Performance correlation of horizontal plastic solar water heaters

Solar Energy. Vol. 12. pp. 183~196.

Pergamon Press, 1968. Printed in Great Britain

PERFORMANCE PLASTIC

CORRELATION SOLAR WATER

OF HORIZONTAL HEATERS

W. H. G O P F F A R T H , R. R. D A V I S O N , W. B. H A R R I S and M. J. BAIRD* (Received 27 June 1968) Abstract- Statistical and experimental methods are used to determine the optical efficiency and the heat lost by radiation and convection in a horizontal plastic solar hot water heater. The heater consisted of a water filled polyethylene bag with a clear top and black bottom. It was insulated on the sides and bottom by styrofoam and covered on the top with one to three layers of clear Tedlar. The effective sun angle, effective sky temperature, ambient temperature and the number of covers are included as variables, The upward heat losses were higher than those predicted by the heat transfer coefficients of Hottel and Woertz. R6sum6- Des m6thodes statistiques et exp~rimentales sont appliqu6es pour d&erminer le rendement optique et la perte de chaleur par radiation et convection dans un chauffe-eau solaire horizontal en plastique. Le chauffe-eau consiste en un sac en poly6thylEne rempli d'eau dont le haut est transparent et le fond noir. Une isolation en "'styrofoam" a 6t6 pos6e sur les c6t6s et au fond, tandis que le haut a 6t6 recouvert d'une h trois couches de Tedlar transparent. L'angle effectif du soleil, la temperature effective du ciel. la temperature ambiante et le nombre de couches sont compris en tant que variables. Les pertes de chaleur vers le haut sont plus 61ev6es que celles prddites par les coefficients de transfert de chaleur de Hottel et Woertz. Resumen- Se emplean mdtodos estadisticos y experimentales para determinar la eficacia 6ptica y la pErdida de calor por radiacion y convecci6n en un calentador de agua solar horizontal pl~istico. El calentador consisti6 de una bolsa de polietileno Ilena de agua. siendo su parte superior transparente y la parle inferior negra. Sus costados y parle inferior estaban aislados con Styrofoam, y la parte superior estaba cubierta con desde una hasta tres capas de Tedlar. En calidad de variables se incluyen el fingulo efectivo del sol. la temperatura efectiva del cielo, la temperatura ambiente, y el nfimero de capas utilizadas. Las prrdidas de calor ascendentes excedieron aquellas prognosticadas pot los coeficientes de traspaso de calor de Hottel y Woerlz.

FOR SOLAR energy to be competitive with conventional heat sources, collector costs must be quite low. Even under ideal conditions it takes several years for a square foot of collector to absorb a dollar's worth of heat. Obviously, the investment per square foot divided by the life in years must be only a fraction of a dollar. The solution can only result when efficient design is combined with cheap, long lasting materials. Quite naturally, there have been many attempts to substitute plastics for glass and metal. These efforts have not been entirely successful because of short material life, but since the solution ultimately will probably depend on finding similar but longer lasting materials, these efforts are justified. In designing any piece of equipment, it is useful to have equations relating performance to the various structural and environmental parameters. For solar water heaters, the principal factors are the internal and external temperatures, the Sun angle, the number and optical properties of the transparent covers and the design of the collector. Some years ago, Hottel and Woertz [1] published an extensive evaluation of flat plate heaters consisting of a blackened metal absorbing surface covered by one to three glass plates. Their results are not directly applicable to the present work because of differences in the design of the plate heaters, but the theory is used to obtain a semiempirical correlation for the performance of plastic heaters. Convection and radiation ~Department of Chemical Engineering, Texas A and M University. College Station, Texas, U.S.A. 183

184

W . H . G O P F F A R T H etal.

losses are separated from absorption and reflection losses by experimental and statistical methods, and the results are correlated with operating variables. The correlation has been used as a standard for evaluating the performance of various solar water heaters of different design [2]. DESCRIPTION A N D OPERATION OF APPARATUS A schematic showing the principal parts of the test heaters is presented in Fig. 1. The collecting element consists of a bag formed by heat sealing one black layer and one clear layer of polyethylene. Before sealing, two thermocouples were permanently installed such that the tips were about ¼in. above the black layer. The bags are equipped with a spout for filling with water. They were filled to a depth of about ½in. in all runs

