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Energy Vol.22. No. 12. pp. 1163-1172, 1997 © 1997ElsevierScienceLtd.All rightsreserved Printed in GreatBritain 0360-5442197$17.00+ 0.00

A COMPUTATIONAL MODEL OF A DOMESTIC SOLAR HEATING SYSTEM WITH UNDERGROUND SPHERICAL THERMAL STORAGE M. INALLI,** M. ONSAL § and V. TANYILDIZI* *Departmentof Mechanical Engineering, Ftrat University,Elazl~, Turkey; ~Departmentof Mechanical Engineering, GaziantepUniversity, Gaziantep, Turkey (Received 31 October 1996)

Abstract--This theoretical study deals with a domestic heating system assisted by solar energy stored in an underground spherical container. The system includes a heat pump. The analytical model employed calculates the water temperature in the storage vessel, as well as the temperature distribution in the surrounding geological structure, by using the monthly-average solar radiation and ambient temperature. Storage temperature, collector efficiency, performance coefficient of the heat pump (COP) and annual solar fraction are computed and presented in various graphs. The importance of seasonal solar energy storing in the ground is demonstrated. © 1997 Elsevier Science Ltd.

1. INTRODUCTION Hot-water production by means solar energy has become widespread. However, the use of solar energy in domestic heating has been minimal in countries such as TUrkiye. Solar energy for heating is only possible seasonably. There are many means of storing seasonal thermal energy. Some of these are boreholes in rock, rock caverns, underground steel tanks, aquifers, duct storage in the ground, underground concrete tanks, open pits on ground and excavated bedrocks. Bankston [1] published a comprehensive review on the subject with emphasis on practical applications in different countries. Brunst0rm et al [2] examined the Lyckebo project, which has been meeting the energy and hot water demands of 550 houses. This system consists of 4320 m 2 of highlyefficient, flat-plate collectors, a rock cavern for seasonal storage of 105 m 3 and a 6 MW electric boiler. Kenisarin et al [3] examined the results of a mathematical model of a centralised solar heating system with seasonal thermal storage applied to the climatic conditions of Tashkent. A site of 50 houses was investigated with a total demand of 1000 MWh/year and a cylindrical reservoir. The slope of flat-plate collectors was fixed at 27 °. The collected portion of solar energy was calculated as 50% for a storage volume of 5000 m 3 and a collector area of 1150 m 2 with selective coating. The Kerava solar village is the first regional building complex in Finland with a combined solar heating and heat pump system using seasonal storage. It comprises 44 flats and is situated 35 km north of Helsinki (60°N). Makinen and Lund [4] have presented findings of cost, performance and a sensitivity analysis for the village. The solar fraction 50% was measured for this system. Lund and Ostman [5] have developed a threedimensional model to predict the performance of seasonal thermal energy storage with vertical tubes charged by solar energy. Their model yielded an annual solar fraction of 70% for a 500-house site with a collector area of 35 m 2 and a rockbed store size of 550 m 3 per house under the Helsinki (60°N) climatic conditions. 13nsal [6] has solved the short- and long-term transient heat-transfer characteristics for spherical containers buried in the ground by using a similarity transformation and Duhamel's superposition principle. Onsal [7] also applied the complex, finite Fourier-transform technique to the thermal energy storage problems, for which the given solutions can be utilized in connection with seasonal thermal energy storage-system simulation studies in cylindrical and spherical coordinates. For the determination of the transient temperature distribution in the geological structure surrounding a spherical container, we have obtained an analytical solution by application of the complex, finite Fourier-transform technique [7]. This solution has been employed to calculate the long-term perform*Author for conespondence. Fax: 90-424-218 19 07; e-mail: [email protected] i 163

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ance of a heating system with a heat pump and an underground container. When the bulk temperature of the container is insufficient to keep the house at the inside design air temperature, a heat pump is used to provide the necessary extra heat flow from the container. 2. DESCRIPTIONOF THE HEATINGSYSTEM The system investigated is shown schematically in Fig. 1. It consists of solar collectors, a spherical underground water-storage vessel, a heat pump, and a house. The heat gained from the solar collectors is transferred to storage throughout the year. Energy stored in the system is only extracted during the heating season. The total annual energy supplied to the system during the whole year is the sum of solar energy and work of the heat pump. A fraction of the energy will be lost into the geological structure and the remaining part extracted for use in the house (the load). At the end of a one-year period, the system will return to its initial conditions. We now analyze the heat-transfer problem. 2.1. Modelling of the heat pump Monthly-average energy requirements for the house are calculated from Qh = (UA)h(Ta - Ta).

