure and flow rate in the residue is carried out from 1st tank (feed inlet end) to Sr -th tank (residue end). Simultaneously, the calculation in the permeate is succeeded from 0-th tank (outlet epoxy sealed end) to Sp -th tank (closed end of hollow-fiber). The time increment t plays an important role in the accuracy of the calculation and we used the dimensionless time increment of 0.03 in all calculations. In the computer simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively. Typical results for dynamic changes in gas flow rates and hydrogen concentration in permeate and residue sides are presented in Fig. 3a–d. In these figures, the relative position is the normalized axial coordinate in the module by the active fiber length, L and the dimensionless time is the normalized time by the residence time based on the feed flow rate in the module. The relative positions of 0 and 1 correspond the permeate and residue ends, respectively. The feed inlet in residue is at the relative position of 0. Of course at the closed epoxy sealed end of permeate the gas flow rate is always zero and the gas flow rate at the residue inlet is constantly 2000 Nm3 h−1 . As shown in Fig. 3a and b, as expected, the gas flow rate in the permeate increases rapidly from the permeate end to the outlet and on the contrary that in the residue decreases from the feed inlet to the residue end. The gas flow rates near the permeate outlet and residue inlet change more quickly as compared with those near the permeate sealed end and residue outlet, respec-
3.2. Dynamic simulation of hydrogen recovery from refinery gases The proposed scheme for dynamic simulation of hollow-fiber membrane modules is examined using examples for hydrogen recovery from refinery gases. The use of membranes for hydrogen separation has been applied commercially in recent years [1]. The high-pressure feed gas is introduced on the shell side and permeate being enriched with hydrogen gas is withdrawn from the inside of the fibers. The operating and feed conditions assumed in process simulation are given in Table 1. The length of epoxy tubesheet is 0.1 m and therefore the permeating length of the hollow fibers in the module is 1.9 m. The kinetic viscosity and density of the permeate gas were assumed to be 2 × 10−5 m2 s−1 and 0.46 kg m−3 , respectively. The permeances of the membrane to gases used in the simulation are presented in Table 2. As described above, the calculations for concentration and flow rate profiles in the membrane module starts with the initial conditions; xi,j = 0, yi,n = 0, Fr j = Fp n = 0 and Pr j = Pp n = Pr in (i = 1, . . ., Cn , j = 1, . . ., Sr and n = 0, . . ., Sp ). The calculation is straightforward and a trial-and-error scheme is not required to obtain the soluTable 2 Permeability used in process simulation. Component
Permeance (Qi ) GPU
H2 (i = 1) CH4 (i = 2) C2 H6 (i = 3) C3 H8 (i = 4)
300 3 2.4 1.92
1GPD = 7.501 × 10−12 Nm3 m−2 s−1 Pa−1 .
Fig. 3. Dynamic performance of hydrogen recovery. Countercurrent operation in which feed is introduced on the shell side and permeate is withdrawn from the inside of the fiber. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively.
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T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
100
600 H2 Recovery [%] (PD) H2 Recovery [%] (No PD) total membrane area [m2] PD: pressure drop in hollow fiber
500
400
80 300 70 200 60
Total membrane area, A [m2]
H2recovery[%]
90
100
50 0
200
400
600
800
1000
0 1200
Inner diameter of hollow fiber, D [μm] Fig. 4. Effect of pressure drop in the permeate (hollow-fiber) side on hydrogen recovery. Countercurrent operation in which feed is introduced on the shell side and permeate is withdrawn from the inside of the fiber. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively.
