dynamic simulation and optimization of a dual-type methanol reactor using genetic algorithms

Research Article

Dynamic Simulation and Optimizationof a Dual-Type Methanol Reactor UsingGenetic Algorithms

In this investigation, a dynamic simulation and optimization for an auto-thermaldual-type methanol synthesis reactor was developed in the presence of catalystdeactivation. Theoretical investigation was performed in order to evaluate theperformance, optimal operating conditions, and enhancement of methanol pro-duction in an auto-thermal dual-type methanol reactor. The proposed reactormodel was used to simulate, optimize, and compare the performance of a dual-type methanol reactor with a conventional methanol reactor. An auto-thermaldual-type methanol reactor is a shell-and-tube heat exchanger reactor in whichthe first reactor is cooled with cooling water and the second one is cooled withsynthesis gas. The proposed model was validated against daily process data mea-sured of a methanol plant recorded for a period of 4 years. Good agreement wasachieved. The optimization was achieve by use of genetic algorithms in two stepsand the results show there is a favorable profile of methanol production ratealong the dual-type reactor relative to the conventional-type reactor. Initially, theoptimal ratio of reactor lengths and temperature profiles along the reactor wereobtained. Then, the approach was followed to get an optimal temperature profileat three periods of operation to maximize production rate. These optimizationapproaches increased by 4.7 % and 5.8 % additional yield, respectively, through-out 4 years, as catalyst lifetime. Therefore, the performance of the methanol reac-tor system improves using optimized dual-type methanol reactor.

Keywords: Catalysts, Dynamic optimization, Genetic algorithms, Methanol, Modeling

Received: October 27, 2007; revised: December 28, 2007; accepted: January 13, 2008

DOI: 10.1002/ceat.200700408

1 Introduction

Methanol is a primary liquid petrochemical that is producedin large scale worldwide. It is used as fuel, as solvent and as abuilding block to produce chemical intermediates. It is pro-duced from synthesis gas in a large scale throughout the world.In the methanol synthesis process, synthesis gas (CO2, CO,and H2) converts to methanol in a tubular packed bed reactor.The synthesis gas is produced from natural gas in the reformersection. Such a reactor usually resembles a vertical shell andtube heat exchanger. A conventional type of methanol reactorincludes tubes that are packed with catalyst pellets. Boilingwater is circulating in the shell side to remove the heat of

exothermic reactions. Methanol synthesis reactions occur in aset of vertical tubes packed by Cu-based catalysts. The reactorthat is presented in this study is a Lurgi-type [1] which in-cludes a shell-and-tube heat exchanger that is installed verti-cally and operates at a pressure in the range of 76–77 bar. Heatof exothermic reactions is removed from tubes by boilingwater, flowing in the shell of the reactor as coolant. The cata-lyst deactivates in the reactor mainly due to thermal sintering,in the course of the process. Thus, reactor operation is a dy-namic state.

The parameters affecting the production rate in an indus-trial methanol reactor are parameters such as temperature andcatalyst deactivation. In the case of reversible exothermic reac-tions such as methanol synthesis, selection of a relatively lowtemperature permits higher conversion but this must be bal-anced against a slower rate of reaction leading to a largeamount of catalyst. To the left of the point of maximum pro-duction rate, increasing temperature improves the rate of reac-tion, which leads to more methanol production. Nevertheless,

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://www.cet-journal.com

Fatemeh Askari1

Mohammad Reza

Rahimpour1

Abdolhossein Jahanmiri1

Ali Khosravanipour

Mostafazadeh1

1 Chemical and PetroleumEngineering Department,School of Engineering, ShirazUniversity, Shiraz 71345 Iran.

–Correspondence: Prof. M. R. Rahimpour ([email protected]),Chemical and Petroleum Engineering Department, School of Engineer-ing, Shiraz University, Shiraz 71345 Iran.

Chem. Eng. Technol. 2008, 31, No. 4, 513–524 513

as the temperature increases, the deterio-rating effect of equilibrium conversionemerges and decreases methanol produc-tion [2].

There are several studies on methanolprocess in the literature. Lange presented areview of methanol synthesis technologies[3]. Rahimpour et al. investigated enhance-ment of methanol production using opti-mized Pd-Ag membrane in a methanolsynthesis reactor [4]. Kordabadi and Ja-hanmiri [5, 6] performed a study on theoptimization of a methanol synthesis reac-tor to enhance overall production and anoptimization investigation on the metha-nol synthesis reactor in the face of catalystdeactivation using multi-objective geneticalgorithms. Recently, Rahimpour and Lot-finejad presented a dynamic model for aPd-Ag membrane dual-type methanol syn-thesis reactor [7].

