Second-Law Analysis of Molten-Salt Thermal Energy Storage in Thermoclines

Transcription

1 Purdue University Purdue e-pubs CTRC Research Pubications Cooing Technoogies Research Center 2012 Second-Law Anaysis of Moten-Sat Therma Energy Storage in Thermocines S. Fueckiger Purdue University S V. Garimea Purdue University, sureshg@purdue.edu Foow this and additiona works at: Fueckiger, S. and Garimea, S V., "Second-Law Anaysis of Moten-Sat Therma Energy Storage in Thermocines" (2012). CTRC Research Pubications. Paper This document has been made avaiabe through Purdue e-pubs, a service of the Purdue University Libraries. Pease contact epubs@purdue.edu for additiona information.

2 Second-Law Anaysis of Moten-Sat Therma Energy Storage in Thermocines* Scott M. Fueckiger and Suresh V. Garimea** Schoo of Mechanica Engineering 585 Purdue Ma, Purdue University West Lafayette, IN USA * Revised for pubication in Soar Energy, October 2011, and in revised form, January 2012 ** Author to whom correspondence shoud be addressed: (765) , sureshg@purdue.edu

3 Abstract The cycic operation of a moten-sat thermocine tank is simuated to investigate the infuence of interna granue diameter and externa convection osses on tank performance. Practica constraints imiting thermocine tank height are taken into account. The authors twotemperature mode, deveoped in earier work (Soar Energy 84: , 2010) for the anaysis of heat transfer and fuid fow in the thermocine tank, is extended to monitor entropy generation and exergy transport. Storage performance is measured in terms of first- and second-aw efficiency definitions, as we as a first-aw efficiency used in conjunction with an outfow temperature criterion. Reducing the diameter of the fierbed granues improves the thermocine tank performance by sustaining higher moten-sat outfow temperatures throughout the discharge phase of the cyce, which resuts in greater operationa efficiency. Externa convection osses strongy infuence entropy generation inside the tank fierbed due to the deveopment of radia temperature gradients and increased irreversibe therma diffusion. Convection osses aso resut in ower tank efficiencies due to the reduction of hot moten sat avaiabe inside the tank. A comparison of the different efficiency definitions empoyed in this work revea that the ad hoc outfow temperature criterion used in past studies provides an overy conservative assessment of thermocine performance. Keywords: soar therma energy, thermocine, moten sat, exergy 2

4 1. Introduction Diminishing fossi-fue reserves and acceerating anthropogenic cimate change via greenhouse gas emissions have renewed goba interest in the conversion of soar energy to eectricity. Large-scae conversion is possibe with concentrating soar power (CSP) pants which focus sunight via heiostats, paraboic troughs, or Fresne refectors (Kob et a., 1991; Mis and Morrison, 2000) onto a tubing network carrying a heat transfer fuid (HTF). The HTF captures the incident radiation as sensibe heat that is then suppied to a steam boier within a Rankine power cyce. CSP technoogies have been demonstrated at both experimenta and commercia scaes, but remain subject to variations in avaiabe sunight due to coud transients, panetary rotation, and seasona changes. The effects act at mutipe time scaes to diminish pant performance and imit the overa reiabiity of eectricity generation from the pant. To match the reiabiity of fossi-fired and nucear power pants, the CSP pant must be augmented with either auxiiary fossi fue suppies or reserves of excess energy coected and stored during periods of high soar irradiance. Fossi-fue backup is undesirabe due to the inherent environmenta impact. In contrast, energy storage decoupes the power cyce from the coection system via sufficient reserves of usefu energy to offset periodic osses in sunight. Energy storage has been studied for CSP appications with sensibe heat, atent heat, and thermochemica reactions (Herrmann and Kearney, 2002; Lovegrove et a., 2004). Of these methods, sensibe heat storage in an excess voume of HTF is preferred due to simpicity and ow reative cost. During periods of high irradiance, excess HTF heated in the coection oop is diverted to a storage voume for ater use. Thermocine tanks offer a ow-cost means of reaizing this concept by storing both the hot and cod fuid inside a singe reservoir. Buoyancy forces, associated with the variation of the HTF density with temperature, maintain stabe therma 3

5 stratification of the two isotherma voumes. A narrow region of arge temperature gradient exists at the interface of these fuid voumes, known as the thermocine or heat-exchange region. The vertica ocation of this heat-exchange region varies in time as the hot HTF is added and then extracted. Cycic operation of the thermocine tank requires fow reversa in order to maintain the therma stratification (fuid fow is downward to charge the tank with hot HTF and subsequenty upward to discharge the hot HTF). When the heat-exchange region reaches the top of the fierbed, the hot suppy of HTF is exhausted and the power cyce is no onger supported by the storage system. Such a tank offers significant financia advantages over an aternative two-tank impementation (Kearney et a., 2003; Eectric Power Research Institute, 2010). Constraints reated to practica deivery of heat to the steam boier restrict the seection of HTF to materias that remain iquid at very high temperatures, i.e., synthetic ois and moten sats. Synthetic ois are an expedient design option as they remain in iquid phase under ambient conditions. However, ow vapor pressures imit operation of ois to temperature beow 400 C, a significant constraint on the CSP pant therma efficiency. Significant gains in efficiency are possibe with a transition to moten nitrate-sat mixtures, which remain iquid up to 600 C. Furthermore, moten sats are ow-cost, non-fammabe, and non-toxic. Commercia sat mixtures incude soar sat (40 wt% KNO 3, 60 wt% NaNO 3 ) and HITEC (35 wt% KNO 3, 40 wt% NaNO 2, 7 wt% NaNO 3, Coasta Chemica Co.). Both mixtures are eutectics and have meting points above ambient; HITEC exhibits a iquid operation range of 142 C to 535 C (Kearney et a., 2003). Freeze prevention and recovery is essentia in a sat-based systems to maintain the sat at eevated temperatures and to minimize component damage in the event of a change of phase. In addition to the HTF, the interior of the thermocine tank is fied with a ow-cost fier materia to act as an additiona storage medium and to reduce the required voume of the more 4

