multi-objective optimization for two catalytic membrane reactors—methanol synthesis and hydrogen...

Chemical Engineering Science 63 (2008) 1428–1437www.elsevier.com/locate/ces

Multi-objective optimization for two catalytic membranereactors—Methanol synthesis and hydrogen production

Shueh-Hen Chenga, Hsi-Jen Chenb, Hsuan Changb,∗, Cheng-Kai Changb, Yi-Ming Chenb

aDepartment of Chemical Engineering, Tunghai University, No. 181, Sec. 3, Taichung Harbor Rd., Taichung 40704, TaiwanbDepartment of Chemical and Materials Engineering, Tamkang University, 151 Ying-Chuan Rd., Tamsui, Taipei 25137, Taiwan

Received 18 July 2007; received in revised form 28 November 2007; accepted 3 December 2007Available online 11 January 2008

Abstract

This paper provides the triple-objective-function optimization results for the catalytic membrane reactors, including one for methanol synthesisand one for hydrogen generation. A 1-D, non-isothermal model, which takes into account the intra-particle diffusion for the catalyst, and theelitist nondominated sorting genetic algorithm (NSGA-II) for the multi-objective optimization are adopted. Optimal solutions for methanolsynthesis and hydrogen generation systems show distinctive feature. One is randomly scattered and the other is linearly spread out in the Paretoplot. Solution characteristics in terms of variable distribution are quite different for the two systems. Device size, including membrane areaand membrane size, shows effects both on the optimal solutions and on the correlation relations between objective functions and variables.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Genetic algorithm; Multi-objective optimization; Catalytic membrane reactor; Methanol synthesis; Carbon dioxide; Hydrogen; Methane

1. Introduction

Packed bed membrane reactor, which incorporates the non-porous or porous membrane into a heterogeneous catalyticreactor, is one of the major manifestations in process intensifi-cation. Product removal or controlled reactant addition throughthe membrane can facilitate the improvement for the reactorperformance. Membrane reactor has gained growing attentionsand been applied in many processes, including organic synthe-sis and inorganic dehydrogenation (Noble and Sterm, 1995).The two processes adopted for optimization study in this paperare methanol synthesis from carbon dioxide/hydrogen andhydrogen generation by steam reforming with catalytic partialoxidation from methane. These processes are significant inview of the utilization of the most important green housegas—carbon dioxide (Skrzypek et al., 1990) and the emergingmost significant energy carrier—hydrogen. This paper studieshydrogen production from methane because natural gas com-prises almost 50% of the world feedstock (Haryanto et al.,2005). Furthermore, the process focused is steam reforming

∗ Corresponding author. Tel.: +886 2 26232094; fax: +886 2 26209887.E-mail address: [email protected] (H. Chang).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.12.005

with catalytic partial oxidation, which is a new development inthe past decade in order to provide energy for the endothermicreactions (De Groote and Froment, 1996; Freni et al., 2000;Zhu et al., 2001; Acvi et al., 2001).

Many experimental and modeling studies on using catalyticmembrane reactors for these two processes have been reported,such as Gallucci et al. (2004), Struis et al. (1996, 2000) andStruis and Stucki (2001) for methanol synthesis and Jin et al.(2000), Basile et al. (2001) and Ji et al. (2003) for hydrogen orsyngas production.

Optimization studies for catalytic membrane reactors in theliterature have been conducted typically by parametric studyand seldom by traditional mathematical search (Sheintuch andDessau, 1996; Diakov and Varma, 2003, 2004). All these stud-ies investigated the optimization for a single performance crite-rion or objective function, such as reactor productivity (rate offormation of desired product), yield, selectivity and conversion.However, for practical purposes, multiple performance criteriaor objective functions might need be taken into account simul-taneously in an optimization task.

