Ganesh Thesis

7/31/2019 Ganesh Thesis

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1. Introduction

Phthalic anhydride (PA) is used widely in industry for the production of a

large variety of products. It is used as chemical intermediates in the production of

plastics from vinyl chloride, and for the production of polyester resins, plasticizers,

dyes, insect repellants, etc. PA is produced by the gas-phase catalytic oxidation of o-

xylene or naphthalene, or a mixture of both. Mixed oxide catalysts, mainly consisting

of V2O5 and TiO2 and packed in several zones in a multi-tubular reactor, are used for

oxidation. The gas leaving the reactor is cooled in one set of switch condensers. The

PA solidifies on the finned tubes. The solidified PA is melted and collected from

these during the heating cycle, during which the exit gas from the reactor is sent to the

other set of cooled switch condensers. The treated gas is then either scrubbed with

water, or catalytically or thermally incinerated (Ullmann, 1992).

The oxidation of o-xylene to PA has been studied by several workers

(Wainwright and Hoffman, 1977; Chandrasekharan and Calderbank, 1979; Skrzypek

et al., 1985) over the years. The earlier workers used the redox model to explain the

oxidation process. Studies conducted by Wainwright and Hoffman (1977) showed

that the redox model could not explain fully all the available experimental data. For

example, it failed to explain the maximum observed by Calderbank (1977) in the rate

of disappearance of o-xylene with the increase in the feed concentration of o-xylene.

Skrzypek et al. (1985) used the Langmuir-Hinshelwood-Hougen-Watson (LHHW)

model to overcome this limitation. In the present study, the latter is used for modeling

PA reactors, which are then optimized using multiple objectives.

Evolutionary techniques, such as genetic algorithm (GA), have been used

successfully to solve single- and multi-objective optimization problems for over a

decade. GA generates a pool of solutions and with the help of probabilistic operators,


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reproduction, crossover and mutation, finds the global optima. Several workers have

developed different adaptations of GA to improve its computational efficiency,

enabling its use for solving complex optimization problems. Deb (2001) and co-

workers developed two versions of the non-dominated sorting genetic algorithm,

NSGA-I and the elitist NSGA-II, to solve problems with two or more objective

functions. A set of non-dominated (equally good) optimal solutions, referred to as

Pareto sets, with a good spread of solutions, is generated using these techniques.

Kasat and Gupta (2003) incorporated the concept of the variable-length jumping gene

(JG; McKlintock, 1987) in the binary-coded NSGA-II and found an eight-fold

reduction in the computational time required for convergence for a computationally

intensive optimization problem, namely, the multi-objective optimization of an

industrial fluidized-bed catalytic cracking unit (FCCU). Bhat (2004) found that the

fixed-length JG adaptation, NSGA-II-aJG, of the binary-coded NSGA-II led to even

faster convergence of the results. Several parallel studies (Simoes and Costa, 1999a,

1999b; Man et al., 2004; Chan et al., 2005a, 2005b; Ripon et al., 2007; Guria et al.,

2005; Bhat, 2007; Agarwal and Gupta, 2007a, 2007b) also indicate efficacies of other

JG adaptations of both the binary-coded as well as the real-coded NSGA-II.

In the present study, multi-objective optimization of two industrial PA reactors

is carried out using NSGA-II-aJG. The second of these problems is particularly

complex, computationally, and a guidedNSGA-II-aJG had to be developed to obtain

optimal solutions.


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2. Formulation

2.1. Reaction Scheme and Mathematical Model

Figure 1 shows the reaction network suggested by Skrzypek et al. (1985). The

reaction model developed by these workers is based on the LHHW kinetics. The six

species, o-xylene (OX), o-tolualdehyde (OT), phthalide (P), phthalic anhydride (PA),

maleic anhydride (MA) and oxygen (O2), are assumed to be adsorbed on the active

sites of the catalyst, while H2O, N2, and the combustion products, COx (a mixture of

CO and CO2), are assumed not to be adsorbed. The surface reaction between the

appropriate species is assumed to be the rate limiting step. The LHHW rate

expressions for this scheme are given in Appendix 1 (Eqs. A1, A2). The values of the

kinetic parameters and the heats of reaction are given in Table 1, while the parameters

characterizing the adsorption are given in Table 2.

The catalyst considered by Skrzypek et al.(1985) was V2O5TiO2 dispersed

on a non-porous carborundum support. The steady state mass and heat balance

equations involving the (external) catalyst surface and the bulk gas are given in

Appendix 1 (Eqs. A3 A11). The temperature of the entire catalyst particle is

assumed to be uniform since the Prater No. is usually low (Elnashaie and Elshishini,

1993). The following assumptions have been used to develop the model for the

kinetics:

a) The reactor operates under steady state conditions;

b) Radial dispersion of mass and heat are negligible;

c) Axial dispersion of mass and heat are negligible;

d) The gas mixture follows the ideal gas law.

The derivations of all the equations are given in Appendix 2.


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Figure 1: Reaction scheme proposed by Skrzypek et al. (1985). Reaction numbers

indicated on the arrows

(MA)

(PA)(P)(OT)(OX)

Phthalic Anhydrideo-Xylene o-Tolualdehyde Phthalide1 4 5

6

7

Maleic Anhydride COx

23 8


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Table 1: Kinetic parameters and heats of reaction (Elanshaie and Elshishini, 1993) for

the scheme of Skrzypek et al. (1985)

Reaction, i

Ei

kJ mol-1

10-15

kio

mol (m3 bed) -1

h-1

(MPa)-2

(-Hi)

kJ mol-1

1 108.36 130 456.05

2 96.65 1.7 1503.7

3 85.35 0.7 3998.8

4 85.35 3.8 414.2

5 93.72 18.0 368.19

6 108.78 84.0 1238.5

7 96.23 28.0 782.4

8 117.72 0.00699 2760.0


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Table 2: Parameters (Elnashaie and Elshishini, 1993) characterizing the adsorption for

the kinetic scheme of Skrzypek et al. (1985)

