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Available online at www.sciencedirect.com

Chemical Engineering and Processing 46 (2007) 1299–1309

A pseudo-dynamic optimization of a dual-stage methanolsynthesis reactor in the face of catalyst deactivation

H. Kordabadi, A. Jahanmiri ∗Chemical and Petroleum Engineering Department, Engineering School, Shiraz University, Shiraz 71345, Iran

Received 17 April 2006; received in revised form 12 October 2006; accepted 12 October 2006Available online 28 December 2006

bstract

This paper presents an optimization investigation on methanol synthesis reactor in the face of catalyst deactivation using multi-objective geneticlgorithms. Catalyst deactivation is a challenging problem in the operation of methanol synthesis reactor and has an important role on productivity ofhe reactor. Therefore, determination of the optimal temperature profile along the reactor could be a very important effort in order to cope with catalysteactivation. Our previous studies clarify the benefits of a two-stage reactor over a single stage reactor. In this study, an optimal temperature trajectory

s obtained for each stage of the corresponding two-stage reactor. Here, steady state optimization is performed in six different activity levels by maxi-

izing the yield and minimizing the temperature of the first stage of the reactor. Multi-objective genetic algorithms are used to solve this two-objectiveptimization. The set of optimal solutions obtained for six activity levels represents an optimal temperature trajectory for each stage, which haseen extended and proposed as a dynamic optimization. This optimization resulted in an additional 3.6% yield, during the course of 4-year process.

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2006 Elsevier B.V. All rights reserved.

eywords: Optimization; Methanol synthesis reactor; Multi-objective genetic a

. Introduction

Methanol is one of the most important products of natural gas.his primary petrochemical is today used to produce a varietyf chemical intermediates. Methanol synthesis consists of threeain parts: synthesis gas preparation, methanol synthesis andethanol distillation.The reactor that serves in our study is a Lurgi-type [1] which

esembles as a shell-and-tube heat exchanger that stands ver-ically and operates at a low pressure of about 75 bar. Theynthesis gas is produced from natural gas in the reformer sec-ion. Methanol synthesis reactions occur in a set of vertical tubesacked by Cu-based catalysts. Heat of exothermic reactions isemoved from tubes by boiling water, flowing in the shell of theeactor as coolant. Catalyst deactivates in the reactor mainly dueo thermal sintering, in the course of process. Thus, reactor oper-tion has a dynamic state. The reactor starts with fresh catalysts

hat end up with low activity so that a design based on steadyperation of the reactor is not economic. Several researchesre reported on methanol process in the literature. Lange pre-

∗ Corresponding author.E-mail address: jahanmir@shirazu.ac.ir (A. Jahanmiri).

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255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2006.10.015

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ented a review of methanol synthesis technology improvement2]. Løvik reviewed some kinetics and deactivation model ofethanol synthesis [3].Two zones could be distinguished in the methanol synthesis

eactor with imprecise transition point. The first zone starts fromeactor entrance and continues to a point that conversion movesoward equilibrium. In this zone the kinetics controls the pro-ess. Later on, control of the process switches to equilibrium.s the catalyst deactivates, the first zone develops forward and

he second zone becomes smaller.Both kinetics of reaction and catalyst deactivation are

trongly related to temperature of catalyst beds. Therefore, theemperature policy of reactor strongly affects the performancef methanol synthesis reactor. Among recent optimizationpproaches, Løvik et al. studied the dynamic modeling, opti-ization and estimation of catalyst deactivation model and

ound an optimal temperature trajectory for a typical methanolynthesis reactor [4]. Ghader determined optimal temperaturef reactor inlets [5]. Jahanmiri and Eslamloueyan presented anptimal temperature profile along methanol synthesis reactor

hrough a steady state optimization [6].

It was shown that an optimal temperature profile could bebtained along the reactor that results in maximum productionate [7]. This optimal temperature profile varies with operation

1300 H. Kordabadi, A. Jahanmiri / Chemical Enginee

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Fig. 1. Typical configuration of the two-stage methanol synthesis reactor.

ime due to catalyst deactivation; so, at different activity levelshere are different optimal temperature profiles.

