Biomedical Engineering Reference
In-Depth Information
This equation is applied for calculating the value of a second
shape factor between two bodies if the first is already known.
A second relation is called the summation rule, which says
the sum of shape factors for the complete environment of a body
equals 1.0:
The net energy exchanged by a surface is the difference between
the radiosity and the irradiation. For a gray surface with α = ε,
and therefore ρ = 1 − ε,
==−=ε+−ε −
Q
A
qJGE
(1
)
GG
(1.88)
b
n
∑
=
(1.85)
which can also be written as
F
1.0.
mk
→
k
=
1
−
−ε ε
EJ
A
=
b
Q
.
(1.89)
(1
)/
The summation rule accounts for the entire environment for an
object.
A limiting case is shown in Figure 1.9 in which
AA
s
This relationship of a flow across a resistance between a poten-
tial difference is shown in Figure 1.10. The format of Equation
1.89 is in terms of a flow that equals a difference in potential
divided by a resistance. In this case, the equation represents the
drop in potential from a black to a gray surface associated with
a finite surface radiation resistance. The equation can be rep-
resented graphically in terms of a steady state resistance. This
resistance applies at every surface within a radiating system that
has non-black radiation properties. Note that for a black surface
for which ε = 1, the resistance goes to zero.
A second type of radiation resistance is due to the geomet-
ric shape factors among multiple radiating bodies. The apparent
radiation potential of a surface is the radiosity. For the exchange
of radiation between two surfaces
A
1
and
A
2
, the net energy flow
equals the sum of the flows in both directions. The radiation
leaving surface 1 that is incident on surface 2 is
>>
. For
this geometry, the reciprocity relation dictates that
F
s
→
n
be van-
ishingly small since only a very small fraction of the radiation
leaving the large surface
s
will be incident onto the small surface
n
. The summation rule then shows that effectively
F
n
→
s
= 1.
The third geometric relationship states that the shape factors
for a surface to each component of its environment are additive.
If a surface
n
is divided into
l
components, then
n
l
=
∑
F
F
(1.86)
mn
→
()
mj
→
j
=
1
where
l
∑
JAF
A
=
A
.
111→
and in like manner, the radiation from 2 to 1 is
n
j
j
=
1
.
JAF
.
Equation 1.86 is often useful for calculating the shape factor for
complex geometries that can be subdivided into an assembly of
more simple shapes.
There exist comprehensive compendia of data for determina-
tion of a wide array of shape factors (Howell, 1982). The reader
is directed to such sources for detailed information. Application
of geometric data to radiation problems is very straightforward.
Evaluation of the temperature, surface property, and geometry
effects can be combined to calculate the magnitude of radiation
exchange among a system of surfaces. The simplest approach is
to represent the radiation process in terms of an equivalent elec-
trical network. For this purpose, two special properties are used:
the irradiation,
G
, which is the total radiation incident onto a
surface per unit time and area, and the radiosity,
J
, which is the
total radiation that leaves a surface per unit time and area. Also,
for the present time it is assumed that all surfaces are opaque (no
radiation is transmitted), and the radiation process is at steady
state. Thus, there is no energy storage within any components of
the radiating portion of the system.
The radiosity can be written as the sum of radiation emitted
and reflected from a surface, which is expressed as
222→
The net interchange between surfaces 1 and 2 is then the sum of
these two flows
JJ
AF
−
1
2
11 2
QJ AF
=
−
JAF
=
.
(1.9 0)
12
→
1112
→
2221
→
1/
→
This process can also be modeled via an electrical network as
shown in Figure 1.11.
Q
E
b
J
1-
ε
ε
A
FIGURE 1.10
Electrical resistance model for the drop in radiation
potential due to a gray surface defined by the property ε.
Q
J
1
J
2
1
A
1
F
1 2
FIGURE 1.11
Electrical resistance model for the radiation exchange
between two surfaces with a shape factor
F
1→2
.
J
=ε +ρ
E G
.
b
(1.87)