Genetic algorithm (GA) is one of the evolutionary algorithmsthat mimic nature’s evolutionary principles. The algorithmcan provide multiple trade-off solutions for multi-objective

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S.-H. Cheng et al. / Chemical Engineering Science 63 (2008) 1428–1437 1429

optimization. The stochastic searching nature of GA makesit particularly suitable for the optimization of systems thatmust be described with complex models, because it allows themodels be solved independently to the optimization algorithm.Nondominated sorting genetic algorithm (NSGA) or its revisedversion (NSGA-II) (Deb, 2001) has been employed for multi-objective optimization for many chemical processes (Rajeshet al., 2000; Oh et al., 2002; Kasat et al., 2002; Nandasana et al.,2003; Nayak and Gupta, 2004; Inamdar et al., 2004; Tarafderet al., 2005; Sarkar and Modak, 2005; Chang and Hou, 2006;Sankararao and Gupta, 2006). However, to the authors’ knowl-edge, multi-objective optimization for catalytic membranereactors using GAs has not been reported yet.

This paper presents the multi-objective optimization resultsfor two important catalytic membrane reactors, methanol syn-thesis and hydrogen generation, respectively. For the givenreactor size, multiple optimal solutions (operating variables)with trade-off relations for triple objective functions, includingmajor feed rate, major product rate and exergy loss are ob-tained. The correlations between objective functions and opti-mal operating variables are analyzed. The effects of membranethicknesses and areas on the optimal solutions are explored. Inthis study, for the membrane reactors, 1-D mathematical modelswith rigorous consideration on the mass diffusion and reaction

r tube

=

0.02

5m

Nafion membrane tube

Feed gas

Purge gas

Cu/ZnO/Al2O3 catalyst

ρs = 2000 kg/m3, εs = 0.53,τs = 4, rs = 2x10-3m

Purge gas

Product gas

L = 10m

εB = 0.4

r1 =

0.0

1m

r2 =

0.0

15m

r3 =

0.0

25m

Feed gas

Purge steamPurge steam

Porous support membrane(km = 0.15 J/(ms K))

Pd membrane

Ni/Al2O3 catalyst(ρs = 2100 kg/m3,εs = 0.53,

τs = 4, rs = 2.5x10-3m)

L = 0.5m

εB = 0.43

Feed gas

Product gas

Product gas

Fig. 1. Catalytic membrane reactor—(a) methanol synthesis and (b) hydrogen generation.

kinetics inside catalyst particles are used. NSGA-II isemployed for searching for the multi-objective optimalsolutions.

2. Mathematical models for catalytic membrane reactors

In this paper, the catalytic membrane reactor model reportedin Ji et al. (2003) is adopted with essential modifications forsimulating methanol synthesis and hydrogen production sys-tems. In the model, plug flow in the bulk phase is assumed;hence there are only temperature and concentration variationsin the axial direction. For industrial packed bed reactors, pres-sure drop commonly amounts to about 10% of the operatingpressure (Bartholomew and Farrauto, 2005), which is not verysignificant when compared to the variation ranges of the totalpressure (Pupper limit/Plower limit) in the optimization problems,i.e., 1.5 times and 2.7 times for the methanol and hydrogensystems. Hence, for simplicity, pressure drop is not consideredin the model. The model also assumes no radial heat trans-fer resistance except the heat conduction resistance within themembrane. The reactor is adiabatic.

The significance of intra-particle diffusion in the catalyst par-ticles has been discussed (Graaf et al., 1990) and examined viamodeling studies (Veldsink et al., 1995; Lommerts et al., 2000).

