Journal of Thermal Science Vol.17, No.1 (2008) 84 89

Received: August 2007 Wen ZENG: Associate professor

www.springerlink.com

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

aSchool of Aero-engine and Energy Engineering, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, China bSchool 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 resi-dence 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 com-bustion 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 selec-tive removal of NOx from exhaust gases[1 3]. The mono-lith reactor contains a large number of small, long mi-cro-channels (in parallel) through which the reacting gas flows. The catalyst is deposited on the wall of the mono-lith reactor either as a porous wash-coat layer or on the wall of the micro-channels[4 6]. 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 com-bustion 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 dimen-sions (radial and axial) with the nonlinear reaction terms appearing in the boundary conditions. Since the solution of such models is time consuming, several different sim-plified models were developed in the past in order to il-lustrate the phenomenon occurring in the monolithic catalysts and to determine the simplest model that retains all the qualitative features of the system[8 11]. Balakotaiah, Gupta and West presented a new simplified model (named as the short monolith or SM model) for a cata-lytic micro-channel that retained all the qualitative steady-state bifurcation features of the full two-dimen-sional 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].

Wen Zeng et al. Bifurcation behaviors of catalytic combustion in a micro-channel 85

Nomenclature B adiabatic temperature rise Pe Peclet number c dimensionless concentration r radial co-ordinate c axial average concentration x axial co-ordinate

Lef fluid Lewis number z dimensionless axial distance P radial Peclet number

Greek letters dimensionless radial distance s transverse Thiele modulusdimensionless axial average temperature dimensionless temperature

Subscriptss 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 s , 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.

Formulation of the SM model

A cylindrical micro-channel on its surface occur a sin-gle first-order exothermic reaction is considered. The physical properties (such as the density, heat and mass diffusivities) are assumed to remain constant and the mi-cro-channel is azimuthal symmetry. With these assump-tions, the steady-state two-dimensional model in dimen-sionless form[12] is given by

2

21 1 1c cf

z P Pe zc (1)

2

21f fLe Le

fz P Pe z

(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,

.22 1f

When the characteristic time for longitudinal diffusion is much smaller compared to that for transverse diffusion, convection and reaction ( , ), the axial gradients within the micro-channel can be ig-nored and the model can be simplified by integrating the equations in the axial direction.

R L 21, 1Pe

Assumed 1

0,c c c z dz ,

1

0, z dz and integrated the two- di-

mensional model from z = 0 to 1 and obtain 1 d d 1

d dc P f c 0 (3)

1 d d 0d d f

P fLe

(4)

with boundary conditions d d 0c , d d 0 , (5a) 0

2 ˆd d , 2sc R c ,2 ˆd d , 2s fB R c Le , (5b) 1

where, ˆ , exp 1R c c r .

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 per-form 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 mi-cro-channel are focused on. Since the steady-state bifur-cation features are dependent on the choice of bifurcation variable, two such choices are considered here. In the first case, the transverse Thiele modulus ( 2

s ) 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 bi-furcation parameter, the bifurcation characters of m and

86 J.Therm. Sci., Vol.17, No.1, 2008

s are presented. In general, there are only two types of bifurcation diagrams of m and s versus 2

s , namely, single-valued and S-shaped diagram. The parameters that determine the shape of the bifurcation diagram are P, Band Lef. Therefore, in this section, the effects of these parameters on the shape of bifurcation diagrams of m

and s are discussed.

Effect of p on bifurcation diagrams of Fig.1 shows bifurcation diagrams of m and s versus

2s for B = 5.0, Lef = 0.3 and two different values of

P(0.1,10.0).

s210-3 10-2 10-1 1000

12345678910111213

B

P=10.0

P=10.0

P=0.1solid

fluid

Lef=0.3 B=5.0

Fig. 1 Effect of P on the bifurcation diagrams of

It can be seen that when , the micro-channel does behave like a homogeneous reactor and both the surface and the bulk fluid have simultaneous igni-tion/extinction. In this case both the fluid and solid (sur-face) temperatures are always close to each other and reach an asymptotic value that is equal to

1P

B for large Thiele modulus. For the case of , there are large radial gradients inside the micro-channel and once the mixed gas on the catalyst surface is ignited, the micro- channel operates in the mass transfer limited regime. It should be noted that at ignition though the surface jumps to a maximum temperature of

1P

fB Le , the fluid mean

temperature goes to Bxm, where xm is the conversion achieved in the mass transfer limited regime for the cor-responding 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.