6 expensive sat. Granuating the fier materia inhibits therma diffusion in the axia tank direction, maintaining therma stratification in the surrounding fuid. The resutant porous assemby is termed a dua-media thermocine tank. Seection of the fier materia is not trivia as repeated therma cycing of the tank may ead to degradation of the granues and potentia entrainment into the fuid fow. Pacheco et a. (2002) screened severa candidate materias for use in moten-sat thermocines through proonged exposure to sat at different therma conditions. Quartzite rock and siica sand were found to be the most resistant to degradation after severa hundred therma cyces. The dua-media thermocine therma storage concept was first demonstrated with a 170 MWh t tank in conjunction with the Soar One piot centra-receiver pant in Barstow, CA (Faas et a., 1986; Radosevich, 1988). The tank interior consisted of Caoria minera oi as the HTF, granite rock as the soid fier, and uage space fied nitrogen. Whie in operation from 1982 to 1986, temperature imitations of the Caoria, couped with insufficient heiostats in the coection fied for excess soar energy capture, reegated the thermocine tank to auxiiary steam generation. Pacheco et a. (2002) ater constructed a sma-scae 2.3 MWh t thermocine at Sandia Nationa Laboratories to vaidate the use of moten sat with quartzite rock and siica sand. A thermocoupe rake imbedded in the tank indicated successfu therma stratification of the sat. The geometric size and high operating temperature associated with physica tanks have argey imited anaysis of the thermocine concept to numerica simuation. Kob (2006) deveoped a CSP pant system mode using TRNSYS commercia software to simuate the theoretica addition of a 30 MWh t thermocine tank to the 1 MW Saguaro paraboic trough pant near Tucson, AZ. The thermocine tank was modeed in TRNSYS as a one-dimensiona component with therma osses enforced at the roof, foor, and surrounding wa. The 5

7 performance of the thermocine mode was vaidated against experimenta data from the Soar One thermocine tank. The resutant system simuation reveaed that thermocine storage in conjunction with an expanded heiostat fied increased the capacity factor of the Saguaro pant from 23% to 42%. Yang and Garimea (2010a) deveoped a computationa fuid dynamics mode of a moten-sat thermocine tank to investigate muti-dimensiona effects. Energy transport inside the tank was soved with a two-temperature mode in order to account for the different transport properties of the fuid and soid. The performance of the mode was vaidated with experimenta tank data reported by Pacheco et a. (2002). The authors conducted a parametric study of tank size, granue diameter, and discharge power to determine optimum discharge conditions. Discharge performance improved with increased tank height and increased storage capacity. Performance aso improved with decreased granue diameter and reduced discharge power, characterized by ow interstitia Reynods numbers. Yang and Garimea (2010b) updated their mode to incude convection osses at the tank wa to the ambient surroundings. In contrast to the resuts obtained under adiabatic-wa conditions, the discharge performance of the thermocine decreased with ow interstitia Reynods numbers. At these ow Reynods numbers, the reduced fuid veocity increased the residence time of the moten sat inside the tank and proonged cooing of sat near the tank wa. This phenomenon is ess pertinent at arger Reynods numbers for which the discharge performance better matched the equivaent adiabatic tank wa resuts. Yang and Garimea (in review) extended the adiabatic mode to study cycic effects of the charge-discharge processes. As with the previous adiabatic discharge mode, performance of the thermocine tank improved with increased tank height and decreased granue diameter. Cyce 6

8 efficiencies associated with the thermocine modes were curve-fit as a function of the interstitia Reynods number and the ratio of moten-sat fow distance to the fier diameter. Fueckiger et a. (2011) updated this cycic investigation to incude effects of convection and radiation osses to the surroundings as we as a composite tank wa for anaysis of therma ratcheting as a potentia faiure mode. In the parameter studies performed of Yang and Garimea (2010a; 2010b; in review), the mode tank size varied over severa fierbed heights from 2.77 m to 56.7 m in order to construct the reported efficiency functions. Practica imits reated to soi bearing capacity were not imposed on the tank height. A recent civi engineering study of Barstow, CA soi found that physica thermocine tanks shoud not exceed 39 feet (11.9 m) (Eectric Power Research Institute, 2010). In addition, Yang and Garimea (2010a; 2010b; in review) measured thermocine tank performance soey on an energy (enthapy) basis. The authors enforced an ad hoc temperature cut-off criterion to the outfow moten-sat discharge to account for the oss of therma quaity. Ony sat outfow at temperatures above 95% of the tota operating range was designated as being usefu for power production. For exampe, if the temperature difference between the hot and cod fuid states was 100 K, ony moten sat discharge within 5 K of the hot temperature imit was designated as usabe for steam generation in the Rankine cyce. An aternative approach to this ad hoc criterion is to measure the quaity of the outfow using the thermodynamic definition of exergy. It is important to note that therma energy storage devices must store not ony energy, but aso exergy (Bejan, 2006). Unike energy, exergy is not subject to a conservation aw and can be destroyed. This destruction is directy proportiona to the generation of entropy associated with the second aw of thermodynamics. In the present work, a numerica simuation of the dua-media thermocine tank is conducted to investigate 7

