Mathematical Modeling of a Porous Medium in Diesel Engines

Abstract

Direct injection diesel engines have high power density with low exhaust emission but suffer from particulate matter (PM). Some new technologies were applied to reduce emissions, but they have not solved the emission problem of diesel engines altogether. The main problem of emissions from diesel engines is the simultaneous process of fuel injection and combustion, so non-homogeneous mixture formation occurs in cylinder space, and non-homogeneity is the main reason for emission generation. The solution to this problem is the separation of injection fuel and combustion processes for homogeneous mixture formation in diesel engines. An applicable practical solution for homogeneous mixture formation is the application of porous media (PM) in diesel engine combustion chambers. PM develops stable ultra-lean combustion and decreases emissions. This chapter has three parts for the mathematical modeling of PM diesel engines. The first part is thermodynamically modeling in a closed cycle. The second is zero-dimensional modeling with the chemical kinetics of PM diesel engines, and the third is three-dimensional CFD modeling with the chemical kinetics of PM diesel engines in open or closed cycle. So, mathematical modeling of PM diesel engines, from simple thermodynamically modeling to complicated 3D modeling, is described in this chapter.

1. Introduction

The target of current diesel engines is low fuel consumption with near-zero emission levels for gaseous and particulate matter components at all operational conditions (engine speed and load). Thus, diesel engines require new concepts for the combustion process. The heterogeneous combustion in diesel engines causes non-uniform heat release in the combustion chamber and in the following: NOx, soot, CO, and UHC formation. Homogeneous combustion can solve this problem. Homogeneous combustion in diesel engines is described as a process of homogeneous mixture formation accompanied by volumetric heat release in the total combustion chamber space with a low-temperature gradient inside the chamber. A possible solution to achieving homogeneous combustion is applying PM inside the combustion chamber. One of the different combustion technologies is inside PM combustion. It is a flameless heat release inside an operating PM followed by homogeneous combustion with near-zero emission [1, 2, 3]. This process is stable combustion with a high-power density in an extensive dynamic range. In a burner with an injection of liquid fuel, high-temperature PM acts like an effective evaporator. The large specific surface area with high heat transfer between fluid and solid phases of PM causes fast vaporization of fuel droplets. The large heat capacity of the PM leads to homogeneous combustion with approximately constant temperature.

Some remarkable features of the PM which attracts its application for combustion chamber of diesel engine, display in Figure 1 [1].

Figure 1.

Remarkable features of PM for application in combustion chamber of diesel engine [1].

Many experimental and numerical research has verified the combination of PM burners with flame stability and low emissions. Such exciting features of combustion inside PM make it plausible for application in diesel engines. However, mixture formation and combustion in a diesel engine are complicated nonstationary high-pressure processes in with a direct spray inside the combustion chamber. So, PM can be applied to solve this problem. Ultra-lean burn combustion can be used in diesel engines with stable flame due to the high volumetric heat transfer coefficient between fluid and solid phases of PM. The large surface area of PM and the high heat capacity of PM can absorb some of the heat released during combustion and transfer it to fresh air during the end of the compression process. Hence, the temperature reduction decreases nitrogen oxides. Also, a reduction in the temperature gradient inside the cylinder leads to a decrease in carbon monoxide formation. Figure 2 displays a view of the PM diesel engine. PM inserts in the cylinder head of a diesel engine, and permanent contact between the in-cylinder mixture and PM (open PM concept) exists [3, 4, 5, 6].

Figure 2.

Diesel engine cylinder head of a with inserted PM [2].

2. Diesel engine concept with inserted PM in head of combustion chamber

Durst and Weclas described the diesel engine concept with new mixture formation and combustion processes in a PM reactor. Application of PM in diesel engines generates a homogeneous and flameless combustion process accompanied by a near-zero emission level. Heat recovery from the last combustion process in PM increases the temperature end of the compression process, resulting in raised thermal efficiency and reduced fuel consumption. Heat absorption in PM leads to a reduction in combustion temperature and near-zero NOx. There is difference between the combustion processes in conventional diesel engines and PM-inserted diesel engines. These processes are liquid fuel injection directly into PM, causing multi-jet splitting for fuel distribution throughout PM volume, fuel vaporization, and mixing with air, accomplished by thermal ignition and heat release [1, 6, 7, 8, 9].

2.1 Diesel PM-engine with concept of an open PM chamber

A diesel PM engine describes as a diesel engine with a homogeneous combustion process in a PM volume. PM engines recognize these processes in PM volume: energy recovery in the cycle, fuel injection in PM, fuel vaporization for liquid fuels, mixing with air, homogenization of air-fuel mixture, self-ignition of the mixture, and homogeneous combustion.

