Section 4.4.10: Towards Process Simulation

Build a material and energy balance model for a tank with water flows in and out.

Tank is assumed to have non-constant cross section: sloping sides.

Gravity outflow model should be modified, and expressed in terms of level instead of holdup. Therefore we need tank geometry equations.

Energy balance introduces energy and enthalpy. Temperature is a more convenient quantity to specify, and to express thermal equilibrium. Hence thermodynamic and physical property equations are needed.

See section 4.4.10.1, `Material and energy balance for a tank'.

Modelling strategy

Write equations which describe the process. Aim for a logical grouping of equations:

unit by unit
within units, e.g.
1. Fundamental balances (conservation)
2. Specific model for unit: `rate equations', geometry,assumptions (well-stirred..)
3. Thermodynamic relations and
  definitions
4. Physical property correlations
5. Specifications (`process constraints')

Choosing specifications

Also tracking whether we have suitable modelling equations.

For each equation

note any new variables
add to list of `unsolved' variables
assume the equation solves for one of the unsolved variables: tick it off and remove it from the `unsolved' list.

Choose something sensible. If there are no suitable `unsolved' candidates, go back to where there was a choice, make a different choice and repeat from there. If making a different choice further back is not possible, consider whether the new equation repeats earlier information.

When all modelling equations have been written there will generally be some `unsolved' variables left. This is because there are usually more variables than equations in the basic model of a unit. The remaining degrees of freedom in the model can be satisfied by specifying the variables which are left on the `unsolved' list.

Dynamic DAE models

For a dynamic model you get a DAE system.

In counting equations and variables exclude the independent variable (time).

Apply the following `ticking off' rule for the ODEs in a DAE system:

For an ODE $\frac{dx_i}{dt} = f({\bf x,y},t)$
always tick off the differential variable x_i.

This ensures that the specifications obtained from the final unsolved variable list do not cause a difficult (high index) DAE system.

Initial conditions

The ticking-off rule for ODEs also means that if we replace the ODEs with `standard' initial conditions at t₀, the AEs can still be ticked off in exactly the same way, because if we replace

$\begin{displaymath}\frac{d{\bf x}}{dt} = {\bf f}({\bf x,y},t) \end{displaymath}$

with

$\begin{displaymath}{\bf x} - {\bf x}(0) = 0 \end{displaymath}$

then the latter standard ICs must be ticked off for the differential variables x.

It is always safe to use standard ICs for a well-behaved DAE, but not always convenient.

E.g. do we really know the initial holdup and internal energy in the tank? Liquid level and temperature might be preferable for ICs.

Alternative ICs

In a DAE system it may be acceptable to give ICs for some of the algebraic variables instead of differential variables, as long as

the number of ICs is the same as the number of ODEs, and
the variables for which ICs are given are linked to the differential variables through the AE part of the model.

Two checks

Having constructed a model, including proposed specifications and initial conditions, you should perform two checks.

Assume fixed values of the differential variables x.
Ensure that the AEs can be solved for the algebraic variables y.
If your ICs are not the standard ones ${\bf x} - {\bf x}_0 = {\bf0}$ then also ensure that for the proposed alternative ICs, the AEs can be solved for the remaining variables, which now include both some algebraic and some differential variables.

Check 1 verifies that the DAE system with standard ICs is well-behaved (not high index).
Check 2 verifies that the proposed ICs are in some sense equivalent to the standard ICs for x.

In the event of a problem ..

If check 1 fails, the specifications and perhaps the modelling equations themselves need to be re-examined.
If check 1 succeeds but check 2 fails, you need different ICs.
The standard ICs can always be used.

An important distinction

It is important not to confuse specifications and initial conditions. Specifications are valid throughout the simulation: for example F₁, T₁, p. ICs are valid only at the starting point t₀: for example L, T.

Never try to give ICs for the same variables as specifications.

Discussion

The tank with material and energy balances is perhaps the simplest non-trivial dynamic process model.

General modelling must also take account of composition dynamics, where there is more than one chemical component.
Thus we should also have equations for the individual component balances. The `well mixed' assumption requires extra equations equating concentrations in the outlet stream to those in the tank.

More complex models (e.g. for an enclosed vessel) should introduce the possibility of multiple phases (e.g. vapour + liquid), with some way of relating them (e.g. phase equilibrium or a mass transfer model).

Then there is an endless range of possible reactor models!

DAE formulation fits well with the `equation-based approach' to process simulation, which at steady state views the process model as a system of NLAEs:

$\begin{displaymath}{\bf g}({\bf y}) = 0 \end{displaymath}$

It is natural to extend such simulators to dynamics by adding discretisations of ODEs. These are often implicit, to cope with potentially stiff dynamic flowsheet simulation problems. Hence the DAE approach.

Conclusion

`Sequential modular simulators' view the process as a collection of unit operations. Each unit is solved independently in sequence (or nowadays sometimes in parallel). This approach may lend itself more to the ODE formulation for some (simple) units, and a DAE approach in others.

The nested iterations of the ODE method may be slower, but also tend to converge more reliably, than the simultaneous solution of the DAE method.

Summary

Dynamic process simulation $\rightarrow$ DAE systems
Use logical techniques for constructing models:
- group equations by unit, and by type within a unit operation,
- notionally solve each equation for a designated variable; always `solve' an ODE for its differential variable.
- Get a sensible set of specifications from unsolved variables.
For DAEs the ICs need not be the differential variables.
Check that the model you develop is well behaved
- with standard ICs (differential variables)
- with the proposed ICs, if different.

Return to Section 4.4 Index
Return to Section 4 Index

Course Organiser

Last modified: Thu Aug 6 10:34:43 BST