- Email: [email protected]
Experimental validation of thermal model of open roof pond
Building and Environment, Vol. 16, No. 2, pp. 93-98, 1981 Printed in Great Britain 0360-1323/81/020093-06 $02.00/0 © 1981 Pergamon Press Ltd.
Experimental Validation of Thermal Model of Open Roof Pond M. S. S O D H A * USHA SINGH* ALOK SRIVASTAVA* G. N. T I W A R I * This note presents a simple periodic thermal model for the performance of an open roof pond. The results of this model are seen to be in good agreement with experiment.
t h r o u g h the roof. S u t t o n [ 7 ] has reported t h a t the surface t e m p e r a t u r e of a roof which would reach 65.56°C, if unsprayed, can be reduced to 42.22°C a n d 39.44°C by m a i n t a i n i n g open roof p o n d of d e p t h 50 m m and 150 m m respectively. In this c o m m u n i c a t i o n , we have developed a thermal model of a n open roof p o n d a n d c o m p a r e d the results given by the model with experiment. An analytical expression for the periodic water temperature, ceiling t e m p e r a t u r e a n d heat flux c o m i n g into the r o o m has been derived; the solar insolation, a t m o s p h e r i c t e m p e r a t u r e a n d enclosure air t e m p e r a t u r e have been assumed to be periodic function of time. T h e analytical results are found to be in good agreement with the observations.
NOMENCLATURE C hi h2 ha h4
hi Hs(t ) K m Mw n p Qc, Qeo Qr~ Q(t) t TR(t ) Tw(t ) TA(t ) x p 09 r~ •2 e
specific heat of concrete roof, kJ kg- 1 "C- 1 16.273 x 10- ~ . h a radiative heat transfer coefficient from water to the ambient, W m - ~ °C convective heat transfer coefficient from water to the ambient, W m - 2 oC - 1 heat transfer coefficient from roof surface to the water, W m - 2 °C inside heat transfer coefficient, W m-2 °C- ~ intensity of solar radiation, W m-2 thermal conductivity of roof material, W m - ~ °C thickness of roof heat capacity of water per unit area, J m - 2 ° C -'~ an integer partial pressure of water vapour at temperature T, pa convective heat transfer from the water to the ambient, W m z evaporative heat transfer from the water to the ambient, W m z radiative heat transfer from the water to the ambient, Wm-2 heat flux coming inside the enclosure through roof per unit time, W m-2 relative humidity time, enclosure air temperature, °C water temperature, °C ambient air temperature, °C axis is vertically downwards density of roof material (kg m - 3) 2n (period)fraction of incident solar energy absorbed by water fraction of incident solar energy absorbed by roof surface. emissivity of water surface.
2. A N A L Y S I S T h e roof p o n d system is schematically represented by Fig. 1. The c o n d u c t i o n of heat t h r o u g h the roof is governed by ~32T
~T
K t3x2 = p c ~ - .
(1)
The energy balance for the water mass m a y be
1. I N T R O D U C T I O N T H E R E D U C T I O N of heat flow t h r o u g h the roof by e v a p o r a t i o n of water has been suggested in m a n y papers [1-7,1; this is of great interest in t h e r m a l design of buildings for h o t a n d dry climates. H o u g h t e n et a/.[6l have reported extensive experimental d a t a o n cooling p r o d u c e d by e v a p o r a t i o n of water film formed on the roof by spray or from the surface of the open roof p o n d ; it was concluded t h a t b o t h the m e t h o d s are a b o u t equally effective in reducing the heat flux
H,(t)
w~_r_ ~"~':..:'~'i~.~'!~~:: :,'~:..~.~ . ~ ~ i'i'~Y~?~'':~!:?''i~?"~!";~:"~:~ ~ i ~ ! ~
"---~]x o ~':
wire x = Ljuncti°n
Unconditioned enclosure
*Centre of Energy Studies, Indian Institute of Technology, New Delhi--110016, India.
Fig. 1. Schematic sketch of open-roof pond. 93
M. S. Sodha, Usha Singh, Alok Srivastava and G. N. Tiwari
94 expressed as
by boundary conditions,
9= -~,/~(1+i)}
dr. Mw "~-=zx "Hs-Qra-Qc.-Qe.
+ h,(T~=o- Tw),
and
(ll)
(2)
where M w is assumed to be constant (in practice much larger than the mass evaporated daily). For a small range of water temperature, the vapour pressure can be assumed to be linear function of temperature i.e.
p=RxT+R 2,
.
The two characteristic lengths of the system are
L1=~-1|--] and
\pc
(3)
where R 1 and R z are constants to be obtained by least square curve-fitting. Thus
Q~.-hl(Ra(Tw-r'TA)+R2(1-r)),
(4)
Q..=h2(Tw - Ta)
(5)
Qc.=h3(Tw- Ta),
(6)
and
t
L2 =//K "~t~l/2
.
(12)
/
For parameters relevant to our system both L1 and
L z are much less than 15cm; hence for transverse dimensions of the system larger than l m the system will essentially be governed by one dimensional considerations. As the solar intensity and ambient air temperature are periodic function of time, water temperature and other parameters will also be a periodic function of time; thus
where h 2 and h 3 are radiative and convective heat transfer coefficients and are given in Appendix 1. Evidently, these coefficients are temperature dependent quantities but since the variations of water temperature and ambient air temperature lie in a small range, these coefficients can be treated as constants. Hence equation (2) can be modified as
Hs(t)=Ho+ ~ H,'exp(ino)t),
n=l
(13)
Ta(t)=Ao+ ~, A.'exp(ine)t),
(14)
n=l
Tw(t)= Wo+ ~. W.'exp(incot),
cto n=l
(15)
TR(t)=Ro+ ~ R.'exp(imm),
(16)
H,=H,.-exp ( - i a n ) ,
(17)
A, = A,," exp ( -
iO'a) ,
(18)
I4I,= W., -exp( - iaw),
(19)
R. = R . . -exp ( - iaR),
(20)
and
dTw
M w ' ~ t =zl"H~-UL(T w-Ta) - hl[Rl(Tw - r" TA) + R2(1 - r)], +hg(T~,=o- Tw)
where
where UL = hz + h3. The energy balance at the surface of the roof may be expressed as -K
(~mX T x=0
='~2
"Hs-h4(Tx=o- Tw).
(8)
The heat transferred to the enclosure air through the roof is given by gJT
- K ~ ~=L= hi(T*=L- TR)"
(9)
The temperature distribution in the roof can be expressed as [8]
where H.., A.., W,,, R,, are amplitudes and an, a A, aw and % are phase terms of solar intensity, ambient air, water and enclosure air temperatures respectively. For daily variation ~o=2n/24 h. Using equations (10) and (15), and the boundary conditions expressed by equations (7-9) and making use of equations (13 16), one obtains the matrices
(UL +hlRI +h4) oh4
T(x,t)=Ax+B+ ~ [C.exp(flx)
n=l
+ D. exp ( - fix)] "exp (incot),
n=l
(7)
(10)
0 -K (K + hiL)
-h41 h4 hi
where A, B, C. and D. are constants to be evaluated *This relation holds good for still air and can be approximately derived from Prasad et al.[4] and Carrier[5].
(21) L hiRo -]
Experimental Validation of Thermal Model of Open Roof Pond 3. EXPERIMENT
where
A small enclosure of inside dimensions 2.2 x 2.2m x 4 0 c m was constructed; the roof was made of concrete of thickness 90mm with a provision to maintain a water pond 300mm deep. The roof was water proofed. The water was filled in the pond on 23 April 1980 and exposed to sun up to 1 May 1980 to attain the steady state condition, the depth of water pond was maintained at 210mm by daily addition of water. The heat flux coming into the room ( - K [AT/AX]) was measured by using a simple technique. Two thermocouple junctions were embedded into the roof at depths 85mm and 90mm respectively (90mm was the thickness of the roof). The temperature as recorded by these two thermocouples was measured by sensitive pontentiometer of least count 0.125°C; the difference of the temperatures gives AT. The heat flux coming into the room is calculated by using formula Q(t)= - K ( A T / A X ) where A X = 5 m m . The air temperature in the enclosure was also measured by a thermocouple while the water temperature was measured by an ordinary mercury thermometer of least count 0.1°C. The variation of solar insolation on a horizontal surface was recorded by Kipp and Zonen pyronometer and is given in Fig. 2. The thermophysical properties of the roof material were measured in the laboratory. The following parameters have been used in numerical evaluation of water temperature, ceiling temperature and heat flux.
zl = rl "Ho + UL "Ao - hi [R2(1 -- r)-- R 1 • r" Ao] and
(iconMw + UL + hlR1 + h4) -- h4
0
-
-h4
-h4 1 (KE+h4) /
(KE - h4)
( K E - h~)exp (-ElL)_]
-(KE+hi)exp(EL ) x
C.
=
z2 ' H .
,
(22)
h-h,R,,J
D.
where
95
Zz=zl "H.+ UL'A.+hl
"R1 "r'A..
The explicit expressions for the water temperature Tw, ceiling temperature T~=L and heat flux entering into the room Q(t) per unit time per unit area are
Tw= Wo + ~, l/V,'exp(inogt),
(23)
n=l ~o
T~:L=A "L+B+ Z . [ C . . e x p (ilL) n=l
+ D . "exp ( - E L ) ] 'exp (incot)
(24)
and
Q(t)=hi[ (AL + B-R°)+.=,~" [C..exp(EL )
1. Roof material .... (concrete)
+D. "exp(-flL)-R,]
q
K = 0.1922 W/m°C
(25)
"exp(inogt)],
p = 1900 kg/m 3
where the constants can be obtained from the matrices.
C = 0.84 x 103 J/kg °C. 1000
o Observed Q (t)
40--
8OO
ii
'~----Hs (t) \ \
/
20
/ /
lytieol
/
\\
iI
~E
50O
~,
0
,0o "0 o
20
o
°
!
\
/
\
/ I / 40
I 0
l 4
I I i t
\ \
l 8
I
l
12
16
Time of the day,
Fig..2. Variation of Q(t) with time: - -
2OO
\ \ \ \
I 20
24
h
analytically o b t a i n e d ; O O O e x p e r i m e n t a l l y o b s e r v e d ; . . . . .
solar intensity.
96
M . S. Sodha, Usha Singh, A l o k Srivastava and G. N. T i w a r i
2. Heat capacity of water per unit area M w = 1,008,951 J / m 2 °C
corresponding to water d e p t h 210 mm. 3. rl =0.05, r~ =0.55. 4. U * = - 9 . 9 9 7 W / m 2 °C (still air). 5. hi =0.012 W / m 2 °C [Dunkle, 9] (still air). 6. h4 = 135.05 W / m 2 °C [ M c A d a m s , 10]. 7. h i = 9 . 2 3 5 W / m 2 °C [ A S H R A E , 11] c o r r e s p o n d i n g to emissivity of the surface 0.9, still air a n d direction of heat flow upward. 8. Thickness of the roof L = 0.09 m. 9. Relative humidity r = 10~o. 10. In the operating water t e m p e r a t u r e range, the following linear relation (obtained by least square curve fitting) is valid, p = 3 2 5 . 1 7 . T - 5 1 5 4 . 8 9 where p is expressed in N / m 2.
t e m p e r a t u r e and heat flux corresponding to the enclosure were numerically evaluated. The Fourier coefficients for solar intensity, a m b i e n t air and enclosure air temperatures were calculated and are given in Tables 1-3 respectively. It is seen t h a t the first six h a r m o n i c s are e n o u g h to reproduce the observed solar intensity, air temperature a n d t e m p e r a t u r e of air in the enclosure. Figures 2-4 represent the periodic variation of water temperature, ceiling t e m p e r a t u r e a n d heat flux coming into the enclosure; the solid curve represents the calculated variation while the experimental observations are illustrated by circled points O . The calculated variation is seen to be in reasonably good agreement with the experimental results: this validates the thermal model of the open roof pond.
Acknowledgement--The authors are thankful to Dr. J. K. Nayak and Mr. Ashvini Kumar for various assistance.
