**3. Demonstration of combustion model**

All mathematical models of combustion engine working cycle can be sorted as follows (Rychter & Teodorczyk, 1990):


Above segmentation defines a model complexity and fidelity in representation of real processes in the model. It is also connected with complication in mathematical tools used for simulation. There are a lot of examples which combines the models according to the above segmentation (Khan et. al., 1973; Patterson, 1994, 1997; Rychter & Teodorczyk, 1990). The fundamental problem in choosing a proper type of the model is to find a compromise between accuracy and intellectual labor involved to describe all physical phenomena. A priority here is the goal of analysis. As a rule, for comparative and/or quantitative research, a simplified model can be used with receiving good results; for qualitative investigations more precise model should be worked out instead.

In a preliminary analysis toward model formulating, a number of physical and chemical processes that occur during the injection, combustion and exhaust pollutant formation were taken into consideration. The latest theoretical and experimental results were regarded. A great effort was made to include to the analysis all phenomena that have a major impact on the various processes modeled, so as their actual nature would be reproduced. Thanks to

Simulation of Combustion Process

thermodynamics for open systems:

U–internal energy of the system [J],

V–system volume [m3], p–system pressure [Pa],

φ–crank angle [deg].

be found in (Woś, 2008)):

for heat fluxes:

for mass transfers:

Q–heat delivered to/derived from the system [J],

H–enthalpy delivered to/derived from the system [J],

where:

**3.1 The model core based on thermodynamic theory** 

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 71

combustion chamber is divided onto two zones: the first one (I) - consisting of the rest of fresh charge, and the second one (II) - drawn by fuel-air mixture boundaries. After short time of autoignition delay τo, the process of evaporated fuel combustion gets set on and runs with the burning rate equals to d*M*h/dφ. It generates a heat flux đ*Q*h/dφ that is supplied into the zone II. Between both zones (I and II), a mass transfer process occurs (d*M*ex/dφ), and between cylinder walls and both zones a heat transfer process occurs also with the rate đ*Q*c/dφ. The whole system gives an elementary mechanical work equals to *p*d*V*/dφ. Except of fundamental combustion reaction, the other free selected ones can be implemented (dissociations, pollutant formation) in order to check various engine output parameters.

In relation of above physical model, a mathematical model of engine working cycle was formulated with taking some indispensable assumptions into consideration. The essential equation for energy conversion inside the cylinder is differential equation of the first law of

dU ðQ dV dH <sup>p</sup> dd dd

Above equation is valid for both zones of the elaborated model, but it must be developed further in order to calculate temperature change in both zones. According to the assumptions taken in the physical model, we can write as follows (detailed evaluation can

II c II h v

*II v ex*

 

> 

ðQ ðQ ðQ ðQ d d dd

*I ex*

*dM dM dM ddd*

 

*dM dM d d*

  

I c I

ðQ ðQ d d

  

(1)

(2)

(3)

this, the model enables to simulate the effect of many structural and operational factors on the engine performance, including detailed emissions.

As a result, a two-zone quasi-dimensional model has been developed. Comparing to singlezone models, the own one is characterized by a much more accurate description of the phenomena in a combustion chamber. This attribute greatly emphasizes the scientific and utilitarian aspect of such a solution. In addition, the model permits to be extended easily of additional, computational blocks. In this respect, the proposed method of analysis exemplifies an important cognitive value and is rarely found in the literature.

Below, an application of worked out, two-zone, quasi-dimensional model of combustion for direct injection diesel engine will be presented. Splitting the combustion chamber into two zones for models of such type of the engines makes a fidelity in representation of phenomena proceeded inside the cylinder much precise, although it complicates mathematics.

Fig. 1. The scheme of physical and chemical processes proceeded in a combustion chamber of direct injection (DI) diesel engine (Woś, 2008)1

As is shown in the Fig. 1, inside the cylinder of volume *V* and pressure *p*, at the end of intake stroke there is a fresh air charge of a mass *M*ch, and at the moment determined by the start of injection angle, an initial fuel quantity begins to be injected. The fuel volume flow rate is d*V*inj/dφ. Here, dφ means the increment of an independent variable that is the crankshaft angle. A part of the fuel begins to evaporate with the rate equals to d*M*v/dφ. It forms the spray cones of total volume *V*II, consisted of fuel-air mixture. Through that, the

<sup>1</sup> The denotations are explained in the chapter body.

combustion chamber is divided onto two zones: the first one (I) - consisting of the rest of fresh charge, and the second one (II) - drawn by fuel-air mixture boundaries. After short time of autoignition delay τo, the process of evaporated fuel combustion gets set on and runs with the burning rate equals to d*M*h/dφ. It generates a heat flux đ*Q*h/dφ that is supplied into the zone II. Between both zones (I and II), a mass transfer process occurs (d*M*ex/dφ), and between cylinder walls and both zones a heat transfer process occurs also with the rate đ*Q*c/dφ. The whole system gives an elementary mechanical work equals to *p*d*V*/dφ. Except of fundamental combustion reaction, the other free selected ones can be implemented (dissociations, pollutant formation) in order to check various engine output parameters.

