View Factors
where y is the inclination angle of the surface, I,+ is the zenith angle of
the sun, AZ is the azimuth angle of the sun with respect to due south,
and AS is the aspect angle (angle between south and the projection onto
the horizontal of the normal to the inclined surface) of the surface. If the
surface is sloped, Fd and Fa usually are less than 1.0; typically Fd =
Fa = (1 + cos y)/2.
It may seem strange that Fr and F, are zero for a plant canopy since
the leaves do receive reflected and emitted radiation from the soil and
from lower layers in the canopy. The important thing to remember here
is that we are not trying to deal with the details of leaf processes when
we compute the absorption of radiation by a canopy. We imagine that we
are far enough from the canopy so that we can treat it as a single, flat
surface which absorbs and emits radiation. Thus we consider the canopy
to be an object with only one side; a practical impossibility, but useful
and consistent conceptually. Later we deal with the details of radiative
exchange by canopy elements.
As stated earlier, Fp for an animal is just equal to APIA, the ratio
of projected area perpendicular to the solar beam to total animal area.
Figure 11.6 shows this ratio for several objects which approximate the
shapes of animals. To use Fig. 11.6, one simply determines the angle
between the longitudinal axis ofthe animal and the solar beam, the general
shape of the animal, and the ratio of length to diameter. It appears that
the view factor should fall in the range 0.1 to 0.3. A sphere has a view
factor for beam radiation of 0.25.
If the effect of shadows are ignored, the diffuse view factors for an
animal suspended above the ground hemisphere and below a sky hemisphere are Fd = Fr = Fa = F, = 0.5 and F, = 1. Both the sky and
the ground "see" more than half the body surface area of the animal, but
the cosine weighting results in the view factor being 0.5. If the animal
is lying on the ground, a large fraction of its surface is not accessible to
radiation, and F,, Fr and F, must be adjusted accordingly.
The view factors for a single leaf are similar to those for an animal.
The view factors for a leaf suspended over the ground and under the sky
hemisphere are F, = 0.5 cose, Fd = Fr = F, = Fg = 0.5, and
F, = 1. Equation (1 1.16) is used to compute 8.
Example 11.3. Find the net radiation for the grass surface in Example 11.2 if the air temperature is 30" C and the grass temperature is
35" C.
Solution. From Example 1 1.2, Sp = 938 W m-2, Sd = 1 10 W mP2,
and Sr = 240 W m-2. From Table A.3, the black body emittances for
30" C and 35" C are 479 and 5 11 W m-2, and the clear sky emissivity
at Ta = 30" C is 0.85. The sky thermal radiant emittance is therefore
La = 0.85 x 479 = 407 W m-2. The ground thermal radiant emittance
is E,OT; = 0.97 x 5 1 1 = 496 W m-2. The shortwave absorbtivity is
equal to 1 - p,. For grass p, = 0.26 (Example 11.2) so a, = 0.74.
where y is the inclination angle of the surface, I,+ is the zenith angle of
the sun, AZ is the azimuth angle of the sun with respect to due south,
and AS is the aspect angle (angle between south and the projection onto
the horizontal of the normal to the inclined surface) of the surface. If the
surface is sloped, Fd and Fa usually are less than 1.0; typically Fd =
Fa = (1 + cos y)/2.
It may seem strange that Fr and F, are zero for a plant canopy since
the leaves do receive reflected and emitted radiation from the soil and
from lower layers in the canopy. The important thing to remember here
is that we are not trying to deal with the details of leaf processes when
we compute the absorption of radiation by a canopy. We imagine that we
are far enough from the canopy so that we can treat it as a single, flat
surface which absorbs and emits radiation. Thus we consider the canopy
to be an object with only one side; a practical impossibility, but useful
and consistent conceptually. Later we deal with the details of radiative
exchange by canopy elements.
As stated earlier, Fp for an animal is just equal to APIA, the ratio
of projected area perpendicular to the solar beam to total animal area.
Figure 11.6 shows this ratio for several objects which approximate the
shapes of animals. To use Fig. 11.6, one simply determines the angle
between the longitudinal axis ofthe animal and the solar beam, the general
shape of the animal, and the ratio of length to diameter. It appears that
the view factor should fall in the range 0.1 to 0.3. A sphere has a view
factor for beam radiation of 0.25.
If the effect of shadows are ignored, the diffuse view factors for an
animal suspended above the ground hemisphere and below a sky hemisphere are Fd = Fr = Fa = F, = 0.5 and F, = 1. Both the sky and
the ground "see" more than half the body surface area of the animal, but
the cosine weighting results in the view factor being 0.5. If the animal
is lying on the ground, a large fraction of its surface is not accessible to
radiation, and F,, Fr and F, must be adjusted accordingly.
The view factors for a single leaf are similar to those for an animal.
The view factors for a leaf suspended over the ground and under the sky
hemisphere are F, = 0.5 cose, Fd = Fr = F, = Fg = 0.5, and
F, = 1. Equation (1 1.16) is used to compute 8.
Example 11.3. Find the net radiation for the grass surface in Example 11.2 if the air temperature is 30" C and the grass temperature is
35" C.
Solution. From Example 1 1.2, Sp = 938 W m-2, Sd = 1 10 W mP2,
and Sr = 240 W m-2. From Table A.3, the black body emittances for
30" C and 35" C are 479 and 5 11 W m-2, and the clear sky emissivity
at Ta = 30" C is 0.85. The sky thermal radiant emittance is therefore
La = 0.85 x 479 = 407 W m-2. The ground thermal radiant emittance
is E,OT; = 0.97 x 5 1 1 = 496 W m-2. The shortwave absorbtivity is
equal to 1 - p,. For grass p, = 0.26 (Example 11.2) so a, = 0.74.