differential evolution strategy for optimization of hydrogen production via coupling of...

Differential Evolution Strategy for Optimization of HydrogenProduction via Coupling of Methylcyclohexane DehydrogenationReaction and Methanol Synthesis Process in a Thermally CoupledDouble Membrane ReactorShahab Amirabadi, Sedigheh Kabiri, Reza Vakili, Davood Iranshahi, and Mohammad Reza Rahimpour*

School of Chemical and Petroleum Engineering, Department of Chemical Engineering, Shiraz University, P.O. Box 71345, Shiraz,Iran

ABSTRACT: The present paper focuses on optimization of a thermally coupled double membrane methanol reactor(TCDMR). This novel configuration is used in order to increase methanol production and achieve pure hydrogen. To reach thisgoal, the exothermic methanol synthesis reaction is coupled with endothermic dehydrogenation of methylcyclohexane to improvethe heat transfer between the endothermic and exothermic sides. Methylcyclohexane (MCH) has been proposed as a potentialcandidate among the cycloalkanes to produce gaseous hydrogen and a liquid aromatic product toluene (TOL). Two differentmembrane layers are assisted in TCDMR to improve mass transfer between the exothermic/endothermic side and bothpermeation sides. A Pd/Ag membrane is used for separation of pure hydrogen from the endothermic side and a hydroxy sodalite(H-SOD) membrane is used for permeation of water from the exothermic side. It has been shown that in situ water removalfrom the exothermic side has certain advantages. H2O removal improves the activity and selectivity control of the methanolsynthesis as well as inhibits the catalyst recrystallization. A steady state heterogeneous model predicts the performance of thisinnovative configuration. The optimization results have been compared with corresponding predictions for a conventionalmethanol reactor, thermally coupled methanol reactor, and thermally coupled double membrane reactor. The differentialevolution (DE) is a simple and efficient global optimization algorithm applied to optimize the thermally coupled doublemembrane reactor considering the summation of methanol, toluene, and H2 recovery yields as the objective function. Thesimulation results demonstrate that lower water production rate in the optimized TCDMR (OTCDMR) caused 13.8%enhancement in methanol yield in comparison with a conventional reactor and the hydrogen recovery permeation side reaches2.15 yields.

1. INTRODUCTION

It is well-known that energy is a basic need in economicdevelopment. There has been an enormous increase in energydemands due to industrial development and population growth.The severe use of fossil fuel as a main energy source has broughtenergy crisis. In order to eliminate fossil fuels dependence,comprehensive research has been carried out on searchingrenewable resources. For the characteristics of unlimited supply,environmentally friendliness, nonemission of greenhouse gases,and high potential energy, hydrogen may be an excellentcandidate for the future energy sector.1

1.1. Hydrogen. Hydrogen is a promising clean fuel andsecondary energy source. The products of hydrogen combustionare water and only a tiny amount of NOxwhich can be reduced byproper techniques such as exhaust gas recirculation (EGR).2

Hydrogen, an energy carrier, is used in heating home and offices,powering zero-polluting vehicles, producing electricity, andfueling aircrafts. In the early 16th century, it was first artificiallyproduced via the mixing of metals with strong acids. Recently,many technologies have been studied for hydrogen productionsuch as: electrolysis with sun and wind power, gasification ofbiomass and coal, metal hydrides, photobiological, watersplitting, and reforming of natural gas.3−5 Another alternativemethod for production of hydrogen is the dehydrogenation ofcyclic hydrocarbons which contains a large amount of hydrogen

(6−8% on weight basis).6−8 In this technique, only H2 anddehydrogenated hydrocarbon are produced. This means that it isfree from any greenhouse gas like CO2 and CO and,consequently, there is no need for hydrogen purification.9,10

1.2. Dehydrogenation of Methylcyclohexane. One ofthe predominant technical issues limiting use of hydrogen energyis its storage. Use of cycloalkanes such as cyclohexane,methylcyclohexane, and decalin has been reported for efficienthydrogen storage. Today, due to health impact concerns,methylcyclohexane (MCH) is proposed as a potential candidateamong the cycloalkanes (e.g., cyclohexane) to produce gaseoushydrogen and a liquid aromatic product toluene (TOL). Thischoice of MCH was made for the following reasons:

• Toluene, a preferable product of the dehydrogenation ofMCH, is safer due to environmental aspects as comparedto benzene, which is the product of cyclohexanedehydrogenation.

• MCH has 6.2 wt % hydrogen content, high boiling point,and higher capacity for storage of hydrogen than other

Received: June 16, 2012Revised: November 28, 2012Accepted: December 13, 2012Published: December 13, 2012

Article

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© 2012 American Chemical Society 1508 dx.doi.org/10.1021/ie301583w | Ind. Eng. Chem. Res. 2013, 52, 1508−1522

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candidates including liquid hydrogen and compressedhydrogen.11,12

• At the same conditions, the conversion of MCH is higherthan cyclohexane conversion.

• The frizzing point of MCH is lower than that ofcyclohexane, so it presents better conditions for use in acold environment.

Besides, MCH is in the liquid phase at room temperature andpressure; it can be used for storage of hydrogen in a stable form asa hydrogen carrier.13 The reaction of dehydrogenation is limitedby thermodynamic equilibrium and highly endothermic whichcan be used for coupling with a suitable exothermic reaction.Coupling strategy is a novel technique receiving considerableattention in the chemical engineering community. For thispurpose, a multifunctional reactor is considered in which bothreactions, exothermic and endothermic, are proceeding simulta-neously in one vessel and the thermal energy of exothermicreaction is used as a heat source of endothermic reactions.14,15

Therefore, it is expected that the energy and materialconsumption as well as environmental impact be reduced.Regarding coupling criteria, the methanol synthesis is anappropriate exothermic reaction to be coupled with dehydrogen-ation of methylcyclohexane.1.3. Methanol Synthesis. Methanol is a vital multifunc-

tional raw chemical that is recovered on a large scale worldwide asa clean-burning fuel, solvent, and as a building block forproduction of chemical intermediates. It has a higher energydensity in comparison with other gaseous fuels, and itsdevelopment and storage is much easier than others. Becauseof these physical and chemical properties, it is widely useddirectly to the fuel cell system called the direct methanol fuel cell(DMFC) which has high energy and wide application among thevarious fuel cells.16−18Methanol has a simple molecule which canbe produced frommany resources, mainly CH4 and synthesis gasconsisting of CO, CO2, H2, and some inert gases. A conventionalfixed-bed methanol synthesis reactor usually consists of a shelland tube heat exchanger. Boiling water exists within the shell andis converted to steam while the exothermic reaction heat ofmethanol synthesis is transferred to it.19,20 Because ofequilibrium nature of the reaction, the conversion of methanolis low in this conventional reactor. Therefore, most of thesynthesis gas should be circulated around the loop, and this posesproblems in operating costs.21 Thus for solving this problem,several techniques have been examined. Most of these efforts arerelated to innovation in configuration of reactor. In the literature,there are many studies whose goals are to improve theproduction rate of methanol in industrial reactors.22−25

