Journal of Membrane Science 497 (2016) 458–471

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Journal of Membrane Science

http://d0376-73

n CorrE-m

journal homepage: www.elsevier.com/locate/memsci

Strategies for the simulation of multi-component hollow fibremulti-stage membrane gas separation systems

Michael Binns a, Sunghoon Lee a, Yeong-Koo Yeo a, Jung Hyun Lee b, Jong-Ho Moon b,Jeong-gu Yeo b, Jin-Kuk Kim a,n

a Department of Chemical Engineering, Hanyang University, Wangsimni-ro 222, Seongdong-gu, Seoul 133-791, Republic of Koreab Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Deajeon 305-343, Republic of Korea

a r t i c l e i n f o

Article history:Received 18 February 2015Received in revised form6 August 2015Accepted 10 August 2015Available online 13 August 2015

Keywords:Multi-stage membranesSimulation strategyNewton–Raphson methodGas separation

x.doi.org/10.1016/j.memsci.2015.08.02388/& 2015 Elsevier B.V. All rights reserved.

esponding author.ail address: jinkukkim@hanyang.ac.kr (J.-K. Ki

a b s t r a c t

Gas separation membranes allow the preferential removal of certain gases from a mixture of gases. If theseparation objective is to obtain high product purity and either high removal efficiency or high productrecovery it is often necessary to implement a multi-stage network of membranes. However, in the lit-erature most modelling approaches consider the simulation of single-stage membranes. Hence, the aimof this study is to identify stable and computationally efficient strategies for simulating complex multi-stage membrane systems. For this purpose a multi-stage membrane modelling framework is developedand six different stable solution strategies are evaluated and compared in terms of the computationaleffort required to solve the resulting sets of equations. These solution strategies vary according to thesequence in which the individual membrane models are solved (sequential and simultaneous ap-proaches) and the manner in which those membrane models are initialised. In all these strategies aNewton–Raphson method is employed to solve the mass balance equations in both single-stage andmulti-stage membrane systems. Comparisons are made using example simulations of 10 different con-figurations of membranes containing 1–4 membranes with different numbers of connections and recyclestreams present.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

A single-stage membrane used for gas separation will typicallyhave a single stream of feed gas and two exit streams called thepermeate and retentate as shown in Fig. 1. The permeate streamcontains gases which have passed through the membrane and theretentate contains any remaining gases from the feed.

For hollow fibre membranes there is a choice to be made as towhether to insert the feed into the bore side (inside the hollowfibre) or the shell side (outside the hollow fibre). In either case thehollow fibres will generally be encased in a module containingmultiple fibres so that the pressures of both sides of the mem-brane are contained and pressurised appropriately. Additionallythere are three basic configurations including co-current, counter-current and cross-flow which vary depending on the direction offlow, location of the exit streams and the influence of permeate-side mixing.

There have been a number of articles which describe metho-dology for the simulation of single-stage membranes. The

m).

equations which are used to describe these systems normally forma boundary value problem, i.e. sets of differential equations withboundary conditions at the inlet and outlets. One of the mostcommonly used and cited method for the simulation of multi-component membrane separations is the work of Pan [1] and theyimplement an iterative method (a shooting-type method [2]) tosolve this boundary value problem where they repeatedly in-tegrate along the membrane until reaching convergence. However,a number of authors have made simplifying assumptions andmodifications which allow the conversion of this problem into aninitial value problem [3,4] which is simpler to solve.

Alternatively, authors such as Coker et al. [5] and Katoh et al.[6] have tackled the boundary value problem directly by dividingthe membrane into a number of sections or tanks which isequivalent to a finite-difference approach to solve the problem.

The methods for obtaining numerical solutions describingthese membrane separations can then be divided into those whichrequire initial values for flow rates, compositions and pressuresinside the membrane such as those of Pan [1] and Coker et al. [5]and other methods which state that no initial values are required[4,7] (methods which require no initial values should convergestarting from a poor/crude initial guess rather than requiring agood initial values). The method of Kaldis et al. [7] uses collocation

Permeate

Retentate

Feed Retentate outlet

Permeate outlet

Fig. 1. Tanks-in-series model representation of a hollow-fibre membrane operated in the counter-current configuration.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 459

together with the Brown method to solve the equations for asingle-stage membrane while the method of Kundu et al. [4]converts the problem in to an initial value problemwhich is solvedthrough dynamic integration using Gear’s method incorporating avariant of Newton’s method. In both cases the authors state thattheir methods are both stable and computationally efficient.

However, the accuracy and computational effort required bythese approaches and by the finite difference methods depend onthe size of the time steps for integration methods, the number ofcollocation points for collocation methods and the number ofsections/tanks for finite-difference methods. These numbers aretypically chosen such that the models are sufficiently accurate toreproduce experimental results. For example Kaldis et al. [7] statethat only 6 collocation points are required to reproduce the ex-perimental results of Pan [1] within an accuracy of 5% while Cokeret al. [5] suggest that 100 finite difference tanks are sufficient formost modelling purposes.

Additionally, a major limitation of most membrane simulationmethodologies is that they consider only the solution of single-stage membranes. This is a significant point because it is knownthat multi-stage membrane systems are normally required to ob-tain high purity and high removal efficiency. For example Baker [8]suggests that due to pressure ratio and selectivity limits in com-mercial membranes a single-stage system may be unable to pro-vide the required separation. Additionally Low et al. [9] show thatfor CO2 capture a single-stage membrane is unable to simulta-neously give high CO2 purity and recovery.

Hence, there are many articles in the literature such as that ofAhmad et al. [10] which consider the simulation and optimisationof multi-stage membrane systems. Although it is possible to usesingle-stage methods to model multi-stage systems through thesequential simulation of the different connected membranes thiscan be a computationally inefficient approach. In particular forcases where one or more of the outlet streams are recycled (e.g. toenhance product recovery) this sequential solution strategy mayrequire a large number of iterations before it converges to asteady-state solution for the whole system. This is shown by Ma-karuk and Harasek [11] who implement a methodology whichsequentially and repeatedly simulates single-stage membranes inorder to obtain solutions for multi-stage membrane systems. Intheir results they show that for cases involving 2 membrane stageswith a recycle thousands of iterations are required to obtain asolution. Hence, single-stage methods can be used to model multi-stage systems but depending on the membrane configuration and

the algorithm used this can be a computationally inefficientapproach.

Ahmad et al. [10] have developed methodology for the simu-lation of single-stage membranes which they use repeatedlywithin a process simulator environment to simulate multi-stagemembrane systems. However, other authors including Khalipouret al. [12] and Kundu et al. [4] have presented methodology forsimulation of single-stage membranes and state that their meth-odologies can be applied to multi-stage membrane systems butthey do not specify how their methodology should be extended tothese multi-stage membrane systems. Hence, it is presumed thatthey use a similar approach as Ahmad et al. [10] repeatedly usingtheir single-stage membrane methodology.

