Modeling Catalytic Converters for Reducing Hydrocarbons
during Cold Start Conditions
Paper # 573
Session #ET-2e
S. Chauhan,
Reader
Department of Chemical Engg. and Tech., Panjab University, Chandigarh- 160014, India
V. K. Srivastava,
Professor and Dean IRD
Chemical Engg. Department
Indian Institute of Technology
Delhi,
Hauz
Khas,
New
Delhi-
110016
India.
ABSTRACT:
Monoliths are used as catalytic converters for treating pollutants coming out of the vehicular
exhaust. Although they have been extensively studied, still there are some areas that need to be
further explored like the cold start period, during which the hydrocarbon emissions are relatively
high. Mathematical modeling has proved to be a good way of analysing the physicochemical
processes occurring in the converter. Modeling is preferred as compared to experimental testing,
the later being very expensive and time consuming. Modeling can be applied for design purposes
and improving the existing converter control strategies. Difficulties encountered during modeling
are due to complexities in reaction schemes and the rate expressions to be used for different
catalytic formulations.
In this paper reduction in the level of hydrocarbon emissions during cold start was analysed.
The transient adiabatic plug flow model accounts for the combined effect of heat transfer, mass
transfer and exothermic chemical reactions which are all strongly coupled. This model considers
for both catalytic and homogenous reactions taking place in the converter assembly. Platinum
used as catalyst helps in early initiation of the catalytic reactions, at a much lower operating
temperature. The catalysts distribution over a large surface area ensures maximum mass transfer
characteristics between the gas phase and the active catalyst. Heat exchange due to convection
between the gas and the solid phase and heat transfer due to conduction in the solid phase are
considered and the effect of radiation is neglected. Also diffusion is neglected due to thin
washcoat. The ordinary differential equations for heat and mass transfer for gas phase so formed
are solved using Runge-Kutta method of fourth order. The partial differential equations for heat
transfer calculating the catalyst temperature are solved by backward implicit scheme. Solutions
derived using both quasi steady and unsteady state are compared and analysed.
INTRODUCTION
In order to reduce the pollutants coming out of the vehicular exhaust, catalytic converters were
first introduced in US some 25 years back. Over the period of time, these converters which are
the principal emission control tools have proved to be an unqualified success. In the automobile
engine combustion of fuel takes place, whereby chemical energy is converted to mechanical
energy. However even in the most efficiently tuned engines this conversion process is never
complete and a considerable amount of polluting exhaust gases are released to the atmosphere.
Release of hydrocarbons (HCs) during cold start of engine is one of the major problems
encountered throughout the world. The HCs form 60-80% of the released pollutants. So after
these exhaust gases come out of the engine they are subjected to catalytic reduction in the
converter.
As these exhaust gases flow through a converter they pass through the monolith channels,
adsorb on the precious metals present in the washcoat and react with the adjacent adsorbed
reactants. These reactions are highly exothermic in nature and start only after the catalyst has
been preheated.1,2 As gases get adsorbed on the catalyst sites, chemical reaction takes place
thereby converting harmful pollutants to relatively less polluting products.
Traditional converter design used empirical approaches that were time consuming and costly.
These approaches could result in over design or under design of converters in terms of noble
metal loading and structural strength. Therefore an analytical approach such as numerical
simulation enables one to explore much possible design options and yields an optimum design.3
Early converters used a palletized catalyst, but they have now been replaced by free-flowing
monolith converters. As compared to the former the later are more compact and being uni-body
can withstand mechanical vibrations caused during motion of the vehicle. Monolith behaviour
during warm up of an automobile from cold start can be adequately predicted by a onedimensional model.2 As compared to two-dimensional model; the one-dimensional model is
simpler and requires less computing time.
Modeling of the behaviour of a three-way monolithic catalytic converter remains a very
difficult challenge, due to the complex competing reactions which are strongly affected by heat
and mass transfer. The need for numerical simulations has induced a lot of research to provide
realistic models of heat and mass transfer as well as reliable kinetic expressions.
