A numerical evaluation of combustion in porous media by EGM (Entropy Generation Minimization)

Energy 35 (2010) 3483e3500

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A numerical evaluation of combustion in porous media by EGM (Entropy Generation Minimization) M. Bidi, M.R.H. Nobari*, M. Saffar Avval Mechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Avenue, P.O. Box: 15875-4413, Tehran, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 October 2009 Received in revised form 15 April 2010 Accepted 29 April 2010 Available online 2 June 2010

Combustion in Porous Media provides interesting advantages compared with the free flame combustion due to the higher burning rates, increased power dynamic range, the extension of lean flammability limits, and the low emissions of pollutants. A numerical code is developed in order to evaluate the effects of different parameters of combustion in porous media. The governing equations including NaviereStokes, the solid and gas energy and the chemical species transport equations are solved using a multistep reduced kinetic mechanism. Flame stabilization and the burner optimization are studied by EGM (Entropy Generation Minimization) method considering the effects of chemical affinities and reaction. It is found that the flames occurring at the upstream half of the porous layer are more stable and more efficient, producing less emissions than those occur at the downstream half of porous layer. Also at a specified equivalence ratio both the heat recirculation efficiency and the Merit number have similar trend by changing the flame location. For a FFL (Fixed Flame Location), there is an optimum value of equivalence ratio at which the burner efficiency is a maximum. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: Combustion Porous media Entropy generation minimization Finite volume

1. Introduction Combustion in porous media differs significantly from free flames due to two main factors including the high surface area of the porous media which provides an efficient heat transfer between the gas and the solids, and well mixing of fuel and oxidant in porous media which augments effective diffusion and heat transfer in the gas phase. These phenomena are an internally selforganized process of heat recuperation. Heat transfer from the high temperature post-flame zone by radiation and conduction through solid medium heats the porous solid in the pre-flame zone, which, in turn, preheats the incoming reactants by convection. This regenerative internal heat feedback mechanism namely heat recirculation results in several interesting characteristics relative to free-burning flames such as higher burning speeds, extension of the lean flammability limits, and the low emission of pollutants. The low emission of porous burners comes from their extended stable operating region allowing operation in the lean equivalence ratios. In other words, the flame stabilization in porous media differs from the

* Corresponding author. Fax: þ98 2166419736. E-mail address: [email protected] (M.R.H. Nobari). 0360-5442/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2010.04.053

conventional free flames primarily due to the heat recirculation provided by the solid matrix. For a certain equivalence ratio, stable operating range is defined as the range between the minimum and the maximum burning velocities at which the flame stabilizes within the porous medium. The lean flammability limit is the lowest equivalence ratio beyond which the flame does not stabilize inside the porous medium for any inlet velocity. Therefore, to extend the flame stability limits, it is necessary to evaluate the burning velocity and its dependence on the heat transfer properties of the porous media. For increasing the burning speeds and extending the flame stability limits in porous media, the heat recirculation has to be increased. Another important characteristic of porous burners is their output radiant flux. Thus an optimum design of a porous burner is a condition at which both the heat recirculation and the output radiant fluxes reach their highest possible values. In the upstream, heat is transferred from the solid matrix to the inlet fueleair mixture (preheat effect) and in the downstream, heat of combustion is transferred from the hot gas products to the solid matrix which in turn radiates to its surrounding. These heat transfer mechanism leads to the increase of temperature up to the adiabatic temperature [1]. Super adiabatic combustion significantly extends the lean flammability limit to the equivalence ratio less than 0.4 [2,3]. The combined effect of lean mixtures and low temperature gradients lead to the low emission of NOx and CO [4].

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M. Bidi et al. / Energy 35 (2010) 3483e3500

Nomenclature chemical reaction affinity, J mol1 Ar Be Bejan number Br Brinkman number modified Brinkman number Brm specific heat capacity, J kg1 K-1 cp Da Darcy number species diffusivity, m2 s1 Dkm Eg dimensionless entropy generation characteristic pore size, m dp h heat transfer coefficient, W m2 K1 hk species specific enthalpy, J kg1 Hv volumetric heat transfer coefficient, W m3 K1 _I rate of irreversibility, W radiant intensity, W m2 sr1 Ih k1 permeability coefficient, m2 k2 inertial loss coefficient, m L porous layer length, m M merit number N total number of involved species Nu Nusselt number P pressure, Pa PPC pores per centimeter Pr Prandtl number available exergy, W Q_ a fuel heating value, J kg1 Qfuel _ radiant heat transfer rate, W Qr radiant heat flux, W m2 q_ 00r qr radiant heat flux vector, W m2 00 _ burner nominal thermal load, W m2 Q th Qrecirc recirculation heat flux, W m2 Rep pore Reynolds number chemical reaction rate, mol m3 s1 Rr Ru universal gas constant, J mol1 K1 b s direction of radiant intensity total entropy generation rate, W K1 S_ gen S000 volumetric entropy generation rate, W m3 K1 gen v true or physical velocity, m s1 Vk species diffusion velocity, m s1

Because of above mentioned advantages, in recent years many researchers have focused on the development of porous media burner technology either experimentally or theoretically. Zhou and Pereira [5] have performed a one-dimensional numerical simulation of methaneeair flame in porous media using a multi-step reaction mechanism. They have shown that NO and CO emission decreases significantly within porous media combustion. A two-dimensional numerical simulation has been performed by Malico and Pereira [6] with a single-step reaction using a multistep chemistry with 26 species and 77 reactions to analyze pollutant formation [7]. Some researchers have studied the effects of various porous media parameters on the combustion characteristics. The study by Sathe et al. [8] has shown the strong effect of convective heat transfer coefficient and radiative properties on the prediction of the gas and solid phase temperature field. Malico and Pereira [9] have performed a two-dimensional numerical study on the porous media combustion using the some approximations for solving the radiative transfer equation. They concluded that the temperature distribution strongly depends on the radiative properties especially in the scattering and that in the absence of radiation the results are

vs Wk xk Yk Yfin ro SL SL0 T

superficial velocity, m s1 species molecular weight, kg mol1 species mole fraction species mass fraction inlet fuel mass fraction outlet radius, m laminar flame speed, m s1 free flame burning speed, m s1 Temperature, K

Greek

e l m mk r se sa s ss s f Fh Fd u_ k u U

emissivity thermal conductivity, W/m-K dynamic viscosity, kg/m-s species chemical potential, J mol1 density, kg/m3 extinction coefficient, m1 absorption coefficient, m1 Stefan Boltzmann constant 5.667  108 scattering coefficients, m1 optical thickness, se L porosity scattering phase function momentum dissipation rate, s2 species molar generation rate, mol/m3-s scattering albedo, ss =se solid angle, sr