This sequential approach has also been used inside optimisa-tion with the Levenberg–Marquardt method for the upgrading ofboth biogas from natural gas [13] and for the extraction of hy-drogen from biomass gasification [14]. However, in both cases it isassumed that a large computational effort was required due to thelarge numbers of iterations required for each simulation.

Additionally a number of authors have suggested methods forthe simultaneous solution of multi-stage membrane systems. Forexample authors such as Qi and Henson [15] and Scholz et al. [16]have included the equations for multi-stage membrane systems asnon-linear constraints inside their MINLP optimisation methods.As these authors do not mention any convergence issues (e.g. dueto numerical stiffness) which might occur at high or low stage cutit is assumed that implicit methods together with damping areused which should be able to handle most cases without difficulty.Also, in both cases these authors have used either simplified orshortcut models and they do not state the computational re-quirements of their multi-stage simulations or of the optimisa-tions which are carried out with these models. Katoh et al. [6] havealso considered the simultaneous solution of multi-stage mem-brane systems, in their case using a dynamic relaxation methodwhich should be a stable method for solving such systems but mayrequire large numbers of small time steps to reach a steady-statesolution. Although the number of steps can often be greatly re-duced using variable step-length methods the number of stepsrequired can still be very large in some cases (depending on thestiffness of the problem, the algorithm used and tolerances spe-cified). Hence, dynamic methods may be computationally slow(generally much slower than steady-state methods), but they areessential if the dynamic behaviour of the system is being studied.

So in summary for the solution of multi-stage membrane

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471460

systems there are three broad classes of solution strategies:

� Dynamic/relaxation-based methods� Sequential -&- repeated use of single-membrane based methods� Simultaneous solution of all membranes

Process simulators which are used to simulate flowsheetscontaining multiple different process units (reactors, separators,heat exchangers, pressure changing units, etc.) can also be used tomodel multi-stage membrane systems. These process simulatorscontain various sequential and simultaneous convergence meth-ods for solving the different models used to represent each unitand the various connections and recycle streams [19]. However,despite the large number of studies which have linked membranesimulation code/programmes with process simulators (e.g. [3,10])none of these articles give serious attention to the method ofconvergence used for multi-stage membrane systems. For exampleAhmad et al. [10] describe a single-stage membrane visual basicroutine which they have used inside Aspen HYSYS

s

for the simu-lation of a number of different multi-stage configurations but donot mention which convergence options they have used withinAspen HYSYS

s

.The choice of convergence methods can have a big effect on the

stability and computational effort required for simulation of multi-stage systems. Hence, for the purpose of design and optimisationof membrane systems which may require large numbers of si-mulations at various different conditions this choice becomes animportant issue (for simulation both inside and outside processsimulators).

The purpose of this article is to give a comparison of differentconvergence strategies for the simulation of multi-stage mem-brane systems. In particular looking at the computational effi-ciency (the CPU time required) and the stability of different in-itialisation and calculation sequences implemented. Hence, thiswork aims to provide guidance as to the most appropriate in-itialisation and convergence strategies which can be used for si-mulation and optimisation of complex multi-stage membranesystems.

2. Methodology

The modelling equations used here are based on those given byKatoh et al. [6] and Coker et al. [5]. Hence a (finite difference type)tanks-in-series model (similar to Katoh et al. [6]) is used here torepresent the two sides of the membrane (see Fig. 1). This can beused to represent any of the common feed flow patterns (cross-flow, co-current, counter-current) or any of the less common flowpatterns such as those using a sweep gas stream on the permeateside. In addition the following assumptions are made:

Assumption 1. Membrane permeances are fixed, independent oftemperature, pressure and gas composition.

Assumption 2. The geometry of the hollow fibres is unaffected byhigh pressures.

Assumption 3. There is no temperature change within themembranes.

Assumption 4. There is no pressure drop on the feed side of thehollow fibres (shell side).

Assumption 5. The bore side pressure drop is described by the

Hagen–Poiseuille equation.

Assumption 6. There is no concentration polarisation.

Assumption 7. The gases behave as ideal gases.

These assumptions largely neglect the influence of non-idealeffects which limit the accuracy of the model at very high pres-sures and for membranes with very high selectivity and per-meances. However, a model based on these assumptions is stillsufficiently accurate for the design of membrane processes. Thishas been shown by various studies which have used similar as-sumptions for the design of membrane processes (e.g. for theupgrading of biogas [16] and for the separation of CO2 from naturalgas [10]). It is also worth noting that the model presented herecould be modified to include these non-ideal effects, for example,through the adoption of the equations presented by Scholz et al.[17].

The approach for simulation of membrane separations requirescalculation of the volumetric flow (at standard conditions) of dif-ferent gases passing through the membrane and here this is basedon Eq. (1)

J Q A P x P y 1i j n i m r j i j p n i n, , , , , ,( )= − ( )

where Qi is the permeance of component i, Prj and Ppn are theretentate and permeate pressures of the jth retentate and the nthpermeate tank and Am is the area of the membrane connectingthose two tanks (see Fig. 1). Also, xij and yin are the mole fractionsof component i in the jth rentate tank and in the nth permeatetank.

In addition to this permeation of gases through the membranethe volumetric flow of gases passed between adjacent tanks mustalso be considered. The total volume flow rate of gases (includingall components) passing from tank j to tank jþ1 is given by Fr,j. Ifthe permeate is operated in a counter-current direction then Fp,nrepresents the volume flow rate of gases passed from tank n totank n�1 (in the co-current direction this would be the volumeflow rate from tank n to tank nþ1). The following set of Eqs. (2)–(5) are written assuming a counter-current flow pattern, to con-vert to either co-current or cross-flow would require only simplemodifications of Eqs. (3) and (5). These equations are used to re-present the separation of a mixture of Cn different gas components(e.g. CO2, CH4, etc.) using a membrane with Sr tanks used to re-present the retentate and Sp tanks used to represent the permeatein the numerical tanks-in-series model. Typically, a hollow fibremembrane module will include multiple membrane fibres (nfibre)and so the total feed should also be divided by the number of fi-bres used. Subsequently Eqs. (2)–(5) are used to simulate a singlefibre and the total outlet flow rates should be calculated by mul-tiplying by the number of fibres used. Additionally, since theseequations are written in terms of the normal volume flow (e.g.N m3 s�1) the feed flow rate provided should be converted tothese units.

For the first tank (tank 1) in the retentate Fr,j�1 and xi,j�1 in Eq.(2) should be replaced by Fr,feed and xi,feed which are the feed flowrate and composition of the gas feed. Also, for the final tank (tankSp) in permeate the terms Fp,nþ1 and yi,nþ1 are set equal to zero astank Sp is at the end of the membrane (in the counter-currentconfiguration).

dxdt

F x J F x

V S/ 2i j r j i j i j n r j i j

r r

, , 1 , 1 , , , ,=

− −( )

− −

dy

dt

F y J F y

V S/ 3i n p n i n i j n p n i n

p p

, , 1 , 1 , , , ,=+ −

( )+ +

Converged?