In this study, an auto-thermal dual-typereactor where the first reactor is cooled by saturated water andthe second reactor is cooled by feed gas has been developed.The dynamic heterogeneous one-dimensional model was con-sidered. The basic structure of the model is composed of heatand mass balance conservation equations coupled throughthermodynamic and kinetic relations, as well as auxiliary cor-relations for predicting physical properties. Due to stark tem-perature effects on methanol synthesis kinetics and catalyst de-activation, optimal temperature is an important factor foroptimal operations of the methanol reactor. The purpose ofthis study is to optimize the methanol synthesis reactor whichconsists of two procedures. An optimization program was de-veloped to obtain more methanol production during a periodof operation. The first approach is to find the optimal ratio ofreactor lengths and optimal cooling temperature profiles alongthe dual-type methanol synthesis reactor to enhance methanolproduction during 1400 days of operation. In second proce-dure, the optimal temperature profiles at three periods of op-eration were performed to reach the maximum methanol pro-duction rate. Optimization tasks were investigated using noveloptimization tools, genetic algorithms. Genetic algorithms areimitations of natural evolution and are believed to be the mostpowerful optimization technique amongst the stochastic meth-ods.

2 Process Description

2.1 Conventional Methanol Reactor (CMR)

Fig. 1 shows the scheme of a single-type methanol reactor[8]. Methanol synthesis is performed by passing a synthesisgas containing hydrogen, carbon dioxide, carbon oxide, andany inert gases at an elevated temperature and pressurethrough several beds of methanol synthesis catalyst. A single-

type (conventional-type and real plant) methanol reactor isbasically a vertical shell-and-tube heat exchanger. The synthe-sis gas is fed to the tube side of the reactor and the coolingwater flows into the shell side in co-current mode. The cata-lyst is packed in vertical tubes and surrounded by boilingwater. The methanol synthesis reactions are carried out overthe commercial CuO/ZnO/Al2O3 catalyst. The heat ofexothermic reactions is transferred to the boiling water andsteam is produced. The product goes to a heat exchangerand its heat transfers to the feed stream. Finally, the coldproduct is transported to the distillation section. In fact,methanol is recovered by cooling the product gas stream tobelow the dew point of the methanol and separation of theproduct as a liquid.

The technical design data of the catalyst pellet and inputdata of the conventional-type methanol reactor are summa-rized in Tabs. 1 and 2.


Figure 1. Schematic of a conventional type methanol reactor [8].

Table 1. Catalyst and reactor specifications [8].

Parameter Value Unit

qs 1770 [kg m–3]

dp 5.47·10–3 [m]

cps 5.0 [kJ kg–1 K–1]

kc 0.004 [W m–1 K–1]

av 626.98 [m2 m–3]

es

s0.123 [–]

Number of tubes 2962 [–]

Tube length 7.022 [m]

514 F. Askari et al. Chem. Eng. Technol. 2008, 31, No. 4, 513–524

2.2 Auto-Thermal Dual-Type Methanol Reactor(ADMR)

Fig. 2 shows the schematic diagram of an auto-thermal dual-type methanol reactor configuration. This process is mainlybased on two-stage reactors consisting of water-cooled andgas-cooled reactors. The synthesis gas is fed to the tube side ofthe gas-cooled reactor (the second reactor). The cold feed syn-thesis gas for the first reactor is routed through tubes of thesecond reactor in a counter-current flow with reacting gas andthen heated by heat of reaction produced in the shell. The out-let synthesis gas from the second reactor is fed to the shell ofthe first reactor (water-cooled) and the chemical reaction isinitiated by the catalyst. The heat of reaction is transferred tothe cooling water inside the tubes of the reactor. At this stage,the synthesis gas is partly converted to methanol in a singlewater-cooled reactor. The methanol-containing gas leaving the

first reactor is directed into the shell of the second reactor. Fi-nally, the product is removed from the downstream of the sec-ond reactor. The large inlet gas pre-heater normally requiredfor synthesis by a single water-cooled reactor is replaced by arelatively small trim pre-heater. As fresh synthesis gas is onlyfed to the first reactor, no catalyst poisons reach the second re-actor. The poison-free operation and the low operating tem-perature results in a virtually unlimited catalyst service life forthe gas-cooled reactor. In addition, reaction control also pro-longs the service life of the catalyst in the water-cooled reactor.If the methanol yields in the water-cooled reactor decreases asa result of declining catalyst activity, the temperature in the in-let section of the gas-cooled reactor will rise with a resultantimprovement in the reaction kinetics and hence, an increasedyield in the second reactor.

