Document [original]

Optimal Operation of a Membrane Reactor Network

Erik Esche,∗,†Harvey Arellano-Garcia,†and Lorenz T. Biegler‡

Chair of Process Dynamics and Operation, Berlin University of Technology, Sekr. KWT-9, Str. des

17. Juni 135, D-10623 Berlin, Germany, and Department of Chemical Engineering, Carnegie

Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA

E-mail: erik.esche@tu-berlin.de

In honor of Günter Wozny’s 65th birthday

Abstract

In this work, a two-dimensional model for a conventional packed-bed membrane reactor

(CPBMR) is presented. The model incorporates radial diffusion and thermal conduction. In

addition, two 10cm long cooling segments for the CPBMR were implemented based on the

idea of a fixed cooling temperature positioned outside the reactor shell. The model is dis-

cretized using two-dimensional orthogonal collocation on finite elements with a combination

of Hermite for the radial and Lagrangian polynomials for the axial coordinate. Membrane

thickness, feed compositions, temperatures at the inlet and for the cooling, diameters, and the

amount of inert packing in the reactor are considered as decision variables. The optimization

results in C2yields of up to 40% with a selectivity in C2products of more than 60%. In addi-

tion, the CPBMR model is integrated into a membrane reactor network (MRN) consisting of

an additional packed-bed membrane reactor with an alternative feeding policy and a fixed-bed

reactor.

Keywords: OCM, Membrane Reactor Network, Orthogonal Collocation, large-scale NLP

∗To whom correspondence should be addressed

†Berlin

‡Pittsburgh

Motivation and Introduction

For remote, isolated wells of natural gas, a combination of steam reforming and Fischer-Tropsch

synthesis is often applied to turn methane into more easily transportable and chemically process-

able hydrocarbons. However, this process demands enormous amounts of energy and has an ef-

ficiency between 25 and 50% depending on reactant compositions and operating conditions.1An

alternative to this process is the oxidative coupling of methane (OCM), which has the potential to

become a key technology in chemical industry.2The OCM process allows for direct production of

alkenes (olefins) or alkanes from methane (CH4). It skips the energy intensive syngas formation

(steam reforming) and could thus potentially be more energetically and economically efficient.

This process offers various opportunities for replacing oil with natural gas.

As part of the Cluster of Excellence “Unifying Concepts in Catalysis” (UniCat)a, a mini-plant is

being built at the Berlin Institute of Technology (Technische Universität Berlin) to investigate the

technical viability of the OCM process on a larger scale. This contribution deals with the mod-

elling and optimization of a part of that mini-plant, namely, a membrane reactor network.

Forthwith, the La2O3/CaO-catalyst is employed for OCM. Given the exothermic nature of OCM

and the undesired simultaneous creation of carbon oxides, any practical application should allow

for good temperature control and low oxygen levels. Several apparatuses like fluidized bed reac-

tors and fixed-bed reactors (FBRs) have been tested therefore, where pellets in the bed carry the

required catalyst. In a classical FBR, the equilibrium composition can ideally be attained at the

outlet and the product streams need to be further processed to extract ethylene and other hydrocar-

bons. A more promising approach has been developed by Lafarga et al. in the form of packed-bed

membrane reactors (PBMRs), which offer additional benefits by gradually feeding oxygen to the

catalyst so as to allow for a higher selectivity in C2products, meaning lower carbon oxide for-

mation.3PBMRs are comparatively simple in their process design, and safer in operation than an

FBR.5They not only offer enhanced catalytic activity and selectivity, but also include the product

separation.

aFor further information on UniCat visit http://www.unicat.tu-berlin.de.

Nevertheless, a permeable membrane implies loss of reactants by diffusion to the non-catalytic

side of the reactor. One measure, which reduces this effect, is the introduction of a recycle stream

feeding a part of a product stream back into the system.71 shows the membrane reactor network

(MRN) proposed by Godini et al.8

Figure 1: Flowsheet of the proposed membrane reactor network. Figure redrawn in accordance

with.8

The network consists of three different types of reactors: a common fixed-bed (FBR) or plug flow

reactor (PFR), a conventional packed-bed membrane reactor (CPBMR) and an alternative, pro-

posed packed-bed membrane reactor (PPBMR).8

The foremost aim of this contribution is not to discuss the optimal structure of a membrane reactor

network for the OCM process, but to model and optimize the operation of the given network.

