Thermal and Exergetic Analysis of Metal Foam-enhanced Cascaded Thermal Energy Storage (MF-CTES)

Transcription

1 Thermal and Exergetic Analysis o Metal Foam-enhanced Cascaded Thermal Energy Storage (MF-CTES) Y. Tian a, C.Y. Zhao b a School o Engineering, University o Warwick, CV4 7AL Coventry, UK Y.Tian.4@warwick.ac.uk b School o Mechanical Engineering, Shanghai Jiaotong University, Shanghai, China Changying.zhao@sjtu.edu.cn Article history: Received 13 August 2012 Revised 9 November 2012 Accepted 9 November 2012 Available online 7 December 2012 Doi: Cited as: Y Tian, CY Zhao. Thermal and exergetic analysis o Metal Foam-enhanced Cascaded Thermal Energy Storage (MF-CTES). International Journal o Heat and Mass Transer 58 (2013): Abstract Metal Foam-enhanced Cascaded Thermal Energy Storage (MF-CTES) has been proposed to solve the problem o poor heat transer during heat exchange process, which is caused by unavoidable decrease o temperature dierences. This paper conducts a theoretical study examining the overall thermal perormance o STES (Single-stage Thermal Energy Storage), CTES (Cascaded Thermal Energy Storage) and MF-CTES, with both heat exchange rate and exergy eiciency being considered. The main indings are: heat exchange rate o STES is improved by CTES (up to 30%), and is urther improved by MF-CTES (by 2 7 times); exergy eiciency o STES cannot be signiicantly improved by CTES (-15% to +30%), nor by MF-CTES; exergy transer rate o STES is increased by CTES (up to 23%), and is urther increased by MF-CTES (by 2 7 times). Keywords: Metal oam; Heat transer enhancement; Exergy; Cascaded Thermal Energy Storage; Brinkman-Forchheimer; Enthalpy method. 1

2 Highlights Metal Foam-enhanced Cascaded Thermal Energy Storage (MF-CTES) has been proposed to enhance energy and exergy eiciency. Three systems are compared: STES (Single-stage Thermal Energy Storage), CTES (Cascaded Thermal Energy Storage) and MF-CTES. Heat transer rate o STES is improved by CTES (up to 30%), and is urther improved by MF-CTES (by 2 7 times). Exergy eiciency o STES cannot be improved signiicantly by CTES (-15% to +30%), nor by MF-CTES; Exergy transer rate o STES is increased by CTES (up to 23%), and is urther increased by MF-CTES (by 2 7 times). 2

3 Table o Contents Nomenclature Introduction Physical problem Mathematical description Exergy analysis Heat transer on the HTF side Heat transer between HTF and -metal oam Heat transer on the -metal oam side Equations o luid dynamics Equations o phase change heat transer The initial and boundary conditions Modelling o metal oam microstructures Numerical procedure Results and discussion Validation Natural convection Comparison o equivalent heat exchange rates among STES, CTES and MF-CTES Comparison o exergy eiciency among STES, CTES and MF-CTES Comparison o exergy transer rate among STES, CTES and MF-CTES Conclusions Acknowledgements Reerences Figure Captions Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig

4 Fig Table Captions Table Table Table

5 Nomenclature a s = speciic surace area m -1 Bi = Biot number: h d/k (dimensionless) c = speciic heat o Heat Transer Fluid (HTF) kj/(kg o C) c p = speciic heat at constant pressure kj/(kg o C) c p.mf = speciic heat o metal oam kj/(kg o C) c p. = speciic heat o Phase Change Material () kj/(kg o C) C = inertia coeicient o luid low in metal oams (dimensionless) d = characteristic length m d = equivalent diameter o metal ibre m d p = equivalent pore size m da = heat transer area element m 2 e = length ratio o cubic juncture node to ligament (dimensionless) g = gravity constant m/s 2 h ex = eective exergy transer rate W/m 2 h = speciic enthalpy o kj/kg h = eective heat transer coeicient or HTF W/m 2 h s = interstitial heat transer coeicient W/(m 2 K) h 1, 2 = system dimensions in y-axis m H L = latent heat kj/kg k = thermal conductivity o luid saturated in metal oams W/(m K) k e = eective thermal conductivity or W/(m K) k = thermal conductivity o W/(m K) k s = thermal conductivity o the metal material in metal oams W/(m K) k se = eective thermal conductivity or metal oam W/(m K) K = permeability m 2 L 1, 2, 3 = system dimension in x-axis m Nu = Nusselt number: h d/ HTF (dimensionless) p = pressure Pa 5

6 Pr = Prandtl number: c p HTF / k HTF (dimensionless) q = heat lux, or eective heat transer rate. W/m 2 R A, B, C, D = thermal resistance (m 2 K)/W Re = Reynolds number: ud/ (dimensionless) R g = ideal gas constant kj/(kg K) s = speciic entropy kj/(kg K) t = time s T a = ambient temperature T = temperature unction o HTF T w = wall temperature T = temperature unction o T MF = temperature unction o metal oam T 0 = HTF inlet temperature T m = melting temperature u = the component o low velocities in x-direction m/s v = the component o low velocities in y-direction m/s V = velocity vector m/s X = speciic anergy kj/kg o C o C o C o C o C o C o C Greek symbols = linear thermal expansion coeicient K -1 = porosity (percentage) η ex = exergy eiciency (percentage) = ratio o ligament radius to ligament length (dimensionless) λ HTF = thermal conductivity o HTF W/(m K) = density kg/m 3 = dynamic viscosity Pa s = Laplace operator 6