This mathematical modeling of permanent contact between working fluid and PM is schematically studied, as illustrated in Figure 3. The PM-combustion chamber is supposed to be inserted in the cylinder head space, and the PM chamber wall is thermally isolated. During the intake process, the PM-heat capacitor has an ignorable effect on the in-cylinder air conditions. Also, during the start of the compression process, a small amount of air is in contact with hot PM. Before TDC, the fuel is injected into PM space, and very fast fuel vaporization for liquid fuels and mixing with air happen in the PM structure. Hence, the fuel is injected close to TDC of the compression process due to high energy storage in PM volume. There is a very complex process during fuel injection, mixture formation, and combustion initiation in the PM structure. The high initial PM temperature (solid-phase temperature of the PM and gas temperature trapped inside PM volume) with mixture formation inside the PM reactor causes self-ignition and volumetric heat release in the combustion process. The reactor heat capacity, pore density, and pore structure can affect the combustion process [2, 3, 4, 5, 6, 7].

Figure 3.

Schematic of a permanent contact PM- diesel engine [8].

Solid phase PM has higher heat capacity than fluid flow and high energy storage capability. Correspondingly, its high surface-to-volume ratio leads to considerable heat exchange. The solid high temperature of PM inside the combustion chamber is a source of the high rate of liquid fuel evaporation and fast mixing with air and self-ignition of the mixture. In diesel engines, mixture formation and combustion simultaneously happen, but in PM engines, these processes occur separately.

The ideal thermodynamic model for combustion in diesel engines is an isobar process, but combustion in PM happens inside diesel engines very fast. Combustion in PM intensifies the reaction rate in a short time scale. Hence, volume change can be considered approximately by a combination of isochoric and isothermal processes.

The last case is illustrated in Figure 3. During the intake process (Figure 3a), PM has not considerably affected the in-cylinder pressure and temperature. Also, during the early compression process, a low quantity of air is in contact with hot PM. In the following compression process, the heat exchange process increases with the motion of the piston to the TDC (Figure 3b). At the TDC, total air is collected in the PM volume. Near the TDC of the compression process, the fuel is injected into the PM volume (Figure 3c), and vaporization of liquid fuel and mixing with air occur very fast in the PM. A volumetric self-ignition of the fuel-air mixture follows flameless combustion by uniform temperature distribution in the PM chamber (Figure 3d). Fuel injection controls combustion initiation timing in the PM volume. The PM structure creates conditions for a homogeneous combustion process and converts the heat into work (Figure 3e). The combustion in a PM can be carried out in the PM volume that cannot occur in the free flame combustion process [1, 2, 3, 4, 5, 6, 7].

2.2 Mathematical thermodynamic modeling of PM diesel engine

For an ideal thermodynamic cycle of conventional and PM diesel engines, a closed cycle is assumed, with working fluid as air with no exhaust gases to the environment. The heat capacity of PM is considerably further than that of fluid. Hence, the solid phase temperature of PM is considered constant during the cycle, and heat exchange between the PM and the working fluid does not affect it. Heat losses of the piston, liner wall, and PM chamber to the environment are ignored, and compression and expansion processes have adiabatically happened.

In the ideal closed cycle energy of a diesel engine (compression ignition engine), combustion occurs at the constant pressure assumed. Heat losses through the combustion chamber to the environment are neglected, and compression and expansion (work) processes happen isentropically. Figure 4a and b shows the P-V (pressure versus volume) and T-S (temperature versus entropy) diagrams of diesel engine closed cycle analysis. The four processes are:

1. process 1 → 2 isentropic compression

2. process 2 → 3 isobar heat addition

3. process 3 → 4 isentropic expansion

4. process 4 → 1 isochoric heat rejection

Figure 4.

(a) P-V diagram of closed cycle of conventional diesel engine and PM engine. (b) T-S diagram of closed cycle of conventional diesel engine and PM engine.

PM diesel engine in the ideal closed cycle energy of the fuel is added to the air in a combination of isochoric and constant temperature. Heat losses through the PM combustion chamber to the environment are ignored, and compression and expansion (work) processes occur isentropically. Figure 4a and b shows a closed cycle’s P-V and T-S diagrams for permanent contact PM diesel-engine analysis. The five processes are:

1. process 1 → 2 isentropic compression

2. process 2 → 3` isochoric heat addition

3. process 3` → 3 isothermal heat addition

4. process 3 → 4 isentropic expansion

5. process 4 → 1 isochoric heat rejection

Compression ratio is defined as r_c, and ρ is the compression ratio during constant temperature heat addition:

$$r_c={\upsilon }_1/{\upsilon }_2$$E1

$$\rho =v_3/v_{3^{\prime }}$$E2

Process 1–2 is isentropic, so:

$$T_2=T_1{\left(r_c\right)}^2$$E3

For PM diesel engines, the energy of fuel is added to air through two process: isochoric process $C_v\left(T_{3^{\prime }}-T_2\right)$ plus isothermal process $R{T}_{3}ln\left(\frac{v_3}{v_{3^{\prime }}}\right)$. Heat loss to environment according to conventional diesel engine is $C_v(T_4-T_2)$.

