Generalized charts of energy storage effectiveness for thermocline heat storage tank design and calibration

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Solar Energy 85 (2011) 2130–2143 www.elsevier.com/locate/solener

Generalized charts of energy storage effectiveness for thermocline heat storage tank design and calibration Peiwen Li a,⇑, Jon Van Lew a, Wafaa Karaki a, Cholik Chan a, Jake Stephens b, Qiuwang Wang c a

Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA b US Solar Thermal Storage LLC, 5151 E. Broadway Blvd. Suite 1020, Tucson, AZ 85711, USA c School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an Shaanxi 710049, China Received 1 July 2010; received in revised form 25 May 2011; accepted 31 May 2011 Available online 25 June 2011 Communicated by: Associate Editor Halime Paksoy

Abstract Solar thermal energy storage is important to the daily extended operation and cost reduction of a concentrated solar thermal power plant. To provide industrial engineers with an effective tool for sizing a thermocline heat storage tank, this paper used dimensionless heat transfer governing equations for fluid and solid filler material and studied all scenarios of energy charge and discharge processes. It has been found that what can be provided through the analysis is a series of well-configured general charts bearing curves of energy storage effectiveness against four dimensionless parameters grouped up from the storage tank dimensions, properties of the fluid and filler material, and operational conditions (such as mass flow rate of fluid and energy charge and discharge periods). As the curves in the charts are generalized, they are applicable to general thermocline heat storage systems. Engineers can conveniently look up the charts to design and calibrate the dimensions of thermocline solar thermal storage tanks and operational conditions, without doing complicated modeling and computations. It is of great significance that the generalized charts will serve as tools for thermal energy storage system design and calibration in energy industry. Ó 2011 Elsevier Ltd. All rights reserved. Keywords: Concentrated solar thermal power; Thermal storage; Thermocline; Energy storage effectiveness; General analysis; Design charts and tools

1. Introduction Power generation using concentrated solar thermal energy is one of the most promising renewable energy technologies (Herrmann and Kearney, 2002). It has received a great amount of research and development work in the last ten years (Gil et al., 2010). In particular, solar trough and solar tower concentrated thermal power generation systems are becoming more and more reliable and matured, and their cost also has been reduced with the increase of productivity and demand (Price et al., 2002). ⇑ Corresponding author. Tel.: +1 520 626 7789; fax: +1 520 621 8191.

E-mail address: [email protected] (P. Li). 0038-092X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2011.05.022

It has been widely accepted that further cost reduction of concentrated solar thermal power systems may be accomplished by adding solar thermal storage system that provides heat for prolonged operation of the power plant and thus increases the operational capacity of the power plant and, at the same time, improves the ability of power dispatch (Pitz-Paal et al., 2007). If an energy-carrying fluid medium in a thermal storage system can be withdrawn at the same temperature at which it had been originally stored, the system has the highest efficiency, or has zero exergy loss from the viewpoint of the second law of thermodynamics (Bejan, 2006). On the basis of this fundamental understanding, the first generation solar thermal storage system was developed in the earlier

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Nomenclature Bi C dr G ⁄ h hp H HCR kr m_ Q Q_ Q* r R Re Sr t T U V z

Biot number, specially defined as h  r/kr heat capacity (J/kg °C) average diameter of rocks (m) mass flux (kg/m2 s) enthalpy of fluid at a location along the main flow direction (J/kg) Heat transfer coefficient (W/m2 °C) corrected heat transfer coefficient (W/m2 °C) overall height of the storage tank (m) dimensionless parameter in heat transfer equations thermal conductivity of rocks (W/m K) mass flow rate (kg/s) energy (J) energy rate or power (W) dimensionless energy or energy ratio average radius of the filler material (m) Radius of the storage tank (m) modified Reynolds number for flow in porous media surface area of solid filler material per unit length of tank (m) time (s) temperature (°C) velocity of heat transfer fluid in the axial direction of a storage tank (m/s) volume of tank (m3) location of a fluid element along the axis of the tank (m)

Greek symbols ar thermal diffusivity of rocks (m2/s)

stage, which included two heat transfer fluid storage tanks, one for hot fluid and the other for cold fluid (Moens et al., 2003). During energy storage process fluid from the cold fluid tank is sent to the solar field to be heated and then stored in the hot fluid tank; while during energy discharge process, fluid from hot fluid tank is pumped out to release heat to the power plant and afterwards flows back to the cold fluid tank. Since the heat transfer fluid is generally expensive (Kearney et al., 2003), it was thus proposed that a single tank be used for thermal storage (Gabbrielli and Zamparelli, 2009). Such a single tank is also named as a thermocline tank which requires hot fluid being on top of cold fluid and essentially stratified during energy charge, storage, and discharge processes. The phenomenon of stratification of fluid by maintaining a temperature gradient is generally referred to as a thermocline. During energy charging processes, hot fluid is charged into the tank from top and at the same time, cold fluid at the bottom of the tank is pumped out to the solar field for absorbing heat.

