Name(s) (up to three students per homework set):
Due:
In class, Friday
March 24, 2017
PSU ID (abc123)
1.
2.
3.
M E 433
Spring Semester, 2017
Homework Set # 7
Professor J. M. Cimbala
For instructor or TA use only:
Problem
Score
Points
1
15
2
25
3
20
4
15
5
25
Total:
100
1.
(15 pts) A nice round-number estimate of all the excess carbon dioxide pumped into the atmosphere in the world due to
burning fossil fuels is 10 Gtonne per year. This is also approximately the amount of CO2 added to the atmosphere each year,
since fossil fuels come from the ground and are no longer part of the carbon cycle, unlike breathing, burning wood, etc.
Suppose we were able to somehow capture all that extra CO2 and convert it into coke (not the soft drink, and not the drug,
but the other coke, which is pretty much pure carbon).
(a) What is coke, and how is it made? What is its density?
(b) Coke is porous, and if we were able to grind it into a powder, we could almost double its density, which would be the
most useful for storage. For consistency in the following calculations, assume the density of coke powder is 1540 kg/m3.
If we converted all of the carbon from the excess carbon dioxide into powdered coke and
put it in a big cubical box, how big (length, width, and height) would the box need to be?
Note: Be careful; you will need to consider only the portion of the CO2 that is carbon, and
assume that all of the carbon is turned into coke.
(c) Repeat for a conical stockpile of powdered coke, assuming that the stockpile is a cone
with an angle of repose of 40o. Calculate the radius and height of the required conical pile.
(d) Is this feasible? What else might you do instead of piling it up?
2.
(25 pts) Reconsider Problem 4 of HW 6. The terrorist reasons that if he releases the poisonous gas at a higher elevation, it
would kill more people. In other words, the hazardous zone would increase in size (area) as compared to the case where the
same gas is released at ground level. In this problem, we will see if he is correct or not. Note: We assume that the crowd
density (number of persons per square meter on the ground) is constant regardless of where the hazardous zone occurs.
(a) Create a plot of the hazardous zone if everything is the same as the previous problem except H = 2 m. Plot both areas
(last week, H = 0, and this week, H = 2 m) on the same plot and compare.
(b) Discuss. Is the terrorist correct? Why or why not?
3.
(20 pts) Particles of density 2010 kg/m3 are being studied by a manufacturer of air pollution control systems (APCSs).
The particles are of various (unknown) shapes and diameters, so the engineers devise an optical technique to measure the
terminal settling velocity Vt, and then plan to calculate the aerodynamic equivalent diameter to use in their APCS
performance simulations. The tests are at STP conditions.
(a) When Vt = 3.76 × 10-4 m/s, calculate the aerodynamic equivalent diameter Dae in microns. Note: You may assume Stokes
flow, but verify that the Reynolds number is small enough for this approximation to be valid. [If not, you will need to
use one of the other curve fits for CD as a function of Re.] Also, you will need to iterate, since Cunningham correction
factor C depends on Dae, which in turn depends on C. [You may do the calculations by hand, but I recommend that you
use Excel or some other computer program to avoid calculation errors and to reduce the monotony of iterating.]
(b) Let us define Dse = spherical equivalent diameter as the diameter of a sphere of actual particle density (here 2010
kg/m3) that would settle at the same (measured) terminal settling speed as the real (non-spherical) particle of the same
density. Calculate Dse in microns for the measured value of Vt given previously.
Note: There is another page. →
4.
(15 pts) As discussed in class, every time we breathe we inhale particles into our lungs. Most of the coarse particles get
trapped in the mucus lining of the bronchial tubes, and are then expelled up to our throats, where the particles are
subsequently swallowed. In this problem, we want to estimate how many particles we swallow in this fashion.
(a) Look up the NAAQS limit for 24-hour exposure of PM10 (coarse particles) in units of µg/m3. State College, PA has
pretty clean air most of the time. Suppose that on a typical day, the PM10 concentration is about 15% of the NAAQS
limit for 24-hour exposure. Here is some more information:
•
•
•
The mean density of the particles is 1000 kg/m3 (people in the air pollution business call this “unit density”).
The mean particle diameter based on mass is measured to be Dp,am (mass) = 4.0 microns.
The collection efficiency of particles that are inhaled into the lungs is 90%. In other words, 90% of the particles are
trapped by the mucus and cilia, and are then swallowed.
• The typical person breathes in about 23,000 L = 23 m3 of air per day.
• Assume a swallowing rate of one swallow every 30 seconds.
Calculate the number concentration of the particles in the inhaled air in units of particles/m3.
(b) Estimate the number of particles that are trapped and swallowed per swallow (give your answer in number of particles
per swallow).
5.
(25 pts) A horizontal elutriator is a simple device that is sometimes used at the inlet to an APCS to remove some of the
larger diameter particles from the air. It consists of parallel horizontal plates, as sketched. Dirty air enters from the left at low
air speed. As the air moves along, particles fall and settle on the plates. We assume that when the particles hit the plate, they
stick there and remain on the plate – they are removed from the air. After some time, the device may get clogged, so it needs
to be washed out. However, there are no moving parts, it does not require electricity, and it is simple and effective at
removing particles from the air. Heavier and/or larger particles settle quickly, while lighter and/or smaller particles take
longer to settle. Therefore we expect the removal efficiency of the device to depend on particle diameter and particle density.
For simplicity, consider STP conditions with
the incoming dirty air consisting of a wellmixed polydisperse aerosol of spherical ash
H
particles of density ρp = 2600 kg/m3 with
diameters ranging from 0.1 to 10 microns.
Air flows at an average speed of U = 0.145
U
L
m/s through the channels, the flow is laminar
in the channels, the length of the horizontal elutriator is 0.845 m and the channel height between plates is H = 1.05 mm. For
best accuracy in your calculations, be sure to include the Cunningham correction factor and the appropriate equation for drag
coefficient (iteration may be required).
(a) Calculate the critical diameter, defined as the minimum particle diameter at which we predict 100% removal through
the elutriator. In other words, particles greater in size than this critical diameter will all settle out and be removed,
whereas particles smaller than this critical diameter will be only partially removed (efficiency less than 100%). You may
have to do some iteration. Your answer should lie between 1 and 2 microns if you do the calculations correctly. Give
your answer to at least 3 significant digits.
(b) Generate a plot of grade efficiency E(Dp), defined as the removal efficiency of aerosol particles as a function of particle
diameter. Note that each particle diameter generates a different grade efficiency since larger particles fall faster than
smaller particles. In your calculations, if E > 100%, set E to 100% since it is impossible to have a removal efficiency
greater than 100%. For consistency, draw your plot with a log axis for Dp (horizontal axis) and a linear vertical axis for
E from 0 to 100%. Use enough data points to make a nice-looking grade efficiency curve.
(c) Briefly comment about this device – when might it be useful, and what are its limitations?