$$q_{\mathit{\text{in}}}=C_v\left(T_{3^{\prime }}-T_2\right)+R{T}_{3}ln\left(\frac{v_3}{v_{3\prime }}\right)$$E4

Heat rejection occurs in a constant volume process:

$$q_{\mathit{\text{out}}}=C_v\left(T_4-T_1\right)$$E5

Engine thermal efficiency is defined according to Eq. (6):

$$\eta =\frac{w}{q_{\mathit{\text{in}}}}=1-\frac{q_{\mathit{\text{out}}}}{q_{\mathit{\text{in}}}}$$E6

Assuming constant specific heat, the thermal efficiency of diesel engine is according to Eq. (7):

$${\eta }_{\mathit{\text{diesel}}}=1-\frac{C_v(T_4-T_1)}{C_p(T_3-T_2)}$$E7

Hence, thermal efficiency of PM diesel engine is according to Eq. (8).

$${\eta }_{\mathit{PM}\mathit{\text{diesel}}}=1-\frac{C_v\left(T_4-T_1\right)}{C_v\left(T_{3^{\prime }}-T_2\right)+R{T}_{3}ln\left(\frac{v_3}{v_{3\prime }}\right)}$$E8

2.3 Mathematical zero-dimensional modeling with chemical kinetics of PM diesel engine

Figure 5 illustrates schematically permanent contact PM diesel engine. It is supposed that PM is inserted inside the cylinder head. All necessary conditions for homogeneous combustion are carried out in the PM combustion chamber [6, 7]. During the intake and early compression process, the PM-heat capacitor has a low effect on the in-cylinder thermodynamic pressure and temperature. Near TDC, the fuel is injected into the PM structure.

Figure 5.

Schematic illustration of a permanent contact PM diesel engine [6].

Furthermore, the fuel is vaporized fast and mixed with air inside the PM. Because of the instant evaporation and combustion of liquid fuel after injection, it is a logical assumption that combustion occurs in the isochoric process. Then, all the combustion products enter the cylinder space instantaneously. So, the energy of the fluid phase of PM is transferred to the in-cylinder volume [1, 6, 7, 8, 9].

For thermodynamic modeling of PM in the combustion chamber, two energy equations are used, a modified gas energy equation and a solid phase energy equation. These assumptions are considered for the modeling process [1]:

1. Combustion process is adiabatic.

2. PM only recuperates the energy of combustion and does not participate in combustion.

3. A single-step or multi-step reaction(s) is considered for modeling of combustion process.

4. Radiation heat loss computes only in solid phase.

5. There is permanent contact between PM and inside mixture.

6. Air-fuel mixture inside PM is premixed in specified temperature and pressure.

7. Heat capacity for both phases of PM depends on temperature.

8. Energy equations for both phases of PM are solved at each crank angle.

9. Species of combustion products with fluid phase temperature inside PM are transferred to in-cylinder volume at each crank angle.

Mixture formation process assumes a homogenous before combustion due to PM space. Combustion happens as a result of self-ignition due to the high initial temperature.

3. PM modeling

3.1 Gas phase energy equation

Combustion inside a chamber is considered as thermodynamically modeling. Hence, gas phase energy equation and reaction rate are solved simultaneously. The closed system first law of thermodynamics (energy equation) for a differential crank angle, dθ, is [10, 11]:

$$\frac{\mathit{dQ}}{\mathit{d\theta }}-\frac{\mathit{dV}}{\mathit{d\theta }}=m\frac{\mathit{du}}{\mathit{dt}}$$E9

The definition of enthalpy h = u + pv and differentiating in constant pressure process gives:

$$\frac{\mathit{dQ}}{\mathit{d\theta }}=m\frac{\mathit{dh}}{\mathit{d\theta }}$$E10

The enthalpy in terms of chemical compositions of gases is:

$$h=\frac{{\sum }_{i=1}^N{N}_i\overline{h_i}}{m}$$E11

where $N_i$ and $\overline{h_i}$ are the moles and molar enthalpy of species i, so:

$$\frac{\mathit{dh}}{\mathit{d\theta }}=\frac{1}{m}\left[\sum _i\left({\overline{h}}_i\frac{d{N}_i}{\mathit{d\theta }}\right)+\sum _i\left(N_i\frac{d{\overline{h}}_i}{\mathit{d\theta }}\right)\right]$$E12