Dt Dz e g gth l P q sr h

time period of energy charging or discharging (s) length of a section or whole of a storage tank (m) porosity of the packed bed energy delivery effectiveness defined by Eqs. (24) or (27) thermal efficiency of a solar power plant viscosity (Pa s) dimensionless time period of energy charge or discharge density (kg/m3) dimensionless parameter dimensionless temperature

Superscript  dimensionless variables Subscripts c energy charge process d energy discharge process exit exit of heat transfer fluid f heat transfer fluid filler for filler material fluid for heat transfer fluid ideal for ideal thermocline (no solid filler material in tank) r filler material (rocks) ref reference time period—a requirement condition of power plant tank for thermocline tank having filler material z location along the axis of the tank

During energy discharging processes, hot fluid in the tank is pumped out from the top, which then releases its heat to the power plant before returning to the tank from the bottom. Essentially both the two-tank and single-tank strategies of thermal energy storage use heat transfer fluid as the heat storage medium. The method of further reducing the use of the high-cost heat transfer fluid has to rely on a secondary energy storage medium, which must be significantly cheaper than the heat transfer fluid (Hasnain, 1998). This mechanism of heat storage using solid material with heat transfer fluid features the third generation of solar thermal energy storage technology that heat transfer fluid serves mainly as the energy carrying medium; while cheaper materials such as rocks, salts, concrete, sands, and even soil serve as energy storage media (Brosseau et al., 2005). Under such a situation, obviously, the heat storage and retrieving process involves heat transfer between the heatcarrying fluid medium and the heat storage solid medium.

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Also, it is important that a single storage tank be used and thermocline phenomenon still be maintained in this third generation of thermal energy storage technique. Due to the existence of thermal interaction between heat transfer fluid (heat carrying medium) and heat storage solid material, a temperature difference between the two is inevitable, and thus the heat-carrying medium can hardly reach the same temperature as it had when it was charged into the tank. Therefore, the actual temperature history of the heat transfer fluid flowing out from the tank during energy charge and discharge can be complicated by the material properties of energy storage medium and energy-carrying fluid, as well as their interaction behavior (characterized by the fluid flow and heat transfer behavior in porous media). Obviously, in order to design such a thermal energy storage system and to size the volume of the tank, one has to analyze the heat transfer behavior of the system. Through the analyses in this study a series of general charts of energy storage effectiveness against a variety of parameters will be configured, which will help industrial engineers in designing and sizing a thermal storage system. The formatting and illustration of the general charts will accommodate full spectrum of selections of properties of solid filler and fluid materials, dimensions of storage tanks, as well as operational conditions. The objective function in the charts is the required quantity of energy delivery in a required time period and a mass flow rate. Since a large number of cases must be analyzed to configure the general charts of energy delivery effectiveness versus variety of parameters of an energy storage system, an effective computational tool must be used. A number of analyses and solutions to the heat transfer governing equations of a working fluid flowing through a fillerpacked bed have been presented in the past (Beasley and Clark, 1984; Schumann, 1929; Shitzer and Levy, 1983; McMahan, 2006; Kolb and Hassani, 2006). As the pioneering work, Schumann (1929) in 1927 presented a set of equations governing the energy conservation of fluid flow through porous media. Schumann’s equations have been widely adopted in the analysis of thermocline heat storage which has solid filler material inside a tank. Schumann’s analysis and solutions were performed for the case of fixed fluid temperature at the inlet to a storage system. Also, the initial temperature in the storage tank is assumed uniform. In most solar thermal storage applications the inlet fluid temperature in charge and discharge processes may vary and the initial temperature can also be strongly nonlinear. To overcome these limitations in Schumann’s analysis, Shitzer and Levy (1983) employed Duhamel’s theorem on the basis of Schumann’s solution to consider a transient inlet fluid temperature of the storage system. However, Shitzer and Levy’s solution still assumed the initial temperature in the tank being uniform. For a heat storage system in a solar thermal power plant, heat charge and discharge are cycled daily. The initial temperature field of a heat charge process is dictated by the most recently completed heat discharge process, and vise versa. Therefore,

non-uniform and nonlinear temperature distribution is typical for both heat charge and discharge processes. To accommodate the non-uniform initial temperature and time-varying inlet fluid temperature, numerical methods to solve the Schumann equations were presented in literature by McMahan (2006), Kolb and Hassani (2006), Pacheco et al. (2002), Van Lew et al. (2009) and Karaki et al. (2010). After a rigorous evaluation and comparison, the most effective and accurate numerical method and the efficient computational schemes given by Van Lew et al. (2009) was used in the current study. 2. Energy storage effectiveness in thermocline tanks 2.1. The scenario of energy charge and discharge for an ideal thermocline tank If an energy-carrying fluid medium in a thermal storage system can be withdrawn at its temperature originally being stored, the system has the highest efficiency, or has zero exergy loss from the viewpoint of the second law of thermodynamics (Bejan, 2006). Such a thermal energy storage system may be idealized by using two separate storage tanks, or by using a single storage tank with an ideal thermal insulation baffle (movable along the height of the tank) in between the hot fluid and cold fluid, as illustrated in Fig. 1. For the single tank in Fig. 1 during energy charging process, hot fluid flows into the tank from top and displaces the cold fluid out of tank from bottom; while during energy discharging process, hot fluid is pumped out from top of the tank and cold fluid is charged in from bottom of the tank. This type of energy storage and delivery using a single tank is named as the ideal thermocline storage. As one single tank is used to achieve both the energy storage and delivery functions, it is obviously more economical than a two-tank thermal energy storage system. The ideal thermocline heat delivery effectiveness may be considered as 1.0 since it has no exergy loss. This is explained in Fig. 2 by showing the fluid exit temperatures during energy charge and discharge processes. In heat

Hot fluid Movable ideal thermal insulation baffle

Cold fluid

Fig. 1. Illustration of an ideal thermocline heat storage tank.

