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ScienceDirect Solar Energy 122 (2015) 783–794 www.elsevier.com/locate/solener

Heat loss from thermal energy storage ventilated tank foundations C. Sua´rez a,⇑, F.J. Pino b, F. Rosa b, J. Guerra b a

AICIA, Andalusian Association for Research & Industrial Cooperation, Camino de los Descubrimiento s/n, Edf. Escuela Superior de Ingenieros de Sevilla, 41092 Seville, Spain b Escuela Superior de Ingenieros, DIE – Grupo de Termotecnia, University of Seville, Avda. Descubrimientos s/n, 41092 Seville, Spain Received 6 July 2015; received in revised form 29 September 2015; accepted 30 September 2015 Available online 10 November 2015 Communicated by: Associate Editor Luisa F. Cabeza

Abstract Thermal energy storage tanks are highly insulated in order to minimize the heat losses through the top and lateral walls and the foundation. Typical tanks of state-of-the-art solar power plants include a ventilation system within the foundation in order to ensure that the working temperature reached in the concrete remains below a maximum allowable value. In the present work, a multilayer analytical model for the estimation of the tank’s bottom heat losses in steady state is developed, including separately the quantiﬁcation of the heat losses due to the ventilation system and the heat loss to the soil. A new correlation for the soil equivalent thermal resistance (which is unknown a priori) is previously obtained using a numerical model which is validated with the results of other authors. A comprehensive parametric analysis of the variables of interest is made and a set of cases covering a wide range of tank geometries, insulation levels, storage temperatures and maximum allowed concrete temperatures are solved. Finally, the obtained results are summarized, providing a quick method for the estimation of the total bottom heat losses and its components (the ventilation heat losses and the heat loss to the soil). These results provide useful information related to the tank foundation design, such as the quantiﬁcation of the evacuated heat due to the ventilation system or the selection of an appropriate bottom insulation thickness level depending on the tank geometry, the storage temperature, the ventilation system and the type of soil. Ó 2015 Elsevier Ltd. All rights reserved.

Keywords: TES; Heat loss; Soil; Foundation

1. Introduction Concentrating solar power (CSP) is unique among renewable energy generators because even though it is variable, like solar photovoltaic and wind, it can easily be highly dispatchable. Most of the currently installed thermal energy storage (TES) systems in utility-scale solar thermal electric plants store energy using sensible heat, employing molten salts in an indirect two-tank design (Kuravi et al., ⇑ Corresponding author. Tel.: +34 954487471.

E-mail addresses: [email protected] (C. Sua´rez), [email protected] (F.J. Pino), [email protected] (F. Rosa), [email protected] (J. Guerra). http://dx.doi.org/10.1016/j.solener.2015.09.045 0038-092X/Ó 2015 Elsevier Ltd. All rights reserved.

2013). The denominated cold and hot tanks are at two different temperature levels depending on the solar power plant (SPP) type (usually 292/386 °C for parabolic trough plants and 292/565 °C for central receiver plants). Even though the tanks are insulated, thermal losses from the tank to the environment (which occur through the tank’s walls, the roof and the foundation), while relatively small, are important and must be analyzed and minimized during the TES design process in order to improve the TES eﬃciency. Heat transfer through the ground has long been recognized as being a substantially more complex problem compared with that through components above ground

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Nomenclature Symbols A tank bottom area (m2) Cp speciﬁc heat capacity (J kg1 K1) D tank diameter (m) N number of nodes Q heat ﬂux (W) q heat ﬂux density (W m2) R tank radius (m) R0 thermal resistance (m2 K W1) T temperature (°C) x radial direction (m) z axial direction (m) Greek letters k material thermal conductivity (W m1 K1) k soil thermal conductivity (W m1 K1) q density (kg m3)

(Anderson, 1991). Due to the coupled nature of the problem, apart from the foundation construction characteristics, the soil properties also play an important role in the foundation heat losses. The thermal conductivity of soil is a major determinant of ground heat transfer: in ﬁrst approximation the heat ﬂow through an insulated ﬂoor is directly proportional to this quantity (Anderson, 1991). A very important number of research eﬀorts on groundcoupled heat losses applied to buildings can be found in the literature. Research on ground-coupled losses applied to buildings started during the 1940s and diﬀerent analytical and semi-analytical methods for determining the earthcontact heat losses can be found in the open literature (Macey, 1949; Delsante et al., 1983; Hagentoft, 1988b; Anderson, 1991; Davies, 1993b). Numerical methods have been also developed to simulate complex systems, where the direct application of analytical solutions was not possible because of the simpliﬁcations that are required in order to produce a solution (Richards and Mathews; 1994; Zoras et al., 2001). Methods for the heat loss calculation of building structures in contact with the ground are also included in diﬀerent design guides, such as ASHRAE (1997), CIBSE (1986), AICVF (1990) or CEN (1992). Contrary to the case of ground-contact building structures applications, much less information can be found in the literature for high temperature tank foundation heat losses. In some works, diﬀerent TES tanks sub-models are used as a part of a global model of a SPP to take into account the tank’s heat losses in the calculations (Powell and Edgar, 2012; Mawire, 2013; Gabbrielli and Zamparelli, 2009; Rovira et al., 2011). However, in those sub-models, in some cases the simpliﬁcations are excessive (for example in Powell and Edgar (2012) it is assumed that no heat transfer occurs from the top or the bottom of each tank). In other cases either only an overall heat transfer

