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4. The Transfer function: 4. The Transfer function What is Transfer Function?
; an algebraic expression for the dynamic relation between the input and output of the process model. It is defined so as to be independent of the initial conditions and of the particular choice of forcing function.
Properties; It can be derived only for a linear differential equation model because Laplace transform can be applied only to linear equations. If model is non-linear, then it must be linearized first.
Advantage; It is easy to interpret and use in calculating output responses for particular input changes.
4.1 Development of Transfer Functions: 4.1 Development of Transfer Functions Example ; Stirred-tank heating system
Energy balance equation. Figure 4.1. Continuous stirred-tank heater.
Slide4: Assumption ; constant liquid holdup and constant inflow(w is constant), a linear model result.
If the process is at steady-state, .
Subtract (4-3) from (4.1).
Define some important new variables(Deviation variables).
By substituting deviation variables for variables, the transfer functions are not related to initial conditions.
Slide5:
Let .
Apply Laplace Transform.
If .
If .
4.2 Property of Transfer Functions: 4.2 Property of Transfer Functions 1. The steady-state output change for a sustained change in input can be calculated directly; we can easily obtain steady state gain K. - Final value theorem.
Steady State gain; the ratio of the output variable change to an input variable change when the input is adjusted to a new value and held there, thus allowing the process to reach a new steady state.
2. The transfer function has the order of the denominator polynomial (in s) as same as the order of the equivalent differential equation.
General n-th order differential equation.
Slide7: Apply Laplace transform to (4.15).
For step input function, obtain the gain K.
Physical Realizability. ( )
- if
for step input change,( ), which has infinite size, should be infinite size at t=0. Therefore, assumption is contradictive
(unrealizable), and .
Slide8: 3. Additive property of the transfer function in parallel processes. 4. Multiplicative property of the transfer function in series processes. Figure 4.3. Block diagram of Multiplicative(series) transfer function model.
Slide9: Example; Two liquid surge tanks in series. Figure 4.4. Schematic diagram of two liquid surge tanks in series. Assumption; The outflow from each tank is linearly related to the height of the liquid(head) in that tank. The two tanks have different cross-sectional areas and
, and that the outflow valve resistance are fixed at and .
Find the transfer function relating changes in outflow from the second tank, , to changes in inflow to the first tank, . Show how this transfer function is related to the individual transfer functions,
and .
and denote the deviations in Tank 1 and Tank 2 levels, respectively.
Slide10: For tank 1.
Putting (4-21) into deviation variable form gives
Apply Laplace transform and rearrange to obtain transfer function.
Similarly, other transfer functions are obtained as follows.
Slide11: The desired transfer function relating the outflow from Tank2 to the inflow to Tank 1 can be derived by forming the product of (4-23) through (4-26).
Figure 4.5. Input-output model for two liquid surge tanks in series.
4.3 Linearization of Non-linear Models: 4.3 Linearization of Non-linear Models Necessity
There is a wide variety of processes for which the dynamic behavior on the process variables in a non-linear fashion.
Laplace transform cannot be applied!
How? Taylor series expansion.
For around the nominal steady-state operating point . Where y is the output and x is input.
At steady state, .
Put (4-29) into deviation variable form.
Slide13: Example; A liquid-level storage system.
Assume that the valve discharge rate is related to the square root of liquid level; .
Derive an approximate dynamic model for this process.
Material balance.
Deviation variables.
Linearize about the steady-state
conditions .
Slide14: Useful results of Taylor series expansion.
(1) , where n is real.
Ex)
(2)
(3)
(4)