A novel underground solar water heater

Ener,qy Confers. Mgmt Vol. 22, pp. 31 to 37, 1982 0196-8904/82/010031.07$03.00/0 Printed in Grea t Britain. All r ights reserved Copyright © 1982 Pe rgamon Press Ltd

A NOVEL U N D E R G R O U N D SOLAR WATER HEATER

M. S. SODHA, N. K. DHIMAN, J. K. NAYAK and G. N. TIWARI Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi. 110016, India

{Received 8 June 1981)

Abstract--This communication presents the performance of a novel solar water heater which consists of a rectangular tank (1 l0 x 80 x l0 cm) made up of G.I. sheet and is provided with inlet and outlet pipes. The tank is buried underground; the top surface of the ground is blackened and glazed so that the surface acts as an absorber of solar radiation while the tank as well as the surrounding earth serves as the storage system. During off-sun shine hours the top surface is covered by a layer of insulation so as to prevent upward loss. An analytical model has been presented to predict the performance of such a system.

Water heater Solar energy Underground Storage

N O M E N C L A T U R E

Ko = thermal conductivity of the soil, kJ/m h °C Po = density of soil, kg/m a Co = specific heat of soil kg/m a

d = depth of water tank, m h2 = heat transfer coefficient between soil (at

x = 0) and water; kJ/m 2 h 'C ha = heat transfer coefficient between soil {at

x = d} and water, kJ/m 2 h C I = depth of burial, m

Mw = heat capacity of water per unit area of the tank kJ/m 2 C

S(t) = solar intensity on the horizontal surface, kJ/m h

t = time coordinate, h T(x, t) = temperature distribution in the soil, at a sur-

face x and at time t, :C T~ = ambient temperature, C T, = sol-air temperature, :C

T~ = water temperature, :C U = overall heat transfer coefficient, kJ/m 2 h °C x = position coordinate, m

r0:t 0 = transmissivity-absorptivity product

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

Water heating has been one of the most successful utilisations of solar energy. Various types of water heaters are in vogue around the world. Amongst them, collector-cum-storage type water heaters have received the widest a t tent ion since these combine collection as well as storage in a single unit. This com- municat ion presents the performance of one type of collection-cure-storage water heater. It consists of a rectangular tank buried in the ground and is filled with water; the top surface of the ground is blackened with b lackboard paint and suitably glazed. Thus this surface serves as the absorber of solar radiation. The absorbed energy is conducted away to the water tank through the ear th and is stored as sensible heat in it. The purpose of burying the water tank in the ground

is to prevent it from rapid temperature fluctuations,

(In fact a large number of storage tanks/systems are kept underground particularly in West Asian deserts where rapid temperature fluctuations occur predomi- nantly.) Further, in addit ion to the solar energy absorbed by the top surface during the day, the system can extract some energy from the surrounding earth during the night due to reradiat ion from earth. Experimental studies carried out with such type of water heater have been reported earlier [1]. The experimental set up consisted of a rectangular tank ( l l 0 x 80 x 10cm) of galvanised iron and is provided with G.I. pipes serving as inlet and outlets for water. The tank, filled with water, was buried under ground. The above experimental observations correspond to the case when the top surface is not covered during night. However, the performance of the system can be enhanced if the top surface is covered by a layer of insulation, during off-sunshine hours, to prevent upward heat loss. This paper presents a thermal model to predict the behaviour of water temperature both during day and night (when top surface is covered). The one dimensional heat conduct ion equation has been solved using appropr ia te boundary conditions to obtain the temperature distr ibution in the ground, and an expression for the water temperature in the tank has been derived as a function of time and depth of the tank from the top surface of the ground. Numerical calculations have been made for parameters corresponding to the experimental observations reported I l l . The thermal model has been found to explain the experimental observations to a satisfactory degree. The effects of variat ion in the thickness of top insulation (used during off sunshine hours), depth of water tank and depth of burial have been demonstra ted in relation to the performance of the system. It is found that the performance of the system is improved significantly when movable insulation is used. A suitable choice of the depth of burial

t.c.~, 22 I c 31

32 SODHA et al.: A NOVEL UNDERGROUND SOLAR WATER HEATER

SOIor rodiatior~ and

_r,o ~T #x x=o

= h2[Tx= 0 -- Tw] (3)

Glon cover

7/ / / f

(a) Ouring sun thine hours

Wooden frame x=-I (Blackened surface)

xsO

x , d

l l l l / / A , ~ j ~ i s g / / / l l l l , , G~ . . . . . • Air gap ~-~ Wooden frame , \ \ \ \ \ \ \ d ~ , d \ \ \ \ \ \ \ \ ~ ..-, (B~c..-..u,~,.)

"';7/- "" • / Water tank

( b l Ouring a l l lun lh ine hours

Fig. 1. Schematic sketch of underground solar water heater. (a) During sunshine hours, (b) during off sunshine

hours.

can enhance the attainable maximum water temperature.

ANALYSIS

The configuration is schematically illustrated in Fig. 1: the plane x = - 1 represents the blackened and glazed surface of the ground while the water tank is kept in the space between the planes x = 0 and x = d. During sunshine hours the system is exposed to solar radiation (Fig. la). Solar Energy is absorbed by the blackened surface (x = - l ) as heat energy: this in turn gets conducted to water in the tank and is stored as sensible heat. However, during the period when the intensity of solar radiation is decreased sub- stantially, the top glazed surface is covered by a layer of insulation (Fig. lb) with a view to reducing the upward loss.

The temperature distribution T(x. t) in the ground is given by the one dimensional Fourier heat conduc- tion equation, viz.

?T t~2T - - = :( (1) ~t ?x 2

where

~t = Ko/poco

subject to the following boundary conditions. At the surface x = - I and at plane x = 0 and x = d, the energy balance conditions respectively are

/

- K o t~T " -~xl. = -~ -- U ( t ) [ T . - T,,= _,] (2)

- K o ~ x T x=a = hs [T " - Tx=d] (4)

where

~oroS(t) T s - - - + T , U(t)

and is known as sol-air temperature. Further T(x, t) is finite x tends to infinity i.e.

T(x, t) is finite for x ---* ~c. (5)

The energy balance for water (neglecting the heat capacity of the material of the tank) can be written as

dTw M,~--'7- = h2[Tx=0 - Tw] - h3[Tw - Tx=d]. (6)

(1/

Since solar radiation and ambient air temperature are periodic in nature, we may express solair temperature as a Fourier series in time

T~ = ~ a. exp (imot). (7) - x

In view of equation (7), the following periodic solutions may be assumed for the temperature distribution in the ground as well as for the water temperature

T ( x , t ) = A x + B

+ ~ [A.exp(flnx) + B .exp( - f l . x ) ] - x

n ; e O

x exp(imot), f o r - l ~ < x ~<0 (8)

and

= c + ~ c, exp ( - fl, x). exp (ino)t).

n * O

fo rd~<x~< ~ (9)

+ ~ :

Tw = ~ T~'exp(ina)t) , (10) - a c

where

/noWoco _~ : f l . = ( l + i ) : q ; % = ~/ 2Ko ; i = \

(11) co = 2n/To: To is the time period.

It may be noted that while writing down the solution for T(x , t ) in the region defined by d ~< x ~< ~, boundary condition (5) has been used. Further char-

SODHA et al.: A NOVEL UNDERGROUND SOLAR WATER HEATER 33

8 0 - -

60

o

20

. / r" ' - \

/ X.,

X..~... - r = . . ~ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ ............. , _ @ . . -

i / ..-/ r/I \ I \x i, ,7 - i ~ - , o

~.:~. ." y

- . . . . . . , ,

_1 I _ .S '~ . . . ._7 ._r , . ,~ . " ' " " " . ~ _

.... . . . . . . . . . . . . . . . . . . . . . . . gour~r ¢oe fficlent$ . . . . . . . 0 0 ~

P

8

e

l o

o , I , i , I , i , I /, 8 IZ I(~ 20 24

Timc {hrs)

Fig. 2. Hourly variation of solar temperature (T~) and over all heat transfer coefficients (U); dashed curves correspond to the data while dotted curves correspond to that calculated from Fourier

coefficients.

acteristic lengths (from equations {1), (8), (9) and (11)) of the problem are

L, = (KoAt/poCo) 1'2 and L 2 = 0Cn I

much smaller than the dimensions of the system for daily variation and ground parameters, thus the one dimensional analysis employed here is justified.

As it has been mentioned earlier, the system is exposed to solar radiation during sunshine hours and is covered during off-sunshine hours. Hence, the upward heat transfer coefficient (U} becomes a time varying quantity. To a good approximation it can be considered to have a constant value during the exposed period and another constant value during the unexposed period. In other words, U(t) can be approximated as a rectangular function as shown in Fig. 2. In view of equation (71, this function can also be expanded as a Fourier series in time

U(t) = ~. U. exp(ino~t). (12) --z

It may be noted that in the Fourier expansions, equations (7) and (121, terms for n = - 5 to n = 5 have been found to be sufficient to reproduce the actual behaviour of the respective functions (Fig. 2). Therefore in all such Fourier expansions, equations (7)-(12), we shall retain terms only for n = - 5 to n = + 5 .

