CHEMICAL ENGINEERINGTRANSACTIONS 
 

VOL. 61, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš 
Copyright © 2017, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-51-8; ISSN 2283-9216 

A Sequential Synthesis Framework for Heat-Integrated 

Complex Reactor Network 

Haotian Ye, Xiong Zou, Xiangqin Li, Hongguang Dong* 

School of Chemical Engineering, Dalian University of Technology, NO.2 Linggong Road, 116024, Dalian, P.R.China 

hgdong@dlut.edu.cn 

Reactor network synthesis plays an important role in finding green and sustainable chemical production 

flowsheet. Previous researches were usually optimized under specific reaction path, leading to a suboptimal 

network structure, and less attention has been paid to heat integration between reactors. However, 

simultaneously optimizing these two problems with multiple reaction paths involved inevitably leads to a 

combinational explosion of computational complexities. Therefore, a hierarchical optimization approach for the 

heat-integrated complex reactor network is proposed in this work. In the first step, the reaction paths tree and 

its relationship with reactor networks are discussed by embedding different reaction pathways into reactor 

network superstructure. The corresponding mathematical model is formulated as a mixed integer non-linear 

programming (MINLP) problem and a number of alternative reactor networks with specified catalysts 

arrangement are obtained by the interactive iteration solving strategy. In the second step, heat integration 

between reactors is achieved by the synthesis of heat exchanger network, which leads to a significant 

decrease in the total energy consumption. Ultimately, a case study for p-xylene production is introduced to 

demonstrate the effectiveness of the proposed method. 

1. Introduction 

The discharge of contaminants in a chemical process is primarily decided by the condition of reactions, which 

mainly depend on the design of reactor network, and the energy consumption indicates the sustainability of a 

chemical process. Optimization of reactor network is to address the problem of reactor network synthesis, 

whose solution approaches can be classified as heuristic, attainable region and superstructure-based 

methods (Peschel et al., 2010). While the most common optimization methods rely on superstructure 

approaches, which have shown great success in reactor network synthesis. Jin S. et al. (2012) proposed a 

double-level optimization method, according to the character of the reactor network model based on CSTR, 

combining linear programming and stochastic optimization approach. Soltani H. and Shafiei S. (2015) applied 

genetic algorithm coupled with quasi linear programming (LP) method for the synthesis of adiabatic reactor 

networks, which converted the MINLP model to a quasi LP. Zhao X. and Marquardt W. (2016) proposed a 

systematic superstructure-based approach to design reactor networks with guaranteed robust stability. 

However, reactor networks were all optimized under a specific reaction path in aforementioned researches, 

usually leading to a suboptimal network structure. Furthermore, heat integration should also be taken into 

consideration for the opinion of energy-saving and emission reduction, while previous researches pay less 

attention to the effect of heat integration on reactor network synthesis (Lavric et al., 2005; Zhang D. and Liu G., 

2016). In most cases both reactor network synthesis and heat exchanger network synthesis lead to MINLP 

problems. Simultaneously optimizing the two problems inevitably causes a serious increase in computational 

complexities. Hence, a hierarchical optimization approach for heat-integrated complex reactor network is 

proposed in this work. Since the chemical reactors are the very core of any chemical process, reaction paths 

tree and its relationship with reactor network is discussed in the first step. By introducing functional catalysts 

arrangement, the reaction paths and reactor network could be incorporated in the modified state space 

superstructure simultaneously. The corresponding mathematical model is formulated as a MINLP problem by 

coupling kinetics in reaction paths and mass/energy balance equations in reactor networks. A number of 

alternative reactor networks with specified catalysts arrangement are obtained by the interactive iteration 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1761165

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Ye H., Zou X., Li X., Dong H., 2017, A sequential synthesis framework for heat-integrated complex reactor network, 
Chemical Engineering Transactions, 61, 1003-1008  DOI:10.3303/CET1761165  

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solving strategy. Then, heat integration between chemical reactors through heat exchangers is employed, 

reactor networks seen as a part of the heat exchanger network, significantly decreasing the total energy 

consumption and preserving their performances in terms of chemical transformation ratios. Finally, a case 

study for p-xylene production is introduced to demonstrate the applicabilities of the proposed sequential 

method. 

2. Problem statement 

2.1 Reactor network 

It was common that the former works considered all reactor units loaded with the same catalysts and just 

maximized the objective function of a single production technology. Through the synthesis of reactor network 

we can finally obtain: (1) the number, type and residence time of reactors; (2) the configuration of reactor 

networks; (3) maximum objective function value. However, in this work simultaneous synthesis has been 

performed. This methodology can be applied to both single reaction path and multiple reaction paths. In the 

later case, important information can also be gained: (1) feasible and activated reaction paths and their 

interconnections; (2) corresponding catalysts arrangement. 

