A publication of
CHEMICAL ENGINEERING TRANSACTIONS
VOL. 35, 2013
The Italian Association
of Chemical Engineering
www.aidic.it/cet
Guest Editors: Petar Varbanov, Jiří Klemeš, Panos Seferlis, Athanasios I. Papadopoulos, Spyros Voutetakis
Copyright © 2013, AIDIC Servizi S.r.l.,
ISBN 978-88-95608-26-6; ISSN 1974-9791
A multi-level mathematical model of the CO catalytic
conversion process
Olena V. Ved *a, Panos Seferlis b, Petro O. Kapustenko a
National Technical University “Kharkiv Polytechnic Institute”, 21 Frunze Str., 61002, Kharkiv, Ukraine
Centre for Research and Technology Hellas, Charilaou Thermi Road – 6th km., 57001 Thermi Thessaloniki, Greece
helen.ved@gmail.com
a
b
This paper presents the description of the conversion of a gas mixture containing carbon monoxide in the
environment of inert gas in the temperature range 400°K–800°K. The mathematical model is based on
three levels of the process conversion of carbon monoxide. The first level describes the chemical kinetics
of surface exothermic catalytic reaction. The second level describes the thermal and hydrodynamic
processes occurring in the boundary diffusion layer. For processes on the third level the main gas flow is
considered. There are developed hydrodynamic and thermal models of multi-component gas flow and the
temperature change in the bulk of the catalyst lattice.
1. Introduction
The reaction of oxidation CO to CO2 with molecular oxygen represents a simple but important reaction. It
possesses the entire set of phenomena inherent in other more complex reactions despite its simple kinetic
mechanism see, e.g., Tuttlies (2004), Bykov et al. (2005) and Deur et al. (2008). It relates to oxidation of
metals in platinum group. This reaction takes place during deactivation of industrial wastes and exhaust
automobile’s gases, see, e.g. Alkemade et al. (2006) and Dvorak et al. (2010). Oxidation of carbon
monoxide takes place in several stages: knocking non-knocking (Ili–Ridil’s and lengmoore-Hanshelwood’s
mechanism) as described by Ivanov et al. (1981). In this work, a three-level descriptive scheme of
catalytical conversion has been used. Two basic possibilities such of conversion are considered. The first
is the possibility under which oxygen, carbon, and carbon dioxide make up the whole gaseous mixture.
The second one is when the above-mentioned gases are found in neutral gas medium. The first level
applies to catalytical surface. It uses standard approach on the presence of double transient scale of
catalytic reaction - fast and slow. In relation to fast transient scale equilibrium is achieved. Chemical
kinetics equation is considered in stationary approximation by intermediary products which are surface
combination with metals on the catalytic surface. Concentration of gaseous mixture components subjected
to conversion plays role in external adiabatically changing parameters in the equation of surface-catalytic
kinetics, as described by Faber (2001) and Nayfen (1973). At the second level, it’s considered that moving
gaseous mixture occupies volume. It may be divided into the core and boundary layer. On this descriptive
level hydrodynamics, heat and mass transfer on the boundary layer are considered. It was discussed by
Tsybenova (2008). For the second description level, first level (surface catalytic reaction) brings boundary
conditions for components’ temperature and their respective concentrations in the gaseous mixture to the
interior boundary of the boundary layers. That catalytic processes takes place on the surface and its
influence on the mixtures’ volume does not distribute further length of free run of the components’
molecules. During description of boundary layers’ simple structure procedure is used involved in ShvetzTarg method, as described by Loytsansky (1962). On the third description level, a model of moving
gaseous mixture is considered with variable viscosity, heat capacity and thermal conductivity on the
interior surface generated by the catalysis’s surface. The mixture is considered as a solid substance with
spatially divided sources and outflow of both thermal and mass transfer nature. The concentration of the
components’ mixture on the outer boundary is corresponding to their boundary layers. It delivers boundary
condition for gaseous mixture subjected to conversion.
2. Mathematical model
The Equations of additional oxidation reaction can be written as follows:


O 2  Pt  2 PtO K1


C O  Pt  PtCO K 2

PtC O  PtO 2 Pt  CO 2

K 3
(1)

