Testing of solar collectors

Solaf Energy, Vol. 20, pp. 293-303. Pergamon Press 1978. Printed in Great Britain

TESTING OF SOLAR COLLECTORS

H. TABOR The Scientific Research Foundation, Jerusalem, Israel

(Received 28 October 1975; in revised form 11 April 1977; received [or publication 27 September 1977)

Abstract--In order that tests on different solar collectors conducted at different times shall be comparable, it is necessary that the test define the governing equation of the collector. This involves measurements at different solar intensities, different operating temperatures and different flow rates.

The paper describes the theory and the experimental procedure which involves connecting, say, four similar collectors in series and measuring the temperature rise across each collector. Since, at any instant of time, the flow rates and the solar intensity are identical for all the collectors, a number of points on the efficiency-temperature curve are obtained from which the characteristic equation of the collectors is readily established. Rules are established to limit the temperature rise permitted and a "standard" day is suggested to provide a quick comparison basis.

INTRODUCTION

The rapid rise in the number of new flat-plate collectors appearing on the market requires some form of unified approach to specifying efficiency that will make possible the comparison between various models and the deter- mination of yield under given climatic and operating regimes.

In a technical note, Doron[1] described methods used in the National Physical Laboratory of Israel in the early sixties.

In one case he measured the "heating" efficiency, i.e. the efficiency when fluid (water) enters the collector at ambient temperature T,.- Ta and leaves at a controlled upper temperature To, the flow being controlled by a thermostatic valve measuring the output temperature.

In the other case, he measured the "boiling" efficiency, i.e. the efficiency when the whole of the collector is at substantially the same temperature by using a fluid with a boiling point at the temperature of interest.

The heating efficiency--which is always higher than the boiling efficiency for the same outlet temperature--is important for collectors used with stratified storage, i.e. where the supply fluid entering the collector is always cold. This is the case for the thermosyphon operation with a "once-through" per day or "single cycle" flow. However, many thermosyphon systems are arranged for faster flow and the fluid traverses the collector several times per day, in which case, at any particular time, the collector operates with only a small temperature difference between input and output [5].

For most applications, it is the "boiling" efficiency that is needed. This is particularly true for installations in- volving a number of collectors in series: each collector having only a small (To- T~) difference.

Since the ultimate aim of the test is to obtain the efficiency under various conditions of temperature rise, solar intensity, etc. the output of the collector is

tThis is desirable in any case because a certificate of test on a single sample is of little value if the manufacturer cannot dupli- cate the product.

generally measured for various flow rates, varying degrees of preheating--to get the operating temperature level in the desired range--and at various solar intensities. This is a long and difficult procedure, since some time must be allowed between each group of measurements in order to eliminate transient effects and it is not easy to control the many parameters.

The present note suggests a simpler and more reveal- ing method of determining the collector efficiency equation, in terms of insolation and mean temperature rise above ambient. It does not attempt to isolate the in- dividual items (heat loss coefficient, optical efficiency, extraction efficiency factor, etc.) that go to make up the total performance. Thermal capacity is not measured so that poor performance under transient conditions cannot be predicted: the tests are conducted under conditions where thermal capacity has almost no effect on the results.

The method is facilitated by having severalt similar collectors (from the same manufacturer) to test. We consider, for convenience, water heaters, though any fluid can be used.

THEORY OF TItE METHOD

The energy H delivered by a collector, at substantially constant temperature, is given by:

H = F(AOaI~ - UA" AT) = mco • (To - Ti)

where Q = insolation intensity in the plane of the collector; A =area of the collector aperture; a// = optical efficiency; U = a heat loss coefficient; AT = mean temperature rise of the collector above ambient= [(T~ + To) /2- T~]; m = mass flow rate of fluid through the collector; c,, = specific heat of the fluid; Ti = inlet temperature; To = outlet temperature; T,~ = ambient temperature; F = extraction efficiency. This allows for the fact that the mean temperature of the plate--which is the source of heat loss--is not the same as the mean temperature of the fluid: F includes--for plate-pipe con- structions--such recognized factors as the plate-pipe

293

294 H. TABOR

efficiency, pipe-plate bond efficiency and the effect of the heat transfer coefficient between fluid and pipe; AT = ( T - T a ) where T is the temperature of any designated point on the collector.

Since a f is a function of the angle of incidence, it may be written (aft)3' where (aft) is the value at normal incidence and 3' is a factor dependent upon the incident angle.

The efficiency E is then:

H F [ ' a M) UAT] m q , ( T o - T 3 -- T 6 = p =- ao

(1)

3' is called the "filter factor"[2] and 3'Q is the "filtered" solar radiation, since we can regard the collector as replaced by a hypothetical collector having constant (aft) and a hypothetical filter outside reducing the incident radiation by a factor 3'.

