This page provides the chapter on radiant heat transfer from the "DOE Fundamentals Handbook: Thermodynamics, Heat Transfer, and Fluid Flow," DOE-HDBK-1012/2-92, U.S. Department of Energy, June 1992.

Other related chapters from the "DOE Fundamentals Handbook: Thermodynamics, Heat Transfer, and Fluid Flow" can be seen to the right.

DOE Handbook: Heat Transfer

Radiant heat transfer is thermal energy transferred by means of electromagnetic waves or particles.

Radiant heat transfer involves the transfer of heat by electromagnetic radiation that arises due to the temperature of a body. Most energy of this type is in the infra-red region of the electromagnetic spectrum although some of it is in the visible region. The term thermal radiation is frequently used to distinguish this form of electromagnetic radiation from other forms, such as radio waves, x-rays, or gamma rays. The transfer of heat from a fireplace across a room in the line of sight is an example of radiant heat transfer.

Radiant heat transfer does not need a medium, such as air or metal, to take place. Any material that has a temperature above absolute zero gives off some radiant energy. When a cloud covers the sun, both its heat and light diminish. This is one of the most familiar examples of heat transfer by thermal radiation.

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.

$$\dot{Q} = \sigma ~A ~T^4$$
(2-12)

where:

 $$\dot{Q}$$ = heat transfer rate (Btu/hr) σ = Stefan-Boltzman constant (0.174 Btu/hr-ft2-°R4) A = surface area (ft2) T = temperature (°R)

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 2-12.

$$\dot{Q} = \sigma ~A ~(T_1^4 - T_2^4)$$

where:

 A = surface area of the first body (ft2) 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.

### Emissivity

Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and are called gray bodies. To take into account the fact that real objects are gray bodies, Equation 2-12 is modified to be of the following form.

$$\dot{Q} = \varepsilon ~\sigma ~A ~T^4$$

where:

ε = emissivity of the gray body (dimensionless)

Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case. Emissivity is a dimensionless number and has a maximum value of 1.0.

Radiative heat transfer rate between two gray bodies can be calculated by the equation stated below.

$$\dot{Q} = f_a ~f_e ~\sigma ~A ~(T_1^4 - T_2^4)$$

where:

 fa = is the shape factor, which depends on the spatial arrangement of the two objects (dimensionless) fe = is the emissivity factor, which depends on the emissivities of both objects (dimensionless)

The two separate terms fa and fe can be combined and given the symbol f. The heat flow between two gray bodies can now be determined by the following equation:

$$\dot{Q} = f ~\sigma ~A ~(T_1^4 - T_2^4)$$
(2-13)

The symbol (f) is a dimensionless factor sometimes called the radiation configuration factor, which takes into account the emissivity of both bodies and their relative geometry. The radiation configuration factor is usually found in a text book for the given situation. Once the configuration factor is obtained, the overall net heat flux can be determined. Radiant heat flux should only be included in a problem when it is greater than 20% of the problem.

Example:

Calculate the radiant heat between the floor (15 ft x 15 ft) of a furnace and the roof, if the two are located 10 ft apart. The floor and roof temperatures are 2000°F and 600°F, respectively. Assume that the floor and the roof have black surfaces.

Solution:

A1 = A2 = (15 ft) (15 ft) = 225 ft2

T1 = 2000°F = 2460°R

T2 = 600°F = 1060°R

Tables from a reference book, or supplied by the instructor, give:

f1-2 = f2-1 = 0.31

$$\begin{eqnarray} Q_{1-2} &=& \sigma A f (T_1^4 - T_2^4) \nonumber \\ &=& \left( 0.174 ~{\text{Btu} \over \text{hr-ft}^2\text{-}^{\circ}\text{R}} \right) (225 ~\text{ft}^2) (0.31) [ (2460^{\circ}\text{R})^4 - (1060^{\circ}\text{R})^4 ] \nonumber \\ &=& 4.29 \times 10^{14} ~\text{Btu/hr} \end{eqnarray}$$