2.2 Heat exchanger network for reactor networks 

When alternative reactor networks are obtained, process streams and activated reactors are identified 

whether they need to be cooled or heated. In this work, reactors are perceived as virtual streams, of which the 

temperature difference between inlet and outlet is tiny, exchanging heat with process streams or hot/cold utility 

through heat exchangers. Thus, the heat capacity flow rates of the process streams and reactors can be 

calculated. Utilities are available and their corresponding temperatures together with costs are given. Also 

given are the heat transfer coefficients for the process streams, reactors, and the hot and cold utilities. 

Furthermore, given are the fixed cost charge for each heat exchanger and the charge coefficients in order to 

calculate the investment of each heat exchanger according to its area. Stage-wise superstructure is adopted 

(Yee and Grossmann, 1990) and the objective of the problem is to achieve economic-optimal heat exchanger 

network, by determining the number of heat exchangers and their respective areas as well as the utility 

consumption. Note that the way heat transfers in reactors and the effects in heat transfer caused by phase 

transformation are ignored. 

3.  Superstructure 

The synthesis strategy is based on such a proposition that the key to couple reaction path synthesis and 

reactor network synthesis is interconnections of reactor modules with different catalysts loaded. Each reactor 

module corresponds to a single production technology and each main reaction takes place only in the reactor 

module with specific functional catalyst loaded. 

S

M

FD1

M

M

M

S

S

S

S
FD2

Ff

CSTR1

CSTR2

CSTR3

PFR3

PFR2

PFR1

 

Figure 1: Schematic initial superstructure of functional catalysts arrangement. 

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Basic types of reactors embedded in this superstructure are: ideal CSTRs and ideal PFRs. Every reaction 

path is guaranteed by a reactor module consisting of a CSTR and a PFR with corresponding functional 

catalyst loaded. Binary variables are introduced and equal to one when CSTR/PFR is activated in the reactor 

module. Considering the differential properties of PFR design equation which in mathematical formulation will 

result in a non-linear programming (NLP) with both algebraic and differential equations, collocation with 

orthogonal polynomials method (Biegler, 2010) is applied to approximate the PFR model. Such approximation 

greatly reduces the complexity of mathematical formulation. Finally the model would be a MINLP problem with 

only algebraic equations.  

The key to build a proper superstructure is that network of reactors and streams must embed all possible 

interconnections both in single unit and among various units. We built a schematic initial superstructure shown 

in Figure 1. From Figure 1, the schematic initial superstructure contains three reactor modules, which means 

desired products can be directed by three kinds of functional catalysts. The interconnections of three modules 

include all possible reaction sequences. In a reactor module, a CSTR and a PFR are interconnected. Besides, 

mixers and splitters are designed in the reactor modules to guarantee the interconnection and self-circulation. 

4. Mathematical model 

I: Equations 

a). Total mass balance 

 n
n

FD FPRO  ( n N ) (1) 

N = {n}, n stands for each component in reaction system. FDn is the pure feed stream of each reactant in mass 

flow, while FPRO is the final product stream in mass flow. 

b). Splitter of each feed stream 

,n n i

i

FD FR   ( ,n N i I  ) (2) 

I = {i}, i stands for number of reactor module. FRn,i is the feed stream of component n from outlet of splitter to 

mixer before each reactor module. 

c). Mixer and splitter prior to each reactor module 

, , ,
( )

n i j i j n i i n
j

FR RR AFIN FRP AIN


   ( , ,i j I n N  ) (3) 

C P

i i i
FRP FRP FRP   ( i I ) (4) 

C in superscripts stands for variables in CSTR reactor. P in superscripts stands for variables in PFR reactor. 

FRPi stands for the mass flow in reactor module i. RRj-i stands for recycle flow from outlet of splitter of reactor 

module j to mixer before reactor module i. AFINj,n is the mass fraction of recycle flow from splitter after reactor 

module j. AINi is the inlet mass fraction of reactor module i.  FRPiC and FRPiP stand for the mass from feed 

and other reactor module flow in CSTR and PFR respectively. 

d). Mixer prior to each reactor 

, , ,

C P C C

i i n i i n i i n
FRP AIN FPC AOUT F AIN   ( ,i I n N  ) (5) 

, , ,

P C P P

i i n i i n i i n
FRP AIN FCP AOUT F AIN   ( ,i I n N  ) (6) 

FPCi and FCPi stand for the mass from PFR in the same reactor module flow in CSTR and mass from CSTR 

in the same reactor module flow in PFR respectively. FiC and FiP stand for the total mass flow in CSTR and 