In previously conducted work by Jašo and Godini et al. , only one-dimensional models for the three

afore-mentioned reactors were applied to investigate the attainable reactor performance.2 9 Their

investigations of the reactor performance included the influence of operating temperature, mem-

brane thickness, methane-to-oxygen ratio at the reactor inlet, overall feed flow rates, gas stream

compositions, and reactor lengths. They saw the temperature rising especially in the FBR by more

than 500 K despite cooling the reactor through its outer shell. Godini et al. proposed the afore-

mentioned feeding policy for the given membrane reactor network.8Their strategy of running a

fixed-bed reactor and a conventional packed-bed membrane reactor alongside each other and con-

necting them both through a proposed packed-bed membrane reactor and recycles allows for an

increase in both yield and selectivity. They started with separate studies of all three reactors by

implementing one-dimensional stationary models to test the effect of oxygen accessibility. The

PPBMR differs from the CPBMR only insofar as methane and oxygen are co-fed to the packed-

bed (tube-side) of the reactor. This new feeding strategy allows for the above-mentioned network

improvements. Their analyses showed an overall yield in C2products of 23.21%, a C2selectivity

of 53.93%, and a methane conversion of 42.66%.

The following chapters present a brief overview on how all three reactors are modeled before

discussing the simulation and optimization of the membrane reactor network.

Derivation of Models

In order to model and optimize the whole network, a model for each of the reactors has been

developed. This section introduces models for all three reactors and outlines how source terms and

transport coefficients are calculated. Moreover, a collocation method for a set of partial differential

equations is discussed.

One- and Two-Dimensional Models: Previously implemented one-dimensional models have

shown higher yields in C2hydrocarbons than physically possible, the focus of this contribution

lies on two-dimensional modelling. The CPBMR is expected to have the largest impact on the

behaviour of the whole network by far. Therefore, a two-dimensional model is implemented for the

CPBMR and one-dimensional models are considered for the other two reactors. All symbols stated

in the following equations are noted and explained in the nomenclature. The one-dimensional

model for the FBR consists of the following differential equations describing concentration and

temperature profiles:

∂ci

∂z=˙cri

,(1)

∂T

∂z=

−kOS ·(T(z)−Tcool)·2+NR

∑

j=1ϕcat ·ρcat ·˙rr j·(−∆RH)·rFBR

ctot ·cp,mix ·uz·rFBR

.(2)

The reactor is heated or cooled through its lateral, outer shell. Tcool is the temperature of the cooling

jacket and kOS the respective heat transfer coefficient.

For the PPBMR the influence of the membrane and the shell-side of the reactor need to be added to

the set of differential equations. Consequently, each side has its own equations for concentrations

and temperatures as follows:

uz,T·π·r2

tube

∂ci,tube

∂z·dz =˙cri·π·r2

tube ·dz −d˙

Ni,diff(z)(3)

uz,S·π·r2

shell −r2

tube·∂ci,shell

∂z·dz =d˙

Ni,diff(z)(4)

∂TT

∂z=

∑

j=1ϕcat ·ρcat ·˙rr j·(−∆RH)·π·r2

tot ·cT

p,mix ·uT

z·π·r2

−

hdiff ·2·π·rS

tot ·cT

p,mix ·uT

z·π·r2

−˙qtrans ·2·π·rS

tot ·cT

p,mix ·uT

z·π·r2

(5)

∂TS

∂z=˙

hdiff ·2·π·rS

tot ·cS

p,mix ·uS

z·π·r2

S−r2

T+˙qtrans ·2·π·rS

tot ·cS

p,mix ·uS

z·π·r2

S−r2

T

−kOS ·TS(z)−Tcool·2·π·rS

tot ·cS

p,mix ·uS

z·π·r2

S−r2

T(6)

2(a) and 2(b) show a sketch of the CPBMR and a differential volume element of its tube-side,

respectively. Hence, the mass balance for the tube-side leads to:

(a) Sketch of the CPBMR for the isothermal model. (b) Differential segment of the tube-side of

the CPBMR for the isothermal model.

Figure 2: Balance volume for the derivation of the isothermal model for the CPBMR.

0=−uz·∂ci(r,z)

∂z+Di,r·∂2ci

∂r2+1

r·∂ci

∂r+˙cri(7)

and for the shell-side respectively:

0=−uz·∂ci(r,z)

∂z+Di,r·∂2ci

∂r2+1

r·∂ci

∂r(8)

The two-dimensional modelling moves the equations for the heat transfer through the outer shell

and the membrane to the boundary conditions. A differential energy balance of the tube-side of

the CPBMR leads to:

ctot ·cp,tot ·uz·∂T

∂z=λ·∂2T

∂r2+1

r·∂T

∂r+

∑

j=1ϕcat ·ρcat ·˙rr j·(−∆RH)(9)

For the description of the shell-side, the reaction term simply needs to be left out:

ctot ·cp,tot ·uz·∂T

∂z=λ·∂2T

∂r2+1

r·∂T

∂r(10)

3 shows the basic idea of the heated or cooled model for the CPBMR. Each 10 cm segment of the

Figure 3: Sketch for the heated or cooled model of the CPBMR.

reactor can be cooled or heated separately through the outer shell according to:

˙qheat/cool(z) = kOS ·T(r=rshell,z)shell −Tcool(11)

=−λG

mix(r=rshell,z)·∂T

∂rshell

(12)

Fluid Properties, Transport Parameters, and Reaction Kinetics: All gases are assumed to

behave as perfect or ideal gases. Correlations published in10 are employed to calculate viscosities,

thermal conductivities of pure components, etc.