Assuming ideal gas relation:

$$\frac{d{\overline{h}}_i}{\mathit{d\theta }}=\frac{\partial {\overline{u}}_i}{\partial T}\bullet \frac{\mathit{dT}}{\mathit{d\theta }}={\overline{c}}_{p,i}\frac{\mathit{dT}}{\mathit{d\theta }}$$E13

where ${\overline{c}}_{p,i}$ is the constant molar pressure specific heat of species i. Concentration of species i is obtained as:

$$\left[X_i\right]=\frac{x_{i}P}{R{T}_g}$$E14

The porosity of PM is φ (0 <φ<1), so the volume of fluid in PM is φV, and the volume of solid is (1-φ) V. So $N_i=\mathit{\phi V}\left[X_i\right]$, with differentiating of this equation:

$$\frac{d{N}_i}{\mathit{d\theta }}=\mathit{\phi V}{\dot{\omega }}_i$$E15

The ${\dot{\omega }}_i$ is the rate of reaction species i.

Gas-phase energy equation by applying PM is determined based on a relation of convective heat transfer between the gas and solid phases of PM [10, 11]:

$$\frac{d{T}_g}{\mathit{d\theta }}=\frac{-h_v\left(T_g-T_s\right)-\phi {\sum }_i\left({\dot{\omega }}_i{h}_i\right)+\phi R{T}_g{\sum }_i{\dot{\omega }}_i}{{\sum }_i\left[\left[X_i\right]\left(c_{p.i}-R\right)\right]}$$E16

$\left(\raisebox{1ex}{$\dot{Q}$}\!\left/ \!\raisebox{-1ex}{$V$}\right.\right)$ is heat loss to the environment.

The empirical relation Eq. (17) is applied to compute the volumetric heat transfer coefficient [1]:

$$h_v=\frac{k_g{\mathit{Nu}}_v}{d^2}$$E17

Volumetric Nusselt number is calculated by relation Eq. (18) [10, 11]:

$${\mathit{Nu}}_v=2+1.11{\mathit{Re}}^{0.60}{\mathit{Pr}}^{0.33}$$E18

Reynolds and Prandtl’s numbers are computed by time-varying conditions of the fluid inside the combustion chamber. Reynolds was calculated according to injection velocity. Eq. (19) is used for the calculation of the Reynolds number [10, 11]:

$$\mathit{Re}=\frac{{\rho }_g{u}_{c}d}{\mu }$$E19

The density of gases from the ideal gas equation:

$$\rho =\frac{P}{T\left(\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{${\mathit{MW}}_{\mathit{mix}}$}\right.\right)}$$E20

$${\mathit{MW}}_{\mathit{mix}}=\sum _i^N{x}_i{\mathit{MW}}_i$$

Relation Eq. (21) applies for calculating velocity where ρ_f is the fuel density, the chamber pressure is p_c, and injection pressure is clarified by p_i [10, 11].

$$u_c=\sqrt{\frac{2\left(p_i-p_c\right)}{{\rho }_f}}$$E21

Eq. (22) shows the calculation of chamber pressure [10, 11]:

$$P=\sum _i\left[X_i\right]R{T}_g$$E22

The modified Arrhenius equation is used for modeling the fuel oxidation as shown in Eq. (23) [1].

$${\dot{\omega }}_{\mathit{\text{Fuel}}}=\frac{d\left[X_{\mathit{\text{Fuel}}}\right]}{\mathit{d\theta }}=-A\mathit{exp}\left(\raisebox{1ex}{$-E_a$}\!\left/ \!\raisebox{-1ex}{$\mathit{RT}$}\right.\right){\left[X_{\mathit{\text{Fuel}}}\right]}^m{\left[X_{\mathit{\text{Oxygen}}}\right]}^n$$E23

A, m, and n are constant coefficients of single-step fuel oxidation Eq. (23), selected as Table 1. After solving coupling Eqs. (16) and (23), the gas temperature, species, and fuel consumption rates are determined. After updating the temperature and concentration of species, the pressure was calculated in Eq. (24) [1].

Fuel	A	Ea/R (K)	m	n
H2	1.8 · 10^13	17,614	1.00	0.50
C3H8	8.6 · 10^11	15,098	0.10	1.65
C8H18	4.6 · 10^11	15,098	0.25	1.50
C10H22	3.8 · 10^11	15,098	0.25	1.50
CH3OH	3.2 · 10^12	15,098	0.25	1.50
C2H5OH	1.5 · 10^12	15,098	0.15	1.60

Table 1.