In this study one-dimensional model is considered to help predict the cold start results at quasi
steady state, along the axial direction in a catalyzed channel for a mixture of fast oxidizing
hydrocarbon propylene and slow oxidizing hydrocarbon propane. Model equations are proposed
both for heterogeneous and homogenous reactions. Also the results derived for quasi steady state
are compared with those derived for unsteady state for the mixture.
CHEMICAL REACTIONS AND KINETICS
Oxidation reactions of propylene and propane present in the exhaust gases are given below:
C3H6 + 4.5 O2  3 CO2 + 3 H2O
C3H8 + 5 O2 3 CO2 + 4 H2O
These gases are assumed to be present in the exhaust in the ratio C3H6/C3H8: ½.4 Both
homogenous as well as heterogeneous phase reactions are considered for both species. In the
heterogeneous phase platinum suspended in aluminia washcoat is considered as the catalyst.
Reaction rates for the above reactions are given below. 5,6,7
Heterogenous reaction:
(-ri)cat(Cs,i ,Ts) = ki-cat exp (-Ei-cat/RTs) Ci
Homogenous reaction:
(-ri)homo(Cg,i ,Tg) = ki-homo exp (-Ei-homo/RTg) Ci Cb
here ‘i’ represents either C3H6 or C3H8, ‘b’ represents O2 and ‘g’ and ‘s’ represent gas and
solid phase. Also subscripts ‘cat’ and ‘homo’ refer to catalytic and homogenous reaction rates
and corresponding parameters.
Values of Rate parameters:
Rate constants:
kC3H6-cat
kC3H8-cat
kC3H6-homo
kC3H8-homo
9.14 X104 cm/s
2.40 X107 cm/s
2.87 X1015 cm3 /g mol s
2.87 X1015 cm3 /g mol s
Activation Energies:
EC3H6-cat
12,000 cal/g mol
EC3H8-cat
21,460 cal/g mol
EC3H6-homo
40,000 cal/g mol
EC3H8-homo
26,700 cal/g mol
THE ONE-DIMENSIONAL MODEL
One-dimensional model is used to simulate the conversion and thermal characteristics of the
adiabatic monolith catalyst. In this model the cylindrical converter is considered. The model
takes into account
-the gas-solid heat and mass transfer
-chemical reaction and related heat release
-axial heat conduction
The radial variations along the circular channel are not considered and only the axial
gradients are accounted for.
Assumptions:






Diffusion in wash coat is neglected,
Negligible axial diffusion of mass and heat transfer in gas phase,
Noble metal concentration is kept constant,
Catalyst does not deactivate,
Monolith is cylindrical with circular cross-section channels,
The heat released by the catalytic reactions inside the washcoat was assumed to be totally
given to the gas phase by convection,
 Heat transfer by radiation within channels and also heat exchange between the substrate
and the surroundings at both inlet and outlet faces of the monolith is neglected,
 Heat conduction inside the gas phase is not accounted and
 Non-uniform flow distributions inside the converter are neglected, as one single channel
represents the entire monolith.
Basic equations:
a) Modeling for a mixture of propylene(C3H6) and propane (C3H8) at Quasi steady state
Quasi-steady gas phase approximation refers to accumulation of mass and energy in the gas
phase being negligible.8 This approach is taken to make the simulations simpler and easier.
At quasi steady state: Tg / t   0
(1)
C
C
g ,1
g ,2
/ t   0
/ t   0
(2)
(3)
where:
Cg , Cs = concentrations in bulk gas phase and at the solid surface (g mole/cm3),
Tg = gas temperature (K)
(i)
Modeling for catalytic reactions only
Mass balances in gaseous phase:
For C3H6 gas: v C g ,1 / x   k g1 S C g ,1  C s ,1   0
For C3H8 gas vC g , 2 / x   k g 2 S C g , 2  C s , 2   0
where:
kg = mass transfer coefficient (cm/s),
(4)
(5)
S = geometric surface area per unit reactor volume (cm2/cm3)
v = average velocity (cm/s).
Mass balances in solid phase:
a r 1cat C s ,1 , Ts   k g1 S C g ,1  C s ,1 
For C3H6 gas
For C3H8 gas
where:
a r 2cat C s , 2 , Ts   k g 2 S C g , 2  C s , 2 
(6)
(7)
a = catalytic surface area per unit reactor volume (cm2/cm3).