Whilst it is often assumed that U is a constant, i.e. that the E, AT relationship is linear, experience shows that this is not so and that U is a slowly rising function of AT, which may be expressed as a polynomial in AT. However, we have chosen to write eqn (1) as

E AT ~ - - = a - b - - (la)f 3' 3'Q

which is found to be a good fit over the normal working range, p is usually in the range 1.1-1.3 for fiat plate collectors.

Equation (1) should therefore be written:

7-- t ----0--J--

The square bracket is called the "retention" efficiency N and we have:

E = F(aft)3"N. (3)

For analytical reasons, it is convenient to write the retention efficiency as:

N = 1 - fP (4)

where ~= (AT/AT,,) , i.e. the normalized value of AT (5)

A T " ' = [ ~ - ~ ] tt~'=[(°tfl)3"Q]'/v[ Uo J (6)

tor is at nearly the same temperature and that we can consider the effective temperature of the collector as the arithmetic mean of the inlet and outlet temperatures. To improve the accuracy in measuring the efficiency we would like (To - T3 to be as large as possible. We now examine how large a temperature difference from inlet to outlet of the collector is permissible without introducing serious errors.

Consider a collector--or a string of collectors in series--with fluid entering at one end and emerging from the other. Taking x as distance along the system we have: energy gained on any strip of length dx and unit width is:

EQ dx = F(dft)3'Q(1 - z p) dx = m G d T = mcpATm dz~

where dT is the temperature rise in the strip, i.e. dAT, i.e.

x = F(aft)3'Q

which, integrated between two values zj and z2 gives:

where

x2 - x, = K ( I z - I,) (7)

f ~ dz lz = 1 - z p (8)

K = m_~ec _ ! 1 F [(a//)yQ] ~p-~)/p' ~ using 6). (9)

When p = 1, the integral in eqn (8) is simply-log (1 - z): for other values of p it is a modified logarithmic function. This is shown in Table 1.

It is interesting to note that if p is known approxima- t e ly -and it does not vary much from one design to another--then the curve of I against the normalized temperature rise z is a general relationship. Since ! is a measure of the length along the collector (eqn 7), this relationship shows the temperature profile along the collector.

Figure 1 shows such a curve. Now the fractional heat loss q from any point along

the collector, i.e. the loss expressed as a fraction of the useful energy reaching and absorbed by the collector, is simply

Clx = Zx p

since, from (2), (b la)= [Uol(aft)]. AT,, is the maximum temperature rise for intensity Q,

corresponding to zero efficiency. An implied assumption so far is that the whole collec-

so that we may, if we wish, plot q vs x, since the x - z relationship has been established.

Between any two points xj and x2, the total fractional heat loss is:

tAppendix 2 discusses other forms of representation. The experimental method discussed here is equally appropriate to the case where p is assumed to be unity over the working range of AT.

SThe bar on AT and z is omitted when dealing with the local values of (T - Ta) and z.

f q dx K f z p dz 1

x dz = K[(I2 - I,) - (z2 - zOl.

Since the area is (x2-xO which, from eqn (7), equals

,f• d z I = 1 _---7-ssrr-

Testing of solar collectors

Table 1.

z = 0.1 .2 .3 .4 .5 .6 ,7 ,8 ,9

F o r p=l . 0 I = .1054 .2231 .3567 .5108 .1i931 .9163 1.2040 1.6094 2.3026 4:

For p=l . 15 I = .1035 .2163 . 34 ]9 .4~48 .6514 .8532 1.1099 1.4677 2.0688

For p~l . 20 I = . 1030 .2145 .3381 .4779 .6404 .8363 1. 0850 1. 4304 2. 0098

295

Z

LO

0 .8

0 6 ¸

0 ,4 ¸

0 2

0 o 4

" d z

f o r p = l . 1 5 =

I.'2 116 2 ' 0 /

Fig. 1.

K(Iz- I,) we have:

Z2 -- Zl True fractional loss per unit area = l 12- I~

and the true retention efficiency 5/, = 1 -fractional loss,

Z2 - - Zj i.e. N, = 12-11' (10)

f

When we use the mean temperature AT, i.e. the mean z = (z2+ zd2) to represent the effective collector temperature, we obtain an approximate retention efficiency N, = 1 - f". To keep the approximation correct to, say within 1 per cent we write:

/ _ Z2 - - Z1

R = N d N , , - - ; - - - - 7 - , / ( 1 - f " ) ~< 1.01. 1 2 - 1 1 /

oz 6.4 d6 o'.8 2

Fig. 2. Max ratio z2/z ~ to satisfy l per cent condition. This condition limits the range of temperature variation (z2 - z,) that should be permitted in any collector under

test, and will be referred to as the "1 per cent condition". The expression for R has been computed and yields

Table 2. This is shown in Fig. 2 and varies very little for values of p between I and 1.3. An alternative--and usually more convenient--method of expressing the condition of eqn ( l l ) is to plot the max value of z2 that falls within the 1 per cent error, for each value of z~. This is shown in Fig. 3.

