NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA

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1 NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA E. P. Francisquetti C. Q. Carpes Graduate Program in Applied Mathematics, Federal University of Rio Grande do Sul, Porto Alegre, RS - Brazil. Av. Bento Gonçalves 9500, , PO Box 15080, Porto Alegre, RS - Brazil. A. L. De Bortoli dbortoli@mat.ufrgs.br Graduate Program in Applied Mathematics and Graduate Program in Chemical Engineering, Federal University of Rio Grande do Sul. Av. Bento Gonçalves 9500, P. O. Box 15080, Porto Alegre, RS, Brazil. Abstract. Porous burners are present in several areas of the industry due to its compact size and high efficiency. However, simulate with precision the phenomena that occur during firing inside the porous medium is very difficult due to the stiffness of the equations of chemical reaction and the non-linearity of the equations of motion. In this work we propose a model able to describe the main characteristics of the flow and predict the formation of pollutants in the combustion process of methane in a porous medium. The equations for the premixed laminar combustion of a stoichiometric methane/air mixture in a porous medium of zirconia is solved. The set of equations consists of the continuity, the momentum, the gas phase energy, solid phase energy and species concentrations. The chemical reaction is modeled by a two-step mechanism, which predicts the formation of some principal combustion products such as the carbon monoxide, the carbon dioxide and the water vapor. The discretization follows the second-order finite difference scheme. The numerical results compare favorably with data found in the literature, showing that the developed model is able to predict the combustion of methane in porous media. Keywords: Combustion, Porous Media, Numerical Simulation, Methane

2 NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA 1 INTRODUCTION The increasing world population requires more efficient use of energy resources to ensure that everyone has access to energy. Energy resources such as oil, gas and coal have been used over the years. Now the goal is to link an adequate supply of energy at affordable prices combined with an environmentally sustainable extraction. For example, the geothermal energy is gaining ground as an alternative sustainable technology, among which there is the wind, the solar or the biomass. However, these different technologies must not be considered isolated, because each has its importance and the combination of these technologies tends to have a more efficient result. Therefore, the development of alternative energy resources is necessary (Kühn, 2004). Combustion in porous media has been studied as an alternative for reducing pollutants produced by burners. During the 80s, for example, (Takeno et al., 1981) proposed a simple way of producing a flame with excess enthalpy by inserting a high-conductivity porous solid into the one-dimensional flame zone to recirculate heat internally through the solid. The results revealed several attractive features of the system, among them the combustion without pollutants formation, generating expectations for a new way of transforming energy. The formation of products depends on the type of fuel and on the physical and chemical mechanisms of combustion. One of the key mechanisms is the thermal formation of NOx, which depends primarily on the temperature of the flame and the availability of oxygen. The radiative heat transfer within the porous medium is far higher than in gases, therefore heat transfer within the porous medium increases, reducing the temperature and lowering the formation of NOx (Ingham and Pop, 2005). The materials used for combustion in the porous medium should resist high temperatures. The materials most commonly used in these applications are alumina ceramics, silicon carbide, zirconia and cordierite. The fig.1 shows the structure of some porous materials used in burners. Figure 1: Example of burners porous materials: zirconia (a), aluminum oxide (alumina) (b) and silicon carbide (c). In addition to the reduced NOx and CO (pollutants) emissions compared to those produced by free flames, porous burners have other advantages such as fuel flexibility, the ability to operate with flame speeds greater than the laminar flame speed, high burning rates, and large flame stability limits. In porous burners, the solid matrix improves the heat recirculation due to the transport