A multi-stage reactor that operates with regard to the opti-al temperature trajectories of each stage is supposed to be

he best idea of implementing optimal conditions. Here, eachtage is identified with its own cooling shell and the correspond-ng temperature. Previous studies indicated the advantages of

two-stage methanol synthesis reactor over the single stageeactor operating in the Shiraz Petrochemical Complex as casetudy [8]. Fig. 1 shows a typical configuration of the two-stageethanol synthesis reactor.Lurgi presented its MegaMethanol technology for methanol

ynthesis reactor using two water-cooled and gas-cooled stagesn order to reduce methanol cost [9]. In this new process design,he coolant temperature of the first stage is higher than that ofhe second stage.

The objective of the current work is to present an optimalethanol synthesis reactor, which is configured with two stages,

nd that coolant of each stage follows an independent tempera-ure trajectory through operation. The lengths of cooling stagesre known according to previous optimization results [8]. There-ore, optimal temperature trajectory of each stage should beiscovered through optimization.

Catalyst activity declines rapidly in the few first months,hile in the following operation time activity value declines

airly gradually. Besides, computation burden of a dynamic opti-ization is quite high. These points direct us to combine steady

tate optimization and multi-objective optimization techniqueso achieve near optimal dynamic conditions and maximizingverall methanol production in this study. Moreover, the results

f this study make dynamic optimization easier with smallerearch space.

In the current study, optimization tasks are handled by steadytate optimization of several activity levels. The set of these

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ring and Processing 46 (2007) 1299–1309

ctivity levels represent the activity trend over catalyst life-time.his optimization is performed mainly to maximize methanolroduction rate; so, maximizing of methanol production rate ishe first objective function. Also, the higher the temperature, theigher rate of catalyst deactivation is observed. To reduce theate of catalyst deactivation, temperature of the reactor shoulde set as low as possible. Thus, minimization of the temperaturef catalyst bed is the second objective in this optimization prob-em. Multi-objective genetic algorithms (MOGAs) are used tonvestigate these tasks.

Genetic algorithms have received considerable attentions as aovel approach to multi-objective optimization problems. Therere some studies that use MOGAs in chemical engineeringroblems. Mitra et al represented a multi-objective dynamicptimization of an industrial Nylon 6 semi-batch reactor usingAs [10]. Rajesh et al. reported results of a multi-objectiveptimization of industrial hydrogen plants [11]. These men-ioned studies used non-dominating sorting genetic algorithmsNSGAs) to find Pareto-optimal solutions [12]. MOGA used inhis study is developed by Fonseca and Fleming on account ofome superiority reported for this kind of MOGAs [13,14].

In the remainder of this paper, a background of MOGAs isiven first, and then the used MOGA is surveyed; later on, theinetics, model of process and its simulation are described; then,ptimization procedure and results are presented. Conclusionppears in the last section.

. Multi-objective genetic algorithms

The genetic algorithms are used to solve multi-objectiveroblems and they are often known as multi-objective geneticor evolutionary) algorithms (MOGAs). In multi-objective opti-ization problems, there usually exist a set of solutions that

annot simply be compared with each other. These trade-offolutions are called non-dominated solutions or Pareto-optimalolutions. Coello et al. surveyed more than twenty GAs-basedethods used in multi-objective optimization problems [15].e classified GAs-based approaches in solving multi-objectiveptimization in as naı̈ve approaches combining objectives into aingle function, non-aggregating approaches that are not Pareto-ased and finally Pareto-based approaches. Several Pareto-basedOGAs are introduced in related literature. Among them, Fon-

eca and Fleming genetic algorithms (FFGAs) are understoods a qualified approach [13]. FFGAs are sorted in ranking Paretoethods because of ranking-based selection [14].A typical structure of FFGAs is given in Fig. 2, where p(t)

tands for population in generation of t. Based on elitism method,areto solutions are saved as E(t) to avoid getting lost and

o display them into next iteration. As it may be seen in theresented algorithm, each objective function is evaluated forndividuals after random initialization of the individuals. Then,he Pareto-solutions are identified. In the following loop, afterecombination of individuals, the objective functions of new

he individuals (offsprings) are evaluated and then Pareto solu-ions will be updated. Fitness assignment and selection are the

ost important steps that often specializes the MOGAs. Here,on-dominated individuals are assigned rank 1 and any other

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s assigned a rank equal to its dominating individuals plus one.election is done by the stochastic universal sampling method,hich is proposed by Baker [16].Because of random errors, GAs often prefer to converge on

single optimal point; so, to handle population diversity andbtain Pareto-based solutions, sharing techniques are used thatodify fitness values to get trade-off solutions. Fonseca andleming have extended a phenotype adaptive sharing method in

heir works to implement on fitness sharing.Chromosome representation is real on bit (bits are filled with

–9 to have strings). Classic two-point crossover and classicutation are utilized here. Sampling method is “universal sam-

ling method”.

. Kinetic consideration, mathematical model andimulation

Methanol synthesis reactor of Shiraz petrochemical complexs chosen as the case study. The current optimization study isased on the deterministic mathematical model of the reactor.he kinetic of such a reaction system and the deactivation modelf catalyst pellet have to be coupled with the mathematical modelo cope with further simulation studies.

.1. Kinetic consideration

In the methanol synthesis, three overall reactions are possible:ydrogenation of carbon monoxide, hydrogenation of carbonioxide and reverse water-gas shift reaction, which follow as

O + 2H2 ↔ CH3OH (1)

O2 + 3H2 ↔ CH3OH + H2O (2)

O2 + H2 ↔ CO + H2O (3)db

ing and Processing 46 (2007) 1299–1309 1301

The kinetics model and the equilibrium rate constants areelected from Graaf’s studies [17,18]. Components of processow are CH3OH, CO2, CO, H2O, H2, CH4, N2.

Deactivation model of CuO/ZnO/Al2O3 catalyst has beennvestigated by several researchers; however the model offeredy Hanken was found to be suitable for industrial applications19]:

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.2. Mathematical model

Methanol reactor is traditionally modeled by a heterogeneousynamic model. In this model, the gradient between solid anduid phases is considered. In addition, in model development,

here are some assumptions as isotherm catalyst pellet, no axialispersion, no viscous flow on catalyst pellets, and also pressureas been considered to drop linearly along the reactor.

The corresponding mass and energy equations for the solidhase are:

sctdyis

dt= kgict(yi − yis) + riρBa, i = 1, 2, ..., N − 1 (5)

BCpsdTs

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ri(−�Hf,i) (6)

hereas yis and Ts are the solid-phase mole fraction and tem-erature, respectively.

The corresponding conservation rules for the fluid phase are:

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hereas yi and T are the fluid-phase variables. The boundaryonditions are:

i = yi0, T = T0 at z = 0; (9)

The initial conditions are:

yi = yssi , yis = yss

is , T = T ss, Ts = T sss , a = 1,

at t = 0; (10)

here,yssi , yss

is are profiles of mole fraction at steady-state, andss and T ss

s are profiles of temperature along the reactor inuid-phase and solid-phase, respectively. The industrial reactorpecifications are tabulated in Table 1.

.3. Simulation

The basic structure of the model is consisted of the partialerivative equations of mass and energy conservative rules ofoth the solid and fluid phase, which have to be coupled with the

1302 H. Kordabadi, A. Jahanmiri / Chemical Engineering and Processing 46 (2007) 1299–1309

Fig. 3. Steady state simulation result for methanol synthesis reactor: (a) temperature and (b) methanol concentration.

Table 1Reactor and catalyst specifications

Specification 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) 5.4T

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Fig. 4. Comparison of dynamic simulation result and plant data for methanolsynthesis reactor.

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coolant temperature trajectories of two cooling shells of the two-

otal molar flow rate per tube (mol/s) 0.65

rdinary differential equation of the deactivation model, and alsoon-linear algebraic equations of the kinetic model and auxiliaryorrelations. This would make a stiff set of partial differentiallgebraic equations. In order to solve the set of reactor modelquations, a steady-state simulation has been used prior to aynamic simulation, and the steady-state simulator gives thenitial values of the dynamic one.

To solve the set of model equations at steady-state conditions,nite difference approximation is applied to the steady-stateodel equations with regard to the consideration of a fresh cat-

lytic bulk with the activity of unity. Gauss–Newton method ispplied to the set of discretized equation in each 30 nodes alonghe reactor.

Temperature and methanol concentration profile of the fluidhase at steady-state condition have been presented in Fig. 3(a)nd (b), respectively.

To solve the set of stiff model equations dynamically, the setf equations have been discretized respect to axial coordinate,nd modifies Rosenbrock formula of order two has been appliedo the discretized equations in each node along the reactor tontegrate the set of equations with respect to time. The pro-ess duration has been considered to be 1200 operating days.ig. 4 compares methanol production rate of the plant with theorresponding value of simulation, and it is obtained that goodgreement could be found between simulation and plant data.

