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CHE426: Problem set #2 (Matlab solutions are not acceptable)
1. Solve the following equations by using Laplace transforms1.
dx
d 2x
+
+x=1
2
dt
dt
dx
d 2x
(b)
+2
+x=1
2
dt
dt
dx
d 2x
(c)
+3
+x=1
2
dt
dt
x(0) = x’(0) = 0 (Note: x’ =
(a)
dx
)
dt
x(0) = x’(0) = 0
x(0) = x’(0) = 0
Use Matlab to plot the behavior of these solutions on a single graph for 0  t  10. Use the
title command to label the graph with your name. What is the effect of the coefficient of
dx/dt?
2. Solve the following differential equations by Laplace transforms1.
d 4x
d 3x
+
= cos t
dt 4
dt 3
dq
d 2q
(b)
+
= t2 + 2t
2
dt
dt
x(0) = x’(0) = x’’’(0) = 0, x’’(0) = 1.
(a)
q(0) = 4, q’(0) = 2
3. Invert the following transforms1.
3s
(a) 2
,
( s  1)( s 2  4)
1
(b)
,
2
s ( s  2s  5)
3s3  s 2  3s  2
(c)
s 2 ( s  1)2
4.2 Two consecutive, first order reactions take place in a perfectly mixed, isothermal
continuous reactor (CSTR).
k2
k1
A
B
C
Volumetric flow rates (F) and density are constant. The reactor operates at steady state. The
inlet stream to the reactor contains only A with CA,in = 10 kmol/m3. If k1 = 2 min-1, k2 = 3
min-1, and F = 0.1 m3/min, find the tank volume that maximized the concentration of
component B in the product stream. Show all your work.
5.2 A tank containing 3.8 m3 of 20% (by volume) NaOH solution is to be purged by adding
pure water at a rate of 4.5 m3/h. If the solution leaves the tank at a rate of 4.5 m3/h, determine
the time necessary to purge 90% of the NaOH by mass from the tank. Assume perfect
mixing. Specific gravity of pure NaOH is 1.22.
6. In tank A are 200 gal of brine containing 80 lbs of dissolved salts. Solution from this tank
runs at a rate of 4 GPM into a second tank, B, which contains initially 100 gal of brine with a
concentration of 0.2 lb/gal of solution. Similarly, solution runs from tank B at the same rate.
Determine the concentration of salt in tank B after 30 minutes.
pure water
4GPM
4GPM
A
7.
4GPM
B
B
C
A
5
=
+
+
s
s  r1
s  r2
s  s  4 s  3
2
In this equation, r1 < r2. Determine B and C.
8. Given f(t) = 3  4(t  1)U(t  1) + 4(t  3)U(t  3), determine f(2) and f(5).
9. Find the Laplace transform of e-2tcos 3t

10. Find the inverse of F (s) =
s3
s  6 s  18
2
11. Figure 6 shows the schematic of a process for treating residential sewage. In this
simplified process, sewage (without bacteria) at a rate of 6000 gal/min is pumped into a wellmixed aeration tank where the concentration of bacteria CB,aration is maintained at 0.25 lb/gal.
The treated sewage is then pumped to a settling tank where the bacterial is separated and
recycled back to the aeration tank. The treated sewage leaving the settling tank has no
bacteria in it while the recycle sewage contains a bacterial concentration of 1.0 lb/gal. Both
the aeration and the settling tanks have the same volume of 5106 gallons. You can assume
the liquid (sewage) density remains constant throughout the process and neglect the mass
loss due to the generation of CO2 leaving the aeration tank.
Air
CO2
Sewage
Qin
Process pump
Qtreated
Aeration tank
Air
Settling
tank
Qout
Recycle pump
Qrecycle
Figure 6 A process for treating residential sewage.
If 6000 gal/min of sewage enters and leaves the treatment facility, determine the two
volumetric flow rates Qtreated and Qrecycle.
References
1. D.R. Coughanowr and S. LeBlanc, Process Systems Analysis and Control, McGraw-Hill,
3nd edition, 2008.
2. Mass Transfer by Hines and Maddox.
3. Process Modeling, Simulation, and Control for Chemical Engineers by Luyben.