Non-convective roof pond for passive space heating

Applied Energy 12 (1982) 21-36

NON-CONVECTIVE

ROOF POND HEATING

FOR PASSIVE SPACE

N. D. KAUSHIK and S. K. RAO

Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi--llO 016 (India)

SUMMARY

This paper reports on an investigation of a passive system comprising a nonconvective pond placed on the roof of a building in order to heat the building. Periodic analysis of the solar heat transfer process in the system, which is exposed to solar radiation and atmospheric temperature on one side and is in contact with room air at constant temperature (corresponding to an air-conditioned room) on the other, indicates that this system provides better thermal storage than a thick concrete roof or the convective roof pond system. In sunny winter climates, such a system can provide 100 % of all heating needs. A viscosity stabilised non-convective pond of shallow depth (lO-15cm) is envisaged to be the most suitable for the present application.

NOMENCLATURE

C Cs p Ps p,, ho h2 h3

Specific heat of the water (J/kg °C). Specific heat of the concrete roof (J/kg °C). Density of the non-convective water (kg/m3). Density of the concrete roof (kg/m3). Heat capacity per unit volume of the convective zone water (J/m 3 °C). Heat transfer coefficient between the surface of the pond and the ambient air (W/m 2 °C). Heat transfer coefficient between the non-convective zone water and the convective zone water (W/m 2 °C). Heat transfer coefficient between the convective zone water and the concrete roof (W/m: °C). 21

Applied Energy 0306-2619/82/0012-0021/$02.75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain

22

h4 K ll lz 13

7

O(t) x t

So S(x, t) Tl(x, t) T~(x, t) TA(t) T~o To~ Tw(t) Two T~,m nj

pAT)

N . D . KAUSHIK, S. K. RAO

Heat transfer coefficient between the concrete roof and the room air at constant temperature (W/m 2 °C). Thermal conductivity of the water (W/m °C). Thermal conductivity of the concrete roof (W/m °C). Depth of the non-convective zone (m). Depth of the convective zone (m). Thickness of the concrete roof (m). Absorptivity of the blackened surface. Humidity of the atmosphere. Heat flux entering the room at a time, t (W/m2). Position co-ordinate (vertically downwards) (m). Time co-ordinate (s). Average value of solar intensity at x = 0 (W/m2). Amplitude of the ruth harmonic of S(x, t) at x = 0 (W/m2). Insolation at a point specified by x and at a time, t (W/mZ). Temperature distribution in the non-convective water (°C). Temperature distribution in the roof slab (°C). Ambient air temperature at time, t (°C). Average value of the ambient air temperature (°C). Amplitude of the ruth harmonic of TA(t) (°C). Convective water temperature (°C). Average water temperature (°C). Amplitude of mth harmonic of Tw(t) (°C). Absorption coefficient (m-1). Fraction of the solar intensity having absorption coefficient, n~. Saturated vapour pressure at a temperature, T.

INTRODUCTION

Passive systems offer simple and inexpensive means for heating/cooling buildings. One such passive system for heating buildings relies on the 'skytherm' process, first demonstrated by Hay and Yellot. 2 In this process, the roof of the building stores solar heat in water bed-like containers or in roof top ponds covering most or all of the ceiling. The solar heated water conducts its warmth through the ceiling and heats the building. During off-sunshine hours, the warm water is covered with an insulation to reduce heat losses. The routine opening and closing of the cover is rather a liability to the user. In this paper, we investigate whether it is possible to passively heat a building by an open, non-convective pond placed on the roof instead of the convective type pond incorporating the movable insulating cover. This arrangement is obviously free from any liability of routine mechanical operation. Previously, salt gradient non-convective ponds were proposed for space

NON-CONVECTIVE ROOF POND FOR PASSIVE SPACE HEATING

23

heating but with different configurations (Rabl and Nielsen, 8 Nielsen7). Nonconvective ponds using viscosity stabilisation appear to be more suitable for the proposed system.

