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# HW1 6130 Solution

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Homework I (due January 21, 2010)

1. Between what two circle numbers on the Porton graticule will the image of a 1-m particle fall when viewed with a 43 objective and a 15 eyepiece? Circle

number 10 has an image diameter of 9.9 mm. (10%)Sol>

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2. (a) A straight fiber has an aspect ratio of 5:1. If the fiber is viewedperpendicular to its axis, what is the maximum ratio of Ferets diameter toprojected area diameter? (b) For particles having a square projected crosssection, what is the ratio of Ferets diameter (averaged over all orientations) to

the projected area diameter? (20%)Sol>

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3. A particle has a smooth, circular cross section of diameter dp. The fractaldimension of the particle's perimeter is to be estimated by fitting a four-sided

and a six-sided regular inscribed polygon to the circle. Based only on thisinformation, what fractal dimension is estimated for this step size range. [Hint:

The length b of the side of a regular inscribed polygon with n sides is b = dp

sin(180o/n).] (15%)

Sol>

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4. Fumed silica produced by aerosol process is an agglomerate following a powerlaw (fractal-like). Professor Ulrich just produced some fumed silica following

an algorithm that generates the structure with Df= 2. The radius of gyration of

the agglomerate, Rg, equals 1 m; the primary particle size, dpo, equals 5 nm;

and the prefactor, A, equals 1.23. (a) What is the total number of primaryparticles in the agglomerate? (b) The agglomerate is heated and coalesces to

form a spherical particle. What is the radius of the particle? (c) Ifdp0 is doubled,what is the new value ofRg? Assume the same growth algorithm for large and

small primary particles; the number of particles remains the same. (d) An

agglomerate with Rg= 0.5 m, composed of the same primary particles (dpo = 5

nm) with the same Dfcollides and sticks to the original Rg= 1 m agglomerate.

What is the new value ofRg, assuming that Dfis preserved? (20%)

Sol>

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5. Measurement from a utility boiler shows the aerosol has a lognormal size

distribution with a geometric mean particle diameter of 0.3 m and g of 1.5. If

the number concentration is 106 #/cc, what is the mass concentration? Particles

are spherical with a density of 4.5 g/cm

3

. (15%)Sol>

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6. You are given the following data, obtained by sequential sieving of a sample ofgranite dust:

Sieve opening (m) Mass captured on Sieve (g)

200 4.0100 21.650 38.4

40 8.0Final Pan 8.0

Total 80.0

Determine the mass median diameter and geometric standard deviation of thisdistribution using log-probability graph paper. Use the appropriate conversion

equation to determine the count median diameter and mass mean diameter.(15%)

Sol>

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Ref> Log-Probability graph

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7. (a) An aerosol is mixed with radon gas, resulting in a surface coating ofradioactive radon decay products on the particles. The aerosol is then divided

into eight aerodynamic size groups, and the radioactivity of each size group ismeasured. How can this information be used to calculate the count median

diameter if the distribution is lognormal? All particles have the same densityand are geometrically similar. Assume the log-probability graph paper is

available. (b) For particles less than 0.05 m, light extinction efficiency is

proportional to dp4. If an aerosol is lognormally distributed with a CMD of 0.01

m and a GSD of 1.8, what is the diameter of average extinction efficiency?

(20%)

Sol>

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8. A powder company generates their powder using an aerosol technique. Themeasurement of the generated powder indicates that 16% of the total particles is

contributed by particles less than 0.3 m and 2.5% by particles larger than 2.4

m. Assuming the powder is lognormally distributed, calculate the dpg and g.

(10%)Sol>

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