3.6 Integrating System

We found in the previous chapter that material balances on liquid surge vessels or gas drums often yielded models with the following form:

Equation 3.32

In the Laplace domain, this is

Equation 3.33

Consider an integrating process initially at steady state, with y(0) = 0.

Step Response

If a step input change of Du is made at t = 0,

and we find the time-domain value

Equation 3.34

That is, the output ramps with a constant slope of kDu.

Impulse Response

If an impulse input of magnitude P is made at t = 0,

then the output immediately changes to a new steady-state value of

Example 3.5: Tank-Height Problem

The mathematical model for a liquid surge tank is (see Example 1.3)

where h is the liquid height, A is the constant cross-sectional area of the tank, F₁ is the inlet flow rate, and F₂ is the outlet flow rate. Assume that the outlet flow rate remains constant at a steady-state value of F_2s. Defining the output and input in deviation variable form as

For a constant cross-sectional area of 10 m², the model is

Step Response

For a step input change of 0.25 m³/min, the output response is

which is shown in Figure 3-7. If the steady-state height is 2 meters, then the height as a function of time is

Impulse Response

For an impulse input of 1 m³, the output response is

which makes sense, because the cross-sectional area is 10 m³.