Energy balance in gas phase:
 C pg v g Tg / x   hS Tg  Ts   0
where:
ρg = gas density (g/cm3),
(8)
Cpg = specific heat of gas (cal/g K),
h = heat transfer coefficient (cal/cm2 s K)
Energy balance in solid phase:
 s  2Ts / x 2   hS Tg  Ts   a H 1  r 1cat C s ,1 , Ts   a H 2  r 2cat C s , 2 , Ts  
C ps  s Ts / t 
(9)
where:
(-H) = heat of reaction (cal/gmole)
s = thermal conductivity of wall (cal/cm s K),
Cps = specific heat of solid (cal/g K),
s = solid density (g/cm3)
t = time (s).
Initial conditions:
C g ,1 0, t   C g0,1
C g , 2 0, t   C g0, 2
Ts x,0  Ts0
Tg 0, t   Tg0
Boundary conditions:
at x  0, Ts / x   0
and at x  L, Ts / x   0
The equations (4), (5) and (8) are ODEs whereas equation (9) is a PDE.
(ii)
Modeling for both catalytic and homogenous reactions
Mass balances in gaseous phase:
For C3H6 gas: v C g ,1 / x    r 1 hom o C g ,1 , Tg   k g1 S C g ,1  C s ,1   0
For C3H8 gas vC g , 2 / x    r 2 hom o C g , 2 , Tg   k g 2 S C g , 2  C s , 2   0
(10)
(11)
(12)
(13)
Mass balances in solid phase:
a r 1cat C s ,1 , Ts   k g1 S C g ,1  C s ,1 
For C3H6 gas
(14)
For C3H8 gas
(15)
a r 2cat C s , 2 , Ts   k g 2 S C g , 2  C s , 2 
Energy balance in gas phase:
 C pg v g Tg / x   hS Tg  Ts    H 1  r 1 hom o C g ,1 , Tg    H 2  r 2 hom o C g , 2 , Tg   0
(16)
Energy balance in solid phase:
 s  2Ts / x 2   hS Tg  Ts   a H 1  r 1cat C s ,1 , Ts   a H 2  r 2cat C s , 2 , Ts  
(17)
C ps  s Ts / t 
Initial conditions:
C g ,1 0, t   C g0,1
C g , 2 0, t   C g0, 2
Ts x,0  Ts0
Tg 0, t   Tg0
Boundary conditions:
at x  0, Ts / x   0
and at x  L, Ts / x   0
The equations (12), (13) and (16) are ODEs whereas equation (17) is a PDE.
(18)
(19)
b) Modeling for a mixture of propylene(C3H6) and propane (C3H8) at Unsteady state
In the unsteady state model the time derivative terms in both gas concentrations and gas
temperatures are accounted for along with those of solid temperature while solving mass and
energy balance equations.
Mass balances in gaseous phase:
For C3H6 gas v C g ,1 / x    r 1 hom o C g ,1 , Tg   k g1 S C g ,1  C s ,1   C g ,1 / t 
(20)
For C3H8 gas vC g , 2 / x    r 2 hom o C g , 2 , Tg   k g 2 S C g , 2  C s , 2   C g , 2 / t 
(21)
Mass balances in solid phase:
a r 1cat C s ,1 , Ts   k g1 S C g ,1  C s ,1 
For C3H6 gas
(22)
For C3H8 gas
(23)
a r 2cat C s , 2 , Ts   k g 2 S C g , 2  C s , 2 
Energy balance in gas phase:
 C pg v g Tg / x   hS Tg  Ts    H 1  r 1 hom o C g ,1 , Tg    H 2  r 2 hom o C g , 2 , Tg 
 C pg  g Tg / t 
(24)
Energy balance in solid phase:
 s  2Ts / x 2   hS Tg  Ts   a H 1  r 1cat C s ,1 , Ts   a H 2  r 2cat C s , 2 , Ts  
C ps  s Ts / t 
Initial conditions:
C g ,1 0, t   C g0,1
C g , 2 0, t   C g0, 2
Tg 0, t   Tg0
Ts x,0  Ts0
Boundary conditions:
at x  0, Ts / x   0
(25)
(26)


at x  L, C g ,1 / x  0
at x  L, C g , 2 / x   0
at x  L, Ts / x   0
at x  L, Tg / x   0
(27)
The equations (20), (21), (24) and (25) form a set of PDEs.
