International Journal of Heat and Mass Transfer 80 (2015) 835–843

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Numerical study of gas separation using a membrane

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.09.0720017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: alo2@lehigh.edu (A. Oztekin).

Nawaf Alkhamis a,b, Dennis E. Oztekin a, Ali E. Anqi a,c, Abdulmohsen Alsaiari a,b, Alparslan Oztekin a,⇑a Lehigh University, Dept. of Mechanical Engineering & Mechanics, USAb King Abdulaziz University, Dept. of Mechanical Engineering, Saudi Arabiac King Khalid University, Dept. of Mechanical Engineering, Saudi Arabia

a r t i c l e i n f o

Article history:Received 6 June 2014Received in revised form 24 September 2014Accepted 26 September 2014

Keywords:Computational fluid dynamicsMembrane modelingGas separationTurbulent flow

a b s t r a c t

Computational fluid dynamics simulations are conducted for multicomponent fluid flows in a channelcontaining spacers. A new and unique model has been presented for the treatment of the membraneboundaries in the separation CO2 from CH4 in a binary mixture. The equation governing the flux throughthe membrane is derived from first principles. The membrane is modeled as a functional surface, wherethe mass fluxes of each species will be determined based on the local partial pressures, the permeability,and the selectivity of the membrane. The approach introduced here is essential for simulating gas–gasseparation. Baseline Reynolds stress, k–x BSL, and large eddy simulation, LES, turbulence models areemployed to study spatial and temporal characteristics of the flow for Reynolds number up to 1000. Itis shown here that the spacers have a strong effect on the membrane performance. The process of sep-arating CO2 from CH4 is improved by the presence of spacers in the membrane system. It is demonstratedthat spacers should be an integral part of the membrane system design in the application of gas–gasseparation.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction In the past, gas–gas separation using a membrane had been

Natural gas consumption has increased significantly in recentyears. The impurities found in raw natural gas, extracted fromunderground, should be minimized to protect pipelines fromcorrosion. Membranes are used to separate these undesired gasses,thereby purifying the natural gas. In order to minimize capitaland operation costs of the purification process, the membraneperformance needs to be enhanced.

In this study, flows of a binary mixture, CH4 and CO2, in a channelbounded by two membranes are studied for a wide range ofReynolds numbers. Cases, with and without spacers of varying sizesand shapes, are considered. The steady flow, bounded by themembrane walls, is characterized by a simple laminar flow modelin the case without any spacers, and by a k–x baseline Reynoldsstress turbulent model in the case with spacers. A unique model ispresented for the treatment of the membrane boundaries; withwhich CO2 absorption and CH4 losses through the membrane arecalculated for both cases. The membrane flux model, derived frombasic principles by the present authors, is necessary to accuratelyrepresents gas separation and is valid for the desalination processin the limit the concentration of the one of the component tendsto zero.

studied extensively by several investigators. Such studies include:improving the permeability and the selectivity of the membrane[1–5]; operating the membrane at the optimum temperature andpressure [6–8]; or improving the separation modules [9–11]. Inthis study, the focus is less on these aspects of separation and moreon enhancing the membrane performance using momentum mix-ing. This is a well-known and studied alternative approach forimproving membrane performance. There have been extensivestudies which show that enhanced momentum mixing in an openchannel improves membrane performance in water treatment.However, the effect of mixing on the membrane’s performance ingas–gas separation has not been studied.

