CH EN 3453 – Heat Transfer
Summary of Heat
Transfer by Radiation
Chapters 12 and 13
Reminders…
•
Homework #12 due today (last one!)
– Turn in by 4:00 PM to ChE main office
– Scores on web site are updated, so you should be able to complete #1(a)
– Sorry ‘bout that part (b) on problem #1
• Consider it a trick question...
•
Final project report due Wednesday by 8:00 PM
– Email the file to [email protected]
– Check the rubric one last time to make sure you have done everything
required
•
Final exam Wednesday, December 17 from 8:00 AM to 10:00 AM
– 50% is review of conduction and convection, mostly multiple choice
– 50% is three calculation problems relating to radiation
•
•
Wednesday: Conduction review
Friday: Convection review
Radiation with Participating Media
(Gaseous Emission and Absorption)
• Gas radiation
– Nonpolar gases (O2, N2) neither emit nor
absorb radiation
– Polar gases (CO2, H2O, hydrocarbons) do
• In most cases, contribution of gas to
radiation can be safely neglected
• Notable exception: H2O and CO2 at high
temperatures (e.g. in combustion gases)
General Considerations
• The medium separating surfaces of an enclosure may affect radiation
at each surface through its ability to absorb, emit and/or scatter
(redirect) radiation.
• Participating media may involve semitransparent solids and liquids, as
well as polar gases such as CO2, H2O, CH4, and O3.
• Radiation transport in participating media is a volumetric phenomenon,
and for polar gases is confined to discrete wavelength bands.
• Beer’s law: A simple relation for predicting the exponential decay of
radiation propagating through an absorbing medium.
...where !𝜆 is the spectral absorption coefficient (m–1)
• Transmissivity and absorptivity of medium of thickness L
Emissivity of Water Vapor
Emissivity of Carbon Dioxide
Pressure Correction
H 2O
CO2
H2O + CO2 Correction
ε g = ε w + ε c − Δε
Gas Radiation - Geometries
Example - Problem 13.126
A gas turbine combustion chamber may be approximated
as a long tube of 0.4-m diameter. The combustion gas is
at a pressure and temperature of 1 atm and 1000°C,
while the chamber surface temperature is 500°C. If the
combustion gas contains 0.15 mol fraction each of
carbon dioxide and water vapor, what is the net radiative
heat flux between the gas and chamber surface, which
may be approximated as a blackbody?
Example: Problem 13.129
Products of combustion (2000 K, 1 atm) flow through a
long, 0.25-m-diameter pipe whose outer surface is black.
Combustion gas contains CO2 and H2O, each at 0.1 atm.
Gas may be treated as air in fully developed flow at 0.25
kg/s. Pipe is cooled by water in cross flow at 0.3 m/s
and 300 K. Determine the pipe wall temperature and
heat flux. Emission from the pipe wall may be neglected.
Review of Radiation
Radiation Spectrum
Intensity vs. Wavelength and Direction
The Solid Angle
Solid Angle Geometry
ω=
A2
A2 cosθ 2
r2
θ2
r
+
A2 cosθ 2
Projected Area
Radiation Heat Transfer
• Energy transfer between two elements A1
and A2
q1− j = I × A1 cosθ1 × ω j −1
=
I × A1 A2 cosθ1 cosθ 2
r2
From Example 12.1...
ω 3−1 = ω 4 −1 =
ω 2 −1
A3
10 −3 m 2
−4
=
2 = 4.00 × 10 sr
2
r
( 0.5m )
A2 cosθ 2
10 −3 m 2 × cos 30°
=
=
= 3.46 × 10 −3 sr
2
2
r
( 0.5m )
Blackbody
• Hypothetical perfect radiative surface
• Absorbs all incident radiation, regardless
of wavelength and direction
• Emits maximum theoretical energy
• Diffuse emitter – Radiation emitted evenly
in all directions
The Planck Distribution
• Emissive power of a blackbody depends
on temperature and wavelength
• Planck figured out this relation
• Plot of E vs. λ looks like this:
NOTES:
• Total power increases
with temperature
• At any given
wavelength the
magnitude of emitted
radiation increases with
temperature
• Wavelength of radiation
decreases with
temperature
• Sun is approximated by
blackbody at 5800 K
• At T < 800 K, most
radiation in infrared
Wien’s Displacement Law
• For a given temperature, spectral emission
goes through a maximum at a given
wavelength.
