Rigorous Treatments of Variable Time Delays

A thermal plant usually consists of various pieces of heat transfer apparatus connected together by pipes. Simulation of the transport delay introduced by these pipes is frequently required. When the fluid velocity through the pipes is constant, the transfer function of the piping lag is easily derived. When the velocity of the fluid is a function of time, however, the transport delay also becomes a function of time, and it is not possible to find a transfer function in the usual way. This has been a source of difficulty to many investigators in the past. A solution to this problem is offered in the present paper. Basically, the approach used is to change the independent variable from time t to integrated flow q. The transport delay in measured in units of q and is shown to be a fixed number in these units. Thus, in terms of the new variable a transfer function can be rigorously defined to which the usual rational function expansions apply. The resulting approximation to the transfer function is then realized in a straightforward manner using standard analog computer elements. In the process of the realization the independent variable is changed back from q to t and thus the original variable-delay characteristic is restored.