Droplet Flash Boiling

Flash boiling is the rapid phase change (vaporization) of a pressurized fluid entering ambient conditions below its vapor pressure.

It has been found that the onset and extent of flash boiling are mainly determined by the superheat $Δ T$ defined as:

Figure 1. EQUATION_DISPLAY

Δ T = T_{d} - T_{sat}

(3125)

where $T_{d}$ is the temperature of an injected droplet and $T_{sat}$ is its saturation temperature at the ambient pressure.

There are three classes of sub-models for flash boiling:

A set of zero-dimensional phenomenological models to simulate the cavitation and nucleation processes inside an injector nozzle, and to provide initial conditions of the injected droplets at the nozzle exit.
A model for enhanced evaporation due to superheating.
A model for droplet breakup as a result of bubble nucleation, growth, or bursting.

These three models can be enabled and disabled separately, according to the conditions being simulated.

In-Nozzle Nucleation

The in-nozzle flash boiling model is activated when the superheat $Δ T$ is greater than a specified threshold value:

Figure 2. EQUATION_DISPLAY

Δ T > Δ T_{threshold}

(3126)

$Δ T$ is set to 10 K by default. The effective droplet velocity $v_{eff}$ at nozzle exit is:

Figure 3. EQUATION_DISPLAY

v_{eff} = \frac{A_{n}}{\dot{m}} (P_{sat} - P_{\infty}) + v_{vena}

(3127)

where:

$A_{n}$ is the nozzle hole area.
$\dot{m}$ is the injection mass flow rate.
$P_{s at}$ is the saturation pressure at the liquid injection temperature.
$P_{\infty}$ is the ambient pressure downstream of the nozzle.
$v_{vena}$ is velocity at vena contracta.

Figure 4. EQUATION_DISPLAY

v_{v ena} = \frac{v_{mean}}{C_{c}} = \frac{\dot{m}}{A_{n} ρ_{f} C_{c}}

(3128)

where $ρ_{f}$ is the density of the injected liquid.

$C_{c}$ is the contraction coefficient:

Figure 5. EQUATION_DISPLAY

C_{c} = {[{(\frac{1}{C_{c 0}})}^{2} - 11.4 \frac{r}{D}]}^{- 0.5}

(3129)

where:

$C_{c 0}$ is the baseline contraction coefficient with a default value of 0.611.
$r / D$ is the roundness ratio at the nozzle inlet, the default value is set to 0.07 [694].

Following the practice outlined by Price et al. [685], this correlation, introduced by Kamoun et al. [669], is used to calculate the spray cone angle $θ$ at the nozzle exit for a fully-flashing spray:

Figure 6. EQUATION_DISPLAY

θ = a β^{2} + b β + c

(3130)

The coefficients $a$ , $b$ and $c$ in Eqn. (3130) are set to -3.208, 366.61, and -10324, respectively, and $β$ is a non-dimensional parameter defined as:.

β = {log}_{10} (\frac{R_{p}^{2} Θ^{3}}{m_{a}^{2}})

where:

$R_{p}$ is the pressure ratio of the saturation pressure and the ambient pressure: $R_{p} = P_{sat} / P_{\infty}$
$m_{a}$ is the atomic mass of the injected liquid.
$Θ$ is the dimensionless surface tension.

$Θ$ is defined as:

Θ = \frac{a_{0} σ}{k_{b} T_{d}}

where:

$a_{0}$ is the molecular surface area, $a_{0} = {(36 π)}^{1 / 3} {v_{l}}^{2 / 3}$ , where $v_{l}$ is the liquid molecular volume.
$σ$ is the surface tension.
$k_{b}$ is Boltzmann's constant.

At the flash boiling condition, vapor is generated inside the nozzle. The vapor volume flow rate ${\dot{V}}_{vap}$ is calculated as:

Figure 7. EQUATION_DISPLAY

{\dot{V}}_{vap} = f V_{b} S_{nozzle} N_{nuc}

(3131)

where:

$f$ is the bubble departure frequency.
Figure 8. EQUATION_DISPLAY

$f = \frac{1.18}{D_{b}} {[\frac{σ g (ρ_{liq} - ρ_{vap})}{ρ_{liq}^{2}}]}^{1 / 4}$

(3132)
- $g$ is gravity.
- $D_{b}$ is the bubble departure diameter:
  Figure 9. EQUATION_DISPLAY
  
  $D_{b} = 2.64 \times 10^{- 5} θ_{c} {(\frac{σ}{g [ρ_{liq} - ρ_{vap}]})}^{0.5} {(ρ^{*})}^{0.9}$
  
  (3133)
  where $θ_{c}$ is the bubble-surface contact angle set to 45.78 (in degrees) as the default.
$V_{b}$ is the bubble volume at departure.
$S_{nozzle}$ is the inner surface area of the nozzle orifice.
$N_{nuc}$ is the nucleation site density per unit surface.