Yes

No

x y FF Pr p p, , , ,Initialise

Specify parameters:

F P Pfeed feed feed outletx

for all tanks

Compute cross membrane fluxes ( J )

F PFr p p, ,Calculate

Calculate dxdt

dydt

and

x x

y y

=

=

+

+

t

t

d d Lninner outer fibre

dxdt

dydt

Fig. 2. Relaxation-based solution strategy for simulation of gas separationmembranes.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 461

F F J4

r j r ji

C

i j n, , 11

, ,

n

∑= −( )

−=

F F J5

p n p ni

C

i j n, , 11

, ,

n

∑= +( )

+=

Eqs. (2) and (3) are ODEs (ordinary differential equations) de-scribing the rate of change of retentate and permeate mole frac-tions and Eqs. (4) and (5) are algebraic expressions for calculationof the total gas flow rates Fr and Fp. If the numbers of tanks on eachside of the membrane (Sr and Sp) are not equal then the crossmembrane volume flow rates Ji,j,n should then be added togetherappropriately in Eqs. (2)–(5) to reflect the flow rate of gases pas-sing between adjacent tanks. For example Katoh et al. [6] re-commend using a greater number of tanks on the permeate sidefor a number of applications. It should also be noted that Eq. (5)will change depending on the direction of the permeate flow rate.

In addition to the gas flow rates this model also requiresknowledge and calculation of pressures on both sides of themembrane. Here it is assumed that there is no pressure drop onthe retentate side while the permeate side change in pressure(from one tank to the next) is calculate based on the Hagen–Poi-seuille relation [5] in Eq. (6).

P L S F d128 / 1/ 1/ 6n n p p n,4( )( )μ πΔ = ( ) ( )

where in this equation L is the effective length of the membrane,μn is the viscosity in the nth tank and d here is the inner diameterof the hollow fibres.

In the work of Katoh et al. [6] the method of calculating visc-osities is not specified while in the work of Coker et al. [5] theyspecify that the mixing rules of Wilke are implemented but theydo not specify their method for the calculation of pure componentviscosities. The pure component viscosities are calculated hereusing the correlations of Lucas and the mixing rules of Wilkefound in the literature [18].

Two different approaches for solving these equations for sin-gle-stage membranes are considered here: a relaxation methodand a Newton–Raphson method. These methods are programmedin MatLab

s

and CPU times are obtained for comparison of differentmethods using an Intel

s

i5 3.40 GHz desktop computer with 8 Gbof memory.

2.1. Relaxation solution strategy for single-stage membranes

This study aims to find steady-state solutions for the above setof differential algebraic equations (DAEs). However, one of themost stable and reliable methods for achieving this is through theuse of relaxation methods where the above equations are nu-merically integrated with respect to time until the system reachesa steady-state. Unfortunately, this often requires a very largenumber of small time steps in order to reach the solution withoutintroducing any large perturbations which cause the method toeither fail or reach an unphysical solution. Hence, this method maybe considered inappropriate for the purposes of optimisation dueto the CPU time required (typically between 5 min and 1 h usingthe approach described here). This issue about heavy CPU times ismentioned by Katoh et al. [6] with regard to the use of small timesteps, however they do not report the CPU times required for theirsimulations.

The relaxation method implemented here follows the proce-dure given in Fig. 2 which is a modification of the explicit Eulermethod of integration. This is a relatively simple method, which ismodified here to account for the algebraic equations for total flowrates and permeate pressures which are recalculated after each

time step.Alternative more complex relaxation strategies such as that of

Kundu et al. [4] could also be implemented using backward dif-ferential methods which simultaneously solve the algebraicequations while integrating the ODEs. This could increase thecomputational efficiency but would require the calculation of aninitial point which satisfies the algebraic equations. However, inthis study the modified explicit Euler method is used because ourgoal when using relaxation is to obtain a solution in a simplemanner. Although explicit methods are less stable than implicitmethods small time steps can be used to make this simpler explicitmethod more stable and reliable.

2.2. Newton–Raphson solution strategy for single-stage membranes

In order to obtain solutions in a quick and efficient manner weconsider the use of a Newton–Raphson algorithm in order to

Evaluate function values

Converged?

Yes

No

x y FF Pr p p, , , ,Initialise

Specify parameters:

F P Pfeed feed feed outletx

for all tanks

d d Lninner outer fibre

Set =

Q

(e.g. using values previously generated by relaxation)

x y FF Pr p p, , , ,( )

Evaluate Jacobian of derivatives

Z

Jac ( Z )

F ( Z )

Gaussian elimination

Jac Z - F ( Z )=

Update including dampingZ,

Z Z + ( damp Z )=

Fig. 3. Newton–Raphson solution strategy for simulation of gas separationmembranes.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471462

directly solve the set of algebraic equations which are satisfied atsteady-state. At steady-state Eqs. (2)–(6) become the followingalgebraic Eqs. (7)–(11) which can be solved directly.

F x J F x

V S0

/ 7r j i j i j n r j i j

r r

, 1 , 1 , , , ,=

− −( )

− −

F y J F y

V S0

/ 8

p n i n i j n p n i n

p p

, 1 , 1 , , , ,=+ −

( )+ +

F F J09

r j r ji

C

i j n, 1 ,1

, ,

n

∑= − −( )

−=

F F J010

p n p ni

C

i j n, 1 ,1

, ,

n

∑= − +( )

+=

P P L S F d0 128 / 1/ 1/ 11n n n p p n1 ,4( )( )μ π= − + ( ) ( )+

These equations are solved using the procedure given in Fig. 3which is the Newton–Raphson method where analytical deriva-tives of Eqs. (7)–(11) are calculated. In this procedure Z representsthe set of unknown variables including x and Fr for each of the Srretentate tanks and y, Fp and Pp for each of the Sp permeate tanks.Also, in Fig. 3 F(Z) is the right hand side of Eqs. (7)–(11) and Jac isthe related Jacobian containing derivatives of the terms in F (i.e.dF/dZ, see [2] for details of the Newton–Raphson method).

This is similar to the method of Coker et al. [5] except that intheir case the pressure drop equation is not solved simultaneously,rather it is updated following the calculation of flow rates (re-quiring the subsequent re-calculation of flow rates), an iterativeprocess which should be repeated until both flow rates andpressure values are converged. Authors including Chowdhury et al.[3] and Kundu et al. [4] have pointed out that this approach(sometimes referred to as a finite difference approach) requiresinitial estimates of flow rates along both sides of the membrane.Additionally estimates of the pressures along both sides of themembrane are also useful. However, as mentioned by Kaldis et al.[7] the sensitivity with respect to initial conditions (stability) is notdiscussed in the work of Coker et al. [5].