(1)

If heat is supplied by a heat pump, then Qh = W( COP) = W/3Th/[Th - Tw].

(2)

Depending on the characteristics of the heat pump and the capacities of heat-exchangers, 0.2 < /3 < 0.3. It is assumed that the energy demand of the house is supplied by a panel heat-exchanger or fan-coil unit. Then Eq. (1) becomes

(3)

Qh = (UA )n¢(Th - Ti).

~

colleetor

Control Unit

Pump

Heat Pump

Pump #

i

GROUND

T(r, t)

,~

i ~

tt~: ~;~ ....... ~

r ....

WATERSTORAGE Fig. 1. Schematicof a domestic heating system with an underground spherical water-stora.gevessel.

A domestic solar heating system with undergroundspherical thermal storage

] 165

Th from Eqs. (1) and (3) is inserted into Eq. (2). After non-dimensionalising the resulting equation, the dimensionless heat pump work is w = WI[(UA)hT=] = (¢b~- ~ba)[u(~ - ~b.) + ~ - ~bw]/{fl [u(~b~ - ~ba) + ~b~+ 1]}.

(4)

In Eq. (4), ~ba and ~bi are known. Hence, the work of heat pump depends only on the dimensionless water temperature ~bw. When the house is perfectly insulated and the heat-exchanger is sufficient, u = 1; otherwise, u < 1. When ~b,~ > [u(thl - ~ba) + ~], circulation of hot water is sufficient and the heat pump does not do work. 2.2. Analysis of the transient heat-transfer problem in the storage and surrounding geological structure The transient temperature in the geological structure is given by the following partial differential equation, initial and boundary conditions: ~2T]~r2 -I- (2/r)OT/Or = (1/a)OT/Ot,

(5)

T(R,t) = T,(t),

(6)

T(o~,t) = T~,

(7)

T(r,O) = T(r, one year).

(8)

The radius of the storage container is R. Water in the storage is assumed to be fully mixed and therefore has a spatially-lumped, time-varying temperature Tw(t). The position of the storage is assumed deep; hence, the farfield temperature around the storage may be taken as constant and equal to the deep-ground temperature T=. The transient temperature of the ground is assumed to have a one-year period beginning in July. In practice, the system may require 5-15 years to become really periodic. The energy balance equation for storage is Q = pwVwc,dTw/dt - kA(OT/Or)lR.t.

(9)

The dimensionless forms of Eqs. (5-9) are ,92~/~x2 = ~b/~l",

(10)

¢(1,r) = 4,w(¢),

(ll)

~,'r)

= 0,

~x,O) = ~ x , Y ) , q = pd~bw/dr - (o~qJ/dx)l,.. + ~Oll.~.

(12) (13) (14)

The specified problem has been solved by Onsal [7]. The following expressions are obtained for the temperature distribution [~xo-)] in the geological structure: 0o

~xo') = ~

~,(x) exp[i21rn~-/Y],

(15)

~,(x) = thw, e x p { - (1 + i)(x[~-~/x/Y)(x-1)},

(16)

I~wn q,(rh - i~h)l(Tl 2 + rl]),

(17)

=

M. Inalli et

! 166

al

Table 1. Physical properties of the geological structure. Ground type

Density(kg/m3)

Thermal conductivity(W/m-K)

Thermal diffusivity(m2/s)

Specific heat(J/kg-K)

Clay Coarse gravel Granite Sand

1500 2050 2640 1500

1.4 0.519 3.0 0.3

!.1 x 10-6 1.39 x 10-7 1.4 x 10-6 2.5 x 10-7

848 1842 811 800

(18) where qn is the complex finite Fourier coefficient corresponding to heat input or output to and from the storage. Details of the calculation are given in Ref. [7]. 2.3. Calculation of the monthly-average value of the net heat gain for the storage In terms of dimensionless quantities, the net heat gain to the storage will be equal to the heat collected by solar collectors minus the energy extracted by the heat pump, i.e. q(~') = qs(l") - qh('r) + w('r)/T,

(19)

here, qh('r) and q~(~') are calculated by applying the degree-day and ~b methods [8], respectively. The

following equations hold for the ~b method:

(20)

qs('r)= CAcFR( ~'ot)H.rdp,

(21)

= exp{[a + b(RnlR)][Xc + c~r~]}.

