Wei Xu a. aSchool of Aero-engine and Energy Engineering, Shenyang Institute of Aeronautical Engineering, Shenyang 110034,. China. bSchool of Energy and ...
Journal of Thermal Science Vol.17, No.1
(2008)
8489
DOI: 10.1007/s11630-008-0084-z
Article ID: 1003-2169(2008)01-0084-06
Bifurcation behaviors of catalytic combustion in a micro-channel Wen Zeng a*
Maozhao Xie b
Hongan Maa
Wei Xua
a
School of Aero-engine and Energy Engineering, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, China b School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China
Bifurcation analysis of ignition and extinction of catalytic combustion in a short micro-channel is carried out with the laminar flow model incorporated as the flow model. The square of transverse Thiele modulus and the residence time are used as bifurcation parameters. The influences of different parameters on ignition and extinction behavior are investigated. It is shown that all these parameters have great effects on the bifurcation behaviors of ignition and extinction in the short micro-channel. The effects of flow models on bifurcation behaviors of combustion are also analyzed. The results show that in comparison with the flat velocity profile model, for the case of the laminar flow model, the temperatures of ignition and extinction of combustion are higher and the unsteady multiple solution region is larger.
Keywords: catalytic combustion, bifurcation theory, short monolith
Introduction Catalytic monoliths are used in automobile converts, power generation, partial oxidation reactions and selective removal of NOx from exhaust gases[13]. The monolith reactor contains a large number of small, long micro-channels (in parallel) through which the reacting gas flows. The catalyst is deposited on the wall of the monolith reactor either as a porous wash-coat layer or on the wall of the micro-channels[46]. While conventional combustion occurs in the presence of a flame, catalytic combustion is a flameless process taking place at lower temperatures, therefore, results in lower emissions of nitrogen oxides[7]. Furthermore, catalytic combustion offers fewer constrains concerning flammability limits and reactor design. These advantages of catalytic combustion permit its potential wide applications. The reactant inside the micro-channel is transported to the surface by transverse diffusion and is carried forward by convection and axial diffusion, thus producing con-
centration gradients in both axial and radial directions. The steady-state behavior of a micro-channel is described by partial differential equations in at least two dimensions (radial and axial) with the nonlinear reaction terms appearing in the boundary conditions. Since the solution of such models is time consuming, several different simplified models were developed in the past in order to illustrate the phenomenon occurring in the monolithic catalysts and to determine the simplest model that retains all the qualitative features of the system[811]. Balakotaiah, Gupta and West presented a new simplified model (named as the short monolith or SM model) for a catalytic micro-channel that retained all the qualitative steady-state bifurcation features of the full two-dimensional model[12]. They used the SM model to study the steady-state bifurcation behavior of the micro-channel and derived many analytical results. However, in their bifurcation analysis, the flat velocity profile was used as the flow model, which is not an ideal flow model for the gas flow in the micro-channel[13].
Received: August 2007 Wen ZENG: Associate professor www.springerlink.com
Wen Zeng et al. Bifurcation behaviors of catalytic combustion in a micro-channel
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Nomenclature B c c Lef P Greek letters Y R Subscripts
s
Peclet number radial co-ordinate axial co-ordinate
adiabatic temperature rise dimensionless concentration axial average concentration fluid Lewis number radial Peclet number
Pe r x z
dimensionless radial distance dimensionless axial average temperature
Gs
T
transverse Thiele modulus dimensionless temperature
surface (or solid phase)
m
mean
In this paper, a complete bifurcation analysis of the short micro-channel by using the laminar flow model instead of the flat velocity profile as the flow model is presented. The square of the transverse Thiele modulus and the residence time is used as bifurcation parameter. The effects of parameters such as Gs , B, Lef and P on steady combustion characteristics are analyzed. Moreover, the effects of the flow models (the flat velocity profile and the laminar flow model) on the bifurcation behaviors and the steady characteristics of catalytic combustion in the micro-channel are presented.
dimensionless axial distance
1
R R Y ¨ R Y , z dz and integrated the two- di0
mensional model from z = 0 to 1 and obtain ¬¯ ¡ 1 d Y dc ° P f Y ¯ 1 c 0 ¢ ± ¡ Y dY dY ®° ¢ ± ¬¯ ¡ 1 d Y dR ° P f Y ¯ R 0 ± ¡ Y dY dY ®° Le ¢ ¡¢ °± f with boundary conditions dc dY 0 , dR dY 0 , Y 0 d c d [ I 2 Rˆ c , T 2 ,
(3) (4)
(5a)
s
Formulation of the SM model A cylindrical micro-channel on its surface occur a single first-order exothermic reaction is considered. The physical properties (such as the density, heat and mass diffusivities) are assumed to remain constant and the micro-channel is azimuthal symmetry. With these assumptions, the steady-state two-dimensional model in dimensionless form[12] is given by sc 1 1 s sc ¬¬ 1 s 2 c f Y (1) Y sz P Y sY sY ®® Pe sz 2 sR Le f 1 s sR ¬¬ Le f s 2R (2) Y sz P Y sY sY ®® Pe sz 2 The boundary conditions and various dimensionless groups appearing in above equations are as shown in [12]. For the case of laminar flow inside the channel, f Y
f Y 2 1 Y 2 .
When the characteristic time for longitudinal diffusion is much smaller compared to that for transverse diffusion, convection and reaction ( R L , Pe 1, G 2 1 ), the axial gradients within the micro-channel can be ignored and the model can be simplified by integrating the equations in the axial direction. 1
Assumed c c Y ¨ c Y , z dz , 0
c , R 2 Le f , where, Rˆ c , R c exp R 1 R r
. dR dY
B ¸ Gs2 ¸ Rˆ
Y 1
(5b)
A comparison with the general model has shown that the results of the SM model presented an ideal agreement that retained all the qualitative steady-state bifurcation features of the full two-dimensional model as shown in [12]. Hence, in this paper, this model is also used to perform a bifurcation analysis of ignition and extinction of catalytic combustion in a short micro-channel. However, differing from [12] in which the flat velocity profile was used as flow model, the laminar flow model is used as our flow model. In this paper, the bifurcation behaviors of a short micro-channel are focused on. Since the steady-state bifurcation features are dependent on the choice of bifurcation variable, two such choices are considered here. In the first case, the transverse Thiele modulus ( Gs2 ) is taken as the bifurcation variable. In the second case, the residence time is taken as the bifurcation variable.
Classification of the bifurcation diagrams with Thiele modulus as the bifurcation parameter With the transverse Thiele modulus taken as the bifurcation parameter, the bifurcation characters of Tm and
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J.Therm. Sci., Vol.17, No.1, 2008
Ts are presented. In general, there are only two types of bifurcation diagrams of Tm and Ts versus Gs2 , namely, single-valued and S-shaped diagram. The parameters that determine the shape of the bifurcation diagram are P, B and Lef. Therefore, in this section, the effects of these parameters on the shape of bifurcation diagrams of Tm and Ts are discussed.
Effect of Lef on the bifurcation diagrams of T Fig.2 shows bifurcation diagrams of T and Ts versus 2 Gs for B = 5.0 and two different values of P (0.1,10.0) and Lef (0.2, 2.0) respectively. 14 12
Effect of p on bifurcation diagrams of T
8
solid
P=0.1
6 fluid
4
P=10.0
2 0 -4 10
P=10.0
10
-3
-2
10 2 Is
6
T
13 12 Lef=0.3 B=5.0 11 10 9 8 7 solid 6 B 5 4 3 2 1 0 -3 -2 10 10
P=10.0
10 T
Fig.1 shows bifurcation diagrams of Tm and Ts versus 2 Gs for B = 5.0, Lef = 0.3 and two different values of P(0.1,10.0).
Le f=0.2 B=5.0
5
P=0.1
10
-1
10
0
10
1
P=0.1
Le f=2.0 B=5.0
fluid
4
fluid
T
P=10.0
solid
3 P=10.0 fluid
2 -1
2
Is
10
0
10
1
Fig. 1 Effect of P on the bifurcation diagrams of T 0
It can be seen that when P 1 , the micro-channel does behave like a homogeneous reactor and both the surface and the bulk fluid have simultaneous ignition/extinction. In this case both the fluid and solid (surface) temperatures are always close to each other and reach an asymptotic value that is equal to B for large Thiele modulus. For the case of P 1 , there are large radial gradients inside the micro-channel and once the mixed gas on the catalyst surface is ignited, the microchannel operates in the mass transfer limited regime. It should be noted that at ignition though the surface jumps to a maximum temperature of B Le f , the fluid mean temperature goes to Bxm, where xm is the conversion achieved in the mass transfer limited regime for the corresponding transverse Peclet number (P). At the same time, for the case of =10.0, compared with P = 0.1, the temperatures of ignition/extinction of the bulk fluid are lowered, and the temperature of ignition of the mixed gas on the catalyst surface remains constant, however, the temperature of extinction of the mixed gas on the catalyst surface is increased.
10
-3
10
-2
10
-1
10
0
2
Is
Fig. 2
Effect of Lef on the bifurcation diagram of T
As showed in Fig.2, for the case of P = 10.0, with the value of Lef increasing, the shapes of the bifurcation diagrams of Tm and Ts change from the S-shape to the single-valued diagram. Moreover, the regions of multiple solutions of Tm and Ts are narrower and the temperatures of ignition and extinction of the bulk fluid remain constant, however, the extinction temperatures of the mixed gas on the catalyst surface is lower. For the case of P = 0.1, with the value of Lef increasing, the bifurcation diagrams of Tm and Ts still remain S-shaped, but the regions of multiple solutions are narrower. Moreover, the temperatures of the ignition and extinction of the bulk fluid and the mixed gas on the catalyst surface remain constant.
Effect of B on the bifurcation diagrams of T Fig.3 shows bifurcation diagrams of Tm and Ts versus
Wen Zeng et al. Bifurcation behaviors of catalytic combustion in a micro-channel
Gs2 for Lef = 0.1 and two different values of P (0.1,10.0) and B (2.0,5.0) respectively.
8
B=2.0 Le f=0.1
P=10.0
T
6
4 P=0.1 solid
2
fluid P=10.0
0 -3 10
20
10
-2
-1
10 I2s
10
0
10
1
B=5.0 Le f=0.1 P=10.0
15
87
Lef = 0.2, B = 3.0 and two different values of Is (0.1,1.0). For the case of 1 P 1 , for arbitrary values of Is, Ts will reach a constant (B), however, for the case of 1 P 1 and Is =1.0, a multiple solutions region of Ts will be observed. When the value of Is reduced to 0.1, the multiple solutions regime of Ts will disappear. The reason can be detected from the physical meaning ofIs. The square of transverse Thiele modulus,Is2, is the ratio of the radial diffusion time to the reaction time. With Is increasing, the micro-channel operates in the mass transfer limited regime. There are large radial temperature gradients inside the micro-channel between the bulk fluid and the mixed gas on the catalyst surface. The process of combustion of the whole mixed gas in the micro-channel will be unstable. With Is reducing to 0.1, the microchannel does behave like a homogeneous reactor and both the surface and the bulk fluid have simultaneous ignition/extinction points, thus the combustion process of the whole mixed gas in the micro-channel will be stable. At the same time, with Is increasing, the temperatures of ignition/extinction of the mixed gas on the catalyst surface are increased.
T
10 10
Is=1.0 B=3.0 Lef=0.2 P=0.1
8
solid
5
fluid P=10.0
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Ts
0 -7 10
6
2
Is
4
Fig. 3 Efect of B on the bifurcation diagram of T 2
10
-7
10
-6
10
-5
-4
10
-3
10 1/P
-2
10
-1
10
10
0
3.5 3
Is=0.1 B=3.0 Le f=0.2
2.5 2 Ts
For the case of P = 10.0, with the value of B increasing, the region of multiple solutions of Tm and Ts are larger and the temperature of the ignition and extinction of the bulk fluid and the mixed gas on the catalyst surface will increased. For the case of P = 0.1, with the value of B decreasing, the bifurcation diagram of Tm and Ts will change from the S-shaped diagram to the single-valued diagram. Moreover, the temperatures of extinction of the bulk fluid and the mixed gas on the catalyst surface will be increased.
Classification of the bifurcation diagrams with residence time as the bifurcation parameter The steady-state behavior becomes rather complex when the residence time is taken as the bifurcation variable. In this section, only the effect of parameter Gs on the bifurcation diagram of Ts is discussed. Fig.4 shows bifurcation diagrams of Ts versus 1/P for
1.5 1 0.5 0 -1 10
100
101 1/P
102
103
Fig. 4 Effect of Is on the bifurcation diagram of Ts
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Effects of flow models on the bifurcation diagram of Ts In this Section, the effects of flow models on the bifurcation diagram of Ts are discussed. For the case of flat velocity profile[12], f Y 1 . The bifurcation diagrams with Thiele modulus as the bifurcation parameter Fig.5 shows the effects of flow models on the bifurcation diagrams of Ts versus Is2 for Lef = 0.3, B = 5.0 and two different values of P (0.1,10.0). 6
of the mixed gas on the catalyst surface will be increased. The bifurcation diagrams with residence time as the bifurcation parameter Fig.6 shows the effects of flow models on the bifurcation diagrams of Ts versus 1/P for Lef = 0.2, B = 3.0 andIs = 1.0. For the laminar flow model, the region of multiple solutions is larger, the peak value of Ts and the temperatures of the extinction of the mixed gas on the catalyst surface are higher than that of the flat velocity profile. At the same time, with the residence time prolonging, for the two cases, Ts will reach the value of B. 10
channel laminar flow
Is=1.0 B=3.0 Lef=0.2
5
8
flat flow
channel laminar flow
6
3
Ts
Ts
4
2
P=0.1 Le f=0.3 B=5.0
1 0 -3 10
flat flow
4
2 10
-2
10
I2s
-1
10
0
10 channel laminar flow
10
-7
10
-6
10
-5
10
-4
-3
10 1/P
10
-2
10
-1
10
0
Fig. 6 Effect of flow models on the bifurcation diagram of Ts (for transverse Peclet)
8
Conclusions
Ts
flat flow
6 4 P=10.0 Le f=0.3 B=5.0
2 0 -3 10
10
-2 2
10
-1
10
0
Is
Fig. 5 Effect of flow models on the bifurcation diagram of Ts 2 (forIs )
For the case when P = 0.1, the difference for the bifurcation diagram of Ts between the flat velocity profile and the laminar flow model is small. However, with the value of P rising, the bifurcation curve of Ts for the laminar flow model will be stabilized farther away from the bifurcation curve for the flat velocity profile and the peak value of Ts will be higher, the region of multiple solutions is larger and the temperatures of the extinction
In this paper, a two-dimensional model (SM model) of a micro-channel with a first-exothermic reaction is analyzed. Bifurcation analysis of ignition and extinction of catalytic combustion in this model is carried out for the condition when the laminar flow model is used as the flow model. The square of transverse Thiele modulus and the residence time is used as bifurcation parameter respectively. The effects of parameters such as B, Lef, P and Is on the bifurcation diagrams are analyzed. Moreover, the effects of the flow models (the flat velocity profile and the laminar flow model) on the bifurcation behavior and the combustion steady character of catalytic combustion in the micro-channel are presented. The following results are obtained in this study: (1) With Thiele modulus as the bifurcation parameter, the bulk fluid and the mixed gas on the catalyst surface present obvious bifurcation character with the minichange in Ts2 and the great factors that affect the bifurcation character are B, P and Lef. The region of multiple solutions is enlarged with Lef decreasing and B, P in-
Wen Zeng et al. Bifurcation behaviors of catalytic combustion in a micro-channel
creasing. (2) With residence time as the bifurcation parameter, the region of multiple solutions is narrower with Is decreasing. (3) With the Thiele modulus as the bifurcation parameter, with the value of P rising, the bifurcation curve of Ts for the case of the laminar flow model is stabilized farther away from the bifurcation curve for the case of flat velocity profile and the peak value of Ts is higher, the region of multiple solutions becomes larger and the extinction temperatures of the mixed gas on the catalyst surface are increased. (4) With residence time as the bifurcation parameter, for the case of the laminar flow model, the region of multiple solutions is larger, the peak value of Ts and the extinction temperatures of the mixed gas on the catalyst surface are higher than those in the case of the flat velocity profile.
Acknowledgments This research is supported by the National Key Basic Research Project of China (No.2001CB209201).
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