An inclined surface situated on the terrestrial plane is characterized by two geo­metrical quantities: inclination b, the inclination of the surface compared to the horizontal, and surface’s azimuth aw, the angle that the projection on the normal to the surface’s horizontal plane has to rotate to superimpose itself on the southern direction. If that rotation is counter clockwise, angle aw is considered to be posi­tive. The angle between the solar rays and the normal to the surface is called the angle of incidence i.

The direct radiation intercepted by a surface is:

Gb = Ibn cos i

The general expression for cos i:

cos i = cos(a — aw )cos a sen b + sen a cos b (17)

or, as a function of the fundamentals angles L, d and h:

cos i = send (sen L cos b — cos L sen b cos aw)

+ cos d cos h(cosL cos b + sen L sen bcos aw)

+ cos d sen b senawsenh (18)

This expression indicates three cases of particular interest:

• For a horizontal surface (b = 0°), we have:

cos i = sen d sen L + cos d cos L cos h = sen a (19)

• For a vertical surface facing south (aw = 0°, b = 90°), we have:

cos i = — sen d cos L + cos d sen L cos h (20)

The introduction of the surface’s azimuth aw results in a remarkable complication when compared with the case where aw = 0°. In that case, using geometrical demon­stration, it can be easily shown that radiation on an inclined surface of angle b at latitude L is equal to the radiation on latitude (L — b), these being surfaces parallels. Therefore, for an inclined surface facing south, we have:

cos i = sen(L — b)sen d + cos(L — b)cos h cos d (21)

Often, it is useful to know when the Sun rises and sets as regards an oriented surface: the surface ‘sees’ the Sun when the angle of incidence is lower than 90° and the solar altitude is more than 0° at the same time. The Sun rises and sets on
the surface connected with the minimum hour angle ha’ (and ht’), between the absolute value which is calculated by ignoring sen a (hour angle of dawn and sun­set on the horizon), and the absolute value is obtained by ignoring cos i (i. e. con­sidering i = 90°).

As a rule, in the northern hemisphere for southward oriented surfaces, when it is winter and days are short, it is sufficient to ignore sen a; however, during sum­mer, when the angle of incidence is more than 90° and the Sun has already risen and before it sets, it is enough to ignore cos i. The general rule, valid only if the surface actually sees the Sun, is given by the following equations:

min ha(a = 0°), ha(i = 90°)

= min ht (a = 0°), ht(i = 90°)|

In the simple case of inclined surfaces facing south, we have: cos i = cos 90° = 0

= sen(L — b)sen d + cos (L — b)cos d cos h

I ha (i = 90°) = I ht (i = 90°) =arcos [ — tg (L — b) tg d] (25)

|ha (a = 0°) = |ht(a = 0°) = arcos [ — tg L tg d] (26)

For northward oriented surfaces, there could be both two dawns and two sunsets (in spring and in summer, in the northern hemisphere) and no dawns and no sun­sets, that is, absence of direct lightening (in autumn and winter).

When the surface is not oriented southward, it is not possible to get simple closed — form expressions for the hour angles on the surface at dawn and sunset.

Assuming that 4o is the instantaneous direct radiation on a horizontal plane, linked to normal direct radiation by the relation:

hn=ho/sena (27)

and applying the (16), we get:

G = /bocos i/sen a = I bo R (28)

where

Rb = cos i/sen a (29)

Rb is the inclination factor for direct radiation; remembering that for a horizontal surface it is sufficient to put b = 0°, expression (28) states that the direct radiation Gb on a surface that is inclined and oriented in any direction is equal to the product of direct radiation Ibo on a horizontal plane and the inclination factor [1, 3].

For a southward oriented surface, we have: