dynamic optimization of a novel radial-flow, spherical-bed methanol synthesis reactor in the...

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 6 2 2 1 – 6 2 3 0

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Dynamic optimization of a novel radial-flow, spherical-bedmethanol synthesis reactor in the presence of catalystdeactivation using Differential Evolution (DE) algorithm

M.R. Rahimpour*, P. Parvasi, P. Setoodeh

Chemical Engineering Department, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz 71345, Iran

a r t i c l e i n f o

Article history:

Received 16 February 2009

Received in revised form

12 May 2009

Accepted 14 May 2009

Available online 27 June 2009

Keywords:

Dynamic optimization

Differential Evolution algorithm

Methanol synthesis

Spherical-bed reactor

Catalyst deactivation

* Corresponding author. Tel.: þ98 711 230307E-mail address: [email protected] (

0360-3199/$ – see front matter ª 2009 Interndoi:10.1016/j.ijhydene.2009.05.068

a b s t r a c t

In this work, a novel radial-flow spherical-bed methanol synthesis reactor has been

optimized using Differential Evolution (DE) algorithm. This reactor’s configuration visual-

izes the concentration and temperature distribution inside a radial-flow packed bed with

a novel design for improving reactor performance with lower pressure drop. The dynamic

simulation of spherical multi-stage reactors has been studied in the presence of long-term

catalyst deactivation. A theoretical investigation has been performed in order to evaluate

the optimal operating conditions and enhancement of methanol production in radial-flow

spherical-bed methanol synthesis reactor. The simulation results have been shown that

there are optimum values of the reactor inlet temperatures, profiles of temperatures along

the reactors and reactor radius ratio to maximize the overall methanol production. The

optimization methods have enhanced additional yield throughout 4 years of catalyst

lifetime, respectively.

ª 2009 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights

reserved.

1. Introduction remove the heat by heat exchangers between the reactors.

Methanol is an important multipurpose based chemical,

a simple molecule which can be recovered from many

resources, predominantly natural gas [1]. Tubular packed bed

reactors (TPBRs) are used extensively in industrial methanol

synthesis [2]. Some potential drawbacks of this type of

reactors are the pressure drop across the reactor, high

manufacturing costs resulting from a large wall thickness and

low production capacity [3]. One potentially interesting idea

for industrial methanol synthesis is the use of a spherical

packed bed reactor (SPBR) [4], which is the subject of this work.

The advantages of this reactor are a small pressure drop, low

manufacturing costs as a result of a small wall thickness, and,

if desired, a high production capacity. It is possible to connect

different spherical reactors in series in a production plant and

1; fax: þ98 711 6287294.M.R. Rahimpour).ational Association for H

The flow in SPBRs is radial so that it offers a larger mean cross-

sectional area and reduced distance of travel for flow

compared to traditional vertical columns. Consequently, the

pressure drop in these radial geometry reactors is reduced

radically. Another characteristic of the spherical radial-flow

reactor (RFR) is that since heat transfer will be dominated by

the spherical geometry, a small hot zone should develop

where reaction occurs. This small reaction zone can provide

several advantages, especially in reversible exothermic

reactions.

The factors affecting the production rate in an industrial

methanol reactor are parameters such as thermodynamic

equilibrium limitations and catalyst deactivation. In the case

of reversible exothermic reactions, such as methanol

synthesis, selection of a relatively low temperature permits

ydrogen Energy. Published by Elsevier Ltd. All rights reserved.

mailto:[email protected]

