Template:Model theory (Text, Mill, Perfect Mixing, Dynamic, Time Step)

The unsteady-state Perfect Mixing differential equation is numerically solved by a discretised time stepping approach (i.e. Euler's method). The change in load mass in a size fraction $\Delta s_{i}$ during a sufficiently small time increment $\Delta t=t_{n+1}-t_{n}$ is:

$\Delta s_{i}=\left(f_{i}-D_{i}s_{i}+\sum _{j=1}^{i-1}A_{ij}R_{j}s_{j}-(R_{i}s_{i}-A_{ii}R_{i}s_{i})\right).\Delta t$

Similarly, for liquids:

$\Delta s_{w}=\left(f_{w}-D_{w}s_{w}\right).\Delta t$

The time stepping approach is a convenient numerical approximation to the solution of the unsteady-state Perfect Mixing population balance differential equation. The approach is, however, subject to several limitations:

• The mass of particles separately discharged from or broken out of a size interval in a time step cannot exceed the mass of particles actually present in that size interval.
• Similarly, the overall maximum discharge flow rate of pulp from the mill cannot exceed the total volume of pulp in the mill in a time step.

The time step size used internally by the model is automatically reduced to ensure the breakage, discharge and pulp flow rate limits per step are not exceeded. This is achieved by computing a number of sequential sub-steps at the reduced internal step size for each requested external step.

This is useful if the either a fixed time step specified by an application is too large (e.g. SysCAD) or a numerical solution is desired in as few steps as possible (e.g. Excel). The automatic time step adjustment is largely invisible to the user and manifests only as a slightly slower execution speed.

The calculated time step size may be overridden with a larger user-specified sub-step count if increased accuracy in the numerical approximation is desired.