9 entropy generation inside the fierbed as we as the recovery of energy and exergy during cycic operation. A parametric study of fierbed granue diameter and externa convection osses assesses the respective infuence on both first- and second-aw cyce efficiencies. These resuts are then compared to the outfow temperature criterion-based efficiency used in the iterature. 2. Numerica Modeing 2.1 Probem Description A schematic diagram of the thermocine storage tank is provided in Figure 1. The tank of diameter d is fied with a porous bed of granuated fier to a height h. Adjacent to the top and bottom of the porous fierbed are two distributors of height h ( h 0. 05h ), free of any fier. Fuid enters and exits the tank through two tubuar ports of diameter d ( d 0. 1d ) extending from the distributors. The open distributor regions serve to diffuse the turbuent tube fow at the distributor inet eveny into the tank fierbed, preventing the formation of radia temperature gradients in the stratified fuid. An aternative form of distribution is though the use of pipe manifods imbedded within the fierbed. Such manifods are not axisymmetric and drasticay increase the computationa resources necessary to mode the thermocine tank. The open distributor regions considered here simpify the numerica approach and maintain axisymmetric fow conditions. HITEC moten sat is seected as the thermocine tank heat transfer fuid. For therma P transport anaysis, the specific heat ( C, ) of HITEC sat is J/kg K. Temperaturedependent functions, derived by Yang and Garimea (2010a) from experimenta data (Coasta Chemica Co.), provide the fuid density, viscosity, and therma conductivity: T T 200 (1) 8

10 T exp nT (2) 4 T T k (3) The porosity of the soid fier is fixed at 0.22 in accordance with experimenta observation for quartzite rock and siica sand mixture (Pacheco et a., 2002). To simpify anaysis of the mixture, the size of the fier is defined with a singe representative granue diameter. The density and specific heat are taken as quartzite rock properties, 830 kg/m 3 and 2500 J/kg K (Specific Heat Capacities of Some Common Substances). The soid therma conductivity is derived from data for quartz materias (Heraeus Base Materias). However, therma diffusion between soid granues is negected in the mode due to the sma contact area and high contact resistance between partices. The soid granues sti infuence diffusion in the surrounding moten sat, characterized by the foowing correation for effective therma conductivity (Gonzo, 2002): exp k eff k (4) 1 where 1 and k k k 2k. s s The veocity of the moving heat-exchange region is infuenced by conduction and convection with the soid fier and is therefore not equa to the veocity of the moten-sat fow, but is aso dependent on the porosity and voumetric heat capacities of the dua storage media. Yang and Garimea (in review) defined the ratio of the heat-exchange zone veocity to motensat superficia veocity ( C 0 ) as foows: C 0 v u, h P, (5) 0, h C P, C 1 scp, s For the properties of the chosen mixture of moten sat and rock, this veocity ratio is Governing Equations 9

11 Governing equations for mass and momentum transport in the moten-sat fuid contained in the thermocine tank are as foows, with the momentum fux in the porous fierbed being governed by Darcy s Law with the Brinkman-Forchheimer extension: t u 0 (6) u uu F P ~ τ g u u u t K K (7) In the momentum transport equation, the stress deviator tensor is defined as ~ ~ ~ τ 2 S 2 tr S, ~ 2 1 where S u u T is the rate of strain tensor. The spatia gradient of the thermocine tank 3 in poar coordinates is e r r e r e x x. The axisymmetric nature of the thermocine geometry eiminates a veocities and functiona dependencies in the circumferentia direction (θ). The inertia coefficient ( F ) and permeabiity ( K ) are determined from the iterature (Krishnan et a., 2004; Beckermann and Viskanta, 1988). Separate energy equations for the moten sat (subscripted ) and fier (subscripted s) are required to mode non-therma-equiibrium conditions between the fuid and soid: C t P, T 1 scp, s t T s C T k T h T T u (8) h i P, T s T eff Equations (8) and (9) remain couped by a voumetric heat transfer coefficient ( h i ) associated with convective heat exchange between the moten sat and soid fier. Interstitia forced convection in the porous media is modeed with the Wakao and Kaguei (1982) correation: Nu i Re Pr 1/ 3 i s (9) (10) 10

12 Therma diffusion in the fierbed is enforced ony in the fuid energy equation via the effective therma conductivity of the dua-media mixture. As noted before, therma diffusion is negected in the soid fier due to contact resistance between partices. Non-dimensiona terms associated with the geometry and moten-sat fow inside the fierbed are defined as: x X u, 0 t 0 r R u, 0 t 0 u 0 t 0 d s t,, t0 T T T T c, h c u 0 d C Re s, Pr P,, k 2 hid s Nui, k u0 Ec, C T T P, h 2 c h 2 Da. K The CFD simuation must aso consider entropy transport for a second-aw anaysis of the thermocine tank mode. Departure from therma equiibrium again requires separate equations for moten sat and soid fier: C P, t n T scp, s n Ts t q T q T q T uc P, nt S gen, s S gen, s (11) (12) Unike the transport variabes considered hitherto, entropy is not a conserved property. Thus an unknown source term ( S gen ) exists in both equations. This voumetric entropy generation arises from irreversibe processes associated with operation of the thermocine tank, i.e., therma diffusion and viscous dissipation. In accordance with the second aw of thermodynamics, entropy generation is aways non-negative. The conservative form of the entropy transport equation is difficut to sove due to the unknown generation term. Instead, entropy transport is recast in a non-conservative form by removing conservation of mass and energy terms imbedded within Equations (11) and (12) 11

