SIMPLIFIED HEAT TRANSFER MODEL FOR STABELIZED PREMIXED FLAMES IN POROUS MEDIA

This work presents a simplified steady state one dimensional heat transfer model for stabilized premixed flames in porous inert media. Two energy conservation equations describe the heat transfer process in solid and fluid regions of a porous burner. The thermophysical properties are considered constant and a plug flow is adopted. The stabilized premixed flame acts as a heat source in a specified section of the domain. The energy conservation equations are discretized by the finite volume method, using upwind scheme on the convective terms and central difference scheme on the diffusive terms. The linear systems of algebraic equations are solved by Tridiagonal Matrix Algorithm (TDMA). The results are compared with experimental and theoretical data. The effects of the porosity, Peclet number and thermal conductivity ratio between the solid and the fluid on temperature fields are depicted. Furthermore, the results reveal that the model is able to represent superadiabatic flames and the heat recirculation process in the porous burner.


INTRODUCTION
There is a growing interest in developing small-dimensioned power sources with high power density to use as batteries of electronic devices.Advances in technology and miniaturization processes have led to the development of a new power generation mechanisms in micro scale.However, miniaturization of combustion devices faces scale problems for power below 1 kW due to problems with high friction between moving parts (LEE, 2009;VEERARA-GAVAN;CADOU, 2010, and VEERARAGAVAN;CADOU, 2011, JU;MARUTA, 2011).
Combustion in porous inert media has been intensively studied over the past years due to the ability of porous burners to operate within a great power range with high thermal efficiency and low pollutant emission.This allows the use of fuels with reduced heating values and leaner mixtures.The heat recirculation process makes possible to achieve superadiabatic flame temperatures (MULLER et al., 2014).Besides, the increased temperatures reached and improved efficiency allows more compact and high power devices.Other researches have shown unique characteristics of stabilized combustion in PIM.The huge internal surface contributes to a very efficient heat transport between the solid and the gas phases (DELALIC et al., 2014).
The flame heat produced increases the flame temperature due to heat recirculation.The heat is recirculated through solid conduction from the matrix in the region of the stabilized flame to the matrix upstream.Solid-to-gas convection upstream of the stabilized flame preheats the incoming reactants, resulting in local temperatures above the adiabatic flame temperature (LEE, 2009, and JU;MARUTA, 2011).
Studies have shown that porous media combustion with stabilized embedded flames emits less quantities of NOx than purely stoichiometric gaseous combustion.In addition, it is possible to have stable combustion when using lean gaseous mixture, with equivalence ratio lower than flammability of free-flame combustion, which implies in reducing the emission of toxic gases to the atmosphere.Furthermore, such burners allow an increase in the combustion rate and thermodynamic efficiency for energy conversion, by raising the temperature in the reaction zone.This greatly increases the energy efficiency of the burners and widens the range of potential fuels (VISKANTA, 2011).
One of the most important problems of porous-medium burners is stabilizing the flame in a specific zone of the inert porous medium.Four different flame control methods have been developed.The first method considers the porous medium formed of two layers, in which the Peclet number is smaller in the first layer than in the second layer.The second method considers cooling of the post combustion zone.
The third method allows retaining the flame in a specific zone of the porous medium by means of a periodic exchange of the mixture inlet and the exhaust of the combustion gases.Finally, the fourth method of flame stabilization uses a porous matrix with non-constant cross-sectional area (BUBNOVICH; HENRÍQUEZ, 2007).
A variety of different porous media are being used to hold combustion of gaseous fuels, including beds of ceramic particles, open-cell ceramic foams (reticulated ceramics), metal and ceramic fiber mats and ported metals or ceramics (VISKANTA, 2011).Different structures can be used, as reticulated foam, spheres, thin wire mesh or honeycomb.Temperature profiles and burning velocities depend strongly on the structural properties of the porous media (AKBARI &HIAHI, 2009, andBARRA et al., 2003).
Thermal resistance, mechanical strength, conductive heat transport and oxidation resistance are some of the desired characteristics for this kind of application.Aluminum oxide (Al2O3), silicon carbide (SiC), and zirconium dioxide (ZrO2) are proposed as suitable materials, while metallic materials were found less suitable for PM because of their inadequate thermal stability and high thermal inertia (ZIABASHARHAGH; MOHAMMADI, 2012).
The porous burners operating integrated to the thermophotovoltaic systems (TPV) produce high density of power and increase the thermal efficiency, allowing the development of more compact combustors.The TPV system include a micro combustor, an infrared emitter and TPV cells located around the micro combustor.The heat is transferred by radiation from the wall of the combustor to the TPV cells, which convert radiation specter photons, the radiated heat, directly into photoelectrons, electric energy (LEE, 2009).Optical filters are installed between the infrared emitter and TPV cells, separated by vacuum to minimize convective and conductive heat transfer.Since TPV systems require a temperature of operation above 1000°C, the ability to achieve superadiabatic temperatures is extremely interesting for this type of application, once less amount of fuel is required.Pereira et al. (2009) and Pereira et al. (2011) the energy equation in its suitable form for both the gas phase and the solid matrix.The two equations are simultaneously solved, once they are coupled through the convection heat exchange between them.
The energy conservation equation for the gas is given by: (1) where: Ɛ is the porosity in the upstream region, θ is the dimensionless temperature of the gas phase, θs is the dimensionless temperature of the solid phase, Pe is the Peclet number, Nu is the Nusselt number and δ is the Dirac function, which is applied on the flame position.
The energy conservation equation for the solid is defined as: (2) where Γ is the thermal conductivity ratio.
The dimensionless position, X, is given by: (3) where d m is the mean pore diameter for porous media.
The dimensionless temperature is calculated as: (4) In Equation ( 4), T n is the non-reacted temperature, T r is the reacted temperature, Cp is the gas specific heat, Y Fn is the mass fraction and Q is the heat of reaction.
A constant volumetric heat transfer coefficient, h v , is estimated using Eq. ( 5) (FU et al., 1998).The volumetric Nusselt number represents the ratio of convective to conductive heat transfer across the boundary and is calculated with correlation defined by Eq. (6).
(5) (6) proposed an approximated solution for the velocity propagation of adiabatic stationary premixed flames in porous inert media, using a one-dimensional model based on the conservation equations of mass, energy and species.Pereira et al. (2009) concluded that the porous medium increases the flame temperature and velocity.This influence is more pronounced for leaner mixtures, higher solid-phase thermal conductivities, lower porosities and lower fuel Lewis numbers.
The present work aims to develop a one-dimensional model, based on energy conservation equations for solid and gas phases, representing the heat transfer process inside a porous burner, which could be used to analyze the heat recirculation and the adiabatic flame effects on the temperature fields.

