Solar energy converters: The relationship between efficiencies and other parameters

Solar Energy Converters: The Relationship Between Efficiencies and Other Parameters B y M . v. M e n t s Research Council of Israel, Jerusalem, Israel

Every comprehensive computation for a particular solar energy device necessitates the use of an enormously large number of parameters and variablesf 5. 8. 9 It is almost impossible to deal with all the scientific, engineering, and cost aspects in a general summary. As a result, every endeavour to calculate the merits of a certain attractive case is per se an isolated treatment that cannot be derived from a generalized theory, and the opposite is also true--such an isolated treatment performed on a special case cannot be applied to a large group of cases. The difficulties in generalizing start with the quantitative variation of the sunshine available. The amount of sunshine varies with the geographical site, with the time of day, and with the time of year. In some cases, sufficient mathematical expedients eventually result in very complicated and cumbersome expressions. But the unpredictable influences of rain, clouds, wind, dirt on the collectors, etc., make the validity of such expressions very doubtful. 6 A second reason for the difficulty in generalizing is the complicated relation describing the heat balance of the collector. In its most simplified form, this relation looks like this:

A closed and c o n v e n i e n t m a t h e m a t i c a l f r a m e work is presented c o n t a i n i n g t h e m a i n descriptive parameters for solar energy converters. Principles for t u r b i n e s driven by s u n - g e n e r a t e d s t e a m are also f o r m u l a t e d . T h e effect o f every p a r a m e t e r and o f c h a n g e s in o n e or m o r e o f t h e m o n o p t i m u m efficiencies or o p e r a t i n g t e m p e r a t u r e s c a n be f o u n d from t h e relations p r e s e n t e d . It is n o t claimed that the physical optimum conditions are also t h e best o n e s e c o n o m i c a l l y . A l t h o u g h u n likely, it is q u i t e possible t h a t c o n d i t i o n s o n b o t h sides of t h e p h y s i c a l o p t i m u m arc b e t t e r from a practical or e c o n o m i c p o i n t of view t h a n t h o s e at t h e physical o p t i m u m itself.

INTRODUCTION The idea of generating power by driving some sort of heat engine with steam produced in a solar collector is almost as old as the machines themselves. However, although it is possible to build such devices, no practical installations have been realized which were also promising from an economical point of view. The difficulties are a result of the following conditions: (a) The peak power of solar radiation at midday is only about 300 Btu per sq ft per hr at favorable localities. (b) The power averaged over a whole day is considerably less than this, even if cloud and rain disturbances are ignored. (c) It is impossible to collect all of the solar radiation, particularly at those elevated collector temperatures needed to generate high pressure steam. According to the second law of thermodynamics, as well as because of practical considerations, as high a steam temperature as possible is required. (d) Where it has proved possible to avoid some of these difficulties by scientific and engineering "tricks," it has nevertheless always seemed that the investment costs of the required device did not warrant development on a really serious scale. (e) Problems connected with storage facilities to cover the nighttime and periods without sunshine have not yet been solved.

s(la(t) = arS(t~(t) - s A ( T Ta) s S ( T - T,) 5 / 4 - s C ( T 4 -

T, 4)

[1]

The left-hand term represents that part of the energy absorbed by the collector that is passed through to the liquid to be heated. The right-hand terms represent, respectively, the solar radiation absorbed by the collector surface and losses due to conduction, convection and radiation. The constants A, B, and C depend on the geometric conditions of the device, on the several material constants of the components, and on the condition of the surroundings. T is the temperature of the collector, and Ta is the temperature of the surrounding air or the "next" item, for example, a glass cover sheet. Information on the use of such equations and the determination of the constants involved can be found in the literature of solar energy. It is the aim of this study to show that when one sets aside cost considerations it is possible to express the efficiency of a solar energy collector as a function only of a "figure of merit" of the device and the working 44

form of an ellipse, a cosine, or something in between. (See Fig. 1.) The ellipse may apply more to cases of heliostatie mounting and the cosine to devices with a fixed (most favorable) tilt. (b) Experience s' 9, i0 also shows that the ultimate curve of the losses as a function of the collector temperature is always smooth and that it can be presented to a very good approximation by the relation: (I~ = PA T4/3 [2]

