The sprays are assumed to be spherical with radius r, even though the assumption is valid for solid particles(liquid droplets) when their shape has no consequence on the combustion.
\frac{\partial f_j}{\partial t} + \nabla_x\cdot(\mathbf{v}f_j) = - \nabla_v\cdot(F_jf_j) - \frac{\partial }{\partial r}(R_jf_j) - \frac{\partial }{\partial T}(E_jf_j) + Q_j + \Gamma_j, \quad j = 1,2,..,M.