Collisional growth theory

Chapter 7 Powder Synthesis with Gas Phase Reactants 

Assuming each particle is the same diameter, the collision frequency, Z, for one particle with its neighbors is given by 

The factor i prevents duplicate counting. As each collision reduces the number of particles by 1 the change in number density is expressed as 

This differential equation has the following solution for the initial condition N = Nq at t = 0  

This equation can be used to calculate the average particle radius, using 

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