/t~lt)

-tp/~

midday - - ~ t

where AT is the temperature difference between collector (T) and surroundings (Ta), and p is a constant of the collector device. It should be mentioned that q,, (or better, s~z) represents the sum of the three losses described in connection with Equation 1. Both Maria Telkes (Fig. 3) I° and in an improved way Tabor (Fig. 5)9 calculated such losses for very special cases, and Fig. 2 and Table I show that Equation [2] applies very well to their results. In Fig. 2 the results of Tabor (Fig. 5) 9 are somewhat rearranged. The losses of four of his collectors (1, 2, 2a, and 4), divided by an arbitrarily accepted radiation maximum of 300 Btu per sq ft per hr are shown as a function of the steam temperature. This ratio is identical with a factor u, later to be defined as the fractional losses of the collector at midday. (See Equations [9], [12], and [13].) The same procedure is followed with the best curve in Telkes (Fig. 1) 1°, indicated by " M . T . " in Fig. 2. In Table I, the "figure of merit," 0, inversely proportional to p in Equation [2], is calculated for every one of the five examples. (See definition, Equation [12].)

**P/~

FIG. 1--Sun radiation as a function of the time of day. 0.9

"M.r~ O.B

]

.,./

q2a"

i;

0.7

/

0.6 0.5

/

/:

Q

/f

0.4

•

.2

*

,. ,,

0.3

O.Z

/__e" •

.o/i

0.1

loo

~oo

300

4oo

soo

TABLE I

600

Tin°F

DETERMINATION OF THE F I G U R E OF M E R I T 0 FOR SOME

COLLECTORS DESCRIBED IN THE LITERATURE~,x°

FI~. 2--Fractional losses at midday of some solar collectors versus steam temperature.

T

temperature of the extracted steam. By multiplying this efficiency, averaged over the whole day, with the over-all efficiency of the turbine, one finds the total efficiency of the combination. Optimum conditions for working temperatures and the total efficiency can then be derived using differential calculus. The mathematical expressions and the treatment as a whole are fairly simple in nature. They provide a method for determining immediately how the situation alters under different conditions or changes in one or more of the describing parameters. This method shows some resemblance to those in other fields, especially in physics, where one also expresses now and then maximum or practically attainable yields as a function of only one or two important parameters. I' The conditions under which this can be done must of course be welldefined and clearly stated. The treatment for solar energy conversion will be based mainly on the following three postulates: (a) Experience I' 4, 6.7 indicates that the sun's radiation during the day as a function of time always has the

T4/3

150 200

340 650

250 300 40O 500 600

1,010 1,400 2,270 3,22O 4,250

Result: 0 =

AT4/8/Ul AT 4/ 8/u~ AT4/*/UMT" AT 4/~u~ i

850 83O

840

1,900

2,00O 2,050 2,10O

2,700 2,650 2,700

5,050 5,00O 5,000 4,800 4,600

2,000

2,650

4,800

2,550

AT4/Vu4

11,000 10,800 11,300 11,100 10,900

11,0o0

(c) The sensible heat needed to warm water to the boiling point will not be taken into account. Including this heat and its consequences in the calculations would give rise only to minor shifts in the results. (See Appendix V.) It is further assumed that the solar energy collector delivers saturated steam and that this steam is superheated by some conventional fuel-fired heater. This procedure is cheaper than superheating the steam with solar energy. The influence on cost of this superheating will not be taken into account, and the estimate of the over-all plant efficiency of the turbine will be based on the data of Elliott (Fig. 1).3 This efficiency is defined by 45

l

I

SIMPLIFICATIONS

~,o

~ turb

AND

IN THE

into

ZO

/ f

1oo

/~'

zoo

400

3oo

500

600

Tin °F.

