**3.1. One-dimensional modeling**

Many researchers studied the combustion and heat transfer in porous radiant burners. The PM was assumed to emit, absorb, and scatter radiant energy. Non-local thermal equilibrium between the solid and the gas is assumed for and combustion was described by a one-step, multi-steps or kinetic reactions. The effect of the optical depth, scattering albedo, solid thermal conductivity, upstream environment reflectivity, and interphase heat transfer coupling on the burner performance can be considered. Also, low solid thermal conductivity, low scattering albedo, and high inlet environment reflectivity produced a high radiant efficiency. The system consisted of a packed bed or foams in which a natural gas–air mixture combusts inside it. Radiative heat transfer in the packed bed or foams was modeled as a diffusion process, and the flow and temperature distribution in the packed bed or foams can be determined. The numerically results usually were compared with available experimental data for a similar system. Subsequently, the one-dimensional predictions of methane/air combustion in inert PM can include full mechanism (49 species and 227 elemental reactions), skeletal mechanism (26 species and 77 elemental reactions), 4-step reduced mechanism (9 species and 1-step global mechanism). The effects of these models on temperature, species, burning speeds and pollutant emissions were examined by researchers. Experimental and numerical investigations found that the flammability limits of the gaseous mixture in PM were more sensitive to their geometric properties than the physical properties.

#### *3.1.1. One-dimensional governing equations*

The following assumptions are made to simplify the problem:


Continuity equation:

532 Numerical Simulation – From Theory to Industry

component fractions.

radiation.

physical properties.

**3. Numerical modeling** 

**3.1. One-dimensional modeling** 

stress and strength retention at the peak regenerator temperature (1673 K), and excellent oxidation resistance. Metallic materials were found less suitable for PM because of their inadequate thermal stability and high thermal inertia. Fe–Cr–Al-alloys and nickel-base alloys were found suitable for some applications but they were said to be comparatively less heat resistant. Structures of ceramic foams with different base materials were observed to possess high porosity, good conduction heat transport, low thermal inertia, low radiation heat transport properties and relatively high pressure drop. The effective thermal conductivity of anisotropic porous composite medium could vary largely with the

The thermophysical properties of the air such as density, thermal conductivity and specific heat are assumed to be functions of the temperature. Usually, the pressure drop through the porous burner is not that high (with high porosity of PM) and its effect on the thermophysical properties can be neglected. In general, the properties of the solid phase may be assumed to be constant and assumed that there is thermal non-equilibrium between the gas and solid phases. Therefore, there are two energy equations to model the energy transport in the system. The porous material can be assumed as a scattering, emitting and absorbing medium. Gaseous radiation is assumed to be negligible compared to the solid

Many researchers studied the combustion and heat transfer in porous radiant burners. The PM was assumed to emit, absorb, and scatter radiant energy. Non-local thermal equilibrium between the solid and the gas is assumed for and combustion was described by a one-step, multi-steps or kinetic reactions. The effect of the optical depth, scattering albedo, solid thermal conductivity, upstream environment reflectivity, and interphase heat transfer coupling on the burner performance can be considered. Also, low solid thermal conductivity, low scattering albedo, and high inlet environment reflectivity produced a high radiant efficiency. The system consisted of a packed bed or foams in which a natural gas–air mixture combusts inside it. Radiative heat transfer in the packed bed or foams was modeled as a diffusion process, and the flow and temperature distribution in the packed bed or foams can be determined. The numerically results usually were compared with available experimental data for a similar system. Subsequently, the one-dimensional predictions of methane/air combustion in inert PM can include full mechanism (49 species and 227 elemental reactions), skeletal mechanism (26 species and 77 elemental reactions), 4-step reduced mechanism (9 species and 1-step global mechanism). The effects of these models on temperature, species, burning speeds and pollutant emissions were examined by researchers. Experimental and numerical investigations found that the flammability limits of the gaseous mixture in PM were more sensitive to their geometric properties than the

$$
\frac{
\partial
}{
\partial t
}
\left(
\rho\_{\mathcal{S}}
\varepsilon
\right) + \frac{
\partial
}{
\partial \mathcal{x}
}
\left(
\rho\_{\mathcal{S}}
u
\varepsilon
\right) = 0.
\tag{2}
$$

