Dynamic optimization of a novel radial-flow, spherical-bed methanol synthesis reactor in the presence of catalyst deactivation using Differential Evolution (DE) algorithm

international journal of hydrogen energy 34 (2009) 6221–6230

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Dynamic optimization of a novel radial-flow, spherical-bed methanol synthesis reactor in the presence of catalyst deactivation 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

article info

abstract

Article history:

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

Received 16 February 2009

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

Received in revised form

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

12 May 2009

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

Accepted 14 May 2009

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

Available online 27 June 2009

catalyst deactivation. A theoretical investigation has been performed in order to evaluate the optimal operating conditions and enhancement of methanol production in radial-flow

Keywords:

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

Dynamic optimization

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

Differential Evolution algorithm

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

Methanol synthesis

optimization methods have enhanced additional yield throughout 4 years of catalyst

Spherical-bed reactor

lifetime, respectively.

Catalyst deactivation

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

1.

Introduction

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

remove the heat by heat exchangers between the reactors. The flow in SPBRs is radial so that it offers a larger mean crosssectional 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

* Corresponding author. Tel.: þ98 711 2303071; fax: þ98 711 6287294. E-mail address: [email protected] (M.R. Rahimpour). 0360-3199/$ – see front matter ª 2009 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2009.05.068

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Nomenclature

r2

a cpg

r3

Di Derj Ed fj Ft DH298i k1 k2 k3 kjg keff Kd Kj Kpj M N P r r1

activity of catalyst specific heat of the gas at constant pressure, J kgmol1 K1 tube outside diameter, m diffusion coefficient of component j in the mixture, m2 s1 activation energy used in the deactivation model J kgmol1 partial fugacity of component j, bar total molar flow rate per tube, mol s1 enthalpy of reaction i at 298 K reaction rate constant for the 1st rate equation, mol kg1 s1 bar1/2 reaction rate constant for the 2nd rate equation, mol kg1 s1 bar1/2 reaction rate constant for the 3rd rate equation, mol kg1 s1 bar1/2 mass transfer coefficient for component j, m s1 conductivity of fluid-phase, Wm1 K1 deactivation model parameter constant s1 adsorption equilibrium constant for component j, bar1 equilibrium constant based on partial pressure for component j number of reactions number of components total pressure, bar radial coordinate m rate of reaction for hydrogenation of CO, kgmol m3 s1

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 gradually towards the exit for increasing thermodynamic equilibrium conversion. Like in the world of modeling, the field of dynamic optimization 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 consideration of costs or profits. Thus an optimum economic design could be based on conditions giving the least cost per unit of

ri R Ri Ro t T TR Tshell Ushell ur V yj

rate of reaction for hydrogenation of CO2, kgmol m3 s1 reaction rate constant for the 3rd rate equation, kgmol m3 s1 reaction rate of component j, kgmol m3 s1 universal gas constant, J kgmol1 K1 inner diameter of reactor, m outer diameter of reactor, m time, s bulk gas phase temperature, K reference temperature used in the deactivation model, K temperature of coolant stream, K overall heat transfer coefficient between coolant and process streams, Wm1 s1 radial velocity of fluid-phase, m s1 total volume of reactor, m3 mole fraction of component j in the fluid-phase, kgmol m3

Greek letters 3 void fraction of catalytic bed void fraction of catalyst 3s n stoichiometric coefficient r density of catalytic bed, kg m3 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)

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. Optimization tasks have been investigated by novel optimization tools, Differential Evolution (DE) algorithm. Optimization of reactor was studied in four approaches. In the first approach,

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2.1.

Mutation

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.

    vi;Gþ1 ¼ xi;G þ K  xr1 ;G  xi;G þ F  xr2 ;G  xr3 ;G

2.

2.2.

