Spray
Spray Equations
This section outlines the modeling and mathematics for the spray routines. Firstly, spray modeling relies on the following assumptions:
Dilute spray: droplet volume inside an Eulerian cell is much smaller than the volume of the gas phase; the droplets can be modeled as Lagrangian point source terms relative to the Eulerian gas phase
Infinite conductivity model: temperature within a droplet is temporally varying but spatially uniform
Onethird rule: the thermophysical properties in the film of an evaporating droplet can be approximated as a weighted average of the state at the droplet surface (weighted as 2/3) and the state of the surrounding gas (weighted as 1/3)
Ideal equilibrium: the liquid and vapor state at the surface of the droplet are in equilibrium
The radiation, Soret, and Dufour effects are neglected
The evaporation models follow the work by Abramzon and Sirignano [2] and the multicomponent evaporation is based on work by Tonini. [1] Details regarding the energy balance are provided in Ge et al. [5]
The subscript notation for this section is: \(d\) relates to the liquid droplet, \(v\) relates to the vapor state that is in equilibrium with the liquid and gas phase, \(L\) relates to the liquid phase, and \(g\) relates to the gas phase. The subscript \(r\) relates to the reference state with which to approximate the thermophysical and transport properties. This reference state is assumed to be in the evaporating film that surrounds the droplet state and is approximated as
where \(A = 1/3\) according the the onethird rule. Additional nomenclature: \(M_n\) is the molar mass of species \(n\), \(\overline{M}\) is the average molar mass of a mixture, \(\mathcal{R}\) is the universal gas constant, \(N_L\) is the number of liquid species, and \(N_s\) is the number of gas phase species. \(Y_n\) and \(\chi_n\) are the mass and molar fractions of species \(n\), respectively. The user is required to provide a reference temperature for the liquid properties, \(T^*\), the critical temperature for each liquid species, \(T_{c,n}\), the boiling temperature for each liquid species at atmospheric pressure, \(T^*_{b,n}\), the latent heat and liquid specific heat at the reference temperature, \(h_{L,n}(T^*)\) and \(c_{p,L,n}(T^*)\), respectively. Note: this reference temperature is a constant value for all species and is not related to the reference state denoted by the subscript \(r\).
The equations of motion, mass, momentum, and energy for the Lagrangian spray droplet are:
where \(\mathbf{X}_d\) is the spatial vector, \(\mathbf{u}_d\) is the velocity vector, \(T_d\) is the droplet temperature, \(m_d\) is the mass of the droplet, \(\mathbf{g}\) is an external body force (like gravity), \(\dot{m}\) is evaporated mass, \(\mathcal{Q}_d\) is the heat transfer between the droplet and the surrounding gas, and \(\mathbf{F}_d\) is the momentum source term. The density of the liquid mixture, \(\rho_d\), depends on the liquid mass fractions of the dropet, \(Y_{d,n}\),
The droplets are assumed to be spherical with diameter \(d_d\). Therefore, the mass is computed as
The procedure is as follows for updating the spray droplet:
Interpolate the gas phase state to the droplet location using a trilinear interpolation scheme.
Compute the boiling temperature for species \(n\) at the current gas phase pressure using the ClasiusClapeyron relation
\[T_{b,n} = \left(\log\left(\frac{p_{\rm{atm}}}{p_g}\right) \frac{\mathcal{R}}{M_n h_{L,n}(T^*_{b,n})} + \frac{1}{T^*_{b,n}}\right)\]The boiling temperature of the droplet is computed as
\[T_{d,b} = \sum^{N_L}_{n=0} Y_{d,n} T_{b,n}\]Since we only have the latent heat at the reference condition temperature, we estimate the enthalpy at the boiling condition using Watson’s law