Gas phase energy equation:

$$\rho\_g \varepsilon\_g \varepsilon\_g \frac{\partial T\_g}{\partial t} + \rho\_g \varepsilon\_g \varepsilon u \frac{\partial T\_g}{\partial \mathbf{x}} + \sum\_i \rho\_g \varepsilon Y\_i V\_i c\_{gi} \frac{\partial T\_g}{\partial \mathbf{x}} + \varepsilon \sum\_i \dot{\alpha}\_i h\_i \mathcal{W}\_i = \varepsilon \frac{\partial}{\partial \mathbf{x}} \left( (k\_g + \rho\_g c\_g D\_{|\cdot|}{}^D) \frac{\partial T\_g}{\partial \mathbf{x}} \right) - h\_v (T\_g - T\_s). \tag{3}$$

Solid phase energy equation:

$$
\rho\_S c\_S (1 - \varepsilon) \frac{\partial T\_s}{\partial t} = k\_s (1 - \varepsilon) \frac{\partial^2 T\_s}{\partial x^2} + h\_v (T\_g - T\_s) - \frac{d q\_r}{d x} = 0. \tag{4}
$$

Species transport equation:

$$
\rho\_g \varrho\_g \varepsilon \frac{\partial Y\_i}{\partial t} + \rho\_g \varepsilon u \frac{\partial Y\_i}{\partial \mathbf{x}} + \frac{\partial}{\partial \mathbf{x}} \left(\rho\_g \varepsilon Y\_i V\_i\right) - \varepsilon \dot{\alpha} b\_i W\_i = 0. \tag{5}
$$

$$
\partial\_i V\_i = - \left( D + D\_{im}^d \right) \frac{1}{X\_i} \frac{\partial X\_i}{\partial \mathbf{x}}.\tag{6}
$$

density:

$$
\rho \text{ = constant density.}\tag{7}
$$

Radiation:

$$q\_r(\mathbf{x}) = -\frac{16}{3} \frac{\sigma T\_s^3}{\beta} \frac{d T\_s}{d \mathbf{x}}.\tag{8}$$

#### *3.1.2 Boundary conditions for one-dimensional model*

The following boundary conditions are considered in the computations:


$$\frac{1}{r^2} \frac{\partial}{\partial t} \left( \varphi \rho\_g \mathcal{L}\_{pg} T\_g \right) + \frac{1}{r^n} \frac{\partial}{\partial r} \left( \varphi \rho\_g \mathcal{L}\_{pg} r^n \upsilon T\_g \right) = \frac{1}{r^n} \frac{\partial}{\partial r} \left( \varphi k\_g r^n \frac{\partial T\_g}{\partial r} \right) - \left( 1 - \varphi \right) h\_\nu \left( T\_g - T\_\mathfrak{s} \right) + \varphi \Delta H\_\mathbb{C} \mathcal{S}\_{fg} \tag{9}$$

$$\frac{1}{2}\frac{\partial}{\partial t}(\rho\_s \mathcal{C}\_s T\_s) + \frac{1}{r^n} \frac{\partial}{\partial r} \left(r^n k\_s \frac{\partial T\_s}{\partial r}\right) - h\_v \left(T\_s - T\_g\right) - \nabla.F. \tag{10}$$

$$\nabla\_\cdot F = -(1 - \omega)(G - 4E\_b),\tag{11}$$

$$
\nabla^2 G = \eta^2 (G - 4E\_b),
$$

$$
\eta^2 = 3\beta^2 (1 - \omega)(1 - \text{g\,\omega}),
\tag{12}
$$

$$\frac{\partial}{\partial t} \left( \rho\_g m\_f \right) + \frac{1}{r^n} \frac{\partial}{\partial r} \left( \rho\_g r^n \upsilon m\_f \right) = \frac{1}{r^n} \frac{\partial}{\partial r} \left( r^n D\_{AB} \rho\_g \frac{\partial m\_f}{\partial r} \right) - \mathsf{S}\_{fg} \ , \tag{13}$$

$$S\_{fg} = f \rho\_g^2 \, m\_f m\_{o\_2} \exp\left(-\frac{E}{RT\_g}\right). \tag{14}$$

$$T\_{g|r=r\_{ln}} = \begin{array}{c} 0 \\ \end{array} \quad \text{at} \qquad \qquad r = r\_{ln} \,.$$

$$\frac{\partial T\_g}{\partial r}\Big|\_{r=r\_{out}} = \begin{array}{c} T\_{ln} \\ \end{array} \quad \text{at} \qquad \qquad r = r\_{out} \,. \tag{15}$$

$$h\_{\rm in} \left[ \left( T\_{g\ ,\rm in} - T\_{\rm s\left|r=r\_{\rm in}\right.} \right) \right] + \sigma \epsilon\_{\rm in} \left[ \left( T\_{\rm in,amb}^4 - T\_{\rm s}^4 \vert\_{\rm s\left|r=r\_{\rm in}\right.} \right) \right] = -k\_s \frac{\partial T\_{\rm S}}{\partial r} \bigg|\_{r=r\_{\rm in}} \quad \text{at} \qquad r = r\_{\rm in} \cdot \frac{\partial T\_{\rm S}}{\partial r}$$