4. N U M E R I C A L R E S U L T S A N D DISCUSSIONS Using the hourly data of solar radiation, atmospheric air t e m p e r a t u r e and r o o m t e m p e r a t u r e for a typical hot sunny day, 2 M a y 1980 at Delhi, the periodic variation of water temperature, ceiling *It may be remarked that although the water temperature is lower than the ambient temperature, it still loses heat by radiation because the sky temperature is lower than the water temperature (being much less than the ambient temperature) for parameters of interest here the heat gained by the water surface by convection is smaller in magnitude than that lost by radiation. Thus the net effect of radiation and convection is loss of heat by the water surface, which is at a lower temperature than the ambient air; this accounts for a negative value of U~.
APPENDIX I The convective, evaporative and radiative heat transfer coefficients can be written as
11/3
I
h3=0.884 (7~w-7"A)+ p w - r ' P a .(Tw+273.15 ) 268.9 × 103/~w h I = 16.273 ×
10 -3 "h 3
and h2 =c .a[(Tw + 273.15 )4 - ( T A + 261.15)4]/[~rw - TA].
Table 1. Fourier coefficients for daily variation of solar intensity on horizontal surface on 2 May 1980 at Delhi n H,, (W m - 2)
0
1
2
3
4
5
6
258.32
399.5025 3.2155
162.3762 0.1571
2.3236 4.3011
26.3634 0.2618
2.4003 0.9606
6.2632 6.1231
an(radian)
Table 2. Fourier coefficients for daily variation of ambient air temperature on 2 May 1980 at Delhi n A,,(°C)
0
1
2
3
4
5
6
39.9580
7.2900 3.7898
2.2607 0.4502
0.6301 1.9115
0.6897 4.0397
0.5407 5.9407
0.2601 1.3900
a A (radian)
Table 3. Fourier coefficients for daily variation of enclosure air temperature on 2 May 1980 at Delhi n R,,(°C) a~ (radian)
0
1
2
3
4
5
6
33.0
3.3243 4.3801
0.9923 5.6530
0.6352 2.0151
0.2385 1.930l
0.2007 2.9305
0.5499 3.4803
,
Experimental Validation of Thermal Model of Open Roof Pond
97
Analytical result
o Experimentally observedc
e
i
l
~
34--
3:5_
32
ti
o
o
31 0 G (.3
0
0
0
0
30
29
I
i
O
Is
4
Time of the day,
Fig. 3. Variation of ceiling temperature with time:
g
24
h
analytically obtained; C) C) © experimentally observed. o
o
o Experimentally observedtemperature
o
29
28
0
0
0
0
0
27
E 26
25 0
0 0
24
I
0
I
4
I
I
8
I
12
Time of the day,
Fig. 4. Variation of water temperature with time: - -
t6
I
20
24
h
analytically obtained; (D C) C) experimentally observed.
REFERENCES
1. H. R. Hay and J. I. Yellot, Natural mr-conditioning with roof pond and movable insulation, ASHRAE Trans. 75, 178 (1969). 2. S.P. Jain and K. R. Rao, Movable roof insulation in hot climates, BMg Res. Practice, 229 (1974). 3. M.S. Sodha, A. K, Khatry and M. A. S. Malik, Reduction of heat flux through a roof by water film, Solar Energy 20, 189 (1978). 4. C.R. Prasad, G. S. Dutt, S. R. C. Sathyanarayan and V. K. Rao, Studies on skytherm cooling. Proc. Ind. Acad. Sci. C2 Part 3, 339 (1979). 5. W. H. Carrier, The temperature of evaporation. Trans. Am. Soc. Heat. Vent. Engrs 24, 24-50 (1968). 6. F. C. Houghten, T. H. Olson and Gutberlet, Summer cooling load as affected by heat gain through dry, sprinkled and water covered roof. ASHVE Trans. 46, 239 (1942). 7. G. E. Sutton, Roof spray for reduction in transmitted solar radiation. Heating-piping and Airconditioning, p. 139 (1950).
98
M. S. Sodha, Usha Singh, Alok Srivastava and G. N. Tiwari 8. J. L. Threlkeld, Thermal Environmental Engineering. Prentice-Hall, Englewood Cliffs, New Jersey (1970). 9. R.V. Dunkle, Solar water distillation: The roof type still and multiple diffusion still. 5th Int. Conf. of Developments in Heat Transfer. University of Colorado (1961). 10. W.H. McAdams. Heat Transmission. McGraw-Hill, New York (1954). 11. American Society of Heating Refrigeration and Air-conditioning Engineers, New York (1967).
Experimental Validation of Thermal Model of Open Roof Pond M. S. S O D H A * USHA SINGH* ALOK SRIVASTAVA* G. N. T I W A R I * This note presents a simple periodic thermal model for the performance of an open roof pond. The results of this model are seen to be in good agreement with experiment.
t h r o u g h the roof. S u t t o n [ 7 ] has reported t h a t the surface t e m p e r a t u r e of a roof which would reach 65.56°C, if unsprayed, can be reduced to 42.22°C a n d 39.44°C by m a i n t a i n i n g open roof p o n d of d e p t h 50 m m and 150 m m respectively. In this c o m m u n i c a t i o n , we have developed a thermal model of a n open roof p o n d a n d c o m p a r e d the results given by the model with experiment. An analytical expression for the periodic water temperature, ceiling t e m p e r a t u r e a n d heat flux c o m i n g into the r o o m has been derived; the solar insolation, a t m o s p h e r i c t e m p e r a t u r e a n d enclosure air t e m p e r a t u r e have been assumed to be periodic function of time. T h e analytical results are found to be in good agreement with the observations.
NOMENCLATURE C hi h2 ha h4
hi Hs(t ) K m Mw n p Qc, Qeo Qr~ Q(t) t TR(t ) Tw(t ) TA(t ) x p 09 r~ •2 e
specific heat of concrete roof, kJ kg- 1 "C- 1 16.273 x 10- ~ . h a radiative heat transfer coefficient from water to the ambient, W m - ~ °C convective heat transfer coefficient from water to the ambient, W m - 2 oC - 1 heat transfer coefficient from roof surface to the water, W m - 2 °C inside heat transfer coefficient, W m-2 °C- ~ intensity of solar radiation, W m-2 thermal conductivity of roof material, W m - ~ °C thickness of roof heat capacity of water per unit area, J m - 2 ° C -'~ an integer partial pressure of water vapour at temperature T, pa convective heat transfer from the water to the ambient, W m z evaporative heat transfer from the water to the ambient, W m z radiative heat transfer from the water to the ambient, Wm-2 heat flux coming inside the enclosure through roof per unit time, W m-2 relative humidity time, enclosure air temperature, °C water temperature, °C ambient air temperature, °C axis is vertically downwards density of roof material (kg m - 3) 2n (period)fraction of incident solar energy absorbed by water fraction of incident solar energy absorbed by roof surface. emissivity of water surface.
2. A N A L Y S I S T h e roof p o n d system is schematically represented by Fig. 1. The c o n d u c t i o n of heat t h r o u g h the roof is governed by ~32T
~T
K t3x2 = p c ~ - .
(1)
The energy balance for the water mass m a y be
1. I N T R O D U C T I O N T H E R E D U C T I O N of heat flow t h r o u g h the roof by e v a p o r a t i o n of water has been suggested in m a n y papers [1-7,1; this is of great interest in t h e r m a l design of buildings for h o t a n d dry climates. H o u g h t e n et a/.[6l have reported extensive experimental d a t a o n cooling p r o d u c e d by e v a p o r a t i o n of water film formed on the roof by spray or from the surface of the open roof p o n d ; it was concluded t h a t b o t h the m e t h o d s are a b o u t equally effective in reducing the heat flux
H,(t)
w~_r_ ~"~':..:'~'i~.~'!~~:: :,'~:..~.~ . ~ ~ i'i'~Y~?~'':~!:?''i~?"~!";~:"~:~ ~ i ~ ! ~
"---~]x o ~':
wire x = Ljuncti°n
Unconditioned enclosure
*Centre of Energy Studies, Indian Institute of Technology, New Delhi--110016, India.
Fig. 1. Schematic sketch of open-roof pond. 93
M. S. Sodha, Usha Singh, Alok Srivastava and G. N. Tiwari
94 expressed as
by boundary conditions,
9= -~,/~(1+i)}
dr. Mw "~-=zx "Hs-Qra-Qc.-Qe.