http://www.elsevier.com/locate/he

Nomenclature

a activity of catalyst

cpg specific heat of the gas at constant pressure,

J kgmol�1 K�1

Di tube outside diameter, m

Derj diffusion coefficient of component j in the

mixture, m2 s�1

Ed activation energy used in the deactivation model

J kgmol�1

fj partial fugacity of component j, bar

Ft total molar flow rate per tube, mol s�1

DH298i enthalpy of reaction i at 298 K

k1 reaction rate constant for the 1st rate equation,

mol kg�1 s�1 bar�1/2

k2 reaction rate constant for the 2nd rate equation,


k3 reaction rate constant for the 3rd rate equation,


kjg mass transfer coefficient for component j, m s�1

keff conductivity of fluid-phase, Wm�1 K�1

Kd deactivation model parameter constant s�1

Kj adsorption equilibrium constant for component j,

bar�1

Kpj equilibrium constant based on partial pressure for

component j

M number of reactions

N number of components

P total pressure, bar

r radial coordinate m

r1 rate of reaction for hydrogenation of CO,

kgmol m�3 s�1

r2 rate of reaction for hydrogenation of CO2,

kgmol m�3 s�1

r3 reaction rate constant for the 3rd rate equation,

kgmol m�3 s�1

ri reaction rate of component j, kgmol m�3 s�1

R universal gas constant, J kgmol�1 K�1

Ri inner diameter of reactor, m

Ro outer diameter of reactor, m

t time, s

T bulk gas phase temperature, K

TR reference temperature used in the deactivation

model, K

Tshell temperature of coolant stream, K

Ushell overall heat transfer coefficient between coolant

and process streams, Wm�1 s�1

ur radial velocity of fluid-phase, m s�1

V total volume of reactor, m3

yj mole fraction of component j in the fluid-phase,

kgmol m�3

Greek letters

3 void fraction of catalytic bed

3s void fraction of catalyst

n stoichiometric coefficient

r density of catalytic bed, kg m�3

Superscripts and subscripts

0 inlet conditions

i reaction number index (1, 2 or 3)

j number of components

s at catalyst surface

ss initial conditions (i.e., steady-state condition)


higher conversion, but this must be balanced against a slower

rate of reaction, which leads to the requirement of a large

amount of catalyst. Up to the maximum production rate point,

increasing temperature improves the rate of reaction, which

leads to more methanol production. Nevertheless as the

temperature increases beyond this point, the deteriorating

effect of equilibrium conversion emerges and decreases

methanol production. Therefore one of the important key

issues in methanol reactor configuration is implementing

a higher temperature at the entrance of the reactor for

a higher reaction rate, and then reducing temperature grad-

ually towards the exit for increasing thermodynamic equilib-

rium conversion.

Like in the world of modeling, the field of dynamic opti-

mization has its own jargon to address specific characteristics

of the problem. Most optimization problems in process

industry can be characterized as non-convex, non-linear, and

constrained optimization problems [5]. For plant optimization

typical optimization parameters are equipment size, recycle

flows and operating conditions like temperature, pressure and

concentration. An optimum design is based on the best or

most favorable conditions. In almost every case, these

optimum conditions can ultimately be reduced to a consider-

ation of costs or profits. Thus an optimum economic design

could be based on conditions giving the least cost per unit of

time or the maximum profit per unit of production. When one

design variable is changed, it is often found that some costs

increase and others decrease. Under these conditions, the

total cost may go through a minimum at one value of the

particular design variable, and this value would be considered

as an optimum. A number of search algorithm methods for

dealing with optimization problems have been proposed in

the last few years in the fields of evolutionary programming

(EP) [6], evolution strategies (ES) [7], genetic algorithms (GAs)

[8] and particle swarm optimization (PSO) [9]. DE algorithm is

a stochastic optimization method minimizing an objective

function that can model the problem’s objectives while

incorporating constraints. The algorithm mainly has three

advantages; finding the true global minimum regardless of the

initial parameter values, fast convergence, and using a few

control parameters. Being simple, fast, easy to use, very easily

adaptable for integrand discrete optimization, quite effective

in non-linear constraint optimization including penalty

functions and useful for optimizing multi-modal search

spaces are the other important features of DE algorithm [10].

In this study, the novel radial-flow spherical-bed methanol

synthesis reactor configuration has been optimized. Optimi-

zation tasks have been investigated by novel optimization

tools, Differential Evolution (DE) algorithm. Optimization of

reactor was studied in four approaches. In the first approach,

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 6 2 2 1 – 6 2 3 0 6223

an optimization program has been developed to obtain more

methanol production through optimum inlet temperature of

reactors. In the second approach, the optimal temperature

profiles along the reactors during the period of operation have

been considered to reach maximum methanol production

rate. In the third approach, the optimum radius ratios of

reactors were obtained for maximum production rate. Finally,

the deactivation parameters were optimized using plant data.

2. Differential Evolution (DE) algorithm

The DE algorithm is a population based algorithm similar to

genetic algorithms using similar operators: crossover, muta-

tion, and selection. The main difference in constructing better

solutions is that genetic algorithms depend on crossover

while DE relies on mutation operation. This main operation is

founded on the differences of randomly sampled pairs of

solutions in the population.

