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CHAPTER 17
17.1 Consider a gray, isothermal and isotropically-scattering medium contained between large, isothermal, gray
plates at temperatures T1and T2, and emittances ǫ1and ǫ2, respectively. Determine the radiative flux between
the plates using the S2-approximation.
dq
dτ=(1 −ω)(4πIb−G),
µ, τ =τL: 4J2=G−q
µ.
For the symmetric S2-approximation (µ=0.57735) this is identical to the P1-approximation with Mark’s
boundary conditions, while for the nonsymmetric S2-approximation (µ=0.5000) it is identical to the two-flux
approximation. Eliminating G, we find
G=4σT4
m−1
1−ω
dq
dτ
with integration constants C1and C2to be determined from the boundary conditions.
429
430 RADIATIVE HEAT TRANSFER
17.2 Consider a large, isothermal (temperature Tw), gray and diffuse (emittance ǫ) wall adjacent to a semi-infinite
gray absorbing/emitting and linear-anisotropically scattering medium. The medium is isothermal (tem-
perature Tm). Determine the radiative flux as a function of distance away from the plate using the S2-
approximation.
Solution
The S2-approximation is given by equations (17.18) and (17.19) with i=1. Alternatively, following Example
17.1, I+and I−may be eliminated in favor of Gand q, leading to
dq
m−G),
ary conditions, while for the nonsymmetric S2-approximation (µ=0.5000) it is identical to the two-flux
approximation. Eliminating G, we find
q′′ =−(1 −ω)G′=(1 −ω)(1/µ2−A1ω)
| {z }
=γ2
q,
q=C1e−γτ +C2eγτ.
aC1=4σ(T4
w−T4
m)−γC1
1−ω,
w−T4
m)
22
ǫ−1+q4−A1ω
1−ω
For ǫ=1 and ω=0 the heat flux goes to q/σ(T4
w−T4
m)=2µ=1.1547 (Symmetric) or 1.000 (Nonsymmetric) as
compared to 1.000 for the exact solution.
CHAPTER 17 431
17.3 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=κ1+κ1x;κ0=0.01 cm−1;κ1=0.0002 cm−2,
where xis measured from plate 1. The medium does not scatter radiation. Determine, for radiative equilib-
rium, the nondimensional heat flux Ψ = q/σ(T4
1−T4
2) by the exact method, and the S2-approximation.
432 RADIATIVE HEAT TRANSFER
17.4 Black spherical particles of 100 µm radius are suspended between two cold and black parallel plates 1 m apart.
The particles produce heat at a rate of π/10 W/particle, which must be removed by thermal radiation. The
number of particles between the plates is given by
NT(z)=N0+ ∆Nz/L,0<z<L;N0= ∆N=212 particles/cm3.
(a) Determine the local absorption coefficient and the local heat production rate; introduce an optical
coordinate and determine the optical thickness of the entire gap.
(b) If the S2-approximation is to be employed, what are the relevant equations and boundary conditions
governing the heat transfer?
(c) What are the heat flux rates at the top and bottom surfaces? What is the entire amount of energy released
by the particles? What is the maximum particle temperature?
Solution
Assuming that the temperature of the particles will be such that all important radiation is for wavelengths
<10 µm we have
0
0
0
=6.660 ξ1+1
2ξ,
and
µq+G=0, τ =τL:−1
µq+G=0or q(1
where the last condition follows from the symmetry of the problem. For the symmetric S2-approximation
(µ=0.57735) this is identical to the P1-approximation with Mark’s boundary conditions, while for the
nonsymmetric S2-approximation (µ=0.5000) it is identical to the two-flux approximation.
0
0
434 RADIATIVE HEAT TRANSFER
17.5 Two infinitely long, concentric cylinders of radii R1and R2with emittances ǫ1and ǫ2have the same constant
surface temperature Tw. The medium between the cylinders has a constant absorption coefficient κand
does not scatter; uniform heat generation ˙
Q′′′ takes place inside the medium. Determine the temperature
distribution in the medium and heat fluxes at the wall if radiation is the only means of heat transfer, using
the S2-approximation.
Solution
For concentric cylinders the S2-approximation follows from equations (17.36) and (17.37) as
dI1
where µ1=−µ2=−µ,w′′
1=w′′
2=2π, and α3/2=α1/2+w′′
1µ1=−2πµ (or α3/2/w′′
i=µ1=−µ2). Adding and
subtracting the equations according to the definitions of Gand qgives
dq
Applying the condition of radiative equilibrium with internal heat generation, this may be rewritten for a
1
τ
d
dτ(τΨ)=1,dΦ
dτ=−1
µ2Ψ,
τ=τ1:b1Φ′= Φ −1, τ =τ2:−b2Φ′= Φ −1.
Integrating:
τΨ = 1
2τ2+C1,Ψ = 1
2τ+C1
τ,
Φ = −1
4µ2τ2−1
µ2C1ln τ+C2.
2−1
4τ
b1
τ1
+b2
τ2
+ln τ2
τ1
436 RADIATIVE HEAT TRANSFER
17.6 An infinite, black, isothermal plate bounds a semi-infinite space filled with black spheres. At any given
distance zaway from the plate the particle number density is identical, namely, NT=6.3662 ×108m−3.
