Computers and Chemical Engineering 29 (2005) 1041–1045

Optimal design of an auto-thermal ammonia synthesis reactor

B.V. Babu∗, Rakesh AngiraDepartment of Chemical Engineering, Birla Institute of Technology and Science (BITS), Pilani, Rajasthan 333031, India

Received 12 March 2002; received in revised form 19 July 2004; accepted 16 November 2004Available online 7 January 2005

Abstract

This paper presents the simulation and optimal design of an auto-thermal ammonia synthesis reactor. The main objective in the optimaldesign of an auto-thermal ammonia synthesis reactor is the estimation of optimal length of reactor for different top temperatures with theconstraints of energy and mass balance of reaction and feed gas temperature and mass flow rate of nitrogen for ammonia production. Thousandsof combinations of feed gas temperature, nitrogen mass flow rate, reacting gas temperature and reactor length are possible. In the presentstudy, NAG subroutine (D02EJF) in MATLAB (with analytical Jacobian) for simulation in combination with Quasi-Newton (QN) methodfor optimization is used to verify the contradictory results reported using gear package of old NAG subroutine (D02EBF), which is nowr acteristico owed anys d for reactorl©

K

1

detpss11p

t

+

aU(

9)thodpar-ig-

ningin theres-ulateof the

m-tivepar-nine,ctor.per-10 m

tem-ortedlso,

0d

eplaced with D02EJF, in combination with simple GA in earlier literature. And the new NAG subroutine shows a unimodal charf the objective function, compared to a multimodal one found in the past study. It is found that new routine (D02EJF) neither shpikes, reverse reaction effect nor the equation became unstable even at a high top temperature of 800 K. New results are obtaineength and optimal cost of reactor.

2004 Elsevier Ltd. All rights reserved.

eywords:Optimization; Ammonia synthesis reactor; Simulation; NAG subroutine D02EJF; Quasi-Newton method

. Introduction

Ammonia is one of the most important chemicals pro-uced as it enjoys the wide use in the manufacture of fertiliz-rs. Hence, modeling and simulation of ammonia manufac-

uring process has received considerable attention among therocess industries. Simulation models for ammonia synthe-is converters of different types have been developed for de-ign and optimization (Annable, 1952; Eymery, 1964; Dyson,965; Murase, Roberts, & Converse, 1970; Singh & Saraf,979; Upreti & Deb, 1997), and control (Shah, 1967) pur-oses.

Murase et al. (1970)computed the optimum temperaturerajectory along the reactor length applying the Pontryagin’s

∗ Corresponding author. Tel.: +91 1596 245073x205/224 (O);91 1596 244977 (R); fax: +91 1596 244183.E-mail addresses:bvbabu@bits-pilani.ac.in (B.V. Babu),

ngira@bits-pilani.ac.in (R. Angira).RL: http://discovery.bits-pilani.ac.in/discipline/chemical/BVb

B.V. Babu).

maximum principle (PMP).Edgar and Himmelblau (198used Lasdon’s generalized reduced-gradient (LGRG) meto arrive at an optimal reactor length corresponding to aticular reactor top temperature of 694 K. However, theynored a term mentioned in Murase’s formulation, pertaito the cost of ammonia already present in the feed gas,objective function. Also the expressions of the partial psures of nitrogen, hydrogen and ammonia, used to simthe temperature and flow rate profiles across the lengthreactor, were not correct.

Upreti and Deb (1997)rectified above stated shortcoings. They used Murase’s formulation with correct objecfunction and correct stoichiometric expressions of thetial pressures of N2, H2, and NH3. They used simple GA icombination with Gear package of NAG library’s subroutD02EBF, for the optimization of ammonia synthesis reaThey obtained mass flow rate of nitrogen, feed gas temature and reaction gas temperature at every 0.01 m ofreactor length. Moreover, there is a contradiction in theperatures and gas flow rate profiles obtained. They repthe profiles that were not smooth as in earlier literature. A