### **3.1 The model core based on thermodynamic theory**

In relation of above physical model, a mathematical model of engine working cycle was formulated with taking some indispensable assumptions into consideration. The essential equation for energy conversion inside the cylinder is differential equation of the first law of thermodynamics for open systems:

$$\frac{\text{d}\mathbf{U}}{\text{d}\,\rho} = \frac{\text{\"d}\mathbf{Q}}{\text{d}\,\rho} - \text{p}\frac{\text{dV}}{\text{d}\,\rho} + \frac{\text{dH}}{\text{d}\,\rho} \tag{1}$$

where:

70 Fuel Injection in Automotive Engineering

this, the model enables to simulate the effect of many structural and operational factors on

As a result, a two-zone quasi-dimensional model has been developed. Comparing to singlezone models, the own one is characterized by a much more accurate description of the phenomena in a combustion chamber. This attribute greatly emphasizes the scientific and utilitarian aspect of such a solution. In addition, the model permits to be extended easily of additional, computational blocks. In this respect, the proposed method of analysis

Below, an application of worked out, two-zone, quasi-dimensional model of combustion for direct injection diesel engine will be presented. Splitting the combustion chamber into two zones for models of such type of the engines makes a fidelity in representation of phenomena

**dVinj**

**zone II**

**NOx dQc** 

**dMv dQh** 

**zone I**

pdV

Fig. 1. The scheme of physical and chemical processes proceeded in a combustion chamber

As is shown in the Fig. 1, inside the cylinder of volume *V* and pressure *p*, at the end of intake stroke there is a fresh air charge of a mass *M*ch, and at the moment determined by the start of injection angle, an initial fuel quantity begins to be injected. The fuel volume flow rate is d*V*inj/dφ. Here, dφ means the increment of an independent variable that is the crankshaft angle. A part of the fuel begins to evaporate with the rate equals to d*M*v/dφ. It forms the spray cones of total volume *V*II, consisted of fuel-air mixture. Through that, the

of direct injection (DI) diesel engine (Woś, 2008)1

1 The denotations are explained in the chapter body.

**dMex** 

**p, VII, TII, MII, UII, RII**

exemplifies an important cognitive value and is rarely found in the literature.

proceeded inside the cylinder much precise, although it complicates mathematics.

**p, VI, TI, MI, UI, RI**

**CO2 O2 H2O <sup>O</sup>N2 C12H22**

the engine performance, including detailed emissions.

U–internal energy of the system [J],

Q–heat delivered to/derived from the system [J],

V–system volume [m3],

p–system pressure [Pa],

H–enthalpy delivered to/derived from the system [J],

φ–crank angle [deg].

Above equation is valid for both zones of the elaborated model, but it must be developed further in order to calculate temperature change in both zones. According to the assumptions taken in the physical model, we can write as follows (detailed evaluation can be found in (Woś, 2008)):

for heat fluxes:

$$\begin{aligned} \frac{\partial \mathbf{Q\_{l}}}{\mathbf{d}\,\rho} &= -\frac{\partial \mathbf{Q\_{c1}}}{\mathbf{d}\,\rho} \\ \frac{\partial \mathbf{Q\_{l}}}{\mathbf{d}\,\rho} &= -\frac{\partial \mathbf{Q\_{c1}}}{\mathbf{d}\,\rho} + \frac{\partial \mathbf{Q\_{h}}}{\mathbf{d}\,\rho} - \frac{\partial \mathbf{Q\_{v}}}{\mathbf{d}\,\rho} \end{aligned} \tag{2}$$

for mass transfers:

$$\begin{aligned} \frac{dM\_I}{d\rho} &= \frac{dM\_{ex}}{d\rho} \\ \frac{dM\_{II}}{d\rho} &= \frac{dM\_v}{d\rho} - \frac{dM\_{ex}}{d\rho} \end{aligned} \tag{3}$$

Simulation of Combustion Process

*i i*

  Ii I

M–mass of the zone [kg],

V–zone volume [m3].

d*T*I/dφ, d*T*II/dφ, i.e.:

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 73

I I

*d T d d*

II II

 

(9)

(10)

 

(11)

(12)

(7)

 

ðQ dV u g p ug d d

 

*i i i*

ðQ dV u g <sup>p</sup> d d

*dM <sup>u</sup> dT h Mg <sup>d</sup> T d*

*i ex I Ii*

R–universal gas constant for a whole zone [J/(kg·K)],

T–average temperature of the zone [K],

transferred mass can be evaluated as follows:

*ex Ii I*

According to the ideal gas law equation of Clapeyron, it also means that:

IIi II II IIi II

The other differentiates, such as đ*Q*I/dφ, đ*Q*II/dφ, d*V*I/dφ, d*V*II/dφ, d*M*v/dφ, can be calculated with use of independent submodels. To resolve above system algebraically, d*M*ex/dφ must be eliminated and expressed by other known components. To do that, we can use an overall assumption that the pressure *p* in both zones is always equal in value:

pI = pII (8)

*I I I II II II I II M RT M R T V V*

where the symbols refer to both zones, such as subscript indicates, and they mean as follows:

Going ahead, at any time the mass of the first zone is the sum of initial mass of fresh air charge *M*ch and mass transferred *M*ex. Similarly, for the second zone it is a mass of evaporated fuel *M*v from which the transferred mass *M*ex must be subtracted. Then

> *v II II I* ch *I I II ex I I II II II I*

To differentiate it relatively to the crank angle variable φ, we receive a formula for

dMex dT dT dV dV dM I II I II v *<sup>f</sup>* , ,, , *d dddd d*

Now, replacing the component d*M*ex/dφ in the system of equations (7) with the above function we receive a new system of two differential equations with only two unknowns:

*I II*

 

 

*dT dT AB C d d dT dT DE F d d*

 

*I II*

component d*M*ex/dφ as a function expressed by the other differentiates:

*<sup>M</sup> R T V M RTV <sup>M</sup> RTV R T V*

*dM <sup>u</sup> dT dM <sup>h</sup> M g <sup>h</sup>*

*ex IIi II <sup>v</sup> ex <sup>i</sup> IIi i v*

and for enthalpy fluxes:

$$\begin{aligned} \frac{dH\_{\parallel}}{d\rho} &= h\_{ex} \cdot \frac{dM\_{ex}}{d\rho} \\ \frac{dH\_{\parallel \parallel}}{d\rho} &= h\_v \cdot \frac{dM\_v}{d\rho} - h\_{ex} \cdot \frac{dM\_{ex}}{d\rho} \end{aligned} \tag{4}$$

where:

I, II–subscripts referred to zone I and II, in order,

Qc–heat of cooling [J],

Qh–heat generated by combustion [J],

Qv–heat consumed by vaporizing fuel [J],

Mex–mass transferred between both zones [kg],

Mv–mass of evaporated fuel [kg],

hex–specific enthalpy of transferred mass; it is specific enthalpy of I or II zone depending on direction of mass flow [J/kg],

hv–specific enthalpy of fuel vapor [J/kg],

Total internal energy of any thermodynamic system can be expressed by multiplying specific internal energy *u* and system mass *M*. Thus, we can also differentiate this multiplication, what gives:

$$\frac{d\mathbf{U}I}{d\boldsymbol{\varrho}} = \frac{d(M \cdot \boldsymbol{u})}{d\boldsymbol{\varrho}} = \boldsymbol{u} \cdot \frac{d\boldsymbol{M}}{d\boldsymbol{\varrho}} + \boldsymbol{M} \cdot \frac{d\boldsymbol{u}}{d\boldsymbol{\varrho}}\tag{5}$$

If we consider that specific internal energy u for various compounds mixture can be calculated: u u <sup>i</sup> g*i i* ; then, assuming *<sup>i</sup> i i dg <sup>u</sup> d* as near to null, we will get:

$$\frac{d\underline{u}}{d\underline{\sigma}} = \frac{dT}{d\underline{\sigma}} \cdot \sum\_{i} \left( \mathbf{g}\_{i} \cdot \frac{\widehat{\boldsymbol{\alpha}} \mathbf{u}\_{i}}{\widehat{\boldsymbol{\alpha}} T} \right) \tag{6}$$

where:

ui-specific internal energy for an "i" component [J/kg],

gi-mass fraction of an "i" component in a whole system [kg/kg],

T-temperature of the system [K].

Substitution of all above equations into the fundamental equation (1) for both zones will give a system of two differential equations with three unknowns: d*T*I/dφ, d*T*II/dφ, and d*M*ex/dφ:

$$\begin{cases} \left[\sum\_{i} \left(\mathbf{u}\_{\text{li}\mathbf{g}\_{\text{li}}}\right) - h\_{ex}\right] \frac{dM\_{ex}}{d\rho} + M\_{l} \sum\_{i} \left(g\_{li} \frac{\partial u\_{li}}{\partial T}\right) \frac{dT\_{l}}{d\rho} = \frac{\partial \mathbf{Q}\_{l}}{\mathbf{d}\rho} - \mathbf{p} \frac{\mathbf{dV}\_{l}}{\mathbf{d}\rho} \\\\ \left[h\_{ex} - \sum\_{i} \left(\mathbf{u}\_{\text{li}\mathbf{g}\_{\text{li}}}\right)\right] \frac{dM\_{ex}}{d\rho} + M\_{\text{II}} \sum\_{i} \left(g\_{\text{li}i} \frac{\partial u\_{\text{li}i}}{\partial T}\right) \frac{dT\_{\text{II}}}{d\rho} = \frac{\partial \mathbf{Q}\_{\text{II}}}{\mathbf{d}\rho} - \mathbf{p} \frac{\mathbf{dV}\_{\text{II}}}{d\rho} - \left[\sum\_{i} \left(\mathbf{u}\_{\text{li}\mathbf{g}\_{\text{II}}}\right) - h\_{\text{v}}\right] \frac{dM\_{\text{v}}}{d\rho} \end{cases} (7)$$

The other differentiates, such as đ*Q*I/dφ, đ*Q*II/dφ, d*V*I/dφ, d*V*II/dφ, d*M*v/dφ, can be calculated with use of independent submodels. To resolve above system algebraically, d*M*ex/dφ must be eliminated and expressed by other known components. To do that, we can use an overall assumption that the pressure *p* in both zones is always equal in value:

$$\mathbf{p}\_{\mathbf{I}} = \mathbf{p}\_{\mathbf{II}} \tag{8}$$

According to the ideal gas law equation of Clapeyron, it also means that:

$$\frac{\mathbf{M}\_I \cdot \mathbf{R}\_I \cdot T\_I}{V\_I} = \frac{\mathbf{M}\_{II} \cdot \mathbf{R}\_{II} \cdot T\_{II}}{V\_{II}} \tag{9}$$

where the symbols refer to both zones, such as subscript indicates, and they mean as follows:

M–mass of the zone [kg],

72 Fuel Injection in Automotive Engineering

*II <sup>v</sup> ex <sup>v</sup> ex*

  . (4)

*dH dM dM h h dd d*

hex–specific enthalpy of transferred mass; it is specific enthalpy of I or II zone depending on

Total internal energy of any thermodynamic system can be expressed by multiplying specific internal energy *u* and system mass *M*. Thus, we can also differentiate this

> *dU d M u dM du* ( ) *u M dd d d*

If we consider that specific internal energy u for various compounds mixture can be

*i i*

Substitution of all above equations into the fundamental equation (1) for both zones will give a system of two differential equations with three unknowns: d*T*I/dφ, d*T*II/dφ, and

*dg <sup>u</sup> d*

*i i i du dT u <sup>g</sup> dd T*

(5)

as near to null, we will get:

(6)

; then, assuming *<sup>i</sup>*

ui-specific internal energy for an "i" component [J/kg],

gi-mass fraction of an "i" component in a whole system [kg/kg],

 

 

*<sup>I</sup> ex ex*

*dH dM <sup>h</sup> d d*

I, II–subscripts referred to zone I and II, in order,

Qh–heat generated by combustion [J],

Mv–mass of evaporated fuel [kg],

direction of mass flow [J/kg],

multiplication, what gives:

calculated: u u <sup>i</sup> g*i i*

T-temperature of the system [K].

where:

d*M*ex/dφ:

Qv–heat consumed by vaporizing fuel [J],

hv–specific enthalpy of fuel vapor [J/kg],

Mex–mass transferred between both zones [kg],

and for enthalpy fluxes:

Qc–heat of cooling [J],

where:

R–universal gas constant for a whole zone [J/(kg·K)],

T–average temperature of the zone [K],

#### V–zone volume [m3].

Going ahead, at any time the mass of the first zone is the sum of initial mass of fresh air charge *M*ch and mass transferred *M*ex. Similarly, for the second zone it is a mass of evaporated fuel *M*v from which the transferred mass *M*ex must be subtracted. Then transferred mass can be evaluated as follows:

$$M\_{ex} = \frac{M\_{\upsilon} \cdot R\_{II} \cdot T\_{II} \cdot V\_I - M\_{\text{ch}} \cdot R\_I \cdot T\_I \cdot V\_{II}}{R\_I \cdot T\_I \cdot V\_{II} + R\_{II} \cdot T\_{II} \cdot V\_I} \tag{10}$$

To differentiate it relatively to the crank angle variable φ, we receive a formula for component d*M*ex/dφ as a function expressed by the other differentiates:

$$\frac{d\mathbf{M}\_{\rm ex}}{d\rho} = f\left(\frac{\mathbf{d}\mathbf{T}\_{\rm I}}{d\rho}, \frac{\mathbf{d}\mathbf{T}\_{\rm II}}{d\rho}, \frac{\mathbf{d}\mathbf{V}\_{\rm I}}{d\rho}, \frac{\mathbf{d}\mathbf{V}\_{\rm II}}{d\rho}, \frac{\mathbf{d}\mathbf{M}\_{\rm v}}{d\rho}\right) \tag{11}$$

