Available online at http://ijcpe.uobaghdad.edu.iq and www.iasj.net 

Iraqi Journal of Chemical and Petroleum 
 Engineering  

Vol.22 No.3 (September 2021) 11 – 17 
EISSN: 2618-0707, PISSN: 1997-4884 

 

Corresponding Authors:  Name: Ghazwan S. Ahmed
 
, Email: ghazwan.s.ahmed@tu.edu.iq, Name: Jasim I. Humadi, Email: 

Jasim_alhashimi_ppe@tu.edu.iq, Name: Ahmad A. Aabid, Email: Ahmadchemical1991@uomosul.edu.iq 
 

IJCPE is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. 

 

Mathematical Model, Simulation and Scale up of Batch Reactor 

Used in Oxidative Desulfurization of Kerosene 

 
Ghazwan S. Ahmed 

a
, Jasim I. Humadi 

b 
and Ahmad A. Aabid 

c 

 
a 
Chemical Engineering Department, College of Engineering, Tikrit University, Iraq 

b
 Petroleum and Gas Refining Engineering Department, College of Petroleum & Minerals Engineering, Tikrit University, Iraq 

c 
Petroleum Reservoir Engineering Department, College of Petroleum &Mining, Mosul University, Iraq 

 

Abstract 

 
   In this paper, a mathematical model for the oxidative desulfurization of kerosene had been developed. The mathematical model and 

simulation process is a very important process due to it provides a better understanding of a real process. The mathematical model in 

this study was based on experimental results which were taken from literature to calculate the optimal kinetic parameters where 

simulation and optimization were conducted using gPROMS software. The optimal kinetic parameters were Activation energy 

18.63958 kJ/mol, Pre-exponential factor  2201.34 (wt)
-0.76636

. min
-1 

 and the reaction order 1.76636. These optimal kinetic parameters 

were used to find the optimal reaction conditions which used to obtain a high conversion (≥ 99%). These optimal reaction conditions 

were reaction temperature 379.4 
o
K and reaction time 160 min. A scale up to batch reactor was conducted using these optimal kinetic 

parameters and optimal reaction conditions and the results showed the best reactor size that can be used at a diameter of 1.2 m. 
     
Keywords: Mathematical model, Optimal kinetic parameters, Optimal reaction condition, Scale up, Batch reactor 
 

Received on 27/05/2021, Accepted on 16/07/2021, published on 30/09/2021 
 
https://doi.org/10.31699/IJCPE.2021.3.2  

 
1- Introduction 
 

   Petroleum fractions contain different types of sulfur 

compounds (thiols, sulfides, and thiophenes) which is 

harmful compound to industry and environment [1].  

   These compounds are undesirable in petroleum 

fractions because they poison the catalyst in refinery and 

cause corrosion problems of internal combustion engines, 

also can emit SO2 in the combustion process [1,2]. Due to 

these disadvantages, many techniques have been 

developed for sulfur removal [3].  

   One of these techniques is the oxidative desulfurization 

(ODS) process which receives more attention because it 

operates under low operation conditions and doesn’t 

consume hydrogen compared with hydrodesulfurization 

(HDS) [3]. Thus, the ODS process was studied by a lot of 

researchers using different catalysts and oxidants under 

different operating conditions and for different petroleum 

fractions [1-6]. Despite this experimental study, the 

simulation process and modeling become necessary 

because they provide a better understanding of the process 

[7].  

   The mathematical model of chemical processes involves 

several simulations and optimization that have many 

advantages such as estimation of the optimal operations 

without making any change on the real process. The 

process model is very efficient and profitable due to huge 

operations, control, safety, and design systems [8].  

   Finding a model which describes the experimental 

process under different process conditions consider a hard 

task challenge due to the experimental process contain 

several types of structure [16]. The mathematical model 

for a batch reactor can be complex due to the effect of 

many factors inside the reactor including mixing 

efficiency, size and shape of catalyst particles, kinetics on 

the catalyst surface, and pore diffusion within the 

framework effective or apparent reaction rate constant 

[17,18]. This study aims to find optimal kinetic 

parameters by applying a mathematical model and 

compared the results from the model with experimental 

results taken from the literature [9]. These kinetic 

parameters were used to find optimum reaction conditions 

which can give high conversion (>99%). Also, this study 

includes the scale-up study of batch reactor based on 

optimal kinetic parameters and optimum reaction 

conditions to find the appropriate volume of the reactor. 