The unknown quantities appearing in equations {8), (9) and (10) can be estimated using equations (2)-(6). Then the water temperature is given by a matrix

equation

[M]x ['N] = [R] {13)

where I'M] is a matrix having the following elements

mjk = U j -k Vk_(s+ tl - Yj_{s+ ll ~jk

for j = 1 to (2s + 1) k = 1 to (j + s); [1 ~< j ~< (s + 1)]

= (j - s) to {2s + 1); l-(s + 2) ~< j <~ (2s + 1)]

= 0 (14)

and IN] and [R] are column vectors defined as

nit = T#~-(,+ u, for 1 ~<j ~< (2s + 1)

rjl =b j_ , ,+ u, for 1 ~<j ~<(2s+ 1)

where

(15)

bJ-(s+l) "~- L U-(s-~l)a-(q-J+l) q=o

j -1 + ~ U~a_Iq_j+,+.~, for 1 ~< j ~< {s + U)

q=l q>j

12s+ 11

= E Utt+J-12s+l)al,-~l (161 q=o

-.t- L Uqa-lq-J+s+l), q=l

f o r ( s + 2 ) ~ < j ~ < ( 2 s + 11.


The V's and Y's are defined as follows

Vj = PJ e x p ( - f l d ) + P~ exp(flfl)

Yj = [PJ exp ( - f l j / ) - P} exp (f l f l)]Koflj

where

PJ = 1 - K~oflj ij(oM,,.

' ( & ;)( P~ = 1 + + ijtoMw

s = 5

and (~ij is Kronecker's delta.

(17)

+ h3Kofl j ~]

h3 q KoflJ_]

113 + Ko#J_J

(18)

N U M E R I C A L RESULTS AND DISCUSSIONS

As has been hinted earlier, the analysis intends to give a theoretical basis to the experimental observations reported ear l i e r [ l ] in connection with a water heater buried underground and to study the effect of movable insulation over it. So numerical calculations for water temperature have been performed corresponding to solair temperature and atmospheric temperature of 26 April 1978 i-1]. In the following calculations the relevant parameters used are

Ko = 1.869 kJ/m h C

= 2.5 × 10 -3 m2/h

Mw = 419.0 kJ/m 2 :C

1 = 0.05 m, 0.10 m

ro:% = 0.5.

The heat transfer coefficients as calculated for the relevant expressions [2. 3] are

h 2 = II 3 = 360.94 kJ/m z h :C.

The Fourier coefficients of sol-air temperature (T~) and heat transfer coefficients (U) are given in Tables 1 and 2 respectively. In Fig. 2 the hourly variations of solair temperature and overall heat transfer coefficient are shown: in each are the dashed curve corresponds to the data used while the dotted curves correspond to the values calculated for the Fourier coefficients. It is quite evident that for a good convergence of the Fourier series, it is sufficient to consider terms from n = - 5 to n = +5.

Since the experiment was carried out without covering the top surface during off sunshine hours, the calculations have been made with a constant value for overall heat transfer coefficient. (U = 57.24 k j/m2 h :C) corresponding to a wind speed of 5 km/h). While solid curve (I) corresponds to the theoretical calculations, the experimental points are shown by O. It is seen that the experimental obser-

Table 1. Fourier analysis of daily variation of solair temperature, 26 April 1978

m Real Imaginary

0 40.4018 - - - 5 - 7.923 - 9.109 - 4 0.485 3.48 - 4 -0.360 0.338 - 2 0.350 -0.2815 - I 0.2095 0.188

1 -0.2095 -0.188 2 0.305 0.2815 3 - 0.0360 - 0.338 4 0.485 - 3.48 5 - 7.923 9.109

vations are in good agreement with the theory presented in this paper. The movable insulation greatly enhances the performance of the water heater by preventing upward loss during off sunshine hours. Curve II of Fig. 3 shows the improvement in the performance of the water heater by movable insulation. It is evident that the swings in water temperature decrease appreciably with the maxima of water temperature occurring 3 h earlier than in the uninsulated case. The smoothing of water temperature and the phase shift of the maxima/minima are also strongly dependent on the depth of soil below which the water tank is buried. Figure 4 shows this effect explicitly. Since the performance of the water heater with movable

Table 2. Fourier Analysis of overall heat transfer coefficient

m Real Imaginary

0 29.9909 - - - 5 -5.1198 3.9218 - 4 -0.0005 0.0002 - 3 10.4117 -4.3139 - 2 0.0005 - 0.0002 - 1 - 34.2998 4.5167

1 -34.2998 -4.5161 2 0.0005 0.0002 3 10.4117 4.3139 4 - 0.0005 - 0.0002 5 - 5.1198 - 3.9298

Table 3. Fourier Analysis of daily variation of solair temperature, 28 January 1980

m Real Imaginary

0 41.5045 - - - 5 -4.956 -3.159 - 4 2.427 1.906 - 3 -0.0718 -0.207 - 2 -0.0948 - 0.449 - 1 - 0.0024 - 0.048

