Perturbation of the Inlet Flow Rate and Control

Suppose that the inlet flow rate was mostly a constant but from time to time it suffered an upset. The upset would either increase the inlet flow or decrease it. If this were to occur, then the level in the tank would either increase or decrease, unless there was some attempt made to change the outlet flow rate. A simple control algorithm would be one in which the exit flow rate is adjusted automatically when there is an upset, either above or below the design flow rate. To do this analysis we need to specify the upset and the system s response to it. 

Let the upset be some additional flow rate over or below the design value. Stated in simple mathematical terms  

For this to be dimensionally consistent, it is clear that the proportionality constant must have dimensions of area per time so that its product with h — h(t) is in dimensions of volume per time. During the upset we are discussing here, the system responds transiently, that is, it goes away from steady state. The governing equation is  

This equation is the one that must be integrated and to do so we must know how the upset behaves in time. Before the upset from t = 0 until some time just below tl, qp is zero and h = h(t)  

The upset begins at time tl, and it instantaneously rises to a value of qp. The upset stays at the level qp from tl until a time just less than t2. At time t2 it falls instantaneously back to zero. This kind of function is called a UnitStep. For the sake of making a graph of this type of disturbance, we plot the UnitStep function and 1 — UnitStep function using the parameters 1 and 6 in order to have a pulse of unit height that is five time-units wide  

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