Subscripts awa area weighted average adb adiabatic g gas phase in inlet section of porous layer mwa mass weighted average in inlet section of porous layer P phase average s solid phase h at a given wave number w wall

not in good agreement with the experimental data. Brenner et al. [10] have reported the laminar and turbulent permeability tensor, effective heat conductivity and emissivity of the porous medium as a function of temperature for Alumina fiber and SiC lamellae structure. Definition of the range of stable combustion in porous media is an important issue which has been studied by different researchers [11e13]. The burning velocity and its effect on the various parameters of the porous media and other physical properties are the main factors in the definition of stable flame inside a porous medium. Using multi-step reaction mechanisms in numerical combustion studies leads to a better prediction of combustion characteristics. Chen et al. [14] have used a multi-step chemistry and achieved temperatures higher than the adiabatic temperatures in porous media. They have shown that the flame thickness in the combustion zone is thicker in the case of multi-step chemistry than the one in the single-step reaction. Hsu and Matthews [15] have obtained better predictions for NO, CO and CO2 by adding 3 more reactions of Zeldovich NOx mechanisms. Zhou and Pereira [16] have compared four reaction mechanisms (detailed mechanism with 277 reactions, Skeletal mechanism with 77 reactions, 4-step

M. Bidi et al. / Energy 35 (2010) 3483e3500

reduced mechanism and a single-step global mechanism) to predict the temperature distribution and Species concentrations. For optimization of energy consumption in systems dealing with large amount of energy, the design methods based on the Second law of thermodynamic are widely used to measure the irreversibility of processes. Conserving useful energy depends on how to design an efficient heat transfer process from thermodynamic point of view. Energy conversion processes are accompanied by an irreversible increase in entropy, which leads to a decrease in exergy. Thus, even though energy is conserved, the quality of the energy decreases by converting it into a different form of energy at which less work can be obtained. Reduced entropy generation will result in more efficient designs of energy systems. Therefore, in recent years, EGM (Entropy Generation Minimization) has become a topic of great interest in the thermo-fluid area. Bejan [17] has focused on the different aspects of entropy generation in applied thermal engineering where the generation of entropy destroys the available work of a system. Consequently, it is of great interest to focus on the irreversibility of the heat transfer and fluid flow processes to better understand the function of associated entropy generation mechanisms. The study by Bejan [18] has also conducted an extensive review on the EGM. Furthermore, many researchers have studied the entropy generation in various flow cases. Yapıcı et al. [19] performed a numerical solution of the local entropy generation rate in a methane-air combustion chamber using a two-step reaction mechanism. They calculated entropy generation rate due to the high temperature and velocity gradients in the combustion chamber for various fuel flow rates, but they did not take into account entropy generation by irreversibility of chemical reactions. A comprehensive equation to determine the rate of local entropy generation in multi-component reacting laminar fluid flow involving heat and mass transfer is formulated by Teng et al. [20]. This equation involves a term for entropy generation due to chemical reactions. In the present study, combustion in two-dimensional porous media is numerically simulated using a multi-step reaction mechanism. Separate energy equations for the gas and solid phases are considered where the radiation effects of the solid phase are included. A second order finite volume scheme is applied to discretize the conservation equations. For the first time, the flame stabilization of a submerged flame in porous media is analyzed by EGM method. Also the effects of porosity, equivalence ratio, and thermal loads on the porous burner performance are studied by EGM method to achieve optimum design conditions. By including the entropy generation due to chemical reactions in the total entropy generation equation, the importance of the EGM method in the flame stabilization in porous media and the efficient design of porous burner are investigated.

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2. Mathematical model 2.1. Governing equations The schematic diagram of the problem under consideration is shown in Fig. 1. A premixed methaneeair gas mixture enters in an adiabatic axi-symmetric duct containing an inert porous layer of length L. The flame may be located either inside or outside the porous layer. The gas-phase boundary conditions extend upstream and downstream of the foam, between x ¼ xin and x ¼ xout , to ensure that the flame inside the foam is least influenced. The Solid matrix is assumed to be gray and gaseous radiation is neglected compared to the solid radiation. Non-local thermal equilibrium between the gas and the solid phase is accounted for considering separate energy equations for each phase. Due to the presence of the solid and fluid phases in the porous media, the solid matrix extracts energy from the reaction zone. It is assumed that the solid is chemically inert. For more accurate simulation of flow in porous media, it is necessary to solve the true or physical velocity formulation. Using the physical velocity formulation, the steady state governing equations in an isotropic porous media can be written as: Continuity equation:

V$ðfr vÞ ¼ 0

(1)

where v is the true or physical velocity vector, f is the porosity, and r is the density of the gas mixture. Momentum equation:

V$ðfr v vÞ ¼ fVP þ V$ðfmVvÞ  ðVPÞP

(2)

where the last term is the pressure drop of the porous matrix which is calculated via:

ðVPÞP ¼

m k1

vs þ

m k2

jvs j

(3)

where vs is the superficial velocity and is related to the physical velocity by vs ¼ f v, k1 and k2 are the permeability and inertial loss coefficients, respectively, and can be estimated as:

k1 ¼

dp f3 f3 ; k ¼ 2 150 ð1  fÞ2 1:75 ð1  fÞ d2p

(4)

where dp is the mean particle diameter (packed beds) or the characteristic pore size (foam matrices). In foam matrix, dp can be calculated as [13]:

pffiffiffiffiffiffiffiffiffiffiffiffi 4f=p ðcmÞ dp ¼ PPC

(5)

where the PPC is the pores per cm of the foam. Gas-phase energy equation:



V$ fr v cp Tg



N X   rYk Vk cpk VTg ¼ V$ flg VTg  f k¼1

f

N X

  hk u_ k Wk þ Hv Ts  Tg

(6)

k¼1

Fig. 1. Schematic of a porous burner.

where subscripts g and s denote the gas and solid phase, respectively, Vk the diffusion velocity, Yk the mass fraction, hk the enthalpy, u_ k the net molar rate of generation, Wk the molecular weight of kth species, and N is total number of involved species. Hv is the volumetric heat transfer coefficient between the solid and the gas phase and it can be calculated using the following relation:

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M. Bidi et al. / Energy 35 (2010) 3483e3500

global rates are presented in Appendix A. The global rate for each reaction is calculated by considering the forward and backward Arrhenius rates of the elementary reactions.

T=Twall (constant) L = 50 ro

Equation of State:

u in , Tin

ro

r¼

r x

Alg Nu d2p

(13)

Wm ¼ 1=

N X Yk Wk

(14)

k¼1

Fig. 2. Schematic diagram of the pipe totally filled with a porous medium.

Hv ¼

PWm Ru T

(7)

The mass diffusion coefficient of each species in the mixture, and other transport and thermodynamic data are calculated by subroutine TRANFIT [23] and thermodynamic database of CHEMKIN-II [24].

For foam, A ¼ 1, the Nusselt number can be calculated as [13]:

Nu ¼

  1:236 Re 0:0426 þ L=dp

(8)

where L is the foam length and Re ¼ r f udp =m. For a packed bed, A ¼ 6ð1  fÞ, the Nusselt number may be defined as:

Nu ¼ 2 þ 1:1Pr1=3 Re0:6 p

(9)

where Pr is the Prandtl number. Solid phase energy equation:

  0 ¼ V:ð1  f ls VTs Þ  Hv Ts  Tg  V$qr

(10)

where qr is the radiant heat flux. The diffusion or Rosseland approximation is used for optically thick media (i.e. thick foams with small pores):

16s Ts3 qr ¼  VT 3k

(11)

where s is the StephaneBoltzmann constant and k is the extinction coefficient with the base value of k ¼ 270 m1 . The extinction coefficient in terms of porosity can be estimated by k ¼ ð3=dp Þð1  fÞ[21]. Species conservation equation is expressed as:

V$ðfr v Yk Þ ¼ V$ðfr Dkm VYk Þþ fu_ k Wk ; k ¼ 1;2;.;N

(12)

where Dkm is the diffusion coefficient of kth species in the mixture. A reduced mechanism consisting of 15 reactions and 19 species is used (Barlow et al. [22]). These reactions with the corresponding

In a reacting fluid flow, irreversibility mainly occurs due to the heat transfer, the viscous effects of the fluid and chemical reactions. The entropy generation rate can be expressed as the sum of contributions of these effects, and thus, it depends functionally on the local values of velocity, temperature, and chemical reaction rates. The volumetric entropy generation rate S000 gen at each point can be calculated as

S000 gen ¼

  S000 gen

  S000 gen

fric

  þ S000 gen

(15)

reac

"    # vTg 2 vTg 2 ð1  fÞls þ ¼ þ heat T0 vx vr T0  2  2 # vTs vTs þ  vx vr

flg

"

m Fd

fric

¼ ð1  f Þ

T0

þf

m k1 T0

jvj2

(17)

1

0.8

0.6

0.6 Morosuk [28]

0.4

Present

Morosuk [28] Present

0.2

0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

(16)

The parameter f is set to unity for flow in the porous medium and to zero for flow in a region without porous material. The heat dissipation term due to friction ðFd Þ for axis-symmetric flow conditions can be expressed as:

0.8

0.4

heat

  þ S000 gen

000 000 where the ðS000 gen Þheat ; ðSgen Þfric ; and ðSgen Þreac represent the entropy generation rates due to the gas and solid heat transfer, fluid friction, and chemical reaction rate, respectively. They may be defined as [25]:

b

1

  S000 gen

r/r0

r/r0

a

2.2. Entropy generation rate

0 0.94

0.95

0.96

0.97

0.98

0.99

1

u / u in Fig. 3. Fully developed velocity and temperature profiles at the exit of a circular duct totally filled with porous medium with Da ¼ 102

M. Bidi et al. / Energy 35 (2010) 3483e3500

1

1

0.8

0.8

0.6

0.6

r/r0

r/r0

a

3487

0.4

Thermal - Morosuk [28] Thermal - Present Mechanical - Morosuk [28] Mechanical - Present Porous - Morosuk [28] Porous - Present

0.2

0 -5 10

10-4

10-3

10-2

10-1

100

101

0.4 Total - Morosuk [28] Total - Present

0.2

0 -1 10

100

101

Eg

Eg 1

b

Eg

0.9 Morosuk [28] Present

0.8

0.7

0

10

20

30

40

50

r / r0 Fig. 4. Rate of entropy generation in a circular duct totally filled with porous medium with Da ¼ 102: (a) profiles at the exit of the pipe, (b) cross sectional average.

Fd ¼ 2

"

vu vx

2  2  2 #   vv v vu vv 2 þ þ þ þ vr r vr vx

(18)

S000 gen

 reac

¼

N X k¼1

As stated by Rakopoulos et al. [34], the entropy generation of chemical reactions can be calculated as follows:



Ar ¼ 

NR 1X Rr Ar T r¼1

n0k mk

(21)

where mk is the chemical potential of kth species, and it is expressed based on the chemical potential at the standard conditions, m0k :

(19)

2000

Equivalence Ratio = 0.9 Flow of the Fuel= 5 lpm

where Rr and Ar are the chemical reaction rate and affinity of rth reaction, respectively. Consider a common equilibrium reaction.

k¼1

v0k Wk 4

N X i¼1

1500

v00k Wk

(20) Temperature (K)

N X

Affinity for this reaction is defined as:

x L

Yapici et al. [19]

1000

Present

Burner Wall Exhaust

Air Inlet Fuel Inlet

ri

ro

500

Combustion Chamber

rf r

0

x

Isolated Wall

Fig. 5. Coordinate system and two-dimensional axi-symmetric model of the free flame burner.

0

0.1

0.2

0.3

0.4

0.5

Axial distance (m) Fig. 6. Centerline temperature distributions of a free flame at the equivalence ratio of 0.9.

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M. Bidi et al. / Energy 35 (2010) 3483e3500

2.2

Present Yapici et al. [19]

2

0.85

M erit No.

Total E ntropy G eneration R ate (W /K)

0.9 2.4

1.8 1.6

0.8 Present

1.4

Yapici et al. [19] 1.2 1 0.5

0.6

0.7

0.8

0.9

1

0.75 0.5

0.6

0.7

Equivalence Ratio

0.8

0.9

1

Equivalence Ratio

Fig. 7. Effect of the equivalence ratio on the burner performance for a free flame.

mk ¼ m0k þ Ru Tlnxk

(22)

Finally the total entropy generation rate over the volume V can be calculated as:

S_ gen ¼

I

S000 gen dV

(23)

V

The Bejan number, Be, is defined as the ratio of entropy generation by heat transfer to the total entropy generation:

  S_ gen heat Be ¼ S_ gen

(24)

The merit number (M) is defined as the ratio of the exergy transferred to the sum of the exergy transferred and exergy destroyed [25].

M ¼

Q_ a

(25)

Q_ a þ I_

_ the rate of total irreversibility which is expressed as: where Iis

I_ ¼ Tamb $S_ gen

    _ Cp mwa ðTout Þmwa Tin Q_ fluid ¼ m The total heat transfer rate is:

    Q_ a ¼ Q_ a wall þ Q_ a fluid

Pn

ðjÞawa ¼

(28)

where Q_ wall is the heat transfer rate from the wall of the combustion chamber to the environment and can be written as

Q_ wall ¼ Awall ðq00 Þawa

(29)

Q_ fluid is the heat transfer rate to the fluid, which can be written as:

2.4

Nondim. Flame Speed (S L/S L0)



2.2 2 1.8 1.6 1.4 1.2 1 Equivalence Ratio = 0.5, Present Equivalence Ratio = 0.7, Present Equivalence Ratio = 0.9, Present Equivalence Ratio = 0.5, Diamantis et al.[13] Equivalence Ratio = 0.7, Diamantis et al.[13] Equivalence Ratio = 0.9, Diamantis et al.[13]

0.8 0.6 0.4

Table 1 Parameters and the base values of physical properties.

