Lesson 19 Radiation I 19.1 Nomenclature E λ,b = spectral emission from a blackbody surface [W/m 2 -μ] E b = emission from a blackbody surface [W/m 2 ] q = radiative heat flux [W/m 2 ] F = fraction of blackbody band emission λ = wavelength in microns [μ=μm] ε = total emissivity, between 0 and 1 (ε=1 → blackbody, ε < 1 → gray body) α =absorptivity, between 0 and 1, α=ε if surface is blackbody or gray body σ = Stefan-Boltzmann Constant = 5.67×10 -8 W/m 2 -K 4 19.2 Fundamental Concepts Radiation Radiation is the phenomenon of electromagnetic (EM) wave propagation. EM-waves transfer heat energy through interactions with solid surfaces or fluids. Unlike conduction and convention, no medium is required to transfer heat by radiation, and it becomes more predominant as temperature increases. Figure 19.1 shows the range of EM waves that we commonly experience in everyday life. Figure 19.1 Electromagnetic Spectrum
A set of notes about radiative heat transfer, corresponding to a course in Heat Transfer taught at the University of Maryland.
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Lesson 19 Radiation I
19.1 Nomenclature
Eλ,b = spectral emission from a blackbody surface [W/m2-μ]
Eb = emission from a blackbody surface [W/m2]
q = radiative heat flux [W/m2]
F = fraction of blackbody band emission
λ = wavelength in microns [μ=μm]
ε = total emissivity, between 0 and 1 (ε=1 → blackbody, ε < 1 → gray body)
α =absorptivity, between 0 and 1, α=ε if surface is blackbody or gray body
σ = Stefan-Boltzmann Constant = 5.67×10-8
W/m2-K
4
19.2 Fundamental Concepts
Radiation Radiation is the phenomenon of electromagnetic (EM) wave propagation. EM-waves
transfer heat energy through interactions with solid surfaces or fluids. Unlike conduction and
convention, no medium is required to transfer heat by radiation, and it becomes more
predominant as temperature increases. Figure 19.1 shows the range of EM waves that we
commonly experience in everyday life.
Figure 19.1 Electromagnetic Spectrum
Differences between Radiation and Convection
Radiation Convection
(a) Medium None needed Needed
(b) Distance Can reach very far Local
(c) Speed Speed of light Relatively low
(d) Math Integral equations Differential equations
(e) q" εσTs4 h(Ts –T∞)
(a) Radiation can travel through the vacuum of space from the sun to the earth, but also
through the air to warm the earth’s surface.
(b) We can see the light emitting from a candle from afar, but we cannot feel it even from
one foot away.
(c) Generally, conduction is the slowest mode of heat transfer.
Radiation = f(λ,ω) λ=wavelength, ω=solid angle. We will not consider ω.
Definitions Relating to λ and ω
Diffuse surface: emits same radiation intensities at different solid angles.
Gray surface: emits same radiation intensities at different wavelengths.
Spectral emissivity: function of wavelength.
Total emissivity: average of spectral emissivity; constant at a given temperature.
Angular emissivity: function of solid angle.
Hemispherical emissivity: average of angular emissivity; constant at a given temperature.
19.3 Blackbody Radiation
Blackbody Surfaces (a) Will absorb all incidental radiation, regardless of λ; nothing is reflected or transmitted.
(b) Emit the maximum amount of radiation for any T (i.e., ε=1).
(c) ≠ black surfaces. Even the sun can be approximated as blackbody.
Planck Blackbody Spectral Emissive Distribution
𝑬𝝀,𝒃 =𝑪𝟏
𝝀𝟓 (𝐞𝐱𝐩 (𝑪𝟐
𝝀𝑻) − 𝟏)
(1)
Where
C1=3.742×108 and
C2 = 1.439×104
Figure 19.2 Blackbody Spectral Emission as a function of λ, parameterized by T.
(a) λmaxT = 2898 (Wien’s Displacement) (1a)
(b) Through evolution, we have adapted to capture solar radiation (T=5800 K) around the
vicinity of the peak of the spectral emission.
(c) In order for our eyes to see emitted lighted (not merely reflected), temperatures must be
very high.
Stefan-Boltzmann Law
𝑬𝒃 = ∫ 𝑬𝝀,𝒃𝒅𝝀 = 𝝈𝑻𝟒∞
𝟎
(2)
Where σ = Stefan-Boltzmann Constant = 5.67×10-8
W/m2-K
4.
The law governing the relationship between the heat flux and the temperature, called the
Stefan-Boltzmann law, is given as
𝒒′′ = 𝜺𝝈𝑻𝟒 = 𝜺𝑬𝒃 and 𝒒 = 𝑨𝒒′′ = 𝑨𝜺𝝈𝑻𝟒 = 𝑨𝜺𝑬𝒃 (2a)
19.4 Fraction of Blackbody Emission
Fraction of Blackbody Emission Frequently we are interested in finding a portion of the emissive radiation between a range of
wavelengths from λ1 to λ2 (partial area under Eλ,b-λ curve). The fraction, F, can be derived as
𝐹𝜆1𝑇→𝜆2𝑇 =∫ 𝐸𝜆,𝑏𝑑𝜆
𝜆2
𝜆1
𝐸𝑏=
∫ 𝐸𝜆,𝑏𝑑𝜆𝜆2
0
𝜎𝑇4−
∫ 𝐸𝜆,𝑏𝑑𝜆𝜆1
0
𝜎𝑇4= 𝐹0→𝜆2𝑇 − 𝐹0→𝜆1𝑇
𝑭𝝀𝟏𝑻→𝝀𝟐𝑻 = 𝑭𝟎→𝝀𝟐𝑻 − 𝑭𝟎→𝝀𝟏𝑻 (3)
Values of F can be found from Table A-2 in Heat Transfer: Lessons with Examples Solved
by MATLAB, by Dr. Tien-Mo Shih.
