Fig. 8. Dependency between diameter dcof the drop leaving the injector and fluid’s surface tension σ at
two spray velocities v (v1 = 200 m/s,v2 = 400 m/s). The x -axis value range
15-30 mN/m corresponds to typical surface tensions of fuels at
temperatures 90 °C (Table 1).
Fig. 8 shows the fuel drop size according to the surface tensions of
fuels at various spray velocities. The figure shows that as the surface
tension increases, the drop size also increases. It is important that
the fuel’s surface tension has a greater effect on drop size at lower
velocities (v1 = 200 m/s) than at higher
velocities (v2 = 400 m/s).
In conclusion, it can be claimed that the fuel drop’s size is
significantly influenced by the velocity v of the fuel spray,
dynamic viscosity μ , density ρ and surface tension factorσ . In order to characterize the drops formed during the
preparation of the air-fuel mixture, these parameters must be viewed
separately and the physical and chemical properties of each fuel shall
be taken into account and projected into the working conditions of a
real engine.
The data in Table 3 takes into account that the temperature of the
sprayed fuel and the density of the spraying environment are comparable
to the actual environment in the engine cylinder. Here the temperature
of the sprayed fuel was chosen to be 90 °C \cite{Zha17}, which corresponds to the
temperature of the working engine. The density of the spraying
environment was 17 kg/m3.
- Mathematical representation of reflexive and stretching
separation
The description of the collision of fuel drops is based on the
assumption that the drops move confluently and collisions only take
place when one fuel drop catches up with another one in the fuel spray.
The movement, collision and separation of drops is described in Fig. 9 \cite{Ash90}.
The calculations are based on the assumption that after the collision of
drops in the fuel spray, the reflexive and stretching separation occur.
Reflexive separation occurs in case of large Weber numbers We and
low values of the impact parameter B . This means either frontal
impact or a similar situation. If the Weber number is large
(>100) and the impact parameter is growing, then stretching
separation shall become dominant after the collision of the drops. The
impact parameter B also determines the number of collisions \cite{Tas14}.
In order to describe these two processes, the kinetic energy of two
colliding drops and the law of the conservation of the surface energy of
the temporarily joined drops shall be used. The Weber number for
separation of drops for the two processes can be described as follows \cite{Ash90}:
\begin{equation}
text{We}_{\text{reflection}}>\frac{3\left[7\left(1+\Delta^{3}\right)^{\frac{1}{3}}-4\left(1+\Delta^{2}\right)\right]\Delta\left(1+\Delta^{3}\right)^{2}}{\Delta^{6}\eta_{1}+\eta_{2}}\nonumber \\
\end{equation}
(18)
which applies to reflexive separation; and
\begin{equation}
text{We}_{\text{stretching}}>\frac{4\left(1+\Delta^{3}\right)^{2}\left[3\left(1+\Delta\right)\left(1-B\right)\left(\Delta^{3}\varphi_{1}+\varphi_{2}\right)\right]^{\frac{1}{2}}}{\Delta^{2}\left[\left(1+\Delta^{3}\right)-\left(1-B^{2}\right)\left(\varphi_{1}+\Delta^{3}\varphi_{2}\right)\right]}\nonumber \\
\end{equation}
(19)
which applies to stretching separation. The dimensionless constants𝜂1 , 𝜂2 and ξ (Table
3) are used for simplifying the calculations and these are obtained as
follows:
\(\eta_{1}=2\left(1-\xi\right)^{2}\left(1-\xi^{2}\right)^{\frac{1}{2}}-1\)(20)
\(\eta_{2}=2\left(\Delta-\xi\right)^{2}\left(\Delta-\xi^{2}\right)^{\frac{1}{2}}-\Delta^{3}\)(21)
\(\xi=\frac{1}{2}B\left(1+\Delta\right)\) (22)
The dimensionless values of φ1 andφ2 are used for describing the stretching
separation and these values denote the respective proportions of spatial
areas in joined drops. The values of φ1 andφ2 , parts of interaction volumesV1i , V2i and interaction
volume Vi can be represented as follows:
\(\varphi_{1}=\left\{\par
\begin{matrix}1-\frac{1}{4^{3}}\left(2\Delta-\lambda\right)^{2}\left(\Delta+\lambda\right)\\
\frac{\lambda^{2}}{4^{3}}\left(3\Delta-\lambda\right)\\
\end{matrix}\right.\ \)(23)
\(\varphi_{2}=\left\{\par
\begin{matrix}1-\frac{1}{4}\left(2-\lambda\right)^{2}\left(1+\lambda\right)\\
\frac{\lambda^{2}}{4}\left(3-\lambda\right)\\
\end{matrix}\right.\ \) (24)
\(V_{\text{ji}}=\varphi_{j}V_{j}\) (25)
\(V_{i}=V_{1i}+V_{2i}\) (26)
The dimensionless value of λ is expressed as follows:
\(\lambda=\left(1-B\right)\left(1+\Delta\right)\) (27)
The selection criterion for the value of φ1 is\(h>r_{1}\) and \(h<r_{1}\) respectively and the selection criterion
for the value of φ2 is \(h>r_{2}\) and\(h<r_{2}\) respectively. The value h marks the interaction
height and is expressed as follows:
\(h=\frac{1}{2}\left(D_{1}+D_{2}\right)\left(1-B\right)\) (28)
The values of r1 and r2express the radii of drops. In order to understand better the equations
18-27, several numeric examples have been given in Table 3.