Fig. 3. (Continued).
tively. The permeate outlet flow rate rapidly increases and reaches the steady-state flow rate of 1283.18 Nm3 h−1 at about = 30. The increase in residue outlet flow rate is slow and the steady-state flow rate of 716.81 Nm3 h−1 cannot be accomplished for < 30. It is seen in Fig. 3c that the change of H2 concentrationin with time in the permeate near the outlet of permeate is more rapid as compared with that near the permeate end. The change in H2 concentration at the closed end of hollow-fiber (y1,0 ) starts at after = 10. The transient of H2 concentration profile in the residue side from the initial steady-state to the new steady-state is seen in Fig. 3d. The H2 concentration in the residue outlet (x1,Sr ) gradually approches the steady-state concentration of 0.2068. The inner diameter of hollow-fiber is changed to examine the effect of pressure drop in the hollow-fiber. In the simulation, the mixing degrees in the shell and fiber sides are represented by 10 and 20 (plug flow) hypothetical tanks, respectively. Fig. 4 depicts the effect of pressure drop in the hollow-fiber (permeate side) on hydrogen recovery. Since the number and thickness of membrane are fixed, the active membrane area decreases with increasing the inner diameter of follow-fiber as reperented by a broken line. The flux of permeate through the membrane is constant with constant membrane thichness. As a result, the total permeation flux through the hollow-fibers decreases as the active membrane area decreases. H2 recover decreases with increasing inner diameter of
hollow-fiber decreasing active membrane area. When the pressure drop for flow in the hollow-fibers is not considered, H2 recovery monotonously increases with decreasing hollow-fiber diameter. The pressure drop in the hollow-fiber causes the depression of H2 recovery and its effect is signuficant for D < 400 m and the maximum H2 recovery is obtained at around D = 200 m. At smaller fibers gas flow rate decreases due to the increase in pressure drop and as a result the H2 recovery decreases. Fig. 5 illustrates effects of mixing degree in hollow-fiber membrane modules. The results for Sr :Sp = 1:1, Sr :Sp = 1:20 and Sr :Sp = 10:20 corresponds to those for the perfect mixing in both the residue and permeate sides, the perfect mixing in the residue side and plug flow in the permeate side and the intermediate mixing degree in the residue side and plug flow in the permeate side, respectively. According to the simulated results in Fig. 5a and b, the number of hypothetical tanks in the permeate side insignificantly affects the dynamic changes in the outlet H2 concentration in the permeate side or the product and H2 recovery. The increase in the number of tanks or decrease in the mixing in the residue side causes quick changes in the outlet H2 concentration in the permeate side or the product and H2 recovery. Effects of mixing degree in the residue side is more significant as compared with that in the permeate side. This is because of the gas transfer from the residue side to the permeate side. The concentration variation in the residue or feed side dominates the driving force for gas transfer and as a result permeate flux through the membrane rather than that in the permeate side. Fig. 5a shows that the decrease of mixing degree in the residue side improves the hydrogen purity. As depicted in the insert of Fig. 5a, overshoots of y1,0 compared with their final steady-state values are found. The value of y1,0 increases, passes through the maximum and then finally approaches to its steady-state values. The overshoot of H2 concentration in the product stream is more considerable at higher mixing degree and for the perfect mixing in both the residue and permeate sides the overshoot is the highest. The steady-state values for Sr :Sp = 1:1, Sr :Sp = 1:20 and Sr :Sp = 10:20 are 96.53, 96.60 and 97.24%, respectively. The overshoots for the systems in which the mixing in the residue side is perfect mixing are more distinguishable as compared with that for the system with imperfect mixing in the residue side. The H2 recovery for imperfect mixing in both the permeate and residue sides deviates rather considerably from that for perfect mixing in both the permeate and residue sides. The transient to the steady-state condition slows down with
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
369
Table 3 Operating and feed conditions assumed in the simulation of methane separation process consisting of two-stage cascade with recycle stream. Feed (Module 1 and 2) Feed flow rate (Fr in ) (Module 1) Fresh feed composition (xi in ) (Module 1)