MODEL
The proposed model is based on the energy conservation equations applied to a premixed flame stabilized inside an inert porous matrix.The thermophysical properties are admitted constant and it is considered that the flame is stabilized in a known position.The proposed model considers a stationary flame within an infinite porous burner, similarly to the condition of a long burner in which the flame stabilizes deeply within the porous medium.
This paper considers that the stabilized flame occurs in the middle of the porous media.The porous media is so-called single-layer when is entirely made of the same material on both sides of the flame position and have the same physical properties as shown in  The numerical model is developed applying The conservation equations were discretized by using finite volumes method with upwind scheme, on the convective terms, and central difference scheme, on the diffusive terms.The linear systems of algebraic equations are solved by Tridiagonal Matrix Algorithm (TDMA).The computational domain is defined between -4 < ζ < 4 (PEREIRA et al., 2009).This domain is divided into 2001 volumes with same size.The combustion happens in an infinitely thin flame sheet and the flame is in ζ = 0.
The non-burnt gas is heated up by the solid phase upstream from the flame and the solid phase is heated up by the burned gas downstream from the flame.At the flame sheet, the gas temperature reaches the superadiabatic peak, and then decreases in the re--equilibrium region to the adiabatic flame temperature, Tr.As a result of the flame sheet assumption, the gas temperature is discontinuous at the flame position.