FzG. 3--Over-all efficiencies of turbines as a function of the saturation temperature before superheating.

3413

[3]

murk. - net heat rate If one recalculates the results of Elliott as a function of the saturation temperature T of steam before superheating (i.e., the collector steam temperature) one gets the ranges in Fig. 3. In the first range, the pressure varies from 100 to 300 psig, in the second from 300 to 500 psig, etc. By consulting the original references given by Elliott, every range could eventually be replaced by isolated dots. It can be shown from Fig. 3 that the over-all turbine efficiency can be determined fairly closely by ~turb. = k" ( T -

To)

k ~ 6.6.10 -~

[4]

the slope k being about 6.6.10 -4, as derived from Fig. 3. The over-all turbine efficiency, as represented by the solid line in Fig. 3, tallies with recent data of Roberts22 (In his Table II, one should consider the saturation temperature belonging to the stated working pressure.) As T~ seems to be in the neighborhood of 70 ° F and as this is about the same as what was termed the "temperature of the surroundings," T~, in the definition above, Equation [4] can also be written as kAT k ~ 6.6-10 -4

~turb.

A B

=

C

The theoretical maximum efficiency of a heat engine, known as the Carnot efficiency, is given by the wellknown formula:

k

T - - T~ T

AT T~ + A T

THEORY

NOMENCLATURE

[5]

~t~ -

USED

The acceptance of the above three postulates does not impose a limitation on the number of examples that can be used. In the first place, the resemblance between the previously accepted quantities and the actual ones is striking, or at least (as far as the third postulate is concerned) fully explained. It is further evident that small deviations from the conditions laid down in the postulates give rise to changes hardly worth mentioning in the end results. (See appendix IV.) Finally, as we shall find in the other assumptions made in this section, too great a precision in this field is of little value, for at least two reasons: (a) The end results will have to be corrected anyway for a number of significant factors, including clouding, superheating, etc., as mentioned by Hottel. 5 (b) The still unsolved problems of steam storage and other unpredictable engineering factors cause an unknown and perhaps serious decrease in the maximum yields obtained by any theoretical computation. Some of the other simplifications used are: (a) Effects of clouding, accumulating dirt or dust on parts of the collector, mirrors, or covering glass sheets, etc., are neglected. (b) The temperature difference between the outside of the collector (fixing the losses) and the generated steam is neglected. (c) The fact that the steam delivery rate at a fixed temperature to the turbine is not constant during the day is not taken into account. This means that either the turbine must be adapted in some hypothetical way to these conditions, or ideal steam storage facilities exist. Some of these assumptions do not present any real limitation, as the results can be adapted to more realistic circumstances by inserting "corrected" or "effective" values for some of the parameters.

3O

10

ASSUMPTIONS

m n p

[6]

Tabor (Fig. 6) 9 and Courvoisier 13 m a d e use of this last

efficiency to determine certain theoretical optimum results. In the following treatment, however, use will be made of the turbine efficiency from Equation [5] instead of Equation [6]. (See Appendix I.)

/5 ¢ q8 qz ~a(t) 46

= a constant describing heat losses by conduction, = a constant describing heat losses by convection, = a constant describing heat losses by radiation, = a constant describing over-all turbine efficiency, °F -1, = a constant used to indicate the relationship between 71.2 and u, = 4/3 m, = a constant describing heat losses in a general form, Btu ft -~ hr -1 °F -4/3 = average power rate, watt ft -~, = midday sun radiation, Btu ft -~ hr -1, = general heat losses, Btu ft -~ hr -~, = usefully absorbed heat, Btu ft -~ hr -1,

=

qn(t)

q,2(t) Qd

Q~ Qb 8

S T Ta Ts

AT ATopt. t

tp u

~opt.

X

Z

¢0

0 ~turb.