 

2- Methods and Methodology 
 

2.1. Process Description 

 

   The simulation results were made on experimental 

results taken from literature. The experimental section is 

consisting of a batch reactor for oxidative desulfurization 

of mercaptans from kerosene using merox on activated 

carbon catalyst in alkaline solution by air as oxidant. 

http://ijcpe.uobaghdad.edu.iq/
http://www.iasj.net/
mailto:ghazwan.s.ahmed@tu.edu.iq
mailto:Jasim_alhashimi_ppe@tu.edu.iq
mailto:Ahmadchemical1991@uomosul.edu.iq
http://creativecommons.org/licenses/by-nc/4.0/
https://doi.org/10.31699/IJCPE.2021.3.2


G. S. Ahmed et al. / Iraqi Journal of Chemical and Petroleum Engineering 22,3 (2021) 11 - 17
 

 

 

12 
 

   In order to easy for matching between experimental and 

simulation results some assumptions are used: 

 

 The gaseous reactant exists in large excess and the 
liquid is saturated with gas at all times. 

 Isothermal and constant pressure in the reactor. 

 Steady state experimental unit. 

 The reactant used in the model section consists of 
butanethiol and behaves as butanethiol in properties 

because it considers the most sulfur compound present 

in kerosene. 

 

   And the reaction can be stated below where thoils 

oxidize on the catalyst using air as an oxidant to produce 

disulfide which can remove easily by the adsorption 

process because it has high polarity. 𝑅𝑆𝐻 +  𝑂2 →
  𝑅𝑆𝑆𝑅      
 

2.2. Mathematical Model 

 

   The mathematical model is a set of algebraic and 

differential equations which can be solved by 

computational technique. The mathematical model of 

chemical processes involves several simulations and 

optimization that have many advantages such as 

estimation of the optimal operations without making any 

change on the real process. It can be developed by the 

formulation of the equations of mass and energy balance. 

A basic mathematical model will include a chemical 

reaction rate and combination mass and heat transfer 

equations with the kinetic equation [10].  

   The mathematical model can be started by making 

material balance about batch reactor concerning 

mercaptan concentration and this lead to [12]: 

 

t=∫
−𝑑𝐶𝑅𝑆𝐻

(−𝑟𝑅𝑆𝐻)

CRSH

CRSHO
                                                                    (1) 

 

   The reaction rate of chemical reaction could be taken 

into account by assuming n
th

 order kinetics. 

 

(-rRSH) = kappCRSH
n
                                                                  (2) 

 

   The apparent kinetic constant was related to intrinsic 

kinetic constant by internal diffusion which was 

represented by the catalyst effectiveness factor (ƞO) as 

follows[11,12]: 

 

kapp = ƞO kin                                                                         (3) 

 

   By substitution of eq. (2) and (3) in eq. (1) and 

integration to this equation we can get the final expression 

which is: 

 

𝐶𝑅𝑆𝐻 = [𝐶𝑅𝑆𝐻𝑂
(1−𝑛)

+ (𝑛 − 1). 𝑡. 𝑘𝑖𝑛 . ƞ𝑜 ]
(

1

(1−𝑛)
)
                            (4) 

 

   The reaction rate constant (kin) can be calculated for 

ODS reaction by using the Arrhenius equation as follows: 

 

kin =ko 𝑒
(

−𝐸

𝑅𝑇
)                                                                         (5) 

   Eq. (4) contain effectiveness factor which can be 

calculated from the following equation that can be used 

for sphere particle [11, 12]: 

 

ƞO=
3(Ф coth(Ф)−1)

Ф2
                                                                   (6) 

 

   Effectiveness factor is a function of Thiele modulus 

which is calculated from the following equation [14]: 

 

Ф=
𝑉𝑝

𝑆𝑝
√

(𝑛+1)

2
 
𝐾𝑖𝑛𝐶𝑅𝑆𝐻

(𝑛−1)

𝐷𝑒
                                                         (7) 

 

   Molecular diffusivity which is present in eq. (7) can be 

estimated from the equation [9]: 

 

De= 7.8*10
-8

 
𝜓𝐵 𝑀𝐵

(ʋ𝑅𝑆𝐻)
0.6

  
𝑇

𝜇
                                                          (8) 

 

Where:𝜓𝐵 Non-aggregation coefficient. (This is equal to 
1 for kerosene) 

The molar volume of solute is calculated from the 

following equation [11, 12]: 

 

ʋ𝑅𝑆𝐻 = 0.285(ʋ𝑐𝑅𝑆𝐻 )
1.048                                                       (9) 

 

   The equations (1) to (9) were coded and simultaneously 

solved by gPROMS software. 

