Template:Model theory (Text, Mill, Perfect Mixing, Population Balance)

The Perfect Mixing model is based on a population balance of particles entering the mill, breaking into smaller sizes, and discharging as product. For a mill operating in steady-state, the diagram in Figure 1 below represents the balance for a given size fraction:

Figure 1. Schematic diagram of the steady-state population balance adopted by the Perfect Mixing model.

The steady-state population balance is formulated mathematically as:

$f_{i}-R_{i}s_{i}+\sum _{j=1}^{i}{A_{ij}R_{j}s_{j}}-p_{i}=0$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $f_{i}$ is the mass flow rate of solids of size interval $i$ in the mill feed
• $p_{i}$ is the mass flow rate of solids of size interval $i$ in the mill product
• $s_{i}$ is the mass of solids of size interval $i$ in the mill load
• $R_{i}$ is the breakage rate of solids of size interval $i$ in the mill load
• $A_{ij}$ is the appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval $j$ into progeny of size interval $i$

As the mill is perfectly mixed, the product is related to the mill contents and discharge rate as:

$p_{i}=D_{i}s_{i}$

where $D_{i}$ is the rate of discharge of solids in size interval $i$ from the mill.

Substitution and rearrangement of the above equations leads to:

$p_{i}={\dfrac {f_{i}+\sum \limits _{j=1}^{i}{A_{ij}{\dfrac {R_{j}}{D_{j}}}p_{j}}}{1+{\dfrac {R_{i}}{D_{i}}}}}$

The mill product can therefore be computed provided a feed rate, appearance function and breakage rate per discharge rate, $R_{i}/D_{i}$, is available. Alternatively, the $R_{i}/D_{i}$ function can be determined from the feed rate, product rate and appearance function.