CHE 441
Answers to problem set #2
(1) Figure 1 shows the schematic of a process for treating residential sewage. In this simplified
process, sewage at a rate of 6000 gal/min is pumped into a well-mixed 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.
1) If 6000 gal/min of sewage enters and leaves the treatment facility, determine the two
volumetric flow rates Qtreated and Qrecycle.
2) If the recycle pump fails so that the flow rate through the process pump is reduced to 6000
gal/min, determine the time it takes before the sewage treated by the process is unsafe to release.
A minimum level of 0.1 lb/gal of bacteria in the aeration tank is necessary to assure safe levels of
sewage in the discharge.
Sewage
Qin
Air
CO2
Process pump
Qtreated
Aeration tank
Air
Settling
tank
Qout
Recycle pump
Qrecycle
Figure 1 A process for treating residential sewage.
1) Determine the two volumetric flow rates Qtreated and Qrecycle.
Qtreated = 8000 gal/min, Qrecycle = 2000 gal/min
2) The time it takes before the sewage treated by the process is unsafe to release = 764 min.
(2)1 The air pollution control equipment on a municipal waste incinerator includes a fabric filter
particle collector (known as a baghouse). The baghouse contains 424 cloth bags arranged in
parallel, that is 1/424 of the flow goes through each bag. The gas flow rate into and out of the
baghouse is 47 m3/s, and the concentration of particles entering the baghouse is 15 g/m3. In
normal operation the baghouse particulate discharge meets the regulatory limit of 24 mg/m3.
1
Davis M.L. and Masten S. J., Printiples of Environmental Engineering and Science, McGraw Hill, 2004, pg. 94
During preventive maintenance replacement of the bags, one bag is inadvertently not replaced, so
only 423 bags are in place.
Calculate the fraction of particulate matter removed and the efficiency of particulate removal
when all 424 bags are in place and the emissions comply with the regulatory requirements.
Estimate the mass emission rate when one of the bags is missing and recalculate the efficiency of
the baghouse. Assume the efficiency for each individual bag is the same as the overall efficiency
for the baghouse.
Solution
Efficiency of particulate removal = 99.84%
Efficiency of bag house with 1 bag missing = 99.605 %
(3) You want to separate all the coal particles having a diameter of 80 μm or larger from a slurry.
To do this, the slurry is pumped into the bottom of the large settling tank as shown. It flows
upward and flows over the top of the tank, where it is collected in a trough. If the solid coal has
SG = 1.4 and the total flow rate is 250 gpm, how big should the tank be?
Feed (Q)
Overflow
Overflow
A
Q/A
Vt
Underflow
Dt = 12.447 ft
(4) A gravity settling chamber consists of a horizontal rectangular duct 6 m long, 3.6 m wide, and
4 m high. The duct is used to trap sulfuric acid mist droplets entrained in an air stream. The
droplets settle out as the air passes through the duct and can be assumed to behave as rigid
spheres. If the air stream has a flow rate of 8.0 m3/s, what is the diameter of the largest particle
that will not be trapped by the duct using quiescent model? (Acid: ρ = 1.75 g/cm3, μ = 3 cP. Air:
ρ = 0.0075 g/cm3, μ = 0.02 cP.)
Solution
D = 0.0136 cm = 136 m
(5) Your assignment is to design a cyclone to remove at least 98% of particles from a flue gas
stream. The flue gas stream flow rate is 5000 m3/min. at 216oC and 1 atm. A particle rate of 3.16
lb/s enters with the flue gas. The flue gas composition and particle size distribution are listed in
Table 1. The particle density is 3400 kg/m3. The maximum pressure drop allowed is 4,000 Pa.
Use Prop (TK program) for viscosity and density of the flue gas.
Table 1 Flue gas composition and particle size distribution
Flue gas species Mole fraction
dp (μm)
wt %
CO2
0.076
19.00
2.55
N2
0.442
41.50
3.04
SO2
0.003
49.00
1.39
H2O
0.374
64.50
9.76
CH4
0.105
83.00
24.01
Total
1.000
120.50
24.40
181.50
24.93
212.00
9.91
Calculate the size of a standard cyclone using the following procedure:
1) Assume a cyclone diameter D = 4 m.
2) Evaluate other cyclone dimensions (W, H, De, Lb, and Lc) using Figure 2, cyclone type (3)
Q
3) Calculate the gas inlet velocity V =
, where Q is the gas volumetric flow rate.
WH
L
1
4) Calculate the number of revolutions (N) in the outer vortex,
N=
[ Lb + c ]
H
2
5) Calculate the “50% cut diameter”, dpc, defined by
1