Increasing methanol yield within a membrane reactor wasstudied by Struis et al.26 Removal of reactants from product gasesmay shift the equilibrium reaction to increase conversion ofchemical reactants to products. In this way using a shell and tubemembrane reactor is an excellent idea.27 Khademi et al. used Pd/Ag membrane for hydrogen separation in coupling of methanoland benzene production.28

1.4. Pd/Agmembrane. Several technologies have been usedfor separation of hydrogen: pressure swing adsorption, solventadsorption, cryogenic recovery, and membrane separation.Among these methods, membrane technology is a promisingchallenge because of its low operating costs and minimizing unitoperations.29 The reactive application of membrane technologyin chemical reaction processes has been recently used to enhancethe yield/selectivity of reactions.30 One of the most important

advantages of membrane reactors is the capability of beating theequilibrium limitation. According to the Le Chatelier−Brownprinciple, removing the products in a reversible reaction shifts thereaction to more production.31 In this way, a variety of materialshave been examined including carbon microporous membranes,ceramic iron-transport membranes, crystalline alloy membranes,and amorphous alloy membranes, etc.32 Among these developedmembranes, Pd is 100% selective for H2 transfer and Pd-basedmembranes have high permeability, good surface characteristicsand resistance to high temperature and corrosion.33 Using Pd/Agmembranethe most commonly used alloy for hydrogenextractioneliminates the dependence of hydrogen purificationunit. Key requirements on Pd-based membranes are low costs,sufficient mechanical strength, and high hydrogen permeabilityas well as good thermal stability.34 Different configurations of Pdmembrane reactors have been described by Tosti et al.35 Lin et al.used membrane reactors for methanol steam reforming.36 Theperformance of Pd/Ag membrane reactors for methanolsynthesis was investigated by Rahimpour and Ghader.37

1.5. H-SODMembrane. Zeolite membranes have been usedin different applications such as separation processes, chemicalsynthesis, sensors, and opto-electronic devices.38 These mem-branes are prepared through different methods including in situhydrothermal synthesis, vapor phase transport method, secon-dary growth method, and embedding microcrystals of zeoliteinto a matrix.39 Hydroxy sodalite is a zeolite-like material whichconsists of the primary building unit. It is a crystallinealuminosilicate with a three-dimensional channel network. Thisclathrasil may be an excellent candidate for separation of smallmolecules such as Helium (2.6 Å) and water (2.7 Å) from a gas orliquid mixture. Also, it has an excellent thermal and mechanicalstability and high selectivity of water to hydrogen.40 H-SOD is100% selective for water transport on the basis of molecularsieving in hybrid processes and especially methanol synthesisprocess. Using water selective membrane increases reactionyields and prevents catalyst deactivation.39 The performance ofalumina H-SOD membranes in the various organic alcoholsdehydrogenation was reported by Khajavi.41 According to the LeChatelier−Brown principle, as products are removed, the systemconsciously restores equilibrium by producing new products.One of the most important byproduct in methanol synthesisreaction is water. So, in situ H2O removal accelerates the watergas shift reaction to enhance CO2 conversion and improvemethanol productivity. In this study, both hydrogen and waterpermselective membranes are used in the thermally couplemethanol reactor. An optimum design for novel thermallycoupled membrane reactors is based on the best and mostfavorable states which can reduce the cost and increase profit.

1.6. Differential Evolution (DE) Algorithm. The mainparameters in plant optimization are dimensions of equipmentand recycle flow and operating conditions such as: temperature,pressure, and concentration. An optimum economic designcould reach to minimum cost per unit of time or maximum profitper unit of production. In recent years, some advanced systems inthe fields of evolutionary programming (EP),42 evolutionstrategies (ES),43 simulated annealing (SA),44 genetic algorithms(GA),45 and differential evolution (DE) have been developed fordealing with optimization problems. DE is an improved versionof GA for faster optimization. Unlike simple GA that uses binarycoding for representing problem parameters, DE uses real valuedparameters. DE has a major advantage over other kinds ofnumerical methods. It only needs information about theobjective function and other accessory properties like differ-

Industrial & Engineering Chemistry Research Article

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entiability and continuity are not required.46 Some of the DEadvantages are the following:

• Finding the true global minimum regardless of the initialparameter values

• Being simple and easy with fast convergence• Needing a few control parameters• Being easily adjustable for integrand discrete optimization• Being quite effective in nonlinear constraint optimization.• Being helpful for optimizing multimodel search spaces.

This stochastic optimization technique has been appliedsuccessfully in a wide range of application areas, including thebatch fermentation process, synthesis, and optimization of heatintegrated distillation system, estimation of heat transferparameters in a trickle bed reactor, optimization of an alkylationreaction, optimization of thermal cracker operation, optimizationof adiabatic styrene reactor, optimization of nonlinear chemicalprocesses, optimization of process synthesis and designproblems, optimal design of shell and tube heat exchangers.22

DE has two main disadvantages; parameter tuning needsnumerous runs, and it is not a suitable solution for problemswhich are very time-consuming. The most excellent controlparameter settings of DE are problem dependent. Brest et al.proposed self-adaptive DE in order to overcome these problemsdealing with DE.46 Optimization of methanol synthesis andcyclohexane dehydrogenation in a thermally coupled reactorusing the DE method is proposed by Khademi et al.22 Also,Rahimpour et al. investigated a dynamic optimization of a novelradial-flow-spherical bed methanol synthesis reactor in thepresence of catalytic deactivation using a DE algorithm.47Thementioned advantages of DE are enough for us to choose it as anoptimization technique in this paper.1.7. Objectives. The object of this study is modeling and

optimization of a thermally coupled double membrane methanolreactor (TCDMR)which is composed of four sides containinga water permeation side, methanol synthesis reaction side, MCHdehydrogenation side, and hydrogen permeation sideusingDE method. The innovations of the process are using the energyefficient concept of the coupling of endothermic−exothermicreactions, using simultaneously two different membranes toselective separation of hydrogen and water removal andenhancement of simultaneous hydrogen and methanol produc-tion in a single reactor. The water removal from exothermic sideby H-SOD membrane improves the methanol production ratewhile the hydrogen removal from endothermic side by Pd/Agmembrane enhances the hydrogen and toluene production rates.The operating conditions of a conventional methanol synthesisreactor have been used as a basis for simulating of TCDMR.Finally, the simulation results of optimized TCDMR(OTCDMR) are compared with the ones in a conventionalreactor, TCMR, and TCDMR.