The bulk-average dynamic behavior of catalyst activity isketched in Fig. 5. The local change of activity along the reactors due to local variation of temperature, which consequentlyffects the catalyst activity of bed.

The detail discussion of the mathematical model of industrialeactors and the behavior of different states have been thoroughlyiscussed in the work done by Rezaie et al. [20] and distribution

sst

Fig. 5. Average catalyst activity in reactor for 1400 days of operation.

f catalyst activity at different operating times in the reactor beds presented by Rahimpour et al. [21]. Methanol and tempera-ure surfaces that they are obtained with dynamic simulation areiven in Figs. 6 and 7. Catalyst bed performance along lengthf reactor and during time is shown with these figures.

. Formulation of the optimization problem

The aim of this optimization problem is to obtain optimal

tage reactor. In design points of view, stages 1 and 2 are theame; although, heat of reaction remove with different coolantemperature from each of stage. From our previous study, the

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Fig. 6. Methanol mole fraction surface.

ength of the first and second stages are set to be 4.4 and 2.6 m,espectively [8]. In order to get optimal temperature trajectories,teady state optimization is carried out at six specific activityevels, independently. The chosen activity values are a = 0.9,.8, 0.7, 0.6, 0.5 and 0.4. These activity levels stand for theynamic operating conditions of the system. Thus, the set ofptimal solutions obtained at these six points could be used asn approximation of the dynamic solution.

Maximizing of methanol production rate is the first objectiveunction. High temperature of catalyst bed increases catalysteactivation; and consequently the overall methanol productionecreases. So, the temperature of catalyst bed must be as lows possible. Our previous study showed that the temperaturef the second stage converges to an acceptable low tempera-ure through maximization, but in the first stage this result isot achieved [8]. Therefore, the temperature of the first stage ofhe reactor should be minimized, employing the second objec-ive function to reduce catalyst deactivation that is shown with2 = Tbed,1.

Since the computer codes are provided to minimize thebjective functions, it is needed to transform the first objectiveunction into one, which is to be minimized. Temperatures of

oolant in each stage are considered as an optimization variable.he mathematical model equations at steady-state conditionre the equality constraints. Løvik used a path constraint forynamic optimization of methanol synthesis reactor to avoid

Fig. 7. Temperature surface of reactor.

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ing and Processing 46 (2007) 1299–1309 1303

evere catalyst deactivation [3]. Similar path constraint has toe used in this optimization problem to ban that optimizationerformance provides maximum solid temperatures, TMAX,bed,bove 543 K as optimal solution. Penalization method is usedn the first objective function to handle this path constraint. Theenalty function depends on violation of constraint and activityevel. The higher the activity levels the more penalty value toeduce probability of surviving. Finally, optimization problems formulated as follows:

in f1 = −FMeOH + 10a(TMAX,bed − 543) (11)

in f2 = Tbed,1 (12)

ubjected to : x(k + 1) = f (x(k), Tshell), x(0) = x0 (13)

20 K < Tshell,1 < 540 K (14)

05 K < Tshell,2 < 525 K (15)

ath constraint : Ts < 543 K (16)

hereas Eq. (13) stands for disceretized model of reactor includ-ng ODEs for each node.

The current optimization study can be investigated throughsingle-objective optimization problem rather than a multi-

bjective one. In such single-objective steady-state optimizationroblem, a soft constraint should be used, taking into account theffect of temperature on catalyst deactivation; however, provid-ng an appropriate soft constraint and its penalty function in ordero make problem converge on a desirable temperature (optimalingle solution) is more difficult than multi-objective imple-entation. Furthermore, MOGAs can provide wide choices asareto-optimal solutions to select more convenient decision vari-bles.

. Results and discussion

Multi-objective optimization is investigated for each activityevel at steady-state condition. The case study is a Lurgi type

ethanol synthesis reactor, operating in Shiraz Petrochemicalomplex. In the two-stage reactor that is considered here has

imilar design to the case study. In the other word, it coulde presumed that both of them are the same, but each stagef two-stage reactor is cooled with different coolant tempera-ure. Total length of two-stage one is equal to case study. Theoolant temperatures are different in the two stages. Referenceemperature is 525 K and all results of optimization are com-ared with the corresponding operating conditions of the casetudy at the reference operation. On GA parameter, populationize is 25 and crossover and mutation probabilities are 0.70 and.02, respectively.

