optimization of methanol synthesis reactor using genetic algorithms

Chemical Engineering Journal 108 (2005) 249–255

Optimization of methanol synthesis reactor using genetic algorithms

H. Kordabadi1, A. Jahanmiri∗

Chemical Engineering Department, Shiraz University, Shiraz, School of Engineering, 987654321, Iran

Received 29 July 2004; received in revised form 16 February 2005; accepted 23 February 2005

Abstract

This paper presents a study on optimization of methanol synthesis reactor to enhance overall production. A mathematical heterogeneousmodel of the reactor was used for optimization of reactor performance, both at steady state and dynamic conditions. Here, genetic algorithmswere used as powerful methods for optimization of complex problems. Initially, optimal temperature profile along the reactor was studied.Then, a stepwise approach was followed to get an optimal two-stage cooling shell to maximize production rate. These optimization problemswere performed through steady-state optimizations with regard to dynamic properties of the process. The optimal reactor with two-stagecooling shell presented higher performances. This optimization approach enhanced a 2.9% additional yield throughout 4 years, as catalystlifetime. Therefore, we can deduce to redesign methanol synthesis reactor with a two-stage cooling shell reactor based on this study.©

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2005 Elsevier B.V. All rights reserved.

eywords:Methanol synthesis; Methanol reactor; Optimization; Genetic algorithms

. Introduction

Methanol is one of the most important petrochemical prod-cts. It is used as a fuel, solvent and as a building block toroduce chemical intermediates. The main step of methanolrocess is methanol synthesis. Methanol synthesis reactorsre designed based on two technologies, high-pressure syn-

hesis operating at 300 bar and low-pressure synthesis oper-ting between 50 and 100 bar[1].

The methanol synthesis reactor studied here was a Lurgi-ype, which is operated in the low-pressure regime[2]. Theynthesis gas—CO2, CO, H2—is produced from natural gasn reformer plant and enters to the reactor. Methanol synthesiseactions occur in a set of vertical tubes packed with CuO-ased catalyst. Heat of exothermic reactions is removed from

ubes using boiling water, which is circulating as a coolant inhe shell of reactor. The methanol synthesis reactor exhibitsdynamic behavior, mainly due to catalyst deactivation. Theperating period of the reactor starts with fresh catalyst and

∗ Corresponding author. Tel.: +98 711 2303071; fax: +98 711 6287294.

ends in a certain low activity (about 0.4) so that operatiothe reactor at that time is not economic. The operating pof the reactor is about 4 years as its catalyst lifetime. Aftercatalyst cycle time, the deactivated catalyst must be repwith a fresh one.

There are several researches on methanol processliterature. Lange presented a review of methanol synttechnologies[1]. Moreover, several studies were reportedkinetic models of methanol synthesis[3,4] and deactivatiomodels regarding effects of temperature and gas comtion [5,6]. Because of severe temperature effects, botmethanol synthesis kinetics and catalyst deactivation,mal temperature policies is a key to optimal operatiomethanol synthesis reactor. Løvik studied dynamic moing and optimization of methanol synthesis reactor andtimation of a catalyst deactivation model[7]. She presentean optimal temperature trajectory along the methanol reand optimal recycling ratio in her work.

There are several aspects in optimization of tubular rtors. Velasco et al. presented optimal inlet temperaturejectories for adiabatic packed reactors in the face of c

E-mail addresses:hojat [email protected] (H. Kordabadi),[email protected] (A. Jahanmiri).

1 Fax: +98 711 6287294.

lyst deactivation[8]. Dixit and Grant studied optimal coolanttemperature in a non-isothermal reactor[9]. Optimization oftubular reactors had focused on reversible and exothermic

385-8947/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
oi:10.1016/j.cej.2005.02.023

250 H. Kordabadi, A. Jahanmiri / Chemical Engineering Journal 108 (2005) 249–255

Nomenclature

Ac cross-sectional area of each tube (m2)av specific surface area of catalyst (m2/m3)a activity of catalystcpg specific heat of the gas at constant pressure

(J/mol)cps specific heat of the solid at constant pressure

(J/mol)ct total concentration (mol/m3)Di tube inside diameter (m)Ed activation energy used in the deactivation

model (J/mol)FMeOH production rate (t/day)Ft total molar flow rate per tube (mol/s)hf gas–solid heat transfer coefficient (W/m2 K)Kd deactivation constant (1/h)k node numberkgi mass transfer coefficient for componenti (m/s)I objective functionP penalty functionri reaction rate of componenti (mol/kg s)T bulk gas phase temperature (K)R universal gas constant (J/mol K)Ts catalyst temperature (K)Tshell temperature of boiling water in the shell side