7 Subscripts e = eective value = HTF e = eective value or MF = metal oam MF- = metal oam and re = reerence value se = eective value or metal oam 7

8 1. Introduction Eicient utilisation o solar energy is increasingly being considered as a promising solution to global warming and a means o achieving a sustainable development or human beings. Because solar energy has low-density and intermittency, Thermal Energy Storage (TES) plays a pivotal role in balancing energy demand and energy supply. TES technologies rely on high-quality Phase Change Materials (s), which should have high heat storage capacity and excellent heat transer perormance. s have received extensive research interest during the last decade, and they were investigated in a variety o applications: energy saving buildings [1], solar still [2], solar cooker [3, 4], industrial waste heat recovery [5] and solar power plants [6]. Most s have large heat storage capacity, ranging rom 90 kj/kg to 330 kj/kg [7], but they suer rom the common problem o low thermal conductivities, being around 0.2 W/(m K) or most parain waxes and 0.5 W/(m K) or most inorganic salts [7]. Low heat transer perormance has been the main actor restricting the application o s in situations which require rapid energy release/storage [8, 9]. Researchers have proposed various methods enhancing heat transer in s, and these include: incorporating high thermal conductivity enhancers into s [10, 11]; adopting porous heat transer media [12 16]; Cascaded Thermal Energy Storage (CTES) [17, 18]. High thermal conductivity enhancers including metal ins, metal beads and metal powders improve heat transer in s, but their enhancement eects are limited to between 60% and 150% [11] which is not high enough or most application requirements. Porous media were also used to enhance heat transer or s, and these include carbon materials and metal oams. Nakaso et al. [15] tested the use o carbon ibres to enhance heat transer in thermal storage tanks, reporting a twoold rise in eective thermal conductivities. Sari and Karaipekli [13] abricated a series o the Parain/EG (Expanded Graphite) composites, and ound that the eective thermal conductivity was increased by 81.2% 272.7% depending on the mass raction o EG added. However, the main disadvantage o EG is its structural discontinuity, which means that heat cannot be transerred very smoothly and eiciently. 8

9 To overcome the structural discontinuity o the Parain/EG composites, metal oams have been investigated, as they have continuous metal matrices with porosity ranging rom 85% to 97%, as well as high thermal conductivities. Zhou and Zhao [16] investigated the Parain/EG and Parain/Metal oam composites to compare their dierent enhancement eect on s, and concluded that both composites increased heat transer rate signiicantly with metal oams showing better perormance. The reason accounting or this was later given in a theoretical analysis by Tian and Zhao [19]: the structures inside the expanded graphite are rather sparse (discontinuous), whilst the metal oam matrices are much more continuous so that they can draw heat much more rapidly than expanded graphite. Cascaded Thermal Energy Storage (CTES), consisting o multiple s with cascaded melting temperatures, has been proposed as a solution to heat transer deterioration, which oten arises when charging/discharging a single-stage storage system. For a single-stage storage system, the temperature o the heat transer luid alls rapidly when transerring heat to the ; as a result, the temperature dierence between heat transer luid and the is signiicantly reduced, which leads to poor heat transer at the end o the storage [20]. The problem is that the is melted rapidly at the entrance part, but much more slowly at the end o the storage. A similar problem occurs or the discharging process: the at the end o the storage might not be used as the temperature o heat transer luid rises. Such problems can be solved by adopting CTES, in which the s with cascaded melting temperatures can help to maintain a relatively high temperature dierence. Gong and Mujumdar [21] investigated a ive-stage s system, and ound a signiicantly improved heat transer (34.7%) compared to the single. Michels and Pitz-Paal [6] investigated a three-stage system, and ound that a higher proportion o s melted and a more uniorm heat transer luid outlet temperature than in the traditional single-stage storage. The study by Michels and Pitz-Paal [6] was based on energy eiciency, not having considered exergy eiciency that represents the utilisable part o energy. Exergy analyses or multiple systems were conducted by Watanabe and Kanzawa [17], and Shabgard et al. [22]. Watanabe and Kanzawa [17] ound increased exergy eiciency by using multiple s, whilst Shabgard et al. [22] ound 9

10 that the multiple s recovered more amount o exergy despite o having lower exergy eiciency at times. A thermal analysis taking exergy into account does not only consider the quantity o energy, but also the quality o energy, and thereore is very important. However, there are only a ew publications addressing exergy issues or CTES; none o these studies has combined CTES with other heat transer enhancement techniques, especially the use o metal oams. This paper aims to investigate, or the irst time, the idea o the metal oam-enhanced CTES system, examining its technical easibility and evaluating its energy and exergy perormance. 2. Physical problem Three systems are presented and compared in this study. They are Single-stage Thermal Energy Storage (STES), Cascaded Thermal Energy Storage (CTES) and Metal Foam-enhanced Cascaded Thermal Energy Storage (MF-CTES), which are illustrated in Figs. 1(a), (b) and (c), respectively. STES is ormed by a single, which is 4 shown in Fig. 1(a). Both CTES and MF-CTES are ormed by staging three s along the HTF (heat transer luid) low direction: 1, 2 and 3 shown in Figs. 1(b) and (c). CTES and MF-CTES are made o the same s, with the only dierence being that MF-CTES uses metal oam to enhance heat transer. The thermo-physical properties o these s used are listed in Table 1 [23]. In Fig. 1, H L (kj/kg) and T m ( ) denote the latent heat and melting temperature, respectively; h and L denote system dimensions. The HTF enters each system rom the let (inlet temperature T 0 = 100 ), and exits rom the right with the outlet temperature T (t) which varies with time. The melting temperature o 4 (in STES) was chosen to have the average value o the melting temperatures o 1, 2 and 3, so that a comparison between the three systems is justiied. The initial temperatures o all three systems are equal to the ambient temperature, which is 20. Other parameters or the systems are given in Table Mathematical description 10