The shell of the CPBMR is void of any internal installations. Hence, it can be assumed that the

radial flux in the shell is due to common gas diffusion. Fuller et al. present a semi-theoretical,

semi-empirical function for the calculation of binary diffusion coefficients.11 Using these binary

diffusion coefficients, Kee et al. introduced mixture averaged diffusion coefficients, which are ap-

plied as diffusion coefficients for the shell-side of the CPBMR.13 The packed-bed in the tube of the

CPBMR impedes the diffusion of gas and the axial flow through the packed-bed of course affects

the radial mass transport. In this contribution, an approach suggested by Tsotsas and Schlünder

et al. is used, which has already been successfully applied to a packed-bed membrane reactor, in

which a radial effective dispersion coefficient is defined for each component as the sum of a molec-

ular and a crossmixing term, where the molecular term may be calculated in accordance with Kee

et al.’s correlation.14 16

In this work, a porous membrane, which allows the permeation of gas, separates shell- and tube-

side of the CPBMR. The flux of any component through the membrane is calculated with the help

of Kundsen’s diffusivity theory as has been experimentally shown by Lafarga et al.17 Diffusion is

assumed to be the only radial transport mechanism. Therefore, the flux through the membrane of

a component ion either side can also be described with Fick’s law.

For the thermal conductivity of the gas mixture in the shell-side, a model presented in18 may be

used. To combine individual thermal conductivities into a single one for the whole mixture, the

rule developed by Wassiljeva, Mason, and Saxena19 is applied. Several approaches exist, which

describe the effective radial thermal conductivity λeff in a packed-bed. A model for the radial ther-

mal conduction published by Bauer and Schlünder20 21 is employed.

Two different types of transport cause a heat flux through the membrane separating shell- and

tube-side of the CPBMR: The diffusive mass transport brings about an enthalpy flow ˙

hmembrane and

conduction of the membrane itself enables heat transfer ˙qmembrane between both sides:

˙qmembrane =kmembrane ·(Ttube(r=rtube,z)−Tshell(r=rtube,z)) (13)

hmembrane =

∑

i=1˙ni,diff ·cp,i·(Ttube(r=rtube,z)−Tshell(r=rtube,z)) (14)

Specchia et al.22published correlations for the calculation of heat transfer coefficients for heat

transfer through walls adjoining catalytic packed-beds. For the one-dimensional case, a different

approach is required as there are no temperature gradients on either side of the membrane. Dixon23

developed correlations for the latter case. Moreover, the kinetic model of Stansch et al. for the

oxidative coupling of methane over a La2O3/CaO-catalyst is used in this work. Their reaction

mechanism is detailed in.24

Orthogonal Collocation for Reactor Models: All differential equations in this contribution are

discretized via orthogonal collocation on finite elements. Third order Lagrangian polynomials are

employed to collocate ODEs (ordinary differential equations) on finite elements using Radau roots

to guarantee the continuity of each variable across finite elements.26

For partial differential equations (PDEs) a combination of Hermite and Lagrangian polynomials is

derived for the descretization. In case second order derivatives appear in a differential equation, the

continuity of first order derivatives across finite elements needs to be ensured. For this application,

Hermite cubic polynomials are of advantage. Their usage guarantees the continuity of the function

itself and its first derivative between two adjoining finite elements. The polynomials employed

forthwith are taken from Finlayson, who used Hermite polynomials for the two-dimensional dis-

cretization of a sphere.28

The basic idea of extending the one-dimensional orthogonal collocation to a second dimension is

to use different functions for each direction, which depend on different variables and then multiply

both of them.

cLH(u,v) = 4

∑

l=1

au,l·`l(u)!· 4

∑

l=1

av,l·Hl(v)!(15)

Equation 15 can also be written as follows where ai,j=au,i·av,j:

cLH(u,v) = a1,1·`1(u)·H1(v)+a1,2·`1(u)·H2(v)+a1,3·`1(u)·H3(v)(16)

+a1,4·`1(u)·H4(v)+a2,1·`2(u)·H1(v)+...+a4,4·`4(u)·H4(v)(17)

The new approximation function, which is basically a surface function, contains 16 coefficients

ai,j, half of which assume the value of the collocated variable at certain collocation positions and

the other half are the respective first, radial derivative. This is depicted in 4.