Calculation techniques for the constants a, b and c and the values o f R , , R and Xc are given in Ref. [8]. 3. NUMERICAL VALUES AND SOLUTION PROCEDURE Black, two-glass, flat-plate solar collectors were used. The parameters are bo = 0.15, (Tot), = 0.76, UL = 4.5 W/m2-K and FR = 0.95. The design heat load of the house is 10000 W. The winter design temperature and the inside design air temperature will be assumed equal to - 12.0°C and 20°C, respectively. Therefore, the UA-value of the house is 10000/[20 - ( - 12)] = 312.5 W/K, which is the ratio of the design heat load of the house to the winter design temperature difference. The problem has been studied for four types of geological structures with the physical properties listed in Table 1. The farfield, deep-ground temperature is assumed to be 10°C. The meteorological data used in the calculations for Elazl~-TUrkiye (38.7°N) are given in Table 2. In the computer program, we first form the monthly-average value of the daily solar radiation on a horizontal surface (see Table 2) and then convert this value to that on a tilted surface facing south by using the method of Lui-Jordan [8]. Iteration is done as follows: An initial storage temperature (e.g., the farfield, deep-ground temperature) is assumed. Then temperature distribution is computed in the presence of solar radiation. Depending on this temperature distribution, the efficiency of the solar collector and if necessary, the work of the heat pump are calculated. Thus, the net energy gain and the bulk temperature of the storage are obtained. A comparison between the initial and computed temperatures Table 2. Meteorological data used in the calculations for Elazli~, TUrkiye (38.7°N). Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Ta (*(2) R(MJ/m2-day)

- 1.3 4.9

0.0 8.0

4.7 10.9

11.8 15.3

17.4 17.2

22.9 21.1

27.2 20.9

2 7 . 0 22.0 19.9 15.6

14.8 10.2

7.8 5.9

1.5 4.6

A domestic solar heating system with underground spherical thermal storage

1167

70

60 "-I"-R

- S~ m

- " 4 k - - R - 7,S m

i"

+ R

40

=10~m

30

¢~

10 Jr 0J

i

I

I

i

I

D

I

I

I

I

i

Mmtth Fig. 2. Annual variation of the water temperature in the storage vessel of radius R buried in clay; the collector slope angle was 38.7°; Ac = 40 m 2.

is performed and if the fractional difference is satisfactorily small (10-5), iteration is terminated. Otherwise, it is continued. Typical iteration numbers, providing exactly the same results, are 30 or 100 depending on the relaxation coefficient. The number of terms used in the Fourier expansion is ten. It has been shown that including more terms has little or no effect on the results. 4. RESULTS AND DISCUSSION

4.1. The effect of the storage volume The annual variation of the storage temperature surrounded with clay is shown in Fig. 2. When the storage volume is increased, the annual average storage temperature and its amplitude decrease. It is seen that the highest temperatures occur during the summer season while the lowest occur towards the end of winter. Fig. 3 shows the variation of the monthly-average collector efficiency for several storage radii. Although the maximum collector efficiency is obtained in September for a container of 10-m radius, the minimum is encountered in the same month for a container of 2.5-m radius. Fig. 4 shows the monthly-average performance coefficient of the heat pump (COP) for the same radii. When R = 2.5, 5.0 and 7.5 m, the heat pump operates between December and April; it operates between November and April when R = 10.0 m. The variation of annual solar fraction (i.e. the rate of solar energy contributing to the heat load) is given in Fig. 5. As the collector area increases, the maximum solar fraction occurs at different storage radii. The optimum values were determined for a storage radius of 10 m at 80 7O #

6O

i-

40 t

O

20 10

~

~. ~-25m ~I--.-R ffi5.0 m ~ R ~7.~m ~ a-10Dm

I I

I

¢

I

t

I

I

I

t

I

I

t

Fig. 3. Annual variation of the collectorefficiencyin the storage vessel of radius R buried in clay; the collector slope angle was 38.7°; Ac = 40 m2.