(t)

,1,2 (t)

~total

~opt. ~hb

[9]

Referring to Postulate (a) and Fig. 1, we assume as border-line cases the following two solar radiation daily distributions: 0.1 (t) = 08V~- -- x 2

z = 4t/tp

[101

and

0.2(t)

=

~. cos

,ot

o~t~ = 2 r

[111

The values of ~, and tp do not necessarily need to be the same in both cases. We further introduce the following two help factors:

,l(t) = c~r[1 - u/%,/1 - xq

f

[15]

tp[4

dt

JO

where tel is the time after which the collector cannot reach the temperature T, and tel is in turn defined by ,l(tcl) = 0 From Equations [10] and [14] it follows that tel (tv/4)~¢/1 -- uS

[71

=

[16] [17]

The integration of Equation [15], making use of the result of Equation [17], yields

The collector efficiency at the time t may now be defined by

71 ~ (ideal)

[141

fo to' nl( t)O.l( t) dt

TREATMENT

qa/qa

[131

The average daily efficiency of the device may be defined by

or

=

[12]

The factor 0 may be termed the "figure of merit" of the collector. It should be noticed that this factor also takes into account the maximum solar radiation. This requires that even a device good from the standpoint of low losses and a good S / s ratio may have a low figure of merit if located at a site of low solar radiation. The factor u simply indicates the fractional losses of the receiver under midday conditions. This can be verified by inserting t = x = 0 in Equations [14] and [14']. One could also first introduce this definition for the factor u and further define the figure of merit 0 with the help of the left-hand side of Equation [13]. This is actually done in making up Table I. If we now investigate the daily distribution as given by Equation [10], it follows from combining Equations [9], [10], and [13] that

From Equations [1] and [21 it follows that a device with a theoretical efficiency of 100 per cent is characterized by a = 1; r = 1; p = 0

~¢o11. ( t )

q,/p

u = AT4/3/0 = p A T 4 1 3 / ( Sar~ ) s q'

~1 =

qo (ideal) = S q,(t) s

aT -8

and

over-all turbine efficiency, the same, for low temperatures, theoretical Carnot efficiency of a turbine, = collector efficiency at the time t, = the same in the case of special radiation distributions, average day efficiency under same conditions, average day efficiency of the combined installation, the same under optimum conditions, collector efficiency of "heating and boiling" installation.

MATHEMATICAL

S.

0 =

8

7/turb.

7}1,2

7/eoU.(t)- a r I 1 - - p A T 4 / 3 / ( a T ~ O , ( t ) ) I

=

~cl,2

,co..

Combining Equations [1], [2], [7], and [8], we obtain, for the collector efficiency,

sun radiation at the time t, Btu ft -2 hr -1, = the same according to a special ellipse distribution, = the same according to a special cosine distribution, ~- daily sun radiation, Btu ft -2, = sensible heat addition, Btu per lb, = heat of evaporation, Btu per lb, ~- total exposed liquid conducting area, ft 2, = total sun collecting area, ft 2, = temperature of steam, °F, -~ temperature of surroundings, °F (°R), -~ sink temperature of turbine, °F, TT ~ , °F, = : t h e same under optimum conditions, °F = time, hr, = cut-off time in case of special distributions, hr, = period; tv/2 is the time of sunshine, hr, = help factor, indicating midday fractional heat loss, dimensionless, the same under optimum conditions, u 3/4 = AT/O 3/4, dimensionless, ~-~ 4t/ tp, dimensionless, = 1 + 2T./O 3/4, dimensionless, integrated absorbtion coefficient, integrated effective transmission or reflection coefficient, circle frequency, hr -1, S= "figure of merit", 0 = a T - G / p , °F 413,

O.(t)

[8]

47

O / T -2 [cos-' U - 11"

u V q- ~ 1

[181

,~T/OS/+IAT

~t,1 = kar[1 -

1.0

"t~T:

[21]

k ~ 6.6" 10-+

t

Efficiency curves of an almost parabolic type are also shown by Baum, 2 without detailed interpretation as to their determination. (See the example at the end of this section.) Differentiating with respect to AT one finds, for the optimum conditions:

o.6

5Top+. = !,++/<2+ ?]op+. = ¼ karO +/4

"',,',.N,. z - ' -

k ~ 6.6" 10-4

--,,,,..'~ I

I

I

1

,

i

'

1,0

:--U

Fro. 4 - - T h e average daily collector efficiency as a f u n c t i o n of t h e p a r a m e t e r u.