Several investigations study the effects of momentum mixingon the membrane performance without considering the masstransport. Karode and Kumar [12] and Saeed et al. [13] considera steady 3D laminar flow model to study the effects of a cylindricalspacer on the pressure drop at low Reynolds number. Fimbres-Weihs et al. [14] and Ranade and Kumar [15] employed a directnumerical simulation (DNS) of the Navier–Stokes equation to studythe effects of a cylindrical spacer on the pressure drop and the dragcoefficient for a wide range of Reynolds numbers reaching above1000. They have reported that the critical Reynolds number forthe onset of transition from steady to unsteady flow occurs ataround 300. Schwinge et al. [16] have studied how the staggeredand the inline cylindrical spacers affect the flow field using direct

Nomenclature

A surface area [m]D diffusion coefficient [m2/s]L channel length [m]

P permeability molem s pa

h iRe Reynolds number [–]Sc Schmidt number [–]S spacing between spacers [m]U average velocity [m/s]Y mass fraction [–]h channel height [m]k turbulent kinetic energy [J/kg]m mass flux [kg/m2/s]Dp pressure difference [Pa]s stress tensorv y-component of the velocity [m/s]y y coordinate [m]�a molar selectivity [–]q density [kg/m3]b⁄, b1, b turbulent model parameters [–]C concentration [mole/m3]J molar flux [mole/m2]M molecular weight [g/mole]

P permeance gm2 s Pa

h iSh Sherwood number [–]

F1 blending function [–]St staggered offset [m]V suction rate [m/s]d spacer diameter [m]hm mass transfer coefficient [m/s]l membrane thickness [m]p pressure [Pa]u x-component of the velocity [m/s]c rate of strain tensorx x coordinate [m]t kinematic viscosity [m2/s]a mass selectivity [–]x specific dissipation rate [1/s]r, k turbulent model parameters [–]

Subscriptsa and b species: CO2 or CH4

i and j index notationCH4 properties of CH4

tot total propertiesT eddy propertiesCO2 properties of CO2

w properties at the membrane

836 N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843

numerical simulations. Ranade and Kumar [17] considered a 3Dk–e turbulent model to study the effects of cylindrical spacers onthe pressure drop at Reynolds numbers above 350. In the presentstudy, so as to model the effects of the membrane, a mass transportequation is introduced and the membrane is modeled as a func-tional surface through which the mass flux is determined by thelocal pressures and the local concentration. Several studies alsointroduced a mass transport equation, but still treated the mem-brane as an impermeable wall. Al-Sharief et al. [18] and Shakaibet al. [19] consider a 3D laminar Navier–Stokes model to studythe effects of cylindrical spacers on the mass transfer in a waterdesalination process. A mass transport equation was also intro-duced by Al-Sharief et al. [18] assumed the mass flux of salt is con-stant on the membrane; an assumption which can be safely madein desalination studies, but not in gas–gas separation. Shakaib et al.[19] assumed constant concentration on the membrane. Severalinvestigations improved upon the idea of treating the membraneas a permeable wall. Pal et al. [20] studied the effect of cylindricalspacers on membrane performance in the food industry. The mem-brane flux was assumed to be constant and a constant concentra-tion on the membrane was assumed on a 2D laminar flow model.Subramani et al. [21] and Guillen and Hoek [22] assumed thatthe flux through the membrane was constant and that the massflux of the solute was constant. Both of them used a 2D laminarflow model to simulate the flow behavior in water treatment. Subr-amani et al. [21] studied two parallel membranes without spacers.Guillen and Hoek [22] studied the effects of different shapes ofspacers on the membrane performance. Not considered in thesepreviously mentioned studies, real separation processes need toconsider the flux through the membrane as a function of the localpressure and osmosis pressure. Lyster and Cohen [23], Villaluengaand Cohen [24] and Fletchera and Wiley [25] considered the localpressure and osmosis pressure in their models. They assumed themass flux of the salt was linearly proportional to that of the water.All of these studies considered a flow with two parallel membraneswithout any spacers. With a similar membrane treatment as whatwas seen in these aforementioned papers [23–25], Fimbres-Weihs

and Wiley [26] used a 3D laminar model to study the effects ofspacer orientation on the membrane performance. Ahmad andLau [27], Lau et al. [28], Ma and Song [29] and Ma et al. [30] usedDNS to solve the Navier–Stokes equation for flow between twomembranes with spacers. Ahmed and Lau studied the effects ofdifferent shapes of spacer attached on the membrane and Lauet al. [28] studied the effects of cylindrical spacers on membraneperformance. Ma and Song [29] and Ma et al. [30] studiedthe effects of varying placements of square spacer on themembrane performance.