THERMAL MODEL

A non-convective pond normally consists of a convective zone at the surface, a nonconvective zone and a second convective zone at the bottom. The upper convective zone is caused by the variations in surface heat input and losses, as well as by windgenerated surface currents and waves; the bottom convective zone is characterised by the time variation of solar intensity absorbed by the pond floor. In our experiments with shallow non-convective ponds (Kaushik and Bansal, 4 Kaushik and RaoS), it was observed that the surface convective zone is rather thin and obscure and hence, for the purpose of thermal modelling, we have considered it insignificant. In view of this, the non-convective roof pond system is as depicted schematically in Fig. 1. Part of the incident solar radiation is reflected at the pond's surface, the remainder entering the pond being attenuated throughout its depth and being almost completely absorbed at the black bottom. The temperature and heat flux distribution in the non-convective and concrete zones are governed by the Fourier heat conduction equation and the appropriate boundary conditions. The convective zone temperature distribution, T,,(t) is governed by energy balance considerations.

SOLAR RADIATION & ATMOSPHERIC AIR

!

I'Y,'::iI =o cO.VEC, E

x"

FT/-/~ / / / / 7/-//I

~

-~.,2

V/ / / / / / /

,.~,.,~.,~

TR ROOM TEMPERATURE

Fig. 1.

Schematic diagram of the non-convective roof pond system.

24

N. D. KAUSHIK, S. K. RAO

[A] Non-convective zone The heat conduction equation is:

K

O2Tl(x,t) OTl(X ,t) dS(x,t) Ox 2 = p C Ot d ax

(1)

The boundary conditions are as follows. At x = 0

K OTt(x't) x=O =h'l[T'a(t)- T,(x=O,t)]

(2)

(see details in Appendix 1). At x = l~

K OTI(X't) I ~X

=h2[Tl(x = l l , t ) _ Tw(t)]

(3)

x=lt

[B] Concrete slab zone The heat conduction equation is:

Ks

d2Tz(x, t) OTz(x, t) ~x 2 - psCs ~t

(4)

The boundary conditions are as follows. At x = l~ + 12 -Ks

O T 2~( x , t) x --h +~2 =

ctS(x = 11 + 12, t) + h3[T.,(t ) - T2(x = 11 + 12, t)]

(5)

At x = 11 + 12 + 13 A

_ K s OT2(x,t)] ~X

Ix=tt +t2+t3

=h4[T2(x = l I + l 2 + l 3 , t ) - TR]

(6)

[C] Convective zone The heat flux and temperature in this zone are governed by the heat balance equation: dTw(t) ~Tt(x, t) I ~T2(x, t)] (7) P J2 dt K ~x x=. + K s Ox x=l~ +t~ + S ( x = 11, t) The nature of the attenuation of solar radiation in the pond water has been expressed as a summation of five terms by Kaushik et al. 3 The observed daily variation of solar intensity and ambient air temperature may be represented by a Fourier series of six terms (Kaushik et al.3). Thus: 6

TA(t) = Tao + ) ' rail

T~,.e i"'a

(8)

NON-CONVECTIVE ROOF POND FOR PASSIVE SPACE HEATING

25

and" 6

5

m=l

j=l

or 6

q-

5

X Someirn~a]E~ ] l j - nei x m=l

1

(9)

j=l

where:

Soo = zS o t

So,, = zSom (r = coefficient o f t r a n s m i s s i o n at the surface = 1 - r e f l e c t i v e losses) a n d : Pl = 0 . 2 3 7

n 1 = 0 . 0 3 2 m -1

P2 = 0 . 1 9 3

n z = 0 . 4 5 0 m -~

p3 = 0 . 1 6 7

n3 = 3 . 0 0 0 m -1

P4 = 0 . 1 7 9

n4 = 35-000 m -1

Ps = 0.224

n 5 = 255.0 m -

Because the a m b i e n t air t e m p e r a t u r e a n d solar i n t e n s i t y are a s s u m e d to be periodic, Tl(x, t), T2(x, t), T'A(t) a n d Tw(t ) will also be p e r i o d i c in n a t u r e . So we have: 5

Tl(x,t) = Ax + B -

Soo~e-"Jx njK

)' j=l

6

5

V S°'P~nje-"':) e '"°~ A"e~'~ + B"e-a'x +/_...a K(fl 2 - n}) J

+ ra=j

(10)

j=l 6

T2(x, t) =

C'x + D + ) '

(C,.e "x + D,.e

(1 1)

m=l 6

' T'a(t ) = TAo +

~

~

n,l= !

, iratot T'.,e

(12(a))

26

N.D.

K A U S H I K , S. K. R A O

and: 6

T~(t) = Two +

Twine ira~t

(12(b)

m=l

where: = (imo~psCs) 1/2

T'ao-

TAo(ho + heCxY) C2he + ho + hlC1 ho + heC 1

and: Tam - T~,.(ho + h~C17) ho + heCl

The heat flux entering the room is'

Q_(t) = - K s

T2(x, t) I~=,,+,~+,3 O~ 6

~.,(C., e.,.,~ +~ + t~). D., e-.~., +t~ + ~)) e~,.o,,t _ TR1 m=l

(13) Equations (10), (11), (12) and (13) specify the temperature and heat flux distribution in the system. Constants involved in these equations may be obtained by substituting the values of T~(x, t), T2(x, t), Tw(t ), etc. in the boundary conditions given by eqns (2), (3), (5), (6) and (7) and considering the time-independent and time-dependent parts separately. This yields two sets of linear simultaneous equations, illustrated by the matrix equations (eqns (14) and (15)), respectively: [X1][C'I] = [Y1]

(14)

[X2][C2] = [Y2]

(15)

where"

[Xl] =

-K -(K+h211) 0

h'1 0 -h 2 0 0 {h3(ll + 12)- Ks} k-4

-K

0

Ks

0 0 h3

0 h2 -h3

0

0

J

NON-CONVECTIVE ROOF POND FOR PASSIVE SPACE HEATING

27

- A -7 B

i

C'

[c'~] =

D

LTwol

(

SooPj 1+

[Y1] =

h' x

-

+ h'l"T'ao

~SooIJ j e - n~l~ + 12)

rR 0 (h', + K / L )

i (h'l (Kflm-- +Kflm) h2)

e 3mh

0 0

[X2] =

(Kflm -

o

h2)e -/~'/'

0 {(h3 _ Ksam ) e~m., +l~)} { - (h 4 + ~ K s ) e ~mu,+l~+l~)} a = K s e~,,,(tl + 12)

0 0

Kflm

em'

Kfl,. e - m,

01

0

h2

0 I(h3 + K s a m ) e -~-", ÷t~} i( Ks % _ h 4 ) e - ~ u , +t~ +t~)} --

-A,. B,. [c~] =

Cm Dm T,,,m h'l T',,,,,- So,,,lajnj ( ~ + nj) 1 (~SomJJ j e - nflll +/2}

[Y2] =

0 -

a,,,.I~j

c

1 + (3~

KS~ m

e

- ~ m ( l l + 12) -

imo~pJ2]

28

N. D. KAUSHIK, S. K. RAO N U M E R I C A L COMPUTATIONS AND RESULTS

Numerical computations have been made for the heat flux Q(t) entering the room, and the convective water temperature, Tw(t), using eqns (12) and (13), respectively. The constants involved in the expressions were obtained by solving the matrix equations (eqns (14) and (15)). The heat transfer coefficients embodied in the above expressions were obtained using the theory of free and forced convection, radiation and evaporation (McAdams, 6 Carrier1). The appropriate equations are given in Appendix 2 wherein we note that the temperature differences (AT) at various interfaces in the system are required for the calculation of the heat transfer coefficients (hr, h 2 and h 3 ) . Initially, these AT values were guessed at and utilised for the calculation of heat transfer coefficients as well as the temperature differences at various interfaces of the system, using eqns (10), (11) and (12). The calculated values of A Twere compared with their assumed values and the difference was minimised by successive iterations. The resulting heat transfer coefficient values used in the present calculations are: h o = 10.89 W/m 2 °C h 2 = 48"28 W/m 2 °C h e = 0-273 W / m

2

°C

h3 =

78"92 W/m 2 °C

h 4 = 6-07 W/m E °C The values of S o, TAo, So,. and Tam in the above calculations were obtained from the observed daily variations of solar intensity and ambient air temperature at New Delhi on the 12th of January, 1974 (a typical winter's day). The harmonic analysis of the hourly data was carried out and the resultant Fourier coefficients are given in Tables 1 and 2. The amplitudes of the harmonics of the solar intensity and ambient air temperature are given by: S'o,, =A',,e -'a'~ (16) Ta". = B~,e -i*m

(17)

TABLE 1 HOURLY FOURIER COEFFICIENTS OF THE SOLAR INTENSITY OF THE 12th OF JANUARY, 1974 AT NEW DELHI rh A " ( W / m 2) a= (radians)

0

1

2

3

4

5

6

160.8336 --

272.1878 3.2684

156.9849 0.2583

46.1790 3.5556

11.3388 3.5440

16.7296 0.6550

2.9876 4.4546

TABLE 2 HOURLY FOURIER COEFFICIENTS OF THE AMBIENT AIR TEMPERATURE OF THE 12th OF JANUARY, 1974 AT NEW DELHI th B" ~= (radians)

0

I

2

3

4

5

6

16.9208 --

6.6114 4.3438

1.3666 1.0355

0-4654 5.0553

1.1417 5.6625

0.2710 0.8052

0.2490 4-8630

NON-CONVECTIVE

ROOF POND

29

FOR PASSIVE SPACE HEATING

13=0.15m Iz =O.01m

I_

11= 0.20m

o

0-06m am

0.0L.m 0.02m "I"

I

18 Locol meon time

I

~24 0m

[a)

"~

60

13:O.20m

t

12:0'01m

E ,~0

~

~

o° _

-20

O.02m

~'-~.

s//

/~

~o~

Ib)

Fig. 2. Daily variations of the heat flux entering the room for different depths of the non-convective zone, corresponding to (a) l z = 0.01 m and 13 = 0.15 m and (b) 11 = 0-01 m and 13 = 0.20 m.

30

N . D . KAUSHIK, S. K. RAO

The values of the other thermophysical parameters of the system are as follows: K = 0-569 W / m °C

Ks =

p = 1000'0 kg/m 3

Ps

= 1853 kg/m 3

C = 4190.0 J/kg °C

Cs

= 800'0 J/kg

Pw = 4190000.0 J/m 3 °C

0-72 W / m °C

°C

= 0.94

=0"9 7 =0-6 11 = 0.0 to 0.40 m

13 = O . 1 5 m and 0 . 2 0 m

/ 2 = 0.0 to 0-40 m

TR = 2 1 ° C

The passive heating performance o f the system m a y be evaluated in terms o f the heat flux entering the r o o m , which is assumed to be at constant temperature (corresponding to an air-conditioned room). Figures 2(a) and (b) exhibit the hourly -o---o---o-

6°I

0.15m

x

x

x

0.20m

•

,;

:

0.25m

12 =0.01

~: &8-E o

36I

c

2

Qmin.

-

,o

0.08

0.12

Depth of non-convective

0.16 zone (rn)

-2L,~

Fig. 3.

Variations of Q .... Q=~, and Q with the depth of the non-convective zone corresponding to fixed values of / 2 (0.01 m) and 13 (0.15m).

31

NON-CONVECTIVE ROOF POND FOR PASSIVE SPACE HEATING 60

13:0,15m IT:Ore ~0

12:0.04m :

20

t )i Local m ( mean a

•

-20

E

13=0A5m

40

/////

\ \

////

20

\\

\

\

\ 0.0s

\ \00,

x 2 12

18 Locat mean

(b)

_201 ~.~[1:0 80 -

24

time

m L1 2 = ° m

--50 ~

~o

~20

40 -

-40 --

~

-

._~ .fl

-

Fig. 4. Variations of the heat flux entering the room. (a) For different depths of the convective zone corresponding to fixed values of l 1 = 0 and l 3 = 0.15 m. (b) For different depths of the non-convective zone corresponding to fixed values of 12 = 0 and l 3 = 0.15 m. (e) For different thicknesses o f the r o o f corresponding to fixed values of It = 0 and 13 = 0.

32

N. D. KAUSHIK, S. K. RAO

variations of the heat flux, Q(t), entering the room for different depths of nonconvective zone. The day under consideration is a typical winter's day (the 12th of January, 1974) in New Delhi. It is seen that, in the absence of a non-convective zone (l I = 0), the heat flux is negative from 00.00 h to 10.00 h, after which it is positive for all remaining hours. The heat flux, Q(t), increases with increase in the depth of the non-convective zone (ll) and becomes positive for all hours for l 1 > 0.06m. A systematic variation of Q . . . . 0 a n d 0min is shown in Fig. 3. This indicates that the system with a non-convective zone of about 0.06 m can provide 100 % of all heating needs and the system with a non-convective zone of larger depth may even meet the total heating needs in harsher winter climates.

56-

12--0.01m

/-.8

40

% 32

g

24

16

,

I

0.18

,

I

0.32

,

I

0.l,8

,

I

0.64

,

I

0.80

Depth of non-convective zone 12 (rn)

Fig. 5.

Variations of Q with the depth of the non-convective zone corresponding to a fixed value of 12 (0.01 m).

33

NON-CONVECTIVE ROOF POND FOR PASSIVE SPACE HEATING

In Figs 4(a), (b) and (c), we illustrate the Q(t) variations caused by the variations of the depth of the convective zone ('/2), the thickness of the roof slab ('/3) and the depth of non-convective zone ('/1), respectively. The increase in '/2 and '/3 improves the load levelling in the heat flux and an increase in '/1 enhances the heat flux and causes passive heating. The depth of the non-convective zone ('/1) governs the solar thermal collection/ storage characteristics of the system. We have, therefore, investigated the optimum size of/1 . Figure 5 illustrates the variation of the average heat flux entering the room with the depth of the non-convective zone. The heat flux increases rather rapidly with / 1 for small values of/1 up to about 0.20 m and tends to steadily reach a limiting value for higher values of l 1.

PRACTICAL REALISATION OF THE SYSTEM

The essential element of the system is the non-convective water pond. A nonconvective pond is capable of collecting and storing solar heat and is often referred to as a solar pond (Tabor, 11 Weinberger14). The more conventional nonconvective pond is the salt gradient stabilised pond. In such ponds, due to the existence of a salt density gradient, there always exists a molecular diffusion that tends to erase the initially formed salt gradient. These ponds therefore require routine repairs of their salt gradients; this is quite a cumbersome job. We therefore consider the salt gradient stabilised non-convective ponds as unsuitable for the present application. Another type of non-convective pond that appears to hold promise for present applications is the viscosity-stabilised pond (Shaffer9). In such a pond, water is made non-convective by the addition of thickeners rather than by a density gradient. Thickeners could be polymer oil/water gels. Possible kinds of

"~

50~-/ /

12 =O.01m 13 =O.15m

40 30

Ii= O.O?m O.05m O.03m O.01m

20 1

0

I

6

I

12

I

18

I

21.

Local mean time Fig. 6.

Daily variations of the convective zone water temperature corresponding to fixed values of / z (0.01 m) and Ia (0-15m).

34

N . D . KAUSHIK, S. K. RAO

thickeners and their behaviour at various pond temperatures are still a subject for detailed investigation. However, it is interesting to note that the pond bottom temperatures involved in the proposed system are rather low (25 °-45 °C). This is illustrated in Fig. 6. Our recent experiments (Kaushik and Rao 4) with several thickeners have indicated that, for such a low range of temperature, poly(vinyl) alcohol and carboxy methyl cellulose are quite appropriate thickeners and that a concentration of 1.5 to 2 ~o serves the required purpose.

ACKNOWLEDGEMENTS The authors are grateful to Professor M.S. Sodha, Mr P.K. Bansal and Mr Subhash Chandra for their co-operation and help.

REFERENCES 1. W. H. CARRIER,The temperature of evaporation, Trans. Am. Soc. Heating and Ventilating Engineers, 24 (1968), pp. 24--50. 2. H. R. HAY and J. I. YELLOT, Natural air conditioning with roof pond and movable insulation. ASHRAE Transactions, 75 (1969), p. 178. 3. N. D. KAUSHIK, P. K. BANSALand M. S. SODr~^, Partitioned solar pond collector/storage system, Applied Energy, 7(2) (1980), p. 169. 4. N. D. KAUSHIK and P. K. BANSAL,Transient behaviour of salt gradient stabilized shallow solar pond. Applied Energy, 10 (1982), pp.47-63. 5. N. D. KAUSHIKand S. K. RAO, Thermal performance of viscosity stabilized solar pond (work in progress). 6. W. H. MCADAMS, Heat transmission, McGraw-Hill, New York, 1954. 7. C. E. NIELSEN,Salinity gradient solarpondsfor passive storage, National Passive Solar Conference, October, 1979. 8. A. RAaL and C. E. NIELSEN, Solar Ponds for space heating, Solar Energy, 17(1) (1975), pp. 1-12. 9. L. H. SnAr'l~R, Viscosity stabilized solar ponds. Phase I, CEM-4200-572, Centre for Environment and Management Inc., 275, Windsor Street, Hartford, CT 06120, USA. 10. M. S. SODHA, N. D. KAUSmK and S. K. RAO, Thermal analysis of three zone solar pond, Int. J. Energy Research (1982) (in press). 11. H. TABOR, Solar ponds, Solar energy, 7(4) (1963), pp. 189-94. 12. J. L. THRELKELD, Thermal environmental engineering, Prentice Hall Inc., 1970. 13. A. VoN KALECSlrqSKY, Uber die ungarischen warmen and heissen Kochsalzseen als naturliche warme--Accumulation Annalen der physik, 4 (1902), p. 409. 14. H. WEXrqBERGER,Physics of solar ponds, Solar Energy, 8(2) (1964), pp. 45-56.

APPENDIX 1

The energy balance equation at the surface is:

K OTx(x, ~ t) x=o = h° [Ta(t) -- TI(X = 0, t)] - rnL

(18)

NON-CONVECTIVE ROOF POND FOR PASSIVE SPACE HEATING

35

where: ho=h,+h

(19)

c

The heat loss, n~, due to evaporation from the pond's surface may be given by the following equation: rnL = h~[ps (Ta) - Yps(TA(t))]

(20)

From the steam tables for the temperature range of interest (15 °C < T 1 < 50 °C): P s ( T ) = 2 9 3 . 3 T - 3911.505

(21)

Therefore: rhL = he[( T 1 - 7 T A)C~ - C2]

(22)

where: C 1 = 293.3

and

C 2 =

3911"505(1 - y)

Therefore: K c3Tl(x' d x t) x=0 = h ' l [ T ' , ( t ) - T l ( X = 0 , t)]

(23)

h'~ = h o + heC17

(24)

where:

and: T'a(t) -

TA(t)[ho + 7heC1] ho + h~C1

C2he + - ho + heC 1

(25)

APPENDIX 2

The combined heat transfer coefficient to the atmosphere by the pond surface may be written as: ho=h,+hc

where the radiation heat transfer coefficient, h,, is given by: h, = e s a [ T ~ - ( T A - 1 2 ) ' ] / [ T s - TA]

(26)

and the convective heat transfer coefficient, h c, is given by (Sodha et al.*°): he= 5.7 + 3.8V where: es=emissivity of the pond surface, a = S t e f a n - B o l t z m a n n constant (W/m 2 K), T s = pond surface temperature (K), T A = ambient air temperature (K) and V = wind velocity (m/s).

36

N. D. K A U S H I K , S. K. R A O

The evaporative heat transfer coefficient, h~, is given by the Lewis equation: h~ =0.013 x

he

The convective heat transfer coefficients, h; (i = 2, 3), may be given by:

(h;L =0.54 K )s

t- ~

\Wjs)

27)

where: L = characteristic dimension (m), g = acceleration due to gravity (m s-2), #x = viscosity (N s/m2), fl = coefficient o f volumetric expansion ( K - ~), Cp = specific

heat ( l k g - i K ) , A T = temperature difference (K), p =density (kg/m3), K = c o n ductivity (W/m K) (fcorresponds to fluid) and the heat transfer coeffÉcient from the concrete roof to the inside air is taken as given by Threlkeld.12