When the same procedure is followed for the second distribution possibility as given by Equation [11], one obtains, respectively: ,2(t) --- aT[1 -- U/COS c0t]

[14']

fo~°'ndt)O.(t) dt "1"12 ~ ,t o

t~2

[15'1

tp/4

[

O,2(t)dt

1 COS- I U --

[17']

,2 = ar[%/a -- u 2 -- u cos-' u]

[18']

EXAMPLE

Baum (Fig. 9) 2 calculated the efficiency of three stations as a function of the steam temperature. As no

Both the results of [18] and [18'] are shown in Fig. 4, together with the curve 1 - u v4. It can be seen, and this is the practical consequence of Postulate (a), that with an uncertainty of only several per cent, one m a y say for every conceivable solar radiation distribution ?]1,2 ~

ar[1 --

u*/4],if u

[23]

We are of the opinion that these simple relations m a y be of great help in rough but quick estimates of various yields in a range of cases. This is true not only for optimum conditions, but also for situations on both sides as well. The relations should further prove helpful in cases where one wants to see quickly what happens when one or more of the parameters of the system are varied. (See appendixes.) Under the loose assumption that it will be very difficult in most practical cases to obtain values of a r in excess of 0.85, the relationship between ?]o,t. and 0, as expressed by Equation [23], is shown in Fig. 5. The efficiencies of the five collectors in Fig. 2 are indicated in this figure. In the diagram, "4*" marks the efficiency of the installation "4" if a r = 0.773 instead of 0.85, as stated by Tabor (Table III).9 The meaning of the other curves and points in Fig. 5 m a y be obtained from Appendixes I and II and the following example.

%.\ l

[22]

•%

'Ttotat

I mj~0

< 0.5

-

S ~

or, again with some optimism, ?]1,2 ~--- ar[1 - -

u m]

[191

( 15

(See Appendix IV and Equations [25] and [25']). By inserting Equation [13] into this result, one obtains m,2 = at[1 -- AT/O3/4] [20]

"~T: 10

B2

/ /

"

"z~

/

~"

-,a,

This relation shows that the approximate dependence of the average daily collector efficiency on the collector temperature and the figure of merit, 0, is very simple indeed. The efficiency of the combined installation (collector and turbine) is determined by the product of both efficieneies m.~ and ?]t,~h., as given by Equations [5] and [20]:

5

r "p"

r

Z/;o

.... •

i

t ,

~000

J i

,

|

,

I

•

|

.

IO,OO0

Oin°F ~

FIG. 5 - - O p t i m u m t o t a l efficiencies as a f u n c t i o n of t h e f i g u r e of m e r i t , 0, of t h e c o l l e c t o r s .

48

APPENDIX

detailed data are furnished, we can attempt only to explain his results according to the above theory. This might be done as follows: From the optimum temperatures of each of his three curves we may deduce the figure of merit, 0, of every station, using Equation [22]. The calculated efficiency for every value of 0 can be read from Fig. 5 and then compared to the values found by Baum 2 himself. The results of this procedure are shown in Table II. The agreement seems very good. (See also Equation [26] in Appendix III). The optimum efficiencies belonging to the three assumed figures of merit of the Baum stations are marked BI, B~, and B3 in Fig. 5.

