Minimizing convective heat losses in flat plate solar collectors

Solur Ener#y Vol 25, pp. 521 526 ~) Pergamon Press Ltd, 1980. Printed in Great Britain

0038-092X/80q201-0521/$02.00/0

MINIMIZING CONVECTIVE HEAT LOSSES IN FLAT PLATE SOLAR COLLECTORS A.

MALHOTRA,

H. P. GARG and USHA RANI

Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 029, India

(Received 31 Jamlary 1980; accepted 1 July 1980) Abstract Based on available correlations relations are found for the local maxima's and minima's in heat transfer as the gap spacing is varied in flat plate solar collectors. These relations can shorten the task of selecting an optimum gap. A criterion is proposed for evaluating the use of alternate mediums in the enclosed space. It is shown that the use of heavy gases such as Argon can result in a 34 per cent reduction in heat losses. Nusselt number correlations of a single gap are extended to a two-cover system. It is found that by using two-covers there is an overall saving of more than 50 per cent in convection losses. It is also found that heat transfer rates in the laminar and turbulent regions are relatively insensitive to the internal spacing of the covers but reduces on changing from the mid-way position in the initial regime. A new type of two cover system is proposed in which the upper space is partially evacuated and it is shown that heat losses can be reduced by 85 per cent on a one-tenth reduction of pressure. Design relations for calculating cover spacings and heat transfer coefficients in this system are derived.

l. I N T R O D U C T I O N

2. C O R R E L A T I O N S

A knowledge of convective heat losses in flat plate solar collector systems is of interest to designers of solar equipment. An early review of literature on convection in enclosed spaces was carried out by T a b o r I l l . More recent studies such as those by O'Toole and Silvesten [2] and Hollands [3] provide engineering correlations of heat transfer through enclosed horizontal layers. Following Clever [-4] and Hollands et aL [5] these have been generalized for tilted layers, thus making their use suitable for flat plate collectors. In spite of the availability of such correlations no definite procedure exists yet for determining a suitable gap spacing to be used in flat plate collectors. Also there is a difference of opinion as to how these correlations should be used in order to evaluate mediums different from air or even air at sub-atmospheric pressures. In the present work the task of selecting gap spacings and evaluating alternate media's has been considerably simplified through location of gap spacings at certain critical conditions of heat losses. Further, correlations applicable for a single cover system are extended to a system with two glass covers. The effect of relative gap spacings and total gap is then analyzed for a two cover system. A new type of two cover system is discussed in which only the upper enclosed space is partially evacuated. It is shown how the individual gaps for this system can be designed to give the best performance. Some preliminary studies on a single cover collector and the effect of alternate mediums has already been discussed in an earlier publication [6]. s.~+256 ,

521

FOR NUSSELT NUMBER

Typically flat plate collectors consist of several feet of a black absorber. To reduce thermal losses one or more glass covers are placed at a height L above the absorber as illustrated in Figs. 1 and 2. Because of temperature difference between the absorber and cover plate heat losses c a n n o t be totally eliminated. The mode of heat transfer is conduction when Rayleigh number, R~t is less than the critical value of 1708. At other Rayleigh numbers convection takes place. A review of correlations quantifying heat transfer in such situations has been carried out by Buchberg et al. [7]. They recommend use of the following three region correlation: Nu = 1 + 1.446(1 - 1708/R~.); 1708 < R~. < 5900 (1) Nu = 0.229(Rfi)°25z' 5900 < R~ < 9.23 x 104

(2)

Nu = 0.157(Rfi)°285; 9.23 x 104 < R~. < 106.

(3)

Another set of correlations as formulated by O'Toole and Silveston is Nu = 0.00238(R~.)°s16; 1700 < R~t < 3500

(4)

Nu = 0.229(R~)°252; 3500 < R~. < l05

(5)

Nu = 0.104(Ra)°3°5(Pr)°°84; 105 < Rh

<

10 9.

(6)

The above correlations have been stated in a generalized form as applicable to tilted layers. Apart from the three region correlation the O'Toole and Silveston correlation is mentioned here because it is useful for fluids other than air when the effect of Prandtl

522

A. MALHOTRA,

H. P. GARG and USHARANI

:.,,,.. HOi'ioil Fig. I. Description of enclosed space.

pper cover Lover

cover

bsorber

\

~ ~/f/

~/~/'~/

~ 0., Tilt ~

Angle

Note

Horizontal

Fig. 2. Description of two cover system.