A set of trade-off solutions (that a solution includes opti-al coolant temperature of Tshell,2 and Tshell,1) is obtained with

ptimization for each activity level. Fig. 8(a) and (b) shows

areto-optimal fronts in objectives space and optimal trade-offolutions of coolant temperatures for three typical activity levels,= 0.8, 0.6 and 0.4. Steady state simulations of Pareto-optimal

olutions shows that optimal solutions do not violate the path

1304 H. Kordabadi, A. Jahanmiri / Chemical Engineering and Processing 46 (2007) 1299–1309

ctive

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Fig. 8. (a) Pareto-optimal fronts in obje

onstrain and consequently they have no contribution to penaltyalue. Therefore, f1 can be either noted by –FMEOH.

As seen in Fig. 8(b), temperature of stage 1 reactor is secondbjective and this objective function is not affiliated with coolantemperature of stage 2; it is the key point of solution scatteringn Pareto set. When operation commences with fresh catalysta = 0.8 or 0.6 similar to which a = 0.9 and 0.7), most of reac-ion is happen in the stages 1 and 2 faces to nearby equilibriumituation; so, stage 2 has little effect on methanol conversionhat this effect is covered by “sharing method (niching)” usedn multi-objective optimization. Thus, optimization convergeso Pareto solutions that coolant temperature of stage 2 is whichields maximum production rate. There is a unique value foroolant temperature providing maximum production at certainime of operation. Consequently, solutions in Pareto set appear

ith unique Tshell,2 and different Tshell,1.In contrast, Tshell,2 contributes to production rate of reactor as

atalyst deactivation develops during operation time (the periodhat activity is about a = 0.4 similar to when a = 0.5). This contri-

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space and (b) pareto-optimal solutions.

ution is so considerable that it could be not covered by “sharingethod”; consequently, fitness sharing leads optimization con-

ergence to Pareto solutions with scattered Tshell,2 similar toshell,1. As a conclusion, the lower the catalyst activity, the morecattered Tshell,2 obtained in Pareto solutions.

Moreover, as catalyst deactivates, the range of optimal vari-bles in Pareto-optimal set open out on account of this facthat the higher the activity, the conversion is more sensitive toemperature of reaction. Moreover, kinetics and equilibrium con-rolling roles along the reactor demonstrate this phenomenon.ecause of catalyst deactivation, the kinetics zone develops into

he second stage of reactor during operation so that after aboutyear conversion is more sensitive to Tshell,2 with comparison

o Tshell,1.The results of these multi-objective optimizations provide

range of choices to select optimum condition. Based on thenowledge of reactor performance during time (that catalysteactivation depends on reactor temperature) and its proper-ies in different point of catalyst tubes and some trial and error,

H. Kordabadi, A. Jahanmiri / Chemical Engineering and Processing 46 (2007) 1299–1309 1305

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titthe activity of the second stages is about 7% higher than in thefirst stage and about 3–4% higher than the reference operationat 525 K.

Fig. 9. Optimal coolant temperature trajectories.

t is possible for a unique decision making that a single solu-ion is selected from each Pareto-optimal solution. Six optimalolutions depict a trend maximizing the overall production ofethanol. Decision-making is an innovative method for this

articular optimization problem that is adopted from systemehavior. For instance, that each temperature has how effectsn later deactivation is a criterion for decision making. Finally,n optimal temperature trajectory is obtained from the set of theix optimal solutions for each stage.

A multi-objective optimization is featured by confliction ofbjective functions. An other essential point about Pareto solu-ions is that bounds of them are different from optimizationariables bounds on account this fact that in the exothermiceaction there is a unique temperature at a certain time of oper-tion (defined activity) that it causes maximum conversion; so,ither lower or higher temperature of reaction yields less con-ersion during reactor performance. This phenomenon depictsub-bounds for Pareto solutions instead of bounds of optimiza-ion variables with respect to reaction properties at each time.

Fig. 9 shows optimal temperature trajectory of each stage.his graph demonstrates the optimal temperature of coolant atach stage. Each temperature in the trends of stages 1 and 2re obtained according to an optimal solutions extracted fromareto set. Each step of these stepwise trajectories proposes anptimal temperature in a period which the activity level is equalo the defined values.

