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Thermal performance of a combined packed bed–solar pond system—a numerical study
PERGAMON
Applied Thermal Engineering 18 (1998) 1207±1223
Thermal performance of a combined packed bed±solar pond systemÐa numerical study F. Al-Juwayhel *, M.M. El-Refaee College of Engineering, Kuwait University, P.O. Box 5969, Kuwait Received 2 November 1997
Abstract Solar ponds have recently become an important source of energy that is used in many dierent applications. The technology of the solar pond is based on storing solar energy in salt-gradient strati®ed zones. Many experimental and numerical investigations concerning the optimum operational conditions and economical feasibility of solar ponds have been published in the last few decades. In the present study a modi®ed solar pond with a rock bed inserted at the bottom is suggested and investigated. In order to conduct this study and predict the thermal performance of the combined system, a onedimensional transient numerical model is developed. The boundary conditions are based on measured ambient and ground temperatures at Kuwait city. The model is validated for standard plain saltgradient solar ponds and is then used to examine the thermal performance of the combined storage system for dierent rock material and bed geometry. It has been shown that the storage temperature is remarkably increased when low thermal diusivity rocks (such as Bakelite) are used in the packed bed. Meanwhile, when high conductive rocks are used, the thermal storage temperature considerably deteriorates and the temperature variations amplitudes are almost ¯attened out. The bed geometry also plays a signi®cant role in the storage process. As expected, an appreciable gain in the storage temperature was obtained for thicker rock beds. Low porosity rock beds, as well, produce higher storage temperatures in the storage zone. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Thermal storage; Solar ponds; Rock bed
Nomenclature [A] a ai,i ÿ 1, ai,i, ai,i + 1
temperature coecient tridiagonal matrix albedo (re¯ection factor) of the pond's surface elements of the coecient matrix [A]
* Author to whom correspondence should be addressed. 1359-4311/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 9 7 ) 0 0 1 0 1 - 4
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d Cp C Ho H0 Ha Hi hc k K K' Kae Kw L1 L2 L3 Lg LRB N Pp Ps Pt qc q qsr qsc qse qex r n i R S T T 8C TÄ t v x
day number starting from January 1 speci®c heat of the brine speci®c heat (air or ground), kJ/kg K global radiation on a horizontal surface, kWh/day average solar intensity, W/m2 available solar energy below the surface just after re¯ection from pond surface, W/m2 transmitted solar energy at cell, W/m2 convective heat transfer coecient, W/m2 K iteration solution level thermal conductivity, W/m K eective thermal conductivity of the brine solution in the, W/m K average eective thermal conductivity (brine and rock), W/m K convective thermal conductivity, W/m K thickness of the UCZ, m thickness of the NCZ, m thickness of the LCZ, m thickness of the ground zone, m thickness of rock bed, m number of cells in the pond-ground model partial pressure of water vapour in the ambient air, mm Hg vapour pressure of water at the surface temperature (Ts), mm Hg atmospheric pressure, mm Hg heat transfer by conduction at any cell boundary, W/m2 heat losses at the pond surface, W/m2 heat losses due to radiation from pond surface, W/m2 heat losses due to convection from pond surface, W/m2 heat losses due to evaporation from pond surface, W/m2 rate of heat extraction in the storage zone (LCZ), W/m2 elements of the right-hand side of vector R explicit right-hand side vector in the temperature algebraic system salinity concentration in weight percent temperature vector in the system [A]T(n + 1) = R temperature, 8C amplitude of storage temperature variation, 8C time from start of operation wind speed, m/s depth below water surface, m
Subscripts a i g 0 max s st sky
air cell ground initial condition maximum surface storage sky
Greek symbols r l eo
density, kg/m2 latent heat of water evaporation, kJ/kg emissivity of water
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223 s o
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Stefan-Boltzmann constant, W/m2 K4 overrelaxation parameter
1. Introduction Salt gradient solar pond (SGSP) technology oers promise as a cost eective method of collecting and storing solar energy. It is a relatively shallow body of water in which a stabilizing salinity gradient prevents thermal convection, thus allowing the pond to act as a trap for solar radiation. The application of solar ponds include space heating, power generation and desalination. The con®guration of a solar pond (Fig. 1) is generally identi®ed by four dierent zones: the upper convective zone (UCZ), the non-convective zone (NCZ), the lower convective zone (LCZ) and the ground zone (GZ). In the upper connective zone (UCZ), a thin convective layer of low salinity water exists. The bottom brines are known as the lower convective storage zone (LCZ). The high salt concentration in the LCZ necessitates a transition gradient zone to the upper fresh water zone (see Fig. 1). The non-convective gradient zone (NCZ) insulates the LCZ from the cooler UCZ. Once a solar pond is established, part of the solar radiation striking the surface penetrates the pond zones to reach the LCZ where it is trapped. Part of this trapped energy is stored in the LCZ and the other part is either transferred back into the water by conduction or lost through the ground and pond sides. For ecient operation of a solar pond, the storage temperature should be high to obtain maximum corresponding Carnot eciency. It is important, as well, to maintain this high storage temperature during all seasons by decreasing the amplitudes of the storage temperature variation. Therefore, most of the investigations were devoted in the last decade to develop storage means that are capable of achieving these goals. The main objective of the present study is to examine a possible enhancement of the thermal performance of a solar pond when a rock-bed is inserted at the bottom of the pond. Although the combined rock bed±solar pond system is simple in terms of the physical concepts involved, its theoretical description and operational characteristics are complicated. Additionally, in spite of the fact that experimental investigation of the combined system is essential, it is still costly and very time consuming. Any alteration of the combined rock bed± solar pond system after it has been established, e.g. an increase or decrease in the porosity of the rock bed, is considered to be a major operation especially for large ponds. Moreover, due to the very long response time of the system, a complete rock bed±solar pond analysis becomes a very much complicated and tedious task. Therefore, theoretical analysis and computer simulation are handy tools for evaluating and studying the in¯uence of a large number of controlling parameters involved in the combined system. The only condition is that the physics of the rock bed and the solar pond should be correctly simulated in the model. Several aspects of the plain solar pond operation have been studied by many investigators. The majority used a one-dimensional model to simulate and analyze the performance of solar ponds [1±16]. However, El-Refaee et al. [17] demonstrated that the two-dimensional eects diminish when the pond aspect ratio exceeds 4. In the present study, a one-dimensional model, which assumes small convective motions and very slow salt diusion, is closer to the real case. Therefore, a one-dimensional, central dierence numerical model is developed in the present
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Fig. 1. Rock bed±salt gradient solar pond (RBSGSP), con®guration and zones.
work to simulate basic concepts of the new solar pond with rock bed (RB) at the bottom. The developed code is ®rst validated against previous predictions for plain salt gradient solar ponds and is then used to study the eects of the controlling parameters of the RB (rock material and geometry) on the thermal performance of the combined system. The following sections detail the mathematical and the numerical modeling, boundary and initial conditions. Predictions are obtained for real operational conditions for two years of simulation.
2. Mathematical modeling In this section, the mathematical formulation that governs the transient thermal behavior of the dierent regions inside the rock bed-solar pond system is presented. 2.1. Absorption of radiation in solar ponds The solar energy transmitted at any depth x below the pond surface is gradually attenuated due to the absorption of the radiation while traveling downward in the pond (Fig. 2). General
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relationships that relate the transmitted solar radiation and the pond distance x are available in the literature [18±20]. Hawlader [21] and many other investigators express the relationship as an exponential function: Hi 1 ÿ bHa eÿgx
x > 0:
1
The value of Ha represents the available solar radiation just below the pond surface; Ha is related to the global radiation Ho on a horizontal surface by the equation: o 1 ÿ a; Ha H where a is the albedo (re¯ection factor) of the surface [22] and is given as a function of the
Fig. 2. Energy ¯uxes for dierent cells in the RBSGSP.
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solar zenith distance. In the present study a simpli®ed attenuation relationship [21, 22, 7] is used to express the transmission depth relation as: Hi Ha 0:36 ÿ 0:08 ln xi ;
2
where xi is the distance from the pond surface to the layer inside the pond. The rate of attenuation is decreasing sharply near the surface due to the full absorption of the infrared (invisible) part of the solar radiation in the absorption surface boundary layer. The Kuwait city meteorological data [23] for Ho, Ta, and Tg are used in the present simulation. The weather data have been averaged for the last few years. Then the data are recast in a simple Fourier three-terms form as: 2pd o d 5:36 1:69 cos ÿ 0:873p H 365 4pd 0:18 cos ÿ 0:814p kWh=day; 365
3
2pd 4pd ÿ p 0:93 cos ÿ 1:17p C; Ta d 27:54 10:23 cos 365 365
4
2pd 4pd ÿ 1:21p 0:46 cos ÿ 1:22p C Tg d 30:35 6:31 cos 365 365
5
2.2. Heat transfer model The solar pond zones are divided into a number of ®nite cells each of depth Dx (Fig. 2). Conservation of energy is then applied on each cell. The controlled salinity gradient inside the pond is kept constant during the operation of the pond (4% in the UCZ and 22% in the LCZ). 2.2.1. Energy conservation for the surface cell From Fig. 2A, the energy balance gives: Dx @T ; Ha ÿ H3=2 ÿ qcs ÿ qL rs Cps 2 @t 3=2
6
where H3/2 is the transmitted solar radiation at the interface layer between the surface cell and the succeeding cell. Similarly, the temporal derivative (@ T/@t)3/2 is calculated at the same above-interface layer. The energy loss qL is due to radiation, convection and evaporation at the surface: qL qsr qsc qse :
7
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The magnitudes of the above terms are calculated as follows: qsc hc Ts ÿ Ta ;
8
where hc=5.7 + 3.8v [18], qse
lhc ps ÿ pp ; 1:6ca pi
9
qsr eo s T4s ÿ T4sky :
10
The sky temperature Tsky can be expressed [24] as: p Tsky Ta 0:55 0:061 pp 0:25 : The heat conducted downward from the surface cell, qcs, is given as: @T qcs ÿK03=2 @x 3=2
2.2.2. Energy conservation for other cells in the UCZ From Fig. 2B, the energy balance for cell B is given as: @T Hiÿ1=2 ÿ Hi1=2 qciÿ1=2 qci1=2 ri Cpi Dx ; @t i where, qciÿ1=2
ÿK0iÿ1=2
qci1=2
ÿK0i1=2
@T @x @T @x
11
12
13
iÿ1=2
; 14
i1=2
The term K' (the eective conductivity of the brine solution in the UCZ) is related to the molecular thermal conductivity of the brine by K' = 30 K. It should be noted that the brine density, thermal conductivity, and speci®c heat are taken to be variable with the temperature and salinity percentage; the relations are given [22] as: ri 998 650Si ÿ 0:4 Ti ÿ 20 "
kg=m3 ;
15
# 343:5 370Si Ti 273 1=3 Ki exp ln 0:24 0:2Si 2:3 ÿ 1ÿ W=m K; 16 647 30Si Ti 273
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Cpi 5:328 ÿ 97:6S2i 404S2i ÿ0:006913 7351Si ÿ 3:15S2i Ti 273 9:6 10ÿ6 ÿ 0:00193Si 0:00823S2i Ti 2732 2:5 10ÿ9 1:666 10ÿ6 Si ÿ 7:125 10ÿ6 S2i Ti 2733
kJ=kg K;
17
where Si is the salinity of the brine at index i. 2.2.3. Energy equation for the cells in NCZ and LCZ Equation (13) is generally applicable for each cell of NCZ and LCZ. In the NCZ the thermal conductivity Ki is calculated by using the relation of equation (16). However, an eective thermal conductivity K' = 5K is used in the LCZ [14]. To account for the extracted heat from the LCZ, equation (13) is modi®ed as: @T 0 ; 18 Hiÿ1=2 ÿ Hi1=2 qciÿ1=2 ÿ qci1=2 ÿ qex ri Cpi Dx @t i where q'ex is the rate of heat extracted per cell in the storage zone (LCZ). 2.2.4. Energy equation for the cells at the interface layer between the LCZ and GZ From Fig. 2C, the energy balance for cell C is given as: @T ; H qc iÿ1=2 ÿ qi1=2 ÿ q0ex ri Cpi Dx @t i where qciÿ1=2 ÿ
qci1=2
K0iÿ1=2
@T @x
19
iÿ1=2
;
@T ÿ Kg : @x i1=2
The energy balance for cell D in the ground zone is given as: @T qciÿ1=2 ÿ qci1=2 rg Cg ; @t i where qciÿ1=2
ÿK0iÿ1=2
qci1=2 ÿKg
@T @t
iÿ1=2
@T : @t i1=2
and
20
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2.2.5. 2.2.6. Energy equation for the cells in the rock-bed (RB) The energy balance for cells in the RB layer is expressed in exactly the same way as the one used for LCZ with the only exception that average eective properties (brine and rocks) rather than the absolute properties are used in the energy equation. The average eective thermal conductivity, speci®c heat and density (Kae, Cpae and rae) for the rock bed layer have been calculated using the following expressions: Kaeff 1 ÿ eKR e K0w ; Cpaeff 1 ÿ eCpR e Cpw ; raeff 1 ÿ erR e rw ; where e is the porosity of the rock bed, and R and w refer to the rock and water (Saline) convective properties, respectively. In order to compensate for the high convection current in the top saline layer of the LCZ, a convective conductivity K'w is assumed to be equal to ®ve times the conductivity of the saline K'w=5Kw. However, its value in the rock bed layer is reduced down to three times the saline conductivity. This is due to the damping eect of the rocks on the convection current in the rock bed, K'w=3Kw. To account for the rock porosity, the absorbed solar energy for each cell in the RB zone is modi®ed by the factor eÿ0.1 [25]. 2.2.7. Energy equation for the cells in the ground zone (GZ) From Fig. 2E, the energy balance for cell E gives: ÿKg
@T @T @T Kg rg Cg Dx : @x iÿ1=2 @x i1=2 @t i
21
2.2.8. Boundary and initial conditions The transient ambient and ground temperatures given by equations (4) and (5) are used as boundary conditions. Initial brine and soil (ground) temperatures are taken to be 258C.