11 3.1. Exergy analysis The entropy change [24] o a thermal system rom state 1 to state 2 can be written as: s s c ln( T / T ) R ln( p / p ) H T (1) 2 1 p 2 1 g 2 1 L m The last term on the right hand side o Eq. (1) stands or the entropy change i a phase change process is involved. phase change temperature. H L (kj/kg) is the phase change enthalpy and T m ( o C) is the The unusable part o energy (i.e. anergy X), depending on the entropy increase (irreversibility), can then be written as [24]: X T s s T c ln( T / T ) R ln( p / p ) H T a 2 1 a p 2 1 g 2 1 L m Thus the proportion o the usable energy can be calculated by Eq. (3): p 2 1 cp T2 T 1 X ex c T T Substituting Eq.(2) into Eq. (3), exergy eiciency is given in Eq. (4): p 2 1 cp T2 T 1 T a c p ln( T2 / T1 ) R g ln( p2 / p1 ) H L T m ex c T T Because most s are incompressible, c T T c T ln( T / T ) T H T ex c T T p 2 1 p a 2 1 a L m p 2 1 This study uses Eq. (5) to obtain the exergy eiciency or all three systems: STES, CTES and MF-CTES. (2) (3) (4) (5) 3.2. Heat transer on the HTF side Considering a charging process, heat lows rom the high-temperature heat transer luid (HTF) to low-temperature s. The thermal resistance o heat transer is made up by two parts: the HTF-side resistance and -side resistance. The approximate value o the Nusselt number Nu on the HTF side can be obtained by employing the Dittus Boelter Equation [25] Nu 0.023Re Pr (6) 11

12 In this study, water was used as HTF. Table 2 gives the operating parameters o all three energy storage systems. With a low velocity o 0.5 m/s, a kinetic viscosity o m 2 /s, a Prandtl number o 3.56 and a characteristic length o 0.01 m, the heat transer coeicient h is given by h = λ HTF Nu/d (7) rom which h = W/m 2. To compare the thermal resistance between the HTF side and side, the Biot number [25] is given by h d Bi 55.9 (8) k This number qualitatively represents how many times larger the thermal resistance is on the s side than on the HTF side. With Bi much greater than 1, the thermal resistance on the HTF side can be reasonably neglected, and this simpliies the ollowing analyses. It should be noted that the obtained Bi number is an approximate value, because this study used rectangular ducts, rather than round ducts which were assumed in the Dittus Boelter Equation. Even allowing or this, the Bi number will still be much greater than 1, so that the thermal resistance on the HTF side is so low that it can be neglected Heat transer between HTF and -metal oam. Perect thermal insulation was assumed in this study, so the heat transer equations can be established based on energy balance: s absorb the same amount o thermal energy as the HTF releases, which is relected in Eq. (9). Due to x u t T x h x t t h T MF h 1 da MF cp. MF (1 ) t t MF c ( h2 h 1) da MF h1 da, Eq. (9) can be re-written as: T h TMF c ( h2 h1) u h1 MF cp. MF (1 ) x t t (9) (10) 12

13 In order to cope with the phase change heat transer problem, the enthalpy method has been employed in this study. The correlation between the enthalpy unction h (x, y, t) and its temperature unction T (x, y, t) is given by: h, h (, c Tm ) c T Tm, h ctm, ctm H L (11) h H L, h ( c Tm H L, ) c 13

14 3.4. Heat transer on the -metal oam side In this section, the governing equations or heat transer on the -metal oam side, including luid dynamics equations, phase change heat transer equations and their initial and boundary conditions will be ormulated. The process o solving complicated equations by numerical methods can be signiicantly simpliied i the physical problem is symmetrical. A symmetrical physical problem requires that the computational domain, the initial and boundary conditions, and the governing equations should all be symmetrical. The computational domain or the present study is: 0 x L 1 L 2 L 3 and h 1 / 2 y h1 / 2 (shown in Fig. 1). Such a rectangular domain is symmetrical with respect to the x-axis. The initial and boundary conditions are discussed later in Subsection 3.4.3, which indicates that the upper part (above x-axis) has identical initial and boundary conditions to the lower part (below x-axis), meaning that the initial and boundary conditions are also symmetrical upon the x-axis. However, the present study takes natural convection into account, in which the gravity and temperature dierencedriven buoyancy are not symmetrical, so the luid dynamics equation in y-direction is not symmetrical with respect to the x-axis. Hence the current physical problem will have to be solved on the whole computational domain Equations o luid dynamics When natural convection takes place, the metal oam still remains stationary, whilst the keeps moving under a buoyancy orce driven by temperature dierence. To tackle with such complicated low in the porous metal oam, a volume-averaging technique has been employed [19, 26, 27], or which the classical Continuity Equation is: V = 0 (12) Here, denotes the volume-averaged value o a certain unction over an REV (Representative Elementary Volume inside metal oams) [19]. The Continuity Equation takes on the ollowing orm under the Cartesian coordinate system: u x v y 0 (13) 14