(K)TR reference temperature used in deactivation rate

(K)t time (s)tgen generation numberUshell boiling water-gas overall heat transfer coeffi-

cient (W/m2 K)x state variableyi bulk gas phase mol fraction for componenti in

gas phaseyis mol fraction componenti on the solid phasez axial reactor coordinate (m)

Greek letters�Hf,i enthalpy of formation of componenti (J/mol)εB void fraction of catalytic bed (m3/m3)

Superscripts and subscripts0 inlet conditionsss initial conditions

reactions so far[10]. Mansson et al. applied optimal controltheory to find an optimal temperature profile along an ammo-nia synthesis reactor, which maximized the concentration ofammonia in effluent stream of the reactor[11].

The objective of the current study was to optimizemethanol synthesis reactor and consisted two different ap-proaches. The first approach was to find the optimal temper-

ature profile along the methanol synthesis reactor to max-imize methanol production rate, while the second one wasan investigation to develop a two-stage reactor with differ-ent coolant temperatures in cooling shells as a realistic ap-proach. The idea of redesigning of methanol synthesis reactorwith a two-stage cooling shell was used to increase the over-all methanol production throughout in 4 years of operation.Both approaches were performed by steady-state optimiza-tion for some certain catalyst activity levels. These differentcatalyst activities were selected to stand for reactor dynamicsin the catalyst lifetime. Genetic algorithms (GAs) were usedfor optimization of the reactor. GAs enable us to solve thisconstrained non-linear problem with numerous variables. Itis believed that GAs could be used as powerful techniques tosolve complex and real-world problems.

Almost since last two decades, GAs have been used in alarge-scale application of engineering problems. Some ap-plications of GAs have been reported in chemical engineer-ing problems such as optimal design, operation and control[12–14].

2. Genetic algorithms

Genetic algorithms (GAs) are a class of non-traditionalstochastic methods solving complex optimization problemso est uralc suit-a s iss ing

tingp db tion( di-v erateo tors.A es. It

f the real world[15,16]. They are optimization techniquhat artificially simulate the gradual adaptation of nathromosomes in the quest of producing better and moreble individuals. A typical structure of genetic algorithmhown inFig. 1, in whichp(t) is a population of solutionseneration oft.

Genetic algorithms begin with a population of staroints initialized randomly (t = 0). Each point is evaluateased on objective function value. Then, a new populaiterationt + 1) is formed according to fitness value of iniduals (selection). Some of these points randomly genffspring through a number of predefined rules or operafter some number of generations, the program converg

Fig. 1. Structure of a typical genetic algorithm.

H. Kordabadi, A. Jahanmiri / Chemical Engineering Journal 108 (2005) 249–255 251

is hoped that the best individuals represents a “near-optimumor reasonable solution”.

3. Kinetic, model and simulation

3.1. Kinetic

In the methanol synthesis, three overall reactions are pos-sible: hydrogenation of carbon monoxide, hydrogenation ofcarbon dioxide that is strongly exothermic and reverse water-gas shift reaction:

2CO + 4H2 ↔ 2CH3OH + H2O

2CO2 + 5H2 ↔ 2CH3OH + H2O

CO2 + H2 ↔ CO + H2O

Kinetic model and the equilibrium rate constants are se-lected from Graaf’s studies[4,17].

The catalyst of low-pressure methanol synthesis isCuO/ZnO/Al2O3 and during the course of process, it is deac-tivated mainly due to thermal sintering. Among several stud-ies, the deactivation model suggested by Hanken was foundappropriate to use[6]:

3

eousm dientb eling,a is nov alystp odela

ε

ρ

w m-p

ε

ε

Table 1Design specifications of industrial methanol reactor

Specifications Value

Number of tubes 2962Length of reactor (m) 7.022Bulk density of bed (kg/m3) 1132Void fraction of bed (m3/m3) 0.39Internal radius of tubes (mm) 38Catalyst diameter (mm) 5.4

whereyi andT are the fluid-phase variables. The boundaryconditions are:

yi = yi0, T = T0 atz = 0 (6)

The initial conditions are:

yi = yiss, yis = yss

is , T = T ss, Ts = T sss , a = 1 at t = 0

(7)

where yssi andyss

is are profiles of mole fractions andT ssandT ss

s profiles of temperature along the reactor in fluid-phase and solid-phase, respectively. The industrial reactorspecifications are demonstrated inTable 1.

3.3. Simulation

The mathematical heterogeneous model involves of a sys-tem of partial differential equations solved in steady stateand dynamic modes. These equations are discretized withrespect to axial coordinate to 30 nodes along the reactor.This provides a set of ordinary differential equations ineach node.