Subscripts avg average b base c concrete cv convective dom domain ext exterior ins insulation max maximum stg storage vent ventilation x_ext x direction, nodes located below exterior air (Tables 2 and 3) x_tank x direction, nodes located below the tank (Tables 2 and 3) x_total x direction, total (Tables 2 and 3)

coeﬃcient is used to take into account the storage tank total heat losses (Mawire, 2013) (without distinguishing the contributions of the diﬀerent parts: wall, top, bottom) or no information is provided in the methodology of obtaining the overall heat transfer (Gabbrielli and Zamparelli, 2009; Rovira et al., 2011). More detailed thermal models to obtain the heat losses in TES tank’s plants can be found in references SchulteFischedick et al. (2008), Zaversky et al. (2013) and Rodrı´guez et al. (2013). In these investigations, a tank based on the geometry and operating conditions of the Andasol-1 commercial trough power plant is analyzed and the heat losses are evaluated. Foundation consist of a thin steel layer followed by a thin layer of dry sand, a foam-glass insulation layer and an air-cooled concrete foundation designed to keep the concrete temperature below a maximum admissible value. Although the tank geometry and the operating conditions used were similar, the reported results in terms of the heat losses are diﬀerent, due to the diﬀerent methodologies and model assumptions used in each model. Particularly, the foundation heat losses obtained using these complete thermal models were in the range of 23–49 kW for the cold tank and 31–62 kW for the hot tank. These diﬀerent results and the lack of speciﬁc studies applied to TES tank foundation heat losses, suggest the necessity of a more accurate calculation methodology of this ground-coupled heat transfer problem. In the present work, the classical problem of ground-coupled conduction is revised for estimating the foundation heat losses for the practical application of TES tanks. 2. Problem definition A description of the studied problem is presented in the next paragraphs.

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2.1. Tank heat losses to the environment Even though TES tanks are typically highly insulated, thermal losses from the tank to the environment occur through the tank’s walls, the roof and the foundation due to the high tanks storage temperatures. These high storage temperatures have also an impact in the foundation construction design. In state-of-the-art TES tanks, cooling pipes are embedded within the foundation in order to ensure that the working temperature reached in the concrete remains below a maximum allowable value. Usually a passive ventilation system is designed, being air the cooling ﬂuid (Zaversky et al., 2013; Rodrı´guez et al., 2013; Relloso and Delgado, 2009), but other design alternatives are also considered (for example: active cooling with water, Gabbrielli and Zamparelli, 2009). In Fig. 1 it is shown a sketch of a slab on ground cylindrical storage tank (left) and its corresponding thermal resistances (right). As a ﬁrst approximation, the corresponding thermal resistances to the aforementioned heat ﬂuxes can be considered in parallel (Gabbrielli and Zamparelli, 2009), in which the top and the wall heat losses are caused by the temperature diﬀerence with the exterior temperature Text and the bottom heat losses are caused by the temperature diﬀerence with the deep-soil temperature Tsoil. Soil temperature distribution is aﬀected by many factors such as the structure and physical properties of the ground, the phreatic level, the ground surface cover or the climate interaction. The earth temperature beyond a depth of 1 m is usually insensitive to the diurnal cycle of air temperature and solar radiation and the annual ﬂuctuation of the earth temperature extends to a depth of about 10 m (Florides and Kalogirou, 2005). In deeper layers, the temperature distribution remains unchanged throughout the year (neglecting the geothermal gradient for very high depths and the changes in the water level caused by rains). The annual average air temperature can be taken as a ﬁrst

785

approach of the deep-soil temperature (Gabbrielli and Zamparelli, 2009).