098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2004.11.010

1042 B.V. Babu, R. Angira / Computers and Chemical Engineering 29 (2005) 1041–1045

Nomenclature

List of symbolsU overall heat transfer coefficient (kcal/m2 h K)S1 surface area of cooling tubes per unit length of

reactor (m)W total mass flow rate (kg/h)Cpf heat capacity of feed gas (kcal/kg K)�H heat of reaction (kcal/kg mole of N2)S2 cross-sectional area of catalyst zone (m2)−dNN2/dx reaction rate (kg moles of N2 /h m3)Cpg heat capacity of reacting gas (kcal/kg K)f catalyst activitypN2, pH2, partial pressure of N2, H2, and NH3pNH3

k1, k2 rate constants

they reported reverse reaction condition at the top tempera-ture of 664 K which was not found in literature earlier. Hence,the present study is carried out in order to take care of theabove deficiencies.

The three-coupled differential equations are solved usingNAG subroutine (D02EJF) in MATLAB (version 5.1). Thestep sizes of 0.01 and 0.001 are used whileUpreti and Deb(1997)used the step size of 0.01 only. Finally, Quasi-Newton(QN) method is used for optimization in combination withabove subroutine, and the results are compared.

2. Problem formulation

Upreti and Deb (1997)in their paper cited Edgar and Him-melblau (1989) for model equations, but did not present them.Unfortunately, Eqs. (c) and (d) on page 545 ofEdgar andHimmelblau (1989)are incorrect where “1.5” is given as thecoefficient. In fact, it should be in the power ofpH2 as stated

in Eq. (3) of this paper. The model equations presented inMurase et al. (1970)also contain the above error. Hence, forthe sake of clarity, it becomes important to give the correctmodel equations. All the corrected modeling equations andconstraints are presented below:

dTf

dx= US1

WCpf(Tg − Tf ) (1)

dTg

dx= − US1

WCpg(Tg − Tf ) + (−�H)S2

WCpg

(−dNN2

dx

)(2)

dNN2

dx= −f

[k1

pN2p1.5H2

pNH3

− k2pNH3

p1.5H2

](3)

where

k1 = 1.78954× 104 exp

(−20800

RTg

)(4)

k2 = 2.5714× 1016 exp

(−47400

RTg

)(5)

The boundary conditions are:Tf =T0 atx= 0;Tg =Tf atx= 0;N0

N2= 701.2 kmol/h m2 atx= 0.

The three inequality constraints that limit the values ofthree of the design variables are as given below:

0

4

0

S i-tr y.F tionsa al too e re-m thesev

tained

Fig. 1. The profiles ob

.0 ≤ NN2 ≤ 3220 (6)

00≤ Tf ≤ 800 (7)

.0 ≤ x ≤ 10.0 (8)

ince the reaction gas temperature (Tg) depends on the nrogen mass flow rate (NN2), feed gas temperature (Tf ) andeactor length (x), explicit bounds onTg are not necessarrom the system model, we have three differential equand four variables, making the degrees of freedom equne. We specify the length of the reactor, calculate thaining variables using the system model and then pass

ariables to the optimization routine.

using subroutine D02EJF.

B.V. Babu, R. Angira / Computers and Chemical Engineering 29 (2005) 1041–1045 1043

Fig. 2. The profiles obtained using D02EJF for different top temperatures (step size = 0.01).

3. Results and discussion

3.1. Temperature and flow rate profiles

The system Eqs.(3)–(5)were solved using NAG subrou-tine (D02EJF) in MATLAB (with analytical Jcobian). It isevident from the graph that the profiles are smooth and thereare no spikes as reported byUpreti and Deb (1997). Fig. 1(a) and (b) show the profiles obtained using small step sizesof 0.01 and 0.001 to ensure that we may not miss spikesif any. But even with this small step size the profiles arevery smooth.Upreti and Deb (1997)reported that the reversereaction predominated the forward reaction at the top tem-perature of 664 K. In the present study, surprisingly, there isno such trend observed in the profiles obtained (for whichUpreti and Deb (1997)gave a very good physical explana-tion) as can be seen inFig. 2 (a). To see the presence of anyreverse reaction effect, the program is executed for temper-atures even below 664 K with interval of 10 K up to 600 K.But no such trend is observed. Typical results obtained at toptemperature of 640 and 600 K are shown inFig. 2 (b) and(c), respectively. As is evident from the figures there is no

reverse reaction effect even at a top temperature as low as600 K.