3. Parametric studyÐresults and discussions In order to determine the eect of the addition of a rock bed layer at the bottom of the lower convective zone (LCZ), a parametric study is carried out, using the numerical modeling described in Appendix A, for dierent values of the controlling variables such as the bed material and geometry (porosity and height of the rock bed). Figure 3 compares the thermal performance between a standard plain solar pond and a Bakelite rock bed±solar pond. The dimensions and properties of the standard solar pond are as follows: UCZ thickness NCZ thickness LCZ thickness
0.3 m 1.3 m 1.4 m
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GZ thickness Cell spacing Time increment Initial brine temperature
3.0 m 5 cm 24 hours 258C
Operation period is 28 months, starting from 1st April . . . Average wind speed 5.0 m/sec Water emissivity 0.97 Surface re¯ection factor 0.05 Thermal extracted loading 10 W/m2 No. of heating days 90 days Ground: Ground thermal conductivity Ground speci®c heat Ground density
Kg=0.85 W/m K Cpg=0.9 kJ/kg K rg=1550 Kg/m3
The rock bed layer dimensions and properties are: Rock Porosity e 30% Rock bed layer depth 30% from the bottom of the LCZ Cp=1.59 kJ/kg K K = 0.232 W/m K r = 1273.5 kg/m3 a = 1.14 10ÿ7 m2/s It is evident from Fig. 3 that a signi®cant increase in the storage temperature is obtained when rock-bed is inserted at the bottom of the regular solar pond. As indicated in the ®gure the storage temperature has increased from 708C to 938C in the second year of the pond operation. The thermal performance of the combined rock bed±solar pond system is, however, in¯uenced by a number of controlling parameters such as the rock material, rock-bed height and porosity. The detailed analysis of the contributions of these parameters are shown next.
3.1. Eect of the rock-bed material Figure 4 shows the temperature variations in the storage zone versus months of the year for three dierent rock materials: Bakelite, Sandstone and Fireclay rock (referred to as clay in the ®gure). Most of which have the characteristics necessary for high thermal storage. It is clear that the storage temperature increases when a low thermal diusivity (a) rock bed (Bakelite) is used. This favorable eect of the thermal diusivity on the storage temperature, however, is not straightforward since the storage temperature still increases even when a rock bed with slightly higher a(1.89 10ÿ7) is used. Therefore, it looks like a itself is not the only controlling factor, but rather the individual values of r, Cp and K who play an additional important role in the thermal storage process (Fig. 5).
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Fig. 3. Eect of inserting a rock bed at the bottom of a solar gradient solar pond.
Fig. 4. Eect of rock material on the thermal performance of the RBSGSP.
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Fig. 5. Comparison between Bakelite rock-bed and other rock-beds.
Fig. 6. Eect of the rock-bed porosity on the thermal performance of the RBSGSP.
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Fig. 7. Eect of the rock-bed thickness on the thermal performance of the RBSGSP.
3.2. Eect of the rock bed porosity, e The eect of the porosity, spaces between rocks, of the rock bed is shown in Fig. 6 for four dierent values of e (20%, 30%, 40% and 50%) and at standard values of the other controlling parameters (L1, L, L3, Qex and LRB). Clearly, the storage temperature increases by the decrease of the porosity. Rock beds of very small porosity, however, have a negative eect on the storage temperature since in this case the rock-bed is considered to be an extension of the ground zone. In this regard we should note that there is an optimum critical value of the porosity that should be used to obtain maximum gain in the storage temperature. This critical value depends on many factors such as the material of the rocks, thickness of the rock layer, heights of the dierent zones of the combined system and the rate of heat extracted.
3.3. Eect of the rock bed thickness, LRB The time-history of the storage temperature for dierent heights of the rock bed and at standard values of the other controlling parameters (L1, L, L3, Qex and e) are illustrated in Fig. 7. As expected, increasing rock layer thickness produces an appreciable increase in the storage temperature. The gain in the storage temperature, however, is not linearly proportional with the rock bed height. Maximum gain in the storage temperature is achieved when the bed height is increased from 40% to 50% of the LCZ. Whereas, less gain is obtained when the bed height is increased from 20% to 30% of the LCZ.
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4. Concluding remarks The conclusions of the present work are summarized in the following: 1. It is concluded that inserting a rock bed (RB) at the bottom of a salt gradient solar pond generally enhances the thermal energy storage in the LCZ. A Bakelite rock bed (a = 1.14 10ÿ7 m2/s) of thickness 30% of the LCZ and porosity of 30% enhances the storage temperature of a standard solar pond (L1=0.3 m, L2=1.3 m, L3=1.4 m, Qex=10 W/m2) by 22%, Fig. 7. 2. Based on the time-history of the storage temperatures (Fig. 6), rock beds of low porosity (30%±20%) have higher maximum storage temperatures than those with low porosity (40%±50%). Decreasing the rock-bed porosity from 30% to 20%, led to an incremental increase in the storage temperature of 3.5%. This incremental increase, however, decreases when the porosity is reduced from 40% to 30% (2.7%) or from 50% to 40% (1.8%). 3. Additional increase in the storage temperature could be achieved when a thicker rock bed with smaller porosity is used, Figs. 6 and 7. An increase of about 13% in the storage temperature is achieved when the height of the RB is increased from 20% to 50% of the LCZ. Whereas, a gain in the maximum storage temperature of about 8% is obtained when the rock bed porosity is decreased from 50% down to 20%, Fig. 6. 4. Rocks of lower thermal diusivity, a, produce higher peak storage temperature in the LCZ. The variation amplitude of the storage temperature, however, increases with the increase of a, Table 1 and Fig. 5. This conclusion means that rocks of low thermal diusivity should be used to enhance the storage performance of the present combined system. 5. Numerical experiments on the combined system show the important role of the thermal conductivity to ¯atten out the variations of storage temperature. Table 2 presents typical values of the storage temperatures and temperature amplitudes for three rocks of dierent thermal conductivity and the same thermal diusivity. As expected, it is evident in the table that packed beds with higher thermal conductivity rocks produce almost ¯at storage Table 1 Rock material
a(m2/s)
Tmax(8C)
TÄ(8C)
Imaginary rock Bakelite Marble
0.0098 10ÿ5 0.0114 10ÿ5 0.1390 10ÿ5
96 94 89
34 36 39
Table 2 a(m2/s) 1.14 10ÿ7 1.14 10ÿ7 1.14 10ÿ7
K(W/m K) 0.232 10.0 30.0
Tmax(8C)
TÄ(8C)
95 82 29
36 34 1
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Table 3 a(m2/s) 1.14 10ÿ7 1.14 10.ÿ7
r(Cp)
K(W/m K)
r(r,kg/m3)
Cp(kJ/kg K)
Tmax(8C)
1000 4380 4380 1000
0.5 0.5
100 4380
4380 1000
100 84
temperature pro®les. The maximum storage temperature, however, is signi®cantly reduced with the increase of the thermal conductivity. 6. The speci®c heat of the rocks has a remarkable eect on the maximum temperature of the storage zone. Two types of rocks of dierent thermal speci®c heats and having the same thermal diusivity and conductivity were used to conduct two numerical experiments. Temperature predictions are listed in Table 3. As is clear in the table, rocks with high speci®c heat predict higher maximum temperature than low thermal speci®c heat rocks.