AND

~LENE

THERMA iNSULAT

OCOUPLES

:ELLS

RIZONTAL TOP

BLE /

THERMu~,uurL~o

COPPER LEADS TO CONTROL PANEL AND RECORDER

Fig. 1. Plastic test cell.

which placed the thermocouple tips near the center of the water layer. After filling, the bags were weighed and their spouts were temporarily sealed with polyethylene tape. The thermal insulating cells for the heaters were constructed of 2 in. thick polystyrofoam and are about 1 in. deep. The Tedlar insulating covers were attached to wooden frames of the same dimensions as the styrofoam box. After the water-filled polyethylene bag was placed clear surface up in the insulating cell, the Tedlar layers were stacked to the desired depth and tightly secured by four bolts and wing nuts. The distance between layers was about ½in. The effective heating area of this unit was 2.17 ft z.

185

Horizontal plastic solar water heaters

Direct and diffuse solar radiation were measured with an Eppley pyrheliometer. The outputs from the pyrheliometer and the thermocouples were fed to a control panel so that any input could be switched to a recorder. A potentiometer-voltage divider was used to reduce the inputs to the scale of the one-millivolt recorder. The thermocouple cold junction was buried in the ground, and a compensation circuit consisting of a battery and voltage divider was used to adjust the reading of a reference junction to that of an accurate glass thermometer maintained at the same temperature. The system is shown in Fig. 2. COLD JUNCTION

REFERENCE THERMOCOUPLE

EPPLY PYRHELIOMETER

SOLAR HEATER

COMPENSATION CIRCUIT

Fig. 2. Measuring circuitry.

After the bag was filled with water, weighed and placed in the test cell, the Tedlar layers were bolted in place, and the system was covered until the sun reached the desired initial angle. The temperature within the cell was recorded. the pyrheliometer was switched on the recorder, the exact time was noted and the cell was uncovered. During the run, the diffuse radiation was measured by casting a shadow which just covered the pyrheliometer element until a minimum was obtained on the recorder. The temperature in the bag was allowed to rise the desired amount (usually lo-30°F) and the cell was covered again. The final time was again carefully noted, not only to obtain the exact length of the run, but also to allow accurate Sun angle calculation. Usually another run was begun shortly after completion of the previous one to obtain data at various temperatures. Most runs were about 20 min in length and the sequence of runs was terminated at about 160°F. To obtain the optical losses of the system. some runs were made as near ambient temperature as possible to minimize other losses.

EQUATIONS

FOR

RADIATION

AND

CONVECTlON

LOSS

Assuming the convection coefficient between any two layers to be equal to the temperature difference to the 4 power (this includes the top layer to the atmosphere) and assuming the emissivity to be the same between any two layers including the top layer and the atmosphere, Hottel and Woertz obtained the following equation for

S.E. Vol.

12 No. 2-D

W. H. G O P F F A R T H et al.

186

upward heat losses. c Qu/.4 = N + I (T~,.-- T,,)5/4+

(T.4r - Te4).

(1)

The effective temperature for radiation was obtained by Brunt [3] as a function of atmospheric temperature and humidity. T~ = T~ [0.55 + 0-33 (P,o) v2] 114.

(2)

Since only initial and final temperatures were taken during each run, some assumptions must be made in using average temperatures, Tae and Tar, for convection and radiation. If it is assumed that the convection coefficient is independent of temperature then

Qc=C

£o

( r - r ~ ) dO.

(3)

=0

If it is also assumed that dT d-0- = a constant

(4)

one obtains upon integration

Qc = C( T a c - T~)O T~c = T2 + T1 2

(5)

Equation (5) was used to calculate Tac in Eq. (1). If the upper heat loss for radiation is given by f0

or = F e c r Jo ( T 4 -- Te 4) dO

(6)

and if Eq. (4) is again assumed, one obtains

Qr = F~o-[ T4r-- -/-~4]0 in which T4r = (T,_,4+ T23TI + T 2 2 T I 2 + T 2 T 1 3 + T14)[5.

(7)

This relation was used for T]r in Eq. ( 1), but for the temperature differences encountered in these experiments, T~c and T~r differ by only a few tenths of a degree. Because of the approximation of Eq. (4) relatively short runs were made. The results can be extended to any time period, however, by numerical or graphical integration of Eq. (1).

