CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

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

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

A Novel Mathematical Method for the Simultaneous 

Optimization of Hydrogen Network and Reactor 

Yingjia Wang, Guilian Liu*, Xiao Tian 

School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi Province, China, 710049 

guilianliui@mail.xjtu.edu.cn 

In refinery, the hydrogen-consuming reactors have a great effect on the hydrogen network integration, their 

optimization should be considered simultaneously. This paper propose a new mathematical programming 

model for the optimization of the hydrogen network. The objective is to target the hydrogen-consuming reactor 

and hydrogen network with the optimal economic performance. In this model, the match between hydrogen 

source and hydrogen sink, sulfur content of the feed oil, hydrogen consumption and the operating conditions 

of the reactor are considered simultaneously. The latter includes the residence time, pressure, temperature 

and hydrogen to oil ratio. Based on this model, the overall optimization of the hydrogen network can be 

achieved easily, as well as the optimal hydrogen distribution network and the optimal operating conditions of 

the reactor. The hydrogen network of a refinery is optimized by the proposed method. 

1. Introduction 

Nowadays, there is a worldwide trend towards processing heavier and high-sulfur crude oil. Refiners are being 

compelled to increase their hydro-processing levels to upgrade these crudes and satisfy market demand and 

environment regulations. Hydrogen is critical to convert poor quality crude oil into desired products, and its 

consumption is demonstrating a significant growth. Better hydrogen management through hydrogen network 

integration is needed for energy savings and economical benefit. 

Catalytic diesel oil hydrogenation is one of the main hydro-processing processes to facilitate the control of the 

product’s sulfur content. The hydrogen-consuming reactor is the key equipment, and its operating parameters 

have a great effect on the hydrogen network integrationt. To decrease the hydrogen consumption and the 

cost, the reactor operating parameters should be considered in the integration of a hydrogen network. 

Alves and Towler (2002) found that the pinch exists in the hydrogen network and developed the iterative 

hydrogen surplus method for targeting the pinch point and the minimum fresh hydrogen consumption. Zhao et 

al. (2006) – working on hydrogen distribution systems, Saw et al. (2011) generalizing on property based 

resource condervation, developed different graphical methods for solving the same problem without iteration. 

Foo and Manan (2006) improved Property Cascade Analysis (PCA) method to Gas Cascade Analysis (GCA) 

method to solve the gas integration problem with purification. Zhao et al. (2007) developed the graphical 

method addressing integration of hydrogen distribution networks with multiple impurities. Mao et al. (2015a, 

2015b) developed an integration method for the hydrogen network with the coupling of sink and source 

streams considered. 

Except graphical methods, mathematical programming methods are also employed to solve hydrogen 

network. Van den Heever and Grossmann (2003) proposed an MINLP model for a hydrogen network 

consisting of five plants, four inter-connected pipelines and 20 customers. Ahmad et al. (2010) develop an 

approach for the design of flexible hydrogen networks. Umana et al. (2014) developed an integrated approach 

for refinery process and hydrogen network design. Zhou et al. (2013) presented a systematic modeling 

methodology for the optimal synthesis of sustainable refinery hydrogen networks accounting both the 

economic and environmental aspect. Deng et al. proposed two different superstructure-based mathematical 

programming models to synthesize the hydrogen network (2014a) and performed comparative analysis 

(2014b). Liao et al. (2016) revealed the impact of a purifier on the hydrogen network from three aspects. 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652067 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Wang Y., Liu G., Tian X., 2016, A novel mathematical method for the simultaneous optimization of hydrogen 
network and reactor, Chemical Engineering Transactions, 52, 397-402  DOI:10.3303/CET1652067   

397



Although the hydrogen network can be integrated by these methods, the optimization of the hydrogen-

consuming reactor's operating parameters is not considered.  