Application of Orthogonal Collocation to Reactor Models: The two-dimensional model for

the CPBMR is discretized using orthogonal collocation as described above. Hence, the CPBMR

displayed in 2(a) needs to be divided into several axial and radial finite elements. The following

scheme is applied to formulate a linearly independent set of equations based on the discretized

differential equations: The assessment of each stand-alone reactor and the network is done using

Figure 4: Depiction of the collocating surface function using Hermite and Lagrangian polynomials

and the meaning of respective collocation variables.

Godini et al.’s8definitions of yield Y, selectivity Sin C2hydrocarbons, and CH4conversion X.

In order to solve the resuling NLP problem , the interior point barrier method implemented in

IPOPT is used. For details see Biegler et al.29 30

Stand-Alone Operation of the CPBMR

Preceding the optimization, extensive simulation studies are carried out. To accurately simulate the

CPBMR, three radial finite elements for the tube-side, two for the shell-side, and twelve axial finite

elements are required. By making each of the aforementioned model parameters in turn dependent

on local concentrations and temperatures, it could be found that all fluid properties and transport

coefficients should in fact not be calculated with averaged concentrations and temperatures, but us-

ing locally dependent values. Relative errors of mass and atom balances are below 10−5. Ignoring

the heat of reaction or the influence of the heat loss through the outer shell can have a big impact on

the performance. Non-isothermal models should be preferred at all times. A comparison between

the two-dimensional and a one-dimensional model shows an overestimation of the reactor perfor-

mance in terms of yield in C2hydrocarbons by as many as 25 percentage points. The full-scale

model for the CPBMR consists of around 160 000 variables. Due to the reaction kinetics and some

of the correlations for transport parameters, the entire system is highly non-linear. Consequently,

a number of measures is tested to improve the convergence behaviour of the entire system such

as avoiding non-differentiable points, scaling of variables (e.g. using natural logarithms), linearis-

ing constraints, increasing the sparsity of matrices, and tuning IPOPT. While the manual scaling

did not yield any actual improvements, especially as there are no practical ways to scale second

order derivatives, tuning IPOPT led to some measurable reductions of the convergence time. By

choosing MA57 from the Harwell Subroutine Library31 as a linear solver and the Metis package

for matrix reordering,32 the convergence time could be reduced to less than 25% in comparison to

the default configuration.

Optimization of the CPBMR:Several operational and geometrical parameters of the CPBMR

can be manipulated. The following 1 contains a comprehensive list of parameters of the CPBMR

that may be modified within given bounds. For matters of problem size, additional possible de-

cision variables are disregarded. Among those are the inlet pressures for shell- and tube-side,

superficial velocities, and selection of the right type of catalyst. Similarly, the reactor length will

be held constant at 20cm, because of the difficulties related to removing or adding an entire heat-

ing/cooling segment, each of which is 10 cm long. The lower and upper bounds noted in 1 state

what should be possible theoretically.

2 shows the configuration and the performance at the starting point (0), an intermediate step (1),

and the final optimization results (2). Land Uassign active lower or upper bounds on decision

variables, which are specified by the user. Figure 5 shows the concentration profiles for ethylene

and ethane and the temperature profile for the final optimization step noted in 2.

Table 1: Operational and geometrical parameters of the CPBMR that may be modified within the

given bounds.

Parameter Symbol Value Lower Upper Unit

bound bound

Geometrical Parameters

Diameter of tube dtube 0.007 0 dshell m

Diameter of shell dshell 0.010 dtube – m

Membrane thickness δmem 50 0.1 100 µm

Catalyst density ρcat 3 600 0 3 700 kg/m3

Catalyst volume fraction ϕcat 0.64 0 1 –

Operational Parameters

Temperature at inlet, shell-side TS

Inlet 1023.15 290 1375 K

Temperature at inlet, tube-side TT

Inlet 1023.15 290 1375 K

Cooling/heating temperature, seg. I TI

h/c1023.15 290 1375 K

Cooling/heating temperature, seg. II TII

h/c1023.15 290 1375 K

Molar fraction of oxygen, shell-side xS

O20.128 0 1 –

Molar fraction of nitrogen, tube-side xT

CH40.170 0 1 –

Table 2: Collection of decision variables for three steps in the optimization of the CPBMR and

respective performance.

No. TS

in TT

in TI

h/c TII

h/c xS

O2xT

CH4

[K] [K] [K] [K] [–] [–]

0 1023 1023 1023 1023 0.128 0.170

1 990L990L990L990L0.149U0.162L

2 970L1013 970L970L0.157U0.128

No. dTdSρcat ϕcat δmem

[mm] [mm] [kg/m3] [–] [µm]

0 7.0 10 3600 0.64 50

1 6.0L7.8L3700U0.70U62.4

2 6.0L7.8L3700U0.70U65U

No. Yield Selectivity Conversion

in C2in C2of CH4

0 0.302 983 0.550 275 0.550 604

1 0.440 726 0.685 512 0.642 914

2 0.468 500 0.632 873 0.740 276

Selectivity Target

While having a high yield in C2products is advantageous, the product gas still needs to be cleaned

of both reactants and side-products, like carbon oxides, before further processing. The selectivity

in C2hydrocarbons is a measure for how many of the reacted methane molecules formed hydro-

carbons and how many carbon atoms went into the formation of carbon oxides: the lower the

selectivity, the more carbon oxides are produced. The selectivities presented in the optimization

results for the stand-alone operation of the CPBMR are already quite high (≥60%). Nevertheless,

it is examined to what extent it is possible to further increase the selectivity for a given optimal

solution by enforcing a lower bound on the selectivity. As a starting point, the intermediate step 1

in 2 is chosen. The yield in C2hydrocarbons at that point lies at roughly 44% while the selectivity

is just above 68.5%.

Apparently, an increase in the selectivity target of one percentage point does not cause the yield to

drop by less than that amount. Yield decreases seem to be getting slowly larger when surpassing

a selectivity of 75%. Only seven of the eleven decision variables stay at their original value com-

paring the starting point to the last step of the selectivity target optimization. The most obvious

movement here is a shift to an even higher methane to oxygen ratio in the packed-bed as both

methane fraction and membrane thickness go up while the oxygen fraction goes down.

Discussion of Results Before proceeding to the next step – the integration of the CPBMR into

the membrane reactor network – a short discussion of all results so far is in order.

The performance of the CPBMR reported herein is – with respect to the yield in C2hydrocar-

bons – better than expected and reaches higher levels than have ever been experimentally found.

In order to simulate and optimize the CPBMR successfully, five radial and twelve axial finite el-

ements are required. However, this system seems to be touching its boundaries in the last few

optimization studies carried out here. It is possible that with an even larger number of radial finite

elements an even better performance with respect to the yield inC2hydrocarbons could be reached.

The incorporation of radial effects into the CPBMR model makes a difference and is vital for ob-

taining more sensible results in comparison to the one-dimensional case. It is exactly this radial

influence that makes the simulation and optimization of the CPBMR complicated as it is mainly

responsible for increasing the number of required variables by a factor of ten.

Overall, the general optimization of the CPBMR has confirmed some of the trends already found

in a rough sensitivity analysis that was carried out on the decision variables:

1. There seems to be a general trend towards a thicker membrane. This obviously reduces the

heat transfer between shell and tube, but the predominant effect seems to be the reduction

of the molar flux of heavier molecules. Oxygen enters the tube through the membrane in

the largest quantities, because of the large concentration difference between shell-side and

tube-side. Consequently, the thicker the membrane the lower the oxygen flux, and thus, the

lower the resulting oxygen concentrations in the reactor tube-side, which apparently ensure

the highest possible yields. The oxygen levels found in simulations and optimizations of the

CPBMR described above range between roughly zero and 500Pa. What is interesting to see

in this context is that there is – even at the outlet of the 20cm long CPBMR – still a positive

C2hydrocarbon formation rate. Common perception was that at that point the potential of

methane conversion should be exhausted.

2. Inlet temperatures of both shell- and tube-side have dropped below the original 1 023.15 K

and the cooling jacket is extracting some 28.5 W from the reactor in addition to the heat

transported away by the shell-side stream while ensuring an almost isothermal temperature

level in the tube-side of the CPBMR, and thus, allowing for optimal operating conditions

along the entire reactor length as shown in Figure 5(c).

3. With respect to the catalytic bed, there seems to be a trend pointing at the minimization of

the actual gas phase and covering as much of the packing with catalyst. However, this trend

should not be overrated. A sensitivity analysis shows that the actual increase in the yield

caused by this trend is comparatively small.

4. Another trend that has reemerged is the increasing dilution of the shell-side gas flow with

nitrogen. This can be understood as a further move towards near-isothermal reactor operation

as the higher dilution eases the exothermic effects of the reactions.

5. Lastly, a steady decline in the diameter of both tube and shell is observed. Obviously, this

again has two beneficial effects: First of all, the total heat caused by the reaction is smaller.

On a smaller diameter the heat transfer through the reactor shell is more effective.

Overall, it appears that the C2hydrocarbon production depends mostly on an optimal temperature

control and the presence of a small amount of oxygen in the packed-bed. Economically speaking,

however, there is a trade-off between a higher yield through dilution, diameter reduction, and the

actual amount ofC2hydrocarbons obtained in a reactor. Smaller reactors and higher dilution would

require more reactors in total and thus more effort when it comes to the actual product separation.

Finally, a few comments need to be made on some numerical issues:

1. After the intermediate step all further attempts to decrease the lower bounds on the temper-

atures have to be abandoned as the optimization just keeps running into either restoration

phase failures or local infeasibilities.