While it is true that methods such as the approach im-plemented here require initial values for flow rates and pressuresalong the membrane, in this study it was found that a single goodsolution can be used as an initial point in order to start the processand can be used to calculate a wide range of different conditionsusing the Newton–Raphson method. Hence, using this singlestarting point the fast Newton–Raphson method can be used re-peatedly within an optimisation procedure to find the optimalprocess conditions.

For the purpose of obtaining this “good” starting point anumber of different initialisation options have been considered.One option would be to obtain this initial starting point using arelaxation-based simulation as described in Section 2.1. However,additional initialisation options can also be considered using theavailable knowledge of the feed (the feed volume flow rate Fr,feed(for a single fibre) and composition xfeed). Hence, the followingdifferent options are considered:1) Initialisation using the relaxation method

x x y x F F

F F i C j S n S

2 , , ,

1, ; 1, ; 1,

i j i feed i n i feed r j r feed

p n r feed n r p

, , , , , ,

, ,

) = = =

= ( = … = … = … )

x x y x F F

F F i C j S n S

3 , , /2,

/2 1, ; 1, ; 1,

i j i feed i n i feed r j r feed

p n r feed n r p

, , , , , ,

, ,

) = = =

= ( = … = … = … )

x x y x F F F i C j

S n S

4 , , , 0 1, ; 1,

; 1,

i j i feed i n i feed r j r feed p n n

r p

, , , , , , ,) = = = = ( = … = …

= … )

x x y F F F F i C j

S n S

5 , 0, , 1, ; 1,

; 1,

i j i feed i n r j r feed p n r feed n

r p

, , , , , , ,) = = = = ( = … = …

= … )

where options 2–5 are simple initial guesses and options 1 requirea relaxation-based simulation.

To quantify the stability of these different initialisation ap-proaches the maximum allowed damping coefficient (damp, asshown in Fig. 3) which can be used to converge to a real physicalsolution is determined. The use of a smaller damping coefficient inthe Newton–Raphson method (in the range 0odampr1 ) allowsfor a more stable convergence by reducing the size of the stepstaken. For example this is useful for cases where very high (0.94)or very low (0.1o) stage cut (¼total permeate outlet flow/totalfeed flow) simulations (or any cases involving numerical stiffness)

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 463

are required. Hence, determining the maximum allowed dampingcoefficient gives some empirical measurement of the stability ofthe numerical method at each set of conditions. The maximumvalue is determined here by testing values 0.1, 0.2, 0.3 …., 1.0 usingand recording the maximum value which converges to the “cor-rect” physical solution. For cases where the simulation does notconverge even with a damping value of 0.1 the maximum dampingvalue is recorded as 0 (although it may be possible to convergeusing an even smaller damping coefficient).

In general a smaller damping factor makes the convergence ofthe procedure more stable however it also increases the compu-tational time (as mentioned by Makaruk and Harasek [11]). Hencea good strategy to employ is to initially set a small damping factor(e.g. 0.1) then when the solution converges past a certain toleranceincrease this factor to 1 in order to accelerate the finalconvergence.

In comparison with the relaxation method, this Newton–Raphson solution strategy is very fast and direct, typically findinga solution using a much smaller number of iterations (compared tothe large number of time steps used for relaxation). The CPU timeinvolved is typically under 1 s without damping or 4–6 s ifdamping is employed.

2.3. Newton–Raphson solution strategies for multi-stage membranesystems

As mentioned in the introduction this article aims to comparedifferent strategies for the simulation of multi-stage membranesystems. In process simulators such as Aspen Plus

s

there are se-quential modular and simultaneous (equation-oriented) con-vergence strategies for the simulation of multi-unit systems [19].The advantages of the sequential approach include simpler

Set Recycle Flow Rates = 0

of 1st Membrane

Converged?

of MembraneNM th

Update Recycle Flow Rates

and Compositions using :

Yes

No

Multi-Membrane Simulation Stra

of 1st M

of NM th

Update Recyc

and Comp

Steady-state Simulation

Steady-state Simulation

Steady-sta

Steady-sta

Sequential

Sequential

Direct substitution / Wegstein /Newton-Raphson / Broyden

Set Recycle

Fig. 4. Multi-stage membran

initialisation and less effort required to formulate the problem [19]but for more complex systems and for optimisation this approachcan become computationally expensive [19]. Alternatively, si-multaneous approaches are can potentially be much faster butrequire good initialisation [19].

As single-stage membrane models are very commonly used inthe literature the simplest method for the solution of a multi-stagesystemwould be a sequential modular approach where the single-stage membrane programme is called repeatedly. However, for thepurpose of achieving faster convergence simultaneous approachescan also be considered at the expense of a more complex in-itialisation and formulation. Hence, a comparison of differentpossible solution strategies is made in this study with the aim tohighlight the most computationally efficient and stableapproaches.

2.3.1. Sequential strategiesWhen there are multiple membranes present it would seem

natural to implement the Newton–Raphson method developed forsingle-stage membranes sequentially for each membrane. Hence,starting with simulation of the 1st membrane connected to thefeed stream, then simulation of membranes connected to the exitstreams of the 1st membrane, then simulation of membranesconnected to the outlets of those membranes and so on. However,this can lead to large numbers of iterations required for con-vergence when there are recycle streams present (requiring re-peated simulation of the membranes, updating the flow rates andcompositions of the recycle streams after each set of sequentialsimulations). The general procedure for this type of simulationstrategy is shown as the “sequential” strategy in Fig. 4. When thereare recycle streams present a number of different strategies can beemployed for the updating of recycle stream properties after each

tegies

Set Recycle Flow Rates = 0

embrane

Membrane

le Flow Rates

ositions

Steady-stateSimultaneous

( including connections and recycles )

te Simulation

te Simulation

Simulation of all NM membranes

Simultaneous

initialisation Simple initialisation

Flow Rates = 0

e simulation strategies.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471464

sequence.

A. Direct substitutionB. WegsteinC. Newton–RaphsonD. Broyden

The four methods considered here include the simple directsubstitution method where the recycle streams are set equal to therelevant calculated membrane outlets. This method is the mostbasic which generally requires large numbers of iterations and sothe other three methods are intended to accelerate convergenceby improving the estimation of these recycle stream conditionsafter each sequence of calculations. In the Wegstein method this isachieved using a simple set of equations which use the valuesobtained from previous iterations to improve the estimation ofthese unknown recycle stream conditions in subsequent iterations.Alternatively the Newton–Raphson method can be implementedfor the calculation/convergence of these recycle streams (a nestedapproach where the Newton–Raphson method is used twice: forthe solution of single-stage membranes and for the solution ofrecycle streams). However, the calculation of the Jacobian for thisNewton–Raphson method requires the calculation of derivativesusing a numerical differences method (not shown in Fig. 4) whichis known to be computationally intensive. For this reason theBroyden method is also considered which includes a simplermethod for the approximate calculation of the Jacobian allowingfor faster iterations. A description of these methods which arecommonly used in process simulators can be found in the book ofTurton et al. [19].

