Download Notes on Radiation - Cambridge University Engineering Department

Cambridge University Engineering Department
Engineering Tripos Part IIA
Module 3A6: Heat and Mass Transfer
Some Notes on Radiation
Dr. Christos Markides
[email protected]
Nomenclature:
h
Plank’s constant [J.s]
co
Speed of light in vacuum [m/s]
k
Boltzmann’s constant [J/K]
σ
Stefan-Boltzmann constant [W/m2.K4]
ω
Solid angle [sr]
r
Spherical coordinate system radius [m]
θ, φ
Spherical coordinate system angles [rad]
Ar
Area of radiation receiving surface [m2]
As
Area of radiation sending surface [m2]
λ
Radiation wavelength [m]
ε
Emissivity/emittance; proportion/fraction of incident irradiation emitted [-]
ρ
Reflectivity/reflectance; proportion/fraction of incident irradiation reflected [-]
α
Absorptivity/absorptance; proportion/fraction of incident irradiation absorbed [-]
τ
Transmissivity/transmittance; proportion/fraction of incident irradiation transmitted [-]
Q
Generic (radiation/radiative/radiant) (energy flow rate/power) [W]
Q’
; generic (radiation/radiative/radiant) flux, or (energy flow rate/power) per unit area [W/m2]
Qλ
; generic spectral (radiation/radiative/radiant) (energy flow rate/power), or (energy flow rate/power)
per unit wavelength [W/m]
Qλ’
G
G’
Gλ
Gλ’
J
J’
; generic spectral (radiation/radiative/radiant) flux [W/m3]
(Irradiation/irradiance) power [W]
; (irradiation/irradiance) flux, or power per unit receiving area [W/m2]
; spectral (irradiation/irradiance) power, or power per unit wavelength [W/m]
; spectral (irradiation/irradiance) flux [W/m3]
(Radiosity/radiant exitance) power [W]
; (radiosity/radiant exitance) flux, or power per unit sending area [W/m2]
Jλ
; spectral (radiosity/radiant exitance) power, or power per unit wavelength [W/m]
Jλ’
; spectral (radiosity/radiant exitance) flux [W/m3]
E
E’
Eλ
(Emissive/emission) power [W]
; (emissive/emission) flux, or power per unit sending area [W/m2]
; spectral (emissive/emission) power, or power per unit wavelength [W/m]
Eλ ’
; spectral (emissive/emission) flux [W/m3]
I
; (radiation/radiative/radiant) intensity, or power per unit solid angle [W/sr]
Iλ
L
Lλ
; spectral (radiation/radiative/radiant) intensity, or intensity per unit wavelength [W/sr.m]
; radiance, or power per unit area per unit solid angle [W/m2.sr]
; spectral radiance, or radiance per unit wavelength [W/m3.sr]
CNM
12 of March, 2008
th
1. PRELIMINARY
For solid bodies at steady state, we must consider the irradiation onto the surface, and the balance of the radiosity
from the surface, the absorption into the solid and the transmission through the solid, with the irradiation either
being reflected, absorbed or transmitted according to, G = Gr + Ga + Gt, or ρ + α + τ = 1. Reflectance can be diffuse
(independent of angle) or specular (mirror-like; angle of incidence equals angle of reflection).
Radiosity
(J)
Irradiation
(G)
Emission
(E)
Reflected
Irradiation
(Gr = ρG)
Absorption
(Ga = αG)
Transmission
(Gt = τG)
Figure 1. Types of radiative heat transfer to, from, into and through a solid surface.
For non-transparent (opaque) solids, the energies must still always balance. There is no radiation being transmitted
through the solid, Gt = 0, or τ = 0, and hence the irradiation onto the surface must always balance the sum of the
radiosity from the surface and the absorption into the solid, G = Gr + Ga, or ρ + α = 1.
The plane angle and solid angle are two derived units in the SI system. The following definitions are taken from NIST:
"The radian is the plane angle between two radii of a circle that cuts off on the circumference an arc equal in length
to the radius". The abbreviation for the radian is rad. There are 2π radians in a circle, i.e. in 360o. The solid angle
extends this concept to three dimensions: "One steradian (sr) is the solid angle that, having its vertex in the centre of
a sphere, cuts off an area on the surface of the sphere equal to that of a square with sides of length equal to the
radius of the sphere". The solid angle is thus defined as the ratio of the spherical area to the square of the radius:
If we divide the surface area of a sphere by the square of its radius, we find that there are 4π steradians of solid
angle in a sphere. One hemisphere has 2π steradians.
1. GENERIC RADIATION (Lλ,rad)
Spectral radiance:
Spectral radiation power:
Spectral radiation flux:
Hemispherical spectral radiation flux:
2. IRRADIATION (Lλ,i)
Hemispherical spectral irradiation flux:
Total irradiation flux:
Diffuse (direction independent) irradiation flux:
3. RADIOSITY (Lλ,e+r = Lλ,e + Lλ,r)
Hemispherical spectral radiosity flux:
Total radiosity flux:
Diffuse (direction independent) radiosity flux:
4. EMMISION (Lλ,e)
Hemispherical spectral emissive flux:
Total emissive flux:
Diffuse (direction independent) emission flux:
5. BLACK BODIES