Single-step oxidation of several fuels [10, 11].

$$P=\sum _i\left[X_i\right]RT$$E24

NASA seven-term polynomials are applied to compute the thermodynamic properties. For the calculation of constant pressure of specific heat, NASA polynomial can be used, which depends on temperature:

$$c_p=R\left(a_1+a_{2}T+a_3{T}^2+a_4{T}^3+a_5{T}^4\right)$$E25

Enthalpy of gas can be determined from [10, 11]:

$$\mathit{h}=\mathit{R}\left({\mathit{a}}_{\mathbf{1}}+\frac{{\mathit{a}}_{\mathbf{2}}\mathit{T}}{\mathbf{2}}+\frac{{\mathit{a}}_{\mathbf{3}}{\mathit{T}}^{\mathbf{2}}}{\mathbf{3}}+\frac{{\mathit{a}}_{\mathbf{4}}{\mathit{T}}^{\mathbf{3}}}{\mathbf{4}}+\frac{{\mathit{a}}_{\mathbf{5}}{\mathit{T}}^{\mathbf{4}}}{\mathbf{5}}+\frac{{\mathit{a}}_{\mathbf{6}}}{\mathit{T}}\right)$$E26

$$\mathit{h}\left(\mathit{T}\right)=∆\mathit{hf}\left(\mathbf{298}\right)+\left[\mathit{h}\left(\mathit{T}\right)-\mathit{h}\left(\mathbf{298}\right)\right]$$

3.2 Solid phase energy equation

Energy equation for the solid phase of PM is:

$$\dot{Q}=m\frac{\mathit{dh}}{\mathit{d\theta }}$$E27

By chain rule differentiable:

$$\frac{\mathit{dh}}{\mathit{d\theta }}=\frac{\partial h}{\partial T_s}\frac{d{T}_s}{\mathit{dt}}=c_s\frac{\mathit{dT}}{\mathit{d\theta }}$$E28

Solid phase volume is (1-φ) V, so:

$$\frac{\dot{Q}}{\left(1-\phi \right)V{\rho }_s}=c_s\frac{d{T}_s}{\mathit{d\theta }}$$E29

where ρ_s is the density of solid phase, and c_s is the specific heat of solid phase.

Due to heat transfer between energy gas and solid phase, $h_v\left(T_g-T_s\right)$ is the heat transfer from the gas phase to solid phase. Also, ${\dot{q}}_r$ is radiation from solid phase to environment. Therefore, Eq. (30) calculates solid temperature.

$$\frac{d{T}_s}{\mathit{d\theta }}=\frac{h_v\left(T_g-T_s\right)-{\dot{q}}_r}{\left(1-\phi \right){\rho }_s{C}_s}$$E30

The solid phase energy is computed based on Eq. (31) [10, 11]:

$$\frac{d{T}_s}{\mathit{d\theta }}=\frac{h_v\left(T_g-T_s\right)-\mathit{\epsilon \sigma }\frac{A}{V}\left(T_s^4-T_0^4\right)}{\left(1-\phi \right){\rho }_s{C}_s}$$E31

where ${\left(\frac{A}{V}\right)}_s$ is the surface area to volume ratio of the PM, and T₀ is the environment’s temperature where radiation loss occurs.

3.3 Solution method of the equations

Three coupled Eqs. (32)–(34) that are relevant to solid and gaseous phase temperature and species concentrations are solved.

$$\frac{d{T}_g}{\mathit{d\theta }}=f\left(\theta ,\left[X_i\right],T_g,T_s\right)$$E32

$$\frac{d{T}_s}{\mathit{d\theta }}=g\left(\theta ,\left[X_i\right],T_g,T_s\right)$$E33

$$\frac{d\left[X_i\right]}{\mathit{d\theta }}=h\left(\theta ,\left[X_i\right],T_g\right)$$E34

Runge–Kutta fourth-order method is applied to solve the fluid and solid phase energy equations in PM and compute the species concentrations. Due to the high rate of combustion, the small crank angle (Δθ) should be considered. The step size for solving Runge–Kutta method was assumed 10⁻⁶ s or 0.1 crank angle that depend on engine speed [10, 11].

$$k_{1,g}=hf\left({\theta }_n,T_g,T_s\right)$$

$$k_{1,s}=hg\left({\theta }_n,T_g,T_s\right)$$

$$k_{2,g}=hf\left({\theta }_n+\frac{h}{2},T_g+\frac{k_{1,g}}{2},T_s+\frac{k_{1,s}}{2}\right)$$

$$k_{2,s}=hg\left({\theta }_n+\frac{h}{2},T_g+\frac{k_{1,g}}{2},T_s+\frac{k_{1,s}}{2}\right)$$

$$k_{3,g}=hf\left({\theta }_n+\frac{h}{2},T_g+\frac{k_{2,g}}{2},T_s+\frac{k_{2,s}}{2}\right)$$

$$k_{3,s}=hg\left({\theta }_n+\frac{h}{2},T_g+\frac{k_{2,g}}{2},T_s+\frac{k_{2,s}}{2}\right)$$

