## At

At where

L = monthly heating load or demand (MJ).

N = number of days in a month.

Ta = monthly average ambient temperature (°C).

Ht = monthly average daily total radiation on the tilted collector surface _ (MJ/m2).

( Ta ) = monthly average value of (Ta), = monthly average value of absorbed over incident solar radiation = S/Ht.

For the purpose of calculating the values of the dimensionless parameters X and Y, Eqs. (11.6) and (11.7) are usually rearranged to read

fr l

The reason for the rearrangement is that the factors FRUL and FR(ra)n are readily available form standard collector tests (see Chapter 4, Section 4.1). The ratio FR /Fr is used to correct the collector performance because the heat exchanger causes the collector side of the system to operate at higher temperature than a similar system without a heat exchanger and is given by Eq. (5.57) in Chapter 5. For a given collector orientation, the value of the factor (toi)/(toi)„ varies slightly from month to month. For collectors tilted and facing the equator with a slope equal to latitude plus 15°, Klein (1976) found that the factor is equal to 0.96 for a one-cover collector and 0.94 for a two-cover collector for the whole heating season (winter months). Using the preceding definition of (ra), we get

If the isotropic model is used for S and substituted in Eq. (11.10), then:

HD (Ta)D

1 + cos(ß)'

Ht (Ta)n

2 J

- cos(ß)"

In Eq. (11.11), the (îa)/(Ta)„ ratios can be obtained from Figure 3.24 for the beam component at the effective angle of incidence, 0B, which can be obtained from Figure A3.8 in Appendix 3, and for the diffuse and ground-reflected components at the effective incidence angles at (3 from Eqs. (3.4a) and (3.4b).

The dimensionless parameters, X and Y, have some physical significance. The parameter X represents the ratio of the reference collector total energy loss to total heating load or demand (L) during the period At, whereas the parameter Y represents the ratio of the total absorbed solar energy to the total heating load or demand (L) during the same period.

As was indicated, /-chart is used to estimate the monthly solar fraction, /, and the energy contribution for the month is the product of / and monthly load (heating and hot water), Li. To find the fraction of the annual load supplied by the solar energy system, F, the sum of the monthly energy contributions is divided by the annual load, given by

The method can be used to simulate standard solar water and air system configurations and solar energy systems used only for hot water production. These are examined separately in the following sections.

### Example 11.1

A standard solar heating system is installed in an area where the average daily total radiation on the tilted collector surface is 12.5 MJ/m2, average ambient temperature is 10.1°C, and it uses a 35 m2 aperture area collector, which has FR(Ta)n = 0.78 and FrUl = 5.56 W/m2-°C, both determined from the standard collector tests. If the space heating and hot water load is 35.2 GJ, the flow rate in the collector is the same as the flow rate used in testing the collector, FR /FR = 0.98, and (ra)/(Ta)n = 0.96 for all months, estimate the parameters X and Y.

Solution

Using Eqs. (11.8) and (11.9) and noting that AT is the number of seconds in a month, equal to 31 d X 24 h X 3600 sec/h, we get

fr l

1.30

35.2 X 109

35.2 X 109

0.28

11.1.1 Performance and Design of Liquid-Based Solar Heating Systems

Knowledge of the system thermal performance is required in order to be able to design and optimize a solar heating system. The /-chart for liquid-based systems is developed for a standard solar liquid-based solar energy system, shown in

Figure 11.1. This is the same as the system shown in Figure 6.14, drawn without the controls, for clarity. The typical liquid-based system shown in Figure 11.1 uses an antifreeze solution in the collector loop and water as the storage medium. A water-to-water load heat exchanger is used to transfer heat from the storage tank to the domestic hot water (DHW) system. Although in Figure 6.14 a one-tank DHW system is shown, a two-tank system could be employed, in which the first tank is used for pre-heating.

The fraction f of the monthly total load supplied by a standard solar liquid-based solar energy system is given as a function of the two dimension-less parameters, X and Y, and can be obtained from thef-chart in Figure 11.2 or the following equation (Klein et al., 1976):

f = 1.029Y - 0.065X - 0.245Y2 + 0.0018X2 + 0.0215Y3 (11.13)

Application of Eq. (11.13) or Figure 11.2 allows the simple estimation of the solar fraction on a monthly basis as a function of the system design and local weather conditions. The annual value can be obtained by summing up FIGURE 11.1 Schematic diagram of a standard liquid-based solar heating system. FIGURE 11.2 The f-chart for liquid-based solar heating systems.

the monthly values using Eq. (11.12). As will be shown in the next chapter, to determine the economic optimum collector area, the annual load fraction corresponding to different collector areas is required. Therefore, the present method can easily be used for these estimations.

### Example 11.2

If the solar heating system given in Example 11.1 is liquid-based, estimate the annual solar fraction if the collector is located in an area having the monthly average weather conditions and monthly heating and hot water loads shown in Table 11.2.

Table 11.2 Average Monthly Weather Conditions and Heating and Hot Water Loads for Example 11.2

Month

Ht (MJ/m2)

Ta (°C)

L (GJ)

January

12.5

10.1

35.2

February

15.6

13.5

31.1

March

17.8

15.8

20.7

April

20.2

19.0

13.2

May

21.5

21.5

5.6

June

22.5

29.8

4.1

July

23.1

32.1

2.9

August

22.4

30.5

3.5

September

21.1

22.5

5.1

October

18.2

19.2

12.7

November

15.2

16.2

23.6

December

13.1

11.1

The values of the parameters dimensionless X and Y found from Example 11.1 are equal to 1.30 and 0.28, respectively. From the weather and load conditions shown in Table 11.2, these correspond to the month of January. From Figure 11.2 or Eq. (11.13), f = 0.188. The total load in January is 35.2 GJ. Therefore, the solar contribution in January is fL = 0.188 X 35.2 = 6.62 GJ. The same calculations are repeated from month to month, as shown in Table 11.3.

It should be noted that the values of f marked in bold are outside the range of the f-chart correlation and a fraction of 100% is used, as during these months, the solar energy system covers the load fully. From Eq. (11.12), the annual fraction of load covered by the solar energy system is

E Li 190.8

 Month Ht (MJ/m2) Ta (°C) L (GJ) X Y f fL January 12.5 10.1 35.2 1.30 0.28 0.188 6.62 February 15.6 13.5 31.1 1.28 0.36 0.259 8.05 March 17.8 15.8 20.7 2.08 0.68 0.466 9.65 April 20.2 19.0 13.2 3.03 1.18 0.728 9.61 May 21.5 21.5 5.6 7.16 3.06 1 5.60 June 22.5 29.8 4.1 8.46 4.23 1 4.10 July 23.1 32.1 2.9 11.96 6.34 1 2.90 August 22.4 30.5 3.5 10.14 5.10 1 3.50 September 21.1 22.5 5.1 7.51 3.19 1 5.10 October 18.2 19.2 12.7 3.25 1.14 0.694 8.81 November 15.2 16.2 23.6 1.76 0.50 0.347 8.19 December 13.1 11.1 33.1 1.37 0.32 0.219 7.25 Total load = 190.8 Total contribution = 79.38

It should be noted that the /-chart was developed using fixed nominal values of storage capacity per unit of collector area, collector liquid flow rate per unit of collector area, and load heat exchanger size relative to space heating load. Therefore, it is important to apply various corrections for the particular system configuration used. 