Exercises
EP314/323
PROCESS DYNAMICS &
CONTROLS
Content:
Solution of ODE
Dynamic Behavior
Order of a Transfer Function Model
High Order Systems
Performance characteristics
Exercise 1: MODELING TOOLS
Q1. Transform the following:
a. sin 2t +
π
4
b. 2 exp
−t
3
Q2. Invert the following transforms:
3
3
b.
a.
c.
2
s+2
s + 4s− 5
2
s 5s+1
Q3. Find 𝑥(𝑠) for the following differential equation:
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Exercise 2: MODELING TOOLS
Q1. Find the final value of the function 𝑥 𝑡 :
a.
s 4 − 6s 2 + 9s − 8
x s =
s s − 2 s 3 + 2s 2 − s − 2
b.
𝑠+3
x s =
𝑠(𝑠 2 + 3𝑠 + 6)
Q2. Sketch a graph and find f(t) for the forcing input:
1 𝑒 −𝑠 𝑒 −3𝑠
a. f s =
+ 2 −
𝑠
𝑠
𝑠
1 − 2𝑒 −𝑠 + 𝑒 −2𝑠
b. f s =
𝑠2
Q3. Determine 𝑓(𝑡) at 𝑡 = 1.5 and at 𝑡 = 3 for the function:
f t = 0.5u t − 0.5u t − 1 + t − 3 u(t − 2)
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Exercise 3:
FIRST-ORDER SYSTEMS
Q1. Derive the transfer function 𝐻(𝑠)/𝑄(𝑠) for the liquidlevel system when the tank level operates about the
steady-state value of ℎ𝑠
(a) ℎ𝑠 = 1 𝑓𝑡
(b) ℎ𝑠 = 3 𝑓𝑡
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Exercise 3:
FIRST-ORDER SYSTEMS
Q2. A tank having a cross-sectional area of 2 𝑓𝑡 2 and a
linear resistance of 𝑅 = 1 𝑓𝑡/𝑐𝑓𝑚 is operating at steady
state with a flow rate of 1 𝑐𝑓𝑚. At time 𝑡 = 0, the flow varies
as shown in Fig.Q2.
(a) Determine 𝑄(𝑡) and
𝑄(𝑠) by combining simple
functions.
(b) Obtain an expression for
𝐻(𝑡).
(c) Determine ℎ(𝑡) at 𝑡 = 2
and 𝑡 → ∞.
Fig.Q2
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Exercise 3:
FIRST-ORDER SYSTEMS
Q3. In the two-tank mixing process shown in Fig. Q3, x
varies from 0 lb salt/ft3 to 1 lb salt/ft3 according to a step
function. Determine a time does the salt concentration in
tank 2 reach 0.6 lb salt/ft3.
Given, the holdup volume of each tank is 6 ft3 .
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Exercise 4:
SECOND-ORDER SYSTEMS
Q1. A step change, 𝑈(𝑠) of magnitude 4 is introduced into
a system having the transfer function:
𝑌 𝑠
10
= 2
𝑈(𝑠) 𝑠 + 1.6𝑠 + 4
Determine:
a. Percent overshoot
b. Rise time
c. Maximum value of 𝑌(𝑡)
d. Ultimate value of 𝑌(𝑡)
e. Period of oscillation
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Exercise 4:
SECOND-ORDER SYSTEMS
Q2. The two tanks transfer function is given by 𝐻2 (𝑠)/𝑄(𝑠).
The system is initially at steady state with 𝑞 = 10 𝑐𝑓𝑚.
The following data apply to the tanks: 𝐴1 = 1 𝑓𝑡 2 , 𝐴2 =
1.25 𝑓𝑡 2 , 𝑅1 = 1 𝑓𝑡/𝑐𝑓𝑚, and 𝑅2 = 0.8𝑓𝑡/𝑐𝑓𝑚.
𝐻2 𝑠
𝑅2
= 2
𝑄(𝑠)
𝑠 + (1 + 𝐴1 𝑅2 )𝑠 + 1
a. If the flow changes from 10 to 11 cfm according to a
step change, determine 𝐻2 (𝑡).
b. Determine 𝐻2 (1), 𝐻2 (3.5), and 𝐻2 (∞).
c. Determine the initial levels (actual levels) ℎ2 (0) in the
tanks.
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