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.
[1][2] The sprays are assumed to be spherical with radius
, even though the assumption is valid for solid particles(liquid droplets) when their shape has no consequence on the combustion.
For liquid droplets to be nearly spherical, the spray has to be dilute(total volume occupied by the sprays is much less than the volume of the gas) and the Weber number
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 (droplets) of chemical species
total species), that one can find with radii between
, located in the spatial range between
, traveling with a velocity in between
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,[5] Williams[1][6] and Tanasawa.
[7][8] 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
(density function is defined without the temperature so accordingly dimensions of
changes) and due to the fact that the mean flow is parallel to
axis, the steady spray equation reduces to where
Integrating with respect to the velocity results The contribution from the last term (spray acceleration term) becomes zero (using Divergence theorem) since
is very large, which is typically the case in rocket motors.
The drop size rate
is well modeled using vaporization mechanisms as where
, 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
, and with a transformed coordinate
If the combustion chamber has varying cross-section area
at the spraying location, then the solution is given by where
are the number distribution and mean velocity at