$$k_{4,g}=hf\left({\theta }_n+h,T_g+k_{3,g},T_s+k_{3,s}\right)$$

$$k_{4,s}=hg\left({\theta }_n+h,T_g+k_{3,g},T_s+k_{3,s}\right)$$

Updated gas and solid phase temperatures of PM can be obtained from Eqs. (35) and (36). An update of species concentration is computed from Eq. (37):

$${\theta }_{n+1}={\theta }_n+∆\theta$$

$$T_{g,n+1}=T_{g,n}+\frac{1}{6}\left(k_{1,g}+2k_{2,g}+2k_{3,g}+k_{4,g}\right)$$E35

$$T_{s,n+1}=T_{s,n}+\frac{1}{6}\left(k_{1,s}+2k_{2,s}+2k_{3,s}+k_{4,s}\right)$$E36

$${\left[X_i\right]}_{n+1}={\left[X_i\right]}_n+h\left({\theta }_n,{\left[X_i\right]}_n,T_{g,n+1}\right)$$E37

4. In-cylinder modeling

4.1 Mass conservation

The total mass of the cylinder is a combination of the energy of the in-cylinder and fluid phase volume of PM:

$${\left(\mathrm{m}\right)}_{\mathit{\text{cylinder}},n}+{\left(\mathrm{m}\right)}_{\mathit{PM}\mathit{\text{fluid}},n}={\left(\mathrm{m}\right)}_{\mathit{\text{cylinder}},n+1}$$E38

4.2 Energy equation

The total energy of the cylinder is a combination of the energy of the in-cylinder and the fluid phase volume of PM:

$${\left(\mathrm{m}{c}_v\mathrm{T}\right)}_{\mathit{\text{cylinder}},n}+{\left(\mathrm{m}{c}_v\mathrm{T}\right)}_{\mathit{PM}\mathit{\text{fluid}},n}={\left(\mathrm{m}{c}_v\mathrm{T}\right)}_{\mathit{\text{cylinder}},n+1}$$E39

$${\mathrm{m}}_{\mathit{cyl},n}{c}_{v,\mathit{cyl},n}{T}_{\mathit{cyl},n}+m_{f_{\mathit{PM}},n}{c}_{v,f_{\mathit{PM}},n}{T}_{f_{\mathit{PM}},n}={\mathrm{m}}_{\mathit{cyl},n+1}{c}_{v,\mathit{cyl},n+1}{T}_{\mathit{cyl},n+1}$$E40

So finally, in-cylinder temperature and pressure were updated according to Eqs. (41) and (42):

$$T_{\mathit{cyl},n+1}=\frac{{\mathrm{m}}_{\mathit{cyl},n}{c}_{v,\mathit{cyl},n}{T}_{\mathit{cyl},n}+m_{f_{\mathit{PM}},n}{c}_{v,f_{\mathit{PM}},n}{T}_{f_{\mathit{PM}},n}}{\left({\left(\mathrm{m}\right)}_{\mathit{\text{cylinder}},n}+{\left(\mathrm{m}\right)}_{\mathit{PM}\mathit{\text{fluid}},n}\right)c_{v,\mathit{cyl},n}}$$E41

$$P_{\mathit{cyl},n+1}=\sum _i{\left[X_i\right]}_{n+1}R{T}_{\mathit{cyl},n+1}$$E42

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, 9, 10, 12, 13, 14]. For modeling the PM reactor, some assumptions were considered:

1. There is non-equilibrium thermal energy between gas and solid phases of PM.

2. The solid phase of PM is homogeneous and isotropic; has a variable property with temperature; and has no catalyst effects.

3. Radiation heat transfer to the environment is considered for the solid phase. Moreover, the effect of its radiation on the evaporation of droplets is neglected.

Considering to the above assumptions, modified governing equations are [8, 9, 10, 12, 13, 14]:

5.1 Continuity equation for species i is

$$\frac{\partial \left({\rho }_i\phi \right)}{\partial t}+\nabla .\left({\rho }_i\mathit{u\phi }\right)=\nabla .⌈\rho {\mathit{\phi D}}_{\mathit{im}}\nabla \left(\frac{{\rho }_i}{\rho }\right)⌉+\phi {\dot{\rho }}_i^c+{\dot{\rho }}^s{\delta }_{i1}$$E43

where the diffusion coefficient D_im is based on kinetic theory of gases. u is the velocity vector, ρ_i is the density of species i, and ρ is the density of mixture. ${\dot{\rho }}_i^c$ is the density of species generation or destruction during combustion, and ${\dot{\rho }}^s$ is the density of spray.