2.2. Bottom heat losses A sketch of the foundation adapted from (Rodrı´guez et al., 2013) is shown in Fig. 2 (left). The tank foundation typically consist of a thin steel layer (slip plate) followed by a thin layer of dry sand, a foam glass insulation layer and an air cooled concrete foundation designed to keep the concrete below a maximum working temperature. The concrete slab is directly cast on ground. Bottom heat losses (also referred to as foundation heat losses in the work) can be characterized by the equivalent thermal circuit shown in Fig. 2 (right), where the slip plate and the dry sand layers are not include, as their thermal resistance is negligible in comparison with the insulation layer. The heat losses through the tank foundation (Qbottom), depend mainly on the next factors: (i) the tank diameter, (ii) the tank storage temperature (Tstg), (iii) the insulation level (insulation thermal resistance R0ins ), (iv) the type of soil (characterized by the soil equivalent thermal resistance R0soil ) and (v) the evacuated heat ﬂux by the cooling system (Qvent) needed to achieve the concrete temperature below the maximum allowed value (Tmax). The concrete thermal resistances over and below the ventilation tubes (R0c1 and R0c2 respectively), as well as the convective resistance R0cv (due to the ﬂuid movement near the tank bottom), also contribute to the tank thermal insulation but less importantly. The cooling system divides the thermal resistance of the foundation into two parts which can be considered to be in series. Due to the high and wide range of storage temperatures, the diﬀerent layers above the ventilated concrete must be design to be able to reach high temperatures. Furthermore, it becomes necessary to take into account the variation of the thermal conductivity with temperature in

Fig. 1. Description of the problem and equivalent thermal resistance network of the tank heat losses.

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Fig. 2. Foundation detail and equivalent thermal resistance network of the bottom heat losses.

Table 1 FoamglasÒ thermal conductivity dependence with temperature. T (°C)

10

40

100

160

220

300

k (W/m K)

0.041

0.046

0.057

0.07

0.085

0.099

the diﬀerent layers’ thicknesses calculation, especially in the insulation layer, as it aﬀects very importantly to the thermal resistance values. In Table 1 it is shown this dependence for the case of foamglasÒ, for diﬀerent temperatures (Foamglas, 2015). In tanks storing molten salts with high melting point temperatures and operating at relatively low temperatures (near the melting point) it is particularly important the determination of the tank bottom temperature Tb. As described by Sua´rez et al. (2015), even though it is expected a mainly homogeneous temperature distribution within the tank, the minimum local temperatures occur near the tank surfaces, and, consequently, the risk of crystallization of the molten salt is higher at these locations. 3. Solution procedure A one-dimensional steady-state multilayer model is used to solve the heat transfer through the tank foundation. Consequently, temperature gradients are considered to exist along only a single coordinate direction (z coordinate), and heat transfer is supposed to occur exclusively

in that direction, being temperatures and heat ﬂuxes independent of time. The equivalent thermal circuit representation is shown in Fig. 3, in which only the most important tank foundation layers from the heat transfer point of view (insulation and concrete) are included. The concrete layer is divided in two parts in order to properly model the evacuated heat through the cooling pipes. The slip plate and the dry sand layers are not included as their thermal resistances are negligible in comparison with the insulation layer, which is the controlling thermal resistance. The characterization and analysis of the thermal bridges situated in the tank foundation boundaries and the modiﬁcations in the phreatic level by rains throughout the year are out of the scope of the present work. Consequently, the thermal resistances included in the model are the convective resistance ðR0cv Þ, the thermal resistances corresponding to the insulation layer ðR0ins Þ and the upper and lower concrete layers (R0c1 and R0c2 ) and the equivalent thermal resistance of the soil layer ðR0soil Þ. The layers’ thicknesses, the temperatures Tstg, Tmax and Tsoil, as well as the convective resistance R0cv are considered as input data. Insulation and concrete thermal conductivities are assumed to vary linearly with temperature, following a known law. The total bottom heat losses (Qbottom), the heat evacuated by the cooling system to the environment (Qvent), and the heat loss to the soil (Qsoil) are the main variables to calculate.

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Fig. 3. Foundation detail and equivalent thermal circuit.

3.1. Bottom heat losses

Qbottom ¼ A

The one-dimensional bottom heat losses (Qbottom) can be expressed in terms of the bottom area A (normal to the direction of heat transfer), the temperature diﬀerence Tstg Tmax and the thermal resistances R0cv and R0ins (Eq. (1)). Alternatively, Qbottom can be expressed in terms of the temperature diﬀerence Tstg Tb and R0cv (Eq. (2)). The insulation thermal resistance R0ins can be calculated as a function of the insulation thickness Lins and the thermal conductivity kins (Eq. (3)). The thermal conductivity kins is evaluated at the insulation average temperature Tins,avg and it is assumed to vary linearly with temperature with known coeﬃcients ko,ins and bins (Eqs. (4) and (5) respectively). Qbottom