The algorithm uses mutation operation as a seek mecha-

nism and selection operation to direct the search toward the

probable regions in the search space. The DE algorithm also

uses a non-uniform crossover that can take child vector

parameters from one parent more often than it does from

others. Using the components of the existing population

members to build trial vectors, the recombination (crossover)

operator efficiently shuffles information about successful

combinations, enabling the search for a better solution space.

An optimization task consisting of D parameters can

be represented by a D-dimensional vector. In DE, initially

a population of NP solution vectors is randomly created. This

population is successfully improved by applying mutation,

crossover, and selection operators. The main steps of the DE

algorithm are given below [10]:

Initialization

Evaluation

Repeat

Mutation

Recombination

Evaluation

Selection

Until (termination criteria are met)

Fig. 1 – Obtaining a new proposal in DE [6].

2.1. Mutation

For each target vectorxi;G, a mutant vector is produced by

vi;Gþ1 ¼ xi;G þ K��xr1 ;G � xi;G

�þ F�

�xr2 ;G � xr3 ;G

�(1)

where i; r1; r2; rr3˛f1; 2;.;NPg are randomly chosen and must

be different from each other. In Eq. (1), F is the scaling factor

which has an effect on the difference vector ðxr2 ;G � xr3 ;GÞ, K is

the combination factor [10].

2.2. Crossover

The parent vector is mixed with the mutated vector to

produce a trial vector uji;Gþ1

uji;Gþ1 ¼�

uji;Gþ1 if�rndj � CR

�or j ¼ rni;

qji;G if�rndj > CR

�and jsrni;

(2)

where j ¼ 1;2;.;D; rj ˛½0;1� is the random number, CR is

crossover constant ˛½0; 1�, and rni ˛ð1;2;.;DÞ is the randomly

chosen index [10].

2.3. Selection

All solutions in the population have the same chance of being

selected as parents independent of their fitness value. The

child produced after the mutation and crossover operations is

evaluated. Then, the performance of the child vector and its

parent is compared and the better one is selected. If the parent

is still better, it is retained in the population.

Fig. 1 shows DE’s process in detail: the difference between

two population members (1, 2) is added to a third population

member (3). The result (4) is subject to crossover with the

candidate for replacement (5) in order to obtain a proposal (6).

The proposal is evaluated and replaces the candidate if it is

found to be better.

3. Model development

Methanol synthesis is generally performed by passing

a synthesis gas comprising hydrogen, carbon oxides, and any

inert gasses at an elevated temperature and pressure through

one or more beds of catalyst, which is often a copper–zinc

oxide catalyst. The following three overall equilibrium reac-

tions are relevant in the methanol synthesis [11]:

Feed

Product

Fig. 2 – Schematic diagram of a methanol synthesis single-

stage spherical reactor.

Feed

Product

Fig. 3 – Schematic diagram of a methanol synthesis three-stage spherical reactor.


COþ 2H24CH3OH (3)

CO2 þ 3H24CH3OHþH2O (4)

COþH2O4CO2 þH2 (5)

A schematic sketch of the spherical reactor is presented

in Fig. 2. The catalyst is situated in the dome and between

two perforated spherical shells. The synthesis gas enters

the reactor between the catalyst bed and the pressure-

resistant reactor wall. It flows steadily from outside

through the catalyst bed into the inner sphere. The gas is

removed from the inner sphere through a tube to the

outlet [12].

Due to small pressure drop and low manufacturing costs

multi-stage, spherical-bed reactors could be utilized instead of

single ones in order to achieve production improvement. The

three-stage configuration is illustrated in Fig. 3.

3.1. Reactor model

The mathematical model corresponding to the spherical

packed bed flow reactor is derived starting from the dynamic

model developed by Rahimpour et al. for tubular flow reactors

Table 1 – Catalyst and reactor specifications [15].

