~ Pergamon 0960-1481(95)00102-6

Renewable Energy, VoL 6, No. 8, pp. 887 891, 1995 Elsevier Science Ltd

Printed in Great Britain 096(~1481/95 $9.50+0.00

A SIMPLE D E S I G N TOOL FOR SOLAR H E A T I N G

J O H N A. P A L Y V O S Department of Chemical Engineering, National Technical University of Athens, Greece GR-157 80

(Received 20 September 1993 ; accepted 22 June 1995)

Abstract--For the purpose of quick-sizing DHW and space heating solar systems of standard design, a simple correlation is derived and an equivalent generalized plot is constructed, on the basis of which the long-term behavior can be predicted. The typical deviation from detailed simulation values (RMS error) is about 5% for various Greek cities and for the allowable range of design parameters.

1. INTRODUCTION

An obvious, yet basic, characteristic of solar systems is the dependence of their performance upon the weather. The relevant weather variables, however, are neither completely random nor deterministic in nature. At best, they could be described as irregular functions of time, regardless of the time scale involved, i.e, whether we are dealing with hourly, daily, monthly, or even yearly values.

It is this irregular behavior of the meteorological variables which makes difficult the study of solar systems, even the simplest ones, such as those for space and water heating, The experimental determination of their performance, for instance, is expensive, time consuming and, in essence, it cannot really provide a definite answer as to what the system's behavior would be, should the weather conditions be different. Thus, the need arises for suitable design tools, typically based on the computer.

The traditional detailed simulations are perhaps difficult or even prohibitive for the average designer of a typical solar installation since, among other things, they require a tremendous amount of hard- to-find data (hourly values), as well as considerable computing power. It has been shown, however, that they can lead to generalized correlations which, in a very computationally easy manner, can predict the long-term behavior of the installation quite success- fully. In order to have wide applicability, such cor- relations should involve only a few dimensionless groups, made up of design parameters and meteoro- logical variables. Following their analytical identi- fication, these groups can be empirically correlated with the results of the detailed simulations, producing a design tool which is slightly degraded but much easier to use.

This paper outlines the derivation of such a simple

tool which is similar to the established prototype, namely, the f-chart [1]. The proposed correlation pro- duces RMS errors of the order of 5%, compared with the detailed simulation itself, for typical installations in various Greek cities. Thus, in view of the limited accuracy of the available meteorological data--typi- cally of the same order of magnitude--such an empiri- cal tool can be considered satisfactory for simple design purposes.

2. THE BASIC EQUATION FOR THE SOLAR FRACTION

For the typical solar space and water heating system under study, the conservative model of a uniform temperature (fully mixed) storage tank is chosen, while thermal losses through the hydraulic portion of the installation are ignored. The relevant heat balance on the tank--which is considered to be indoors--is of the form

dT~ (Mcp)s d i - = Qu(t) - OL(t) + O,~(t), ( l )

where M, cp, and T~ are the mass, heat capacity, and tank fluid temperature, while Qu(t), QL(t), and Qn(t) are the rates of collector gain, supply to the load, and auxiliary energy source complement, respectively [2]. From this balance equation simple design correlations can be derived, which can successfully predict the long-term behavior of the heating system.

How efficient a typical system will be depends, mainly, upon the design parameters, the heating load and, of course, the weather. Although in a given year the meteorological fluctuations will definitely not fol- low the typical time-series used in the detailed simu- lations, the long-term predictions based on such rep- resentative data produce reasonable results [3]. This

887

888

suggests that for a predetermined period, the per- formance of the system could be correlated with the total insolation and the heat load for this period, as well as with the design parameters.

With simplicity in mind, eq. (1) is integrated over a time period At. If this period is long enough, e.g. a month, then the left-hand side term--which describes the internal energy change in the storage tank--is small compared with the right-hand side energy rate quantities and, hence, can be omitted [4,5]. Thus, the solar fraction, f, defined as f = 1 -Qn/QL, can finally be approximated by the expression

AcFRNtiT(z~) AoFRV f: (T~-- T.)dt, (2)

in which QL is the (integrated) supply of heat to satisfy the load, A~, UL, and (z~) are the area, the total heat loss coefficient, and the monthly mean transmittance- absorptance product of the collector, FR is the col- lector heat removal factor, and T, is the ambient tem- perature [2]. The product NHv, on the other hand, expresses the monthly mean total radiation incident on the tilted plane of the collector.