+ h,(T~=o- Tw),
and
(ll)
(2)
where M w is assumed to be constant (in practice much larger than the mass evaporated daily). For a small range of water temperature, the vapour pressure can be assumed to be linear function of temperature i.e.
p=RxT+R 2,
.
The two characteristic lengths of the system are
L1=~-1|--] and
\pc
(3)
where R 1 and R z are constants to be obtained by least square curve-fitting. Thus
Q~.-hl(Ra(Tw-r'TA)+R2(1-r)),
(4)
Q..=h2(Tw - Ta)
(5)
Qc.=h3(Tw- Ta),
(6)
and
t
L2 =//K "~t~l/2
.
(12)
/
For parameters relevant to our system both L1 and
L z are much less than 15cm; hence for transverse dimensions of the system larger than l m the system will essentially be governed by one dimensional considerations. As the solar intensity and ambient air temperature are periodic function of time, water temperature and other parameters will also be a periodic function of time; thus
where h 2 and h 3 are radiative and convective heat transfer coefficients and are given in Appendix 1. Evidently, these coefficients are temperature dependent quantities but since the variations of water temperature and ambient air temperature lie in a small range, these coefficients can be treated as constants. Hence equation (2) can be modified as
Hs(t)=Ho+ ~ H,'exp(ino)t),
n=l
(13)
Ta(t)=Ao+ ~, A.'exp(ine)t),
(14)
n=l
Tw(t)= Wo+ ~. W.'exp(incot),
cto n=l
(15)
TR(t)=Ro+ ~ R.'exp(imm),
(16)
H,=H,.-exp ( - i a n ) ,
(17)
A, = A,," exp ( -
iO'a) ,
(18)
I4I,= W., -exp( - iaw),
(19)
R. = R . . -exp ( - iaR),
(20)
and
dTw
M w ' ~ t =zl"H~-UL(T w-Ta) - hl[Rl(Tw - r" TA) + R2(1 - r)], +hg(T~,=o- Tw)
where
where UL = hz + h3. The energy balance at the surface of the roof may be expressed as -K
(~mX T x=0
='~2
"Hs-h4(Tx=o- Tw).
(8)
The heat transferred to the enclosure air through the roof is given by gJT
- K ~ ~=L= hi(T*=L- TR)"
(9)
The temperature distribution in the roof can be expressed as [8]
where H.., A.., W,,, R,, are amplitudes and an, a A, aw and % are phase terms of solar intensity, ambient air, water and enclosure air temperatures respectively. For daily variation ~o=2n/24 h. Using equations (10) and (15), and the boundary conditions expressed by equations (7-9) and making use of equations (13 16), one obtains the matrices
(UL +hlRI +h4) oh4
T(x,t)=Ax+B+ ~ [C.exp(flx)
n=l
+ D. exp ( - fix)] "exp (incot),
n=l
(7)
(10)
0 -K (K + hiL)
-h41 h4 hi
where A, B, C. and D. are constants to be evaluated *This relation holds good for still air and can be approximately derived from Prasad et al.[4] and Carrier[5].
(21) L hiRo -]
Experimental Validation of Thermal Model of Open Roof Pond 3. EXPERIMENT
where
A small enclosure of inside dimensions 2.2 x 2.2m x 4 0 c m was constructed; the roof was made of concrete of thickness 90mm with a provision to maintain a water pond 300mm deep. The roof was water proofed. The water was filled in the pond on 23 April 1980 and exposed to sun up to 1 May 1980 to attain the steady state condition, the depth of water pond was maintained at 210mm by daily addition of water. The heat flux coming into the room ( - K [AT/AX]) was measured by using a simple technique. Two thermocouple junctions were embedded into the roof at depths 85mm and 90mm respectively (90mm was the thickness of the roof). The temperature as recorded by these two thermocouples was measured by sensitive pontentiometer of least count 0.125°C; the difference of the temperatures gives AT. The heat flux coming into the room is calculated by using formula Q(t)= - K ( A T / A X ) where A X = 5 m m . The air temperature in the enclosure was also measured by a thermocouple while the water temperature was measured by an ordinary mercury thermometer of least count 0.1°C. The variation of solar insolation on a horizontal surface was recorded by Kipp and Zonen pyronometer and is given in Fig. 2. The thermophysical properties of the roof material were measured in the laboratory. The following parameters have been used in numerical evaluation of water temperature, ceiling temperature and heat flux.
zl = rl "Ho + UL "Ao - hi [R2(1 -- r)-- R 1 • r" Ao] and
(iconMw + UL + hlR1 + h4) -- h4
0
-
-h4
-h4 1 (KE+h4) /
(KE - h4)
( K E - h~)exp (-ElL)_]
-(KE+hi)exp(EL ) x
C.
=
z2 ' H .
,
(22)
h-h,R,,J
D.
where
95
Zz=zl "H.+ UL'A.+hl
"R1 "r'A..
The explicit expressions for the water temperature Tw, ceiling temperature T~=L and heat flux entering into the room Q(t) per unit time per unit area are
Tw= Wo + ~, l/V,'exp(inogt),
(23)
n=l ~o
T~:L=A "L+B+ Z . [ C . . e x p (ilL) n=l
+ D . "exp ( - E L ) ] 'exp (incot)
(24)
and
Q(t)=hi[ (AL + B-R°)+.=,~" [C..exp(EL )
1. Roof material .... (concrete)
+D. "exp(-flL)-R,]
q
K = 0.1922 W/m°C
(25)
"exp(inogt)],
p = 1900 kg/m 3
where the constants can be obtained from the matrices.
C = 0.84 x 103 J/kg °C. 1000
o Observed Q (t)
40--
8OO
ii
'~----Hs (t) \ \
/
20
/ /
lytieol
/
\\
iI
~E
50O
~,
0
,0o "0 o
20
o
°
!
\
/
\
/ I / 40
I 0
l 4
I I i t
\ \
l 8
I
l
12
16
Time of the day,
Fig..2. Variation of Q(t) with time: - -
2OO
\ \ \ \
I 20
24
h
analytically o b t a i n e d ; O O O e x p e r i m e n t a l l y o b s e r v e d ; . . . . .
solar intensity.
96
M . S. Sodha, Usha Singh, A l o k Srivastava and G. N. T i w a r i
2. Heat capacity of water per unit area M w = 1,008,951 J / m 2 °C
corresponding to water d e p t h 210 mm. 3. rl =0.05, r~ =0.55. 4. U * = - 9 . 9 9 7 W / m 2 °C (still air). 5. hi =0.012 W / m 2 °C [Dunkle, 9] (still air). 6. h4 = 135.05 W / m 2 °C [ M c A d a m s , 10]. 7. h i = 9 . 2 3 5 W / m 2 °C [ A S H R A E , 11] c o r r e s p o n d i n g to emissivity of the surface 0.9, still air a n d direction of heat flow upward. 8. Thickness of the roof L = 0.09 m. 9. Relative humidity r = 10~o. 10. In the operating water t e m p e r a t u r e range, the following linear relation (obtained by least square curve fitting) is valid, p = 3 2 5 . 1 7 . T - 5 1 5 4 . 8 9 where p is expressed in N / m 2.
t e m p e r a t u r e and heat flux corresponding to the enclosure were numerically evaluated. The Fourier coefficients for solar intensity, a m b i e n t air and enclosure air temperatures were calculated and are given in Tables 1-3 respectively. It is seen t h a t the first six h a r m o n i c s are e n o u g h to reproduce the observed solar intensity, air temperature a n d t e m p e r a t u r e of air in the enclosure. Figures 2-4 represent the periodic variation of water temperature, ceiling t e m p e r a t u r e a n d heat flux coming into the enclosure; the solid curve represents the calculated variation while the experimental observations are illustrated by circled points O . The calculated variation is seen to be in reasonably good agreement with the experimental results: this validates the thermal model of the open roof pond.
Acknowledgement--The authors are thankful to Dr. J. K. Nayak and Mr. Ashvini Kumar for various assistance.
4. N U M E R I C A L R E S U L T S A N D DISCUSSIONS Using the hourly data of solar radiation, atmospheric air t e m p e r a t u r e and r o o m t e m p e r a t u r e for a typical hot sunny day, 2 M a y 1980 at Delhi, the periodic variation of water temperature, ceiling *It may be remarked that although the water temperature is lower than the ambient temperature, it still loses heat by radiation because the sky temperature is lower than the water temperature (being much less than the ambient temperature) for parameters of interest here the heat gained by the water surface by convection is smaller in magnitude than that lost by radiation. Thus the net effect of radiation and convection is loss of heat by the water surface, which is at a lower temperature than the ambient air; this accounts for a negative value of U~.