5.2 Gas phase momentum equation

$$\frac{\partial \left({\rho }_{g}u\right)}{\partial t}+\nabla .\left({\rho }_g\mathit{uu}\right)=-\nabla P-\nabla \left(\frac{2}{3}{\rho }_{g}k\right)+\nabla .\sigma +F^s-\left(\frac{∆P}{∆L}\right)$$E44

F^s is the momentum source of the liquid fuel injection term, and the term $\left(\frac{∆P}{∆L}\right)$ on the right-hand side of Eq. (43) is the pressure drop source by PM where Ergan equation is used [8, 9, 10, 12, 13, 14].

$$\left(\frac{∆P}{∆L}\right)=\left(\frac{\mu }{\alpha }u+c_2\frac{1}{2}{\rho }_g\left|u\right|u\right)$$E45

$$\alpha =\frac{{d_p}^2}{150}\frac{{\epsilon }^3}{{\left(1-ϵ\right)}^2}$$

$$c_2=\frac{3.5}{d_p}\frac{\left(1-\epsilon \right)}{{\epsilon }^3}$$

5.3 Gas phase energy equation with gaseous fuel injection

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\mathit{\phi \rho }{c}_p{T}_g\right)+\nabla .\left(\mathit{\phi \rho }{c}_p{T}_{g}u\right)+\phi \sum _i{\dot{\omega }}_i{h}_i{W}_i=-\mathit{\phi P}\nabla .u+\phi A_0\mathit{\rho ϵ}+\left(1-A_0\right)\sigma :\nabla u+\phi \nabla .\\ \left(\left(k_g+{\rho }_g{c}_g{D}_{\Vert }^d\right)\nabla T_g\right)-h_v\left(T_g-T_s\right)+{\dot{Q}}^s\end{array}$$E46

where T_g is the temperature of gas, φ is the PM porosity, u is the velocity vector, h_i is the enthalpy of species i, k_g is the fluid thermal conductivity, D_||^d is the thermal dispersion coefficient along the length of the PM, and h_v is the volumetric heat transfer coefficient. The term ($\phi \nabla .\left(\left(k_g+{\rho }_g{c}_g{D}_{\parallel }^d\right)\nabla T_g\right))$ 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 ($h_v\left(T_g-T_s\right)$) 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, 11, 12, 13, 14].

5.4 Gas phase energy equation with liquid fuel injection

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\mathit{\phi \rho }{c}_p{T}_g\right)+\nabla .\left(\mathit{\phi \rho }{c}_p{T}_{g}u\right)+\phi \sum _i{\dot{\omega }}_i{h}_i{W}_i=-\mathit{\phi P}\nabla .u+\phi A_0\mathit{\rho ϵ}+\left(1-A_0\right)\sigma :\nabla u+\phi \nabla .\\ \left(\left(k_g+{\rho }_g{c}_g{D}_{\Vert }^d\right)\nabla T_g\right)-h_{\mathit{gs}}\left(T_g-T_s\right)+\left(1-\delta \right)h_{\mathit{gl}}{A}_p\left(T_g-T_l\right)-\delta {\dot{m}}_p{H}_{\mathit{gl}}\end{array}$$E47

$$\mathrm{\delta }=\left\{\begin{array}{c}0T_g<T_{\mathit{sat}}\\ 1T_g=T_{\mathit{sat}}\end{array}\right.$$

The term $h_{\mathit{gs}}\left(T_g-T_s\right)$ is added to represent volumetric convective heat transfer between gas and solid phases of PM. The term $\left(1-\delta \right)h_{\mathit{gl}}{A}_p\left(T_g-T_l\right)$ 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. A_p is the droplet surface area, and T_l is the liquid droplet temperature. Figure 6 illustrates schematic heat transfer between gas, liquid and solid phases in PM space.

Figure 6.

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, 11, 12, 13, 14].

$$h_{\mathit{gs}}=\frac{6\phi }{{d_p}^2}{k}_g\left(2.0+1.1{\mathit{Re}}^{0.6}{\mathit{Pr}}^{0.33}\right)$$E48

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, 16, 17].

$$h_{\mathit{gl}}=\frac{k_g}{d_p}\left(2.0+0.6{{\mathit{Re}}_p}^{0.5}{\mathit{Pr}}^{0.33}\right)$$E49

5.5 Solid phase energy equation

$$\frac{\partial }{\partial t}\left(\left(1-\phi \right){\rho }_s{c}_s{T}_s\right)=\nabla \bullet \left[k_s\left(1-\phi \right)\nabla T_s\right]+h_{\mathit{gs}}\left(T_g-T_s\right)-\nabla .q_r$$E50