Differential Evolution (DE) algorithm

The DE algorithm is a population based algorithm similar to genetic algorithms using similar operators: crossover, mutation, 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 mechanism 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)

For each target vectorxi;G , a mutant vector is produced by (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 ;G Þ, K is the combination factor [10].

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 reactions are relevant in the methanol synthesis [11]:

Feed

Product

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

Fig. 2 – Schematic diagram of a methanol synthesis singlestage spherical reactor.

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Feed

Product

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

CO þ 2H2 4CH3 OH

(3)

CO2 þ 3H2 4CH3 OH þ H2 O

(4)

CO þ H2 O4CO2 þ H2

(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 pressureresistant 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

[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 relations, 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 considered as we study a realistic reactor. The energy and mass balances can be written as [15]: 3

n  X vyj 1 v 2 ¼ 2 yij r r ur yj þ 3s ð1  3Þa vt r vr i¼1

j ¼ 1; 2; .; N and i ¼ 1; 2; .; M

ð1  3ÞrCpc

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

(6)

n X   vT 1 v DHi r ¼ 2 rur r2 Cp T  Tref þ 3s ð1  3Þa vt r vr i¼1

(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 1 – Catalyst and reactor specifications [15]. Parameter 3

rs (kg m ) dp (mm) Cps (kJ kg1 K1) 3 3s aap (m2 m3) Cpg (kJ kg1 K1) V (m3)

Spherical reactor Inner radius Outer radius

Value

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

1770 5.47 5.0 0.5 0.4 626.98 2.98 80

Feed conditions

One-stage (m)

Two-stage (m)

Three-stage (m)

1.0 2.72

1.0 2.19

1.0 1.94

Value

Composition (mol%) CH3OH CO2 CO H2O H2 N2 CH4

0.5 9.4 4.6 0.04 65.9 9.33 10.26

Total molar flow rate (kmol s1) Inlet temperature (K) Pressure (bar)

2 503 76.98

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

17.7892

P (bar)

T (K)

H2 (mol%)

CO (mol%)

CH3OH (mol%)

CO2 (mol%)

H2O (mol%)

N2 (mol%)

CH4 (mol%)

81.95

313

77.53

3.9695

0.4114

2.5197

0.0806

3.4285

12.05

3.3.

The boundary conditions are as follows: vyj vT ¼ 0; ¼0 vr vr r ¼ Ro ; yj ¼ yj0 ; T ¼ T0

r ¼ Ri ;

(8)

   da Ed 1 1 a5 ¼ Kd exp  R T TR dt

The initial conditions are: t ¼ 0;

3.2.

yi ¼

yss i ;

Deactivation model

Catalyst deactivation model for the commercial methanol synthesis catalyst was adopted from Hanken [17].

ss

T¼T ; a¼1

(9)

Reaction kinetics

Reactions (3)–(5) are not independent and therefore one is a linear combination of the other ones. Kinetics of the lowpressure 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.

(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: TR ¼ 513 K, Ed ¼ 91270 J mol1, and Kd ¼ 0.00439 h1.

Steady State Condition 0.1573 0.1573 0.1573

Objective Function

Feed (kmol s1)

0.1572 0.1571 0.1571 0.1571 0.157 0.1569 0.1569

Outlet composition (mol%)

Homogenous model

Hartig et al. data

Relative error (%)

First reactor H2 CO CH3OH CO2 H 2O N2 CH4

78.006 3.113 1.740 2.009 0.558 3.693 12.978

76.550 3.321 1.613 2.131 0.532 3.510 12.340

1.902 6.281 7.826 5.705 4.964 5.195 5.172

Second reactor H2 CO CH3OH CO2 H 2O N2 CH4

77.088 2.396 3.043 1.694 0.960 3.800 13.355

75.640 2.612 2.806 1.811 0.913 3.591 12.620

1.915 8.249 8.479 6.471 5.152 5.809 5.828

Third reactor H2 CO CH3OH CO2 H 2O N2 CH4

76.580 1.801 4.212 1.476 1.233 3.915 14.009

75.010 2.017 3.705 1.630 1.141 3.652 12.840

2.093 10.709 13.700 9.442 8.146 7.182 9.104

0

10

20

30

40

50

60

70

80

90

100

Iteration Fig. 4 – Objective function values for steady-state optimization.

Optimum Reactor inlet Temperature 520 515 510

Temperature (K)

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

505 500 495 490 485 480

First Reactor Second Reactor Third Reactor

475 470

0

200

400

600

800

1000

1200

Time (day) Fig. 5 – Dynamic optimal inlet temperatures for (a) first stage (b) second stage and (c) third stage.

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The partial differential equations (PDEs) of the onedimensional 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 achieved. 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.

Table 5 – Optimal values of reactors’ inlet temperatures (K). Time (day) 0 100 200 300 400 500 600 700 800 900 1000 1100 1200

4.

First reactor

Second reactor

Third reactor

471.71 476.53 484.64 489.44 493.03 495.62 497.35 499.16 500.60 501.87 502.97 503.89 504.82

503.74 505.84 509.62 511.51 512.97 514.11 514.82 515.45 515.98 516.50 517.03 517.40 517.79

502.07 503.70 507.92 509.88 511.04 512.14 513.08 513.64 514.17 514.66 515.19 515.55 515.89

Optimization and results 4.1.

The technical design data of the catalyst pellet and input data of the reactor have been summarized in Tables 1 and 2, respectively.

a

For this optimization study, inlet temperature of reactor is variable because the design conversion of reversible

b

Optimum mole fraction from First reactor outlet With Optimization Without Optimization

0.042

With Optimization Without Optimization

0.06

Methanol mole fraction

0.04 0.038 0.036 0.034 0.032 0.03

0.058 0.056 0.054 0.052 0.05 0.048 0.046

0.028

0.044 0

200

400

600

800

1000

0.042

1200

0

200

400

Time (day)

600

800

Time (day)

c

Optimum mole fraction from Third reactor outlet 0.08 With Optimization Without Optimization 0.075

Methanol mole fraction

Methanol mole fraction

Optimum mole fraction from Second reactor outlet 0.062

0.044

0.026

Optimization of the reactor inlet temperatures

0.07

0.065

0.06

0.055

0.05

0

200

400

600

800

1000

1200

Time (day) Fig. 6 – Optimal methanol mole fraction for (a) first stage (b) second stage and (c) third stage.

1000

1200

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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 lifetime 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.

Table 6 – Additional yield achieved for output methanol mole fractions with optimal temperature profiles. Reference

Optimum

Additional yield (%)

0.0673 0.0668 0.0663

0.0698 0.0694 0.0688

3.71 3.89 3.83

Activity ¼ 0.9 Activity ¼ 0.7 Activity ¼ 0.5

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

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

b

Optimum Temperatue Profile - First Reactor (Activity = 0.9)

Optimum Temperatue Profile - Second Reactor (Activity = 0.9)

600

550

590

545

580

540

Temperatue (K)

570 560 550 540 530

535 530 525 520 515

520

510

510 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

505

1.9

1

1.1

1.2

1.3

Radius (m)

c

1.4

1.5

1.6

1.7

1.8

Radius (m) Optimum Temperatue Profile - Third Reactor (Activity = 0.9)

540 535

Temperatue (K)

Temperatue (K)

a

4.2.

530 525 520 515 510 505

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Radius (m) Fig. 7 – Optimal temperature profiles at a [ 0.9 for (a) first stage (b) second stage and (c) third stage.

1.9

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Methanol Production vs. Time

Average of Square Absolute Error in Methanol Production

310 Plant Data Hanken Deactivation Model Optimized Deactivation Model

54

Methanol Production (ton)

52

Objective Function

50 48 46 44

40

280

270

250 0

10

20

30

40

50

0

200

400

Fig. 8 – Objective function values for optimization of deactivation parameters.