Think of a deep, closed cavity with a small opening through which radiation (photons) can pass.

The body absorbs all irradiation, reflects none and transmits none.

The body is a diffuse emitter.

The body emits the maximum possible heat for a body at that temperature.

No reflected radiation (no reflected irradiation):
or
, and
or

No transmitted radiation (no transmitted irradiation):
, and

All incident radiation (all irradiation) is absorbed:
, and

or
or
The spectral emission intensity is defined as the black body spectral intensity:
, and

Plank’s spectral distribution is,
where h and k are Plank’s and Boltzmann’s constants. The final expression is almost redundant, since all heat flow
rates equal heat fluxes multiplied by the radiation sending surface area As. From this point onwards, we will deal
only with fluxes.
This distribution is, clearly, a function of two variables, λ and T. For a black body at a given temperature, we can find
the wavelength associated with the maximum (peak) emission by partially differentiating this expression with
respect to λ (while T is kept constant), and setting this equal to zero:
The final expression above is now, conveniently, a function of a single variable, λmaxT. The exponential term is small,
for all but very high λ and T (or products of λT), as shown in the table below:
T [K]
10
λ [μm]
100
1000
10000
0.01
0.00E+00 0.00E+00 0.00E+00 3.20E-63
0.1
0.00E+00 0.00E+00 3.20E-63 5.63E-07
1
0.00E+00 3.20E-63 5.63E-07 2.37E-01
10
3.20E-63 5.63E-07 2.37E-01
Table 1. The value of
-
for various λ and T.
Note that λmax is usually a short wavelength, becoming even shorter as T increases for the same value of λmaxT.
Then, the Wien displacement law states that, approximately:

Similarly, the Wien distribution law is a simplification to Plank’s spectral distribution, based on the value of the
exponential term being much greater than unity, which occurs when
, or
, as discussed in
the preceding sections:

Four more expressions that you might encounter are,
and:
These expressions are convenient, more so than the previous ones, because (as with the final expression for the
peak emission wavelength, λmax) the two independent variables always appear paired together. So, we can think of
λT as a single variable, and tabulate values of the expressions for various values of λT.

The Stefan-Boltzmann law is,
where σ is the Stefan-Boltzmann constant.
The Stefan-Boltzmann law is obtained by integration over all wavelengths (frequencies). Physically, this means that
the black body is able to emit (a heat flow rate according to Plank’s spectral distribution, or the Wien distribution
law) at all wavelengths.

