- Email: [email protected]
A simple design tool for solar heating
~
RenewableEnergy, VoL 6, No. 8, pp. 887 891, 1995
Pergamon
Elsevier Science Ltd Printed in Great Britain 096(~1481/95 $9.50+0.00
0960-1481(95)00102-6
A SIMPLE D E S I G N TOOL F O R S O L A R 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--Forthe 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 hardto-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 successfully. In order to have wide applicability, such correlations should involve only a few dimensionless groups, made up of design parameters and meteorological variables. Following their analytical identification, 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 produces 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--typically of the same order of magnitude--such an empirical 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 follow the typical time-series used in the detailed simulations, the long-term predictions based on such representative data produce reasonable results [3]. This
887
J. A. PALYVOS
888
suggests that for a predetermined period, the performance 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~)
AoFRVf: (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 transmittanceabsorptance product of the collector, FR is the collector heat removal factor, and T, is the ambient temperature [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 integral cannot be readily evaluated. In order to circumvent this dead-end, dimensional analysis will be used in the following paragraph, in an effort to empirically 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 temperature (for which monthly values are available, hence, it can be integrated), a dimensionless temperature T* is defined as T * = (Ts-~)/(Tref--Ta), with Tref a reference temperature, and t* a dimensionless 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
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 following set of dimensional parameters--which all affect the solar fraction--can be isolated : (Ac,/tT, QL, UL, AT, At). A dimensionallyequivalent but useful, for comparison purposes, set is the following (AcFR, NHT(Z~), QL, UL, AT, At), in which the mean temperature difference, AT = ( T r a - 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 AcF'R (z~)HTN At AT UL
QL
L 2 0 0 0 0 2
M 0 1 0 0 1 1
t 0 - 2 1 0 --3 -2
T 0 0 0 1 -- 1 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 specially developed interactive computer program uses this information to produce all the possible dimensionless 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 '
f
T*(Tref- Ta) dt _
dT*
,
thus producing on the right-hand side of eq. (2)
which focusses on the collector, as it simply represents the ratio of its (long-term) energy gain to its corresponding 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 for solar heating The three energy quantities--collector gain, collector 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 characteristic energy quantity common to both dimensionless 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.
889
under study, namely that of the standard configuration 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 following was considered to be the best :
(y)2 (y)3
-0.0983Y2-5.12 ~
+3.33 ~
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
O.O0 0,10 0,20 0.30 0,40 0.50 0.60 3.00 ~ I = ~ = ~ n r ~ n n F n ~ n ~ t n ~ = n n ~ ; = n ~ n = ~ = n n ~ n n = [ ~ = = ~ w ~ = = r ~ w ~ n ] ~
0.70
0.B0
0.90
3,00
d 2,00
2.00
if LL ~ 1.oo
~ ~
~
o.a
0.3
O.O0' InJ 0.0
r n~lnnlllnl=n,
O, 10
,lal,
0,20
,laltannlaa=l
0.30
~ ~
0.9
b.8 0.7
1.00
0.8
0.4 ===all jl=Lnlaaal,n
0.40
(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 correlation 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 0.0430.024 with corresponding correlation coefficients in the range 96.3-98.9%. Obviously, care should be exercised in using too many terms in the regression equation, 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 o n - - i n the complexity 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
4. AN EMPIRICAL DESIGN TOOL
Z
.
0.50
j~lnnrLlala== uaonll=nalJajltlaoaannn
0.60
0.70
O.BO
0,00 0.90
(T-~)gTN/ U L(T~,-T.)At Fig. 1. Generalized diagram for the design of heating systems using liquid heat transfer and storage media.
J.A. PALYVOS
890
0.0
Table 1. Range of the design variables Parameter
Range
(~) At UL /~
0.7~.85 30-100 (m2) 2-5 (W/m2) 30-60 (degrees) 150-400 (W/K)
UA
10.0
........
2.0
, .........
4.0
~. . . . . . . . .
6.0
, ........