Appendix A A.1. Numerical modeling The governing partial dierential equations (6), (18), (19) and (21) are parabolic in time and elliptic in space. An implicit central dierencing scheme is used to approximate the spatial derivatives, and forward dierencing is used to approximate the temporal derivatives. For the surface cell A (Fig. 2A) equation (4) is approximated as: ! T n1 ÿ T n1 Dx 0 2 1 K3=2 1 ÿ aH0 1 ÿ 0:36 0:08 ln 2 Dx ÿ T n1 ÿ ÿ hc T n1 a 1
n r n 1 Cpi
Dx T1 ÿ 2
n ln hc P n s ÿ Pp
n 1:6C n 2 Pt T2 n1 ÿ T1
2Dt
ÿ eo s T41 ÿ T4sky n
T2 n
! ;
A1
where (n + 1) denotes the implicit new time level and (n) denotes the old time level. This ®nitedierence equation could be recast into the following form: a11 T n1 a12 T n1 r n 1 2 1 ;
a11
K3=2 n n Dx ; ÿ h c r1 Cp 4Dt Dx
A2 A2a
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a12
r n 1
K3=2 n n Dx ÿ r1 Cp1 4Dt Dx
n l n hc P n s ÿ Pp n 1:6C n 2 Pt
eo s T41
n ÿ hc T n1 ÿ r n i Cpi a
A2b
ÿ
T4sky n
Dx a ÿ 1H0 1 ÿ 0:36 0:8 ln 2
Dx T1 T2 n : 2 2Dx
A2c
Similarly, the ®nite-dierence equations for the rest of the cells in the brine and soil are obtained by using the same dierence scheme. The obtained system of dierence equations is a tridiagonal algebraic system of the form: n1 n ai;i1 T n1 ai;iÿ1 T n1 iÿ1 ai;i Ti i1 ri ;
A3
where ai,i ÿ 1, ai,i and ai,i + 1 are known elements of the coecient tridiagonal matrix [A], and is the right-hand-side vector of the linear system [A] [ T](n + 1)=R(n). The index i varies r(n) i from one up to N, where N is the total number of cells in the brine and soil. This linear system is solved numerically for the unknown T(n + 1) by the Gauss-Seidel relaxation procedure. The solution is unconditionally stable because all nonlinear terms are delayed by one time step. The following iterative loop for each time step is written as: ri;n ai;iÿ1 k1 ÿ T ÿ T k A4 T~ k1 i;n1 ÿ i1;n1 ; ai;i ai;i iÿ1;n1 k k ~ k1 T k1 i;n1 Ti;n1 o Ti;n1 ÿ Ti;n1 ;
A5
where k is the iteration solution level and o is the overrelaxation parameter (o = 1.3). Iterations seize when the iteration convergence limit is reached.
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[11] T.S. Jayadev, J. Henderson, Salt concentration gradient solar pond and optimization, Annual Report, Solar Energy Research Institute, Golden, Colorado, USA. [12] K.A. Meyer, Numerical model to describe the layer behaviour in SGSP, J. Solar Energy Eng 105 (1983) 341± 347. [13] Y.S. Cha, W.W. Schertz, Modeling of surface convective layer of SGSP, J. Solar Energy Eng 104 (1982) 293± 298. [14] M.A. Hassab, I.A. Tag, I.A. Jassim, F.Y. Al-Juburi, Solar pond design for Arabian Gulf conditions, Applied Energy 28 (1987) 191±213. [15] W.A. Kamal, A parametric study of the small-scale solar pond (SSSP), Energy Convers, Mgmt 26 (314) (1986) 337±345. [16] T. A. Newell, Investigation of salt strati®ed solar pond operational characteristics, PhD thesis, MIE Department, University of Utah, USA, 1980. [17] M.M. El-Refaee, R.R. Mansour, F. Al-Juwayhel, Transient performance of a two-dimensional salt gradient solar pondÐa numerical study, International Journal of Energy Research 20 (1996) 713±731. [18] Y.Y.N. Kishore, Y. Joshi, A practical collector eciency equation for non-convecting solar ponds, Solar Energy 33 (1984) 391±395. [19] C.L. Gupta, R.R. Isaac, S.M. Patel, Design procedure for a solar pond, Regional Journal of Energy, Heat and Mass Transfer 2 (1980) 193±203. [20] J.R. Hull, K.V. Liu, Y.S. Cha, W.T. Sha, J. Kamal, C.E. Nielson, Dependence of ground heat losses upon solar pond size and perimeter insulation: Calculated and experimental results. Solar World Forum. Proceedings of International Solar Energy Society Congress, Brighton, England. 1981, pp. 661±666. [21] M. Hawlader, An analysis of the non-convective solar pond, Solar Energy 27 (3) (1980) 195±198. [22] M.A. Darwish, Solar ponds: A seasonal solar energy storage system for process heat application. 1st Saudi Engineering Conference, Jeddah. L: 1983, pp. 1±8. [23] M.I. Safar, Dust and climatological data in Kuwait. State of Kuwait, Directorate General of Civil Aviation, Meteorological Department, Kuwait, 1988. [24] H. Buchberg, J.R. Roulet, Simulation and optimization of solar collection and storage for house heating, Solar Energy 12 (1968) 13±50. [25] F. Al-Juwayhel, M.M. El-Refaee, Packed bed±salt gradient solar pondÐexperimental and numerical investigations. Final Report, Project Em070. Kuwait University, 1997. To be submitted. [26] A. Akbarzadeh, R.W.G. MacDonald, Y.E. Warg, Reduction of surface mixing in solar ponds by ¯oating rings, Solar Energy 31 (4) (1983) 377±380. [27] A.A. Al-Homoud, A. Al-Mara®e, A. Al-Kandari, E. Abou-Seido, The eect of gradient build-up time on the pond performance, International Journal of Ambient Energy 12 (1991) 1±7. [28] A.A. Al-Homoud, A. Al-Mara®e, Al-Kandari, A, Temperature distribution measurements for dierent operating conditions. Unpublished internal report, Kuwait Institute of Scienti®c Research (KISR), 1990. [29] M.S. Hispsher, R.F. Boehm, Heat transfer considerations of a non-convecting solar pond heat exchanger. Presented at ASME Winter Annual Meeting, Paper No. 76-WA/Sol-4, 1976. [30] J.R. Hirschmann, Salt ¯ats as solar heat collectors for industrial purposes, Solar Energy 13 (1970) 83±97. [31] J.R. Hull, Y.S. Cha, W.T. Sha, W.W. Schertz, Construction and ®rst year's operational results of the ANL Research Salt Gradient Solar Pond, Proceeding of AS/ISES 5 (1982) 197±202. [32] C.E. Nielson, J. Kamal, The 400 m2 solar pond: One year of operation, Proceedings of ASES 4 (1981) 758± 762. [33] A. Rabl, C.E. Nielson, Solar ponds for space heating, Solar Energy 17 (1975) 1±12. [34] R.A. Tybout, A recursive alternative to Weinberger's model of solar pond, Solar Energy 11 (2) (1967) 109±111. [35] D.D. Weeks, H.C. Bryant, What happens when a solar pond boils?. International Solar Energy Conference, Solar Rising, Philadelphia AS-ISES (1981). [36] Z.M. Zhang, Y.F. Wang, A study on the thermal storage of the ground beneath solar ponds by computer simulation, Solar Energy 44 (5) (1990) 243±248.
Applied Thermal Engineering 18 (1998) 1207±1223
Thermal performance of a combined packed bed±solar pond systemÐa numerical study F. Al-Juwayhel *, M.M. El-Refaee College of Engineering, Kuwait University, P.O. Box 5969, Kuwait Received 2 November 1997