As seen, there is a sharp increasing trend of each stage inbout the first year of operation, due to sharp catalyst deacti-ation. In contrast, the coolant temperature of each stage hasust one stepwise increase in the other 3 years. These trendsre in agreement with the comments reported in the literaturehat coolant temperature of has to be increased during operationn order to compensate the catalyst deactivation effects (Løvik,001). The optimal temperature trajectories of coolant streamay be illustrated in Fig. 9. The corresponding operation time

f each activity level is obtained through dynamic behavior of

atalyst deactivation, which is presented in Fig. 5.

As it may be seen in Fig. 9, there are similar trends in the tem-erature trajectories of two stages. In both of them, temperature

Fig. 10. Temperature surface in the reactor.

f coolant increases with time. There is also an approximatelyonstant difference between coolant temperatures of the twotages so that the two stages of cooling shell could be named asigh-temperature and low-temperature stages.

The main goal of this optimization problem is to enhance theverall production. Therefore, the results of these steady-stateptimizations offered for the optimal temperature trajectoriesf coolant are used for dynamic optimal solutions. Dynamicimulation shows that the two-stage reactor operating in opti-al operation conditions yields 3.6% higher overall production

ompared to the single stage reactor operating at reference theemperature. Fig. 10 shows the optimal temperature surface inhe reactor during its process duration and also respect to axialoordinate. This surface demonstrates the trend of reactor tem-erature caused by the two high and low temperature coolinghells and the effect of applying the coolant temperature policiess resulted in Fig. 9.

Fig. 11 shows the activity surface according to time and reac-or length. As it is observed, the lowest activity level is observedn the middle of the reactor that is exposed to higher tempera-ures all the time. As it is seen, at the end of catalyst life time,

Fig. 11. Catalyst activity surface.

1306 H. Kordabadi, A. Jahanmiri / Chemical Enginee

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rdmcreactor gives about 0.86 million dollars higher profit than opti-mal single stage reactor operating under optimal temperaturetrajectory over 4 years. On the other hand, capital cost of theredesigning for new configuration is negligible compared to the

Fig. 12. Optimal production rate throughout catalyst life-time.

Fig. 12 shows optimal production rate with handling of opti-al temperature trajectories and production rate of reference

eactor during operation time. In the first days of operationptimal is less than reference; but, in following time optimumurpass the reference so that by passing of time more profitsould be obtained instead of reference one. As seen, there areome rough fluctuations for optimal production rate. These ariserom change of temperature of coolant in the first stage or in theecond stage of reactor. According to Fig. 9, optimal operationolicy recommends that temperature of each stage varies inde-endently; therefore, trend of production rate emerges as whats observed in Fig. 12.

Comparison between the results of maximizing the methanolroduction, using a single-objective and a multi-objective opti-ization for the two-stage reactor shows that the advantages

f multi-objective instead of single-objective optimization. Theingle objective optimization causes 3.1% additional productionor this two-stage reactor, while the multi-objective optimizationesulted in 3.6% additional methanol production.

Fig. 13(a) and (b) compares the optimal temperature tra-ectories obtained by single objective and multi-objectiveptimizations. As may be seen, in the both stages, tempera-ures obtained by single objective optimization are higher thanemperature chosen among Pareto solutions of multi-objectiveptimization in most parts of operating time; however, in theast year, optimal temperatures are alike. It seems that the forcef multi-objective optimization is in term of lower temperaturesausing less deactivation rate in the first stage.

To clarify the benefits of employing a two-stage reactornstead of the conventional single stage reactor of Lurgi, thewo-stage and the single stage reactors are compared with eachther at the same condition. Therefore, the single stage reactor isptimized to get an optimal coolant temperature trajectory. Theethanol yield of case study, which is operating with optimal

emperature trajectory increases by 1.8% over 4 years, comparedo the operation at reference temperature. On the other words,

peration of the reactor with regard to the optimized temperaturerajectories results in 1.8% higher yield for the two stage one,hen compared to a conventional single stage reactor.

Fg(

ring and Processing 46 (2007) 1299–1309

To clarify the benefits of employing a two-stage reactornstead of the conventional single stage reactor of Lurgi, the two-tage and the single stage reactors are compared with each othert the same condition. In the other words, case study and repre-ented design are compared in the optimal operation conditiono show profits of two-stage design. Therefore, the single stageeactor is optimized to get an optimal coolant temperature trajec-ory. The methanol yield of optimal case study increases by 1.8%ver 4 years, compared to the operation at reference tempera-ure. Consequently, optimal two-stage reactor gives 1.8% moreroduction instead of case study working in optimal thermalonditions.