Horizontal plastic solar water heaters OPTICAL

187

EFFICIENC1ES

Solar heater losses may be separated into optical losses, resulting from reflection and absorption by the covers and by reflection from the absorber, and thermal losses, resulting from conduction, convection and radiation. The optical losses depend only on the incident radiation and the optical properties of the heater. Hottel and Woertz calculated optical efficiencies vs Sun angle from the transmittance and refractive index of the glass and the absorptivity of the blackened surface. It would be difficult, however, to calculate the absorptivity of the water-filled, clear-topped, black-bottomed polyethylene bag which served as the absorber in these experiments; so the optical efficiency of the entire system was obtained experimentally. A number of runs (Table 1) were made in which the initial temperature of the heater was near or below ambient and the final temperature only a little above ambient so Table 1. T r a n s m i t t a n c e - a b s o r p t i o n product data (ambient runs) Angle o f

Energy

incidence

collected

Total solar input

(B.t.u./hr ft ~)

(B.t.u./hr ft 2)

(B.t.u./hr ft-")

(B.t.u./hr ft z)

(°F)

(°F)

292 282 303 236 213 150 191 149 121

47 43 54 47 43 32 55 27 23

240 232 239 180 162 110 136 88 61

98 87.5 96.5 87 98.5 75.5 91 76-3 91

117.6 100.8 111.8 93.8 110 73.4 94-3 85.5 94.3

99 100.5 96 98 78.5 89.5 50-5 58 58.3

123 122-2 114 118 83.3 93-1 57.4 75.2 72-1

94.5 96.5 98 89 84-5 92 65.3 47 59-8

98.1 110.8 110.4 97-3 98 98 65.3 48-5 71.3

~°)

Diffuse solar Corrected Ambient input energy collected temperature

Average heater temperature

One Tedlar cover 3.3 41.5 47 53.5 58 60 61 74 76

217 218 222 173 149 112 133 79 56

Correlation equations Fig. 3 E = 15.7 + 78"0(T~t)

Fig. 4 E = 25"9+ 68"3( Treo

Two Tedlarcovers 34 43 29.5 33 69-5 51 57 55 64.5

205 186 221 212 78 173 132 168 116

286 261 295 286 147 248 197 246 199

' 43 43 47 43 39 37 126 35 31

221 201 232 225 81 175 136 177 124

Correlation equations Fig. 3

E = l l'3 + 84"4( Tr~t) T h r e e Tedlar covers 40 169 35 203 33.5 210 59.5 117 53 145 42 239 72 74 70 76 57.5 149

251 289 291 202 239 348 147 159 241

E= 51 47 47 43 35 63 39 31 39

Fig, 4 16.3+79.5(T,et 170 209 215 120 150 242 74 76 153

Correlation equations E=

Fig. 3 14.2+79.1 (Tr,,~)

Fig. 4 E = 22.2 + 67-7( Trio

188

W . H . G O P F F A R T H eta/.

that the loss to the surroundings would be small, and the difference between the pyrheliometer reading and the heat collected would be optical losses. Actually the average temperature (Eq. (5)) was usually a little above ambient, and a correction was made using Hottel and Woertz overall heat transfer coefficients. Radiation consists of both diffuse and direct components, and it is necessary to separate the two for each run. The diffuse component for radiation for a clear sky is equivalent to a single source at an angle of 58 deg from vertical. To obtain the optical efficiency for the diffuse component, ambient runs were made at Sun angles near 58 deg for one, two and three Tedlar layers. For other ambient runs. the optical efficiency of the direct component was determined from HmEm -- HoEo E =

H

(8)

in which Era, Eo and E are the measured, diffuse and direct optical efficiencies respectively. Transmission through transparent covers can be calculated by the equation 1--R T~ef = 1 + ( 2 N - - 1)R (9) in which R is given by R= sin2(i--r) ~ tan2(i--r) (10) 2 sinZ(1 + r ) 2 tan2(i+r)" The relation between i, the angle of incidence, and r. the angle of refractance, is n sin i = n' sinr. (11) The absorptivity of the collector is also a function of the angle of incidence. Examination of experimental data obtained by Yellot[4] indicated that a function similar to Tref could be used to represent these data. Therefore. it seemed possible that an overall optical correlation might be obtained in terms of Tre f. The optical efficiences were fitted by a linear regression to the equation (12)

E = A +BTref.

The results are shown in Figs. 3 and 4 and Table 1. j(...,) c3 0 no. Z 0 I--Z nO

mr*' I

I-W(..~ Z~ F~-h ~O Z

I-30

40 ANGLE

5O

60

70

80

OF INCIDENCE

Fig. 3. Transmittance- absorption product- used with Eq. (15).

Horizontal plastic solar water heaters

189

90

F,j 80 C~ n~ Z 0

~g

°} ~ ¢n

!\/, oo7.:

70

6O

I

~a (,') z

40

30

40

50

60

70

BO

A N G L E OF I N C I D E N C E

Fig. 4. Corrected transmittance-absorptionproduct, used with Eq. (16).

CALCULATION AND CORRELATION OF UPWARD HEAT LOSS The heat losses by convection and radiation fr.om the top of the collector were obtained from the difference between the heat collected and heat lost by other causes. The total solar energy converted to heat is equal to the incident radiation times the mean optical efficiency.

Qa__v= HmEr, A

(13)

in which E,, is obtained by rearranging Eq. (8). Then

atria =

Qa,.- Q~ol- Qtb A

(14)

in which Qco~.the net heat gained per hour by the water in the collector, is obtained from the product of the temperature rise and the water weight divided by the run time in hours. The bottom loss Q~, is a small correction calculated from the thermal conductivity of the insulation. Data were obtained for a wide variety of Sun angles, ambient temperatures and heater temperatures, with one, two and three Tedlar coverings. The results were correlated with Eq. (1). Until all data were discarded in which the sun was obscured by clouds during any part of the run, the correlation was poor. This is not surprising, but probably it is particularly the result of the failure of the assumption of linear variation of temperature with time and of the wide fluctuation in the effective Sun angle. Some of the data retained were collected with clouds in view though not obscuring the Sun. Even this could cause error by changing the effective angle of the diffuse radiation. The 101 clear sky data in Table 2 were correlated by the following expression Qu~a

=

4 __ -5"84 0"813(Tac--Ta) 5j4 0"477(T,.. Te 4 ) + N+I N+I

(15)

190

W.

+.=

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Horizontal plastic solar water heaters

.

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191

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l'+-,l ~ ,

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192

W . H .

G O P F F A R T H

et al.

~ l l P l i

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Horizontal plastic solar water heaters

193

The following statistical parameters were obtained: standard deviation, 6.88 B.t.u./hr ft2; multiple correlation coefficient, 0.978; and an average error for the 101 points of 36 per cent. The magnitude of the last value is deceptive. Many data points have small values of QtttA such that small errors of magnitude cause large per cent errors. For the 79 points having values of Qlt~a larger than 10 B.t.u./hr ft:, the average error is 17 per cent. Though Eq. (15) correlates the data fairly well, it is obvious that the constants do not give actual values for the convection and radiation losses. Furthermore. the optical efficiencies were taken from Fig. 3 in which it was assumed that Te was equal to T,,. Even at ambient temperature, there are still losses due to radiation. Using the last term in Eq. (15), the ambient runs were corrected for radiation losses and recorrelated to obtain the results in Fig. 4. U sing this correlation, the Ott/,4 values for 60 of the 101 data points were recalculated. These data give the following results

att/A

=

0"504 --5"68+~i(Tac--

Ta)

514 0"988tr 4 +--N-~'i- (T~r-- Te4).

(16)

The multiple correlation coefficient is 0-97 and the standard deviation is 8.6 B.t.u./hr ft z. The constant term should be zero, but in view of the assumptions made in deriving the correlation equation, exact compliance cannot be expected. The coefficient for the radiation term is about what would be expected for Tedlar which, being slightly transparent to infrared, would have a higher value than glass. The convection coefficient is higher than that usually accepted. The average error for all 60 points is about 30 per cent, but omitting the four data for which Q~t~.~is less than 10 B.t.u./hr ft 2 reduces the average error to 11.5 per cent. Use of Eq. (16) to further correct the ambient runs for radiation loss did not improve the correlation. Figure 5 compares the results obtained by Fig. 3 and Eq. (15) and by Fig. 4 and Eq. (16), with the results of Hottel and Woertz. The results are not really comparable because of different construction and because the Hottel and Woertz U values were for collectors tilted at 30 deg, which gives lower values for C, the convection term: but the value of C in Eq. (16) is nearly twice the recommended value for horizontal plates. The effect of humidity through its influence on the effective sky temperature (Eq. (2)), which was not considered in the Hottel and Woertz correlation, is shown. It is recommended that Fig. 4 and Eq. (16) be used since the constants are more realistic and the statistical parameters similar. Because of the negative constant term, both correlations will give low results when the heater temperature is near the sky and ambient temperatures. The following example illustrates the use of Fig. 4 and Eq. (16) to calculate the performance of a solar heater under given thermal and solar conditions. At 2:20 p.m. (2:03 solar time) on 20 September 1963, in Bryan, Texas, the solar radiation was 329 B.t.u./hr ft ~, of which 34 B.t.u. were diffuse and 204 were direct. The ambient temperature was 92°F, and the wet-bulb temperature of 62°F gave an effective sky temperature of 60.3°F. The angle of incidence was calculated to be 53 deg. A solar heater having a single Tedlar cover was at a temperature of 146-8°F. From Fig. 4, the optical efficiency of the direct component is found to be 80.2: for the diffuse component it is 77.4. From

