Accuracy of the European solar water heater test procedure. Part 1: Measurement errors and parameter estimates

Solar Energy Vol. 47, No. I, pp. 1-16, 1991

Printed in the U.S.A.

0038-092X/91 $3.00 + ,00 Copyright © 1991 Pergamon Press plc

ACCURACY OF THE EUROPEAN SOLAR WATER HEATER TEST PROCEDURE. PART 1: MEASUREMENT ERRORS AND PARAMETER ESTIMATES B. B O U R G E S , * A. RABL, t B. LE1DE,* M. J. CARVALHO,* and M. COLLARES-PEREIRA ¢ *35760 St.-Gr6goire, France, *Centre d'Energ&ique, Ecole des Mines, 75272 Paris Cedex 06, France, tDept, de Energias Renovaveis, LNETI, 1699 Lisboa, Portugal Abstract--The Collector and System Testing Group (CSTG) of the European Community has developed a procedure for testing the performance of solar water heaters. This procedure treats a solar water heater as a black box with input-output parameters that are determined by all-day tests. In the present study we carry out a systematic analysis of the accuracy of this procedure, in order to answer the question: what tolerances should one impose for the measurements and how many days of testing should one demand under what meteorological conditions, in order to be able to guarantee a specified maximum error for the long term performance? Our methodology is applicable to other test procedures as well. The present paper (Part 1) examines the measurement tolerances of the current version of the procedure and derives a priori estimates of the errors of the parameters; these errors are then compared with the regression results of the Round Robin test series. The companion paper ( Part 2 ) evaluates the consequences for the accuracy of the long term performance prediction. We conclude that the CSTG test procedure makes it possible to predict the long term performance with standard errors around 5% for sunny climates ( 10% for cloudy climates). The apparent precision of individual test sequences is deceptive because of large systematic discrepancies between different sequences. Better results could be obtained by imposing tighter control on the constancy of the cold water supply temperature and on the environment of the test, the latter by enforcing the recommendation for the ventilation of the collector.

1. INTRODUffrlON The basic decision criteria for the purchase of a solar water heater are the cost of the system and the value of the energy delivered by the system over its lifetime. To determine the latter, a standard test procedure is necessary. Such a procedure should satisfy a number of requirements: • it should allow prediction of long-term average performance for any set of reasonable operating conditions • it should have sufficient accuracy to permit a meaningful ranking o f candidate systems • it should not impose a significant cost burden. Several approaches are possible[I,2 ]. In the United States, for example, the ASHRAE-95 test has become an accepted standard [ 3 ]; however, by itself it does not permit extrapolation of the performance beyond the very specific conditions of the test. To overcome this limitation, Klein and his colleagues [ 4,5 ] have proposed a method where the annual performance is calculated by means of F - C H A R T or a reduced version of TRNSYS, using as input a simplified characterization

This work has been carried out within the framework of the European Solar Collector and System Testing Group under contract 33.34-87-12 E D I S P between the Joint Research Centre, Ispra, and LNETI, Lisbon, with subcontract to the Centre d'Energ6tique, ARMINES, Ecole des Mines. M. Collares-Pereira is now at Centro para a Conservaqao de Energia, Estrada de Alfragide, Praceta 1, Alfragide, 2700 Amadora, Portugal

of the components (with four effective parameters: collector area, collector gain coefficient F r ( r a ) n , collector loss coefficient, and tank volume) as determined by the ASHRAE-95 test. In Europe the Solar Collector and System Testing G r o u p [ 6 ] has been following a track that looks quite different because no reference is made to c o m p o n e n t parameters such as a collector heat loss coefficient; rather the system is characterized from the start as a black box with input-output parameters that are determined by all-day tests. This method is very flexible, allowing extrapolation to different utilization conditions while taking stratification effects into account. Whereas the development of the European C S T G test procedure is c o m p l e t e [ 7 ] , a systematic analysis of measurement errors and their consequences for the uncertainty of the long term performance is still lacking. This question of accuracy is crucial if one wants to know what accuracy requirements to impose for the measurements and how m a n y days of testing to dem a n d under what meteorological conditions. Filling that gap is the goal of the present study. The analysis of measurement errors is presented in Part 1, and the consequences for the long term performance prediction are discussed in the companion paper, Part 2 in this same issue o f Solar Energy[ 8 ]. These papers are a shortened version of a more complete report [ 9 ] that can be obtained from the authors. Even though we discuss only the C S T G procedure explicitly, our methodology can also be applied to other test procedures.

B. BOURGESel al. Quite generally there are four basic types of errors that affect the prediction of long term system performance: • measurement errors, both systematic and random, of the short-term test • errors of the mathematical model used for analyzing the test data and predicting the long-term performance • differences between the equipment tested and the equipment actually used, including its degradation over time • uncertainties about the operating conditions, i.e., the actual weather and the actual water use during the life of the system; Since our primary goal is the analysis of measurement errors, we focus on the first type. Our framework would allow analysis of the other types, and we briefly address the question of model errors, as well. We develop the error analysis by way of examining the results of the CSTG Round Robin tests. Under the Round Robin program the same solar water heater has been tested by several laboratories to see how consistent the results are. Eight laboratories have participated; their location is shown on the map of Fig. 1. In this paper they will be referred to by the three-letter abbreviations listed under Fig. 1. Round Robin data are available for three heaters which we designate the inirials: BP = BP-Calpak; HE = Helioduc; IC = integrated collector-storage system. In this publication we present the results for the BP system; the complete report [9 ] contains analogous resuits for HE and IC. The BP system has a collector area of 3.2 m 2, a tank volume of Vs = 0.16 m 3, and a

tank heat loss coefficient Us = 2.2 W/C. A fuller description of these systems can be found in the documents of the CSTG [ 6 ]. 2. THE CSTG TEST PROCEDURE 2.1. The linear i n p u t - o u t p u t m o d e l

The basic ingredient of the European CSTG test procedure is the linear input-output model for the daily performance of solar water heaters. According to this model the heat Q [in MJ] added to the tank during the day is a simple linear function Q = ao + auH + arT

with T = ( T a - T,)

(1)

of three variables: H = daily total solar irradiation on collector aperture [ MJ 2/ m]; Ta = daytime average ambient temperature [C ]; and Ts = temperature of storage tank at start of day [C], with three coefficients, t~o, a n and aT. Storage is assumed well mixed at the start of the day. The relation is valid only for positive Q. A sample plot of test data will be presented in Fig. 6. This model evolved during the initial years of the CSTG program, from the analysis of measured as well as simulated data [ 2,6,10,11 ]. The model is very plausible: a n represents the ability to capture and convert solar radiation, c~r represents the heat losses from collector and tank during the day, and a o accounts for

Fig. 1. Geographic distribution of the laboratories participating in the CSTG Round Robin program: CON = Conphoebus, 1-95127 Catania, Italy; CST = CSTB, F-06561 Valbonne, France; DSE = Danish Solar Energy Testing Laboratory, DK-2630 Tasstrup, Denmark; INT = Inst. Nacional de Technica Aerospacial, Madrid, Spain; ITW = Inst. f'dr Thermodynamik und W~raetechnik, Universit~itStuttgart, GFR; JRC = Joint Research Center, 1-21020 Ispra, Italy; LNE = LNETI, Lisbon, Portugal; TNO = Inst. of Applied Physics, Delft, Netherlands.