   The mathematical model was used to estimate the 

optimal kinetic parameters by minimizing the absolute 

error between experimental and predicted data.  

   In order to calculate the optimal kinetic parameters 

using the gPROMS software we must reduce the value of 

sum of square error (SSE) below [13]: 

 

SSE = ∑ (𝐶𝑅𝑆𝐻𝑒𝑥𝑝 − 𝐶𝑅𝑆𝐻𝑝𝑟𝑒𝑑 )
2𝑖

1                                              (10) 

 

Where ( i) represent the number of runs. 

The mathematical model was containing several constant 

parameters which showed in Table 1. 

 

Table 1. Values of constant parameters used in ODS 

model 

Value Unit Symbol Parameter 

Co=0.0193 Wt% Co 
Initial 
concentration 

T1=303, T2=313, 
T3=323 

o
K T1, T2, T3 Temperature 

Tim1=5, Tim2= 

30, Tim3=60 
Tim4=90, 

Tim5=120 

Min 

Tim1, Tim2, 

Tim3, Tim4, 

Tim5 

Batch Time 

8.314 
J/mol.

 

o
K 

R Gas constant 

96.38 g/cm
3 

ʋ𝑅𝑆𝐻 Molar volume 

0.4 Mm Rp Particle radius 

 

 

 

 

 

 



G. S. Ahmed et al. / Iraqi Journal of Chemical and Petroleum Engineering 22,3 (2021) 11 - 17
 

 

 

13 
 

2.3. Scale up of Batch Reactor 
 

   The model of batch reactor was developed by making 

material balance about reactor and energy balance for 

reactor and jacket which can be used to control the 

temperature of the reactor. 

Material and energy balance will lead to the following set 

of ordinary differential equations [15]: 

 

Material balance: 

 
𝑑𝑐

𝑑𝑡
= −𝑘𝑖𝑛  𝐶

𝑛                                                                     (11) 

 

Energy balance: 

 

Reactor: 
𝑑𝑇

𝑑𝑡
=  

𝑈𝐴

𝜌𝑟 𝐶𝑝𝑟 𝑉
 (𝑇𝑗 − 𝑇) +

∆𝐻 𝑟𝑅𝑆𝐻  𝑉

𝜌𝑟 𝐶𝑝𝑟 𝑉
+

𝜇 𝑁2 𝑉

𝜌𝑟 𝐶𝑝𝑟 𝑉
                              (12) 

 

Jacket: 

 
𝑑𝑇𝑗

𝑑𝑡
= −  

𝑈𝐴

𝜌𝑗 𝐶𝑝𝑗 𝑉𝑗
 (𝑇𝑗 − 𝑇) +

𝐹

𝑉𝑗 
(𝑇𝑗 − 𝑇𝑛 )                                  (13) 

 

   The following equations were acquired to solve these 

differential equations: 

 

A=𝜋 𝐷 𝐿                                                                           (14) 

 

V=
𝜋

4
 𝐷2𝐿                                                                           (15) 

 

𝑉𝑗 =
𝜋

4
 (𝐷𝑗

2 − 𝐷2)𝐿                                                             (16) 

 

L=1.5D                                                                             (17) 

 

   And 

 

 𝐷𝑗 = 1.3𝐷                                                                        (18) 

 

   The viscosity of kerosene is used as a function of 

temperature which is estimated from the experimental 

result in literature: 

 

𝜇 = 0.016𝑇 − 3.548                                                    (19) 
 

   The model of scale up of batch reactor contains several 

numerical values which showed in Table 2. 

