Module 4. Radiation heat transfer

Lesson 22

BASIC LAWS OF THERMAL RADIATION, PLANK’S LAW

22.1  Basic Laws of Thermal Radiation

22.1.1  Plank’s law

The radiant power of a body E is defined as the amount of energy emitted by unit area per unit time for electromagnetic waves of a length ranging from λ = 0 to λ = ∞. However, for a detailed study of a phenomenon it is also important to know the law of radiant energy distribution in respect to wavelengths at different temperatures, E_λ = f( λ,T). The value E_λ is the radiant power of the body for wavelengths ranging from λ to λ+d λ referred to the interval of wavelength d λ under consideration. Hence,

and this value is called spectral intensity or simply radiation intensity, expressed in kcal/m²-hr-micron or kcal/m³-hr.

Plank has succeeded in determining theoretically the law governing the change in radiation intensity as a function of wavelength for a black body:

Where λ = wavelength, m;

           T = absolute temperature of body, ^oK;

e  = natural logarithm base;

           c₁ = constant, equal to 3.17 x 10^-16 kcal/m²-hr;

           c₂ = constant, equal to 1.44 x 10^-2 m-^oK.

Plank’s law illustrates that at λ = 0 radiation energy is zero. E_0λ first increases with λ and reaches its maximum at a certain λ_m, then diminishes and drops again to zero at λ = ∞. With the increase in temperature, the maximum of radiant energy shifts towards the shorter waves. Wien’s law relates T and λ_m by the following expression:

λ_mT= 2.9 [mm-°K].         

The total amount of energy emitted by waves of all lengths is obviously

The value E₀ is called integrated or total radiation, though it is nothing but the radiant power of a black body.

Further at temperatures common in engineering the energy of visible radiation (λ = 0.4-0.8) is negligible compared with infrared radiation (λ = 0.8-40.0).

For real bodies, the variation in radiation intensity with the wavelength and temperature can be determined only by an experimental study of their spectra. Bodies whose radiation spectrum is continuous and the curve E_λ = f(λ) is similar to the corresponding curve of a black body at the same temperature, i.e.  = const for all wavelengths, are called gray bodies. Experiments reveal that most technical materials are gray bodies.