3 Mathematical Model of the Reactors

3.1 First Reactor

The reactor simulation includes steady state and dynamicmodels. The steady state simulation illustrates reactor perfor-mance with a fresh catalyst in the absence of catalyst deactiva-tion. Methanol reactors are modelled by a dynamic heteroge-neous model, which is a conventional model for a catalyticreactor with heat and mass transfer resistances. The balancestypically account for accumulation, convection, and transportto the solid phase. In this model, the following assumptionswere considered: one-dimensional plug flow in the shell andtube sides, the non-ideality of gas is neglected, the axial disper-sion is neglected here, and the heat loss by a coolant is consid-ered. The mass and energy balances for the solid phase are ex-pressed by:

i = 1,2,...,N–1 (1)

esct∂yis

∂t� kgictav�yi � yis� � g riqBa

qBcpsesdTs

dt� avhf �T � Ts� � qBa

�N

i�1

g ri��DHf �i� �2�

where yis and Ts are the solid-phase mole fraction and temper-ature, respectively.1) The following two conservation equationsare written for the fluid phase:

i = 1,2,...,N–1

eBct∂yi

∂t� � Ft

Ac

∂yi

∂z� avctkgi�yis � yi� �3�

eBctcpg∂T

∂t�� Ft

Accpg

∂T

∂z�avhf �Ts � T��pDi

AcUc�Tc �T� (4)


Table 2. Input data of the reactor [8].

Feed conditions Value Unit

Composition (mol %)

CH3OH 0.50 [–]

CO2 9.40 [–]

CO 4.60 [–]

H2O 0.04 [–]

H2 65.90 [–]

N2 9.30 [–]

CH4 10.26 [–]

Total molar flow rate per tube 0.64 [mol s–1]

Inlet temperature 503 [K]

Temperature of coolant 524 [K]

Pressure 76.98 [bar]

Figure 2. Schematic of an auto-thermal dual-type methanol reac-tor.

–1) List of symbols at the end of the paper.

Chem. Eng. Technol. 2008, 31, No. 4, 513–524 Dynamic optimization 515

where yi and T are the fluid-phase mole fraction and tempera-ture, respectively.

For heterogeneous models, several extra correlations for esti-mation of mass and heat transfer coefficients are used.

3.2 Second Reactor

The mass balance for the gas phase, and mass and heat balancefor the solid phase are the same as for the first reactor but withdifferent initial conditions. The energy balance for the gasphase in the second reactor is as follows:

CtCpg∂Ttube

∂t�� Ftube

AtubeCPtube

∂Ttube

∂z� pDi

AtubeUtube�T�Ttube� (5)

The boundary conditions are as follows:

z = 0; yi = yis, T = T0 (6)

while the initial conditions are:

t = 0; yi = yiSS, yis = yis

SS, T = TSS, TS = TSSS, a = 1 (7)

4 Reaction Kinetics

The three main reactions that occur in the methanol reactorare: the hydrogenation of CO, the hydrogenation of CO2, andthe reversed water-gas shift reaction:

CO + 2 H2 ↔ CH3OH DH298 = –90.55 kJmol–1 (8)

CO2 + 3 H2 ↔ CH3OH + H2O DH298 = –49.43 kJmol–1 (9)

CO2 + H2 ↔ CO + H2O DH298 = +41.12 kJmol–1 (10)

Reactions (8)–(10) are not independent, so that one is a lin-ear combination of the other ones. Kinetics of the low-pressuremethanol synthesis over commercial CuO/ZnO/Al2O3 catalystshas been widely investigated. In this current study, the rate ex-pressions have been adopted from Graaf et al. [9]. The rateequations combined with the equilibrium rate constants [10]provide enough information about the kinetics of methanolsynthesis. The correspondent rate expressions due to the hy-drogenation of CO, CO2, and the reversed water-gas shift reac-tions are:

r1 � k1KCO�fCOf 3�2H2

� fCH3OH��f 1�2H2 KP1��

�1 � KCOfCO � KCO2fCO2

��f 1�2H2

� �KH2O�K1�2H2

�fH2O�(11)

r2 � k2KCO2�fCO2f 3�2H2

� fCH3OHfH2O��f 3�2H2

Kp2��1 � KCOfCO � KCO2

fCO2��f 1�2

H2� �KH2O�K1�2

H2�fH2O�

(12)

r3 � k3KCO2�fCO2

fH2� fH2OfCO�KP3�

�1 � KCOfCO � KCO2fCO2

��f 1�2H2

� �KH2O�K1�2H2

�fH2O�(13)

The reaction rate constants, adsorption equilibrium con-stants, and reaction equilibrium constants which occur in theformulation of kinetic expressions are tabulated in Tabs. 3–5,respectively.

