Fig. 4. Schematic drawing of hybrid breakout model \cite{Kim09}.
There are several hybrid breakout models: WAVE-RT \cite{Rei86,Rei87,ImK11,Pat98,Ric97,Bea99}, WAVE-TAB \cite{ORo87,Mar13,Pil87,Tay63}, WAVE-DDB \cite{Nic69,Ibr93},
WAVE-ACT \cite{Som10}, etc. Various models have been compared in the overview
article \cite{Bra14}. The relations used in this study are given according to the
WAVE-RT model which was used most widely in research. According to the
original WAVE model, the surface of the fluid leaving the injector
develops Kelvin-Helmholtz instability, which lead to the emerging of
sinusoidal surface waves. These waves lead to the separation of the
unstable part of fluid from the spray, which leads to the generation of
drops. According to the WAVE model \cite{ORo87,Nic69}, the drop growth speedΩKH and corresponding wavelengthΛKH is represented as follows:
\(\frac{\Lambda_{\text{KH}}}{r}=9,02\frac{\left(1+0,45Z^{0,5}\right)\left(1+0,4T^{0,7}\right)}{\left(1+0.87\text{We}_{g}^{1,67}\right)^{0,6}}\)(9)
\(\Omega_{\text{KH}}\left(\frac{\rho_{f}r^{3}}{\sigma_{f}}\right)^{0.5}=\frac{(0,34+0,38\text{We}_{g}^{1,5})}{\left(1+Z\right)\left(1+1,4T^{0,6}\right)}\)(10)
The relations 9 and 10 contain members which are expressed as follows:
\(d_{c}=2B_{0}\Lambda_{\text{KH}}\) (11)
\(\tau_{\text{KH}}=\frac{3,726B_{1}r}{\Omega_{\text{KH}}\Lambda_{\text{KH}}}\)(12)
where We and Re are the Weber number and Reynolds number.
While the Weber number determines the nature of drops after the possible
coalescence of drops, then the Reynolds number characterizes the
distribution of drops in a gas environment.
\(Z=\frac{\sqrt{\text{We}_{f}}}{\text{Re}_{f}}\) (13)
\(T=Z\sqrt{\text{We}_{g}}\) (14)
\(\text{Re}_{f}=\frac{\rho_{f}\text{vr}}{\mu_{f}}\) (15)
were ρf and ρg are the
densities of fluid and gas (kg/m3); v – fluid
velocity (m/s). In this context, the value of v can be considered
equal to velocity of the spray leaving the injector, r – radius
of fluid spray leaving the injector (m), µf –
dynamic viscosity of fluid (Pa·s), σf andσg – surface tension of fluid and gas (N/m).
The physical and chemical properties of fuel affect the fuel drop size
in the fuel spray. Thus, their influence has been described in detail in
the following Fig. 5–8. In this research, the range of variated
parameters are chosen accordingly to describe the diesel engine work
mode. Values of the fuels parameters, by example dynamic viscosity
µf, density etc. are used on condition of the engine.
The following relations are for finding the diameterdc of the drop leaving the injector’s spray and
drop breakout time τKH \cite{ORo87,Nic69}:
\(\text{We}_{f}=\frac{\rho_{f}v^{2}r}{\sigma_{f}}\) (16)
\(\text{We}_{g}=\frac{\rho_{g}v^{2}r}{\sigma_{f}}\) (17)
where B0 and B1 are
empirical constants with values B0 = 4.5 andB1 = 40. Various sources \cite{ORo87,Liu93,Gon92} give different values to
the constants B0 and B1 .
The values of B1 are usually within the range
1-60 depending on the characteristics of the injector.