All these above equations (1)-(19) are solved in dimensionless form using the following
expressions:
C1  C g ,1 / C g0,1 ,
Tg'  Tg / Tg0 ,
z  x/ L
t '  t / t0
C2  C g , 2 / C g0, 2 ,
Ts'  Ts / Tg0 ,
All the ODEs in cases a) (i) and (ii) are solved by Runge-Kutta method of fourth order, whereas
the PDEs in cases a) (i) and (ii) and b) are solved by backward implicit scheme.9,10
In Figure 1. I = 1 denotes the entry point, I = 2 to M-1 donate the grid points in between entry
and one grid point earlier to the exit and I = M denotes the grid point at the exit of the reactor, J
levels show the increment in time used in the backward implicit scheme. By applying the BCs a
tridiagonal-banded matrix is formed. The matrix has M×M components. Hence more computer
storage is required. This M×M component matrix is transferred to M×3 components as shown in
Figure 2. This way a lot of storage space is saved and it helps in reducing the time as iterations
are done again and again. The method is used for solving the tridiagonal-banded matrix.9 Effect
of grid sizes has been studied.
Figure 1: Backward Implicit Scheme for calculating concentration and temperature variation with
time.
J=N
I=1
Time
I = M= (N-1) t’ (i.e., J = N)
T's varies from I = 1,M
z
t’
J = N-1
I=1
z
I = M= (N-2) t’ (i.e., J = N-1)
Time
T's varies from I = 1,M
t’
J = N-2
Time
I=1
Time
I = M= (N-3) t’ (i.e., J = N-2)
T's varies from I = 1,M
z
---
---
---
---
---
---
J=2
I=1
z
I = M Time = 0+t’ (i.e., J = 2)
T's varies from I = 1,M
t’
J=1
I=1
z
Initial time, t’ = 0 (i.e., J = 1)
I = M T's is same from I = 1,M
z = grid length in axial direction
t’ = grid time (time step)
Length of Converter [-]
DISCUSSION OF RESULTS:
a) Effect of homogenous and heterogeneous reactions on the conversion of
propylene
Hydrocarbons propylene and propane gases react with oxygen to form carbon dioxide and
water vapour. These reactions can occur in both homogenous as well as heterogeneous phases
depending upon the operating conditions. Heterogeneous reactions due to the presence of
catalyst start at a lower temperature. In an earlier work, a mixture of propylene and propane
exhausts gases entering at temperature 390oC and upto 800oC the effect of homogenous
reactions was found to be negligible.11 So in this work relative effect of both homogenous and
catalytic reactions was studied at temperatures 900oC and 1000oC. The results so obtained are
discussed with help of Figures 2, 3 and 5 representing concentrations of propane and Figures 4
and 6 representing concentrations of propylene.
Figure 2 represents propane gas concentration in axial direction for a mixture of exhaust gases
propane (2000ppm) and propylene (4000ppm) introduced in the converter at 9000C. It is found
that conversion of propane increases slightly in the case of catalytic-homogenous reactions taken
together as compared to catalytic reaction only. At time 3.20 the concentrations of propane are
0.3083 and 0.3013 for catalytic reaction only and 0.3069 and 0.2995 for homogenous-catalytic
reactions both for axial distances 0.40 and 1.00 respectively. Homogenous kinetics gives slightly
more conversion although not much effect is felt at these parameter values (and also at lower
values).
Figure 2: Comparison of Concentration variation with Axial Length for
Catalytic and both Catalytic and Homogenous reactions for Propane Gas at
2000ppm, exhaust gases at 9000C.