The aforementioned studies were focused on water desalinationwith membranes. The concentration of the salt in saline water(molesalt/molewater) varies from 0.002 to 0.04. The salt flux on themembrane is negligible when compared with the water flux. Forthe mass transport through the membrane, the salt flux can beassumed to be linearly proportional to the water flux. However,neither assumption can be used in the gas–gas separation wherethe concentrations of both species are comparable. In a model forgas–gas separation, the flux of both species has to be consideredin the momentum and mass transports through the membrane.In the present study, the model presented considers these effects.This makes the model presented unique in both its ability to accu-rately model gas–gas separation flows and in its adaptability toother membrane flow applications. There exists no previous modelwhich could study these flows without making some of the afore-mentioned assumptions about the membrane.

2. Governing equations

Steady two dimensional flows of CH4 and CO2 binary mixture ina channel bounded by two parallel membranes are studied here forReynolds number from 200 to 1000 using a k–x baseline Reynoldsstress turbulence model. Transient two dimensional simulationsare also conducted using both large eddy simulations (LES) and ak–x BSL turbulence model for Re = 800 to investigate the effectof vortex shedding on the mass transport through the membrane.

N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843 837

The aspect ratio of the computational domain is L/h = 120, where Lis the channel length and h is the gap between two parallel mem-branes. The Reynolds number based on the inlet condition isRe = Uh/t, where U is the average fluid speed and t is the kinematicviscosity. The schematic of the computational domain is illustratedin Fig. 1.

The equations governing the laminar fluid motion inside thechannel between membranes without the spacers is the followingsimplification of the Navier–Stokes equation

@uj

@xj¼ 0 ð1Þ

@ui

@tþ uj

@ui

@xj¼ � 1

q@p@xiþ v @

2ui

@x2j

ð2Þ

Here j is the summation index and i can be either 1 or 2. ui is thefluid velocity vector, p is the pressure, xi is the spatial coordinate,t is the time, t is the kinematic viscosity, and q = qa + qb is the den-sity of the binary mixture where qa and qb are the density of species‘‘a’’ and ‘‘b’’, respectively. x � x1 is the stream-wise direction, y � x2

is the span-wise direction; likewise, u � u1 is the stream-wise com-ponent of the velocity vector and v � u2 is the span-wise compo-nent of the velocity vector. The density of the mixture is assumedto be constant in Eq. (2). The mass transport equation for species‘‘a’’ is of the form:

@qYa

@tþ uj

@qYa

@xj¼ D

@2qYa

@xj@xjð3Þ

where D is the diffusion coefficient and Ya = qa/q is the mass frac-tion of species ‘‘a’’ in a binary mixture of CH4 and CO2. The totaldensity in Eq. (3) is not treated as a constant.

Fimbres-Weihs et al. [14], Ranade and Kumar [15], and Schwingeet al. [16] documented that the flow past arrays of spacers becomestransient and the Karman vortex shedding behind the spacers cre-ates large and small eddies for values of Re above 300. It has beenshown, by Menter [31], that the k–x BSL turbulence model predictsflow in the near and outer wake regions better for a large range ofReynolds numbers. This method is an improved version of the k–xmodel and it captures separation better than both the original k–xand k–e models. k–x BSL turbulent model is employed to simulateflow past arrays of spacers for the range of Re considered here. Thek–x BSL turbulence model yields

@u1

@tþ uj

@u1

@xj¼ � 1

q@p@xiþ @

@xjðmþ tTÞ

@u1

@xj

� �ð4Þ

where u = u1, v = u2, x = x1 and y = x2. tT = k/x is the eddy viscosityand is determined using

