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HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPRÉM 
Vol. 36(1-2) pp. 143-147 (2008) 

IMPROVING THE MATHEMATICAL MODEL OF A HETEROCATALYTIC 
TUBE REACTOR 

T. VARGA , G. RÁDI, T. CHOVÁN 

University of Pannonia, Department of Process Engineering, H-8200 Veszprém Egyetem u. 10., HUNGARY 
E-mail: vargat@fmt.uni-pannon.hu 

 

In recent decades the applicability and usefulness of mathematical models in the solution of many industrial problems 
have been shown. There are many cases when the correct answer can not be given because of the complexity of the 
problem however CFD approaches can be applied to dig deeper and improve the reliability of results. In this work the 
improvement of the first-principle model of a tubular fixed-bed reactor is shown applying COMSOL Multiphysics. 
Mathematical models of tubular fixed bed catalytic reactors to simulate their operation are well known and the 
formulation of such models is well documented. In most cases the aim of modeling is the development of the simpliest 
model which can be applied to solve the modeling task. However, nowadays these modeling problems can be more 
complicated and require more accurate models, e.g. in case of model based diagnostic systems, early detection systems. 
After the model with the adequate complexity was developed it must be solved with the necessary accuracy. In general, 
as the complexity of the model increases, so does the chance of encountering numerical difficulties. Our purpose is to 
apply the most appropriate software methodology to solve problems at each level of the model hierarchy, therefore a 
proccess model of an industrial heterocatalytic reactor has been extended with integrating transport processes on micro 
level. The idea is the integration of a model of a single catalyst pellet implemented in COMSOL Multiphysics into the 
steady-state process simulator of the reactor developed in MATLAB. The proposed approach makes possible to develop 
the model of a whole catalyst bed from the particle models. The first step to accomplish our purpose is the 
implementation of the pellet model in the applied CFD code. Then the pellet model can be integrated into the process 
simulator at the higher model hierarchy level. 

Keywords: heterocatalytic tube reactor, mechanistic modeling, computational fluid dynamics, catalyst pellet 

Introduction 

The aim of all production technologies is formation of a 
more valuable material through some chemical reactions. 
The reactor is a place where reactions generally take 
place. Hence, a reactor must be able to convert the 
necessary amount of raw materials into products with 
desired quality in given amount of time. A chemical 
technology can not be imagined without a reactor 
consequently the design process of a reactor has a really 
important role in generating a well profitable technology. 
Furthermore the planing of instrumentation and 
optimization of operating variables is also crucial. Main 
steps in development of a well functioning reactor are 
the following:  

● defining desired performance of reactor operation; 
● collecting all a prior information and physical 

constraints; 
● determining and/or investigating velocity field in 

the reactor; 
● assigning all operating variables. 
 
Nowadays development of flow field in process 

equipments is a key issue in planning of process unit. 

With a well designed flow field the overall process 
performance and safety of operation can be increased at 
the same time. To support this step, different modeling 
softwares and flow sheet simulators have been developed 
in last decades. As computational capacity of computers 
has being increased the role of computational fluid 
dynamics (CFD) in applied modeling technics and 
methods has significantly grown since the first computer 
was switched on [1]. 

CFD is a collection of numerical solvers for partial 
differential equations. With applying CFD codes more 
complicated problems in more complex structures can 
be investigated in details than with other modeling 
technics. Coupling of mathematical models of different 
physical and chemical processes with efficient numerical 
solvers makes possible to analyze chemical engineering 
and other kind of engineering problems in much more 
details. Of course, as in other modeling technics, CFD 
models must be validated before it can be used to 
predict the behavior of invesigated system. Validation 
of models is based on comparing simulation results with 
measured variables, i.e. developing an adequate CFD 
model, because of the complexity of model, needs well-
designed and precisely carried out experiments. 



 

 

144

In recent years many papers applying CFD technics 
to analyze some phenomenon or to design a reactor 
have been published. The most frequently investigated 
phenomenon are the transport processes between a 
stationary and a moving phase, or two moving phases, 
such as heat transfer [2] and mass transfer [3] in a 
fluidized bed. Another interesting field of application of 
CFD models is the modeling of bubble columns [4-8]. 
CFD models can be applied not only in reactor and 
column design but in solving other kind of engineering 
problems, e.g. designing a protective suit [9]. These 
examples show that CFD is capable to simulate transport 
processes and it can be applied as a part of complex 
mathematical model. 