1430 S.-H. Cheng et al. / Chemical Engineering Science 63 (2008) 1428–1437

Table 1Reactions for methanol synthesis and hydrogen production

Methanol synthesis Hydrogen generation

1. CO + 2H2 ↔ CH3OH 1. CH4 + 2O2 ↔ CO2 + 2H2O2. CO2 + H2 ↔ H2O + CO 2. CH4 + H2O ↔ CO + 3H2

3. CO2 + 3H2 ↔ CH3OH + H2O 3. CO + H2O ↔ H2 + CO2

4. CH4 + 2H2O ↔ CO + 4H2

For that diffusion, Veldsink et al. (1995) concluded that Fick’smodel is satisfactory when compared to other more complexmodels. In this study, the kinetics of the multiple reactions andthe intra-particle diffusion are taken into account in solving thecontinuity equations for the catalyst particles. However,the catalyst particles are assumed isobaric and isothermal andthe total pressure and temperature equal to that of the bulkphase. Founded on the low Prater number and the low gas filmresistance, Elnashaie and Elshishini (1993) and Nandasanaet al. (2003) rationalized the assumption that the temperatureof the catalytic particle is uniform at the temperature of thegas outside. For the methanol system and the hydrogen systemstudied in this paper, the conservative estimates of the Praternumber are in the order of 10−1 and 10−2, respectively. Thegas film resistance is expected to be low too due to the highgas flow rates.

2.1. Methanol synthesis

For methanol synthesis, the configuration and typical indus-trial dimensions (Struis and Stucki, 2001) of the membrane re-actor are depicted in Fig. 1(a). The Nafion� membrane tubestudied by Struis et al. (1996, 2000) and Struis and Stucki(2001) is adopted. The reaction kinetics is based upon thatreported by Graaf et al. (1988) for low-pressure (15–50 bar)methanol synthesis on a Cu/ZnO/Al2O3 catalyst. The kineticrelations are widely accepted and thought to be more fun-damental because the effects of other mechanisms such asthe intra-particle diffusion resistance have been excluded. Thereactions involved are listed in Table 1.

The mass and energy balance equations for the bulk gas inthe tube (reaction) side are

dFi

dz= (1 − �B)Ac�s

a�(De,i/rs)(dps,i/d�)|�=1

(−10−5RT tube)− AmNi ,

i = 1 . . . nc, (1)

Ni = P 0i exp(−EA,i/RT )

�(ptube,i − pshell,i ), i = 1 . . . nc,

(2)

T =Tshell + Ttube

2, (3)

dTtube

dz= 1∑nc

i=1FiCpi

{−

nc∑i=1

[(dFi

dz+ AmNi

)· Hi

]−q

},

(4)

q = Amkm

(r2 − r1)(Ttube − Tshell). (5)

Boundary conditions:

z = 0, Fi = Fi |z=0, Ttube = Tin. (6)

For the shell (sweeping gas) side, the equations are

dGi

dz= AmNi, i = 1 . . . nc, (7)

dTshell

dz= Am + ∑nc

i=1NiHi∑nc

i=1GiCpi

. (8)


z = 0, Gi = Gi |z=0, Tshell = Tin. (9)

The mass balance equation for the catalyst particle is

1

�2

d

d�

(De,i�

2 dps,i

d�

)= −10−5RT tube�sr

2s

NR∑k=1

(�i,kRs,k).

(10)


� = 0,dps,i

d�= 0, � = 1, ps,i = ptube. (11)

The determination of effective diffusivities in the catalystparticle considers both molecular diffusion and Knudsen dif-fusion. The molecular diffusivities are estimated using Fulleret al.’s correlation (Poling et al., 2001). The pore sizes forestimating Knudsen diffusivity are based on the distributionreported in Xu and Froment (1989). The permeability coeffi-cients are from Struis et al. (1996).

The bulk phase mass and energy balance differential equa-tions, Eqs. (1), (4), (7) and (8), are integrated using Gear’smethod with tolerance of 1 × 10−6. The mass balance differ-ential equations describing the concentration profiles withinthe catalyst particles, Eq. (10), are solved using Global SplineOrthogonal Collocation method (Villadsen and Michelsen,1978) and Broyden method (Press et al., 1992). The Splinemethod is used because of the stiff nature of the intra-particleconcentration profiles near the catalyst surface. The model hasbeen checked with the experimental data for packed bed reac-tors with and without membrane reported in Struis and Stucki(2001). Typical intra-particle concentration profiles are shownin Fig. 2, which depicts the stiff concentration gradients nearthe catalyst surface.

