**3.3 Fuel evaporation model**

Fuel evaporation process is a predecessor of combustion, which begins to run just after the start of injection. The evaporation rate is the function of numerous factors, both spray surroundings and the mere fuel parameters. Nevertheless, the temperature inside the combustion chamber is the most important here. For a single fuel droplet, a relationship between size decreasing and evaporation intensity is known (Kowalewicz, 2000):

$$\mathbf{d}\rho^2 - \mathbf{d}^2 = \mathbf{K} \cdot \mathbf{t} \tag{27}$$

where:

78 Fuel Injection in Automotive Engineering

p, q–the exponents that correspond with adopted definition of droplet mean diameter [–];

*i i i n d*

2

2

3

3

*i i i n d*

*n*

*i i i i n d*

3

*n d* 

*n d* 

*i i i n d*

*n*

*i i i i n d*

*n d* 

*n* 

 

 

3,2 2 *i i i i n d*

0.121 qVf 0.131 (26)

*d*

2,0

2,1

<sup>3</sup> 3,0

3,1

*d*

*d*

*d*

*d*

*d*

the values of p and q and formula shape for various definitions is given in Table 3,

**Equivalent mean diameter of droplets p q Calculation formula** 

3 2

Table 3. The list of chosen definition formulas for calculation of mean diameter of droplets

For combustion engine research area, the most usefulness definition is this one, given by Sauter formula *d*3,2 (Table 3). It allows the most accurate rendering of the phenomena, where evaporation, heat and mass transfer, and combustion proceeds and is strictly crucial. Since the equation of SMD definition can be used only for research of mere injection process, comparative studies give different empirical formulas for calculations with using other parameters. For example, Hiroyasu and Katoda (Hiroyasu & Katoda, 1976) elaborated the experimental formula which is convenient to use in engine fuel injection and combustion studies. The equation, which has been consequently used by other researchers (Benson et.

dp,q–theoretical equivalent mean diameter of droplets in a spray jet [mm],

Arithmetic d10 1 0 1,0

di–an actual diameter of droplet in spray jet [mm],

Areal d20 2 0

Areal comparative d21 2 1

Volumetric d30 3 0

Volumetric comparative (Probert) d3 3 1

(also SMD - Sauter mean diameter) d32

al., 1979; Heywood, 1988) is following:

d3,2–Sauter mean diameter [μm],

where:

d3,2 = A· Δp−0.135 ρ<sup>g</sup>

A–a constant for specific sprayer type [–]; for hole sprayer: A = 23.9,

in a spray jet (Orzechowski & Prywer, 1991)

Volumetric-areal (Sauter)

ni–the number of droplets of the actual diameter di [–].

where:

K–evaporation intensity factor that depends on temperature of surrounding where the fuel is injected [mm2/s],

t–evaporation time [s],

d0–initial diameter of droplet [mm],

d–diameter of droplet after the time *t* [mm].

An evaporation intensity factor *K* is the function of temperature and can be derived from experimental measurements. The equation (27) allows calculating the time of complete droplet evaporation by assuming *d* = 0. Also the total mass flux of the fuel vapor coming from a single droplet can be determined as follows:

$$
\dot{m}\_v = \frac{\pi \cdot K \cdot d\_0 \cdot \rho\_f}{6} \tag{28}
$$

where:

*mv* –fuel vapor mass flux generated by evaporating single droplet of initial diameter *d*<sup>0</sup> [g/s],

ρf–fuel density [g/mm3],

K, d0–the same values as for equation (27).

The mass flux of fuel vapor coming from the entire spray jet depends on the numbers of droplets and their size distribution (atomization spectrum). Exact quantitative calculations are practically impossible here. Hence, the averaging equivalent values must be considered including droplet mean diameter and the number of droplets in accordance with actual fuel volume injected. From the droplet equivalent size theory we can estimate the number of

Simulation of Combustion Process

where:

where:

t–time [s],

n–crankshaft rotational speed [1/min],

Vf(φ)–instantaneous volume of liquid fuel in the stream jet [m3], –remaining denotations are as same as in equations (28)-(30).

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 81

Instantaneous volume of liquid fuel in the stream jet depends on fuel injection rate and fuel

*<sup>f</sup> <sup>v</sup> inj v inj*

By resolving equations (30) and (31) that are related each to other we get the rate of evaporated fuel as the function of crankshaft rotation angle. When the calculations exceed the moment of an autoignition, combustion will start and consequntly, the equation (31) will

The chemistry of combustion and formation of different compounds can be included into the overall structure of the presented model. It will be shown on the example of NO formation, where the two of reversible Zeldovich's reactions will be analyzed (Zeldovich et.