2. The above noted optimization formulations and tasks required, in total, nearly three months

to get to the last step.

3. For the last few tasks, the changes made to the variable bounds have to be chosen very

carefully and the increases consequentially become ever smaller.

It should, be noted that all the conclusions so far should be handled with care. It is still questionable

how accurate the model is and most of all to what extend Stansch’s kinetics are in fact applicable

in a conventional packed-bed membrane reactor.

There are of course a number of inaccuracies in the implemented reactor model apart from the

margins of error of all applied transport and fluid parameter correlations. One issue, in particular,

has to be revisited: One reason for the excellent performance of the CPBMR might be a question-

able applicability of the kinetics developed by Stansch et al.24 Their kinetics have been formulated

based on experimental data from the application of the OCM process in a microcatalytic fixed-bed

reactor. As their reactor does not allow for continuous oxygen injection along the reactor length,

the entire amount needs to be fed with methane. This obviously means that the oxygen concentra-

tions at the inlet of the fixed-bed will always be higher than in a PBMR. Consequentially, Stansch

et al. claim validity of their kinetics for oxygen partial pressures ranging from 1 kPa to 20kPa.

This can lead to some minor trouble in a fixed-bed reactor whenever oxygen is consumed by the

reaction mechanism and drops below 1 kPa, but it is almost certainly an issue from inlet to out-

let in a PBMR. In the conventional feeding-mode, no oxygen is being injected to the tube-side of

the membrane reactor. The only oxygen in the reactor tube-side arrives there by permeating the

membrane from the shell-side. Accordingly, the oxygen partial pressure will always stay at quite

low levels. In fact, it has been observed that whenever methane conversion is close to or larger

than 50% in the CPBMR model, the oxygen level at every single collocation position is well below

1 kPa ranging from 0 to 500 Pa. As the parameters of the kinetics were not fitted for this range

of partial pressures, it can easily be imagined that this leads to an overestimation (or possibly un-

derestimation) of the reactor performance. Given how a PBMR works, there is however no way

to guarantee oxygen levels of more then 1kPa in the fixed-bed – at least with this kinetic system.

A closer look at the formation rates of all components for those low partial pressures of oxygen

shows indeed a maximum for the formation of C2hydrocarbons for temperatures above 1000K,

well below 1000 Pa of oxygen. For details on this behaviour see Figure 6 This, by no means,

invalidates the kinetic system, but shows that a thorough experimental investigation is required.

Operation of the Membrane Reactor Network

This section deals with the simulation and optimization of the membrane reactor network shown

in Figure 1. After a brief description of the implementation of the one-dimensional models for

the FBR and the PPBMR, this part goes on by presenting some details on how those two and the

CPBMR are going integrated into the MRN. Lastly, details on the attempted general optimization

of the MRN will be presented. In addition, the assumption, that one-dimensional models for both

FBR and PPBMR suffice, is revisited by comparing their results against two-dimensional models.

Implementation of Models for FBR and PPBMR:The FBR is expected to be considerably

shorter than the CPBMR. Hence, only one heating/cooling segment is introduced. Methane and

oxygen need to be fed to the reactor at the same inlet. Consequently, the concentration profiles can

be expected to be steeper and temperature hot spots could be more of a problem. This basically

means that the length of individual finite elements needs to be a lot smaller and that more axial

finite elements are required in comparison to the CPBMR. Keeping the temperature in the reactor

in check is a bit more of a challenge compared to the CPBRM as feed dilution with nitrogen

gas of more than 80% together with a catalyst dilution of one to four was necessary in previous

work.33 It appears that the heat transfer through the outer shell is even more important in the FBR

compared to the CPBMR. It is found that especially oxygen disappears quite quickly, when the

temperature level gets out of control. Moreover, the axial derivative becomes so large that for

longer finite elements negative oxygen values are unavoidable as the collocation is incapable of

accurately following that decline. Apart from being unacceptable, the temperature increase is also

contradictory to the aim of achieving high yields in C2hydrocarbons. For a feed dilution of 85%,

methane conversion climbs to roughly 62.5%, however, selectivity is so low that there is close to

no yield in C2products at all. For a 87% dilution, this is fairly different: Methane conversion is

half as high at 33.8% and the yield in C2around 7.6%. The sudden formation of hot spots needs to

be taken into account as this requires a denser discretization. As a starting point, 1 cm of reactor

length will be discretized with 100 axial finite elements each 10−4m long. The higher oxygen

levels in the tube-side of the PPBMR cause the same trouble as in the FBR, meaning that yet again

a higher number of finite elements is required, which need to be quite short. The PPBMR in the

MRN sits right behind the FBR. Some additional oxygen and nitrogen is added to the flow leaving

the FBR before entering the PPBMR, but for now the concentrations of the flow leaving the FBR

will simply be reused for the inlet of the tube of the PPBMR. The required dilution with nitrogen

found here is obviously quite high. This is, however, not necessarily unexpected. The best yield

in C2hydrocarbons experimentally reported so far requires a dilution of methane with helium of

98% while allowing for a yield of 35% and a selectivity of 54% in a membrane reactor using a

Bi1.5Y0.3Sm0.2O3−∆-catalyst,.35 Figures 7 and 8 show the concentration profiles for the reaction

zones of both reactors at a feed dilution of 87%.