@k@tþ uj

@k@xj¼ @

@xjðmþ rtTÞ

@k@xj

� �þ snj

@un

@xj� b�kx ð5Þ

Fig. 1. The schematic

@x@tþ uj

@x@xj¼ @

@xjðmþ rtTÞ

@x@xj

� �� b1x2 þ k

xk

snj@un

@xj

þ 2ð1� F1Þrx

@k@xj

@x@xj

ð6Þ

Here k is the turbulent kinetic energy, x is the specific dissipationrate, and s is the residual stress tensor. The constants are selectedto be r = 0.5, b⁄ = 0.09, b1 = 0.075 and k = 0.556 [31]. The detaileddescription of the blending function, F1, is given in [31]. In Eq. (5)and (6) both j and n are summation indices. The species equationfor the k–x model is of the form:

@qYa

@tþ uj

@qYa

@xj¼ @

@xjðDþ rDTÞ

@qYa

@xj

� �� ScTbx ð7Þ

where DT is the eddy diffusion coefficient and ScT is the turbulentSchmidt number. ScT and b are selected to be 0.85 and 0.09 in thepresent work.

In order to capture the effects of vortex shedding on themembrane performance, transient simulations are conductedusing the transient k–x BSL model and the LES turbulence model.The equations governing the LES model for incompressible flowsare obtained by filtering the Navier–Stokes equation [32–36]

@ui

@xi¼ 0 ð8Þ

@ui

@tþ @uiuj

@xj¼ � 1

q@p@xiþ 2v

@cij

@xj� @sij

@xjð9Þ

where ui and p are the filtered velocities and pressure fields. cij isthe rate of strain tensor and sij is the residual stress tensor, whichrepresents interactions across both large and small scales.

At the inlet, the flow is fully developed. On the surface of themembrane, the no slip boundary condition is imposed on thevelocity field. The suction rate is calculated from local pressureand concentration. A detailed description of the membrane modeland the boundary conditions is given in the next section.

3. Membrane modeling and boundary conditions

The molar flux through a membrane for a species ‘‘a’’ is given by

Ja ¼Pa

lPð1Þa � Pð2Þa

� �¼ Pa

lDPa ð10Þ

where Ja is the molar flux of species ‘‘a’’ per unit area extracted fromthe feed flow, l is the thickness of the membrane, Pa is the molar

permeability of species ‘‘a’’, DPa Pð1Þa � Pð2Þa

� �is the partial pressure

difference of species ‘‘a’’ across the membrane. The mass flux (m)per unit area through the membrane can be determined as

m ¼ ma þmb ¼ Ma Ja þMb Jb ¼ MaPa

lDpa þMb

Pb

lDpb ð11Þ

of the geometry.

Fig. 2. Meshing geometry near the first cylinder.

Fig. 3. Velocity profile obtained using different meshes.

838 N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843

where ma and mb are mass flux of species of ‘‘a’’ and ‘‘b’’. Eq. (11) canbe rewritten in terms of molar selectivity, a ¼ Pa=Pb, as

m ¼ MbPb

lMa

MbaDptot þ 1�Ma

Mba

� �Dpb

� �ð12Þ

Here Dptot = Dpa + Dpb is the total pressure difference across themembrane. With the definition of the permeance of species of ‘‘b’’Pb = Mb Pb=l, the mass selectivity a = Pa/Pb, and the relation betweenthe partial pressure and the total pressure Dpa = Ya Dptot Eq. (12)yields

m ¼ PbDptot½aþ ð1� aÞYb� ð13Þ

The suction rate through the membrane can then be determined as

Vw ¼mq¼ PbDptot

q½aþ ð1� aÞYb� ð14Þ

The boundary conditions imposed on the velocity field are: aty/h = 0 (lower membrane; see Fig. 1) u = 0 (no-slip) and v = �Vw

(suction); at y/h = 1 u = 0 and v = Vw; and no-slip and no-penetra-tion at the surface of spacers. Here the suction rate Vw dependson the selectivity of the membrane, the permeability of the mem-brane, the pressure drop across the membrane, and the mass frac-tions of the species in the mixture, as depicted in Eq. (14).