15 Here, u and v denote the components o the velocity V in x-direction and in y- direction respectively. Based on the Brinkman-Forchheimer extended Darcy model [26, 27], the Momentum Equations are given by: u u u p u v = t x y x 2 2 C u u u u u 2 2 x y K K (14) v v v p u v = t x y y 2 2 v v C v ( ) 2 2 v v g T Tre x y K K Here, g denotes the gravity constant, denotes the porosity o the metal oam, (15) denotes the dynamic viscosity o the, denotes the density o the, K is the permeability coeicient [26, 27], C denotes the inertial actor or luid low in metal oams [19, 26, 27], and denotes the thermal expansion coeicient o the. The low resistances consist o three parts: irstly, the irst-order resistance (Darcy term) which is denoted by the third terms on the right hand side o Eq. (14) and Eq. (15); secondly, the second-order resistance (Forchheimer correction term) which is denoted by the ourth terms on the right hand side o Eq. (14) and Eq. (15); thirdly, the Brinkman viscous resistance which is denoted by the second terms on the right hand side o Eq. (14) and Eq. (15). The last term on the right hand side o Eq. (15) represents the buoyancy orce caused by temperature dierences inside the, and it is the driving orce o natural convection. The intensity o natural convection mainly depends on two actors: driving orce and resisting orce. The driving orce increases with increasing temperature dierences, whilst the resisting orce can be reduced by decreasing the viscosity ( ) o the used. With ixed temperature dierences, larger viscosity results in a weaker natural convection. With ixed viscosity, larger temperature dierences result in a stronger natural convection. Eqs. (13) (15) are used to describe the buoyancy-driven luid low, but they also hold true when natural convection does not 15

16 take place, which is just a special case when is ininite. The present study treats the non-convection heat transer region as a special case o natural convection ( ), so that all cases can use the same equations thus simpliying the subsequent simulation work. When implementing numerical simulation, the program can automatically make the ollowing judgement: i the is still in solid state, its viscosity will be assigned an ininite value to ensure the absence o natural convection; once the inishes melting and becomes liquid, the real value o its viscosity will be assigned, so that the buoyancy orces can be precisely decided Equations o phase change heat transer In order to cope with the phase change heat transer problem, the Enthalpy Method [19] has been employed in this study. The correlation between the enthalpy unction h (x, y, t) and its temperature unction T (x, y, t) is given by Eq. (11). Under the Cartesian coordinate system, the Energy Equations or the and metal oam can be written as: 2 2. (1 ) T MF T MF MF MF cp MF k T se h 2 2 s as T MF T t x y 2 2 h T T T T cp. u v k e 2 2 t x y x y h a T T s s MF (16) (17) Here, kse denotes the eective thermal conductivity o metal oam (without embedded), k e denotes the eective thermal conductivity o the porous (when taking o the metal oam matrix). Their method o calculation is given in subsection hs denotes inter-phase heat transer coeicient between metal ligaments and, and as is speciic surace area o the metal oam. Their values are obtained by employing the model by Calmidi and Mahajan [27], the detail o which is given in subsection In Eq. (16) and Eq. (17), the irst and second terms on the right hand side represent heat conduction and inter-phase heat transer, respectively. The second 16

17 term on the let hand side o Eq. (17) represents the convection term or, which equals zero beore natural convection occurs ( u v 0) The initial and boundary conditions The governing equations in this study are Eqs. (10), (11) and (13) (17). Their initial conditions are given by: u v 0 (18) t 0 t0 T T 20 C (19) t0 MF t0 T 100 C (20) t 0 The boundary conditions are: u, v u, v u, v u, v 0 (21) x 0 xl 1 x L 1L 2 x L 1L 2L 3 u v 0 y h1 /2 y h1 /2 (22) T 100 C (23) x 0 Equations (21) and Eq. (22) give the non-slip boundary conditions o velocities. Eq. (23) gives the HTF temperature T at its let boundary. T is a unction o only horizontal coordinate x and time t, because the thermal resistance o HTF at the y- direction can be neglected when Bi is much greater than 1, as discussed in Section 3.2. The heat released rom HTF is transerred to and metal oam, but the percentage between and metal oam needs to be careully decided or an accurate calculation result. Calmidi and Mahajan [27] used an explicit presumption to decide the percentage o the heat absorbed by and metal oam at their common boundary, with being k / ( k k ) 100% and metal oam being k / ( k k ) 100%. Such e se e se se e presumption can make the simulation simpler and quicker, but meanwhile it results in inaccuracy. Exact percentages between and metal oam should be decided by an implicit relationship, which was given by Tian and Zhao [19] and Zhao et al. [28], shown in Eqs. (24) (27): T T T x y y MF c ( h2 h 1) u k se k e y h1 /2 y h1 /2 (24) 17

