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Dynamic modeling and optimization of a novel methanolsynthesis loop with hydrogen-permselective membranereactor

P. Parvasi, A. Khosravanipour Mostafazadeh, M.R. Rahimpour*

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 12 January 2009

Received in revised form

19 February 2009

Accepted 21 February 2009

Available online 7 April 2009

Keywords:

Dynamic optimization

Reactor loop

Pd–Ag membrane

Catalyst deactivation

Differential evolution method

* Corresponding author. Tel.: þ98 7112303071E-mail address: [email protected] (

0360-3199/$ – see front matter ª 2009 Interndoi:10.1016/j.ijhydene.2009.02.062

a b s t r a c t

In this paper, typical and Pd–Ag membrane methanol loop reactors have been analyzed. In

the proposed models all basic equipments in the methanol loop were included. Detailed

dynamic models described by set of ordinary differential and algebraic equations were

developed to predict the behavior of the overall processes. The conventional model was

validated against plant data, and then the results of the hydrogen-permselective

membrane loop are compared with the conventional model. Using this novel model,

diffusion by membrane tubes compensates reduction of production rate due to catalyst

deactivation. By use of the membrane model, dynamic optimization of temperatures was

performed for improving overall methanol production. Here, differential evolution (DE)

method was applied as powerful method for optimization of procedure. Optimal inlet

temperatures of membrane tube, steam drum and both of them were determined. The

optimization approaches enhanced additional yield throughout 4 years of operation as

catalyst lifetime. Therefore, the methanol synthesis loop can be deduced to redesign based

on this study.

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

reserved.

1. Introduction The dynamic simulation of methanol synthesis processes,

Many chemical process systems consist of a reactor and

a separation unit. These unit operations are considered to be

the core of a chemical process. The behavior of reactor–

separator–recycle systems is relevant for integrating concep-

tual design and plant wide control at an early stage of

conceptual design, when the recycle structure of the flow

sheet is established. At this point, the reactor is the first unit to

be considered in detail because the chemical species present

in the reactor effluent determine the separation section.

Hence, reactor modeling, sizing, and control are considered

before separation is addressed.

; fax: þ98 7116287294.M.R. Rahimpour).ational Association for H

in particular, has a wide range applications including; the

start-up and shut-down investigations, system identification,

safety, control, optimization, and transient behavior and

operability studies [1]. The dynamic simulation is preferred to

steady-state simulations in operability studies since the

former provides a realistic description of the transient states

of the loop owing to the fact that the numerical solution

strategies employed in dynamic models are more robust than

the solution of a steady-state model.

The application of membrane conversion technology in

chemical reaction processes is now mainly focused on reac-

tion systems containing hydrogen and oxygen, and is based

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

mailto:[email protected]

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http://www.sciencedirect.com

Nomenclature

a activity of catalyst,

Ac cross area of reactor (m),

cpg specific heat of the gas at constant pressure

(J kgmol�1 K�1),

cps specific heat of the solid at constant pressure

(J kgmol�1 K�1),

Cj concentration of component j in the fluid phase

(kgmol m�3),

Cjs concentration of component j in the solid phase

(kgmol m�3),

ct Total concentration (mol m�3),

dp particle diameter (m),

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),

hf gas–solid heat transfer coefficient (W m�2 K�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

(mol kg�1 s�1 bar�1/2),

k3 reaction rate constant for the 3rd rate equation

(mol kg�1 s�1 bar�1/2),

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

keff conductivity of fluid phase (W m�1 K�1),

Kd deactivation model parameter constant (s�1),

Kj adsorption equilibrium constant for component j

(bar�1),

Kpi equilibrium constant based on partial pressure for

component i,

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


r3 reaction rate constant for the 3rd rate equation


ri reaction rate of component i (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),

Ts temperature of solid phase (K),

Tshell temperature of coolant stream (K),

ushell overall heat transfer coefficient between coolant

and process streams (W m�1 s�1),

ur radial velocity of fluid phase (m s�1),

V total volume of reactor (m3),

z axial reactor coordinate.

Greek letters

3 void fraction of catalytic bed,

3s void fraction of catalyst,

v stoichiometric coefficient,

h catalyst effectiveness factor,

r density of catalytic bed (kg m�3),

rs density of catalyst (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).


on inorganic membranes such as Pd and ceramic membranes

[2]. In many hydrogen-related reaction systems, Pd–alloy

membranes on a stainless steel support were used as the

hydrogen permeable membrane [3]. It is also well known that

the use of pure palladium membranes is hindered by the fact

that palladium shows a transition from the a-phase

(hydrogen-poor) to the b-phase (hydrogen-rich) at tempera-

tures below 300 �C and pressures below 2 MPa, depending on

the hydrogen concentration in the metal. Since the lattice

constant of the b-phase is 3% larger than that of the a-phase,

this transition leads to lattice strain and, consequently, after

a few cycles, to a distortion of the metal lattice [4]. Alloying the

palladium, especially with silver, reduces the critical

temperature for this embitterment and leads to an increase in

the hydrogen permeability. The highest hydrogen perme-

ability was observed at an alloy composition of 23 wt% silver

[5]. Palladium-based membranes have been used for decades

in hydrogen extraction because of their high permeability and

good surface properties and because palladium, is 100%

selective for hydrogen transport [6]. These membranes

combine excellent hydrogen transport and discrimination

properties with resistance to high temperatures, corrosion,

and solvents. Key requirements for the successful develop-

ment of palladium-based membranes are low costs as well as

permselectivity combined with good mechanical, thermal and

long-term stability [7]. These properties make palladium-

based membranes such as Pd–Ag membranes very attractive

for use with petrochemical gases. A thin palladium or palla-

dium-based alloy layer is prepared on the surface or inside the

pores of porous supports. Many researchers have developed

supporting structures for palladium or palladium-based alloy

membranes. The materials in commercial use for porous

supports are: ceramics, stainless steel and glass. The

membrane support should be porous, smooth-faced, highly

permeable, thermally stable and metal adhesive [8].

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. For plant optimization,

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 ) 3 7 1 7 – 3 7 3 3 3719

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 algorithms methods for

dealing with optimization problems have been proposed in

the last few years in the fields of evolutionary programming

(EP) [9], evolution strategies (ES) [10], genetic algorithms (GA)

[11] and particle swarm optimization (PSO) [12].

DE algorithm is a stochastic optimization method mini-

mizing 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,

veryeasilyadaptable for integerand discreteoptimization,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 [13].

Several works have been performed on application of Pd–Ag

membrane reactors. Rahimpour and Ghader [14] investigated

Pd–Ag membrane reactor performance for methanol

synthesis. They considered steady-state homogeneous model

for methanol reactor. Rahimpour and Lotfinejad [15] presented

dynamic model for studying Pd–Ag dual-type membrane

reactor for methanol production. They showed methanol

production can be increased in membrane dual-type reactor.

Rahimpour and Lotfinejad [16] compared co-current and

counter-current modes of operation for a membrane dual-type

methanol reactor. Khosravanipour Mostafazadeh and Rahim-

pour [17] proposed a Pd–Ag membrane catalytic bed for

naphtha reforming. Rahimpour et al. [18] suggested a new

approach to improve the methanol production in an industrial

single methanol synthesis reactor by applying selective

permeation of hydrogen from synthesis gas and adding it to the

reaction side. They considered quasi-steady-state model for

simulation of membrane methanol reactor and also they

modeled single reactor without considering the loop. Rahim-

pour and Alizadehhesari [19] developed fluidized-bed

membrane reactor for methanol synthesis. Recently, Rahim-

pour and Alizadehhesari [20] developed a model of membrane

methanol reactor for increasing carbon dioxide removal.

Rahimpour and Elekaei Behjati presented a novel fluidized-bed

hydrogen-permselective membrane reactor [21]. Iulianelli

et al. [22] investigated CO-free hydrogen production by steam

reforming of acetic acid in a Pd–Ag membrane-assisted reactor.

The goal of this study was to perform the AASR reaction in a Pd–

Ag MR in order to study the acts for hydrogen selectivity,

hydrogen yield and CO-free hydrogen recovery by varying the

mode of operation, the reaction pressure and the sweep factor.

Tostia et al. [23] performed a study on design and process of Pd

membrane reactors. In that research, the permeator tube was

planned that permits the free elongation and contraction of the

palladium alloy tube keeping away from any mechanical

stress. The different patterns of Pd membrane reactors applied

for separating pure hydrogen are explained and a membrane

process for producing highly pure hydrogen from ethanol

reforming is also implemented. Rahimpour [24] studied on

hydrogen production in a fluidized-bed membrane reactor for

naphtha reforming. In aforesaid work, a novel fluidized-bed

membrane reactor (FBMR) for naphtha reforming in the pres-

ence of catalyst deactivation has been proposed. In this reactor

configuration, a fluidized-bed reactor with permselective Pd–

Ag (23 wt% Ag) wall to hydrogen has been used. This work

showed how FBMR can be useful for catalytic naphtha

reforming by enhancement of aromatic production, increase of

catalyst activity and hydrogen production. Gallucci et al. [25]

presented co-current and counter-current modes for ethanol

steam reforming in a dense Pd–Ag membrane reactor. In their

work a conventional and a palladium membrane reactor

packed with a CO-based catalyst was modeled and the results

for both co-current and counter-current modes of operation

are showed in terms of ethanol conversion and molar fraction

versus temperature, pressure, the molar feed flow rate ratio

and axial coordinate. Molaei Dehkordi and Memari [26] did

a compartment model for methane steam reforming in

a membrane bubbling fluidized-bed reactor. A compartment

model for methane steam reforming was performed to illus-

trate the flow pattern of gas contained by the dense region of

a membrane fluidized-bed reactor, in the bubbling configura-

tion both with (adiabatic) and without (isothermal) inflowing

oxygen.

Also several researches were performed on reactor

modeling and optimization of methanol synthesis. Rahim-

pour and Elekaei Behjati [27] simulated and optimized

membrane dual-type methanol reactor. Parvasi et al. [28]

simulated a dynamic methanol loop in the presence of cata-

lyst deactivation. Askari et al. [29] optimized dual-type

methanol reactor using genetic algorithm.

The previous studies focused on optimization and appli-

cation of membrane reactor for methanol production without

considering the role of the loop of methanol synthesis. The

purpose of this work is to study the typical and membrane

synthesis loop with exothermic, high-pressure gas phase

chemical reactor systems for methanol production and finally

optimization of membrane loop parameters using DE method

as a strong method of optimization.

2. Description of methanol synthesis loops

2.1. Conventional process

Fig. 1 shows a typical real methanol synthesis loop in Shiraz

Petrochemical Complex. Methanol synthesis is generally

performed by passing synthesis gas comprising hydrogen,

carbon oxides and any inert gasses like nitrogen at an elevated

temperature and pressure through one or more beds of

a methanol synthesis catalyst, which is often a copper-con-

taining composition [28]. The following three overall equilib-

rium reactions are relevant in the synthesis:

Recycle Flow (FR)

Purge (FP)


COþ 2H24CH3OH (1)

Forth Boundary

Third Boundary

Product (FM

)

Feed (Fff)

CO2 Injection (FCO2

)

(Ff)

(Fe)

(FR2

)

Second BoundaryFirst Boundary

Fig. 2 – A membrane methanol synthesis loop.

CO2 þ 3H24CH3OHþH2O (2)

COþH2O4CO2 þH2 (3)

Methanol is generally recovered by cooling the product gas

stream to below the dew point of the methanol and separating

off the product as a liquid. The process is often operated in

a loop; thus the remaining unreacted gas stream is usually

recycled to the synthesis reactor as part of the synthesis gas

via a circulator. In addition to compressing the reactants, the

recycle also needs to be recompressed due to pressure losses

in the synthesis loop. Fresh synthesis gas, termed make-up

gas, is added to the recycled unreacted gas to form the

synthesis gas stream. A purge stream is taken from the

circulating gas stream to prevent a high concentration of inert

gases accumulating in the recycle loop.

2.2. Membrane reactor loop

Fig. 2 shows the novel schematic diagram of hydrogen

permeable membrane reactor loop which simulated in this

work. This process is similar to typical loop, but in the

membrane loop, recycle gas is flowed through membrane

tubes of reactor in co-current mode with reacting gas mixture

stream in the shell side. Outlet gas stream from membrane

tubes is added to the fresh synthesis gas stream to form the

feed of reactor.

3. Methanol synthesis loop models

In this section, first of all the models representing the

processes behaviors are needed. Such typical loop model is

then confronted with plant data to assess its model validity,

which in general requires changes in the model. After

several iterations we end up with a validated model and

compared with membrane model, which is membrane

model ready for use within dynamic optimization. The

methanol synthesis loop has five unknown streams that are

feed reactor stream, output reactor stream, and crude

methanol stream, purge stream and recycle stream. Mole

fractions of all components are unknown but the specifi-

cations of the recycle stream are similar to purge stream.

Solving of heat exchanger model requires guess of mole

Recycle Flow (FR)

Purge (FP)

Product (FM

)

Forth Boundary

Third Boundary

Feed (Fff)

CO2 Injection (FCO2

)

(Ff)

(Fe)

Second Boundary

First Boundary

Fig. 1 – Typical synthesis loop: the boundaries are shown

for material balances.

fractions of recycle and reactor feed streams and it

continues with trial and error.