The boundary conditions imposed on the mass fraction alongthe membrane is determined from the conservation of massapplied on the surface

Ja ¼ �D@Ca

@y¼ � 1

MaD@

@yðqYaÞ ð14Þ

where Ca is the concentration of species ‘‘a’’. The gradient of themass fraction of species ‘‘a’’ at the membrane wall is related to

@Ya

@y¼ @

@yqa

qa þ qb

� �¼ 1

q2 qb@qa

@y� qa

@qb

@y

� �ð15Þ

As shown in Eq. (15) the variation in total density is accounted for.Using Eq. (10) and (15) the boundary condition for the mass fractionof species ‘‘a’’ along the surface of the lower and the upper mem-brane is

DdYa

dy¼ DptotPb

qð1� aÞYaYb ð16Þ

In the field of membrane science, the mass flux across the mem-brane is modeled by the classical equation:

�D@qa

@y¼ RVwqa ð17Þ

Here R is the rejection rate of species ‘‘a’’. It can be seen that thisnew model, Eq. (16), and the common flux model, Eq. (17), areequivalent in the case were the rejection rate is

R ¼ aaþ ð1� aÞYb

ð18Þ

This shows that the new model is consistent with the classicalmodel for mass flux across a membrane. Typically, in desalinationstudies, the rejection rate of salt, R, can be assumed to be constantsince the concentration of salt is very small (Yb ? 0) and Rapproaches to 1. However, for membranes used in the gas separa-tion process; the concentration of both species is comparable andso the rejection rate must be treated as a function of the mass frac-tions and the membrane properties.

The mass transfer coefficient, the corresponding local Sherwoodnumber, and the friction factor are determined from

hmðxÞ ¼D @Ya

@y

ðYm � YwÞ; Sh ¼ hmH

Dand f ¼ 2

qU2 sw ð19Þ

where hm is the mass transfer coefficient at x, Ym is the bulk massfraction of the species ‘‘a’’, Yw is the mass fraction on the membraneof species ‘‘a’’, Sh is the local Sherwood number, f is the local frictionfactor and sw is the wall shear stress.

4. Numerical method

Commercial software, CFX 14.5, is employed to solve the conti-nuity, the momentum and the mass transport equations. The CELexpression language, that is built in CFX, is used to model the mem-brane. CFX employs a fully-coupled solver for the velocity and pres-sure fields for both steady and transient simulations. Thepermeance of CO2 is PCO2 = 9.8 � 10�6 g

m2 s pa and the membrane

mass selectivity is a = PCH4=PCO2 = 0.0086. The spacer diameter tohydraulic diameter is d/h = 0.5. The total density of the mixture istreated as constant in the momentum and continuity equations.Seventeen cylinders placed in the computational domain as spacersfor S/d = 10 and thirty four cylinders for S/d = 5. The Schmidt

Fig. 4. Local value of Sherwood number as a function of x/h for Re = 1000. Solid linedenotes the results of the present numerical simulation while dashed line denotesthe Sh for the fully-developed laminar flow.

Table 1Comparison between the predicted and experimental results.

Re Cd experiments [37] Cd k–x BSL – present % Difference in Cd

250 1.95 1.93 % 1.2500 2.16 2.06 % 4.6

1000 2.37 2.10 % 7.11500 2.22 2.48 % 10.4

Fig. 6. The suction rate along the membrane for Re of (a) 200 and (b) 1000.

N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843 839

number Sc = t/D = 1.5 and the mass fraction of CH4 at the inlet isselected to be YCH4 = 0.7 and a total pressure difference across themembrane is chosen as Dptot = 7 MPa. Two different arrangementsof spacers are considered: the inline case, spacers are positioneddirectly behind one another, and the staggered case, a series of spac-ers which are unaligned. Circular and triangular cross-sectionedspacers are used in the simulations for both inline and staggeredcases. The spacing between the cylinders is selected to be S/d = 10and 5. For the staggered arrangement, the offset of the center ofthe cylinders, St, is selected such that St/d = 1. Fig. 2 depicts themesh structure near the first cylinder in the channel. It should benoted that the mesh spacing is finer inside the boundary layer.The same structure of mesh is used through the computationaldomain.

Fig. 5. The velocity vector and the contours of concentration for Re = 500 and S/d = 10. Images on the left denote the velocity vector field and images on the right denote thecontours of concentration. Images at the top and in the middle denote the inline and staggered geometries of circular spacers and the images at the bottom denote the inlinegeometry of triangular spacers.

Fig. 7. The normalized mass change of (a) CH4 and (b) CO2.

Fig. 8. The local value of the Sherwood number for Re = 1000.

Fig. 9. The Sherwood number as a function of Reynolds number.

840 N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843

4.1. Grid selectivity

In order to test the spatial convergence, simulations are con-ducted using 1.2 � 106, 1.8 � 106, and 2.1 � 106 cells. Fig. 3 showsthe steady velocity profile at x/h = 6.25 in the middle between twospacers for inline cylinder arrangement with S/d = 5 at Re = 1000.The velocity profiles, predicted using three different mesh sizes(1.2 � 106, 1.8 � 106 and 2.1 � 106), are nearly identical. The aver-age value of the drag coefficient calculated for different mesh sizesare 1.405, 1.396 and 1.385, respectively for the 1.2 � 106, 1.8 � 106

and 2.1 � 106 meshes. That ensures that the grid independency issatisfactory. The results presented in the present study are

Fig. 10. The averaged value of the friction factor as a function of Reynolds number.

N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843 841

obtained using 1.2 � 106 cells for both steady and transient simu-lations. In order to obtain temporal convergence a time step of8.0 � 10�6 for the LES and 2.5 � 10�5 for the transient k–x BSLmodel is used. These time steps are small enough to capture theperiodic nature of the Karman vortex street. The iterations forthe steady flow simulations end when the residuals in flow proper-ties are all less than 1e�6. Similarly, the iterations occurring withineach time step end as their residuals become less than 1e�6.

5. Results and discussion

The separation model in this work is validated by comparingthe predicted results against benchmark results. The membraneis considered to be a permeable wall and the flow is consideredto be laminar in the channel without any spacers. At high Reynoldsnumbers, the flow is expected to behave similar to that of flow nearan impermeable wall; this is because the suction rate is very smallcompared with main stream. The local value of the Sherwood num-ber (Sh) is calculated and compared against the Sherwood numberfor a fully developed flow, as shown in Fig. 4. The result predictedhere for the developing flow tends to the fully developed value ofSh asymptotically with normalized downstream position; validat-ing the membrane model. In order to validate the k–x BSL modelemployed here simulations are conducted for flow past a

Fig. 11. (a) The velocity vectors, (b) the contours of concentration, (c) the lift coefficientThe images are around (1) the first and the second cylinder, (2) the sixth and the seven

rectangular cross-sectioned cylinder for 250 6 Re 6 1500. The pre-dicted values of average drag coefficient agree well with thosemeasured by Davis et al. [37], as shown in Table 1. This illustratesthat the k–x BSL turbulence model is a good choice to simulate theflow in the geometry considered here for the range of Re of interest.

Fig. 5 shows the steady state velocity vector and the contours ofconcentration of CH4 around first two spacers in the channel forRe = 500 and S/d = 10. The velocity and the concentration fieldsare shown for inline and staggered arrangements of cylindricalspacers and the inline arrangement of triangle spacers. Thesteady-state flow structure illustrated in these images is a timeaveraged representation of what is, in reality, a periodic flow.The real flow structure is the classic case of a flow around a bluntbody, in which vortices form on alternating sides behind the bodyand shed periodically. This flow structure behind the body is calledthe Karman vortex street. There are notable differences in the flowstructures between these images. Large and small eddies are pres-ent in the wake of each spacer and the middle region toward themembrane. The momentum mixing, caused by the Karman vortexstreet, results in more uniform distribution of concentration andthe enhancement in the mass transfer through the membranes.The vortex shedding will also cause an increase in the pressuredrop across the system, this must be taken into consideration ifsuch a design is to be implemented in a real system.

vs time and (d) the power spectrum density of the lift coefficient predicted by LES.th cylinder, and (3) the last cylinder.

Fig. 12. Sherwood number comparison between LES and transient k–x BSL.

842 N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843

Fig. 6 shows the suction rate along the membrane for Re = 200and 1000 with S/d = 10. On Fig. 6, results for both the inline andstaggered arrangements of circular spacers and the inline arrange-ment of the triangular spacers are displayed. Fig. 6 also shows thesuction rate along the membrane for flows without the spacers forcomparison purposes. The suction rate along the membraneincreases as Re is increased. The spacer’s influence on the mass fluxthrough the membrane is more pronounced for high Reynoldsnumber flows, as shown in Fig. 6. The effect of the spacers on themembrane performance is not as strong at low Re flows for allgeometries considered here. The suction rate fluctuates along themembrane. It can be inferred from the data, that a local maximumforms near the spacer and a local minimum forms near the middlepoint between the spacers.

In order to determine membrane performance, the normalizedmass change of the CO2 absorbed and the CH4 lost is determinedfrom

Dma ¼massa;in �massa;out

massa;inð22Þ

where massa is the mass of species ‘‘a’’ in a binary mixture of CH4

and CO2. The subscripts ‘‘in’’ and ‘‘out’’ denote the mass of each spe-cies at the inlet and the outlet of the channel.

Fig. 7 depicts the normalized mass change of CH4 and CO2

across the channel as a function of Re. For comparison purposes,Fig. 7 displays both the normalized mass change of the inlineand staggered orientations. The mass of absorbed CO2 and lostCH4 for the inline and the staggered geometries with S/d = 10 and5 are compared against those without the spacer. As is seen, theeffect of vortex shedding on the CO2 extracted through the mem-brane is captured by the time averaged representation obtainedfrom the steady state simulations. The presence of spacers in thechannel enhances the passage of CO2 through the membrane,while reducing the percentage mass loss of CH4 through the mem-brane. This clearly shows that the spacers enhance the perfor-mance of the membrane and the improvement is greater athigher Reynolds number.

The Sherwood number is a better indicator of the membraneperformance since it eliminates the effects of the concentrationconditions selected at the inlet and the length of the test section.Fig. 8 shows the local value of the Sherwood number as a functionof x. The Sherwood number is calculated for the flow without thespacers and for the inline geometry of circular spacers withS/d = 10. The Sherwood number fluctuates in the flow directionwith a local maxima occurring near the spacers and a local minimaoccurring near the midpoints between spacers. The average valueof the Sherwood number in the case with spacers is more thanthree times greater than in the case without the spaces. This isattributed to the mixing caused by the Karman vortex street.

Sherwood number is averaged from x/h 20 to 80 and plotted asa function of Reynolds number in Fig. 9. Both the inline and stag-gered arrangements are displayed and compared with the nospacer flow case. The Sherwood number is elevated by the pres-ence of spacers at all Re. The degree of enhancement in the mem-brane performance is much greater at higher Re flows. For both theinline and the staggered geometry, the mass transport through themembrane increases as the spacing between blockages is reduced.Also the staggered geometry provides higher mass transport com-pared to the inline geometry at high Reynolds number. The shapeof spacer has little influence at low Re, but it has a stronger influ-ence at high Re. Triangular spacer leads to a higher mass fluxthrough the membrane compared to the circular spacers at high Re.