T stg T max ¼A 0 Rcv þ R0ins

ð1Þ

Qbottom

T stg T b ¼A R0cv

ð2Þ

R0ins

Lins ¼ k ins ðT ins;avg Þ

T ins;avg ¼

T b þ T max 2

k ins;avg ¼ k o;ins ð1 þ bins T ins;avg Þ

Qsoil ¼ A

T max T c1c2 R0c1

T c2soil T soil R0soil

Qvent ¼ Qbottom Qsoil

ð4Þ ð5Þ

ð8Þ ð9Þ

Concrete thermal resistances R0c1 and R0c2 must be evaluated as a function of each thickness (Lc1 and Lc2) and each material thermal conductivity kc1 and kc2 (Eqs. (10) and (11)). The thermal conductivity is calculated at each concrete average temperature Tc1,avg and Tc2,avg and it is assumed to vary linearly with temperature, with known coeﬃcients ko,c and bc (Eqs. (12)–(15) respectively). R0c1 ¼

Lc1 k c1 ðT c1;avg Þ

ð10Þ

R0c2 ¼

Lc2 k c2 ðT c2;avg Þ

ð11Þ

T max þ T c1c2 2 T c1c2 þ T c2soil T c2;avg ¼ 2 k c1;avg ¼ k o;c ð1 þ bc T c1;avg Þ

ð14Þ

k c2;avg ¼ k o;c ð1 þ bc T c2;avg Þ

ð15Þ

T c1;avg ¼ ð3Þ

ð7Þ

ð12Þ ð13Þ

Eqs. (1)–(5) represent a system of 5 equations with 5 unknown variables (Qbottom, Tb, Tins,avg, kins,avg, R0ins ), which can be solved using an iterative procedure. Without the loss of generality, other layers could be easily added in the mathematical model in case of diﬀerent foundations.

Eqs. (6)–(15) represent an indeterminate system of 10 equations with 11 unknown variables (Qsoil, Qvent, Tc1c2, Tc2soil, Tc1,avg, Tc2,avg, kc1,avg, kc2,avg, R0c1 ; R0c2 and R0soil ). The soil equivalent thermal resistance R0soil , which is unknown a priori, must be calculated separately in order to close the equations system.

3.2. Heat loss to the soil and ventilation heat losses

3.3. R0soil correlation

Similarly, the heat loss to the soil (Qsoil) can be related to the bottom area A, the temperature diﬀerence Tc1c2 Tsoil and the thermal resistances R0c2 and R0soil (Eq. (6)), where Tc1c2 is related to the bottom losses through Eq. (7). Alternatively, the heat loss to the soil Qsoil can be expressed in terms of the temperature diﬀerence Tc2soil Tsoil and R0soil (Eq. (8)). Performing an energy balance on the node associated with the concrete ventilation, ventilation heat losses are equal to the diﬀerence between the total bottom losses and the heat loss to the soil (Eq. (9)). Qsoil ¼ A

T c1c2 T soil R0c2 þ R0soil

ð6Þ

A computational ﬂuid dynamics (CFD) model is developed to analyze the conduction phenomenon that takes place in the soil and to derive a correlation for the soil equivalent thermal resistance R0soil . Representative boundary conditions for a state-of-the-art thermal storage tank foundation are used in the simulations, covering a comprehensive set of operating conditions, diﬀerent types of soils, tank diameters and maximum allowed concrete temperatures. 3.3.1. Description of the numerical model The geometry and the model boundary conditions are described in Fig. 4. An enlarged domain in depth (zdom) and far-ﬁeld dimensions (xdom) is discretized using a 2D

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Fig. 4. Computational domain and boundary conditions.

geometry. Taking advantage of the axis symmetry, only a half of the enlarged computational domain underneath the tank has been modeled. A constant uniform temperature (equal to Tmax) has been imposed for the soil surface in contact with the concrete (Tc2soil) and a convective boundary condition for the external air (Text, hext) is imposed for the exterior surface. A constant uniform temperature distribution condition equal to the deep ground soil temperature (Tsoil) is imposed for the bottom of the domain and an adiabatic condition is imposed for the side far-ﬁeld condition. CFD results were obtained by solving the energy equaÒ tion using the commercial software Ansys Fluent (Ansys, 2011), based on a ﬁnite volume approach. A 2D steady-state calculation was performed in the closed domain using constant properties for soil. The energy equation with its associated boundary conditions was iteratively solved till convergence of its residuals (convergence parameter equal to 1014) and until all the variables of interest (temperatures at diﬀerent points and heat ﬂux through the foundation) reached a constant value. Energy quantities were approximated by the second-order-upwind spatial discretization scheme. The calculations have been performed in double precision, so that ensuring an accurate energy balance, independently from the boundary conditions. The total number of cells ranges between 6.5 105 and 1.4 106 depending on the tank diameter. 3.3.2. Validation of the numerical code The numerical results obtained by Hagentoft (1988a) for the steady-state heat losses in a circular slab are used as a benchmark to validate the numerical model of the present

work. They considered a circular slab of radius R and insulation thickness Lins with thermal conductivity kins supported directly upon the soil (thermal conductivity k), as shown in Fig. 5. The temperature above the slab is Ti and the temperature at the deep-ground temperature is Tsoil. Hagentoft (1988a) proposed the following expression for the calculation of Qbottom: Qbottom ¼ k ðT i T soil Þ R hsc ðd=RÞ