Effect of Lef on the bifurcation diagrams of Fig.2 shows bifurcation diagrams of and s versus

2s for B = 5.0 and two different values of P (0.1,10.0)

and Lef (0.2, 2.0) respectively.

2s

10-4 10-3 10-2 10-1 1000

2

4

6

8

10

12

14

P=10.0

P=10.0

P=0.1solid

fluid

Lef=0.2 B=5.0

2s

10-3 10-2 10-1 100 1010

1

2

3

4

5

6

P=0.1

P=10.0

solid

fluid

fluid

Lef=2.0 B=5.0

Fig. 2 Effect of Lef on the bifurcation diagram of

As showed in Fig.2, for the case of P = 10.0, with the value of Lef increasing, the shapes of the bifurcation dia-grams of m and s change from the S-shape to the sin-gle-valued diagram. Moreover, the regions of multiple solutions of m and s are narrower and the temperatures of ignition and extinction of the bulk fluid remain con-stant, 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 dia-grams of m and s still remain S-shaped, but the regions of multiple solutions are narrower. Moreover, the tem-peratures of the ignition and extinction of the bulk fluid and the mixed gas on the catalyst surface remain con-stant.

Effect of B on the bifurcation diagrams of

Fig.3 shows bifurcation diagrams of m and s versus

Wen Zeng et al. Bifurcation behaviors of catalytic combustion in a micro-channel 87

2s for Lef = 0.1 and two different values of P (0.1,10.0)

and B (2.0,5.0) respectively.

2s

10-3 10-2 10-1 100 1010

2

4

6

8 P=10.0

P=10.0

P=0.1solid

fluid

B=2.0 Lef=0.1

2s

10-7 10-6 10-5 10-4 10-3 10-2 10-1 1000

5

10

15

20P=10.0

P=10.0

P=0.1solid

fluid

B=5.0 Lef=0.1

Fig. 3 Efect of B on the bifurcation diagram of

For the case of P = 10.0, with the value of B increas-ing, the region of multiple solutions of m and s are lar-ger 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 m and s 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 vari-able. In this section, only the effect of parameter s on the bifurcation diagram of s is discussed.

Fig.4 shows bifurcation diagrams of s versus 1/P for

Lef = 0.2, B = 3.0 and two different values of s (0.1,1.0). For the case of 1 P 1 , for arbitrary values of s, s

will reach a constant (B), however, for the case of 1 1P and s =1.0, a multiple solutions region of s

will be observed. When the value of s reduced to 0.1, the multiple solutions regime of s will disappear. The reason can be detected from the physical meaning of s.The square of transverse Thiele modulus, s

2, is the ratio of the radial diffusion time to the reaction time. With sincreasing, the micro-channel operates in the mass trans-fer 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 s reducing to 0.1, the micro- channel 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 s increasing, the temperatures ofignition/extinction of the mixed gas on the catalyst sur-face are increased.

1/P

s

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

2

4

6

8

10

s=1.0 B=3.0 Lef=0.2

1/P

s

10-1 100 101 102 1030

0.5

1

1.5

2

2.5

3

3.5

s=0.1 B=3.0 Lef=0.2

Fig. 4 Effect of s on the bifurcation diagram of s

88 J.Therm. Sci., Vol.17, No.1, 2008

Effects of flow models on the bifurcation dia-gram of s

In this Section, the effects of flow models on the bi-furcation diagram of s are discussed. For the case of flat velocity profile[12], .1f

The bifurcation diagrams with Thiele modulus as the bifurcation parameter

Fig.5 shows the effects of flow models on the bifurca-tion diagrams of s versus s

2 for Lef = 0.3, B = 5.0 and two different values of P (0.1,10.0).

2s

s

10-3 10-2 10-1 1000

1

2

3

4

5

6channel laminar flow

flat flow

P=0.1 Lef=0.3 B=5.0

2s

s

10-3 10-2 10-1 1000

2

4

6

8

10channel laminar flow

flat flow

P=10.0 Lef=0.3 B=5.0

Fig. 5 Effect of flow models on the bifurcation diagram of s

(for s2)

For the case when P = 0.1, the difference for the bi-furcation diagram of s between the flat velocity profile and the laminar flow model is small. However, with the value of P rising, the bifurcation curve of s for the laminar flow model will be stabilized farther away from the bifurcation curve for the flat velocity profile and the peak value of s will be higher, the region of multiple solutions is larger and the temperatures of the extinction

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 bifurca-tion diagrams of s versus 1/P for Lef = 0.2, B = 3.0 and s = 1.0.