Should it be found that a collector under test does not satisfy the 1 per cent condition, then we can either preheat the inlet water--at the same flow rate--or increase the flow rate: both these actions have the effect of reducing the z:/zj ratio.

The essence of the proposed test method, which involves having a number of collectors--say four--connected in series, is that we obtain a number of points on the E - A T curve for the same value of Q so that the

(lrl)

Table 2.

z2~ T o / M a x p e r m i s s i b l e t e m p e r a t u r e range Zo/Z i = i J T i

such that 7z o r ~ -~ may be used to de te rmine retent ion eff ic iency within 1%.

T = 0.2 0 .3 0 .4 0 .5 0 .6 0.7 0 .8

~'|'0 ~T i

- - ~--< 4 . 7 2 . 3 1 . 6 7 1 . , t ] 1 . 2 6 1 . 1 5 1.08

296

I.O

0.8

0.6

0 .4

0 . 2

o 6,8

Fig. 3. Max z 2 for given z~ to satisfy 1 per cent condition.

H. TABOR

constants a, b and p of eqn (la) are readily determined. Thus, apart from the correction term % the efficiency under any conditions is established. The only parameter that has to be controlled during the entire test is the flow rate.

We consider Fig. 1 which shows the value of z vs I (i.e. Table 1 for p = 1.15). It represents the length along the collector---or a number of collectors in series--since I = (l/K)x.

For the purpose of our discussion, the length is divided into four equal parts representing 4 collectors in series.

Let us adjust the value of K (by adjusting the flow rate m) such that the value of zo for the last collector outlet is, say, 0.827, i.e. the scale has a length of 1.6 units.

Then the xlK values for the four collectors, and the corresponding z values are:

However, if we increase the flow rate so that the last output temperature z is, say, 0.7, i.e. I = 1.1, we find:

Table 3.

x._ 0 .275 .55 .825 1.10 K

z 0 .25 .44 .582 .70

and the 1 per cent condition is just satisfied. At all higher flow rates, i.e. for lower upper temperatures, the 1 per cent rule is always satisfied.

Thus provided that we set the flow rate so that temperature z on the last collector output is less than 0.7 (for 4 collectors in seriest) we can safely use the mean :? values or mean AT values as representative of the effective temperature of the collector in calculating the efficiency from eqn (4) or (la).

Since we measure ATand not L we should know AT,.. This could be determined by stopping the flow and determining the maximum temperature rise. Note that we do not need to know AT,. accurately, as the values of z are required only to check the 1 per cent condition and not to determine efficiencyA: However, in practice, we do not need to measure ATm at all: we compute it § from eqn (6) after the characteristics a, b and p have been determined and then simply check the z values for the 1 per cent condition.

Before describing the experimental procedure, we make some comments about the factor 3'. This factor has been discussed in Ref. [2] and Hottel in Ref. [3] shows variation of a3 for various angles of incidence for flat plate collectors with two or three sheets of glass and a black painted absorber.

What emerges is that, for flat plate collectors, the departure of 3' from unity is small up to incidence angles of about 500 and that the values of y to this point are almost independent of whether there are one, two or three glass covers or whether a black painted or selective absorber is used.

Collector No. 1 Collector No. 2 Collector No. 3 Collector No. 4

X 0 0.4 0.8 1.2

K z 0 0.343 0.575 0.730

0.1715 0.459 0.6525 0.7785

1.6

0.827

From Fig. 3, we find that for each collector, the temperature range is too high to satisfy the 1 per cent condition for :~.

i'For 3 collectors in series, the upper permissible z value is 0.58 unless a spare collector is used as a preheater. Its characteristics are not important and no measurements are made on it.

SThis is not so for determining "heating" efficiency, as shown later.

§This is probably more accurate than the measured value, since A T,, is a mathematical quantity defined by eqn (la) and this equation may not hold for efficiencies near zero..

IAppendix 3 indicates the effect of wind--and how efficiency, determined at one wind velocity, can be estimated at another. A test certificate should state the wind velocity or that the results have been adjusted to a standard wind value. It should also state the mean ambient temperature at the teime of test.

With a likely error of less than 1% we can assume:

Incident angle = 0 15 ° 300 450 3~ = 1.0 0.99 0.97 0.935

EXPERIMENTAL PROCEDURE

The testing must be conducted on a clear day, with as little wind as possible, i.e. less than say 3 m/see.¶ It is limited to incidence angles of +45 °, which, in practice means somewhat less than +3 hours about solar noon for a collector facing the equator and tilted to the horizontal approximately by the latitude angle. This is the angle recommended for the test.

(1) Four collectors are connected in series--the outlet

Testing of solar collectors 297

of one collector being connected to the inlet of the next. If the supplier has not provided four (identical) collectors, a smaller number may be tested. In this case, the flow rate must be increased to prevent the final output temperature being too high. Thus for 3 collectors, final Zo ~< 0.58: for two collectors, final Zo ~< 0.44 (see Table 3). However, if we wish to test these collectors at higher z,, values, we can use a collector from another batch as a preheater. All that is required is to check that the 1 per cent condition is adhered to, using Fig. 3.

(2) Cold water--approximately at ambient temperature from an insulated elevated tank--to provide a substantially constant head--is fed into the bottom of the first collector. It passes out of the upper end and then to the bottom of the second collector, etc. The intercon- necting pipes are insulated but, as a precaution, ther- mometers or thermocouples are fitted at all inlets and outlets.t

(3) The water flow rate is set by a valve in the cold line to provide a flow rate that will heat the water to about the desired output temperature, i.e. zo ~< 0.70. The setting is not changed during the test.

(4) The system is allowed to warm up during the morning on a clear day and measurements are made during the three hours before and after solar noon. Since the morning readings are taken with increasing value of Q and the afternoon readings with decreasing values, any lag in response due to thermal capacity will tend to be cancelled out.:~

(5) Simultaneous measurements of ambient temperature, inlet and outlet temperatures, water flow rate and the solar incidence in the plane of the collector are made. The number of such measurements can be quite small: this depends upon their accuracy.

If the angle of incidence of the solar beam at any time is not known by calculation, it should be measured from time to time. This can conveniently be done by measuring the shadow length of a pencil held perpendicular to the plane of the collector. High accuracy is not needed.

ANALYSIS OF RESULTS

The measured results are plotted as in Fig. 4, i.e. the inlet and outlet temperatures less ambient, vs time, and yQ vs time.

Now E - H___ m c e ( T o - Ti) Q Q

or Ely = mce(T° - Ti) 3'O (12)

LO.

0.9.

0.8,

0 . 7

0 6 ,

60-

50"

40-

<~ 30]

I0

Time

YQP ~\

I I AT inlel 1

4 I() ;2 14 15 to tb

Fig. 4.

peratures. Also

a--~ = L + r , _ L = a t . + a r o - , 2 2

for each collector so that the AT values are also obtained.

If we consider two times t, and tb near solar noon, corresponding to the same yQ value yQ~, we now have eight points marked yj for which the Ely and AT values are known.

We can choose another value of Q, say Q2 lower than Q~ (but not too far removed in time that the y value is uncertain) and obtain a secon__d set of eight points Y2.

We now plot Ely vs AT (Fig. 7). With y and Q constant for all the first eight points, we could extrapolate backwards to determine the value of a, which is the intercept on the Ely ordinate (i.e. AT= 0). The following technique is used to locate a. An important feature to note is that at any value of AT, the thermal loss of efficiency, i.e. the term bATP/yQ is pro- portional to 1/yQ i.e. all intercepts such as NA, NB on Fig. 7 must be in the ratio yQ2/yQI. If we use the symbol G to represent the values of Ely, we have:

Since mcp is known for any given flow rate, we can at once determine Ely for any time ordinate for the 4 collectors from the values of the inlet and outlet tern-

fThe pipes or insulation should be painted white to reduce any solar collection. For well-insulated pipes a single thermocouple can be used to measure the output of one collector and the input to the next.

:[:The transient response time constant is approximately MI(U + 2m G) hr, where M is the heat capacity of the collector per unit area. Practical values are in the order of 0.1 hr.

bAT" a - G ~ = - -

(yO),

bAT" a - G2 = -

(yOh"

This leads at once to

a= G , ( y Q h - G~(yQ)2 = G,R - G2 (yQ), - (yQ)2 R - I

298

where

R = (yQ),/(yQ)2.

Since R is a constant for the two curves, we can take n lines like NAB to obtain a series of values of a. The average value will be:

GjR - G2 O

R - 1

where

We can thus draw the line El3' = a with a high measure of confidence. The constants b and p can now be determined from a plot of

E bAT f' a

3'Q

But even without determining the values of b and p we can, having located the line 17,/3, = a, determine the efficiency curve for any value of (TQ) from a single set of results--such as the (3'Q), data--by using the intercept ratio principle referred to above.

If R,, = (3"Q,)/(3"Q) then,

E - = G = G i R o - a ( R , , - 1 ) . Y

All the terms on the right hand side are known so that a complete plot of Ely vs AT at intensity Q is obtained.

Note that even if we are not interested in the value of p, we should recognize that in practice, it is not u__nnity. Thus whilst many workers suggest plotting E vs AT/Q, it is clear that as p # 1, the points for different Q values cannot fall on a single curve.

Thus in seeking to determine a, b and p from experimental points, it is preferable to plot efficiency against AT for each value of intensity in order not to add additional scatter.

As an example, for a collector satisfying the equation

E AT T M

-- = 0.783 - 0.00281 - - "r O

(Ely) was plotted vs (ATI3"Q) for two values of TQ of 0.906 and 0.625.

