Facilities for Partial Differential Equations

Partial Differential Equations In the introduction it was stated that, for simulation of complete engineering systems rather than components within a system, it was necessary to convert partial differential equations to ODEs or DAEs. This gives rise to the concept of the lumped parameter model of a component. It is assumed that it is possible to take a single representative value of a variable such as pressure. Within the component instead of assuming that

P ≡ P(t,x,y,z)

we assume

P ≡ P(t)

This lumped parameter assumption is very successful but there are occasions when the assumption is clearly invalid. A good example of this occurs with hydraulic pipes. The assumption is clearly invalid in the following two cases:

• The pipe is very long. For example if the diameter is 10mm and the length is 30m variations of pressure with position can be extreme.

• Wave effects are significant. Normally if the time taken for a pressure wave to travel along the pipe is τ seconds and τ>tfin/100, wave effects can greatly modify system behavior. Since the lumped parameter assumption does not permit accurate representation of wave effects in pipes, geometrical variation of pressure and flow rate must be considered.

The best solution in these situations is to compromise and assume that variations of variables with respect to position within the length of the pipe are significant but variations across a section are not. In terms of equations we assume

P ≡ P(t,x)

where x is measured along the length of the pipe.

To fully describe the process of converting hydraulic pipe flow equations to a form suitable for classic ODE solvers is beyond the scope of this document. However, the process can be illustrated by taking a simpler example of heat conduction along an insulated rod with governing equation

where K depends on thermal conductivity, density and specific heat capacity and will be assumed constant. To convert this to an ODE we introduce a collection of node points along the rod and store a temperature Ti at each node. We require an expression for the derivatives of each temperature with respect to time. To do this we approximate the previous equation as follows:

Note that we are making simple approximations for derivatives in the spatial dimension but allowing the standard integrator to take care of the time dimension. This method is known as the method of lines and is a very useful one for converting partial differential equations to ODEs. It is capable of extension to 2 or even 3 spatial dimensions but in such cases in is normally better to use specialist partial differential equation software.

For hydraulic pipes the situation is more complex because partial differential equations for both pressure and flow rate must be employed. However, the method is still applicable and is used in the more complex Simcenter Amesim hydraulic pipe models. Normally interlaced meshes for pressure and flow rate and upwind differences are employed. The interested reader is referred to [5].