This work aims to integrate the hydrogen network with the hydrogen-consuming reactor to increase its 

economical performance. The relation among hydrogen consumption and operating parameters of reactor is 

deduced. A mathematical model is established with the constraints on the match between hydrogen source, 

hydrogen consumption and the operating conditions of the reactor considered. Based on this model, the 

hydrogen network and the operating parameters of the reactor a refinery is optimized simultaneously. 

2. Relation among the hydrogen consumption and operating parameters of reactor 

Multiple reactions occur in the hydrogen-consuming reactor. In this work, the hydro-desulfurization reaction is 

mainly considered, while hydrogen consumption of the other reactions is taken as a constant. The hydrogen 

consumed in the reactor can be determined by Eq(1). H0 denotes the hydrogen consumed in other reactions. 

0 0
( ) /

r oil S S S
H f C C M H  

 (1) 

Based on the reaction kinetics, the hydro-cracking desulfurization reaction rate is related to the sulfur content 

in the reaction system, and is shown by Eq(2). According to the Arrhenius equation, the reaction rate constant, 

k, can be determined by Eq(3). Pre-exponential factor, 
0

k , is relevant to the hydrogen pressure, the rate of 

hydrogen to oil, as shown by Eq(4). Based on Eqs(1) - (4), the relation among sulfur content and the operating 

parameters of reactor is deduced, as shown by Eq(5) 

n

s

dS
kC

d


 
(2) 

 0 exp /k k RT T 
 

(3) 

20
[P ] [H/ O]

H
k A

 


 
(4) 

2

/1 1

0
C C ( 1) [P ] [H/ O] a

E RTn n

s s H
n A e

 


 
  

 
(5) 

Since sulfur is mainly converted to H2S in the reactor, the hydrogen consumption of the reactor can be 

calculated by Eq(6).  

( )( )
1/ (1 )

1/ (1 )

0 0

0

( 1)
=

n
n

oil s s

r

s

f C C n k
H H

M

t
-

-
- + -

+

 

(6) 

3. Mathematical Model 

3.1 Problem statement 
In a hydrogen network with NC sources and NK sinks, sink SKq and source SRp are coupled as they are 

connected by a hydrogen-consuming reactor. Each hydrogen source with given flow rate (FSRi), and hydrogen 

concentration (CSRi) can be allocated to every hydrogen sink. For each sink, there are the limit on its inlet 

hydrogen concentration (CSKj) and total inlet flow rate, (FSKj). The objective is to target the hydrogen network 

with the minimum cost and the optimal operating parameters of the hydrogen-consuming reactor. 

3.2 Objective function 
The objective is to minimize the total annual cost, O, which includes both the reactor investment cost and the 

hydrogen utility cost, as shown by Eq(7) 

Objective function:  Min (O) (7) 

O a I b hus= ´ + ´  (8) 

Where b, hus, a and I denote the hydrogen price, hydrogen utility consumption, annualisation factor and total 

investment cost, respectively. And a is given by Eq(9). In this equation, i is the fractional interest rate per year, 

and n is the number of years. 

(1 )

(1 ) 1

n

n

i i
a

i

 


 
 

(9) 

3.3 The reactor investment cost 
The space time of reactor is defined by Eq(10). The catalyst is set as loaded within 50 % of the reactor volume 

in this work. The height and volume of reactor are determined by Eq(11) and Eq(12), respectively. 

R
V

F
 

 
(10) 

16H R  (11) 
2

V R H   (12) 

398



Where H and R, are the height and radius, V, VR, and 
 
are the volume of the reactor, the catalytic bed 

volume, space time of the reactor, respectively. 

The reactor cylinder wall thickness is calculated by Eq(13). 15CrMo stainless steel is chosen as the cylinder 

material. The welded joints coefficient, ϕ, is taken as 1 (double-welded butt joints, 100 % non-destructive 

testing). From the chemical equipment design manual, allowable stress, [σ]t,
 
is taken as 100 MPa. And the 

designed thickness is shown by Eq(14). In this equation, C1
 
and C2 are

 
the negative deviation and the steel 

corrosion allowance, and are taken as 0.9 mm and 1mm, respectively. 

