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

  • One-third 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

\[ \begin{align}\begin{aligned}T_r &= T_d + A (T_g - T_d)\\Y_{r,n} &= Y_{v,n} + A (Y_{g,n} - Y_{v,n})\end{aligned}\end{align} \]

where \(A = 1/3\) according the the one-third 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:

\[ \begin{align}\begin{aligned}\frac{d \mathbf{X}_d}{d t} &= \mathbf{u}_d,\\\frac{d m_d}{d t} &= \sum^{N_L}_{n=0} \dot{m}_n,\\m_d \frac{d Y_{d,n}}{d t} &= \dot{m}_n - Y_{d,n} \frac{d m_d}{d t},\\m_d \frac{d \mathbf{u}_d}{d t} &= \mathbf{F}_d + m_d \mathbf{g},\\m_d c_{p,L} \frac{d T_d}{d t} &= \sum^{N_L}_{n=0} \dot{m}_n h_{L,n}(T_d) + \mathcal{Q}_d.\end{aligned}\end{align} \]

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, \(Y_{d,n}\),

\[\rho_d = \left( \sum^{N_L}_{n=0} \frac{Y_{d,n}}{\rho_{L,n}} \right)^{-1}\]

The droplets are assumed to be spherical with diameter \(d_d\). Therefore, the mass is computed as

\[m_d = \frac{\pi}{6} \rho_d d_d^3\]

The procedure is as follows for updating the spray droplet:

  1. Interpolate the gas phase state to the droplet location using a trilinear interpolation scheme.

  2. Compute the boiling temperature for species \(n\) at the current gas phase pressure using the Clasius-Clapeyron 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

    \[h_{L,n}(T^*_{b,n}) = h_{L,n}(T^*) \left(\frac{T_{c,n} - T^*}{T_{c,n} - T^*_{b,n}} \right)^{-0.38}\]
  3. 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 Clasius-Clapeyron 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

    \[p_{{\rm{sat}},n} = d 10^{a - b / (T_d + c)}\]
  4. 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 one-third rule and the remaining reference mass fractions are normalized gas phase mass fractions to ensure they sum to 1

    \[\begin{split}Y_{r,n} = \left\{\begin{array}{c l} \displaystyle Y_{v,n} + A (Y_{g,n} - Y_{v,n}) & {\text{If $Y_{v,n} > 0$}}, \\ \displaystyle\frac{1 - \sum^{N_L}_{k=0} Y_{v,k}}{1 - \sum^{N_L}_{k=0} Y_{g,k}} Y_{g,n} & {\text{Otherwise}}. \end{array}\right. \; \forall n \in N_s.\end{split}\]
  5. The average molar mass, specific heat, and density of the reference state in the gas film are computed as

    \[ \begin{align}\begin{aligned}\overline{M}_r &= \left(\sum^{N_s}_{n=0} \frac{Y_{r,n}}{M_n}\right)^{-1},\\c_{p,r} &= \sum^{N_s}_{n=0} Y_{s,n} c_{p,g,n}(T_r),\\\rho_r &= \frac{\overline{M}_r p_g}{\mathcal{R} T_r}.\end{aligned}\end{align} \]
  6. Transport properties are computed using the reference state: dynamic viscosity, \(\mu_r\), thermal conductivity, \(\lambda_r\), and mass diffusion coefficient for species \(n\), \(D_{r,n}\).

  7. 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

    \[(\rho D)_r = \sum_{n=0}^{N_L} (\rho D)_{r,n}^*\]
  8. 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

    \[{\rm{Re}}_d = \frac{\rho_r d_d \left\|\Delta \mathbf{u}\right\|}{\mu_r}\]
  9. The mass source term is modeled according to Abramzon and Sirignano (1989). The following non-dimensional 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

      \[ \begin{align}\begin{aligned}B_M &= \displaystyle\frac{\sum^{N_L}_{n=0} Y_{v,n} - \sum^{N_L}_{n=0} Y_{g,n}}{1 - \sum^{N_L}_{n=0} Y_{v,n}}\\B_T &= \left(1 + B_M\right)^{\phi} - 1\end{aligned}\end{align} \]


      \[\phi = \frac{c_{p,r} (\rho D)_r {\rm{Sh}}^*}{\lambda_r {\rm{Nu}}^*}\]

      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

      \[ \begin{align}\begin{aligned}\dot{m}_n &= 0\\\mathcal{Q}_d &= \pi \lambda_r d_d (T_g - T_d) {\rm{Nu}}_0\end{aligned}\end{align} \]
  10. 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

    \[ \begin{align}\begin{aligned}S_{\rho} &= \mathcal{C} \sum^{N_L}_{n=0} \dot{m}_n,\\S_{\rho Y_n} &= \mathcal{C} \dot{m}_n,\\\mathbf{S}_{\rho \mathbf{u}} &= \mathcal{C} \mathbf{F}_d,\\S_{\rho h} &= \mathcal{C}\left(\mathcal{Q}_d + \sum_{n=0}^{N_L} \dot{m}_n h_{g,n}(T_d)\right),\\S_{\rho E} &= S_{\rho h} + \frac{1}{2}\left\|\mathbf{u}_d\right\| S_{\rho} + \mathcal{C} \mathbf{F}_d \cdot \mathbf{u}_d\end{aligned}\end{align} \]


    \[\mathcal{C} = -\frac{N_{d}}{V_{\rm{cell}}},\]

    \(N_{d}\) 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, specify USE_PARTICLES = TRUE and SPRAY_FUEL_NUM = N where N 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 or peleLM.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 of NC7H16 and NC10H22 will be used.

    • The liquid fuel species names are specified using particles.fuel_species = NC7H16 NC10H22. The number of fuel species listed must match SPRAY_FUEL_NUM.

    • Many values must be specified on a per-species basis. Following the current example, one would have to specify particles.NC7H16_crit_temp = 540. and particles.NC10H22_crit_temp = 617. to set a critical temperature of 540 K for NC7H16 and 617 K for NC10H22.

    • 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 called SP3, we would put particles.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., where SP will refer to a fuel species name




Default Value


Names of liquid species




Name of gas phase species to contribute


Inputs to fuel_species


Liquid reference temperature




Critical temperature




Boiling temperature at atmospheric pressure




Liquid \(c_p\) at reference temperature




Latent heat at reference temperature




Liquid density




Liquid thermal conductivity (currently unused)



Liquid dynamic viscosity (currently unused)



Couple momentum with gas phase




Evaporate mass and exchange heat with gas phase




Fix particles in space




\(N_{d}\); Number of droplets per parcel




Output ascii files of spray data




Particle CFL number for limiting time step




Ascii file name to initialize droplets



  • 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 Clasius-Clapeyron 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 1.152E-2 4.123E-7
    particles.SP_lambda = 7.243 1.223 4.223E-8 8.224E-9
    particles.SP_mu = 7.243 1.223 4.223E-8 8.224E-9

    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:

const bool init_parts)
  int num_jets = 1;
  std::string jet_name = "jet1";
  m_sprayJets[0] = std::make_unique<SprayJet>(jet_name, Geom(0));

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:





Jet center location



Jet normal direction



Jet velocity magnitude



Jet diameter



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



Temperature of the injected liquid



Mass fractions of the injected liquid based on particles.fuel_species

Yes, if SPRAY_FUEL_NUM > 1


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



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)

start_time and end_time

Beginning and end time for jet



Droplet diameter distribution type: Uniform, Normal, LogNormal, Weibull, ChiSquared



9 Demonstration of injection angles. \(\phi_J\) varies uniformly from \([0, 2 \pi]\)

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 over-injecting 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:

  1. 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

    \[m_{\rm{inj}} = \dot{m}_{\rm{inj}} \Delta t + m_{\rm{acc}}\]
  2. The time period for the current injection event is computed using

    \[t_{\rm{inj}} = \Delta t + t_{\rm{acc}}\]
  3. 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.

  1. 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 over-injecting in the future. A balance is necessary: the higher the minimum number of parcels, the less likely to over-inject 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]























RungeHep RungeDec RungeMix








10 Droplet diameter, temperature, and n-octane mass fraction comparisons with Figure 4.33 in [1]


11 Droplet diameter and temperature comparisons with [2]


12 Droplet diameter and temperature comparisons with [3]