$$h\_{\rm out} \left[ \left( T\_{\rm out,in} - T\_{\rm s|r=r\_{\rm out}} \right) \right] + \sigma \epsilon\_{\rm out} \left[ \left( T\_{\rm out,amb}^4 - T\_{\rm s}^4 \right)\_{\rm s|r=\rm out} \right) \right] = -k\_s \frac{\partial T\_{\rm s}}{\partial r} \Big|\_{r=r\_{\rm out}} \quad \text{at} \quad r = r\_{\rm out}. \tag{16}$$

$$m\_f = \begin{array}{c} m\_{f,in} \\ \end{array} \quad \text{at} \qquad r = r\_{in}$$

$$\frac{\partial m\_f}{\partial r} = \begin{array}{c} \mathbf{0} \\ \end{array} \quad \text{at} \qquad r = r\_{out}. \tag{17}$$


$$\frac{\partial(\wp \rho\_l)}{\partial t} + \nabla. (\wp \rho\_l \text{ } \mu \text{ }) = \nabla. \left[ \rho \text{ } \wp \text{ } D\_{\text{im}} \nabla \left( \frac{\rho\_l}{\rho} \right) \right] + \varphi \text{ } \phi\_l^c + \phi^s \text{ } \delta\_l \text{ }, \tag{18}$$

$$\frac{\partial(\rho \, u)}{\partial t} + \nabla. \left(\rho \, u \, u\right) = -\frac{1}{a^2} \nabla P - \nabla \left(\frac{2}{3} \, \rho \, k\right) + \nabla. \sigma + F^S - \left(\frac{\Delta P}{\Delta L}\right). \tag{19}$$

$$
\left(\frac{\Delta p}{\Delta L}\right) = \left(\frac{\mu}{\alpha}\,\,u + \,c\_2\,\,\frac{1}{2}\,\,\rho\,\,|u\,|\,u\right). \tag{20}
$$

$$\alpha = \frac{d^2}{150} \frac{\epsilon^3}{(1 - \epsilon)^2} \quad , \quad c\_2 = \frac{3.5}{d} \frac{(1 - \epsilon)}{\epsilon^3} \,. \tag{21}$$

$$\frac{\partial}{\partial t} \left( \varphi \rho c\_p T\_g \right) + \nabla \cdot \left( \varphi \rho c\_p T\_g u \right) + \varphi \sum\_l \dot{\omega}\_l \, H\_l W\_l = -\varphi P \, \nabla \cdot u + \varphi A\_0 \rho \epsilon + (1 - A\_0) \sigma \colon \nabla u$$

$$\varphi \nabla \cdot \left( (k\_g + \rho\_g c\_g D\_{\parallel}^d) \nabla T\_g \right) - h\_\nu \{ T\_g - T\_\mathcal{s} \} + \dot{Q}^s \,. \tag{22}$$

$$D\_{\parallel}^{d} = 0.5 \,\, a\_{g} Pe \,\, \tag{23}$$

$$\mathbf{N}\mathbf{u}\_{\mathbf{v}} = \mathbf{2} + \mathbf{1}.\mathbf{1}\,\mathbf{R}\mathbf{e}^{0.6}\,\mathbf{Pr}^{0.33},\tag{24}$$

$$h\_{\nu} = \frac{6\rho}{d^2} \,\, k\_g N u\_{\nu} \tag{25}$$

$$\frac{\partial}{\partial t} \left( (1 - \varphi) \, \rho\_s \, c\_s \, T\_s \right) = \nabla \cdot \left[ k\_s \, (1 - \varphi) \nabla T\_s \right] + h\_v \left( T\_g - T\_s \right) - \nabla \, q\_r. \tag{26}$$

$$\frac{\partial}{\partial t} \{\boldsymbol{\varrho}\boldsymbol{\rho}\boldsymbol{Y}\_{l}\} + \nabla. \{\boldsymbol{\varrho}\boldsymbol{\rho}\boldsymbol{Y}\_{l}\boldsymbol{u}\} + \nabla. \{\boldsymbol{\varrho}\boldsymbol{\rho}\boldsymbol{Y}\_{l}\boldsymbol{v}\_{l}\} - \boldsymbol{\varrho}\ \boldsymbol{\dot{\omega}}\_{l}\boldsymbol{W}\_{l} = \mathbf{0},\tag{27}$$

$$\psi\_l = \begin{pmatrix} -\{D + D\_{m\parallel}^d\} \frac{1}{X\_l} \ \nabla X\_{l\prime} \end{pmatrix} \tag{28}$$

$$Pe = \frac{\rho\_{c\_{p\parallel}|u\rangle}}{k\_g} \,\,\,\tag{29}$$

Numerical Simulation of Combustion in Porous Media 539

(35)

Chemical mechanism for oxidation of methane fuel is considered chemical production rate:

1

effects of turbulence on combustion the common Eddy-Dissipation model can be used.

1 CH4 + 2 O2 →CO2 + 2 H2O 2 O2 + 2 N2 →2 N + 2 NO 3 2 O2 + N2 →2 O + 2 NO 4 N2 + 2 OH →2 H + 2 NO

Spalding suggested that combustion processes are best described by focusing attention on coherent bodies of gas, which squeezed and stretched during their travel through the flame. This model relates the local and instantaneous turbulent combustion rate to the fuel mass fraction and the characteristic time scale of turbulence. The application of this model requires adjustment of a specific coefficient and ignition time to match the experimental combustion rate with the computational combustion rate. In combustion chamber of engine, high turbulence intensity exists and hence combustion for such device lies in the flameletsin-eddies regime. The intrinsic idea behind the model is that the rate of combustion is determined by the rate at which parcel of unburned gas are broken down into the smaller ones, such that there is sufficient interfacial area between the unburned mixture and hot gases to permit reaction and also the turbulence length scale which is quite important can determine the turbulent burning rates. In this case, the coefficients of model were

A general chemical equilibrium reaction with *v′i,s* and *v′'i,s* representing the stoichiometric

Number Equation

5 H� ⇆2H 6 O� ⇆2O 7 N� ⇆ 2N 8 O+H⇆OH 9 O� + 2 H�O ⇆ 4 OH 10 O� + 2 CO ⇆ 2 CO�

determined from experimental analysis of conventional engine.

coefficient of reaction and products for the chemical species *Mi* 

**Table 1.** Kinetic and equilibrium reactions

**5. Foundation of reaction** 

**5.1. Chemical equilibrium** 

*i* 

*NR*

'' ' , , ,

*i ki ki i*

and are stoichiometric coefficients. Combustion process includes ten equations and twelve species. These equations are presented in Table .1, which includes one-step reaction for methane fuel and equations 2-4, are Zeldovich mechanism for NO formation. Rate of reactions are computed with Arrhenius method. But for six other equations that reaction rate is very quick relative to last four equations hence, equilibrium reactions are considered. In order to consider

 *v vR* 

( )

$$D\_{m1}^{d} = 0.5 \, D\_{lm} P e\_{l} \,\tag{30}$$

*Xi* is molar fraction of species *i, Pe* is Peclet number, *d* is diameter of sphere in packed bed, is species dispersion coefficient, *Yi* is mass fraction of species *i* and *vi* is diffusion velocity of species *i* in the mixture.

#### Turbulence model

Since there is no model presented for simulation of turbulent compressible-flow in PM by any researcher. Hence the basic *κ-* ε equations were used without any modification.

The transport equation for *κ* turbulent kinetic energy:

$$\frac{\partial(\rho k)}{\partial t} + \nabla. (\rho u \, k) = -\frac{2}{3} \left[ \rho \,\kappa \nabla. u + \,\,\sigma : \nabla u + \nabla. \left[ \left( \frac{\mu}{\rho r\_k} \right) \nabla k \right] - \rho \varepsilon + \mathcal{W} \,\, ^{s} \right. \tag{31}$$

where σ is stress tensor. With a similar consideration dissipation rate, *ε:* 

$$\frac{\partial \left(\rho \varepsilon \right)}{\partial t} + \nabla \cdot \left(\rho u \varepsilon \right) - \left(\frac{2}{3} \mathbb{C} \in \mathbb{c}\_1 - \mathbb{C} \in \mathbb{s}\_3\right) \rho \varepsilon \nabla \cdot \mu + \nabla \cdot \left[\left(\frac{\mu}{\text{Pr}\_{\varepsilon}}\right) \nabla \cdot \varepsilon \right] + \frac{\mathsf{E}}{k} \left[\mathbb{C} \in \mathbb{c}\_1 \sigma : \nabla \mu - \mathbb{C} \in \mathbb{c}\_2 \ \rho \varepsilon + c\_s \dot{\mathcal{W}}^s \right],\tag{32}$$

The parameters*,* ,, and are constant whose values are determined from experiments and some theoretical considerations and is viscous stress tensor.