2[ ]

c i

t

c

P D

P


 



 

(13) 

2 1d
C C C      

 (14) 

Where 
c

P  and 
i

D denote the pressure and inner diameter of the column, respectively. 

Based on the chemical containers and equipment handbook, the mass per meter can be calculated by Eq(15). 

According to Eq(10) - Eq(15), the investment of the reactor can be calculated by Eq(16).  
6

7850 ( ) 10G DN  


      (15) 

3 3 3
0.622158 (3321.21 1.9)(166.11 1.9)in D     

 
(16) 

Where D denotes the material price of reactor cylinder. 

3.4 Constraints of the hydrogen network 

3.4.1 Hydrogen sink constraints 
For every sink, the total flowrate and purity of the source matching it should be greater than its requirements, 

as shown by Eq(17) and Eq(18). 

 ,i j SKj
i

F F

 
(17) 

   
    

   
 , , SK/ii j SR i j j

i i

F c F c

 

(18) 

3.4.2 Hydrogen source constraints 
The hydrogen in a source can either be supplied to sinks or be sent to the fuel gas. The total amount of gas 

supplied to the sinks should not be larger than that available from the source, i.e. 

 , ii j SR
j

F F

 
(19) 

3.4.3 Coupled stream constraints 
A pair of coupled sink and source streams, SKq and SRp, are connected by a hydrogen-consuming reactor, 

and their hydrogen content satisfies the mass balance shown by Eq(20). 

 
,i q q p pr SK SK SR SR

H F C F C
 

(20) 

For the hydrogen-consuming reactor, the inlet hydrogen, which is the coupled sink, is not affected by the 

reactor operating parameters, and its flowrate and concentration are generally constant. The coupled source 

is separated from the outlet stream of the reactor. Generally, its concentration is not affected by the reactor's 

operating parameters, as the separator can accommodate the variation of the inlet feed stream. However, its 

flowrate change along the reactor's operation parameters, and can be calculated by Eq(21) 




p p ,i

q

q

SK SK r

SR

SR

F C H
F

C
 

(21) 

3.4.4 Reactor constraints 

To satisfy the requirement on the product composition, there is the limitation on the maximum allowable sulfur 

content of the reactor effluent. To obtain the proper product, there is the limitation on the minimum hydrogen 

consumption. With P denotes the minimum hydrogen consumption, this constraint is shown by Eq(22).  

( )( )
1/ (1 )

1/ (1 )

0 0

0

( 1)
n

n

oil s s

s

f C C n k
H P

M

t
-

-
- + -

+ ³

 

(22) 

4. Case study 

In a refinery of China, there are 9 sources (SR3 is hydrogen utility) and 10 sinks, as shown in Table 1. By the 

hydrogen surplus method, the minimum hydrogen utility flow rate is identified to be 131.71 mol/s.  

For this system, the hydrogen sink constraints and hydrogen source constraints can be determined according 

to Eqs(17), (18), (19). Based on the flowchart of the system, it is identified that the hydrogen sink, SK2, and 

399



hydrogen source, SR9, is coupled. They are connected by the catalytic cracking diesel oil reactor with 0.8 

million annual processing ability. So the coupled streams constraint is shown by Eq(23), and the hydrogen 

source SR9 also accords with the hydrogen source constraints. 

Table 1: Data of current hydrogen system network  

Sources  Purity, % Flow rate, mol/s Sinks Purity, % Flow rate, mol/s 

SR1 99.99 56.80 SK1 91.33 318.09 

SR2 93.61 320.36 SK2 90.80 124.96 

SR3 90.80 371.45 SK3 90.80 33.52 

SR4 86.63 4.54 SK4 90.00 102.24 

SR5 84.91 1,204.19 SK5 89.06 0.23 

SR6 84.19 456.68 SK6 84.91 1,136.02 

SR7 83.95 45.44 SK7 84.19 454.41 

SR8 72.11 942.90 SK8 83.31 51.12 

SR9 50.02 12.50 SK9 80.00 45.44 

   SK10 72.11 908.82 

 

2 2

9

9

SK SK r

SR

SR

F C H
F

C

-
=

 

(23) 

For this reactor, when the temperature, pressure, the hydrogen to oil ratio are set as 643.15 K, 10 MPa and 

1,000, respectively, the reaction kinetics is shown by Eq(24) 
0.373 0.373

0
0.06665

s s
C C 

 
 

 
(24) 

Besides, the sulfur content in feedstock is 28,300 μg/g, the density of crude is 0.9260 g/cm3, and the flowrate 

of crude is 91.32 t/h. According to Eq(22), the reactor constraint is determined, as shown by Eq(25). 

  
1/0.373

4 0.373

0 0 0
7.9271 10 0.06665

r s s
H C C H P


 

     
 

(25) 

To satisfy requirement on the sulfur content of product, 30.772 μg/g, P is set as 112.41 mol/s.  

The objective function and all constraints are programmed in GAMs software, and the model is solved by 

MINOS solver. The results shows that the optimal space time in the reactor is 3.852 h, the hydrogen utility is 

141.832 mol/s, and the annual cost is 63.77 million dollars per year.  

For the case, the optimal space time and corresponding annual cost is illustrated in Figure 1. The relation 

between annual cost and space time is almost linear. Furthermore, the variation of HUS and total cost along 

the Hr can be identified by this method, and are shown by Figure 2 and Figure 3, respectively. Figure 2 shows 

that increasing the hydrogen consumption of the reactor will result in the hydrogen utility consumption 

increases linearly. However, the relationship between the total cost and Hr is not linear; the growth rate 

increases as Hr increases, as shown by Figure 3. Therefore, lower hydrogen consumption is a better choice. 

63

63.5

64

64.5

65

65.5

66

0 2 4 6 8 10

O
/(

$
/y

)

/(h)
 

141.78

141.8

141.82

141.84

141.86

141.88

112.38 112.4 112.42 112.44

/(mol/s)
r

H

H
u

s
/(

m
o

l/
s
)

 
Figure 1: annual cost versus  diagram Figure 2: hydrogen utility versus Hr diagram 

According to Eqs(3) and (4), Eq(26) can be obtained. This equation shows that the reaction rate constant, k, 

refers the impact of temperature, pressure and hydrogen to oil ratio. Therefore, these parameters can be 

optimised simultaneously. So the reaction constraint is transformed to Eq(27). In addition, the boundary of k 

and τ are added, which are declared by Eq(27) and Eq(28), respectively. 

2

/
k ( 1) [P ] [H/ O] a

E RT

H
n A e

  
 

 
(26) 

  
1/0.373

4 0.373

0 0
7.9271 10

s s
C C k P


 

   
 

(27) 

400



0.001 0.4k   (28) 

0.1 2   (29) 

With the objective function and all constraints programmed in GAMs software and the model solved by MINOS 

solver, the optimal space time is identified to be 1.314 h and the optimal k is 0.195. In this case, the hydrogen 

utility consumption is 141.832 mol/s, and the cost is 62.75 M$/y. 

61.6

62.4

63.2

64

64.8

65.6

66.4

112.39 112.41 112.43 112.45

$/y

$/y

/(mol/s)rH

Simultaneous

Optimisation

Single 

optimisation 

            

O
/(

$
/y

)

 

62.72

62.8

62.88

62.96

63.04

0.19 0.24 0.29

O
/(

$
/y

)

k/(    )
1

h


 

Figure 3: the annual cost versus Hr diagram Figure 4: the annual cost versus k diagram 

Similar with that shown in Figure 2, the hydrogen network utility monotonically increases along that of the 

hydrogen consumption in the reactor. The variation trend of cost along Hr is also illustrated in Figure 3. This 

figure shows that the cost increases with Hr, and the growth rate first decreases and then increases. 