2.3.2. Simultaneous strategiesAlternatively, the Newton–Raphson method can be extended to

solve all the membranes simultaneously, including all connectionsand recycles. This is known as the equation-oriented approachwhen used in process simulators [19]. In this way a system withmultiple membranes can be simulated in a quick and efficientmanner provided that it is appropriately initialised.

This multi-stage membrane Newton–Raphson method followsthe same procedure shown in Fig. 3, except that now there are oneset of Eqs. (7)–(11) included for each membrane (with additionalsets of equations for each of the multiple tanks used in the modelfor each membrane). Hence, if there Nm membranes used to se-parate a gas mixture with CN components, using Sr tanks to modelthe retentate and Sp tanks to model permeate, the total number ofequations to be solved will be Nm� [(CN� (SrþSp))þSrþSpþSp].Additionally there will now be an augmented set of unknownvariables Z*¼(Z1, Z2 … ZNm) and subsequently an augmented Ja-cobian of derivatives Jacbian* containing terms corresponding toeach membrane and to the various connections and recycles.

Z

Z

1

2 =

= ( )x y F F P1 1 1 1 1r p p, , , ,

( xMembrane 1

Membrane 2

R

P

P

1

2

1

Fig. 5. Example of a multi-stage membrane system w

Considering for example the 3-stage membrane system shownin Fig. 5 it can be seen that there are three sets of unknownvariables (Z1, Z2 and Z3) which need to be calculated based onthree sets of equations (F(Z1), F(Z2) and F(Z3)). Also it should benoted that for the first tank Eqs. (7) and (9) must be modified toinclude: the gas feed from the recycles added to the feed andpassed into membrane 1 (F(Z1)), the outlet of membrane 1 passedinto membrane 2 (F(Z2)) and the outlet of membrane 2 passed intomembrane 3 (F(Z3)). For this example the resulting set of equationswhich are solved as part of the Newton–Raphson method areshown in matrix format in Fig. 6 including derivative terms foreach of the three membranes in addition to derivatives related tothe various connections. It should be noted that the Newton–Raphson method is robust and it will still converge to a solutioneven if these derivatives related to connections between mem-branes and recycles are neglected and set equal to zero. In fact,setting these derivatives equal to zero can increase the stability ofthe method since it is effectively removing terms which have thepotential to introduce stiffness into the problem (i.e. due to thedifferent scales of operating conditions inside the different con-nected membranes). However, the inclusion of these terms canalso accelerate convergence reaching a solution using feweriterations.

The stability and rate of convergence of this simultaneous ap-proach depend on:

� the method used for initialisation.� the level of damping employed in the multi-stage Newton–

Raphson method.� the inclusion or exclusion of connection/recycle terms in the

Jacobian.

Here two different initialisation strategies are considered:

1. Sequential initialisation2. Simple initialisation

In the sequential initialisation approach each membrane is si-mulated in sequence as shown in the “Sequential initialisation”strategy in Fig. 4 providing initial values for the recycle streamswhich are subsequently used in the simultaneous methodologydescribed above. This provides a “good” initial point from whichthe simultaneous method can be used with or without damping.Alternatively, a “Simple initialisation” method can also be con-sidered in which all the recycle streams are set to zero flow rateand these initial values are used directly in the simultaneousmethodology. This gives a lower quality initial point which meansthat the method may fail to converge. However, this can be com-pensated for by using a small damping value inside the multi-stage Newton–Raphson method and/or by excluding (setting to

Z 3 =

)y F F Pr p p, , , ,

( )x y F F Pr p p, , , ,

2 2 2 2 2

3 3 3 3 3

Membrane 3

R

R

P3

2

3

ith three membranes including recycle streams.

ZJacobian

=

ZF

Z

Z

Z ZF

ZF

ZF

Connection TermsMembrane 1

Membrane 1 Membrane 1

Membrane 2

Membrane 2 Membrane 3Connection Terms

Membrane 2 Membrane 3

Recycle Terms Recycle Terms

Membrane 1

Membrane 2

Membrane 3

Derivatives

Derivatives

Derivatives

1

2

3

1

2

3

Fig. 6. Newton–Raphson method for simultaneous solution of multi-stage membrane system (in this case using 3 counter-current membranes with recycles).

Table 1Parameters used for validation cases.

Numberof fibres

Effectivelength(m)

Outerdiameter(μm)

Inner dia-meter(μm)

Feedpressure(kPa)

Permeateoutletpressure(kPa)

Case 1a 20 0.15 200 80 3528.0 92.8Case 2b 368 0.25 160 80* 690.0 100.0

* Assumed value.a Pan [1].

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 465

zero) the connection/recycle terms inside the Jacobian (e.g. see theconnection/recycle terms in Fig. 6).

Hence, two different multi-stage simulation strategies arerecommended:

E. Simultaneous – Sequential initialisation(without damping and including all connection/recycle terms inthe multi-stage Jacobian)

F. Simultaneous – Simple initialisation(with damping and excluding all connection/recycle terms in

b Feng et al. [20].

Table 2Membrane permeance values and feed composition used in validation.

Permeances (GPU)

the multi-stage Jacobian)Where strategy E takes advantage of the good initialisation

point provided by the sequential initialisation and hence does notrequire any further steps to enhance stability. While Strategy Fstarts from a simpler initial point and uses both the stability en-hancing modifications mentioned above.

N2 O2 CO2 CH4 C2H6 C3H8

Case 1a 0 0 40.0478 1.1124 0.3059 0.0596Case 2b 1.8 9.3 0 0 0 0

Feed composition (% volume)

N2 O2 CO2 CH4 C2H6 C3H8

Case 1a 0 0 48.5 27.9 16.26 7.34Case 2b 79.5 20.5 0 0 0 0

a Pan [1].b Feng et al. [20].

3. Single-stage membrane model validation and stabilityanalysis

To validate the above equations and modelling approach twodifferent sets of experimental data from the literature are con-sidered. This validation involves the comparison between modeland experimental results from single-stage membranes. Hence, anadditional assumption made here is that these models can beextended to multi-stage systems without loss of accuracy. Firstly,considering the removal of CO2 from natural gas and the experi-ments carried out by Pan [1]. Secondly, considering the separationof air and related experiments carried out by Feng et al. [20].

The details of the experimental conditions are given inTables 1 and 2 including the feed and permeate outlet pressures,the numbers of hollow fibres, the membrane geometries, thepermeances of each component and the feed composition.

For the case of removal of CO2 from natural gas Pan [1] givesresults based on a counter-current flow pattern provided at dif-ferent stage cuts (stage cut¼Permeate flow/Feed flow). For the air-separation case Feng et al. [20] give results based on both counter-current and co-current flow patterns.

The model results calculated with the procedures in Figs. 2 and3 (single-stage membrane relaxation and steady-state strategies)are used to simulate both cases and the comparisons of experi-mental values from the literature and model predictions are givenin Figs. 7 and 8. It can be seen that the model very accurately

predicts the outlet concentrations of all components for thecounter-current simulations of both cases. For the air-separationco-current configuration the model clearly shows a different resultfrom the counter-current case and despite the small deviationfrom experiment the model can still be used to accurately predictmembrane performance with reasonable accuracy.

For the case given by Pan [1] the stabilities of the 5 differentsingle-stage initialisation options mentioned in Section 2.2 aretested. For this purpose the maximum damping coefficient is de-termined for each of the 5 initialisation options for a range ofdifferent feed flow rates / stage cuts (as discussed in Section 2.2).This is shown in Fig. 9 where it can be seem that initialisationoptions 4 and 5 did not converge for any of the conditions tested(possibly requiring a damping coefficient smaller than the mini-mum 0.1 tested here) suggesting that these two options are the

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.765

70

75

80

85

90

95

100

Stage cut

Per

mat

e co

mpo

sitio

n (%

vol

)

CO2 (model)

CO2 (experiment)

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

5

10

15

20

25

Stage cut

Per

mea

te c

ompo

sitio

n (%

vol

)

CH4 (model)

CH4 (experiment)

C2H6 (model)

C2H6 (experiment)

C3H8 (model)

C3H8 (experiment)

Fig. 7. Comparison of model predictions against experimental data of Pan [1].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.982

84

86

88

90

92

94

96

98

100

Stage cut

N2

com

posi

tion

of re

tent

ate

outle

t (%

vol

)

Counter−current (model)Counter−current (experiment)Co−current (model)Co−current (experiment)

82 84 86 88 90 92 94 96 98 10020

30

40

50

60

70

80

90

100

N2 composition of rententate outlet (%vol)

N2 re

cove

ry

Counter−current (model)Counter−current (experiment)Co−current (model)Co−current (experiment)

Fig. 8. Comparison of model predictions against experimental data of Feng et al.[20].

Stage cut0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Max

imum

dam

ping

coe

ffici

ent

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Initialisation option 1Initialisation option 2Initialisation option 3Initialisation option 4Initialisation option 5

Fig. 9. Comparison of the maximum damping coefficient for different initialisationoptions used in the single-stage Newton–Raphson membrane solution method.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471466

least stable. The most stable initialisation options are two of thesimpler options (2 and 3) which are shown to require little or nodamping allowing the maximum damping coefficient to be usedfor a wide range of conditions. However, all methods are shown torequire smaller damping coefficients in order to converge at veryhigh stage cuts. In principle the initialisation using a solution fromthe relaxation method calculated at conditions close to those beingsimulated with the Newton–Raphson method should be a verystable approach. However, it is shown here in Fig. 9 that the in-itialisation using a single solution from the relaxation method isless stable at feed conditions with stage cut much higher andmuch lower than that used to provide the initialisation. So forexample initialisation option 1 which is calculated here at condi-tions giving stage cut¼0.53 is less stable for much lower andmuch higher stage cut simulations. Hence, for this reason andbecause the relaxation method requires more computational effortit is only recommended if convergence issues are found usingoptions 2 or 3 at some certain feed conditions.

4. Example calculations: multi-stage CO2 removal from naturalgas

In order to compare the computational efficiency of the sixdifferent strategies considered here (four different sequentialmethods and two different simultaneous methods) example cal-culations have been performed using the ten different membraneconfigurations shown in Fig. 10.

For these example calculations a natural gas feed has been

selected with composition given by Pan [1] and with a flow rate of500 Nm3/s (a flow rate similar to that of flue gases from powerplants, another example where membranes are being considered[21]). In addition the feed pressures of all membranes have beenchanged to 400 kPa and the permeate outlet pressures of allmembranes have been changed to 100 kPa. Also, isothermal con-ditions have been assumed with a temperature of 10 °C. Pressurechanging units have not been included in Fig. 10 and it is assumedthat the necessary compressors are used to create this pressuredifference across each of the membranes. In a subsequent stepafter the convergence of membrane simulations the costs andenergy requirements of the required compressors and vacuumpumps can be calculated if required (e.g. for the design of amembrane system).

Given these feed conditions, pressures and using the per-meances given by Pan [1] it is necessary to estimate the requiredmembrane area which is required to give a significant separationof the feed gas. Here the significant separation mentioned can be

Fig. 10. Ten configurations used to test multi-stage membrane simulation strategies.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 467

defined as giving a stage cut (fraction of total gas feed permeatingthrough the membrane) between 0.1 and 0.9 and hence avoidingextreme situations where all or none of the gas is permeatingthrough the membrane. For this purpose a modified version of Eq.(1) can be used to obtain a crude estimate of the membrane arearequired based on these conditions (Eq. (12)).

AF

Q Pm

N m s

GPU 7.501 10 Pa 12estimatefeed

max

23 1

12

( )( ) =( )⋅ × Δ ( ) ( )

−

−

In this equation Qmax is the largest magnitude permeance value(from the set of components in the feed) in units of GPU, ΔP is thepressure difference across the membrane in Pa, Ffeed is the feed

flow rate in N m3 s�1 and Aestimate is an estimate of the membranearea required in m2. This equation will give a good “crude” initialestimate provided that the component of the gas mixture havingthe largest magnitude permeance makes up a large fraction of thetotal feed. Alternatively for cases where this is not valid Eq. (12)can be modified to include the average permeance (Qav) of the gasfeed (weighted according to the component fractions) as shown inEqs. (13) and (14).

AF

Q Pm

N m s

GPU 7.501 10 Pa 13estimatefeed

av

23 1

12

( )( ) =( )⋅ × Δ ( ) ( )

−

−

Table 3Membrane areas used for the simulation of 10 different membrane configurations.

Area (m2)

“small” membranes case (m2) “large” membranes case (m2)

Memb – 1 5.548�106 1.123�107

Memb – 2 5.548�105 1.123�106

Memb – 3 5.548�105 1.123�106

Memb – 4 5.548�104 1.123�105

Memb – 5 5.548�104 1.123�105

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471468

QQ x

QGPU

14av

iC

i i feed

iC

i

1 ,

1

n

n( ) =

∑

∑ ( )=

=

Hence, for the conditions in these example calculations theestimated area required is approximately 5.548�106 m2 based onEq. (12) or 1.123�107 m2 based on Eqs. (13) and (14). Based on thefibre geometry the number of fibres required can then be simplycalculated. As mentioned above these are crude estimates and theexact membrane area and/or pressure values should be chosenfollowing simulation and optimisation using the modellingmethodology presented here. However, these crude estimates areuseful as a starting point for such simulation and optimisationstudies.