ð16Þ

where d is the soil equivalent insulation thickness deﬁned by: k d ¼ Lins ð17Þ k ins and hsc is the heat loss factor, which depends on the parameter d/R. For slabs with thick insulations (d/R > 0.6), the heat loss factor can be approximated with an error below a 3% in comparison with numerical results with the expression: p ð18Þ hsc ðd=RÞ ¼ d=R þ 4=3p An analysis of the amount of soil to model (far-ﬁeld distance xdom and deep ground distance zdom) and the number of nodes is developed for two insulated circular slabs of 5 m radius over a clay soil (k = 1.5 W/m K). Temperature diﬀerence is ﬁxed at 15 °C and the considered thermal insulations are 12 cm and 24 cm for cases 1 and 2 respectively (kins = 0.04 W/m K). The results included in Table 2 indicate that an enlarged domain of 26 26 m2 or 52 52 m2 (approximately 5 and 10 times de slab radius respectively) were enough to

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Fig. 5. Benchmark problem for the validation of the numerical code.

Table 2 Soil domain size sensitivity analysis. Case

xdom (m)

zdom (m)

Ntotal

Nx_tank

Nx_ext

Nx_total

Nz

Qcase_1 (W)

Qcase_2 (W)

1 2 3

6 26 52 Hagentoft (1988a)

6 26 52 –

4200 28500 72500 –

60 60 60 –

10 130 230 –

70 190 290 –

60 150 250 –

267.4033 270.2406 270.1951 266.8600

158.4632 159.4162 159.3997 158.8900

Table 3 Soil domain mesh sensitivity analysis. Case

xdom (m)

zdom (m)

Ntotal

Nx_tank

Nx_ext

Nx_total

Nz

Qcase_1 (W)

Qcase_2 (W)

1 2 3

52 52 52 Hagentoft (1988a)

52 52 52 –

4256 18125 72500 –

15 30 60 –

58 115 230 –

73 145 290 –

62 125 250 –

259.1025 270.5343 270.1951 266.8600

155.6946 159.523 159.3997 158.8900

properly model the problem, with errors below 1.3% and 0.4% in comparison with the reference results respectively. Similarly, the results included in Table 3 suggest that a mesh density of 145 125 nodes or 290 250 nodes were adequate, with errors below 1.6% and 0.5% in comparison with the reference results respectively. As the total number of nodes is acceptable from a computational cost point of view, an enlarged domain of 52 52 m2 and a mesh density of 290 250 nodes are selected for computational model. A CFD model with similar characteristics is used to solve the conductive problem in the soil underneath the TES tanks (but diﬀerent boundary conditions). 4. Results and discussion Firstly, using the described CFD model, a correlation for the soil equivalent thermal resistance is obtained numerically. Then, the described methodology is directly applied to typical state-of-the-art TES tanks and the results are discussed in detail and summarized, providing a quick method for the estimation of the total bottom heat losses and its components (the ventilation heat losses and the heat loss to the soil).

4.1. R0soil correlation results A set of numerical simulations with diﬀerent boundary conditions is made in order to obtain a relation that correlates the equivalent soil resistance R0soil with the main problem parameters: the soil thermal conductivity and the tank diameter. In Table 4 a summary of the 9 studied cases is shown, covering diﬀerent types of soils (clay, sand and rock) and tanks diameters (ranging from 20 to 60 m). The soil temperature and the exterior convective boundary conditions are ﬁxed in the calculations (Tsoil = 10 °C, hext = 5 W/m2 K, Text = 10 °C). Simulation results revealed that, in concordance with Hagentoft (1988a), R0soil depends mainly on the parameter R/k. Linear and power family functions (Eqs. (19) and Table 4 Studied cases for the obtention of R0soil correlation. Variable

Value 1

Value 2

Value 3

Total values

k (W/m K) D (m) Total cases

1.5 20 –

2.0 38.5 –

3.5 60 –

3 3 9

2

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of 7.7 equivalent hours (Relloso and Delgado, 2009). The cold tank (14 m height and 38.5 m diameter) operated at 292 °C and had a passive ventilation system through foundations in order to decrease concrete temperature (Relloso and Delgado, 2009). The tank foundation consists of an insulation layer of foam-glass and a naturally air ventilated concrete layer. A linear dependence of the thermal conductivity with temperature for the insulation layer (based in the values in the values of Table 1, Eq. (21)) and for the concrete layer (obtained from Kodur (2014) for a siliceous aggregate type, Eq. (22)) are considered in the calculations:

Fig. 6. R0soil correlation results in comparison with the simulation results.