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M. Laalli et al 10

8. IR=2.Sm nR=&0m

6

IR=7.Sm

(.1

E R m 10.0m 4

0

~

Mor~ Fig. 4. Monthly-average coefficient of performance of the heat pump in the storage vessel of radius R buried in clay; the collector slope angle was 38.7°; Ac = 40 m 2.

100 90

7O

50 - - d k - - R m7.5 m

4o

R m 10.0m

30 10

I

I

t

I

I

20

30

40

50

60

Area of collector, nf Fig. 5. The variation of annual solar fraction in the storage vessel of radius R buried in clay; the collector slope angle was 38.7 °.

60

--4k---Clr/

I 1Oo Mmth Fig. 6. Annual variation of the water temperature in the storage vessel of 5-m radius for different geological structures; the collector slope angle was 38.7°; Ac : 40 m 2.

A domestic solar heating system with underground spherical thermal storage

1169

70

at

O

80

40

3O

Fig. 7. Annual variation of the collector efficiency in the storage vessel of 5-m radius for different geological structures; the collector slope angle was 38.7°; Ac = 40 m2. a collector area of 10 m 2, a radius of 7.5 m for a collector area of 20 m 2, and a radius of 5 m for collector areas of 30, 40, 50, or 60 m 2. 4.2. Effect of the geological structure surrounding the storage The effects of the geological structure on the bulk temperature of the water, monthly-average collector efficiency, monthly-average performance coefficient of the heat pump (COP), and annual solar fraction are shown in Figs 6-9, respectively. Sand-surrounded storage systems have the highest temperatures whereas granite-surrounded systems have the lowest values as is expected on the basis of the thermal conductivities for these media. The highest COP is obtained when the storage is surrounded by sand. In general, geological structures with higher conductivities give the best results for collector areas smaller than 18 m 2. For greater collector areas, the average storage temperature increases and the storage system loses some of its energy to the ground which is regained in part. The existence of the intersection point in Fig. 9 is due to this mechanism. 4.3. The effect of collector slope In Fig. 10, the annual solar fraction vs the collector slope is given for different geological structures. In granite, the collector slope has only a small effect on the annual solar fraction. For different storage radii, the annual solar fraction vs collector slope is shown in Fig. 11. Because of higher efficiency, the optimum collector slope is found to be greater than the latitude angle for a storage radius of 2.5 m. For a 5-m radius, the optimum slope is equal to the latitude angle. It is less than the latitude angle for radii of 7.5 and 10 m. 10

m(:~/ QSana

mGmra) mO(m~ g m ~

Fig. 8. Monthly-averagecoefficientof performance of the heat pump in the storage vessel of 5-m radius for different geological structures; the collector slope angle was 38.7°; A~ = 40 m2.

! 170

M. Inalli ct al

IO0 go

70

60

c~

.--I---~d ~Gr~te

e,-

=

40 30i

10

Comomv~

i

I

i

I

t

20

30

40

50

60

A r e a of c o b ( : t o r , n'F Fig. 9. Variation of the annual solar fraction in the storage vessel of 5-m radius for different geological structures; the collector slope angle was 38.7 °.

100

i

9O 80

~ C ~ -4l--.Smd

60

~Grm~ •- - I I - - C o m e

5

0 0

10

m.~

~ 20

30

40

50

go

70

80

gO

Fig. 10. The effect of collector slope on annual solar fraction in the storage vessel of 5-m radius for different geological structures; Ac = 40 m 2.

gO

60

~RaT.Sm --Ill--R • 10.0m

501

0

10

I 20

30

I 40

I go

I go

I 70

I 80

I 90

e,,ok:~ mpL W Fig. I 1. The effect of collector slope on annual solar fraction in the storage vessel of radius R buffed in clay; Ac ffi 40 m 2.

