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Unformatted text preview: HW1 461 1/3 Student Name: ________________________________________ HW #1 CHE 461 Due Mon. April 5, 2010 1.(8) Given a truncated conical tank
where the flow out F0 is a function of the height of liquid:
when the units of h are ≡ ft of liquid in tank
Assume density is constant and R = tank radius at bottom of tank =
F0 (t ) = 0.2 h(t ) π and H = height of cone shape = 3 ft 3 2.56 Liquid volume in the tank ( r = radius of upper circular surface of liquid in ft. ): Vliquid (t ) = π R2 H
3 − π r2 ( H − h)
3 with hmax = 2.56 ft 1. Use the relationship between r(t) and h(t) to show/derive the following relationship: 6π π Vliquid (t ) = α h(t ) + β h (t ) + γ h (t )
2 where α = π R 2 , β = 3 −π R 2
, and γ =
3H 2 2. Write a dynamic mass balance and linearize the ODE about the nominal steady state where F i = 0.2 ft3/min is
the steady-state input which gives a steady-state output: h = ________ ft.
3. Use the Laplace transform to obtain the transfer function model G (s) = h '( s )
, where h '( s ) = Laplace transform of ⎡ h (t ) − h ⎤
F 'i ( s ) What is the gain of G(s) : _________________________________ (Note: give value and units)
What is the time constant of G(s) : _________________________________ (value and units)
Show on a sketch where the pole of G(s) is in the complex plane (show numerical value of location) 4. Using the nonlinear model, what is the maximum steady-state inlet flow rate Fi that can be used without
overflowing ( h > hmax) the tank ?
What is the maximum steady-state Fi using your transfer function (linear) model ?
_______________________________ ft3/min hw1 461 2/3 2.(6) The figure below was made by an approximate simulation of the unit step setpoint change response for each
4e −0.05 s
of four "real" PID controllers with the following process: G ( s ) =
( s + 1)( 0.5s + 1)
The Base Case controller had a gain of 1.0, an integral time constant of 0.5 time units and a derivative time constant
of 0.2 time units. Copy both the MATLAB files ( hw1.mdl, sfunykl2.p) from the CHE461 web page
( http://classes.engr.oregonstate.edu/cbee/spring2010/che461 ) , modify the controller blocks to be the four cases
below to make a figure similar to the one below. Draw an arrow to each curve on your MATLAB Figure and label
the arrow with the letter (A,B,C,D) which corresponds to the controller description:
A - Base Case PID - default settings in hw1.mdl
B - Same as A except “controller gain” is cut in half
C - Same as A except “integral time constant” is doubled (integral "action" is halved)
D - Same as A except “derivative time constant” is doubled (derivative "action" is doubled) 2a) Print and hand-in the plot you obtain using YOUR hw1.mdl (the Simulation Parameters and plotting block are
already set for you, but you need to change the controller parameters as described above).
2b) Discuss the effects the change in each of the three PID controller parameters (Cases B,C and D) had on the
response curve before the change (A = Base Case). y ' , Process Output Variable Series=Real PIDs, alpha=0.05 (Eqn 8-15, pg 193 SEM2)
0 0 1 2 3
Time 4 5 6 HW1 461 3/3 3.(6) a) Calculate the closed-loop transfer function for Example 1 in the "Fundamentals of Simple Feedback
Control" handout when Gc = 2.5 instead of 1.25 in the handout. new (Y/Ysp)closed-loop , servo =
Complete the table and then compare (use complete sentences) the values of the parameters of the your closed-loop
transfer function with the original ones.
Gain Gain of Time constant of
(Y/Ysp)closed-loop , servo (Y/Ysp)closed-loop , servo 1.25 (old) 0.8 0.8 2.50 (new) b) Calculate the closed-loop transfer function for Example 2 in the "Fundamentals of Simple Feedback Control"
handout when Gc = 2.5 instead of 1.25 in the handout. new (Y/D)closed-loop , regulatory = Complete the table and then compare (use complete sentences) the values of the parameters of the closed-loop
transfer function with the original ones. Also sketch (in the complex plane) the locations of the poles and zero of the
two transfer functions of this part (b). Then write a statement comparing the locations of the (old) zero+poles
versus the locations of the (new) zero+poles to describe the effect of increasing the controller gain on closed-loop
Gain Gain of
(Y/D)closed-loop Slower time
(Y/D)closed-loop Faster time
(Y/D)closed-loop τz for zero of
(Y/D)closed-loop 1.25 (old) 0.2 2.0 0.8 4 2.50 (new) ...
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This note was uploaded on 04/17/2010 for the course CHE 461 taught by Professor Staff during the Winter '08 term at Oregon State.
- Winter '08