Now, replacing the component d*M*ex/dφ in the system of equations (7) with the above function we receive a new system of two differential equations with only two unknowns: d*T*I/dφ, d*T*II/dφ, i.e.:

$$\begin{cases} A \cdot \frac{dT\_I}{d\rho} + B \cdot \frac{dT\_{II}}{d\rho} = C\\ D \cdot \frac{dT\_I}{d\rho} + E \cdot \frac{dT\_{II}}{d\rho} = F \end{cases} \tag{12}$$

Simulation of Combustion Process

*II*

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 75

According to the physical model layout that is shown in Fig. 1, the geometry of fuel injection sprays defines the volume of the zone of fuel-air mixture signed as zone II. A following

Spray tip penetration *s* and tip angle *2*α are the principal parameters. They allow calculating the volume of zone II (*V*II) by multiplying the volume of elementary spray and number of

2 3

 

 

*s*

 

(14)

(15)

Equation (14) expresses an instantaneous volume of the zone II that varies within the injection duration. According to the assumptions made, the change of zone II volume (*V*II)

1 1 ( )

Empirical formulas have been used for further analysis. They are based on criterion numbers that are widely used in fluid mechanics. And so, the relationship of spray tip penetration *s* and

*<sup>r</sup> inj*

 

*u*

(16)

(17)

*i s v ctg ctg*

*dV ds i s d d ctg ctg*

*ctg ctg*

1 () 1 2 () ( ) () 1 3 1 13 1

 

1 11 ( ) <sup>3</sup> 1 1

2

2

VII(φ)–total volume of combustion zone II at specified crankshaft angle position [m3],

*s s Vi s*

2 3

 

 

2 3

 

1 1

2 3

 

1 1 ( ) ( ) 1 1

*ctg ctg ctg*

simplified single spray cone geometry model has been adapted (Fig. 2).

sprays generated by the injector, i.e. the number of holes in the sprayer *i*:

3

 

where the symbols used in above equations, both (14) and (15), mean:

s(φ)–spray tip penetration at specified crankshaft angle position [m], vs(φ)–spray tip velocity at specified crankshaft angle position [m/deg],

velocity *v*s is given by the following formulae (Orzechowski & Prywer, 1991):

*s*

*v*

<sup>0</sup> <sup>1</sup> ( ) <sup>2</sup> <sup>6</sup>

*d w*

 *u*

*a n*

<sup>0</sup> ( ) 2 2 () *<sup>r</sup> <sup>s</sup>*

*d a s*

*ds d w*

*i s*

*II*

i–number of holes in the sprayer [–],

α–a half of spray tip angle [rad],

 

just depends on a spray tip penetration increasing d*s*/dφ:

dVII /dφ –change of total volume of zone II [m3/deg],

φ–independent variable: crankshaft angle position [deg].

where A, B, C, D, E, F contains expressions of known variables, which can be evaluated by use of independent submodels and/or separated formulas. In this shape of the system, the unknowns d*T*I/dφ, d*T*II/dφ can not be calculated numerically yet. Such methods need explicit from of the equations. To get it, the system (12) has to be transformed (solved algebraically) relatively to d*T*I/dφ, d*T*II/dφ, which mean the variables now. For instance, applying the method of Cramer determinants we will get:

$$\begin{cases} \frac{dT\_I}{d\rho \rho} = \frac{B \cdot F - E \cdot C}{B \cdot D - E \cdot A} \\\\ \frac{dT\_{II}}{d\rho \rho} = \frac{C \cdot D - A \cdot F}{B \cdot D - E \cdot A} \end{cases} \tag{13}$$

Above computer simulation friendly form of equations can be already implemented into the numerical calculation package and allow program running. Obviously, this core model has to be added with necessary sub-models describing other phenomena like heat transfer, fuel injection, fuel atomization and evaporation, ignition delay, combustion rate, combustion products formation, etc. The chosen ones will be shown further.

#### **3.2 Fuel injection model**

Modeling of the fuel injection in the combustion chamber space is one of the most difficult issues in all simulation works regarding the processes in reciprocating combustion engines. This is caused mainly by limited expertise knowledge in this field. Thus, in simulation works covering the combustion chamber space, a representation level of fuel injection submodel, as well fuel evaporation and combustion one is assumed with consideration of total accuracy of the whole model. More complex, spatial mathematical description should be used only in the cases where the injection process is the main essence of modeling.

In the current study, a number of simplifying assumptions in description of the fuel injection process have been made. Nevertheless, they were tailored to the level of accuracy in whole zero-dimensional model layout. According to the preliminary analysis, it has been assumed that the distribution of fuel density in the sprays generated is the same in all directions; next, the shape of sprays is characterized by a constant tip angle, and spray microstructure is described by the mean droplet diameter according to Sauter definition (SMD – Sauter Mean Diameter) and it is uniform throughout the entire space of fuel jet.