Parameter Value

rs (kg m�3) 1770

dp (mm) 5.47

Cps (kJ kg�1 K�1) 5.0

3 0.5

3s 0.4

aap (m2 m�3) 626.98

Cpg (kJ kg�1 K�1) 2.98

V (m3) 80

Sphericalreactor

One-stage(m)

Two-stage(m)

Three-stage(m)

Inner radius 1.0 1.0 1.0

Outer radius 2.72 2.19 1.94

[13], making the corresponding changes for a spherical

geometry. In this study, homogeneous one-dimensional

models have been considered. The basic structure of this

model is composed of heat and mass balance conservation

equations coupled through thermodynamic and kinetic rela-

tions, as well as, auxiliary correlations for predicting physical

properties.

In this simple model we assume that gradients of

temperature and concentrations between catalyst and gas

phases can be ignored and the equations for the two phases

can be combined [14]. The general fluid-phase balance is

a model with the balances typically account for accumulation,

convection, and reaction. In the current work, axial dispersion

of heat is neglected and the heat loss by a coolant is consid-

ered as we study a realistic reactor. The energy and mass

balances can be written as [15]:

3vyj

vt¼ �1

r2

v

vr

�r2uryj

�þ 3sð1� 3Þa

Xn

i¼1

yijr

j ¼ 1;2;.;N and i ¼ 1;2;.;M (6)

ð1� 3ÞrCpcvTvt¼ �1

r2

v

vr

�rurr

2Cp

�T� Tref

��þ 3sð1� 3Þa

Xn

i¼1

DHir (7)

where T and yj are, respectively, the temperature and

concentration of component j in the fluid-phase and a is the

activity of catalyst.

Table 2 – Input data for first reactor [15].

Feed conditions Value

Composition (mol%)

CH3OH 0.5

CO2 9.4

CO 4.6

H2O 0.04

H2 65.9

N2 9.33

CH4 10.26

Total molar flow rate (kmol s�1) 2

Inlet temperature (K) 503

Pressure (bar) 76.98

Table 3 – Specification data for reactor feed [12].

Feed (kmol s�1) P (bar) T (K) H2 (mol%) CO (mol%) CH3OH (mol%) CO2 (mol%) H2O (mol%) N2 (mol%) CH4 (mol%)

17.7892 81.95 313 77.53 3.9695 0.4114 2.5197 0.0806 3.4285 12.05

0.157

0.1571

0.1571

0.1571

0.1572

0.1573

0.1573

0.1573

Ob

jective F

un

ctio

n

Steady State Condition


The boundary conditions are as follows:

r ¼ Ri;vyj

vr¼ 0;

vTvr¼ 0

r ¼ Ro; yj ¼ yj0 ; T ¼ T0

(8)

The initial conditions are:

t ¼ 0; yi ¼ yssi ; T ¼ Tss; a ¼ 1 (9)

3.2. Reaction kinetics

Reactions (3)–(5) are not independent and therefore one is

a linear combination of the other ones. Kinetics of the low-

pressure methanol synthesis over commercial CuO/ZnO/

Al2O3 catalysts has been widely investigated. In the current

work, the rate expressions have been selected from Graaf et al.

The corresponding rate expressions due to the hydrogenation

of CO, CO2, and the reversed water–gas shift reactions are

given in Appendix A [16].

The reaction rate constants, adsorption equilibrium

constants, and reaction equilibrium constants, which occur

in the formulation of kinetic expressions, are tabulated in

Appendix A, respectively.

Table 4 – Comparison of simulation results with Hartiget al. [12] data.

Outlet composition(mol%)

Homogenousmodel

Hartiget al. data

Relativeerror (%)

First reactor

H2 78.006 76.550 1.902

CO 3.113 3.321 6.281

CH3OH 1.740 1.613 7.826

CO2 2.009 2.131 5.705

H2O 0.558 0.532 4.964

N2 3.693 3.510 5.195

CH4 12.978 12.340 5.172

Second reactor

H2 77.088 75.640 1.915

CO 2.396 2.612 8.249

CH3OH 3.043 2.806 8.479

CO2 1.694 1.811 6.471

H2O 0.960 0.913 5.152

N2 3.800 3.591 5.809

CH4 13.355 12.620 5.828

Third reactor

H2 76.580 75.010 2.093

CO 1.801 2.017 10.709

CH3OH 4.212 3.705 13.700

CO2 1.476 1.630 9.442

H2O 1.233 1.141 8.146

N2 3.915 3.652 7.182

CH4 14.009 12.840 9.104

3.3. Deactivation model

Catalyst deactivation model for the commercial methanol

synthesis catalyst was adopted from Hanken [17].

dadt¼ �Kdexp

��Ed

R

�1T� 1

TR

��a5 (16)

where TR, Ed and Kd are the reference temperature, activation

energy, and deactivation constant of the catalyst, respectively.