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Texit (top) TH

Texit (bottom) TH

TL TL t (hour)

t

(a)

Δt

(b)

Δt

Fig. 2. Illustration of temperature variation of the fluid flowing out of an ideal thermocline storage tank. (a) Heat charge process. (b) Heat discharge process.

charge process, hot fluid charges in from top of the tank. The temperature of flowing-out fluid at the bottom is shown in Fig. 2a. When hot fluid is withdrawn from top of the tank, cold fluid enters in the tank from bottom, and the same high-temperature fluid is discharged out as shown in Fig. 2b. The ideal heat rate of delivery from an ideal thermocline tank at a required mass flow rate and a period (both are operational conditions required by the power plant) is defined as: Q_ P Q_ T ¼ ¼ m_  C f ðT H  T L Þ gth

ð1Þ

where Q_ T is thermal energy rate, Q_ P is electrical power, and gth is thermal efficiency in thermal power plant. The ideal thermocline energy delivery is calculated as: QT ¼ Dt  Q_ T

ð2Þ

and the volume of ideal thermocline storage tank is _ f V ideal ¼ Dt  m=q

ð3Þ

2.2. Energy storage effectiveness in a thermocline tank having filler material When a solid filler material is packed in a heat storage tank, as shown in Fig. 3a, it leaves a void volume of eVideal, where Videal is the volume of the tank and e the void fraction. Under the same mass flow rate as established by the system requirements of the power plant, the real fluid velocity in the charge/discharge processes of the tank (assuming to have the same dimensions from that of Videal) with filler material will be higher than that in an ideal thermocline tank. Therefore, during a discharge process, the temperature of the fluid flowing out from top will decrease after a time when the pre-existing hot fluid is completely discharged; from then on the hot fluid discharged out from top originates from cold fluid coming into the tank that has been heated by the filler material. The more the discharge process progresses, the more the temperature of the discharged fluid will decrease, as illustrated in Fig. 3b. When hot fluid is charged into the tank, the pre-existing cold fluid in the tank is displaced out from the bottom of

(a) Texit (top) TH

Texit (bottom) TH Tcut-off

Tcut-off

TL

TL t

t ( ε Δt)

Δt

(b)

( ε Δt)

Δt

(c)

Fig. 3. Illustration of temperatures of heat transfer fluid during energy charge and discharge processes from a thermocline tank with filler material. (a) Thermocline tank with filler material. (b) Heat discharge from top. (c) Heat charge from top.

the tank in the meantime. After the pre-existing cold fluid is discharged, any further discharged fluid will be the fluid which enters the tank at high temperature and gives its energy to the cold filler material. Since the hot fluid cannot give the entirety of its thermal energy to the cold filler material, the discharged fluid temperature at the tank bottom will gradually increase. The more the charge process progresses, the more the temperature of the discharged fluid from the bottom of the tank will increase. Fig. 3c schematically illustrates this temperature variation of the discharged fluid from the bottom of a tank. Although the discharged fluid has temperature degradation/drop, as shown in Fig. 3b, a heat storage tank having filler material is still being considered as a cost-effective and economical technology, since the heat transfer fluid is expensive and shall not be used as the major heat storage medium. Therefore, the significance of using filler material is to displace and minimize the use of heat transfer fluid in a heat storage tank. No matter whether a filler material is being used or not in a storage tank, a power plant operation requires a specified period of heat transfer fluid discharge time under a specified mass flow rate. Therefore, minimizing the temperature degradation by proper design and operation of a thermal storage tank becomes the key issue of thermocline energy storage, which is the focus of this study. Before going into mathematical analysis about the heat transfer and energy conservation of fluid and filler material

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in a thermocline tank, it is worthwhile to observe the following three scenarios of energy storage when a tank of Videal is filled with a filler material: (1) The energy charge period is the same as that of required discharge period. In this case, using the heat storage tank having filler material one can never achieve the ideal quantity of heat discharge. This is because the energy charged into the tank cannot be larger than the ideal amount of energy, as seen from illustration in Fig. 3c, and therefore, the energy discharge process cannot achieve the ideal quantity of energy, as seen from illustration in Fig. 3b. (2) There is a longer period of energy charge, Dtcharge, than the period of required energy discharge, Dtdischarge. From the above analysis and its conclusion, it is obvious that in order to get a required amount of energy discharge, using an increased charge time is inevitable. However, even if a longer heat charging time is applied, there is still the possibility that an ideal quantity of heat discharge cannot be achieved under the particular conditions that the filler material has (qC)filler 6 (qC)fluid. This is because that in such a situation, there is still no extra amount of energy, compared to that of the ideal case, being stored in the tank, and the best case scenario is that a tank is fully charged in a long time. On the other hand, the energy discharge process is not comparable to that of an ideal thermocline tank, and thus the energy discharged cannot approach the ideal quantity, as seen from the illustration in Fig. 3b. (3) Having a filler material with (qC)filler > ( qC)fluid together with a heat charging period, Dt charge, longer than the required heat discharging period, Dtdischarge, is the prerequisite for a heat storage tank with filler material to deliver heat that approaches the ideal heat delivery seen in an ideal thermocline tank. A larger (qC)filler compared to (qC)fluid allows more energy than the required discharge energy being stored, if a longer period of charge than that of discharge is applied. In conclusion, only when the two conditions—(qC)filler > (qC)fluid and Dtcharge > Dtdischarge are both satisfied is it possible that the storage tank at a volume of Videal can contain more energy than the required amount for discharging, and thus it is possible to approach an ideal amount of energy delivery. It may be observed from the above three scenarios that a greater heat storage capacity than that in the ideal thermocline case together with a longer heat charge period than discharge period, Dt charge > Dtdischarge, must be satisfied in a thermocline system (having filler material) for it to deliver the amount of energy comparable to an ideal thermocline system. The requirement regarding the heat storage capacity can be mathematically expressed as: ðqCÞfiller V tan k ð1  eÞ þ ðqCÞfluid V tan k e > ðqCÞfluid V ideal

