**6.2 Radiation model**

Because of the high temperature of the combustion zone and solid phase, radiation heat loss should be considered. Absorption coefficient of the solid phase of PM is higher than that of the gas phase; hence, gas phase radiation loss in analogy with solid phase radiation loss can be ignored. Several relations for modeling of radiation intensity were presented in the literatures. The heat source term ∇*:qr*, due to radiation in the solid phase of PM in Eq. (54), is computed by the Rosseland method [18].

$$q\_r = -\frac{16}{3} \frac{\sigma\_b T\_s^3}{\beta} \nabla T\_s \tag{55}$$

### **6.3 Gas injection model**

Gaseous fuel (methane, propane, hydrogen) is directly injected into high-temperature of the PM volume. The gaseous fuel injection model's detail can be found in Ref [5]. The model is for simulating transient direct injection of gaseous fuel into the combustion chamber using a logically refined computational grid [8–10, 12–14].

#### **6.4 Liquid fuel injection**

The essential dynamics of a fuel spray and its interactions with an in-cylinder flow are very complex problems. To compute the mass, momentum, and energy exchange among spray and gas, the distribution of drop sizes, velocities, and temperatures should be determined. In many sprays, drop Weber numbers are more significant than unity, and drop oscillations, distortions, and breakup must be calculated. Drop collisions and coalescence in a diesel engine can be significant in many engine sprays [19–22]. Jet interaction with the PM has four phases [3]:

Phase A: free jet formation from outlet of the nozzle to PM surface.

Phase B: multi-jet splitting as a result of jet interaction with PM surface.

Phase C: liquid fuel distribution in the PM space.

Phase D: liquid fuel passes through the PM space.

Therefore, due to four phase of interaction, particles motion and energy equation need to be modified. **Figure 7** displays schematic modeling of liquid-fuel jet impingement with PM.

#### **6.5 Particles motion equation**

With the injection of liquid droplets, drag force is applied to particles. By impingement of liquid droplets on PM, more drag force is applied to droplets. Hence, the drag coefficient should be modified with available correlation for the

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

**Figure 7.** *Displays schematic modeling of liquid-fuel jet impingement with PM [1].*

impingement of liquid droplets on a single cylinder. Eq. (55) shows the equation of droplets' motion. μ is the viscosity, ρ<sup>p</sup> is the density of liquid fuel, and Rep is the Reynolds number of droplets based on velocity difference among droplets and incylinder fluid. Cd is the modified drag coefficient of the surface of droplets [23–25].

$$\frac{du\_p}{dt} = \frac{18\,\mu}{\rho\_p d\_p^2} \frac{C\_D Re\_p}{24} \left(u - u\_p\right) \tag{56}$$

$$C\_D = \begin{cases} -0.44 + 0.001 \quad Re\_p + \frac{5.64}{Re\_p} + \frac{5.04}{Re\_p} & Re\_p < 400\\ 1.1 & \text{Re}\_p \ge 400 \end{cases}$$

### **6.6 Energy equation for particle (heating-evaporation equation)**

The droplet temperature is computed according to a convective heat transfer between the gas and solid phases of PM and latent heat transfer among the droplets with solid and fluid phases for PM. The total convective heat transfer coefficient from in-cylinder gas to liquid droplet is calculated by Eq. (56). Heat transfer coefficient among liquid droplets and solid phase of PM is computed by consideration of two phenomena. The first term is heat exchange among multi-jet splitting with the solid phase of PM modeling by jet impingement of liquid spray on a hot wall. This heat transfer coefficient was calculated by Eq. (57) and was derived by Rosenow [15]. The second term is heat exchange among the solid phase of PM and gas phase flow in PM that is modeled by heat transfer hot wall to the vaporized fuel-air mixture. This heat transfer coefficient was computed by Eq. (58) and derived by McAdams [11, 20]. The heat exchange process of liquid droplets and gas phase is highly complex in PM-volume and has not been clearly understood. That which correlation (57) or (58) is dominated during heat transfer, and what is a portion of Eqs. (57) and (58) in heat transfer from the solid phase of PM to the gas phase of PM and liquid droplets. Hence, the random number α, where α∈0,1, is inserted in Eq. (59), which is generated by the programming language in each time step. This random number determined the portion of each term in Eqs. (57) and (58) on the total heat transfer coefficient (Eq. (59)). The effect of radiation heat transfer through gas and solid phases of PM on the temperature of liquid droplets has been ignored due to a low volume of liquid droplets [10–16].

$$m\_p \mathcal{C}\_p \frac{dT\_p}{dt} = (\mathbf{1} - \delta) h\_{\mathcal{S}} A\_p \left( T\_{\mathcal{S}} - T\_l \right) + (\mathbf{1} - \delta) h\_{\mathcal{S}} A\_p (T\_s - T\_l) + \delta \dot{m}\_p H\_{\mathcal{S}l} \tag{57}$$

$$h\_{sl-1} = \frac{v\_0 H\_{\rm kg}}{T\_s - T\_l} \exp\left[\mathbf{1} - \left(\frac{T\_s}{T\_{sat}}\right)^2\right] \tag{58}$$

$$h\_{sl-2} = 0.023 \, \frac{k\_{\rm g}}{d} \, Re^{0.8} Pr^{0.4} \tag{59}$$

$$h\_{sl} = a \, h\_{sl-1} + (1 - a)h\_{sl-2} \tag{60}$$