The chosen numerical values for these parameters are:

TR¼ 513 K, Ed¼ 91270 J mol�1, and Kd¼ 0.00439 h�1.

0 10 20 30 40 50 60 70 80 90 1000.1569

0.1569

Iteration

Fig. 4 – Objective function values for steady-state

optimization.

0 200 400 600 800 1000 1200470

475

480

485

490

495

500

505

510

515

520

Time (day)

Tem

peratu

re (K

)

Optimum Reactor inlet Temperature

First ReactorSecond ReactorThird Reactor

Fig. 5 – Dynamic optimal inlet temperatures for (a) first

stage (b) second stage and (c) third stage.

Table 5 – Optimal values of reactors’ inlet temperatures(K).

Time (day) First reactor Second reactor Third reactor

0 471.71 503.74 502.07

100 476.53 505.84 503.70

200 484.64 509.62 507.92

300 489.44 511.51 509.88

400 493.03 512.97 511.04

500 495.62 514.11 512.14

600 497.35 514.82 513.08

700 499.16 515.45 513.64

800 500.60 515.98 514.17

900 501.87 516.50 514.66

1000 502.97 517.03 515.19

1100 503.89 517.40 515.55

1200 504.82 517.79 515.89


4. Optimization and results

The technical design data of the catalyst pellet and input data

of the reactor have been summarized in Tables 1 and 2,

respectively.

0 200 400 600 800 1000 12000.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042

0.044

Time (day)

Meth

an

ol m

ole fractio

n

Optimum mole fraction from First reactor outlet

With OptimizationWithout Optimization

0 200 4000.05

0.055

0.06

0.065

0.07

0.075

0.08

Tim

Me

th

an

ol m

ole fractio

n

Optimum mole fraction

a

c

Fig. 6 – Optimal methanol mole fraction for (a) fir

The partial differential equations (PDEs) of the one-

dimensional model of spherical reactor were solved by means

of the orthogonal collocation numerical method [18,19].

Numerical solution was performed by Rahimpour et al. [20].

For more accuracy of homogeneous model, the results of

steady-state predictions for spherical reactors are compared

with Hartig et al.’s data [12] and a good agreement was ach-

ieved. Tables 3 and 4 show these results.

Differential Evolution algorithm is applied to determine the

optimal reactor operating conditions for methanol production

process. The goal of this work is to maximize the methanol

production during 1200 days of operation.

In this study, the optimization of reactor was investigated

using four approaches: optimal reactor inlet temperature

approach, optimal temperature profile approach, optimal

catalyst deactivation parameter approach, and optimal radius

ratio of reactors.

4.1. Optimization of the reactor inlet temperatures

For this optimization study, inlet temperature of reactor is

variable because the design conversion of reversible

0 200 400 600 800 1000 12000.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06

0.062

Time (day)

Meth

an

ol m

ole fractio

n

Optimum mole fraction from Second reactor outlet


600 800 1000 1200e (day)

from Third reactor outlet


b

st stage (b) second stage and (c) third stage.

Table 6 – Additional yield achieved for output methanolmole fractions with optimal temperature profiles.

Reference Optimum Additional yield (%)

Activity¼ 0.9 0.0673 0.0698 3.71

Activity¼ 0.7 0.0668 0.0694 3.89

Activity¼ 0.5 0.0663 0.0688 3.83


exothermic reactions often closely approach the upper

boundary imposed by thermodynamics [21].

In the case of reversible exothermic reactions, selection of

a relatively low temperature permits higher conversion, but this

must be balanced by the slower rate of reaction resulting in the

need for a large amount of catalyst. Hence, the reactor inlet

temperatures in three-stage reactors can be adjusted (at optimal

temperatures) to maximize methanol mole fraction at reactor

outlet. Therefore, according to deactivation rate, dynamic

optimal temperatures are obtained through catalyst lifetime.