In this paper the concept of a detailed mathematical 
model of an industrial used heterocatalytic reactor will 
be introduced. The former version of reactor model was 
applied in a Operator Support System [10] to support 
process operators in choosing appropriate values of 
important operating variables to keep the production in 
safe and efficient regime. To improve the accuracy of 
our reactor model mass and heat transfer processes 
between solid catalyst pellets and moving gas are 
considered. In previous investigations a small difference 
was noticed between the characteristic of measured and 
calculated temperature profiles but model relability 
couldn’t be furher improved without increasing model 
complexity. However, to improve our quasi-single phase 
model solid phase and all coupled transport processes 
are considered.  

Our idea is the integration of the model of a single 
catalyst pellet implemented in COMSOL Multiphysics 
into the steady-state process simulator of the reactor 
developed in MATLAB. The proposed approach makes 
possible to build up catalyst bed from particles which 
have different size. The first step to accomplish our 
purpose is the implementation of the pellet model in the 
applied CFD code. Then pellet model can be integrated 
into the process simulator at the higher model hierarchy 
level. After short introduction regarding the most 
frequently applied CFD codes and some CFD application 
examples in model development the investigated case 
study is presented, than simulation results obtained at 
this level of work are evaluated and finally conclusions 
are drawn and future steps are outlined. 

Case study 

Our approach is to apply the most appropriate software 
to solve problems at each level of model hierarchy, 
therefore a mathematical model of an industrial 
heterocatalytic reactor was developed with integrating 
processes at the level of thermodynamic phases. 

Reactor model 

The investigated, vertically positioned reactor contains a 
large number of tubes filled with catalyst as shown in 
Fig. 1. A highly exothermic reaction takes place as 

reactants are rising up in tubes passing through the fixed 
bed of catalyst particles while generated heat by reaction 
is removed through tube walls by cooling media. The 
reaction has equalibrium and since temperature is 
increasing in reactor more and more product decomposed. 
Due to the highly exothermic reaction it has a great 
chance to develop hot spots somewhere in catalyst bed 
which increase the rate of catalyst ageing. Direction of 
the flow of cooling media is not important in this 
process because there is only 7-8 K difference between 
the inlet and outlet temperature of cooling media. 
Selection of appropriate operating conditions is important 
to avoid the development of reactor runaway and to 
increase lifetime of catalyst. 

The investigated reactor is a heterocatalytic tube reactor 
where gas phase is the moving phase while catalyst is 
the stationary phase. In our previous work a quasi-single 
phase mathematical description of the reactor was 
introduced [10]. Since model accuracy couldn’t be 
improved without increasing model complexity the 
quasi-single phase must be divided into solid and gas 
phases as it shown in Fig. 2. State variables are calculated 
in both phases which are coupled by mass and heat 
transport processes. As it can be seen at components 
level in Fig. 2 the reaction is considered only in solid 
phase so the produced heat by the highly exothermic 
reaction is removed by cooling media in jacket.  

 

coolant in

coolant out

reagents

product

x = 0

x = L

( )Wout,W B,T

( )Win,W B,T

( )out,Gout,Giout,Gout,G p,c,B,T

( )in,Gin,Giin,Gin,G p,c,B,T  
Figure 1: Simplified scheme of reactor 

 
As it can be seen in the model structure in Fig. 2 

process variables calculated in two phases in the tubes. 
Thermodynamic phases are coupled by mass and heat 
transport processes. Steps of the assumed overal process 
are summarized in Fig. 3. First steps, the adsorption of 
reagents take place simultanously and it is called 
competitive adsorption because two components are 
competing to get to the same active sites on the surface 



 

 

145

of the catalyst. After both reagents adsorb a second order 
reaction takes place and generates the product which 
leaves the surface of catalyst by desorption process. 
Developed model calculates changes only along the 
reactor, i.e. model is one dimensional and it doesn’t 
consider the diffusivity along the radius of catalyst bed. 
Before the introduction of model equations some 
simplifications must be considered to simplify the real 
problem. These simplifications are summarized as follows: 

● gas and solid phases are considered in tubes, 
which are connected by the mass and heat transport 
processes; 

● highly exothermic, second order reaction takes 
place in solid phase; 

● diffusivity along radius of the reactor is neglected; 
● all catalyst pellets have the same size and spherical 

shape; 
● to calculate pressure drop in catalyst bed a 

modified Ergun-equation is applied. 
 