ð4Þ

Here, Vtan k is the volume of the actual storage system (having filler material), which can be larger or smaller than Videal, depending on the following two situations. When there is (qC)filler  (qC)fluid the V tank can be smaller than Videal, which could be the case of using phase change material for the filler material. For most sensible heat thermal storage system (with filler material) that have (qC)filler not greatly larger than (qC)fluid or even smaller, V tank should be typically larger than Videal. The mathematical analysis hereafter will investigate and quantitatively show the above discussed phenomena for a storage tank with filler material. 3. Heat transfer and energy storage modeling and computation 3.1. Energy conservation in fluid and rocks Through energy balance analysis for fluid and rocks, governing equations for the temperatures of fluid and rocks will be constructed. Shown in Fig. 4 is a one-dimensional control volume of size dz in a rock-packed flow bed. Both the energy conservation in the fluid and in the rocks in the control volume will be examined. For convenience of analysis, the positive direction of coordinate z is set always identical to the fluid flow direction. In an energy charge process hot fluid flows into the tank from the top, and thus z = 0 is at the top of the tank. In a heat discharge process, cold fluid flows into the tank from bottom and this makes z = 0 for the bottom of the tank. Several assumptions are typically made to simplify the analysis of a thermocline (with filler material) heat storage process: (1) A uniform radial distribution of the fluid flow and rocks through the storage tank is assumed to reduce the analysis to a one-dimensional problem along the axis, z, of the storage tank. (2) The contact between rocks is point contact and therefore heat conduction between rocks are negligible. (3) It has been confirmed that the Peclet number of fluid flow in a thermocline tank is large (Pe  100), and therefore the heat conduction in the axial direction in the fluid is negligible (Kays et al., 2005). (4) It is assumed that lumped heat capacitance method is applicable to the transient heat conduction in a single rock;

z H

dz 2R

Fig. 4. One control volume/element for energy balance analysis.

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U qf C f epR2 H hS r

ð11Þ

however, if the Biot number (Incropera and Dewitt, 2002) of the rock, due to its property and heat transfer with fluid, is large, a correction of large Biot number effects is considered later. (5) There is no heat loss from the storage tank to the surroundings; this applies to both the energy charge and discharge process as well as the resting time in between a charge and a discharge. Heat loss from a thermal storage tank is inevitable; however, one needs to decide the dimensions of a storage tank firstly in order to find the heat loss. Later on, to compensate for the heat loss will result in a larger volume of the heat storage tank and a longer heat charge period. A simple way of refining the design is to increase both the heat charge time and tank size with a percentage equal to the ratio of heat loss versus the projected heat delivery. Nevertheless, to make this study focused, the current work concentrates on the determination of the dimensions of a storage tank without considering heat loss. This assumption also assists the connection of the results from a heat charging process to be the initial condition of the following on discharging process, and visa versa. By doing so, multiple cyclic energy charges and discharges in the actual operation can be simulated from the current modeling analysis. Based upon the above modeling assumption (1), the crosssectional area of the tank seen by the fluid flow is assumed constant at all points along the axis of the tank, and is: af ¼ epR

2

ð5Þ

The thermal energy balance of the fluid in the control volume dz is: @T f @t

ð6Þ

where the average fluid velocity in the packed bed is: U¼

m_ qf af

ð7Þ

With substitution for the definition of enthalpy change, ⁄z+dz  ⁄z = CfoTf, and rearrangement of Eq. (6), the energy balance equation for fluid becomes: hS r @T f @T f þU ðT r  T f Þ ¼ 2 @t @z qf C f epR

In the equation, Sr is the heat transfer surface area of rocks per unit length of the tank, which is a function of the dimensions of filler material. Based upon the modeling assumption (2), Sr is calculated as: fs pR2 ð1  eÞ r

Sr ¼

ð8Þ

h ¼ 0:191

Re ¼

4Grchar lf

G¼

m_ epR2

ed r 4ð1  eÞ

ð16Þ

For the energy balance of the filler material (rocks) in a control volume dz as shown in Fig. 4, it is understood that the filler delivers or takes heat to or from the passing fluid at the cost of a change in the internal energy of the filler. The equation is: hS r ðT r  T f Þdz ¼ qr C r ð1  eÞpR2 dz

@T r @t

ð17Þ

With substitution of dimensionless variables given in Eq. (9), the above governing equation for filler turns to be

z ¼ z=H t ¼ t=ðH =U Þ

ð9cÞ ð9dÞ

H CR ¼

where

ð15Þ

and rchar is defined as the characteristic radius by Nellis and Klein (2009) (sometimes defined as the hydraulic radius):

where

ð10Þ

ð14Þ

where G is the mass flux of fluid through the porous bed,

ð9aÞ ð9bÞ

The dimensionless governing equation for heat transfer fluid is finally shown to be

ð13Þ

where the Reynolds number is the modified Reynolds number for porous media, defined as (Nellis and Klein, 2009):

hf ¼ ðT f  T L Þ=ðT H  T L Þ hr ¼ ðT r  T L Þ=ðT H  T L Þ

@hf @hf 1 þ ¼ ðhr  hf Þ @t @z sr

_ f 0:278 2=3 mC Re Pr epR2

@hr H CR ¼ ðhr  hf Þ  @t sr

Introducing the following dimensionless variables,

ð12Þ

where fs is 3.0 assuming the fillers are spheres of a radius of r. The heat transfer coefficient h (W/m2 °C) between solid and fluid in porous media in the above equations was based upon the analysis provided by (Nellis and Klein, 2009).