AT - - = 0 20

Q=0.906 ~ % = 78.3 70.55 Y

Q=0.625 L % = 78.3 70.85 y Difference in percentage = 0 0.3 points

H. TABOR

The use of E vs AT/Q or Ely vs AT/3"Q is legitimate for a narrow range of Q values and many workers record the range of Q when presenting the results in this form.

The method described here allows the efficiency curve for any standard Q value to be determined, even if no measurements were conducted at this Q value. This makes it possible to compare collectors from different manufacturers for any defined solar intensity.

cOMPARATIVE BEHAVIOUR--"THE STANDARD" PERFORMANCE

The analytical form of eqn (la) is that which would be used by an engineer designing a complete system. By writing a suitable program, he can insert radiation and ambient temperature data, information on storage and heat load, and the collector equation, and hence determine overall performance. However, for less sophisti- cated applications, an engineer or purchaser would like some simple index to define the performance of a collector for comparative purposes.

The simplest solution is to agree on a single_standard Q value called Qs and then present the E - A T curve for this value. It is proposed that Qs be taken as 0.63 kW/m 2 (542kcal/m2hr or 200 Btu/ft%hr--dependent upon the units used for determining (b) and 3' is taken as unity. The reasons are explained in Appendix 1 where daily average efficiencies are determined for clear days. Except for a constant, the daily efficiency can be represented approximately by an equation of the form of eqn (la) if we use for Q the value Q~ proposed.

Thus summarizing, we can write the efficiency of a collector at uniform temperature in general form as

E bAT p "/ "/O

For a quick comparison between different collectors, for example, to compare relative output (on a clear day) against cost, we can use a standard value Q, and write:

k ( a - AT"~ = b Q , /

where Qs -- 0.63 kW/m 2, and k = 0.92. If the collector is to be used over a wide temperature

range, we cannot use the mean ATand must return to the form

E - - = N ya

where N = (z2 - Zl/12- Ii) as given in eqn (10)

40 60 80 100

61.57 52.06 42.19 32.04

62.23 53.09 43 .61 33.86

0.66 1.03 1.42 1.82

°C/kW m- 2


1.0

0.8

OE,

0.4

0.2

0.2 0 4 0.6 0.8

Zo

Fig. 5. N h Retention et~ciency for heating.

1.0

,0 l

0.8 - ~

0.6.

\ \

o.4. ~ ~

Q2~.

o.~ 0;4 o~6 o;s x=o,/5

Fig. 6. Retention efficiency for standard day, --, Zero heat capacity; - - - , heat capcity = 1.5 Btu/hr ft 2.

L ~ dz I~ = 1 - z p'

For the special case where the cold end of the collector is at ambient temperature, i.e. z~ = 0, we have

Z Nh -- ~.

This is the "heating" retention efficiency. This has been plotted in Fig. 5 for two values of p, i.e. p- l .0 and p-l.2. For other values of p it is sufficiently accurate to inter- polate. The value of z is given by eqn (5) and AT,, determined from eqn (6) using the values of a, b and p that have been determined.

NOTE ON EFFECT OF ATMOSPHERIC CONDITIONS The text has called for testing on a clear day with little

wind. Several workers have proposed conducting the entire test under a transparent canopy thereby eliminating entirely the effect of wind. This involves technical difficulties such as ensuring that the canopy is sufficiently remote as not to be heated by the collector and that its optical properties are uniform so as not to distort radiation reaching the collector and the pyranometer. Whilst complete absence of wind would improve the accuracy for intercomparison of collectors tested at different times, it does not help much in knowing how the collector would behave in real use. Other workers in the field have suggested the use of a blower to produce a standard wind speed, but this is also difficult for large area collectors or especially for the present system that involves testing several collectors at once.

Appendix 3 suggests a method of estimating the effect of wind speed.

It is shown that eqn (la) should be written:

where

E _ _ b A T " ( I + ~-a y) (la, b) 1 , (2

C6 V y = ~ V

O

6

_E 7"

O

N • q . . . . . .

Fig. 7.

C6 and C7 are dependent upon the AT range, and on the net radiation exchange with the sky. If the testing has been carried out at some (low) value of V, say V, then it is shown that, with the aid of a test at another value of V2, it is possible to extrapolate back to V = 0.

Estimates for a typical fiat plate collector show the efficiency at AT= 20°C to drop about 0.3 per cent for a wi_nd change from 0 to 4 m/sec: the drop is 5.7 per cent at AT= 40°C and 20.2 per cent at AT= 60°C (see Appendix 3).

REFERENCES 1. B. Doron, Testing of solar collectors. Solar Energy 9, 103-4

(June 1965). 2. H. Tabor, Mirror boosters for solar collectors. Solar Energy

10, 111-118 (July 1966). 3. H. C. Hottel, Space heating with solar energy, p. 63 M.I.T.,

Massachusetts (1954).