The term (1-ϕ) is due to the solid volume of PM. T_s is the solid phase temperature, k_s is the thermal conductivity of solid phase of PM, ρ_s is the density, and c_s is the specific heat of solid phase of PM. q_r 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, 9, 10, 12, 13, 14]:

$$\frac{\partial \left(\mathit{\rho k}\right)}{\partial t}+\nabla .\left(\mathit{\rho u}k\right)=-\frac{2}{3}\mathrm{\rho }\mathrm{\kappa }\nabla .u+\sigma :\nabla u+\nabla .\left[\left(\frac{\mu }{{\mathit{Pr}}_k}\right)\nabla k\right]-\mathit{\rho \epsilon }+{\dot{W}}^s$$E51

with a similar one for the dissipation rate ε:

$$\frac{\partial \left(\mathit{\rho \epsilon }\right)}{\partial t}+\nabla .\left(\mathit{\rho u}\epsilon \right)-\left(\frac{2}{3}{\mathrm{c}}_{{\mathrm{ϵ}}_1}-{\mathrm{c}}_{{\mathrm{ϵ}}_3}\right)\mathrm{\rho }\mathrm{\epsilon }\nabla .u+\nabla .\left[\left(\frac{\mu }{{\mathit{Pr}}_{\epsilon }}\right)\nabla ϵ\right]+\frac{\mathrm{ϵ}}{\mathrm{k}}\left[{\mathrm{c}}_{{\mathrm{ϵ}}_1}\sigma :\nabla u-{\mathrm{c}}_{{\mathrm{ϵ}}_2}\mathit{\rho \epsilon }+c_s{\dot{W}}^s\right]$$E52

6. Equation of state

$$P={\rho }_{g}RT/\overline{W}$$E53

6.1 Combustion model

Chemical mechanism for oxidation of Decane or other hydrocarbon fuels is considered. ώ_i chemical production rate:

$${\dot{\omega }}_i=\sum _{i=1}^{\mathit{NR}}\left(v_{k,i}^{\prime \prime }-v_{k,i}^{\prime }\right)R_i$$E54

$v_{k,i}^{\prime \prime }$ and $v_{k,i}^{\prime }$ are stoichiometric coefficients. The Arrhenius model calculates reaction rates. For other equations, the reaction rate is very quickly relative to the main equations; equilibrium reactions are considered. In order to calculate the effects of turbulence on combustion, Spalding’s eddy-breakup model is considered. The idea of the eddy-breakup model is that the combustion rate is computed by the rate at which large parcels of unburned gas are broken down into smaller particles. The turbulence length scale is significant in determining turbulent burning rates [8, 9, 10, 12, 13, 14].

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 $\nabla .q_r$, due to radiation in the solid phase of PM in Eq. (54), is computed by the Rosseland method [18].

$$q_r=-\frac{16}{3}\frac{{\sigma }_b{T_s}^3}{\beta }\nabla T_s$$E55

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, 9, 10, 12, 13, 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, 20, 21, 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.

Figure 7.

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

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 impingement of liquid droplets on a single cylinder. Eq. (55) shows the equation of droplets’ motion. μ is the viscosity, ρ_p is the density of liquid fuel, and Re_p is the Reynolds number of droplets based on velocity difference among droplets and in-cylinder fluid. C_d is the modified drag coefficient of the surface of droplets [23, 24, 25].

$$\frac{d{u}_p}{\mathit{dt}}=\frac{18\mu }{{\rho }_p{d}_p^2}\frac{C_D{\mathit{Re}}_p}{24}\left(u-u_p\right)$$E56

$$C_D=\left\{\begin{array}{ll}-0.44+0.001{\mathit{Re}}_p+\frac{5.64}{{\mathit{Re}}_p}+\frac{5.04}{{{\mathit{Re}}_p}^{0.28}}& {\mathit{Re}}_p<400\\ 1.1& {\mathit{Re}}_p\ge 400\end{array}\right.$$

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, 11, 12, 13, 14, 15, 16].

$$m_p{C}_p\frac{d{T}_p}{\mathit{dt}}=\left(1-\delta \right)h_{\mathit{gl}}{A}_p\left(T_g-T_l\right)+\left(1-\delta \right)h_{\mathit{sl}}{A}_p\left(T_s-T_l\right)+\delta {\dot{m}}_p{H}_{\mathit{gl}}$$E57

$$h_{\mathit{sl}-1}=\frac{v_0{H}_{\mathit{lg}}{\rho }_l}{T_s-T_l}\mathit{exp}\left[1-{\left(\frac{T_s}{T_{\mathit{sat}}}\right)}^2\right]$$E58

$$h_{\mathit{sl}-2}=0.023\frac{k_g}{d}{\mathit{Re}}^{0.8}{\mathit{Pr}}^{0.4}$$E59

$$h_{\mathit{sl}}=\alpha h_{\mathit{sl}-1}+\left(1-\alpha \right)h_{\mathit{sl}-2}$$E60

7. Conclusions

This chapter illustrates the mathematical modeling of PM diesel engines: thermodynamically modeling, zero-dimensional closed cycle modeling with chemical kinetics of PM diesel engine, and three-dimensional CFD modeling with PM diesel engine chemical kinetics. So, mathematical modeling of PM diesel engines, from simple thermodynamically modeling to complicated 3D modeling, has been described.