'
8.0
"~
8.0
A
6.0
0
A
~
o
i
r
e
10.0
10.0
~
*
8.0
-
6.0
n
8 •
i
~:~ 4.0
4.0
a
2.0 enables estimation of the monthly solar fraction, for the given design parameters and the local meteorological 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.
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 thermal 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 increasi n g - b u t not linear--function of both the total collector 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 simulation. The deviations from the 45 ° line--which are of the order of 5% suggest that the empirical tool underestimates the solar fraction. The overall agreement 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,
~
0.0
,,,= ....
0.0
* o° * = I_ariso
"
o o o o o
Hania
I,l=,,===,l,=l,,i,,,I,=,,,=,,=llPlllll
2.0
4.0
f-Simulotion
6.0
2.0 0.0
B.O
10.0
xlO
Fig. 2. Comparison of simple design tool predictions with detailed simulation results for solar space heating systems.
however, were made for a number of Northern European locations [10]. The success of the empirical method confirms the prediction that the long-term performance of the heating system is not very sensitive to the hourly fluctuations 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 cp f FR HT M N QL Qu Qn t /ref t* 72a Trof T~ T*
area of collector (m 2) heat capacity of fluid (J/kg K) solar fraction collector heat removal factor daily global insolation on the tilted collector plane (kJ/m 2) storage tank fluid mass (kg) number of collector panels heated space losses (kJ) collector useful energy gain (kJ) auxiliary heat (kJ) time reference time dimensionless time ( = t/tref) ambient temperature (°C) reference temperature (°C) storage tank (uniform) fluid temperature (°C) dimensionless temperature (= (Ts- T,)/ (Tref - T.))
A simple design tool for solar heating UL collector overall t h e r m a l loss coefficient ( W / m 2) UA heating load (k J) X f-chart dimensionless g r o u p Y f-chart dimensionless g r o u p Greek symbols collector plate a b s o r p t a n c e /~ collector tilt (degrees) r collector cover t r a n s m i t t a n c e Superscripts rate q u a n t i t y m e a n value.
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).
891
3. 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 performance prediction methodology for integral collection-storage solar domestic hot water systems. Presented 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.
RenewableEnergy, VoL 6, No. 8, pp. 887 891, 1995
Pergamon
Elsevier Science Ltd Printed in Great Britain 096(~1481/95 $9.50+0.00
0960-1481(95)00102-6
A SIMPLE D E S I G N TOOL F O R S O L A R 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--Forthe 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 hardto-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 successfully. In order to have wide applicability, such correlations should involve only a few dimensionless groups, made up of design parameters and meteorological variables. Following their analytical identification, 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 produces 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--typically of the same order of magnitude--such an empirical 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 follow the typical time-series used in the detailed simulations, the long-term predictions based on such representative data produce reasonable results [3]. This
887
J. A. PALYVOS
888
suggests that for a predetermined period, the performance 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~)
AoFRVf: (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 transmittanceabsorptance product of the collector, FR is the collector heat removal factor, and T, is the ambient temperature [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 integral cannot be readily evaluated. In order to circumvent this dead-end, dimensional analysis will be used in the following paragraph, in an effort to empirically 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 temperature (for which monthly values are available, hence, it can be integrated), a dimensionless temperature T* is defined as T * = (Ts-~)/(Tref--Ta), with Tref a reference temperature, and t* a dimensionless 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
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 following set of dimensional parameters--which all affect the solar fraction--can be isolated : (Ac,/tT, QL, UL, AT, At). A dimensionallyequivalent but useful, for comparison purposes, set is the following (AcFR, NHT(Z~), QL, UL, AT, At), in which the mean temperature difference, AT = ( T r a - 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 AcF'R (z~)HTN At AT UL
QL
L 2 0 0 0 0 2
M 0 1 0 0 1 1
t 0 - 2 1 0 --3 -2
T 0 0 0 1 -- 1 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 specially developed interactive computer program uses this information to produce all the possible dimensionless 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 '
f
T*(Tref- Ta) dt _
dT*
,
thus producing on the right-hand side of eq. (2)
which focusses on the collector, as it simply represents the ratio of its (long-term) energy gain to its corresponding 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 for solar heating The three energy quantities--collector gain, collector 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 characteristic energy quantity common to both dimensionless 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.