RESULTS
Initially, in order to validate the proposed numerical solution, porous medium and gas-phase temperature distributions are compared with a reference data (PEREIRA et al., 2009).
Table 1 shows the values of the model properties and parameters, considering the combustion of a lean methane-air mixture with equivalence ratio Ф = 0.8.Transport and geometric properties of the solid--phase are typical of porous burners and the gas-phase properties were approximated by the air properties evaluated at 1300 K. where λ g is the gas thermal conductivity and Pr is the Prandtl number.
Equations ( 1) and ( 2) are coupled through the Nusselt number and the temperature difference between the solid and gas phases.Since there is no heat loss, it indicates that the heat released during the combustion process is used for heating the gas phase or it is transported to the solid matrix that conducts it to the preheating zone.
The mean pore diameter is used as a parameter of the porous matrix and is empirically modeled (FU et al., 1998).The correlation presented in Eq. ( 7) uses a uniform pore distribution model.The equation is suitable for the properties range used at this work. (7 where ϕ is the linear pore density. The studied cases consider that ϕ = 50 ppi and the flame velocity is constant and equal to 0.82 m/s (PEREIRA et al, 2009).The Peclet number represents the ratio between advection and diffusive heat transport rate.It is an important parameter for flame stabilization, since it considers both matrix critical properties, as the porous size, and mixture properties, as the flame speed.This parameter is defined as: where ρ g is the gas density and SF is the flame speed.
The thermal conductive ratio is given by: (9) and the dimensionless length is written as: (10) The boundary conditions defined state that both gas and solid surfaces are at the same temperature, when distanced from the upstream flame and achieve thermal development far away from the downstream the stabilized flame.
(11) (12) Figures 3 and 4 show the simulation results and experimental data (MOSER et al., 2010).In these figures, experimental and numerical results are very close in the vicinity of the flame, proving the efficiency of the model.However, the error between experimental data and the numerical solution increases downstream, because the real burner is finite, but the model considers an infinite burner.So, in real burner, heat is lost at the end of the burner, reducing the temperature.
Moreover, for lower equivalence ratios and flame speeds the results are even more similar.This can be explained by the properties selected for the calculations.Once the gas phase properties were considered at 1300 K, it is reasonable to expect that this approximation provides better results for leaner mixtures.
Fig. 2 shows that the proposed model, developed in this paper, can correctly describe the behavior of gas--phase temperature and porous media temperature in a porous combustor, including the superadiabatic flame temperature.As expected, in the preheating zone (-4 < ζ < 0), heat is transferred from the solid to the gas due to heat recirculation, improving the combustion rate and thermodynamic efficiency.It allows burning lean mixtures, reducing the required fuel amount.
Figure 2 also shows that the dimensionless flame temperature has a peak value equals to 1.221, which corresponds to 2371 K, according to Eq. ( 4).This result indicates that the superadiabatic flame temperature was achieved.After this peak, the temperature stabilizes at θ=1.As imposed by the model, the flame is stabilized in the middle of the porous burner.
The numerical solution was also compared to Moser et al. (2010) that presents experimental results for lean and rich premixed ethanol in an Alumina (Al2O3) and Zirconia (ZrO2) porous media.In these experiments the equivalence ratio, Ф, is an important parameter.The temperature in the porous media along axial position was computed, considering rich and lean ethanol-air mixtures under different flame speed conditions.Then lean mixture cases with Ф=0.55 and Pe=8.37 and with Ф=0.55 and Pe=8.60 and rich mixture cases with Ф=1.9 and Pe=7.24 and with Ф=1.9 Pe=6.34 are evaluated, considering the parameters in Table 2.  Another important parameter is the porosity.Dividing Eqs. ( 1) and ( 2) by Ɛ, it is possible to identify the terms influenced by the porosity.They are the coupling terms between the solid and gas phases and the heat conduction term in the solid phase.Nevertheless, the porosity influences the mean pore diameter and, consequently, the Nusselt number.
Figure 7 shows that the maximum flame temperature for the gas-phase is achieved with matrix porosity equal to 0.8, a typical burners condition.The nonmonotonic behavior of the maximum flame temperature with the porosity could be explained, observing that increasing the porosity the heat conduction decreases in the solid phase, but the Nusselt number increases.A parameter analysis could be performed to evaluate the effects of the thermal conductivity ratio, Γ, the matrix porosity, Ɛ, and the Peclet number, Pe.Such analysis, can be used to select ideal conditions of the burner for different applications.
Figure 5 shows the influence of thermal conductivity ratio on dimensionless gas temperature.It is shown that higher values of Γ increase the peak flame temperature from ϴmax=1.22,Γ=60, to ϴmax=1.31,Γ=120.It can be explained, since higher values of thermal conductivity improves the heat conduction to the preheating zone, resulting in higher flames temperature.Furthermore, similarly, in Fig. 6, the porous medium presents higher temperatures downstream the flame location and lower temperatures at the pre-heating zone, meaning that the heat exchange to the gas-phase was increased.Figure 10 shows the influence of the Peclet number on the solid temperature.It is observed that the temperature in the solid phase after the flame position ζ > 0 is increased for lowest Peclet number.This result is reasonable because it was shown that gas--phase temperature at this region is also the highest.