In this study, the objective function used to determine

optimized inlet temperature of reactors is defined as the sum

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

510

520

530

540

550

560

570

580

590

600

Optimum Temperatue Profile - First Reactor

(Activity = 0.9)

Radius (m)

Tem

peratu

e (K

)

a

1 1.1 1.2 1.3 1.4505

510

515

520

525

530

535

540

Optimum Temperatu

(Activ

Rad

Tem

peratu

e (K

)

c

Fig. 7 – Optimal temperature profiles at a [ 0.9 for (a

of mole fraction of methanol component at the reactor outlet.

Fig. 4 shows the objective function values for steady-state

optimization.

Optimal inlet temperature profiles through catalyst life-

time for three-stage reactors are shown in Fig. 5.

Values of optimal inlet temperatures for some days are

reported in Table 5.

Optimal reactor inlet temperatures should be increased

during the catalyst lifetime to compensate for the reduction of

production rate due to catalyst deactivation. For instance, this

optimization approach enhanced a 30% additional yield for final

product at first reactor. Fig. 6 shows the difference between

methanol mole fractions at reactor outlet for this activity.

4.2. Optimal temperature profile approach

From a theoretical point of view, there is an optimal

temperature profile along the methanol synthesis reactor,

which maximizes methanol production rate as reported in

the literature for exothermic reactors. The optimal profile

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9505

510

515

520

525

530

535

540

545

550

Optimum Temperatue Profile - Second Reactor

(Activity = 0.9)

Radius (m)

Tem

peratu

e (K

)

b

1.5 1.6 1.7 1.8 1.9

e Profile - Third Reactor

ity = 0.9)

ius (m)

) first stage (b) second stage and (c) third stage.

0 10 20 30 40 50 6038

40

42

44

46

48

50

52

54

Iteration

Ob

jective F

un

ctio

n

Average of Square Absolute Error in

Methanol Production

Fig. 8 – Objective function values for optimization of

deactivation parameters.

0 200 400 600 800 1000 1200250

260

270

280

290

300

310Methanol Production vs. Time

Time (day)

Meth

an

ol P

ro

du

ctio

n (to

n)

Plant DataHanken Deactivation ModelOptimized Deactivation Model

Fig. 9 – Comparison of Hanken and optimized deactivation

models with observed methanol production rate (ton/day).


changes during operation because of catalyst deactivation

and therefore, it is not unique over different time intervals

[22]. In this study pseudo-dynamic optimization was used

instead dynamic optimization. According to deactivation

rate, three activity levels (a¼ 0.9, 0.7 and 0.5) were chosen to

study optimal inlet temperatures. These values reflect

dynamic properties of reactor operation and give some

information about variation of optimal temperatures

through catalyst lifetime [23]. In this approach, the objective

function is similar to the objective function of the previous

section (Table 6).

Fig. 7 shows optimal temperature profile along the meth-

anol synthesis reactors for activity level of 0.9.

Table 8 – Values of methanol production predicted byoptimal deactivation model compared with Hanken’smodel.

Time(day)

Plant data(ton/day)

Hanken’smodel

(ton/day)

Percenterror withplant data

Newmodel

(ton/day)

Percenterror withplant data

100 296.50 293.69 0.95 290.26 2.10

4.3. Optimization of catalyst deactivationrate parameters

The deactivation of low-pressure methanol synthesis catalyst

has been investigated by Kuechen and Hoffmann [24]. In their

experiments, the proportion of hydrogen was kept constant

whilst the CO:CO2 ratio was varied in order to make a clear

distinction between completely different behavior of the

catalyst due to variations in the CO:CO2 ratio. Therefore,

deactivation kinetics is a function of temperature and the

fugacity ratio of carbon monoxide to carbon dioxide as illus-

trated in the following equation:

Table 7 – Optimal values of deactivation model.

Ed 111542.07 J mol�1

Kd 0.4301156 h�1

n 4.000021

m 0.000960

dadt¼ �

�COCO2

�m

Kd exp

�Ed

R

�1T� 1

TR

�an (17)

The optimization of parameters (Ed, Kd, n and m) was

performed by DE algorithm and results of new model for

one-stage spherical reactor for feed flow rate 2.22 kgmol s�1

were compared with Hanken’s model and plant data of

reactor. In this study, the objective function, which is used

to determine optimized deactivation parameters, is defined

as the average of square absolute error in methanol

production between plant data and model results through

catalyst lifetime. Fig. 8 shows the objective function values

for this optimization.