 2
9 W
 , where  is the gas viscosity.
dpc = 



2

NV



p
g 


6) Calculate the collection efficiency, i, of each particle size
i =
1
1  (d pc / d pi ) 2
7) Calculate the overall efficiency, o, of the cyclone: o =
 
i i
, where i is the mass fraction
of particles in the ith size range. If o < 0.98, assume another diameter and repeat step 2-7.
8) Calculate the pressure drop across the cyclone
P = 8gV2
HW
De2
D(m) =5
efficiency = 0.988, Dp(N/m2) = 1726.58
>>
(6) A stream containing 7.5 kg/s water, 6.0 kg/s acrylonitrile, and 25 ppm NH3 is separated into
an acrylonitrile layer and an aqueous layer in a decanter. A negligible amount of water is
dissolved in the acrylonitrile layer, but 0.068 mass fraction acrylonitrile is found in the aqueous
layer. Both the aqueous layer and the acrylonitrile layer from the decanter contain ammonia, with
the concentration in the aqueous layer higher by a factor of 4.3.
(a) Determine the amount (kg/s) of acrylonitrile in the aqueous layer.
(b) If the aqueous layer contains 0.50 kg/s acrylonitrile, determine the concentration
(ppm) of NH3 in the acrylonitrile layer.
Solution
(a) Determine the amount (kg/s) of acrylonitrile in the aqueous layer.
0 kg wa ter
Acrylonitrile la ye r
7.5 kg wate r
Aq ue ous layer
0.068 = m a /(m a + 7.5)
a = 547 kg
(b) If the aqueous layer contains 0.50 kg/s acrylonitrile, determine the concentration (ppm) of
NH3 in the acrylonitrile layer.
Ca = 8.4586 ppm
(7) A well-mixed sewage lagoon (a shallow pond) is receiving 430 m3/d of sewage out of a sewer
pipe. The lagoon has a surface area of 105 m2 and a depth of 1.0 m. The pollutant concentration
in the raw sewage discharging into the lagoon is 180 mg/L. The organic matter (pollutant) in the
sewage degrades biologically in the lagoon according to first-order kinetics. The reaction rate
constant is 0.007 /d. Assuming no other water losses or gains, find the steady state concentration
of the pollutant in the lagoon effluent.
Solution
3
430 m /da y
5
180 mg/L
Surfa ce a rea = 10 m
1.0 m
2
3
430 m /da y
68.497 mg/L
(8) A dust-bearing gas stream enters a sedimentation chamber at a rate of 10 m3/s with a particle
loading of 0.25 kg/m3. The volume of the sedimentation chamber is 200 m3 with a height of 2 m.
If the terminal velocity of the particle is 0.05 m/s, determine the particle loading (kg/m3) of the
gas stream leaving the chamber using
a) Continuous quiescent sedimentation,
b) Complete-mix sedimentation,
c) Plug flow vertical mixing sedimentation.
Solution
Continuous quie sc ent
sedimenta tion
Ab = Volume/height = 200/2 = 100 m2
a) Continuous quiescent sedimentation,
0.125 kg/m3
b) Complete-mix sedimentation,
0.167 kg/m3
c) Plug flow vertical mixing sedimentation.
0.152 kg/m3
Comp lete -m ix
sedimenta tion
Plug flo w vertica l mixing
sedimenta tion