differential evolution (de) strategy for optimization of hydrogen production, cyclohexane...

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Differential evolution (DE) strategy for optimization ofhydrogen production, cyclohexane dehydrogenation andmethanol synthesis in a hydrogen-permselective membranethermally coupled reactor

M.H. Khademi, M.R. Rahimpour*, A. Jahanmiri

Chemical Engineering Department, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz 71345, Iran

a r t i c l e i n f o

Article history:

Received 26 June 2009

Received in revised form

15 December 2009

Accepted 17 December 2009

Available online 20 January 2010

Keywords:

Methanol synthesis

Dehydrogenation of cyclohexane

Recuperative coupling

Hydrogen-permselective membrane

Hydrogen production

Optimization

Differential evolution

* Corresponding author. Tel.: þ98 711 230307E-mail address: [email protected] (

0360-3199/$ – see front matter ª 2009 Publisdoi:10.1016/j.ijhydene.2009.12.080

a b s t r a c t

In this work a novel reactor configuration has been proposed for simultaneous methanol

synthesis, cyclohexane dehydrogenation and hydrogen production. This reactor configu-

ration is a membrane thermally coupled reactor which is composed of three sides for

methanol synthesis, cyclohexane dehydrogenation and hydrogen production. Methanol

synthesis takes place in the exothermic side that supplies the necessary heat for the

endothermic dehydrogenation of cyclohexane reaction. Selective permeation of hydrogen

through the Pd/Ag membrane is achieved by co-current flow of sweep gas through the

permeation side. A steady-state heterogeneous model of the two fixed beds predicts the

performance of this configuration. A theoretical investigation has been performed in order

to evaluate the optimal operating conditions and enhancement of methanol, benzene and

hydrogen production in a membrane thermally coupled reactor. The co-current mode is

investigated and the optimization results are compared with corresponding predictions for

a conventional (industrial) methanol fixed bed reactor operated at the same feed condi-

tions. The differential evolution (DE), an exceptionally simple evolution strategy, is applied

to optimize this reactor considering the mole fractions of methanol, benzene and hydrogen

in permeation side as the main objectives. The simulation results have been shown that

there are optimum values of initial molar flow rate of exothermic and endothermic stream,

inlet temperature of exothermic, endothermic and permeation sides, and inlet pressure of

exothermic side to maximize the objective function. The simulation results show that the

methanol mole fraction in output of reactor is increased by 16.3% and hydrogen recovery in

permeation side is 2.71 yields. The results suggest that optimal coupling of these reactions

could be feasible and beneficial. Experimental proof-of-concept is needed to establish the

validity and safe operation of the novel reactor.

ª 2009 Published by Elsevier Ltd on behalf of Professor T. Nejat Veziroglu.

1. Introduction resources, predominantly natural gas. It is produced from

Methanol is an important multipurpose base chemical,

a simple molecule which can be recovered from many

1; fax: þ98 711 6287294.M.R. Rahimpour).hed by Elsevier Ltd on be

synthesis gas on a large scale worldwide. Synthesis gas

consists of H2, CO2, CO and some inert components. Methanol

conversion in a conventional fixed bed methanol reactor is

half of Professor T. Nejat Veziroglu.

mailto:[email protected]

http://www.elsevier.com/locate/he

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 5 ( 2 0 1 0 ) 1 9 3 6 – 1 9 5 0 1937

low due to the equilibrium nature of the reaction. Therefore,

most of the synthesis gas must be circulated around the loop

and this poses problems in operating costs. The methanol

synthesis reaction is exothermic and the total moles reduce as

the reaction proceeds. The temperature and pressure of

reaction are 495–535 K and 5–8 MPa, respectively [1].

Fig. 1(a) shows the schematic diagram of a conventional

methanol synthesis reactor. A conventional methanol reactor

is basically a vertical shell and tube heat exchanger. The

catalyst is packed in vertical tubes and surrounded by the

boiling water. The methanol synthesis reactions are carried

out over a commercial CuO/ZnO/Al2O3 catalyst. The heat of

exothermic reactions is transferred to the boiling water and

steam is produced. A two-dimensional steady-state simula-

tion of a single stage conventional type methanol reactor

shows that properties of reactor are not varying in the radius

of catalyst tube [2]. Recently, a dual-type reactor system

instead of a single-type reactor was developed for methanol

synthesis by Rahimpour et al. [3–5]. The dual-type methanol

reactor is an advanced technology for converting natural gas

to methanol at low cost and in large quantities.

Also several researches were performed on reactor modeling

and optimization of methanol synthesis. A dynamic simulation

and optimization for an auto-thermal dual-type methanol

synthesis reactor was developed by Askari et al. [6] in the

presence of catalyst deactivation using genetic algorithm. Also,

Rahimpour and Elekaei [7] present a study on optimization of

Product

Synthesis Gas from Reformer

Pro

Synthesis Gas from Reformer

Cyclohexane

Sweep Gas

H2 + Sweep Gas

a

b

Fig. 1 – A schematic diagram of (a) a conventional methanol synt

configuration.

a membrane dual-type methanol reactor in the presence of

catalyst deactivation using genetic algorithm. A theoretical

investigation has been performed in order to evaluate the

optimal operating conditions and enhancement of methanol

production in a membrane dual-type methanol reactor. In the

last few years, a Differential Evolution (DE) algorithm for dealing

with optimization problems has been proposed. DE algorithm is

a stochastic optimization method minimizing an objective

function that can model the problem’s objectives while incor-

porating constraints. The algorithm mainly has three advan-

tages; finding the true global minimum regardless of the initial

parameter values, fast convergence, and using a few control

parameters. Being simple, fast, easy to use, very easily adapt-

able for integrand discrete optimization, quite effective in non-

linear constraint optimization including penalty functions and

useful for optimizing multi-modal search spaces are the other

important features of DE algorithm. Recently, a dynamic opti-

mization of a novel methanol synthesis loop with hydrogen-

permselective membrane reactor is presented by Parvasi et al.

[8] using DE method. Also, Rahimpour et al. [9] studied

a dynamic optimization of a novel radial-flow, spherical-bed

methanol synthesis reactor in the presence of catalyst deacti-

vation using DE algorithm.