18 T T T (25) MF y h1 /2 y h1/2 w At the upper boundary, Eq. (24) relects energy conservation between HTF and -metal oam. Here, another restrictive condition is rom the temperature continuity both metal oam and should have the same temperature as the wall temperature at their common boundary, as shown in Eq. (25). Such a combined implicit boundary condition shown in Eq. (24) and Eq. (25) can achieve better accuracy due to its avoidance o extra presumption. It needs to be noted that Tw in Eq. (25) does not need to be pre-determined, the presence o which is only to show that TMF T. In Eqs. (24) and (25), the number o variables ( T MF and T ) is equal to the number o equations, so the equations are solvable and the solution is inite. Similarly, the boundary conditions or the lower boundary have been obtained as ollows: T T MF c ( h2 h 1) u kse k e T x y y MF y h1 /2 y h1 /2 w y h1 /2 y h1 /2 (26) T T T (27) The energy conservation at the lower boundary is shown in Eq. (26), with the temperature continuity condition being given in Eq. (27). Due to perect thermal insulation, all our horizontal boundaries are adiabatic, giving: T T T T 0 x x x x x0 xl 1 xl 1L 2 xl 1L 2 L 3 (28) T MF T MF T MF T MF 0 x x x x x 0 xl 1 xl 1L 2 xl 1L 2 L 3 (29) Modelling o metal oam microstructures There are still several important parameters or metal oam microstructures that need to be determined or solving the governing equations: Eqs. (10), (11) and (13) (17). These include: permeability, inertial actor, pore size, metal ibre diameter, eective thermal conductivity, surace area density, and inter-phase heat transer coeicient. The determination o these parameters is complicated and strongly depends on special microstructures inside metal oams. Several existing models presented by previous 18

19 researchers are employed in this study to obtain these parameters. For simplicity, this subsection only gives the computational ormula or eective thermal conductivity, permeability, inertial actor, surace area density and inter-phase heat transer coeicient. The detailed derivation o all other parameters is given in Calmidi [26] and Zukauskas [29]. Calmidi and Mahajan [27] presented a 2D simpliied model o eective thermal conductivity or metal oams, which gave good agreement with test data. However the real microstructures in metal oams are three-dimensional, and thereore a 3D model is preerred in order to get improved accuracy. In this paper, a 3D structured model presented by Boomsma and Poulikakos [30] has been used to deal with the eective thermal conductivity o metal oams. A tetrakaidecahedron [31] was used in their model to approximate metal oam cells, because that is the polyhedron with the minimal surace energy this is relevant because metal oam cells tend to shrink to the minimal surace when being manuactured by oaming processes. Figure 2 shows the structure o a tetrakaidecahedron, which is a ourteen-ace polyhedron comprising six squares and eight hexagons [32]. By using such a polyhedron approximation, Boomsma and Poulikakos [30] obtained a good agreement between model predictions and experimental data on metal oams with porosities rom 88% to 98%. Their model is shown in Eq. (30): k e 2 2 R R R R A B C D (30a) R R R R A B C D 4 (30b) 2 e 2 (1 e) k s e (1 e) k e 2 2 (30c) 2 2 e 2 e ks 2 e 4 ( e 2 ) e k e s s e k e e k (30d) 2e (30e) e k e k 19

20 3 2 2 (5 8) e e 2 e (30) e In Eqs. (30a) to (30g), R, A R B, RC and RD (30g) are the calculated thermal resistances o our dierent layers inside a tetrakaidecahedron cell. The eective thermal conductivity k e is a result o these our layers being placed in parallel [30]. kse and k e, which are two important parameters in Eq. (16) and Eq. (17), can be also calculated by assigning k 0 and ks 0 in Eq. (30) respectively. From itting experimental data, Calmidi and Mahajan [27] also obtained the empirical ormula or permeability and inertial actor calculations o metal oams. Since their results showed good agreement with experimental data, this present paper has employed their ormula, with Eq. (31) showing permeability and Eq. (32) showing inertial actor respectively: d K (31) d p d p Here, d p C d (32) d p denotes the equivalent diameter o metal oam cells, which can be calculated i knowing the pore density: d p = (meter)/pore density. Pore density is dimensionless and usually measured in ppi (pores per inch). d denotes the equivalent diameter o metal oam ibres, calculated rom d d p 3 1 e (1 )/0.04 Calmidi [26]. To give more accurate calculating results, Eq. (33) has taken into account the non-circular shape o metal ibres by introducing a shape actor. The surace area density o metal oams a s is deined as the total surace area (m 2 ) o metal ibres within unit volume o metal oam matrix (m 3 ), and it can be obtained by (33) 20