Fig. 2. A model for single fuel spray geometry (Woś, 2008)

where A, B, C, D, E, F contains expressions of known variables, which can be evaluated by use of independent submodels and/or separated formulas. In this shape of the system, the unknowns d*T*I/dφ, d*T*II/dφ can not be calculated numerically yet. Such methods need explicit from of the equations. To get it, the system (12) has to be transformed (solved algebraically) relatively to d*T*I/dφ, d*T*II/dφ, which mean the variables now. For instance,

> *dT B F E C d BD EA dT C D A F d BD EA*

(13)

Above computer simulation friendly form of equations can be already implemented into the numerical calculation package and allow program running. Obviously, this core model has to be added with necessary sub-models describing other phenomena like heat transfer, fuel injection, fuel atomization and evaporation, ignition delay, combustion rate, combustion

Modeling of the fuel injection in the combustion chamber space is one of the most difficult issues in all simulation works regarding the processes in reciprocating combustion engines. This is caused mainly by limited expertise knowledge in this field. Thus, in simulation works covering the combustion chamber space, a representation level of fuel injection submodel, as well fuel evaporation and combustion one is assumed with consideration of total accuracy of the whole model. More complex, spatial mathematical description should be used only in the cases where the injection process is the main essence of modeling.

In the current study, a number of simplifying assumptions in description of the fuel injection process have been made. Nevertheless, they were tailored to the level of accuracy in whole zero-dimensional model layout. According to the preliminary analysis, it has been assumed that the distribution of fuel density in the sprays generated is the same in all directions; next, the shape of sprays is characterized by a constant tip angle, and spray microstructure is described by the mean droplet diameter according to Sauter definition (SMD – Sauter Mean Diameter) and it is uniform throughout the entire space of fuel jet.

s

r 2

*I*

*II*

applying the method of Cramer determinants we will get:

products formation, etc. The chosen ones will be shown further.

r

r

Fig. 2. A model for single fuel spray geometry (Woś, 2008)

**3.2 Fuel injection model** 

According to the physical model layout that is shown in Fig. 1, the geometry of fuel injection sprays defines the volume of the zone of fuel-air mixture signed as zone II. A following simplified single spray cone geometry model has been adapted (Fig. 2).

Spray tip penetration *s* and tip angle *2*α are the principal parameters. They allow calculating the volume of zone II (*V*II) by multiplying the volume of elementary spray and number of sprays generated by the injector, i.e. the number of holes in the sprayer *i*:

$$\begin{split} V\_{II}(\boldsymbol{\varphi}) &= i \cdot \left[ \frac{1}{3} \pi \cdot \left( \frac{s(\boldsymbol{\rho})}{c \text{tg } a + 1} \right)^2 \cdot s(\boldsymbol{\rho}) \cdot \left( 1 - \frac{1}{c \text{tg } a + 1} \right) + \frac{2}{3} \pi \cdot \left( \frac{s(\boldsymbol{\rho})}{c \text{tg } a + 1} \right)^3 \right] = \\ &= i \cdot \frac{1}{3} \pi \cdot s(\boldsymbol{\rho})^3 \cdot \left[ \frac{1}{\left( c \text{tg } a + 1 \right)^2} + \frac{1}{\left( c \text{tg } a + 1 \right)^3} \right] \end{split} \tag{14}$$

Equation (14) expresses an instantaneous volume of the zone II that varies within the injection duration. According to the assumptions made, the change of zone II volume (*V*II) just depends on a spray tip penetration increasing d*s*/dφ:

$$\begin{aligned} \frac{dV\_{II}}{d\varphi} &= \mathbf{i} \cdot \boldsymbol{\pi} \cdot \mathbf{s}(\boldsymbol{\varphi})^2 \cdot \left[ \frac{1}{\left( \operatorname{ctg} \, a + 1 \right)^2} + \frac{1}{\left( \operatorname{ctg} \, a + 1 \right)^3} \right] \cdot \frac{d\mathbf{s}}{d\boldsymbol{\varphi}} = \\ &= \mathbf{i} \cdot \boldsymbol{\pi} \cdot \mathbf{s}(\boldsymbol{\varphi})^2 \cdot \left[ \frac{1}{\left( \operatorname{ctg} \, a + 1 \right)^2} + \frac{1}{\left( \operatorname{ctg} \, a + 1 \right)^3} \right] \cdot v\_s(\boldsymbol{\varphi}) \end{aligned} \tag{15}$$

where the symbols used in above equations, both (14) and (15), mean:

VII(φ)–total volume of combustion zone II at specified crankshaft angle position [m3],

dVII /dφ –change of total volume of zone II [m3/deg],

i–number of holes in the sprayer [–],

s(φ)–spray tip penetration at specified crankshaft angle position [m],

vs(φ)–spray tip velocity at specified crankshaft angle position [m/deg],

α–a half of spray tip angle [rad],

φ–independent variable: crankshaft angle position [deg].

