The bed material in a rotary kiln is characterized by two distinct regions:the relatively thinner active region in which the particles roll down the sloping bed surface and the much thicker stagnant region in which the particles are carried upward by the rotating wall. According to this characteristic

a single particulate trajectory model (PTM) is developed starting from intrinsic mechanism of granular flow.A simplified calculation equations involving MRT(mean residence time) and MVF (material volumetric flow) is derived

on the basis of both the vector analysis on axial displacement of individual particles in a single excursion and the statistical-averaged analysis on all excursions.Two modified coefficients ε t and ε f are proposed to improve model validity under practical cases such as irregular municipal solid wastes (MSW) existence and internal structure usage. The newly derived equations of PTM are compared with those proposed by previous workers to show their advantages. Finally

an optimal model for both operation and design of a special rotary kiln pyrolyser is presented and the corresponding optimization solution is given in detail.