Measurement errors and parameter estimates threshold effects (finite thermostat differentials for turning collector flow on and off). However, a detailed interpretation of a n and ar in terms of aperture area and heat loss coefficient is not straightforward. The term a t ( Ta - Ts) accounts only for that part of the heat losses that is due to the starting temperature Ts. The remaining heat losses, due to warm-up of the system during the day, are contained in the aHH term. Obviously a model with daily averages only cannot account for phenomena of shorter time scale. In particular it does not allow for the possibility that the performance can depend on the distribution of solar radiation during the day. As an extreme example compare two days, one with constant irradiance, the other with high irradiance during a short period only, the daily total H being the same and equal to the threshold. Then the nonuniform day can deliver a significant amount of useful energy, while nothing comes from the uniform day, because the dominant heat loss, i.e., from the collector, is incurred when and only when the collector is turned on. Likewise it is easy to see that the performance, for the same value of H, tends to decrease as the length of the day increases; this effect becomes more pronounced for latitudes further away from the equator. In practice the accuracy of eqn ( 1 ) seems to be on the order of 5% for typical operating conditions. Unfortunately a systematic analysis of this question has never been carried out; at the present time it is not clear how much of the scatter of data points is due to measurement errors and how much due to deficiencies of the model. A recent paper by Ernst et a/.[12] is an important step in this direction. It suggests that the length of day effect may be important, but the study is too limited in scope to permit definitive conclusions. 2.2. The test procedure When applying eqn ( 1) to a test procedure, the main difficulty lies in the determination of Q and of Ts in stratified tanks. The use of a large number of temperature probes was rejected as being too intrusive and

Shaded AmbientAir TemperatureSensor Shaded (Adjacentto collector) Pyranometer / Pyranometer~ j,/

Aneoter \

too difficult to implement. Also, knowing Q is not enough. The performance under real conditions depends on the amount of energy left in the tank after draw-off, including mixing during draw-off. Thus it is desirable to determine not only Q but also the details of the draw-off. The CSTG committee has developed a method that tests the solar water heater as a black box; it is completely nonintrusive. To minimize uncertainties due to stratification, the following steps are taken. At the start of the day the system is filled with water at uniform temperature. At the end of the day three tank volumes are withdrawn while carefully monitoring the temperature and controlling the flow rate. The test setup is shown in Fig. 2. The essential elements added to the solar water heater are • a cold water supply with tight control of temperature and flow rate • a temperature probe at the cold water inlet of the tank • a temperature probe at the hot water outlet of the tank • a recirculation pump between outlet and inlet, to permit good mixing of the tank (this pump is needed for step l of following paragraph if system with thermosyphon is to be tested). As summarized very briefly in Fig. 3, the essential steps of the CSTG test procedure are the following: 1. Preconditioning. Before the start of the all-day test, water at constant temperature T~ is circulated through the system (with collector shielded from sun) until the entire system is at uniform temperature Ts = T~. 2. Exposure. From 6:00 to 18:00 (solar time, and regardless of sunset; this reduces the dependence on length of day) the system is allowed to function normally, and the solar radiation H during this period is recorded. The wind speed should not exceed the range of 3-8 m/sec, and use of fans is recommended to maintain 3-5 m/sec. No water is withdrawn during this time.

Temperature Transducer

--1

Insulation I% Hot Water v Draw-off

......

, , ~ Tank- Mixing

~ ~.~

,~ / / Fan (If necessary)

~

~J~ o.w

~ ~ ] ~

System

Bleed-off Pipe

Pump

--

Temperature

Transducer

ShadedAmbientNr TemperatureSensor (Adjacentto store)

Cold wEr

l

F~

F~ow

t

~ter Cont,o,er /

Three Way Valve

/

Inlet Temperature

Controller

Fig. 2. Test loop for CSTG test procedure. From CSTG [ 7 ].

PP~Y

4

B. BOURGESet al.

6 Hours

Solar Noon

BeforeNoon

Sunrise Shade Collecto¢

6 Hours After Noon

3 rids
E

Sunset

Shade CoRector G~¢~

I~ 600 l/hi

.-_

•

Heater

Inlet Ten~erature Contro~

Ventilator

PRE-CONOITIONING Flush system with water at Tc

Temperature ------'-- z--T,,-I Controller DRAW-OFF 3~s of water at 600 ilhr, replace with water at T©

OPERATION Record hourly average values of Ta and u

RECORD HOURLY AVERAGE VALUES Of: Gl.ox~. ANO GOW,aS~EON PLANE OF COLLECTOR

Fig. 3. Summary of CSTG test procedure. From CSTG[7]. 3. Draw-off. At 18:00 three tank volumes are withdrawn at constant flow rate of 0.600 m3/h, while replenishing the tank with cold water at constant temperature Tc (same as at start of day). The temperature To,t during this withdrawal is monitored, as function Toot(V) of the withdrawn volume V. This temperature profile is called "draw-off temperature profile." 4. Loss coefficient o f storage. The heat loss coefficient of the tank, Us, is determined in a separate test by measuring the temperature decrease during the night after the tank has been filled with hot water. Of course such a curt resume cannot do justice to all the subtleties of the full procedure, for which the reader is referred to the respective documents [6,7 ]. The detailed specifications for instrumentation and setup are crucial. In addition, the procedure recommends the determination of two further draw-off profiles to determine the amount of mixing in the tank. However, as shown in Section 3.2 of the companion paper Part 2, the sensitivity of the long term performance to uncertainties in the draw-off profile is so low that we believe these additional profiles to be unnecessary. The draw-offprofile can be presented as a normalized dimensionless function

f(v)

=

[Tout(V ) - -

Tc]/[Tout(V=O)- T c ] .

(2)

Hypothetical forms o f f ( V ) are sketched in Fig. 4. If the tank is at uniform temperature before the drawoff, its energy content relative to Tc is Qo = V~pcp[To.t(V=O)- To].

Combining the last four equations one obtains 3,(V/V,) = 1 -

f(V')dV'/Vs.

(6)

Two limiting cases are of special interest: perfect mixing and perfect stratification. For the latter it is obvious that f ( V ) equals unity for V < Vs and zero for V > Vs, as shown in Fig. 4. For "r(V/Vs) that implies {Io-V/V~ 3"(V/Vs)str

for for

=

O V~

(7)

For a perfectly mixed tank one can easily derive the result 3"(V/V~)mix = e x p ( - V / V ~ ) .

(8)

Figure 5 shows the function 3"(V/Vs) for perfect mixing and perfect stratification, together with measured data for the BP system. As the profile can vary quite markedly from day to day, we indicate the maximum and minimum measured at each V in addition to the average. We have not found any systematic pattern for the day-to-day variation, except for some slight trend during overcast days ( H below 10 MJ/m2). In Fig. 5 only days with H above 13 M J / m 2 have been included. In Section 3.2 of Part 2 we show that uncertainties in 3"(V/V~) have only a small effect on the long-term performance.

