Advanced Building Simulation

74 Degelman
atmosphere is dependent on two factors: the distance the solar beam has to penetrate
the atmosphere (known as the air mass) and the degree of sky obscuration (defined
by the atmospheric turbidity).
The radiation values are typically segregated into two components—the direct and
diffuse. The direct normal insolation utilizes a fairly well-known equation. The generally
accepted formula for direct normal solar radiation is
I DN�I o exp[�a/sin �] (3.13)
where I DN is the direct normal insolation; I o, the apparent solar constant; a, the atmospheric
extinction coefficient (turbidity); �, the solar altitude angle; and 1/sin(�) is
referred to as the “air mass.”
The insolation value will be in the same units as the apparent solar constant
(usually in W/m 2 or Btu/h per sq. ft.) The apparent solar constant is not truly constant;
it actually varies a small amount throughout the year. It varies from around
1,336W/m 2 in June to 1,417W/m 2 in December. This value is independent of your
position on earth. A polynomial equation was fit to the values published in ASHRAE
Handbook of Fundamentals (ASHRAE 2001: chapter 30, table 7):
I o (W/m 2 )�1,166.1�77.375 cos(�)�2.9086 cos 2 (�) (3.14)
The average value of the apparent solar constant is around 1,167; whereas the
average value of the extraterrestrial “true solar constant” is around 1,353W/m 2 . This
means that the radiation formula (Equation (3.13)) will predicts a maximum of 86%
of the insolation will penetrate the atmosphere in the form of direct normal radiation
(usually referred to as “beam” radiation).
Everything in Equation (3.13) is deterministic except for a, which takes on a
stochastic nature. The larger portion of work is in the establishment of a value for a,
the atmospheric extinction coefficient (or turbidity). This variable defines the amount
of atmospheric obscuration that the sun’s ray has to penetrate. The higher value for
a (cloudier/hazier sky), the less the radiation that passes through. ASHRAE publishes
monthly values for a, but these are of little value because they are only for clear days.
In the simulation process, it is necessary to utilize an infinite number of a values so
that the sky conditions can be simulated through a full range of densely cloudy to
crystal clear skies.
Fortunately, the methods presented here require that only one value be required to
do an hour-by-hour analysis of solar radiation intensities for an entire month. This
one value is the average daily solar radiation on a horizontal surface (H). Liu and
Jordan (1960) have shown that with the knowledge of this one value, one can predict
how many days there were during the month in which the daily solar radiation
exceeded certain amounts and have produced several cumulative distribution curves
to show this. Through the use of such a statistical distribution, the local sky conditions
for each day can be predicted and thus the hourly conditions for each day can
also be calculated. Their research results are shown in a set of cumulative distribution
curves. These curves (Figure 3.8) show the distribution of daily clearness indices
(K T) when the monthly overall clearness index (K – T) is known. The Liu–Jordan curves
are exceptionally adept for simulation work. In effect, the simulation process works