(20) respectively) of the variable R/k were explored to best ﬁt the simulation results. R k b R ¼ a2 k

R0soil ¼ a1

ð19Þ

R0soil

ð20Þ

The correlation results are compared to the simulation results in Fig. 6 (the 45° red1 solid line indicates a perfect ﬁt between them). The best ﬁt constant for the linear law was found to be a1 = 0.39, giving an average error of 7.5% and a maximum error of 15.0%. An excellent agreement between the correlation and the simulation results was found for the power law with constants a2 = 0.40 and b = 0.85, giving an average error of 0.9% and a maximum error of 1.3%. This last correlation is used hereafter in the calculations. Once the correlation for the equivalent thermal resistance R0soil is obtained as a function of the geometry (tank radius) and the type of soil (soil thermal conductivity), the system of equations formed by Eqs. (1)–(15) can be solved analytically using an iterative procedure. 4.2. Parametric analysis of the bottom heat losses for TES tanks applications Following the methodology described in chapter 3, the problem is solved using as a reference in the calculations the cold tank characteristics of the Andasol-1 SPP (Table 5), representative for a state-of-the-art TES tanks for parabolic trough applications. Andasol-1, with a turbine output power of 49.9 MWe and a storage capacity 1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.

k ins ðT Þ ¼ 0:0363 ð1 þ 0:0059 T Þ

ð21Þ

k c ðT Þ ¼ 1:7000 ð1 0:0005 T Þ

ð22Þ

The variation of the bottom heat losses (and its components) is examined for the relevant variables of the problem: type of soil, diameter, insulation thickness, maximum temperature and storage temperature. 4.2.1. Parametric analysis – type of soil Fig. 7 depicts the variation of the bottom heat losses as a function of the soil thermal conductivity. As the storage temperature (Tstg) and the maximum concrete temperature (Tmax) are ﬁxed, the total bottom heat losses (Qbottom) are not aﬀected by the soil thermal conductivity k. Nevertheless, the weight of its components Qvent and Qsoil are altered (higher k values imply an approximate linear increment in the Qsoil component and a consequently decrement in the Qvent component). The results show that for the cold tank (Qbottom 57 kW) the ventilation heat losses represented the 51%, 40% and 9% for clayey, sandy and rocky soils respectively. 4.2.2. Parametric analysis – diameter The bottom heat losses are proportional to the square of the tank diameter (bottom area), as shown in Fig. 8. While the heat loss to the soil increase approximately linearly with D, the ventilation heat losses increase quadratically with D, being the result the aforementioned quadratic trend of the total bottom losses. For small tanks with diameters below a certain value D* (19 m in the ﬁgure), the ventilation heat losses are null (the concrete maximum temperature is below the required design temperature without needing the cooling ventilation) and the bottom heat losses trend is linear. Additionally, in Fig. 9 the results in terms of the bottom heat ﬂux density (heat ﬂux per bottom area) is shown. It is observed a constant trend for diameters above the critic one D* (with higher contribution of the ventilation component for higher diameters). That implies that, for example, for the Andasol-1 38.5 m diameter cold tank with 57 kW bottom heat losses (or equivalently 49 W/m2), to increase the diameter up to 50 m, would increase the bottom heat losses up to 96 kW. The heat ﬂux density would remain the same (49 W/m2), but with higher ventilation

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Table 5 Andasol-1 geometry data and operating conditions. Tank

D (m)

Insulation (cm)

Concrete (cm)

Tstg (°C)

Tmax (°C)

Tsoil (°C)

k (W/m K)

Cold

38.5

30 (foam glass)

45 (heavy weight)

292

90

10

2

Fig. 7. Parametric analysis type of soil, Andasol-1 (D = 38.5 m, Lins = 30 cm, Tmax = 90 °C, Tstg = 292 °C).

Fig. 10. Parametric analysis insulation thickness, (D = 38.5 m, k = 2 W/m K, Tmax = 90 °C, Tstg = 292 °C).

the ventilation heat losses are aﬀected by the insulation thickness Lins. The reduction in Qbottom is higher in the ﬁrst added insulation cms. The thicker the insulation layer is, the less decrease in the bottom heat losses is by further adding insulation. For example, for the cold tank illustrated in the ﬁgure (57 kW with 30 cm insulation layer), the results show that while removing 10 cm of insulation would increase the bottom heat losses 26 kW, adding 10 cm of insulation would only decrease them 14 kW. Fig. 8. Parametric analysis diameter, bottom heat ﬂux (k = 2 W/m K, Lins = 30 cm, Tmax = 90 °C, Tstg = 292 °C).