Also,Upreti and Deb (1997)reported that the three differ-ential Eqs.(3)–(5)become unstable at the top temperature of706 K. However, using the above stated numerical method,it is found that the equations are not unstable even at a toptemperature as high as 800 K (Fig. 2 (d)). It may be notedthat Upreti and Deb (1997)used NAG library’s subroutineD02EBF (which is now replaced by D02EJF (NAG, 2004)).The possibility of using wrong equation byUpreti and Deb(1997)is explored (i.e. taking ‘1.5’ as coefficient, as referredin the first paragraph of Section2 in this paper) and nei-ther spikes nor reverse reaction effect were found even withthis.

3.2. Optimization of reactor length

Fig. 3a shows the variation of objective function with re-actor length (without any constraint onTf ). The value of ob-jective function increases with reactor length. Though to anaked eye, the profit function (objective function) appearsto be monotonously increasing, magnification of highlighted

1044 B.V. Babu, R. Angira / Computers and Chemical Engineering 29 (2005) 1041–1045

Fig. 3. The variation of objective function with reactor length.

Table 1Optimum reactor length and objective function

Parameters Methods

PMP (Muraseet al., 1970)

LGRG (Edgar &Himme-lblau, 1989)

D02EBFa with GA(Upreti & Deb, 1997)

D02EJFa with QN D02EJFb with QN

Optimumx (m) 5.18 2.58 5.33 6.586 6.586Objective function (million $/year) Not reported 1.29 4.23 5.0 5.0

a Step size = 0.01.b Step size = 0.001.

part clearly indicates that there exists a single maximum asshown inFig. 3(b).

It is evident fromFig. 3that the objective function profile issmooth and unimodal. Therefore, any simple gradient-basedmethod can be used for optimization as against the popula-tion based stochastic optimization technique such as geneticalgorithm (GA) being used byUpreti and Deb (1997). In thepresent paper Quasi-Newton method is used.

Table 1shows the results obtained and the comparisonwith those obtained byMurase et al. (1970), Edgar andHimmelblau (1989)andUpreti and Deb (1997)using Pon-tryagin’s maximum principle, Lasdon’s generalized reduced-gradient (LGRG) method and genetic algorithm, respectively.From Table 1, it is clear that the optimum reactor length is6.586 m using new NAG subroutine D02EJF. Also, we ob-serve that an optimum reactor length of 2.58 m is reported byEdgar and Himmelblau (1989)and 5.18 m byMurase et al.(1970), both of which are wrong due to the errors in their prob-lem formulations as pointed out byUpreti and Deb (1997). Anoptimum reactor length of 5.33 m and the corresponding ob-jective function value is 4.23× 106 $/year, reported byUpretiand Deb (1997)are also not correct as found in the presentstudy. In addition, backward differentiation formula (BDF)

method (implemented in the NAG subroutine D02EJF) is atrusted method for solution of ODE. Hence, the correct op-timum reactor length can be considered as 6.59 m with anobjective function value of 5.00× 106 $/year.

4. Conclusions

In the present study, new NAG subroutine D02EJF (withanalytical Jacobian) for solving model equations in MAT-LAB and Quasi-Newton method for optimization are usedfor the optimal design of an auto-thermal ammonia synthesisreactor. Results indicate that the profiles of temperatures andflow rate are smooth and there is neither spikes nor reversereaction effect. Also the equations did not become unstableeven at a top temperature of 800 K. A possible error in theintegrator of the NAG subroutine used in the past study isidentified. And the new NAG subroutine shows a unimodalcharacteristic of the objective function, compared to a mul-timodal one found in the past study. The optimum reactorlength depends upon the top temperature. At the top temper-ature of 694 K, the reactor length of 6.59 m was found to givethe optimum objective function value of 5.0× 106 $/year.

B.V. Babu, R. Angira / Computers and Chemical Engineering 29 (2005) 1041–1045 1045

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