rchar ¼

hz   hzþdz Þ þ hS r ðT r  T f Þ dz qf epR2 U ð ¼ qf C f epR2 dz

sr ¼

qf C f e qr C r ð1  eÞ

ð18Þ

ð19Þ

In the charge and discharge processes, the rocks and heat transfer fluid will have a temperature difference at any local location. Once the fluid comes to rest upon the completion of a charge or discharge process, the fluid will equilibrate with the local rocks to reach the same temperature. The energy balance of this situation at a local location is:

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eqf C f T f initial þ ð1  eÞqr C r T rinitial ¼ eqf C f T final þ ð1  eÞqr C r T final

ð20Þ

Here, the initial temperatures of rocks and fluid are from the results of their respective charge or discharge process. The final temperatures of rock and fluid are the same after equilibrium is reached. According to the assumption of no heat loss from the storage tank, it can be seen that the equilibrium temperature at the end of one process (charge or discharge) will necessarily be the initial condition of the next process in the cycle. This connects the discharge and charge processes so that an overall periodic result can be obtained. The initial temperatures of rock and fluid in the storage tank should be known. Also, the inlet fluid temperature is known as a basic boundary condition, with which the rock temperature at inlet location z = 0 can be easily obtained from Eq. (18). Therefore, as the boundary conditions, both temperatures of fluid and rock at the inlet z = 0 are known. 3.2. Correction to lumped capacitance approximation for heat transfer in rocks The modeling presented in Section 3.1 used lumped capacitance method to treat the heat transfer inside rocks. This method actually ignored the resistance of heat conduction inside a rock. This will result in discrepancy which makes the calculated energy going into or coming out from a rock higher than that in the actual physical process. From Incropera and DeWitt (2002) it is known that when the Biot number of the heat transfer of a solid object is larger than 0.1, the lumped capacitance assumption will result in increased inaccuracy for the heat transfer and energy conservation analysis of a rock in a fluid. In order to correct the lumped capacitance approximation, Jeffreson (1972) proposed to give a correction to the convective heat transfer coefficient between rocks and fluid in the form of: 1 1 þ Bi=5 ¼ hp h

Fig. 5. Comparison of Jeffreson correction model, exact transient heat conduction solution, and lumped capacitance solution for a solid object heat transfer in fluid.

dimensionless energy, Q*, in Fig. 5 is the ratio of energy going in or out from the rock compared to the ideal energy going in or out from a rock. The energy going in or out from a rock is counted based on the change of internal energy of the rock. The ideal energy going in or out from a rock is the internal energy change of a rock assuming its temperature completely changed from initial temperature to the fluid temperature around it. Parameters of rocks and heat transfer coefficient used for the comparison of the three methods in Fig. 5 were listed in Table 1. The Bi number of the case is 2.54, which is close to the situation of the rocks in typical thermocline systems. As shown in Fig. 5, the results from Jeffreson correction model agreed with the exact transient heat conduction solution very well. Therefore, the Jeffreson correction, Eq. (21), of the heat transfer coefficient between rocks and fluid in the thermocline system is recommended. 3.3. Numerical method of solution to the governing equations

ð21Þ

where hp is the new heat transfer coefficient to replace the h obtained in Eq. (13). The corrected heat transfer coefficient hp will be used in equations wherever h is needed. Any other terms and properties in all governing equations remain unchanged. The Jeffreson correction allows for the thermocline model to remain in a one-dimensional system yet increases the accuracy of the results by accounting for the internal thermal gradient in the packed bed filler material. A comparison of the results of dimensionless energy going in or out from a single rock of the filler using exact transient heat conduction solution from Incropera and DeWitt (2002), using the lumped capacitance method (Incropera and DeWitt, 2002), and using the corrected lumped capacitance method by introducing a corrected heat transfer coefficient by Eq. (21) were given in Fig. 5. The definition of the

Using the approach of method of characteristic Eqs. (10) and (18) can be solved numerically with minimum computation time and high accuracy. Details of the method of characteristics and numerical computation are given in Van Lew et al. (2009) and Polyanin (2002). However, the key steps of solution to the governing equations are introduced in the following section. Table 1 Properties of the rock and heat transfer coefficient used in Section 3.2. Parameter

Value

Unit

dr h kr qr Cr ar

0.04 355.1 2.8 2630 775 1.374  106

m W/m2 K W/m K kg/m3 J/kg K m2/s

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Therefore the numerical computation takes minimum computing time, which is much more efficient than all other methods seen in McMahan (2006), Kolb and Hassani (2006), and McMahan et al. (2007). 4. General charts of energy storage effectiveness in thermocline tank having filler material 4.1. Energy storage effectiveness

Fig. 6. The matrix of nodes assignment for method of characteristic solution for Eqs. (10) and (18).

As shown in Fig. 6, the length of the storage tank is discretized (represented by node number i) with the maximum grid number of M, and time length is discretized (represented by node number j) with the maximum number of N. The fluid temperature at inlet is known as a function of time. The rock temperature at the inlet is calculated based on Eq. (18), for which an explicit finite difference equation is obtained hr1;jþ1  hr1;j H CR ¼ ðhr1;jþ1  hf1;jþ1 Þ Dt sr

ð22Þ

where the fluid inlet condition hf1;1 to hf1;N and initial condition of rocks hr1;1 are both known and therefore, hr1;2 up to hr1;N can be solved explicitly. Therefore, in the matrix shown in Fig. 6 temperatures of fluid and rocks at nodes #1,1 to #1,N and #1,1 to #M,1 are known. Using the method of characteristics, the study in Van Lew et al. (2009) obtained the following algebraic equation matrix to solve the temperatures of rocks and fluid at the node #2,2. 2 3   " #" #  Dt Dt Dt h 1  þ h f r 1 þ Dt  1;1 1;1 2sr hf2;2 2sr 2sr 2sr 6 7    5 ¼4 Dt H CR Dt   H Dt H Dt h CR CR  H CR 1 þ r 2;2 þh h 1 2s 2s r

r

f2;1

2sr

r2;1

2sr

ð23Þ

This matrix in Eq. (23) includes an implicit condition of discretization of the time and length so that Dz* = Dt*. Since the matrix only has two equations, Cramer’s rule can be applied to obtain the solution of hf2;2 and hr2;2 efficiently. Further, the matrix can be easily applied to other nodes by marching the time and space steps as given in Fig. 6. While the marching of Dz* steps is limited to z* = 1, the marching of time Dt* has no limitation. The error of such an implementation is not straightforwardly analyzed here in this study, but the formal accuracy of method of characteristics is on the order of O(Dt*2) as described in Polyanin (2002). It is important to note that all coefficient terms in Eq. (23) are one-time determined from Dz*, Dt*, sr, and HCR.

As was discussed in Section 2.2 for a thermocline storage tank with filler material, the energy delivered in the required time period at a required mass flow rate is always less than that of the ideal energy delivery in an ideal thermocline tank. With the solution of the governing equations, the discharged fluid temperature from a storage tank can be obtained. If the required heat discharging period is tref,discharge, an energy storage effectiveness can be defined as: R tref ;discharge ½T f ðz ¼ H ; tÞ  T C dt g¼ 0 ð24Þ ðT H  T C Þ  tref ;discharge where the numerator represents the energy discharged from the actual tank, and the denominator represents the energy discharge from an ideal thermocline tank. The dimensionless form of the required time period of energy discharge is defined as: tref ;discharge Pd ¼ ð25Þ H =U Similarly a dimensionless form of the time period of energy charge is defined as: tcharge Pc ¼ ð26Þ H =U Substitute the dimensionless energy discharge period Pd into Eq. (24), there is Z Pd 1 g¼ hf ðz ¼ 1; t Þdt ð27Þ Pd 0 It is now known that g is essentially the function of the following three parameters—Pc/Pd, sr, and HCR. While the dimensionless time period of discharge process and the mass flow rate must be prescribed, the dimensionless time period of energy charge period is the variable that will be searched to determine the targeted value of g, which should approximately equal to 1.0. It has been concluded in Section 2.2 that a longer energy charge time than that of energy discharge time is always needed in order to achieve the energy delivery effectiveness of 1.0. As the results of an example from the current computation, Fig. 7 shows the variation of temperatures of fluid flowing out from the tank in an energy charge and following-up discharge process, respectively. The energy charge process needs a longer time than that of the energy discharge process. Since more energy is charged into the tank than needed for discharge, the temperature degradation of

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when the results of cyclic charge and discharge are independent of the most initial condition in the tank. A most initial condition is assumed where the temperatures of rocks and fluid are both hf = hr = 1 and the program then calculates the temperature distribution during a discharge. The temperature results after a discharge are loaded as the initial condition of the next charge process and the computation is repeated; similarly the results after a charge are loaded into the next discharge process. In the case where the fluid energy density (qfCf) is much lower than the rock energy density (qrCr), the discharge–charge cycle must be repeated many times to reach the end of the computation. The graphs in Fig. 9 clearly agree with the following conclusions that have already been drawn from the qualitative analysis and discussion in Section 2.2: Fig. 7. Dimensionless temperatures of the fluid flowing out from a tank in a charge and a following-on discharge (the case has HCR = 0.1, sr = 0.04, Pd = 4.0, and Pc/Pd = 1.54).

the discharged fluid is not significant, which yield an energy discharge efficiency close to 1.0. 4.2. Energy storage effectiveness chart for tank size design It is convenient that a general series of charts of energy delivery effectiveness g versus Pc/Pd, sr, and HCR be provided for the design of a storage tank. Since g is a function of three variables, the general charts can be configured in different ways. Illustrated in Fig. 8 is a configuration of the charts for a given Pd, in which multiple graphs, each has a specific sr, are provided. In each graph, multiple curves, each has a given HCR, for the energy storage effectiveness g versus Pc/Pd is provided. To have a full spectrum of the charts, more graphs in the same configuration as shown in Fig. 8 must be provided at variety of different values of Pd. Fig. 9 shows some results from the computation in this study, the graphs can form a general chart for a constant Pd of 4.0. In these graphs every data point in the curves were based on the final heat discharge process, which occurs at the end of several cycles of charge and discharge,

1.0

τr1

HCR-n CRn

τr2 HCR-1 CR1

η

τr3 τr4

0.0

τr5 0.5

Πc/Πd

2.0 τri

Fig. 8. Configuration of the charts including multiple graphs of g versus Pc/Pd at a fixed Pd.

(1) The energy delivery effectiveness does not reach 1.0 if Pc/Pd 6 1.0. (2) A decrease of sr corresponds to an increase of the volume of storage tank. Therefore with a decrease of sr the effectiveness g can approach 1.0 more easily under the same values of HCR and Pc/Pd. For example, at a ratio of Pc/Pd = 1.5, the energy delivery effectiveness easily approximates to 1.0 when srchanged from 0.2 to 0.01. (3) A decrease of HCR corresponds to the case that (qrCr) is increasing relatively to (qfCf), and therefore the energy storage capability is improved so that g  1.0 is also easier to achieve. This is seen in every graph in Fig. 9. (4) There are cases that no matter what ratios of Pc/Pd are used, the energy storage effectiveness can hardly approach 1.0. This occurs when either sr or HCR are too large, which physically corresponds to small tank size and qrCr 6 qf Cf, respectively.