Situation 1 describes the frontal impact (B = 0) of two drops
with equal diameters. The Weber number values calculated according to
equations 18 and 19 show that if Wereflection> 4.9, then reflexing separation takes place with no
stretching separation occurring. If none of the separations occur
according to the calculations of Table 3, then it must be either the
bouncing or coalescence of the drops. This model does not discuss these
cases further.
If the value of the impact parameter B is greater (B =
0.20), then it corresponds to a situation in which two drops collide
under conditions similar to a frontal impact. In such cases the
reflexive separation starts to occur from the valueWereflection > 19.3 onwards and
stretching separation Westretching >
167.8.
In situation 3 the drops nearly graze each other (B = 0.80).
Reflexive separation does not occur in this situation. Stretching
separation will occur already with smaller Weber numbers
(Westretching > 4.2).
Situation 4 constitutes a frontal impact of two drops (B = 0),
whereby one of the drops has twice the diameter of the other one (size
ratio of colliding drops is γ = 0.5). Reflexive separation will
occur staring from the value Wereflection> 30.8 and stretching separationWestretching > 38.7. In comparison
to situation 1, the greater values of the Weber numbers are caused by
the fact that the larger drop swallows the smaller one. In case of lowerWe values; surface tension causes the domination of coalescence.
In situations 5 and 6 the size ratio of colliding drops is stillγ = 0.5, but the impact parameter has been increased to B= 0.20 and B = 0.80 respectively. Reflexive separation does not
occur in any of the situations. In situation 5, the stretching
separation will start occurring from Westretching> 153.4 onwards and in situation 6Westretching > 5.4.
It should be noted that in case of stretching separation, the
interaction height h and interaction volumeVi are much smaller than in case of reflexive
separation. In case of reflexive separation, the total volume of joined
drops is equal to the interaction volume.
The separation volume coefficient Cv is
introduced to determine the volume of the fluid separating from two
colliding drops and it is defined as the ratio of the volume separating
from the two drops and the interaction volume. It is presumed \cite{KoG05} thatCv is equal to the energy needed for the
separation and the total energy of the two colliding drops:
\(C_{v}=\frac{\text{KE}_{\text{separation}}-\text{PE}_{\text{coalescence}}}{\text{KE}_{\text{separation}}+\text{PE}_{\text{coalescence}}}\)(29)
KEseparation describes the separation kinetic
energy and PEcoalescence the surface tension
energy which is needed to sustain the coalescence of the two drops. In
case of reflexive separation, the KEseparationand PEcoalescence can be presented as follows:
\begin{equation}
text{KE}_{\text{separation}}=\sigma\pi D_{2}^{2}\left[\left(1+\Delta^{2}\right)-\left(1+\Delta^{3}\right)^{\frac{2}{3}}+\frac{\text{We}}{12\Delta\left(1+\Delta^{3}\right)^{2}}\left(\Delta^{6}\eta_{1}+\eta_{2}\right)\right]\nonumber \\
\end{equation}
(30)
\(\text{PE}_{\text{coalescence}}=0.75\sigma\pi\left(D_{1}^{3}+D_{2}^{3}\right)^{\frac{2}{3}}\)(31)