λT is given in units of μ-K.
At λT=50000, F≈1.
Example Over all λ, a blackbody (ε=1) will emit a radiative flux 𝑞𝑏 = 𝜀𝜎𝑇4 = 𝜎𝑇4. Over a range of
λ, the flux will be 𝑞𝜆1→𝜆2= 𝑞𝑏 ∗ 𝐹𝜆1𝑇→𝜆2𝑇.
19.5 Summary of Equations
Planck Blackbody Spectral Emissive Distribution (1)
𝐸𝜆,𝑏 =𝐶1
𝜆5 (exp (𝐶2
𝜆𝑇) − 1)
C1=3.742×108
C2 = 1.439×104
Wien’s Displacement (1a) λmaxT = 2898 [μ-K]
Stefan-Boltzmann Law (2, 2a)
𝐸𝑏 = ∫ 𝐸𝜆,𝑏𝑑𝜆 = 𝜎𝑇4∞
0
𝑞′′ = 𝜀𝜎𝑇4 = 𝜀𝐸𝑏 and 𝑞 = 𝐴𝑞′′ = 𝐴𝜀𝜎𝑇4 = 𝐴𝜀𝐸𝑏
σ = Stefan-Boltzmann Constant = 5.67×10-8
W/m2-K
4
ε = total emissivity, between 0 and 1 (ε=1 → blackbody, ε < 1 → gray body)
α =absorptivity, between 0 and 1, α=ε if surface is blackbody or gray body
Fraction of Blackbody Emission (3) 𝐹𝜆1𝑇→𝜆2𝑇 = 𝐹0→𝜆2𝑇 − 𝐹0→𝜆1𝑇
Lesson 20
Radiation II
20.1 Nomenclature
E = radiative flux emitted by a plate
f1 = fraction of solar radiation received by earth = 0.0021%
G = incoming radiative flux arriving at the plate
m = plate mass
ελ = spectral emissivity
ε = total emissivity, between 0 and 1 (ε=1 → blackbody, ε < 1 → gray body)
α =absorptivity, between 0 and 1, α=ε if surface is blackbody or gray body
ρ = reflectivity (not density!)
τ = transmissivity (not shear stress!)
20.2 Emissivity
Spectral Emissivity Emissivity is the one radiative property that is unrelated to incoming fluxes; it is a function of
the λ and T of the emitting source. Let us define the spectral emissivity as
𝜺𝝀 =𝑬𝝀(𝝀, 𝑻)
𝑬𝝀,𝒃(𝝀, 𝑻)
(1a)
The total emissivity, ε, can be derived from this equation with integration over all λ and
making use of the fraction of blackbody emission, F:
𝜺 =𝑬(𝑻)
𝑬𝒃(𝑻)=
∫ 𝑬𝝀(𝝀, 𝑻)𝒅𝝀∞
𝟎
𝝈𝑻𝟒=
∫ 𝜺𝝀𝑬𝝀,𝒃(𝝀, 𝑻)𝒅𝝀∞
𝟎
𝝈𝑻𝟒
= 𝜺𝝀𝟏→𝝀𝟐𝑭𝝀𝟏𝑻→𝝀𝟐𝑻 + 𝜺𝝀𝟐→𝝀𝟑
𝑭𝝀𝟐𝑻→𝝀𝟑𝑻 + ⋯
(1b)
From Eq. (1b) we can also readily find the total radiative flux emitting from a body:
𝑬(𝑻) = 𝒒𝒔′′ = 𝜺𝝈𝑻𝒔
𝟒 (2)
20.3 Three Other Radiative Properties
Absorptivity, Reflectivity, and Transmissivity As with Eq. (1a) and Eq. (1b), the same can be applied for
1. absorptivity α (propensity to absorb radiation),
2. reflectivity ρ (propensity to reflect radiation),
3. transmissivity τ (propensity to let radiation transmit through).
𝜶𝝀 =𝑮𝝀,𝜶
𝑮𝝀 → 𝜶 =
𝑮𝒂(𝑻)
𝑮(𝑻)=
∫ 𝑮𝝀,𝒂𝒅𝝀∞
𝟎
∫ 𝑮𝝀𝒅𝝀∞
𝟎
=∫ 𝜶𝝀𝑮𝝀𝒅𝝀
∞
𝟎
∫ 𝑮𝝀𝒅𝝀∞
𝟎
(3a,b)
𝝆 =∫ 𝝆𝝀𝑮𝝀𝒅𝝀
∞
𝟎
∫ 𝑮𝝀𝒅𝝀∞
𝟎
(4)
𝝉 =∫ 𝝉𝝀𝑮𝝀𝒅𝝀
∞
𝟎
∫ 𝑮𝝀𝒅𝝀∞
𝟎
(5)
Note that these three properties all related to incoming radiation. If there is no mention of a
radiative heat source, then these values cannot be found. Even if incoming fluxes are the
same, the resulting α, ρ, and τ may be different depending on source temperature. See
example 20-5.
20.4 Gray Surfaces
Gray Surfaces Gray surfaces are defined as surfaces whose radiative properties are independent of
wavelength. Hence, for gray surfaces, all spectral properties are (1) constant and (2) equal
to the total properties.
Therefore, if G is the total incoming radiation, then it follows that