However, the radius of the suspended spheres diminishes monotonically away from the surface as
a=aoe−z/L;ao=10−4m,L=1 m
(a) Determine the absorption coefficient as a function of z(you may make the large-particle assumption).
(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 S2-approximation, set up the boundary
conditions and solve for heat flux and temperature distribution (as a function of z).
w
438 RADIATIVE HEAT TRANSFER
17.7 Consider two parallel black plates, both at 1000 K, that are 2 m apart. The medium between the plates emits
and absorbs (but does not scatter) with an absorption coefficient of κ=0.05236 cm−1(gray medium). Heat is
generated by the medium according to the formula
˙
Q′′′ =CσT4,C=6.958 ×10−4cm−1,
where Tis the local temperature of the medium between the plates. Assuming that radiation is the only
important mode of heat transfer, determine the heat flux to the plates using the (symmetric) S2-approximation.
C1=4σT4
0.1(4κ/C−1) cos 0.05τL−√3 sin 0.05τL
=4×5.670 ×10−8×10004W/m2
=9030.4W
m2
CHAPTER 17 439
17.8 A furnace burning pulverized coal may be approximated by a gray cylinder at radiative equilibrium with
uniform heat generation ˙
Q′′′ =0.266 W/cm3, bounded by a cold black wall. The gray and constant absorption
and scattering coefficients are, respectively, 0.16 cm−1and 0.04 cm−1, while the furnace radius is R=0.5 m.
Scattering may be assumed to be isotropic. Using the S2-approximation,
(a) set up the relevant equations and their boundary conditions;
(b) calculate the total heat loss from the furnace (per unit length);
(c) calculate the radial temperature distribution; what are centerline and the adjacent-to-wall temperatures?
(d) qualitatively, if the extinction coefficient is kept constant, what is the effect of varying the scattering
coefficient on (i) heat transfer rates, (ii) temperature levels?
and (17.37) as
µ1
dI1
dτ+µ1
2τI1−α3/2
2τw′′
1
I2+I1=S,
dI2
dτ+q
µ=0,
τ=τR:G=q/µ;τ=0 : G,qfinite.
Applying the condition of radiative equilibrium with internal heat generation, this may be rewritten as, with
Sfor isotropic scattering from equation (14.5),
1
d
Q′′′
2
βτ=1
2˙
which could also have been found from a simple energy balance since all generated heat must leave any
(sub-)cylinder of radius r:
2πrLq(r)=πr2L˙
Q′′′,or q(r)=1
2˙
Q′′′r.
440 RADIATIVE HEAT TRANSFER
(c) Integrating equation (17.8-B),
˙
Q′′′
µq(R)=1
2µ
βτR=G=C2−1
4µ2
βτ2
C2=˙
Q′′′
βτR 1
2µ+τR
4µ2!.
From equation (17.8-A)
Q′′′
Q′′′
=1.6625 89
9W/cm2,
since τR=(0.16 +0.04) cm−1×50 cm =10. Thus,
CHAPTER 17 441
17.9 Estimate the radial temperature distribution in the sun. You may make the following assumptions:
(i) The sun is a sphere of radius R;
(ii) As a result of high temperatures in the sun, the absorption and scattering coefficients may be approxi-
mated to be constant, i.e. κν, βν=const ,f(ν, T,r) (free-free transitions!);
(iii) As a result of high temperatures, radiation is the only mode of heat transfer;
(iv) The fusion process may be approximated by assuming that a small sphere at the center of the sun releases
heat uniformly corresponding to the total heat loss of the sun (i.e., assume the sun to be concentric spheres
with a certain flux at the inner boundary r=ri).
(a) Relate the heat production to the effective sun temperature Teff=5777 K.
(b) Would you expect the sun to be optically thin, intermediate, or thick? Why? What are the prevailing
boundary conditions?
(c) Find an expression for the temperature distribution (for r>ri) using the S2-approximation.
(d) What is the surface temperature of the sun?
CHAPTER 17 443
17.10 Repeat Problem 17.9 but replace assumption (iv) by the following: The fusion process may be approximated
by assuming that the sun releases heat uniformly throughout its volume corresponding to the total heat loss
of the sun.
17.3
τ=τR:I1=0,or G−q/µ =0.
τ2
dτ(τ2q)=(1 −ω)(4σT4−G)=˙
β=Q
β.4
3πR3=3σT4
τR
,(17.10-A)
dG
dτ=−1
µ2q=−3q,(17.10-B)
where µ=1/√3=0.57735 for the symmetric S2-approximation and we have also assumed that scattering is
isotropic. Thus, 4σT4=G+˙
Q′′′/κ and
Q′′′
3πr3=4πr2q(r),or q(r)=1
3˙
Integrating equation (17.10-B) gives
G=−Z˙
Q′′′r dτ=C2−1
2
˙
Q′′′
βτ2.
4σT4=˙
κ+˙
β"τR
√3
+1
2τ2
R 1−τ2
τ2
R!#,
4σT4=3σT4
+1
√3
+τR