2. PROCESS DESCRIPTION2.1. Conventional Methanol Reactor (CMR). A conven-

tional methanol reactor (CMR) is a fixed-bed type resembling avertical shell and tube heat exchanger. The commercial CuO/ZnO/Al2O3 catalyst is packed in vertical tubes, and boiling wateris circulating in the shell side. Methanol synthesis reactions takeplace inside the exothermic compartment. Generated heattransfers to boiling water. So, steam is produced andconsequently, the exothermic side is cooled. The specificproperties of the reactor, catalyst, and feed composition areshown in Tables 1 and 2.

2.2. Thermally Coupled Methanol Reactor. Figure 1illustrates a conceptual schematic of thermally couple methanolreactor (TCMR) configuration with two sides: an exothermicside and an endothermic side. Catalytic dehydrogenation ofmethylcyclohexane (MCH) to toluene is assumed to take placein the endothermic side while methanol synthesis occurs in theexothermic side. Methanol synthesis is considered to supply heatfor the dehydrogenation reaction of MCH in order to producehydrogen. The related operating conditions of the endothermicside are shown in Table 3.

2.3. Thermally Coupled Double Membrane Reactor.Figure 2 illustrates schematic diagram of a thermally coupleddouble membrane methanol reactor (TCDMR) which issimulated for simultaneous methanol and hydrogen production.The design of the TCDMR is such that the inner permeationside, the exothermic side, the endothermic side, and the outerpermeation side are made-up in one tubular reactor, respectively.The synthesis gas is fed to the exothermic side, and methanolsynthesis reactions take place over commercial catalyst CuO/ZnO/Al2O3. The produced water diffuses through the H-SODmembrane layer to the inner penetration side. Sweeping gasflows through the inner permeation side and transmits thepermeated water. Dehydrogenation of MCH is assumed to occurin the endothermic side in order to absorb generated heat fromthe exothermic side and cool it. Hydrogen and toluene areproduced during MCH dehydrogenation reaction over Pt/Al2O3catalyst. The wall of the endothermic side is covered with a Pd/Ag membrane so that pure hydrogen can penetrate from theendothermic side into the outer permeation side. Sweeping gas is

Table 1. Feed Composition in Methanol Synthesis Process(Exothermic Side)

parameter value

feed composition (mole fraction)CH3OH 0.0050CO2 0.0940CO 0.0460H2O 0.0004H2 0.6590N2 0.0930CH4 0.1026total molar flow rate (mol s−1) 0.64inlet pressure (bar) 76.98

Table 2. Characteristics of the Catalyst in Exothermic Sideand the Reactor Design Specification in the Exothermic Side

parameter value

catalyst particledensity (kg m−3) 1770particle diameter (m) 5.47 × 10−3

heat capacity (kJ kg−1 K−1) 5.0thermal conductivity (W m−1 K−1) 0.004specific surface area (m2 m−3) 626.98ratio of void fraction to tortuosity of catalyst particle 0.123length of reactor (m) 7.022number of tubes 2962bed void fraction 0.39density of catalyst bed (kg m−3) 1140tube inner diameter(m) 3.8 × 10−2

tube outer diameter (m) 4.2 × 10−2

wall thermal conductivity (W m−1 K−1) 48



fed to this side and carries the permeated hydrogen. The inputdata and operating conditions are the same as those of theTCMR. The operating conditions for both inner and outerpermeation sides are tabulated in Table 4.

3. REACTION SCHEME AND KINETICS3.1. Methanol Synthesis. Three overall reactions are

possible in the methanol synthesis. These reactions arehydrogenation of carbon monoxide, hydrogenation of carbondioxide, and reverse water-gas shift reaction as follows:

+ ↔ Δ = −HCO 2H CH OH 90.55 kJ/mol2 3 298(1)

+ ↔ +

Δ = −H

CO 3H CH OH H O

49.43 kJ/mol2 2 3 2

298 (2)

+ ↔ + Δ = +HCO H CO H O 41.12 kJ/mol2 2 2 298(3)

The pressure and temperature range of reactions are 5−8 MPaand 495−535 K, respectively.48 As seen, CO2 is one of rawchemical in methanol production. In other words, the usefulconversion of CO2 into methanol is a smart method to protectthe global environment.49,50 The rate of methanol synthesisreactions are as follows:51

=−

+ + +r

k K f f f f K

K f K f f K K f

[ / ]

(1 )[ ( / ) ]

P1

1 CO CO H3/2

CH OH H1/2

1

CO CO CO CO H1/2

H O H1/2

H O

2 3 2

2 2 2 2 2 2

(4)

=−

+ + +r

k K f f f f f K

K f K f f K K f

[ / ]

(1 )[ ( / ) ]

P2

2 CO CO H3/2

CH OH H O H3/2

2

CO CO CO CO H1/2

H O H1/2

H O

2 2 2 3 2 2

2 2 2 2 2 2

(5)

=−

+ + +r

k K f f f f K

K f K f f K K f

[ / ]

(1 )[ ( / ) ]

P3

3 CO CO H H O CO 3

CO CO CO CO H1/2

H O H1/2

H O

2 2 2 2

2 2 2 2 2 2

(6)

The kinetic parameters are shown in Table 5.3.2. Dehydrogenation of Methylcyclohexane. Hydro-

gen is expected to be an energy source for the near future. It hasbeen chosen as a renewable and alternative energy source. Thedistribution and storage of hydrogen are serious issues.Methylcyclohexane dehydrogenation/toluene hydrogenationhas been proposed especially for hydrogen long-term storageand long-distance transportation. Toluene is an environmentallysafety product of a dehydrogenation reaction in comparison withbenzene or naphthalene.52 The dehydrogenation reaction ofMCH is as follows:

Figure 1. Schematic diagram of TCMR.