Abstract Solar ponds have recently become an important source of energy that is used in many dierent applications. The technology of the solar pond is based on storing solar energy in salt-gradient strati®ed zones. Many experimental and numerical investigations concerning the optimum operational conditions and economical feasibility of solar ponds have been published in the last few decades. In the present study a modi®ed solar pond with a rock bed inserted at the bottom is suggested and investigated. In order to conduct this study and predict the thermal performance of the combined system, a onedimensional transient numerical model is developed. The boundary conditions are based on measured ambient and ground temperatures at Kuwait city. The model is validated for standard plain saltgradient solar ponds and is then used to examine the thermal performance of the combined storage system for dierent rock material and bed geometry. It has been shown that the storage temperature is remarkably increased when low thermal diusivity rocks (such as Bakelite) are used in the packed bed. Meanwhile, when high conductive rocks are used, the thermal storage temperature considerably deteriorates and the temperature variations amplitudes are almost ¯attened out. The bed geometry also plays a signi®cant role in the storage process. As expected, an appreciable gain in the storage temperature was obtained for thicker rock beds. Low porosity rock beds, as well, produce higher storage temperatures in the storage zone. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Thermal storage; Solar ponds; Rock bed
Nomenclature [A] a ai,i ÿ 1, ai,i, ai,i + 1
temperature coecient tridiagonal matrix albedo (re¯ection factor) of the pond's surface elements of the coecient matrix [A]
* Author to whom correspondence should be addressed. 1359-4311/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 9 7 ) 0 0 1 0 1 - 4
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F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
d Cp C Ho H0 Ha Hi hc k K K' Kae Kw L1 L2 L3 Lg LRB N Pp Ps Pt qc q qsr qsc qse qex r n i R S T T 8C TÄ t v x
day number starting from January 1 speci®c heat of the brine speci®c heat (air or ground), kJ/kg K global radiation on a horizontal surface, kWh/day average solar intensity, W/m2 available solar energy below the surface just after re¯ection from pond surface, W/m2 transmitted solar energy at cell, W/m2 convective heat transfer coecient, W/m2 K iteration solution level thermal conductivity, W/m K eective thermal conductivity of the brine solution in the, W/m K average eective thermal conductivity (brine and rock), W/m K convective thermal conductivity, W/m K thickness of the UCZ, m thickness of the NCZ, m thickness of the LCZ, m thickness of the ground zone, m thickness of rock bed, m number of cells in the pond-ground model partial pressure of water vapour in the ambient air, mm Hg vapour pressure of water at the surface temperature (Ts), mm Hg atmospheric pressure, mm Hg heat transfer by conduction at any cell boundary, W/m2 heat losses at the pond surface, W/m2 heat losses due to radiation from pond surface, W/m2 heat losses due to convection from pond surface, W/m2 heat losses due to evaporation from pond surface, W/m2 rate of heat extraction in the storage zone (LCZ), W/m2 elements of the right-hand side of vector R explicit right-hand side vector in the temperature algebraic system salinity concentration in weight percent temperature vector in the system [A]T(n + 1) = R temperature, 8C amplitude of storage temperature variation, 8C time from start of operation wind speed, m/s depth below water surface, m
Subscripts a i g 0 max s st sky
air cell ground initial condition maximum surface storage sky
Greek symbols r l eo
density, kg/m2 latent heat of water evaporation, kJ/kg emissivity of water
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223 s o
1209
Stefan-Boltzmann constant, W/m2 K4 overrelaxation parameter
1. Introduction Salt gradient solar pond (SGSP) technology oers promise as a cost eective method of collecting and storing solar energy. It is a relatively shallow body of water in which a stabilizing salinity gradient prevents thermal convection, thus allowing the pond to act as a trap for solar radiation. The application of solar ponds include space heating, power generation and desalination. The con®guration of a solar pond (Fig. 1) is generally identi®ed by four dierent zones: the upper convective zone (UCZ), the non-convective zone (NCZ), the lower convective zone (LCZ) and the ground zone (GZ). In the upper connective zone (UCZ), a thin convective layer of low salinity water exists. The bottom brines are known as the lower convective storage zone (LCZ). The high salt concentration in the LCZ necessitates a transition gradient zone to the upper fresh water zone (see Fig. 1). The non-convective gradient zone (NCZ) insulates the LCZ from the cooler UCZ. Once a solar pond is established, part of the solar radiation striking the surface penetrates the pond zones to reach the LCZ where it is trapped. Part of this trapped energy is stored in the LCZ and the other part is either transferred back into the water by conduction or lost through the ground and pond sides. For ecient operation of a solar pond, the storage temperature should be high to obtain maximum corresponding Carnot eciency. It is important, as well, to maintain this high storage temperature during all seasons by decreasing the amplitudes of the storage temperature variation. Therefore, most of the investigations were devoted in the last decade to develop storage means that are capable of achieving these goals. The main objective of the present study is to examine a possible enhancement of the thermal performance of a solar pond when a rock-bed is inserted at the bottom of the pond. Although the combined rock bed±solar pond system is simple in terms of the physical concepts involved, its theoretical description and operational characteristics are complicated. Additionally, in spite of the fact that experimental investigation of the combined system is essential, it is still costly and very time consuming. Any alteration of the combined rock bed± solar pond system after it