• Wien figured this one out:
• This maximum is indicated by the dashed
line in Figure 12.12
Stefan-Boltzmann Law
• If one were to integrate any of the curves
shown in Figure 12.12 over the entire
range of wavelengths, one would get the
total emissive power for a blackbody:
Eb =
∫
∞
0
Text
C1
dλ
λ 5 ⎡⎣ exp ( C2 / λT ) − 1⎤⎦
= σT 4
• The Stefan-Boltzmann constant σ is:
σ = 5.670 × 10–8 W/m2·K4
Band Emission
• Amount of total emitted radiation depends on
range of wavelengths of emission
• Effective emissivity determined by integrating
over wavelengths
• Table 12.1, column “F” provides fraction of total
integrated area to a given wavelength
Example: 12.29
• The spectral, hemispherical emissivity of
tungsten may be approximated by the
distribution given below. What is the total
hemispherical emissivity when the filament
temperature is 2900 K.
Radiation Transfer Types
• Emission (E)
– Associated with energy transfer due to
surface temperature
• Irradiation (G)
– Radiation incident onto a surface
– Irradiation can have three fates:
• Absorption by the surface
(α = absorptivity = fraction of G absorbed)
• Reflection by the surface
(ρ = reflectivity = fraction of
• Transmission through the material
(τ = transmissivity = fraction transmitted)
Irradiation onto a Surface
• Irradiation can have three fates:
– Absorption by the surface
(α = absorptivity = fraction of G absorbed)
– Reflection by the surface
(ρ = reflectivity = fraction of G reflected)
– Transmission through the material
(τ = transmissivity = fraction of G transmitted)
• Sum of α + ρ + τ = 1
Radiosity (J)
• Total radiation leaving a surface.
• Sum of emission plus reflected portion of
irradiation.
Example - Problem 12.52
Consider an opaque, diffuse surface for which
the spectral absorptivity and irradation are
shown below. What is the absorptivity of the
surface for the prescribed irradiation. If the
surface is at 1250 K, what is its emissive power?
View Factors
• Fraction of radiation from surface i that is
captured by surface j
N
• Summation rule:
∑F
j =1
• Reciprocity:
ij
=1
Ai Fij = A j Fji
Review: Radiation between Surfaces
Space
resistance
Surface
resistance
Review: Two-Surface Enclosure
Surface
Space
Surface
resistance resistance resistance
Radiation Shield
Reradiating Surface
“Direct Method” for Solving Networks
• Useful for systems with >2 surfaces
• Balance radiant energy around each surface
node i :
• Solve system
of equations
Multimode Heat Transfer
Example - Problem 13.66
Two parallel, aligned disks 0.4 m diameter
and 0.1 m apart are located in a large room
with walls at 300 K. One of the disks is at
500K with emissivity of 0.6 while the
backside of the second disk is well insulated.
What is the temperature of the insulated
disk?
Radiation with Participating Media
(Gaseous Emission and Absorption)
• Gas radiation
– Nonpolar gases (O2, N2) neither emit nor
absorb radiation
– Polar gases (CO2, H2O, hydrocarbons) do
• In most cases, contribution of gas to
radiation can be safely neglected
• Exception:
Emissivity of Water Vapor
Emissivity of Carbon Dioxide
H2O + CO2 Correction
ε g = ε w + ε c − Δε
Gas Radiation - Geometries