13 (Bejan, 1986). The resutant equations are reorganized and combined into a singe reation for the net entropy generation inside the porous media: S gen k eff T T T 1 (13) The compete derivation of Equation (13) is provided as an appendix. Destruction of exergy is proportiona to the generation of entropy by the foowing equation; the associated reference temperature (T 0 ) is equa to the defined ambient temperature: X T 0 (14) dest S gen 2.3 Boundary Conditions To charge the thermocine tank, hot moten sat enters the top distributor port at a fixed veocity and temperature: d U d 2 u e h z T T h Simutaneous to the hot infow, cod moten sat exits the tank via the bottom distributor. Due to transient density variations of the moten sat inside the thermocine tank, the mass and voumetric fow rate of the outfow are not known a priori. To discharge the tank, fow is reversed such that cod moten sat enters the tank at the bottom distributor port at a fixed veocity and temperature: d U d 2 u e c z T T c The entering veocity of the cod moten sat is proportionay ess than the prior hot infow due to the increased density of the cod sat, baancing the charge and discharge processes of the thermocine tank. 12

14 The no-sip condition is enforced at the interna wa of the tank. Therma osses aong the externa wa surface are governed by forced convection associated with wind veocities: T k r w h w T T 0 (15) The convection coefficient ( h w ) aong the tank wa is determined using the Churchi and Bernstein (1977) correation as a function of wind veocity and air properties at fim temperature. Radiation to the environment is negected as the tank surface is assumed to be reradiating. 2.4 Soution Procedure The mode fierbed and distributor geometries are discretized into a structured nonuniform mesh, with a maximum non-dimensiona ce size of ΔX = ΔR = 0.1. The governing mass, momentum, and energy equations of the moten-sat are discretized with the finite-voume method and soved with the commercia computationa fuid dynamics (CFD) software, FLUENT (FLUENT Documentation). Spatia discretization of the interna convective fuxes is performed with a second-order upwind scheme. Transient discretization is performed with a first-order impicit formuation and a non-dimensiona time step of Δτ = 1.2x10-4. Grid and time-step independence were previousy verified by Yang and Garimea (in review). Pressure-veocity couping is achieved with the PISO agorithm (Issa, 1986). Equation (9) for soid fier temperature and Equation (13) for fierbed entropy generation rate are soved through user-defined functions (UDF). The soution at each time step is considered converged when a dimensioness residuas reduce to ess than At the start of the computation, the entire thermocine tank geometry is initiaized to the cod moten-sat temperature. The tank is charged with hot moten sat for 12 hours and then discharged for another 12 hours to simuate a fu day of operation. It shoud be noted that initia isotherma condition is never reaized in a physica thermocine tank. Thus successive cycing of 13

15 the tank mode is necessary to achieve the appropriate initiaization. The tank then converges to a periodic temperature response with each charge and discharge process. After convergence to periodicity, the performance of the simuated thermocine tank is measured with three efficiencies: first-aw, second-aw, and the ad hoc outfow temperature criterion appied by Yang and Garimea (2010a). The first- and second-aw efficiencies reate the discharge of energy ( E ) and exergy ( X ) to the preceding charge of energy and exergy, defined as foows: E out,dis I (16) Ein,chg X out,dis II (17) X in,chg In contrast, the outfow temperature criterion efficiency is defined by rejecting a moten-sat energy with a non-dimensiona temperature ( ) beow 0.95: E 0.95 dis out, I (18) Ein,chg By definition, the outfow temperature cutoff-based efficiency is aways ess than the first-aw efficiency in Equation (16). The energy and exergy entering the thermocine during charge are both known according to the fixed veocity boundary conditions. As previousy mentioned, the characteristics of the discharge outfow are not known a priori. To assess the therma usefuness of the moten sat eaving the tank, additiona UDFs in the CFD mode record the energy and exergy discharged from the tank distributor ports after every time step. The sums of the vaues determine the tota energy and exergy deivered from the thermocine outfow. Simiar UDFs monitor entropy generation inside the fierbed. 14

