0.723

0.294

1.841

Finally the useful gain (UG) of the collector for that hour is (April has 30 days):

The results for the other hours are shown in Table 11.10. The useful gain for the month is equal to 427.6 MJ/m2.

Table 11.10 Results for All Hours in Example 11.11

Table 11.10 Results for All Hours in Example 11.11

Hour |
h (°) |
e (°) |
Ke |
xc |
rd |
r |
Rh |
kT |
Xm |
g |
$ |
UG |

8-9 |
52.5 |
54.3 |
0.929 |
0.294 |
0.080 |
0.075 |
0.898 |
0.591 |
1.841 |
5.289 |
0.723 |
25.11 |

9-10 |
37.5 |
40.1 |
0.969 |
0.159 |
0.100 |
0.101 |
1.036 |
0.635 |
1.776 |
3.464 |
0.845 |
47.54 |

10-11 |
22.5 |
26.7 |
0.988 |
0.144 |
0.114 |
0.120 |
1.149 |
0.666 |
1.737 |
2.802 |
0.859 |
64.93 |

11-12 |
7.5 |
16.2 |
0.996 |
0.102 |
0.122 |
0.132 |
1.293 |
0.682 |
1.747 |
2.953 |
0.900 |
84.90 |

12-13 |
-7.5 |
16.2 |
0.996 |
0.080 |
0.122 |
0.132 |
1.310 |
0.682 |
1.753 |
3.049 |
0.921 |
88.01 |

13-14 |
-22.5 |
26.7 |
0.988 |
0.250 |
0.114 |
0.120 |
1.127 |
0.666 |
1.728 |
2.676 |
0.760 |
56.34 |

14-15 |
-37.5 |
40.1 |
0.969 |
0.355 |
0.100 |
0.101 |
1.062 |
0.635 |
1.787 |
3.695 |
0.669 |
38.59 |

15-16 |
-52.5 |
54.3 |
0.929 |
0.544 |
0.080 |
0.075 |
1.093 |
0.591 |
1.936 |
14.63 |
0.525 |
22.20 |

Total = |
427.6 |

Although the $ curves method is a very powerful tool, caution is required to avoid possible misuse. For example, due to finite storage capacity, the critical level of collector inlet temperature for liquid-based domestic solar heating systems varies considerably during the month, so the $ curves method cannot be applied directly. Exceptions to this rule are air heating systems during winter, where the inlet air temperature to the collector is the return air from the house, and systems with seasonal storage where, due to its size, storage tank temperatures show small variations during the month.

As indicated in Section 11.2.2, the use of $ curves involves a lot of calculations. Klein (1978) and Collares-Pereira and Rabl (1979b; 1979c) simplified the calculations for systems for which a critical radiation level can be used for all hours of the month.

Daily utilizability is defined as the sum for a month over all hours and all days of the radiation on a tilted surface that is above a critical level, divided by the monthly radiation. This is given in Eq. (11.30). The critical level, Itc, is similar to Eq. (11.36), but in this case, the monthly average (toi) product must be used and the inlet and ambient temperatures are representative temperatures for the month:

In Eq. (11.41), the term (ra)/(Ta)n can be estimated with Eq. (11.11). The monthly average critical radiation ratio is the ratio of the critical radiation level, Itc, to the noon radiation level for a day of the month in which the total radiation for the day is the same as the monthly average. In equation form,

FrUlTj Ta)

C rnRnH rnRnKTH0 The monthly average daily useful energy gain is given by

Daily utilizability can be obtained from Eq. (11.34).

It should be noted that, even though monthly average daily utilizability reduces the complexity of the method, calculations can be still quite tedious, especially when monthly average hourly calculations are required.

It is also noticeable that the majority of the aforementioned methods for computing solar energy utilizability have been derived as fits to North American data versus the clearness index, which is the parameter used to indicate the dependence of the climate. Carvalho and Bourges (1985) applied some of these methods to European and African locations and compared results with values obtained from long-term measurements. Results showed that these methods can give acceptable results when the actual monthly average daily irradiation on the considered surface is known.

Examples of this method are given in the next section, where the $ and /-chart methods are combined.

11.3 THE /-CHART METHOD

The utilizability design concept is useful when the collector operates at a known critical radiation level during a specific month. In a practical system, however, the collector is connected to a storage tank, so the monthly sequence of weather and load time distributions cause a fluctuating storage tank temperature and thus a variable critical radiation level. On the other hand, the /-chart was developed to overcome the restriction of a constant critical level but is restricted to systems delivering a load near 20°C.

Klein and Beckman (1979) combined the utilizability concept described in the previous section with the /-chart to produce the !>, /-chart design method for a closed loop solar energy system, shown in Figure 11.10. The method is not restricted to loads that are at 20°C. In this system, the storage tank is assumed to be pressurized or filled with a liquid of high boiling point so that no energy dumping occurs through the relief valve. The auxiliary heater is in parallel with the solar energy system. In these systems, energy supplied to the load must be above a specified minimum useful temperature, Tmin, and it must be used at a constant thermal efficiency or coefficient of performance so that the load on the solar energy system can be estimated. The return temperature from the load is always at or above Tmin. Because the performance of a heat pump or a heat engine varies with the temperature level of supplied energy, this design method is not suitable for this kind of application. It is useful, however, in absorption refrigerators, industrial process heating, and space heating systems.

The maximum monthly average daily energy that can be delivered from the system shown in Figure 11.10 is given by

This is the same as Eq. (11.43), except that $ is replaced with $max, which is the maximum daily utilizability, estimated from the minimum monthly average critical radiation ratio:

C,min rnRnKTH0 {

Klein and Beckman (1979) correlated the results of many detailed simulations of the system shown in Figure 11.10, for various storage size-collector area ratios, with two dimensionless variables. These variables are similar to the ones used in the /-chart but are not the same. Here, the /-chart dimensionless parameter Y (plotted on the ordinate of the /-chart) is replaced by $maxY, given by

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