Tube side 2300 Nm3 h−1 0.796CH4 (i = 1), 0.193CO2 (i = 2), 0.021H2 (i = 3) 0.75 MPa 0.13 MPa 0.89 MPa 0.13 MPa 40 ◦ C 175/110 m 0.95 m 290,000 0.15 m
Feed pressure (Pr in ) (Module 1) Permeate pressure (Pp out ) (Module 1) Feed pressure (Pr in ) (Module 2) Permeate pressure (Pp out ) (Module 2) Feed temperature (Module 1 and 2) Fiber OD/ID (Module 1 and 2) Fiber length (L) (Module 1 and 2) Number of fibers (Module 1 and 2) ID of membrane modules (Module 1 and 2)
gases are presented in Tables 3 and 4, respectively. It can be seen in Table 4 that the membrane used is more permeable to CO2 and H2 than to CH4 . The production of this membrane separation process is a high purity less permeable gas, CH4 . The configurations of two membrane modules connected in series are same. The modules are operated with the feed entering on the tube side. Unlike the hydrogen recovery process discussed above, the high-pressure feed gas is fed into the fiber and the permeate is withdrawn from the shell side of the module. Since as noted above the mixing in the shell side is more well-mixed than that in the fiber side, we have the relationship of Sr ≥ Sp . The recovery of CH4 and CO2 concentration in the product are specified to be larger than 95% and less than 3%, respectively. Some of the governing equations for the hollow-fiber membrane module operated countercurrent in which the feed is introduced on the fiber side and the permeate is withdrawn from the shell side are different from a set of the equations presented above, Eqs. (1)–(11). The governing equations for this configuration are as follows: Fig. 5. Effects of mixing degree in hollow-fiber membrane modules on dynamic performance of hydrogen recovery. Countercurrent operation in which feed is introduced on the shell side and permeate is withdrawn from the inside of the fiber (, Sr :Sp = 1:1; complete mixing in both residue and permeate sides, , Sr :Sp = 1:20; complete mixing in the residue side and the plug-flow in the permeate side, 䊉, Sr :Sp = 10:20; intermediate mixing in the residue side and plug flow in the permeate side).
decreasing the mixing degree in the residue side. The steady-state H2 recoveries for Sr :Sp = 1:1, Sr :Sp = 1:20 and Sr :Sp = 10:20 are 79.79, 74.33 and 88.68%, respectively. Fig. 5b illustrates the reduction of mixing degree in the residue improves the H2 recovery but that in the permeate side reduces the H2 recovery. 3.3. Dynamic simulation of two-stage membrane separation with recycle stream for methane recovery Recycling allows an increase in the performance of separation. The effect of recycle streams on the dynamic performance of membrane separation processes is examined using separation of CH4 from ternary CH4 /CO2 /H2 mixture containing 78.6 mol% CH4 , 19.3 mol% CO2 and 2.1 mol% H2 . The processing cost is dominated by the cost of CH4 lost in the permeate. Some of the lost CH4 can be recovered by utilizing two or more membrane modules connected in series and provided with recycle streams. The configuration examined in this study is a two-stage cascade with residue recycle (Fig. 6). In this cascade arrangement, the permeate from the 1st module after compression is sent as feed to the 2nd module while the residue from the 2nd module is recycled and mixed the feed to the 1st module. The residue from the 1st module is then the product stream. The operating and feed conditions assumed in the simulation and permeances of the membrane to
1. The rate-of-permeation equations for component i (i = 1, . . ., Cn ) from j-th tank in residue (j = 1, . . ., Sr ) to n-th tank in permeate (n = 1, . . ., Sp ): Ji,j = Qi Am (Pr j xi,j − Pp n yi,n )
(12)
Here n is the smallest integer in j(Sr /Sp ). 2. The rate of change in concentration of component i (i = 1, . . ., Cn ) for j-th tank (j = 1, . . ., Sr ) in residue can be written by Eq. (5). For the 0-th tank (the inlet epoxy tubesheet) and (Sr + 1)-th tank (the outlet epoxy tubesheet) in residue, Eq. (5) yields dxi,0 = dt
and dxi,Sr +1 = dt
Fr in xi,in − Fr 0 xi,0 Ve
(13)
Fr Sr xi,Sr − Fr Sr +1 xi,Sr +1 Ve
(14)
respectively. 3. The rate of change in concentration of component i (i = 1, . . ., Cn ) for n-th tank (n = 1, . . ., Sp−1 ) in permeate is given by Eq. (2). The Table 4 Permeability used in the simulation of methane separation process consisting of two-stage cascade with recycle stream. Component
Permeance (Qi ) GPU
CH4 (i = 1) CO2 (i = 2) H2 (i = 3)
5.8 107 540
370
T. Katoh et al. / Separation and Purification Technology 76 (2011) 362–372
Fig. 6. Two membrane stages in series with recycle stream for methane recovery.