4.3. Chart for calibration analysis of a given storage tank Designers for a thermal storage system often need to calibrate or confirm that a given storage tank can satisfy an energy delivery requirement. As the dimensions of the storage tank and the power plant operational conditions are known, the value of sr is essentially given. This requires that the general charts be configured in the form as shown in Fig. 10. Engineers can look up different energy discharge time from multiple graphs and essentially look up the energy storage capability of a given tank. Based on computation results, Fig. 11 gives four representative graphs, each having multiple curves for the energy delivery effectiveness versus the charge and discharge time ratio. The series of graphs are for a fixed value of sr = 0.03, and each graph has a fixed Pd. The results are calculated in the same manner as discussed for Fig. 9 for cyclic charges and discharges. The size parameters of a storage tank are included in the parameter, sr. Under the same value of sr, a smaller Pd

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Fig. 9. Computed results of multiple graphs of energy storage effectiveness versus Pc/Pd at Pd = 4.0.

means smaller required energy storage, and therefore, the energy delivery effectiveness can reach to 1.0 easily for most cases of HCR. At larger values of Pd, a larger ratio of Pc/ Pd is needed in order to achieve the energy delivery effectiveness of 1.0. 1.0

Πd1

HCR-n CRn

5. Applications of the charts for design of thermocline storage tanks 5.1. Sizing a thermocline storage tank To decide the dimensions of a thermocline storage tank, the required operational conditions from the power plant include: the electrical power, the thermal efficiency of the power plant, the extended period of operation based on stored thermal energy, the required high temperature of heat transfer fluid from the storage tank, the low temperature of fluid returned from the power plant, the properties of the heat transfer fluid and the solid filler material

η

Πd2 HCR-1 CR1

Πd3 Πd4

0.0

Πd5 0.5

Πc/Πdi

2.0 Πdi

Fig. 10. Illustration of a configuration of the charts for energy delivery effectiveness using multiple graphs at different Pd but at a constant sr.

including the nominal radius of fillers, as well as the packing porosity in a thermocline tank. The design analysis using the general charts provided in the present study will include the following steps:

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Fig. 11. A series charts of energy effectiveness versus Pc/Pd at different Pd and a constant value sr = 0.03.

(1) Decide an ideal volume for an ideal thermocline tank.The total thermal energy is related to the required volume of an ideal thermocline tank in the form of: fluid  C fluid ðT H  T L Þ QT ¼ V ideal  q ¼ m_ fluid C fluid ðT H  T L ÞDtdischarge

ð28Þ

Once the volume of the ideal thermocline tank is determined, a chosen diameter, R, and corresponding height, H, of the tank can be decided, which will be used in the first trial for energy storage effectiveness analysis. Following these dimensions, the parameters—Pd, sr, HCR for a thermocline tank with filler material, can be decided, where Pd is determined by setting tref,discharge equal to Dtdischarge. (2) Look up the design charts (Fig. 9) and see if an energy storage effectiveness of 1.0 can be achieved with the parameters decided in step (1). It might be common that the energy effectiveness cannot be close to 1.0 during the first trial. This is because the first

trial uses dimensions from an ideal thermocline storage tank. However, with the results from the first trial one can predict and guess a new sr and a new Pd for the next trial. In fact, the increase of the height of the storage tank can result in both decrease of sr and Pd in the same proportion. (3) If the effectiveness from step (1) cannot approach 1.0 even if a large Pc/Pd is chosen, one has two ways to improve the effectiveness during the second trial, which are to decrease HCR or decrease both sr and Pd in the same proportion. However, HCR is determined by properties of fluid and filler material, which has very limited options, and therefore decrease of sr and Pd is inevitable. The decrease of sr is actually due to the increase of the height of a storage tank. This means that to get an effectiveness of 1.0 one has to increase the height of the storage tank. Because the height of tank is included in both sr and Pd, the decrease of sr must be accompanied by the same proportional decrease of Pd.