Equation of state:

$$P = \ \rho R \ T\_g \ / \ \mathcal{W} \tag{33}$$

*R* is universal gas constant, average molecular weight of mixture, *P* is pressure inside the combustion chamber and the PM.

#### Radiation model

Due to extreme temperature of combustion zone and solid phase, radiation heat transfer is very important. Gas phase radiation in comparison with solid phase radiation that has a high absorption coefficient, is negligible. Several relations for modeling of radiation heat transfer and derived radiation intensity are presented. The heat source term, due to radiation in solid phase that appears in Eq. 26, can be calculated from Rosseland model.

$$q\_{r} = \ -\frac{16}{3} \frac{\sigma \ T\_{s}^{3}}{\beta} \ \nabla T\_{\rm Sv} \tag{34}$$

where *σ* is Boltzmann constant and *β* is extinction coefficient.

Combustion model

Chemical mechanism for oxidation of methane fuel is considered chemical production rate:

$$o o\_i = \sum\_{i=1}^{NR} (\stackrel{\cdots}{v\_{k,i}} - \stackrel{\cdots}{v\_{k,i}}) R\_{i,} \tag{35}$$

and are stoichiometric coefficients. Combustion process includes ten equations and twelve species. These equations are presented in Table .1, which includes one-step reaction for methane fuel and equations 2-4, are Zeldovich mechanism for NO formation. Rate of reactions are computed with Arrhenius method. But for six other equations that reaction rate is very quick relative to last four equations hence, equilibrium reactions are considered. In order to consider effects of turbulence on combustion the common Eddy-Dissipation model can be used.


**Table 1.** Kinetic and equilibrium reactions

538 Numerical Simulation – From Theory to Industry

�

���

The transport equation for *κ* turbulent kinetic energy:

�� + ∇. (�� �) = − �

species *i* in the mixture.

�(��)

 

combustion chamber and the PM.

Turbulence model

Equation of state:

Radiation model

Combustion model

�� (���� ) + ∇. (���� � ) + ∇. (���� �� ) − � ��

�� = −�� + ���

�� =

� � � ��

� �� |�| � ��

*Xi* is molar fraction of species *i, Pe* is Peclet number, *d* is diameter of sphere in packed bed, is species dispersion coefficient, *Yi* is mass fraction of species *i* and *vi* is diffusion velocity of

Since there is no model presented for simulation of turbulent compressible-flow in PM by

� � � ∇. � + � � ∇� + ∇. �� �

*<sup>s</sup> u CC u C u C cW*

The parameters*,* ,, and are constant whose values are determined from experiments and

*R* is universal gas constant, average molecular weight of mixture, *P* is pressure inside the

Due to extreme temperature of combustion zone and solid phase, radiation heat transfer is very important. Gas phase radiation in comparison with solid phase radiation that has a high absorption coefficient, is negligible. Several relations for modeling of radiation heat transfer and derived radiation intensity are presented. The heat source term, due to radiation in solid phase that appears in Eq. 26, can be calculated from Rosseland model.

> �� = − �� � � �� �

 

any researcher. Hence the basic *κ-* ε equations were used without any modification.

where σ is stress tensor. With a similar consideration dissipation rate, *ε:* 

*t k*

some theoretical considerations and is viscous stress tensor.

where *σ* is Boltzmann constant and *β* is extinction coefficient.

 1 3 1 2 <sup>2</sup> . . . : <sup>3</sup> Pr

� �� = 0, (27)

� ∇�� − �� + �� � , (31)

*s*

, (32)

 ���, (28)

, (29)

� = 0.� �����, (30)

���

� = �� �� � �� , (33)

� ∇��, (34)

Spalding suggested that combustion processes are best described by focusing attention on coherent bodies of gas, which squeezed and stretched during their travel through the flame. This model relates the local and instantaneous turbulent combustion rate to the fuel mass fraction and the characteristic time scale of turbulence. The application of this model requires adjustment of a specific coefficient and ignition time to match the experimental combustion rate with the computational combustion rate. In combustion chamber of engine, high turbulence intensity exists and hence combustion for such device lies in the flameletsin-eddies regime. The intrinsic idea behind the model is that the rate of combustion is determined by the rate at which parcel of unburned gas are broken down into the smaller ones, such that there is sufficient interfacial area between the unburned mixture and hot gases to permit reaction and also the turbulence length scale which is quite important can determine the turbulent burning rates. In this case, the coefficients of model were determined from experimental analysis of conventional engine.
