AME 517, Prof. P. Ronney

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_bh ≈ 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 - don’t 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)

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

· Radiative Properties of a Particle Cloud

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

· 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

Approximate Solution Methods (Ch. 14)

· Optically Thin Approximation – appropriate for very little absorption, t_L << 1, radiation acts like volumetric source or sink of constant magnitude throughout the volume

· Optically Thick Approximation (Diffusion Approximation) – appropriate for very strong absorption, t_L >> 1, radiation acts like conduction

· 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 4π steradians into 2 parts, an intensity I⁺ over the 2π steradians in the +z direction, an intensity I^- over the 2π steradians in the -z direction.

· 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.

Final exam from a previous year

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 PDR™. When a very small amount of PDR™ 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 PDR™, and the combustion temperature is unchanged. How would each of the following be changed with PDR™ 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 (Q’’’ = 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/m². The back side of this plate is insulated, so that all 1000 W/m² 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 didn’t get a reasonable answer to c, make up a reasonable guess for T₂ 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 don’t need to re-do the analysis.)