Component, j( )adsjH

kJ mol-1

Kio

(MPa)-1

o-xylene (OX) 27.70 0.987

o-tolualdehyde (T) 30.96 1.905

maleic anhydride (MA) 24.27 5.724

phthalide (P) 16.74 3.760

phthalic anhydride (PA) 24.31 3.819

oxygen (O2) 0.00 0.199


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The properties of the individual components, like the viscosity, thermal

conductivity and the specific heat of the vapor phase are estimated using the equations

given in Daubert and Danner (1989). These properties are estimated using the group

contribution method (Poling et al., 2001) for OT and P. The binary diffusion

coefficients are computed using the Wilke and Lee equation (Poling et al., 2001). The

effective diffusion coefficients for any component, i, in the gas mixture are computed

using the Wilke equation (Elnashaie and Elshishini, 1993). The mass transfer

coefficient, kg,i, of component, i, through the gas film surrounding the catalyst

particle, and the film heat transfer coefficient, h, are computed using the equations

proposed by Wakao and Kaguei (1982) and Wakao et al. (1978), respectively. The

properties of the gas mixture, e.g., viscosity (Wilke equation inPoling et al., 2001),

thermal conductivity (Wassiljewa equation in Poling et al., 2001), and the specific

heat, are computed using the component values and the compositions. The detailed

correlations used can be supplied on request.

Eqs. A1A23 in the Appendix comprise a complete set of coupled equations.

Some of these (for the gas phase) are ordinary differential equations of the initial

value kind (ODE-IVP), while those for the catalyst particles are non-linear algebraic

equations. These need to be solved simultaneously. The concentrations of all the

species and the temperature are known at z = 0. These are used to compute the

concentrations and the temperature at the catalyst surface at that axial location, using

the modified Powell hybrid algorithm (program DNEQNF in the IMSL library). The

effectiveness factors of all the reactions are assumed to be unity at this stage (this is

required since several concentrations are zero at the entry point), and the ODE-IVPs

are integrated using Gears method (code DIVPAG in the IMSL library). This gives

the concentrations and the temperature at location, dz. The gas phase values at dz (or


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any general zi) are used to compute the concentrations and temperature at the solid

surface at that axial location, the effectiveness factors of the reactions are then

evaluated and the ODE-IVPs further integrated for a small interval, dz. This is

continued till the end of the reactor.

2.2. Multi-objective Optimization

Two industrial reactor systems are optimized in this study. These problems are

now described.

2.2.1. Reactor System-1

Reactor System-1 has been suggested by Sato et al. (1981). In this system, o-

xylene is oxidized using air. The reactor operates at atmospheric pressure, and

consists of several identical parallel tubes. Some amount of recycled process gas and

steam are also mixed with the (original) feed so as to reduce the concentration of

oxygen in the feed to the reactor to about 10 % volume by volume. This ensures that

the hot spot is within the safe limit of about 510oC (Reuter et al., 2004; the auto-

ignition temperature of PA is 580oC; Ullmann, 1992). A severe decrease in the PA

yield (due to formation of COx) and in the operating life of the catalyst also result

above about 500oC (Reuter et al., 2004). The use of recycle and steam permits the use

of as high a concentration of the original feed as 85 g o-xylene/m3

air at NTP. The

exothermic heat of reaction is removed by a eutectic mixture of molten sodium and

potassium nitrates and nitrites (Kirk-Othmer, 1982) flowing co-currently outside the

tubes (referred to as the shell side). Because of the symmetry involved, a single tube

is studied.

Sato et al. (1981) reported the use of several zones (layers) of the catalyst, in-

between which there is no catalyst and the process gas just exchanges heat with the


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jacket fluid and cools down. One of these configurations, using two layers of lengths,

L1 and L2 (see Figure 2a), has been used in the present study. This configuration has

been selected since the high activity catalyst of Skrzypek et al. (1985) is being used.

Details of the reactor are given in Table 3.

For the optimization of Reactor System-1, we assume a fixed value of the

mass flux, G, inside the tube. One important objective function is the total amount of

catalyst used. The yield, XPA, of PA is a natural choice of the second objective. Since

the latter is related to the amount of steam produced, no additional objective on the

steam production need be considered. The decision variables are taken as

u [Rgas, H2Oin, TF,in, Tc,in, m, Dp, L1, L2]T (1)

In Eq. 1, Rgas and H2Oin are the volumes of recycle gas (assumed to be pure nitrogen

after post-reactor treatment) and steam added (at actual conditions) per m3

of air at

NTP in the incoming mixture (called original feed) of o-xylene and air, TF,in and Tc,in

are the inlet temperatures of theprocess gas (after mixing of the original feed with the

recycle stream and steam) and coolant, m is the kg of coolant per kg of the process

gas, and Dp is the diameter of the catalyst particles.

A reasonable two-objective optimization problem for Reactor System-1 is

written mathematically as:

Max I1(u) XPA (2a)

Min I2(u) L1 + L2 Lcat (2b)

subject to:

constraints:

Tmax = 510oC (2c)

Lcat 3 m (2d)


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Figure 2: Reactor Systems - (a) 1 and (b) 2

S4

S3

S2

S1

L1

L2

L3

L4

L5

L9

L1

L2

Coolant

(a) (b)


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Table 3: Details of the Reactor Systems 1 and 2

Parameter Value Reference

(a) Reactor System1

Tube length, L 3 m Sato et al. (1981)

Tube diameter, D 20 mm Sato et al. (1981)

OX concentration in the

(original) feed

85 g (m3

air at

NTP)-1

Sato et al. (1981)

Mass flux, G 19,455 kg m-2

h-1

-

(b) Reactor System2

Tube length, L 3.5 m -

Tube diameter, D 25 mm -

Size of the catalyst particle, Dp 3 mm -


h-1

-

(c) Details of the Catalyst (both systems)

Surface area, as 1000 m2

kg-1

Skrzypek et al. (1985)

Shape Spherical -

Bulk density, b 980 kg m-3

Skrzypek et al. (1985)

Void fraction of bed, b 0.42 Leva (1959)

(d) Heat transfer coefficient

Overall heat transfer coefficient, U 110 W m-2

K-1

Elnashaie and

Elshishini (1993)


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Bounds on the decision variables:

1 Rgas 3.6 m3/(m

3of air at NTP) (2e)

0 H2Oin 1.8 m3/(m

3of air at NTP) (2f)

147o

C TF,in 247o

C (2g)

347oC Tc,in 447

oC (2h)

1 m 25 kg coolant/kg process gas (2i)

0.002 Dp 0.008 m (2j)

0.01 L1 1.5 m (2k)

0.4 L2 2.4 m (2l)

The lower limit on the feed input temperature is set as 147oC because the boiling

point of o-xylene is 144oC. According to the Coastal Chemical Company

(http://www.coastalchem.com), heat transfer salts can be used for a wide range of

temperatures. The eutectic mixture of sodium and potassium nitrates and nitrites has a

low melting point of 142oC and can be used up to temperatures as high as 538

oC.