3 E. P. Francisquetti, C. Q. Carpes and A. L. De Bortoli of energy by radiation and conduction, which are orders of magnitude higher than in free flames (Barra et al., 2003); (Mendes et al., 2008); (Hackert et al., 1999); (Brenner et al., 2000). There are many configurations for porous burners, but the design most promising consists of two sections where properties such as porosity, permeability, and thermal conductivity of the solid are different. This configuration is based on the idea that the actual speed of the flame inside the porous matrix is determined by different properties of the medium (Barra et al., 2003). Generally, the porous burners can be classified into two groups: stabilized flame burners within the porous matrix and burners with flame stabilization near the surface with part of combustion taking place outside the porous matrix (Brenner et al., 2000). The porous burners can be used in domestic and industrial applications, such as for gas turbines, manufacturing tempered glass, steel, aluminum, paper and for drying paints, among others (Ingham and Pop, 2005). In this article, we obtain the numerical solution for premixed combustion of a stoichiometric misture of methane/air in porous medium of zirconia, neglecting the gas phase radiation and we analyze the thermal behavior and the generation of the combustion products. The combustion wave is first initiated by premixed combustion at the left end of the reactor, and then it is propagated slowly along the porous medium. The flow interacts with the solid matrix releasing heat proportionally to the temperature difference between the gas and the solid. The chemical reaction is modeled by a two-step mechanism, which predicts the formation of only the principal combustion products such as the carbon monoxide, the carbon dioxide and the water vapor. The set of equations consists of the continuity, the momentum, the gas phase energy, solid phase energy and species concentrations. The discretization follows the Gauss-Seidel finite difference scheme of second order with the TVD (Total Variational Diminishing) scheme to avoid spurious oscillations. 2 GOVERNING EQUATIONS For the premixed laminar combustion of methane is considered a mechanism of two steps. The reactive flow interacts with the solid matrix, releasing heat. The heat released is assumed to be proportional to the temperature difference between the gas and the solid. The porous material is considered to be inert, and the gas phase radiation is neglected. To reduce the number of variables of the problem, the equations are written in their dimensionless form. Furthermore, this form allows to compare similar physical phenomena, even though they have different spatial and temporal scales. Thus, the set of dimensionless equations is composed by the momentum equation, gas phase energy, solid phase energy and mass fraction of species and are given by: Momentum equation for the x axis ( u t + u u x + v u ) = φ 2 P y x + 1 ( 2 ) u Re x + 2 u 2 y 2 Momentum equation for the y axis ( v t + u v x + v v ) = φ 2 P y y + 1 ( 2 ) v Re x + 2 v 2 y 2 (1) (2)

4 NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA Gas phase energy equation T g t + u T g x + v T ( g 1 2 ) [ T g = y ReP r x + 2 T g + HeY 2 y 2 F Y O Da exp Ze(1 T ) ] 1 α(1 T ) + ξ(t g T s ) (3) Solid phase energy equation T s t = 1 ( 2 ) T s ReP r x + 2 T s + ξ(t 2 y 2 g T s ) (4) Fuel mass fraction equation Y F t + u Y F x + v Y F y = 1 ScRe Oxigen mass fraction equation Y O t + u Y O x + v Y O y = 1 ScRe Products mass fraction equation Y P t +u Y P x +v Y P y = 1 ScRe ( 2 y F x 2 ( 2 y O x 2 ( 2 y P x y F y 2 ) DaY F Y O exp [ Ze(1 T ) ] 1 α(1 T ) ) [ + 2 Y O DaY y 2 F Y O exp Ze(1 T ) ] 1 α(1 T ) + 2 y P y 2 ) ( ) YF +DaY F Y O s Y O (5) (6) [ exp Ze(1 T ) ] (7) 1 α(1 T ) where φ is the porosity of the medium, u and v are the velocity components along x and y Cartesian coordinates, t is the time, and P is the fluid pressure. The variables Y F, Y O e Y P represent the mass fractions of the fuel, of the oxygen and of the products respectively; s is the stoichiometric ratio between the fuel and the oxygen, and ω is the reaction rate. In addition, ξ h vl V ρc p, Da Aρ2 λ s Y Ou T b exp( β) (ρ u u s c p ) 2, He h cy F n (T b T u )c p. (8) β E α T b T u, Ze αβ (9) RT b T b In these relations Re is the Reynolds number, Sc is the Schmidt number, and P r is the Prandtl number. The α corresponds to the non-dimensional parameter of the heat release, Ze is the Zel dovich number and Da is the Damköhler number. For the chemical part, the two-step mechanism for the complete combustion of methane is given by the following reactions: CH O 2 CO + 2H 2 O (10) CO O 2 CO 2. The rates of these reactions are given, respectively, by (Turns, 2000) ( ω 1 = exp ) RT (11) (12)

5 E. P. Francisquetti, C. Q. Carpes and A. L. De Bortoli ( ω 1 = exp ) RT where R is the gas constant. (13) 3 SOLUTION PROCEDURE The finite differences method of second order is used for the discretization of the set of governing equations. For the spatial derivatives in the x and y directions, the discretization is given by central differences as ( ) f x ( ) f y = f i+1,j f i 1,j 2 x = f +1 f 1 2 y Here, the approximations of spatial derivatives of first order in the x direction are different because the standard numerical methods produce solutions with excessive oscillations. Thus, the TVD (Total Variation Diminishing) scheme is employed with the objective to eliminate spurious oscillations. The TVD property ensures that the total variation of the error in the solution procedure does not grow in the forward direction of time. For the TVD scheme consider the following approximations of first and second orders using the finite difference method for ( ) f, respectively, given by (Tannehill et al.,1997) x (14) (15) f l = f f i 1,j x (16) f h = f i+1,j f i 1,j (17) 2 x Therefore, a hybrid approximation may be derived for the first order derivative in the x direction ( f ) = f l x Φ(r i )(f l f) h (18) where Φ(r i ) is called flux limiter and is given here by (Tannehill et al., 1997) as r i = f f i 1,j f i+1,j f The second-order spatial derivatives are approximated by the central finite differences method ( 2 ) f x 2 ( 2 f y 2 ) (19) = f i+1,j 2f + f i 1,j ( x) 2 (20) = f +1 2f + f 1 ( y) 2 (21)

6 NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA where x = x i+1 x i 1 and y = y j+1 y j The computational domain is a rectangle of dimensions 6 1 and the mesh contains non equally spaced points as shown in the Fig. 2. The mesh is refined in the x direction. The distance between the nodes decreases at a rate of 10% for x between 2.5 to 3.5 and from 3.5 to 4.5 the distance between two nodes increases at the same rate. Outside this range, the grid spacing is constant. Figure 2: Computational mesh used for the numerical solution of the methane combustion in porous media. With the purpose to compare the results of this model with the results of the one-dimensional model of (Barra et al., 2003), the boundary and initial conditions adopted here are: Initial condition (t = 0) u = 0, v = 0, T = 0, Y P = 0, Y F = , Y O = Boundary conditions For t > 0, x = 0 and y > 0 For t > 0, x = 6 and y > 0 u = 1, v = 0 T = 1 Y P = 0 Y F = Y O = u = 0 x v = 0 x T = 0 x Y F x = 0 Y O x = 0 Y P x = 0

7 E. P. Francisquetti, C. Q. Carpes and A. L. De Bortoli For t > 0, x > 0, y = 0 and y = 1 For t > 0, x = 6 and y > 0 u = 0, u=0 v = 0 v = 0 T = 0 y T = 0 y Y F y = 0 Y F y = 0 Y O y = 0 Y O y = 0 Y P y = 0 Y P y = 0 The Fig. 3 shows the domain of the problem to be solved, the boundary conditions of the variables involved and the centerline (line AB) where the data are collected for comparison. Figure 3: Domain and boundary conditions for the premixed combustion of methane and air in porous media. 4 RESULTS AND DISCUSSION To validate the model, the obtained results were checked with data present by (Barra et al., 2003), whose one-dimensional model is composed by the conservation equations for mass, gas energy, solid energy and gas species. The results presented in this work are collected along the centerline of the domain, line AB. All data presented corresponds to a time equivalent of 2 seconds, that is, the data represent the system state 2 seconds after the ignition of the flame. The Fig. 4 presents the mass fraction of the methane (fuel) along the centerline of the cavity. The flame surface is located around x = 3.5, that is the place where the fuel begins to be consumed and where the chemical reaction occurs more intensively. In this model, we used the two-step mechanism for the complete combustion of methane. The first step (Eq. 10) corresponds to the formation of carbon monoxide, which is consumed in the second reaction (Eq. 11). The mass