Multifunctional reactors integrate, in one vessel, one or

more transport processes and a reaction system (Agar [10];

Zanfir and Gavriilidis [11]) and are widely used in industries as

process intensification tools. These multifunctional reactors

BoilingWater

PureMethanol

Steam Drum

To Distillation Unit

duct

PureMethanol

Benzene

Hydrogen

Separator

To Distillation Unit

hesis reactor and (b) a membrane thermally coupled reactor


make the process more efficient and compact and result in

large savings in the operational and capital costs (Dautzen-

berg and Mukherjee [12]). A multifunctional reactor can be

used, for example, for coupling exothermic and endothermic

reactions. In it, an exothermic reaction is used as the heat

producing source to drive the endothermic reaction(s). In the

last years promising concepts for the recuperative coupling of

exothermic and endothermic reactions have been published

(Itoh and Wu [13]; Kolios et al. [14]; Moustafa and Elnashaie

[15]; Fukuhara and Igarashi [16]; Ramachandran et al. [17,18];

Abo-Ghander et al. [19]). Also, Khademi et al. [20] optimized

the methanol synthesis reaction and cyclohexane dehydro-

genation in a thermally coupled reactor using differential

evolution (DE) method. From these previous studies, coupling

of endothermic and exothermic reactions may enable both

the concentration and temperature profiles along the reactor

to be manipulated, shifting the conversion of thermodynam-

ically limited reactions to higher values, and efficiently using

the heat liberated by an exothermic reaction to provide the

endothermic heat requirements of the other reaction [21].

Chemical reactants may be shifted to products due to

thermodynamic equilibrium by removing the reactants from

product gases [22]. By insertion of Pd/Ag membrane in

a packed-bed reactor, optimal concentration and temperature

profiles via controlled dosing of reactant along the reactor can

be created, so that large improvements in conversion and

selectivity could be achieved and also pure hydrogen can be

produced [3,23–26]. Recently, hydrogen energy has been

attracting much attention due to its potential to reduce envi-

ronmental burdens and the viewpoint of energy security [26].

At present, hydrogen is produced almost entirely from fossil

fuels such as natural gas, naphtha, and coal. In such cases,

however, the same amount of carbon dioxide is released during

the production of hydrogen as that formed by direct combus-

tion of those fuels. Dehydrogenation reactions are an attractive

alternative for hydrogen production because it has essentially

zero CO2 impact giving a positive environmental contribution.

In the present work, a catalytic dehydrogenation reaction in

the endothermic side is used instead of the cooler-water in the

methanol synthesis reactor. The dehydrogenation reaction

chosen is the catalytic dehydrogenation of cyclohexane to

benzene. Fig. 1(b) shows a schematic diagram for the co-

current mode of a membrane thermally coupled reactor

configuration with three sides. The first side is an exothermic

side, where methanol synthesis takes place on the CuO/ZnO/

Al2O3 catalyst. The second side is an endothermic side, where

dehydrogenation of cyclohexane to benzene takes place on the

Pt/Al2O3 catalyst. The sweep gas flows through the third side

(permeation side) which selectively removes the hydrogen by

permeation through the Pd/Ag membrane. Heat is transferred

continuously from the exothermic reaction to the endo-

thermic reaction. The clear advantages of this integrated

catalytic membrane reactor include: achieving a multiple

reactants multiple products configuration, production of pure

hydrogen and possibility of achieving higher degree of in-situ

energy integration between the coupled endothermic dehy-

drogenation reaction and the exothermic synthesis reactions.

One of the important key issues in methanol reactor

configurations is implementing a higher temperature at first

parts of reactor for higher kinetics constants and then

reducing temperature gradually at the end parts of reactor for

increasing thermodynamics equilibrium. In this novel inte-

grated catalytic membrane reactor, this important key issue

achieves and leads to increase the methanol production at the

same operating conditions.

In our previous work [27], a distributed mathematical

model for membrane thermally coupled reactor that is

composed of three sides is developed for methanol synthesis

and cyclohexane dehydrogenation. The effect of various key

operating variables on the performance of the reactor is

numerically investigated. While the optimal operating condi-

tions can increase the production and reduce the operating

cost so, the purpose of this study is optimization of membrane

thermally coupled reactor using DE method, as a strong opti-

mization method. The inlet temperature of all sides, initial

molar flow rate of exothermic and endothermic sides, and

inlet pressure of exothermic side have been considered as

decision variables to reach maximum mole fraction of meth-

anol, benzene and hydrogen in the permeation side.

Rigorous mathematical models are excellent tools for the

exploration of the basic characteristics of such novel

configurations. Such an exploration can achieve consider-

able savings in money and time during the expensive stage of

pilot plant development. The continuous development of the

model in conjunction with the pilot plant optimal utilization

can also achieve considerable benefits on the road towards

the successful commercialization of such efficient novel

configurations.

The paper is organized as follows: reactions scheme and

kinetics are shown in Section 2. Mathematical model and

concepts about DE are explained in Sections 3 and 4 respec-

tively, followed by optimization of membrane thermally

coupled reactor in Section 5. Numerical solution is presented

in Section 6 with results and discussion in Section 7 and

conclusions are drawn in Section 8.

2. Reaction scheme and kinetics

2.1. Methanol synthesis

In the methanol synthesis, three overall reactions are

possible: hydrogenation of carbon monoxide, hydrogenation

of carbon dioxide and reverse water–gas shift reaction, which

are as follows:

COþ 2H24CH3OH DH298 ¼ �90:55 kJ=mol (1)

CO2 þ 3H24CH3OHþH2O DH298 ¼ �49:43 kJ=mol (2)

CO2 þH24COþH2O DH298 ¼ þ41:12kJ=mol (3)

Reactions (1)–(3) are not independent so that one is a linear

combination of the other ones. In the current work, the rate

expressions have been selected from Graaf et al. [28]. The rate

equations combining with the equilibrium rate constants [29]

provide enough information about kinetics of methanol

synthesis. The corresponding rate expressions due to the

hydrogenation of CO, CO2 and reversed water–gas shift reac-

tions over commercial CuO/ZnO/Al2O3 catalysts are:


r ¼k1KCO

hfCOf 3=2

H2� fCH3OH=f

1=2H2

KP1

i� �h � � i (4)
1
1þ KCOfCO þ KCO2fCO2

f 1=2H2þ KH2O=K

1=2H2

fH2O

r2 ¼k2KCO2

hfCO2

f 3=2H2� fCH3OHfH2O=f

3=2H2

KP2

i�


�hf 1=2H2þ�

KH2O=K1=2H2

�fH2O

i (5)

r3 ¼k3KCO2

hfCO2

fH2� fH2OfCO=KP3

i�


�hf 1=2H2þ�

KH2O=K1=2H2

�fH2O

i (6)

The reaction rate constants, adsorption equilibrium

constants and reaction equilibrium constants which occur in

the formulation of kinetic expressions are tabulated in Table 1,

respectively.

2.2. Dehydrogenation of cyclohexane

Hydrogen is an optimum large-scale fuel for the future,

although there remains some problems in transport and long-

term storage. One has to develop the use of alternative fuels

that are easily transformed into hydrogen and that can be

stored in liquid form, and, thus, more safely and economi-

cally. One of these fuels is cyclohexane.