21 assuming that all metal ibres have an ideal cylindrical shape (a shape actor was also introduced by Calmidi and Mahajan [27] to consider the non-circularity): a s (1 )/ d 1 e (34) d p h s is the inter-phase heat transer coeicient between the metal oam struts and, and also needs to be determined. Because the metal oam struts were assumed to have the shape o cylinders, the value o h s can be approximately calculated by the empirical ormulae or the low across a bank o cylinders [29]: Nu Nu Nu h d (35a) s Re Pr, 1 Re 40 s d d k h d 0.52Re Pr, 40 Re 1000 (35b) s s d d k h d (35c) s 0.26Re 0.6 Pr 0.37, 1000 Re s d d k 4. Numerical procedure A FVM-based (Finite Volume Method) program has been developed by the authors to solve Eqs. (10), (11) and (13) (17), which are the governing equations o the current physical problem. The program was compiled and executed under the workspace o Visual Fortran. Coupled heat conduction and natural convection equations were solved simultaneously by employing the SIMPLER algorithm (Semi-Implicit Method or Pressure Linked Equations Revised) [33] in a non-uniorm mesh ( ). The PLS (Power Law Scheme) [34] was employed to discretise convection-diusion terms to save computing time whilst ensuring high accuracy. In the x-direction (total length: 10.5 m), 1200 uniorm grids were used, with each grid m in length, whilst in the y- direction (total length: 0.02 m), 200 grids were used, with each grid m in length. Mesh independency was examined, and it was ound that the dierence between the mesh and the mesh was only 0.17%, meaning a iner mesh is not needed. Due to dierent convergence rates in the three metal-oam samples, the 21

22 optimised time step was ound to be s or the metal oam o 95% porosity and 10 ppi, s or the metal oam o 95% porosity and 30 ppi, and s or the metal oam o 85% porosity and 30 ppi. Time step independency was also examined, and it was ound that or the metal oam o 95% porosity and 10 ppi, the dierence between s and s was 0.23%; or the metal oam o 95% porosity and 30 ppi, the dierence between s and s was 0.22%; or the metal oam o 85% porosity and 30 ppi, the dierence between s and s was 0.23%. Numerical simulations were set to stop when the dierence between two consecutive iterations was less than 10-4 (i.e. 0.01%). The program was run on the high perormance HP Z1 Workstation powered by the quad-core Intel Xeon processor and 8GB RAM (Random Access Memory). Total computational time was 41.5 hours, 72.3 hours and hours or 95% porosity and 10 ppi, 95% porosity and 30 ppi, and 85% porosity and 30 ppi, respectively. The numerical programming needs to ensure that natural convection only takes place at the grids where the is in its liquid state and does not take place at the grids where the is still in its solid state. This is realised by only assigning the real viscosity value to the grids where the is liquid whilst assigning a viscosity with the value o to the grids where the is still solid. 5. Results and discussion 5.1. Validation. To veriy the numerical simulation or its accuracy and correctness, an experiment has been designed, in which a single-stage was embedded in metal oam, shown in Fig. 3. The test section comprised a piece o rectangular copper oam (with the dimension o mm) with parain wax RT58 embedded in it. According to the provider RUBITHERM, the thermo-physical properties o RT58 are melting temperature: 48-62ºC, latent heat o usion: 181 kj/kg, speciic heat: 2.1 kj/kg, dynamic viscosity: Pa s, thermal conductivity: 0.2 W/(m K), thermal expansion coeicient: K -1. The metal oam was sintered onto a thin copper plate rom the bottom side or better thermal contact. Attached to the copper plate was an electrical heater, made o 22

23 lexible silicon with adjustable heat lux, providing continuous and uniorm heat lux or the and metal oam. The heater input power could be precisely controlled and measured by a Variac and an electrical power meter (Hameg HM8115-2, accuracy ±0.5%). This allowed the heat lux used in the test to be calculated through dividing the input power by the surace area o the copper plate. In this test, nine thermocouples (accuracy ±0.1ºC) were placed at dierent locations (y = 8 mm, 16 mm and 24 mm respectively, 3 thermocouples were used or each place to get more reliable readings) inside the to monitor the transient temperature variation. Three thermocouples were placed on the copper plate to record the plate temperatures (y = 0 mm). Here, y denotes the distance between dierent locations and the heating plate. Although perect insulation cannot be guaranteed in the test, the underneath o the heating surace was insulated with Armlex insulation material and other suraces were insulated by acrylic sheets which were transparent or observation during the tests. The temperatures and the input power were automatically recorded by a data acquisition system. From previous work by the authors [9], the overall uncertainty o the test was estimated at 6.67%. The numerical results and the corresponding experimental data are compared in Fig. 4 or y = 0 and 8 mm. Both numerical results and experimental data show that the begins to melt around t = 1200s and inish phase change around t = 4000s. There is good agreement between numerical results and experimental data, and the most probable reason or the small discrepancies between them is that it has been assumed in the model that the has a ixed melting point, similarly to crystal materials. In practice, it is important to note that RT58 melts in a temperature range o 48-62ºC according to RUBITHERM. As shown in Fig. 4, the temperatures increase more slowly ater melting begins, because the heat provided is mainly used or phase change rather than increasing sensible heat. Ater the state o RT58 has become ully liquid (when temperatures are higher than 62ºC), its temperatures begin to increase more rapidly again, because the heat provided is now all used or increasing sensible heat o the. 23