With plain carbon steel, the heat transfer salts can be operated up to 454oC

(http://www.coastalchem.com). Hence the upper bound on the coolant temperature is

set as 447oC (the coolant temperatures are observed to lie below 454

oC with this

choice).

The constraints in Eq. 2 are taken care of by the use of penalty functions. We

do not wish to have solutions with yields of PA below 1.1, since these indicate

incomplete oxidation of o-xylene. Similarly, we need to use penalties if the randomly

generated values of L1 and L2 lead to a violation of Eq. 2d, or if the maximum

temperature of the process gas exceeds 510oC. The modified problem can be rewritten

as:


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Max I1(u) = XPA +

22

catPA1 2 3

LXw 1 w 1 w

1.2 3.0

+ +

(3a)

Min I2(u) = Lcat +

22

catPA1 2 3

LXw 1 w 1 w

1.2 3.0

+ +

(3b)

subject to:

Eqs. 2e 2l (3c)

The weightages, w1 w3, are assigned as follows:

If XPA 1.1, w1 = -1000, else w1 = 0 (4a)

If Lcat 3m, w2 = 0, else w2 = -1000 (4b)

If Tmax

510oC, w

3= 0, else w

3= -1000 (4c)

This problem can now be solved for Reactor System-1.


The second industrial reactor (similar to a unit in India) involves nine

layers/zones of catalyst (Fig. 2b), and there is no mixing of recycle gas or steam at the

feed end. The catalyst of Skrzypek et al. (1985) is used. The diameter of the catalyst

particle is assumed constant at 3 mm, a typical industrial value. Details of this reactor

configuration are given in Table 3.

A meaningful two-objective optimization problem for this system can be

written as:

Max I1(u) = XPA (5a)

Min I2(u) = Lcat9

i

i 1

L=

(5b)

subject to:

constraints:

Tmax = 510oC (5c)


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Total length of reactor tube = 3.5 m (5d)

Li 0.01 m; i = 1, 2, . . . , 9 (5e)

Bounds on decision variables:

65 cin 85 g OX/m3 of air at NTP (5f)

147 oC TF,in 297oC (5g)

337 oC Tc,in 447oC (5h)

1 m 25 kg coolant/kg process gas (5i)

0.2 Si 0.45 m; i = 1, 2, . . ., 7 (5j)

0.1 S8 0.45 m (5k)

none on Li (5l)

The upper bound on the feed temperature has been increased from 247oC for

the first reactor system to 297oC since several optimal solutions for the second reactor

system occurred at the upper bound of 247oC. An extra constraint has been put on the

minimum length of each catalyst zone. Again, in order to ensure near-complete

conversion of o-xylene, and to take care of the constraints, the objective functions are

reformulated using penalty functions as:

Max I1(u) = XPA +

22

catPA1 2 3

LXw 1 w 1 w

1.2 3.6

+ +

(6a)

Min I2(u) = Lcat +

22

catPA1 2 3

LXw 1 w 1 w

1.2 3.6

+ +

(6b)

subject to:

8 8

9 i i

i 1 i

L 3.5 L S 0=

=

(6c)

Eqs. 5e 5l (6d)

The weightages in the penalty terms are given by

If XPA 1.1, w1 = -500; else w1 = 0 (7a)


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If L9 0 m, w2 = -3000; else w2 = 0 (7b)

If Tmax 510oC in bed i; i = 1, 2, . . ., 9; w3 = 0; else w3 = -3000 + 250(i-1)

(7c)

If Li 0.01 m; i = 1, 2, . . ., 9; w3 = 0; else w3 = -300 (7d)

The decision variables for this problem would be

u = [cin, TF,in, Tc,in, m , L1, L2, . . . , L8, S1, S2, , S8]T (8)

In Eq. 8, cin is the mass of OX per m3

of air at NTP, and S1 S8 are the spacing

between the catalyst layers (Fig. 2b). L9 is computed from the total length of the

catalyst tube.

We found that the code for the standard NSGA-II-aJG did not converge to

optimal (Pareto) solutions for this problem. This was because of the extreme

sensitivity of the gas phase temperature to values of L i. A guided NSGA-II-aJG

procedure had to be developed to avoid this problem. This is now described.

The number of decision variables for this problem is reduced from twenty in

Eq. 8 to the following twelve:

u = [cin, TF,in, Tc,in, m , S1, S2, , S8]T (9)

Upper estimates, L1,estimate, L2,estimate, . . . , L8,estimate, of the lengths of the first

eight catalyst zones are provided to the code. After the mapping of any chromosome,

the code calls the subroutine for reactor simulation, which integrates the model

equations. The integration for any catalyst zone, i, is stopped as soon as the

temperature goes past the limiting value of 510oC. This gives improved values, Li, of

the lengths of the catalyst zones (the values of Si to be used for this chromosome are

the same as generated in the NSGA-II-aJG code). It is to be noted that the Gear

subroutine used comes out of the integration after every specified value ofzi to store

the results, and so these improved values of Li would violate the temperature


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constraint. The correct estimates of Li to be used for final simulation should, thus, be

Li - zi (L9 is computed as the balance of the total length of the reactor tube). These,

values, along with the twelve values of the decision variables (Eq. 9) are again sent to

the subroutine for reactor simulation, and the values of the modified objective

functions in Eq. 6 obtained. The guided NSGA-II-aJG works well and gives us the

desired Pareto solutions rapidly, with values of XPA of around 1.1 kg PA produced/kg

OX consumed. The above procedure helps search for solutions in the decision-

variable space where the values of XPA are high while keeping the temperature of the

gas phase within its upper bound. These solutions are missed by the original

algorithm, at least for the several attempts made us. The reason for the success of the

guided algorithm is that XPA increases with the gas phase temperature, and we need to

have the highest possible temperature while having intermediate cooling.

This problem can now be solved for Reactor System-2.


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3. Results and Discussions

3.1. Model Development

The model is first checked for its validity. Elnashaie and Elshishini (1993)

have provided plots of the gas phase temperature and the conversion of o-xylene as a

function of the axial location in a reactor tube, using the kinetic expressions

developed by Skrzypek et al.(1985) It was observed that three additional model

parameters were needed to obtain good agreement between model predictions and the

plots of Elnashaie and Elshishini (1993). These are multiplying factors, TP1 TP3, for

the viscosity, specific heat and the thermal conductivity of the gas mixture:

mix = mix, correl TP1 (9a)

Cp,mix = Cp,mix,correl TP2 (9b)

mix = mix,correl TP3 (9c)

In Eq. 9, subscripts mix and mix,correl indicate mixture properties to be used, and

values estimated by the correlations (as described earlier). SGA-II-aJG has been used

to obtain the three multiplying parameters that minimize the sum of square errors

between one set of results (for the feed mole fraction of OX of 0.0092) of Elnashaie

and Elshishini (1993) and model predicted values. Figure 3 shows the agreement of

the tuned model with this set of data to be quite good. The tuned model was also

found to predict results for two othervalues of the mole fraction of o-xylene in the

feed (0.0042 and 0.0027), even though the latter were not used for tuning (see Fig. 3).

This lends credence to the fact that the model is trustworthy. The tuned values of the

three multiplying factors are given in Table 4.


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Axial position (m)

0.0 0.1 0.2 0.3 0.4

Conversionofo-xylene

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Axial position (m)

0.0 0.1 0.2 0.3 0.4

Gastemp

erature(oC)

355

360

365

370

375

380

Figure 3: Results of model tuning. Results of Elnashaie and Elshishini (1993) for

input mole fractions of OX of: 0.0092,+:0.0042and :0.0027, respectively;

: Results using the model tuned on data for 0.0092


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Table 4: Details for tuning the model

Parameter Value

(a) Reactor details (Elnashaie and Elshishini, 1993)

Tube diameter 0.026 m

Tube length 0.365 m

mole fractions of o-xylene in feed 0.0092, 0.0042, 0.0027


h-1

Feed/(constant) coolant temperature 360oC/360

oC

Catalyst Skrzypek et al.(1985)

(b) Values of the tuned multiplying factors

TP1 0.986

TP2 0.982

TP3 0.984


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Table 5: Computational parameters used in NSGA-II-aJG and guided NSGA-II-aJG

for Reactor Systems-1 and 2, respectively

Parameter

Reactor

System-1

Reactor

System-2

Number of decision variables 8 12

Population size 100 100

Chromosome length 160 240

Cross-over probability 0.95 0.95

Mutation probability 0.048 0.025

Jumping gene probability 0.5 0.5

Fixed length for jumping gene 5 4

L1,estimate - 1 m

L2,estimate L8,estimate - 0.2 m

zi - Li,estimate/2000


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Figure 4(A) (Fig. 4 continued)

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Totalcatalystlength(m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Recyclegas(m3/m3a

iratNTP)

1.5

2.0

2.5

3.0

3.5

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Steaminput(m3/m3a

iratNTP)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

(a)

(b)

(c)

(d)Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Gasinputtemperature(oC)

160

180

200

220

240

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Coolantflow

rate(kg/kgfeed)

5

10

15

20

25

(e)

(f)

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Coolantinputtemperature(

oC)

360

380

400

420

440

B

A

C

A

C


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Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Catalystdiameter(m)

0.002

0.004

0.006

0.008

(h)

(g)

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

L1

(m)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(i)

Yield of PA

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

L2

(m)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Figure 4(B)

Figure 4: Pareto optimal set and decision variables for Reactor System-1


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Interestingly, the optimal value of L1 remains almost unchanged (Fig. 4h), and the

increase in XPA is associated almost solely with the increase of L2.

In Fig. 4a, point B corresponds to XPA of 1.151 while point A, to an XPA of

1.173. The values of the decision variables for these two points are given in Table 6.

Figure 5 shows the profiles of the temperature, OX conversion and the various mole

fractions for these two points. It is interesting to note that an intermediate zone having

no catalyst is required in these reactors for cooling of the gas, so that its temperature

does not go above the limiting value of 510oC. A decision maker can use these results

to decide the preferred solution/operating point, using his or her intuition.

It is interesting to compare the results for two adjacent points, A and C, in

Figs. 4a and d. These points are associated with almost the same values of XPA and

Lcat. The values of the decision variables for these points are given in Table 6. We

observe that the values of the feed temperature, coolant flow rate and the steam input

(and so the concentration of oxygen) are quite different for these points. The relative

insensitivity of XPA and Lcat to these decision variables is responsible for the scatter.

More meaningful and useful results without such scatter can be generated by

parameterizing the decision variables, as done by Sareen and Gupta (1995) for an

industrial nylon-6 reactor.


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Table 6: Values of the decision variables corresponding to points A, B and C in Figs.