2.2. Hydrogen production

For hydrogen production, the configuration and size of themembrane reactor are depicted in Fig. 1(b), followed as thatreported in Ji et al. (2003). The kinetics developed by De Grooteand Froment (1996) for partial oxidation of methane on anNi/Al2O3 catalyst is used. The reactions involved are listed inTable 1. The mathematical model equations are the same asthose listed for the methanol synthesis system, except that the


p i /

p i, s

0.95

1.00

1.05

1.10

1.15

1.20

CH3OHH2OCOH2CO2N2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ξ

Fig. 2. Intra-particle concentration profiles—methanol synthesis system.

ξ0.4

p i /

p i, s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

CH4

H2O

CO

H2

CO2

O2

N2

0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 3. Intra-particle concentration profiles—hydrogen generation system.

catalysts are in the shell side and the sweeping steam flows inthe tube side. Besides, since the Pd membrane is assumed tobe 100% selective for hydrogen, the permeation flux equations(Eq. (2)) must be modified into Eq. (12). The permeablity ofhydrogen is based on Basile et al. (2001).

NH2 = P 0H2

exp(−EA,H2/RT )

�(√

pshell,H2 − √ptube,H2 ) for H2,

Ni = 0 for other species i. (12)

The model has been checked with the literature reportedresults for a simulated membrane reactor (Ji et al., 2003) and anindustrial packed bed reactor (De Groote and Froment, 1996).Typical intra-particle concentration profiles are shown in Fig. 3,which depicts the stiff concentration gradients near the cata-lyst surface as in Fig. 2 for the methanol synthesis system.The profiles of H2, CO and H2O in Fig. 3 are convex or con-cave near the surface rather than monotonic, which must becaused by the complexity from the four simultaneous reversiblereactions.


3. Optimization problems

For a given-sized reactor, the critical performance indexes arethe major product flowrate (methanol or hydrogen), the majorreactant feed rate (hydrogen or methane), and the exergy loss ofthe reactor. In this study, for the two catalytic membrane reac-tors with given dimensions depicted in Fig. 1, the optimizationproblem is to determine the operating conditions that maximizethe major product rate and minimize both the major reactantrate and the exergy loss. While defining the objective functions,these performance indexes are normalized by some fixed ref-erence quantities, respectively. The normalization is simply toadjust the three objective functions to be all in the range of 0and 1. Since the GA method does not involve weighting andcombining the objective functions, normalization of the objec-tive functions does not influence the optimization results. Themaximization for the major product rate can be converted intothe minimization of (1− the normalized major product rate) asshown in Eq. (13). The optimization problem is hence to min-imize the following three objective functions simultaneously:

OF1 = 1 − (F_CH3OH/F_CH3OHref) or

OF1 = 1 − (F_H2/F_H2,ref), (13)

OF2 = F_H2/F_H2,ref or OF2 = F_CH4/F_CH4,ref ,

(14)

OF3 = EXL/EXLref . (15)

The exergy loss is calculated by

EXL = EXshell_in_stream + EXtube_in_stream

− EXtube_out_stream − EXshell_out_stream. (16)

The stream exergy includes the mechanical, thermal and chem-ical components of the exergy when reaching equilibrium witha reference environment defined at 1 bar, 298.15 K and ambi-ent environment composition. The definition and calculation ofexergy are referred to Bejan et al. (1996).

The optimization variables with the allowable variationranges are listed in Table 2. For methanol synthesis system,the variables are the feed gas rate in terms of gas hourly spacevelocity (GHSVF ), ratio of hydrogen to carbon dioxide in thefeed stream (FH2,in/FCO2,in), feed stream temperature (Tt,in),and the reaction pressure (PT,t ). The GHSV is defined at 1 barand 273.15 K. The sweeping gas, which is nitrogen, is assumed

Table 2Decision variables

Methanol synthesis system Hydrogen generation system

Variable Range Variable Range

PT,t (bar) 20–30 PT,s (bar) 1.5–4Tt,in (K) 443–500 Ts,in (K) 700–900GHSVF (h−1) 3000–12000 FCH4,in (mol/s) 0.5–1.5FH2,in/FCO2,in 2–5 FH2O,in (mol/s) 0.25–2.25

FO2,in (mol/s) 0.2–2.1GH2O,in (mol/s) 7.5–45

Table 3GA parameters

Np 50Gen_Set 200pc 0.85pNote1

m 0.05�Note2

c 10�Note2

m 20

Note: 1. Increasing of pm from 0.05 to 0.1 results in significant increase ofcomputation time. 2. Following the values used by Tarafder et al. (2005).