number has to be considered. Its completely Polynomial type representation is also of advantage in extending these correlations for a two cover system as described later. A difference between the two sets of correlation is that the basis of demarcating three regions in eqns (1)-(3) appears to be empirical whereas in the O'Toole and Silverton correlations it is based on flow mechanisms. For most calculations however the three region correlation should be preferred because of its greater accuracy and smooth matching between regions.

or in other words, until Rayleigh number reaches the value of 4038.87. The O'Toole and Silveston correlation on the other hand suggests that this decrease starts immediately upon initiation of Laminar convection at Ra = 3500. In Region IV of Fig. 3 the mode of heat transfer is turbulent convection. This region is represented by a very gradual decrease of conductance (or heat transfer coefficient) with gap spacing L(h~l/L°'~'*5). The earlier view as typified by the McAdam correlation [8] was that conductance remains constant in the turbulent region. At a certain point c on the curve the heat transfer rate falls below the previously attained minima at a. The heat transfer correlations can be used in a straight forward manner to obtain values of gap spacing and heat transfer coefficients at three points of interest a, b, c. These are summarized in Table 1. Point b of the three region correlation was located at a Rayleigh number of 4038.87 as already described. Point c is located by equating the heat transfer coefficient at that point as found from eqn (3) to the value of heat transfer coefficient at a where Nusselt number is equal to unity. Value of Lc was not determined from the O'Toole and Silveston correlation because of a discontinuity in this correlation at this point. Such a tabulation is useful in designing flat plate collectors. It remains for the designer to estimate the range of AT's over which useful heat collection takes place and then to calculate values of La, Lb, Lc over this range. A gap can then be selected so that the peak at b is avoided. Calculation of Lc is useful because it indicates a possible design point sufficiently removed from the peak and leading to a reasonably low value of h.

4. SELECTION O F ALTERNATE M E D I U M S 3. HEAT LOSSES IN A SINGLE COVER SYSTEM

The configuration for this system is as depicted in Fig. 1. The effect of gap spacing on the conductance h is illustrated in Fig. 3. A sharp decrease of conductance in Region I of Fig. 3 represents pure conduction characterized by Nu = 1. When Rayleigh number reaches a critical value of 1708 convection is initiated leading first to an increase in h. This is the initial regime of heat transfer shown as Region II in Fig. 3. This initial increase is followed by a decrease in the value of h, thus leading to a peak at b. The value of Ra number at this peak can be discovered in the following manner. Equation (1) can be rewritten as h-

C L

D L4

The use of air at lowered pressures or of alternate fluids to reduce convection losses in fiat plate collectors has been speculated before, An examination of Table 1 immediately suggests a criterion for the selection of an alternate fluid. It can be seen that the heat transfer coefficients at minima and maxima as well as at c are proportional to KF. Hence computation of this factor alone is an indication of relative merits of different fluids. For example the computed values of

_L[. _I. iiiia.=

I I

I I

~

)

I

I

(7)

I I

i

" ~

I

CONDUCTION

ff

[NIT AL

=

LAMINAR TURBULENT

where C = 2.446 and D=

2470/~2K

AToflp 2 cos B

differentiating eqn (6) with respect to L implies that h increases with gap space as long as

D > CL3/4

o

ai--t-T

. . . . .

I I

I

I I I I I I

I I I

GAP

~

-

SPACING) L

Fig. 3. Schematic depicting effect of gap spacing on conductance.

Minimizing convective heat losses

523

Table 1. Location of critical gap spaces on the heat conductance ha THREE REGION .08L,KF CORRELATION O TOOLE AND SILVESTON

•084 KF

ko

hb

Lb

hc

Lc

11.95/F .115KF

15.92lF .08/, KF

I1.93/F .120KF

15.18/F 0.8/, KF

76.67/F

CORRELATION

K F for some common gases at AT = 5 0 K and a mean temperature of 350 K are H Air CO2 Ar

75.05 39.31 41.72 27.55.

treated in a later section by methods similar to that of this section.

5. HEAT LOSSES IN A TWO COVER SYSTEM

On the basis of the preceding calculation it can be seen that use of light gases such as hydrogen will result in a 91 per cent increase in heat transfer whereas heavy gases such as argon can cut down heat losses by up to 34 per cent. It may be mentioned that a comparison on the present basis is a more valid practice as compared to a previous study by Cobble [8] in which heat transfer coefficients are compared at fixed gap spaces. This is because length scales involved in different fluids are different and these should be accounted for as in the present work. The importance of this can be realized by looking at Fig. 3. Suppose air and CO2 were compared at a Rayleigh number near the peak in the CO2 curve it will appear that this fluid gives a much poorer performance, a result that is misleading. The present method overcomes the problem of deciding at what Rayleigh numbers or gap spaces different fluids should be compared. The complete variation of h with gap-space L is illustrated in Fig. 4 drawn on the basis of the three region correlation. The influence of Prandtl number on different gases is neglected here since as suggested by the O'Toole and Silveston correlation this is expected to be within 1 per cent. Figure 4 is drawn for tilt angle B = 0 and AT = 50°C but effect of other tilt angles and temperature differences can be seen from Figs. 5 and 6 at three locations a, b, c of Fig. 3. Influence of pressure reduction on heat transfer is

