Population Balances


Population balances belong to a branch in the sciences that deals with any particulate flow. It is a statement about the conservation of the population or number of particles present in a system. The word balance is used instead of conservation to stress the fact that particles undergo chemical and physiological changes. For example, particles can react with other particles leading to new chemical compounds. Also, particles undergo nucleation, growth, aggregation, and breakage.

The analysis of systems with particles aims at addressing the behavior of the population of particles and its environment. The population is usually described by the density of an extensive particle property such as the number of particles. Sometimes, it is convenient to use other extensive properties such as mass or volume of particles.

Population balances are an essential ingredient in a variety of disciplines such as physics and chemistry. Biology also employs population balances to study the behavior of cells of various kinds.

A system that contains particles is usually referred to as a disperse phase system or particulate system regardless of the density or role of particles in them.

In the framework of population balances, we are mainly concerned with systems consisting of particles disperesed in an environmental phase. We refer to the environmental phase as the continuous phase. For example, consider the transport of sand particles by air on a windy day. In case, the sand particles consistute the disperse phase, while the air constitutes the continuous phase.

Mathematical Formalism

In this section, we present the mathematical formulation for deriving the population balance equation.

Internal and External Coordinates

In assessing a disperse phase system, one is often concerned with the properties of the particles in that system. Such properties include their position vector $\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$ as well as intrinsic properties such as characteristic length $l$ and other quantities associated with a given particle. Therefore, if a particle has $n$ intrinsic quantities associated with it, it is customary to refer to those as $x_1, x_2, x_3, \cdots, x_n$ or more conveniently by the vector $\mathbf{x} = \left(x_1, x_2, x_3, \cdots, x_n \right)$. In general, a given particle will vary in both $\mathbf{r}$ and $\mathbf{x}$. To make the analysis clear, we split this variation and distinguish between external and internal coordinates. External coordinates will refer to the position vector $\mathbf{r}$ of a particle while internal coordinates will refer to the state quantities $\mathbf{x}$ of a particle. Note that while the external coordinates form an orthogonal basis, internal coordinates do not necessarily form one.

The combined external-internal coordinate system is conveniently referred to as the state space. Particles will convect about this state space as their internal properties change as well as the external environment conditions vary.

Number Density Function

The description of a population of particles is best achieved introducing the number density function $\mathcal N$. The number density function is defined as the average number of particles per unit volume of state space. For example, when tracking particle sizes in a plug flow reactor, the number density function depends on the axial position in the reactor $x$ as well as the particle size $r$. Then, $\mathcal N \equiv \mathcal N(x,r)$ and its units are number per unit state space volume.

One Dimensional Population Balance Equation

Consider the growth of a population of particles uniformly distributed in physical space. This is the case for example in a stirred tank crystallizer with a highly supersaturated solution. The internal coordinate of interest in this case is the particle size denoted by $r$. As the particles grow, they can be thought of as moving along the particle size dimension. In essence, growth is equivalent to convection in internal coordinates: as particles grow, they move from one particle size to another at a rate equal to the growth rate [[G \equiv $\dot {r} $]].

Consider now an arbitrary interval $[a,b]$ on the internal coordinate dimension. The rate of change of the TOTAL number of particles inside this interval is given by

\begin{align} \frac{\partial }{\partial t}\int_{a}^{b} \mathcal N(r,t) \text{ d} r= G(a,t) \mathcal{N}(a,t) - G(b,t)\mathcal{N}(b,t) \end{align}

Remember that the rate of change of a quantity inside a specified region of space is equal to the difference between the incoming and outgoing flowrates into this region.

To make further headway, we cast the right-hand-side of ([[eqref one_d_population_balance]]) in integral form and rewrite ([[eqref one_d_population_balance]]) as

\begin{align} \int_{a}^{b} \left[ \frac{\partial \mathcal N(r,t)}{\partial t} + \frac{\partial }{\partial r}G(r,t)\mathcal{N}(r,t) \right] \text{ d} r=0 \end{align}

Now, since the interval $[a,b]$ is arbitrary, the above integral holds for an arbitrary region of internal coordinate space. Therefore, for the integral to identically zero on an arbitrary region, the integrand must vanish identically. This leads

\begin{align} \frac{\partial \mathcal N(r,t)}{\partial t} + \frac{\partial }{\partial r}G(r,t)\mathcal{N}(r,t) =0 \end{align}

Equation 3 is a continuity equation for the number density function.

Alternate Derivation:

For an alternate derivation, consider a differential element of size $\mathrm{d}x$ in the internal coordinate space. The quantity of interest in this control volume is $\mathcal N(r,t)$.