1 - 0.0024 0.048 2 - 0.0748 0.449 3 - 0.0718 0.207 4 2.427 - 1.906 5 -4.956 3.159


t.5

=,

~.o

Water depth :10 em

Ground depth: 5 cm

Insulation thickness =5 cm

0 0 0 Experimental points

T Without insulation

11 With in sulo hon

I = I = I ~ I ~ I i I /. 8 12 16 20 24

Fig. 3. Hourly variation of water temperature (April 26, 1978). (I) Uninsulated case, (Ill insulated case.

insulation is found to be better, the rest of the calculations refer only to that type of water heater. Again, water heating is required mostly during the winter season. Hence, the following calculations correspond to a typical winter day at Delhi (28 JanuarY 1980). The Fourier coefficients of sol-air temperature corresponding to this date are shown in Table 3 and the hourly behaviour is shown in Fig. 4. In it, water tem- peratures are plotted for three different depths (0,0, 0.05 and 0.1 m) of the tank from the top surface. It is

seen that, in addition to a slight shift in the maxima/ minima of temperature, there occurs a smoothing of the curve as depth increases. It is interesting to note that the water temperature does not decrease appreciably during off-sunshine hours unlike the uninsulated case. Hence, the present system can supply hot water at the maximum temperature even at midnight when there is no solar radiation. A suitable choice of the depth of the tank from the top surface can change the attainable maximum water temperature.

Fig.

8o

55

/

Effect o~ ground depth u~

43 X 5.0 cm

H I0.0 cm

Water depth = 10 Cm

InsulQtlon thickness ;5 Cm

z9 I I , I i I , l ~ I i o ; 8 12 16 20 2;

Ttr~e (hrs)

4. Hourly variation of water temperature (28 January 1980) for different depth of burial.


- I

5t

i E

&3

Ef fec t Of wafer depth

1 2 0 cm

n 5 0 cm

m t O 0 cm

Ground depth : . cm

{nsuiotion th ickness : $ cm

0 /. 8 12 t6 20 2L

Time (h rs )

Fig. 5. Hourly variation of water temperature (28 January 1980) for different depth of water tank.

Figure 5 demonstrates the effect of water tank depth on water temperature for a constant depth of burial (0.05 m). It is seen that swings are greatly reducod as water depth increases though there is a substantial decrease in the maxima/minima. Figure 6 shows the effect of changing the duration and also timing of covering the top surface of the water heater by movable insulation. It is found that, out of the various combinations studied, the combination 8 am-6 pm is found to be the best one. That means if

the movable insulation is used from 6 pm to 8 am, then water temperature will be large throughout the day (curve II). But it may be noted that this behaviour cannot be generalised since it is strongly dependent upon the nature of solar radiation and atmospheric temperature.

However, given the date of use, one can calculate the duration of useful period and hence can suitably cover the water heater by movable insulation. Figure 7 shows the effect of the change in thickness of insula-

InSolation durat ion

/ i a . _ 6 . . ~9 ~ / i lo ,, - 6 ,,

j Iv" 8 ,, - ¢ . , ,

/ Woter depth = I0 cm

InsulQtion thickness =5 ¢ m .

=7 I I I I = I I l I I I I /. 8 12 16 ZO 2~

Time (hrs)

Fig. 6. Hour ly variation of water temperature (28 January 1980) ]'or different duration of the use of movable insulation.


• - H

o

~t E

@

I .?

d ¢

Ground depth * S cat

4s j I I I I 1 i I , I I I 4 8 II II 20 II,

Time (hrs)

Fig. 7. Hourly variation of water temperature (28 January 1980) for different thicknesses of movable insulation.

tion for a fixed depth of burial and water depth. This enables one to choose the right thickness of insulation for covering the water heater. A layer of 0.05 m insulation seems to be good enough for normal use.

C O N C L U S I O N

Thus, we may conclude that the thermal model can explain to a satisfactory degree the variation of water temperature in the tank. The system can supply hot water at the maximum temperature even at midnight when there is no solar radiation. According to the requirement the depth of burial of the water tank can be calculated. It should be noted that, when the top surface of earth is blackened and glazed, there will be moisture deposition on the bottom surface of the

glazing (experienced in the experiment). This, in turn, prohibits the easy transmittance of solar radiation. One way of avoiding this difficulty could be the use of some transparent thin sheet on the top surface. This at least would block the moisture evaporation from the blackened surface to the bottom surface of glazing. More experimental studies are necessary to remove this difficulty.

R E F E R E N C E S

[I] J. K. Nayak. S. C. Kaushik, G. N. Tiwari, M. S. Sodha and D. P. Sabberwal, Proc. naln Solar Ener.qy Cone. India, Bhat'nagar, pp. 219 (1978).

[2] W. C. McAdam, Heat Transmission. McGraw Hill, New York (1954).

[3] J. K. Threlkeld, Thermal Em'ironmental Engineerin.q. Prentice Hall, New Jersey (1970).

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A novel underground solar water heater