300 H. TABOR

4. H. Tabor, Solar energy collector design. Bull. Res. Counc. Israel, 5C No. 1, 5-27 (Nov. 1955). (Reprinted in Trans. Int. Con[. Solar Energy, Tucson, Arizona II, 1A).

5. H. Tabor, A note on the thermosyphon solar hot water heater. Bull. No. 17 Comples 33--41 (Dec. 1%9).

6. R. Bliss, Atmospheric radiation near the surface of the ground, A summary for engineers. Solar Energy 5, 103-120 (July-Sept. 1961).

A P P E N D I X l

Average daily efficiency By applying the instantaneous efficiency equation at each

instant of time during the day when Q and 3, are known, we can determine the instantaneous yield and hence the daily efficiency. This technique has been described in Refs. [2 and 4] and is sketched here.

(a) We first replace the daily Q curve by "filtered" sunshine 3,Q which we write Q,.

(b) The efficiency E t for the collector subject to this radiation is:

Et=a-~t-~-TP =a[1-a~t -~P]=a[I- -~t t ]=aN

where Q~ = (bla)" AT p and is termed the "cut-off-intensity", i.e. intensity at which the efficiency is zero. The energy H delivered at any instant of time is:

H = EtQ ! = alQ t - Q~]

and the total daily heat collected is:

f H dt = a f (Ol- O~) dt for positive values

= aN f Q/dt 3

where

(Q,-Ox)dt (AD

f Qxdt

(c) We draw a number of lines at various heights Qx on the Qs--time graph and take the area above the line as a fraction of the total area. This gives the mean retention efficiency /q. Then the mean efficiency, in terms of the filtered sunshine is E = aN r, whilst in terms of the original solar radiation Q, becomes

= aFtlq (A2)

where F t is the filter factor

f O , dt i.e. f(vO)dt

for the wllole day. Note that (A1) can be written:

Ffl~l f (Ql-O~)dt - (Ala)

f Qdt

which is determined by the process in (c) above, except that the area above the Q~ line is expressed as a fraction of the original total insolation area.

tCorresponding approximately to a peak of l kW/m 2 at noon.

It is convenient to define a mean cut-off ratio )( = QxlQ, where 0 is the mean insolation during the day, and to determine N in terms of )(. The advantage here is that as .-~ is a ratio, we get the same/q values (for a given .~) for different da){s, if the shape of the radiation curve is the same: and we chose Q rather than Qr, ax because it is found that Ff/Q for a given Q~IQ value is less sensitive to shape than FtN for a given value of Q~lQm,x.

The process described in (c) has been applied to a "standard" day (clear equinox day, with the collector facing the equator and tilted at the latitude angle, with sky radiation taken as 15 per cent of global).

The results, shown in Fig. 6, depend upon whether allowance is made for the heat capacity of the collector or not: this is assumed to delay the start of pumping until the collector reaches the design temperature. Figure 6 thus shows two cases; one for a collector having negligible heat capacity and one for a collector having a heat capacity of 31 kJ/°C, m 2. (Most real collectors will lie between these limits).

Over a limited range of, say )( = 0.1-0.45, we can approximate the mid position of the two lines by:

F~N = 0.92 - 0.847X : 0.92(1 - 0.92)().

This is a general equation. To obtain the efficiency /~, we must multiply by the value of a, which is a characteristic of the particular collector. Then

= 0.92a (1 - 0.92)() = 0.92a -[1 - 0.92 b A T p / L a ~ J'

If we compare this with the instantaneous efficiency (taking 3,=1)

E = a ( I --~-}bAT~

we can, except for a the factor 0.92 outside the bracket say that the instantaneous efficiency equation can be used to represent the average daily efficiency if we chose for the value of Q a value (~10.92.

We arbitrarily take a representative value of 0 = 0.58 kW/m2t " (498 kcal/m =, hr or 184 Btu/ft 2, hr) so that @0.92 which we call the "standard intensity" Qs is 0.63 kW/m 2 (542 kcal/m 2, hr or 200 Btu/ft 2, hr).

It is therefore proposed that the E vs ATcurve for Q = Qs be used when comparing collectors. If we agree to define the day chosen in this treatment as a standard day we can then write

O,

as the expected efficiency on the standard day. As a first approximation, we can use this form for any clear day by writing it as

Q

where the (~ is the mean intensity for the day. This approximation assumes that the shape of the solar radiation curve is not very different from that of the standard day and that differences in the 3, function are of secondary importance.

A P P E N D I X 2

Form of the E vs AT equation We have written eqn (la) as:

E - b ATP - -= a (la) Q

which is found to be a reasonably close approximation over the usual working range. However, it could have been written in several other forms which give reasonable approximations. For


example, one form is:

AT E = a - ( bo + b , A-T~ . - ~ .

Due to the departure from linearity being small for most collectors, both forms can be made to fit reasonably well.