5 Catalyst Deactivation Model

The catalyst deactivation model for the commercial methanolsynthesis (CuO/ZnO/Al2O3) was presented by Hanken [11].

da

dt� �Kd exp

�Ed

R

1

T� 1

TR

� �� a5 �14�

where TR, Ed, and Kd are the reference temperature, activation en-ergy, and deactivation constant of the catalyst, respectively. Thenumerical value of TR is 513 K, of Ed is 91270 J mol–1 and of Kd is0.00439 h–1 [12]. Although other deactivation models were inves-tigated by other authors, the above model was fitted with indus-trial operating conditions; that is, the model is the only candidatefor the simulation and modeling of such industrial plants.

6 Numerical Solution

The governing equations of the model form a system ofcoupled equations comprising algebraic, partial differential,and ordinary differential equations. This system of equationsis solved using a two-stage approach consisting of a steady-state identification stage followed by a dynamic solution stage.


Table 3. Reaction rate constants [10].

k � AexpB

RT

� �A B

k1 (4.89 ± 0.29) · 107 –113000 ± 300

k2 (1.09 ± 0.07) · 105 –87500 ± 300

k3 (9.64 ± 7.30) · 1011 –152900 ± 11800

Table 4. Adsorption equilibrium constants [10].

K � A�expB

RT

� �A B

KCO (2.16 ± 0.44) · 10–5 46800 ± 800

KCO2 (7.05 ± 1.39) · 10–7 61700 ± 800

(KH2O/KH2

1/2) (6.37 ± 2.88) · 10–9 84000 ± 1400

Table 5. Reaction equilibrium constants [10].

KP � 10A

T�B

� �A B

KP1 5139 12.621

KP2 3066 10.592

KP3 –2073 –2.029


Knowledge of the steady-state condition of the methanol re-actor in the presence of catalyst deactivation, in principle, issimply a matter of determining the concentration and temper-ature profiles along the reactor axis at time zero. This is ac-complished by setting all time derivatives of the states equal tozero and considering the activity of the fresh catalyst. In thisway, the initial conditions for temperature and concentrationare determined for dynamic simulation. After rewriting themodel equations at steady-state conditions, a set of differentialalgebraic equations (DAEs) is obtained. To solve this system ofequations, backward finite difference approximation is ap-plied. Doing so, the DAEs change to a non-linear algebraic setof equations. The reactors are then divided into 30 separatesections and the Gauss-Newton method is used to solve thenon-linear algebraic equations in each section. The result ofthe steady-state simulation is used as initial conditions fortime-integration of dynamic state equations in each nodethrough the reactor. In counter-current mode, the temperatureof feed synthesis gas to the water-cooled reactor is unknown,while the temperature of feed to the second reactor is known.The shooting method applies the boundary value problem toan initial value problem. The solution is commenced by gues-sing a value for the temperature of the feed synthesis gas. Thewater- and gas-cooled reactors are divided into several nodes.The Gauss-Newton method is then used to solve the non-line-ar algebraic equations in each node. At the end, the calculatedvalues of the temperature of a fresh feed synthesis gas streamare compared with the actual value and this procedure is con-tinued until the specified final values are achieved.

The set of dynamic equations consists of simultaneous or-dinary and partial differential equations due to the deactiva-tion model and conservation rules respectively, as well as thealgebraic equations due to auxiliary correlations, kinetics andthermodynamics of the reaction system. After discretization ofpartial differential equations (PDEs) on the nodes of a one-di-mensional mesh in the axial direction, a system of ordinarydifferential equations (ODEs) is obtained for each node. Onecharacteristic feature of the system is its stiffness. The stiffnessoriginates from the fact that: (1) the dynamics of the deactiva-tion of a catalyst particle is much more rapid than those of thetemperature and concentration of both phases, (2) the spatialdiscretization causes stiffness because of local variations in therate of kinetic and transfer processes. Finally, (3) the rate ofchemical reactions and the deactivation process present in thesystem can be very different from each other. The modifiedRosenbrock formula of order two, which is an iterative multi-stage procedure, was used and the system of model equationswas conveniently converged. It is noted that this method is wellsuited for the system of stiff equations [13].