24 5.2. Natural convection. Natural convection was examined by numerical simulations. Figure 5 shows the low proiles o natural convection or CTES (Cascaded Thermal Storage System) at dimensionless time = 10 when all three s have inished their melting processes. Dimensionless time is deined as the real time divided by a reerence time which equals to the melting time o 2. Three dotted squares in Fig. 5 denote the rectangular enclosures containing 1 (let), 2 (middle) and 3 (right) respectively. Inside each, two eddies are ormed: the larger eddy (clockwise) is situated near the let bottom corner whilst the smaller eddy (anti-clockwise) is situated near the right top corner. It is reasonable to have such two eddies in each, or the ollowing reasons: the and HTF temperatures decrease along the x-axis (the HTF low direction), so the on the let has lower density and thereore moves upward whilst the on the right has higher density and thereore moves downward. The larger eddy is caused by temperature dierences, and is the dominating eddy. The smaller eddy seems to have been ormed by the wake low o the dominating eddy, and is the non-dominating eddy. None o these eddies is situated near the let top corner, because the near the upper boundary has higher temperature (closer to HTF) lacking o driving orces or natural convection to take place. It can also be noted in Fig. 5 that the dominating eddies, the ones near the let bottom corner, tend to become smaller in size along the x-axis, the reason can be attributed to the act that the temperature dierences, which are the driving orces o natural convection, get smaller along the x-axis. The numerical simulation also examined natural convection or MF-CTES (Metal Foam-enhanced Cascaded Thermal Energy Storage). However, the low velocities caused by buoyancy orce are ound to be rather low, with an order o magnitude o 10-4 m/s. At irst sight, this may seem surprising, but it is still believed to be reasonable, or the ollowing reason [19]. The buoyancy orce term g T, which drives natural convection, has an order o magnitude o 10 2, but in the main drag orce term u K (i.e. Darcy term), K has an order o magnitude o According to Equilibrium o Forces, drag orce should have a similar order o magnitude to buoyancy orce, and thereore u should have an order o magnitude o The s used in this study has 24

25 high dynamic viscosity o Pa s (1000 times higher than air) and low thermal expansion coeicient o 1.1*10-4 K -1 (30 times lower than air), so these special physical characteristics result in the velocity driven by buoyancy orce being insigniicant in this case. Natural convection thereore ails to produce dominant inluence on heat transer or MF-CTES. The similar suppression o natural convection was also ound by Stritih [10], who added thirty-two metal ins into to enhance heat transer. However, he ound that the addition o metal ins did not have the desired eects on heat transer enhancement during melting, with the reason being that natural convection was signiicantly suppressed by metal ins and the Rayleigh number in his study was not suiciently high to overcome the large low resistance Comparison o equivalent heat exchange rates among STES, CTES and MF-CTES. Equivalent heat exchange rates were also obtained rom numerical simulations or STES, CTES and MF-CTES. Figure 6 shows the comparison o equivalent heat exchange rates between STES and CTES. It indicates that CTES using cascaded arrangement o s enhances heat transer by up to 30% (overall). However, it should be noted that the equivalent heat exchange rates o CTES system is lower than that o STES when 2 inishes melting or two reasons. Firstly the temperatures in CTES increase rapidly (sensible heat) when 2 (50 ) inishes phase change. Secondly, when the temperatures in CTES rise rapidly, the temperatures in STES have kept relatively constant because 4 (55 ) is still in melting process (latent heat). The rapidly-rising temperatures in CTES have caused the decrease o temperature dierences between s and HTF, resulting in a lower heat transer perormance. The phase change regions o 1, 2, 3 and 4 can also be seen in Fig. 6, which are represented by the horizontal lines. The igure also indicates that 3 uses the most time to inish phase change, because temperature dierences or 3 are much smaller than all other s. As temperature dierences decrease, 2 and 3 take more time to be melted than 1. Figure 7 compares the equivalent heat exchange rates o MF-CTES (Metal Foamenhanced Cascaded Thermal Storage) among three copper-oam samples, the properties 25

26 o which are listed in Table 3. Figure 7 indicates that MF-CTES enhances heat transer by 2 7 times and reduces melting time by 67 87% compared to CTES. Sample C (85% porosity) has better heat transer perormance than Sample A and B (both 95% porosity). This is reasonable because the ormer has more solid structures, which results in higher eective thermal conductivity; thus it can transer heat lux more eiciently to s through the metal oam skeleton. Sample B (30ppi pore density) has better heat transer perormance than Sample A (10ppi pore density). This is also reasonable because higher pore density results in larger contact area between s and metal ligaments so that more heat can be transerred. It can thereore be concluded that the metal-oam samples with lower porosity and higher pore density have better heat transer perormance than the ones with higher porosity and lower pore density. In summary, CTES enhances heat exchange rate by up to 30% compared to STES; MF-CTES enhances heat exchange rate by 2 7 times compared to CTES, depending on the properties o metal-oam samples (porosity, pore density and k s ) Comparison o exergy eiciency among STES, CTES and MF-CTES. Exergy eiciencies were examined by numerical simulations or STES, CTES and MF-CTES. The comparison o exergy eiciencies between STES and CTES is shown in Fig. 8. CTES does not always have higher exergy eiciency than STES (-15% to +30%). The exergy eiciency o CTES is lower than that o STES in early stages beore 4 starts to melt, because 1 and 2 in CTES have delayed temperature rise due to their latent heat. Since lower temperatures mean lower quality o energy, CTES has lower exergy eiciency at this time. However, the situation is changed when 2 inishes phase change and 4 starts phase change. From this time on, the temperatures in CTES begin to increase rapidly (sensible heat) whilst the temperatures in STES keep relatively constant (latent heat), which leads to CTES having a higher exergy eiciency than STES. Figure 9 compares the exergy eiciencies o MF-CTES among three dierent copper-oam samples, indicating that Sample A, B and C all have roughly the same exergy eiciency as CTES. It means metal oams cannot urther improve exergy eiciency or CTES, but they can help CTES to inish melting more quickly by having a 26

