Section 4.4.2: The Richness of Dynamics

A steady state model is described by AEs, a dynamic model by ODE(s) or DAEs.
A dynamic model requires initial condition(s) (ICs) for the differential variable(s)
More constraints are usually required in a dynamic model than in a steady state model of equivalent scope.In the tank model
- steady state only one of the flows (in or out) may be constrained.
- dynamic model both flows are constrained.
A dynamic model usually contains more detail (more variables) than a steady state model of equivalent scope. In the tank model
- introduce the holdup in the dynamic model.
- may have to introduce an extra stream to provide a correct mass balance if the tank overflows.

Avoid infeasible specifications

If we specify a constant outflow greater than the inflow (also constant) the tank may empty.

A careless simulation could then lead to the holdup becoming negative.

Care must be taken to arrange process specifications which affect dynamics so that they do not lead to infeasible situations.

A more difficult example

Tank Outlet flow proportional to $\sqrt M$
.. from Bernoulli, if

liquid has constant density
tank has constant cross sectional area

Dynamic material balance

$\begin{displaymath}dM/dt = a_{1} - k \sqrt M \end{displaymath}$

Can we solve it analytically?

$\begin{displaymath}dM/dt = a_{1} - k \sqrt M \end{displaymath}$

$\begin{displaymath}dM / ( a_{1} - k \sqrt M ) = dt \end{displaymath}$

$\begin{displaymath}\int_{M_{0}} dM / ( a_{1} - k \sqrt M ) = \int_{0} dt = t \end{displaymath}$

Needs a change of variable.

Details in section 4.4.2.1 if interested.

Result:

$\begin{displaymath}k^{2}t/2 = k (\sqrt M_{0} - \sqrt M) - a_{1} ln (\frac{a_{1} - k \sqrt M}{a_{1} - k \sqrt M_{0}} ) \end{displaymath}$

We get t as function of M, but we want M as function of t: Awkward!

Summary

Dynamic models of process systems $\rightarrow$ ODEs or DAEs
An initial condition is needed for each differential variable
Some simple process models can be solved exactly
Most need numerical solution

Next - Section 4.4.3: Numerical Solution of an ODE
Return to Section 4.4 Index
Return to Section 4 Index

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Last modified: Tue Aug 4 12:36:06 BST