The friction factor is calculated along the surface of the mem-brane with and without the spacers. The average value of the fric-tion factor as a function of the Re is depicted in Fig. 10 for variousgeometries. Among the geometries considered here, the pressure

drop is the highest in case of the staggered cylindrical spacers withS/d = 5 while it is the lowest in the case of the inline cylindricalspacers with S/d = 10. The operation of gas–gas separation requiresa high pressure (on the order of several MPa) and the pressure ofthe gas leaving the separation is module is very high as well (alsoon the order of several MPa). The additional pressure drop causedby having spacers has no noticeable consequences (on the order ofkPa). Even when the process uses an energy recovery system, theenergy losses caused by the spacers will be negligible. As a result,the performance of the membrane should be measured mostly bythe increase in mass flux through the membrane without majorconsideration of the pressure drop.

In order to understand the influence of the temporal structureof the flow on the membrane performance transient simulationsare performed using transient k–x and LES turbulence modelingfor Re = 800. The flow past the circular spacers in an inline geome-try with S/d = 10 is simulated. The true effects of the Karman vortexstreet on the mass flux though the membrane can be determinedfrom these simulations. LES provides more details about the flowstructure produced by the Karman vortex street in the wake ofthe spacers. Fig. 11 illustrates the velocity vectors and the contoursof the CH4 concentration at t = 0.021 s. The images are shown atvarious locations of the flow domain: near the first and the secondcylinder (see Fig. 11a1 and b1), near the sixth and the seventh cyl-inder (see Fig. 11a2 and b2), and near the last cylinder (seeFig. 11a3 and b3). The effect of the mixing is clearly visible in theconcentration field. The velocity field is similar around all cylin-ders. The images clearly indicate that the mixture becomes CH4

richer at downstream locations. The lift force signature of the firstcylinder is shown in Fig. 11c. The periodically shedding swirlingvortices cause the lift force exerted by the fluid on the first spacerto fluctuate in time with a well-defined frequency of 668 Hz, asindicated by the power spectral density of the lift coefficient –see Fig. 11d. The Strouhal number of about 0.2 predicted by bothLES and the k–x model agrees well with that measured by [38].Fig. 12 compares the local value of the Sherwood number predictedby the LES and the transient k–x BSL model. The local value of the

N. Alkhamis et al. / International Journal of Heat and Mass Transfer 80 (2015) 835–843 843

Sh is calculated: a bit downstream the first cylinder (x/h = 5.5), atthe midpoint between the first and the second cylinder (x/h = 7.5), and a bit upstream of the second cylinder (x/h = 9.5). TheSh is plotted in Fig. 12 as a function of time. Even though the Shpredicted by LES and the k–x BSL method differs, the mass fluxof CO2 through membranes predicted by these methods agreeswithin 0.03%.

Computational study of three dimensional effects on the mem-brane performance is currently being conducted. The results ofthese studies will be published on a full paper.

6. Conclusion

Computational fluid dynamics simulations are conducted toinvestigate gas–gas separation using a membrane. The CO2 andCH4 binary mixture is considered in the mass transport through amembrane. A unique model is presented in which the membraneis modeled as a functional surface. The mass flux of each speciesis determined based on the local partial pressures of each species,the permeability, and the selectivity of the membrane. Both steadyand transient simulations are conducted by employing k–x andLES turbulence modeling for Re up to 1000. The effect of the shape,spacing and the configuration of spacers on membrane perfor-mance is studied. The amount of CO2 extracted increases and theamount CH4 lost decreases when blockages are placed inside thechannel. Such performance enhancement is more pronounced athigher flow rates. Staggering the spacers, increasing the frequencyof the spacers, and introducing triangular shaped spacers increasesthe mass transport through the membrane. The pressure dropincreases with the spacers, but such increase in the pressure dropis very small compared to the operating pressure of the membrane.Hence, its effect on the membrane performance is not significant.This study shows that spacers should be an important part ofmembrane systems used in gas–gas separation.

Three dimensional and transient effects in these flows, espe-cially with spacers, can be important. The present authors are cur-rently considering such effects. Also being considered by thepresent investigators are flows in membranes supported by porousmedia.

Conflict of interest

None declared.

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