Values of optimal deactivation parameters are reported in

Table 7.

Fig. 9 illustrates the results of methanol production, in

which new catalyst deactivation model is compared with

Hanken’s model.

200 302.60 291.47 3.68 286.29 5.39

300 284.33 289.95 1.98 283.43 0.32

400 277.90 288.78 3.92 281.12 1.16

500 278.20 287.80 3.45 279.09 0.32

600 278.00 286.95 3.22 277.31 0.25

700 274.00 286.19 4.45 275.73 0.63

800 268.10 285.51 6.50 274.22 2.28

900 275.50 284.89 3.41 272.73 1.00

1000 274.58 284.31 3.54 271.33 1.18

0 200 400 600 800 1000 1200220

230

240

250

260

270

280

290

300

310Methanol Production vs. Time

Time (day)

Meth

an

ol P

ro

du

ctio

n (to

n)

Raduis Ratio = 2Raduis Ratio = 3Raduis Ratio = 4Raduis Ratio = 5

Fig. 10 – The effects of reactor radius ratio on methanol

production rate (ton/day).


The methanol production of the one-stage reactor

achieved by optimized deactivation model and Hanken’s

model is compared with plant data [25] in Table 8.

Table A.1 – Reaction rate constants.

k ¼ A expð BRTÞ A B

k1 (4.89� 0.29)� 107 �113,000� 300

k2 (1.09� 0.07)� 105 �87,500� 300

k3 (9.64� 7.30)� 1011 �152,900� 11800

Table A.2 – Adsorption equilibrium constants.

k ¼ A expð BRTÞ A B

K (2.16� 0.44)� 10�5 46,800� 800

4.4. Optimization of reactor radius ratio

In this section, inner and outer radius ratio of one-stage

reactor is variable. We try to find the optimum radius ratio,

which maximizes the final product. Results show that

increasing the radius ratio increases methanol production and

compensates catalyst deactivation effect (Fig. 10). However, it

is not clear if there is a unique optimum value for radius ratio.

In Fig. 10, the radius ratio increases from 2 to 5 to restore

methanol production. The objective function was similar to

the objective function of the first optimization subsection, and

the chosen value for the inner radius was 1 m.
CO
KCO2 (7.05� 1.39)� 10�7 61,700� 800

ðKH2O=K0:5H2Þ (6.37� 2.88)� 10�9 84,000� 1400

Table A.3 – Reaction equilibrium constants.

Kp ¼ 10ðAt�BÞ A B

Kp15139 12.621

Kp23066 10.592

Kp3�2073 �2.029

5. Conclusion

In this study, a radial-flow spherical-bed methanol synthesis

reactor has been optimized to maximize methanol production

yield. The optimization problem includes four approaches:

optimal reactor inlet temperature approach, optimal temper-

ature profile approach, optimal catalyst deactivation param-

eter approach, and optimal radius ratio of reactors. Results of

optimization procedures yield high additional methanol

production during the operating period. A homogeneous

model was used in the optimization investigation. Dynamic

optimization is the method of choice for solving the

constrained non-linear problem under study. Dynamic opti-

mization was implemented using Differential Evolution (DE)

algorithm, which is a powerful optimization technique with

a reasonably low computational complexity. Also, the

proposed method provides guidelines to perform similar

designs for methanol synthesis.

Appendix AAuxiliary correlations

To complete the simulation, auxiliary correlations should be

added to the model.

A.1. Reaction kinetics

R1 ¼k1KCO

hfCOf 1:5

H2� fCH3OH=

�f 0:5H2

KP1

�i� �h � � i (A.1)

1þ KCOfCO þ KCO2fCO2

f 0:5H2þ KH2O=K0:5

H2fH2O

R2 ¼k2KCO2

hfCO2

f 1:5H2� fCH3OHfH2O=

�f 1:5H2

KP2

�i�


�hf 0:5H2þ�

KH2O=K0:5H2

�fH2O

i (A.2)

R3 ¼k3KCO2

hfCO2

fH2� fH2OfCO=KP3

i�


�hf 0:5H2þ�

KH2O=K0:5H2

�fH2O

i (A.3)

A.2. Reaction constants

r e f e r e n c e s

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