0.2

Materials

SiC

k2 (m)

5.6  103

Porous length L (mm) PPC Porosity dp (mm) k1 (m2)

30 8 0.90 1.11 8.7  107

l (WmK1) k (m1)

1.0 270 o(105) 0.8 0.5e0.91

Hv (Wm3K1)

e

Equivalence ratio

(33)

2.6



Tamb ðTout Þmwa

n 1X jA A j¼1 j j

where Aj is the cell face area and A the total averaging area. 3

   Q_ a fluid ¼ Q_ fliud 1 

(32)

where jj , rj and Vj are the selected field variable, density, and cell volume, respectively. The area weighted average of a quantity is computed by:

2.8

(27)

jj rj Vj j ¼ 1 rj Vj

j¼1

ðjÞmwa ¼ Pn

(26)

Tamb ðTwall Þawa

(31)

Subscripts “mwa” and “awa” in these equations represent mass weighted average and area weighted average of the relevant quantities, respectively. The mass weighted average of a quantity is represented by:

where Q_ a is the rate of exergy transfer accompanying the energy transfers at the rates of ðQ_ a Þfluid and ðQ_ a Þwall given as [25]:

   Q_ a wall ¼ Q_ wall 1 

(30)

0 0.005

0.01

0.015

0.02

0.025

0.03

0.035

Axial Coordinate (m) Fig. 8. Burning velocity as a function of the flame position for the three equivalence ratios. SL0 is the laminar flame velocity of a free flame at the same equivalence ratio, which equals 4.04, 19.44 and 33.88 cm/s for equivalence ratios of 0.5, 0.7 and 0.9 respectively.

M. Bidi et al. / Energy 35 (2010) 3483e3500

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Fig. 9. Contours of the gas and the solid temperature at the equivalence ratio of 0.5 for various flame positions: (a) SL/SL0 ¼ 1.53, (b) SL/SL0 ¼ 2.56, (c) SL/SL0 ¼ 1.55.

It is important to note that: (1) The analysis of entropy generation is performed at the domain of 0  x  xout , i.e. from the inlet gas boundary to the porous exit surface. This is done because at real applications, a heat exchanger is usually located after a porous exit surface, but here the optimization of porous burner efficiency is the main concern. Thus ðTout Þmwa is the mass weighted average temperature at the porous exit surface. (2) At the entropy generation domain of interest ð0  x  xout Þ, due to the wall insulation, ðQ_ a Þwall becomes zero and is not included in the Merit number calculations. To obtain the total entropy generation rate, first it is necessary to solve the governing conservation equations given in Section 2.1,

and then the volumetric local entropy generation rate can be calculated using the local velocities, reaction rates, and temperatures obtained from the calculations of the governing conservation equations. The total entropy generation rate over the volume can be obtained by numerical integration.

3. Numerical method 3.1. Solution procedure The governing equations are discretized in a non-uniform structural mesh and are solved by a finite volume method.

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M. Bidi et al. / Energy 35 (2010) 3483e3500

a

b

2000

2000

Equivalence Ratio=0.5

Equivalence Ratio =0.5 S /S =2.56

S /S =1.53

T ad

1000

T T T T

500

0

0

0.01

0.02

0.03

0.04

Present Present Diamantis et al. [13] Diamantis et al. [13]

0.05

0.06

Axial Coordinate (m)

c

Tad

1500

Temperature (K)

Temperature (K)

1500

1000

T T T T

500

0

0

0.01

0.02

0.03

0.04

Present Present Diamantis et al. [13] Diamantis et al. [13]

0.05

0.06

Axial Coordinate (m)

2000 Equivalence Ratio =0.5 S /S =1.55

Tad

Temperature (K)

1500

1000

T T T T

500

0

0

0.01

0.02

0.03

0.04

Present Present Diamantis et al. [13] Diamantis et al. [13]

0.05

0.06

Axial Coordinate (m) Fig. 10. Centerline temperature distributions at the equivalence ratio of 0.5 for various flame positions: (a) SL/SL0 ¼ 1.53, (b) SL/SL0 ¼ 2.56, (c) SL/SL0 ¼ 1.55.

Grid independency is tested using different mesh sizes and finally a non-uniform mesh of 300  150 is selected, in which the typical distances between grid points lay between 0.2 and 1 mm. Diffusive fluxes are discretized using a central

difference scheme and convective fluxes are evaluated using the Beam-Warming scheme. The following cubic flux limiter is used to maintain Monotonicity during the numerical simulation:

Fig. 11. Final field of temperatures after increasing the flame speed of Fig. 3(a) by 5%.

M. Bidi et al. / Energy 35 (2010) 3483e3500

b

0.25

Equivalence Ratio =0.5 SL/S L0 = 1.53

CH4 O2

M ass Fraction

0.2

0.15

0.25

NO

0.1

0.05

CH4 O2 CO2

Equivalence Ratio =0.5 SL/S L0 = 1.55

0.2

CO2 CO H2O

M ass Fraction

a

3491

CO H2O

0.15

NO

0.1

0.05

0

0 0.01

0.02

0.03

0.04

0.05

0.01

0.06

Axial Coordinate (m)

0.02

0.03

0.04

0.05

0.06

Axial Coordinate (m)

Fig. 12. Centerline profiles of the species mass fractions at equivalence ratio of 0.5 for various flame positions: (a) near the entrance SL/SL0 ¼ 1.53, (b) near the exit SL/SL0 ¼ 1.55. The “NO” and “CO” mass fractions are multiplied by 1000 and 10 respectively.

¼

4ri þri 4þri þri

1

ri  2 ri > 2

(34)

where ri represents the ratio of successive gradients on the ith solution cell. For any independent variable, 4i , it can be expressed as:

a

Equivalence Ratio = 0.9 S /S =1.11

Temperature (K)

2500 Equivalence Ratio = 0.9 S L/S L0 =1.58

2000

1500

1000

Tgas Present Tsol Present Tgas Diamantis et al. [13] Tsol Diamantis et al. [13]

500

1500

1000

Tgas Present Tsol Present Tgas Diamantis et al. [13] Tsol Diamantis et al. [13]

500

0

0.01

0.02

0.03

0.04

0.05

0.06

0

0

0.01

0.02

0.03

0.04

0.05

0.06

Axial Coordinate (m)

Axial Coordinate (m)

c

(35)

Tadb

2000

0

4i  4i1 4iþ1  4i

To prevent the bandwidth increase of the coefficient matrix, contribution of higher order fluxes is accounted by differed correction strategy:

b

2500 T

ri ¼

Temperature (K)

 

j ri

2500 Equivalence Ratio = 0.9 S L/S L0 =1.09

Tadb

Temperature (K)

2000

1500

1000

Tgas Present Tsol Present Tgas Diamantis et al. [13] Tsol Diamantis et al. [13]

500

0

0

0.01

0.02

0.03

0.04

0.05

0.06

Axial Coordinate (m) Fig. 13. Centerline temperature distributions at equivalence ratio of 0.9 for various flame positions: (a) SL/SL0 ¼ 1.11, (b) SL/SL0 ¼ 1.58, (c) SL/SL0 ¼ 1.09.