16 3. Resuts and Discussion 3.1 Tank Sizing The aspect ratio of the thermocine tank geometry was reported by Yang and Garimea (2010a) to have a strong infuence on the storage performance. For equa interna tank voume, ta and narrow tanks perform consideraby better than short and wide tanks due to increased therma stratification of the moten sat in the former case. For a tank of fixed height, increasing the diameter does not infuence therma stratification but ony scaes the maximum interna energy content of the moten sat and fier. Since the tank height is indeed constrained by the soi bearing capacity previousy mentioned, thermocine tank performance optimization by aspect ratio is not practicay reevant. In the current work, the fierbed height in the thermocine tank is fixed at 12 m. The tank diameter is aso fixed at 12 m (arger diameters woud not provide any further insight into storage performance but woud increase computation time). The thermocine tank design agorithm and efficiency mode proposed by Yang and Garimea (in review) determine the moten-sat veocity for the fixed fierbed size. Equating the non-dimensiona fierbed height H h u t to the ratio of heat-exchange zone veocity to moten-sat veocity ( 0 0 C 0 ) and fixing the effective granue diameter ( d ) to 5 cm, the fierbed Reynods number and ength ratio ( ) s are and 194.4, respectivey. For these vaues, the cacuated 95% outfow temperature cyce efficiency is Once the fierbed size is fixed as discussed above, the remaining parameters governing thermocine tank storage performance incude the granue diameter and externa osses. Yang and Garimea (2010a) found reduced granue diameter improved the thermocine performance 15

17 by reducing the axia span of the heat-exchange region. However, anaysis of the granue diameter was imited to a minimum effective diameter of 5 cm. For the moten-sat veocity and fierbed porosity under consideration, anaysis of Darcy s aw with the Kozeny-Carman equation reveas that a further reduction in granue diameter is attainabe without arge increases in pressure drop. Thus the simuation of the thermocine tank is extended to incude smaer granue diameters of 2 mm and 1 cm. (The resutant Reynods numbers and ength ratios are outside the imits of the Yang and Garimea (in review) cycic performance mode, precuding a priori cacuations of efficiency). Assuming the thermocine tank is to be instaed at the ocation of previous CSP faciities (i.e., in Barstow, CA), the oca wind speeds often exceed 10 m/s (Nationa Cimate Data Center). Thus convection may ead to substantia osses of stored energy inside the thermocine tank to the environment. Tank wa conditions associated with no osses (Nu w = 0) or wind speeds of 11.1 m/s (Nu = 4260) are considered to assess the infuence of externa convection. In conjunction with the three granue diameters of interest, a tota of six thermocine cases are simuated. As shown in Equation (13), entropy generation in the porous fierbed is a function of both therma diffusion and viscous dissipation. The importance of viscous dissipation ( ) with respect to fuid energy transport and entropy generation may be assessed with the nondimensiona term N, defined as (Nied, 2000): N Ec Pr Da (19) For porous media fows with N much ess than unity, viscous effects are negigibe. Given the geometric scae of the thermocine tank combined with the ow moten-sat veocities, the 16

18 simuated thermocine tanks feature N vaues on the order of Thus viscous effects are inconsequentia and omitted from the entropy generation anaysis. 3.2 Temperature and Veocity Fieds Profies of the moten-sat temperature aong the thermocine axis during a chargedischarge operation are potted in Figure 2 for the three adiabatic cases. As with previous studies, the reduced granue diameter (d s = 2 mm) enabes a thinner heat-exchange region compared to the arger granue diameter (d s = 5 cm). At the midpoint of the charging process (τ = 0.5), the heat-exchange region extends from a non-dimensiona fierbed height ( x / h ) of 0.1 to 0.6 for the 2 mm granues. In contrast, this region extends from 0.1 to 0.75 for the 5 cm granues. Comparison of the tanks with d s = 2 mm and d s = 1 cm reveas simiar engths of the heatexchange region, impying a practica imit to therma stratification in the dua-media mixture. The benefits of a reduced heat-exchange region for the finer fierbed granues incude greater energy content stored at the end of the charge (τ = 1) as we as hotter outfow temperatures during the subsequent thermocine discharge (1 < τ < 2). Temperature contours and fow streamines in the moten sat are potted in Figure 3 for the thermocine tank undergoing discharge (1 < τ < 2) with d s = 1 cm. As the discharge progresses in time, hot moten sat fows out of the top distributor port unti it is exhausted at the end of the cyce (τ = 2). The infuence of externa convection (Nu w = 4260) is apparent in the non-uniform fow streamines. In the adiabatic scenario, highy organized axia veocities are sustained inside the fierbed and resut in perfect vertica stratification of the hot and cod moten sat regions (Yang and Garimea, 2010b). Externa convection disturbs this stratification by inducing secondary buoyancy forces in the sat cooed near the tank wa. The resutant fow reversa disrupts the streamines by introducing radia veocities in the fierbed, seen throughout 17

19 the discharge process in Fig. 3. Externa osses aso generate arge vortices inside the bottom distributor, shown at τ = 1.5 and τ = Outfow Temperature Profies The outfow moten-sat temperature histories associated with tank discharge are potted in Figure 4 for a six simuation cases. The three adiabatic cases sustain the hottest outfow temperatures throughout the discharge, expected as no stored energy is ost to the surroundings. In the non-adiabatic cases, osses to the surroundings cause sma but immediate decreases in outfow temperature foowing the onset of tank discharge. This decine ater becomes significant in the ast haf of the process (τ > 1.5) as the heat-exchange region reaches the top distributor port. Externa convection reduces the interna energy content of the thermocine tank, protracting the heat-exchange region and causing eary depetion of the hot sat suppy. Thus the usabiity of the thermocine tank for steam generation in a CSP pant decreases. As with the disruption of the temperature and veocity fieds, oss of therma quaity magnifies as convection at the tank wa is increased. Throughout most of the tank discharge, smaer granues sustain hotter outfow temperatures compared to arger granues for equivaent tank wa conditions. The thinner heatexchange region associated with the smaer granue diameter enabes the presence of a arger voume of hot moten sat inside the tank, extending the duration of high-temperature outfow. However, thinner heat-exchange regions aso impy arger temperature gradients which cause a more rapid decine in outfow temperature when the hot suppy is exhausted. In the non-adiabatic tanks, this decine eventuay eads to coder outfow temperatures for d s = 2 mm compared to d s = 5 cm. However, this occurs near competion of the discharge. Thus the 2 mm granues remain preferabe for sustaining high therma quaity outfow. 18

