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

The dynamic 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 unsteady-state, the diagram in Figure 1 below represents the balance for a given size fraction:

Figure 1. Schematic diagram of the population balance adopted by the dynamic Perfect Mixing model.

The dynamic population balance is described mathematically as:

${\frac {ds_{i}(t)}{dt}}=f_{i}-D_{i}s_{i}+\sum _{j=1}^{i-1}A_{ij}R_{j}s_{j}-(R_{i}s_{i}-A_{ii}R_{i}s_{i})$

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

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 feed rate of solids in size interval $i$
• $p_{i}$ is the mass product rate of solids in size interval $i$
• $s_{i}$ is the mass of solids in the mill load in size interval $i$
• $R_{i}$ is the breakage rate of solids in the mill load in size interval $i$
• $D_{i}$ is the rate of discharge from the mill of solids in size interval $i$
• $A_{ij}$ 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 $i$

Unlike the steady-state version, the load component $s_{i}$ cannot be eliminated from the equation, nor can the $R_{i}$ and $D_{i}$ components be combined into a single $R_{i}/D_{i}$ term. Therefore, the breakage and discharge rates $R_{i}$ and $D_{i}$ must be specified separately as inputs to the dynamic model.

Finally, an unsteady-state mill simulation must also consider the retention of liquids in the load:

${\frac {ds_{\rm {w}}(t)}{dt}}=f_{\rm {w}}-D_{\rm {w}}s_{\rm {w}}$

where:

• $s_{\rm {w}}$ is the load mass of water in the mill
• $f_{\rm {w}}$ is the mass feed rate of water into the mill
• $D_{\rm {w}}$ is the discharge rate of water from the mill, normally assumed to equal the value of $D_{i}$ at the finest size interval.