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     i  h  þ Jðri Þ F H 4*iþ1=2  F L 4*iþ1=2 F 4iþ1=2 ¼ F L 4iþ1=2 (36) where Fð4Þ is the flux of independent variable 4 and superscripts H and L denote the higher and lower order fluxes, respectively. Pressure and the velocity are coupled by the SIMPLE algorithm of Patankar [26]. Using a multi-step reaction mechanism, the resulting system of algebraic equations becomes stiff, therefore, SIP (Strongly Implicit Procedure) of Stone [27] is used for the solution of the systems of algebraic equations. For combustion simulation, the following two modes are considered in the current developed code: (1) FFL (Fixed Flame Location) mode: A fixed location in the computational domain is taken into account as the flame location where a specified temperature is assigned to calculate the flame speed in an iterative manner using the following relation: I

r Yf ;in SL Ain ¼

u_ fuel Wfuel dV

(37)

V

where Yf ;in is the inlet fuel mass fraction and SL is the flame velocity. The iteration proceeds until the flame velocity is calculated by the order of 106 digits accuracy. (2) CFV (Constant Flame Velocity) at the inlet (mode: A pre-specified value for the flame velocity is defined at the inlet boundary, then by applying appropriate initial guesses for the temperature and species mass fractions, the flame is stabilized anywhere in the computational domain. For a certain flame velocity value, the flame location may be altered by different initial guesses. If the flame velocity is outside of the acceptable range for a given equivalent ratio, it means that the flame cannot be stabilized inside the computational domain (flame quenching). 3.2. Boundary conditions At the inlet of the computational domain (x ¼ 0), the species mass fractions and the temperature of the gas phase are defined, while at the outlet of the computational domain, all gradients of the gas phase are set to zero. At the inlet surfaces of the foam, i.e. at xin:

Tg jx¼x ¼ Tg jx¼xþ in in ug jx¼x ¼ fug jx¼xþ in

in

(38)

validation of the current developed code is carried out considering a non-reactive flow within a porous medium and the conventional free flame combustion. The entropy generation rate of a laminar flow in a circular duct filled with a porous medium is studied to compare the numerical results obtained with the corresponding available data reported by Morosuk [28] who has applied the EGM method for the optimization of the heat transfer efficiency in the pipes and channels filled with a partial or full porous medium. For this reason, a circular duct fully filled with a porous medium shown in Fig. 2 is considered, where a uniform flow with the velocity of uin and the temperature of Tin enters a pipe with the constant temperature at the wall, Twall, and the radius of ro . To achieve fully developed conditions, the pipe length is taken into account as L ¼ 50ro. Axi-symmetric boundary conditions are adopted at r ¼ 0, i.e. v ¼ 0 with the gradients of u and T in the r direction set to be zero. Considering the governing non-dimensional parameters including Brinkman number, Br ¼ m u2in =lðTw  Tin Þ, modified Brinkman number, Brm ¼ BrðTin =DTÞ, Darcy number, and dimensionless temperature, Da ¼ k1 =r02 , q ¼ ðT  Tin Þ=ðTw  Tin Þ, the numerical results obtained at Da ¼ 102 and Brm ¼ 0:1 are shown in Fig. 3a and b where the fully developed velocity and temperature profiles at the exit of the pipe are compared with the results reported by Morosuk [28]. As is evident from the figures, the current results are in a very good agreement with the previous results [28]. Furthermore, the contributions of various mechanisms on the total rate of entropy generation at the exit of the pipe are shown in Fig. 4a. Since the velocity gradients are negligible at the central part of the pipe, the mechanical entropy generation rates due to dissipation is negligible in this region and increases near the wall with increasing the velocity gradients. The porous medium friction term is the dominant mechanism at the central part of the pipe. In this problem, the entropy generation due to heat transfer can be neglected without introducing noticeable error. The cross sectional average of the total entropy generation variation along the pipe is shown in Fig. 4b. The entropy generation asymptotically decreases to a constant value at the fully developed region. The predictions are in a very good agreement with the results of Morosuk [28]. The entropy generation rates and the merit number are studied in a conventional free flame burner to compare the numerical results obtained with the corresponding available results reported by Yapıcı et al. [19]. They have performed a numerical solution of the local entropy generation rate in a methaneeair combustion chamber using a two-step reaction mechanism by neglecting the entropy generation rates of the chemical reactions. The schematic

Vk jx¼x ¼ f Vk jx¼xþ ðk ¼ 1; 2; .; KÞ in

in

The similar boundary conditions are applied at the outlet surface. The boundary conditions for the solid temperature are obtained by:

    vTs 4 at x ¼ xin ¼ ð1 fÞh Ts Tg þð1 fÞes Ts4 TN vx     vTs 4 ð1 fÞls at x ¼ xout ¼ ð1 fÞh Ts Tg þð1 fÞes Ts4 TN vx ð39Þ

ð1 fÞls

4. Results and discussions 4.1. Validation of the solution procedures for evaluating the entropy generation rates There is no previous study on the EGM of combustion in porous media. However, there are some studies concerning with the EGM in non-reactive flows within porous media [28e32] along with some studies about EGM for the free flames [19,20]. Therefore, the

Table 2 Comparison of the performance parameters for the flames at the equivalence ratio of 0.5 stabilized at difference locations. Parameter

Unit

Value for Flame 1 (upstream)

Value for Flame 2 (downstream)

Flame location Flame Speed Ratio (SL/SL0) Thermal load Heat recirculation flux Output radiant flux Heat Recirculation Efficiency Output Radiant Efficiency Gas Output Temperature (Mass weighted average)ðTout Þmwa Available Exergy ðQ_ a Þfluid Entropy generation rate S_ gen Irreversibility I_

m e kW/m2 kW/m2 kW/m2 % % K

0.012 1.53 104.35 31.32 28.12 30 26.95 988

0.030 1.55 105.71 30.23 24.52 28.6 23.2 1257

W W/K W e e

82.0 0.82 246 0.25 0.81

114.0 1.35 404 0.22 0.77

Merit No. Bejan No.