7 Optimization and Results

7.1 Model Validation

7.1.1 Steady-State Model Validation

Validation of the steady state model was conducted by com-parison of model results with plant data at time zero for a con-

ventional dual-type reactor under the design specificationsand input data tabulated in Tabs. 1 and 2, respectively. Themodel results and the corresponding data of the plant are pre-sented in Tab. 6. It was observed that the steady state modelperformed satisfactorily well under industrial conditions and agood agreement between plant data and simulation data ex-isted.

7.1.2 Dynamic Model Validation

In order to verify the agreement of the dynamic model withplant data, the mathematical model was validated with histori-cal process data [8] for a conventional reactor under the designspecifications and input data tabulated in Tab. 1 and 2, respec-tively. The predicted results of production rate and the corre-sponding observed data of the plant are presented in Tab. 7. Itwas observed that the model performed satisfactorily well un-der industrial conditions and a good agreement between daily-observed plant data and simulation data existed.


Table 6. Comparison of the predicted component mole fractionand temperature with plant data [8].

Component Inlet(plant)

Outlet(plant)

Outlet(Model)

Devi %

CO2 9.45 8.1 7.61 6.05

CO 4.95 2.38 2.56 –7.56

H2 56.17 48.97 46.12 5.82

CH3OH 0.31 5.02 4.45 11.35

Temperature (K) 503 528 524.4 0.68

Table 7. Comparison between predicted methanol productionrate and plant data [8].

Time (day) Plant (ton/day) Model (ton/day) Devi%

0 295.0 308.80 4.67

100 296.5 297.03 0.18

200 302.6 289.10 –4.46

300 284.3 283.09 –0.44

400 277.9 278.19 0.10

500 278.2 274.03 –1.50

600 253.0 270.41 6.88

700 274.0 267.19 –2.48

800 268.1 264.30 –1.65

900 275.5 261.67 –5.02

1000 274.6 259.25 –5.58

1100 262.9 257.02 –2.24

1200 255.2 255.18 –0.05


7.2 Optimization

A genetic algorithm is applied to determine the optimal reac-tor operating conditions for the methanol production process.Genetic algorithms are mathematical optimization methodsthat simulate a natural evolution process. The goal of this workis to maximize the methanol production during 1400 days ofoperations. When a single-type reactor is converted to two re-actors, the ratio of lengths is very important for better results.In the upper section of the methanol reactor, the reaction ki-netics is controlling while in the second section where the reac-tion has decreased to the equilibrium value, the equilibrium iscontrolling. Therefore, we assumed the reactor is broken intwo sections without any changes in its construction. Tempera-ture is also an important parameter and changes during thetime and reactor length and for exothermic reactions, an opti-mized temperature exists for further production. It has a directeffect on catalyst activity. Optimization of the reactor wasstudied in two steps. In the first investigation, the optimizationof the temperatures of feed (cooling gas) and cooling saturatedwater, and the length ratios of the reactors were done and inthe second step, the optimal profiles for saturated water andfeed gas temperatures were determined in three steps duringthe time of operations.

7.2.1 First Approach of Optimization

It is well known that for exothermic reactions, there is an opti-mal temperature profile along the methanol synthesis reactorwhich maximizes methanol production rate [14]. In this ap-proach, the optimal temperatures of feed (cooling gas), coolingsaturated water, and the optimal ratio of reactor lengths aredetermined. The temperature of cooling water was bound be-tween 520 K and 535 K, and the temperature of cooling gaswas bound between 480 K and 510 K. Another variable wasthe ratio of the lengths of two reactors. The objective functionwas to maximize the methanol production rate. The values ofthese parameters should determine the best boundaries be-tween the kinetic and thermodynamic regions. The constraintis the temperature of catalyst beds which should be less than543 K along the reactor because at temperatures higher than543 K, the catalyst will be deactivated [15].

This constraint was stated with the penalty function andequals “10 (Ts – 543)” in order to obtain a reasonable solution.Thus, the optimization problem is formulated as below:

Max f = Production + 10 (Tcat – 543)

Path constraint: Tcat < 543 (K) (15)

The results of the optimization are summarized in Tab. 8. Inthis table, Tshell, TF, and Lratio are the temperature of coolingwater, temperature of feed, and the ratio of reactor lengths, re-spectively. The simulation of a dual-type reactor was con-ducted by using the optimization results in Tab. 8. Fig. 3shows the objective function at optimized and non-optimizedconditions as a function of time.