\[h_{L,n}(T^*_{b,n}) = h_{L,n}(T^*) \left(\frac{T_{c,n}  T^*}{T_{c,n}  T^*_{b,n}} \right)^{0.38}\]Compute the latent heat of the droplet using
\[h_{L,n}(T_d) = h_{g,n}(T_d)  h_{g,n}(T^*) + h_{L,n}(T^*)  c_{p,L,n}(T^*) (T_d  T^*) \,.\]and the saturation pressure using either the ClasiusClapeyron relation
\[p_{{\rm{sat}}, n} = p_{\rm{atm}} \exp\left(\frac{h_{L,n}(T_d) M_n}{\mathcal{R}} \left(\frac{1}{T^*_{b,n}}  \frac{1}{T_d}\right)\right)\]or the Antoine curve fit
\[p_{{\rm{sat}},n} = d 10^{a  b / (T_d + c)}\]Estimate the mass fractions in the vapor state using Raoult’s law
\[ \begin{align}\begin{aligned}Y_{v,n} &= \frac{\chi_{v,n} M_n}{\overline{M}_v + \overline{M}_g (1  \chi_{v,{\rm{sum}}})} \; \forall n \in N_L\\\chi_{v,{\rm{sum}}} &= \sum^{N_L}_{n=0} \chi_{v,n}\\\chi_{v,n} &= \frac{\chi_{d,n} p_{{\rm{sat}},n}}{p_g}\\\chi_{d,n} &= \frac{Y_{d,n}}{M_n}\left(\sum^{N_L}_{k=0} \frac{Y_{d,k}}{M_k}\right)^{1}\\\overline{M}_v &= \sum^{N_L}_{n=0} \chi_{v,n} M_n\end{aligned}\end{align} \]If \(\chi_{g,n} p_g > p_{{\rm{sat}},n}\), then \(\chi_{v,n} = Y_{v,n} = 0\) for that particular species in the equations above, since that means the gas phase is saturated. The mass fractions in the reference state for the fuel are computed using the onethird rule and the remaining reference mass fractions are normalized gas phase mass fractions to ensure they sum to 1
\[\begin{split}Y_{r,n} = \left\{\begin{array}{c l} \displaystyle Y_{v,n} + A (Y_{g,n}  Y_{v,n}) & {\text{If $Y_{v,n} > 0$}}, \\ \displaystyle\frac{1  \sum^{N_L}_{k=0} Y_{v,k}}{1  \sum^{N_L}_{k=0} Y_{g,k}} Y_{g,n} & {\text{Otherwise}}. \end{array}\right. \; \forall n \in N_s.\end{split}\]The average molar mass, specific heat, and density of the reference state in the gas film are computed as
\[ \begin{align}\begin{aligned}\overline{M}_r &= \left(\sum^{N_s}_{n=0} \frac{Y_{r,n}}{M_n}\right)^{1},\\c_{p,r} &= \sum^{N_s}_{n=0} Y_{s,n} c_{p,g,n}(T_r),\\\rho_r &= \frac{\overline{M}_r p_g}{\mathcal{R} T_r}.\end{aligned}\end{align} \]Transport properties are computed using the reference state: dynamic viscosity, \(\mu_r\), thermal conductivity, \(\lambda_r\), and mass diffusion coefficient for species \(n\), \(D_{r,n}\).
It is important to note that PelePhysics provides mixture averaged mass diffusion coefficient \(\overline{(\rho D)}_{r,n}\), which is converted into the binary mass diffusion coefficient using
\[(\rho D)_{r,n} = \overline{(\rho D)}_{r,n} \overline{M}_r / M_n.\]Mass diffusion coefficient is then normalized by the total fuel vapor molar fraction
\[(\rho D)^*_{r,n} = \frac{\chi_{v,n} (\rho D)_{r,n}}{\chi_{v,{\rm{sum}}}} \; \forall n \in N_L\]and the total is
\[(\rho D)_r = \sum_{n=0}^{N_L} (\rho D)_{r,n}^*\]The momentum source is a function of the drag force
\[\mathbf{F}_d = \frac{1}{2} \rho_r C_D A_d \left\\Delta \mathbf{u}\right\ \Delta \mathbf{u}\]where \(\Delta \mathbf{u} = \mathbf{u}_g  \mathbf{u}_d\), \(A_d = \pi d_d^2/4\) is the frontal area of the droplet, and \(C_D\) is the drag coefficient for a sphere, which is estimated using the standard drag curve for an immersed sphere
\[\begin{split}C_D = \frac{24}{{\rm{Re}}_d}\left\{\begin{array}{c l} 1 & {\text{If Re$_d$ < 1}}, \\ \displaystyle 1 + \frac{{\rm{Re}}^{2/3}_d}{6} & {\text{Otherwise}}. \end{array}\right.\end{split}\]The droplet Reynolds number is defined as
\[{\rm{Re}}_d = \frac{\rho_r d_d \left\\Delta \mathbf{u}\right\}{\mu_r}\]The mass source term is modeled according to Abramzon and Sirignano (1989). The following nondimensional numbers and factors are used:
