Williams–Boltzmann equation

The Williams–Boltzmann equation, also known as the Williams spray equation is a kinetic equation modeling the statistical evolution of evaporating or burning droplets or solid particles in a fluid medium. It was derived by Forman A. Williams in 1958. The Williams–Boltzmann equation must be solved concurrently with the hydrodynamic equations such as the Navier–Stokes equations with forcing terms accoutning for the presence of sprays.

Mathematical description

Consider a spray of liquid droplets or solid particles with $M$ chemical species, all of which are assumed spherical in shape with radius $r$; the spherical assumption can be relaxed if needed. For liquid droplets to be nearly spherical, the spray has to be dilute (total volume occupied by the droplets is much less than the volume of the ambient fluid) and the Weber number $We=2r\rho _{g}|\mathbf {v} -\mathbf {u} |^{2}/\sigma$, where $\rho _{g}$ is the gas density, $\mathbf {v}$ is the spray droplet velocity, $\mathbf {u}$ is the gas velocity and $\sigma$ is the surface tension of the liquid spray, should be $We\ll 10$.

The droplet/particle number density function for a $j$-th chemical species is denoted by $f_{j}=f_{j}(r,\mathbf {x} ,\mathbf {v} ,T,t)$ such that

$f_{j}(r,\mathbf {x} ,\mathbf {v} ,T,t)\,dr\,d\mathbf {x} \,d\mathbf {v} \,dT$

represents the probable number of droplets/particles of chemical species $j$ (of $M$ total species), that one can find with radii between $r$ and $r+dr$, located in the spatial range between $\mathbf {x}$ and $\mathbf {x} +d\mathbf {x}$, traveling with a velocity in between $\mathbf {v}$ and $\mathbf {v} +d\mathbf {v}$ and having the temperature in between $T$ and $T+dT$ at time $t$. Then the spray equation for the evolution of this density function is given by

${\frac {\partial f_{j}}{\partial t}}+\nabla _{x}\cdot (\mathbf {v} f_{j})+\nabla _{v}\cdot (F_{j}f_{j})=-{\frac {\partial }{\partial r}}(R_{j}f_{j})-{\frac {\partial }{\partial T}}(E_{j}f_{j})+Q_{j}+\Gamma _{j},\quad j=1,2,\ldots ,M.$

where

$F_{j}=\left({\frac {d\mathbf {v} }{dt}}\right)_{j}$ is the force per unit mass acting on the $j^{\text{th}}$ species spray (acceleration applied to the sprays),

$R_{j}=\left({\frac {dr}{dt}}\right)_{j}$ is the rate of change of the size of the $j^{\text{th}}$ species spray,

$E_{j}=\left({\frac {dT}{dt}}\right)_{j}$ is the rate of change of the temperature of the $j^{\text{th}}$ species spray due to heat transfer,

$Q_{j}$ is the rate of change of number density function of $j^{\text{th}}$ species spray due to nucleation, liquid breakup etc.,

$\Gamma _{j}$ is the rate of change of number density function of $j^{\text{th}}$ species spray due to collision with other spray particles.

A simplified model for liquid propellant rocket

This model for the rocket motor was developed by Probert, Williams and Tanasawa. It is reasonable to neglect $Q_{j},\ \Gamma _{j}$, for distances not very close to the spray atomizer, where major portion of combustion occurs. Consider a one-dimensional liquid-propellent rocket motor situated at $x=0$, where fuel is sprayed. Neglecting $E_{j}$(density function is defined without the temperature so accordingly dimensions of $f_{j}$ changes) and due to the fact that the mean flow is parallel to $x$ axis, the steady spray equation reduces to

${\frac {\partial }{\partial r}}(R_{j}f_{j})+{\frac {\partial }{\partial x}}(u_{j}f_{j})+{\frac {\partial }{\partial u_{j}}}(F_{j}f_{j})=0$

where $u_{j}$ is the velocity in $x$ direction. Integrating with respect to the velocity results

${\frac {\partial }{\partial r}}\left(\int R_{j}f_{j}\,du_{j}\right)+{\frac {\partial }{\partial x}}\left(\int u_{j}f_{j}\,du_{j}\right)+[F_{j}f_{j}]_{0}^{\infty }=0$

The contribution from the last term (spray acceleration term) becomes zero (using Divergence theorem) since $f_{j}\rightarrow 0$ when $u$ is very large, which is typically the case in rocket motors. The drop size rate $R_{j}$ is well modeled using vaporization mechanisms as

$R_{j}=-{\frac {\chi _{j}}{r^{k_{j}}}},\quad \chi _{j}\geq 0,\quad 0\leq k_{j}\leq 1$

where $\chi _{j}$ is independent of $r$, but can depend on the surrounding gas. Defining the number of droplets per unit volume per unit radius and average quantities averaged over velocities,

$G_{j}=\int f_{j}\,du_{j},\quad {\bar {R}}_{j}={\frac {\int R_{j}f_{j}\,du_{j}}{G_{j}}},\quad {\bar {u}}_{j}={\frac {\int u_{j}f_{j}\,du_{j}}{G_{j}}}$

the equation becomes

${\frac {\partial }{\partial r}}({\bar {R}}_{j}G_{j})+{\frac {\partial }{\partial x}}({\bar {u}}_{j}G_{j})=0.$

If further assumed that ${\bar {u}}_{j}$ is independent of $r$, and with a transformed coordinate

$\eta _{j}=\left[r^{k_{j}+1}+(k_{j}+1)\int _{0}^{x}{\frac {\chi _{j}}{{\bar {u}}_{j}}}\,dx\right]^{1/(k_{j}+1)}$

If the combustion chamber has varying cross-section area $A(x)$, a known function for $x>0$ and with area $A_{o}$ at the spraying location, then the solution is given by

$G_{j}(\eta _{j})=G_{j,o}(\eta _{j}){\frac {A_{o}{\bar {u}}_{j,o}}{A{\bar {u}}_{j}}}\left({\frac {r}{\eta _{j}}}\right)^{k_{j}}$.

where $G_{j,0}=G_{j}(r,0),\ {\bar {u}}_{j,0}={\bar {u}}_{j}(x=0)$ are the number distribution and mean velocity at $x=0$ respectively.

See also

• Boltzmann equation
• Spray (liquid drop)
• Liquid-propellant rocket
• Smoluchowski coagulation equation