**5.3 Gas phase energy equation with gaseous fuel injection**

$$\begin{split} \frac{\partial}{\partial t} \left( \varrho \rho c\_p T\_\xi \right) + \nabla \cdot \left( \varrho \rho c\_p T\_\xi u \right) + \varrho \sum\_i \dot{\alpha}\_i h\_i W\_i &= -\varrho P \nabla \cdot u + \varrho A\_0 \rho c + (1 - A\_0) \sigma : \nabla u + \varrho \nabla \cdot u \\ \frac{\partial}{\partial t} \left( \left( k\_\xi + \rho\_\xi c\_\xi D\_\| \right) \nabla T\_\xi \right) - h\_\nu \left( T\_\xi - T\_\nu \right) + \dot{Q}' \end{split} \tag{46}$$

where *Tg* is the temperature of gas, φ is the PM porosity, *u* is the velocity vector, *hi* is the enthalpy of species *i*, *kg* is the fluid thermal conductivity, *D|| <sup>d</sup>* is the thermal

dispersion coefficient along the length of the PM, and *hv* is the volumetric heat transfer coefficient. The term (*φ*∇*: kg* <sup>þ</sup> *<sup>ρ</sup>gcgDd* ∥ � �∇*Tg* � �<sup>Þ</sup> is added to the energy equation of conduction heat transfer in the fluid phase of PM and longitudinal dispersion of mixture in PM. The term (*hv Tg* � *Ts* � �) is added to clarify convective heat transfer between gas and solid phases of PM. Heat exchange between solid and gas phases is computed according to convective heat transfer derived by Wakao and Kaguei to estimate heat transfer between packed beds and fluid [10–14].

### **5.4 Gas phase energy equation with liquid fuel injection**

$$\frac{\partial}{\partial t} \left( \rho \rho c\_p T\_\mathcal{g} \right) + \nabla \cdot \left( \rho \rho c\_p T\_\mathcal{g} u \right) + \rho \sum\_i \dot{\omega}\_i h\_i W\_i = -\rho P \nabla \omega + \rho A\_0 \rho c + (1 - A\_0) \sigma : \nabla u + \rho \nabla \cdot \mathbf{J}$$

$$\frac{\partial}{\partial t} \left( \left( k\_\mathcal{g} + \rho\_\mathcal{g} c\_\mathcal{g} D\_\parallel^d \right) \nabla T\_\mathcal{g} \right) - h\_\mathcal{g} \left( T\_\mathcal{g} - T\_\mathcal{s} \right) + (1 - \delta) h\_\mathcal{g} A\_p \left( T\_\mathcal{g} - T\_\mathcal{l} \right) - \delta \dot{m}\_p H\_{\text{gl}} $$

$$\delta = \begin{cases} 0 & T\_\mathcal{g} < T\_{\text{sat}} \\ 1 & T\_\mathcal{g} = T\_{\text{sat}} \end{cases}$$

The term *hgs Tg* � *Ts* � � is added to represent volumetric convective heat transfer between gas and solid phases of PM. The term 1ð Þ � *δ hglAp Tg* � *Tl* � � is the heat transfer among the gas phase and liquid fuel droplets where liquid droplets are lower than saturation temperature of liquid fuel. δ is the Kronecker delta function for sensible energy of fuel droplets and latent heat of vaporization. *Ap* is the droplet surface area, and *Tl* is the liquid droplet temperature. **Figure 6** illustrates schematic heat transfer between gas, liquid and solid phases in PM space.

The heat transfer between the gas and solid phases is calculated according to Eq. (47). Wakao and Kaguei derived that heat transfer between solid phase and hot gas inside PM [10–14].

**Figure 6.** *Schematic heat transfer between gas, liquid and solid phases in PM space.*

*Mathematical Modeling of a Porous Medium in Diesel Engines DOI: http://dx.doi.org/10.5772/intechopen.108626*

$$h\_{\rm gv} = \frac{6\rho}{d\_p} k\_{\rm g} \left(2.0 + 1.1 \, Re^{0.6} Pr^{0.33} \right) \tag{48}$$

The heat transfer among the gas phase and liquid droplets is computed according to Eq. (48). Correlation (48) was derived by Ranz and Marshal for heat transfer among liquid droplets and gas phase during spray [15–17].

$$h\_{\rm gl} = \frac{k\_{\rm g}}{d\_{\rm p}} \left( 2.0 + 0.6 \left. Re\_{\rm p} \right. \right.^{0.5} Pr^{0.33} \right) \tag{49}$$

#### **5.5 Solid phase energy equation**

$$\frac{\partial}{\partial t}((\mathbf{1} - \varrho)\rho\_s c\_s T\_s) = \nabla \cdot \left[k\_s(\mathbf{1} - \varrho)\nabla T\_s\right] + h\_{\mathbb{B}}\left(T\_{\mathbb{g}} - T\_s\right) - \nabla \cdot q\_r \tag{50}$$

The term (1-ϕ) is due to the solid volume of PM. *Ts* is the solid phase temperature, *ks* is the thermal conductivity of solid phase of PM, ρ<sup>s</sup> is the density, and cs is the specific heat of solid phase of PM. *qr* is the radiation heat loss of solid. Considering the high capacity of the solid phase of PM and the low volume of liquid droplets, the effect of heat transfer between the solid phase and liquid droplets on the energy equation of the solid phase is neglected.

#### **5.6 Turbulence model**

Standard κ- ε equations without modifications were used. The transport equation for *κ* turbulent kinetic energy [8–10, 12–14]:

$$\frac{\partial(\rho k)}{\partial t} + \nabla.(\rho u \, k) = -\frac{2}{3} \,\rho \,\text{\(\mu\)} \,\text{\(\mu} + \sigma : \nabla u + \nabla.\left[\left(\frac{\mu}{Pr\_k}\right) \nabla k\right] - \rho \epsilon + \dot{W} \tag{51}$$

with a similar one for the dissipation rate *ε*:

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \nabla.(\rho u \,\varepsilon) - \left(\frac{2}{3}\mathbf{c}\_{\varepsilon\_1} - \mathbf{c}\_{\varepsilon\_3}\right)\rho \,\varepsilon \nabla.u + \nabla.\left[\left(\frac{\mu}{Pr\_{\varepsilon}}\right)\nabla\varepsilon\right] + \frac{\epsilon}{\mathbf{k}}\left[\mathbf{c}\_{\varepsilon\_1}\sigma : \nabla u - \mathbf{c}\_{\varepsilon\_2}\rho\varepsilon + c\_{\varepsilon}\dot{\mathcal{W}}^t\right].\tag{52}$$