The reaction scheme for the dehydrogenation of cyclo-

hexane to benzene is as follows.

C6H124C6H6 þ 3H2 DH298 ¼ þ206:2 kJ=mol (7)

The following reaction rate equation of cyclohexane, r4, is

used [30]:

r4 ¼�k�

KPPC=P3H2� PB

�

1þ�

KBKPPC=P3H2

� (8)

Table 1. – The reaction rate constants, the adsorptionequilibrium constants, and the reaction equilibriumconstants for methanol synthesis and dehydrogenationof cyclohexane reactions.

A B

Methanol synthesis reaction

k ¼ AexpðB=RTÞk1 (4.89 � 0.29)�107 �63,000 � 300

k2 (1.09 � 0.07)�105 �87,500 � 300

k3 (9.64 � 7.30)�106 �152,900 � 6800

K ¼ A expðB=RTÞKCO (2.16 � 0.44)�10�5 46,800 � 800

KCO2 (7.05 � 1.39)�10�7 61,700 � 800

ðKH2O=K1=2H2Þ (6.37 � 2.88)�10�9 84,000 � 1400

KP ¼ 10ðA=T�BÞ

KP1 5139 12.621

KP2 3066 10.592

KP3 �2073 �2.029

Cyclohexane dehydrogenation reaction

k ¼ AexpðB=TÞk 0.221 �4270

KB 2.03 � 10�10 6270

KP 4.89 � 1035 3190

where k, KB, and Kp are, respectively, the reaction rate

constant, the adsorption equilibrium constant for benzene,

and the reaction equilibrium constant that are tabulated in

Table 1. pi is the partial pressure of component i in Pa. The

reaction temperature is in the range of 423–523 K and the total

pressure in the reactor is maintained at 101.3 kPa. The catalyst

for this cyclohexane dehydrogenation reaction is Pt/Al2O3 [31].

3. Mathematical model

Fig. 2 shows a schematic diagram of the co-current mode for

a membrane heat-exchanger reactor configuration. A one-

dimensional heterogeneous model, which is a conventional

model for a catalytic reactor with heat and mass transfer resis-

tances, has been developed for this reactor in order to determine

the concentration and temperature distributions inside the

reactor. In this model the following assumptions are used:

� The gas mixture is an ideal gas in both catalytic reactor

sections.

� Both sections of the reactor are operated at steady-state

conditions.

� Radial variations in both beds are negligible (one-dimen-

sional model).

� With due attention to high gas velocity, axial diffusion of

mass and heat are negligible in both sections.

� Bed porosity in axial and radial directions is constant.

� Laminar plug flow is employed in both endothermic and

exothermic sides.

� The chemical reactions are assumed to take place only in the

catalyst particles and homogenous reactions are neglected.

� Heat loss to surrounding is neglected.

To obtain the mole balance equation and the energy balance

equation, a differential element along the axial direction inside

the reactor was considered. The balances typically account for

convection, transport to the solid phase, diffusion through the

membrane and reaction. The mass and energy balances for

solid and fluid phases in all sides, hydrogen permeation in Pd/

Ag membrane, pressure drop equation and boundary condi-

tions are summarized in Table 2. In equations (9) and (10), h is

effectiveness factor (the ratio of the reaction rate observed to

the real rate of reaction), which is obtained from a dusty gas

model calculations [28]. In equation (12), the positive sign is

used for the exothermic side and the negative sign for the

endothermic side. In equations (11) and (12), b is equal to 1 for

the endothermic side and 0 for the exothermic side. In equa-

tions (13) and (14), b is equal to 1 for hydrogen component and

0 for the sweep gas. In hydrogen-permeation equation, PH2 is

hydrogen partial pressure in Pa. The pre-exponential factor P0

above 200 �C is reported as 6.33 � 10�8 mol m�2 s�1 Pa�1/2 and

the activation energy Ep is 15.7 kJ mol�1 [32–34].

3.1. Auxiliary correlations

To complete the simulation, auxiliary correlations should be

added to the model. In the heterogeneous model, because of

transfer phenomena, the correlations of estimation of heat

and mass transfer between two phases and estimation of

Sweep gas

Permeation of hydrogen

Permeation side

Permeation sideC6H12

C6H6

H2

ArC6H12

Ar

CH3OHCO2

COH2OH2

N2

CH4

CH3OHCO2

COH2OH2

N2

CH4

Heat transfer

Endothermic side

Endothermic side

Exothermic side

Sweep gas H2

Fig. 2 – A schematic diagram of the co-current mode for a membrane heat-exchanger reactor configuration.


physical properties of chemical species and overall heat

transfer coefficient between two sides should be considered.

The correlations used for physical properties, mass and heat

transfer coefficient are summarized in Table 3. For the heat

Table 2. – Mass and energy balances for solid and fluid phasesbalances for permeation side, hydrogen permeation in Pd/Ag mconditions.

definition

Mass and energy balances for solid phase

Mass and energy balances for fluid phase

Mass and energy balances for permeation side

Hydrogen permeation in Pd/Ag membrane (Sievert’s law)

Pressure drop (Ergun momentum balance)

Boundary conditions

transfer coefficient between bulk gas phase and solid phase

(hf) in the exothermic and endothermic side, the heat

transfer coefficient between gas phase and reactor wall is

applicable.

in the exothermic and endothermic sides mass and energyembrane, pressure drop equation and boundary

Equation

avcjkgi;j

�yg

i;j � ysi;j

�þ hri;jrb ¼ 0 (9)

avhf

�Tg

j � Tsj

�þ rb

XN

i�1

hri;j

��DHf ;i

�¼ 0 (10)

�Fj

Ac

vygi;j

vzþ avcjkgi;j

�ys

i;j � ygi;j

�� b

JH2

Ac¼ 0 (11)

� Fj

AcCg

pj

vT gj

vzþ avhf

�T s

j � T gj

�� pDi

AcU1�2

�T g

2 � Tg1

�

� bJH2

Ac

Z T3

T2

CpdT� bpDi

AcU2�3

�T g

2 � T g3

�¼ 0 ð12Þ

�F3

vygi;3

vzþ bJH2

¼ 0 (13)

�F3Cgp3

vT g3

vzþ bJH2

Z T3

T2

CpdTþ pDiU2�3

�T g

2 � T g3

�¼ 0 (14)

JH2¼

2pLP0exp��Ep

RT

�

ln�

DoDi

� � ffiffiffiffiffiffiffiffiffiffiPH2 ;2

p�

ffiffiffiffiffiffiffiffiffiffiPH2 ;3

p �(15)

dPdz¼ 150

ð1� 3Þ2mug

33d2p

þ 1:75ð1� 3Þu2

gr

33dp(16)

z ¼ 0; ygi;j ¼ y g

i0;j; T gj ¼ Tg

j0; P gj ¼ P g

j0 (17)

Table 3. – Physical properties, mass and heat transfer correlations.