Table 3. Operating Conditions and Properties of the Catalystfor Dehydrogenation of Methylcyclohexane to Toluene inTCMR

parameter value

feed composition (mole fraction)C7H14 0.12C7H8 0.0H2 0.0Ar 0.88total molar flow rate (mol s−1) 0.1inlet temperature (K) 503inlet pressure (Pa) 8 × 105

shell inner diameter (m) 0.07bed void fraction 0.39particle diameter (m) 3.55 × 10−3



↔ + Δ = +HC H C H 3H 205 kJ/mol7 14 7 8 2 298 (7)

The product of this reversible reaction can be easily hydro-genated and recycled. Pt/Al2O3 based catalyst is the most widelyreported catalyst for this reaction. Using metal oxides as supportencourages the selectivity through strong metal supportinteraction.53 The rate of above reaction is as follows:

= −⎡

⎣⎢⎢

⎤

⎦⎥⎥r kp

p p

K p1MCH MCH

tol H3

eq MCH

2

(8)

Where PMCH and Keq are partial pressure (atm) and equilibriumconstant (atm3), respectively. The k parameter is reactionconstant and obtained by following equation.54

= − −

=· ·

=

⎜ ⎟⎛⎝⎜

⎛⎝

⎞⎠⎞⎠⎟k A

ER T

A

ER

exp1 1

650

20.46mol

g h atm

26540(K) (9)

4. MATHEMATICAL MODELThe following assumptions are considered during the modelingof TCDMR with the cocurrent flow regime:

• The case is studied at steady state conditions. So, all timevariants are set to zero and the catalyst activity isconsidered to be 1.

Figure 2. Schematic diagram of cocurrent mode for a thermally coupled double membrane reactor configuration.



• One dimensional heterogeneous model has beendeveloped. Hence, the energy and mass balance equationshave been wrote only in axial direction for both solid andfluid phases.

• Plug flow pattern is employed in each side of reaction.Since no back mixing is assumed for fluid passing throughthe plug flow reactor, conversion and outlet temperaturesare easily obtained with integrating differential equations.

• The gases considered to be ideal so that thermodynamicproperties of gas phases can be described by ideal gas law.

• Axial diffusion of mass and heat are negligible.• Radial heat and mass diffusion in both beds are neglected.• Bed porosity in axial and radial direction is constant.• The reactor is assumed to be adiabatic. Therefore, heat

loss to the surroundings is neglected.• Axial diffusion of water through the H-SOD membrane

can be neglected in comparison with radial diffusion.• Ideal selectivity of water to hydrogen is assumed for H-

SOD membrane.• Concentration polarization for Pd/Ag membrane is

negligible.

To obtain the mole and energy balance equations, thedifferential element along axial direction with the length of Δz istaken into consideration. The mass and energy balances arewritten for fluid phase, solid phase, and both permeation sides.The results are presented in Table 6.In eqs 10 and 11, η is effectiveness factor, which is obtained

from a dusty gas model calculation.51 In eq 13, the negative sign isused for endothermic reaction and positive sign is used forexothermic reaction. In eqs 12 and 13, β and φ are considered 1and 0 for the endothermic side and 0 and 1 for the exothermicside. The inner permeation side, the exothermic side, theendothermic side, and the outer permeation sides are numberedfrom 1 to 4. In hydrogen permeation equation, PH2

is hydrogenpermeation pressure in pascal, Di and Do are the outer and innerpermeation diameter of the Pd/Ag layer. The pre-exponentialfactor P0 above 200 °C is reported as 6.33× 10−8 mol m−2 s−1

Pa−0.5, and the activation energy Ep is15.7 kJ/mol. Equation 19 isabout the water permeation in H-SOD membrane, and in thisequation, QH2O is water permeation, which is constant withrespect to temperature and indicates specific property of themembrane. The H2O permeation is in average of 10−6 and 10−7

mol m−2 s−1 Pa−1.

5. DIFFERENTIAL EVOLUTION (DE) METHODThis method is one of the simple powerful techniques for fastoptimization. It is obvious that, while a strategy works well for aspecific problem, it might not be the best choice for the otherones. In addition, the strategy and key parameter to be acceptedfor a given problem are to be determined by trial and error.Basically, in the DE method, the weighted difference betweentwo population vectors added to a third vector. In DE, there arethree key parameters including: NP, the population size, CR,crosses over constant, and F, weight applied to randomdifferential (scaling factor). Choosing these parameters dependson the specific problem applied and is often difficult, but Priceand Storn have given some simple guidelines to make it easy.55

Generally, NP should be about 5−10 times the dimension of theproblem. As for F, it lies in the range of 0.4−1, and for CR, it liesin the range of 0.1−1. Normally, CR should be as large aspossible. In this study, the chosen strategy is DE/best/1/bin andNP, F, and CR are 100, 0.8, and 1, respectively.56

6. OBJECTIVE FUNCTION AND CONSTRAINTSIn this study, the summation of the outlet methanol and tolueneyields from reaction sides and outlet hydrogen yield from thepermeation side is considered as the objective function. Theobjective function is defined as follows:

= + +j Y Y YCH OH C H H3 7 8 2 (22)

In this study, six decision variables should be controlled which arethe inlet temperature of water permeation side (T1), inlettemperature of exothermic side (T2), inlet temperature ofendothermic side (T3), the inlet temperature of hydrogenpermeation side (T4), initial molar flow rate of exothermic side(F2), and initial molar flow rate of endothermic side (F3).Temperature is the main parameter which controls thethermodynamic equilibrium because its profile changes duringthe reactor length and it can directly affect the reaction rates andcatalytic activity. Since optimization of inlet temperature is anappropriate method for saving energy and reducing the size ofpreheater, this parameter is selected as a decision variable. Themain reason to develop an initial optimal flow rate in both

Table 4. Characteristics of TCDMR and Properties of BothInner and Outer Permeation Sides

parameter value

inner permeation sidefeed composition(mole fraction)Ar (sweep gas) 1H2 0total molar flow rate (mol/s) 0.1inlet temperature (K) 375.458inlet pressure (bar) 1outer permeation sidefeed composition (mole fraction)Ar (sweep gas) 1H2O 0total molar flow rate (mol/s) 0.1inlet temperature (K) 527.1944inlet pressure (bar) 1characteristics of TCDMRinner tube or feed synthesis gas side diameter (m) 0.038second tube or endothermic side diameter (m) 0.0533third tube or endothermic side diameter (m) 0.804outer tube or permeation side diameter (m) 0.0886length of reactor (m) 7.022

Table 5. Reaction Rate Constants, Adsorption EquilibriumConstants, and Reaction Equilibrium Constants for MethanolSynthesis

k = A exp(B/RT) A B

k1 (4.89 ± 0.29) × 107 −63000 ± 300k2 (1.09 ± 0.07) × 105 −87500 ± 300k3 (9.64 ± 7.30) × 106 −152900 ± 6800K = A exp(B/RT)KCO (2.16 ± 0.44) × 10−5 46800 ± 800KCO2

(7.05 ± 1.39) × 10−7 61700 ± 800

K K( / )H O H1/2

2 2(6.37 ± 2.88) × 10−9 84000 ± 1400

KP = 10(A/T−B)

KP1 5139 12.621KP2 3066 10.592KP3 −2073 −2.029



exothermic and endothermic sides is the dependence oftransferred heat from exothermic side to endothermic side onthe ratio of the exothermic to endothermic side flow rates. Theinlet pressure of inner permeation side is considered to be 101.3kPa and is not selected as a decision variable. The ranges ofdecision variables are

< <T298 5351 (23a)

< <T495 5352 (23b)

< <T480 5303 (23c)

< <T495 5354 (23d)

< <F0.1 1.52 (23e)

< <F0.2 0.53 (23f)

The lower bound on inlet temperature of the waterpermeation side is set at the environmental temperature (298K), and the upper bound of inlet temperature of the exothermicside is selected for its upper bound. The inlet temperature of theexothermic side must be chosen in order to guarantee theoccurrence of methanol synthesis reactions. Therefore, the lowerbound of inlet temperature of the exothermic side is set at 495 K.Deactivation of catalyst starts at high temperature. Hence, 535 K

is chosen for upper bound of inlet temperature of exothermicside. Two domains for lower and upper bounds for the initialmolar flow rate of the exothermic and endothermic sides areselected with no prior intention. In order to make a driving forceto transfer heat from the exothermic side, its temperature shouldbe above the temperature of the endothermic side. Thus, threeconstraints are also considered for optimization:

<T T3 2 (24a)

< <T495 5352 (24b)

< <T480 5303 (24c)

Constraints must be considered during the optimizationprocedure. The penalty function method is used for insertingthese constraints and automatically eliminating unacceptableresults. Our penalty parameter in this work is 107 while this valuedepends on the order of magnitude of the decision variables inthe problems. Finally, the objective function to be maximized is

∑= − +f j Gpen i2

(25)

Where

= −G T Tmax{0, ( )}1 3 2 (26a)

Table 6. Mass and Energy Balance and Boundary Conditions for Solid and Fluid Phases in Four Sides of TCDMR

parameter value

mass and energy balance for solid phase η ρ− + =a c k y y r( ) 0v j i j i j i j i j bg , ,g

,s

, (10)

∑ρ η− + −Δ =−

a h T T r H( ) ( ) 0v f j j bi

N

i j f ig s

1, ,

(11)

mass and energy balance for fluid phaseβ φ−

∂

∂+ − − − =

F

A

y

za c k y y

J

AJ( ) 0j

c j

i jv j gi j i j i j

c j,

,g

, ,s

,g H

,H O

2

2 (12)

∫

∫

π

β βπ

φπ

φ

−∂∂

+ − ± −

− − − − −

−

=

− −

F

AC

T

za h T T

DA

U T T

J

AC T

DA

U T TD

AU T T

J C T

( ) ( )

d ( ) ( )

d

0

j

c jpj

jv f j j

i

c j

c j T

T

pi

c j

i

c j

T

T

P

,

gg

s g

,3g

2g

H

, ,3 4 3

g4g

,1 2 1

g2g

H O

2

3

4

2 1

2

(13)

outer permeation sideβ−

∂

∂+ =F

y

zJ 0i

4,4g

H2 (14)

∫β π−∂∂

+ + − =−F CTz

J C T D U T Td ( ) 0pT

T

p i4 4g 4

g

H 3 4 3g

4g

23

4

(15)

inner permeation sideφ−

∂

∂+ =F

y

zJ 0i

1,1g

H O2 (16)

∫φ π−∂∂

+ + − =−FCTz

J C T D U T Td ( ) 0pT

T

p i1 1g 1

g

H O 1 2 2g

1g

21

2

(17)

hydrogen permeation in Pd/Ag membrane π=

−

−( )( )

JLP

P P2 exp

ln( )

E

RT

DD

H

0

H ,2 H ,32

p

o

i

2 2

(18)

water permeation in H-SOD membrane = −J AsVr

Q P P( )H O H O H Oexo

H Operm

2 2 2 2 (19)

pressure drop μϕ

εε

ρϕ

εε

= − + −Pz d

QA d

QA

dd

150 (1 ) 1.75 (1 )

s2

p2

2

3c s p

3

2

c2

(20)

boundary condition = = = =z T T y y P P0, , ,j j i j i j j jg

0g

,g

0,g g

0g

(21)



= −G Tmax{0, (495 )}2 2 (26b)

= −G Tmax{0, ( 535)}3 2 (26c)

= −G Tmax{0, (480 )}4 3 (26d)

= −G Tmax{0, ( 530)}5 3 (26e)

DE is used for solving this optimization problem.

7. NUMERICAL SOLUTIONThe set of ordinary differential equations are developed forTCDMR modeling. The energy and mass balances forendothermic, exothermic, and permeation sides of the reactorare written. These equations have to be coupled with nonlinearalgebraic equations of the kinetics model. In this problem, this setof equations can be converted to nonlinear algebraic equationsby using backward finite difference approximation method. Thereactor length is then divided into 100 separate sections thenthese equations are solved in each section, simultaneously.

8. RESULTS AND DISCUSSIONThe validity of the methanol synthesis side model has beenchecked by comparing the simulated results with observed plantdata. Table 7 presents comparison between model prediction

and plant data. As it can be seen, in all cases the experimental dataare in good agreement with simulation data. Thus, thementionedmodel predicts almost accurately under the special case ofindustrial conditions.In this section, optimal operating conditions in the thermally

coupled double membrane reactor are analyzed. The predictedmole fractions and conversion profiles for all components arepresented. Furthermore, the temperature and pressure dropdistributions in both endothermic and exothermic sides havebeen shown. For calculation of methanol yield, methylcyclohex-ane conversion, and hydrogen recovery yield, the followingoperating variables are used:

=+

F

F Fmethanol yield

CH OH,out

CO,in CO ,in

3

2 (27)

=−F F

FMCH conversion

C H ,in C H ,out

C H ,in

7 14 7 14

7 14 (28)

=F

Fhydrogen recovery yield H ,3

C H

2

6 12 (29)

8.1. Base Case. The optimization calculations should bebased on specified conditions in order to set a base case. Theoperating conditions of all sides of the reactor are displayed in

Tables 1−4. Operating conditions for the methanol synthesisside are the same as the conditions used by Rezaie et al.23 In theexothermic side, the inlet composition of the methanol synthesisreaction is typical of industrial methanol synthesis process. Theinlet mole fraction of methylcyclohexane, in the endothermicside, is similar to that presented by Maria et al.54 Therefore, thebase case helps consider the condition at which the generatedheat from methanol synthesis consumed by endothermic MCHdehydrogenation reaction. This process happen in a highertemperature at first parts of exothermic side for higher kineticsconstants and then temperature is reduced at the end parts ofreactor for increasing thermodynamic equilibrium. Thiscondition allows comparison of an optimized thermally coupleddouble membrane reactor with a conventional methanol reactor.