M. Bidi et al. / Energy 35 (2010) 3483e3500

3493

Fig. 14. Contours of the entropy generation rates at equivalence ratio of 0.5 for the flame speed of SL/SL0 ¼ 1.53 stabilized at the upstream half of the porous layer: S1 ¼ gas heat transfer,S2 ¼ solid heat transfer, S3 ¼ dissipation friction, S4 ¼ porous media friction, S5 ¼ reaction and S6 ¼ total entropy generation rate (W/K).

diagram of the burner is shown in Fig. 5. At the burner wall, natural convection is assumed. Taking into account the same physical conditions, the centerline temperature distribution is shown in Fig. 6 for volumetric flow rate of 5 L per minute and the equivalence ratio of 0.9. The present result predicts a lower peak flame temperature mainly affected by the multi-step reaction mechanism used in the present study rather than the two-step reaction mechanism used by Yapıcı et al. [19]. The effect of the equivalence ratio on the total entropy generation rate and the Merit number is shown in Fig. 7. To compare the predictions with the results of Yapıcı et al. [19], first the rate of

entropy generation due to chemical reactions is neglected and the relevant results are shown in Fig. 7a. As is evident from the figure, the variation trend of the current results is similar the previous study [19] with a maximum deviation of 10 percent due to the different chemical mechanisms used here. 4.2. Flame characteristic The basic structure of the submerged flame inside the ceramic foam with large pores is represented in Table 1 [13]. Running the code in the first mode FFL using the temperature of 1200 K at the

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a

b

220 Equivalence Ratio = 0.5

34 Equivalence Ratio = 0.5

200

32

160

30

E fficiency (% )

Q th, Q recirc, Q r,out (kW /m2)

180

140 Thermal Load Heat Recirculation Flux Output Radiative Flux

120 100 80

Heat Recirculation Efficiency Output Radiant Efficiency

28 26 24

60 40

22

20 0 0.2

0.3

0.4

0.5

20 0.2

0.6

0.3

(x-xin)/L 1

d

Equivalence Ratio = 0.5

0.95

0.6

0.5

0.6

0.4

0.35

0.9

0.85

0.8 0.2

0.5

Equivalence Ratio = 0.5

M erit N o.

T otal E ntropy G eneration R ate (W /K)

c

0.4

(x-xin)/L

0.3

0.25

0.3

0.4

0.5

0.6

(x-xin)/L

0.2 0.2

0.3

0.4

(x-xin)/L

Fig. 15. Effect of the flame location on the burner performance at the equivalence ratio of 0.5.

flame location results in the flame velocities as shown in Fig. 8. The burning velocities are normalized by the laminar burning velocity of the free flame at the same equivalence ratios, SL0 , i.e. 4.04, 19.44, and 33.88 cm/s for equivalence ratios of 0.5, 0.7, and 0.9, respectively. Similar to the results represented by Diamantis et al. [13], for a constant equivalence ratio, the flame reaches at its maximum possible speed when it is stabilized approximately in the middle of the ceramic. Extinction occurs close to the either end of the porous layer in the case of lean flames but the flames with equivalence ratios greater than 0.7 can be stabilized anywhere along the foam. Another important result is that the burner nominal thermal load, 00 Q_ h ¼ r SL Yfin Qfuel , varies by the location of flame stabilization (Yfin and Qfuel are the inlet mass fraction and the adiabatic low heating value of the fuel respectively). The temperature contours of the solid and the gas phase for the equivalence ratio of 0.5 are shown in Fig. 9aec. As is evident from the figure, the flame thickness is larger on the central axis than the other locations, resulting in the higher propagation speed of the flame front at the center. For further investigation, the axial temperature variation of the gas and the solid phase are presented in Fig. 10aec. The results predict higher centerline temperatures and higher burning velocity in comparison with the results of Diamantis et al. [13] where they considered one-dimensional solution. This is because of no-slip conditions at the walls in the 2-D simulation. Upstream of the flame, the solid temperature is higher than the gas temperature and downstream of the flame zone the temperature becomes lower than the gas phase. As seen from Fig. 10b, the flame temperature of the gas phase becomes locally

higher than the adiabatic flame temperature, Tadb, near the middle section of the foam. This state is referred to as “super-adiabatic combustion” of porous burners in literatures. In Fig. 9a the flame is located close to the entrance of the porous media, which is stable also after increasing the velocity by 5% as shown in Fig. 11. When the flame is considered at the end of the porous media (Fig. 9c), the similar increase of the velocity causes the flame exit the computational domain. The species mass fractions along the axis of symmetry for the above mentioned two cases are plotted in Fig. 12a and b. As can be seen from the figure, the amount of CO mass fraction is higher in the second flame than the first one. This means less amount of heat release in the second flame. Therefore, fully submerged flames located at the upstream half of the porous medium are more stable, producing less emission. In the next section, the performance of these flames is studied from the EGM point of view. Temperature profiles of the gas and solid phases along the axis of symmetry at the equivalence ratio of 0.9 are shown in Fig. 13ae c for different flame locations. Comparing with the results of the equivalence ratio of 0.5, the temperature increases by about 30% at the equivalence ratio of 0.9. 4.3. Flame stability analysis As stated previously, the heat recirculation in the preheat zone is the most important mechanism which increases the flame speed and the flame stability within the porous medium.

M. Bidi et al. / Energy 35 (2010) 3483e3500

a

b

900 Equivalence Ratio = 0.7

3495

28 Equivalence Ratio = 0.7

800

E fficiency (% )

Q th, Q recirc, Q r,out (kW /m2)

26 700 600 Thermal Load Heat Recirculation Flux Output Radiative Flux

500 400

24 Heat Recirculation Efficiency Output Radiant Efficiency

22

300 20 200 100

18 0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.1

0.2

c

d

5 Equivalence Ratio = 0.7

0.4

0.5

0.6

0.4

0.5

0.6

0.5 Equivalence Ratio = 0.7

4.5 4

0.45

3.5

M erit N o.

T otal E ntropy G eneration R ate (W /K)

0.3

(x-xin)/L

(x-xin)/L

3

0.4

2.5 2

0.35

1.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.3

0

0.1

(x-xin)/L

0.2

0.3

(x-xin)/L

Fig. 16. Effect of the flame location on the burner performance at the equivalence ratio of 0.7.

Also it can be emphasized that the optimum design of a porous burner is accessible by enhancing both the heat recirculation and output radiant flux. For this reason, two types of efficiencies are defined in the literatures consisting of the heat recirculation and the output radiant efficiencies. Heat recirculation in the preheat zone is the sum of the solid conduction and solid to solid radiation which balances by the heat removal of the gases by convection. The heat recirculation efficiency is a measure of the heat transfer amount to the gas in the preheat zone which is defined as.

Heat recirculation efficiency ¼

  00 4 Q_ r; out ¼ q_ 00r; out þ ð1  fÞes Ts;4 out  Tamb

(42)

00

where Q_ r; out is the output radiant flux, and q_ 00r; out is the outcome of the solution of equations (10)e(11). Lets compare the above two mentioned flames (the stabilized flame at the upstream half and the stabilized flame at the downstream half of the porous matrix). Performance characteristics for these flames are summarized in Table 2. The amount of the heat recirculation and the heat recirculation efficiency are higher in the upstream flame than the one in the downstream. In fact, in this

solid to gas convection in preheat zone 00 Q_

(40)

th

00

where Q_ th ¼ r SL Yfin Qfuel is the burner nominal thermal load which is defined previously. At a specified equivalence ratio, the increasing of the heat recirculation efficiency leads to the increase of the burning speed and the flame stability [8,12,13,33]. The output radiant efficiency is defined as:

