As in Example 11.13, add the effect of a load heat exchanger to the performance for the month of June for the system in Example 11.12. The effectiveness of the heat exchanger is 0.48 and its capacitance is 3200 W/°C.


Here we need to assume a storage tank temperature increase of 5°C due to the action of the load heat exchanger. From Eq. (11.45),




The Kt in June is 0.70; therefore, the coefficients are A = —1.5715, B = 0.0871, and C = 1.0544. From Eq. (11.34a) $max = 0.387. As the use of heat exchanger increases the tank temperature, we need to assume a new tank temperature, as in the previous example. Let us assume a tank temperature of 77°C. For this temperature, from Eq. (11.51), Qs = 0.99 GJ and the total load is 16.20 + 0.99 = 17.19 GJ. Therefore, as in previous example,

_ 50x0.82 X 0.89 X 30 X 29.2 X 106 X 0.812 17.19 X 109

X, = AcFRUL(100)A/ = 50 X 5.92 X 100 X 24 X 30 X 3600 = 4 46 L 17.19 X109 .

From Eq. (11.48), fTL = 0.54. Then, we have to check the increase of storage tank temperature assumption. From Eq. (11.54),

From Eq. (11.34a) by trial and error the new value of XC original of 0.39. From Eq. (11.45),

0.47 from the and


0.82 X 0.89x0.122x0.921 X 0.70 X 41.71 X 106 X 0.47


5.92 X 3600

The average tank temperature for the losses is then equal to (75 + 78.6)/2 = 76.8°C. This is effectively the same as the one originally assumed, so no iteration is required. From Eq. (11.53), f fTL




Finally, we also need to check the assumption of storage tank temperature increase (5°C) due to the action of the load heat exchanger. From Eq. (11.55).

AT = fL / AtL = 0.51 X 16.2 X 109 /(10 X 30 X 3600) = 50C


0.48 X 3200

Because this is the same value as the original assumption, no iterations are required and the calculations are complete. Therefore, the solar fraction for June dropped from 64% to 51% due to the presence of the load heat exchanger. This substantial drop in performance is due to the increase of tank temperature and the increased tank losses at the higher temperature.

Klein and Beckman (1979) also performed a validation study to compare the results of the present method with those obtained by the TRNSYS program. Comparisons between the /-chart estimates and TRNSYS calculations were performed for three types of systems: space heating, air conditioning (using a LiBr absorption chiller operated at Tmin = 77°C), and process heating applications (Tmin = 60°C). The comparisons show that, although there are some particular circumstances in which /-chart will yield inaccurate results, the method can be used to predict the performance of a wide variety of solar energy systems.


Passive solar energy systems are described in Chapter 6, Section 6.2. It is of interest to the designer to be able to estimate the long-term performance of passive systems. In this way, the designer could evaluate how much of the absorbed energy cannot be used because it is available at a time when the loads are satisfied or exceed the capacity of the building to store energy.

The unutilizability method is an extension of the utilizability method that is suitable for direct gain, collector storage walls, and active collection with passive storage (hybrid) systems. These are treated separately in the following sections. The unutilizability (called UU) method, developed by Monsen et al. (1981; 1982), is based on the concept that a passively heated building can be considered a collector with finite heat capacity. As in the case of the f- and $ chart methods, the estimations are carried out on a monthly basis and the result is to give the annual auxiliary energy required. The building thermal load is required for the present method. For this purpose, the methods presented in Chapter 6, Section 6.1, can be used. These vary from the detailed heat balance and transfer function methods to the simple degree day method.

11.4.1 Direct Gain Systems

The utilization of large glazed areas and massive thermal storage structures in passive heating systems is an effective and simple means of collecting and storing solar energy in buildings. The method of analysis for this type of system is presented by Monsen et al. (1981). The design of such a passive system cannot be based on a fixed design indoor temperature, as in active systems. The monthly energy streams of a direct gain structure are shown in Figure 11.11. As can be seen, the energy absorbed by the passive system is expressed as


Ar = area of the collector (receiving) window (m2).

(Ta) = product of monthly average value of window transmissivity and room absorptivity.

