Kirchhoff's Law considers that the two bodies are kept in a furnace held at a constant temperature of TK. Assume that, of the two bodies one is the black body and the other is a non-black body, i.e., the body having 'a' value of less than one.Â
Both the bodies will eventually attain the temperature of TK and the bodies neither become hotter nor cooler than the furnace.
At this condition of thermal equilibrium, each body absorbs and emits thermal radiation at the same rate. The rate of absorption and emission for the black body will be different from that of the non-black body.
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Kirchhoff's Law DerivationÂ
Let the area of the non-black body and black body be A1 and A2 respectively. Let T be the rate at which radiation falls on bodies per unit area and E1 and EB be the emission powers ( emissive power is the total quantity of radiation energy emitted by a body per unit area per unit time) of non-black and black bodies respectively. Â
At thermal equilibrium, absorption and emission and emission rates are equal, thus,
Ia1 A1 = A1E1
Ia1 = E1 -------1.0
And Iab A2 = A2 Eb
Ia b = Eb -------2.0
From equations (1.0) and (2.0), we get
kirchhoff's law formula
E1/A1 = Eb/ab
Where a1 and ab = absorptivities of non-black and black bodies respectively.
If we introduce a second body (non-black) then for the second non-black body, we have
I A3 a3 = E2 A3
I a2 = E2 ------3.0
Where a2 and E2 are the absorptivities and emissive power of the second non-black body.
Combining equations (1.0) (2.0) and (3.0), we get
Kirchhoff's law of radiation equationÂ
E1/a1 = E2/a2 = Eb/ab = Eb ---------(4.0)
As the absorptivity of the black body is 1.0
Kirchhoff's law of radiation statement
It states that at thermal equilibrium the ratio of the total emissive power to its absorptivity is the same for all bodies. Equation (4.0) is a mathematical statement of Kirchhoff Law.
The emissivity 'e' of anybody is defined as the ratio of the total emissive power E of the body to that of a black body Eb at the same temperature. The emissivity depends on the temperature of the body only.
e = E/Eb
Since E/a is constant for all bodies
E/a = Eb/ab
e = E/Eb = a/ab
But ab = 1 (for black body)
e = a ------5.0
Thus, when anybody is in thermal equilibrium with its surroundings, its emissivity and absorptivity are equal. Equation (5.0) may be taken as another statement of Kirchhoff Law.
Monochromatic emissive power
It is the radiant energy emitted from a body per unit area per unit time per unit wavelength about the wavelength λ. It is denoted by Eλ. It has the units of W/(m2. μm).
Total emissive power
It is the total quantity of radiant energy of all wavelengths emitted by the body per unit area per unit of time. It is denoted by the symbol E. It has units of W/m2.
For the entire spectrum of radiation from a surface, it is the sum of all the monochromatic radiation from the surface.
E = 0∫infit Eλ dλ
Monochromatic emissivity
It is the ratio of the monochromatic emissive power of a body to that of a black body at the same wavelength and temperature.
eλ = Eλ/Eb λ
Gray bodyÂ
A body having the same value of monochromatic emissivity at all wavelengths is called a gray body.
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Gray body properties
A gray body is one whose emissivity is independent of wavelength.Â
The adjective monochromatic indicates the quantity defined for a particular wavelength/single wavelength.Â
Monochromatic property refers to a single wavelength and total property is the sum of monochromatic values of the property.
Monochromatic values are not important to the direct solution of engineering problems.Â
Take these Notes is, Orginal Sources:Â Unit Operations-II, KA Gavhane
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