Empirical formulas have been used for further analysis. They are based on criterion numbers that are widely used in fluid mechanics. And so, the relationship of spray tip penetration *s* and velocity *v*s is given by the following formulae (Orzechowski & Prywer, 1991):

$$\mathbf{s}(\boldsymbol{\varphi}) = \sqrt{\frac{d\_r \cdot w\_0}{\sqrt{2} \cdot a\_u} \cdot \frac{1}{6 \cdot n} \cdot \left| \boldsymbol{\varphi} - \boldsymbol{\varphi}\_{inj} \right|}\tag{16}$$

$$w\_s(\rho) = \frac{ds}{d\rho} = \frac{d\_r \cdot w\_0}{2 \cdot \sqrt{2} \cdot a\_u \cdot s(\rho)}\tag{17}$$

Simulation of Combustion Process

For high backpressure (M = 0.0095 - 0.028)

> C1= 2.72 k= −0.21 l= 0.16 m= 1

penetration (Orzechowski & Prywer, 1991)

following formula:

α–a half of spray tip angle [rad],

(Orzechowski & Prywer, 1991)

We, Lp, M–same numbers as for equations (20)-(22),

C, k, l, m–experimental constants (see Table 2).

For high backpressure (M = 0.0095 – 0.028)

> C= 0.0089 k= 0.32 l= 0.07 m= 0.5

them can be described according to the general formula:

hence:

where:

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 77

Table 1. The values of experimental constants *C1, k, l, m* used for calculation of spray tip

The spray tip angle, similarly to the spray tip penetration, is the function of parallel known parameters, i.e. densities of fuel and cylinder charge (ρf, ρg), fuel absolute viscosity and surface tension (ηf, σf), an initial velocity of fuel spray tip left the injector (w0), diameter of sprayer holes (dr) and time (t). Excepting an initial phase o the injection, the spray tip angle does not change, so the effect of time axis can be neglected. Further analysis is based on the

tg α = C· Wek · Lpl · Mm (23)

Table 2. The values of experimental constants *C, k, l, m* used for calculation of spray tip angle

The last but not least of the analyzed parameters that is essential for this methodology is microstructure parameter of the spray jet, i.e. mean diameter of droplets. It means an equivalent average value respecting the whole spectrum of diameters of actual droplets generated by the injector. There are a few definitions of equivalent droplet size. Each of

*<sup>i</sup> <sup>i</sup> p q p q <sup>q</sup>*

 

*p*

*i i n d*

*n d*

,

*d*

For low backpressure (M = 0.0014 - 0.0095)

> C1= 0.202 k= −0.21 l= 0.16 m= 0.45

α = arctg (C· Wek · Lpl · Mm) (24)

For low backpressure (M = 0.0014 – 0.0095)

> C= 0.0028 k= 0.32 l= 0.07 m= 0.26

(25)

where:

s(φ), vs(φ)–spray tip penetration [m] and velocity [m/s] at crankshaft angle position φ,

dr–spraying hole diameter [m],

n–crankshaft rotational speed [1/min],

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

φinj–crankshaft angle position at start of injection [deg],

w0–an initial velocity of fuel spray tip left the injector [m/s],

$$
w\_0 = \mu \cdot \sqrt{\frac{2 \cdot \Delta p}{\rho\_f}}\tag{18}$$

μ–flow factor of the injector holes [–]; μ ≈ 0.7 according to (Orzechowski & Prywer, 1991),

Δp–pressure drop inside the sprayer [Pa],

ρf–fuel density [kg/m3],

au–factor of free-stream turbulence in the spray tip layer [–],

$$\mathbf{a}\_u = \mathbf{C}\_1 \cdot \mathbf{W} \mathbf{e}^k \cdot \mathbf{L} \mathbf{p}^l \cdot \mathbf{M}^m \tag{19}$$

We–dimensionless Weber criteria number [–],

$$\text{We} = \frac{\rho\_f \cdot w^2 \cdot d\_r}{\sigma\_f} \tag{20}$$

Lp–dimensionless Laplace criteria number [–],

$$\text{L.p} = \frac{\rho\_f \cdot \sigma\_f \cdot d\_r}{\eta\_f^2} \tag{21}$$

M–air to fuel density ratio [–],

$$M = \frac{\rho\_g}{\rho\_f} \tag{22}$$

w–relative velocity of fuel droplets inside the spray jet [m/s]; w = w0,

dr–diameter of sprayer holes [m],

σf–fuel surface tension [N/m],

ρf–fuel density [kg/m3],

ρg–cylinder charge density (air density) [kg/m3],

ηf–fuel absolute viscosity [kg/(m·s)],

C1, k, l, m–experimental constants (see Table 1).