On the base of chemical kinetic theory, the formula to calculate the NO formation rate

<sup>1</sup> 2 [O] [N ] *dn <sup>k</sup>*

1 2

NO

k1–kinetic constant of the first Zeldovich reaction in forward direction [m3/(mole·s)],

*dV <sup>M</sup> V VV*

*pK T V <sup>V</sup>*

<sup>1</sup> <sup>6</sup> 2 3,2

 

 

*f*

  O + N2 NO + N (33)

N + O2 NO + O (34)

*V dt* (35)

(32)

( ) () () ()

() () ( ) <sup>10</sup> 6

*n d*

*II f*

vaporization speed. It can be described by the following differential equation:

*inj*

dVf /dφ–change of liquid fuel volume in the stream jet [m3/deg],

–remaining denotations are as same as in equation (31).

cover the additional component describing fuel vapor loss due to its burning.

*d*

<sup>V</sup> ( ) inj –volumetric fuel injection rate [m3/deg],

**3.4 Models for formation of chemical compounds** 

according to the above reaction scheme is following:

V–volume of reaction zone [m3], n NO–mole number of NO [mole],

al., 1947, as cited in Heywood, 1988; Kafar & Piaseczny, 1998):

droplets of Sauter mean diameter *d*3,2 covered by the spray jet consisting of the liquid fuel of a volume *V*f:

$$\mathbf{x} = \frac{\mathbf{6} \cdot V\_f}{\pi \cdot d\_{3,2}^3} \tag{29}$$

where:

x–the number of droplets of Sauter mean diameter *d*3,2 inside the spray jet [–],

Vf–the volume of injected and atomized fuel [mm3],

d3,2–Sauter mean diameter (SMD) [mm].

The mass flux of fuel vapor that comes from the entire stream jet is calculated as follows:

$$\dot{M}\_v = \dot{m}\_{\upsilon}^{\prime} \cdot \propto \frac{\pi \cdot p\_1 K \cdot d\_{3,2} \cdot \rho\_f}{6} \cdot \frac{6 \cdot V\_f}{\pi \cdot d\_{3,2}^3} = \frac{p\_1 K \cdot \rho\_f \cdot V\_f}{d\_{3,2}^2} \tag{30}$$

where:

*Mv* –total mass flux of fuel vapor [g/s],

' *m <sup>v</sup>* –mass flux of fuel vapor comes from evaporation of single droplet located in a cloud of droplets [g/s],

x–the number of droplets in the stream jet [–],

p1–factor correcting vaporization intensity of droplets (*K*) located in the cloud; typical value of *p*1 is p1 ≈ 0.8-0.9,

K, d32, Vf, ρf – the same values as for equations (27)-(29).

The value of *p*1, according to (Kowalewicz & Mozer, 1977; Kucharska-Mozer, 1975; Mozer, 1976), respects the impact of cloud of droplets on the single evaporating droplet. It decreases the value of evaporation intensity factor *K*, because the close presence of other evaporating droplets slows down vaporization due to local temperature lowering.

Finally, it was assumed that the change of droplets size produces the same effect as the change of quantity of droplets of unchanged diameter. Then, the instantaneous fuel vaporization speed (total fuel vapor flux) can be expressed by differential equation:

$$\frac{dM\_v}{d\rho} = \frac{p\_1 K \left(T\_{ll}(\rho)\right) \cdot \rho\_f \cdot V\_f(\rho)}{6 \cdot n \cdot d\_{3,2}^2} \cdot 10^{-6} \tag{31}$$

where:

dMv/dφ–instantaneous fuel vaporization speed [kg/deg],

φ–independent variable: the current crankshaft angle position [deg],

K(TII(φ))–evaporation intensity factor as the function of the zone II temperature [mm2/s],

n–crankshaft rotational speed [1/min],

Vf(φ)–instantaneous volume of liquid fuel in the stream jet [m3],

–remaining denotations are as same as in equations (28)-(30).

Instantaneous volume of liquid fuel in the stream jet depends on fuel injection rate and fuel vaporization speed. It can be described by the following differential equation:

$$\begin{split} \frac{dV\_f}{d\rho} &= \dot{V}\_{\text{inj}}(\rho) - \dot{V}\_v(\rho) = \dot{V}\_{\text{inj}}(\rho) - \frac{\dot{M}\_v(\rho)}{\rho\_f} = \\ &= \dot{V}\_{\text{inj}}(\rho) - \frac{p\_1 K \left(T\_{II}(\rho)\right) \cdot V\_f(\rho)}{6 \cdot n \cdot d\_{3,2}^2} \cdot 10^{-6} \end{split} \tag{32}$$

where:

80 Fuel Injection in Automotive Engineering

droplets of Sauter mean diameter *d*3,2 covered by the spray jet consisting of the liquid fuel of

<sup>6</sup> *Vf <sup>x</sup> d*

The mass flux of fuel vapor that comes from the entire stream jet is calculated as follows:

*<sup>p</sup> K d V pK V M mx*

*m <sup>v</sup>* –mass flux of fuel vapor comes from evaporation of single droplet located in a cloud of

p1–factor correcting vaporization intensity of droplets (*K*) located in the cloud; typical value

The value of *p*1, according to (Kowalewicz & Mozer, 1977; Kucharska-Mozer, 1975; Mozer, 1976), respects the impact of cloud of droplets on the single evaporating droplet. It decreases the value of evaporation intensity factor *K*, because the close presence of other evaporating

Finally, it was assumed that the change of droplets size produces the same effect as the change of quantity of droplets of unchanged diameter. Then, the instantaneous fuel

<sup>1</sup> <sup>6</sup> 2 3,2 () () <sup>10</sup>

 

(31)

vaporization speed (total fuel vapor flux) can be expressed by differential equation:

6 *II f f <sup>v</sup> dM pK T V d n d*

K(TII(φ))–evaporation intensity factor as the function of the zone II temperature [mm2/s],

6 '

6

1 3,2 1

3 2 3,2 3,2

 

*f f ff*

(30)

*d d*

x–the number of droplets of Sauter mean diameter *d*3,2 inside the spray jet [–],

Vf–the volume of injected and atomized fuel [mm3],

*v v*

d3,2–Sauter mean diameter (SMD) [mm].

*Mv* –total mass flux of fuel vapor [g/s],

x–the number of droplets in the stream jet [–],

K, d32, Vf, ρf – the same values as for equations (27)-(29).

droplets slows down vaporization due to local temperature lowering.

φ–independent variable: the current crankshaft angle position [deg],

dMv/dφ–instantaneous fuel vaporization speed [kg/deg],

3 3,2

(29)

a volume *V*f:

where:

where:

'

where:

droplets [g/s],

of *p*1 is p1 ≈ 0.8-0.9,

dVf /dφ–change of liquid fuel volume in the stream jet [m3/deg],

<sup>V</sup> ( ) inj –volumetric fuel injection rate [m3/deg],

–remaining denotations are as same as in equation (31).

By resolving equations (30) and (31) that are related each to other we get the rate of evaporated fuel as the function of crankshaft rotation angle. When the calculations exceed the moment of an autoignition, combustion will start and consequntly, the equation (31) will cover the additional component describing fuel vapor loss due to its burning.

#### **3.4 Models for formation of chemical compounds**

The chemistry of combustion and formation of different compounds can be included into the overall structure of the presented model. It will be shown on the example of NO formation, where the two of reversible Zeldovich's reactions will be analyzed (Zeldovich et. al., 1947, as cited in Heywood, 1988; Kafar & Piaseczny, 1998):

$$\text{NO} + \text{N}\_2 \leftrightarrow \text{NO} + \text{N} \tag{33}$$

$$\text{N} + \text{O}\_2 \leftrightarrow \text{NO} + \text{O} \tag{34}$$

On the base of chemical kinetic theory, the formula to calculate the NO formation rate according to the above reaction scheme is following:

$$\frac{1}{V} \cdot \frac{dn\_{\rm NO}}{dt} = 2k\_1 \cdot [\rm O] \cdot [\rm N\_2] \tag{35}$$

where:

V–volume of reaction zone [m3],

n NO–mole number of NO [mole],

t–time [s],

k1–kinetic constant of the first Zeldovich reaction in forward direction [m3/(mole·s)],

Simulation of Combustion Process

**4. Conclusion** 

stratified charge engines.

**5. References** 

978-

008-0227-20-7, Oxford

0-07-028637-X, New York, NY

*Trans,* SAE Inc., Warrendale, PA

131-141, ISSN 1231-4005, Warsaw-Gdańsk, Poland

*Simposium on Combustion Processes*, Cracow, Poland

*Eksploatacji*, No.1, pp. 163-173, ITE, Radom, Poland

Institute of Aviation, Warsaw, Poland

spalania). WNT, ISBN 83-204-2496-8, Warsaw, Poland

Technical Paper 730169, *SAE Trans,* SAE Inc., Warrendale, PA

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 83

The mathematical model of combustion that is presented in brief in this chapter consists of a lot of phenomena. Here, the most important like the energy conversion, fuel injection and NOX formation are presented. Many of physical and chemical events occurred in the actual engine have been omitted in the model or considered in a reduced form. It is because of impossibility in their exact mathematical representation. Surely, it influences model accuracy, but can be partially compensated by pre-calculation parametric estimation process. This way of model validation shows the disadvantage, i.e. it has to be anew performed for each engine taken under simulation. Nevertheless, the presented model can be a valuable research tool to be used for extensive studies on combustion in all types of

Benson, R.S. & Whitehouse, N.D. (1979). *Internal Combustion Engines*. Pergamon Press, ISBN

