Physics 311 - Fall 97

Problem set 4

Due Thursday, Oct. 2

Here are two questions taken from How things work. The idea is that you should be able to give a qualitative answer, as expected in Bloomfield's course, as well as a more quantitative one, based on what you have learned so far in this course.

1. Use the concept of buoyancy to explain why air over a fire rises.

   a.

   Give a qualitative answer.

   b.

   Suppose you are burning wood at the rate of 5 kg per hour. At what rate are you producing energy (in watts) if the burning wood releases 3500 kcalories per kg?

   c.

   How much hot air goes up the chimney (in cubic meters per second) if the air is heated by 200 kelvin? In an open fireplace, most heat goes up the chimney.

2. Many grocery stores display frozen foods in bins that are open at the top. Why doesn't the warm air enter the bins and melt the food?

   a.

   Give a qualitative answer.

   b.

   Determine the heat diffusion constant in air, D, from dimensional analysis. Argue that D can depend on the speed of air molecules, v, and on the mean free path between collisions, l, but not on the mass of the molecules, m, (except that m affects v). This is enough to find D, up to a constant which we take to be 1, but is actually more like 1/3. (If you worry D can depend also on the mean time between collisions, , note that v = l/tau, up to a factor of order 1.)

   c.

   What is v? As you recall, the mean kinetic energy per molecule at temperature is (3/2)k<sub>B theta. From this, find v for the N and O molecules. Which molecule moves faster? What is the typical value of v at room temperature (averaged over molecular species, if you wish)?

   d.

   Next, what is l? As we discussed in class, it obeys the relation , where n is the number density of molecules and is the collision cross-section, about equal to , with d the diameter of a molecule. Give numbers for n and l at room temperature, assuming m.

   e.

   You can now obtain a formula for D in terms of and d. Check the dimensions. Compute the numerical value of D at room temperature and pressure and check with the value in Table 6.6 of the PQRG. Is it much different at the temperature of frozen food?

   f.

   Knowing D obtain a formula for the thermal conductivity of air. Check with the numerical value in the PQRG.

   g.

   Finally, compute the R value of a 10 cm layer of undisturbed cold air near the top of an open frozen food bin.

3. Gradient, divergence, and curl. A few simple (I hope) exercises to refresh your memory.

   a.

   If is a scalar function, show that .

   b.

   If is a vector function, show that .

   c.

   Suppose that in two dimensions

   

   Calculate . Express your result in polar coordinates .

   d.

   Plot the vector field which you calculated in (c).