has been established, e.g. an increase or decrease in the porosity of the rock bed, is considered to be a major operation especially for large ponds. Moreover, due to the very long response time of the system, a complete rock bed±solar pond analysis becomes a very much complicated and tedious task. Therefore, theoretical analysis and computer simulation are handy tools for evaluating and studying the in¯uence of a large number of controlling parameters involved in the combined system. The only condition is that the physics of the rock bed and the solar pond should be correctly simulated in the model. Several aspects of the plain solar pond operation have been studied by many investigators. The majority used a one-dimensional model to simulate and analyze the performance of solar ponds [1±16]. However, El-Refaee et al. [17] demonstrated that the two-dimensional eects diminish when the pond aspect ratio exceeds 4. In the present study, a one-dimensional model, which assumes small convective motions and very slow salt diusion, is closer to the real case. Therefore, a one-dimensional, central dierence numerical model is developed in the present
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Fig. 1. Rock bed±salt gradient solar pond (RBSGSP), con®guration and zones.
work to simulate basic concepts of the new solar pond with rock bed (RB) at the bottom. The developed code is ®rst validated against previous predictions for plain salt gradient solar ponds and is then used to study the eects of the controlling parameters of the RB (rock material and geometry) on the thermal performance of the combined system. The following sections detail the mathematical and the numerical modeling, boundary and initial conditions. Predictions are obtained for real operational conditions for two years of simulation.
2. Mathematical modeling In this section, the mathematical formulation that governs the transient thermal behavior of the dierent regions inside the rock bed-solar pond system is presented. 2.1. Absorption of radiation in solar ponds The solar energy transmitted at any depth x below the pond surface is gradually attenuated due to the absorption of the radiation while traveling downward in the pond (Fig. 2). General
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
1211
relationships that relate the transmitted solar radiation and the pond distance x are available in the literature [18±20]. Hawlader [21] and many other investigators express the relationship as an exponential function: Hi 1 ÿ bHa eÿgx
x > 0:
1
The value of Ha represents the available solar radiation just below the pond surface; Ha is related to the global radiation Ho on a horizontal surface by the equation: o 1 ÿ a; Ha H where a is the albedo (re¯ection factor) of the surface [22] and is given as a function of the
Fig. 2. Energy ¯uxes for dierent cells in the RBSGSP.
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solar zenith distance. In the present study a simpli®ed attenuation relationship [21, 22, 7] is used to express the transmission depth relation as: Hi Ha 0:36 ÿ 0:08 ln xi ;
2
where xi is the distance from the pond surface to the layer inside the pond. The rate of attenuation is decreasing sharply near the surface due to the full absorption of the infrared (invisible) part of the solar radiation in the absorption surface boundary layer. The Kuwait city meteorological data [23] for Ho, Ta, and Tg are used in the present simulation. The weather data have been averaged for the last few years. Then the data are recast in a simple Fourier three-terms form as: 2pd o d 5:36 1:69 cos ÿ 0:873p H 365 4pd 0:18 cos ÿ 0:814p kWh=day; 365
3
2pd 4pd ÿ p 0:93 cos ÿ 1:17p C; Ta d 27:54 10:23 cos 365 365
4
2pd 4pd ÿ 1:21p 0:46 cos ÿ 1:22p C Tg d 30:35 6:31 cos 365 365
5
2.2. Heat transfer model The solar pond zones are divided into a number of ®nite cells each of depth Dx (Fig. 2). Conservation of energy is then applied on each cell. The controlled salinity gradient inside the pond is kept constant during the operation of the pond (4% in the UCZ and 22% in the LCZ). 2.2.1. Energy conservation for the surface cell From Fig. 2A, the energy balance gives: Dx @T ; Ha ÿ H3=2 ÿ qcs ÿ qL rs Cps 2 @t 3=2
6
where H3/2 is the transmitted solar radiation at the interface layer between the surface cell and the succeeding cell. Similarly, the temporal derivative (@ T/@t)3/2 is calculated at the same above-interface layer. The energy loss qL is due to radiation, convection and evaporation at the surface: qL qsr qsc qse :
7
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
1213
The magnitudes of the above terms are calculated as follows: qsc hc Ts ÿ Ta ;
8
where hc=5.7 + 3.8v [18], qse
lhc ps ÿ pp ; 1:6ca pi
9
qsr eo s T4s ÿ T4sky :
10
The sky temperature Tsky can be expressed [24] as: p Tsky Ta 0:55 0:061 pp 0:25 : The heat conducted downward from the surface cell, qcs, is given as: @T qcs ÿK03=2 @x 3=2
2.2.2. Energy conservation for other cells in the UCZ From Fig. 2B, the energy balance for cell B is given as: @T Hiÿ1=2 ÿ Hi1=2 qciÿ1=2 qci1=2 ri Cpi Dx ; @t i where, qciÿ1=2
ÿK0iÿ1=2
qci1=2
ÿK0i1=2
@T @x @T @x
11
12
13
iÿ1=2
; 14
i1=2
The term K' (the eective conductivity of the brine solution in the UCZ) is related to the molecular thermal conductivity of the brine by K' = 30 K. It should be noted that the brine density, thermal conductivity, and speci®c heat are taken to be variable with the temperature and salinity percentage; the relations are given [22] as: ri 998 650Si ÿ 0:4 Ti ÿ 20 "
kg=m3 ;
15
# 343:5 370Si Ti 273 1=3 Ki exp ln 0:24 0:2Si 2:3 ÿ 1ÿ W=m K; 16 647 30Si Ti 273
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F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
Cpi 5:328 ÿ 97:6S2i 404S2i ÿ0:006913 7351Si ÿ 3:15S2i Ti 273 9:6 10ÿ6 ÿ 0:00193Si 0:00823S2i Ti 2732 2:5 10ÿ9 1:666 10ÿ6 Si ÿ 7:125 10ÿ6 S2i Ti 2733
kJ=kg K;
17
where Si is the salinity of the brine at index i. 2.2.3. Energy equation for the cells in NCZ and LCZ Equation (13) is generally applicable for each cell of NCZ and LCZ. In the NCZ the thermal conductivity Ki is calculated by using the relation of equation (16). However, an eective thermal conductivity K' = 5K is used in the LCZ [14]. To account for the extracted heat from the LCZ, equation (13) is modi®ed as: @T 0 ; 18 Hiÿ1=2 ÿ Hi1=2 qciÿ1=2 ÿ qci1=2 ÿ qex ri Cpi Dx @t i where q'ex is the rate of heat extracted per cell in the storage zone (LCZ). 2.2.4. Energy equation for the cells at the interface layer between the LCZ and GZ From Fig. 2C, the energy balance for cell C is given as: @T ; H qc iÿ1=2 ÿ qi1=2 ÿ q0ex ri Cpi Dx @t i where qciÿ1=2 ÿ
qci1=2
K0iÿ1=2
@T @x
19
iÿ1=2
;
@T ÿ Kg : @x i1=2
The energy balance for cell D in the ground zone is given as: @T qciÿ1=2 ÿ qci1=2 rg Cg ; @t i where qciÿ1=2
ÿK0iÿ1=2
qci1=2 ÿKg
@T @t
iÿ1=2
@T : @t i1=2
and
20
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1215
2.2.5. 2.2.6. Energy equation for the cells in the rock-bed (RB) The energy balance for cells in the RB layer is expressed in exactly the same way as the one used for LCZ with the only exception that average eective properties (brine and rocks) rather than the absolute properties are used in the energy equation. The average eective thermal conductivity, speci®c heat and density (Kae, Cpae and rae) for the rock bed layer have been calculated using the following expressions: Kaeff 1 ÿ eKR e K0w ; Cpaeff 1 ÿ eCpR e Cpw ; raeff 1 ÿ erR e rw ; where e is the porosity of the rock bed, and R and w refer to the rock and water (Saline) convective properties, respectively. In order to compensate for the high convection current in the top saline layer of the LCZ, a convective conductivity K'w is assumed to be equal to ®ve times the conductivity of the saline K'w=5Kw. However, its value in the rock bed layer is reduced down to three times the saline conductivity. This is due to the damping eect of the rocks on the convection current in the rock bed, K'w=3Kw. To account for the rock porosity, the absorbed solar energy for each cell in the RB zone is modi®ed by the factor eÿ0.1 [25]. 2.2.7. Energy equation for the cells in the ground zone (GZ) From Fig. 2E, the energy balance for cell E gives: ÿKg
@T @T @T Kg rg Cg Dx : @x iÿ1=2 @x i1=2 @t i
21
2.2.8. Boundary and initial conditions The transient ambient and ground temperatures given by equations (4) and (5) are used as boundary conditions. Initial brine and soil (ground) temperatures are taken to be 258C.
3. Parametric studyÐresults and discussions In order to determine the eect of the addition of a rock bed layer at the bottom of the lower convective zone (LCZ), a parametric study is carried out, using the numerical modeling described in Appendix A, for dierent values of the controlling variables such as the bed material and geometry (porosity and height of the rock bed). Figure 3 compares the thermal performance between a standard plain solar pond and a Bakelite rock bed±solar pond. The dimensions and properties of the standard solar pond are as follows: UCZ thickness NCZ thickness LCZ thickness
0.3 m 1.3 m 1.4 m
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GZ thickness Cell spacing Time increment Initial brine temperature
3.0 m 5 cm 24 hours 258C
Operation period is 28 months, starting from 1st April . . . Average wind speed 5.0 m/sec Water emissivity 0.97 Surface re¯ection factor 0.05 Thermal extracted loading 10 W/m2 No. of heating days 90 days Ground: Ground thermal conductivity Ground speci®c heat Ground density
Kg=0.85 W/m K Cpg=0.9 kJ/kg K rg=1550 Kg/m3
The rock bed layer dimensions and properties are: Rock Porosity e 30% Rock bed layer depth 30% from the bottom of the LCZ Cp=1.59 kJ/kg K K = 0.232 W/m K r = 1273.5 kg/m3 a = 1.14 10ÿ7 m2/s It is evident from Fig. 3 that a signi®cant increase in the storage temperature is obtained when rock-bed is inserted at the bottom of the regular solar pond. As indicated in the ®gure the storage temperature has increased from 708C to 938C in the second year of the pond operation. The thermal performance of the combined rock bed±solar pond system is, however, in¯uenced by a number of controlling parameters such as the rock material, rock-bed height and porosity. The detailed analysis of the contributions of these parameters are shown next.
3.1. Eect of the rock-bed material Figure 4 shows the temperature variations in the storage zone versus months of the year for three dierent rock materials: Bakelite, Sandstone and Fireclay rock (referred to as clay in the ®gure). Most of which have the characteristics necessary for high thermal storage. It is clear that the storage temperature increases when a low thermal diusivity (a) rock bed (Bakelite) is used. This favorable eect of the thermal diusivity on the storage temperature, however, is not straightforward since the storage temperature still increases even when a rock bed with slightly higher a(1.89 10ÿ7) is used. Therefore, it looks like a itself is not the only controlling factor, but rather the individual values of r, Cp and K who play an additional important role in the thermal storage process (Fig. 5).
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
Fig. 3. Eect of inserting a rock bed at the bottom of a solar gradient solar pond.
Fig. 4. Eect of rock material on the thermal performance of the RBSGSP.
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Fig. 5. Comparison between Bakelite rock-bed and other rock-beds.
Fig. 6. Eect of the rock-bed porosity on the thermal performance of the RBSGSP.
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
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Fig. 7. Eect of the rock-bed thickness on the thermal performance of the RBSGSP.
3.2. Eect of the rock bed porosity, e The eect of the porosity, spaces between rocks, of the rock bed is shown in Fig. 6 for four dierent values of e (20%, 30%, 40% and 50%) and at standard values of the other controlling parameters (L1, L, L3, Qex and LRB). Clearly, the storage temperature increases by the decrease of the porosity. Rock beds of very small porosity, however, have a negative eect on the storage temperature since in this case the rock-bed is considered to be an extension of the ground zone. In this regard we should note that there is an optimum critical value of the porosity that should be used to obtain maximum gain in the storage temperature. This critical value depends on many factors such as the material of the rocks, thickness of the rock layer, heights of the dierent zones of the combined system and the rate of heat extracted.