K 4

C O  PtO Pt  CO 2

O 2  2CO  2CO 2




K1 , K 2 , K 3 , K 4 – the reaction rate constants with Arrenius’ form. Mathematical model of
concentrated chemical kinetics, corresponding to Eq(1), is a system of two ordinary differential equations.
This model describes the change in concentration of adsorbed on the surface Pt substances and the
surface oxygen:
Where
x
t
y
t


2

 2


 2 k1 a s 1  x  y  2 k1 x  k 3 xy  k 4 b s x ,

(2)




 k1 b s 1  x  y  k 2 y  k 3 xy ,
Where x, y– surface concentration PtO and PtCO respectively; as – concentration of oxygen;bs – the
concentration of carbon monoxide. Values of the rates of production CO2 defined as follows equation for
stationary conditions of the reaction Eq(1):


c  k 3 xy  k 4 bs x.
(3)
Of the conditions of material balance implies that the production CO2 can be expressed in terms of
concentration CO, As follows:


Value
x



k 2 k 3 b0 s  c x 1  x

 k 4 b0 s  c x.


k 2 b0 s  c  k 2  k 3 x
vs 

x



(4)
is the solution of the following equation:

  
  

2

 2
2 k1 a s k 2
 1 x







k 2 bs  k 2  k 2 k 3 b s 1  x  k 4 b s  2 k1 x k 2 bs  k 2

.
(5)
The solution of this Eq(5) in a low pressure gas mixture can be applied in a wide temperature range of
pressures and temperatures. Mathematical model includes stability equations for intermediate adsorbed
substances’ concentration at the catalytic surface of carbon monoxide and carbon dioxide, equation for
laminar hydrodynamics, heat transfer and mass transfer close to catalytic surface.
During construction, combined cell method for mutually penetrating continuums is used, while the
calculation itself goes back to Rele’s method. The classical Stephan-Maxwell equation is used in
arrangement of expressions for concentrations of streams in multi-components’ mass transfer tasks as
described by Kondepudi and Prigigine (1999). Coefficient of steam diffusion, which is part of the equation,
depends on temperature in accordance to Sazerland’s law as was discussed by Missenard (1965). In
order to draw correlation for mixtures’ viscosity and thermal conductance from temperature and
concentration, the Mionar model was used. This model was chosen by its simplicity and such conditions
that in this model, mutual influence of the gaseous mixtures’ components with each other is accounted
through changes in molecule’s free path. In this case, partial pressure for both carbon monoxide and
dioxide is smaller. Equilibrium equation for intermediate products’ chemical kinetics at the catalytic surface
has such constant rate of reaction that it’s possible to solve relative concentration in a clear way. Laminar
hydrodynamic equation interfaced with both heat and mass transfer may be solved and velocity profile,
temperature and gaseous mixture’s concentration derived as a function of the respective thickness of
boundary layer. As a result, the following simultaneous equations are derived for conversion process
description of gaseous mixture with molecular oxygen, carbon monoxide and dioxide in the presence of
neutral gas within the temperature interval of 400 K to 800 K:
 * T *

T
U

U
w


 T0  T

x
1   w 

T
T0  T  1   w   w

1

T


 T0  T
T


T

 w 
1  
 T 
2

 
U
 C


x
3
1
3

 C0  C


 Ux

1
3
U
  C
x

U

2  T0
x

2
2
w

  w 
T  T0  T  1 
  
 T


  T  T0  T  
1  * T *
U
  T

 Cp  2U 

 T 
U 


3
x
T
x
x

T

2
 *T *
C0  C

 



 2 
C
(6)

 
 2  r  C T0
(7)
T
 
C T0
(8)
C
Equations Eq(6) – Eq(8) are connected simultaneous equations for thickness of hydrodynamic boundary
layer –  w ; temperature boundary layer – T , diffusion boundary layer –  C for carbon monoxide. Values

 w , T ,  C , depend on temperature T0 and T , concentration C0 and C  at the surface and exterior
boundary layer respectively
U
 C
  C  C  ,  * T * Cp  U  T    T   T  T  
0
0