The monthly average energy absorbed S is given by Eq. (3.1) by replacing the hourly direct and diffuse radiation terms with the monthly average terms, as indicated just after the equation in Chapter 3, given by

7~r k ,—\ n ,—„ Î1 + cos(B) HbRb (ra)B + Hd (ja)D ---

The energy lost through the building envelope is shown in Figure 11.11 as the load L. This is estimated by considering that the transmittance of the glazing is 0, given by


(UA)h = product of the overall heat transfer coefficient and the area of the building structure, including the direct gain windows (W/°C). Tb = indoor base temperature (°C).

When solar energy is not enough to supply the load, auxiliary energy, Qaux, must be provided. Also, there might be excess absorbed solar energy, above what is required to cover the load, that cannot be stored and must be dumped, denoted by Qd. Sometimes during a month, sensible heat may be stored or removed from the building structure, provided it has thermal capacitance, called stored energy, not shown in Figure 11.11.

Two limiting cases here need to be investigated separately. In the first, we assume an infinite storage capacity, and in the second, zero storage capacity. In the first case, during a month, all the energy absorbed in excess of the load is stored in the building structure. The infinite capacitance of the building structure implies a constant temperature of the conditioned space. This stored energy is used when required to cover the load, thus it offsets the auxiliary energy, given by the monthly energy balance, as

The plus superscript in Eq. (11.59) indicates that only positive values are considered. Additionally, no month-to-month carryover is considered.

For the second limiting case, the building structure has zero storage capacity, and any energy deficits are covered with auxiliary energy, whereas any excess solar energy must be dumped. The temperature of the building is again constant but this time is due to the addition or removal of energy. The rate of energy dumped can be obtained from an instantaneous energy balance, given by

Similar to the case of solar collectors in an active solar energy system, a critical radiation level can be defined as the level at which the gains are equal to the losses, given by

Because we have zero storage capacity, any radiation above this critical level is unutilizable and is dumped. Therefore, the dumped energy for the month, Qd, is given by qd,z = A (Ta) f (It - Ac)+ dt (11.62)


Over a month, Itc can be considered to be constant, and from Eq. (11.61), its monthly average value is given by

Energy below Itc is useful, whereas energy above Itc is dumped. Equation (11.62) can be expressed in terms of the monthly average utilizability given by Eq. (11.30), and QD may be written as

It is worth noting that, for a passive solar heating system, $ is a measure of the amount of solar energy that cannot be used to reduce auxiliary energy, and it may be called unutilizability.

Using a monthly energy balance, the amount of auxiliary energy required by the zero storage capacity building can be estimated as being equal to the load plus dumped energy minus the absorbed solar energy, given by

Therefore, Eqs. (11.59) and (11.65) give the limits on the amount of auxiliary energy of a real building. Correlations have been developed by Monsen et al. (1981; 1982) in terms of the fraction of the load covered by solar energy. Similar to the active solar energy systems, the solar fraction is equal to f = 1 - (Qaux/L). For these correlations, two dimensionless parameters are specified, X and Y. The X dimensionless parameter is the solar-load ratio, defined as (Monsen et al., 1981):

For the infinite capacitance system, by dividing all terms of Eq. (11.59) by L, X is equal to the solar fraction, fi, given by f = X = 1 - ^^ (11.67)

For the zero capacitance case, q

Replacing Qaux,z from Eq. (11.65), we get fz = 1 (11.68)

The Y dimensionless parameter is the monthly ratio of the maximum storage capacity of the building to the solar energy that would be dumped if the building had zero thermal capacitance. It is therefore called the storage-dump ratio, given by (Monsen et al., 1981):


Cb = effective thermal capacitance, i.e., mass times heat capacity (J/°C). ATb = the difference of upper and lower temperatures, i.e., the temperature range the building is allowed to float (°C).

The two limiting cases have Y values equal to 0 for the zero storage capacity building and infinity for the infinite storage capacity building. The values of the effective thermal capacitance, given by Barakat and Sander (1982), are equal to 60 kJ/m2-°C for light construction buildings, 153 kJ/m2-°C for medium construction buildings, 415 kJ/m2-°C for heavy construction buildings, and 810 kJ/m2-°C for very heavy construction buildings.

Finally, the correlation of the monthly solar fraction is given in terms of X, Y, and $ and is given by (Monsen et al., 1981):

/ = min{ PX + (1 - P)(3.082 - 3.142$)[1 - exp(-0.329X)], 1} (11.71a) where

The auxiliary energy can be calculated from the solar fraction as

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