flux through the membrane Ji,n in Eq. (2) may be written as: Sr /Sp ·n
Ji,n =
Ji,j
(15)
j=1+Sr /Sp ·(n−1)
4. The changes in gas flow rate between the adjacent tanks in permeate (n = 1, . . ., Sp ) and residue (j = 1, . . ., Sr ) sides can be calculated by Eqs. (8) and (9). 5. The fiber-side pressure change can be written as: Pr,j = Pr,j−1 + Pj where Pj = 4
16 DUj /
(16)
L/Sr D
Uj2 2
(17)
Fig. 7 illustrates dynamic performance of methane recovery with the operating and feed conditions given in Table 4. The flow
rate ratio of the Residue 2 to Feed 0 is 0.46. The concentration profiles of CH4 and CO2 in the product stream or the residue side for Module 1 are presented in Figs. 7a and b, respectively. In the computer simulation, for both Module 1 and Module 2 the numbers of hypothetical tanks in the shell and fiber sides are 10 and 20, respectively. The mixing degree in the fiber side is assumed to be close to the plug-flow. It can be found in Fig. 7a that the concentration of slow permeating gas, CH4 , along the residue increases more quickly near the feed inlet as compared with that at the residue outlet. For < 40 the outlet CH4 concentration in the residue (x1,Sr+1 ) from Module 1 is almost zero. This indicates that the feed stream takes about 40 times of the average residence time (40) to reach the residue outlet. The CH4 concentration in the product stream (relative position of 1) rapidly increases after = 40 and approaches to its steady-state concentration of 98.04%. It is interesting that the maximum CH4 concentration in the central region of the fiber can be observed. However, the distribution of CH4 is rather uniform through the hollow-fiber. As shown in Fig. 7b, there is an initial sharp increase in the concentration of fast permeating gas, CO2 ,
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near the residue inlet from the initial value of 0 for short time but the increase in x2,j near the residue outlet is considerably slow. The CO2 concentration in the residue outlet of Module 1 slowly increases from 0 and after a certain time reaches the steady-state concentration of 0.0194 which is about 1/10 of the inlet concentration. A very significant distribution in CO2 concentration from the inlet of the feed to outlet can be found. The overshoots of the variations for CH4 and CO2 concentrations with time are seen near the residue inlet at around = 20. Fig. 7c depicts the dynamic increase of CH4 concentration profile in the residue of Module 2. The increase of x1,j in Module 2 is somewhat slower as compared with that in Module 1. As the time progress the CH4 concentration at the residue outlet of Module 2 (x1,Sr+1 ) approaches to the steady-state of 0.7710. It can be seen in Fig. 7d that CO2 concentration in the residue outlet of Module 2 (x2,Sr+1 ) is zero for < 40 and monotonously increases to the steady-state of 0.2248. Since CO2 concentration in the residue outlet of Module 1 is rather lower than that in the feed, as expected that of Module 2 is higher than that in the feed. In Module 2 the variations for CH4 and CO2 concentrations with time have not their overshoots. Effect of recycle ratio is examined by varying the pressure of the feed to the 2nd stage module or Permeate 1. The increase in the pressure of feed to the 2nd module increases the flow rate of Permeate 2 and as a result decreases that of Residue 2 or the recycle ratio. Fig. 8 shows the performance characteristics of a two-stage
Fig. 7. (Continued).