4a and d

Value/Decision Variable Point A Point B Point C

XPA 1.151 1.173 1.152

Recycle gas [m3/(m3 air at NTP)] 3.134 3.183 3.051

Steam input [m3/(m

3air at NTP)] 0.974 1.014 0.687

Feed/coolant temperature (oC)

224.51/

445.21

223.57/

446.31

201.41/

445.22

Coolant flow rate (kg/kg process

gas)

4.2 4.879 4.67

Catalyst diameter (m) 0.0075 0.0075 0.0076

Catalyst lengths, L1/L2 (m) 0.57/1.001 0.571/1.937 0.571/0.989


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26

(a)Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Temperature(oC)

200

250

300

350

400

450

500

550

Gas

Coolant

(b)Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Temperature(oC)

200

250

300

350

400

450

500

550

Gas

Coolant

Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5


0.0

0.2

0.4

0.6

0.8

1.0

(c)Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5


0.0

0.2

0.4

0.6

0.8

1.0

(d)

Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Molefraction

0.000

0.001

0.002

0.003

0.004

0.005

0.006

(e) (f)Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Molefraction

0.000

0.001

0.002

0.003

0.004

0.005

0.006

OX

COx

PA

OX

PA

COx



27/58

27

Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mole

fraction

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

(g)Axial position (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Mole

fraction

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

(h)

OT

MA

P

OT

MA

P

Figure 5(B)

Figure 5: Profiles corresponding to point A (a, c, e and g) and point B (b, d, f and h),

in Figure 4a


28/58

28


The computational parameters required in the guided NSGA-II-aJG for this

problem are first determined by trial-and-error so as to give the best solutions. The

final values of these parameters are given in Table 5. Fig. 6 shows how the optimal

solutions evolve over the generations for Reactor System-2. Converged Pareto

solutions are observed at the end of about 45 generations. A very sharp increase in Lcat

is observed in the converged Pareto set after XPA of about 1.166. A decision maker

may like to select this point as the preferred solution (operating point) since the

catalyst length almost doubles beyond this point to give a further increase of XPA from

about 1.166 to 1.1708.

Fig. 7a shows the Pareto-optimal solution for Reactor System-2. One of the

objective functions improves at the cost of the other along the Pareto front. Figs. 7b

m show the associated decision variables. Figs. 7n v show the lengths of the nine

catalytic zones. It is observed that even though there is a considerable amount of

scatter in Figs. 7b u, the increase in XPA in Fig. 7a depends mainly on L9. A similar

inference was deduced for Reactor System-1 too, where the increase of the length of

the last zone of the catalyst was associated with an increase of XPA. The values of the

decision variables and the lengths of the catalyst zones are given in Table 7 for points

A and B of Fig. 7a.

Figure 8 shows the profiles of the temperature, conversion and the mole

fractions of the several species corresponding to these two points. It is interesting to

note that the longer ninth catalyst zone (Fig. 7v) leads to a higher increase in the yield

of PA for point B (in Fig. 7a)


29/58

29

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

Gen = 1 Gen = 2

Gen = 3 Gen = 4

Gen = 5

Gen = 10

Figure 6(A) (Figure 6 continued)


30/58

30

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Gen = 15 Gen = 20

Gen = 25 Gen = 30

Gen = 35 Gen = 40

Figure 6(B) (Figure 6 continued)


31/58

31

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

Gen = 45

Figure 6(C)

Figure 6: Evolution of the Pareto set over the generations for Reactor System-2


32/58

32


Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcatalys

tlength(m)

0.4

0.6

0.8

1.0

1.2

Yield of PA

1.10 1.12 1.14 1.16 1.18

O-xylenefeedconcentration

(gOX/m3a

iratNTP)

65

70

75

80

85

Yield of PA

1.10 1.12 1.14 1.16 1.18

Gasinputtemperature(oC)

160

180

200

220

240

260

280

Yield of PA

1.10 1.12 1.14 1.16 1.18

Coolantinputte

mperature(oC)

340

360

380

400

420

440

Yield of PA

1.10 1.12 1.14 1.16 1.18

Coolantflow

rate(kg/kgprocessgas)

5

10

15

20

25

Yield of PA

1.10 1.12 1.14 1.16 1.18

S1

(m)

0.20

0.25

0.30

0.35

0.40

0.45

(a)

(b)

(c)

(d)

(e)

(f)

A

B


33/58

33

Yield of PA

1.10 1.12 1.14 1.16 1.18

S2(

m)

0.20

0.25

0.30

0.35

0.40

0.45

Yield of PA

1.10 1.12 1.14 1.16 1.18

S3

(m)

0.20

0.25

0.30

0.35

0.40

0.45

Yield of PA

1.10 1.12 1.14 1.16 1.18

S4

(m)

0.20

0.25

0.30

0.35

0.40

0.45

Yield of PA

1.10 1.12 1.14 1.16 1.18

S5(

m)

0.20

0.25

0.30

0.35

0.40

0.45

Yield of PA

1.10 1.12 1.14 1.16 1.18

S6

(m)

0.20

0.25

0.30

0.35

0.40

0.45

Yield of PA

1.10 1.12 1.14 1.16 1.18

S7

(m)

0.20

0.25

0.30

0.35

0.40

0.45

(g)

(h)

(i)

(j)

(k)

(l)

Figure 7(B) (Fig. 7 continued)


34/58

34

Yield of PA

1.10 1.12 1.14 1.16 1.18

S8(

m)

0.1

0.2

0.3

0.4

Yield of PA

1.10 1.12 1.14 1.16 1.18

L1

(m)

0.1

0.2

0.3

0.4

0.5

Yield of PA

1.10 1.12 1.14 1.16 1.18

L2

(m)

0.01

0.02

0.03

0.04

0.05

Yield of PA

1.10 1.12 1.14 1.16 1.18

L3(

m)

0.01

0.02

0.03

0.04

0.05

Yield of PA

1.10 1.12 1.14 1.16 1.18

L4

(m)

0.00

0.01

0.02

0.03

0.04

0.05

Yield of PA

1.10 1.12 1.14 1.16 1.18

L5

(m)

0.01

0.02

0.03

0.04

0.05

(m)

(n)

(o)

(p)

(q)

(r)

Figure 7(C) (Fig. 7 continued)


35/58

35

Yield of PA

1.10 1.12 1.14 1.16 1.18

L6(

m)

0.01

0.02

0.03

0.04

0.05

Yield of PA

1.10 1.12 1.14 1.16 1.18

L7

(m)

0.01

0.02

0.03

0.04

0.05

Yield of PA

1.10 1.12 1.14 1.16 1.18

L8(

m)

0.02

0.04

0.06

0.08

0.10

Yield of PA

1.08 1.10 1.12 1.14 1.16 1.18

L9

(m)

0.0

0.2

0.4

0.6

0.8

1.0

(s)

(t)

(u)

(v)

Figure 7(D)

Figure 7: Pareto optimal set and corresponding results for the decision (and other)

variables for Reactor System-2


36/58

36

Table 7: Values of the decision (and some other) variables corresponding to point A

and B in Fig. 7a.

