# 978-0123869449 Chapter 15 Part 2

Document Type

Homework Help

Book Title

Radiative Heat Transfer 3rd Edition

Authors

Michael F. Modest

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CHAPTER 15 391

Tmax =2172 K,

392 RADIATIVE HEAT TRANSFER

15.12 Do Problem 15.10 using the Milne-Eddington approximation.

394 RADIATIVE HEAT TRANSFER

15.13 Do Problem 15.10, using the exponential-kernel approximation.

Note: The necessary exact integral relations have been given in Problem 14.3.

0.6−15

2−6

2 cosh 15

8+sinh 15

8

˙

Q′′′

396 RADIATIVE HEAT TRANSFER

15.14 Consider (a) two parallel plates, (b) two concentric spheres. The bottom/inner surface needs to dissipate a

heat ﬂux of 30 W/cm2and has a gray-diﬀuse emittance ǫ1=0.5. The top/outer surface is at T2=1000 K with

ǫ2=0.8. The medium in between the surfaces is gray and nonscattering (κ=0.1 cm−1), has a thickness of

L=5 cm and is at radiative equilibrium. Determine the temperature at the bottom/inner surface necessary to

dissipate the supplied heat for the two diﬀerent cases (the radius of the inner sphere is R1=5 cm), using the

Milne-Eddington approximation. Compare with the results of Problem 14.12.

CHAPTER 15 397

15.15 A material produces an amount of heat that is constant per unit volume, i.e., ˙

Q′′′ =const. This heat

production needs to be removed by thermal radiation. It is proposed to grind up the (fixed volume of) material

into small particles, which are to be suspended evenly between two cold plates of (identical) emissivity ǫ.

Since it is important to keep the overall temperature level in the particles as low as possible, should the

particles be ground as fine as possible, as large as possible, or does some optimum radius exist? What is the

optimum particle size, and what is the maximum temperature if this size is employed? You may assume

one-dimensional parallel plates with a constant volume fraction of particles, black particles with relatively

large size parameters, and you may use the Schuster-Schwarzschild approximation.

CHAPTER 15 399

15.16 Do Problem 15.15 using the Milne-Eddington (diﬀerential) approximation.

=fv

CHAPTER 15 401

15.18 Consider parallel, black plates, spaced 1 m apart, at constant temperatures T1and T2. Due to pressure

variations, the (gray) absorption coeﬃcient is equal to

κ=κ0+κ1z;κ0=0.01 cm−1;κ1=0.0002 cm−2,

where zis measured from Plate 1. The medium does not scatter radiation. Determine, for radiative equi-

librium, the nondimensional heat ﬂux Ψ = q/σ(T4

1−T4

2) by (a) the exact method, (b) the regular diﬀusion

approximation, (c) the diﬀusion approximation with jump boundary conditions, (d) the two-ﬂux method, (e)

the diﬀerential approximation, and (f) the kernel approximation.

(a) Exact: From Table 14.1 Ψ = 0.3900.

(b) Regular diﬀusion: From equation (15.16)

(c) Diﬀusion with jump condition: Again,

2q=σT4

τ=τL:−1

2q=σT4

2−σT4(τL)=σT4

1−C+3

4qτL.

Subtracting gives

τ=0 : 2q=4σT4

1−G=4σT4

1−C,

τ=τL:−2q=4σT4

2−G=4σT4

1−C+4qτL,

CHAPTER 15 403

15.19 An infinite, black, isothermal plate at 1000 K bounds a semi-infinite space filled with black spheres of uniform

radius a=100 µm. The particle number density is maximum adjacent to the surface, and decays exponentially

away from the surface according to

NT=N0e−Cz;N0=108m−3,C=πm−1.

(a) Determine the absorption and extinction coeﬃcients as functions of z.

(b) Determine the optical coordinate as a function of z. What is the total optical thickness of the semi-infinite

space?

(c) Assuming that radiative equilibrium prevails and using the Milne-Eddington approximation, set up the

boundary conditions and solve for heat ﬂux and temperature distribution (as a function of z).

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