The /-chart design method is used to quickly estimate the long-term performance of solar energy systems of standard configurations. The input data needed are the monthly average radiation and temperature, the monthly load required to heat a building and its service water, and the collector performance parameters obtained from standard collector tests. A number of assumptions are made for the development of the /-chart method. The main ones include assumptions that the systems are well built, system configuration and control are close to the ones considered in the development of the method, and the flow rate in the collectors is uniform. If a system under investigation differs considerably from these conditions, then the /-chart method cannot give reliable results.
It should be emphasized that the /-chart is intended to be used as a design tool for residential space and domestic water heating systems of standard configuration. In these systems, the minimum temperature at the load is near 20°C; therefore, energy above this value of temperature is useful. The /-chart method cannot be used for the design of systems that require minimum temperatures substantially different from this minimum value. Therefore, it cannot be used for solar air conditioning systems using absorption chillers, for which the minimum load temperature is around 80°C.
It should also be understood that, because of the nature of the input data used in the /-chart method, there are a number of uncertainties in the results obtained. The first uncertainty is related to the meteorological data used, especially when horizontal radiation data are converted into radiation falling on the inclined collector surface, because average data are used, which may differ considerably from the real values of a particular year, and that all days were considered symmetrical about solar noon. A second uncertainty is related to the fact that solar energy systems are assumed to be well built with well-insulated storage tanks and no leaks in the system, which is not very correct for air systems, all of which leak to some extent, leading to a degraded performance. Additionally, all liquid storage tanks are assumed to be fully mixed, which leads to conservative long-term performance predictions because it gives overestimation of collector inlet temperature. The final uncertainty is related to the building and hot water loads, which strongly depend on variable weather conditions and the habits of the occupants.
Despite these limitations, the /-chart method is a handy method that can easily and quickly be used for the design of residential-type solar heating systems. When the main assumptions are fulfilled, quite accurate results are obtained.
Although the /-chart method is simple in concept, the required calculations are tedious, particularly the manipulation of radiation data. The use of computers greatly reduces the effort required. Program F-Chart (Klein and Beckman, 2005), developed by the originators of TRNSYS, is very easy to use and gives predictions very quickly. Again, in this case, the model is accurate only for solar heating systems of a type comparable to that which was assumed in the development of the /-chart.
The /-chart program is written in the BASIC programming language and can be used to estimate the long-term performance of solar energy systems that have flat-plate, evacuated tube collectors, CPCs, and one- or two-axis tracking concentrating collectors. Additionally, the program includes an active-passive storage system and analyzes the performance of a solar energy system in which energy is stored in the building structure rather than a storage unit (treated with methods presented in the following section) and a swimming pool heating system that provides estimates of the energy losses from the swimming pool. The complete list of solar energy systems that can be handled by the program is as follows:
• Pebble bed storage space and domestic water heating systems.
• Water storage space and/or domestic water heating systems.
• Active collection with building storage space heating systems.
• Direct gain passive systems.
• Collector-storage wall passive systems.
• Pool heating systems.
• General heating systems, such as process heating systems.
• Integral collector-storage domestic water heating systems.
The program can also perform economic analysis of the systems. The program, however, does not provide the flexibility of detailed simulations and performance investigations, as TRNSYS does.
In the previous section, the f-chart method is presented. Due to the limitations outlined in Section 11.1.4, the f-chart method cannot be used for systems in which the minimum temperature supplied to a load is not near 20°C. Most of the systems that cannot be simulated with f-chart can be modeled with the uti-lizability method or its enhancements.
The utilizability method is a design technique used for the calculation of the long-term thermal collector performance for certain types of systems. Initially originated by Whillier (1953), the method, referred to as the Q-curve method, is based on the solar radiation statistic, and the necessary calculations had to be done for hourly intervals about solar noon each month. Subsequently, the method was generalized for time of year and geographic location by Liu and Jordan (1963). Their generalized Q-curves, generated from daily data, gave the ability to calculate utilizability curves for any location and tilt by knowing only the clearness index, Kt. Afterward, the work by Klein (1978) and Collares-Pereira and Rabl (1979a) eliminated the necessity of hourly calculations. The monthly average daily utilizability, $, reduced much of the complexity and improved the utility of the method.
The utilizability method is based on the concept that only radiation that is above a critical or threshold intensity is useful. Utilizability, Q, is defined as the fraction of insolation incident on a collector's surface that is above a given threshold or critical value.
We saw in Chapter 3, Section 3.3.4, Eq. (3.61), that a solar collector can give useful heat only if solar radiation is above a critical level. When radiation is incident on the tilted surface of a collector, the utilizable energy for any hour is (It - Itc)+, where the plus sign indicates that this energy can be only positive or zero. The fraction of the total energy for the hour that is above the critical level is called the utilizability for that hour, given by
Utilizability can also be defined in terms of rates, using Gt and Gtc, but because radiation data are usually available on an hourly basis, the hourly values are preferred and are also in agreement with the basis of the method.