Table 4Effects of varying GA parameters

Np Gen_Set pc Avg. of OF1 Avg. of OF2 Avg. of OF3

50 200 0.85 0.510 0.485 0.40350 100 0.85 0.523 0.498 0.40650 200 0.9 0.538 0.471 0.38820 100 0.9 0.557 0.499 0.41820 200 0.9 0.545 0.482 0.400

to have a fixed flow rate in terms of GHSV of 2000, a fixedpressure of 1 bar, and the same temperature as the reactantfeed gas. For the hydrogen generation system, the variablesare the feed gas rate in terms of methane, steam and oxy-gen flow rate (FCH4,in, FH2O,in, FO2,in), sweeping steam flowrate (GH2O,in), feed stream temperature (Ts,in), and the reactionpressure (PT,s). The pressure of the sweeping steam is at 1 barand the temperature is the same as that of the reactant feed gas.

4. GA procedure

The optimization problem is solved using real coded NSGA-II (Deb, 2001). Each solution (a set of decision variables)generated in the GA procedure is passed on to the reactor math-ematical model for solving the reactor performances. In casethat any solution is infeasible, the solution is discarded and anew solution is generated. An infeasible solution means thatconvergence is not achieved in solving the model equations us-ing Gear’s method with the specified tolerance for the set ofdecision variables.

The values of the six GA parameters needed are listed inTable 3. For the hydrogen system, the effects of varying Np,Gen_Set, and pc are summarized in Table 4. The results are veryclose in terms of range and distribution of the Pareto points;hence the average values of the objective functions are used forcomparison as listed in the table. Similar results are obtainedfor methanol synthesis system. Although the Pareto solutionsfor methanol system looks scattering, as can be seen in Fig. 4,further varying GA parameters does not generate significantimprovements. Hence, the parameters in Table 3 are adopted.

5. Optimization for methanol synthesis system

For comparison with the multi-objective optimization results,individual parameter studies are conducted for a base case first.The effects of optimization variables on methanol formation


0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

Obj

ectiv

e fu

nctio

n

OF1

OF2

OF3

Chromosome number

Fig. 4. Multi-objective optimization results—methanol synthesis system.

0.0

0.2

0.4

0.6

0.8

1.0

0.20.3

0.40.5

0.60.7

0.80.9

0.1

0.2

0.3

0.40.5

OF

3

OF1OF2

δ = 315μm, Am/Am, base = 1






Fig. 5. Effects of device size on optimal solutions—methanol synthesis. system.

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25Chromosome number

30 35 40 45 50

Obj

ectiv

e fu

nctio

n

OF1

OF2

OF3

Fig. 6. Multi-objective optimization results—hydrogen generation system.


1.52

2.53

3.54

4.5

010

2030

400

PT, s (bar)

1020

3040

Ts, in (K)

0.5

0.7

0.9

1.1

1.3

1.5

FCH4, in (mol/s)

0.25

0.75

1.25

1.75

2.25

010

2030

40

FH2O, in (mol/s)

0.2

0.5

0.8

1.1

1.4

1.72

010

2030

40

FO2, in (mol/s)

7.515

2.2530

3.7545

GH2O, in (mol/s)

Chr

omos

ome

Num

ber

Chr

omos

ome

Num

ber

Chr

omos

ome

Num

ber

010

2030

40C

hrom

osom

e N

umbe

r

900

850

800

750

700

Chr

omos

ome

Num

ber

010

2030

40C

hrom

osom

e N

umbe

r

Fig.