8.2. Simulation and Optimization. As it is mentioned,maximization of yields of methanol, toluene, and hydrogen wasconsidered as an objective function. In order to reach this goal,decision variables ought to be optimized. The differentialevolution method is used to determine the optimal operatingconditions. Table 8 shows results of the optimization for a

thermally coupled double membrane reactor. The simulation of athermally coupled double membrane reactor is carried out byusing optimization outcomes, and the results are shown invarious figures.

8.3. Mole Fraction Behavior. 8.3.1. Exothermic Side.Figure 3a illustrates the mole fraction profile of methanol alongthe reactor, at steady state for exothermic side of OTCDMR incomparison with the one in CMR, TCMR, and TCDMR. Figure3b and c presents similar results for CO and water. As can beseen, the figures show no considerable difference between theoutputs of conventional reactor and TCMR, while the molefractions of component in the middle of these two reactors aredifferent. This is attributed to delay in thermodynamicequilibrium due to decreasing temperature in the shell side ofTCMR by dehydrogenation of MCH. It should be said that thereaction kinetic is the controlling feature in the upper section ofthe reactor while the equilibrium manages the situation in othersections. Above investigation indicates that TCMR configurationcan be improved by inserting two membranes for water andhydrogen removal from exothermic and endothermic sides,respectively. According to the Le Chatelier−Brown principle,removal of products in a reversible reaction shifts the reaction tomore production. When there is a difference between waterpartial pressure in the exothermic side and inner permeation side,water can continuously pass through H-SOD membrane. Sincewater vapor is catalyst poisoning agent in methanol synthesisprocess, its removal increases catalyst lifetime. It can beunderstood from Figure 3a that there is a slight differencebetween trends in conventional methanol reactor and TCDMRowing to the in situ water removal by H-SOD membrane whichshifts themethanol synthesis reaction and increases themethanol

Table 7. Comparison between Simulation and Plant Data forConventional Methanol Reactor20

plantdata

simulationresults

relative errors(%)

output composition (mol %)CH3OH 4.00 4.32 7.78CO2 1.03 0.93 −9.79CO 0.93 0.90 −3.33H2O 1.24 1.38 10.49H2 75.01 74.82 −0.25outlet temperature (K) 527 525.40 −0.30

Table 8. Optimized Parameters for the TCDMR

parameter value

inlet temperature of inner permeation side (T1) 375.45inlet temperature of exothermic side (T2) 535inlet temperature of endothermic side (T3) 529.07inlet temperature of outer permeation side (T4) 527.19initial molar flow rate of exothermic side (F2) 1.196initial molar flow rate of endothermic side (F3) 0.15objective function 2.578



production. In other words, higher CO2 conversion to CH3OH isone of the main advantages of TCDMR. The water removal rateis very low at the entrance of the TCDMR configuration, and as aconsequence, the CH3OH mole fraction does not changesignificantly in this part. The highest mole fraction profile isrelated to OTCDMR, which is due to lower temperature in the

exothermic side caused by utilizing sweeping gas in the inner tubeside. As seen in Figure 3b and c, the production of CO and waterare restrained along TCDMR and OTCDMR.

8.3.2. Endothermic Side. Figure 4a and b depict the hydrogenand toluene mole fraction trajectories in the endothermic side as

a function of the reactor dimensionless length. The mole fractionof hydrogen and toluene are increased as the reaction scheme. Asmall difference can be seen between the TCDMR and TCMRprofiles as a consequence of utilizing Pd/Ag and H-SODmembranes. Because of hydrogen partial pressure differencebetween endothermic side and permeation side, hydrogen cancontinuously penetrate through the Pd/Ag membrane from theendothermic side into the outer permeation side. H2 removalaccelerates the dehydrogenation of MCH; therefore more H2and toluene are produced. Moreover, in situ water removalincreased the exothermic methanol synthesis reaction rateleading to transfer more heat to the endothermic side, andtherefore, the MCH dehydrogenation reaction rate is increased.These figures prove that the OTCDMR provides a morefavorable profile. This means that utilizing optimized operatingparameters lead to efficient coupling of the reactions andseparation.

8.3.3. Outer Permeation Side. Figure 5 demonstrates the H2mole fraction in the outer permeation sides of the OTCDMRand TCDMR. As seen, the mole fraction of hydrogen in the

Figure 3. Comparison of (a) CH3OH, (b) CO, and (c) H2O molefractions between the exothermic sides of CMR, TCMR, TCDMR, andOTCDMR.

Figure 4. Comparison of (a) H2 and (b) C7H8 mol fraction betweenendothermic sides of the TCMR, TCDMR, and OTCDMR.



OTCDMRprofile is higher than that for the TCDMR.Hydrogenpermeation through the membrane layer increases the hydrogenmole fraction in the outer permeation side. The higher sweep gasflow rate in the outer permeation side decreases H2 partialpressure in this side and leads to higher hydrogen permeationthrough the Pd/Ag membrane.8.4. Thermal Behavior. Figure 6a and b present the