0.33
t’=2.80
t’=3.00
0.31
t’=3.20
Conc
0.29
Increasing Time
[-]
0.27
t’=3.40
0.25
____ Catalytic alone
- - - Homogenous and Catalytic both
0.23
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Axial length [-]
Figure 3 represents propane gas concentration in axial direction for a mixture of exhaust gases
with increased inlet concentrations of both propane (3000ppm) and propylene (6000ppm)
introduced in the converter at 9000C. In comparison to Figure 2 slightly more conversion is
observed here for catalytic-homogenous reactions on increasing the inlet concentration of the
gases. At times 3.10 the concentrations of propane for catalytic reaction only are 0.2806 and
0.2722 and for catalytic-homogenous reactions are 0.2747 and for 0.2654 for axial distances 0.40
and 1.00 respectively. Higher concentration of exhaust gases gives more conversion.
Figure 4 represents propylene gas concentration in axial direction for a mixture of exhaust
gases propane (3000ppm) and propylene (6000ppm) introduced in the converter at 9000C. It is
Figure 3: Comparison of Concentration variation with Axial Length for Catalytic and
both Catalytic and Homogenous reactions for Propane Gas at 3000ppm, exhaust gases
at 9000C.
0.34
0.33
t’=2.30
0.32
t’=2.60
t’=2.90
t’=2.90
0.31
Increasing Time
0.3
Conc
[-]
0.29
0.28
t’=3.10
____ Catalytic alone
0.27
t’=3.10
- - - Homogenous and Catalytic both
0.26
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Axial length [-]
found that not much increase in conversion of propylene occurs when catalytic-homogenous
reactions are compared to catalytic reaction only. At times 3.10 the concentrations of propylene
for catalytic reaction only are 0.3489 and 0.2587 and for catalytic-homogenous reactions are
0.3349 and 0.2451 for axial distances 0.40 and 1.00 respectively. This is due to less effect of
homogenous reactions observed for propylene kinetics at these parameter values.
Figure 4: Comparison of Concentration variation for Propylene Gas with
Axial Length for Catalytic and both Catalytic and Homogenous reactions for
propylene gas at 6000ppm in mixture of Exhaust Gases at 9000C.
0.7
t’=2.30
0.6
t’=2.60
Increasing Time
0.5
t’=2.90
Conc
[-]
0.4
0.3
t’=3.10
____ Catalytic alone
- - - Homogenous and Catalytic both
0.2
0
0.1
0.2
0.3
0.4
0.5
Axial length [-]
0.6
0.7
0.8
0.9
1
Figure 5 represents propane gas concentration in an exhaust gas mixture of propane
(3000ppm) and propylene (6000ppm) for both catalytic reaction only and catalytic-homogenous
reaction at an increased inlet exhaust gas temperature of 1000oC. It is found that for the same
time variation the conversion of propane is more for homogenous-catalytic reactions. At axial
distance 0.20 the change in concentration for times 2.70 and 2.80 are 0.3137 and 0.2901 for
catalytic reactions only and 0.3065 and 0.2728 for homogenous-catalytic reactions. This is due to
the fact that more conversion occurs because of homogenous reactions.
Figure 6 represents propylene gas concentration in an exhaust gas mixture of propane
(3000ppm) and propylene (6000ppm) for both catalytic reaction only and catalytic-homogenous
reaction at an inlet exhaust gas temperature of 1000oC. It is found that conversion of propylene is
Figure 5: Comparison of Concentration variation with Axial Length for Catalytic and
both Catalytic and Homogenous reactions for Propane Gas at 3000ppm and 1000oC.
t’=2.50
t’=2.50
t’=2.60
t’=2.60
t’=2.70
0.32
0.3
t’=2.70
Increasing Time
0.28
Conc
0.26
[-]
t’=2.80
t’=2.80
0.24
____ Catalytic alone
- - - Homogenous and Catalytic both
0.22
0
0.1
0.2
0.3
0.4
0.5
Axial length [-]
0.6
0.7
0.8
0.9
1
Figure 6: Comparison of Concentration variation with Axial Length for Catalytic and
both Catalytic and Homogenous reactions for Propylene Gas at 6000ppm and 1000oC.