 

Table 2. Constants used in scale up model 
Value Unit Symbol Parameter 

4.2 kJ/kg K Cpj Jacket solution heat 

capacity 

1000 Kg/𝑚3 𝜌𝑗 Jacket solution 
density 

2.01 kJ/kg K Cpr Reactor solution heat 

capacity 

815.5 Kg/𝑚3 𝜌𝑟 Reactor solution 
density 

2000 RPM
 

N Impeller speed 

225.406 kJ/mol H Heat of reaction 

24 KW/m
2
 
o
K U Overall heat transfer 

coefficient 

420 
o
K Tn Input steam 

temperature 

3- Results and Discussion 
 

3.1. Kinetic Parameters Estimation 

 

   The optimal kinetic parameters which were calculated 

from the model by minimizing the function (SSE) were 

shown in Table 3. 

 

Table 3. Optimal kinetic parameters predicted from the 

model 
Unit Value Parameter 

- 1.76636 𝑛  
kJ/mol 18.63958 E 
(wt)

-0.76636
 . min

-1 
2201.34

 
ko 

 

3.2 Simulation and Experimental Results 

 

   The experimental and simulation results are present in 

Table 4. The simulation results were obtained from the 

model by gPROMS software. While the comparison 

between experimental and simulation results was shown 

in Fig. 1 to Fig. 3. 

 

Table 4. Experimental and simulation results 
Experimental 

conversion 

(%) 

Experimental 

concentration 

(ppm) 

Simulation 

Conversion 

(%) 

Concentration 

by simulation 

(ppm) 

Batch 

Time 

(min) 

Temperatur

e 

(K) 

21.24 152 25.30 144.17 5 303 

66.06 65.5 69.79 58.29 30 303 

83.67 31.5 83.65 31.55 60 303 

88.34 22.5 89.21 20.82 90 303 

92.48 14.5 92.12 15.20 120 303 

26.16 142.5 30.20 134.70 5 313 

73.83 50.5 75.11 48.02 30 313 

88.34 22.5 87.11 24.87 60 313 

92.48 14.5 91.66 16.09 90 313 

93.52 12.5 93.98 11.61 120 313 

30.82 133.5 35.30 124.87 5 323 

80.05 38.5 79.56 39.44 30 323 

90.67 18 89.80 19.67 60 323 

92.74 14 93.50 12.53 90 323 

94.81 10 95.35 8.96 120 323 

 

 

 
Fig. 1. Comparison between experimental and simulation 

results at T=303 
o
K 



G. S. Ahmed et al. / Iraqi Journal of Chemical and Petroleum Engineering 22,3 (2021) 11 - 17
 

 

 

14 
 

 
Fig. 2. Comparison between experimental and simulation 

results at T=313 
o
K 

 

 
Fig. 3. Comparison between experimental and simulation 

results at T=323 
o
K 

 

   Fig. 1 to Fig. 3 above shows the effect of time on the 

conversion at different temperatures in addition to 

comparing experimental and simulation results. From 

these figures, we can observe that the conversion was 

increased by increasing the temperature and time, 

increasing temperature will increase the number of 

attractive molecules, and this leads to an increase in the 

conversion. Also, increasing the temperature will increase 

the rate constant according to the Arrhenius equation, and 

this lead to an increase in the reaction rate while 

increasing the time will increase the contact time between 

the reacting materials and the active sites of the catalyst 

[12, 13]. Also, these figures showed the comparison 

between experimental and simulation results and we can 

see good matching between the simulation and 

experimental results because a few assumptions are used 

in this work and this assumption is used to ease the 

solving of the model where reducing the number of 

assumptions will increase the matching between 

experimental and simulation results. 

 

 

 

 

 

 

 

 

3.3. Optimal Reaction Conditions 

 

   After getting the optimal kinetic parameters, these 

optimal parameters can be used in the model to find the 

optimal reaction conditions which can be used to get a 

high conversion (≥ 99%) and these conditions are shown 

in Table 5. 

 

Table 5. Optimal reaction conditions for high conversion 
Parameter Value Unit 

Temperature 379.4 K 

Time 160 Min 

Conversion 99.135 % 

 

3.4 Scale up of Batch Reactor 

 

   The scale up of the batch reactor is studied under 

different reactor sizes to observe the effect of size on the 

concentration and temperature in the model.  
   The scale up model is used in gPROMS software to 

study the output temperature and concentration under 

different reactor sizes and these values were presented in 

Tables 6 and 7 below. 

