Black Body Radiation Formula and Calculator - Heat Transfer

Heat Transfer Engineering | Thermodynamics

Radiation, Black Body Equation and Calculator

Bodies under thermal agitation induced by temperature emit thermal radiation in the form of electromagnetic waves ranging in wave length from the long infrared to the short ultraviolet. Radiation emitted from a body can travel undiminished through a vacuum or through gases with relatively little absorption. When radiation is intercepted by a second body, part may be absorbed as thermal energy, part may be reflected from the surface, and part may be transmitted still in electromagnetic wave form through the body as in the case of glass.

Consider a body in space receiving radiant energy from some source. If all the incident radiation is absorbed with zero energy being reflected or transmitted, it is a perfect absorber and called a "blackbody". There are no perfect absorbers in nature although some bodies come very close to exhibiting blackbody characteristics. The ratio of the amount of energy absorbed by an actual body to that by a thermal "blackbody" is called the "absorptivity". In the absence of conduction and convection, a body at thermal equilibrium which receives radiation must necessarily emit radiant energy equal to that absorbed. Hence a body which is a good receiver or absorber is a good radiator or emitter. The ratio of the amount of radiant energy emitted by an actual body to that emitted by the ideal blackbody is called the "emissivity" and is numerically equal to the absorptivity. Its numerical value is always less than unity. The emissivity of polished copper, for example, is 0.023, whereas that of oxidized cast iron may be as high as 0.95.

Preview: Radiation, Black Body Calculator

A body that emits the maximum amount of heat for its absolute temperature is called a black body. Radiant heat transfer rate from a black body to its surroundings can be expressed by the following equation.

Q = σ A T⁴ - equation 1

Where:

Q = Heat transfer rate (Btu/hr)
σ = Stefan-Boltamann constant
A = Surface area (ft²)
T = Absolute temperature (°R)
°R = °F + 459.67
Stefan-Boltzman Constant = 5.67 x 10^-8 Watts-m^-2-K^-4
                                          = 0.174 x 10^-8 BTU/Hr-ft²-R⁴

For actual bodies, equation (1) must be modified for departure from ideal blackness and, since the net exchange of radiant energy between two bodies is usually required, it must be modified depending on the geometry of the system. The general equation for the net rate of exchange of radiant heat between two non-black bodies is:

q_r = F_e F_a A σ (T₁⁴ - T₂⁴) - equation 2

Where:

F_e = emissivity factor to allow for departure from black body conditions
F_a = configuration factor based on the geometry of the system (not all of the radiation emitted by a body may be intercepted by the second body)
σ = Stefan-Boltamann constant
A = Surface area (ft²)
T₁ and T₂ are the temperatures of the hot and cold bodies respectively (°R).

The net radiation between two bodies is thus proportional to the difference in the fourth powers of the absolute temperatures, whereas conduction and convection in general are proportional to the difference in the first powers of the temperatures.

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Two black bodies that radiate toward each other have a net heat flux between them. The net flow rate of heat between them is given by an adaptation of Equation:

Q = σ A (T₁⁴ - T₂⁴) - equation 3

Where:

A = Surface area of the first body (ft²)
σ = Stefan-Boltamann constant
T₁ = Temperature of the first body (°R)
T₂ = Temperature of the second body (°R)

All bodies above absolute zero temperature radiate some heat. The sun and earth both radiate heat toward each other. This seems to violate the Second Law of Thermodynamics, which states that heat cannot flow from a cold body to a hot body. The paradox is resolved by the fact that each body must be in direct line of sight of the other to receive radiation from it. Therefore, whenever the cool body is radiating heat to the hot body, the hot body must also be radiating heat to the cool body. Since the hot body radiates more heat (due to its higher temperature) than the cold body, the net flow of heat is from hot to cold, and the second law is still satisfied.