0.7
Increasing Time
0.6
t’=2.50
t’=2.50
0.5
t’=2.60
Conc
t’=2.60
[-]
0.4
t’=2.70
t’=2.70
____ Catalytic alone
0.3
t’=2.80
- - - Homogenous and
t’=2.80
Catalytic both
0.2
0
0.2
0.4
Axial length [-]
0.6
0.8
1
more for homogenous-catalytic reactions compared to catalytic reactions. At axial distance 0.20
the change in concentration for times 2.70 and 2.80 are 0.5186 and 0.4352 for catalytic reactions
only and 0.4910 and 0.3919 for homogenous-catalytic reactions. This is due to the fact that more
conversion of propylene occurs because of homogenous reactions.
b) Comparison of quasi steady state and unsteady state modeling for a
mixture of propylene and propane gases present in the exhaust.
For a mixture of both propane and propylene both quasi steady state and unsteady state models
are considered and a comparison between the results derived from these two models is made.
Three ordinary differential equations and one partial differential equation (3-ODEs and 1-PDE)
represent quasi steady state, whereas the unsteady state system is represented by four partial
differential equations (4-PDEs). Both heterogeneous and homogenous reaction terms are
incorporated in the rate expressions for propylene and propane oxidation reactions. Results of
gas concentrations for the two gas system have been shown in Figures 7 and 8.
Figure 7 shows the comparison of solutions obtained for concentration of propylene along the
converter length with respect to time for quasi steady state and unsteady state models. Results are
derived for dimensionless times 9.00, 11.00 and 11.30. By using the methods of solution,
insignificant changes in concentration are found. At time 11.00 the concentrations are 0.4759,
0.3608 and 0.2880 by using 3-ODEs and 1-PDE and 0.4816, 0.3663 and 0.2920 by using 4-PDEs
at axial lengths 0.30, 0.60 and 0.90 respectively. At time 11.30 the concentration is 0.2719 by
using 3-ODEs and 1-PDE and 0.2721 by using 4-PDEs at axial distance 0.70. Both methods are
giving almost same results.
Figure 7: Comparison of Concentration variation with Axial Length for Propylene Gas in Quasi steady
state (3- ODEs and 1-PDE) and Unsteady state (4-PDEs) systems.
0.7
0.6
t’=9.00
Increasing
Time
0.5
0.4
Conc
[-]
t’=11.0
0.3
- - - -4-PDEs
____3- ODEs and 1-PDE
t’=11.30
0.2
0
0.2
0.4
0.6
0.8
1
Axial length [-]
Figure 8 shows the comparison of solutions obtained for concentration of propane along the
converter length with respect to time for quasi steady state conditions and unsteady state
conditions in a mixture of propylene and propane exhausts. Results are derived for dimensionless
times 9.00, 11.00 and 11.30. Concentration of propane gas calculated in axial direction with
variation of time shows insignificant change in concentration in both cases. At time 11.00 the
Figure 8: Comparison of Concentration variation with Axial Length for Propane Gas in
Quasi steady state (3- ODEs and 1-PDE) and Unsteady state (4-PDEs) systems.
0.34
t’=9.0
0.32
Increasing
Time
t’=11.0
0.3
t’=11.30
0.28
Conc
[-]
- - - 4-PDEs
____3- ODEs and 1-PDE
0.26
0
0.2
0.4
0.6
0.8
1
Axial length [-]
concentrations are 0.3117, 0.2998 and 0.2918 by using 4-PDEs and 0.3103, 0.2975 and 0.2889
by using 3-ODEs and 1-PDE at axial lengths 0.30, 0.60 and 0.90 respectively. At time 11.30 the
concentration is 0.2750 by using 3-ODEs and 1-PDE and 0.2780 by using 4-PDEs at axial
distance 0.70, showing similar results.
CONCLUSIONS:
While considering the effect of homogenous reactions it was seen that they have insignificant
effect during warm-up conditions. Even though the activation energy of propane is much lower
for homogenous reactions still there is not much increase the overall conversion due to very low
concentration of the gases.
Comparison of modeling results for quasi steady state model with those using unsteady state
model was made for propylene and propane mixture. Analyzing these results it was observed that
there is an insignificant change in the concentrations of propylene and propane gases for the two
cases. Both proposed models represent the physical solutions. Numerical method of solution also
gives same results.
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KEYWORDS
Catalytic Converters, Modeling, Hydrocarbons Oxidation, Monoliths, One-dimensional model