For simulation purposes these two areas (calculated by eitherEq. (12) or Eqs. (13) and (14)) are used for the 1st membraneconnected to the feed. Each subsequent downstream membrane isarbitrarily given an area 90% smaller than the preceding upstreammembrane (to reflect the fact downstream membranes will typi-cally have feed flow rates smaller than those fed to the connectedupstream membranes). Hence, two cases have been consideredusing the different membrane areas for the simulation of the dif-ferent configurations as shown in Table 3. A “small” membranescase based on areas calculated with Eq. (12) and a “large” mem-branes case based on areas calculated with Eqs. (13) and (14).

Considering the two different sets of membrane areas, each ofthe ten configurations were simulated within MatLab

s

using anIntel

s

i5 3.40 GHz desktop computer with 8 Gb of memory fol-lowing the strategies given in the methodology section. In parti-cular the following 6 simulation strategies are tested:

A. Sequential – Direct substitutionB. Sequential – WegsteinC. Sequential – Newton–RaphsonD. Sequential – BroydenE. Simultaneous – Sequential initialisationF. Simultaneous – Simple initialisation

For configurations 1–4 where there are no recycle streams andso simulation strategies A–E operate in exactly the same mannergiving identical CPU times (strategies A–E vary only when thereare recycle streams present). The simultaneous solution step instrategy E is not needed if there are no recycle streams present andin such situations the system is solved using only the sequentialinitialisation. The remaining 6 configurations include recyclestreams and these are simulated using strategies A–F until theyconverge to a steady-state which satisfies the mass balances insideall membranes, as well as all connections and recycles.

In all strategies the single-stage membrane simulations arecarried out using damping together with initialisation using option3 from Section 2.2. Inside each single-stage membrane simulationdamp is initially set equal to 0.1 and later changed to 1 when themethod reaches a certain convergence criteria (norm (ΔZ) dividedby the number of unknown variableso0.1; where the norm of avector B is defined as norm(B)¼(B12þB2

2þ….)1/2). Completeconvergence is assumed for single-stage membranes when thisvalue becomes smaller than 1�10�6.

As mentioned in Section 2.3.1 the details of strategies A–D canbe found in the literature [19]. For these calculations in the Weg-stein method used in strategy B the acceleration parameter isconstrained between �5 and 0.

Strategy E is implemented using single-stage membrane si-mulations together with damping as described above for the se-quential initialisation. Additionally, the multi-stage Newton–Raphson method is implemented in this strategy without damping(damp¼1) and including all connection/recycle terms inside the

Jacobian. This is found to be a stable strategy because the se-quential initialisation provides a good initialisation.

Alternatively, strategy F uses only a simple initialisation of therecycle streams which is a relatively poor initial starting point.Hence this strategy is implemented initially using damping in themulti-stage Newton–Raphson method and excluding the connec-tion/recycle terms from the Jacobian. As with the single-stagedamping damp is initially set equal to 0.1 and later changed to1 when the method reaches a certain convergence criteria (norm(ΔZ*) divided by the number of unknown variables o0.1). Aftermeeting this criterion the connection/recycle terms are also re-inserted into the Jacobian to accelerate convergence. Completeconvergence for the multi-stage simulations (in strategies E and F)is assumed when norm (ΔZ*) divided by the total number of un-knowns in all membranes is less than 1�10�6.

The above strategies (A–F) were found to be stable convergingto the same physical solutions for each of the ten configurations.However, it is worth noting that there are a number of alternativesstrategies which are less stable. In particular it was found that:

Strategy D using a unity matrix as an initial Jacobian (a com-mon practise when using the Broyden method) leads to con-vergence problems and hence the implementation used here in-volves the calculation of a Jacobian for the first step using a nu-merical difference method.

All strategies using single-membrane Newton–Raphson meth-od failed without damping (i.e. if damp¼1 starting from the initialpoint).

Also, the multi-stage Newton–Raphson method used in strat-egy F failed to converge for all cases without damping and forsome cases if the connection/recycle terms in the Jacobian areincluded from the start.

Example simulation results are given for configuration 9 inFig. 11. These example results show that with this configuration avery high purity of CO2 is achieved with a flow rate of approxi-mately 10% of the feed.

The resulting CPU times required by each strategy for each ofthe 10 configurations are shown in Figs. 12 and 13 for the se-quential and simultaneous strategies using the “small”membranescase and in Figs. 14 and 15 for the sequential and simultaneousstrategies using the “large” membranes case. It is clear from thesefigures that while the sequential strategies are shown to be fast forconfigurations 1–4 which do not contain any recycles they are anorder of magnitude slower than the simultaneous strategies forconfigurations 5–10.

Considering the different sequential strategies the Wegsteinand Broyden methods are shown to give the best computationalefficiencies for these cases. The Newton–Raphson method is slowhere mainly because of the computational effort required for thecalculation of the Jacobian derivatives using numerical differenceswhich increases the computational effort required, in particular forcases with multiple recycles. The direct substitution method is alsoshown to be relatively slow, due to the large number of iterationsrequired.

The two simultaneous methods are shown to give relatively

Membrane 1

Membrane 2

Membrane 3

500

0.4850

0.2790

0.1626

0.0734

467

0.2998

0.1743

0.0786

0.4472

47

0.9989

0.0011

0.0000

0.0000

14

0.9536

0.0394

0.0065

0.0006

F

==

==

= 453

0.4318

0.3078

0.1794

0.0810

N m s3 -1

400 kPa=P

F

==

==

= N m s3 -1

400 kPa=P

F

==

==

= N m s3 -1

400 kPa=P

R

1

2

2

3RR

P

P

P1

3

143

0.9621

0.0322

0.0052

0.0005

F

==

==

= N m s3 -1

400 kPa=P

F

==

==

= N m s3 -1

kPa=P 100

F

==

==

= s190

0.9711

0.0245

0.0040

0.0003

N m3 -1

kPa=P 100

F

==

==

=x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

sN m3 -1

kPa=P 100

Feed

x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

x

xx

x

C

C

C

C

O

H

H

H

2

2

3

6

8

4

400 kPa

400 kPa

Fig. 11. Sample calculation results based on configuration 9 shown in Fig. 10.

Membrane configuration1 2 3 4 5 6 7 8 9 10

CP

U ti

me

(sec

onds

)

100

101

102

103

104

Sequential - Direct substitutionSequential - WegsteinSequential - Newton-RaphsonSequential - Broyden

Fig. 12. Comparison of CPU times required for the simulation of 10 different con-figurations the “small” membranes using different sequential strategies.