 

   From these tables, the best value for diameter was at 

D=1.2 m  due to that when D=1.2 m, the value of 

concentration reach to minimum value and the 

temperature profile were more stable from the other value 

to diameter, in addition, consider the nearest value to the 

optimal temperature of high conversion. So that, from this 

study for scale up to batch reactor used here the diameter 

will be 1.2 m which can be depending it other dimensions 

of the reactor. 

 

Table 6. Effect of the reactor size on concentration profile 

(concentration in ppm). 

Time(min) D=1(m) D=1.2(m) D=1.4(m) D=1.6(m) D=1.8(m) D=2(m) 

0.0 193 193 193 193 193 193 

10.0 78.94 83.33 86.88 89.83 92.27 94.35 

20.0 34.42 37.29 39.96 42.40 44.64 46.68 

30.0 19.18 20.68 22.15 23.60 25.00 26.35 

40.0 12.56 13.39 14.23 15.08 15.92 16.77 

50.0 9.09 9.57 10.08 10.61 11.13 11.68 

60.0 7.01 7.31 7.63 7.98 8.32 8.69 

70.0 5.65 5.84 6.06 6.29 6.53 6.78 

80.0 4.70 4.83 4.98 5.14 5.31 5.50 

90.0 4.00 4.09 4.19 4.31 4.44 4.58 

100.0 3.47 3.53 3.60 3.70 3.79 3.89 

110.0 3.05 3.09 3.15 3.22 3.29 3.37 

120.0 2.72 2.74 2.79 2.84 2.90 2.96 

130.0 2.44 2.46 2.49 2.53 2.58 2.63 

140.0 2.22 2.22 2.25 2.28 2.32 2.36 

150.0 2.03 2.02 2.04 2.07 2.10 2.14 

160.0 1.86 1.85 1.87 1.89 1.92 1.95 

 

    
 



G. S. Ahmed et al. / Iraqi Journal of Chemical and Petroleum Engineering 22,3 (2021) 11 - 17
 

 

 

15 
 

Table 7. Effect of the reactor size on temperature profile 

(temperature in K) 

Time(min) D=1(m) D=1.2(m) D=1.4(m) D=1.6(m) D=1.8(m) D=2(m) 

0.0 300.0 300.0 300.0 300.0 300.0 300.0 

10.0 354.04 348.53 343.94 340.10 336.84 334.05 

20.0 372.58 368.49 364.48 360.72 357.25 354.07 

30.0 378.79 376.62 374.04 371.30 368.53 365.82 

40.0 380.73 379.87 378.45 376.71 374.77 372.71 

50.0 381.19 381.11 380.46 379.45 378.20 376.75 

60.0 381.14 381.51 381.34 380.84 380.07 379.11 

70.0 380.90 381.56 381.70 381.51 381.09 380.48 

80.0 380.60 381.47 381.80 381.82 381.63 381.27 

90.0 380.27 381.33 381.78 381.94 381.90 381.71 

100.0 379.94 381.15 381.71 381.96 382.03 381.96 

110.0 379.60 380.97 381.62 381.94 382.07 382.09 

120.0 379.26 380.78 381.51 381.88 382.08 382.15 

130.0 378.91 380.59 381.39 381.82 382.05 382.17 

140.0 378.56 380.39 381.28 381.75 382.01 382.16 

150.0 378.20 380.20 381.16 381.67 381.97 382.14 

160.0 377.85 380.00 381.04 381.60 381.92 382.11 

 

4- Conclusions 
 

   In this study, a mathematical model was developed to 

simulate experimental results. The mathematical model is 

used to find the optimal kinetic parameters based on 

experimental results and then used them to find the 

optimal reaction conditions which can give high 

conversion. The optimal kinetic parameters and the 

optimal reaction conditions which was presented in this 

study is used to the scale up of batch reactor . The results 

showed that the best size to reactor which achieve the 

high conversion and temperature nearest to optimal value 

was at reactor diameter of 1.2 m. 

 
Nomenclature 

 

CRSH : Mrcaptan concentration, Wt%. 

CRSHO : Initial concentration of mercaptan, Wt%. 

T: Temperature, 
o
K. 

t:  time, min 

Kapp :  Apparent rate constant, (wt)
-0.76636

 . min
-1 

. 