8 NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA fraction of the carbon monoxide along the centerline of the cavity is presented in the Fig. 5. The concentration of carbon monoxide reaches its maximum near the region x = 3.5 and from this point his concentration begins to decrease. Figure 4: Mass fraction of fuel along the centerline of the cavity. Figure 5: Mass fraction of carbon monoxide along the centerline of the cavity. The figures 6 and 7 show the mass fraction of the oxygen and the mass fraction of the water vapor, respectively. For x < 3.5 the quantity of oxygen practically does not change; after that the ignition temperature is reached, the combustion occurs and the oxygen is consumed while water vapor is formed. For 3 < x < 4 we see a similar profile of mass fractions for both species, where there is a certain symmetry with respect to the y axis. Note that for x = 3, for example, oxygen begins to be consumed near the bottom and near the top of the domain. 5 CONCLUSIONS In the present work we developed a numerical scheme for the solution of the premixed combustion of methane and air in a porous medium of zirconia considering the two-step mechanism for the combustion of methane. The physical model assumes that the flow is steady, laminar, incompressible and two-dimensional. The model was based on the solution of the continuity, momentum,

9 E. P. Francisquetti, C. Q. Carpes and A. L. De Bortoli Figure 6: Profile of the mass fraction of the oxygen inside the cavity. Figure 7: Profile of the mass fraction of the water vapor inside the cavity. gas phase energy, solid phase energy and mass fraction of species. The discretization followed the central finite difference scheme with the TVD scheme to avoid spurious oscillations. The numerical tests compare well with data found in the literature (Barra et al., 2003). The principal contribution of this work is the development of a numerical model for the simulation of the reactive flow in porous media able to predict the formation of products of methane using the two-step mechanism. ACKNOWLEDGEMENTS This research is being developed at UFRGS. Carpes gratefully acknowledges the financial support from CAPES, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, while the other authors gratefully acknowledge the CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil. Prof. De Bortoli gratefully acknowledges the sponsor of the CNPq under the process /

10 NUMERICAL SOLUTION FOR THE COMBUSTION OF METHANE IN POROUS MEDIA REFERENCES Barra, A., Diepvens, J., & Henneke, M., Numerical Study of the Effects of Material Properties on Flame Stabilization in a Porous Burner. Combustion and Flame, vol. 134, pp Brenner, G., Pickenäcker, K., Pickenäcker, O., Trimis, D., Wawrzinek, K. & Weber, T., Numerical and Experimental Investigation of Matrix Stabilized Methane/Air Combustion in Porous Inert Media. Combustion and Flame, vol. 123, pp Hackert, C. L., Ellzey, J. L., & Ezekoye, O. A., Combustion and Heat Transfer in Model Two-Dimensional Porous Burners. Combustion and Flame, vol. 116, pp Ingham, D., Pop, I., Transport Phenomena in Porous Media III. Pergamon. Kühn, M., Reactive Flow Modeling of Hydrothermal Systems. Lectures Notes in Earth Sciences. Springer. Mendes, M. A. A., Pereira, J. M. C., & Pereira, J. C. F., A Numerical Study of the Stability of One-Dimensional Laminar Premixed Flames in Inert Porous Media. Combustion and Flame, vol. 153, pp Takeno, T., Sato, K., & Hase, K., A Theoretical Study on an Excess Enthalpy Flame. Eighteenth Symposium (International) on Combustion, pp Tannehill, J. C., Anderson, D. A., Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer. Taylor & Francis. Turns, S. R., An Introduction to Combustion. McGraw-Hill.

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