7.D

istr

ibut

ion

ofva

riab

les—

hydr

ogen

gene

ratio

nsy

stem

.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0

0.2

0.4

0.6

0.8

1.0

0.20.3

0.40.5

0.60.7

OF3

OF1

OF2

δ=10μm, Am/Am, base=1

δ=10μm, Am/Am,base=5




Fig. 8. Effects of device size on optimal solutions—hydrogen generation system.

and exergy loss have been analyzed. The feed rate (GHSVF )results in reversed-direction effects on methanol formation andexergy loss. However, the other three variables, reactant inlettemperature (Tt,in), reactant ratio (FH2,in/FCO2,in), and reactionpressure (PT,t ), cause identical-direction effects on methanolformation and exergy loss. When feed rate is increased, themethanol formation is increased. Due to the increase of reactionextent and the exothermic nature of the main reactions, theproduct stream temperature will increase and further results inthe decrease of exergy loss.

The optimal solutions obtained from triple-objective-function optimization are presented on the Pareto plot as inFig. 4. The solutions scatter irregularly. As expected, the trade-off relation between OF1 (converse of methanol product rate asdefined in Eq. (13)) and OF2 (hydrogen feed rate) are straight.Similar trade-off, although not as rigid, is observed for OF2and OF3 (exergy loss). However, no obvious trade-off can bepresumed for OF1 and OF3.

To investigate the effect of device size on the optimalsolutions, membrane area and membrane thickness are variedto implement the GA optimization. Fig. 5 shows the optimalsolutions for cases with membrane thickness varying from 80to 315 �m and membrane area varying from 1 to 5 times ofthat of base case. When decreasing the membrane thickness,the optimal solutions are shifted toward the direction of highermethanol production but also higher hydrogen consumptionand exergy loss. When the thickness is reduced from 315 to150 �m, the extent of that effect is not as great as when re-duced from 150 to 80 �m. The effects of increasing membranearea are similar to that of decreasing membrane thickness.

For the base case and the cases with different device param-eters, correlation analysis is conducted for decision variablesand objective functions. Methanol production rate is directlycorrelated to feed rate and reaction pressure, but inversely

correlated with reaction temperature, and its dependence onthe H/C ratio (FH2,in/FCO2,in) is relatively low. Hydrogen feedrate is not closely related to any variable, except the feed gas rateas can be expected. Exergy loss is highly negative correlated toreaction pressure, but only slightly correlated to the other threevariables For each objective function, the correlation directionswith some of the variables are contrary for different cases.

6. Optimization for hydrogen production system

The individual parameter studies for the base case are an-alyzed. The feed rate (GHSVF ) results in increases of bothhydrogen product rate and exergy loss. For steam feed rate, amaximum exists for the hydrogen product rate. The increasingof steam feed rate results in decreasing of exergy loss. However,both effects are not considerable. The increase of oxygen feedrate will cause increases of hydrogen product rate and exergyloss first and followed by decrease effects on both. Reactionpressure shows almost no effect on exergy loss, but its increasecauses increases of hydrogen product rate. Reaction temper-ature increase results in linear increase of hydrogen productrate but linear decrease of exergy loss. The effects of sweepingsteam are small.

The optimal solutions obtained from triple-objective-function optimization are presented on the Pareto plot as inFig. 6. Different from the above methanol synthesis system,all solutions lay linearly on the Pareto plot. Linear trade-offrelations are found. For the optimal solutions, the distributionsof variables are shown in Fig. 7. Some variables widely spreadout, such as methane feed rate, steam feed rate, and sweepingsteam rate. Some others fall in a narrow range, such as the re-action temperature, oxygen feed rate, and oxygen to methaneratio. The reaction pressure is distinctive that it concentrates ontwo regions, one is around 4 bar and the other is within 2–3 bar.