The configuration for this system is as depicted in Fig. 2. Correlations for a two cover system based on total spacing L and overall temperature difference AT can be obtained from correlations already described in the following manner. The convective heat loss q can be expressed variously as: q = h A T = h l A T l = h2AT2. Now assuming that expressed as:

(8)

the nusselt number

Nu = C(Ra)"

can be (9)

eqns (8) and (9) can be combined and after some algebraic simplification yield:

AT,

-

h~

\AT2/

-

(10)

Here r is the non-dimensionalized spacing of the lower glass cover, L1/L. Equation (10) can be rewritten as: AT2

__ ( ~ ) ( 1 -

3n)/(l +n)

AT~

= P'

say.

(11)

Assumptions embodied in eqn (11) are that all properties are evaluated at the overall mean temperature and that mechanisms of heat transfer are similar on both sides of .the center cover. Our estimate of error incurred because of the first assumption is less than 5 per cent for a AT of 100°C and smaller at lower values of AT. The validity of the second

10 =o

% 8 .d

6

z

/.

o

z

Hydrogen

_o

0.9Corbon Dioxide

Z (..)

2

~ A r g o n I

I

I

I

2

/-.

6

8

GAP

10

SPACING,, L cm.

Fig. 4. Conductance variation with gap spacing for different gases,

0.80.7-

o

0.6--

8

o.s

Z

0

1-0

o

I 20

I ~,0

TILT ANGLE.,B

I 60

(deg)

Fig. 5. Influence of tilt angle on conductance expressed as a ratio to the conductance when B = 0.

A. MALHOTRA,H. P. GARGand USHA RANI

524 (:3

DJ (.9 Z

O Z O

1.1

1.0- - ~ 0.9 0.8 0.7 0.6 10

I

I

I

30

50

70

TEMPERATURE

9

DIFFERENCE /',T°C

Fig. 6. Influence of AT on conductance expressed as a ratio to the conductance when AT = 50cC.

assumption, that is similar heat transfer mechanism, can only be ensured when r = 0.5. The likelihood of violating this assumption increases as the deviation of r increases from the central value of 0.5. Further noting that AT = AT~ + AT2 and making use o f e q n (11) both AT1 and ATE can be expressed as a function of P and AT. This makes it possible to express the value of h in terms of these latter quantities through the use of eqn (8). In non-dimensionalised form the result is / 1 Nu = C ( R a ) " / - - ]

V+"/ 1 \ /--/

\1 + P /

(12)

\r 1-a./"

Complete algebraic details of this derivation are not recorded here but can easily be recovered from the indicated procedure. A task that remains is to select a polynomial type of correlation for use in the previous derivation. Equations (2) and (3) of the three region correlation are already polynomials and eqn (1) can easily be replaced by the O'Toole and Silveston equation (4). There is no significant error (less than 3 per cent) in this practice. The values of n and c for use in eqn (12) are indicated in Table 2. The indicated Rayleigh number ranges are designated to provide a smooth matching of Nusselt number between various regions. They are 16 times as high as for a single cover in order to account for the fact that actual Rayleigh number governing heat transfer is 1/16th of the overall value (for r = 0.5).

that the laminar and turbulent Nusselt numbers increase on displacing the central cover away from the mid-position but the initial Nusselt number decreases. Further the laminar and turbulent modes of heat transfer are relatively insensitive to this factor but a saving in heat loss can be achieved in the initial regime by displacing the center glass cover towards the upper or lower plate. Calculations for Table 3 are made in the narrow range of r > 0.3 because as already mentioned eqn (12) applies only for similar heat transfer modes in both spaces, a situation which will rarely occur at extreme values of r. Calculations for differing modes of heat transfer are rather complex and not reported here because of their limited usefulness. Our preliminary calculations indicate that there is no significant advantage in enforcing such a situation. The obvious value of r = 0.5 can be used in solar collectors unless it can be established that most heat collection takes place in the range 27328 < Ra < 52544 required for the initial regime.

