---
tag: 6Q7H2FQ12Q
kind: example
name: Population balance equation
msc: [45K05, 35Q92]
instance_of: [TH4NGC4016]
sources:
  - url: https://en.wikipedia.org/wiki/Population_balance_equation
same_as:
  wikidata: Q7229796
  wikipedia: 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|TH4NGC4016]].

Check in two special cases.
With $G = 0$ and $S = 0$ only aggregation remains, and the equation is the [[Smoluchowski coagulation equation|9JRNCRZ97G]] 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.

## Remarks

In general the internal coordinate can be a vector of several properties, such as size, composition and age, and the particles can also move in physical space.
Moment methods turn the equation into ordinary differential equations for the moments $\int v^k n \, dv$ when the kernels allow it.
Such equations are used for crystallization, emulsions, aerosols and cell populations.

## Explanations

### L1 human

A population balance keeps track of how many particles of each size there are.
Particles grow, which moves them to larger sizes; they stick together, which creates big particles from small ones; and they break, which creates small particles from big ones.
Growth gives a derivative in the size variable, while sticking and breaking give integrals over the sizes of the partners.