Environmental Engineering Reference
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and the amount of energy per unit surface area incident on the surface (irradiation H).
Both in general depend on wavelength and on direction, but here, we consider the
simple case of no wavelength dependence (gray surface) and no direction dependence
(diffuse surface). The quantities to be determined in surface-to-surface radiative heat
transfer problems are the surface temperature and the surface radiative heat flux
(defined to be positive when the net balance is that energy is leaving the surface).
In any specific application, the bounding walls of a system are subdivided into parts
that have constant material properties and have either constant temperature or constant
heat flux. Assuming that the walls are opaque, i.e., no radiative energy is transmitted
through the wall, the material properties determining the surface-to-surface transfer
are emissivity
. In the presence of transmission (e.g., in case
of a glass window), also the transmissivity would have to be considered. In the
absence of transmission of radiation, the part that is not absorbed is reflected with
reflectivity
ε
e
and absorptivity
α
ρ
=1
−
α
. For a gray diffuse surface,
α
=
ε
e
, which is known as Kirchhoff
'
s
law. The emissive power of a surface is given by
E
=
ε
e
E
b
, where
E
b
is the emissive
power of an ideal surface absorbing all incident radiation, called black surface, which
is given by
T
4
E
b
=
σ
ð
Eq
:
4
:
10
Þ
with
σ
being the Stefan
-
Boltzmann constant. The part of the irradiation that
is absorbed is given by
H. The radiative energy leaving the surface is called
radiosity, denoted J, and is given by the sum of emitted and reflected radiation:
J=
α
)H. The radiative heat flux leaving the surface can be written in
two different ways. In the first view, it is the balance between emitted and
absorbed radiation: q =
ε
e
E
b
+(1
−
α
H. In the second view, it is the balance between
the energy leaving the surface and the energy approaching the surface: q = J
ε
e
E
b
−
α
H.
In order to describe the radiative heat transfer between different parts of the solid
boundaries of a certain domain, it is convenient to divide the boundary in N
s
surfaces with, by assumption, uniform properties. The areas A
i
of the surfaces
and their emissivity
−
,N
s
) characterize the problem. The solution
of the heat transfer problem involves both temperature T
i
and heat flux
q
i
(i = 1,
…
ε
e
,
i
=
α
i
(i = 1,
…
,N
s
). In general, for each surface, either temperature or heat flux is known and
the other is to be determined. The radiative energy leaving surface A
j
is given by
A
j
J
j
, and the part of this that is intercepted by surface A
j
is given by the product
F
i−j
A
j
J
j
. Here, F
i−j
is the view factor or configuration factor defined as the frac-
tion of the radiative energy leaving the surface number
i
that is intercepted by
the surface number
j
. Here, intercepted indicates that the radiation is contributing
to the irradiation H
j
and not that the radiation is absorbed by the surface number
j
. The view factors are function of the relative position and orientation of the
surfaces. The values of the configuration factors have to be determined before
the surface heat transfer problem can be solved. Many view factors are available
from databases, and unknown view factors can often easily be determined from
known view factors using so-called view factor algebra (see Modest, 2003).
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