Biomedical Engineering Reference
In-Depth Information
emissiv it y, ε. In general, radiation properties are functions of the
radiation wavelength and for many practical systems can change
significantly over the thermal spectrum. Thus,
10
9
Visible spectrum
10
8
10
7
10
6
10
5
ET
E
(, )
(, )
.
λ
λ
(1.74)
λ
ελ =
(, )
T
λ
5,800 K
T
λ
,
b
2,000 K
1,000 K
10
4
An idealized real surface has radiant properties that are
wavelength independent and is termed a gray surface. For these
conditions,
10
3
10
2
800 K
10
1
ET
ET
()
()
ET
T
()
.
(1.75)
10
0
ε= =
σ
()
T
4
b
500 K
10
-1
100 K
10
-2
A gray surface has the effect of decreasing the magnitude of the
curves in Figure 1.7 by a constant factor over all wavelengths.
In addition to emission, surfaces are continually bombarded
by thermal radiation from their environments. The net radiant
flux at a surface is the difference between the energies received
and lost. As with emission, the surface radiation properties
play an important role in determining the amount of energy
absorbed by a surface. A black surface absorbs all incident radi-
ation, whereas real surfaces absorb only a fraction that is less
than one. The total radiant flux onto a surface from all sources
is called the
radiosity
and is denoted by the symbol
G
[W/m
2
].
In general, the incident radiation will be composed of many
wavelengths, denoted by
G
λ
. A surface can have three modes of
response to incident radiation: the radiation may be absorbed,
reflected, and/or transmitted. The fractions of incident radia-
tion that undergo each of these responses are determined by
three dimensionless properties: the coefficients of absorption,
α, reflection, ρ, and transmission, τ. Conservation of energy
applied at a surface dictates that the relationship among these
three properties must be
10
-3
50 K
10
-4
0.1
0.2
0.4 0.6
1
2
46 10
20
40 60
100
λ(µm)
FIGURE 1.7
Spectral blackbody emissive power as a function of sur-
face temperature and wavelength.
temperature there is an intermediate wavelength for which
E
λ,
T
has a maximum value, and this maximum increases mono-
tonically in magnitude and occurs at shorter wavelengths with
increasing temperature. Wien's displacement law, Equation 1.72,
describes the relationship between the absolute temperature and
the wavelength at which maximum emission occurs:
max
λ=µ⋅
T
2898[mk].
(1.72)
Equation 1.71 is integrated over the entire emission spec-
trum to obtain the expression for the total emitted radiation,
Equation 1.70.
α+ρ+τ=
1.
(1.76)
∞
Figure 1.8 illustrates these phenomena for radiation incident
onto a surface that is translucent, allowing some of the radia-
tion to pass through. All three of the properties are wavelength
dependent.
2
2
π
hc
hc
kT
∫
o
o
4
ET
()
=
d
λ=σ
T
(1.73)
b
−
5
λ
exp
1
0
λ
Equation 1.73 represents the area under an isothermal curve in
Figure 1.7 depicting the maximum amount of energy that can
be emitted from a surface at a specified temperature. This set
of equations provides the basis for quantifying the temperature
effect on thermal radiation. It applies to idealized, black surfaces.
ρ
G
λ
λ
(
λ
)
G
λ
1.2.4.2 Surface Effects
Next we will consider the effect of real, rather than idealized,
surface properties on thermal radiation exchange. Real surfaces
emit less than blackbody radiation at a given temperature. The
ratio of real to black radiation levels defines a property called the
α
λ
(
λ
)
G
λ
τ
λ
(
λ
)
G
λ
FIGURE 1.8
Absorption, reflection, and transmission phenomena
for a surface irradiated with a multi-wavelength incident radiation,
G
λ
.