Kin :  Intrinsic rate constant, (wt)
-0.76636

 . min
-1

. 

Ko:  Pre-exponential factor, (wt)
-0.76636

 . min
-1

. 

E:  Activation energy, kJ/mol. 

R:  Gas constant, J/mol 
o
K. 

n:  Order of reaction. 

ƞO :  Effectiveness factor. 

Ф:  Thiele modulus. 

(-rRSH):  Reaction rate, Wt%/sec. 

De:  Effective diffusivity, Cm
2
/sec. 

𝜓𝐵 :  Non-aggregation coefficient. 
𝑀𝐵:  Molecular weight of solvent, g/gmol. 
ʋ𝑅𝑆𝐻 :  Molar volume of mercaptan, m

3
/mol. 

ʋ𝑐𝑅𝑆𝐻 :  Critical molar volume of mercaptan, m
3
/mol. 

𝜇:  Viscosity, Pa.s. 
Vp :  Particle volume, m

3
. 

Sp:  External surface area of particle, m
2
. 

U:  Overall heat transfer coefficient, KW/m
2
 
o
K. 

𝜌𝑗  :  Density of solution in jacket, Kg/m
3
. 

𝐶𝑝𝑗  :  Heat capacity of solution in jacket, kJ/kg 
o
K. 

 𝑉𝑗 :  Jacket volume, m
3
. 

𝜌𝑟 :  Density of solution in reactor, Kg/m
3
. 

𝐶𝑝𝑟  :  Heat capacity of solution in reactor, kJ/kg 
o
K. 

 𝑉:  Reactor volume , m
3
. 

D:  Reactor diameter, m. 

L:  Reactor height, m. 

Dj :  Diameter of jacket, m. 

F:  Steam flow rate, kg/s. 

Tn:  Temperature of input steam, 
o
K. 

Tj:  Jacket temperature, 
o
K. 

N:  Impeller speed, RPM. 

∆𝐻:  Heat of reaction, kJ/mol. 
SSE:  Sum of square error 

 

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نموذج رياضي و محاكاة وتوسيع نطاق المفاعل الدفعي المستخدم في عملية ازالة 
 باالكسدة من الكيروسينالكبريت 

 
   3 احمد عبد السالم عابد و 2 , جاسم ابراهيم حمادي1غزوان صالح احمد 

 

 قسم الهندسة الكيمياوية, كلية الهندسة, جامعة تكريت, العراق 1
 قسم هندسة عمليات تكرير النفط والغاز, كلية هندسة النفط والمعادن, جامعة تكريت, العراق 2

  قسم هندسة المكامن النفطية, كلية هندسة النفط والتعدين, جامعة الموصل, العراق 3

 
 الخالصة

 
في هذا البحث ، تم تطوير نموذج رياضي إلزالة الكبريت باالكسدة من الكيروسين. يعد النموذج الرياضي    

الحقيقية. اعتمد النموذج الرياضي في وعملية المحاكاة عملية مهمة للغاية نظًرا ألنه يوفر فهًما أفضل للعملية 
هذه الدراسة على نتائج تجريبية مأخوذة من األدبيات من أجل حساب المعامالت المثلى لحركية التفاعل. كانت 

 2201.34كيلوجول / مول ، وعامل ما قبل األسي  18.63958المعامالت الحركية المثلى هي طاقة التنشيط 
. تم استخدام هذه المعامالت الحركية المثلى إليجاد ظروف 1.76636بة التفاعل ومرت  1-. دقيقة0.76636-( )وزن

 هي المثلى التفاعل ظروف كانت(. ٪99 ≤التفاعل المثلى التي تم استخدامها للحصول على تحويل عالي )
تخدام توسيع نطاق للمفاعل الدفعي باس إجراء تم. دقيقة 160 التفاعل وزمن كلفن 379.4 التفاعل حرارة درجة

هذه المعامالت الحركية المثلى وظروف التفاعل المثلى وأظهرت النتائج أفضل حجم مفاعل يمكن استخدامه 
 م . 1.2بقطر 

 
 الكلمات الدالة: نموذج رياضي, افضل المعامالت الحركية, افضل الظروف التشغيلية, توسيع نطاق, مفاعل دفعي