2
x

x

DT
2
C

(9)
T
T
Equation Eq(9) describe changes in temperature for gaseous mixture T 
concentration C  in a porous catalyst substances’ mixture.
T0  T


and carbon dioxide
  ,
C T0 
 T C0  C  
C
2 T0 
2 DT0 
rC T0
(10)
Equation Eq(10) connects between each other temperature and concentration at the catalytic surface and
in the stream at gaseous mixture’s pores.
3
 
 T0   * 
   *
T

T * C 
T0  C 
 T0  2 ,


T *
3
T * C 
T

 C

T * C   T

T
  *



T  C  T *
 
0 1,85
T 
T 2 ,

 DT0   DT *  


T*
T *


3
T * C 
2 ,
  T0    * 


T  C

(11)
 1,85
   DT *   TT * 
,D T
3
T 2


T*




(12)
Equations Eq(11) – Eq(12) specifies dependence of gaseous mixture transport coefficients from
temperature T0 and T  .
k   k  b  C0   xs  1  xs 
C T0   2  3 H
 k 4  bH  C0   xs
k2  bH  C0   k2  k3 xs
(13)
Equation Eq(14) describes production velocity of carbon dioxide xs , oxygen’ concentration aH and carbon
monoxide bH respectively in gaseous mixture at the entrance in catalytic volume.
me
xs 
A
T0
1  me
ne

B
T0
B
1  ne T0
2a H  C 0
A
T0
2a H  C 0
2a H  C 0
2a H  C0   bH  C0 

2a H  C0 
bH  C0 3
bH  C 0
2a H  C0  2a H  C0   bH  C0 
bH  C0 3
(14)
Equation Eq(15) describes superficial oxygen’ concentration, that depends on values m , n , A , B . These
values are derived from rate constant of reaction for carbon monoxide k1 , k 2 , k3 , k 4 and activation
energy of superficial reaction.
3. Results and discussion
In this system of equations Eq(6) – Eq(14) are unknown quantities: w, T, c, T0, C0, T∞, C∞. The system
Eq(6) – Eq(14) study is difficult. To simplify using the following approach: instead of Eq(6) – Eq(14) a
system of equations of the conjugate heat – and mass exchange consisting of two equations for the
unknowns T∞, C∞. In the approximation that uses the volume and concentration of heat sources with a
capacity equal to the capacity of these surface sources. The solution of such a system and finding the
values T∞(x), C∞(x) and substituting in the other equation Eq(6) – Eq(14) the equation is not, and the
formulas for calculating the values w, T, c, T0, C0. The system of equations for T∞, C∞ surround source is:
  c  c
 ,
U
x
V

 * *

   T  C U


T

T

x

r  c

V
KP
S1
(15)
T   T e ,
where V– the volume per unit area of the catalyst surface; K – coefficient of heat transfer between the
catalytic environment and the environment that surrounds it. If the value of the power source of thermal
conductivity and diffusion, taken on the catalyst surface is expressed in terms of their value in the core with
the Taylor expansion, we obtain the following system of equations:
 c  

 T    T



,

 
 

 
 T    T


c
T 
r
2
 T


c 
c
2
 c  

 T    T



,

D 
 

D 
 T    T


c


(17)
For thickness of the boundary layers δT, δc we get the following equation:

 T
T 
*
*

 T
  c  T  U

 T

x
c 
,

 c
,


c

U 
x
D
(18)
Eliminating from Eq(21) and Eq(22) all unknown except ΔT we get the following equation of the form:
3
T 2 
r i
2 T