4.2.4. Parametric analysis – maximum temperature Fig. 11 shows the variation of the bottom heat losses with the designed maximum allowed temperature in the concrete foundation (Tmax). It is observed that lower values of Tmax cause an increase in the bottom heat losses. Even though a reduction in the design temperature Tmax, provokes a decrease in the loss of heat to the soil (lower temperature diﬀerence between the concrete and the soil), at the same time higher ventilation heat losses are required to decrease the concrete temperature. The observed trends are approximately linear. As an example, for the cold tank (57 kW with Tmax = 90 °C), the results show that the

Fig. 9. Parametric analysis diameter, bottom heat ﬂux density (k = 2 W/ m K, Lins = 30 cm, Tmax = 90 °C, Tstg = 292 °C).

component contribution (53% with the 50 m diameter versus 42% with the original diameter). 4.2.3. Parametric analysis – insulation thickness Fig. 10 shows the decrease in the bottom heat losses as the insulation layer increases. As the soil temperature (Tsoil) and the maximum concrete temperature (Tmax) are ﬁxed, the soil heat losses (Qsoil) remain constant and only

Fig. 11. Parametric analysis maximum temperature, (D = 38.5 m, k = 2 W/m K, Tmax = 90 °C, Tstg = 292 °C).

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bottom heat losses are decreased approximately 2.0 kW for each 10 °C Tmax is increased (ventilation heat losses are decreased 6.2 kW and soil losses are increased 4.2 kW). 4.2.5. Parametric analysis – storage temperature The cold and hot tanks storage temperatures depend on the SPP type and on its design. These operating temperatures are typically 292/386 °C for parabolic trough plants (such as Andasol-1) and 290/565 °C for central receiver plants (for example Solar Two), but other storage ﬂuid temperatures are also possible. Figs. 12 and 13 show the variation of the bottom heat losses per square meter with the aforementioned working temperatures. For the Solar Two tanks it is assumed the same input data than the Andasol-1 plant, except the diameter (11.6 m). The insulation thicknesses considered in the calculations are ﬁxed to 30 cm in all cases. As expected, higher values of Tstg cause an increase in the bottom heat losses. Due to the eﬀect of the reduction in the thermal conductivity at higher temperatures, the reduction is not purely linear and a quadratic trend is observed. As the soil temperature (Tsoil) and the maximum concrete temperature (Tmax) are ﬁxed, the total soil heat losses (Qsoil) remains constant and only the ventilation heat losses are increased with higher storage temperatures. For low storage temperatures (temperatures below a certain value T stg ), the concrete maximum temperature is itself lower than the required design temperature without

Fig. 12. Parameter analysis storage temperature, (D = 38.5 m, k = 2 W/m K, Tmax = 90 C, Lins = 0.3 m).

needing the cooling ventilation and the bottom heat losses trend is linear (T stg is 225 and 360 °C for Andasol-1 and Solar Two tanks respectively). This situation, which occurs in the case of the cold tank of the Solar Two plant, is caused by the high insulation level. 4.3. Estimation of the loss of heat from a ventilated tank foundation Using the methodology described in the present work, a set of diﬀerent cases were solved (35,868 cases in total) covering a wide range combination of parameters (outlined in Table 6). The objective of the calculations was to obtain the total bottom heat losses and the fractions of its components. A ﬁxed concrete thickness of 45 cm, a convective coeﬃcient of 5 W/m2 K and a soil temperature of 10 °C are used in all the calculations. The thermal conductivity is assumed in all cases to vary linearly according to Eqs. (21) and (22). 4.3.1. Total bottom heat losses In Fig. 14 the results of the total bottom losses per square meter are shown as a function of the insulation thickness for typical temperature ranges in parabolic trough and tower applications. In the case of the cold tank, a storage temperature of 292 °C is assumed for both applications. Four diﬀerent values for the maximum concrete temperature are plotted in the ﬁgure for each storage temperature. Once the bottom heat ﬂux density qbottom is obtained for a given insulation thickness, storage temperature and concrete temperature, it is straightforward to obtain the bottom heat losses Qbottom by multiplying qbottom by the tank’s base area. In most practical situations, the majority of the parameters that aﬀect the bottom heat losses are ﬁxed due to plant requirements (tank geometry, storage temperature and maximum allowed concrete temperature) or to the tank environment (type of soil and exterior temperature). Fixing these variables, the insulation thickness appears to be the only design parameter that can be changed to reduce the bottom heat losses. Nevertheless, this reduction is not linear with the insulation thickness. It is observed that the reduction in qbottom is higher in the ﬁrst added insulation cms, and that, the thicker the insulation layer is, the less decrease in the bottom heat losses occurs by further Table 6 Studied cases for the estimation of the tank foundation heat losses.