The heat conductance curve The effect of gap spacing on conductance h is compared with that of a single cover in Fig. 7. The minima and maxima in heat transfer occur at larger gap spaces in case of a two cover system. This is because convection now takes places over smaller gaps and over smaller individual temperature differences. There is an overall saving of more than 50 per cent in convective heat losses with two covers. As with a single cover system heat losses can be reduced if the gap spacing is increased sufficiently. However it must be remembered that gap spacing cannot be increased indiscriminately because of increased costs and shading effects. Shading influences are considered in detail by Nahar and Garg [10].

Table 3. Influence of internal spacing, r on Nusselt number expressed as a ratio to the Nusselt number when r = 0.5 Nu/Nur = 0-5 r

INITIAL

LAMINAR

TURBULENT

Internal spacing of covers

0.3

0.786

1.018

1.013

The internal spacing of covers is characterized by the quantity r. Its effect on increase or decrease of heat transfer, can be determined from eqn (12). Some calculated values are given in Table 3. It can be seen

0.4

0.952

1.004

1.004

0.45

0.990

1.002

1.002

0.5

1.000

1.000

1.000

Table 2. Constants for eqn (12) c

n

.00238

0.816

27328 <~Ra ( 52544

LAMINAR

.229

0.252

52544 ( Ro < lZ.77xtO S

TURBULENT

.157

0.287

14.77x 10S
INITIAL

Rayleigh number rQnge for r = 0.5

< 16x106

Minimizing convective heat losses

oO 6 ~E

..... - BOTH LINES

.g

525

~" ~[.u 2.0 S'

SINGLE COVER TWO COVERS

._T

h ...... k1 1.5 _-- . . . . . . L2

3

z

/

2 o z o

o 0

I

I

I

i

I

2

,~

6

B

10

1

/

/../"~

/./ '

~

0.5

oL'-' o.o

GAP SPACING,L. cm

~

/.

Fig. 7. Conductance variation with gap spacing for a two cover and a single cover system.

I

I

I

I

J

2

3 "

5

G 7

I

I

I

B 9 11)

PRESSURE RATIO Pot m/Pt

Fig. 8. Effect of pressure ratio on conductance and gap spacings. 6. PARTIALLY EVACUATED COLLECTORS Heat losses can be cut down by reducing the pressure of the enclosed air in flat plate and tubular collectors [11]. The heat transfer coefficient under the most influential conditions is proportional to the quantity K F as discussed earlier. Since density is proportional to pressure other properties remaining unaffected, the heat transfer coefficient of a single cover system will change in the ratio (P/Patm)2/3 on partial evacuation. However the fabrication of such a collector will probably be difficult, A system which appears to be more attractive from a practical point of view is one with two transparent covers in which only the upper enclosed space is evacuated. New systems of this type can be constructed or existing systems converted with the availability of hollow transparent cover units; the upper and lower surfaces of this unit acting as the two covers. Such collectors can be designed in a way that for a certain temperature difference AT both the air enclosures are kept at conditions corresponding to point a of Fig. 3. This point is selected because otherwise in this case the total gap may become unduly large. For this collector system the heat transfer coefficient under reduced pressures can be found from the following relation. h hatm

-

1 [1

q'-

(Patm/p)l/2] 4/3

These results are illustrated in Fig. 8. It can be seen that the upper evacuated gap has to be increased faster than the lower gap. In actual practice A T does not remain constant but the most judicious practice still appears to be to design such systems for a chosen AT and consider the variations as a necessary deviations from an optimum. 7.

CONCLUDING REMARKS

Convective heat losses in several conceivable situations of flat plate solar collectors were considered in this work. Cumbersome algebraic details were either left out or included in the appendix in order to make the presentation smoother. Complications in treatment such as effect of property variations have also not been included in the treatment so as not to distract from the main thrust. The present study will help in calculating heat losses and gap spaces for a single and double cover system. The study would be very useful for deducing the effects of alternate mediums and in designing partially evacuated systems. It is hoped that the present work will provide helpful guidelines to flat plate collector designers concerned with heat losses in their systems.

(13)

NOMENCLATURE where hatm is the heat transfer coefficient for a single cover system filled with air at atmospheric pressure at the same AT. Derivation of eqn (13) and the following eqns (14) and (15) are provided in the Appendix. A reduction in pressure must be accompanied by a change in the total gap as well as the internal spacing in order to prevent heat transfer from entering into the conduction regime. These can be found from the following relations: L Zatm

--

1 24/3 [1 + (Patm/p)l/2] 4/3

(14)

and r = 1/[1 + (e, tm/P)l/2].

