Source terms for aggregation and coalescence

Consider a population of particles characterized by an NDF that is a function of time, physical space, and particle mass, i.e. n(f,x, p). We define as and p as the masses of the particles that, after sticking together, will result in a particle of mass p. In this case, is a single conservation statement, namely p = + p, resulting in a Jacobian equal to unity. The final source term (written here in terms of the kernel) is 

If the particle density is constant, this description is equivalent to one based on the particle volume. On the other hand, consider a system with two internal coordinates (such as, for example, particle mass pi and particle surface area p2) that are additive during an aggregation event p = + p, where p = ( pi, p2), = and p = ( pi, p2). This 

Ip = [ 3 ( p3ji/3 nonlinear transformation results in a Jacobian equal to p[ p -. The source term for aggregation is then 

As a final example, let us consider a system of coalescing droplets characterized by particle mass and particle velocity Vp. In this case, the continuity statements are written in terms of particle mass and momentum, respectively, as p = + p and pVp = v + pVp or, equivalently, as p = p - and Vp = ( pVp - vp/( p - p. The Jacobian for this nonlinear transformation is 

As we will discuss next, the coalescence kernel used in Eq. (5.164) normally depends on the velocity difference yS( p vp p,Vp) = S( p p, Vp - Vp ). Since the different kernels are primarily functions of the particle size dp and some additional variables, only these functional dependences will be highlighted and discussed. 

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