194

W. H. GOPFFARTH

CT-

eral.

To) OF

Fig. 5(a).

(T-T,) Fig. 5(b).

OF

Horizontal plastic solar water heaters

195

I00 TO = 7 0 *F 3 LAYERS 8C

,.

6C

4-;

'

,

'

i

tc+~

~Yl

i

I

i

i

I.

-~ 4-m

40

vy-'t

S

i

i

h

i

i

i

J~ ~0

60

80

I00

[

t

J

T - T o ) *F

Fig. 5(c). Fig. 5(a-c). Comparison of Eqs. (15) and (16) with the Hottel and Woertz heat transfer coefficients. a = Eq. (15) and 40% humidity b = Eq. (15) and 80% humidity c = Eq. (16) and 40% humidity d = Eq. (16) and 80% humidity.

these values, the total radiation being absorbed is calculated to be 190.6 B.t.u./hr ft ~. Substituting the appropriate temperatures in Eq. (16) gives Qw~ = - 5 - 7 + n.~oa_~_~.:(_146.8 - 92) sl4 .+ o. 171 × 10-8(626.84 _ 520.34 ) 2

(17)

Qlt~a = - 5 " 7 + 21"5 + 69"3 = 85" 1 B.t.u./hr ft 2.

(18)

or

Bottom loss through the insulation was 6.1 B.t.u./hr ft 2. The energy collected should then equal the energy absorbed minus that lost. or 1 9 0 . 6 - 8 5 . 1 - 6 . 1 = 99.4. The measured value was 93. T h e calculated efficiency is (99-4/239) × 100 = 41-6 per cent. Acknowledgements-The support of this work by the Ot~ce of Saline Water, Department of the Interior. Washington, D.C. is acknowledged. The authors appreciate the many helpful suggestions of Dr, George O. G. LSf. The use of the digital computer of the Texas A & M Experiment Station is also appreciated. NOMENCLATURE area in flz A . B constants in Eq. (12) C convection coefficient in Eq. ( I ) in B.t.u./(hr)(flz)(°F) E. E,,. Et, direct, total and diffuse optical efficiencies F~ radiation coefficient in Eq. ( 1). dimensionless H. H,,. Ho direct, total and diffuse radiation received on a horizontal surface in B.t.u./(hr)(fl 2) N number of covers yielding two reflecting surfaces t l . 11 ' refractive index of air and cover material respectively A

196

W.H.

Pw Qc, Qr Qav QcoJ Q~a, Qtt R T, T~, T2 T~ T~r T~ Te T~f 0 Gr [ 1] [2] [3] [4]

G O P F F A R T H etat.

partial pressure of water in the atmosphere, in. of Hg heat lost by convection and radiation in B.t.u./(hr)fft 2) solar energy available after transmission through covers in B.t.u./hr solar energy collected in B.t.u./hr bottom and upper heat losses from solar heaters in B.t.u./hr reflectivity of cover material as a fraction of the incident radiation instantaneous, initial and final solar heater temperature °R average solar heater temperature for convection loss °R average solar heater temperature for radiation loss °R ambient temperature °R effective sky temperature °R transmittance efficiency of cover allowing for reflection time in hours Stefan-Boltzman constant 0.173 x l0 -s B.t.u./hr ft 2 °R 4

REFERENCES H. C. Hottel and B. B. Woertz, Trans. A m. Soc. mech. Engrs 64, 91 ( ! 942). W. B. Harris, R. R. Davison and D. W. Hood. Solar Energy 9. 193 (1965). D. Brunt, Supplement to Q. Jl. R. met. Soc. 66, (1940). J. I. Yellott, Trans. Am. Soc, mech. Engrs 79, 1349 (1957).