(3) Energy Available = Area under Curve

The energy withdrawn in a volume element d V is T

dQ = dVpcp[To.t(V) - To].

(4)

~ .¢' J "%

I

For future reference we need to know the fraction 3'(V/ II,) of the heat remaining in the tank after withdrawing a volume V. Since the energy withdrawn is the integral of dQ from V' =0 to V'= V one can write 3"(II/Vs) as 3"(V/Vs) = 1 --

dQ/Qo.

(5)

I = Ideal System 2 = System with little mixing in tank 3 = System with complete

I out.

~. I

2~ ]

"x

o ~ ~,.

I"'- .......

"1"

Tc

0

-

',1 ,~

mi,ino intank

1

4 = System with draw-off through

a heat exchanger

t"

.......

V$

", ~-

~'~ ":-2.2....................... .

Volume

2v s

- ..,......

3Vs

Fig. 4. Draw-off temperature profiles (theoretical). From CSTG[7].

Measurement errors and parameter estimates

5

• (Vl/V s] .8

.6

0

.4-

.2

0

0

.~

1

115

2

215

VI/V s 3.5

Fig. 5. Data for draw-offtemperature profile of BP system ( maximum, average and minimum observed for BPLNE series on days with H > 13 M J/m2). Profiles for perfect mixing and for perfect stratification are shown as solid lines.

Five quantities are measured during the test: H = daily irradiation on collector aperture (approx. +_3%); Ta = ambient temperature (+_0.5 K); Tin = temperature at inlet of storage, (+_0.1 K ) ; To,t = temperature at outlet of storage, (+_0.1 K ) ; and I? = flow rate during draw-off (_+ 1% ). The numbers in parentheses are the tolerances for the measurements as specified by [ 7], with which the tolerances reported by the individual laboratories are essentially in agreement. The insolation is to be measured with a first class pyranometer (according to the terminology of the World Meteorological Organization). Our estimate of +_3% for the tolerance is a simplification: the true tolerance specification depends on the precise operating conditions. Measuring ambient temperature during sunny days is problematic, even if one had the most precise instrument, perfectly shielded from radiation. In fact, T~ is not well defined when the air is not at uniform temperature, being heated by convection from various irradiated surfaces at different temperatures. Ta should be the temperature of the air surrounding the solar water heater. Obviously the collector and its surroundings are in full sunlight, and the various surfaces do not have the same temperature because of differences in absorptance, emittance, and geometry. A further error arises from the fact that the relation between sky temperature and air temperature varies from day to day* while the model represents these temperatures

by a single quantity Ta. These are the d o m i n a n t errors in the variable T = Ta - Ts of the input-output model, and they are difficult to avoid in outdoor tests. We believe that these errors can a m o u n t to several degrees K. They can be reduced by using a radiation shield for the thermometer, as well as a fan next to the collector, as the C S T G procedure recommends. But this ventilation of the collector, has not been made obligatory, and not all tests have used it. Another error source arises from a loophole in the C S T G specifications for the temperature Tc of the cold water supply. Whereas the required accuracy of the measurement of T~, in quite stringent, no explicit tolerance limit is imposed for the constancy of T,. between morning and evening.t The specification of +_0.25 K for constancy of the water supply concerns only the period of the draw-off itself. Maintaining close tolerance of Tc can be difficult and costly if the temperature of the water mains fluctuates significantly. Data on this point are not generally provided by the participating laboratories, but we have seen quite a few cases where Tc varies by more than 1 K. Finally there may be errors due to greater or lesser deviations from the C S T G procedure. For example, at least one of the laboratories did not have the means to purchase enough automatic valves; as a result, on days when nobody was there to drain the cold water supply pipe before the start of the draw-off, some of the water entering storage was above the nominal cold water temperature because it had been sitting in the pipe all day, exposed to ambient. In most laboratories several systems have been

* In degrees K the ratio of sky temperature and air temperature ranges from about 0.9 on clear days to 1.0 on cloudy days. This effect is partially compensated by the elevated temperature of the ground in front of the collector and by the limited view factor for the coldest part of the sky.

t The only reference to this question is the statement "Water replacing this should be at the temperature Tc defined during the preconditioning of the system" concerning the cold water during draw-off [lst paragraph of Section A5.2.6 of CSTG, 1989].

3. MEASUREMENT ERRORS

3.1. Error sources

6

B. BOURGESet aL

tested at the same time and with a single test setup, in which case the draw-off must be done sequentially. Since the draw-off takes on the order of an hour (at 0.60 m 3 / h ) , there could be significant differences due to heat losses from storage between the nominal end of the test (18:00) and the actual start of the draw-off. For instance with typical values of Us = 2 W / m 2 K and A T = 25 K between storage and ambient, a onehour delay causes a loss of 2 W / m 2 K × 25 K × 3600 s = 0.2 M J, easily one or two percent of the daily harvest. The precise details of the installation of the temperature sensors in the water heater can differ from one laboratory to another, and differences in flow patterns can cause differences in temperature readings. Furthermore, it is impractical to completely control the albedo of the ground and the shades cast by buildings or trees, or to assure that collector and pyranometer see exactly the same radiation. Likewise, differences in wind-induced convection are difficult to eliminate completely in an outdoor setting, even if one places a fan in front of the collector. Even if all such variations would be or could be reported, quantification of their effects would remain elusive. In the following we will try, as much as possible, to evaluate the effects of the various error sources. In order to understand differences between laboratories it will be necessary to distinguish random errors and systematic errors. A systematic error affects an entire series of measurements in a regular manner, for instance, as a uniform bias in some variable. If a systematic error is known it can be compensated. Random errors vary during the series, without any regular pattern. Systematic errors are reproducible, random errors are not. The distinction may not always be clear; there might be a pattern in the errors even if it is u n k n o w n to the observer. Also, systematic errors of individual data can look like random errors when data are combined. For example, the R o u n d Robin results from different laboratories can be combined and analyzed as a single series (as we shall do in Section 4); then the systematic errors of the individual laboratories will contribute to the r a n d o m error of the combined series. 3.2. Effect on coefficients o f input-output model The measurement errors affect the parameters ao, a n and a r a n d the function 3'. We postpone discussion of errors in 3' and in the storage heat loss coefficient Us to Part 2 on long-term performance [ 8 ]: 3" because its errors will turn out to have negligible effect, and Us because it is measured at night in a separate test and its errors are therefore independent from the other parameters. As the parameters ao, aH and aT are determined by linear regression of eqn ( 1 ), their errors are due to errors in Q. The latter is determined by measuring the energy withdrawn during draw-offduration t3v [in see]

Q3v = ff3v ("OCp(Tout - Ti.)dt.

(9)

The flow rate being constant, this can be written in terms of the average temperatures measured during the draw-off Q3v = llpcp(Tout- Tin)t3v,

(10)

where the total volume withdrawn is three times the volume Vs of the storage tank IYt3v = 3Vs.