Parameter Equation Reference

Component heat capacity Cp ¼ aþ bTþ cT2 þ dT3

Mixture heat capacity Based on local compositions

Viscosity of reaction mixtures Based on local compositions

Mixture thermal conductivity Lindsay and Bromley [35]

Mass transfer coefficient between gas and solid phases kgi ¼ 1:17 Re�0:42 Sc�0:67i ug � 103 Cussler [36]

Re ¼ 2Rpug

m

Sci ¼ m

rDim�10�4

Dim ¼ 1�yiPi¼j

yiDij

[37]

Dij ¼10�7T3=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=Miþ1=Mj

p

Pðv3=2ciþv2=3

cjÞ2

Reid et al. [38]

Overall heat transfer coefficient1U ¼ 1

hiþ Ai lnðDo=DiÞ

2pLKwþ Ai

Ao

1ho

Heat transfer coefficient between gas phase and reactor wall hCprmðCpm

K Þ2=3 ¼ 0:458

3Bðrudp

mÞ�0:407 [39]

Heat transfer

coefficient between sweep

gas and reactor

wall in the permeation side

hDHK ¼ 0:023ðruDH

mÞ0:8ðCpm

K Þ0:3

[40]


4. Differential evolution

Differential evolution (DE) method, a recent optimization

technique, is an exceptionally simple and easy to use evolu-

tion strategy, which is significantly fast and robust in

numerical optimization and is more likely to find a function’s

true global optimum [41]. A basic version of DE consists of

following steps [42,43]:

1. Input: dimensions of problem (D) and key parameters (the

population size NP, the scaling factor F, and the crossover

constant CR).

2. Generate initial population in upper (UB) and lower (LB)

bounds of each decision variable for NP times in an array.

3. Evaluate the objective function (performance index) for

each individual including penalty terms.

4. Perform mutation, crossover operation to obtain the trial

vector for each target vector in the population.

a) For each vector Xt (target vector), select three distinct

vectors Xa, Xb and Xc randomly from the population

array other than vector Xt. and create mutant vector as

mutant vector ¼ Xc þ F(Xa � Xb)

b) Perform crossover for each target vector with its mutant

vector to create a trial vector as below.

for k ¼ 1 to D

If ((random no. (0, 1)<CR) or k ¼ D)

trial vectork ¼mutant vectork

else

trial vectork ¼ target vectork

end for

5. Check whether the parameters of trial vector are within

the bounds. If an individual of this trial vector is outside

the bounds, then this parameter is assigned a value

randomly within the associated bounds.

6. Evaluate the cost for trial vector as in step 3.

7. Each trial or target vectors which has better performance

index is transferred as a member to the next generation.

8. Repeat steps 4–7 until the termination criterion is satisfied

(Section 4.2).

4.1. Different strategies of DE

Different strategies can be adopted in DE algorithm depending

upon the type of problem for which DE is applied. The strat-

egies can vary based on the vector to be perturbed, number of

difference vectors considered for perturbation, and finally the

type of crossover used. The following are the 10 different

working strategies proposed by [44]: (1) DE/best/1/exp, (2) DE/

rand/1/exp, (3) DE/rand-to-best/1/exp, (4) DE/best/2/exp, (5)

DE/rand/2/exp, (6) DE/best/1/bin, (7) DE/rand/1/bin, (8) DE/

rand-to-best/1/bin, (9) DE/best/2/bin, (10) DE/rand/2/bin.

The general convention used above is DE/x/y/z. DE stands

for differential evolution, x represents a string denoting the

vector to be perturbed, y is the number of difference vectors

considered for perturbation of x, and z stands for the type of

crossover being used (exp: exponential; bin: binomial). In this

study, the sixth strategy of DE, i.e., DE/best/1/bin is used.

Hence, the perturbation can be either in the best vector of the

previous generation or in any randomly chosen vector. Simi-

larly for perturbation either single or two vector differences

can be used. For perturbation with a single vector difference,

out of the three distinct randomly chosen vectors, the

weighted vector differential of any two vectors is added to the

third one. Similarly for perturbation with two vector differ-

ences, five distinct vectors, other than the target vector, are

chosen randomly from the current population. Out of these,

the weighted vector difference of each pair of any four vectors

is added to the fifth one for perturbation. In exponential

crossover, the crossover is performed on the D variables in one

loop until it is within the CR bound. The first time a randomly

picked number between 0 and 1 goes beyond the CR value, no

crossover is performed and the remaining D variables are left


intact. In binomial crossover, the crossover is performed on

each of the D variables whenever a randomly picked number

between 0 and 1 is within the CR value. So for high values of CR,

the exponential and binomial crossovers yield similar results.

In binomial case, the last variable always comes from random

noisy vector to ensure that the random noisy vector is different

from the target vector, and hence the above procedure is

applied up to D � 1 variables. The strategy to be adopted for

each problem is to be determined separately by trial and error.

A strategy that works out to be the best for a given problem

may not work well when applied for a different problem.

4.2. Choosing NP, F, and CR

Choosing NP, F, and CR depends on the specific problem

applied, and is often difficult. But some general guidelines are

available. Normally, NP should be about 5–10 times the number

of parameters in a vector. As for F, it lies in the range 0.4–1.0.

Initially F ¼ 0.5 can be tried then F and/or NP is increased if the

population converges prematurely. A good first choice for CR is

0.1, but in general CR should be as large as possible [41]. The

stopping criteria may be of two kinds. One may be some

convergence criterion that states that the difference between

the best and the worst evaluations of objective function in

current generation should be less than some specified value

[42]. The other may be an upper bound on the number of

generations. The stopping criteria may be a combination of the

two as well. The best combination of these key parameters of

DE for each of the strategies mentioned earlier is again

different. Price and Storn [44] have mentioned some simple

rules for choosing the best strategy as well as the corresponding

key parameters. In this study, the population size is chosen 100,

the scaling factor 0.8 and the crossover constant 1.0. Among

DE’s advantages are its simple structure, ease of use, speed, and

robustness. Already, DE has been successfully applied for

solving several complex problems and is now being identified

as a potential source for accurate and faster optimization.