temperature profiles for a conventional methanol reactor,TCMR, TCDMR, and OTCDMR on the exothermic andendothermic sides, respectively. The highest temperature profileis related to the exothermic side where heat is generated. Part ofthe generated heat is used for driving the endothermic reactionand the rest is used for increasing temperature of mixtures in bothreaction sides and also the inner permeation side in TCDMR.The dehydrogenation side temperature is always lower than theexothermic side temperature; this temperature difference causesa driving force for heat transfer between two sides. Theexothermic side temperature of TCMR and TCDMR increasessmoothly, and a hot-spot develops as demonstrated in Figure 6abecause the generated heat on the exothermic side is morecompared with the consumed heat on the endothermic side atthe entrance of these coupled reactors. In fact, the temperatureprofile on the exothermic side of the TCDMR is expected to behigher than the CMR and TCMR profiles due to water removalwhich leads to shifting the methanol synthesis reaction to moreproduction. Consequently, more reaction heat is released andmore heat is consumed in the endothermic side which causes thetemperature increase. However, since the generated heattransfers to both the endothermic side and inner permeationside, the temperature profile of the TCDMR is lower comparedwith other reactors. According to the Le Chatelier−Brownprinciple, temperature reduction in the exothermic reversiblereactions leads to shifting the reaction in the exothermicdirection. Figure 6b shows the temperature profile of thedehydrogenation side along the endothermic sides ofOTCDMR, TCDMR, and TCMR. At the reactor entrance, thedifference between endothermic and exothermic temperatures islow. Therefore, the lowest heat transfer amount takes placebetween these two sides at this section of OTCDMR. It shouldbe noted that the outlet temperature of the endothermic andexothermic sides is nearly the same (530 K).8.5. Pressure Drop. Pressure drop is mainly determined by

the term of viscous energy loss. The Ergun equation (which is eq

20) is usually used to calculate pressure drop through the catalystpacked bed. Figure 7a and b shows pressure drop along theOTCDMR and TCMR for exothermic and endothermic sides,respectively. Since the temperature profile of the OTCDMR ishigher compared with temperature profiles of TCMR in bothsides, the density of gas phase and consequently the velocity andrelated viscose losses of OTCDMR become higher. Thus,pressure drop profile through this reactor is higher than TCMR.

8.6. Conversion and Yield Changes. Figure 8 shows thevariation of rates of reactions for OTCDMR. It is obvious that thereaction rates decreased along the reactor. One important pointin this figure is that the predominant reactions are hydrogenationof CO and hydrogenation of CO2. At the entrance of reactor thedifference between rates of reactions is higher than the one at theend part of reactor.Figure 9a and b compares the conversion profiles of H2 and

CO on the exothermic sides of the CMR, TCMR, TCDMR, andOTCDMR. Figure 9a shows the conversion profile of H2. It canbe seen that H2 conversion of CMR and TCMR reactors are thesame at the end of the reactors. Its profile for TCDMR is slightlyhigher than the one in CMR and TCMR, which is related to insitu water removal leading to an increase the H2 consumptionand also because of the lower temperature profile of the TCDMRon the exothermic side. The highest H2 conversion profile is

Figure 5. Comparison of hydrogen mole fraction between outerpermeation sides of the TCDMR and OTCDMR.

Figure 6. (a) Temperature profile of the exothermic sides of the CMR,TCMR, TCDMR, and OTCDMR. (b) Temperature profile of theendothermic sides of the TCMR, TCDMR, and OTCDMR.



related to the OTCDMR configuration in which H2 conversionreaches 23.04%. As seen in Figure 9b, the CO conversion profilein the TCMR and TCDMR have the same trends and are lowerthan conventional profiles. The lowest conversion profile isachieved in the OTCDMR because CO production is suppressed

along this configuration, and also, the rate of production of CO inthe reverse water−gas shift (WGS) reaction is lower than otherreactions on the exothermic side (see Figure 8). The C7H14conversion profile is illustrated in Figure 9c. It can be understoodfrom this figure that the conversion profile of the TCDMR ishigher than TCMR due to utilizing Pd/Ag membrane; in otherwords, the H2 removal from endothermic side shifts the

Figure 7. (a) Pressure drop along the exothermic sides of the TCMRand OTCDMR. (b) Pressure drop along the endothermic sides of theTCMR and OTCDMR.

Figure 8. Variation of rate of reactions for exothermic and endothermicsides in the OTCDMR.

Figure 9. Conversion profiles for (a) H2 and (b) CO along the CMR,TCMR, TCDMR, and OTCDMR for the exothermic side and (c)C7H14 along the TCMR, TCDMR, and OTCDMR for the endothermicside.



dehydrogenation reaction of MCH to products and MCHconversion will be increased. The highest profile is related toOTCDMR, which is due to its higher temperature profile on theendothermic side compared to other reactors. Increasingtemperature in the endothermic reversible reaction leads toshifting the reaction in the endothermic direction. MCHconversion reaches 78.96% in this type of reactor.Figure 10a−c demonstrates desired product yields such as

methanol, toluene, and H2. Figure 10a shows the productionyield of methanol which is the main desired product. Fortunately,its yield increased in the OTCDMR as a result of improvement ofheat transfer between exothermic and endothermic sides and alsowater permeation from the exothermic side to the innerpermeation side. Figure 10b and c depict the toluene andhydrogen recovery trajectories. As it can be seen, the yields oftoluene and H2 in the TCMR and TCDMR have approximatelythe same trends. Toluene and hydrogen yields reach 0.79 and2.15 in the OTCDMR.The performance of the CMR, TCMR, TCDMR, and

OTCDMR in the methanol synthesis process and dehydrogen-ation of MCH to toluene in order to produce methanol and purehydrogen is summarized in Table 9. The superiority of theOTCDMR is distinguished from this table. Obviously, hydrogenrecovery, toluene, and methanol yield increase in the TCDMR.In the OTCDMR, methylcyclohexane conversion, methanolyield, and hydrogen recovery yield reach 78.96%, 0.4122, and2.1571, respectively.Simulation results signify an increment of about 6.26 and

6.05% in methanol yield of the TCDMR in comparison with theTCMR and CMR, respectively. In addition, an 8.125%enhancement in MCH conversion is achieved for the TCDMRin comparison with the TCMR.8.7. Production Rate. Figure 11a and b presents the

comparison of methanol and H2 recovery production rate in theCMR, TCMR, andOTCDMR. Interestingly, the production rateof these desired products increased in the OTCDMR. Improve-ments of 27.93% and 53.99% in methanol production rate areachieved for the OTCDMR in comparison with the CMR andTCMR. Besides, an increment of about 41.47% in the hydrogenrecovery production rate for the OTCDMR in comparison withthe TCMR can be seen. According to these figures, theOTCDMR is an efficient configuration for production of purehydrogen and methanol.