Variable Point A Point B Variable Point A Point B

XPA 1.166 1.1708 S7 (m) 0.357 0.208

Concentration of OX

(g OX/m3

of air at NTP)

72.814 80.648 S8 (m) 0.435 0.275

Gas input temperature (oC) 274.52 285.5 L1 (m) 0.291 0.243

Coolant input temperature

(oC)

388.01 387.6 L2 (m) 0.012 0.010

Coolant flow rate (kg/kg

process gas)

6.087 12.022 L3 (m) 0.011 0.011

S1 (m) 0.356 0.356 L4 (m) 0.011 0.012

S2 (m) 0.363 0.367 L5 (m) 0.016 0.015

S3 (m) 0.345 0.348 L6 (m) 0.011 0.011

S4 (m) 0.436 0.343 L7 (m) 0.019 0.019

S5 (m) 0.245 0.206 L8 (m) 0.027 0.024

S6 (m) 0.369 0.263 L9 (m) 0.205 0.798


37/58

37

Axial position (m)

0 1 2 3

Temperature(oC)

250

300

350

400

450

500

550

Gas

Coolant

Axial position (m)

0 1 2 3

Temperature(oC)

250

300

350

400

450

500

550

Gas

Coolant

Axial position (m)

0 1 2 3


0.0

0.2

0.4

0.6

0.8

1.0

Axial position (m)

0 1 2 3


0.0

0.2

0.4

0.6

0.8

1.0

Axial position (m)

0 1 2 3

Molefraction

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Axial position (m)

0 1 2 3

Molefraction

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

(a) (b)

(c) (d)

(e) (f)

OX

PA

COx

OX

PA

COx



38/58

38

Axial position (m)

0 1 2 3

Mole

fraction

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Axial position (m)

0 1 2 3

Mole

fraction

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

(g) (h)

OT OT

PMA

P

MA

Figure 8(B)

Figure 8: Profiles corresponding to point A (a, c, e and g) and point B (b, d, f and h),

in Figure 7a


39/58

39

Figure 9 shows the effect of the mutation and crossover probabilities on the

Pareto optimal solution for Reactor System-2. Multi-objective optimization of this

system is computationally time consuming. Hence, the right set of values of the

parameters is needed for the guided NSGA-II-aJG to obtain a smooth Pareto-optimal

solution in the least number of generations. The Pareto solutions obtained for the

reference values of mutation and crossover probabilities give better solutions, at least

in the high PA yield region.

The effect of varying the number of catalyst zones in Reactor System-2 is

shown in Figure 10. It is observed that seven zones of catalyst can give almost the

same high yields of PA, as do nine zones. However, the total catalyst length is lower

in the lower PA yield region, where the preferred solution may be located. Figure 11

shows the effect of the total length of the reactor tube for a seven catalyst zone

Reactor System-2. It is observed that slightly larger values of Lcat are indicated when

shorter reactor length of 3 m is taken, than when a 3.5 m tube is used.

It will be interesting to carry out multi-objective optimization of this reactor

with the number of catalyst zones and the length of the reactor tube as decision

variables. The former would require varying lengths of the chromosomes in the

population, depending on the number of catalyst zones. The codes of the various

adaptations of NSGA-II-JG do not permit this at present, and is being attempted

currently.


40/58

40

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

pc

= 0.95

pm = 0.025 (ref)

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

pc

= 0.95

pm

= 0.04

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

pc = 0.95

pm = 0.01

Yield of PA

1.10 1.12 1.14 1.16 1.18

Totalcataly

stlength(m)

0.4

0.6

0.8

1.0

1.2

pc = 0.95

pm = 0.025 (ref)

Yeild of PA

1.10 1.12 1.14 1.16 1.18


0.4

0.6

0.8

1.0

1.2

pc

= 0.90

pm = 0.025

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.2

0.4

0.6

0.8

1.0

1.2

pc

= 0.98

pm = 0.025

(a) (d)

(b) (e)

(c) (f)

Figure 9: Pareto-optimal solutions for different mutation and crossover probabilities

for Reactor System-2 at the 45th

generation


41/58

41

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.2

0.4

0.6

0.8

1.0

1.2

Figure 10: Pareto-optimal solutions for Reactor System-2 with different number of

catalyst zones; : 9 catalyst zones (ref) and : 7catalyst zones.


42/58

42

Yield of PA

1.10 1.12 1.14 1.16 1.18


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 11: Pareto-optimal solutions for Reactor System-2 with 7 catalyst zones and

two different lengths of the reactor tube: : L = 3.5 m and : L = 3.0 m


43/58

43

4. Conclusion

Multi-objective optimization problems have been solved for two industrial

phthalic anhydride units using a general reactor model. Adaptations to NSGA-II-aJG

have been developed so as to obtain converged solutions more rapidly than possible

otherwise. It is observed that the productivity of PA can be easily enhanced by a

proper choice of decision variables.


44/58

44

5. Appendix

5.1. Appendix 1

Complete set of equations for oxidation of o-xylene to phthalic anhydride

(adapted from Elnashaie and Elshishini, 1993)

(a) LHHW rate expressions for the ith

reaction in Fig. 1:

1 1= v OXr k Y

3 3= v OXr k Y

4 4=

v OTr k Y

5 5=

v Pr k Y

6 6=

v OXr k Y

7 7=

v OTr k Y

8 8=

v PAr k Y (A1)

Here

2

2 2

2

1=

+ + + + + +

O

v

OX OX OT OT MA MA P P PA PA O O

t

Y

K Y K Y K Y K Y K Y K Y P

ii io

g

Ek k exp

R T

=

; i = 1, 2, . . . , 8

ads

j

j jo

g

HK K exp

R T

=

; j = OX, OT, P, PA, MA, O2 (A2)

2 2= v OXr k Y


45/58

45

(b) Steady state mass and heat balance equations between the bulk gas and the outer

surface of the impervious catalyst:

( )1 2 3 6, , ,

1

+ + + =

sg OX s OX b OX s t

b

r r r r k a Y Y P

(A3)