\[ \begin{align}\begin{aligned}F(B) &= (1 + B)^{0.7}\frac{\log(1 + B)}{B}\\F_2 &= \max(1, \min(400, {\rm{Re}}_d)^{0.077})\\{\rm{Pr}}_r &= \frac{\mu_r c_{p,r}}{\lambda_r}\\{\rm{Sc}}_r &= \frac{\mu_r}{(\rho D)_r}\\{\rm{Sh}}_0 &= 1 + (1 + {\rm{Re}}_d {\rm{Sc}}_r)^{1/3} F_2\\{\rm{Nu}}_0 &= 1 + (1 + {\rm{Re}}_d {\rm{Pr}}_r)^{1/3} F_2\\{\rm{Sh}}^* &= 2 + \frac{{\rm{Sh}}_0  2}{F(B_M)}\\{\rm{Nu}}^* &= 2 + \frac{{\rm{Nu}}_0  2}{F(B_T)}\end{aligned}\end{align} \]The Spalding numbers for mass transfer, \(B_M\), and heat transfer, \(B_T\), are computed using
\[ \begin{align}\begin{aligned}B_M &= \displaystyle\frac{\sum^{N_L}_{n=0} Y_{v,n}  \sum^{N_L}_{n=0} Y_{g,n}}{1  \sum^{N_L}_{n=0} Y_{v,n}}\\B_T &= \left(1 + B_M\right)^{\phi}  1\end{aligned}\end{align} \]where
\[\phi = \frac{c_{p,r} (\rho D)_r {\rm{Sh}}^*}{\lambda_r {\rm{Nu}}^*}\]Note the dependence of \({\rm{Nu}}^*\) on \(B_T\) means an iterative scheme is required to solve for both. The droplet vaporization rate and heat transfer become
\[ \begin{align}\begin{aligned}\dot{m}_n &= \pi (\rho D)_{r,n}^* d_d {\rm{Sh}}^* \log(1 + B_M). \; \forall n \in N_L\\\mathcal{Q}_d &= \pi \lambda_r d_d (T_g  T_d) {\rm{Nu}}^* \frac{\log(1 + B_T)}{B_T}\end{aligned}\end{align} \]If the gas phase is saturated for all liquid species, the equations for heat and mass transfer become
\[ \begin{align}\begin{aligned}\dot{m}_n &= 0\\\mathcal{Q}_d &= \pi \lambda_r d_d (T_g  T_d) {\rm{Nu}}_0\end{aligned}\end{align} \]
To alleviate conservation issues at AMR interfaces, each parcel only contributes to the gas phase source term of the cell containing it. The gas phase source terms for a single parcel to the cell are
\[ \begin{align}\begin{aligned}S_{\rho} &= \mathcal{C} \sum^{N_L}_{n=0} \dot{m}_n,\\S_{\rho Y_n} &= \mathcal{C} \dot{m}_n,\\\mathbf{S}_{\rho \mathbf{u}} &= \mathcal{C} \mathbf{F}_d,\\S_{\rho h} &= \mathcal{C}\left(\mathcal{Q}_d + \sum_{n=0}^{N_L} \dot{m}_n h_{g,n}(T_d)\right),\\S_{\rho E} &= S_{\rho h} + \frac{1}{2}\left\\mathbf{u}_d\right\ S_{\rho} + \mathcal{C} \mathbf{F}_d \cdot \mathbf{u}_d\end{aligned}\end{align} \]where
\[\mathcal{C} = \frac{N_{d}}{V_{\rm{cell}}},\]\(N_{d}\) is the number of droplets per computational parcel, and \(V_{\rm{cell}}\) is the volume for the cell of interest. Note that the cell volume can vary depending on AMR level and if an EB is present.
Spray Flags and Inputs
In the
GNUmakefile
, specifyUSE_PARTICLES = TRUE
andSPRAY_FUEL_NUM = N
whereN
is the number of liquid species being used in the simulation.Depending on the gas phase solver, spray solving functionality can be turned on in the input file using
pelec.do_spray_particles = 1
orpeleLM.do_spray_particles = 1
.The units for PeleLM and PeleLMeX are MKS while the units for PeleC are CGS. This is the same for the spray inputs. E.g. when running a spray simulation coupled with PeleC, the units for
particles.fuel_cp
must be in erg/g.There are many required
particles.
flags in the input file. For demonstration purposes, 2 liquid species ofNC7H16
andNC10H22
will be used.The liquid fuel species names are specified using
particles.fuel_species = NC7H16 NC10H22
. The number of fuel species listed must matchSPRAY_FUEL_NUM
.Many values must be specified on a perspecies basis. Following the current example, one would have to specify
particles.NC7H16_crit_temp = 540.
andparticles.NC10H22_crit_temp = 617.
to set a critical temperature of 540 K forNC7H16
and 617 K forNC10H22
.Although this is not required or typical, if the evaporated mass should contribute to a different gas phase species than what is modeled in the liquid phase, use
particles.dep_fuel_species
. For example, if we wanted the evaporated mass from both liquid species to contribute to a different species calledSP3
, we would putparticles.dep_fuel_species = SP3 SP3
. All species specified must be present in the chemistry transport and thermodynamic data.
The following table lists other inputs related to
particles.
, whereSP
will refer to a fuel species name
Input 
Description 
Required 
Default Value 


Names of liquid species 
Yes 
None 

Name of gas phase species to contribute 
Yes 
Inputs to


Liquid reference temperature 
Yes 
None 

Critical temperature 
Yes 
None 

Boiling temperature at atmospheric pressure 
Yes 
None 

Liquid \(c_p\) at reference temperature 
Yes 
None 

Latent heat at reference temperature 
Yes 
None 

Liquid density 
Yes 
None 

Liquid thermal conductivity (currently unused) 
No 


Liquid dynamic viscosity (currently unused) 
No 


Couple momentum with gas phase 
No 


Evaporate mass and exchange heat with gas phase 
No 


Fix particles in space 
No 


\(N_{d}\); Number of droplets per parcel 
No 


Output ascii files of spray data 
No 


Particle CFL number for limiting time step 
No 


Ascii file name to initialize droplets 
No 
Empty 
If an Antoine fit for saturation pressure is used, it must be specified for individual species,
particles.SP_psat = 4.07857 1501.268 78.67 1.E5
where the numbers represent \(a\), \(b\), \(c\), and \(d\), respectively in:
\[p_{\rm{sat}}(T) = d 10^{a  b / (T + c)}\]If no fit is provided, the saturation pressure is estimated using the ClasiusClapeyron relation; see
Temperature based fits for liquid density, thermal conductivity, and dynamic viscosity can be used; these can be specified as
particles.SP_rho = 10.42 5.222 1.152E2 4.123E7 particles.SP_lambda = 7.243 1.223 4.223E8 8.224E9 particles.SP_mu = 7.243 1.223 4.223E8 8.224E9
where the numbers represent \(a\), \(b\), \(c\), and \(d\), respectively in:
\[ \begin{align}\begin{aligned}\rho_L \,, \lambda_L = a + b T + c T^2 + d T^3\\\mu_L = a + b / T + c / T^2 + d / T^3\end{aligned}\end{align} \]If only a single value is provided, \(a\) is assigned to that value and the other coefficients are set to zero, effectively using a constant value for the parameters.
Spray Injection
Templates to facilitate and simplify spray injection are available. To use them, changes must be made to the input and SprayParticlesInitInsert.cpp
files. Inputs related to injection use the spray.
parser name. To create a jet in the domain, modify the InitSprayParticles()
function in SprayParticleInitInsert.cpp
. Here is an example:
void
SprayParticleContainer::InitSprayParticles(
const bool init_parts)
{
int num_jets = 1;
m_sprayJets.resize(num_jets);
std::string jet_name = "jet1";
m_sprayJets[0] = std::make_unique<SprayJet>(jet_name, Geom(0));
return;
}
This creates a single jet that is named jet1
. This name will be used in the input file to reference this particular jet. For example, to set the location of the jet center for jet1
, the following should be included in the input file,
spray.jet1.jet_cent = 0. 0. 0.
No two jets may have the same name. If an injector is constructed using only a name and geometry, the injection parameters are read from the input file. Here is a list of injection related inputs:
Input 
Description 
Required 