Solution of the steady-state model was implementedon steady-state optimizations, as well as to determinet re-a odelw theo oft with“ owni nolc ofp odsw ethodo ua-t

r isc plexo r ofs s. Ina lysta t iss

da

dt= −Kd exp

(−Ed

R

(1

T− 1

TR

))a5 (1)

.2. Model

Here, optimizations are investigated with heterogenodel. In the heterogeneous model development, the graetween solid and fluid phases is considered. In the modxial dispersion is neglected, and it is assumed that thereiscous flow on the catalyst pellets and also isotherm catellet is considered. The equations of heterogeneous mre as below. Solid-phase equations:

sctdyis

dt= kgi(yi − yis) + riρBa i = 1,2, . . . , N − 1 (2)

BcpsdTs

dt= avhf (T − Ts) + ρBa

N∑i=1

ri(−�Hf,i) (3)

hereyis andTs are the solid-phase mole fraction and teerature, respectively. Fluid-phase equations:

Bct∂yi

∂t= − Ft

Ac

∂yi

∂z− avctkgi(yi − yis)

i = 1,2, . . . , N − 1 (4)

Bctcpg∂T

∂t= − Ft

Ac

∂T

∂z+ avhf (Ts − T )

+ πDi

Ac

Ushell(Tshell − T ) (5)

he concentrations and temperature profile along thector at zero time (initial conditions). Steady-state mas obtained by elimination of all time-derivatives inriginal ordinary differential equations in each node

he reactor. These algebraic equations are solvedGauss–Newton” method. Results of simulation are shn Fig. 2a and b for profiles of temperature and methaoncentration along the reactor. To solve the systemartial differential algebraic equations, different methere tested and it was observed that Rosenbrock mf order 2 was more efficient for such set of stiff eq

ions.In Fig. 3, the predicted production rate of the reacto

ompared with plant data of Shiraz Petrochemical Comver a period of about 1200 operating days. The erroimulation was found to be less than 5% in most caseddition, dynamic simulation showed a decline in catactivity (of middle point of the reactor for sample) as ihown inFig. 4.


Fig. 2. Steady-state simulation result for methanol synthesis reactor: (a) temperature; (b) methanol concentration.

Fig. 3. Comparison of dynamic simulation result and plant data for methanolsynthesis reactor.

Fig. 4. Average catalyst activity in reactor for 1400 days of operation.

4. Optimization and results

Methanol synthesis reactor of Shiraz Petrochemical Com-plex was chosen as a case study. The coolant temperature waabout 525 K. Therefore, all of the results were compared withoperation conditions in 525 K as reference. In this study, theoptimization of reactor was investigated in two approaches,optimal temperature profile approach and optimal two-stagecooling shell approach. Steady-state model was used to evalu-ate objective function, while dynamic simulation was used toevaluate overall methanol production throughout in 4 years.

4.1. Optimal temperature profile approach

From a theoretical point of view, there is an optimal tem-perature profile along the methanol synthesis reactor, whichmaximizes methanol production rate as reported in literaturefor tubular and exothermic reactors[10]. Because of catalystdeactivation, this optimal profile changes during operation sothat there is not a unique optimal temperature profile in dif-ferent times. Therefore, according to deactivation rate shownin Fig. 4, three activity levels equal toa= 0.9, 0.7 and 0.5were chosen to study optimal temperature profile. These val-ues stand for dynamic properties of reactor operation and givesome information about variation of optimal profile throughcatalyst lifetime.

As mentioned in modeling and simulation, the reactor wasdiscretized with respect to axial coordinate in 30 nodes. Themost accurate optimal temperature profile was achieved withoptimization of 30 parameters standing for coolant tempera-ture in 30 nodes. These temperatures were bounded between510 K and 535 K. The objective function was to maximizethe methanol production rate. The equations of steady-statemodel are the equality constraints. Here, there is just onepath constraint that states temperature of catalyst beds—as astate variable—which must be less than 543 K along the re-actor[7]. It was attempted to avoid dramatically catalyst de-activation. Because optimization problem was implementedw hasew of thefl altym t thesi tionp

M

s

x

5

w ti-m ined

s

ith heterogeneous model, temperature of the solid pas considered as constraint rather than temperatureuid phase. This constraint was implemented with penethod. Penalty function was taken zero provided tha

olution satisfies constraint and equals to “10 (Ts− 543)”n order to discard the violator solution. Then, optimizaroblem was formulated as below:

ax I = FMeOH − P thatP = 10(Ts − 543)

ubjected to

(k + 1) = f (x(k), Tshell), x(0) = x0;

10(K)< Tshell < 535(K); Ts < 543(K);

hereP is the penalty function. Through optimization, opal temperature profiles of coolant in the shell were obta


Fig. 5. Optimal coolant temperature profile along the cooling shell of reactor.