A domestic solar heating system with undergroundspherical thermal storage

1171

5. CONCLUSIONS The computational model presented in this study may be used at low cost to determine the longterm performance of a spherical thermal energy storage vessel. It is seen from the present results that the storage temperature, collector efficiency, performance coefficient of the heat pump, and annual solar fraction indicate the technical feasibility of seasonal energy storage in the deep ground, possibly in existing wells. It is shown that a collector area greater than 60 m2/house does not yield a significant advantage for the climatic conditions of TUrkiye. Our results agree qualitatively with the Kerava solarvillage experiment [4]. REFERENCES 1. Bankston, C. A., in Advances in Solar Energy. Plenum Press, New York, 1988, pp. 352-444. 2. BrunstrOm, C., Larsson, M., Hoist, P., Zinko, H. and Hillstrllm, C. G., The Lyckebo Project--a Swedish central solar heating plant with seasonal storage, Report no. UL-FUD-B 85: 13, 1985. 3. Kenisarin, M. M., Lund, P. D. and Karabaev, M. K., Applied Solar Energy, 1988, 24, 52. 4. Makinen, R. and Lund, P. D., Kerava solar village--assisted heat pump system with long-term heat storage, Report TKK-F-A521-Otaniemi, 1983. 5. Lund, P. D. and Ostman, M. B., Solar Energy, 1985, 34, 351. 6. Onsal, M., Proceedings of the XXI International Symposium--Heat and Mass Transfer in Building Material and Structure, Dubrovnik, Yugoslavia, 4-8 September 1989. 7. Onsal, M., in Recent Advances in Fourier Analysis and Applications, ed. J. S. Byrnes and S. L. Byrnes. Klewer Academic Publishers, Dordrecth, The Netherlands, 1990, p. 59. 8. Duffle, J. A. and Beckman, W. A., Solar Engineering of Thermal Processes. John Wiley and Sons, New York, 1980. 9. Kakaq, S. and Yener, Y., Heat Conduction, Middle East Technical University, Ankara, Publication number: 60, 1979. NOMENCLATURE A = Surface area of the spherical storage vessel Ac = Collector surface area bo = Collector constant c = Specific heat of the geological structure cw = Specific heat of water C= ll(4~rRk T®) COP = Performance coefficient of the heat pump F~ = Collector heat-transfer factor H = monthly-average dally radiation on a horizontal surface I:Ir = monthly-average dally radiation on a tilted surface facing south k = Thermal conductivity of the geological structure P = (Pw cw)/(3pc) q = Dimensionless net heat rate, Ql(4~r

RkT~) qn = Dimensionless rate of heat loss from the house q, = Complex transform of the dimensionless energy transferred to the storage qs = Dimensionless solar beat-gain rate Q = Net energy input to the storage system Qh = Monthly-average heat load of the house r = Radial coordinate R = Radius of the spherical storage t = Time T = Temperature of the geological structure surrounding the thermal storage

Ta = Outside air temperature Th = Temperature of the fluid at the load-side heat exchanger T~= Inside design air temperature Tw = Temperature of water in the storage vessel T~-- Farfleld deep-ground temperature u=-- ( UA)hI( UA)he u~= Collector heat-loss coefficient ( UA )h = UA-value of the house (UA)he = UA-value for the load-side heat exchanger vw= Volume of the spherical storage vessel W-- Dimensionless heat-pump work W = Heat-pump work x - - rlR Y = Dimensionless time for one year, a(one year)/R2 a = Thermal diffusivity of the ground /3 = Heat-pump coefficient *It, -02= Defined in Eq. (18) ~b= Dimensionless temperature, (T T~)/T® ~ba= Dimensionless ambient temperature, (Ta - T~)/T® ~b~ = Dimensionless inside design air temperature, ( T~ - T~)/T® ~b~,= Dimensionless water temperature of the spherical storage, (Tw - T®)/T® ~bwn= Complex transform of the dimensionless water temperature in the spherical storage (b = The concept of monthly-average daily utilizability y = (47rRk)/( UA)h

1172

M. Inalli et al

~,= x 4gx) p = Density of the geological structure Pw = Water density ¢ = Dimensionless time 0n = Complex transform of the dimensionless geological structure temperature

(ca) = Monthly-average collector transmittance-absorbance product (va)~ = Value of ca for beam radiation in the normal direction with respect to the collector surface