Membrane configuration1 2 3 4 5 6 7 8 9 10

CP

U ti

me

(sec

onds

)

5

10

15

20

25

30

35

Simultaneous - Sequential initialisationSimultaneous - Simple initialisation

Fig. 13. Comparison of CPU times required for the simulation of 10 different con-figurations the “small” membranes using different simultaneous strategies.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 469

fast convergence for all 10 configurations and for both of the dif-ferent membrane area cases (“large” and “small”). However, it isshown that the strategy with sequential initialisation method isfaster for most of the configurations with the exception of con-figuration 6. It is presumed that the simple initialisation approachis faster only for this configuration because configuration 6 has avery small recycle flow rate compared to the other configurations.Hence, the simple initialisation which uses a starting point withzero flow rate is a good initial point for this configuration (underthe operating conditions used here).

For all other configurations the sequential initialisation ap-proach is shown to give slightly faster convergence for the “small”membranes case. Considering the “large” membranes case (ex-cluding configuration 6) the sequential initialisation is shown to

give significantly faster convergence for configurations which in-clude one or more recycle streams. Hence, the simultaneousstrategy with sequential initialisation is considered to be the mostcomputationally efficient approach for use in the design and op-timisation of complex multi-stage membrane systems.

5. Discussion and conclusions

A modelling framework including equations necessary to si-mulate membrane separations is presented here based on ex-pressions taken from the existing literature. However, the existingliterature typically does not include details of the strategies em-ployed for the solution of such membrane systems and in parti-cular strategies for the solution of multi-stage membrane systemshave only been considered by a small number of different authors.

Hence, in this article six different strategies are considered forthe solution of multi-stage membrane systems. In the simpler

Membrane configuration1 2 3 4 5 6 7 8 9 10

CP

U ti

me

(sec

onds

)

100

101

102

103

104

Sequential - Direct substitutionSequential - WegsteinSequential - Newton-RaphsonSequential - Broyden

Fig. 14. Comparison of CPU times required for the simulation of 10 different con-figurations the “large” membranes using different sequential strategies.

Membrane configuration1 2 3 4 5 6 7 8 9 10

CP

U ti

me

(sec

onds

)

5

10

15

20

25

30

35

40

45

50

55

Simultaneous - Sequential initialisationSimultaneous - Simple initialisation

Fig. 15. Comparison of CPU times required for the simulation of 10 different con-figurations the “large” membranes using different simultaneous strategies.

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471470

sequential strategies any existing modelling methodology for thesolution of single-stage membranes can be implemented re-peatedly until an overall converged solution is obtained. However,it is shown in this study that while these strategies can obtainfeasible solutions, they may require a large computational effort(e.g. using large numbers of iterations). Four different sequentialstrategies are considered and it is found that among these se-quential strategies the Wegstein and Broyden methods gave thebest performance (lowest CPU times) for the set of example cal-culations tested.

The more complex simultaneous strategies are shown to givemuch faster (approximately 10 times faster) convergence thansequential strategies for the simulation of multi-stage membraneconfigurations containing recycles. Two different simultaneousstrategies are considered here which vary according to the waythey are initialised and the stability enhancing modificationswhich are implemented. The strategy (strategy E: Simultaneous –

sequential initialisation) giving the best overall performance in-volves the sequential initialisation of individual membranes fol-lowed by the simultaneous solution of all membranes, connectionsand recycles. This strategy does not require any stability enhancing

modifications in the simultaneous solution step as the sequentialinitialisation provides a good initialisation. Alternatively, a secondstrategy (strategy F: Simultaneous – simple initialisation) is alsoconsidered in which sequential initialisation is bypassed and thesimultaneous method is implemented starting from a relativelypoor initial point. Hence, this strategy requires stability enhancingmodifications (damping and excluding connection/recycle termsfrom the Jacobian of the multi-stage Newton–Raphson methodemployed) which allow the strategy to converge but also slowdown the rate of convergence. In the set of example calculationsperformed it was found that strategy E required lower CPU timesfor nine out of the ten membrane configurations tested. Strategy Fwas found to be faster only for the configuration and conditionswhere there is a relatively small recycle flow rate in which case thesimple initialisation gives a better starting point.

For these reasons (fast and stable solution strategy) it is re-commended that the simultaneous strategy with sequential in-itialisation should be used for simulation and optimisation ofmulti-stage membrane systems. This strategy and modelling fra-mework can be extended to any number of membranes with anynumber of physically feasible complex connections and recycles.Due to the shorter simulation time optimisation (which may re-quire thousands of simulations carried out at different conditions)also becomes substantially quicker and should give results withina shorter timeframe.

This simultaneous strategy is also important when consideringthe simulation of membranes connected to chemical or bio-chemical reactors, distillation columns, condensers and variousother pieces of equipment that need to be simulated at a gasprocessing plant (or refinery or petrochemical plant). If the modelequations are available for those other units then the same si-multaneous strategy with sequential initialisation can be extendedto include all the process units. Alternatively, the membrane net-work can be simulated in sequence with the other processeswhich may be simpler to implement (but potentially requiringmore computational effort to converge).

Acknowledgements

This work was supported by the Korea CCS R&D Center (KCRC)grant funded by the Korea government (Ministry of Science, ICT &Future Planning) (No.2014M1A8A1049338)

Nomenclature

Am membrane effective surface area between two ad-jacent tanks (m2)

Cn number of components (dimensionless)d and dinner inner diameter of the hollow fibre (m)douter outer diameter of the hollow fibre (m)damp damping parameter (dimensionless)F total volumetric flow rate to adjacent tank on the

same side of the membrane (N m3 s�1)F set of functions including Eqs. (7)–(11) (for all

membrane tanks)J Volumetric flow rate across the membrane

(N m3 s�1)L Effective length of the hollow fibre (m)Nm number of membrane stages (dimensionless)nfibre number of hollow fibre tubes (dimensionless)P pressure (Pa)Qi permeance of component i (N m3 m�2 s�1 Pa�1)

M. Binns et al. / Journal of Membrane Science 497 (2016) 458–471 471

S number of tanks used for modelling the retentate/permeate (dimensionless)

t time (s�1)T temperature (K)V volume of the retentate/permeate (m3)xi retentate mole fraction of component i

(dimensionless)yi permeate mole fraction of component i

(dimensionless)Z set of unknown variables (x, y, Fr, Fp, Pp) (various

units)

Greek characters

μn dynamic viscosity of the gas mixture in the nthtank of the permeate (Pa s)

ΔPn Change in pressure between tanks nþ1 and n (Pa)

Subscripts

i ith componentj jth permeate tankn nth retentate tankp permeater retentatefeed membrane feed (in the retentate)total total gas flow (considering all the membrane fibres

combined)outlet membrane outlet (in the permeate)

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[3] M.H.M. Chowdhury, X.S. Feng, P. Douglas, A new numerical approach for a

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