CONCLUSION
The proposed model provided properly numerical solutions to the temperature curves in the solid and gas phases of a porous burner, predicting the superadiabatic flame temperature and the heat recirculation process.Moreover, the results agree with other theoretical results and with experimental data.
However, as expected, when the hypotheses of the model are disrespected, the model fails.It was observed when the real burner is simulated and the energy lost at the end of the burner is not considered in the model.In this case, the error in the temperature Fig. 8 indicates that, for regions downstream the flame positions, the porosity influence is reduced.However, in the preheating zone, the lowest temperature field was obtained when Ɛ=0.9, since, under such condition, the heat conduction in the solid became worst.The Peclet number take into account the flame velocity and is an important parameter for any burner design.In this way, to analyze the influence of the Peclet number, cases with the parameters presented in Tab. 1 are considered.
Figure 9 shows that the lowest Peclet number provides greater gas temperatures until the flame position.This indicates that heat recirculation and heat exchange at the preheating zone is more effective, and consequently, the superadiabatic flame temperature is also the highest.At this condition, the numerical model predicts θmax=1.30.For the slowest flame speed condition, with Pe=14.84, it was obtained θmax=1.15.
increases with the proximity of the exit region of the burner.
Furthermore, the performed parameter analysis shows that the increasing thermal conductivity ratio the superadiabatic flame temperature is also increased, since the heat recirculation is improved.So, the downstream the flame the temperature in the solid increases, but upstream the flame the temperature in the solid decreases.
The porosity effects on the superadiabatic temperature revealed a nonmonotonic behavior, because this parameter influences the conduction in the solid phase, but also influences the convection in the gas phase.These combined heat transfer process indicates the existence of an optimum condition to maximize the flame temperature.
The Peclet number is linked to the flame speed and is a key parameter to control the combustion the porous burner.The superadiabatic flame temperature decreases when the Peclet number increases, since convection is improved and the heat flows quickly far away from the flame.Conversely, the temperature in the solid became higher when the Peclet number is increases.

Fig. 1 .
The parameter ζ represents the non-dimensional position from the flame location.Negative values of the parameter ζ represent the preheating zone and positive values represent the combustion zone and further.

Figure 1 :
Figure 1: Schematic diagram for a porous burner single--layer

Figure 2 -
Figure 2 -Dimensionless temperature in the porous burner.

Figure 5 -
Figure 5 -Conductivity ratios effects on dimensionless gas temperature.

Figure 6 -
Figure 6 -Conductivity ratios effects on dimensionless solid temperature.

Figure 9 -
Figure 9 -Peclet number effects on dimensionless gas temperature.

Figure 10 -
Figure 10 -Peclet number effects on dimensionless solid temperature.

Table 1 -
Parameters of the porous burner (theoretical model).

Table 2 -
Parameters of the porous burner (experimental model).