3.3. Eect of the rock bed thickness, LRB The time-history of the storage temperature for dierent heights of the rock bed and at standard values of the other controlling parameters (L1, L, L3, Qex and e) are illustrated in Fig. 7. As expected, increasing rock layer thickness produces an appreciable increase in the storage temperature. The gain in the storage temperature, however, is not linearly proportional with the rock bed height. Maximum gain in the storage temperature is achieved when the bed height is increased from 40% to 50% of the LCZ. Whereas, less gain is obtained when the bed height is increased from 20% to 30% of the LCZ.
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4. Concluding remarks The conclusions of the present work are summarized in the following: 1. It is concluded that inserting a rock bed (RB) at the bottom of a salt gradient solar pond generally enhances the thermal energy storage in the LCZ. A Bakelite rock bed (a = 1.14 10ÿ7 m2/s) of thickness 30% of the LCZ and porosity of 30% enhances the storage temperature of a standard solar pond (L1=0.3 m, L2=1.3 m, L3=1.4 m, Qex=10 W/m2) by 22%, Fig. 7. 2. Based on the time-history of the storage temperatures (Fig. 6), rock beds of low porosity (30%±20%) have higher maximum storage temperatures than those with low porosity (40%±50%). Decreasing the rock-bed porosity from 30% to 20%, led to an incremental increase in the storage temperature of 3.5%. This incremental increase, however, decreases when the porosity is reduced from 40% to 30% (2.7%) or from 50% to 40% (1.8%). 3. Additional increase in the storage temperature could be achieved when a thicker rock bed with smaller porosity is used, Figs. 6 and 7. An increase of about 13% in the storage temperature is achieved when the height of the RB is increased from 20% to 50% of the LCZ. Whereas, a gain in the maximum storage temperature of about 8% is obtained when the rock bed porosity is decreased from 50% down to 20%, Fig. 6. 4. Rocks of lower thermal diusivity, a, produce higher peak storage temperature in the LCZ. The variation amplitude of the storage temperature, however, increases with the increase of a, Table 1 and Fig. 5. This conclusion means that rocks of low thermal diusivity should be used to enhance the storage performance of the present combined system. 5. Numerical experiments on the combined system show the important role of the thermal conductivity to ¯atten out the variations of storage temperature. Table 2 presents typical values of the storage temperatures and temperature amplitudes for three rocks of dierent thermal conductivity and the same thermal diusivity. As expected, it is evident in the table that packed beds with higher thermal conductivity rocks produce almost ¯at storage Table 1 Rock material
a(m2/s)
Tmax(8C)
TÄ(8C)
Imaginary rock Bakelite Marble
0.0098 10ÿ5 0.0114 10ÿ5 0.1390 10ÿ5
96 94 89
34 36 39
Table 2 a(m2/s) 1.14 10ÿ7 1.14 10ÿ7 1.14 10ÿ7
K(W/m K) 0.232 10.0 30.0
Tmax(8C)
TÄ(8C)
95 82 29
36 34 1
F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
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Table 3 a(m2/s) 1.14 10ÿ7 1.14 10.ÿ7
r(Cp)
K(W/m K)
r(r,kg/m3)
Cp(kJ/kg K)
Tmax(8C)
1000 4380 4380 1000
0.5 0.5
100 4380
4380 1000
100 84
temperature pro®les. The maximum storage temperature, however, is signi®cantly reduced with the increase of the thermal conductivity. 6. The speci®c heat of the rocks has a remarkable eect on the maximum temperature of the storage zone. Two types of rocks of dierent thermal speci®c heats and having the same thermal diusivity and conductivity were used to conduct two numerical experiments. Temperature predictions are listed in Table 3. As is clear in the table, rocks with high speci®c heat predict higher maximum temperature than low thermal speci®c heat rocks.
Appendix A A.1. Numerical modeling The governing partial dierential equations (6), (18), (19) and (21) are parabolic in time and elliptic in space. An implicit central dierencing scheme is used to approximate the spatial derivatives, and forward dierencing is used to approximate the temporal derivatives. For the surface cell A (Fig. 2A) equation (4) is approximated as: ! T n1 ÿ T n1 Dx 0 2 1 K3=2 1 ÿ aH0 1 ÿ 0:36 0:08 ln 2 Dx ÿ T n1 ÿ ÿ hc T n1 a 1
n r n 1 Cpi
Dx T1 ÿ 2
n ln hc P n s ÿ Pp
n 1:6C n 2 Pt T2 n1 ÿ T1
2Dt
ÿ eo s T41 ÿ T4sky n
T2 n
! ;
A1
where (n + 1) denotes the implicit new time level and (n) denotes the old time level. This ®nitedierence equation could be recast into the following form: a11 T n1 a12 T n1 r n 1 2 1 ;
a11
K3=2 n n Dx ; ÿ h c r1 Cp 4Dt Dx
A2 A2a
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F. Al-Juwayhel, M.M. El-Refaee / Applied Thermal Engineering 18 (1998) 1207±1223
a12
r n 1
K3=2 n n Dx ÿ r1 Cp1 4Dt Dx
n l n hc P n s ÿ Pp n 1:6C n 2 Pt
eo s T41
n ÿ hc T n1 ÿ r n i Cpi a
A2b
ÿ
T4sky n
Dx a ÿ 1H0 1 ÿ 0:36 0:8 ln 2
Dx T1 T2 n : 2 2Dx
A2c
Similarly, the ®nite-dierence equations for the rest of the cells in the brine and soil are obtained by using the same dierence scheme. The obtained system of dierence equations is a tridiagonal algebraic system of the form: n1 n ai;i1 T n1 ai;iÿ1 T n1 iÿ1 ai;i Ti i1 ri ;
A3
where ai,i ÿ 1, ai,i and ai,i + 1 are known elements of the coecient tridiagonal matrix [A], and is the right-hand-side vector of the linear system [A] [ T](n + 1)=R(n). The index i varies r(n) i from one up to N, where N is the total number of cells in the brine and soil. This linear system is solved numerically for the unknown T(n + 1) by the Gauss-Seidel relaxation procedure. The solution is unconditionally stable because all nonlinear terms are delayed by one time step. The following iterative loop for each time step is written as: ri;n ai;iÿ1 k1 ÿ T ÿ T k A4 T~ k1 i;n1 ÿ i1;n1 ; ai;i ai;i iÿ1;n1 k k ~ k1 T k1 i;n1 Ti;n1 o Ti;n1 ÿ Ti;n1 ;
A5
where k is the iteration solution level and o is the overrelaxation parameter (o = 1.3). Iterations seize when the iteration convergence limit is reached.
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