**Final Exam Study Guide**

**December 10, 2009**

__Format of the exam__

The final exam will be open book. You may use any reference materials you
want, but no laptop computers are allowed. The exam will have both analytical/numerical problems and
short-answer questions, covering conceptual knowledge.

__Material covered__

There will be a question on surface-to-surface
radiation, but the exam will emphasize the material on radiation in
participating media (Chapters 9, 10, 11, 13 and 14). This material includes:

** **

Equation of Radiative
Transfer in Participating Media (Ch. 9)

- Attenuation by Absorption and Scattering
(simple)
- Augmentation by Emission and Scattering (emission
is simple but strange, scattering is hard because you have to integrate
over all incoming directions)
- Radiative transfer equation (conservation of
energy along a ray, including emission, scattering and absorption)
- Incident Radiation Function (G) (sum of all
radiation incident on a point)
- Radiative Heat Flux (
**q**) (sum of all radiation crossing a particular plane) - Divergence of the Radiative Heat Flux
(conservation of radiant energy at a point, integrated over all incoming
and outgoing rays)

Radiative Properties of
Molecular Gases (Ch. 10)

- Atomic and Molecular Spectra – basic
physics of radiation, quantization of energy states, radiation bands
- Radiation from a single line – Doppler,
collision and combined (Voight) broadening
- Narrow Band Models – net effect of many
lines over a narrow band whose width in wavenumber space is small enough
that I
_{b}_{h}Å constant; different narrow-band models make different assumptions about the distribution of line strengths and spacings, but all assume the same width for each line. - Wide Band Models - donÕt bother looking at individual lines, look at net
effect of an entire absorption band with a presumed k
_{h}(h) (e.g. exponential) - Total Emissivity and Mean Absorption
Coefficient (look at net effect of all bands for a particular molecule.)

Radiative Properties of
Particulate Media (Ch. 11)

á Absorption and Scattering from a Single Sphere
(Mie scattering, very complicated)

á Radiative Properties of a Particle Cloud

á Radiative Properties of Small Spheres (Rayleigh
Scattering, simple case of Mie scattering where particle size is much smaller
than the wavelength of radiation)

á Radiative Properties of Large Spheres (simple case
of Mie scattering where particle size is much larger than the wavelength of
radiation, just treat the particle like you would a macroscopic surface as in
Chapters 5 and 6)

Exact Solutions For
One-Dimensional Gray Media (Ch. 13)

- General Formulation for a Plane-Parallel
Medium (leads to exponential integrals)
- Radiative Equilibrium of a Nonscattering
Medium (leads to integral equation (Eq. 13.41))
- Radiative Equilibrium of a Scattering Medium
(unless the scattering is isotropic, much more complicated than
non-scattering medium because of the direction-dependence on scattering)
- Plane Medium with Specified Temperature Field

á Optically Thin Approximation – appropriate
for very little absorption, t_{L}
<< 1, radiation acts like volumetric source or sink of constant magnitude
throughout the volume

á Optically Thick Approximation (Diffusion
Approximation) – appropriate for very strong absorption, t_{L} >> 1, radiation acts like
conduction

á Schuster-Schwarzschild Approximation (Two-flux
method) – approximate method when t_{L} not close to optically thin or thick limit – break
up the intensity I(W) over all 4¹
steradians into 2 parts, an intensity I^{+} over the 2¹
steradians in the +z direction, an intensity I^{-} over the 2¹
steradians in the -z direction.

á Milne-Eddington Approximation (Moment Method) –
another approximate method when t_{L} not close to optically thin or thick limit – take
0^{th} and 1^{st} moment of the radiative transfer equation,
close the system at the 2^{nd} order moment; results very similar to
two-flux method.

1. (30 points, 3 points each part) Answer the following questions about
the relation between the blackbody emissive powers and radiative fluxes for N
diffusely or specularly reflecting surfaces as given in the text (NO CREDIT FOR
YES OR NO ANSWERS WITHOUT EXPLANATION):

a)
Does radiant energy
need to be conserved?

b)
Can there be
conductive or convective heat loss/gain to/from one or more of the surfaces?

c)
Can there be a
radiatively participating gas between the surfaces?

d)
Can there be a radiatively
non-participating gas between the surfaces?

e)
Do the surfaces have
to be flat?

f)
Do the surfaces have
to be opaque (transmittance = 0)?

g)
Do the surfaces have
to be black (e = 1 for all
wavelengths), gray (e = same for all
wavelengths, not necessarily 1) or can e vary with wavelength?

h)
Is there a limit on
N?

i)
Do all the surfaces
have to be able to see each other, or can some be "hidden" from
others?

j)
Does the temperature
of each surface have to be uniform?

2. (30 points, 7.5 points each part) Ronney Gas and Oil Co. claims to have
invented a new fuel additive, called PDRª. When a very small amount of PDRª is added to a fuel, *the
width of every emission-absorption line in the combustion products is doubled*, that is, b_{c} ¨ 2b_{c}, b_{L} ¨ 2b_{L} (assume only collision broadening
is important, neglect Doppler broadening.) The line strengths (S) are unchanged by PDRª, and the
combustion temperature is unchanged.
How would each of the following be changed with PDRª fuel additive? In particular, would each of the following
increase, decrease or stay the same, and if it did change would it increase or
decrease by more than, less than or almost exactly a factor of 2?

a) The total radiant emission from the combustion
products in a very small volume of gas (optically thin limit)

b) The total radiant emission from the combustion
products in a very large volume of gas (optically thick limit)

c) The total radiant emission from the combustion
products (not in the optically thin or thick limit, somewhere in between the
two.)

d) The Planck mean absorption coefficient of the
combustion products

3. (40 points) A gray gas with absorption coefficient k = 100 m^{-1} fills the space between two
black parallel plates 0.01 m apart.
The gas does not scatter radiation, and there is no heat generation
within the gas (QÕÕÕ = 0).
Conduction and convection heat transfer can be neglected. Plate 1 has a temperature of 300K. A heater is imbedded in plate 2; the
heat flux from this heater is 1000 W/m^{2}. The back side of this plate is insulated, so that all 1000
W/m^{2} must be transferred through the gas to Plate 1.

a) (5) What would
the temperature of plate 2 be if there were no gas between the plates?

b) (5) With the gas between
the plates, would the optically thin or optically thick approximation (or
neither) be valid for this problem?

c)
(15) What is the
temperature of plate 2 with the gas between the plates? Use the two-flux method.

d)
(10) For the
temperature computed in part c), what would the heat transfer between the
plates be if you used the exact solution from chapter 13? (If you didnÕt get a reasonable answer
to c, make up a reasonable guess for T_{2} and proceed.)

e)
(5) If half of the
gas were removed, so that now k = 50 m^{-1}, and an isotropically scattering medium with s_{s} = 50 m^{-1} were added, for
the conditions of part c) would the temperature of plate 2 increase, decrease
or remain the same? (You can
explain physically without using equations, you donÕt need to re-do the
analysis.)