889
under study, namely that of the standard configuration 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 following was considered to be the best :
(y)2 (y)3
-0.0983Y2-5.12 ~
+3.33 ~
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
O.O0 0,10 0,20 0.30 0,40 0.50 0.60 3.00 ~ I = ~ = ~ n r ~ n n F n ~ n ~ t n ~ = n n ~ ; = n ~ n = ~ = n n ~ n n = [ ~ = = ~ w ~ = = r ~ w ~ n ] ~
0.70
0.B0
0.90
3,00
d 2,00
2.00
if LL ~ 1.oo
~ ~
~
o.a
0.3
O.O0' InJ 0.0
r n~lnnlllnl=n,
O, 10
,lal,
0,20
,laltannlaa=l
0.30
~ ~
0.9
b.8 0.7
1.00
0.8
0.4 ===all jl=Lnlaaal,n
0.40
(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 correlation 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 0.0430.024 with corresponding correlation coefficients in the range 96.3-98.9%. Obviously, care should be exercised in using too many terms in the regression equation, 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 o n - - i n the complexity 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
4. AN EMPIRICAL DESIGN TOOL
Z
.
0.50
j~lnnrLlala== uaonll=nalJajltlaoaannn
0.60
0.70
O.BO
0,00 0.90
(T-~)gTN/ U L(T~,-T.)At Fig. 1. Generalized diagram for the design of heating systems using liquid heat transfer and storage media.
J.A. PALYVOS
890
0.0
Table 1. Range of the design variables Parameter
Range
(~) At UL /~
0.7~.85 30-100 (m2) 2-5 (W/m2) 30-60 (degrees) 150-400 (W/K)
UA
10.0
........
2.0
, .........
4.0
~. . . . . . . . .
6.0
, ........
'
8.0
"~
8.0
A
6.0
0
A
~
o
i
r
e
10.0
10.0
~
*
8.0
-
6.0
n
8 •
i
~:~ 4.0
4.0
a
2.0 enables estimation of the monthly solar fraction, for the given design parameters and the local meteorological 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.
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 thermal 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 increasi n g - b u t not linear--function of both the total collector 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 simulation. The deviations from the 45 ° line--which are of the order of 5% suggest that the empirical tool underestimates the solar fraction. The overall agreement 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,
~
0.0
,,,= ....
0.0
* o° * = I_ariso
"
o o o o o
Hania
I,l=,,===,l,=l,,i,,,I,=,,,=,,=llPlllll
2.0
4.0
f-Simulotion
6.0
2.0 0.0
B.O
10.0
xlO
Fig. 2. Comparison of simple design tool predictions with detailed simulation results for solar space heating systems.
however, were made for a number of Northern European locations [10]. The success of the empirical method confirms the prediction that the long-term performance of the heating system is not very sensitive to the hourly fluctuations 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 cp f FR HT M N QL Qu Qn t /ref t* 72a Trof T~ T*
area of collector (m 2) heat capacity of fluid (J/kg K) solar fraction collector heat removal factor daily global insolation on the tilted collector plane (kJ/m 2) storage tank fluid mass (kg) number of collector panels heated space losses (kJ) collector useful energy gain (kJ) auxiliary heat (kJ) time reference time dimensionless time ( = t/tref) ambient temperature (°C) reference temperature (°C) storage tank (uniform) fluid temperature (°C) dimensionless temperature (= (Ts- T,)/ (Tref - T.))
A simple design tool for solar heating UL collector overall t h e r m a l loss coefficient ( W / m 2) UA heating load (k J) X f-chart dimensionless g r o u p Y f-chart dimensionless g r o u p Greek symbols collector plate a b s o r p t a n c e /~ collector tilt (degrees) r collector cover t r a n s m i t t a n c e Superscripts rate q u a n t i t y m e a n value.
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).
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3. 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 performance prediction methodology for integral collection-storage solar domestic hot water systems. Presented 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.