The general transport equation for particles of size j can be written as
 | (19–100) |
In the above equation, the right-hand side represents the net generation rate of particles of size j due to processes such as nucleation, collisions leading to coagulation, and gas-particle interactions as described in Particle Inception through Particle Depletion . The particle velocity U j is considered to be sum of the velocities due to gas flow, particle diffusion, and thermophoresis and is written as
 | (19–101) |
The particle diffusion and thermophoretic velocities, denoted by V D and V T respectively in Equation 19–101 , are expressed as a function of particle number density and gas temperature gradient, respectively. Thus,
 | (19–102) |
In the above equations, δj and Θj are diffusivity and thermophoretic coefficient of particle of size j whereas η and ρ are viscosity and density of the surrounding gas. The diffusivity and thermophoretic coefficients are, in general, functions of the Knudsen number regime. The general expressions for these can be written as
 | (19–103) |
Where the slip correction factor S is given by
 | (19–104) |
In the above equation α is the Knudsen number = 2λ/D j and the gas mean free path is written as
 | (19–105) |
Ansys Chemkin uses values of 1.257, 0.4, and 0.55 for constants A1, A2, and A3, respectively.
The thermophoretic coefficient is written as
 | (19–106) |
In the above equation, C R represents the ratio of gas to particle thermal conductivity. Values of constants C S , C T , and C M used by Ansys Chemkin are 1.17, 2.18, and 1.14, respectively.
The exact form of transport equation for particles depends on the reactor model. For example, for the closed and open 0-D reactors and Plug-flow reactor, diffusion and thermophoresis are not applicable. On the other hand, for flame simulators all terms are retained. Additionally, while the sectional method uses the complete forms of various terms described above, some simplifying approximations are invoked when the method of moments is used. These are explained in the following section.