Output radiant efficiency ¼

00 Q_ r; out 00 Q_

(41)

th

00

where the output radiant flux, Q_ r; out , is the energy emitted from the exit burner surface given by

large pore, the low extinction coefficient foam, the radiation has a greater contribution on the heat recirculation than the conduction. Therefore, the flame location is important from the standpoint of radiant energy exchange and determining the maximum allowable heat release. When the flame is stabilized near the inlet, the upstream radiant flux is larger than the downstream radiant flux due to the higher peak flame temperature (Fig. 10) and the less shielding between the flame and the upstream edge. As mentioned above, the higher heat recirculation efficiency in the upstream flame leads to the higher stability of the flame. Now consider a flame close to the entrance of the foam. If, for some reason, the flow speed is slightly increased, the flame will

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a

b

2000 Equivalence Ratio = 0.9

24 Equivalence Ratio = 0.9

1800

22

1400 1200

E fficiency (% )

Qth, Q recirc, Q r,out (kW /m2)

1600

Thermal Load Heat Recirculation Flux Output Radiative Flux

1000 800

20

18

600

16

Heat Recirculation Efficiency Output Radiant Efficiency

400 200

14 0

0.1

0.2

0.3

0.4

0.5

0

0.6

0.1

0.2

(x-xin)/L 5

d

Equivalence Ratio = 0.9

0.4

0.5

0.6

0.6 Equivalence Ratio = 0.9

4.5 4

0.55

3.5

M erit N o.

T otal E ntropy G eneration R ate (W /K)

c

0.3

(x-xin)/L

3 2.5 2

0.5

0.45

1.5 1

0.4 0

0.1

0.2

0.3

0.4

0.5

0.6

(x-xin)/L

0

0.1

0.2

0.3

0.4

0.5

0.6

(x-xin)/L

Fig. 17. Effect of the flame location on the burner performance at the equivalence ratio of 0.9.

migrate downstream where its burning velocity is higher (Fig. 8) and a ‘restoring’ force brings the flame backwards to its original position. If the fuel mass fraction increases slightly, the equivalence ratio and flame speed will increase, resulting in the flame to move upstream where the burning velocity is lower to provide a stable flame. By the same physical logic, it can be concluded that a downstream flame will be unstable due to the slight perturbations causing the flame to move downstream where the burning speed is lower and leads the flame migrate all the way until the exit. As is evident from Table 2, the trends of the Merit number and the recirculation efficiency are similar. This is because the increasing of the heat recirculation leads to the reduction of the temperature gradients and chemical affinities. Therefore, the total entropy generation rate decreases by the reduction of both the heat transfer and the reaction terms. This phenomenon is accompanied by the Merit number increase which is an indicator of more stable flame. Another interesting result evident from Table 2 is that the output radiant heat flux and output radiant efficiency of the upstream flame are also greater than those of the downstream flame. This is because the closer the flame is to the exit, the lower the solid temperature is (Figs. 10 and 13) and a smaller length of ceramic radiates in contrast with the stabilized flames close to the entrance. 4.4. Burner optimization by EGM method The optimization of the burner from the EGM point of view investigates the minimum total rate of generated entropy or the

maximum Merit number by considering one of the effective parameters involved in the burner performance as a variable and the others constant. The contours of entropy generation rates are shown in Fig. 14 for the first case. By looking at the entropy generation contours, it is found that the chemical reactions play an important role in the total entropy generation rates. Based on the numerical results obtained, the entropy generation due to dissipation (friction part) can be neglected comparing with the thermal and reactive entropy generation rates. It has to be emphasized that in porous media the momentum dissipation rate is larger than the non-porous region. First, the two flames at near the entrance and near the exit of the porous medium mentioned above are compared for the equivalence ratio of 0.5. Due to the less production of CO and NO in the flame near the entrance (Fig. 12), the amount of chemical affinity is lower comparing with the flame near the exit. Some other important parameters of these two flames are compared at the Table 2. It is not surprising that for the first flame, the available exergy of the fluid is lower than that of the second flame owing to its lower mass weighted outlet temperature. In contrast with the lower fluid exergy, the Merit number of the first flame is higher than the second one. This results from its lower entropy generation rate, indicating its higher efficiency. The Merit number is a good indicator of the flame performance, and the definition of the combustion efficiency as the ratio of the fluid heat release (Eq. (30)) to the nominal thermal load is not an appropriate indicator of combustion performance. Based on these reasons, the Merit number must be considered as the performance indicator of the flame.

M. Bidi et al. / Energy 35 (2010) 3483e3500

F lame S peed R atio (S L/S L0)

a

3497

2.6

2.4

2.2

2

1.8

1.6

1.4 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Equivalence Ratio

b

c

3000

Heat Recirculation Efficiency Output Radiant Efficiency

30

2000

E fficiency (% )

Q th, Q recirc, Q r,out (kW /m2)

32

Thermal Load Heat Recirculation Flux Output Radiative Flux

2500

34

1500

1000

28 26 24 22 20

500 18 0 0.4

0.6

0.8

1

16 0.4

1.2

0.5

0.6

Equivalence Ratio

e

8

0.8

0.9

1

1.1

1.2

1

1.1

1.2

0.6

7 6

0.5

5

M erit N o.

T otal E ntropy G eneration R ate (W /K)

d

0.7

Equivalence Ratio

4

0.4

3 2

0.3

1 0 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Equivalence Ratio

0.2 0.4

0.5

0.6

0.7

0.8

0.9

Equivalence Ratio

Fig. 18. Effect of the equivalence ratio on the burner performance (flame is stabilized at the middle of the foam).

The effect of the flame location on the burner performance at the equivalence ratio of 0.5 is shown in Fig. 15(a)e(d). A dimensionless distance is defined as x ¼ ðx  xin Þ=L, where xin and L are the location of the inlet section and length of the porous matrix, respectively, as shown in Fig. 1. As the flame moves downstream, the burning speed increases (Fig. 8), leading to the increasing of the thermal load. The increasing of the thermal load and flame temperature with the reduction of radiation losses at the inlet surface leads to the increasing of the heat recirculation in the preheat zone until the location of x ¼ 0.45. Beyond this location, as the flame moves further downstream, due to the less shielding between the flame and the downstream edge, the downstream

radiant flux becomes greater than the upstream one. Thus the heat recirculation efficiency reaches its maximum value at the location about x ¼ 0.45. Increasing the thermal load causes the increase of the available exergy of the flame owing to the higher energy credits. Although the increasing of the heat recirculation decreases the temperature gradients, the reaction entropy term is a dominant term in the total entropy generation rates, increasing the irreversibility as the flame moves downstream. In fact, the increasing of the thermal load and burning speed are the direct results of higher reaction rates. Since the rate of available exergy growth is greater than the rate of the irreversibility growth, the Merit number increases until the

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Fig. 19. Total entropy generation rate and Merit number as a function of porosity (flame is stabilized at the middle of the foam).

location at about x ¼ 0.45. Hence, at the equivalence ratio of 0.5, the flame has the highest heat recirculation efficiency, the best stability condition, and the best second law performance at the location of x ¼ 0.45.