( )1 4 7, , ,

1

=

sg OT s OT b OT s t

b

r r rk a Y Y P

(A4)

( )2, , ,

1

=

sg MA s MA b MA s t

b

rk a Y Y P

(A5)

( )4 5, , ,

1

=

sg P s P b P s t

b

r rk a Y Y P

(A6)

( )5 6 7 8, , ,

1

+ + =

sg PA s PA b PA s t

b

r r r r k a Y Y P

(A7)

( )3 8, , ,

8

1

+ =

sg COx s COx b COx s t

b

r rk a Y Y P

(A8)

( )2 2 2

1 2 3 4 5 6 7 8

, , ,

4.5 8.5 3 2 5.5

1

+ + + + + + + =

sg O s O b O s t

b

r r r r r r r r k a Y Y P

(A9)

( )2 2 2

1 2 3 4 5 6 7 8

, , ,

3 5 3 2 2

1

+ + + + + + + =

sg H O s H O b H O s t

b

r r r r r r r r k a Y Y P

(A10)

( )

8

,

1

( )

1

i i s

is s

b

H r

ha T T

=

=

(A11)

The above coupled equations are solved using the DNEQNF program in the IMSL

library.

(c) Effectiveness factors:


46/58

46

The effectiveness factor of each reaction is determined as:

( )( )

, ,

, ,

,

,

i s s i s

i

i b i b

r T Y

r T Y = (A12)

while the componenteffectiveness factors are written as:

( )

( )1 2 3 6

1 2 3 6

+ + +=

+ + +s

OX

b

r r r r

r r r r

( )

( )1 4 7

1 4 7

=

s

OT

b

r r r

r r r

( )

( )2

2

sMA

b

r

r =

( )

( )4 5

4 5

sP

b

r r

r r

=

( )

( )5 6 7 8

5 6 7 8

sPA

b

r r r r

r r r r

+ + =

+ +

( )

( )3 8

3 8

sCOx

b

r r

r r

+=

+(A13)

(d) Steady state balance for the bulk gas phase:

( ), 1 1, 2 2, 3 3, 6 6,

= + + +

OX b

b b b b

F

YdG r r r r

dz M (A14)

( ), 1 1, 4 4, 7 7,

=

OT b

b b b

F

YdG r r r

dz M (A15)

( ),

2 2,

=

MA b

b

F

Yd

G rdz M (A16)

( ), 4, 5 5,

=

P b

b b

F

YdG r r

dz M (A17)


47/58

47

( ), 5 5, 6 6, 7 7, 8 8,

= + +

PA b

b b b b

F

YdG r r r r

dz M (A18)

( ), 3 3, 8 8,8

= +

COx b

b b

F

YdG r r

dz M (A19)

( )2 , 1 1, 2 2, 3 3, 4 4, 5 5, 6 6, 7 7, 8 8,4.5 8.5 3 2 5.5

= + + + + + + +

O b

b b b b b b b b

F

YdG r r r r r r r r

dz M

(A20)

( )2 , 1 1, 2 2, 3 3, 4 4, 5 5, 6 6, 7 7, 8 8,3 5 3 2 2

= + + + + + + +

H O b

b b b b b b b b

F

YdG r r r r r r r r

dz M

(A21)

( ) ( ) ( )8

,

1

4pFref i i i b c

iF

CdG T T H r U T T

dz M D

=

=

(A22)

( ) ( )

=

pC

c C ref C

C

Cdm T T DU T T

dz M (A23)

The above set of equations is solved using the program, DIVPAG in the IMSL library.

The yield of PA, XPA, is defined by

XPA (kg PA produced)/(kg o-xylene consumed)

The concentrations of the two species are used to evaluate the yield.


48/58

48

5.2. Appendix 2

Derivations

(a) L H rate expression:

Consider reaction 1:

o-Xylene + O2 o-Tolualdehyde + H2O

Step 1: Adsorption of o-xylene on the catalyst active site:

OX + lOXa

OXd

k

k

OXl

where, l is a vacant active site and OXl is an active site with o-xylene adsorbed on it,

kOXa is the adsorption rate constant and kOXd is the desorption rate constant.

The rate equation for this step can be written as:

OXlOXa OX,s l OXd OXl

dCk P C k C

dt= (B1)

where, POX,s is the partial pressure of o-xylene just adjacent to the catalyst surface, C l

is the concentration of the vacant active sites and COXl is the concentration of the

active sites with o-xylene adsorbed on them.

Similarly for oxygen:

2

2 2 2 2

O l

O a O ,s l O d O l

dCk P C k C

dt= (B2)

where,2O ,s

P is the partial pressure of oxygen just adjacent to the catalyst surface and

2O lC is the concentration of the active sites with oxygen adsorbed on them.

Step 2: Surface reaction between the adsorbed species (OXl and O2l):

OXl + O2l

*1k

2l + H2O

Water formed by the reaction is not adsorbed on the catalyst surface.


49/58

49

The surface reaction rate equation can be written as:

2

*

1 1 OXl O lr k C C= (B3)

The surface reaction between the adsorbed species is taken as the rate limiting step

(Skrzypek et al., 1985).

Hence 2O lOXl

dCdC0

dt dt= = .

Thus:

OXaOXl OX,s l OX OX,s l

OXd

kC P C K P C

k= =

2

2 2 2 2

2

O aO l O ,s l O O ,s l

O d

kC P C K P C

k= = (B4)

Similarly for all the other adsorbed species:

OTl OT OT,s lC K P C=

MAl MA MA,s lC K P C=

Pl P P,s lC K P C=

PAl PA PA,s lC K P C= (B4)

where, Ki are the adsorption equilibrium constants.

