**RADIATION
**

By R.L. Snyder and K.T. Paw U

Copyright - Regents of the University of
California

Created - June 28, 2000

Last Revision –June 13, 2001

*Electromagnetic
radiation is one method for transfer of energy without the need for a
medium.
The radiation emitted can be modeled as a function of frequency or
wavelength
using Planck’s function, where h and k are constants and c is the speed
of
light. Recall that:** *

Speed of light = wavelength x
frequency *
***( c = **

_{}

*h*=6.626 x 10^{-34} J s
Planck’s
constant

*k* = 1.3806 x 10^{-23} J K^{-1}^{ }Boltzmann’s constant

*c* = 3.0 x 10^{8} m s^{-1} Speed of light

The energy emitted by a black body
source is
a function of the 4^{th} power of the temperature in Kelvin
following
the Stefan-Boltzmann law.

The maximum wavelength (l_{max}) for
energy emitted depends on the temperature as defined by Wein’s law,
where the
wavelength is in mm and the temperature is in Kelvin.

**Solar**
mm

**Earth**
mm

The Stefan-Boltzmann law applies to
‘black’
bodies, which emit the maximum possible energy at all wavelengths. For
‘gray’ bodies,
which are not perfect emitters, an emissivity factor (0<e£ 1.0) is included.

For a black body, *e* = 1.0. For a gray body *e* < 1.0.

According to Kirchoff’s
Law, a body that is a good emitter at any given wavelength is
also a good
absorber at that wavelength.

*Emissivity
=
Absorptivity*

** **

**RADIATION
SYMBOLS AND TERMS**

Direct Solar Radiation (*Q** _{s}*) –
short wave band radiation energy flux density (W m

Diffuse Solar Radiation (*q*) – diffuse or scattered short wave band
radiation energy flux density
(W m^{-2}) received at the surface

Upward Long Wave Radiation (*L _{u}*) – long wave radiation (W m

Downward Long Wave Radiation (*L _{d}*) – long wave radiation (W m

Net Radiation (*R _{n}*) – the net amount of total radiation (W m

*R _{n}* = (1 -

* *

**DIRECT BEAM
RADIATION**

Outside of the Earth’s atmosphere,
the flux
density of solar radiation is called the ‘solar constant’ (*Q _{c}* = 1367 W m

As the radiation passes through the
atmosphere, some of the radiation is scattered and the flux density
that is
transmitted can be estimated using an extinction equation

where *x* is the path length
of the
direct radiation through the atmosphere and *k** _{a}* is the extinction coefficient, which
depends on
turbidity of the atmosphere. The flux density can also be determined
using a
transmission coefficient (

For our purposes we will use the
transmission
equation to avoid the need to determine path length through the
atmosphere.
Note that Q is the flux density flow of energy from
the sun (J s^{-1})
per unit area perpendicular to the sun’s rays. If the energy is
received by a
surface that is not perpendicular to the sun’s rays, a correction for
the angle
of incidence is required.

*To determine the flux density
received
by an object, the flow ***( or
flux) of direct beam short wave radiation from the sun is
divided by the
projection area (A_{p} = the projected area of an object that
is
perpendicular to the sun’s rays). On the other hand, the flow or flux
of the
direct radiation (R_{ab}) is calculated as:**

*R _{ab}* =

If we know *R _{ab}*,
then the amount of energy received per unit surface area of the object
is
calculated using the "interception factor" (

The surface area of an object is
estimated
using geometry, but determining *A _{p}* is often
difficult.
However, it is relatively simple to determine

*A _{p}* =

where *a* is the zenith
angle. This relationship comes from trigonometry where the cos(*a*) accounts for the fact that the length of
one dimension of the
horizontal shadow area is increased by the factor cos(*a*).

The zenith angle (a) is calculated using the formula

_{}

where *f* is the site latitude
in radians, *d* is the angle of the sun relative to the
equatorial
plane, *w* is the angular velocity of Earth’s
rotation (15^{o}
h^{-1 }= 0.2618 rad h^{-1}) and *t* is the time
in hours
relative to solar noon.

*t* = *h* – 12

where *h* is military time (*h*
=
0-24 starting at midnight). Using the above relationships, we can
determine the
flux density of direct radiation received per unit surface area as

**DIFFUSE
RADIATION**

Diffuse radiation is energy that is
scattered
by the atmosphere. Some of this radiation is received by the surface,
but it is
not direction dependent. Also, during clear days, the diffuse radiation
is
relatively constant during daytime at about 15% of the peak clear sky
radiation
until *a* > 75^{o}. For *a* > 75^{o} the diffuse radiation
decreases approximately
linearly with *a* from 75^{o} to 90^{o}.
During cloudy
conditions, the diffuse radiation is more complicated and measurements
are
recommended.

**Radiation on a
horizontal
surface**

When the direct solar energy is
received by a
horizontal surface, the horizontal area (*A _{h}*) is
equal to the object area (

The total short wave (solar)
radiation
received is the sum of the direct (*Q _{s}*) and
diffuse (

*R _{ns}* = (

Where 0 £ *a* £ 1.0 is the albedo (reflection)

**NET RADIATION**

*Net radiation is the net
amount of energy
from both short and long wave radiation that is absorbed by a surface. *

where is the direct beam and *q* is the diffuse solar radiation received by
an object and *L _{d}* and

_{}

_{}

where *e*_{a} is the emissivity of the sky, *e* is the emissivity of the surface, *T _{a}*
is the effective atmosphere temperature, and

*L _{u}* = -

*L _{d}* = (1 -

where *c*
is the fraction cloud cover.
The –9.0 W m^{-2}
is included in the right-hand expression to account for the difference
in cloud
base and weather station temperature. It is assumed that the colder
cloud base
will emit radiation at about 9.0 W m^{-2} less than if it were
at the
weather station temperature. In reality, this is an empirical
coefficient and
it varies depending on the actual cloud base temperature. The cloud
cover (*c*) is typically estimated as a function of
the ratio of actual (*R _{s}*) to maximum possible solar radiation (

For a grass surface*, *

*e** **»** *0.98

and for clear sky,

_{}

where *e* is the vapor
pressure in
Pascals (Pa) measured at screen height

_{}

Here, *T _{d}* is
the dew point temperature (

REVIEW OF RADIATION
TERMS

A = surface area of object

Q_{c}
= energy flux density on a surface normal to the rays w/o an atmosphere

Q
= energy flux density on a surface normal to the rays with an atmosphere

A_{p}
= area of object that intercepts radiation (normal to the rays)

A_{p}/A
= interception factor

A_{h}
= area of object shadow on a horizontal surface

A_{h}/A
= shape factor

R_{ab}
= Q A_{p} = energy flux or flow to the object

Q_{s}
= R_{ab}/A = Q (A_{p}/A) energy flux to an object
divided by
the surface area of the object

a = zenith angle is the
angle from a line
that is perpendicular to a flat (horizontal) surface

For
flat (horizontal) surfaces, A_{p}/A = cos(a)