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 (a ¼ 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 methanol synthesis reactors for activity level of 0.9.

4.3. Optimization of catalyst deactivation rate 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 illustrated in the following equation:

Table 7 – Optimal values of deactivation model. 111542.07 J mol1 0.4301156 h1 4.000021 0.000960

600

800

1000

1200

Time (day)

60

Iteration

Ed Kd n m

290

260

42

38

300

Fig. 9 – Comparison of Hanken and optimized deactivation models with observed methanol production rate (ton/day).

m 

  da CO Ed 1 1 an Kd exp ¼  R T TR dt CO2

(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 s1 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.

Table 8 – Values of methanol production predicted by optimal deactivation model compared with Hanken’s model. Time Plant data Hanken’s Percent New Percent (day) (ton/day) model error with model error with (ton/day) plant data (ton/day) plant data 100 200 300 400 500 600 700 800 900 1000

296.50 302.60 284.33 277.90 278.20 278.00 274.00 268.10 275.50 274.58

293.69 291.47 289.95 288.78 287.80 286.95 286.19 285.51 284.89 284.31

0.95 3.68 1.98 3.92 3.45 3.22 4.45 6.50 3.41 3.54

290.26 286.29 283.43 281.12 279.09 277.31 275.73 274.22 272.73 271.33

2.10 5.39 0.32 1.16 0.32 0.25 0.63 2.28 1.00 1.18

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Methanol Production vs. Time

Appendix A Auxiliary correlations

310

Methanol Production (ton)

300 290

To complete the simulation, auxiliary correlations should be added to the model.

280 270

A.1.

260 250 Raduis Ratio = 2 Raduis Ratio = 3 Raduis Ratio = 4 Raduis Ratio = 5

240 230 220

0

200

400

600

800

1000

1200

Time (day) 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.

4.4.

Optimization of reactor radius ratio

Reaction kinetics

h i  k1 KCO fCO fH1:5  fCH3 OH = fH0:5 KP1 2 2 h  i  R1 ¼  1 þ KCO fCO þ KCO2 fCO2 fH0:5 þ KH2 O =K0:5 H2 fH2 O 2

(A.1)

 h i k2 KCO2 fCO2 fH1:5  fCH3 OH fH2 O = fH1:52 KP2 2  h  i R2 ¼  þ KH2 O =K0:5 1 þ KCO fCO þ KCO2 fCO2 fH0:5 H2 fH2 O 2

(A.2)

h i k3 KCO2 fCO2 fH2  fH2 O fCO =KP3  h  i R3 ¼  þ KH2 O =K0:5 1 þ KCO fCO þ KCO2 fCO2 fH0:5 H2 fH2 O 2

(A.3)

A.2.

Reaction constants

Table A.1 – Reaction rate constants. B Þ k ¼ A expðRT

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.

5.

A

B 7

k1 k2 k3

(4.89  0.29)  10 (1.09  0.07)  105 (9.64  7.30)  1011

113,000  300 87,500  300 152,900  11800

Table A.2 – Adsorption equilibrium constants. B Þ k ¼ A expðRT

KCO KCO2 0:5 Þ ðKH2 O =KH 2

A

B

(2.16  0.44)  105 (7.05  1.39)  107 (6.37  2.88)  109

46,800  800 61,700  800 84,000  1400

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 temperature profile approach, optimal catalyst deactivation parameter 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 optimization 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.

Table A.3 – Reaction equilibrium constants. A

Kp ¼ 10ð t BÞ K p1 K p2 K p3

A

B

5139 3066 2073

12.621 10.592 2.029

references

[1] Khademi MH, Jahanmiri A, Rahimpour MR. A novel configuration for hydrogen production from coupling of methanol and benzene synthesis in a hydrogenpermselective membrane reactor. Int J Hydrogen Energy, 2009;34:5091–107.

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