(11)

Uncertainties in cp and t3v are negligible. Thus the error AQav is due to the errors ATout and ATin in the temperatures and the error A I?in the flow rate during drawoff

AQ3v = Af'Ocp ( Tout - L . ) t3v -+ ~f'pcp(AT~out -- A~/~in)t3v .

(12)

Using eqns (10) and ( 11 ) this can be written as AQ3v = Q3vAIY/V + 3 VsDcp( A Tou t - ATin ).

(13)

If the temperature of the water mains were perfectly constant, and if all of the energy could be withdrawn in three volumes, then this would be the error AQ. But in reality the average temperature of the tank after draw-off Te3v is not exactly equal to the temperature Ts at the start of the day, for two reasons: first because of changes in cold water supply temperature (the above mentioned loophole), and second because of incomplete extraction. The latter effect can be seen in the value of'), at V = 3 Vs. In theory, eqn (8) for a perfectly mixed tank should be the upper limit for the energy remaining in the tank. This implies 3"(3) < e x p ( - 3 ) _-_ 0.05. Actual data vary greatly from one system to another: for HE 3'(3) is indistinguishable from zero, for BP it ranges from 2% to 6%, and for IC it is l0 to almost 40% (in the IC system the heat is extracted indirectly via heat exchanger, much of the heat can remain in the tank, and eqn (8) is not relevant). For a system whose 3'( 3 ) is larger than a few percent, the CSTG procedure can lead to sizable errors (but one wonders whether a system with such inefficient heat extraction could survive in the market place, anyway). Adding a term V~pcp( T~ - T,3v) to AQav or eqn (13) we find that the total error AQ is A Q = Q3vAII/~V + 3C(ATout - ATin) + C(Ts - Te3v),

(14)

with C = Vspcp. Note that the t e r m (ATou t -- ATin ) is multiplied by 3C because it is an error in the measurement of the

Measurement errors and parameter estimates temperature during withdrawal of 3 volumes. The term ( Ts - Te3v) is multiplied only by Cbecause it represents a shift in the value of the temperature used as basis for computing the energy Q in the storage volume. In practice the errors are uncorrelated and the linear sum in eqn (14) is to be replaced by the root of the quadratic sum. Before presenting results it is appropriate to distinguish the effects of random and of systematic errors. If the error AQ were entirely random and unbiased, characterized by an rms (root mean square) deviation tr

: V E { A Q 2} ,

(15)

and if there were no errors in the independent regression variables H and T = ( Ta - Ts), then the resulting errors in ao, an and a r would be given by the variancecovariance matrix as described in Section 4. However, in the case of the Round Robin tests, one could expect several of the errors to be systematic within each laboratory, while others are likely to be random. Pyranometer errors tend to be systematic. (Note that a bias due to dirt accumulation on the cover has almost no net effect on the test results, if collector and pyranometer are always cleaned together.) Temperature measurements are likely to have random errors (due to the irregularities of turbulent flow) in addition to systematic ones (due to instrument bias). Errors in Ta and in ( Ts - Te3v) are partly systematic, partly random. This distinction between random and systematic errors is necessary to understand the scatter of the results from different laboratories. To obtain the systematic errors in ao, an and a r we note that the true relationship Q = ao + a n H + a r T ,

with T = ( T a - Ts),

(16)

is estimated in terms of measured quantities (indicated with a tilde) as

Aao + a H A t t + A a H H + a r A T + A a r T = [ao + OHH + a r T ] A I ? / V

+ 3C(ATo.t - A ~ . ) + C(Ts - Tear).

(19)

Even though this is only a single equation, one can extract a separate equation for each of the three a parameters if one notes that the a are determined by fitting data over a range of H and of T. In order for eqn (19) to hold over a range of values of H and of T, the sum of the coefficients of H must vanish separately, and likewise for T. For the coefficients of T this implies aar = arAV/K

(20)

if AT is independent of T. For AH one can assume that the error is a fixed percentage (All~H) of H; then one obtains A a n = a H [ ( A H / H ) + (AI?/V)],

(21)

and, for Aao, Aao = a o A f ' / V + C ( G - Te3v)

+ 3C(ATo~t - A~Pi.) - a r A T .

(22)

The A H / H term in A a n is very plausible: it represents the change in slope due to a change in the scale of the H-axis. The A I?/V term contributes to all three coefficients because it represents a change in the scale of the dependent variable. The asymmetry between A a n and Aar arises from the different behavior of insolation and temperature errors; AH is proportional to H whereas AT is independent of T. If the measurements have known systematic biases A//, AI?, AT, etc., the corresponding bias in the coefficients a can be calculated by eqn (20) to (22). But in general only approximate upper limits of the biases are known, not their precise values and signs, For a general estimate of systematic errors, one should therefore assume probabilistic distributions and add the terms quadratically. Then eqns (21 ) and (22) are replaced by

0 = Q + AQ = &o + &./~ + &rT

Aa2n = a H [ ( A H / H ) 2 + (AI?/V)2],

(23)

= (ao + Aao) + (all + Aa/~)(H + AH)

+ (at+ Aar)(T+

AT).

(17)

and Aa2o = [ ~ o A I ? / V ] 2 + [ C ( T , - Te3v)] 2

To lowest order these two equations imply + [3C(AiPout - A ~ . ) ] 2 + [ a r A T ] z

A Q = Aao + a H A I t + A a H H + arAT + AarT.

(24)

with, as before, (18) C = Gpc,.

This can be equated with AQ ofeqn (14). Furthermore the term Q3v in the latter can be replaced by Q since we are only interested in the lowest order. Thus, we obtain the equation

In Table 1 we list numerical results for the BP system, for a variety of combinations of( T~ - Te3v), ( A 7~out - ATi,), and AT = (ATa - ATs), centered around

8

B. BOUROESet al. Table 1. Measurement errors and their effect on AQ and on Aao, evaluated for C = V~ocp = 0.67 M J / K and a r = 0.40 M J / K (close to BP system). Contribution of A V / V < 1% has been neglected. (a) errors due to (T~ - Teav) and (ATo= - A~Pi,). The quadratic sum o=, eqn (25), equals AQ. (b) combination of typical values of g~ from (a) with errors due to (AT, -- ATe). Quadratic sum a2, eqn (26), indicates systematic error Aao if only systematic error contributions are included. If calculated with random errors, ~2 is the a priori estimate of the effective standard error of the regression tr temperature

(Ts-Te3 v ) [C]

errors

e n e r g y errors

(ATout-ATtn) [C]

C(Ts-Te3 v ) [MJ]

quadratic

3C(6~ut-ATtn) [MJ]

sum

~1 [MJ]

0.5 0.5

0.1 0.2

0.33 0.33

0,20 0.40

0.39 0.52

1.0 1.0

0.1 0.2

0.67 0.67

0.20

0.40

0.70 0.78

1.5 1,5

0.1 0.2

1.00 1.00

0.20 0.40

1.02 1.08

(a) ~1

[ &Ta-ATs )

¢2/0

¢T [ ATa-hTs )

at Q=IOMJ [g]

~/Q at Q=2ONJ IX]

[MJ]

[C]

[MJ]