(15)

B F g h K L Nu P Pr q r R,a p fl /~ AT

tilt angle defined as [PrATgflp 2 cos B///2] 1'3 acceleration due to gravity conductance or heat transfer coefficient thermal conductivity gap spacing between the top-most cover and absorber plate Nusselt number, hL/K pressure Prandtl number heat flux internal spacing of covers, L1/L Rayleigh number generalized for tilted layers A TgflL 3p 2 P r cos//12 density coefficient of thermal expansion dynamic viscosity temperature differences between top-most cover and absorber plate

526

A. MALHOTRA, H. P. GARG and USHA RANI

Subscripts atm value when enclosed air is at atmospheric pressure a,b,c value at corresponding point of Fig. 3 1, 2 value for the upper and lower enclosed gaps respectively in a two cover system

or,

AT2 _ ( p , ~ 2 A~I

(A2)

\P2/

and hence AT REFERENCES

l. H. Tabor, Radiation, convection and conduction coefficients in solar collectors. Bull. Res. Conc. of Israel. 6C, 155-176 (1958). 2. J. L. O'Toole and P. L. Silveston, Correlations of convective heat transfer in confined horizontal layers. Chem. Engn 9 Pro O. Syrup. Ser. 57(32), 81-86 (1961). 3. K. G. T. Hollands, Convectional heat transport between rigid horizontal boundaries after instability. Phys. Fluids 8, 389-390 (1965). 4. R. M. Clever, Finite amplitude longitudinal convection rolls in an inclined layer. J. Heat Transfer, Trans ASME, Set. C 95(3), 40%408 (1973). 5. K. G. T. Hollands, T. E. Unny, G. D. Raithby and L. Konicek, Free convective heat transfer across inclined air layers. J. Heat Transfer, Trans ASME, Set. C 98(2), 189-193 (1976). 6. A. Malhotra, H. P. Garg and Usha Rani, The effect of gap-spacing on convective losses in flat-plate collectors. Proc. Natl Solar Eneroy Cony. 1979, Bombay, 13-15 Dec. 1979. 7. H. Buchberg, I. Carton and D. K. Edwards, Natural convection in enclosed spaces, a re-view of application to solar energy collection. J. Heat Transfer, Trans. A S M E 98(2), 182-188 (1976). 8. W. H. M. C. Adams, Heat Transmission, 3rd Edn. McGraw-Hill, New York (1954). 9. M. M. Cobble, Minimizing convection losses in solar collectors. Proc. Conf. Physics of Solar Eneroy, Benghazi, Arab Development Institute Publication, Tripoli, pp. 137-140 (1977). 10. N. M. Nahar and H. P. Garg, Free convection and shading due to gap spacing between absorber plate and cover glazing in solar energy flat-plate collectors. Appl. Eneroy To be published (1980). 11. K. L. Moan, Evaluation of an all-glass, enacuated, tubular, non-focussing, nontracking solar collector array. 1st Ann. Prog. Rep. 1 July 1976-31 Auo. 1977, pp. 84 Owens Illinois Inc., Toledo (1977).

The overall heat transfer coefficient h is h-

h 1h 2

hi + h2

APPENDIX

0.084KF1 - 0.084KF2

0.084KF1F 2

F1 + F2

(A4)

hal m = 0.084KF.

(A5)

Dividing eqn (A4) by (A5) h

1

hatm

F/F 2 + F/FI 1

(A6)

m

Substituting results of eqns (A3) in (A6) and simplifying: h

1

hat m

[1 -~- ( p l / P 2 ) l / 2 ] 4/3'

(A7)

The total gap L is I 1.95

I

1.95

L m El -~- L2 = r~r~ + 2KF--

(AS)

At atmospheric pressure LI = Lz and AT 1 = ATa = AT~2, 2 x 11.95 Lat m = 2 L 1

-

(A9)

KF

Dividing eqn (AS) by (A9) and simplifying L

1

L~tm

2.52

[ l + (pl/P2)l/2] 4:3

(AI0)

also

(ATIp2) 1/3

\Pl/

(All)

therefore,

For a two cover partially evacuated system and designed to operate at conditions corresponding to point 'a' of Fig. 3, -

-

For a single cover system operating at atmospheric pressure

L2 - F1

hi AT2 hE -- AT1

AT -= 1 + (p2/pl) 1'2. (A3) 6"/'2

-- 1 + (pl/p2)l'2;

AT1

(ATIp~) 1/3 (AT2R2) 1/3

(A1)

L1 1 r - L1 + L~ - [1 + (pl/p2) 1/2] "

(AI2)

Noting that P,/P2 is equal to Patm/P eqns (A7), (A10) and (All) can directly be written as eqns (13)-(15) of the main text.