 T

 * *  T 
  c T U

x
1
2


 T  



 T


1
T 2




r i 
 T


2  *
*  T

c
T
U



x
1
2





0
(19)
The set of equations Eq(16) – Eq(19) allows to find the approximate dependence of the : w, T, c, T0, C0,
T∞, C∞ on the ultimate location x along the surface of the catalyst. While in the second equation of the
system K=0, this corresponds to a thermally insulated system. Such a case can solve a system of
equations Eq(16) just by expressing values in C∞ to T∞. The value of the source c also becomes a
function of temperature T∞ and the longitudinal coordinate x along the surface of the catalyst in the
equation for the temperature T∞ obviously not included. The equation for T∞ integrated in quadrature. In a
more realistic case, when there is an external heat sink system Eq(21) can be reduced to a single equation
for the temperature, which is related to the Abel’s and Riccati’s equations. When solving equations in the
case of thermal insulation or external heat sink is a problem the lack of information about the values of the
rate constants in expression for the source. At low pressure these values are well defined, and in the
normal pressure is greatly reduced. Below it is assumed that under normal pressure in the expression for
the source of the second term is dominated by the rate constant with k+4, the value of the crust at low and
normal pressures differ by about seven orders of magnitude. This fact can be formally taken into account
the multiplier ξ=10-7. Another problem complicating the solution – the system Eq(21) by reducing it to a
single equation for the temperature T∞(x) lies in the fact that at high temperatures on the catalyst surface
are significant radioactive heat exchange. In this case, the dominance of the neutral background gas in the
mixture of the mixture can be considered almost transparent to radiant heat so that the radioactive heat
transfer plays an important role for the heat transfer to the environment. Having said that, the heat transfer
coefficient K can be written as follows:
1
K 
1
i

 ct

ct
,
1
(25)
 ec   er
Where αi – coefficient of heat transfer from the environment to the catalytic wall defining this
environment;  ct – the thickness of the wall;  ct –thermal conductivity of the wall;  ec – coefficient of heat
transfer from the wall to the environment by convection mechanism;  er – coefficient of heat transfer from
the wall to the environment by radiation mechanism. Including all of the above into the equation for the
temperature T∞(x) after excluding the value c∞(x) from the system Eq(21). The equation becomes:
T

 u T
x
u
S

T    
T
 v
 

 Tn u        Te

 
u        Te

 T 
u    
 
   u        Te
  Tn 
u    
 


V U
  k 4  r  bn

*
   c  T
*
,
v

S   k4

,
V U
 v
 u  






U


 v
u  Te




    T T  T ,
e





(26)
KP
,
*
 S1    c   T
Where S is the area of the side surface of the catalytic environment; V is the volume of the catalytic
environment; bn is initial concentration of carbon monoxide;S1 is cross–sectional area of the catalytic
environment; P is the girth catalytic environment; α, β are parameters equal 0,6∙10-3, 0,22 respectively.
Solving Eq(26) after substituting in the first equation of Eq(21) we can find the dependence of the CO2 in
the gas mixture. Equation Eq(26) is used to describe the additional oxidation catalytic converter in the
exhaust gas.
4. Conclusions
The physical model of the oxidation of carbon monoxide on a catalytically active surface of the
solid support, which includes three stages, is created. In the first stage, the process of chemical
transformations with the formation of intermediates between the catalytically active centers of media and
products of incomplete oxidation of carbon, followed by additional oxidation acts on similar active centers
is considered. A mathematical model that describes the physical processes of flow conversion also has
three levels of description. The first level describes the chemical kinetics of the surface of the catalytic
reaction in the approximation of the separation of the kinetics of fast and slow effects. The second level of
description is associated with the concept of the boundary layer. In the studied model the boundary layer is
a thin layer of variable thickness, in which every characteristic value of the gas mixture is changed from its
value at the surface to that away from the surface. At the third level of description the hydrodynamic and
thermal model is developed. The physical model of each stage is represented by systems of equations.
Kinetic equations describe the interactions of the reacting molecules of the gas medium with a catalytically
active centers, mass transfer and thermal processes. Preliminary laboratory researches have confirmed
the adequacy of the proposed mathematical model of the physical data kinetic, hydro and thermal studies.
This has enabled using proposed model to calculate the parameters of the catalytic converter exhaust
gases of internal combustion engine with spark-ignition volume of 1,1 dm3. Converter of exhaust gases of
internal combustion engine was made and its quality of cleaning of exhaust gases is investigated. Tests
showed 100% conversion of carbon monoxide and other saturated and unsaturated hydrocarbons in the
converter.
Acknowledgements
Financial support from the EC Project DISKNET (FP7-PEOPLE-2011-IRSES-294933) is sincerely
acknowledged.
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