[MJ]

0.4 0.4 0.4

0,5 1,0 1.5

0.20 0.40 0.50

0.45

4.5

2.2

0.57 0.72

5.7 7.2

2.8 3.5

0.7 0.7 0.7

0.5 1.0 1.5

0.20 0.40 0.60

0.73 0.81 0.92

7.3 8.1 9.2

3.6 4.0 4.6

1.0 1.0 1.0

0.5 1.0 1.5

0.20 0.40 0.60

1.02 1.08 1.17

10.2 10.8 11.7

5.1 5.4 5.8

(b)

what we believe to be probable values. Note that for the C S T G tolerance o f I A ~?/V [ < 1% the contribution of A I?/Vis negligible relative to the other terms; therefore we do not even include it in the table. Part (a) of the table concerns the terms (T~ - Te3v) and (ATout A ~ . ) ; both temperatures and the corresponding energy terms are listed. The last column shows their quadratic sum o~ -

o, = V [ C ( T s - Te3v)] 2 + [3C(AT~out- A~n)] 2-

(25) This is the error AQ, to the extent that A I ? / V is negligible. Part (b) of the table takes typical values for o~ as suggested by Part (a) and combines them quadratically with possible errors in A T = (ATe - ATe); the result is shown in the last c o l u m n as o2 o2 = 1/o 2 + [aT(ATe - ATe)] 2.

(26)

This is the systematic error A a o , to the extent that A l ? / V i s negligible, ol and a2 have several interpretations. o,, i.e., the error of Q, is partly systematic, partly random, reflecting the composition o f the temperature errors. T o find the random error of Q only the random temperature errors are used in e l , and analogously for the systematic errors--although we find it difficult to decide how m u c h to assign to each category. In any case, the d o m i n a n t term is due to ( Ts - Te3v), and it

can have a large systematic component due to the lack of control of the cold water supply temperature between morning and evening, which causes a fairy regular pattern of warm-up during the day in some laboratories. This effect can easily account for 0.5 MJ of systematic differences from one laboratory to another (for the BP system a l K difference contributes 0.67 M J ) . In general our estimates for o~ are on the order of 0.5 to 1 MJ for the water heaters of the R o u n d Robin program. For the calculation of the systematic errors of the a, only the systematic errors are to be used in eqns (20) to (26). The systematic error in a r is very small (1%). The systematic error in a n is on the order of 3%, essentially the (systematic) error of the pyranometer. The systematic error in a o could be on the order of 0.5 to l MJ, in large part due to errors in ambient temperature (with some question mark about the magnitude of its systematic c o m p o n e n t ) . The last colu m n s of Table 1 (b) express o2 as relative error, for Q = l0 and 20 MJ. The relative error is at least 2%, which justifies our neglect of the 1% error in flow rate. Coming back to eqn (15), we note that a, (calculated with random errors only) is an a priori estimate o f the standard errors of the residuals of the regression to be performed in Section 4 if there are no random errors in the independent variables H and T = (Ta Ts). In most regression analyses random errors in the independent variables are assumed to be negligible. But for the R o u n d Robin tests Ta can have large rand o m errors. U n d e r certain conditions such errors can be combined with the error of the dependent variable; -

Measurement errors and parameter estimates these conditions, discussed more fully in Section 2.14 of Draper and Smith [ 13 ], appear to hold in our case. Then the standard error of the regression is the quadratic sum of the error in Q and the error in T, the latter multiplied by a t . That is the quantity a2 ofeqn (26). It is in effect the a priori estimate of the standard error of the regressions in Section 4 (apart from possible contribution of errors of the input-output model). The values in Table I(B) suggest something on the order of 0.5 to 1 MJ, possibly with a sizable contribution from uncertainties in T~.

TEST

between the measured value Q~ and the regression Qi = Oto + otnHi +

,

x =

"

'

1

H.

Ot ----

The coefficients ao, a n and ar are determined by a least-squares fit of the Q data. In order to be able to state the results, we need a brief summary of the required statistical tools; for details we refer to the book of Draper and Smith [ 13 ]. The measured quantities, for day i, are the insolation Hi, the difference T~ = T~,~ - T,,~between ambient temperature and starting temperature of storage, and the energy output Q~. They are related by

(27)

aH

,

and

,

t

(31)

=

o~ T

are matrices of dimension (nX1),

(nX3),

(3×1),

and

(nX1),

respectively. The least-squares estimate is found by minimizing the sum of the squares of the errors e~. In vector notation this means setting the derivative of e re with respect to a equal to zero, where the superscript T designates the transpose (obtained by interchanging rows and columns). One obtains immediately the result & = (XrX)-LXrQ,

where t~ is the error, i.e. the difference

(28)

Oi

X rX =

(30)

where

4.1. Regression formulas

ei = O i -

(29)

Q = Xa + e

DATA

Qi = oto + OtHHi + OtTTi + ei

arTi

at the measured H, and 7",.. It is convenient to rewrite eqn (27) in vector notation as

Q =

4. STATISTICAL ANALYSIS OF ROUND ROBIN

9

(32)

the tilde indicating that this is not the true value (which is unknown), but the least-squares estimate. It is easy to see that X r x can be written in the form

n

nB

n;P

nfi

n(/q 2 + o-~)

n(I]73 + COV( H , T ) ) ]

n~ n ( H T + C O V ( H , T ) )

n(~2 + ~ )

(33)

j

with

I~ =

1_

C O V ( H , T ) = 1 ~ (Tj - T ) ( H , - 1~) n

~ Hi = average of H during test,

(34)

ni=l

= covariance of H and T,

= rurcrHar

n

7~ = _1 ~ T~ -- average of T during test, n i=l

i

(38)

ni_l

(35)

with

rHr = correlation coefficient of H and T. While the true errors t are as unknown as the true a, they can be estimated as the residuals

n

n i=t

=Q-Q = variance of H during test,

/'1 i= I

= variance of T during test,

with

(~=X&.

(39)

(36)

(37)

Critical examination of the residuals is one of the most important parts of the job: they contain crucial evidence for or against a model. The variance a 2 of the residuals is the expectation value of e re; it is estimated by

B. BOURGESet al.

10 ~.2 =

1

n-3

~] ~.z

(40)

where the n in the denominator is reduced by 3 because the data have been used to estimate 3 parameters before ;2 could be calculated. The standard error of the regression ~ is one possible measure of the goodness of a fit. Another is the coefficient R 2, also known as square of the multiple correlation coefficient. R 2 indicates how much of the variation of the data Q, about the mean (~ is explained by the correlation. The closer R 2 is to unity, the better. As for errors of the parameter estimates, & is an unbiased estimate of a, i.e., its expectation value is a. Its variance-covariance matrix V ( a ) , defined as expectation value of (& - a ) r ( & _ a ) , can be shown to be [13] V ( a ) = 0"2(xTx) -1,

(41 )

and it is estimated by using b2 instead offf 2. The standard errors of the components of a are the square roots of the diagonal elements of V ( a ) . For example, the standard error of ao is estimated as

~o

=

~V[(xTx)-'],,

(42)

with the 11 element of ( X r X ) -~ . A complete error analysis, as presented in Part 2, requires all six of the independent elements of X rX or its inverse, because the parameter estimates are correlated. The correlation between the errors is determined by the off-diagonal elements of V ( a ) ; for instance the correlation coefficient between errors of ao and a u is given by V(~),2

r~o~n = - -

•

4.2. Regression results Here we present the results for the BP system. Those for the other two systems (HE and IC) can be found in the full report [ 9 ]. To designate the test sequences we use the following nomenclature: the first two letters designate the system, the next three the laboratory (as per code of Fig. 1 ), and if there is more than one sequence for one system and laboratory a n u m b e r is added. For example, BPITW3 is the third test sequence of the BP system at the Institut fiir T h e r m o d y n a m i k u n d W[irmetechnik in Stuttgart. The combination of all sequences for a system carries the label ALL. We have also created a subset of each ALL sequence by rejecting days that appear less than fully reliable; those sequences are designated by AL. The dates for the test sequences are shown in Table 2, together with the regression results.