The total number of active sites, CT, on the catalyst surface is:

2T l OXl OTl MAl Pl PAl O lC C C C C C C C= + + + + + +

Equation (B4) gives:

2 2T l OX OX,s OT OT,s MA MA,s P P,s PA PA,s O O ,sC C 1 K P K P K P K P K P K P = + + + + + + (B5)


50/58

50

Using equations (B4) and (B5) in equation (B3) gives:

( )2 2

2 2

* 2

1 OX O T OX,s O ,s

1 2

OX OX,s OT OT,s MA MA,s P P,s PA PA,s O O ,s

k K K C P Pr

1 K P K P K P K P K P K P=

+ + + + + +

(B6)

Dividing both numerator and denominator by 2tP gives:

2

2 2

1 OX,s O ,s

1 2

OX OX,s OT OT,s MA MA,s P P,s PA PA,s O O ,s

t

k Y Yr

1K Y K Y K Y K Y K Y K Y

P

=

+ + + + + +

(B

7)

Rate expressions for the other reactions are derived in the same manner.

(b) Steady state component balance and heat balance for the gas phase:

Assumptions:

i) The reactor operates at steady state conditions.

ii) Radial dispersion of mass and heat are negligible.

iii) Axial dispersion of mass and heat are negligible.

iv) The gaseous components follow ideal gas law.

Component balance in bulk phase:

Component j IN Component j OUT = Rate of removal of j

( ) ( ) ( ) ( ) ( )2 2 2s sj b j b jz z z

b b

u u C R C R R R z+

=

(B8)

where,j,produced

j 3

bed

molR

m .s

Dividing throughout by 2R z and taking limits as z 0 , gives:

( )j sj

d C uR

dz= (B9)


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51

Substitutingj

j s

F

GYC u

M= in the above equation gives:

( )j Fj

d Y / MG R

dz= (B10)

Heat Balance in bulk phase:

Heat IN Heat OUT + Heat transferred from catalyst = Heat removed by jacket

( )[ ] ( )[ ]

( ) ( ) ( )[ ]

2 2s sF FF b ref F b ref z z z

F b F b

2

j j j c

j

u uCp CpR T T R T T

M M

H r R z U 2 R z T T

+

+

= (B11)

Dividing throughout by 2R z and taking limits as z 0 , gives:

[ ]

( ) ( )

FF s ref

F

j j j c

j

Cpd u T T

M 2UH r T T

dz R

= (B12)

Substitutingj sC u G= gives:

[ ] ( ) ( )F ref j j j cjF

Cpd 4UG T T H r T T

dz M D

=

(B14)

Heat balance for coolant:

Heat IN Heat OUT + Heat transferred from bulk phase = 0

[ ] [ ] ( )[ ]c c ref c c ref cz z zmCp T T mCp T T U D z T T 0+ + = (B15)

Dividing throughout by z and taking limits as z 0 , gives:

[ ]( ) ( )c c ref

cd Cp T Tm U D T T

dz = (B16)


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52

5.3. Appendix 3

Kinetic and Transport properties

(i) Binary Diffusivity: (Poling et al., 2001)

Wilke and Lee equation:

( )0.5 3 1.5ABAB 0.5 2

AB AB D

3.03 0.98/ M 10 TD

PM

=

(C1)

where,

1

AB

A B

1 1M 2

M M

= +

;

A BAB

2

+ = ; 1/ 3A B b,i, 1.18V = (

o

A ) ;

( ) ( ) ( ) ( )D B * * **

A C E G

exp DT exp FT exp HTT = + + + ;

*

AB

kTT =

;

0.5

AB A B

k k k

=

; 1.15T

k

= and

P Pressure, bars; Vb molar volume at normal boiling point, cm3/gmol; Tb

normal boiling point, K; A = 1.06036, B = 0.1561, C = 0.193, D = 0.47635, E =

1.03587, F = 1.52996, G = 1.76474, H = 3.89411

(ii) Effective Diffusivity: (Elnashaie and Elshishini, 1993)

Wilke Equation:

nj

effj 1i,mix i ijj i

Y1 1

D 1 Y D=

=

(C2)


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53

(iii) Mass transfer coefficient between catalyst particle and gas phase: (Wakao and

Kaguei, 1982)

*

g,i p 0.6 1/ 3

ieff

i,mix

k D2.0 1.1Re Sc

D= + (C3)

where,p

mix

GDRe =

and mix

eff

mix i,mix

ScD

=

(iv) Viscosity of gaseous mixture: (Poling et al., 2001)

ni i

mix ni 1

j ijj 1

Y

Y=

=

=

(C4)

where,

20.5 0.25

i i

j j

ij 0.5

i

j

M1

M

M8 1

M

+ =

+

(v) Thermal Conductivity of gaseous mixture: (Poling et al., 2001)

Wassiljewa equation:

ni i

mix ni 1

j ij

j 1

Y

Y A=

=

=

(C5)

where, Aij = 1.0, if i = j

else( ) ( )

20.5 0.25

*

tri trj i j

ij 0.5

i

j

1 / M / MA

M8 1M

+ =

+

; * 1.0 ;

( ) ( )

( ) ( )j r,i r,itri

trj i r, j r , j

exp 0.0464T exp 0.2412T

exp 0.0464T exp 0.2412T

=

;

1/ 63

c,i i

i 4

c,i

T M210

P

=

and

Tc,i critical temperature, K; Pc,i critical pressure, bar


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54

(v) Heat transfer coefficient between catalyst particle and bulk phase: (Wakao et al.,

1978)

p 0.6 1/ 3

mix

hD2.0 1.1Re Pr= +

(C6)

where,pmix mix

mix

CPr

=

(vi) Specific heat of the gas mixture:

n

pmix i pi

i 1

C YC=

= (C7)

(vii) For components like o-xylene, maleic anhydride, phthalic anhydride, combustion

products, oxygen, nitrogen and steam, the values of the vapor phase viscosity, thermal

conductivity and specific heat are obtained from Daubert and Danner (1989).

(viii) For o-tolualdehyde and phthalide, the group contribution method is used for the

property estimations. Specific heat, Cp; boiling point, Tb; critical constants, Tc, Pc and

Vc; and acentric factor, , are calculated using the Constantinou and Gani property

functions from the group contributions (Poling et al., 2001). With these values, the

vapor phase viscosity and thermal conductivity of OT and P are estimated using the

Chung et al. method given in Poling et al. (2001). This method was first tested on

components whose viscosity and thermal conductivity were known, viz, PA and OX.

The values were found to be almost the same as that obtained from Daubert and

Danner (1989).


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55

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