In general we can write:

~ = a - ~ f(~'--T).

We see that if we use two known yQ values at a single A T value to determine a, as indicated in the text, this value of a is correct whatever the/orm o/the/unction/(AT) which form need not be known.

APPENDIX 3

Effect o/outdoor conditions The window of a horizontal collector loses energy to the

surroundings by radiation and convection. The Iongwave radiation loss q,, for a body at ambient tem-

perature T, is given by:

qr = eR (A3.1)

where ~ = emissivity of the body; R = longwave net radiation to the sky for a horizontal plane at ambient temperature.

The value of R is primarily a function of ambient temperature and the relative humidity for clear skies/6]. (For cloudy skies, R vanishes, but our measurements are conducted under clear sky conditions.) For the usual values of ambient temperature and humidity, R is in the range 20-30 Btu hr -~ ft -2, as seen from Re/. [6]. In SI units, R is in the range 60--100 W/m 2.

The values of R are for horizontal planes facing a hemispheri- cal sky. If the plane is tilted by an angle 0 to the horizon, it "sees" a fraction / of the sky where

f 1 +cos 0 . = ~ . ' f

Equation (A3.1) should therefore be written:

q, = cIR. (A3.2)

If the body is at temperature T~, we have to add a radiation term 4~rT,,38T, where

o- = Stephan-Boltzman constant

8T=(T,-T~); T,.=~(Te+ To).

However, as this term is included in the surface loss coefficient as given in the A S H R A E Guide (1972) p. 348 which may ,be represented by:

q~ = (Cj + C2V)AT (A3.3)

we have, total heat loss H is:

H = qr + qc = ~/R + (C~ + C 2 V)A T = ~/R + XA T (A3.4)

where

X = Cj + CzV= C~(1 + C3V)

where

tThis assumes a dark ground: the NBS proposed testing pro- cedures call for a low albedo from the ground in front of the collector.

C3 = C J Q .

The A S H R A E Guide shows Ct -1 .6 , C2-0.22 for smooth surfaces, when V is in mph and qc is in Btu hr -~ ft 2. In SI units, Ci-0 .009kW, m-2°C -1 and C2-0.0028kW, m 2°C-Im Is. Whence C~ 40.3 m -j s. Equation (A3.4) may be written:

H = ~/Tr + X ~ = h ~ = 1 ~ (3.6) ro

where h o is an effective outdoor conductance, and r o the effective outdoor resistance for the upper face of the collector.

Let the conductance from the absorber to the upper window be U v and the resistance r e = (l/Ug).

The upward heat coefficient U, from absorber to air is:

1 ho v~ ro + r~ l + hor~

Also

8T r,, I - - = =

AT ro+rg l+h,,rg

Substituting for ho, the quantity [(~/R/6T) + X] and eliminating 8T leads to:

(z + x ) v , u~ = u~ + x (A3.7)

where

Z = -~-~:". (A3.8) AT

For zero wind, we have:

Z + X o or U U"°X" U"° = Ug + Xo ~ Z + X o - U,o (A3.9)

where the subscript o on U and X indicates zero wind. The total loss coefficient U for the collector is:

U = U~ + U# (A3.10)

where U d is the loss through the rear insulation and is assumed to be almost independent of the wind. For convenience, we write:

ua = pu.o (A3.] I)

and

U = U u +pU, o and Up = U~o(l +p). (A3.12)

The change in U (due to the wind) is:

aV=AU.

i.e.

A U _ 1 Uu

U o 1 + p Uuo"

(A3.5) From eqn (A3.9) we find that

au . (U.o - z ) ( X - Xo) U.o x ( z + x o ) - U.o(X-Xo)

[uo - (1 + p)z]Q v = (1 + p)(Q + Q V)(Z + CO - UoQ V

using (A3.5) and (ALl 1), i.e.

302 H. TABOR

AU,,= C4{Uo-(I+o)Z]V ( 3 . 1 4 ) 1

U° l+ C3[1 U° CI)]V (1 + p)(g +

where

Then

c3 (A3.15) C4 = (I + o)(Z + C,);

AU U. - - =

Up (l+p)U~o CdUo - (1 + p)Z]V = y

(I + p)(Z + CO V (A3.16)

where

Then

C3 C~ = (1 + o)2(Z + C,)

U U ~ = I + Y .

In the expression (la) for efficiency, we have the b term propor- tional to U, i.e. we can write, to allow for the effect of wind.

The expression for y in (A3.16) includes U,, which we do not know but, from the derivation of (la), we see that:

u bAT p-~

U°= F

and this quantity should be substituted in the expression for y. In general, we do not know the values of F or p for a collector

under test, so we write (A3.16) as:

Thus if the experiments are carried out at a (low) velocity VI, and we require the results for zero velocity, we determine the efficiency at some higher value of V 2.