Table 2. Round Robin test sequences for BP system start

sequence

end

season a) of days

numberb) (MJ]

~ [)

R2

BPCON BPCST BPDSE BPINT

16,05.87 21,11.86 10,09.87 09,11.87

29,05.87 10.05.87 05,10,87 12,11.87

summer spring fall winter

9 26 17 4

0.641 0.893 0.723 1.750

0.994 0,988 0.982 0.844

BPITWl BPITW2 BPITW3

22,04.87 31,05.87 20,10.87

12,05.87 22.08.87 12,11.87

summer summer winter

16 36 15

0.640 0.845 0.640

0.994 0.988 0.979

BPJRCl BPJRC2 BPLNE BPTNO

16.09.87 01.07.87 27,06.87 13.08.87

01,04.87 03.07.87 19.07.87 23.09.87

spring summer summer fall

16 3 21 35

0.752 c) 1.483 1.316

0.994

BPALL BPAL BPITW12

a l l BP s e q u e n c e s 198 subset (most r e l i a b l e data) of BPALLd) 120 BPITWl and BPITW2 52

1.835 1.621 0.780

0.947 0.962 0.990

(MJ] BPCON

[MJ)

[m 2]

(43)

(TaOffaH

[m2] (MJ/K] [MJAK)

[)

[)

0.820 0.969

[]

BPCST BPDSE BPINT

-1.72 -0.87 -1.25 -4.97

0.64 0.55 0.67 10.68

1.04 1.03 1.04 1.21

0.04 0.03 0.04 0.53

0.66 0.67 0.27 0.51

0.03 0.08 0.03 0.58

-0.92 -0.94 -0.86 -0.99

-0.33 0.18 0.47 -0.27

0.16 -0.30 -0.05 0,13

8PITW1 BPITW2 BPITW3

-3.31 -3.52 -0.86

0.74 0.39 0.35

1.02 1.05 0.94

0.03 0,02 0.04

0.35 0.37 0.25

0,04 0.02 0.03

-0.95 -0,91 -0.55

0.70 -0.33 0.21

-0.62 0.16 0.63

BPJRC1 BPjRC2 BPLNE BPTNO

-2.31 c) 1.91 -0.57

0.48

1.04

0.02

0.57

0.05

-0.55

-0.63

-0.17

1.79 0.69

0.74 1.07

0.10 0.04

0.27 0.40

0.08 0,03

-0.98 -0.65

0.06 0.37

-0.16 -0.04

BPALL BPAL BPITW12

-1.21 -2.19 -3.38

0.34 0.37 0,30

1.00 1.04 1.04

0.02 0.02 0.02

0.34 0.38 0.37

0.02 0.02 0.02

-0.92 -0.92 -0.93

-0.15 0.17 -0.17

0.07 -0.16 0.13

(a) Approximate assignment to seasons, defined here as winter = 21 Oct.-21 Feb., summer = 21 Apr.21 Aug., spring/fall = rest; (b) Days with Q < 0 have been removed (5 days from BPITWI, l day from BPITW2, 5 days from BPITW3, and 1 day from BPTNO; (c) Not enough days for regression; (d) BPALL without BPJRC2, BPITW3, BPTNO, BPINT, BPLNE.

Measurement errors and parameter estimates a)

Qi(HI,T I )

)0

|M J ]

25

II

'S'

20

a 15 < T i

i5

S < T t < 15

•*

o -S < T t < o-15

5

5

< -5

0

T t <-15

-5

< T|

x

I0 o

x x

i .........

~. . . . . . . . . I0

HI i .........

5

b)

Q|(HI'T=O) [MJ]

¢ used •

suppressed

x

[ H J / m 21

i ......... i ......... i ......... r 15 20 25 30

o

28:30 26 :

24: 2220t8.

16 14. 12

g 2

4

6

8

l0

c)

18 20 22

24 26 2B 30

--L--

o uged 0

12 14 16

suppressed

:I 1 ......................... . . . . . . . . . . . . . 2221111197531 53197531

I ......................... 13579

1 1 I I 1222 13579135

Ti

tK]

Fig. 6. Round Robin data for BP system, all series combined (= BPALL). Points with Q _< 0 are shown but not used in the regression. (a) Raw data, grouped by intervals of T. The straight lines show the regression, evaluated at T = 0 and at T = 10 K. (b) Data reduced to T = 0, according to eqn (44). (c) Data reduced to H = 0, according to eqn (45).

Figure 6 ( a ) shows, for the BPALL sequence, the data points Qi versus Hi, grouped by intervals of Ti. The distribution of the residuals can be seen more clearly if one adjusts the data Q~ to T = 0 by plotting the quantity Q~ - arTi which is equal to Qi - arTi = O_(Hi, T = 0 ) + ei,

(44)

since 0, = Q_(Hi,Ti) = O.(Hi, T = 0 ) + arT~. This is shown in Fig. 6 ( b ) . The analogous adjustment to H =0 Q i - a n H i = Q ( H = 0 , Ti) + ei,

(45)

can be seen in Fig. 6(c). Points with Qi < 0 are shown with a special symbol; we have not included them in the regression, because they fall outside the scope of the i n p u t - o u t p u t model (except for a passive system like the IC). Suppressing these points may introduce a slight bias: near values o f H and T w h e r e the energy vanishes according to the regression, one rejects points with negative error and keeps points with positive error. Such bias could be avoided by raising the threshold for rejection but then one loses some good data points. A more rigorous ap-

proach would employ a model with two different straight lines, representing two different heat loss coefficients, one when the collector is turned on, the other when it is off. In any case we did not judge this problem important enough. The main conclusion to be drawn from Fig. 6 is that the distribution of residuals is quite uniform, consistent with the basic hypotheses of linear least-squares regression. There are no obvious discrepancies that would invalidate the use of the linear i n p u t - o u t p u t model. O f special interest is the question whether the residuals follow a normal (Gaussian) distribution. While not necessary for the regression itself, it would permit more definitive conclusions in terms of confidence intervals of t and F tests. The frequency histograms of the residuals, in the full report [9], show that normality is indeed well satisfied by BP and HE, although less well for IC. For most of the sequences the correlations are good in the sense that R 2 is above 0.90. But the standard error of the regression b is around l M J, about twice as large as the priori estimate of Section 3. In general one would expect the errors to decrease as the n u m b e r of days, and the range of H and T are increased. To test the effect of the range of H and T, we plot, in Fig. 7 the parameter estimates and their

B. BOURGESel al.

12 a) o~H + crmH 1.2 El

O 2 Im 1

1.0

0.8

-r

"i 10 LNE20 TN~,DSEAti I CST

0.6

b) ~O 4- ~

l0

[MJl

[

MJ/) ]

0

-10

~INT

rm

[ M J/m 2 ]

-20 i

c) aLF -+ W~T

csT2

0.8

2O

30

cON

__

[ MJ/K ] 06

|

JRCl

~J~INT TNO ~

ITW12\ .=1

~

0,4.