APPENDIX I The convective, evaporative and radiative heat transfer coefficients can be written as
11/3
I
h3=0.884 (7~w-7"A)+ p w - r ' P a .(Tw+273.15 ) 268.9 × 103/~w h I = 16.273 ×
10 -3 "h 3
and h2 =c .a[(Tw + 273.15 )4 - ( T A + 261.15)4]/[~rw - TA].
Table 1. Fourier coefficients for daily variation of solar intensity on horizontal surface on 2 May 1980 at Delhi n H,, (W m - 2)
0
1
2
3
4
5
6
258.32
399.5025 3.2155
162.3762 0.1571
2.3236 4.3011
26.3634 0.2618
2.4003 0.9606
6.2632 6.1231
an(radian)
Table 2. Fourier coefficients for daily variation of ambient air temperature on 2 May 1980 at Delhi n A,,(°C)
0
1
2
3
4
5
6
39.9580
7.2900 3.7898
2.2607 0.4502
0.6301 1.9115
0.6897 4.0397
0.5407 5.9407
0.2601 1.3900
a A (radian)
Table 3. Fourier coefficients for daily variation of enclosure air temperature on 2 May 1980 at Delhi n R,,(°C) a~ (radian)
0
1
2
3
4
5
6
33.0
3.3243 4.3801
0.9923 5.6530
0.6352 2.0151
0.2385 1.930l
0.2007 2.9305
0.5499 3.4803
,
Experimental Validation of Thermal Model of Open Roof Pond
97
Analytical result
o Experimentally observedc
e
i
l
~
34--
3:5_
32
ti
o
o
31 0 G (.3
0
0
0
0
30
29
I
i
O
Is
4
Time of the day,
Fig. 3. Variation of ceiling temperature with time:
g
24
h
analytically obtained; C) C) © experimentally observed. o
o
o Experimentally observedtemperature
o
29
28
0
0
0
0
0
27
E 26
25 0
0 0
24
I
0
I
4
I
I
8
I
12
Time of the day,
Fig. 4. Variation of water temperature with time: - -
t6
I
20
24
h
analytically obtained; (D C) C) experimentally observed.
REFERENCES
1. H. R. Hay and J. I. Yellot, Natural mr-conditioning with roof pond and movable insulation, ASHRAE Trans. 75, 178 (1969). 2. S.P. Jain and K. R. Rao, Movable roof insulation in hot climates, BMg Res. Practice, 229 (1974). 3. M.S. Sodha, A. K, Khatry and M. A. S. Malik, Reduction of heat flux through a roof by water film, Solar Energy 20, 189 (1978). 4. C.R. Prasad, G. S. Dutt, S. R. C. Sathyanarayan and V. K. Rao, Studies on skytherm cooling. Proc. Ind. Acad. Sci. C2 Part 3, 339 (1979). 5. W. H. Carrier, The temperature of evaporation. Trans. Am. Soc. Heat. Vent. Engrs 24, 24-50 (1968). 6. F. C. Houghten, T. H. Olson and Gutberlet, Summer cooling load as affected by heat gain through dry, sprinkled and water covered roof. ASHVE Trans. 46, 239 (1942). 7. G. E. Sutton, Roof spray for reduction in transmitted solar radiation. Heating-piping and Airconditioning, p. 139 (1950).
98
M. S. Sodha, Usha Singh, Alok Srivastava and G. N. Tiwari 8. J. L. Threlkeld, Thermal Environmental Engineering. Prentice-Hall, Englewood Cliffs, New Jersey (1970). 9. R.V. Dunkle, Solar water distillation: The roof type still and multiple diffusion still. 5th Int. Conf. of Developments in Heat Transfer. University of Colorado (1961). 10. W.H. McAdams. Heat Transmission. McGraw-Hill, New York (1954). 11. American Society of Heating Refrigeration and Air-conditioning Engineers, New York (1967).