Radiation

Radiation is the mode of heat transfer across a system boundary due to a temperature difference in the form of electromagnetic waves as a result of change in the electronic configurations of the atoms and molecules. Radiative heat transfer does not require a medium to pass through; thus, it is the only form of heat transfer present in vacuum.

Electromagnetic Spectrum

All types of electromagnetic waves are classified in terms of wavelength and are propagated at the speed of light (c = 3 x 10^8 m/s).

Electromagnetic Spectrum

Stefan-Boltzman Law

The emissive power of a black body is proportional to absolute temperature to the fourth power.

\[ \boxed{E_b = σ T^4}\]

where:          Eb = Emissive Power, the gross energy emitted from an ideal surface per unit area, time 

  σ = Stefan Boltzman constant, $5.67X10^{-8} W/m^2 K^4$

T = Absolute temperature of the emitting surface, K.

Plank’s Law

\[E_{bλ} = \frac{C_1}{λ^5(e^{\frac{C_2}{λT}}-1)}\]

$C_1 = 2 \pi h c^2 = 3.742 × 10^8 Wμm^4 / m2 $

$C_2 = h c/k = 1.439 × 10^4 μm-K $

$E_{bλ}$ = Monochromatic (single wavelength)  Emissive Power of a black body

c = speed of light = 3x 10^8 m/s

h = Plank's constant = 6.625 x 10^-34 J-s

λ = Wavelength, μm

k = Boltzmann constant = 1.3805 x 10^-23 J/K

T = Absolute temperature, K

An implicit definition Monochromatic (single wavelength) Emissive Power, $E_{bλ}$  is given by
\[E_b = \int^∞_0 E_{bλ}. dλ\]  

• We can obtain Stefan-Boltzmann Law by integrating Plank's law over all wavelengths.

Wien Displacement Law

It state that the product of the temperature(T) of a black body and the wavelength(λm) at which the maximum value of monochromatic emissive power occurs, is constant.

\[λ_m T = Constant = 2897  μm.K\]

Some Definitions

Black Body

A black body is defined as a perfect emitter and absorber of radiation. It absorbs all incident radiation regardless of wavelength and direction.

Diffuse Surface

A surface is said to be diffuse if its properties are independent of direction.

Gray Surface

A surface is said to be gray if its properties are independent of wavelength.

Radiation Intensity (I)

It is defined as the energy emitted from an ideal body, per unit projected area on a plane normal to the direction of radiation, per unit time, per unit solid angle. 

\[I = \frac{dq}{dA cos θ d \Omega}\]

Emissivity (ε)

Real surfaces have emissive power, E somewhat less than that of ideal surface (black body). It is defined as the ratio of Emissive power of any body to the Emissive power of a black body of equal temperature.

\[ε = \frac{E}{E_b}\]

Note:

• For a black body ε = 1, for white body ε = 0.
• Value of ε varies from 0 to 1.
• Emissivity may vary with temperature and wavelength.

Receiving Properties

Irradiation(G)

Total incident radiation on a surface from all directions per unit time and per unit area of surface.

1. The fraction of irradiation absorbed by the surface is called the absorptivity(α),
2. the fraction reflected by the surface is called the reflectivity (ρ), and
3. the fraction transmitted is called the transmissivity (τ). For opaque body τ =0.

\[G_{abs} + G_{ref} + G_{tr} = G\]

\[\boxed{α + ρ + τ = 1}\]

Radiosity (J)

It is defined as the radiation energy leaving from a surface per unit time, per unit area of surface. 

\[J = E +  ρG\]

Note:  For a black body J = Eb, since it absorbs all incident radiation and hence no reflection.

Kirchhoff's Law

The emissivity of a body is equal to its absorptivity whenever body is in thermal equilibrium with its surroundings. We can say a good absorber is a good emitter.

\[\boxed{ α = ε}\]

Intensity of Radiation and Lambert's Cosine Law

According to Lambert's Cosine Law, the radiant intensity observed from an ideal ideal diffuse surface is directly proportional to the cosine of the angle of emission (θ).

\[I = I_n cos(θ)\]

• Emissive power of a black body is $\pi$ times the intensity of radiation.

\[E_b = \pi I_b\]

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