Fig. 13. Parameter analysis maximum k = 2 W/m K, Tmax = 90 °C, Lins = 0.3 m).

temperature,

(D = 11.6 m,

Variable

Range

Increment

Cases

D (m) k (W/m K) Storage T (°C) Concrete T (°C) Insulation thickness (m) Total cases

10–70 1–4 292, 386, 565 70–100 0.10–0.70 –

10 0.5 – 10 0.01 –

7 7 3 4 61 35,868

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793

Fig. 14. Summary of the results. Total bottom heat loss density as a function of the insulation thickness for diﬀerent Tstg and Tmax values. The % decrease in Qbottom by adding 1 additional cm of insulation is indicated along the x axis.

Table 7 Cold and hot tanks bottom heat ﬂux density for diﬀerent insulation thicknesses. Lins (m)

Cold tank (292 °C) qbottom (W/m2)

Hot tank (386 °C) qbottom (W/m2)

Hot tank (565 °C) qbottom (W/m2)

0.15 0.20 0.30 0.40 0.50 0.60 0.70

93 71 49 37 30 25 21

150 117 80 61 49 41 35

284 222 155 118 96 80 69

As an illustrative example, in Table 7 the results of qbottom are summarized for diﬀerent insulation thicknesses for cases in which the maximum allowed concrete temperature is ﬁxed to 90 °C. Inversely, the insulation thicknesses required to achieve a certain range of values of qbottom are included in Table 8 (again Tmax is ﬁxed to 90 °C). 4.3.2. Loss of heat to the soil and ventilation heat losses The total bottom heat losses are the sum of two components, the loss of heat to the soil and the heat evacuated to the environment by the ventilation system:

adding 1 more cm. This reduction in qbottom is quantiﬁed in the ﬁgure in diﬀerent percentage ranges (denoted by vertical dashed lines). For example, starting from a thickness insulation of 30 cm, the addition of 1 more cm will have an impact in the decrease of qbottom of only a 2–3% (taking the 30 cm case as the reference), and starting from a thickness insulation of 48 cm the addition of 1 more cm will have an impact in the decrease of qbottom below a 1.5–2% (taking the 48 cm case as the reference).

Qbottom ¼ Qsoil þ Qvent

ð23Þ

In Fig. 15, the results of the bottom heat losses to the soil per square meter are shown as a function of the parameter R/k for the analyzed values of the maximum concrete temperature (which is ﬁxed by the ventilation system). The results exhibit a hyperbolic pattern. The heat loss to the soil decreases asymptotically for high values of the parameter R/k (cases corresponding to high diameters tanks and/or low thermal conductivity soils) and vice versa.

Table 8 Cold and hot tanks bottom insulation thicknesses for diﬀerent heat ﬂux density ranges. Cold tank (292 °C) 2

Hot tank (386 °C) 2

Hot tank (565 °C)

qbottom (W/m )

Lins (m)

qbottom (W/m )

Lins (m)

qbottom (W/m2)

Lins (m)

<100 70–100 50–70 35–50 25–35 22–25 <22

>0.14 0.14–0.21 0.21–0.30 0.30–0.42 0.42–0.60 0.60–0.70 >0.70

<160 120–160 90–120 70–90 50–70 35–50 <35

>0.14 0.14–0.20 0.20–0.27 0.27–0.35 0.35–0.49 0.49–0.7 >0.7

<300 250–300 200–250 150–200 100–150 70–100 <70

>0.14 0.14–0.18 0.18–0.23 0.23–0.31 0.41–0.48 0.48–0.7 >0.7

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794

Fig. 15. Summary of the results. Heat loss density to the soil.

As the heat loss to the soil only depends on the concrete and soil’s temperatures and thermal resistances, the insulation thickness and the storage temperature do not have any inﬂuence in the results. Finally, once the bottom heat losses and the heat losses to the soil are obtained, it is straightforward to determinate the ventilation heat losses using Eq. (23). 5. Conclusions The loss of heat through ventilated TES tanks foundations has been analyzed for typical operating conditions in state-of-the-art SPPs using a one-dimensional steadystate multilayer model. The foundation ventilation system, which is designed to establish a temperature in the concrete layer below a maximum temperature, extracts to the environment a fraction of the total bottom losses and the rest is conducted through the soil. A CFD model is developed to derive a correlation for the soil equivalent thermal resistance as a function of the soil conductivity and the tank radius. The obtained correlation is then used in the mathematical model, which is directly applied to typical stateof-the-art TES tanks. The results are discussed in detail through a comprehensive parametric analysis and summarized graphically and in tables, providing a quick method for the estimation of the total bottom heat losses and its components (the ventilation heat losses and the heat loss to the soil). The obtained results provide important information related to the tank foundation design, such as the quantiﬁcation of the evacuated heat due to the ventilation system or the selection of an appropriate bottom insulation thickness level depending on the tank geometry, the storage temperature, the ventilation system and the type of soil. References AICVF (Association des Ingenieurs de Climatisation et de Ventilation de France) Chauﬀagge-Calculs des Deperditions et Charges Thermiques d’Hiver, Pyc Edition, AICVF 1990, Paris.

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