PFR respectively. AINi,nC and AINi,nP are the inlet mass fraction of CSTR and PFR in reactor module i, 

respectively. AOUTi,nC and AOUTi,nP are the mass fraction of outlet of CSTR and PFR after reaction 

respectively. 

e). Splitter after each reactor 

C C

i i i
F FCP FF  , 

P P

i i i
F FPC FF   ( ,i j I ) (7) 

FFiC and FFiP stand for the mass flow outlet of reactor module i from CSTR and PFR respectively. 

f). Splitter after each reactor module 

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, , , ,
( )

C C P P

i i n i i n i j i n i i n

j

FF AOUT FF AOUT RR AFIN RM AFIN


    ( , ,i j I n N  ) (8) 

RMi stands for the stream from reactor module i to final mixer. 

h). Final mixer 

,
( )

i i n n

i

RM AFIN FPRO APRO   ( ,i I n N  ) (9) 

APROn is the mass fraction for component n in final product stream FPRO. 

h). Kinetic equations 

Reaction rate expressions: 

, ,
( , )

rp i i n i
R f A T  ( , ,i I n N rp RP   ) (10) 

RP = {rp}, rp stands for every feasible reaction path. This equation expresses a general reaction rate. Rrp,i is 

mass based general reaction rate for reaction path rp in reactor module i. Similarly, Ai,n and Ti are general 

outlet mass fractions and outlet temperature of CSTR and PFR. 

Arrhenius equation: 

k0 exp[ / ( )]
rp rp rp rp

k Ea RT    ( rp RP ) (11) 

Trp is the temperature of reaction rp. In the same reactor module there will be different reactions in progress, 

but we want the temperature to be the optimal only for main reaction. Consequently, the value of Trp should be 

equal to the suitable value for main reactions. In other words, Trp=Ti in the same reaction module.  

j). Mass balance for reactors 

Mass balance for CSTR: 

, , , ,

C C C C

i n i n i rp i n

rp

AOUT AIN t R    ( , ,i I n N rp RP   ) (12) 

Mass balance for PFR: 

,

P

i sk i

sk

t t  ( ,i I sk SK  ) (13) 

, , , , , , , , ,i n sk i n sk i sk k rp i n sk k

k

AOUT AIN t R    ( , , , ,rp RP i I n N sk SK k K     ) (14) 

SK = {sk}, sk is the number of finite element of a PFR. K = {k}, k is the number of internal collocation point in a 

finite element. ti,sk stands for the residence time of reactants in element sk of reactor module i. Ωk is a 

orthogonal polynomial to approximate the PFR. 

II. Objective function 

max ( ) ( )
P C P C

tp i i i i i

i i

TAP aFPRO APRO b z z c V V ACHEN         ( ,tp i N ) (15) 

Maximum Total Annualized Profit (TAP) is set as objective function in this work. a and ci stand for the profit 

coefficient of target product (tp) and cost charge coefficient of reactor respectively. b stands for the fixed cost 

charge of each reactor, while ziP and ziC are binary variables. ViP and ViC stand for the reactants mass in PFR 

and CSTR respectively. ACHEN is the annualized cost of heat exchanger network. 

III. Heat balance 

0
/ ( )

n n n n
QF FD MW CP T T     ( n N ) (16) 

, , ,
( )

C C C

i i i rp i n rp n

n rp

Q F t R H    ( , ,i I n N rp RP   ) (17) 

, , , , , ,
( )

P P

i i i sk k rp i n sk k rp n

sk n rp k

Q F t R H      ( , , , ,rp RP i I n N sk SK k K     ) (18) 

QFn is the energy needed by feed component n, while QiC and QiP stand for the heat generated in CSTR and 

PFR in reactor module i. MWn and CPn are the molecular weight and heat capacity of component n. T is the 

temperature of reactor modules, while ΔHrp stands for the heat of reaction rp. 

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IV. Solving strategies 

Both of the reactor network model and heat exchanger network model result in MINLP problems. In solving 

steps we combined deterministic algorithm with random disturbance. In order to improve the global optimality 

of solutions, we followed the strategy of random disturbance initializations as well as multi-iteration and 

screened local optimal solutions. 