5. Optimization of membrane thermallycoupled reactor

In this study, maximization of the objective namely,

summation of output mole fractions of methanol, benzene

and hydrogen in permeation side is considered. The objective

is as follows:

J ¼ yCH3OH þ yC6H6þ yH2

(18)

Six decision variables namely, inlet temperature of

exothermic side T01, inlet temperature of endothermic side T02,

initial molar flow rate of exothermic side F01, initial molar flow

rate of endothermic side F02, inlet pressure of exothermic side

P01, and inlet temperature of permeation side T03 are consid-

ered for optimization. For tubular and exothermic reactors,

temperature and pressure are two important parameters that

change during the reactor length and they have a direct effect

on thermodynamics equilibrium and catalyst activity. The

main reason to develop an inlet optimal temperature in all

sides is energy saving and reducing in size of pre-heater. While

the transferred heat from the exothermic side to endothermic

side is dependent on the ratio of the exothermic-to-endo-

thermic side flow rates, these two parameters are selected as

a decision variable. Jeong et al. [31,45] reported that the

cyclohexane dehydrogenation reaction occurs in 101.3 kPa,

hence the inlet pressure of endothermic side is considered

equal to this value and is not selected as a decision variable.

The bounds of decision variables are:

495 < T01 < 535 K (19)

423 < T02 < 523 K (20)

0:05 < F01 < 1:0 mol=s (21)

0:05 < F02 < 1:0 mol=s (22)

50 < P01 < 80 bar (23)

298 < T03 < 535 K (24)

To ensure that the temperature of synthesis gas at the

reactor inlet, T01 is not too low for the methanol synthesis

reaction to occur, the lower bound on inlet temperature of

exothermic side is set at 495 K. At high temperatures catalyst

starts to deactivation [1], hence an upper bound of 535 K is

chosen for inlet temperature of exothermic side.

While the activity of the Pt/Al2O3 catalyst in the tempera-

ture range of 423–523 K was evaluated in a conventional

packed bed and no significant deactivation was found after at

least 3 days of use [31], the bounds for the inlet temperature of

endothermic side, T02 is chosen. The lower and upper bounds

for the initial molar flow rate of exothermic and endothermic

side have been selected with no prior intention. The pressure

of methanol synthesis reaction is 50–80 bar [1], therefore this

domain is considered for the inlet pressure of exothermic side.

The environment temperature (298 K) is selected as the lower

bond for inlet temperature of permeation side, and its upper

bound is the same as upper bound for inlet temperature of

exothermic side.

Three constraints are also considered for optimization:

T2 < T1 (25)

495 < T1 < 535 K (26)

423 < T2 < 523 K (27)

The constraints on temperature (Eq. (25)) are based on the

minimum temperature required in order to make a driving

force for heat transfer from the solid wall.

The optimization problem considered above is then refor-

mulated so as to include the constraints. Penalty function

method is employed for handling constraints. The constraints

in equations (25)–(27) are incorporated into the objective

function (Eq. (18)) using penalty functions. This method

involves penalizing the objective function in proportion to the

extent of constraint violation (i.e., the penalty function takes

a finite value when a constraint is violated and a value of zero

when constraint is satisfied). In the present case, we used 107

as our penalty parameter. But this value depends on order of

Table 4. – Comparison between simulation and plant datafor conventional methanol synthesis reactor.

Reactor inlet Reactor outlet

Exp. Calc.

Composition (mol %)

CO2 3.45 2.18 2.43

CO 4.66 1.44 1.52

H2 79.55 75.71 76.54

CH4 11.72 12.98 12.96

N2 0.032 0.16 0.035

H2O 0.08 1.74 1.47

CH3OH 0.032 5.49 5.05

Feed flow

rate (mol s�1)

0.565 0.510 0.511

Temperature (K) 503 528 524.1

Pressure (bar) 61.83 61.16 61.59


magnitude of the variables in the problem; it may change from

problem to problem.

The objective function considered for minimization,

finally, is thus:

Minimize

f ¼ �Jþ 107X5

i¼1

G2i (28)

where

G1 ¼maxf0; ðT2 � T1Þg (29)

G2 ¼maxf0; ð495� T1Þg (30)

G3 ¼maxf0; ðT1 � 535Þg (31)

G4 ¼maxf0; ð423� T2Þg (32)

G5 ¼maxf0; ðT2 � 523Þg (33)

The resulting optimization problems are solved using the

proposed algorithm, DE.

6. Numerical solution

The formulated model composed of 11 ordinary differential

equations and the associated boundary conditions lends itself

to be an initial value problem. The algebraic equations in the

model incorporate the initial conditions, the reaction rates, the

ideal gas assumption, as well as aforementioned correlations

for the heat and mass transfer coefficients and the physical

properties of fluids. These equations along with the discretized

ordinary differential equations using backward finite differ-

ence form a set of non-linear algebraic equations. The reactor

length is then divided into 100 separate sections and the Gauss–

Newton method in MATLAB programming environment is used

to solve the non-linear algebraic equations in each section.

7. Results and discussion

The model of methanol synthesis side was validated against

conventional methanol synthesis reactor for a special case of

constant coolant temperature under the design specifications.

The comparison between simulation and plant data for

conventional methanol synthesis reactor is shown in Table 4.

It was observed that the model performed satisfactorily well

under special case of industrial conditions and the observed

plant data were in good agreement with simulation data.

In this section, optimal operating conditions in the co-

current membrane thermally coupled reactor are analyzed

and the predicted mole fraction, conversion and temperature

profiles are presented. The performance of the thermally

coupled membrane reactor is analyzed, using different oper-

ating variables, for methanol yield, cyclohexane conversion

and hydrogen recovery yield as follows:

Methanol yield ¼ FCH3OH; out

FCO; in þ FCO2 ; in(34)

Cyclohexane conversion ¼ FC6H12 ; in � FC6H12 ; out

FC6H12 ; in(35)

Hydrogen recovery yield ¼ FH2 ;3

FC6H12 ; in(36)

The reaction scheme for cyclohexane dehydrogenation

indicates that if all cyclohexane is converted to benzene and

hydrogen, then ideally, the hydrogen yield reaches the value of

three.

7.1. Base case

In order to establish a reference point, so that the influence of

various parameters can be evaluated, calculations are first

carried out for a ‘‘base case’’ and the operating conditions used

for all sides of the reactor are given in Table 5. Operating

conditions for the methanol synthesis side are similar to those

used by Rezaie et al. [46]. The inlet composition of the methanol

synthesis reaction is typical of industrial methanol synthesis

process. It corresponds to a hydrogen:carbon-dioxide ratio of 7

having small amount of CH3OH, CO and H2O together with inert

gases of CH4 and N2. On the endothermic side, the inlet mole

fraction of cyclohexane that is diluted with argon is the same as

that presented by Jeong et al. [31]. Thus, the base case aims to

investigate the situation when the cyclohexane dehydrogena-

tion is used to consumption the generated heat from methanol

synthesis and to cool down it, resulting in a higher temperature

atfirstpartsofexothermicsidefor higherkineticsconstantsand

then reducing temperature gradually at the end parts of reactor

for increasing thermodynamics equilibrium which is similar to

the temperature profile along a tube filled with catalyst within

a methanol conventional reactor. This allows comparison of the

methanol synthesis process in the optimized membrane ther-

mally coupled reactor (OMTCR) with conventional methanol

reactor (CMR) for similar thermal behavior. The simulation

results of the membrane reactor in the endothermic side are not

compared with any reference case.

