CHAPTER 19
19.1 A semi-infinite, gray, isotropically scattering medium, originally at zero temperature, is subjected to colli-
mated irradiation with a constant flux qonormal to its nonreflecting surface. Set up the integral relationships
governing steady-state temperature and radiative heat flux within the medium, assuming radiative equilib-
rium.
19.2
L
ql
ql = 100W/m2
45° 45°
= 0,
s
σ
= 1
κ
∋
= 0.2 cm–1
In a greenhouse a layer of water (thickness L=5 cm) is resting on
top of a black substrate. The water is loaded with growing organisms
that scatter light isotropically but do not absorb (σs=0.2 cm−1). The
water layer is illuminated by two long growth-enhancing lights, fitted
with reflector shields that make the light essentially parallel each light
delivering a heat flux of ql=100 W/m2(per unit area normal to the
light rays). Using the exact method, calculate energy generated within
the water and the radiative heat flux absorbed by the black surface in
the zone between the lights, where the heat transfer is essentially one-
dimensional. Emission from the water and substrate are negligible.
Hint: Use Figs. 3-16 and 19-4.
(10.61).
(ii) Alternatively, evaluating ˆ
qc·ˆ
kwe find that Gcand ˆ
qc·ˆ
kare the same as in Example 19.1 if qsis replaced
448 RADIATIVE HEAT TRANSFER
19.3 Reconsider the medium described in Example 19.1. Rather than being bounded by a cold black surface at the
bottom, the layer is now exposed to the nonparticipating gas as well as to solar irradiation (using mirrors)
on both sides. Determine radiative heat flux and its divergence within the layer in terms of the function
Φ(τL, ω, µo, τ) given in Example 19.1.
CHAPTER 19 449
19.4 Solve Problem 19.1 using the P1-approximation.
and (19.20) become
dqd
dτ=(1 −ω)(4σT4−Gd)+ωGc,
Substituting,
dGd
dτ=−3q+3qoe−τ,
Gd=(5 −3e−τ)qo,
and
dqd
dτ=−dqc
dτ=qoe−τ=(1 −ω)(4σT4−Gd)+ωqoe−τ,
450 RADIATIVE HEAT TRANSFER
19.5 The starship Enterprise is hitting a Klingon cruiser with its phaser gun. The armament of the cruiser is a
partially reflecting material that, after some irradiation, partly evaporates, forming a protective gas layer
above the surface. Assuming that the surface is at evaporation temperature Tev and has an emittance ǫ,
the gas has an absorption coefficient κ1and a thickness L, determine the fraction of the heat flux that hits
the Klingon ship. Under these conditions you may assume the effects of conduction and convection to be
negligible (but not reradiation from the gas). Use the P1-approximation.
4/ǫ +3τL
CHAPTER 19 451
19.6 Reconsider Problem 19.5. After further irradiation, the surface material starts to disintegrate, spewing
particulate material into the gas layer. If we make the assumption that the debris has an absorption coefficient
κpand (isotropic) scattering coefficient σsp, how does this modify the surface irradiation?
Problem 19.5,
q=[5−(8/ǫ −3)e−τL]qph −4σTev
.
452 RADIATIVE HEAT TRANSFER
19.7 Consider a semi-infinite gray medium with a nonreflecting surface. The medium is cold, absorbs (absorption
coefficient κ) and scatters isotropically (scattering coefficient σs). Collimated radiation obeying the relation
qc=qo(1 −cos ατx)ˆ
k
shines normally onto the medium as shown. Determine the reflectivity of the medium (i.e., the fraction of the
irradiation leaving the interface in the opposite direction), using the P1-approximation.
Hint: To solve the two-dimensional governing equation, set Gd(τx, τz)=G1(τz)+G2(τz) cos ατx.
z, z
τ
x, x
τ
qc (x)
τ
