Page 191 - Mechanical Engineers' Handbook (Volume 4)

180 Heat-Transfer Fundamentals

T 4
e b
where e is the total emissive power and is the Stefan-Boltzmann constant, which has the
b
2
4
4
2
value 5.729 10 8 W/m K (0.173 10 8 Btu/hr ft R ).
Planck’s Distribution Law
The temperature amount of energy leaving a blackbody is described as the spectral emissive
power, e , and is a function of wavelength. This function, which was derived from quantum
b
theory by Planck, is
2 C
e b 1
[exp(C / T) 1]
5
2
2
2
where e b has a unit W/m m (Btu/hr ft m).
2
Values of the constants C and C are 0.59544 10 16 W m (0.18892 10 8
2
1
2
4
Btu m /hr ft ) and 14,388 m K (25,898 m R), respectively. The distribution of the
spectral emissive power from a blackbody at various temperatures is shown in Fig. 19, which
shows that the energy emitted at all wavelengths increases as the temperature increases. The
maximum or peak values of the constant temperature curves illustrated in Fig. 20 shift to
the left for shorter wavelengths as the temperatures increase.
The fraction of the emissive power of a blackbody at a given temperature and in the
wavelength interval between and can be described by
1
2
ed ed
1
1
2
F T T T 4 0 b 0 b F o T F o T
1
1
2
2

4
where the function F o T (1/ T ) e b d is given in Table 16. This function is useful
o
for the evaluation of total properties involving integration on the wavelength in which the
spectral properties are piecewise constant.
Wien’s Displacement Law
The relationship between these peak or maximum temperatures can be described by Wien’s
displacement law,
max T 2897.8 m K
or
max T 5216.0 m R
3.2 Radiation Properties
While to some degree, all surfaces follow the general trends described by the Stefan-
Boltzmann and Planck laws, the behavior of real surfaces deviates somewhat from these. In
fact, because blackbodies are ideal, all real surfaces emit and absorb less radiant energy than
a blackbody. The amount of energy a body emits can be described in terms of the emissivity
and is, in general, a function of the type of material, the temperature, and the surface con-
ditions, such as roughness, oxide layer thickness, and chemical contamination. The emissivity
is, in fact, a measure of how well a real body radiates energy as compared with a blackbody
of the same temperature. The radiant energy emitted into the entire hemispherical space
4
above a real surface element, including all wavelengths is given by e T , where is
less than 1.0 and is called the hemispherical emissivity (or total hemispherical emissivity to