Considering an average methanol production rate of 270 met-ic tons per day, and taking into account the value of 120 USollars for each ton [22], the optimization brings a benefit 1.7illion dollars profit over 4 years of operation compared to the

onventional process. Moreover, optimal two-stage methanol

ig. 13. Comparison between coolant temperature trajectories obtained by sin-le objective optimization and two-objective optimization. (a) High temperaturefirst stage) and (b) low temperature (second stage).

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ptimization benefits. The reader should note that the case wetudied was a small size plant. For a Lurgi-type reactor withapacity of 5000 ton per day, the profits could increase up to 12illion dollars over 4 years of operation.From the view point of industrial aspects, it is better to replace

he objective functions with more economical ones. Since theodification supported by this study do not has significant capi-

al and energy cost, their application will be valuable. Although,ore optimization variables and economical objective functions

ield more accurate optimal design and operation strategies forwo-stage methanol synthesis reactor.

. Conclusion

In this study, the optimal temperature policy for a two-stageethanol synthesis reactor has been investigated by the use ofulti-objective genetic algorithms. In this optimization prob-

em, production yield is maximized and temperature of therst stage is minimized by steady state optimization at sixpecific activity levels. The temperature of the first stage is min-mized to reduce the rate of catalyst deactivation. In this study,

ulti-objective genetic algorithms help us to select convenientolutions from Pareto sets, with regard to catalyst deactivationn preparation for steady state optimization. The set of optimalolutions gives an optimal temperature trajectory for each stagef the reactor.

By applying the optimized policies of the coolant temperaturef both the states in dynamic simulation, the overall productionf two-stage reactor increases by 3.6% compared to the conven-ional process, and it increases by 1.8% compared to operationf the case study at its optimal temperature trajectory. The profitf this optimal operation is 1.7 million dollars compared to theeference operation and about 0.9 million dollars compared tohe optimal single stage reactor. The profit would proportion-lly increase with the increase in the plant capacity. In case thathe size of methanol plant increases to 5000 tons per day of

ethanol, the profit obtained by optimization would exceed to2 million dollars during the course of process.

ppendix A

activity of catalystv specific surface area of catalyst pellet (m2 m−3)c cross section area of each tube (m2)i inner area of each tube (m2)o outside are of each tube (m2)t total concentration (mol m−3)pg specific heat of the gas at constant pressure (J mol−1 K)ps specific heat of the solid at constant pressure

(J mol−1 K)p particle diameter (m)i tube inside diameter (m)

ij binary diffusion coefficient of component i in j (m2 s−1)o tube outside diameter (m)im diffusion coefficient of component i in the mixture

(m2 s−1)

TTuu

ing and Processing 46 (2007) 1299–1309 1307

d activation energy used in the deactivation model(J mol−1)

1 first objective function2 second objective functioni partial fugacity of component i (bar)MeOH Production rate (ton day−1)t total molar flow per tube (mol s−1)f gas–solid heat transfer coefficient (W m−2 K−1)i heat transfer coefficient between fluid phase and reactor

wall (W m−2 K−1)o heat transfer coefficient between coolant stream and

reactor wall (W m−2 K−1)Hf,i enthalpy of formation of component i (J mol−1)H298 enthalpy of reaction at 298 K (J mol−1)

KH2O/K1/2H2

gi mass transfer coefficient for component i (m s−1)1 reaction rate constant for the first rate equation

(mol kg−1 s−1 bar−1)2 reaction rate constant for the second rate equation

(mol kg−1 s−1 bar−1/2)3 reaction rate constant for the third rate equation

(mol kg−1 s−1 bar−1)conductivity of fluid phase (W m K−1)

d deactivation model parameter constant (s−1)i adsorption equilibrium constant for component i

(bar−1)Pi equilibrium constant for component iw thermal conductivity of reactor wall (W m K−1)

length of reactor (m)i molecular weight of component i (g mol−1)

number of components used in the model (N = 6)total pressure (bar)

a atmospheric pressure (bar)universal gas constant (J mol−1 K−1)

i reaction rate of component i (mol kg−1 s−1)1 rate of reaction for hydrogenation of CO