The place and the dimensions of the ranges in Fig. 3 seem to indicate that the value of the assumed turbine efficiency (Equations [4] and [5]) is on the high side for low temperatures. By "low temperatures,' we mean those, say, below 450 ° F (400 psig). In order to show how the situation changes if a more realistic value is accepted in this region, the optimum conditions will be recalculated for a turbine efficiency given by ~'turb. = 1.0.10-3AT -- 0.13 [5'] shown by the dotted line in Fig. 3. (This treatment shows the flexibility of the theory set up in the previous section.) Differentiation with respect to AT of the product of the efficiencies as given by Equations [5'] and [20] enables us to determine the optimum conditions as in all previous cases. The result is:

TABLE II TENTATIVE DETERMINATION OF THE FIGURE OF MERIT 0 FOR THE THREE RUSSIAN COLLECTORS2

ATopt.(°C) aTopt.(~F) Bj B2 B3

340 270 180

610 485 325

0

~opt. (0/0)

~Baum(%)

13,090 9,500 5,600

17.0 13.5 9.1

16.4 14.4 9.5

APPENDIX

ATopt.

T. +

The result given by Equation [23"] is shown in Fig. 5 under "low temp." APPENDIX

AT

hi.2 = ar[1 -- 5T/O ~/4]

[6]

~/o,t. (Carnot)

+ %/T,~ -[- TaOa/4

= ar[Z - V / ~ ]

[20]

Uovt. = (½)4/3 _- 0.40

Uopt. (Carnot) = ½ (1 - ~o,t.)

[25']

as derived from Equations [13] and [22], and ?]opt./ATopt. ~

constant ~ 2.8.10 -4

[26]

as derived from Equations [22] and [231. (See Table II.) Finally, it is worthwhile to compute the power rate of the whole installation by taking into account the total available solar radiation. It appears from the measurements of Ashbel I that for Jerusalem one may say, to a very good approximation ~8 ~ 1.4 cal/(cm 2) (min) ~ 310 B t u / ( f t 2) (hr)

[27]

and

[22']

tp/4 = 6½ hr (yearly average)

[23']

[28]

During the whole year, t J 4 varies between 5 and 7 hours. (See Equations [10] and [11].) For a heliostatic installation, it follows, by integrating Equation [10] and inserting the values from Equations [27] and [28] that

where Z = 1 --}- 2 T J g / 4

III

In order to give a clearer conception of the relation between the different magnitudes, the following equations may be added to those in the previous sections:

By combining the efficiencies from Equations [5] and [20], the derived results were obtained and dealt with briefly. By writing down the product of the efficiencies from Equation [6] and [20] one obtains results for an ideal Carnot machine driven by solar generated steam. Optimum results can be obtained by differentiating with respect to AT. The mathematical manipulations to be carried out are very simple and will not be repeated here. In spite of their low practical value, the results are given below: ATo,t. (Carnot) = - T ,

[22"]

~opt. (low temp.) = "r03/4 4 • 10-211 - 1~/e3/4] 2 [23"]

In the previous sections the following three important efficiencies were considered: ~t~b. ~ 6.6.10-4AT [5] hi. -

(low temp.) = !A3/4 2v ~- 65

and

I

AT

II

[24] [25]

The result given by Equation [23'] is also shown in Fig. 5 by the dotted line "Carnot," for T, = 530 ° R. The probable efficiencies of the four collectors (1, 2, 2a, and 4; marked 1~, 2o, 2a~, and 4~) on this "Carnot" curve, agree fairly well with the results in Tabor (Fig. 6), 9 especially if we take into account that the numerical conditions underlying his and our own calculations are not quite the same.

1

Qd ~-~ 310.6½.2 Jo ~

-- x2 dx ~.~ 3150 Btu/ft 2

[29]

Averaged over the natural day of 24 hours, this means that the rate is 3150 /5 ~ ~ 3413 49

1000 2--T" ~ 387 watt/ft 2

[30]

For a fixed-tilt installation it follows in the same way, using Equation [11], that Q'~ ~ 310.6½ • 4 ~ 2550 B t u / f t 2

tion [20]. Assuming some self-evident practical conditions, the above relation can be worked out, and, multiplying it with the over-all turbine efficiency from Equation [5] one finds

[29']

7r

and accordingly

1 + _ Qh_ P ' ~ 31n w a t t / f t 2

~/total,J,b =lcarO 3/4

[30']

