One-Axis and Two-Axis Tracker Insolation. Compare the 40º latitude, clear-sky insolation on a collector at solar noon on the summer solstice for a two-axis tracking mount versus a single-axis polar mount. Ignore ground reflectance.

Question

One-Axis and Two-Axis Tracker Insolation. Compare the 40º latitude, clear-sky insolation on a collector at solar noon on the summer solstice for a two-axis tracking mount versus a single-axis polar mount. Ignore ground reflectance.

Accepted Answer

1. Two-Axis Tracker: To find the beam insolation from (7.21) [latex]I_{B} \ = \ Ae^{-km}[/latex], we need the air mass ratio m, the apparent extraterrestrial flux A, and the optical depth k. To find m, we need the altitude angle of the sun. Using (7.7) with a solstice declination of 23.45º,

[latex]\begin{matrix} \beta _{N} \ = \ 90º \ − \ L \ + \ \delta \ = \ 90 \ − \ 40 \ − \ 23.45 \ = \ 73.45º \ ext{Air mass ratio} \ m \ = \ \frac{1}{\sin \ \beta} \ = \ \frac{1}{\sin \ 73.45º} \ = \ 1.043 \quad \quad \end{matrix}[/latex]

From Table 7.6, or Eqs. (7.22), (7.23) and (7.28), we find A = 1088 W/m², k = 0.205, and C = 0.134. The direct beam insolation on the collector is therefore

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

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

[latex]C \ = \ 0.095 \ + \ 0.04 \ \sin \ \left[\frac{360}{365} \left(n \ - \ 100 ight) ight][/latex] (7.28)

[latex]I_{BC} \ = \ I_{B} \ = \ Ae^{-km} \ = 1088 \ \left({W}/{m^{2}} ight) \ \cdot \ e^{−0.205x 1.043} \ = \ 879 \ {W}/{m^{2}}[/latex]

Using (7.35) the diffuse radiation on the collector is

[latex]\begin{matrix} I_{DC} & = \ C I_{B} \ \left[\frac{1 \ + \ \cos\left(90º \ - \ \beta ight)}{2} ight] \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ & = \ 0.134 \ \cdot \ 879 \ \left[\frac{1 \ + \ \cos\left(90º \ - \ 73.45º ight)}{2} ight] \ = \ 115 \ {W}/{m^{2}} \end{matrix}[/latex]

The total is [latex]I_{C} \ = \ I_{BC} \ + \ I_{DC} \ = \ 879 \ + \ 115 \ = \ 994 \ {W}/{m^{2}}[/latex]

2. One-Axis Polar Tracker: The beam portion of insolation is given by (7.38)

[latex]I_{BC} \ = \ I_{B} \ \cos \ δ \ = \ 879 \ {W}/{m^{2}} \ \cos \left(23.45º ight) \ = \ 806 \ {W}/{m^{2}}[/latex]

The diffuse portion, using (7.39), is

[latex]\begin{matrix} I_{DC} & = \ C I_{B} \ \left[\frac{1 \ + \ \cos\left(90º \ - \ \beta \ + \ \delta ight)}{2} ight] \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ & = \ 0.134 \ \cdot \ 879 \ {W}/{m^{2}} \ \left[\frac{1 \ + \ \cos\left(90 \ - \ 73.45 \ + \ 23.45 ight)}{2} ight] \ = \ 104 \ {W}/{m^{2}} \end{matrix}[/latex]

The total is [latex]I_{C} \ = \ I_{BC} \ + \ I_{DC} \ = \ 806 \ + \ 104 \ = \ 910 \ {W}/{m^{2}}[/latex]

The two-axis tracker provides 994 W/m², which is only 9% higher than the single-axis mount.

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).

Question 7.12: One-Axis and Two-Axis Tracker Insolation. Compare the 40º la......

Related Answered Questions

Average Monthly Insolation on a Tilted Collector. Average horizontal insolation in Oakland, California (latitude 37.73ºN) in July is 7.32 kWh/m²-day. Estimate the insolation on a south-facing collector at a tilt angle of 30º with respect to the horizontal. Assume ground reflectivity of 0.2. ...

Verified Answer:

Verified Answer:

Verified Answer:

Beam Insolation on a Collector. In Example 7.8, at solar noon in Atlanta (latitude 33.7º) on May 21 the altitude angle of the sun was found to be 76.4º and the clear-sky beam insolation was found to be 902 W/m². Find the beam insolation at that time on a collector that faces 20º toward the ...

Verified Answer:

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. ...

Verified Answer:

Where in the World Are You? With your watch set for Pacific Standard Time (PST), you travel somewhere and when you arrive you note that the upper limb sunrise is at 1:11 A.M. (by your watch) and sunset is at 4:25 P.M. It is July 1^st (δ = 23.1º, E = −3.5 min). Where are you? ...

Verified Answer:

Sunrise in Boston. Find the time at which sunrise (geometric and conventional) will occur in Boston (latitude 42.3º) on July 1 (n = 182). Also find conventional sunset. ...

Verified Answer:

Solar Time to Local Time. Find Eastern Daylight Time for solar noon in Boston (longitude 71.1 ºW) on July 1^st. ...

Verified Answer:

Estimate the insolation available on a clear day in January on a south-facing collector with a fixed, 30º tilt angle at the site having the sun path and obstructions diagram shown in Fig. 7.15. ...

Verified Answer:

Tilt Angle of a PV Module. Find the optimum tilt angle for a south-facing photovoltaic module in Tucson (latitude 32.1º) at solar noon on March 1. ...

Verified Answer: