In combustion, the Williams spray equation, also known as the Williams–Boltzmann equation, describes the statistical evolution of sprays contained in another fluid, analogous to the Boltzmann equation for the molecules, named after Forman A. Williams, who derived the equation in 1958.
Mathematical description
The sprays are assumed to be spherical with radius, even though the assumption is valid for solid particles when their shape has no consequence on the combustion. For liquid droplets to be nearly spherical, the spray has to be dilute and the Weber number, where is the gas density, is the spray droplet velocity, is the gas velocity and is the surface tension of the liquid spray, should be. The equation is described by a number density function, which represents the probable number of spray particles of chemical species , that one can find with radii between and, located in the spatial range between and, traveling with a velocity in between and, having the temperature in between and at time. Then the spray equation for the evolution of this density function is given by where
This model for the rocket motor was developed by Probert, Williams and Tanasawa. It is reasonable to neglect, 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, where fuel is sprayed. Neglecting and due to the fact that the mean flow is parallel to axis, the steady spray equation reduces to where is the velocity in direction. Integrating with respect to the velocity results The contribution from the last term becomes zero since when is very large, which is typically the case in rocket motors. The drop size rate is well modeled using vaporization mechanisms as where is independent of, but can depend on the surrounding gas. Defining the number of droplets per unit volume per unit radius and average quantities averaged over velocities, the equation becomes If further assumed that is independent of, and with a transformed coordinate If the combustion chamber has varying cross-section area, a known function for and with area at the spraying location, then the solution is given by where are the number distribution and mean velocity at respectively.