Figure 10. EQUATION_DISPLAY

N_{nuc} = \frac{1}{D_{b}^{2}} N_{nuc}^{*} F (ρ^{*})

(3134)

The non-dimensional nucleation density $N_{nuc}^{*}$ in Eqn. (3134) is:

Figure 11. EQUATION_DISPLAY

N_{nuc}^{*} = {(\frac{D_{b} (T_{f} - T_{s}) ρ_{vap} H_{L}}{2 σ T_{s}})}^{y}

(3135)

where:

$T_{f}$ is the temperature of the injected liquid.
$H_{L}$ is the vaporization latent heat.
$y$ is a model constant is set to 4.4 [685].

The property density function $F (ρ^{*})$ in Eqn. (3134) is defined as:

F (ρ^{*}) = 2.157 \times 10^{- 7} {(ρ^{*})}^{- 3.12} {(1 + 0.0049 ρ^{*})}^{4.13}

By knowing the volume flow rate of vapour from Eqn. (3131), the diameter of injected droplets can be calculated based on the asymmetric vapor film assumption outlined in Price et al. [685].

Taking into account liquid evaporation inside a nozzle, and assuming adiabatic nozzle walls, the temperature of injected droplets $T_{d}$ at nozzle exit can be obtained as

T_{d} = T_{f} - \frac{1}{C_{p l}} [\frac{ρ_{vap} {\dot{V}}_{vap} H_{L}}{\dot{m}} + \frac{1}{2} (v_{eff}^{2} - v_{mean}^{2})]

Droplet Superheat Evaporation

Droplet superheat evaporation takes place when the superheat $Δ T$ is greater than a user-specified threshold value ${Δ T}_{threshold}$ :

Figure 12. EQUATION_DISPLAY

Δ T > {Δ T}_{threshold}

(3136)

In addition to the standard evaporation model considering heat transfer from the ambient gas to the droplets, evaporation due to flash boiling is modelled by the correlation of Adachi et al. [685] as

{\dot{m}}_{s h} = \frac{A_{d} h Δ T}{H_{L}}

where $Δ T$ is the superheat and $h$ is the heat transfer coefficient defined as:

\begin{matrix} h = h_{1} Δ T^{0.26} & 0 < Δ T \leq 5 \\ h = h_{2} Δ T^{2.33} & 5 < Δ T \leq 25 \\ h = h_{3} Δ T^{0.39} & 25 < Δ T \end{matrix}

The model coefficients $h_{1}$ , $h_{2}$ , and $h_{3}$ are set to 760, 27, and 138000, respectively.

Following Schmehl et al. [697], a minimum value of $h$ is used to account for the heat condition inside the droplet:

Figure 13. EQUATION_DISPLAY

h_{\min} = k / (D_{p} / 2)

(3137)

where $k$ is the thermal conductivity of the droplet and $D_{p}$ is the droplet diameter.

Droplet Thermal Breakup

Senda’s bubble nucleation and growth model [701] is adopted to simulate droplet breakup due to bubble nucleation and growth inside a droplet. In Senda’s model, vapor bubbles are first generated by nucleation in droplets. The number of bubble nuclei $N$ can be obtained by:

Figure 14. EQUATION_DISPLAY

N = C_{1} \exp (\frac{- C_{2}}{Δ T}) (\frac{π}{6} D_{p}^{3})

(3138)

where $C_{1}$ and $C_{2}$ are model constants, with default values of $1.11 \times 10^{12}$ and 5.28, respectively.

Bubble growth is modeled by solving the Rayleigh-Plesset equation:

Figure 15. EQUATION_DISPLAY

R \frac{d R}{d t} + \frac{3}{2} {(\frac{d^{2} R}{d t^{2}})}^{2} = \frac{1}{ρ} (P_{w} - P_{\infty})

(3139)

where

Figure 16. EQUATION_DISPLAY

P_{w} = P_{v} + (P_{r 0} + \frac{2 σ}{R_{0}}) {(\frac{R_{0}}{R})}^{3 n} - \frac{2 σ}{R} - \frac{4 μ_{liq}}{R} (\frac{d R}{d t}) - \frac{4 k}{R^{2}} (\frac{d R}{d t})

(3140)

where:

$P_{v}$ is the fuel saturation pressure at droplet temperature.
$P_{r 0}$ is the pressure at the nozzle orifice.
$R$ is the bubble radius.
$R_{0}$ is the initial bubble radius, set to 10 μm by default.
$n$ is the polytropic index for bubble growth. The default value is 1.
$σ$ is the surface tension.
$μ_{liq}$ is the liquid viscosity.
$k$ is the surface viscosity coefficient and is set to $1.2 \times 10^{- 5} N \cdot s/m$ in Sedna et al. [701]. As discussed in Ida & Sugiya [667], a larger value of surface viscosity coefficient can damp oscillation of the solution when solving Eqn. (3139).

Droplet break up due to flash boiling occurs only when the bubble volume fraction $ϵ$ is greater than a threshold value $ϵ_{max}$ :

Figure 17. EQUATION_DISPLAY

ϵ = \frac{V_{bubble}}{V_{bubble} + V_{liquid}} > ϵ_{max}

(3141)

The value of $ϵ_{max}$ is set to 0.55 as in Kawano et al. [670]. When a droplet breaks up, the number of child droplets is set to twice the number of bubble nuclei as predicted by Eqn. (3138).