20 3.4 Entropy Generation For a second-aw perspective on the simuated thermocine tank performance, entropy generation inside the fierbed is potted in Figure 5 as a function of non-dimensiona cyce time for d s = 1 cm. Energy generation is normaized with respect to the maximum (fina) vaue of the adiabatic cyce. As iustrated with the semi-og pot, generation is strongy infuenced by the incusion of externa convection osses and resuts in more than an order-of-magnitude increase in generation. This trend is true for a three granue diameters. Cooing of the sat near the wa deveops radia temperature gradients inside the thermocine tank, providing an additiona pathway for irreversibe mixing and heat transfer between the stratified hot and cod regions. The magnitude of the temperature gradient vector increases and more entropy is generated. Comparison of entropy generation between different granue diameters shows negigibe variation reative to the infuence of externa convection. 3.5 Thermocine Efficiency Thermocine tank efficiencies associated with the discharge performance are compared in Tabe 1 for each case. Incuded in the tabe is the first-aw efficiency (ratio of energy discharged to energy charged), the second-aw efficiency (ratio of exergy discharged to exergy charged), and the outfow temperature criterion-based efficiency (ratio of energy discharged with Θ > 0.95 to energy charged). As with the other thermocine tank metrics, tanks with adiabatic wa conditions yied the best performance for a three efficiency definitions. Externa osses reduce the energy and exergy content of the moten-sat discharge outfow, preventing fu recovery of the energy and exergy suppied during the charge process. Reduced granue diameter aso improves efficiency by sustaining more of the moten sat voume inside the tank at the hot temperature for eventua discharge. 19

21 The second-aw efficiency of the thermocine tank cyce is ess than the first-aw efficiency, as expected for a therma energy storage device. This disparity remains minima for the three adiabatic tank wa scenarios but increases with the incusion of convection osses. As before, exergy transport accounts for not ony energy but aso the usabiity of that energy. The second-aw efficiency is therefore more sensitive to the effects of convection and provides a better indication of the resutant tank performance. Among the three metrics, energy efficiency in conjunction with a 95% outfow temperature criterion yieds the owest performance vaue. For the adiabatic tank with d s = 2 mm, the outfow temperature remained above the 95% temperature threshod throughout the discharge, thus yieding an outfow temperature criterion-based efficiency equa to the pure first aw-efficiency. For the adiabatic tank with d s = 5 cm, the outfow temperature criterion-based efficiency is This vaue exceeds the efficiency predicted by the Yang and Garimea (in review) cycic efficiency mode (0.790) discussed in Section 3. It shoud be noted that Yang and Garimea monitored the transport of moten sat at the imits of the dua-media fierbed, inside the distributor region. In the current investigation, tanks exposed to wind osses experience recircuation zones at this interface, adding arge uncertainties to the anaysis of energy and exergy exchange with the thermocine (shown in Fig. 3). The infow and outfow of the thermocine tank are instead evauated at the distributor ports of reduced diameter, where recircuation does not occur, to avoid this uncertainty. Incusion of the distributer regions effectivey increases the height of the tank and resuts in a thermocine cyce efficiency greater than the mode prediction. As with the second-aw efficiency, the outfow temperature criterion-based efficiency is subject to arge decreases for non-adiabatic tanks. Externa osses generate sma but immediate 20

22 drops in outfow temperature at the star of discharge, as seen in Fig. 4. This temperature drop off inhibits sustained deivery of hot moten sat with Θ > 0.95, resuting in temperature-criterion efficiencies as ow as Because of this sharp decine compared to the previous first- and second-aw efficiency definitions, the Θ > 0.95 stipuation for moten sat outfow is an overy conservative thermocine tank design metric. As indicated by the second-aw efficiency, motensat outfow beow this temperature remains serviceabe for the power bock. Instead of rejecting this coder outfow, the Rankine cyce shoud be compiant with the ower-quaity moten sat to continue steam generation and power output at reduced therma efficiency. A practica exampe of this capabiity is the power bock within SEGS VI paraboic trough pant, which can operate with the HTF being as much as 90 K beow the nomina design point (Kob, 2011). 4. Concusions Thermocine tanks are a ow-cost therma energy storage option for arge-scae CSP pants. Numerica simuation of a dua-media thermocine tank is performed to investigate the effects of different granue diameters and non-adiabatic boundary conditions aong the tank wa. Performance of the thermocine tank is assessed with three separate metrics: outfow temperature with discharge time, entropy generation inside the fierbed, and net cyce efficiency. Efficiency of the tank is measured in terms of moten-sat energy, exergy, as we as energy subject to an outfow temperature criterion. Thermocine tanks fied with sma granues exhibit narrower heat-exchange regions reative to tanks with arge granues. This improves the therma stratification of the moten sat and yieds higher outfow temperatures during discharge. The thermocine efficiency (both firstand second-aw) is consequenty greater for the smaer granue diameter. However, a trade-off 21