(mol kg−1 s−1)2 rate of reaction for hydrogenation of CO2

(mol kg−1 s−1)3 reversed water-gas shift reaction (mol kg−1 s−1)e Reynolds numberp particle diameter [m]c Schmidt number

time (s)bulk gas phase temperature (K)

bed,1 average temperature of the first stage catalyst bed (K)R reference temperature used in the deactivation model

(K)s temperature of solid phase (K)sat saturated temperature of coolant at operating pressure

(K)shell temperature of coolant stream (K)

shell,1 coolant temperature in first cooling stage (K)shell,2 coolant temperature in second cooling stage (K)

superficial velocity of fluid phase (m s−1)g linear velocity of fluid phase (m s−1)

1 ginee

U

y

y

xz�

Gε

ε

μ

ν

ρ

ρ

τ

A

f

B

a

•

•

•

ab1s

B

aism

t

k

[

of

wa

wtebe

h

TTplo

R

[[

[[[

[

308 H. Kordabadi, A. Jahanmiri / Chemical En

Shell overall heat transfer coefficient between coolant andprocess streams (W m−2 K−1)

i mole fraction of component i in the fluid phase(mol mol−1)

is mole fraction of component i in the solid phase(mol mol−1)state variableaxial reactor coordinate (m)

z incremental axial length (m)

reek lettersB void fraction of catalytic bed (m3 m−3)S porosity of catalyst (m3 m−3)

viscosity of fluid phase (kg m−1 s−1)ci critical volume of component i (cm3 mol−1)B density of catalytic bed (kg m−3)g density of fluid phase (kg m−3)

turtuosity factor of catalyst

ppendix B

Some other equations and correlations that they are requiredor modeling and simulation are given in this appendix.

.1. Kinetics model

The kinetic rate expressions based on Graaf’s experimentsre as below:

Hydrogenation of carbon monoxide:

r1 = k1KCO[fCOf3/2H2

− fCH3OH/(f 1/2H2

KP1)]

(1 + KCOfCO + KCO2fCO2 )[f 1/2H2

+ kfH2O](A1)

Hydrogenation of carbon dioxide:

r2 = k2KCO2 [fCO2f3/2H2

− fCH3OHfH2O/(f 3/2H2

KP2)]

(1 + KCOfCO + KCO2fCO2 )[f 1/2H2

+ kfH2O](A2)

Reversed water-gas shift reaction:

r3 = k3KCO2 [fCO2fH2 − fH2OfCO/KP3]

(1 + KCOfCO + KCO2fCO2 )[f 1/2H2

+ kfH2O](A3)

Other equations and correlations for reaction rate constants,dsorption equilibrium constants and equilibrium constantsased on partial pressure could be found in Graaf et al. (1988,990) providing enough information about kinetics of methanolynthesis.

.2. Auxiliary correlations

To complete the simulation, auxiliary correlations should be

dded to the model. For the heterogeneous model, the concerns how to estimate the overall heat transfer coefficient betweenhell and tube, and the correlations for estimation of heat andass transfer between two phases should be considered too.

[

[

ring and Processing 46 (2007) 1299–1309

The mass transfer coefficient of each component, betweenhe bulk of gas phase and the solid phase:

gi = 1.17 Re−0.42 Sc−0.67i ug × 103 (A4)

More details could be found in reference published by Cussler23].

The overall heat transfer coefficient between the boiling waterf shell side and the gas phase in the tube side is given by theollowing correlation:

1

Ushell= 1

hi+ Ai ln(Do/Di)

2πLKw+ Ai

Ao

1

ho(A5)

here hi is the heat transfer coefficient between the gas phasend reactor wall and is obtained by the following correlation:

hi

Cpρμ

(Cpμ

K

)2/3

= 0.458

εB

(ρudp

μ

)−0.407

(A6)

here in the above equation, u is superficial velocity of gas andhe other parameters are those of bulk gas phase and dp is thequivalent catalyst diameter. ho is the heat transfer coefficient ofoiling water in the shell side which is estimated by the followingquation:

o = 7.96(T − Tsat)3(

P

Pa

)0.4

(A7)

and P are temperature and pressure of circulating boiling water,sat the saturated temperature of boiling water at the operatingressure of shell side and Pa is the atmospheric pressure. Theast term of the above equation has been considered due to effectf pressure on the boiling heat transfer coefficient.

eferences

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gineer

[

[

[

[

H. Kordabadi, A. Jahanmiri / Chemical En

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[

[

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