This basic relation, however, is not in a position to directly produce values forf. Since the temperature T~ is a complicated function of T~, QL, and Hv, the inte- gral cannot be readily evaluated. In order to cir- cumvent this dead-end, dimensional analysis will be used in the following paragraph, in an effort to empiri- cally relate the solar fraction to suitable (and naturally appearing) dimensionless groups.

3. DIMENSIONAL STUDY OF THE BASIC EQUATION

The first term on the right-hand side of eq. (2) is the Y group of the f-chart, with FR instead of Fk since, for simplicity, the collector-tank heat-exchanger has been omitted [6]. In order to isolate the ambient tem- perature (for which monthly values are available, hence, it can be integrated), a dimensionless tem- perature T* is defined as T * = (Ts-~) / (Tref- -Ta) , with Tref a reference temperature, and t* a dimen- sionless time, t* = t/tra, with t~f a reference time (T~f can be 100°C and tref 1 day). Integration by parts [2] transforms the integral of eq. (2) into

f T*(Tref- Ta) dt

_ d T * ,

thus producing on the right-hand side of eq. (2)

J. A. PALYVOS

the second f-chart dimensionless group, namely X [6]. The groups X and Y which emerged from this

inspectional analysis, however, are not the only ones possible. With reference to the basic eq. (2), the fol- lowing set of dimensional parameters--which all affect the solar fraction--can be isolated :

(Ac,/tT, QL, UL, AT, At).

A dimensionally equivalent but useful, for comparison purposes, set is the following

(AcFR, NHT(Z~), QL, UL, AT, At),

in which the mean temperature difference, AT = (Tra - 7~a) is used, as well as the dimensionless quantities FR, N, and ~ as factors, since they do not alter the dimensional form of the respective variables. The specific combination of FR, N, and ~ with the particular dimensional quantities Ac and HT, simply aims at a direct comparison with the traditional groups.

For the above set, the dimensional matrix is of the form

L M t T AcF'R 2 0 0 0

(z~)HTN 0 1 - 2 0 At 0 0 1 0 A T 0 0 0 1 UL 0 1 --3 -- 1 QL 2 1 - 2 0

In it, the elements represent the exponents in the dimensional representation of the respective variables. For example, the area Ac has dimensions [L] 2, hence the value 2 for the respective first row-first column element.

Based on a well-known algebraic method [7], a spe- cially developed interactive computer program uses this information to produce all the possible dimen- sionless groups involved [2]. For the situation at hand, three possible complete sets emerge, each having two independent dimensionless groups--as expected on the basis of the Pi-theorem. The various groups involved are the well known X and Y groups of the f-chart, as well as a third one,

VL(Tref-- ~)At '

which focusses on the collector, as it simply represents the ratio of its (long-term) energy gain to its cor- responding thermal losses. The X and Y groups, on the other hand, represent the ratios of the collector losses and the collector gain, respectively, over the heating load of the installation [1].

A simple design tool

The three energy quantities--collector gain, col- lector losses, heating load--emerge in the groups of the three sets in a symmetric manner. That is, each one of the three quantities in turn appears on both dimensionless groups of the particular set. In this way, any one of the three possible combinations can form the basis for an empirical correlation for the solar fraction, the emphasis being each time on the charac- teristic energy quantity common to both dimen- sionless groups.

The first set, that of X and Y (i.e. the f-chart set) is still the set of choice, as it puts the emphasis on the behavior of the system in relation to the heating load. Of the remaining two sets, that of X and the third group puts the emphasis on the negative aspects of the operation, i.e. on its thermal losses, while that of Y with the third group features the collector's positive role, i.e. its energy gain. For the purposes of this paper, the third set will be used to develop a simple design tool, completely analogous to the f-chart for liquid systems. It should be noted at this point that the third group is actually Y/X, a legitimate combination emerging repeatedly upon execution of the computer program in automatic mode, in which case it actually performs all the possible row interchanges in the dimensional matrix.

4. AN EMPIRICAL DESIGN TOOL

The final stage of the procedure leading to the desired design tool, that is to a simple expression for the solar fraction, is that of fitting the relevant model equation to the proper data. The latter are generated by repeated detailed simulations of the typical system

for solar heating 889

under study, namely that of the standard con- figuration using a liquid heat transfer and storage media [6]. The simulations, performed via a modified form of TRNSYS [8], were run in such a way that the design parameters were varied within specific ranges (cf. Table 1), which obviously establishes the range of validity of the design tool itself.

Of the various regression equations tested, the fol- lowing was considered to be the best :

(y)2 (y)3 -0 .0983Y2-5.12 ~ +3.33 ~ . (4)

In it, as already noted, X and Y are the f-chart groups. The standard deviation o f f is 0.082 while the cor- relation coefficient is 99.3%. Moreover, on the basis of the t-distribution, all the terms of the correlation (4) are significant.

Other regression models used, with fewer terms, produced standard deviations in the range 0.043- 0.024 with corresponding correlation coefficients in the range 96.3-98.9%. Obviously, care should be exer- cised in using too many terms in the regression equa- tion, as there is a tendency nowadays to use the power of modern computers for "overfitting" to the available data. That is, from a certain point on- - in the com- plexity of the regression equation--the model sought ends up fitting to the noise of the data. At any rate, the form of eq. (4) was chosen also for the purpose of comparison with the literature [6].

On the basis ofeq. (4), Fig. 1 was constructed, which

O.O0 3.00

d 2,00

Z i f

LL ~ 1.oo

O.O0' 0.0

0,10 0,20 0.30 0,40 0.50 0.60 0.70 0.B0 0.90 ~ I = ~ = ~ n r ~ n n F n ~ n ~ t n ~ = n n ~ ; = n ~ n = ~ = n n ~ n n = [ ~ = = ~ w ~ = = r ~ w ~ n ] ~ 3,00

~ ~ 0.9 ~ ~ ~ 0.8 o.a b.8 0.7

0 . 4 0 . 3

2.00

1.00

InJ r n ~ l n n l l l n l = n , , l a l , , l a l t a n n l a a = l ===al l j l = L n l a a a l , n j ~ l n n r L l a l a = = u a o n l l = n a l J a j l t l a o a a n n n 0,00

O, 10 0,20 0.70 O . B O 0.90 0.30 0.40 0.50 0.60

( T-~)gTN / U L (T~, -T. ) At

Fig. 1. Generalized diagram for the design of heating systems using liquid heat transfer and storage media.

890 J.A. PALYVOS

Table 1. Range of the design variables

Parameter Range

(~) 0.7~.85 At 30-100 (m 2) UL 2-5 (W/m 2) /~ 30-60 (degrees) UA 150-400 (W/K)

enables estimation of the monthly solar fraction, for the given design parameters and the local meteoro- logical quantities. It should be noted that the curves of Fig. 1 and eq. (4) should be used for calculations covering the entire year, in which case the results are more reliable. Although the design parameters should conform to the constraints of Table 1, if a point falls outside the range of the curves in Fig. 1, extrapolation can give satisfactory results.

0.0 2.0 4.0 6.0 8.0 10.0 1 0 . 0 . . . . . . . . , . . . . . . . . . ~ . . . . . . . . . , . . . . . . . . 1 0 . 0

8 . 0 ' ~ 8 . 0

o ~ i r e 8 i •

"~ A * - 6.0 6.0

0 A n

~:~ 4.0 4.0

a

2.0 ~ * o ° * = I_ariso 2.0 " o o o o o Hania

0 . 0 , , , = . . . . I , l = , , = = = , l , = l , , i , , , I , = , , , = , , = l l P l l l l l 0 . 0

0 . 0 2 . 0 4.0 6 .0 B.O 1 0 . 0

f - S i m u l o t i o n x l O

Fig. 2. Comparison of simple design tool predictions with detailed simulation results for solar space heating systems.