Generally the fresh feed of methanol unit operation has H2,

CO2 and CO that make synthesis gas, although fresh feed has

consisted of CH4 and N2 that not incorporate in reaction of

methanol production. Specification of fresh feed reported in

Table 1.

Although the dehydration of synthesis gas done, the trace

of water remained in synthesis gas. The mole fraction of

methanol is account in fresh feed stream in other to the

problem is generalized. Therefore, all streams consist of

CH3OH, CO2, CO, H2O, H2, N2 and CH4. For simulation of loop

and obtained the specification of streams, the under cases

must be recognized.

The typical methanol synthesis loop has five unknown

streams: feed reactor stream, output reactor stream, and

crude methanol stream, purge stream and recycle stream.

Mole fractions of all components are unknown too but the

specification of the recycle stream is similar to purge stream.

The recycle ratio is constant and fresh feed value is known,

the value of FR calculated through FR¼ Fff� (Recycle Ratio). In

this situation reactor feed stream value calculated by adding

FR and Fff streams. FR2 is the outlet stream of shell side which

goes to the mixer. As the above mentioned, totally 31

unknown parameters (three streams and 28 mole fractions)

must be calculated by solving the material balance over mixer,

reactor model and separator model.

3.1. Material balance for first boundary (mixer)

Mass balances for typical synthesis loop can be written as:

Fff

�yiff

�þ FR

�yiR

�� Ff

�yif

�¼ 0 for i ¼ 1; 3; 4;5;6 (4)

Fff

�y2ff

�þ FR

�y2R

�� Ff

�y2f

�þ FCO2

¼ 0 (5)

X7

i¼1

yif � 1 ¼ 0 (6)

Table 1 – Specification of fresh feed [30].

Fff

(kmol s�1)P (bar) T (�C) H2

(mol %)CO CO2 CH4 N2 H2O

1635.23 75.13 67 64.20 14.66 12.67 4.37 2.52 0.20


For hydrogen permeable membrane synthesis loop, Eqs. (4)

and (5) can be written as below:

Fff

�yiff

�þ FR2

�yiR2

�� Ff

�yif

�¼ 0 for i ¼ 1;3;4;5; 6 (7)

Fff

�y2ff

�þ FR2

�y2R2

�� Ff

�y2f

�þ FCO2

¼ 0 (8)

3.2. Simulation of heat exchanger (second boundary)

There are no phase changes in the heat exchanger and this

unit is small in compare with reactor unit. Therefore, a lump

model is used for this unit.

It is modeled by assuming the cold gas has a temperature

TC in the heat exchanger inlet, which heat is transferred at

a rate QC from the hotter tube metal at temperature TM.

QC ¼ UCACðTM � TCÞ (9)

The hot gas flows counter-currently and has a temperature

TH in the heat exchanger. Heat is transferred from the hot gas

into the tube metal at a rate QH.

QH ¼ UHAHðTH � TMÞ (10)

Finally, each stream of the cold and hot sides of the

exchanger is described by an energy balance because both

temperatures can change with time. Cold stream flow rate (FC)

is a function of time too.

ravCPVH

dTH

dt¼ ðFHCPHTHÞL�ðFHCPHTHÞ0�QH (11)

ravCPVC

dTC

dt¼�FCðtÞCPCTC

�L��FCðtÞCPCTC

�0�QC (12)

in above equations heat capacity (CP) is evaluated by following

expression.

CP

R¼ Aþ B

2ðTin þ ToutÞ þ

C3

�T2in þ TinTout þ T2

out

�þ DðTinToutÞ

(13)

Heat capacity coefficients A, B, C and D for all components in

heat exchanger streams are given in Appendix A.

3.3. Simulation of methanol synthesis reactor (thirdboundary)

A typical methanol reactor handles the process of conversion

of synthesis gas (CO2, CO and H2) to form methanol. Such

a reactor usually resembles a vertical shell and tube heat

exchanger. The tubes are packed with catalyst pellets and

boiling water is circulating in the shell side to remove the heat

of exothermic reactions. In this study, homogeneous one-

dimensional models have been considered.

3.3.1. Reactor modelIn these simple models we assume that gradients of tempera-

ture and concentrations between the phases can be ignored and

the equations for the two phases can be combined. 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 reactors.

For typical reactor, energy and mass balances can be written as:

3Bctvyi

vt¼ �Ft

Ac

vyi

vzþX

j

hrjrBa i ¼ 1; 2;.;7 (14)

3BctcpgvTvt¼�Ft

Ac

vTvzþpDi

Ac

UshellðTshell�TÞþhrBaXN

i¼1

ri

��DHf ;i

�(15)

where T and yi are the temperature and concentration of

component i in the fluid phase, a is the activity of catalyst and

h is effectiveness factor. The procedure of catalyst effective-

ness factor calculation has been reported in Ref. [28].

Boundary and initial conditions are as follows:

z ¼ 0; yi ¼ yi0; T ¼ T0 (16)

t ¼ 0; yi ¼ yssi ; T ¼ Tss; a ¼ 1 (17)

For hydrogen permeable membrane reactor, energy and

mass balances can be written as follows.

3.3.1.1. Reaction side.

3Bctvyi

vt¼ �Fr

t

Ac

vyi

vzþX

j

hrjrBa i ¼ 1; 2; 3;4;6;7 (18)

3BctvyH

vt¼ �Fr

t

Ac

vyH

vzþX

j

hrjrBaþ aH

Ac

� ffiffiffiffiffiffiptH

p�

ffiffiffiffiffiffiprH

p �(19)

3BctcpgvTvt¼ �Fr

t

Ac

vTvzþ pDri

Ac

UshellðTshell � TÞ

þ pDro

Ac

UtubeðTtube � TÞ þ hrBaX3

i¼1

ri

��DHf ;i

�ð20Þ

where Fr is total molar flow rate of gas in reaction side for each

tube. aH is hydrogen permeation rate constant. rB is density of

bed and prH and pt

H are partial pressures of hydrogen in reaction

and tube side. Ac is cross-sectional area of reaction side. Dri

and Dro are inner diameter of reaction side and outer diameter

of tube side, respectively.


z ¼ 0; yi ¼ yi0; T ¼ T0 (21)

t ¼ 0; yi ¼ yssi ; T ¼ Tss; a ¼ 1 (22)

3.3.1.2. Tube side. The mass balance equation is written only

for hydrogen in the tube side:

ctvyH

vt¼ �Ft

t

Ac

vyH

vzþ aH

Ac

� ffiffiffiffiffiffiprH

p�

ffiffiffiffiffiffiptH

p �(23)

ctcpgvTtube

vt¼ �Ft

t

Ac

vTvzþ pDro

Ac

UtubeðTtube � TÞ (24)


z ¼ 0; yH ¼ yHR; T ¼ T0 (25)

t ¼ 0; yH ¼ yssHR; T ¼ Tss (26)

where yHR is the concentration of hydrogen in the recycle

stream.