0.2

LNE ALL

00 ¸ -iO

IT~ gee O +T [KI

0

Fig. 7. Parameter estimates for BP system and their standard deviation, as function of meteorological conditions during the test series. For each series the length of the bar indicates one standard deviation a above and below the respective central value. (a) a n +- a~n vs./t + an (same labels as (b)). (b) a o -+ a=o vs. /t -+ an (same abscissa as (a)). (c) ar -+ a.r vs. ~P-+ crr.

standard errors as a function of the test conditions. Parts (a), for a n , and (b), for a o , have not only the same format but the same abscissa (the insolation). Each test sequence is represented by a point, located at/-I and &n (part a ) o r / t and &o (Part b); its label is shown in part (b) only. In part (a) each point is the center of a cross whose vertical length equals -+~n and whose horizontal length equals -+an, the standard deviation of H. To avoid excessive cluttering, a n is shown only in part (a), the sequence identification only in part (b), both having the same format. Part (c) conveys the same information for the dependence of ~ r on ~P and a t . One can see the expectation borne out by the data. For instance the BPINT1 sequence has the smallest range of insolation values and the largest errors b~o and ~,,n. Sequences with large ~r such as BPTNO 1 and BPCONI have small errors in a t . Another purpose of these graphs is to check whether the parameter es-

timates might vary with/1or T. Such variations could arise from nonlinearities not taken into account by the input-output model, e.g., change of collector operating time with H. Within the accuracy of the results, Fig. 7 shows no significant deviations from linearity: the estimates are independent o f / t and ~P. The values of the parameters, together with their standard errors, are listed in Table 2. The correlation coefficients for the standard errors are also provided, as calculated according to eqn (43). For most of the data sets the coefficient r , o , n is close to -0.9, which means that there is a strong negative correlation between the errors oftxo and o f c t n . This is to be expected, especially if there are insufficient data points at low values of H. A low intercept ao can be compensated by a high slope a n , or vice versa, for regressions that fit the data comparably well. By contrast, for the other parameter pairs the correlation coefficients do not show any definite trend, ranging from -0.63 to +0.70 for

251

13

Measurement errors and parameter estimates

(a) Q(H,T=O)

[MJ] 20

S

x BPCON BPCST

BPDSE

15

~

10 5

/

~

.

~f'~"

BPINT

0

"'=

/.~++~+,,~

A

J

/

~ BPITWl WcBPITW2

/ . ~ ~

1~)BPITW3 I-I BPJRCl -t- BPLNE

/~l~/,w ~

4>BPTNO

0

H [MJlm2] -5 . . . . . . . . .

I ' ' '

. . . .

' ' I ' ' ' ' '

. . . .

I0

5

I . . . . . . . . .

15

I . . . . . . . . .

20

I . . . . . . . . .

25

(b) Q(H,T=O) [MJ] 25-

I

30

/ .7~J.~/'~

201510

;4"~ .

t

1:1: BPITWl ~k BPITW2

~

...-~,~:';~""~'*"

~

BPITW'3

0 H [MJ/m 2 ] I . . . . . . . . .

I . . . . . . . . .

I . . . . . . . . .

5

18

I . . . . . . .

15

i r & . . . . . . . .

20

i . . . . . . . . .

25

i

.30

(c) Q(H,T=O) 25.

J Y ~ " ~ "~/"

[MJ] 2015

H [MJ/m2 ] -5

i . . . . . . . . .

i ,,

5

i1

l,

ii

i i . . . . . . . . .

10

i,

15

. . . . . . . .

i . . . . . . . . .

20

i . . . . . . . . .

25

i

30

Fig. 8. The regression equations for each of the BP test series, plotted as Q ( H , T=0) vs. H. (a) All BP series. (b) The BPITW series. (c) The winter series (21 Oct.-21 Feb.).

r ~ o , r a n d from - 0 . 5 2 to + 0 . 6 3 for r,H~r. But in any case m o s t o f the correlation coefficients are very different from zero. For that reason the effect o n collected energy c a n n o t be calculated with the c u s t o m a r y form u l a for uncorrelated errors, a n d instead the m o r e complicated analysis in Section 3 o f Part 2 m u s t be employed. Finally, as a m e a n s to detect systematic differences, we c o m p a r e the regressions for the individual test sequences with each other, as plot o f Q versus H at T = 0 in Fig. 8. Part ( a ) shows all BP sequences together. Even t h o u g h a p r o p e r c o m p a r i s o n should include the confidence intervals, as discussed (e.g., D r a p e r a n d S m i t h [ 1 3 ] , p. 210), we do n o t show t h e m because the

graphs are already loaded. Also, the standard error o f each regression gives a good indication, a n d in any case the m o s t m e a n i n g f u l indicator of the accuracy is the error a o in the resulting long-term p e r f o r m a n c e prediction, a q u a n t i t y which we do present for each sequence in Section 3 of Part 2. F o r m o s t regressions, except B P L N E , * the slopes in Fig. 8 are approximately the same. T h a t the slopes should be close to each other, with the bulk o f the differences showing u p in the c o n s t a n t t e r m a o , w a s * The tests of the BP system at LNE have been repeated recently, with smaller errors and in closer agreement with the sequences of the other laboratories.

B. BOURGESet al.

14 (d)

/

Q(H,T=O) 25[MJ]

20-

j

15.

'°o

10. 5 0. -5 I . . . . . . . . .

I . . . . . . .

5

(e)

Q(H,T=O)

',1

. . . . . . . . .

10

I . . . . . . . . .

15

I . . . . . . . . .

I . . . . . . . . .

20

25

I

30

25 20 15 10 5 0 #

H [MJ/m2 ]

-5 i .........

I .........

5

i .........

10

i .........

15

i .....

20

,,,,i

.........