Fig. 6. Optimal temperature profile in cooling shell and in catalyst beds ata= 0.5.

for three different activities. These profiles are presented inFig. 5. All of them have a peak near the entrance, and then, de-cline toward the end of reactor. Furthermore, there are somedifferences among them. The lower the activity the optimaltemperature profile is obtained in higher temperature and thetemperature declines more gradually.

Optimal temperature profile of coolant causes an optimaltemperature profile in the tubes of the reactor.Fig. 6 showsoptimal temperature profile along the cooling shell and inthe tube fora= 0.5. As temperature of coolant declines, thereactions shift to the equilibrium slowly and consequently,that temperature of the catalyst bed declines gradually. Thiseffect of optimal coolant temperature could be observed onmole fraction profiles along the reactor.Fig. 7 shows that

Table 2Additional yield achieved with optimal temperature profiles

Activity Reference (t/day) Optimum (t/day) Additional yield (%)

a= 0.9 304 321 5.6a= 0.7 296 307 3.7a= 0.5 278 286 2.8

methanol mole fraction along the reactor for reference andoptimal temperature ata= 0.5. Higher methanol mole frac-tions were observed in the reactor. The differences betweenmethanol mole fractions in optimal and reference status in-crease in the earlier lengths of reactor and start to decreasetoward the end of reactor. There are similar results for othertwo activity levels ofa= 0.9 and 0.7.

To show the effect of optimal temperature profile ofcoolant on the reactor performance, methanol productionrates were evaluated. The results are given inTable 2, andas it is seen, the lower the activity, the lower additional yield.

4.2. Optimal two-stage reactor approach

Optimal temperature profiles give us some useful informa-tion about reactor operating condition. The results show thatit is possible to achieve more production rate with a suitabletemperature strategy in the reactor. An instantaneous sug-gestion is to design a multi-stage cooling shell for methanolsynthesis reactor so that optimal temperature profile can beachieved by using different temperature profiles along theshell. These optimal temperatures do dynamically change.

To provide optimal multi-stage cooling shell, the bestnumber of cooling stages, the best length of stages, and thebest temperature in each stage must be available. This ap-p ctiv-i n oft ints,t as thefi n then ndedb

und.T for at s op-t onds tioni ults.T shelli ngtha

inT ad-d .A areh eachs arec erved
Fig. 7. Methanol mole fraction along the reactor.
roach, as former, was implemented for three different aty values to get qualified solutions for dynamic operatiohe reactor. Here, the objective function, equality constraemperatures bound and path constraint are the samerst approach. Length of cooling stages is searched withiode number—as an optimization variable—and is bouetween 1 and 29.

As first step, the best number of stages should be foherefore, optimal temperatures and length of stages

hree-stage cooling shell is needed. Several runs showimizations converge on a solution that length of the sectage becomes almost zero for all three activity. Optimiza

nvestigation for a four-stage reactor shows similar resherefore, it means a reactor with a two-stage cooling

s the best choice and in the second approach optimal lend temperatures of each stage should be searched.

Optimization result of two-stage reactor is shownable 3. Optimal temperatures, length of first stage anditional yield are given inTable 3for a= 0.9, 0.7 and 0.5s seen, for all activities, temperatures of the first stageigher than temperatures of second stage. Moreover, intage, coolant temperatures of different activity levelslose to each other and significant differences are obs


Table 3Optimization results of two-stage cooling shell reactor

Activity T of firststage (K)

T of secondstage (K)

Length of firststage (m)

Additionalyield (%)

a= 0.9 532 515 2.1 4.8a= 0.7 532 516 3.26 3.1a= 0.5 534 518 4.7 2.6

among the optimal lengths of stages regarding different ac-tivity levels. In addition, the results show that the higher theactivity, the shorter the length of the first stage and more theadditional yield. The temperature and the length of stages arethe fittest solution to optimal temperature profile.

In spite of significant additional yield, the result shownin Table 3is not useable from engineering point of view be-cause of different lengths of cooling stages. Surely, dynamicoptimization is the best performance to find optimal lengthof stages and optimal temperature of stages; but dynamic op-timization has high computing cost.

As Fig. 4 shows, in the most catalyst lifetime, activitylevel is less than 0.6; so, we base our decision mainly on theoptimal solution ofa= 0.5. According to the length obtainedfrom optimization at different activity levels (Table 3), and thecorresponding operating time duration of the process betweendistinct activity levels, the optimal length was modified usingoperating time-based weight factors (obtained fromFig. 4).After this modification, the optimal lengths of first and secondsteps changed to 4.45 and 2.56 m, respectively.

In this last step of second approach, optimization was in-vestigated to find the optimal temperature of each stage forthree activity levels. Optimal temperatures of each stage andadditional yield achieved for optimal two-stage reactor aregiven inTable 4. As seen, optimization converges on signifi-cant lower temperature in both stages, especially fora= 0.9.T t ac-t thee s thec nd re-a ieldio eo e. Int stagei

ady-s alystl d on

TO

A

aaa

Fig. 8. Typical design of the two-stage methanol synthesis reactor.

Fig. 9. Optimal coolant temperature trajectory in each stage of cooling shellduring operating time.

the results shown inTable 4, the temperature trajectory usedin dynamic simulation is shown inFig. 9. These trajecto-ries are presented according to activity value during opera-tion. Simulation results show this presented new design ofmethanol synthesis reactor with two-stage cooling shell andtemperature trajectory according toFig. 9provides 2.9% ad-ditional production throughout 4 years. As seen inFig. 9,reactor is constructed of a high-temperature stage and a low-temperature stage that each stage tracks an independent step-wise temperature trajectory. Moreover, optimal temperatureof methanol synthesis reactor in earlier operating time is moreimportant than the later time.

Fig. 10 shows optimal temperature surface in catalystbeds. This surface describes effects of coolant temperature

he considerable low temperature in period that catalysivity is a= 0.9, avoids dramatic catalyst deactivation inarlier times that deactivation rate is very fast and affectonversion in later operating period. These acceptable alistic solutions are obtained with a small difference in y

mprovement rather results inTable 3. A typical configurationf the two-stage reactor is represented inFig. 8. As seen, thutput of the first reactor is the input of the second stag

his two-stage reactor, the coolant temperature of eachs controlled with different steam pressures.

In order to evaluate the effect of these optimal stetate solutions on total methanol production during catifetime, 1400 days, dynamic simulation was used. Base

able 4ptimization results of two-stage reactor

ctivity T of firststage (K)

T ofsecondstage (K)

Additionalyield (%)

= 0.9 526 511 4.6= 0.7 531 515 2.9= 0.5 533 519 2.5

Fig. 10. Optimal temperature surface in reactor.


Fig. 11. Average activity in first and second stages of reactor during oper-ating period.

Table 5Typical GAs used in this optimization problem

v Psize C Pc (%) Pm (%) tg

First approach 30 250 6 80 0.03 80Second approach I 3 25 2 70 0.01 25Second approach II 2 20 2 70 0.01 15

v, the number of optimization variables;Psize, the population size;C, thenumber of crossover points;Pc andPm, the crossover and Mutation proba-bility; and tg, the generation number.

switching between first and second stages, also temperaturevarying along time on catalyst beds.

Fig. 11 shows average catalyst activity in the first andsecond stages. As seen, in about first 600 days average activityin the second stage is more than the first one; but, in followingdays of 4 years they converge to each other so that at theend of operating period, average activity level in both stagesenclose to each other in abouta= 0.4. It is shown that differentoperating temperatures do not affect on final activity level.

A quite simple GA was used in this study. The programof optimization ran for several times and based on differentsets of GAs parameters generally, real encoding (real genes),classic multi-point crossover, classic mutation and roulettewheel method of selection were used in this study. Although,a suitable set of GA parameters were used in each step ofoptimization according to the optimization variables and theother properties of the optimization problem. Typical sets ofGA parameters are available inTable 5.

5. Conclusion

In this study, methanol synthesis reactor has been opti-mized to maximize methanol production yield. Optimization

problem includes two approaches. In the first approach, op-timal temperature profile along the reactor is studied for dif-ferent activity level. In the second approach, a reactor withoptimal two-stage cooling shell and optimal temperature tra-jectory during this time is obtained. This new design yields2.9% additional methanol production during operating pe-riod. Mathematical heterogeneous model is used in optimiza-tion investigation.

In this study, GAs are used as powerful optimization tech-niques which give good solutions for this constrained non-linear problem. Although dynamic optimization is the bestinvestigation for this problem, this study possesses importantresults. This paper allows us to perform on a similar designfor methanol synthesis reactor and the solutions prepare agood starting point to implement dynamic optimization withreduced computing cost.

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optimization of methanol synthesis reactor using genetic algorithms

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