Jet center location 
Yes 

Jet normal direction 
Yes 

Jet velocity magnitude 
Yes 

Jet diameter 
Yes 

\(\theta_J\); Full spread angle in degrees from the jet normal direction; droplets vary from \([\theta_J/2,\theta_J/2]\) 
Yes 

Temperature of the injected liquid 
Yes 

Mass fractions of the injected
liquid based on

Yes, if


\(\dot{m}_{\rm{inj}}\); Mass flow rate of the jet 
Yes 

Sets hollow cone injection with angle \(\theta_J/2\) 
No (Default: 0) 

\(\theta_h\); Adds spread to hollow cone \(\theta_J/2\pm \theta_h\) 
No (Default: 0) 

\(\phi_S\); Adds a swirling component along azimuthal direction 
No (Default: 0) 

Beginning and end time for jet 
No 

Droplet diameter distribution
type: 
Yes 
Care must be taken to ensure the amount of mass injected during a time step matches the desired mass flow rate. For smaller time steps, the risk of overinjecting mass increases. To mitigate this issue, each jet accounts for three values: \(N_{P,\min}\), \(m_{\rm{acc}}\), and \(t_{\rm{acc}}\) (labeled in the code as m_minParcel
, m_sumInjMass
, and m_sumInjTime
, respectively). \(N_{P,\min}\) is the minimum number of parcels that must be injected over the course of an injection event; this must be greater than or equal to one. \(m_{\rm{acc}}\) is the amount of uninjected mass accumulated over the time period \(t_{\rm{acc}}\). The injection routine steps are as follows:
The injected mass for the current time step is computed using the desired mass flow rate, \(\dot{m}_{\rm{inj}}\) and the current time step
\[m_{\rm{inj}} = \dot{m}_{\rm{inj}} \Delta t + m_{\rm{acc}}\]The time period for the current injection event is computed using
\[t_{\rm{inj}} = \Delta t + t_{\rm{acc}}\]Using the average mass of an injected parcel, \(N_{d} m_{d,\rm{avg}}\), the estimated number of injected parcels is computed
\[N_{P, \rm{inj}} = m_{\rm{inj}} / (N_{d} m_{d, \rm{avg}})\]
If \(N_{P, \rm{inj}} < N_{P, \min}\), the mass and time is accumulated as \(m_{\rm{acc}} = m_{\rm{inj}}\) and \(t_{\rm{acc}} = t_{\rm{inj}}\) and no injection occurs this time step.
Otherwise, \(m_{\rm{inj}}\) mass is injected and convected over time \(t_{\rm{inj}}\) and \(m_{\rm{acc}}\) and \(t_{\rm{acc}}\) are reset.
If injection occurs, the amount of mass injected, \(m_{\rm{actual}}\), is summed and compared with the desired mass flow rate. If \(m_{\rm{actual}} / t_{\rm{inj}}  \dot{m}_{\rm{inj}} > 0.05 \dot{m}_{\rm{inj}}\), then \(N_{P,\min}\) is increased by one to reduce the likelihood of overinjecting in the future. A balance is necessary: the higher the minimum number of parcels, the less likely to overinject mass but the number of time steps between injections can potentially grow as well.
Spray Validation
Single Droplet Tests
Single droplet tests are performed and compared with computational or experimental results published in literature. These tests are setup in PeleProduction/PeleMPruns/single_drop_test
. To run a test case, simply open Validate.py
and set the case name from the table below
case = TestCaseName()
then do python Validate.py
.
The following table details the parameters of each test:
Test Case Name 
\(T_g\) [K] 
\(p_g\) [bar] 
\(T_d\) [K] 
\(d_d\) [um] 
\(\Delta u\) [m/s] 
Ref 


1000 
1 
300 
200 
6.786 


1500 
10 
300 
100 
15 


348 
1 
294 
1334 
3.10 


273 
1 
272 
500570 
2.5 