Biomedical Engineering Reference
In-Depth Information
The three properties are defined according to the fraction of
radiosity that is absorbed, reflected, and transmitted:
Since the surface areas of the interior bodies are very small in
comparison to the enclosure area, their influence on the radia-
tion field is negligible. Also, the radiosity to the interior bodies
is a combination of emission and reflection from the enclosure
surface. The net effect is that the enclosure acts as a blackbody
cavity regardless of its surface properties. Therefore, the radios-
ity within the enclosure is expressed as
G
G
()
()
λ
λ
,
abs
αλ=
()
(1.77)
λ
λ
λ
G
G
()
()
λ
λ
,
ref
ρλ=
()
(1.78)
λ
λ
GE
( .
bs
=
(1.8 0)
λ
G
G
()
()
.
λ
λ
For thermal equilibrium within the cavity, the temperatures of
all surfaces must be equal.
T
n
-1
=
T
n
=
T
.. =
T
s
. A steady state
energy balance between absorbed and emitted radiation on one
of the interior bodies yields
λ
,
tr
τλ=
()
(1.79)
λ
λ
It is well known that there is a very strong spectral (wave-
length) dependence of these properties. For example, the green-
house effect occurs because glass has a high transmissivity (τ)
at relatively short wavelengths in the visible spectrum that are
characteristic of the solar flux. However, the transmissivity is
very small in the infrared spectrum in which terrestrial emis-
sion occurs. Therefore, heat from the sun readily passes through
glass and is absorbed by interior objects. In contrast, radiant
energy emitted by these interior objects is reflected back to the
source. The net result is a warming of the interior of a system
that has a glass surface exposed to the sun. The lens of a camera
designed to image terrestrial sources of thermal radiation, pre-
dominantly in the infrared spectrum, must be fabricated from a
material that is transparent at those wavelengths. It is important
to verify whether the spectral dependence of material surface
properties is important for specific applications involving ther-
mal radiation.
An additional important surface property relationship is
defined by
Kirchhoff's law
, which applies for a surface that is in
thermal equilibrium with its environment. Most thermal radia-
tion analyses are performed for processes that are steady state.
For the surface of a body
n
having a surface area
A
n
, at steady
state the radiation gained and lost is balanced so that the net
exchange is zero. To illustrate, we may consider a large isother-
mal enclosure at a temperature,
T
s
, containing numerous small
bodies, each having unique properties and temperature. See
Figure 1.9.
α− ⋅ =
(1. 81)
()()0.
nn ns
GA
ET A
n
The term for radiosity may be eliminated between the two fore-
going equations:
ET
()
.
ns
n
ET
()
=
(1.82)
bs
α
This relationship holds for all of the interior bodies. Comparison
of Equations 1.75 and 1.82 shows that the emissivity and absorp-
tivity are equal:
(1.83)
α=ε
,
or
α =ε
λ
.
λ
The general statement of this relationship is that for a gray
surface, the emissivity and absorptivity are equal and indepen-
dent of spectral conditions.
1.2.4.3 Surface Geometry Effects
The third factor influencing thermal radiation transfer is the
geometric sizes, shapes, and configurations of body surfaces in
relation to the aggregate surfaces in the environment. This effect
is quantified in terms of a property called the shape factor, which
is solely a function of the geometry of a system and its environ-
ment. By definition, the shape factor is determined for multiple
bodies, and it is related to the size, shape, separation, and orien-
tation of the bodies. The shape factor
F
m
→
n
is defined between
two surfaces,
m
and
n
, as the fraction of energy that leaves sur-
face
m
that is incident onto the surface
n
. It is very important to
note that the shape factor is directional. The shape factor from
body
m
to
n
is probably not equal to that from body
n
to
m
.
Values for shape factors have been compiled for a broad range
of combinations of size, shape, separation, and orientation and
are available as figures, tables, and equations (Howell 1982;
Siegel and Howell 2002). There are a number of simple relations
that govern shape factors and that are highly useful in working
many types of problems. One is called the reciprocity relation:
G = E
b
(
T
s
)
T
s
A
n
A
n
-1
E
n
E
n
+1
E
n
-1
A
n
+1
A
s
FIGURE 1.9
Steady state thermal radiation within a large isothermal
enclosure containing multiple small bodies.
AF AF
.
nn mmmn
=
(1.8 4)
→
→