27 much higher heat exchange rate. As seen in Fig. 8 and Fig. 9, all metal-oam samples inish melting more quickly than CTES, with Sample C being the quickest one. In summary, CTES does not always have higher exergy eiciency than STES (-15% to +30%); MF-CTES cannot urther improve exergy eiciency or CTES, but can help CTES to inish melting more quickly by having higher heat exchange rates (melting time reduced by 67 87%) Comparison o exergy transer rate among STES, CTES and MF-CTES. The Second Law o Thermodynamics states that q cannot be 100% converted into electricity, meaning that heat exchange rate cannot relect the real eiciency o a TES system. Thus, a new concept o exergy transer rate h ex has been proposed in this study to evaluate the overall thermal perormance o STES, CTES and MF-CTES. h ex = q η ex (W/m 2 ) (36) h ex is the eective exergy transer rate, representing how much useul thermal energy is transerred rom HTF to s during charging processes. Eective exergy transer rates h ex were obtained by numerical simulations or STES, CTES and MF-CTES. Figure 10 shows the comparison o h ex between STES and CTES system, it can be concluded that CTES nearly always produces higher exergy transer rate (up to 23%) than STES. It needs to be noted that CTES delivers slightly lower exergy transer rate than STES, only when 1 starts phase change and ater 4 inishes phase change. There are two probable reasons or this. Firstly, when 1 starts its phase change, CTES had lower exergy eiciency than STES despite o CTES having slightly higher heat exchange rate than STES. Secondly, ater 4 inishes its phase change, the heat exchange rate o STES is higher than CTES due to the long-time delay o temperature rise (latent heat o 4), but the exergy eiciency STES is much lower than CTES (shown in Fig. 8) due to its low temperatures ater phase change. Figure 11 compares the eective exergy transer rate (h ex ) o MF-CTES among three dierent copper-oam samples, indicating that Sample C has the highest h ex than Sample A and B, and that all metal-oam samples have much higher h ex (by 2 7 times) than CTES. 27

28 In summary, CTES nearly always has higher exergy transer rates (up to 23%) than STES; MF-CTES can urther increase exergy transer rates o CTES by 2 7 times. 6. Conclusions The numerical results have shown good agreement with experimental data. Natural convection exists in all the three cases studied: STES, CTES and MF-CTES. Due to low thermal expansion coeicients and high viscosities o s used, natural convection was ound to be rather weak in MF-CTES, since metal oams have large low resistance. CTES enhances heat exchange rate by up to 30% compared to STES. MF-CTES enhances heat exchange rate by 2 7 times compared to CTES, depending on the properties o metal-oam samples (porosity, pore density and metal thermal conductivity). Simulation results indicate that the metal oams with lower porosity and higher pore density have better heat transer perormance than the ones with higher porosity and lower pore density. CTES does not always have higher exergy eiciency than STES (-15% to +30%). MF-CTES cannot urther improve exergy eiciency or CTES, but can help CTES to inish melting more quickly by having higher heat exchange rates (melting time reduced by 67 87%). CTES nearly always has higher exergy transer rate (up to 23%) than STES. MF-CTES can urther increase exergy transer rate o CTES by 2 7 times. Acknowledgements This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC grant number: EP/F061439/1), and the National Natural Science Foundation o China (NSFC grant number: ). The authors are grateul to Proessor Keith Godrey or his valuable advice that signiicantly improved this study. Acknowledgement also goes to the two anonymous reviewers or their proessional review comments which have urther improved the quality o this study. 28

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32 Figure Captions Figure 1: STES, CTES and MF-CTES. Figure 2: Tetrakaidecahedron [32]. Figure 3: The experimental test rig. Figure 4: A comparison between numerical results and experimental data. Figure 5: Flow proiles o natural convection or CTES. Figure 6: A comparison o equivalent heat exchange rates q (W/m 2 ) between STES and CTES. Figure 7: A comparison o MF-CTES equivalent heat exchange rates q (W/m 2 ) among three dierent metal-oam samples. Figure 8: A comparison o exergy eicencies η ex (%) between STES and CTES. Figure 9: A comparison o MF-CTES exergy eicencies η ex (%) among three dierent metal-oam samples. Figure 10: A comparison o h ex between STES and CTES. Figure 11: A comparison o h ex among dierent metal-oam samples or MF-CTES. 32

33 y T 0 O HTF 4 H L4, T m4 HTF T (t) h 1 h 2 (a) STES x y T 0 O Heat Transer Fluid (HTF) H L1, T m1 H L2, T m2 HTF H L3, T m3 T (t) h 1 h 2 L 1 L 2 L 3 (b) CTES x y T 0 Heat Transer Fluid (HTF) thermal insulation T (t) O h 1 h 2 1 H L1, T m1 2 H L2, T m2 3 H L3, T m3 L 1 L 2 L 3 (c) MF-CTES x Fig. 1. STES, CTES and MF-CTES. 33

34 (a) A single tetrakaidecahedron; (b) Three tetrakaidecahedrons lapped together Fig. 2. Tetrakaidecahedron [32]. (a) Experimental setup Fig. 3. The experimental test rig. (b) Test section 34