Schematic diagram of membrane reactor has been shown

in Fig. 3.

3.3.2. Hydrogen permeation through the palladiummembraneIn Eqs. (19) and (23), aH is hydrogen permeation rate constant

and is defined as [31]:

aH ¼2pLP

ln

�Ro

Ri

� (27)

where Ro and Ri stand for outer and inner radius of Pd–Ag

layer. The permeability of hydrogen through Pd–Ag layer as

a function of temperature is as follows [32,33]:

P ¼ P0 exp

��Ep

RT

�(28)

where the pre-exponential factor P0 above 200 �C is reported

as 6.33� 10�8 (mol m�2 s�1 Pa�1/2) and activation energy Ep is

15.7 kJ mol�1 [33,34]. For pure palladium layer, Eq. (27) changes

to Eq. (29) as follows [34]:

aH ¼2pL

ln

�Ro

Ri

� DC0ffiffiffiffiffiP0

p (29)

where P0 is 1.013� 103 Pa, C0 is defined as a standard

concentration of hydrogen in the palladium which is at

equilibrium with the hydrogen gas phase at P0. The value of

1280 mol m�3 was reported for C0 at 473 K, D is the diffusion

coefficient for hydrogen in palladium [35], and L is the length

of the reactor. In the range of 140–310 �C, the diffusion coef-

ficient is defined as: D (m2 h�1)¼ 8.25� 10�4 exp (�21,700/RT ).

3.3.3. Reaction kineticsEqs. (1)–(3) are not independent so that one is a linear combi-

nation of the other ones. Kinetics of the low-pressure meth-

anol synthesis over commercial CuO/ZnO/Al2O3 catalysts has

been widely investigated. In the current work, the rate

expressions have been selected from [36]. The correspondent

rate expressions due to the hydrogenation of CO, CO2 and the

reversed water–gas shift reactions are given in Appendix A.

The reaction rate constants, adsorption equilibrium

constants and reaction equilibrium constants which occur in

Fig. 3 – Schematic of hydrogen permeable membrane

reactor [18].

the formulation of kinetic expressions are tabulated in

Appendix A, respectively [37].

3.3.4. Deactivation modelCatalyst deactivation model for the commercial methanol

synthesis catalyst was adopted from [38].

dadt¼ �Kd exp

��Ed

R

�1T� 1

TR

��a5 (30)

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

energy and deactivation constant of the catalyst, respectively.

The numerical value of TR is 513 K, Ed 91,270 J mol�1 and Kd

0.00439 h�1.

3.4. Simulation of methanol synthesis separator (fourthboundary)

Flash separator is a cylindrical tank of known dimensions to

which reactor outlet stream is introduced at known pressure

and temperature in thermodynamic equilibrium. The

entering stream reaches this equilibrium by means of an

expansion caused by a valve placed just before the tank. This

equilibrium allows the separation of the reactor outlet stream

in two phases, and the liquid in equilibrium is extracted from

the bottom while the vapor is extracted from the top. To give

a qualified estimate of the amount of methanol in the liquid

phase, thermodynamic models are required.

There are two traditional classes of thermodynamic

models for phase equilibrium calculations: one is liquid

activity coefficient and the other is equation of state models.

Activity coefficient models can be used to describe mixtures of

any complexity, but only as a liquid well below its critical

temperature. Due to the simplicity and the accuracy of pre-

dicting K-values, cubic of equations of state (CEOS) is widely

used in refinery and petroleum reservoir industries for the

prediction of phase behavior. Equation of state is a mathe-

matical relation between volume, pressure, temperature, and

composition is called the equation of state and most forms of

the equation of state are of the pressure-explicit type. Many

equations of state have been proposed, but most of them are

essentially empirical in nature.

Peng and Robinson [39] proposed the following CEOS,

which is the most widely used equation in chemical engi-

neering thermodynamics. It is known to give slightly better

predications of liquid densities than other two parameters

EOS such as Soave–Redlich–Kwong EOS. The familiar PR EOS is

formulated as:

P ¼ RTv� b

� avðvþ bÞ þ bðv� bÞ (31)

where a and b are the two parameters of PR EOS.

In the mixing rules proposed by Van der Waals, following

linear mixing rules are adapted for parameter b:

b ¼X

i

yibi (32)

Parameter a is evaluated by the following expression:

a ¼X

i

Xj

yiyj

ffiffiffiffiffiffiffiffiaiaj

p �1� kij

�(33)

Initialization

Evaluation

Repeat

Mutation

Recombination

Evaluation

Selection

Until (termination criteria are met)


The compressibility factor expression for vapor and liquid

phases derived from the PR EOS is as follows:

Z3 þ ðB� 1ÞZ2 þ�A� 2B� 3B2

�Zþ

�B2 þ B3 � AB

�¼ 0 (34)

The fugacity coefficient expression for component i derived

from the PR EOS is as follows:

RT ln fi ¼bi

bðZ� 1Þ � lnðZ� BÞ þ A

2ffiffiffi2p

B��

2a0

a� bi

b

�

�

264ln

Zþ�

1�ffiffiffi2p �

B

Zþ�

1þffiffiffi2p �

B

375 (35)

where:

a0 ¼X

j

yiyj

ffiffiffiffiffiffiffiffiaiaj

p �1� kij

�(36)

Nikos and coworkers [40] proposed following equation that

is applicable for multi-component phase equilibrium calcu-

lations where the system under study contains N2, CO2 and

CH4. Binary interaction relation has a function of pressure,

temperature and acentric factor [41].

kij ¼ d2T2rj þ d1Trj þ d0 (37)

In the above equation i represents of N2, CO2 or CH4 and j

represents of other components. d0, d1 and d2 are given in

Appendix A.

In the separator process pressure is high, therefore

following correlation was used for effect of pressure. For

N2 – other component:

k0ij ¼ kij

�1:04� 4:2� 10�5 P

�(38)

For CO2 – other component:

k0ij ¼ kij

�1:044269� 4:375� 10�5 P

�(39)

in Eqs. (38) and (39) P in psia.

4. Differential evolution algorithm (DEA)

The DE algorithm is a population based algorithm like genetic

algorithms using the 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 mech-

anism 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. By use of 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, a population of

NP solution vectors is randomly created at the initiate. This

population is successfully improved by applying mutation,

crossover and selection operators. The main steps of the DE

algorithm are given below [13]:

4.1. Mutation

For each target vector xi,G, a mutant vector is produced by

vi;Gþ1 ¼ xi;G þ K��xr1 ;G � xi;G

�þ F�

�xr2 ;G � xr3 ;G

�(40)

where i; r1; r2; r3˛f1;2;.;NPg are accidentally 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, and K is the combination factor [13].

4.2. 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;

(41)

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 [13].

4.3. Selection

All solutions in the population have the same possibility of being

selected as parents without dependence of their appropriate-

ness 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.

5. Numerical solution

Fig. 4 shows trend of solving the model:

1. Get input data for calculation that these data consist of

composition and flow rates of fresh feed stream, recycle

ratio, reactor pressure, input temperature of reactor feed

stream and separator pressure.

Input Data

(1)

Guess Unknown Flow rates

& Compositions (2)

Guess XCO

& XCO2

(3)

Solve Equs. 4 to 6(4)

Solve Model of Heat Exchanger

(5)

Solve Model of Reactor

(6)

Solve Model of Separator

(7)

Clculate New Values for XCO

& XCO2

and Compute Error (8)

Replace New Values of

XCO

& XCO2

in Equs.

YES

NO

0.001Error

Data

Output

Fig. 4 – Flowchart of solving the model.

Table 3 – Calculated data from steady-state simulation.

Components Reactor outlet Recycle Product

CH3OH (mol %) 6.02 0.41 68.02

CO2 (mol %) 9.50 12.00 1.11

CO (mol %) 2.30 4.03 0.03

H2O (mol %) 2.40 0.05 22.25

H2 (mol %) 37.40 39.12 0.37

N2 (mol %) 21.00 19.18 3.00

CH4 (mol %) 21.40 25.20 5.02


2. Initial guess for all unknown parameters. In this section

guess values for all composition and flow rates that these

values optimized with trial and error.

3. Initial guess for xCO and xCO2 . With guess CO2 conversion

and CO conversion in reactor, model solved and new values

for xCO and xCO2 obtained. Progress of computer program-

ming is reach to end while difference of new xCO and xCO2

values and old values is approach to zero.

Table 2 – Design data from PFD (conventional loop) [30].

Components Reactor outlet Recycle Product

CH3OH (mol %) 5.82 0.38 67.54

CO2 (mol %) 10.73 12.68 1.21

CO (mol %) 2.88 3.56 0.03

H2O (mol %) 2.48 0.04 30.06

H2 (mol %) 33.75 37.85 0.46

N2 (mol %) 21.44 22.62 0.05

CH4 (mol %) 22.61 22.86 0.54

4. Solving equations of mixer that with solving these equa-

tions reactor input flow rate and compositions are

corrected.

5. Solve model of heat exchanger. Identifying the steady-state

condition of the methanol heat exchanger, in principle, is

simply a matter for determining temperatures at reactor

and separator inlet. This is accomplished by setting the all

time derivatives equal to zero in energy balance equations.

ðFHCPHTHÞL�ðFHCPHTHÞ0�QH ¼ 0 (42)

�FCðtÞCPCTC

�L��FCðtÞCPCTC

�0�QC ¼ 0 (43)

When the above equations combined with boundary

condition to the heat exchanger, an algebraic system of

equations obtained. This non-linear algebraic system of

equations should be solved. Gauss–Newton’s method is used

to solve the non-linear equations. In above equations CPCand

CPHare the molar heat capacity of the mixture that computed

by CPmixture¼P

i

ðyiCPiÞ. While CPi

is molar heat capacity of pure

component. In this model the inlet temperature of hot stream

is unknown which this value is calculated by reactor model

later, therefore this temperature guess and then corrected by

trial and error.

6. Solve model of reactor. With known reactor input flow rate

and compositions, material balance equations for each

component and energy balance equation for gas mixture

were written and solved in each node of reactor. To solve

this set of equations, backward finite difference approxi-

mation is applied here. Then the reactor is divided into 30

separate nodes.

7. Solve model of separator. When output reactor stream flow

rate and components are known, flash calculation is used

in separator to obtain flow rates and components of

upstream and downstream.

8. Computation of xCO and xCO2 . Now obtain new values for xCO

and xCO2 with under equations and compare with old values.

xCO2new¼

�Ff � yfð2Þ � Fe � yeð2Þ

��

Ff � yfð2Þ� (44)

xCOnew¼

�Ff � yfð3Þ � Fe � yeð3Þ

��

Ff � yfð3Þ� (45)

0 1 2 3 4 5 6 70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Reactor Length (m)

Meth

an

ol m

ole fractio

n

0 1 2 3 4 5 6 70.02

0.025

0.03

0.035

0.04

0.045

Reactor Length (m)

CO

m

ole fractio

n

0 1 2 3 4 5 6 70.09

0.095

0.1

0.105

0.11

0.115

0.12

Reactor Length

CO

2 m

ole fractio

n

0 1 2 3 4 5 6 7500

505

510

515

520

525

530

535

Reactor Length (m)

Tem

peratu

re (K

)

a b

cd

Fig. 5 – Steady-state results for methanol reactor, (a) CH3OH, (b) CO, (c) CO2, and (d) temperature.

Table 4 – Comparison between predicted methanolproduction rates by typical model with plant data.

Time(day)

Plant(tone day�1)

Typicalmodel

(tone day�1)

Errorpercent

0 295.00 319.23 8.21

100 296.50 299.84 1.13

200 302.60 289.37 4.37

300 284.30 279.27 1.77

400 277.90 274.11 1.36

500 278.20 270.48 2.77

600 253.00 267.50 5.73

700 274.00 264.89 3.32

800 268.10 262.54 2.07

900 275.50 260.39 5.48

1000 274.60 258.41 5.90

1100 262.90 256.56 2.41

1200 255.20 254.84 0.14


6. Optimization and results

6.1. Model validation

6.1.1. Steady-state model validationThe results of steady-state modeling are compared with the

data reported in Ref. [30]. Table 2 presents the industrial data

and Table 3 shows the predicted values from simulation

results. Tables 2 and 3 show the simulation results have good

agreements with design data.

Fig. 5 demonstrates the steady-state model results for three

important species (Fig. 5(a)–(c)) and temperature of reaction side

(Fig. 5(d)). This figure shows that the methanol mole fraction

increases along the reactor and the mole fractionsof COand CO2

decrease along the reactor. Also the temperature sharply

increases and then smoothly decreases along the reactor.

6.1.2. Dynamic model validationDynamic model of typical synthesis loop validation was

carried out by comparison of model results with the historical

process data over a period of 1200 operating days under the

design specifications and input data [42]. Comparison

between predicted methanol production rates with plant data

is shown in Table 4.

As can be seen from Table 4, there is a systematic deviation

(or bias error) from the experimental values which is due to

the fact that simulation result has a good agreement with

experimental data.

Fig. 6 shows the activity of catalyst over an operation

period of 1200 days. As shown in this figure, the activity of the

fresh catalyst declines markedly during operation.

The predicted results of concentrations of components at

reactor outlet as a function of time are presented in Fig. 7(a)–(c).

0 200 400 600 800 1000 12000.4

0.5

0.6

0.7

0.8

0.9

1

Time (day)

Catalyst activity

Fig. 6 – Catalyst activity as a function of time.

Table 5 – Comparison between predicted methanolproduction rates by typical and membrane models.

Time(day)

Typical model(tone day�1)

Membranemodel

(tone day�1)

ProductionImprovement (%)

0 319.23 331.97 3.99

100 299.84 320.57 6.91

200 289.37 316.01 9.21

300 279.27 306.70 9.82

400 274.11 302.31 10.29

500 270.48 298.76 10.45

600 267.50 293.40 9.68

700 264.89 293.94 10.97

800 262.54 294.41 12.14

900 260.39 294.90 13.25

1000 258.41 295.41 14.32

1100 256.56 295.93 15.35

1200 254.84 296.46 16.33


Because of the main product is methanol and the main reac-

tants are carbon monoxide and carbon dioxide, changing of

theses components is shown. This figure demonstrates

methanol mole fraction decreases and CO, CO2 mole fraction

increases during the time. Also Fig. 7(d) shows the outlet

temperature profile during the time and illustrates the outlet

0 200 400 600 800 1000 12000.048

0.05

0.052

0.054

0.056

0.058

0.06

0.062

Time (day)

Meth

an

ol m

ole fractio

n

Reactor outlet composition

0 200 400 600 800 1000 12000.095

0.1

0.105

0.11

0.115

0.12

Time (day)

CO

2 m

ole fractio

n


a b

c d

Fig. 7 – Reactor outlet as a function of time, (a) C

temperature increases in one-third of section of operation time

but it will be constant at the rest of time approximately.

6.2. Comparison and contrast

In Table 5, the results of outlet methanol production in

membrane system were compared with the results of typical

0 200 400 600 800 1000 12000.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Time (day)

CO

m

ole fractio

n


0 200 400 600 800 1000 1200525.75

525.8

525.85

525.9

525.95

526

Time (day)

Te

mp

eratu

re (K

)

Reactor outlet temperature

H3OH, (b) CO, (c) CO2, and (d) temperature.

0 200 400 600 800 1000 120036.5

37

37.5

38

38.5

39

39.5

40

40.5

41

41.5

Time (day)

Pressu

re (b

ar)

Partial pressure of hydrogen in tubePartial pressure of hydrogen in shell

Fig. 8 – Comparison of hydrogen partial pressures in

membrane reactor at tube side and shell side.

Table 6 – Predicted methanol production rates bymembrane models.

Time (day) Membrane model(tone day�1)

1300 296.72

1400 297.81

1500 297.98

1600 297.89

1700 297.73

1800 297.56

1900 297.39

2000 297.23


methanol synthesis loop. In this study 10 mm value was

chosen for membrane thickness.

As can be seen from this table, performance of methanol

reactor system improved when a membrane was used in

a conventional-type methanol reactor. Also the results

demonstrate that after 600th day of operation, methanol

production increases due to the difference effect of partial

pressures among tube side and shell side overcomes catalyst

deactivation impact. Fig. 8 shows the partial pressure differ-

ences over an operation period of 1200 days. As revealed in

this figure, the partial pressure differences are increased

slowly. Fig. 9 also presents the hydrogen permeation flux in

tube side. In fact, hydrogen permeation rises as time goes on.

Fig. 9(a) shows the hydrogen permeation flux in the reactor,

while Fig. 9(b) presents the hydrogen permeation flux in one

tube of reactor. In fact, because of catalyst deactivation, the

hydrogen consumption declines as time goes on, and the

concentration of hydrogen in the loop increases and thus,

partial pressure of tube side increases slowly. This effect is not

clear in Fig. 8 but it is shown in Fig. 9(a) obviously.

0 200 400 600 800 1000 120032

34

36

38

40

42

44

46

Time (day)

Hyd

ro

gen

p

erm

eatio

n flu

x (to

ne/ d

ay m

2)

a

Fig. 9 – Hydrogen permeation flux (a) in r

To investigate the effect of partial pressures among tube

side and shell side on methanol production, dynamic behavior

of membrane loop extended to 2000 days. New results

demonstrate which after 1500th day’s methanol production

declines which is due to the fact that the catalyst deactivation

effect overcomes on partial pressure differences. The

predicted results of methanol production are presented in

Table 6.

On the other hand, the optimization of membrane loop

reactor was investigated and optimal inlet temperatures to

the reactor were found.

Also CO and CO2 conversions are calculated by Eqs. (44) and

(45). Values for CO and CO2 conversions in steady-state mode are

0.486 and 0.249 in typical loop and 0.622 and 0.278 for membrane

reactor, respectively. It is observed that the conversions in

membrane reactor are greater than typical reactor.

6.3. Optimization of the membrane inlet temperature

In this section, membrane inlet temperature (tube side

temperature) considered variable for optimization study. In

this case, selection of relatively low temperature in tube side

permits higher heat transfer between tube side and shell side

and therefore higher methanol production in shell side, but

this must be balanced the lower hydrogen permeation in

membrane tube [43,44] and therefore, lower methanol

production in reaction side. Hence, the membrane inlet

0 200 400 600 800 1000 1200Time (day)

Hyd

ro

gen

p

erm

eatio

n flu

x (to

ne/ d

ay m

2)

0.0125

0.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

0.017

b

eactor and (b) in one tube of reactor.

0.4 0.5 0.6 0.7 0.8 0.9 1520

530

540

550

560

570

580

590

600

610

620

Catalyst activity

Tem

peratu

re (K

)

Optimal inlet temperatures for membrane tube

Fig. 10 – Optimal temperature trajectory of permeation side

at three periods of catalyst life.

0.4 0.5 0.6 0.7 0.8 0.9 1521

522

523

524

525

526

527

528

529

Catalyst activity

Tem

peratu

re (K

)

Optimal temperatures for steam shell

Fig. 12 – Optimal steam drum temperature trajectory at

three periods of catalyst life.


temperature can be adjusted (at optimized temperature) to

maximize methanol mole fraction at reactor outlet as the

catalyst is deactivated. According to deactivation rate, three

activity levels equal to 1, 0.8 and 0.6 were chosen to study

optimal inlet temperatures. These values stand for dynamic

properties of reactor operation and give some information

about variation of optimal temperatures through catalyst

lifetime [45]. The results of optimal temperature trajectory of

membrane side (sweeping gas) at three periods of catalyst

lifetime are shown in Fig. 10.

The optimal tube inlet temperature increases versus

catalyst life to be balanced with higher hydrogen permeation

flux. Following figure shows methanol mole fractions with

optimal membrane inlet temperature. Fig. 11 contrasts the

methanol mole fraction in membrane reactor during the time

at optimal and non-optimized inlet tube temperatures. This

0 200 400 600 800 1000 12000.068

0.07

0.072

0.074

0.076

0.078

0.08

0.082

0.084

Time (day)

Meth

no

l m

ole fractio

n

Without optimizationWith optimized tube inlettemperature

Fig. 11 – Methanol mole fraction at optimal tube inlet

temperatures.

figure proves the higher production rate in optimized

membrane system during the time.

6.4. Optimization of the cooling water inlet temperature

The results of optimal steam drum temperature trajectory at

three periods of catalyst life are shown in Fig. 12. The optimal

steam drum temperature was enhances as time goes on to be

balanced with higher hydrogen permeation flux. Also Fig. 13

explains methanol mole fractions with optimal steam drum

temperature. This figure compares the methanol mole frac-

tion in membrane reactor during the time at optimal and non-

optimized steam drum temperatures. This figure shows the

higher production rate in optimized membrane system during

the time.

0 200 400 600 800 1000 12000.068

0.07

0.072

0.074

0.076

0.078

0.08

0.082

Time (day)

Meth

an

ol m

ole fractio

n

Without optimizationWith optimized steam drumtemperature

Fig. 13 – Methanol mole fraction at optimal steam drum

temperatures.

0.4 0.5 0.6 0.7 0.8 0.9 1460

470

480

490

500

510

520

530

540

550

Catalyst activity

Tem

peratu

re (K

)

Steam drumInlet membrane tube

Fig. 14 – Optimized temperature of inlet membrane tube

and steam drum as a function of activity.

75 80 85 90 95 100330

335

340

345

350

355

360

365

370

375

380

Pressure (bar)

Meth

an

ol p

ro

du

ctio

n rate (to

ne/d

ay)

Fig. 16 – The effects of increased membrane inlet pressure

on methanol production at steady-state condition.


6.5. Complete optimization

In this approach, the optimal temperatures of both inlet

membrane tube and steam drum are determined. The objec-

tive function was to maximize the methanol production rate.

The values of these parameters should determine the best

500505510515520525530535

Catalyst b

ed

tem

peratu

re (K

)

0 200 400600 800 1000

1200

0 200 400600 800 10001200

02

467.022

02

46

7.022

0

0.02

0.04

0.06

0.08

0.1

0.12

Time (d

ay)

a

c

Reactor length (m

)

Time (d

ay)

Reactor length (m

)

Meth

an

ol m

ole fractio

n

Fig. 15 – Optimized plots of bed dynamics for (a) methanol mole

of reactor bed, and (d) catalyst activity.

boundaries between the kinetic and thermodynamic regions.

This optimization step enhanced 40% additional yield for final

product. These values are shown by Fig. 14.

Fig. 15 shows the optimal methanol mole fraction

surface, temperatures surfaces for membrane tube and

480

490

500

510

520

530

540

Mem

bran

e tu

be

tem

peratu

re (K

)

Catalyst activity

0 200 400 600 800 10001200

0 200 400 600 800 10001200

0.40.50.60.70.80.9

1

02

467.022

02

467.022

b

d

Time (d

ay)

Reactor length (m

)

Time (d

ay)

Reactor length (m

)

fraction, (b) temperature of membrane tube, (c) temperature

0 1 2 3 4 5 6 70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Length (m)

Meth

an

ol m

ole fractio

n

0 1 2 3 4 5 6 70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Length (m)

Meth

an

ol m

ole fractio

n

Membrane ReactorTypical Reactor

a b

Membrane ReactorTypical Reactor

Fig. 17 – Comparison of methanol mole fraction in typical and membrane loops for similar fresh feed condition, (a) at 1st

day, and (b) at 1200th day.


catalyst bed, and also 3D plot of catalyst activity,

respectively.

Fig. 15(a) depicts that methanol production increases along

the reactor and decreases during the time. Also it can be seen,

declining total conversion in the reactor arises from deacti-

vation of the catalyst as time goes on and also decreasing

methanol production is sharper at the commencing operation

time. It can be understood from Fig. 15(b) which the optimal

temperature of tube side increases along reactor length and

enhances as time goes on. Fig. 15(c) also shows the 3D profile

of temperature of reaction side. This figure shows the

temperature sharply increases at the initial of reactor and

then it get smooth and slowly increases during the time.

Finally, Fig. 15(d) shows catalyst activity decreases during the

time and reactor length as well.

6.6. Extra works

In other optimization procedure, we have attempted to find

optimum membrane inlet pressure to obtain maximum

final product. The results illustrate that the increased

membrane inlet pressure able to increase methanol

production but it doesn’t have optimum value. Fig. 16

shows the pressure increases from 75 to 100 bar to restore

methanol production.

On the other side, Rahimpour et al. [18] shows for single

methanol reactor after reactions go to completion quickly,

reaction rates decrease till reach to equilibrium. Therefore

reactor length should be divided into two sections. In the

first section, reactions are carried out so quickly that

reaction is controlling while in the second section, reaction

rates decrease and equilibrium is controlling. The main

duty of membrane is reducing length of equilibrium

controlling section that causes extension of reaction

controlling side. In fact, the membrane just effects on the

length of equilibrium controlling section by removal of

specific component and shifts reactions to the right hand

side and does not effect directly on reaction controlling

section. Therefore, to optimize length of membrane in loop

study, length of both reaction and equilibrium controlling

sections kept variables which effective membrane length

obtain. The results show that for membrane loop reactor,

membrane length doesn’t have optimum value because

reactor feed composition varied with typical loop model.

Fig. 17 shows the comparison of methanol production for

typical and membrane loop in the beginning and at the end

of catalyst lifetime. Fig. 17(a) shows the difference of

methanol mole fraction at membrane reactor and conven-

tional reactor enhances along the reactor. Fig. 17(b) depicts

the difference of methanol mole fraction at membrane

reactor and conventional reactor at the end of catalyst

lifetime is more than the difference for fresh catalyst.