Integration of FBR,PPBMR, and CPBMR:For practical reasons, the network has basically

only two different feed streams: the first containing methane, the second oxygen. Both gases will

be diluted with nitrogen. This means that streams 2, 11, and 14 (see 1) consist of the same molar

fractions of oxygen and nitrogen, streams 1, 9, and 6 of the same molar fractions of methane and

nitrogen. As a starting point and to get a good match with the previously done simulations of FBR

and PPBMR, each stream is diluted to a molar fraction of nitrogen of 87%. Because of recycle

stream number 7 from PPBMR shell-side to CPBMR, tube-side superficial velocities in both FBR

and PPBMR are reduced to 0.4 m/s to ensure that the same can stay below or at 1 m/s in the

CPBMR. The heating/cooling temperature in the CPBMR is slightly decreased to 950 K to prevent

possible problems as a consequence of recycle stream no. 7 from the shell-side of the PPBMR to

the tube-side of the CPBMR. Similarly, as a further precaution, the shell-side inlet temperature of

the PPBMR is decreased to 900 K. Both stream 14, which is initally set to zero, and recycle stream

12, which will be activated to just 5% of its possible flow, can increase the oxygen concentration

in the tube-side of the PPBMR, and thus, strengthen exothermic reactions. 3 contains the results

of the network simulation for the configuration described above.

Table 3: Results of the MRN simulation.

Component Yield in Selectivity in Conversion of

C2Products C2Products Methane

FBR 0.047 666 0.194 696 0.244 823

PPBMR 0.008 562 0.142 233 0.060 195

CPBMR 0.418 325 0.603 931 0.692 669

MRN 0.293 901 0.430 539 0.682 637

This point is obviously far from being an optimal solution as both methane conversion and yield in

C2hydrocarbons of the network are lower than those for the individual CPBMR. Nevertheless, it is

a good starting point in order to show that the network model actually works. The CPBMR in the

network is probably close to the optimal solution found in the stand-alone optimization, because of

the high dilution with nitrogen required by the other two reactors. The reduction of the superficial

velocity in the FBR causes the temperature in that particular reactor to drop very quickly, thus,

reducing the reaction rates to nearly zero after the first 2mm of reactor length. The situation in

the PPBMR is quite similar, although the reaction rates do not become completely zero before the

reactor end.

General Optimization of the MRN:For the general optimization of the MRN, all geometrical

and operational parameters mentioned for the CPBMR can be manipulated. In addition, more

or less the same parameters are relevant for the other two reactors. The network itself offers

some additional decision variables through the manipulation of recycle streams and the two feed

streams. Initial sensitivity analyses at the afore-mentioned starting point show the appearance of

numerous (local) infeasibilities brought along by the additional two reactors. Generally speaking,

only the improvements described for the CPBMR above led to any improvements in the yield ofC2

hydrocarbons for the entire network. The sensitivity analysis would imply removing the additional

reactors. However, this could simply be because of the excellent performance of the CPBMR at

the starting point.

Further Investigation of FBR and PPBMR:The investigation on the CPBMR shows the ne-

cessity of its corresponding two-dimensional model. Given the size and complexity of the model

for the MRN, only one-dimensional models are first used for the additional reactors. In order to

further examine the implications of this simplification, two-dimensional models for both FBR and

PPBMR are implemented. Even after a few millimetres of reactor length, the results of one- and

two-dimensional models deviate by several percentage points. The simultaneous feeding of oxy-

gen and methane to the catalytic packed-bed leads to the formation of a hot-spot in the reactor

center, which cannot be seen in the one-dimensional case. In the case of the FBR, it might be

possible to tune the behaviour of the 1D system to the 2D, provided that the temperature can be

controlled more effectively. The latter is not possible for the PPBMR. The diffusive flux through

the membrane, yet again, necessitates the second dimension. A two-dimensional model for the

entire MRN amounts more than half a million variables and cannot be solved in a timely manner

using available hardware.