Besides, the relationship between the optimal space time and the annual cost is similar with Figure 1. The 

optimal k and annual cost is shown in Figure 4. As the optimal k increases, the annual cost increases slowly 

and then grows fast, which indicates that a lower k is better. 

SR1

SR7

SR9

SR2

SR6

SK1
318.09

(91.3)

SK2
124.96

(90.8)

SK4
102.24

(90.0)

SK3
33.52

(90.8)

56.80

(99.99)

320.36

(93.61)

456.68

(84.19)

2.067

(50.02)

45.44

(83.95)

SR3
141.873

(90.80)

SR4
4.54

(86.63)

SR5
1204.19

(84.91)

SR8
942.90

(72.11)

SK5
0.23

(89.06)

SK6
1136.02

(84.91)

SK8
51.12

(83.31)

SK7
454.41

(84.19)

SK9
45.44

(80.00)

28.540

222.154

SK10
908.82

(72.11)

28.260

60.4300.09514.90822.773

18.965 18.645 11.891 16.391 0.033 75.556 0.392

4.540

76.905 54.936 6.655 57.523 0.034 493.595 453.952 32.674 7.710 20.208

443.261 13.419

0.068 7.820 29.731 7.820

50.751 4.676 7.652 879.821

0.066 0.066 0.066 0.066 0.066 0.066 0.352 0.347 0.971 10.433

  To the fuel 

gas system
 

Figure 5: the optimal hydrogen distribution network for when the product sulfur content is limited to 4.66 μg/g 

For single optimisation and simultaneous optimization, the variation of costs are shown in Table 2. The results 

show that the simultaneous optimization is better than single optimization. When the hydrogen consumption is 

112.43 mol/s, the cost decreases by 3.81 % to 63.01 M$. By this model, the hydrogen distribution network 

with the minimum cost can also be identified. That for the sulfur content of 4.66 μg/g is shown in Figure 5. 

5. Conclusion 

This paper proposed a new mathematical programming model for the simultaneous optimization of the 

hydrogen network and the hydrogen-consuming reactor. This model can be applied to identify the minimum 

401



annual cost which includes the investment cost for the reactor and the operation cost for the hydrogen utility, 

as well as, to determine the optimal space time, temperature, hydrogen pressure and the rate of the hydrogen 

to oil. The hydrogen network and the catalytic cracking diesel oil reactor of an refinery are optimized by the 

proposed model. When the hydrogen consumption of the reactor is set as 112.41 mol/s, the optimal space 

time is 3.852 h. The results also show that as the hydrogen consumption increases, the hydrogen utility is in 

nearly linear growth, while the growth rate of total cost increases. In the simultaneous optimization of space 

time and k, the optimal space time is identified to be 1.314 h and the optimal k is 0.195. The trend of hydrogen 

utility is similar to that with only space time optimized, while the growth rate of cost decreases first and then 

increases. Compared with that with only space time optimized, the results of the latter is more economical. 

When hydrogen consumption is set as 112.43 mol/s, the potential saving of total cost is 3.81 %. 

Based on the proposed method, multiple reactions/reactors can be considered with the operating parameters 

of each reactor optimized simultaneously. This will be studied in the future work. 

Table 2: The comparison between single optimization and simultaneous optimization  

Maximum sulfur 

content in product  

Hydrogen 

consumption 

Total cost (106$) Decrement 

rate (%)  is optimized Both  and k optimized  

30.72 112.41 63.77 62.75 1.60 

17.27 112.42 64.18 62.77 2.80 

4.66 112.43 65.50 63.01 3.81 

Acknowledgments 

Financial supports provided by 973 programs (2012CB720500) and the National Natural Science Foundation 

of China (21476180) are gratefully acknowledged. 

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