7. Conclusion

One potentially interesting idea for industrial methanol

synthesis is using optimal membrane reactor. Performance of

methanol reactor system improved when a membrane was

used in a conventional-type methanol reactor. The results

demonstrate that after 600th day of operation, methanol

production increases and after 1500th day of operation

methanol production declines. Also in this study, a membrane

single-type methanol synthesis loop was optimized dynami-

cally; a mathematical heterogeneous model was used in

optimization and the optimal temperatures for a membrane

methanol synthesis loop were obtained by the use of defer-

ential evolution algorithms (DEA), as powerful optimization

techniques which give good solutions for these constrained

non-linear problems. Overall methanol production

throughout 4 years of catalyst life was considered as optimi-

zation criterion to be maximized. The optimization included

three approaches: in the first approach, optimal permeation

side (tube side) temperature was determined. For second step

of optimization, optimal steam drum (cooling water)

temperature was computed. Finally, optimal temperatures of

both steam drum and cooling gas were calculated. Conse-

quently, this new design leads us to optimal operation policy

and yields 40% additional methanol production during oper-

ating period.


Appendix A. Heat capacity constants

Heat capacity constants

Table A.1.1. Heat capacity constants [46].Comp A B� 103 C� 106 D� 10�5

CH3OH
3.249 0.422 0.0 0.083
CO2
3.376 0.557 0.0 �0.031
CO
3.457 1.045 0.0 �1.157
H2O
1.072 9.081 �2.164 0.0
H2
3.280 0.593 0.0 0.040
N2
3.211 12.216 �3.45 0.0
CH4
3.470 1.450 0.0 0.121
Reaction kinetics [36]

k1KCO

24fCOf1:5

H2�fCH3OH�

f0:5KP1

�35

R1 ¼H2�

1þ KCOfCO þ KCO2fCO2

�f 0:5H2þ�

KH2O

K0:5H2

�fH2O

� (A.2.1)

R2 ¼

k2KCO2

24fCO2

f1:5H2�fCH3OHfH2O�

f1:5H2

KP2

�35

�1þ KCOfCO þ KCO2

fCO2

�f 0:5H2þ�

KH2O

K0:5H2

�fH2O

� (A.2.2)

R3 ¼k3KCO2

fCO2

fH2�fH2O

fCOKP3

��

1þ KCOfCO þ KCO2fCO2

�f 0:5H2þ�

KH2O

K0:5H2

�fH2O

� (A.2.3)

These expressions have been selected from [36] were pre-

sented for low-pressure methanol synthesis up to 50 bar and it

is suitable model for methanol synthesis. Methanol synthesis

in 75 bar is also in low-pressure process category. In fact, in

the previous research [47], the authors have shown that

partial pressure can be used instead of partial fugacity in these

conditions, because all fugacity coefficient approach to unity.

This result shows that pressure sensitively of Graaf’s equa-

tions can be expanded to 75 bar.

Reaction constants [37]

Table A.2.1. Reaction rate constants.k ¼ A expðB=RTÞ 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� 11,800
Table A.2.2. Adsorption equilibrium constants.k ¼ A expðB=RTÞ A B

KCO
(2.16� 0.44)� 10�5 46,800� 800
KCO2
(7.05� 1.39)� 10�7 61,700� 800
ðKH2O=K0:5H2Þ
(6.37� 2.88)� 10�9 84,000� 1400
Table A.2.3. Reaction equilibrium constants.KP ¼ 10ðA=T�BÞ A B

KP1
5139 12.621
KP2
3066 10.592
KP3
�2073 �2.029
Binary interaction relations constants [40]

For N2 – other component:

d0 ¼ 0:1751787� 0:7043 Log�uj

�� 0:862066

�LogðujÞ

2

d1 ¼ �0:584474þ 1:328 Log�uj

�þ 2:035767

�LogðujÞ

2

d2 ¼ 2:257079þ 7:869765 Log�uj

�þ 13:50466

�LogðujÞ

2þ8:3864�LogðujÞ

3For CH4 – other component:

d0 ¼ �0:01664� 0:37283 Log�uj

�þ 1:31757

�LogðujÞ

2

d1 ¼ 0:48147þ 3:35342 Log�uj

�� 1:0783

�LogðujÞ

2

d2 ¼ �0:4114� 3:5072 Log�uj

�� 0:78798

�LogðujÞ

2For CO2 – other component:

d0 ¼ 0:4025636þ 0:1748927 Log�uj

�

d1 ¼ �0:94812� 0:6009864 Log�uj

�

d2 ¼ 0:741843368þ 0:441775 Log�uj

�

Auxiliary correlations

To complete the simulation, auxiliary correlations should be

added to the model. In the case of homogeneous model, the

only main concern is estimation of overall heat transfer

coefficients between coolant – reaction bed sides and reaction

bed – membrane tube sides.

Coolant – reaction bed sides heat transfer coefficientThe overall heat transfer coefficient between the circulating

boiling water of the shell side and the gas phase in the reac-

tion side is given by the following correlation.

1Ushell

¼ 1hReaction bed

þAi ln

�do

di

�

2pLKW

þ Ai

Ao

1hcoolant

(A.4.1)

where the heat transfer coefficient between the gas phase in

the reaction side and reactor wall is obtained by the following

correlation [48].

hReaction bed ¼�

0:4583B

��

dpur

m3B

��0:407

��

Cpru3B

�� Prð�2

3 Þ (A.4.2)

In the above equation, u is superficial velocity of gas and the

other parameters are those of bulk gas phase and dp is the

equivalent catalyst diameter, K is thermal conductivity of gas,

r, m are density and viscosity of gas, respectively, and eB is void

fraction of reaction bed.

In Eq. (A.4.1), heat transfer coefficient of boiling water in

the coolant side which is estimated by the following equation

[49].

hcoolant ¼ 7:96� ðTsat � TsÞ3��

PPa

�0:4

(A.4.3)


T and P are temperature and pressure of boiling water in the

shell side, Tsat is the saturated temperature of boiling water at

the operating pressure of shell side and Pa is the atmospheric

pressure. The last term of the above equation has been

considered due to effect of pressure on the boiling heat

transfer coefficient.

Reaction bed – membrane tube sides heat transfer coefficientThe overall heat transfer coefficient between the gas phase in

the membrane tube and the gas phase in the reaction side is

given by the following correlation.

1Utube

¼ 1hmembrane tube

þAi ln

�do

di

�

2pLKmembrane

þ Ai

Ao

1hReaction bed

(A.4.4)

where the heat transfer coefficient between the gas phase in

the tube side and membrane wall is obtained by the following

correlation [49].

hmembrane tube ¼�

Kgas

di

�� 0:012�

�Re0:87 � 280

�� Pr0:4 (A.4.5)

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