-

APPENDIX

77L~ = ar[1 -- u m] ntotal = ko~r[1 - AT~/3//O~]AT

In (1 - , )

v = u 3/4 = 5 T / O 3'~

(n+l

[19'] [21']

and

Topt.,bh =

0 . 5 7 0 ~/4

~opt..bh = 1.20" ~' karO 3/4

[22'"] [23"]

k ~ 6.6" 10-~ REFERENCES

n=-~m

1. Ashbel, 1), Solar radiation and soil temperature. Hebrew University Pr., 1942. (In Hebrew.) 2. Baum, V. A. ; Apar,asi, R. R. ; and Garf, B. A., "High-power solar installations. ' J. Solar Energy Sci. Eng. 1(1) : 6-12, Jan. 1957. Fig. 9. 3. Elliott, Rodger D., "Economics of steam-temperature selection in nuclear power plants." Nucleonics 10(2): 57-61, Feb. 1952. 4. Daniels, Farrington and Duffle, John A., Solar energy research. Madison, Univ. of Wisconsin Pr., 1955. p. 35. 5. Hottel, H. C., "Residential uses of solar energy." (In:

Reference to Fig. 3 shows, especially at values of u = 0.4, 40, as in Equation [25'], that m is so close to 3/4 that the results are hardly affected. The same is true for slight deviations from the power 4/3 in Equation [2]. On the other hand, it is possible to apply differential calculus to Equation [21] for every power of both factors AT. This means that the theory could be further evaluated for other forms of loss curves.

Proceedings of the World Symposium on Applied Solar Energy. SRI, 1956. p. 104.) 6. Kassander, A. Richard, "Fuel for solar furnaces." J. Solar Energy Sci. Eng. 1(2, 3) : 44-47, April-July 1957. 7. Villena, Leonardo, "Solar energy in Spain." J. Solar Energy Sci. Eng. 1(4): 32-34, Oct. 1957. 8. Morse, Roger N., "Solar water heaters." (In: Proceedings of the World Symposium on Applied Solar Energy. SRI,

V

If the sensible heat addition is to be included in the calculations, the composite collector efficiency nhb is given by Qh + Qb

[13"]

It can be seen that Equation [21"] leads to Equation [21], if Qh/Qo = 0. Only if Qh/Qb ~ 1 ( T ~ 600°F) will the results from Equation [21"] start to differ noticeably from those of Equation [21]. In that case, Qh/Qb = 1, and vopt. from Equation [21"] proves to be 0.57, so that both the optimum values for AT and ntot,~ are shifted towards

Differentiating this relation with respect to AT, the factor 4 in Equation [23] is replaced by

1956. p. 192.) 9. Tabor, H., "Solar energy collector design." Bull. Res. Counc. Israel 5C : 5-27, Nov. 1955. 10. Telkes, M., "Solar thermoelectric generators." J. App. Phys. 25: 765-777, June 1954. p. 770, Fig. 1. 11. Clark Jones, R., "The ultimate sensitivity of radiation detectors." J. Opt. Soc. Am. 3701): 886, Nov. 1947 and 39(5): 352-53, May 1949. 12. Roberts, H. E., "Trends in power generation." Nucleonics 16(7) : 76-79, July 1958. 13. Courvoisier, P., "The solar engine: an analysis." Mech. Eng. 79(5) : 445-47, May 1957.

[311

f dq_~ + Qb "qh

-

where

one would obtain, instead of Equation [21],

n~b -

+

IV

It might seem that the choice of the powers 4/3 and 3/4 in both Equations [2] and [19] (see Fig. 3) is somewhat artificial, as these values lead to the power unity for AT in Equations [20] and [21]. It can be shown, however, that another power in the neighborhood of unity for AT in Equations [20] and [21] gives rise only to minor differences in the results. If Equation [19] is rewritten as

APPENDIX

[21"]

Qb

"qb

as stated by Tabor (Equation 3 e ) ) Indices h and b refer to "heating" and "boiling" respectively, and ~h is dependent on AT, as in Equa-

50