membrane module are plotted as a function of recycle ratio defined as the ratio of the recycled residue flow rate from the 2nd stage (Residue 2) to the flow rate of fresh feed (Feed 0). The numbers of hypothetical tanks in the shell and fiber sides are assumed to be
20 99.66 (CH4)
90 10 80
70 0.34 (CO2 )
60
50
■
CH4 recovery
● ▲
CH4 purity
50.0 (CH4 recovery)
0 0.1 (single stage) Fig. 7. Dynamic performance of methane recovery. Countercurrent operation in which feed is introduced to the inside of hollow-fibers and permeate is withdrawn from the shell side. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug-flow), respectively.
0
0.2
CO2 concentration
CO2 concentration in Residue 1(mol%)
CH4 recovery and CH4 purity in Residue 1 (%)
100
-10 0.3
0.4
0.5
0.6
Recycle ratio (-)
Fig. 8. Effects of recycle ratio on CH4 recovery and CH4 purity and CO2 concentrations in the residue from the 1st membrane module (Residue 1) or product of the membrane separation. In the simulation, the numbers of tanks in the shell and fiber sides are 10 and 20 (plug flow), respectively.
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10 and 20, respectively. For the single membrane stage, as plotted on the left-hand y-axis in Fig. 8, the concentrations of CH4 and CO2 in the outlet residue are 99.66 and 0.34%. The CH4 recovery is 50%. According to the specifications the purity of CH4 is larger than 95% and the product must contain <3% CO2 . Certainly the specifications are satisfied by the single stage operation. However, the CH4 recovery of 50% is very low. With the recycle ratio the CH4 purity slightly decreases and the corresponding CO2 concentration increases but the CH4 recovery given by the solid line significantly is improved. For instance, at the recycle ratio of 0.46 the CH4 recovery is 95.78% with the CH4 purity of 98.04% and the CO2 concentration of 1.94%. It should be noted here that commercial multistage cascade arrangement may be expensive due to compression of permeate between stages. 4. Conclusions We developed a new simulation model available to predict the dynamic performance of hollow-fiber gas separation membrane modules. A tanks-in-series model was used to describe nonideal mixing flows in the membrane module. The relaxation method is applied to solve the governing simultaneous ordinary differential equations for transport across the membrane, mass balance and pressure distributions in a hollow-fiber gas separation module. It is a straightforward scheme and includes no trial-and-error procedures. In the relaxation method, the changes in compositions, flow rates and pressures of the tanks in both the residue and permeate sides with time are pursued from the initial steady-state based on the uniform profiles of concentrations and pressures and stagnant fluids. After certain time the gas flow rates, compositions and pressures in both the residue and permeate sides reaches the new steady-state. The validity of the proposed simulation model and scheme was proven using the experimental data and simulation results for hydrogen gas separation in the stream from hydrotreater, hydrogen recovery from purge gas of ammonia plant and air separation reported in the literature. Using the proposed simulation model the dynamic performance of membrane gas separation processes, hydrogen recovery process and two-stage methane separation process with residue recycle from the second stage to feed, is examined by varying the operating conditions, i.e., flow pattern (countercurrent and concurrent), the bulk mixing degree (perfect mixing, plug flow and intermediate mixing), pressure drop and recycle ratio. The computational results indicate that effects of mixing degree in the feed side is more significant as compared with that in the permeate side and less mixing in the feed side results in higher performance. It is possible that the perfect mixing and plug-flow assumptions result in underestimation and overestimation of the dynamic performance of hollow-fiber membrane module, respectively. For the two-stage CH4 separation with recycle stream for methane recovery, the residue recycle to feed is found to improve CH4 recovery efficiency. The proposed simulation model considering nonideal mixing in membrane modules and simulation scheme provide more reliable
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