Fig. 4 presents the methanol mole fraction and temperatureprofiles of conventional (CMR), auto-thermal dual-type

(ADMR), and optimized auto-thermal dual-type (OADMR)methanol reactors on the 1st (fresh catalyst) and 1400th day ofoperation. The input temperatures of ADMR are the same asCMR. As seen from Fig. 4a), on the first day, the methanolmole fraction in the first reactor at OADMR is higher thanADMR and CMR, but in the second stage reactor, the metha-nol mole fraction is slightly lower than the conventional reac-tor. Furthermore, the temperature of the catalytic bed(Fig. 4b)) in the dual-type reactor is higher than a normalpacked bed reactor except at 1 m at the end of reactor. On theother hand, from Figs. 4c)–4d) (the 1400th day), methanolmole fraction and temperature in an optimized dual-type anddual-type reactors are greater than in a single-type reactor.With deactivation of the catalyst during this time, the heatgeneration increases. On the other side, the feed gas has a low-er ability to cool than cooling water. Therefore, when the reac-tants enter the second reactor, the temperature increases. Infact, differences between methanol mole fractions in two kindsof reactors (CMR & OADMR) increases as time goes on (seeFig. 5).

Fig. 5 contains the methanol production during 1400 daysof operation for conventional, dual-type and optimized dual-type methanol synthesis reactors. The optimal methanol pro-duction in dual-type reactor is more than dual-type and con-ventional reactor, except at initial days of operation. Also, thedifferences of methanol production rates will increase duringthe time. The mean production of optimized auto-thermaldual-type methanol reactor is 4.7 % more than conventionalmethanol reactor.

In Fig. 6, the 3D profiles of methanol mole fraction, temper-ature of the reactor bed, temperatures of coolants, and activityof the catalyst along the optimized reactor and during the time


Table 8. The optimized parameters for first approach.

Tshell (K) TF (K) Lratio

529 505 1

70

75

80

85

90

95

100

105

110

0 200 400 600 800 1000 1200 1400

Time (day)

Ob

jecti

ve F

un

cti

on

Figure 3. Objective functions as a function of time. The solidline is OADMR and the dashed line is ADMR.


are shown. Fig. 6a) shows the profile of methanol molefraction as a function of reactor length and time. It can beunderstood from this figure that the methanol mole frac-tion increases during the reactor length in both reactors,but the rate of increasing in the first reactor is more thanin the second one. Similar to steady-state simulation, dy-namic simulation shows that methanol mole fraction in-creases along the reactor, although the rate of conversiondecreases. Decline of total conversion in the reactor arisesfrom deactivation of the catalyst as time goes on.

The optimal temperature surface of the catalyst bed isdemonstrated in Fig. 6b). The inlet temperature of the firstreactor increases over this time period. Maximum temper-ature is observed at the initial length of the second reactor;moreover, this maximum value increases over time and itmoves along the second reactor. Throughout this period,after maximum temperature, the adjusted temperature isobserved. Temperature of the second reactor increases asthe catalyst deactivates so that it peaks after inlet of thesecond reactor in the last operating times.


0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 5 6 7

Reactor length (m)

Meth

an

ol m

ole

fra

cti

on

CMR 1st dayOADMR 1st dayADMR 1st day

(a)

500

505

510

515

520

525

530

535

540

0 1 2 3 4 5 6 7

Reactor length (m)

T/K

CMR 1st day

OADMR 1st day

ADMR 1st day

(b)

0

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6 7

Reactor length (m)

Meth

an

ol m

ole

fra

cti

on

CMR 1400th day

OADMR 1400th day

ADMR 1400th day

(c)

500

505

510

515

520

525

530

535

540

0 1 2 3 4 5 6 7

Reactor length (m)

T/K

CMR 1400 day

OADMR 1400th day

ADMR 1400th day

(d)

Figure 4. Comparison of (a) methanol mole fraction, (b) temperature of reactive bed on the 1st day, and (c) methanol mole fraction, (d) tem-perature of reactive bed on the 1400th day of operation for the conventional methanol reactor and optimized dual-type methanol reactor.

Figure 5. Comparison of the production rate of methanol over time forconventional and optimized auto-thermal dual-type methanol reactors.


Fig. 6c) shows the surface temperature profiles of coolants.The horizontal surface shows the temperature of cooling satu-rated water and the other profile demonstrates the tempera-ture of cooling gas which enters the tube side of the second re-actor. In the first reactor – due to vaporization of saturatedliquid water to saturated water vapor – the temperature doesnot change but the temperature of cooling gas increases overtime and reactor length. The 3D profile of catalyst activity atoptimal conditions are shown in Fig. 6d). As seen from thisfigure, the activity decreases over time. Fresh catalyst deacti-vates so fast that the rate of deactivation decline over time. Ac-cording to the reaction, from the entrance of the first reactorto the outlet of the second reactor, chemical kinetics is trans-ferred towards chemical equilibrium conditions. Because ofdeactivation of the catalyst and depression of conversion, thehot spot moves along the reactor as time goes on.