For the same value of AT and yQ we have:

Gi E~ a_bATP[1 + C6VI ] = 7 = S0-L l+C~v4

G 2=-=a-bE1 ATP[I+ C6V2 ] Y Y L 1 + C7V2]'

Whence it is readily proven that, for zero wind, we obtain:

(Gi - G2) VI Go = G1"¢"

. F I + C T V I T V2- v , [ ~ ]

The value of C6 need not be known and it is sufficiently accurate to assume C7 - 0.2.

Note that, at high values of ATand high wind speeds, the value of y is not negligible so that the effect on efficiency can be considerable. We consider an actual case.

a=0.783, b=0.00282, p = l , l l , F=0.925, U o = b ~ ° . l l =

0.00305A T o..

Efr = 0.06 kW/m 2, yQ = 0.63 kW/m 2, p = 0.25

We obtain the following table:

A T = 20 40 60°C Up = 0.0042 0,00458 0.00979

Z = ~ ' / - - 0.0030 0,0015 0.00075 AT C6 = 0.005 0.044 0.0687 C7 = 0.219 0.200 0.188

For V = 2mls Y = 0.011 0.070 0.110 V = 4mls 0.017 0.110 0.173 V = 6mls 0.020 0.135 0.215 V = 8mls 0.023 0.153 0.244

C6V Y I +C7 V

r where C6 and C7 are weak functions of AT. Since C7< C3 and C a - 0.3 m- i s from the ASHRAE reference, C7 will be of the order of 0.2. For small values of V we can write, with little error,

C6 V Y - I + 0 . 2 V'

This is useful when it is not possible to carry out the calibration at zero wind as recommended in the text,

t i t is interesting to note that for the value of AT that makes Z = Up/(1 + p), the wind has no effect. The physical explanation is that the heat lost by net radiation R is just balanced by heat coming up from the absorber, the window then being at ambient temperature. In real collectors this effect is often obscured by the solar radiation absorbed in the window material.

The corresponding efficiencies are:

A T = 20 40 60°C For V = 0 m/s Ely = 0.6585 0.5143 0.3616

2 0.6572(0.2) 0.4956(3.6) 0.3152(12.8) 4 0.6564 (0.3) 0.4849 (5.7) 0.2887 (20,2) 6 0.6561 (0.4) 0.4781 (7.0) 0.2710 (25.1) 8 0.6557 (0.4) 0.4733 (8.0) 0.2588 (28.4)

(The numbers in brackets are the decrease in efficiency expressed as a percentage of the V = 0 values).

Post review note. Several changes and corrections have been introduced since the paper was originally presented at the ISES Meeting (July 1975). (i) Figure 4 in th___e original showed the ordinate as Ely: it should have been AT. (ii) It was suggested originally to use a double flow rate in the afternoon to yield more points on the upper end of the efficiency curve. However, this leads to rather small (T o - Ti) values across each collector so that uncertainties in measuring these quantities can introduce relatively large errors. (The use of a single flow rate virtually eliminates heat capacity errors under clear sky testing).

Resumen--Por cuanto el ensayo de colectores realizado en distintos momentos debe ser comparable, es necesario que este defina la ecuaci6n de los mismos. Esto involucra la medici6n a diferentes intensidades solares, diferentes temperaturas de operaci6n y diferentes regimes de flujo.

E1 artfculo describe la teorfa y el procedimiento experimental que considera a cuatro colectores conectados en serie, midi6ndose el incremento de temperatura a tray& de cada uno. Por cuanto a cada instante el r~gimen de flujo y la intensidad solar es identica para todos ellos se obtiene un ntimero de puntos de la curva de rendimiento vs


temperatura que facilita el establecimiento de la ecuacion caracterfstica de los colectores. Se establecen reglas para limitar el incremento de temperatura permitido y se sugiere un dia "normal" para proveer una base rfipida de comparaci6n.

R~sum~--Pour que des essais sur des capteurs diff~rents, conduits h des Epoques diffErentes, puissent ~tre comparables, il est n~cessaire de dEfinir l'¢quation de fonctionnement du capteur. Ceci meten jeu des mesures h diff6rentes intensitEs solaires, h diff6rentes temperatures de fonctionnement et avec diff~rents debits.

Cet article dEcrit la thEorie et la procedure expErimentale qui met en jeu, par exemple, quatre capteurs similaires, connectEs en sErie, dont on mesure I'acroissement de temperature h travers chaque capteur. Puisqu'~ chaque instant le debit et le rayonnement solaire sont identiques pour tousles capteurs, plusieurs points de la courbe rendement-tempErature sont obtenus h partir desquels l'6quation caract6ristique du capteur peut ~tre immEdiatement obtenue. On 6tablit des r6gles pour limiter l'61~vation de temperature permise, et on suggbre on jour "standard" pour fournir une base de comparaison rapide.

SE Vol. 20, No. 4--8

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Testing of solar collectors