23 exists between thermocine performance and increased pumping power to overcome the reduced fierbed permeabiity. An economic and experimenta study is recommended to optimize granue diameter between storage efficiency and pressure drop. Entropy generation inside the fierbed is predominanty a function of the therma boundary conditions at the tank wa. The presence of externa convection introduces radia temperature gradients, eading to increased irreversibe therma diffusion and greater entropy generation. Externa convection aso reduces the thermocine tank efficiency due to the oss of hot moten sat avaiabe for discharge. This reduction is most severe in the outfow temperature criterion-based efficiency, but this is demonstrated to be an overy conservative performance metric. Determination of both the first and second-aw tank efficiency eiminates the need for ad hoc therma anaysis. Combining this anaysis with a power bock mode wi compement these mode efficiencies and increase practica understanding by reating the outfow temperature degradation during discharge to CSP pant performance. Acknowedgement The authors woud ike to thank Professor Zhen Yang of Tsinghua University for hepfu discussions reated to evauation of thermocine cyce efficiency. 22

24 References Beckermann C., Viskanta, R., Natura convection soid/iquid phase change in porous media. Int. J. Heat Mass Transfer 31, Bejan, A., Convection Heat Transfer. John Wiey & Sons, Hoboken, New Jersey. Bejan, A., Advanced Engineering Thermodynamics, 3 rd Ed. John Wiey & Sons, Hoboken, New Jersey. Churchi, S.W., Bernstein, M., A correating equation for forced convection from gases and iquids to a circuar cyinder in crossfow. J. Heat Transfer 99, Eectric Power Research Institute. Soar Thermocine Storage Systems: Preiminary Design Study. Pao Ato, CA: Evans, L.C., Entropy and partia differentia equations. ast accessed October Faas, S.E., Thorne, L.R., Fuchs, E.A., Gibertsen, N.D., MWe soar therma centra receiver piot pant: therma storage system evauation fina report. Sandia Nationa Laboratories Report, SAND Fueckiger, S., Yang, Z., Garimea, S.V., An integrated and mechanica investigation of moten-sat thermocine energy storage. App. Energy 88, FLUENT Documentation, Fuent Inc. Gonzo, E.E., Estimating correations for the effective therma conductivity of granuar materias. J. Chem. Eng. 90, Heraeus Base Materias. ast accessed October Herrmann, U., Kearney, D.W., Survey of therma energy storage for paraboic trough power pants. J. So. Energy Eng. 124,

25 HITEC Heat Transfer Sat. Coasta Chemica Co., L.L.C., Brenntag Company, ast accessed on October Issa, R.I., Soution of impicity discretized fuid fow equations by operator spitting. J. Comput. Phys. 62, Kearney, D.W., Herrmann, U., Nava, P., Key, B., Mahoney, R., Pacheco, J., Cabe, R., Potrovitza, N., Bake, D., Price, H., Assessment of a moten sat heat transfer fuid in a paraboic trough soar fied. J. So. Energy Eng. 125, Kob, G.J., Apert, D.J., Lopez, C.W., Insights from the operation of Soar One and their impications for future centra receiver pants. So. Energy 47, Kob, G.J., Performance anaysis of thermocine energy storage. ISEC Proceedings of the ASME Internationa Soar Energy Conference, Denver, Coorado. Kob, G.J., Evauation of annua performance of 2-tank and thermocine therma storage systems for trough pants. J. So. Energy Eng. 133, Krishnan, S., Murthy, J.Y., Garimea, S.V., A two-temperature mode for anaysis of passive therma contro systems. J. Heat Transfer 126, Lovegrove, K., Luzzi, A., Sodiani, I., Kreetz, H., Deveoping ammonia based thermochemica energy storage for dish power pants. So. Energy 76, Mis, D.R., Morrison, G.L., Compact inear Fresne refector soar therma powerpants. So. Energy 68, Nationa Cimate Data Center, Nationa Oceanic and Atmospheric Administration, ast accessed October

26 Nied, D.A., Resoution of a paradox invoving viscous dissipation and non-inear drag in a porous medium. Transport Porous Media 41, Pacheco, J.E., Showater, S.K., Kob, W.J., Deveopment of a moten sat thermocine therma storage system for paraboic trough pants. J. So. Energy Eng. 124, Radosevich, L.G., Fina report on the power production phase of the 10 MWe soar therma centra receiver piot pant. Sandia Nationa Laboratories Report, SAND Specific Heat Capacities of Some Common Substances. The Engineering Toobox, ast accessed October Wakao, N., Kaguei, S., Heat and Mass Transfer in Packed Beds. Gordon and Beach, New York. Yang, Z., Garimea, S.V., Therma anaysis of soar therma energy storage in a motensat thermocine. So. Energy 84, Yang, Z., Garimea, S.V., Moten-sat therma energy storage in thermocines under different environmenta boundary conditions. App. Energy 87, Yang, Z., Garimea, S.V. Cycic operation of moten-sat therma energy storage in thermocines for soar power pants. Energy Conv. Mgmt, in review. 25