7.2. Simulation and optimization

With due attention to subjects of Section 5, the optimization

approach is maximization of the mole fractions of methanol,

Table 5. – Operating conditions for methanol synthesisprocess (exothermic side), dehydrogenation ofcyclohexane to benzene (endothermic side) andpermeation side.

Parameter Value

Exothermic side

Gas phase

Feed composition (mole fraction)

CH3OH 0.005

CO2 0.094

CO 0.046

H2O 0.0004

H2 0.659

N2 0.093

CH4 0.1026

Catalyst particle

Density (kg m�3) 1770

Particle diameter (m) 5.47 � 10�3

Heat capacity (kJ kg�1 K�1) 5.0

Thermal conductivity (W m�1 K�1) 0.004

Specific surface area (m2 m�3) 626.98

Ratio of void fraction to tortuosity of catalyst particle 0.123

Length of reactor (m) 7.022

Bed void fraction 0.39

Density of catalyst bed (kg m�3) 1140

Tube inner diameter (m) 3.8 � 10�2

Tube outer diameter (m) 4.2 � 10�2

Wall thermal conductivity (W m�1 K�1) 48

Endothermic side

Gas phase

Feed compositiona (mole fraction)

C6H12 0.1

C6H6 0.0

H2 0.0

Ar 0.9

Inlet pressurea (Pa) 1.013 � 105

Particle diameterb (m) 3.55 � 10�3

Bed void fraction 0.39

Shell inner diameter (m) 6 � 1�2

Permeation side

Feed composition (mole fraction)

Ar (sweep gas) 1.0

H2 0.0

Total molar flow rate (mol s�1) 1.0

Inlet pressure (Pa) 1.013 � 105

Membrane thickness (m) 6 � 10�6

Thermal conductivity of membrane (W m�1 K�1) 153.95

Shell inner diameter (m) 8 � 10�2

a Obtained from Jeong et al. [31].

b Obtained from Markatos et al. [47].

Table 6. – The optimized parameters for membranethermally coupled reactor.

Parameter Value

Inlet temperature of exothermic side (T01), K 495.9

Inlet temperature of endothermic side (T02), K 476.4

Initial molar flow rate of exothermic side (F01), mol s�1 0.255

Initial molar flow rate of endothermic side (F02), mol s�1 0.082

Inlet temperature of permeation side (T03), K 535.0

Inlet pressure of exothermic side (P01), bar 80.00

Objective function 0.195


benzene and hydrogen in permeation side through the optimal

initial molar flow rate of exothermic and endothermic stream,

inlet temperature of exothermic, endothermic and permeation

sides, and inlet pressure of exothermic side. Differential

evolution method is applied to determine the optimal reactor

operating conditions for hydrogen, methanol and benzene

production process in a membrane thermally coupled reactor.

The results of the optimization for membrane thermally

coupled reactor (MTCR) (using Differential evolution method

and MATLAB programming) are summarized in Table 6.

According to Le Chatelier’s principle, when an independent

variable of a system at equilibrium is changed, the equilibrium

shifts in the direction that tends to reduce the effect of the

change. When pressure of the methanol synthesis reaction is

increased, the equilibrium shifts in the direction to increase

the methanol production. Therefore, the optimal inlet pres-

sure of exothermic side is 80 bar (upper bound of pressure).

The simulation of membrane thermally coupled reactor is

carried out using optimization results in Table 6 and the results

of this simulation are shown in several figures. Fig. 3(a)–(e)

shows the comparison of mole fraction of components in

exothermic side of optimized membrane thermally coupled

reactor (OMTCR) with conventional methanol reactor (CMR).

Fig. 3(a) illustrates the mole fraction profile of methanol along

the reactor, at steady-state for exothermic side of OMTCR and

CMR. Fig. 3(b)–(e) presents similar results for other components.

The important point as illustrated in these figures is a reaction

kinetic controlling in the upper sections of reactor while in

other sections, the rate of reactions has decreased to its equi-

librium value and equilibrium is controlling.

As it can be seen in Fig. 3(a), the comparison of methanol

mole fraction in exothermic side of OMTCR with CMR shows

that the methanol mole fraction in output of OMTCR is

increased by 16.3%. This novel configuration leads to delay in

thermodynamic equilibrium, while CMR reaches to equilib-

rium in the first half of the reactor. Fig. 4 is simultaneous plot

of mole fraction for cyclohexane, benzene and hydrogen in

the endothermic side of OMTCR along the reactor axis.

Hydrogen permeation from endothermic side to separation

side results in shifting of the reaction to right and increases

the benzene mole fraction and its purity. Fig. 5 shows profile of

hydrogen mole fraction along the reactor axis in the perme-

ation side of OMTCR. While there is a difference between

hydrogen partial pressure in the endothermic and permeation

side, hydrogen can continuously pass from the endothermic

side into the permeation side. Therefore, the hydrogen mole

fraction in the permeation side should increase along the

reactor length.

Fig. 6(a)–(c) shows axial temperature profiles for CMR and

OMTCR in exothermic side, endothermic side and permeation

side, respectively. In addition, the highest temperature is

observed at the exothermic side, since this is where heat is

generated. Part of this heat is used to drive the endothermic

reaction and the rest to heat the reaction mixtures in both

sides. The temperature of the dehydrogenation side is always

lower than that of the exothermic side in order to make

a driving force for heat transfer from the solid wall. Along the

exothermic side of OMTCR, temperature increases smoothly

and a hot spot develop as demonstrated in Fig. 6(a) and then

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

HC

3n

oit

ca

rf

elo

m H

O

Dimensionless length

OMTCR

CMR

0 0.2 0.4 0.6 0.8 1

0.07

0.075

0.08

0.085

0.09

0.095

OC

2n

oit

ca

rf

elo

m


OMTCR

CMR

0 0.2 0.4 0.6 0.8 1

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

noi

tc

ar

f el

om

OC


OMTCR

CMR

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

H2

noi

tc

ar

f el

om

O


OMTCR

CMR

0 0.2 0.4 0.6 0.8 1

0.58

0.6

0.62

0.64

0.66

H2

noi

tc

ar

f el

om


OMTCR

CMR

a b

c d

e

Fig. 3 – Comparison of (a) CH3OH, (b) CO2, (c) CO, (d) H2O and (e) H2 mole fraction along the reactor axis between exothermic

side of OMTCR and CMR.


decreases to 495 K. Note that in the exothermic side, the outlet

temperature is the same as the inlet temperature. According to

Le Chatelier’s principle, temperature reduction in the second

half of the OMTCR leads to shift the reaction in the direction of

methanol production. The maximum temperature of OMTCR

moves towards the right with respect to CMR. This leads to

increase of kinetic duration of methanol synthesis reaction

and consequently delay in thermodynamic equilibrium.

