his is a mechanical engineering HW

In class, Friday April 7, 2017

Name(s) (up to three students per homework set): PSU ID (abc123)

1.

2.

3.

M E 433 Spring Semester, 2017

Homework Set # 8

Professor J. M. Cimbala

For instructor or TA use only: Problem Score Points

1 25 2 30 3 25 4 20

Total: 100 1. (25 pts) Heather notices some flexible square tubing at a hobby shop, as sketched to the right, and comes up with an idea

for a simple particle-removal system when air pollution is bad outside yet we still need some fresh air supplied into our home. Based on material she learned in our class, she designs a coil of the square tubing (painted blue to be more attractive) as sketched below right. The idea is that outside (polluted) air gets sucked into the coil, particles are removed by inertial separation, and the cleaned air is supplied to the house. The inside dimension of the tubing is 2.00 inches, and the radius of her coil (from the centerline to the middle of the tube) is chosen to be 0.29 m. She does some simple fluid mechanics calculations and determines that a small fan at one end of the tubing should allow air to flow through the tubing at about 12.0 m/s. What she does not know is how long the coil has to be; in other words, how many loops the coil needs to have in order to effectively clean the air. Heather needs to do some analysis, and your job is to help her. Let’s analyze the case where the mass concentration of PM2.5 in the outside air is four times the 24- hour NAAQS (cj = 4*35 µg/m3 = 140 µg/m3). The particles are estimated to have an average density of 1600 kg/m3. Assume STP conditions for simplicity, and assume well-mixed settling since the flow is turbulent. Also neglect gravity and assume particles stick to the wall when on contact. (a) Consider 2.5 micron particles as the average particle diameter. How many loops are needed in the coil to reduce the

mass concentration of 2.5 micron particles by a factor of four (to get it down to the NAAQS for PM2.5)? Note that a loop is defined as 360o around, and there are about 4 loops in the picture.

(b) If Heather were to build this device with the number of loops calculated in Part (a), plot the grade efficiency curve E(Dp) for particles in the range 0.1 to 10 microns. Also calculate the “cut diameter” Dp,cut at which the removal grade efficiency is exactly 50%.

(c) If Heather’s room is 6 by 8 by 2.4 m, and indoor air quality experts like to have at least one room air exchange per hour, how many of these devices would she need to install? Is this feasible and would it help her lungs? Discuss.

2. (30 pts) Reconsider the Gaussian plume of Homework Set 5, Problem 4. At that time, we considered gaseous air

pollution. Here, suppose the air pollution from the stack consists of lead-containing particles of mean diameter 10 microns, and density 7000 kg/m3. The source strength is still 2.8 g/s. All else is the same as the previous problem (atmospheric conditions, stack height, buoyancy, wind speed, etc.). We consider here only ground absorbing conditions, since it is assumed that the particles stick to the ground on contact. For the air, assume STP air properties. (a) Calculate the terminal settling velocity Vt of these particles. This is critical to the rest of the problem, so make sure your

answer is between 0.02 and 0.03 m/s before continuing. (b) As a test case, calculate the centerline mass concentration of these particles on the ground at a distance of 2.5 km from

the stack. Based on the NAAQS limit for lead, and the percentage of lead in the air pollution particles, the safe limit is determined to be 1.4 µg/m3. [Note: As a check before continuing, your answer should lie between 2 and 3 µg/m3.]

(c) Plot an area on the ground (y vs. x) downwind of the plant in which the ground-level mass concentration of these lead- containing particles exceeds the safe limit. This will be considered the “hazardous area”, in which people living there are exposed to potentially harmful concentrations of lead in the air. Be sure to show your work (algebra) along with your final equation. As a check before continuing, calculate the y values at x = 2.5 km. You should get +/- values of y between 200 and 300 m (0.200 to 0.300 km). Then calculate y for various values of x, and generate and print out a plot of the hazardous area. For consistency, plot x (km) and y (km) and adjust the scales as necessary to show a nice plot. Briefly comment on your results.

Note: There is another page. →

3. (25 pts) In this problem, we perform some simplified calculations to predict the removal efficiency of a cyclone due to particles being flung against the outer wall. Sketched is the top view of our home-made cyclone (this is not a standard Lapple cyclone). Dusty air at 80oC and atmospheric pressure enters from the right at average speed 12.6 m/s, and then the air spins around as shown. The air contains particles between 1 and 300 microns, and the density of each individual particle is 2300 kg/m3. The middle radius rm of the particles is 0.325 m, and the inlet channel height is W = 0.139 m. We set r = rm = constant as a first-order approximation since W is small compared to rm, as discussed in class, and to avoid having to integrate. On average, the air spins around 2.0 times before being sucked out of the central pipe. Particles experience inertial separation as they circle around, and some of them hit the outer wall. We assume that any particles that hit the outer wall are removed from the flow (they either stick to the wall or fall down to the bottom of the cyclone). As discussed in class, many additional particles are removed as they try to make the sharp turn into the outlet pipe, but we are not considering those particles in our simplified calculations. (a) For a particle of diameter 10 microns, and assuming laminar settling, calculate the radial “settling” velocity vr (in m/s)

and predict the removal grade efficiency at this diameter (in %). Caution: You will need to iterate as in previous problems to calculate the settling velocity, but replace gravity with the centrifugal acceleration as discussed in class. Be sure to include the Cunningham correction factor and use the appropriate equation for drag coefficient. For large radial acceleration, Re can be > 5. The air is not at STP, so use the air table provided on the course website to find the appropriate air viscosity and density. Your answer should lie between 60% and 70%. Repeat for well-mixed flow. Your answer should lie between 40% and 50%.

(b) Once you are confident that your equations and solution are correct for the 10 micron particle, repeat for particles ranging from 1 to 300 microns, and plot the grade efficiency curve E(Dp) for both the laminar and well-mixed models, using as many diameters as needed to create a nice-looking plot. Also find two diameters: 1. The smallest diameter for which the laminar model predicts 100% removal. 2. The “cut diameter” Dp,cut at which the removal grade efficiency is exactly 50% for the well-mixed case.

(c) Which of the models (laminar or well-mixed) is most likely to be most accurate in this particular case? Why? Discuss our approximations. Would the actual cyclone have a better or worse removal efficiency than our simple predictions?

4. (20 pts) In class we discussed several equations and approximations for calculating the terminal settling velocity Vt of

particles in quiescent air. Consider unit density particles settling in quiescent air at STP conditions. On the same plot, draw three curves of Vt vs. Dp: • Best estimate: Use appropriate equation for CD, depending on Reynolds number; apply Cunningham correction factor,

and iterate as necessary to obtain the best possible estimate of drag coefficient and terminal settling velocity. • Stokes approximation with Cunningham correction: Use CD = 24/Re regardless of the value of Re; apply

Cunningham correction factor and iterate as necessary to obtain the Stokes flow estimate of terminal settling velocity. • Best estimate of CD, but no Cunningham correction: Use appropriate equation for CD, depending on Reynolds

number; ignore the Cunningham correction factor (set C = 1), and iterate as necessary to obtain the best estimate of terminal settling velocity when C is ignored.

For consistency, make your plot a log-log plot, with Dp ranging from 10-3 to 103 microns on the horizontal axis, and Vt ranging from 10-8 to 101 m/s on the vertical axis. For full credit, print out your plot, and make sure the axes and data sets are clearly labeled with a legend or other label. Finally, summarize and comment: Specifically, for what range of particle diameter is it okay to ignore the Cunningham correction factor? For what range of particle diameter is it okay to use the Stokes approximation?

U in W

rm

U out

Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.