- Linear State Space Models
- Introduction to Laplace Transforms
- Transfer Functions
- First-Order Behavior
- Integrating System
- Second-Order Behavior
- Lead-Lag Behavior
- Poles and Zeros
- Processes with Dead Time
- Padé Approximation for Dead Time
- Converting State Space Models to Transfer Functions
- Matlab and Simulink
- Student Exercises
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:
In the Laplace domain, this is
Consider an integrating process initially at steady state, with y(0) = 0.
If a step input change of Du is made at t = 0,
and we find the time-domain value
That is, the output ramps with a constant slope of kDu.
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 F2s. Defining the output and input in deviation variable form as
For a constant cross-sectional area of 10 m2, the model is
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
Figure 3-7. Step response of a liquid surge tank. Deviation variables.
For an impulse input of 1 m3, the output response is
which makes sense, because the cross-sectional area is 10 m3.