At the entrance of dehydrogenation side in OMTCR, the

temperature increases and a hot spot form and then the

temperature decreases (see Fig. 6(b)). As can be seen in

Fig. 6(c), temperature profile in permeation side of OMTCR

decreases along the reactor length. Also, temperature differ-

ence between exothermic and endothermic sides is shown in

this figure.

Fig. 7(a) shows the variation of rate of reaction for both sides

of OMTCR. Comparing the values for the reaction rates present

in the exothermic side, it can be seen that the predominant

reactions are hydrogenation of CO and hydrogenation of CO2;

however water–gas shift reaction can be neglected, its contri-

bution being significant. At the reactor entrance, the rate of

cyclohexane dehydrogenation increases rapidly which is due

to increase in temperature of endothermic side (see Fig. 6(b)).

Fig. 7(b) illustrates the variation of the generated and

0 0.2 0.4 0.6 0.8 1

0

0.02

0.04

0.06

0.08

0.1

noi

tc

ar

f el

om


C6H

12

C6H

6

H2

Fig. 4 – Profiles of cyclohexane, benzene and hydrogen

mole fraction along the reactor axis in the endothermic

side of OMTCR.

0 0.2 0.4 0.6 0.8 1

490

500

510

520

530

540

K ,e

ru

ta

re

pm

eT


OMTCR

CMR

0 0.2 0.4 0.6 0.8 1

470

480

490

500

510

520

530

K ,e

ru

ta

re

pm

eT


OMTCR

CMR

514

521

528

535

K ,e

dis

noi

ta

em

re

p f

o e

ru

10

15

20

25

K ,e

cn

er

ef

fid

er

ut

ar

e

a

b

c


consumed heat flux from the exothermic and the endothermic

reaction, respectively, and also transferred heat into endo-

thermic side, and transferred heat from permeation and

exothermic side along the reactor. In the first half of the

reactor, methanol reaction proceeds faster than dehydroge-

nation and as a result more heat is produced by the exothermic

reaction than consumed by the endothermic one. The excess

heat raises the temperature of the system in the first half of the

reactor as illustrated by the temperature profile in Fig. 6(a).

In this region, the generated heat flux is higher than the

consumed one. The system heats up and a peak in the

generated heat flux is observed. Afterward, the generated heat

flux decreases rapidly, mainly due to fuel depletion. The

opposite situation occurs when the consumed heat flux is

higher than the generated one. If the consumed heat flux is

higher than the generated one, the system starts to cool down

resulting to low temperature, which in turn decreases both

reaction rates. Thus, after a certain position along the reactor

(dimensionless length ¼ 0.25 in Fig. 7(b)) the generated heat

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

0.025

H2

noi

tc

ar

f el

om


Fig. 5 – Profile of hydrogen mole fraction along the reactor

axis in the permeation side of OMTCR.

0 0.2 0.4 0.6 0.8 1

500

507


ta

re

pm

eT

0 0.2 0.4 0.6 0.8 1

0

5

pm

eT

Fig. 6 – Variation of temperature for (a) exothermic side,

and (b) endothermic side of OMTCR and CMR and (c)

permeation side of OMTCR and temperature difference

between exothermic and endothermic sides along the

reactor axis.

flux becomes lower than the consumed one, which coincides

with a hot spot development (see Fig. 6(a) and (b)).

The transferred heat into the endothermic side is

summation of transferred heat from the permeation side by

convection and hydrogen diffusion, and transferred heat from

-1.5

-0.5

0.5

1.5

2.5

3.5

4.5


ml

om

,n

oit

ca

er

fo

et

aR

3-

s1

-

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

Rate o

f reactio

n fo

r cyclo

hexan

e

deh

yd

ro

gen

atio

n, m

ol m

-3s

-1

Hydrogenation of CO

Hydrogenation of CO2

Water-gas shift

(a)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


W,

xul

ft

ae

H

Generated heat in exothermic side

Consumed heat in endothermic side

Transferred heat into endothermic side

Transferred heat from permeation side

Transferred heat from exothermic side

a

b

Fig. 7 – Variation of (a) rate of reaction for both sides and (b) generated and consumed heat flux in exothermic and

endothermic side respectively and also, transferred heat into endothermic side, and transferred heat from permeation and

exothermic side along the reactor for the OMTCR.


the exothermic side. Along the reactor length, the heat values

consumed by the endothermic side and transferred into the

endothermic side are close to each other. This demonstrates

the efficient thermal communication between the all sides,

and which is due to high solid wall and membrane thermal

conductivity. At the reactor entrance, the transferred heat into

the endothermic side is upper than the consumed heat by the

endothermic side, which is due to high temperature differ-

ence. A decrease in the transferred heat from exothermic side

is observed near the reactor entrance, and is associated to the

low temperature difference between exothermic and endo-

thermic sides in that region as shown in Fig. 6(c). This

reduction leads to decrease in the transferred heat into

endothermic side. A maximum in the reaction heat fluxes

consumed and transferred into the endothermic side are

located at the same axial position, namely 0.25. After this

position along the reactor, the consumed heat by the dehy-

drogenation side becomes larger than the transferred heat

into the endothermic side and the system starts to cool down

(see Fig. 6(b)). As can be seen in Fig. 6(c), in the permeation

side, the temperature profile decreases smoothly which is due

to variation of transferred heat from the permeation side (see

Fig. 7(b)).

In Fig. 8(a) and (b), respectively, methanol yield in the

exothermic side of OMTCR and CMR, cyclohexane conversion

in the endothermic side and hydrogen recovery yield in the

permeation side of OMTCR along the reactor are shown. In

OMTCR, cyclohexane reaches 99.9% conversion, and meth-

anol 0.4773 and 0.4182 yields in the exothermic side of OMTCR

and CMR, respectively. The comparison of methanol yield in

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

dlei

Yl

on

ah

te

M


OMTCR

CMR

0

0.2

0.4

0.6

0.8

1


noi

sr

ev

no

ce

na

xe

hol

cy

C

0 0.2 0.4 0.6 0.8 1

0

0.6

1.2

1.8

2.4

3

dlei

yy

re

vo

ce

rn

eg

or

dy

H

a

b

Fig. 8 – Variation of (a) methanol yield in exothermic side of

OMTCR and CMR, (b) cyclohexane conversion in

endothermic side and hydrogen recovery yield in

permeation side of OMTCR along the reactor length.


