# 978-0123869449 Chapter 19

Document Type

Homework Help

Book Title

Radiative Heat Transfer 3rd Edition

Authors

Michael F. Modest

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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.

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

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