Table 1. The values of experimental constants *C1, k, l, m* used for calculation of spray tip penetration (Orzechowski & Prywer, 1991)

The spray tip angle, similarly to the spray tip penetration, is the function of parallel known parameters, i.e. densities of fuel and cylinder charge (ρf, ρg), fuel absolute viscosity and surface tension (ηf, σf), an initial velocity of fuel spray tip left the injector (w0), diameter of sprayer holes (dr) and time (t). Excepting an initial phase o the injection, the spray tip angle does not change, so the effect of time axis can be neglected. Further analysis is based on the following formula:

$$\text{tg } \mathbf{a} = \mathbf{C} \cdot \mathbf{W} \mathbf{e}^k \cdot \mathbf{L} \mathbf{p}^l \cdot \mathbf{M}^m \tag{23}$$

hence:

76 Fuel Injection in Automotive Engineering

s(φ), vs(φ)–spray tip penetration [m] and velocity [m/s] at crankshaft angle position φ,

0

*We*

*Lp*

w–relative velocity of fuel droplets inside the spray jet [m/s]; w = w0,

2

*<sup>p</sup> <sup>w</sup>* 

μ–flow factor of the injector holes [–]; μ ≈ 0.7 according to (Orzechowski & Prywer, 1991),

au = C1· Wek · Lpl · Mm (19)

 

*M*

2 *f r f w d*

2 *f f r f*

> *g f*

*d*

*f*

(18)

(20)

(21)

(22)

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

φinj–crankshaft angle position at start of injection [deg],

w0–an initial velocity of fuel spray tip left the injector [m/s],

au–factor of free-stream turbulence in the spray tip layer [–],

where:

dr–spraying hole diameter [m],

n–crankshaft rotational speed [1/min],

Δp–pressure drop inside the sprayer [Pa],

We–dimensionless Weber criteria number [–],

Lp–dimensionless Laplace criteria number [–],

M–air to fuel density ratio [–],

dr–diameter of sprayer holes [m], σf–fuel surface tension [N/m],

ηf–fuel absolute viscosity [kg/(m·s)],

ρg–cylinder charge density (air density) [kg/m3],

C1, k, l, m–experimental constants (see Table 1).

ρf–fuel density [kg/m3],

ρf–fuel density [kg/m3],

$$\mathfrak{a} = \operatorname{arctg} \left( \mathbf{C} \cdot \mathbf{W} \mathbf{e}^{\mathbf{k}} \cdot \mathbf{L} \mathbf{p}^{\mathbb{I}} \cdot \mathbf{M}^{\mathbb{m}} \right) \tag{24}$$

where:

α–a half of spray tip angle [rad],

We, Lp, M–same numbers as for equations (20)-(22),

C, k, l, m–experimental constants (see Table 2).


Table 2. The values of experimental constants *C, k, l, m* used for calculation of spray tip angle (Orzechowski & Prywer, 1991)

The last but not least of the analyzed parameters that is essential for this methodology is microstructure parameter of the spray jet, i.e. mean diameter of droplets. It means an equivalent average value respecting the whole spectrum of diameters of actual droplets generated by the injector. There are a few definitions of equivalent droplet size. Each of them can be described according to the general formula:

$$d\_{p,q} = r^{-q} \sqrt{\sum n\_i \cdot d\_i^p} \tag{25}$$

Simulation of Combustion Process

with experiments.

where:

where:

[g/s],

ρf–fuel density [g/mm3],

is injected [mm2/s], t–evaporation time [s],

d0–initial diameter of droplet [mm],

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

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

from a single droplet can be determined as follows:

**3.3 Fuel evaporation model** 

Δp–fuel injection overpressure [MPa], ρg–density of cylinder charge [kg/m3],

qVf–amount of a single fuel injection volume [mm3].

in Direct Injection Diesel Engine Based on Fuel Injection Characteristics 79

The presented methodology for calculation of fuel spray jet parameters is based on extensive experimental studies. Thus it is expected to provide a good consistence of calculated results

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

d02 − d2 = K · t (27)

K–evaporation intensity factor that depends on temperature of surrounding where the fuel

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

*v*

*m*

0 6

*K d*

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

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

*f*

(28)

between size decreasing and evaporation intensity is known (Kowalewicz, 2000):

where:

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

p, q–the exponents that correspond with adopted definition of droplet mean diameter [–]; the values of p and q and formula shape for various definitions is given in Table 3,

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

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


Table 3. The list of chosen definition formulas for calculation of mean diameter of droplets in a spray jet (Orzechowski & Prywer, 1991)

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

$$\mathbf{d}\_{3,2} = \mathbf{A} \cdot \Delta \mathbf{p}^{-0.135} \text{ } \mathbf{p}\_{\mathbb{R}} \, ^{0.121}\mathbf{q} \text{v}^{0.131} \tag{26}$$

where:

d3,2–Sauter mean diameter [μm],

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

Δp–fuel injection overpressure [MPa],

ρg–density of cylinder charge [kg/m3],

qVf–amount of a single fuel injection volume [mm3].

The presented methodology for calculation of fuel spray jet parameters is based on extensive experimental studies. Thus it is expected to provide a good consistence of calculated results with experiments.