5. RESULTS

The relevant parametric study using the proposed model is fast, easy, and produces results as expected. For instance, the annual performance of the heating system is a linear and decreasing function of the ther- mal losses--as expressed by the quantity FRUL--and a linear and increasing function of the product FR(~). A higher collector tilt, on the other hand, also increases the solar fraction linearly, for the usual range of tilt angles.

The annual performance appears to be an increas- i n g - b u t not linear--function of both the total col- lector area and the tank's storage capacity. In both cases the relevant curves (of f vs Ac or M/Ao) are almost logarithmic, thus implying the diminishing returns of adding an excessive number of panels or of increasing excessively the storage tank capacity.

Calculations for various parameter combinations and for three widely scattered Greek cities are shown in Fig. 2, which is a plot of the solar fraction as predicted by the simple design tool just described vs the solar fraction as predicted by the detailed simu- lation. The deviations from the 45 ° line--which are of the order of 5% suggest that the empirical tool underestimates the solar fraction. The overall agree- ment of the predicted values, however, is in general satisfactory and similar to that of the f-chart, which also is noted for its conservative predictions [9]. As a matter of fact, its mean error (5.3%) is analogous to the error of f-chart predictions (6.1%) which,

however, were made for a number of Northern Eur- opean locations [10].

The success of the empirical method confirms the prediction that the long-term performance of the heat- ing system is not very sensitive to the hourly fluc- tuations of the solar flux, the other weather variables, and the system temperatures. Thus, the performance can indeed be predicted using mean meteorological data and system characteristics.

NOMENCLATURE

Ac area of collector (m 2) cp heat capacity of fluid (J/kg K) f solar fraction

FR collector heat removal factor HT daily global insolation on the tilted collector

plane (kJ/m 2) M storage tank fluid mass (kg) N number of collector panels

QL heated space losses (kJ) Qu collector useful energy gain (k J) Qn auxiliary heat (k J)

t time /ref reference time t* dimensionless time (= t/tref) 72a ambient temperature (°C)

Trof reference temperature (°C) T~ storage tank (uniform) fluid temperature (°C)

T* dimensionless temperature (= (Ts- T,)/ (Tre f - T.))

UL collector overall the rmal (W/m 2)

UA heat ing load (k J) X f-chart dimensionless group Y f-chart dimensionless group

Greek symbols collector plate absorp tance

/~ collector tilt (degrees) r collector cover t ransmi t tance

Superscripts rate quant i ty m e a n value.

A simple design tool for

loss coefficient 3.

REFERENCES

1. W. A. Beckman, S. A. Klein and J. A. Duffle, Solar Heating Design by the f-chart Method. Wiley (1977).

2. J. A. Palyvos, Simple design of solar space heating systems, N.T.U. Report 911 (1991) (in Greek).

solar heating 891

S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar heating systems. Solar Energy 18, 113 (1976).

4. G. O. G. Lof and R. A. Tybout, Cost of house heating with solar energy. Solar Energy 19, 253 (1973).

5. A. Zollner, S. A. Klein and W. A. Beckman, A per- formance prediction methodology for integral col- lection-storage solar domestic hot water systems. Pre- sented at the ASME/Solar Energy Conf., Knoxville (March 1985).

6. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes, 2nd Edition, Ch. 20. Wiley (1991).

7. L. Brand, The pi theorem of dimensionless analysis. Arch. Rat. Mech. Anal. 1, 35 (1957).

8. TRNSYS, a transient simulation program, version 11.1. Engineering experiment station report 38-1 I, University of Wisconsin-Madison (1981).

9. J. A. Duffle and J. W. Mitchell, f-chart : predictions and measurements. Trans. ASME, J. Solar Energy Engng 105, 3 (1983).

10. B. L. Evans, W. A. Beckman and J. A. Duffle, f-chart in European climates. Presented at the 1st EC Conf. on Solar Heating, Amsterdam, 1984.