7.2.2 Second Step of Optimization

The activity of the catalyst decreases sharply in several of theinitial days of operation and in the residual operating time theactivity decreases slowly. Because of deactivation of the catalystduring this time, the optimal temperature of the bed should beconsidered as a function of time. On the other hand, the tem-perature of the bed can be controlled by coolants. For this pur-pose, in this part, optimization trends are performed by dy-namic optimization with changing temperatures of coolants atthe 200th and 700th days of operations. This method illus-trates the activity behavior during the catalyst lifetime.

This optimization was done to maximize methanol produc-tion rate. There are 6 parameters which include three parame-ters for cooling feed temperatures and three parameters forcooling water temperatures. The reactor lengths are the same


Figure 6. 3D profiles of (a) methanol mole fraction, (b) temperature of reactor bed, (c) temperatures of coolants, and (d) catalyst activityalong reactor length and time at first optimization.


as for the first optimization and the formulation of the optimi-zation problem is the same as Eq. (15). In this part, the limita-tions are as below.

470 (K) < TFeed1 < 490 (K) (16a)

480 (K) < TFeed2 < 500 (K) (16b)

490 (K) < TFeed3 < 510 (K) (16c)

515 (K) < TShell1 < 525 (K) (17a)

520 (K) < TShell2 < 535 (K) (17b)

525 (K) < TShell3 < 540 (K) (17c)

The results of optimal temperature trajectory at 3 periods oftime are shown in Fig. 7.

Fig. 8a) and 8b) show the optimal temperature surface forcoolants and catalyst beds, respectively. In Fig. 8a), three sur-faces for coolants with sudden changes at the 200th and the700th day of experiment are shown. In this figure, increasingthe temperature of the cooling gas along the reactor over threetime intervals is observed. Also, in each profile of cooling gas,the slight increasing in temperature is observed during overtime. Fig. 8b) demonstrates the temperature of the beds whichsuddenly rises at the 200th and 700th days of operation. Thefirst period of time consists of 200 operating days. The temper-ature of processing gas at the entrance of the first reactorincreases by about 3 to 5 K, and over a period of 500 dayslater, the temperature increases with a slow rate. Over thelast 700 days, because of the increasing cooling water tempera-ture over this period, the temperature increase is not detect-able.

The optimal activity profile in the reactor is shown inFig. 8c). Although the reactor in a third period of operation(700 days) operates at 534 K, because the temperature in afirst period of the process is 519 K, the catalyst activity ishigher than for the previous approach throughout the timesintervals. Fig. 8d) shows the 3D plot for methanol mole frac-tion. As can be seen from this figure, changing temperaturesof coolants cause the sudden increase in methanol mole frac-tion.

The sensitivity of the optimization results for ratio of reac-tors lengths is shown in Fig. 9. As seen, during the time theproduction rate of methanol for Lratio =1 is more than theother ratios (0.6 and 1.5). Also we see when the ratio ofthe lengths is equal to 1.5 the production rate is higher thanthe ratio is 0.6.

For comparison of the optimized dual-type reactor with theconventional methanol reactor, methanol production rate overa period of 4 years is shown in Fig. 10a). An additional yieldof 5.8 % is obtained in the production rate of the optimizeddual-type methanol reactor compared to the conventionalmethanol reactor. The differences of methanol production inthe two types of reactors increase over time. Because the dual-type reactor operates at 519 K, in a first interval of 200 days,comparing 525 K in a conventional type of reactor, the catalystis deactivated more slowly in the residual days of operationand the production rate increases. In fact, the selection of 3temperatures during this time causes sudden increases in pro-duction rate at the 200th and 700th days of operation.

Fig. 10b) presents 1.2 % additional yield in production rateof the second optimization compared to the first optimization.Sometimes the production rate in the first optimized reactor ishigher than the other and vice versa, but in general, the pro-duction rate of the second optimized reactor is higher thanthat of the first.

8 Conclusion

The methanol production reactions are stronglyexothermic and the catalyst is deactivated over time.Therefore, the development of an auto-thermal two-stage methanol reactor could pave the way to increas-ing the methanol production in the methanol synthesisprocess. One potentially interesting idea for industrialmethanol synthesis is using an optimal auto-thermaldual-type reactor. In this investigation, an auto-ther-mal dual-type methanol synthesis reactor was mod-elled and optimized dynamically to maximize metha-nol production rate. The optimization method used isbased on genetic algorithms (GAs). Overall productionthroughout 4 years of catalyst life was considered asoptimization criterion to be maximized; also, 3 vari-ables which are length ratio of the reactors, feed andcooling water temperatures, are tuned. Optimizationincludes two procedures. In the first approach, the ra-tio of reactor lengths and temperature profile alongthe reactor were optimized. Useful results were ob-tained from optimization, that is, yields optimal values


Figure 7. Temperature of coolants as a function of time for the second stepof optimization.


for temperatures and reactor length ratio. In the second ap-proach, based on the results, optimization was followed by an-other task that optimal behaviors to which feed and coolingwater temperatures are concluded. The results lead us to opti-mal operation policy yields of 4.7 % and 5.8 % in additionalmethanol production during operating time for the first andsecond optimization approaches, respectively. A comparison ofcalculated temperature profile of the catalyst along the lengthsof the reactors shows the extremely favorable temperature pro-file over the optimal auto-thermal dual-type reactor system.The favorable temperature profile of the catalyst along the re-actor in the optimized reactor results in increased productionrate in this system.

Symbols used

Ac [m2] cross section area of each tubeAsell [m2] cross section area of shella [–] activity of catalystav [m2 m–3] specific surface area of catalyst

pelletcPg [J mol–1] specific heat of the gas at

constant pressurecPs [J mol–1] specific heat of the solid at

constant pressureDi [m] tube inside diameterEd [J mol–1] activation energy used in the

deactivation model


Figure 8. Plots of bed dynamics for (a) temperatures of coolants, (b) temperature of reactor bed, (c) catalyst activity, and (d) methanolmole fraction for second optimization.


Ft [mole s–1] total molar flow per tubefi [bar] partial fugacity of component ihf [W m–2 K–1] gas-solid heat transfer coefficientKd [s–1] deactivation model parameter

constantKi [bar–1] adsorption equilibrium constant

for component iKPi [–] equilibrium constant based on

partial pressure for component ik1 [mol kg–1 s–1 bar–1/2] reaction rate constant for the 1st

rate equationk2 [mol kg–1 s–1 bar–1/2] reaction rate constant for the

2nd rate equation

k3 [mol kg–1 s–1 bar–1/2] reaction rate constant for the3rd rate equation

kgi [m s–1] mass transfer coefficient forcomponent i

L [m] length of reactorN [–] number of componentsP [bar] total pressurePa [bar] atmospheric pressureR [J mol–1 K–1] universal gas constantri [mol kg–1 s–1] reaction rate of component ir1 [mol kg–1 s–1] rate of reaction for

hydrogenation of COr2 [mol kg–1 s–1] rate of reaction for

hydrogenation of CO2

r3 [mol kg–1 s–1] reversed water-gas shift reactionT [K] bulk gas phase temperatureTR [K] reference temperature used in

the deactivation modelTs [K] temperature of solid phaseTsat [K] saturated temperature of boiling

water at operating pressureTshell [K] temperature of coolant stream,

in first reactorTtube [K] temperature of coolant stream,

in second reactort [s] timeUshell [W m–2 K–1] overall heat transfer coefficient

between coolant and processstreams

U [m s–1] superficial velocity of fluid phaseyi [mol mol–1] mole fraction of component i in

the fluid phaseyis [mol mol–1] mole fraction of component i in

the solid phasez [m] axial reactor coordinate


Figure 9. Sensitivities of methanol production rate to ratio of re-actor length.

(a) (b)

Figure 10. (a) Comparison of methanol production rate between CMR and ADMR, (b) Comparison of methanol production rate betweentwo kinds of optimization procedure.


Greek letters

DHf,i [kJ mol–1] enthalpy of formation ofcomponent i

DH298 [kJ mol–1] enthalpy of reaction at 298 KeB [–] void fraction of catalytic bedes [–] void fraction of catalystm [–] stoichiometric coefficientqB [kg m–3] density of catalytic bedg [–] catalyst effectiveness factor

Superscripts and subscripts

f feed conditionsin inlet conditionsout outlet conditionsk reaction number index (1, 2, or 3)s at catalyst surfacess initial conditions (i.e., steady-state condition)tube tube side

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dynamic simulation and optimization of a dual-type methanol reactor using genetic algorithms

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