In order to solve the moments of the particle size distribution function, transport equations (or conservation equations) of the size moments are required. The transport equations of the moments for the moments of the particle size distribution function can be derived from those for the discrete number density function.

According to the size moment definition given by Equation 19–2 , the transport equation of the r -th size moment can be derived by applying to each term of Equation 19–129 . The result of this operation leads to the transport equation for

(19–107)

, , and are the production terms of the r -th particle size moment contributed by nucleation, coagulation, and surface reactions, respectively. The definitions of these terms are given in Equation 19–42 , Equation 19–65 - Equation 19–67 , and Equation 19–76 . As mentioned earlier, somewhat simplified expressions are used for particle diffusivity and the thermophoretic coefficient when using the method of moments. Specifically, the particle diffusivity given in Equation 19–107 can be rewritten in terms of the particle size class as

(19–108)

Similarly, thermophoretic velocity is expressed as

(19–109)

Now Equation 19–107 can be expressed entirely in terms of size moments and the final form of the r -th size moment equation becomes

(19–110)