**5. Mathematical three-dimensional CFD modeling with chemical kinetics of PM diesel engine**

The 3D computational domain is composed of structured, unstructured, or hybrid meshes. The computational domain of the PM engine is a combination of in-cylinder volume and PM reactor. For the in-cylinder space, original governing equations are applied, but the governing equations need to be modified to simulate PM volume. Momentum, gas phase energy, and chemical species continuity equation are modified. Also, a new equation for solid phase energy equation with a radiation model is derived [8–10, 12–14]. For modeling the PM reactor, some assumptions were considered:


Considering to the above assumptions, modified governing equations are [8–10, 12–14]:

### **5.1 Continuity equation for species i is**

$$\frac{\partial(\rho\_i \rho)}{\partial t} + \nabla.(\rho\_i u \rho) = \nabla. \left[ \rho \rho D\_{im} \nabla \left( \frac{\rho\_i}{\rho} \right) \right] + \rho \left. \dot{\rho}\_i^{\varepsilon} + \dot{\rho}^{\varepsilon} \delta\_{i1} \tag{43}$$

where the diffusion coefficient Dim is based on kinetic theory of gases. u is the velocity vector, ρ<sup>i</sup> is the density of species i, and ρ is the density of mixture. *ρ*\_ *c <sup>i</sup>* is the density of species generation or destruction during combustion, and *ρ*\_ *<sup>s</sup>* is the density of spray.

#### **5.2 Gas phase momentum equation**

$$\frac{\partial \left(\rho\_{\mathcal{g}} u\right)}{\partial t} + \nabla \cdot \left(\rho\_{\mathcal{g}} u u\right) = -\nabla P - \nabla \left(\frac{2}{3} \rho\_{\mathcal{g}} k\right) + \nabla \cdot \sigma + F' - \left(\frac{\Delta P}{\Delta L}\right) \tag{44}$$

F<sup>s</sup> is the momentum source of the liquid fuel injection term, and the term <sup>Δ</sup>*<sup>P</sup>* Δ*L* � � on the right-hand side of Eq. (43) is the pressure drop source by PM where Ergan equation is used [8–10, 12–14].

$$
\left(\frac{\Delta P}{\Delta L}\right) = \left(\frac{\mu}{a}u + c\_2 \frac{1}{2}\rho\_\varrho |u|u\right) \tag{45}
$$

$$
a = \frac{d\_p^{-2}}{150} \frac{\epsilon^3}{(1 - \epsilon)^2}
$$

$$
c\_2 = \frac{3.5}{d\_p} \frac{(1 - \epsilon)}{\epsilon^3}
$$