25

t

30

Fig. 8. (Contd.). (d) The fall/spring series (22 Feb.-20 Apr. and 22 Aug.-20 Oct.). (e) The summer series (21 Apr.-21 Aug.).

anticipated by the analysis of Section 3.2. However, the differences in a o , as large as _+1.5 MJ, are somewhat larger than expected from the analysis of measurement errors, as summarized in Table 1(b). To see whether seasonal differences might account for some of the differences, we have repeated the same graph in parts ( b ) - ( e ) , with all the regression lines but labels only for certain subgroups. Part (b) shows the sequences for ITW (Stuttgart), the only laboratory to have tested the same system during three seasons. Parts ( c ) - ( e ) separate the sequences according to season, defined as "winter" = 2 1 Oct.-2 1 Feb., "summer" = 21 Apr.-21 Aug., "fail/spring" = the rest, a division of the year into three equal parts spaced symmetrically around equinox. The BP summer sequences do indeed lie mostly on the low side, consistent with the expected length-of-day effect. As for the other systems [ 9 ], the HE winter sequences are also consistent with this trend; for IC the trend is opposite, although questionable given the limited data base and the intrinsic uncertainties associated with this collector design. The length-of-day effect may be real, but firm conclusions remain elusive in view of the experimental

uncertainties. Temperature errors may offer a more plausible explanation.

5. CONCLUSIONS Based on an examination of the tolerances specified for the measurements of the CSTG test procedure, we have derived a priori estimates of the errors to be expected for the parameters of the linear input-output model. For a possible explanation of laboratory-tolaboratory differences of the results, we have distinguished random and systematic errors. While we find that the tolerances are, for the most part, well chosen, two important sources of uncertainty remain: (i) the determination of ambient temperature Ta; and (ii) the difference ( Ts - Te3v) between the storage temperature T, at the start of the test and the temperature Tear of storage after withdrawal of 3 tank volumes. Errors due to ambient temperature are difficult to quantify and even more difficult to eliminate. Basically, the temperature of the environment seen by the solar water heater is ill-defined, resulting from a variety of different radiant and convective effects. With indoor tests one could reduce the scatter of the data, but part of the problem would remain anyway: in actual use a solar collector will always be outdoors. The CSTG procedure tries to minimize these uncertainties by de-

Measurement errors and parameter estimates m a n d i n g , quite correctly, a radiation shield for the t h e r m o m e t e r a n d a ventilation system beside the collector to e n h a n c e a n d standardize the convective heat loss. This latter p o i n t has n o t always been followed d u r i n g the R o u n d R o b i n program, a n d its neglect can be a significant source of uncertainty. T h e u n c e r t a i n t y in ( T~ - Te3v) depends o n the constancy o f the cold water supply t e m p e r a t u r e T~. While the spirit o f the C S T G procedure d e m a n d s that T~ be constant, the text fails to impose a limit for the variation o f Tc between m o r n i n g a n d evening. Several laboratories have, in fact, avoided the high cost of stringent t e m p e r a t u r e control, a n d we have seen variations as large as several degrees. For consistency with the other accuracy requirements, we would r e c o m m e n d that a tolerance limit o f at most 0.5 K be included in the procedure. A s s u m i n g reasonably tight control of the uncertainties due to T~ a n d T~, the analysis of m e a s u r e m e n t errors indicates t h a t the standard error a of the regression should be a r o u n d 0.5 M J for the systems tested u n d e r the R o u n d R o b i n program. But in reality the standard errors of the actual test results are about twice as large, as we have f o u n d by carrying o u t a systematic regression analysis of the R o u n d R o b i n test results. There are significant differences ( u p to a b o u t 1 M J ) between laboratories, a n d in some cases even between different test series of the same system at the same laboratory. The explanation for the large errors appears to reside mostly in the uncertainties of T~ a n d To, a n d to some extent in various a n d sundry little deviations from the C S T G procedures that usually pass unreported. A n o t h e r part of the discrepancies m a y reside in shortcomings o f the i n p u t - o u t p u t model itself. Therefore we have repeated the analysis with simulated test sequences supplied by the Institut f'fir T h e r m o d y n a m i k u n d W~irmetechnik, Universit~it Stuttgart. T h e results imply that a significant part o f the scatter o f the data arises from the i n p u t - o u t p u t model itself. In addition there seems to be a slight seasonal bias, b u t it is only a few percent, too small to explain all the differences between sequences that we have observed. A more systematic study with simulated data is needed to settle this question. In this paper we have performed a n error analysis of the test results. T h e ultimate accuracy criterion is, of course, the accuracy to which the long-term perform a n c e can be predicted. T h a t is the subject of the c o m p a n i o n paper.

NOMENCLATURE

C = V~pcp cp COV(H,T) f(V) H

storage capacity, MJ / K specific heat of water, k J / k g . K covariance of H and T draw-off temperature profile daily total solar irradiation on collector aperture, MJ/m 2 n number of days in sequence

15

Q heat added to the tank during the day, MJ

Q3v energy withdrawn during draw-off Q rnr R2 T 7~ Ta Tc T~3v Ti, To~t T~ t3v

U.~ V Vs V(a) X

matrix of measured values Qi correlation coefficient of H and T, eqn ( 38 ) square of multiple correlation coefficient (Ta - T~) average of T during test daytime average ambient temperature C temperature of cold water supply, C average temperature of the storage after withdrawing 3 V, temperature at inlet of storage temperature at outlet of storage temperature of storage tank (assumed well mixed ) at start of day, C draw-offduration, s storage heat loss coefficient, W / K flow rate during draw-off, m3/s volume of the storage tank, m 3 variance-covariance matrix of the a matrix of data for Hi and T,

Greek ao coefficient of input-output model, MJ an coefficient of input-output model, m 2 aT coefficient of input-output model, MJ / K 7(V/Vs) = fraction of energy remaining in storage after withdrawing V ~i residual = Qi - 0r = difference between measured value Qi and regression ()i p density of water, kg/m 3 ~ao standard error of ao, MJ era, standard error of aM, m 2 ~ar standard error of at, M J / K ,j error combination ofeqn (25), MJ -2 error combination ofeqn (26), MJ o ~ = standard error of regression ( Qi vs. Hi and T/) ~r2 variance of the residuals ~; cr2 variance of H during test ~ variance of Tduring test [ ]+ indicates that only positive values are counted: A designates errors, e.g., AH is error of H; subscript i designates a particular day, e.g., H~ = solar radiation of day i; tilde designates estimated values of a parameter, e.g., b is estimate of~r.

REFERENCES

1. W. Gillett and J. E. Moon, Solar collectors: test methods and design guidelines, Reidel, Dordrecht ( 1985 ). 2. J. Adnot, B. Bourges, L. Kadi, and B. Petlportier, Models of the thermal performance of solar water heaters, Report to JRC within framework of CSTG, Centre d'Energ&ique, Ecole des Mines, Paris (March 1986). 3. ASHRAE Standard 95-1981, Methods of testing to determine the thermal performance of solar domestic hot water systems, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1791 Tullie Circle, N.E., Atlanta, GA ( 1981 ). 4. A. H. Fanney and S. A. Klein, A rating procedure for solar domestic water heating systems, A S M E J. of Solar Energy Engineering 105, 430 (1983). 5. B. V. Minnerly, S. A. Klein, and W. A. Beckman, A rating procedure for solar domestic hot water systems based on ASHRAE-95 test results, Report Solar Energy Laboratory, University of Wisconsin, Madison, Wl. Solar Energy (in press).

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B. BOURGESel al.

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