27 Appendix Entropy generation in an unconsoidated porous medium In accordance with the second aw of thermodynamics, entropy transport in a differentia contro voume is governed by the foowing equation: s t q T q T us S gen (A1) The transport equation is recast in non-conservative form by formuating the transient and convection terms as a materia derivative of entropy. Equation (A1) is reorganized to sove for the voumetric entropy generation (Bejan, 1986; Evans, 2004): S gen Ds q Dt T q T (A2) For incompressibe materias, the above materia derivative converts to a materia derivative of temperature. The divergence of the heat fux over temperature is aso expanded to give: S gen C T P DT Dt q q T 2 T T q T (A3) For a porous medium, separate entropy generation equations must be considered for the iquid and soid phases. In the case of a moten-sat thermocine, the soid fier is granuar or unconsoidated. Due to contact resistance between partices, therma diffusion is negected in the soid equation. The soid region sti contributes to therma diffusion in the iquid region through an effective therma conductivity (Gonzo, 2002). The energy source term in both equations represents convective heat exchange between the iquid and the fier: S gen, C T P, DT Dt q q T T T 2 hi ( Ts T ) T (A4) C DT h ( T T ) P,s s i s S gen, s 1 s Ts Dt T (A5) s 26

28 To further simpify the resutant entropy equations, the energy transport equations beow are reconsidered: C t P, t T s 1 C P, u C T q h T T ) h ( T i s P, T ) i ( s (A6) (A7) As with entropy transport, the fuid energy transport equation is recast in non-conservative form to generate a materia derivative of enthapy. As an incompressibe fuid, this materia derivative is again converted to a materia derivative of temperature. DT CP, q hi ( Ts T ) (A8) Dt T s C h ( T T ) 1 s P,s i s t (A9) The resutant energy equations are incorporated into the previous entropy equations, shown beow: S gen, q T T 2 T (A10) S gen, s 0 (A11) In the iquid region, the resut is simpy the dot product of the heat fux and temperature gradient over the temperature squared pus the effects of viscous dissipation. In contrast, the soid region is reveaed to be isentropic. This resut is an artifact of the mode assumption that therma diffusion between the soid granues is negigibe. Thus the net voumetric entropy generation in an unconsoidated porous medium is governed by the foowing: S gen k eff T T 2 2 T 0 (A12) 27

29 It shoud be noted that the derivation of Equation (A12) did not require any assumptions reated to the materia properties of the iquid region. Functiona dependencies on temperature (inherent to the iquid moten sat) are effectivey removed by mass and energy conservation. 28

30 Nomencature C P specific heat, J/kg K d diameter of thermocine tank, m d diameter of inet and outet ports, m d s diameter of fier granues, m e r unit vector in the r-direction, - e θ unit vector in the θ-direction, - e x unit vector in the x-direction, - E energy (enthapy), J F inertia coefficient, - h height of thermocine fierbed, m h height of tank distributors, m h i interstitia convection coefficient, W/m 3 K h w externa convection coefficient, W/m 2 K k therma conductivity, W/m K K permeabiity, m 2 r S radia tank ocation, m entropy, J/K S gen entropy generation, J/K t 0 haf-cyce period, s 29

31 T u temperature, K veocity vector, m/s u 0 moten-sat veocity at the top of the fierbed, m/s v x X veocity of heat-exchange zone, m/s axia tank ocation, m exergy, J Greek ε porosity, - η efficiency, - μ viscosity, Pa s ρ density, kg/m 3 Φ viscous dissipation function, 1/s 2 Subscript c h s w ow inet discharge temperature high inet charge temperature moten sat soid fier at the tank wa 0 reference state I first aw 30

32 II second aw 31

33 Tabe Tabe 1 Summary of thermocine tank efficiencies (first aw, second aw, and temperature criterion) for different externa oss conditions (Nu w ) and granue diameter (d s ). Case d s [cm] Nu w I II I,

34 List of Figures Figure 1 Schematic iustration of a thermocine tank therma energy storage system. Figure 2 Moten-sat temperatures aong axis of an adiabatic thermocine tank during a chargedischarge cyce for different granue diameter: (a) d s = 2 mm, (b) d s = 1 cm, (c) d s = 5 cm. Figure 3 Moten-sat temperature contours and veocity fieds of a thermocine tank during the discharge process in the presence of externa convection (Nu w = 4260) at the tank wa. The fierbed region extends from 0 to 1 aong the non-dimensiona tank height and is composed of 1 cm granues. Figure 4 Outfow temperature history of thermocine tank discharge for different tank wa conditions and granue diameter. Figure 5 History of entropy generation inside thermocine tank fierbed during a chargedischarge cyce for adiabatic and externa oss conditions (d s = 1 cm). 33

35 Figure 1 Schematic iustration of a thermocine tank therma energy storage system. 34

36 (a) 35

37 (b) 36

38 (c) Figure 2 Moten-sat temperatures aong axis of an adiabatic thermocine tank during a chargedischarge cyce for different granue diameter: (a) d s = 2 mm, (b) d s = 1 cm, (c) d s = 5 cm. 37

39 Figure 3 Moten-sat temperature contours and veocity fieds of a thermocine tank during the discharge process in the presence of externa convection (Nu w = 4260) at the tank wa. The fierbed region extends from 0 to 1 aong the non-dimensiona tank height and is composed of 1 cm granues. 38

40 Figure 4 Outfow temperature history of thermocine tank discharge for different tank wa conditions and granue diameter. 39

41 Figure 5 History of entropy generation inside thermocine tank fierbed during a chargedischarge cyce for adiabatic and externa oss conditions (d s = 1 cm). 40