The effects of device size on the optimal solutions are shownin Fig. 8. Reducing membrane thickness or increasing mem-brane area will both result in shifting the optimal solutionstoward the direction of higher hydrogen production but alsohigher methane consumption and exergy loss. In the figure,two solution points with 5 �m thickness and 5 times area laydistinctly lower than the other ones and they have markedlylower exergy losses. These two solutions fall in the first frontas the other ones and hold the special character of using theclose to lower limit pressure and the close to higher limittemperature. These solutions only appear in the case of verythin membrane thickness (5 �m) and very large membrane area(5 times area). Therefore, they should represent the optimalsolutions unique to the device with intensive membrane sepa-ration capacity.

With respect to the correlation analysis, hydrogen pro-duction rate is positively correlated to most of the decisionvariables, except for some variables in certain cases. For exam-ple, oxygen feed rate and reaction temperature in the case of5 �m thickness and 5 times area. The exceptional correlationdirection corresponds to the evolving of the above-mentionedunique solutions. Because of the linear trade-off relations be-tween objective functions, the correlation characteristics to thedecision variables for all three objective functions are verysimilar.

7. Conclusions

Using NSGA-II, multiple solutions with trade-offs amongtriple objective functions are obtained for catalytic membranereactors for methanol synthesis as well as hydrogen genera-tion, respectively. Characteristics of the optimal solutions arerevealed too.

Two systems show distinctive features. For methanol syn-thesis, the solutions are more randomly scattered in both thesolution space and the Pareto plot. Trade-off relation betweenmajor product rate and exergy loss is relatively weak. For hy-drogen generation, the solutions are linearly spread out in thePareto plot. Binary trade-off relations among the three objec-tive functions are all linear. The distribution analysis indicatesthat certain narrow-range values should be adopted for somevariables.

For both systems, the device size, i.e., membrane thicknessand membrane area, affects the optimal solutions as well as thecorrelation characteristics between the objective functions andvariables.

Compared to the individual parameter sensitivity study,multi-objective optimization provides much more informationon the optimal solutions and correlation relations betweenobjective functions and variables.

Notation

a� catalyst particle area per unit mass, m2/kgAc reactor cross section area, m2

Am membrane area per unit reactor length, m

CP,i heat capacity of component i, J/(mol K)De,i intra-particle effective diffusivity, m2/sEA activation energy of membrane

permeability, J/molEX exergy rate, J/sEXL exergy loss rate, J/sF feed side flow rate, mol/sG permeate side flow rate, mol/sGen_Set number of generation set for optimization

iterationGHSV gas hourly space velocity, h−1

Hi molar enthalpy of component i, kJ/moli component ikm heat conduction coefficient across mem-

brane, J/(m s K)L reactor length, mnc component numberNi membrane permeation flux of component

i, mol/(m2 s)NP population sizeOFk objective function kpc crossover probabilitypi bulk gas phase partial pressure of compo-

nent i, barpm mutation probabilityps,i intra-particle partial pressure of compo-

nent i, barP 0 pre-exponential factor of membrane

permeability, J/molq heat flux across membrane per unit reactor

length, J/mR gas constantRs,k reaction rate of the kth reaction at catalyst

surface, mol/(kgcat s)r1 inside radius of membrane tube, mr2 outside radius of membrane tube, mr3 inside radius of shell, mrpore,j jth size pore radius of catalyst,mrtube membrane tube radius, mrs catalyst particle radius, mTshell reactor shell side temperature, KTtube reactor tube side temperature, Kz axial direction coordinate, m

Greek letters

� membrane thickness, m�B reactor bed porosity�s catalyst particle porosity�c probability distribution index for

crossover�m probability distribution index for mutation�i,k stoichiometric coefficient of component i

in the kth reaction� intra-particle dimensionless radius�s catalyst density, kg/m3

standard deviationB bed tortuosity


Subscripts

i component iin inletF feeds shell or surfaceT totalt tube

Acknowledgment

The authors acknowledge the financial support of NationalScience Council of Taiwan.

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multi-objective optimization for two catalytic membrane reactors—methanol synthesis and hydrogen...

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