Heywood, J.B. (1988). *Internal Combustion Engines Fundamentals*. McGraw Hill Co., ISBN

Hiroyasu, H. & Kadota, T. (1976). Models for Combustion and Formation of Nitric Oxide

Kafar, I. & Piaseczny, L. (1998). Mathematical Model of Toxic Compound Emissions in

Khan, I.M., Greeves, G. & Wang C.H.T. (1973). Factors Affecting Smoke and Gaseous

Kowalewicz, A. & Mozer, I. (1977). Method for cylinder pressure and temperature rate

Kowalewicz, A. (2000). *Fundamentals of Combustion Processes* (in Polish: Podstawy procesów

Kucharska-Mozer, I. (1975). *Simulation of cylinder pressure and temperature in direct injection* 

Kuszewski, H. & Szlachta, Z. (2002). Ecological point of fuel spray jet shaping in

Mozer, I. (1976). *Simulation of cylinder pressure and temperature in direct injection diesel engine* 

and Soot in Direct Injection Diesel Engines. SAE Technical Paper 760129, *SAE* 

Exhaust Gases Produced by Marine Engines. *Journal of KONES*, Vol.5, No.1, pp.

Emissions from Direct Injection Engines and a Method of Calculation. SAE

calculation in compression-ignition engine based on fuel injection characteristics (in Polish: Metoda określania przebiegu ciśnień i temperatur w cylindrze silnika wysokoprężnego na podstawie charakterystyki wtrysku). *5th International* 

*diesel engine based on fuel injection characteristics. Part I.* (In Polish: Metoda symulacji przebiegu temperatur i ciśnień w cylindrze silnika wysokoprężnego z wtryskiem bezpośrednim na podstawie charakterystyki wtrysku). Report No.3.41.113,

compression-ignition engine (in Polish: Ekologiczny aspekt kształtowania własności strugi rozpylanego paliwa w silniku wysokoprężnym). *Problemy* 

*based on fuel injection characteristics. Part II.* (In Polish: Metoda symulacji przebiegu temperatur i ciśnień w cylindrze silnika wysokoprężnego z wtryskiem

[O],[N2]–molar concentration of O-atoms and N2-molecules inside the reaction zone [mole/m3].

It proves that the formation rate is controlled by the first Zeldovich reaction. Atoms of oxygen come mainly from dissociation process O2 ↔ 2 O, and their concentration can be calculated as follows:

$$\mathbf{[O]} = \left(\mathbf{K}^c{}\_O \cdot \left[\mathbf{O}\_2\right]\right)^{\frac{1}{2}}\tag{36}$$

where:

Kc O–equilibrium constant of oxygen dissociation reaction referred to the concentration [mole/m3],

[O],[O2]–molar concentration of O-atoms and O2-molecules inside the reaction zone [mole/m3].

Finally, the NO formation rate formula (35) takes a following shape (all denotations are as same as above):

$$\frac{1}{V} \cdot \frac{dn\_{\rm NO}}{dt} = 2k\_1 \cdot K\_{\rm O}^{c} \frac{1}{2} \cdot [\rm O\_2]^{\frac{1}{2}} \cdot [\rm N\_2] \tag{37}$$

The same formula can express a mass flux of NO in kilograms, so as to be used directly in the model differential equation system:

$$\frac{d\mathbf{M}\_{\rm NO}}{d\rho} = \frac{\mu\_{\rm NO} \cdot V}{6000 \cdot n} \cdot \left[ 2k\_1 \cdot K^c\_\odot \stackrel{1}{\sim} [\mathbf{O}\_2]^{\frac{1}{2}} \cdot [\mathbf{N}\_2] \right] \tag{38}$$

where:

µ NO–molar mass of nitric oxide [g/mole]; µ NO = 30.0061,

n–engine crankshaft speed [rev/min],

–remaining denotations are as same as above.

The constants *Kc* O and *k*1 can be gathered from the bibliography sources (Heywood, 1988; Rychter & Teodorczyk, 1990), and are equal to:

$$k\_1 = 7,6 \cdot 10^7 \cdot \exp\left(\frac{-38000}{T}\right) \left[\frac{\text{m}^3}{\text{mole} \cdot \text{s}}\right] \tag{39}$$

$$\begin{bmatrix} 5 + 0.310805 \ln(T) \cdot \frac{12954}{T} \cdot \\ + 1,07083 - 0.738336 \cdot 10^{-4} \cdot T + \\ + 0,344645 \cdot 10^{-8} \cdot T^2 \end{bmatrix}$$

$$K^c\_{\text{O}} = \frac{10^6}{\overline{R}T} \tag{100} \\ \left[ \frac{\text{mole}}{\text{m}^3} \right] \tag{40}$$