For a stabilized flame near the inlet of the foam, as the flame moves downstream, the output radiant flux increases due to decreasing the radiation losses at the inlet surface until the location of x ¼ 0.4 Fig. 15a and b. Beyond this location, as the flame moves

M. Bidi et al. / Energy 35 (2010) 3483e3500

further downstream, the solid temperatures become lower and a smaller length of ceramic radiates, leading to decreasing the output radiant flux. These trends are consistent with the results obtained by Sathe et al. [8,12] and Dimantis et al. [13]. Although the heat recirculation efficiency and the Merit number have an optimum value at the location about x ¼ 0.45 for the equivalence ratio of 0.5, the output radiant flux reaches its maximum value at the location of x ¼ 0.4. Therefore, the optimum location of the flame in a porous burner is a trade off between the heat recirculation (which is equivalent to the second law efficiency) and the output radiant efficiency. Similar trends are investigated at equivalence ratios of 0.7 and 0.9 in Figs. 16(aed) and 17(aed), where the optimum location is shifted to downstream as the equivalence ratio increases. At the higher equivalence ratios, temperatures, burning speeds, and thermal loads are higher, leading to increasing the radiation effect which in turn enhances the heat recirculation and the output radiative fluxes. Again it is emphasized that at a specified equivalence ratio, the trends of the heat recirculation efficiency and the trends of the Merit number are similar to each other. Hence the EGM method can be used to optimize the flame location from the stability point of view, but the complete optimization involves augmentation of the radiant output flux. The effect of equivalence ratio on the combustion heat transfer behavior within a porous region is shown in Fig. 18(a)e(e). Here, the flame location is fixed at the middle of the porous matrix and the relevant flame speed (thermal load) is calculated at different equivalence ratios. The numerical results obtained indicate that the maximum second law efficiency occurs at the equivalence ratio of about 0.9. Less than 0.9, the effect of low combustion temperature and more than 0.9, the incomplete combustion effects on the flame performance reduce the Merit number. The thermal load, the recirculation heat, and the output radiant flux increase with the increase of the equivalence ratio. However, the heat recirculation and the radiant output efficiencies decrease due to the higher growth rate of the thermal load in comparison with the growth rates of the heat recirculation and output radiant fluxes. These trends are consistent with the results obtained by Bara et al. [33]. Reducing the heat recirculation efficiency leads to the reduction of the effective flame speed ratio. Also the available exergy increases with equivalence ratio due to increasing the thermal load and energy contents. Furthermore, the rate of the irreversibility increases due to increasing both the chemical affinities and the temperature gradients. Up to the equivalence ratio of 0.9, the increment rate of available exergy becomes larger than that of the rate of irreversibility, but beyond that it is reversed. The outcome of the above physical reasoning leads to the Merit number increase up to 0.9. However, at various equivalence ratios, there is no certain relation among the flame stability, the heat recirculation [8,33], and the Merit number. Depending on the optimization goal, there are different choices of equivalence ratios. To enhance the thermal load and the output radiant flux, the equivalence ratio must be increased, but to enhance the flame speed ratio and to minimize the rate of entropy generation, the equivalence ratio must be decreased. However, to achieve the best burner performance from the second law point of view, there is an optimum value for the Merit number. Fig. 19 shows the effect of porosity on the burner performance at the equivalence ratios of 0.5, 0.7 and 0.9. As is evident from this figure, the total entropy generation rate is increased by increasing the porosity, however, there is an optimum point for porosity at which the Merit number is maximum, indicating the best efficiency point of combustion in porous media. The optimum porosities are 0.86, 0.88 and 0.90 for the equivalence ratios of 0.5, 0.7 and 0.9, respectively. At very lower porosities than the optimum value, the

3499

efficiency of the flame decreases due to high energy transmission from the flame to the solid matrix. At larger porosities than the optimum value, the flame efficiency decreases owing to the reduction of preheating effect where the structure of flame becomes similar to the free flame combustion. 5. Conclusions Combustion in a two-dimensional axis-symmetric porous burner is numerically simulated considering separate energy equations for the gas and the solid phase. The chemical species transport equations are solved using a multi-step reduced kinetic mechanism. Flame stabilization and the burner optimization are studied by the EGM method. The effects of the chemical reaction rates which play an important role in the generation of total entropy rates are considered. Also, the numerical results indicate that the effects of momentum dissipation rate are negligible in the total entropy generation rate. It is found that the Merit number is an appropriate indication of combustion performance. The results show that the flames occurring upstream half of the porous layer are more stable, more efficient, and producing less emissions than those occur at the downstream half of the porous layer. At a specified equivalence ratio, the trends of the heat recirculation efficiency and the trends of the Merit number are similar to each other. Hence the EGM method can be used to find the optimum flame location in a porous burner form the stability point of view, but the complete optimization is the trade off between the maximum heat recirculation and the maximum output radiant efficiencies. Increasing the equivalence ratio leads to increase of thermal load, the heat recirculation, and output radiant fluxes, but it decreases the efficiencies of the heat recirculation and output radiant as well as the effective flame speed. Furthermore, the irreversibility and the available exergy increase by increasing the equivalence ratio and there is an optimum equivalence ratio at the maximum Merit number. Appendix A Reduced Kinetic Mechanism (Barlow et al. [22])

No.

Reaction

Global rates for reduced mechanism (1/cm3 s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

H þ ½O2 4 OH H2 þ ½O2 4 H þ OH HO2 4 ½O2 þ OH ½O2 þ H2O2 4 OH þ HO2 ½O2 þ ½ C2H2 4 H þ CO CH3 þ CO þ C2H4 4 ½O2 þ CH4 þ 1.50C2H2 ½ O2 þ 2CH3 4 H2 þ CH4 þ CO ½ O2 þ CH3 4 H þ CH2O ½ O2 þ CH4 4 OH þ CH3 ½ O2 þ CO 4 CO2 ½ O2 þ C2H6 4 CH4 þ CH2O H þ OH 4 H2O H þ CH4 þ NO þ HCN 4 ½ O2 2CH3 þ N2 H þ ½ O2 þ CH4 þ HCN 4 2CH3 þ NO ½ O2 þ CH4 þ NH3 þ HCN 4 H2O þ 2CH3 þ N2

3.7229E-10 4.1280E-11 5.6583E-10 2.0553E-13 1.4143E-11 1.4083E-12 2.6626E-11 3.2685E-12 2.7449E-11 8.6956E-14 1.1656E-12 9.1663E-11 1.1917E-12 1.5580E-12 1.0105E-13

References [1] Babkin VS. Filtrational combustion of gases. Present state of affairs and prospects. Pure Appl Chem 1993;65:335e44. [2] Hsu PF, Hoewll JR, Mettews RD. Experimental and numerical study of premixed combustion within nonhomogeneous porous ceramics. ASME J Heat Transfer 1993;115:744e50. [3] Contarin F, Barcellos WM, Saveliev AV, Kennedy LA. Energy Extraction from a porous media Reciprocal flow burner with embedded heat exchangers. ASME J Heat Transfer 2005;127:123e9.

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