OMTCR with CMR shows that the methanol yield in output of

OMTCR is increased by 14.13%. Also, as it can be seen in

Fig. 8(b), hydrogen recovery of OMTCR is 2.71 yields. Overall,

the optimized operating parameters for this case lead to effi-

cient coupling of the two reactions.

8. Conclusion

Optimization of methanol synthesis reaction coupled with

dehydrogenation of cyclohexane to benzene by means of

indirect heat transfer and hydrogen production in a catalytic

membrane thermally coupled reactor was studied by a one-

dimensional model. The reactor consists of two separated side

for exothermic and endothermic reactions and a permeation

side for hydrogen recovery. A base case was generated

considering similar operating conditions to industrial meth-

anol reactor. Differential evolution (DE) method, an excep-

tionally simple evolution strategy, is applied to determine the

optimal reactor operating conditions. Maximization of outlet

mole fractions of methanol, benzene and hydrogen in the

permeation side is considered as the objective function. It is

shown that suitable amount of initial molar flow rate in the

exothermic and endothermic side, inlet temperature of all

sides and inlet pressure of exothermic side can provide the

necessary heat to heat up the mixtures and to drive the

endothermic process at the same time. The short distance

between the heat sink and transferred heat increases the

efficiency of heat transfer. This new configuration represents

some important improvement in comparison to conventional

methanol reactor as follows: reduces the size of the reactors;

lower outlet temperature of product stream in exothermic

side and so enhances the equilibrium conversion; methanol

mole fraction in output of reactor is increased by 16.3%;

produces pure hydrogen in the permeation side; increase in

kinetics duration of methanol synthesis reaction and delay in

thermodynamics equilibrium; benzene is also produced as an

additional valuable product; and autothermality is achieved

within the reactor. In the optimized reactor, cyclohexane

reaches 99.9% conversion in the endothermic side, methanol

0.4773 yields in the exothermic side and hydrogen recovery

2.71 yields in the permeation side. The comparison of meth-

anol yield in OMTCR with CMR shows that the methanol yield

in output of OMTCR is increased by 14.13%. The results indi-

cate that methanol synthesis reaction and cyclohexane

dehydrogenation in a membrane thermally coupled reactor is

feasible and beneficial.

Nomenclature

av specific surface area of catalyst pellet, m2 m�3

Ac cross section area of each tube, m2

Ai inside area of inner tube, m2

Ao outside area of inner tube, m2

c total concentration, mol m�3

Cp specific heat of the gas at constant pressure, J mol�1

dp particle diameter, m

Di inside diameter, m

Dij binary diffusion coefficient of component i in j,

m2 s�1

Dim diffusion coefficient of component i in the mixture,

m2 s�1

Do outside diameter, m

DH hydraulic diameter, m

Ep activation energy of permeability, kJ mol�1

fi partial fugacity of component i, bar

F total molar flow rate, mol s�1

G mass velocity, kg m�2 s�1

hf gas–solid heat transfer coefficient, W m�2 K�1

hi heat transfer coefficient between fluid phase and

reactor wall in exothermic side, W m�2 K�1

ho heat transfer coefficient between fluid phase and

reactor wall in endothermic side, W m�2 K�1

DHf,i enthalpy of formation of component i, J mol�1

JH2 hydrogen-permeation rate in Pd/Ag membrane,

mol m�1 s�1

k rate constant of dehydrogenation reaction,

mol m�3 Pa�1 s�1

k1 rate constant for the 1st rate equation of methanol

synthesis reaction, mol kg�1 s�1 bar�1/2


k2 rate constant for the 2nd rate equation of methanol

synthesis reaction, mol kg�1 s�1 bar�1/2

k3 rate constant for the 3rd rate equation of methanol

synthesis reaction mol kg�1 s�1 bar�1/2

kg mass transfer coefficient for component i, m s�1

K conductivity of gas phase, W m�1 K�1

KB adsorption equilibrium constant for benzene, Pa�1

Ki adsorption equilibrium constant for component i in

methanol synthesis reaction, bar�1

Kp equilibrium constant for dehydrogenation reaction,

Pa3

Kpi equilibrium constant based on partial pressure for

component i in methanol synthesis reaction

Kw thermal conductivity of reactor wall, W m�1 K�1

L reactor length, m

Mi molecular weight of component i, g mol�1

N number of components (N ¼ 6 for methanol

synthesis reaction, N ¼ 3 for dehydrogenation

reaction)

P permeability of hydrogen through Pd/Ag membrane,

mol m�2 s�1 Pa�1/2

P0 pre-exponential factor of hydrogen permeability,

mol m�2 s�1 Pa�1/2

P total pressure (for exothermic side: bar; for

endothermic side, Pa

Pi partial pressure of component i, Pa

r1 rate of reaction for hydrogenation of CO, mol kg�1 s�1

r2 rate of reaction for hydrogenation of CO2,

mol kg�1 s�1

r3 rate of reversed water-gas shift reaction,

mol kg�1 s�1

r4 rate of reaction for dehydrogenation of cyclohexane,

mol m�3 s�1

ri Reaction rate of component i (for exothermic

reaction: mol kg�1 s�1; for endothermic reaction:

mol m�3 s�1)

R universal gas constant, J mol�1 K�1

Rp particle radius, m

Re Reynolds number

Sci Schmidt number of component i

T temperature, K

u superficial velocity of fluid phase, m s�1

ug linear velocity of fluid phase, m s�1

U overall heat transfer coefficient between exothermic

and endothermic sides, W m�2 K�1

vci critical volume of component i, cm3 mol�1

yi mole fraction of component i, mol mol�1

z axial reactor coordinate, m

Greek letters

aH hydrogen-permeation rate constant,

mol m�1 s�1 Pa�1/2

m viscosity of fluid phase, kg m�1 s�1

r density of fluid phase, kg m�3

rb density of catalytic bed, kg m�3

s tortuosity of catalyst

Superscripts

g in bulk gas phase

s at surface catalyst

Subscripts

0 inlet conditions

B benzene

C cyclohexane

i chemical species

j reactor side (1: exothermic side, 2: endothermic side,

3: permeation side)

k reaction number index

r e f e r e n c e s

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http://www.ICSI.berkeley.edu/~storn/code.html

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