_______________________
Last Name, First
CHE426: Problem set #3
1. An isothermal, first-order, irreversible reaction A  B with rate constant k takes place in
the liquid phase in a constant-volume reactor. Due to imperfect mixing, the reactor system
can be modeled as a two-tank system with back mixing as shown in the sketch below3.
F + FR
V1
CA1
F
CA0
FR
V2
CA2
F
CA2
Assuming F and FR are constant, write the differential equations describing the concentration
CA1 and CA1 as function of time. Do not solve the equations.
Solution
V1
dCA1
= FCA0 + FRCA2  (F + FR)CA1  V1kCA1
dt
V2
dCA2
= (F + FR)CA1  FCA2  FRCA2  V2kCA2
dt
2. If a forcing function f(t) has the Laplace transform
f(s) =
e s  e 2 s e 3s
1
+

s
s2
s
Plot f(t) for 0  t  5. Note: The unit step function u(t  2) is represented by Matlab as
heaviside(t-2).
Solution
3. Solve the following equation for y(t)
2

t
0
y( )d =
dy
+ 3y
dt
y(0) = 1
Solution
y(t) = 2e-2t  e-t
4. Express the function given in Fig. 3-4 in the t-domain and the s-domain
Figure 3-4
Solution
f=u(t-1)+(t-2)u(t-2)-(t-3)u(t-3)-u(t-3)-(t-5)u(t-5)
f(s) =
e s e2 s e3s e3s e5 s
 2  2 

s
s
s
s
s
5. Sketch the following functions:
a) f(t) = u(t)  2u(t  1) + u(t  3)
b) f(t) = 3tu(t)  3u(t  1)  u(t  2)
Solution
a) f(t) = u(t)  2u(t  1) + u(t  3)
b) f(t) = 3tu(t)  3u(t  1)  u(t  2)
6. Determine f(t) at t = 1.5 and at t = 3 for the following function:
f(t) = 0.5 u(t) − 0.5 u(t  1) + (t  3) u(t  2)
Solution
f (1.5) = 0 and f (3) = 0.
7. Find and sketch the solution to the following differential equations using Laplace
Transforms.
a) y′ + y =δ (t) , y(0) = 0
b) y′+ y =δ (t −1), y(0) = 0
Solution
a) y′ + y =δ (t) , y(0) = 0
y(s) =
1
 y(t) = e-t u(t)
s 1
b) y′+ y =δ (t −1), y(0) = 0
e s
y(s) =
 y(t) = e-(t-1) u(t  1)
s 1
8. For the following transforms, find lim f(t).
t 
1
s ( s  1) 2
1
(b) f(s) =
s ( s  1) 2
(a) f(s) =
Solution
(a) f(s) =
1
s ( s  1) 2
lim f(t) = lim sf(s) = lim
t 
s 0
(b) f(s) =
s 0
1
s ( s  1) 2
1
=1
( s  1) 2
Final value theorem does not apply, because of pole in the RHP (s=1).
f (t)→∞ as t →∞ , because of exp(t) terms in the solution.
9. Develop the dynamic model equations for the continuous stirred tank (CST) thermal mixer
shown in Figure 3.9.
FC
(F1)spe c
FT
F1
F2
T1
T2
TT
T
Figure 3.9 Schematic of a CST thermal mixing process.
The process parameters and variables are defined as:
F1: mass flow rate of stream 1 (initially 5 kg/s)
F2: mass flow rate of stream 2 (5 kg/s)
M: mass of liquid in the mixer (100 kg) = constant (perfect level control)
T1: temperature of stream 1 (25oC)
T2: temperature of stream 2 (75oC)
t: time (s)
v: the time constant for the flow controller on stream 1 (2 s)
Ts: the time constant for the temperature sensor on the product stream (6 s)
At time equal to 10 seconds, a step change in the special flow rate for stream 1 is made from
5 kg/s to 4 kg/s. Plot the product stream temperature and the sensor temperature from 0 to
100 s. Use Matlab to plot and label the graph with your name using the Title command.
Solution
M
dT
= F1T1 + F2T2  (F1 + F2)T
dt
Sensor model:
dTs
1
=
(T  Ts)
dt
 Ts