9. CONCLUSIONOptimization of the methanol synthesis reaction coupled withthe dehydrogenation of methylcyclohexane to toluene in athermally coupled double membrane reactor was studied. Thereactor consists of four sides: the inner permeation side,exothermic side, endothermic side, and outer permeation side,respectively. Twomembranes are used in this configuration. Oneis an H-SOD membrane for water removal from the exothermicside, and the other one is a Pd/Ag membrane for hydrogenseparation from the endothermic side. Methylcyclohexane(MCH) was proposed as a potential candidate among thecycloalkanes (e.g., cyclohexane) to produce gaseous hydrogenand a liquid aromatic product, toluene (TOL). The differentialevolution (DE) algorithm, stochastic optimization method, isapplied to adjust the optimal reactor operating conditions.Maximization of outlet methanol, toluene, and hydrogenrecovery yield is assumed as the objective function. Six decisionvariables, namely inlet temperature of the water permeation side,inlet temperature of the exothermic side, inlet temperature of the

endothermic side, the inlet temperature of the hydrogenpermeation side, initial molar flow rate of the exothermic side,and initial molar flow rate of the endothermic side, areconsidered for optimization. This novel configuration showssome important advantages in comparison with a conventionalreactor and a thermally coupled methanol reactor as follows:

Figure 10. (a) CH3OH production yield for the exothermic side and (b)C7H8 and (c) H2 production yields for the endothermic side.



• The reactor size is reduced.• Pure hydrogen is produced in the permeation side by using

a Pd/Ag membrane.• A lower water production rate reduces catalyst recrystalli-

zation.• Toluene as an additional valuable product has been

produced.• The rate of methanol synthesis increases due to water

removal from the exothermic side to the inner permeationside leading to a shifting thermodynamic equilibrium.

Simulation results show that the methanol yield of theTCDMR is increased by 6.26 and 6.05% in comparison with theTCMR and CMR, respectively. Moreover, a 8.125% enhance-ment inMCH conversion is seen for the TCDMR in comparisonwith the TCMR. In the optimized reactor, methanol yield in theexothermic side and hydrogen yield in the permeation sidereached 0.4122 and 2.1571, respectively. The superiority of theOTCDMR is distinguished from simulation results.

■ APPENDIX: AUXILIARY CORRELATIONSThe mass transfer coefficients between the gas and solid phaseshave been taken from the work of Cussler.57

= ×− −k Re Sc u1.17 10i ig0.42 0.67

g3

(A1)

μ=Re

R u2 p g

(A2)

μρ

=× −Sc

D 10iim

4(A3)

The diffusivity of gaseous component is given by

=−

∑ =

Dy1

ii

i jy

D

m i

ij (A4)

=+

+

−

DT M M

P v v

10 1/ 1/

( )iji j

i j

7 3/2

c3/2

c3/2

(A5)

Where, Dij is the binary diffusivities using the Fuller−Schetter−Giddins method that is reported by Reid and his co-workers.58Miand νi are themolecular weight and critical volume of componenti. The overall heat transfer coefficient, in all sides, is given by thefollowing correlation.

π= + +

U hA D D

LKAA h

1 1 ln( / )2

1

i

i o i

w

i

o o (A6)

Where hi is the convection heat transfer coefficient between thegas phase and the reactor wall which is estimated by the followingequation.59

ρμ

μ

ερ

μ=

−⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

hC

C

K

ud0.458

p

p2/3

B

p0.407

(A7)

■ AUTHOR INFORMATIONCorresponding Author*Tel.: +98 711 2303071. Fax: +98 711 6287294. E-mail address:[email protected] authors declare no competing financial interest.

■ NOMENCLATUREav = specific surface area of catalyst pellet, m2 m−3

Ac = cross-sectional area of each tube, m2

Ao = inside area of inner tube, m2

C = total concentration, mol/m3

Cp = specific heat of the gas at constant pressure, j/moldp = particle diameter, mDi = tube inside diameter, mDij = binary diffusion coefficient of component i in j, m2 s−1

Dim = diffusion coefficient of component i in the mixture, m2

s−1

Do = tube outside diameter, mDsh = shell inside diameter, mf i = partial fugacity of component i, barF = total molar flow rate, mol s−1

G = mass velocity, kg m2 s−1

hf = gas−solid heat transfer coefficient, W m−2 K−1

hi = heat transfer coefficient between the fluid phase andreactor wall on the exothermic side, W m−2 K−1

Table 9. Comparison of the Performance of the Reactors

conversion yield

reactor C7H14 CH3OH C7H8 H2 recovery

OTCDMR 0.789 0.412 0.790 2.157TCDMR 0.686 0.378 0.687 1.766TCMR 0.630 0.354 0.631 1.893CR 0.355

Figure 11.Comparison of (a) methanol flow rates in the CMR, TCMR,and OTCDMR and (b) hydrogen flow rates in the TCMR andOTCDMR.



mailto:[email protected]

ho = heat transfer coefficient between the fluid phase andreactor wall on the endothermic side, W m−2 K−1

ΔHf,i = enthalpy of formation of component i, J mol−1

JH = permeation rate of hydrogen through the Pd/Agmembrane, mol m−1 s−1

JH2O = permeation rate of water through the H-SODmembrane, mol m−3 s−1

k = rate constant of dehydrogenation reaction, mol m−3 Pa−1

s−1

k1 = rate constant for the first rate equation of the methanolsynthesis reaction, mol kg−1 s−1 bar−1/2

k2 = rate constant for the second rate equation of the methanolsynthesis reaction, mol kg−1 s−1 bar−1/2

k3 = rate constant for the third rate equation of the methanolsynthesis reaction, mol kg−1 s−1 bar−1/2

kgi = mass transfer coefficient for component i, m s−1

K = conductivity of fluid phase, W m−1 K−1

Ki = adsorption equilibrium constant for component i, bar−1

Kp = equilibrium constant for dehydrogenation reaction, Pa3

Kpi = equilibrium constant based on partial pressure forcomponent i in methanol synthesis reaction, −Kw = thermal conductivity of reactor wall, W m−1 K−1

L = reactor length, mMi = molecular weight of component i, g mol−1

N = number of components, −P = total pressure, barPi = partial pressure of component i, Par1 = 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

R = universal gas constant, J mol−1 K−1

Rp = particle radius, mRe = Reynolds number, −Sci = Schmidt number of component i, −T = temperature, Ku = superficial velocity of fluid phase, m s−1

ug = linear velocity of fluid phase, m s−1

U = overall heat transfer coefficient between exothermic andendothermic sides, W m2 K−1

vci = critical volume of component i, cm3 mol−1

yi = mole fraction of component i, mol mol−1

z = axial reactor coordinate, mGreek Lettersμ = viscosity of fluid phase, kg m−1 s−1

ρ = density of fluid phase, kg m−3

ρb = density of catalytic bed, kg m−3

τ = tortuosity of catalyst, −Superscriptsg = in bulk gas phases = at surface catalyst

Subscripts0 = inlet conditionsi = chemical speciesj = reactor side

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