Direct Beam Radiation at the Surface of the Earth. Find the direct beam solar radiation normal to the sun’s rays at solar noon on a clear day in Atlanta (latitude 33.7º) on May 21. Use (7.22) and (7.23) to see how closely they approximate Table 7.6.

Question

Direct Beam Radiation at the Surface of the Earth. Find the direct beam solar radiation normal to the sun’s rays at solar noon on a clear day in Atlanta (latitude 33.7º) on May 21. Use (7.22) and (7.23) to see how closely they approximate Table 7.6.

[latex]A \ = \ 1160 \ + \ 75 \ \sin \ \left[\frac{360}{365} \left(n \ - \ 275\right)\right] \quad \left({W}/{m^{2}}\right)[/latex] (7.22)

[latex]k \ = \ 0.174 \ + \ 0.035 \ \sin \ \left[\frac{360}{365} \left(n \ - \ 100\right)\right][/latex] (7.23)

Accepted Answer

Using Table 7.1 to help, May 21 is day number 141. From (7.22), the apparent extraterrestrial flux, A, is

[latex]\begin{matrix} A & = \ 1160 \ + \ 75 \ \sin \ \left[\frac{360}{365} \left(n \ - \ 275 ight) ight] \ = \ 1160 \ + \ 75 \ \sin \ \left[\frac{360}{365} \left(141 \ - \ 275 ight) ight] \ & = \ 1104 \ {W}/{m^{2}} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ \end{matrix}[/latex]

(which agrees with Table 7.6).

From (7.23), the optical depth is

[latex]\begin{matrix} k & = \ 0.174 \ + \ 0.035 \ \sin \ \left[\frac{360}{365} \left(n \ - \ 100 ight) ight] \quad \quad \quad \quad \quad \ & = \ 0.174 \ + \ 0.035 \ \sin \ \left[\frac{360}{365} \left(141 \ - \ 100 ight) ight] \ = \ 0.197 \end{matrix}[/latex]

(which is very close to the value given in Table 7.6).

From Table 7.2, on May 21 solar declination is 20.1º, so from (7.7) the altitude angle of the sun at solar noon is

[latex]\beta _{N} \ = \ 90º \ − \ L \ + \ \delta \ = \ 90 \ − \ 33.7 \ − \ 20.1 \ = \ 76.4º[/latex]

The air mass ratio (7.4) is

[latex]m \ = \ \frac{1}{\sin \ \beta} \ = \ \frac{1}{\sin \left(76.4º ight)} \ = \ 1.029[/latex]

Finally, using (7.21) the predicted value of clear sky beam radiation at the earth’s surface is

[latex]I_{B} \ = \ Ae^{−km} \ = \ 1104 \ e^{− 0.197 imes 1.029} \ = \ 902 \ {W}/{m^{2}}[/latex]

TABLE 7.6 Optical Depth k, Apparent Extraterrestrial Flux A, and the Sky Diffuse Factor C for the $21^{st}$ Day of Each Month
Month:	Jan	Feb	Mar	Apr	May	June	July	Aug	Sept	Oct	Nov	Dec
A (W/ $m^{2}$ ):	1230	1215	1186	1136	1104	1088	1085	1107	1151	1192	1221	1233
k:	0.142	0.144	0.156	0.180	0.196	0.205	0.207	0.201	0.177	0.160	0.149	0.142
C:	0.058	0.060	0.071	0.097	0.121	0.134	0.136	0.122	0.092	0.073	0.063	0.057
Source: ASHRAE (1993).

TABLE 7.1 Day Numbers for the First Day of Each Month
January	n = 1	July	n = 182
February	n = 32	August	n = 213
March	n = 60	September	n = 244
April	n = 91	October	n = 274
May	n = 121	November	n = 305
June	n = 152	December	n = 335

Question 7.8: Direct Beam Radiation at the Surface of the Earth. Find the ......

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