A steady-state model based on the overall rate of all 

mass transport processes and reaction with detailed 
description of applied simplifications can be found in 
[10]. That model was solved in MATLAB and calculated 
temperature profiles were compared to measured ones 
in order to determine missing model parameters. It was 
concluded that accuracy of developed model was 
appropriate, although some minor differences could be 
experienced between characteristics of calculated and 
measured profiles. To further improve model accuracy, 
overall rate was substituted for by a CFD model of a 
catalyst pellet to simulate basic steps introduced in Fig. 3. 
Steady-state reactor model is not introduced in this 
paper, a detailed description can be found in [10].  
 

Reactor

Tube(s) Jacket

Solid phase

A B C X A B C

Mass&Heat 
transport

Reaction(s)

Heat 
transport

Liquid phase

W

Gas phase

 
Figure 2: Structure of two phases model 

 
Our concept is the integration of a catalyst pellet 

model implemented in COMSOL Multiphysics into the 
reactor simulator developed in MATLAB. The first step 
to accomplish our objective is the implementation of 
pellet model in the applied CFD code. As it can be seen 
in Fig. 2 processes at the micro level can be simulated 
and developed CFD model can be integrated into higher 
hierarchy level. 

COMSOL is more than just a software which uses a 
CFD code because mathematical models of many 
processes in different area of science are collected and 
classified; consequently the necessary time to solve an 

engineering problem is less than with applying other 
CFD software. COMSOL applies the finite element 
method [1] to discretize the investigated object and to 
solve model. 

 

 
Figure 3: Basic steps of the assumed overall process 

Catalyt pellet model in COMSOL Multiphysics 

The first step of model building in COMSOL is the 
selection of the necessary correlations from the model 
collection. Of course there is a possibility to introduce 
new correlations. To investigate complex geometries an 
interactive graphical interface helps the user to define it 
or the user can import it from a CAD software.  

Our COMSOL model contains only one catalyst 
pellet with the narrow surroundings of the particle. The 
surrounding is flowing around the pellet and the transport 
processes make the connection between two phases. 
MATLAB simulator runs this simulator and builds the 
whole catalyst bed. Difference between each pellet is 
considered by changing boundary conditions before 
solving the pellet model. To describe the whole catalyst 
bed the MATLAB simulator of the reactor evaluates 
boundary conditions based on the position of the acual 
pellet in the catalyst bed. Because of the axial symmetry 
of tubes and the spherical symmetry of the catalyst 
pellet the COMSOL model is two dimensional. 

Based on the introduced simplifications and the 
introduced pellet model the following equations were 
applied to calculate the state variables in both phase. All 
component mass balances in the gas phase contain terms 
to calculate convective and conductive transport processes 
and naturally the mass transport process between the 
solid and gas phase: 

( ) ( ),ccA
dx

cd
D

dx
cBd G

i
S
i

GS
i

GS
2

G
i

2
G
i

G
i

G

−⋅β⋅+−=
⋅  

where  
BG – flow rate of gas phase;  
ci

G – concentration of i-component in gas phase;  
i = {A; B; C}; 

x – reactor length;  
Di

G – diffusion coefficient of i-component in gas phase; 
AGS – interface area between the gas and solid phase;  
βi

GS – mass transfer coefficient between the gas and 
solid phase of i-component;  

ci
S – concentration of i-component in solid phase.  



 

 

146

Of course the convective term in solid phase is missing: 

( ) ,rVccA
dx

cd
D i

SG
i

S
i

GS
i

GS
2

S
i

2
S
i ⋅ν⋅+−⋅β⋅−=  

where  
Di

S – diffusion coefficient of i-component in solid 
phase;  

VS – the volume of solid phase;  
νi – the stoichiometric coefficient of i-component;  
r – reaction rate which is calculated by the following 

correlation: 

,
K

cc
ccekr

S
X

S
CS

B
S
A

TR
E

0
S

A

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛ ⋅
−⋅⋅⋅= ⋅

−
 

where  
k0 – preexponential factor;  
EA – activation energy;  
R – ideal gas constant;  
TS – temperature of solid phase;  
cx

S – concentration of active sites on catalyst;  

K – reaction eqaulibrium constant which is a function 
of the temperature.  

 
As it can be seen in the last equation the effect of 
temperature on reaction rate is considered therefore 
changes of temperature of gas and solid phase are 
calculated too. Terms in both equations are the same as 
those considered in component balances: 

( ),TTA
dx

Td
dx

dT
Bc

GSGSGS

2

G2
G

G
GG

p
G

−⋅α⋅+

+⋅λ−=⋅⋅⋅ρ  

where  
ρG – density of gas phase;  
cG

G – heat capacity of gas phase;  
TG – temperature of gas phase;  
λG – heat conduction coefficient in gas phase;  
αGS – heat transfer coefficent between the gas and 

solid phase. 

  

  
a) Temperature in catalyst pellet and in surroundings b) The concentration of A component in catalyst pellet 

and in surroundings 

  
c) Concentration of B component in catalyst pellet and 

in surroundings 
d) Concentration of C component in catalyst pellet and 

in surroundings 

Figure 4: Changes in the value of state-variables along the investigated part of reactor in steady-state 



 

 

147

And finally the heat balance of solid phase: 

( ) ( ) ,rSGSGSGS2
S2

S ΔHrVTTαA
dx

Td
λ −⋅⋅+−⋅⋅−=⋅  

where  
λS – heat conduction coefficient in solid phase;  
∆Hr – reaction heat.  

 
This is just a short description of model equations to 
show the structure of pellet model. All physical properties 
are calculated as the function of temperature. These 
eqautions were implemented in COMSOL to develop a 
pellet model which replaces the reaction rate of overall 
catalyst processes of earlier developed reactor simulator. 
Integration of reactor simulator made in MATLAB and 
the pellet model implemented in COMSOL is just a 
vision yet, still as it can be seen in next chapter that 
pellet model gives good results.  

Results and discussions 

To check the applicability of the developed COMSOL 
Multiphysics pellet model in our modeling approach, 
the value of state variables in front of reactor were given 
as boundary conditions in the solution of the pellet 
model. Generated results are plotted in Fig. 4.  

In Fig 4a temperature surface can be seen and as it 
was expected the highly exothermic reaction taking 
place in catalyst increases the temperature of pellet and 
it warms up the flowing gas. Due to the diffusion in 
catalyst pellet the concentration of reagents decrease in 
the direction to the centre of the pellet (check Fig. 4b 
and Fig. 4c.) while the concentration of the product 
changes in the opposite direction as shown in Fig. 4d.  

In this model all components can be connected to the 
same active sites on the catalyst but in reality there is a 
competition between components and the rate and the 
equalibrium of the adsorption determine the concentration 
in the catalyst. Since the temperature profile and properties 
of inlet and outlet flow are measured during the reactor 
operation the catalyst pellet model can not be validated 
without integrating it into the reactor simulator.  

As it can be seen in Fig. 4d the outlet concentration 
of product from the investigated part of the reactor is 
very low and it suggests that the parameters applied to 
calculate the rate of adsorption must be modified. Still 
the characteristic of changes in case of all the calculated 
state variables are the same as we expected. 

Conclusions and future work 

To follow an operation of a tube reactor usually process 
operators do not have any more information but some 
temperature measurements along the reactor and 
composition of inlet and outle flow. Hence, a reactor 
simulator can be very useful to follow the operation and 
to help process operators, e.g. to avoid development of 
reactor runaway or to optimize reactor operation. To 
achieve this aim an adequate reactor model with correct 
model parameters is needed. The paper proposed a possible 
way to improve a former developed mathematical model 
of a heterocatalytic tube reactor applied in the process 
industry to detect reactor runaway.  

At this phase of our work applicability of COMSOL 
Multiphysics in modeling a catalyst pellet and its narrow 
surroundings is investigated. The developed model and 
the applied software can be applied to generate and 
simulate state variables in the whole catalyst bed.  

Integration of pellet model into an earlier developed 
reactor simulator is our next step. After the success 
integration a model validation must be performed 
comparing the calculated temperature profiles with the 
measured ones. Finally the accurate model can be built 
into an operator support system to improve reliablility 
of runaway detection. 

 

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