Many bodies actually only emit over a limited band of wavelengths. To describe this we define band emission to
be,
where the values of
can also be found in tables. Here,
or the Wien distribution law:
6. REAL SURFACES

Emissivity:

Directionally averaged emissivity:

Total hemispherical emissivity:
can be either from Plank’s spectral distribution,

Absorptivity:

Reflectivity:

Transmissivity:

Irradiance, absorption, reflection and transmission:

Absorptivity, reflectivity and transmissivity:

Opaque media have:
, and

Kirchoff’s law:
7. GRAY SURFACES

Given diffuse irradiation and emission, and from Kirchoff’s law:

In addition, if either: (i) αλ and ελ are independent of λ as well, and hence constants, or (ii) the irradiation field
corresponds to that of a black body at the same temperature T as the surface, then,
the latter coming from the fact that for these two surfaces,
,
,
but also the definitions of α, αλ, ε and ελ:
This relationship applies for any range of wavelengths:
8. VIEW FACTORS
Surface “i” emits radiative intensity Li, part of which will reach surface “j”,
and for a diffuse surface “i”:
The term
can be thought of as the common projection of the solid angle area associated with
surface “j” and surface area of surface “i” along the direction of a straight line joining the two.
Integrating for the total heat flow rate from “i” to “j”,
and for a diffuse surface “i” over which Ji’ is uniform over the surface areas:
Similarly, the heat flow rate from surface “j” to surface “i” is:
Hence, define the view factor as,
such that the reciprocity relation below, holds:
The net heat flow rate from surface “i” to surface “j” is:
9. RADIATION EXCHANGE FROM SURFACES
We consider now the radiative heat transfer to and from a gray body. Assume that the surface of the body is uniform
in temperature, and that all its emissive and reflective properties are also uniform. Furthermore, assume that the
heat fluxes are also uniform over the surface, and that the surface is diffuse.
We return to Figure 1.
Radiosity
(J)
Irradiation
(G)
Emission
(E)
Reflected
Irradiation
(Gr = ρG)
Absorption
(Ga = αG)
Transmission
(Gt = τG)
The net power from the surface is equal to the radiosity minus the irradiation:
Now, the radiosity is made up of emission and reflected irradiation,
so that:
If there is no transmission, τ = 0, and if α = ε, ρ + ε = 1. The equation above for the net power becomes,
or, in terms of the net flux:
10. RADIATIVE HEAT TRANSFER RESISTANCE NETWORKS
Finally, we must approach the problem of radiative heat transfer between two or more bodies. We return to the two
bold equations for the net Q from a surface “i” to a surface “j” in terms of the view factor and the emissivities of the
surfaces. They both look identical to Ohm’s law for electrical resistance networks, if one is careful to employ, Ji, Jj and
Eb as potentials and the expressions in the denominators as the resistances, as illustrated below:
Eb1’ = σT14
J1’
J2’
Eb2’ = σT24
Figure 2. Full, radiative heat transfer between two surfaces.
The above is the standard way of approaching the full, radiative heat transfer between two surfaces. For three
surfaces:
Eb1’ = σT14
J1’
J2’
I1
I2
J3’
Eb3’ = σT34
Figure 3. Full, radiative heat transfer between three surfaces.
Three simplifications can be made. Depending on the physical situation of the heat transfer:
1. If one of the three bodies is black, e.g. “3”, then ε3 = 1, and consequently Eb3 = J3.
Eb2’ = σT24
2. Similarly, if one of the bodies is insulated such that an adiabatic boundary condition applies, e.g. “3”, then no
heat flow/current can flow out of the circuit at Eb3, no current can flow through R3, and hence Eb3 = J3.
3. If one the bodies is a perfect reflector, e.g. “3”, then ε3 = 0, and there can be no current though R3, because
R3 is infinite, even though Eb3 ≠ J3. The current flowing from J1 to J3, gets completely diverted towards J2.
In all of the above cases, the current through the two view-factor resistances must be equal, I1 = I2.
Note that, for the above approach to be valid, our assumptions of all surfaces being gray, diffuse, uniform in
temperature, with uniform emissive and reflective properties, and uniform heat fluxes over those surfaces, must
hold.