Utilizability for a single hour is not very useful, whereas utilizability for a particular hour of a month having N days, in which the average radiation for the hour is It, is very useful, given by
In this case, the average utilizable energy for the month is given by NIt Such calculations can be done for all hours of the month, and the results can be added up to get the utilizable energy of the month. Another required parameter is the dimensionless critical radiation level, defined as
For each hour or hour pair, the monthly average hourly radiation incident on the collector is given by
The ratios r and rd can be estimated from Eqs. (2.83) and (2.84), respectively.
Liu and Jordan (1963) constructed a set of $ curves for various values of Kt . With these curves, it is possible to predict the utilizable energy at a constant critical level by knowing only the long-term average radiation. Later on Clark et al. (1983) developed a simple procedure to estimate the generalized $ functions, given by
The monthly average hourly clearness index, kT, is given by kT = L (11.27)
T lo and can be estimated using Eqs. (2.83) and (2.84) as
I r - r H kT = = — Kt = = [a + p cos(h)]KT (11.28)
h rd rd Ho where a and (3 can be estimated from Eqs. (2.84b) and (2.84c), respectively. If necessary, Ho can be estimated from Eq. (2.79) or obtained directly from Table 2.5.
The ratio of monthly average hourly radiation on a tilted surface to that on a horizontal surface, Rh, is given by
The $ curves are used hourly, which means that three to six hourly calculations are required per month if hour pairs are used. For surfaces facing the equator, where hour pairs can be used, the monthly average daily utilizability, presented in the following section can be used and is a more simple way of calculating the useful energy. For surfaces that do not face the equator or for processes that have critical radiation levels that vary consistently during the days of a month, however, the hourly $ curves need to be used for each hour.
As can be understood from the preceding description, a large number of calculations are required to use the $ curves. For this reason, Klein (1978) developed the monthly average daily utilizability, concept. Daily utilizability is defined as the sum over all hours and days of a month of the radiation falling on a titled surface that is above a given threshold or critical value, which is similar to the one used in the $ concept, divided by the monthly radiation, given by
The monthly utilizable energy is then given by the product NHt The value of $ for a month depends on the distribution of hourly values of radiation in that month. Klein (1978) assumed that all days are symmetrical about solar noon, and this means that $ depends on the distribution of daily total radiation, i.e., the relative frequency of occurrence of below-average, average, and above-average daily radiation values. In fact, because of this assumption, any departure from this symmetry within days leads to increased values of This means that the $ calculated gives conservative results.
Klein developed the correlations of $ as a function of KT, a dimensionless critical radiation level, Xc, and a geometric factor R/Rn. The parameter R is the monthly ratio of radiation on a tilted surface to that on a horizontal surface, Ht/H, given by Eq. (2.107), and Rn is the ratio for the hour centered at noon of radiation on the tilted surface to that on a horizontal surface for an average day of the month, which is similar to Eq. (2.99) but rewritten for the noon hour in terms of rdHD and rH as
where rdn and rn are obtained from Eqs. (2.83) and (2.84), respectively, at solar noon (h = 0°). It should be noted that Rn is calculated for a day that has a total radiation equal to the monthly average daily total radiation, i.e., a day for which H = H and Rn is not the monthly average value of R at noon. The term Hd/H is given from Erbs et al. (1982) as follows.
11.9514K3 + 9.3879K-4 for KT < 0.715 for Kt > 0.715
2.5557K2 + 0.8448KT
The monthly average critical radiation level, Xc, is defined as the ratio of the critical radiation level to the noon radiation level on a day of the month in which the radiation is the same as the monthly average, given by
The procedure followed by Klein (1978) is, for a given KT a set of days was established that had the correct long-term average distribution of KT values. The radiation in each of the days in a sequence was divided into hours, and these hourly values of radiation were used to find the total hourly radiation on a tilted surface, It. Subsequently, critical radiation levels were subtracted from the It values and summed as shown in Eq. (11.30) to get the $ values. The $ curves calculated in this manner can be obtained from graphs or the flowing relation
$ = exp j
A + B
A north-facing surface located in an area that is at 35°S latitude is tilted at 40°. For the month of April, when H = 17.56 MJ/m2, critical radiation is 117 W/m2, and pG = 0.25, calculate $ and the utilizable energy.
For April, the mean day from Table 2.1 is N = 105 and 6 = 9.41°. From Eq. (2.15), the sunset time hss = 83.3°. From Eqs. (2.84b), (2.84c), and (2.84a), we have a = 0.409 + 0.5016 X sin(hss - 60) = 0.409 + 0.5016 X sin(83.3 - 60) = 0.607
ß = 0.6609 - 0.4767 X sin(hss - 60) = 0.6609 - 0.4767 X sin(83.3 - 60) = 0.472