For the laminar flow model, the region of multiple so-lutions is larger, the peak value of s 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, s will reach the value of B.

1/P

s

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

2

4

6

8

10

channel laminar flow

flat flow

s=1.0 B=3.0 Lef=0.2

Fig. 6 Effect of flow models on the bifurcation diagram of s(for transverse Peclet)

Conclusions

In this paper, a two-dimensional model (SM model) of a micro-channel with a first-exothermic reaction is ana-lyzed. 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 re-spectively. The effects of parameters such as B, Lef, Pand s on the bifurcation diagrams are analyzed. More-over, the effects of the flow models (the flat velocity pro-file and the laminar flow model) on the bifurcation be-havior and the combustion steady character of catalytic combustion in the micro-channel are presented. The fol-lowing 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 mini- change in s

2 and the great factors that affect the bifurca-tion 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 89

creasing.(2) With residence time as the bifurcation parameter,

the region of multiple solutions is narrower with s de-creasing.

(3) With the Thiele modulus as the bifurcation pa-rameter, with the value of P rising, the bifurcation curve of s 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 s is higher, the region of multiple solutions becomes larger and the ex-tinction 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 s and the extinction temperatures of the mixed gas on the catalyst surface are higher than those in the case of the flat veloc-ity profile.

Acknowledgments

This research is supported by the National Key Basic Research Project of China (No.2001CB209201).

References

[1] C.T. Goralski, L.D.Schmidt. Modeling Heterogeneous and Homogeneous Reactions in the High-Temperature Catalytic Combustion of Methane. Chemical Engineering Science, vol.54, pp. 5791 5807, (1999).

[2] O. Deutschmann, F.Behrendt and J.Warnatz. Formal Treatment of Catalytic Combustion and Catalytic Con-version of Methane. Catalysis Today, vol.46, pp. 155 163, (1998).

[3] S. Bhattacharyya and R.K.Das. Catalytic Control of Automotive NOx: A Review. International Journal of En-ergy Research, vol.23, pp.351 369, (1999).

[4] L.L.Raja, R.J.KEE, O.Deutschmann, J.Warnatz and L.D.Schmidt. A Critical Evaluation of Navier-Stokes, Boundary-Layer, and Plug-Flow Models of the Flow and Chemistry in a Catalytic-Combustion Monolith. Catalysis Today, vol.59, pp.47 60, (2000).

[5] O.Deutschmann, R.Schwiedernoch, L.I.Maier and. Chat-terjee. Natural Gas Conversion in Monolithic Catalysts: Interaction of Chemical Reactions and Transport Phe-nomena. Study in Surface Science and Catalysis, vol.136, pp.251 258, (2001).

[6] R.Schwiedernoch, S.Tischer, O.Deutschmann and J.Warnatz. Experimental and Numerical Investigation of the Ignition of Methane Combustion in a Plati-num-Coated Honeycomb Monolith. Proceedings of Combustion Institute, vol.29, pp. 1005 1011, (2002).

[7] R.E.Hayes and S.T.Kolaczkowski. Introduction to Cata-lytic Combustion. Gordon and Breach Science, Amster-dam, 1997.

[8] L.L.Hegedus. Temperature Excursions in Catalytic Monoliths. A.I.Ch.E.Journal, vol.21, pp.849 853, (1975).

[9] G.Damk hler. Influences of Diffusion, Fluid Flow and Heat Transport on the Yield in Chemical Reactors. Der Chemie-Ingeniur, vol.3, pp.359 485, (1937).

[10] L.C.Young and B.A.Finlayson. Mathematic Models of Monolith Catalytic Converter. A.I.Ch.E.Journal, vol.22, pp. 331 353, (1976).

[11] R.H.Heck, J.Wei and J.R.Katzer. Mathematical Modeling of Monolithic Catalysts. A.I.Ch.E.Journal, vol.22, pp. 477 483, (1976).

[12] V.Balakotaiah, N.Gupta and D.H.West. A Simplified Model for Analyzing Catalytic Reactions in Short Mono-liths. Chemical Engineering Science, vol.55, pp. 53675383, (2000).

[13] N.Gupta, V.Balakotaiah and D.H.West. Bifurcation Analysis of a Two-dimensional Catalytic Monolith Reac-tion Model. Chemical Engineering Science, vol.56, pp.1435 1442, (2001).