Conclusions and Outlook

Simulations carried out as part of this work show that using a two-dimensional instead of just a one-

dimensional model for the CPBMR is necessary and makes quite a difference at higher methane

conversion rates, although previously carried out research in this field36 suggested the difference

maybe rather small. The advantages of the CPBMR in contrast to the FBR have become fairly

obvious here. As was proven in the selectivity target investigation, it is possible to ensure both

high yields of more than 40% and selectivities of more than 70% at the same time. The fairly small

influx of oxygen through the membrane prevents side-reactions and helps keep the temperature

increase in check at the same time. In addition, the influence of a heating/cooling system on the

CPBMR has been tested. The configuration implemented here in combination with the feed dilu-

tion allows for almost isothermal temperature profiles in the reactor. These very helpful operating

conditions can, however, not be implemented in FBR and PPBMR. The higher oxygen concen-

trations in the catalytic bed lead to barely controllable temperature spikes causing the oxygen to

react fairly quickly and causing low yields in C2hydrocarbons. Further studies show that the radial

temperature dependence cannot a priori be neglected in the FBR and is almost certainly an issue

in the PPBMR because of the insulating effect of the shell-side.

All the previously drawn conclusions were made under the assumption of applicability for Stan-

sch’s kinetics. However, all results obtained for the CPBMR lie in a range for which Stansch et

al. do not claim validity for their kinetics. The partial pressure of oxygen simulated in the tube-side

of the CPBMR lies well below their lower bound of 1000 Pa between 0 and 500.

Therefore, an important topic for our current research work represents an examination of the appli-

cability of the reaction kinetics applied here. Experiments are on the way so as to test the operating

conditions with high yields and selectivities and to distinguish to what extent they are reasonable.

For example, Schomäcker et al.37 discover that lattice oxygen of a vanadium oxide catalyst plays

a greater role at low oxygen pressures. Therefore, the intermediate reduction of the catalyst might

influence the selectivity of the reaction mechanism.

(a) Ethane concentrations (b) Ethylene concentrations

Figure 5: Ethane, ethylene and temperature profiles for the last step of the CPBMR stand-alone optimization.

Figure 6: Rate of formation of C2products for various partial pressures of oxygen and different

temperature levels.

Figure 7: Concentration profiles for the FBR using a feed dilution of 87%.

Figure 8: Concentration profiles for the PPBMR tube-side using a feed dilution of 87%. For details

on which line symbolizes which component, please refer to 7.

Notation

Nomenclature

Symbol Meaning Unit Explanations/Comments

cconcentration mol/m3

cpspecific heat capacity kJ/kg K

˙cr component rate mol/m3s

henthalpy flux W/m2area specific enthalpy flow

kheat transfer coefficient W/m2K

`Lagrangian polynomial –

˙qheat flux W/m2area specific heat flow

rradius, radial coordinate m

˙rr reaction rate mol/m3s,

mol/g s

conversion rate of reactions,

differs between gas phase and

surface reactions

ttime s

uzsuperficial velocity m/s

zaxis, axial coordinate m

Ddiffusion coefficient m2/s

HHermite polynomial –

Nmolar flow mol/s

Ttemperature K

ϕvolume fraction –

λthermal conductivity W/m K

ρdensity kg/m3

i,j,kindex variables walk through components or

reactions

rradially

cat catalyst

cool variable belongs to heat-

ing/cooling system

diff diffusion

mix mixture

shell,Sshell-side

tot total e.g. sum or average over all

species

trans transfer e.g. heat transfer through a

membrane

tube,Ttube-side

Ggas

OS outer shell

CPBMR coventional packed-bed

membrane reactor

packed-bed membrane reac-

tor with a conventional feed-

ing policy

FBR fixed-bed reactor

MRN membrane reactor network network consisting of FBR,

CPBMR, and PPBMR

PBMR packed-bed membrane reac-

tor

PPBMR proposed packed-bed mem-

brane reactor

membrane reactor with an al-

ternative feeding policy

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List of Figures

1 Flowsheet of the proposed membrane reactor network. Figure redrawn in accor-

dance with.8...................................... 3

2 Balance volume for the derivation of the isothermal model for the CPBMR. . . . . 6

3 Sketch for the heated or cooled model of the CPBMR ................ 7

4 Depiction of the collocating surface function using Hermite and Lagrangian poly-

nomials and the meaning of respective collocation variables. . . . . . . . . . . . . 10

5 Ethane, ethylene and temperature profiles for the last step of the CPBMR stand-

aloneoptimization .................................. 22

6 Rate of formation of C2products for various partial pressures of oxygen and dif-

ferenttemperaturelevels................................ 23

7 Concentration profiles for the FBR using a feed dilution of 87%. . . . . . . . . . . 24

8 Concentration profiles for the PPBMR tube-side using a feed dilution of 87%. For

details on which line symbolizes which component, please refer to 7. . . . . . . . . 24

List of Tables

1 Operational and geometrical parameters of the CPBMR that may be modified within

thegivenbounds.................................... 12

2 Collection of decision variables for three steps in the optimization of the CPBMR

and respective performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Results of the MRN simulation............................ 18