Population balance equation

Example

Let $n(v, t) \ge 0$ be the number density of particles of volume $v > 0$ at time $t$ in a well-mixed vessel. Let $G(v)$ be the growth rate of a particle of volume $v$, and let $\beta(v, w) = \beta(w, v) \ge 0$ be the aggregation kernel. Let $S(v) \ge 0$ be the breakage rate. Let $b(v \mid w)$ be the density of fragments of volume $v$ produced when a particle of volume $w > v$ breaks. A population balance equation with growth, aggregation and breakage is $$\frac{\partial n}{\partial t} + \frac{\partial}{\partial v}\big(G(v) n\big) = \frac12 \int_0^v \beta(v - w, w) \, n(v - w, t) \, n(w, t) \, dw - n(v, t) \int_0^\infty \beta(v, w) \, n(w, t) \, dw + \int_v^\infty b(v \mid w) \, S(w) \, n(w, t) \, dw - S(v) \, n(v, t).$$ The left side is a transport equation in the internal coordinate $v$. The right side lists birth and death of particles by aggregation and by breakage. It contains partial derivatives in $t$ and $v$ and integrals of $n$ in $v$, so it is an integro-partial differential equation.

Check in two special cases. With $G = 0$ and $S = 0$ only aggregation remains, and the equation is the Smoluchowski coagulation equation in the volume variable. With $\beta = 0$, $S = 0$ and constant $G = g$, the equation is $\partial_t n + g \, \partial_v n = 0$. It is solved by $n(v, t) = n_0(v - g t)$ for any initial profile $n_0$ that vanishes for $v \le 0$. Every particle grows at the same rate, and the profile moves to the right.