5. Case study 

P-xylene is an important intermediate of chemical manufacture process. Two core reaction units are xylene 

isomerization and toluene selective disproportionation. However, compared with toluene disproportionation 

and xylene isomerization, alkylation is an irreversible reaction so that by this reaction path we can achieve 

higher utilization ratio of aromatic rings. Besides, disproportionation of toluene generates equimolar products 

of benzene and xylene, which shows a larger optimization space of benzene utilization. An extra reactor 

module of benzene conversion can be added to joint production technology. Thus we have constructed a 

reactor network superstructure containing four reaction pathways: toluene alkylation, toluene 

disproportionation, xylene isomerization and benzene alkylation. Note that different reaction path belongs to 

different reactor module. Module 1: isomerization (Xu O. and Su H., 2007); Module 2: disproportionation (Xu O. 

and Su H., 2007); Module 3: toluene alkylation (Wang T. et al., 2011); Module 4: benzene alkylation (Zhao B. 

and Guo X., 2013). 

 

Feed conditions (mass flow) : 46 t/h for toluene; 32 t/h for methanol; 39 t/h for benzene. 

Reaction temperature: 623.15 K 

Profit of p-xylene: 200 $/t 

Working time: 8000 h/y 

Cost of Reactors ($/y) = 5000+c[Reactants mass (t)]. (c=250 for module 1, 2, 4; c=300 for module 3) 

Cost of Heat Exchangers ($/y) = 5000+150[Area (m2)] 

Cost of Cooling/ Heating Utility = 20/ 100 $/(kW·y) 

Temperature of Cooling/ Heating Utility: inlet 300 K, outlet 320 K/ inlet 680 K, outlet 680 K. 

heat transfer coefficient: 1 kW/(m2·K) for feed, cold utility; 5 kW/(m2·K) for hot utility; 2 kW/(m2·K) for reactors. 

 

After computation and iteration the maximum objective function value is TAP=12,072,200 $/y, which saves 

1,381,269 $/y compared with the reactor network without heat-integration. The corresponding reactor network 

configuration and heat exchanger network configuration are shown in Figure 2. All the reactors activated are 

PFR with residence time 0.133 h (disproportionation, PFR2), 500 h (toluene alkylation, PFR3), 32 h (benzene 

alkylation, PFR4) respectively. In this structure, (1) isomerization is not activated because of the feed condition 

and the cost of the reactors; (2) toluene alkylation is activated because of its irreversibility and high selectivity 

for P-xylene, although its reaction rate is much smaller than that of disproportionation (shown in Table 1); (3) 

disproportionation and benzene alkylation connects with each other because either product of one module 

serves as the reactant of the other module; (4) From Table 1, we can see the optimal reactor network has 

smaller heat-integration cost than that of suboptimal reactor networks, which proves we can carry out reactor 

network synthesis and heat-integration sequentially in cases like this. 

 

PFR4

PFR2
FT=16.0t/h

FB=35.2t/h

PFR3

FT=30.0t/h
FM=32.0t/h

F=117.0t/h

F=16.0t/h

F=117.0t/h

FT=46t/h

FB=39t/h
FM=32t/h

FB=3.8t/h F=65.8t/h

aPX=0.13

F=7.0t/h

     

FEEDT

623.15

3930.6kW

PFR2

PFR3

PFR4

623.15

623.15 623.15

623.15 623.15

623.15 293.15613.15

4604.8kW

2799.9kW

163.6kW

143.9kW

623.15 293.15613.15
122.8kW

623.15 293.15367.68613.15
114.1kW

FEEDM

FEEDB

850.0kW

6306.4kW

 

Figure 2: Resulting process flowsheet for the case 

 

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Table 1: Comparison between different reactor networks 

 Optimal reactor network Suboptimal reactor 

network 

Optimal reactor network 

without module 3 

Annualized profit ($/y) 

(with utilities free) 

12,333,732 12,330,104 11,194,016 

Annualized profit ($/y) 

(without heat-integration) 

10,690,931 10,687,123 9,586,977 

Heat integration cost ($/y) 261,532 261,722 219,481 

Total Annualized Profit ($/y) 12,072,200 12,068,382 10,974,535 

6. Conclusion 

In this contribution we considered a methodology for simultaneous optimization of reactor network synthesis 

and reaction paths synthesis with heat integration. An initial reactor network superstructure without heat 

exchanging units is constructed. In each reactor module within superstructure a CSTR and a PFR are 

interconnected and are selectively activated by binary variables. A number of alternative reactor networks with 

specified catalysts arrangement can be obtained from the superstructure. Then, heat integration of chemical 

reactors through heat exchangers is employed, trying to decrease the total energy consumption, while 

preserving their performance. This methodology is tested by a p-xylene production scenario and final result 

proves the feasibility of the sequential solving strategy with total annualized profit for more than 12 M$/y and 

1.4 M$/y decrease in the total energy consumption. 

Acknowledgments  

This work is supported by the National Natural Science Foundation of China (No.21276039). 

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