A simple theoretical model, the Kulikovskiy--Sveshnikova--Beghin (KSB) model, is outlined, describing the motion of a particle cloud moving down an incline. This model includes both the entrainment of surrounding ambient fluid and the entrainment of particles from the slope and is equally valid for Boussinesq and non-Boussinesq flows. However, this model can predict physically impossible densities when there is significant particle entrainment. Modifications to the model are proposed which eliminate this problem by including the entrained snow volume. With the modified model, physically realistic mean densities are predicted which have a significant impact on the Richardson number--dependent ambient entrainment. The improvements are illustrated by comparing analytical solutions to the original and the modified KSB equations for the case of a particle cloud traveling on a slope of constant angle, with constant ambient fluid and particle entrainment. Solving the modified model numerically, predictions are compared with data from several large powder snow avalanches at the Swiss Vallie de la Sionne avalanche test site. The modified KSB model appears to capture the dynamics of the avalanche front well; however, problems remain with relating the theoretical geometry to a real avalanche geometry. The success of this model in capturing the front dynamics shows that with careful assumptions that reflect the physics, it is possible to describe aspects of complex flows such as powder snow avalanches with simple models.