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Although the situation of selecting a smaller sr in step (3) is common, the opposite scenario of selecting a larger sr may also occur. This is corresponding to the case that the energy storage effectiveness gcan approach 1.0 at Pc/Pd slightly larger than 1.0. This also means that the storage tank may be oversized. One can increase sr and Pd in the same proportion to achieve an effectiveness of 1.0, but at the expense of a larger Pc/Pd than before. Physically, this scenario corresponds to the situation that one can use a smaller tank but at a longer charge time to obtain the same amount of required energy delivery. This may happen if the HCR is very small, which is due to a much larger value of(qrCr) in relation to (qfCf). 5.2. Find the energy storage capacity and the period of operation for a given tank If the dimensions of a storage tank and the operational conditions of power plant, including the mass flow rate and the high and low fluid temperatures, are given, it is the task of designers to find a proper time period of energy charge that can satisfy the needed operation time of the power plant. The known parameters will be sr and HCR at a required operation period of Pd. A series of graphs (as given by Fig. 11) for the given sr must be looked up. From the set of graphs one needs to identify a particular graph for the given Pd. In such a particular graph, the energy storage effectiveness versus Pc/Pd at various values of HCR can be easily found. If at the required Pd the energy storage effectiveness cannot approach 1.0 at any value of Pc/Pd, a new Pd must be selected. In this way the storage tank is calibrated for whether or not it is suitable for the required Pd. 5.3. An example of tank size and operation design The following example shows the sizing procedure of a storage tank. A solar thermal power plant has 1.0 MW electrical power output at a thermal efficiency of 20%. The heat transfer fluid used in the solar field is TherminolÒ VP-1. The power plant requires the high and low fluid temperatures of 395 °C and 310 °C, respectively. River rocks are used as the filler material and the void fraction of packed rocks in the tank is 0.33. Required time period of energy discharge is 1.0 h, the storage tank diameter is 6 m. The rock diameter is 4 cm. Making use of Eq. (28), the above given details on the power plant, as well as the properties of TherminolÒ VP1, we can find a necessary mass flow rate of 23.76 kg/m3, and an ideal tank height of 4 m to deliver the required power. Using Eqs. (12) and (20) we find a heat transfer coefficient to be 32.05 W/m2 K. With this information, the values of HCR, Pd, and sr are 0.451, 3.03, and 0.06, respectively. Given in Fig. 12 is a family of charts for Pd = 3 and sr = 0.06 at various values of HCR and Pc/ Pd. Fig. 12 shows that at given value of HCR there is no

Fig. 12. Energy effectiveness versus Pc/Pd at Pd = 3 and sr = 0.06.

time ratio Pc/Pd that will deliver an effectiveness value of 1.0 from this tank of an ideal volume. One option to deliver more power and approach an effectiveness of 1.0, is to increase the height of the storage tank. When the height is increased to 6 m, the values of Pd and sr change to 2.03 and 0.0404, respectively. Fig. 13 gives the family of curves for Pd = 2 and sr = 0.04. In this figure, it is seen that at HCR of 0.45, when the time period ratio, Pc/Pd, is approximately 1.5 an effectiveness of 0.99 is possible. This should be acceptable. A design engineer may feel that such a large period of charge is not acceptable. In that case, the designer can increase the height once again. When the height of the storage tank is increased to 8 m, the values of Pd and sr become 1.52 and 0.0303, respectively. Fig. 14 shows the family of curves for Pd = 1.5 and sr = 0.03. In Fig. 14 we clearly see that when Pc/Pd > 1.25, an effectiveness of almost 1.0 is possible for all values of HCR. This above example demonstrates the power of having a data library containing the most common combinations of

Fig. 13. Energy effectiveness versus Pc/Pd at Pd = 2 and sr = 0.04.

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Fig. 14. Energy effectiveness versus Pc/Pd at Pd = 1.5 and sr = 0.03.

HCR, Pd, and sr. The design engineer of this 1.0 MWe power plant thermocline solar thermal storage has been given a clear idea of the impact of design choices and feasibility of the overall performance of the storage system without the need for complicated and lengthy calculations in a simulation. Again, it is the expectation of the authors that the currently proposed general charts be made available as a design tool in concentrated solar power industry. As to the calibration application, using generalized charts to find the energy storage capacity and the period of operation for a given tank, the procedures described in Section 5.2 is straightforward and easy to follow. It may not be necessary to show an example here. Finally, it is to be noted that heat loss from a thermal storage tank is inevitable. After the tanks size being decided, it may be necessary to estimate a heat loss. To compensate the heat loss, a larger volume of the heat storage tank and a longer heat charge period may be needed. The current authors propose that a simple way of refining the design is to increase both the heat charge time and tank size with a percentage equal to the ratio of heat loss versus the projected heat delivery. 6. Concluding remarks Thermocline solar thermal energy storage technology is receiving increased attention in the solar energy industry. The energy storage capacity and energy delivery effectiveness in a thermocline tank is governed by the energy conservation equations of both solid filler material and heat transfer fluid. As has been discussed in this paper that the two governing equations are partial differential equations and solving them will need numerical computations, which may not be convenient to engineers to repeatedly solve when they design or calibrate thermocline storage tanks. To provide a generalized tool for reference in designing a thermal storage tank, this study gave a comprehensive analysis to the thermocline energy storage and delivery performance. The study used dimensionless heat transfer

governing equations for fluid and solid filler material and could analyze the energy charge and discharge effectiveness in general. Consequently this paper proposed a series of general charts illustrating the energy storage effectiveness as a function of four dimensionless parameters that are grouped and comprised of all the given parameters of a thermocline tank including: properties of solid filler material and heat transfer fluid, dimensions of the storage tank, mass flow rate of heat transfer fluid, filler material size and the void fraction of packing in the tank, and energy charge and discharge time period. The charts are generalized and are applicable to any cases of thermocline thermal storage. Engineers can conveniently look up the general charts and decide the dimensions of thermocline solar thermal storage tanks as well as operational conditions without doing complicated modeling computations. It is of great significance to reducing the work load when they design and calibrate thermal energy storage systems. The authors expect the currently proposed general charts be widely adopted as a benchmark design tool for application in concentrated solar power industry. Finally due to the limited space of this paper, it does not provide a large number of the graphs for the general charts proposed in the work. However upon request, the current authors will provide a larger number of graphs and charts covering a wide range of parameters for industrial application. Acknowledgements The authors are grateful to the supported by the US Department of Energy, National Renewable Energy Laboratory, under DOE Award Number DE-FC3608GO18155, and US Solar Thermal Storage LLC. References Beasley, D.E., Clark, J.A., 1984. Transient response of a packed bed for thermal energy storage. International Journal of Heat and Mass Transfer 27 (9), 1659–1669. Bejan, Adrian, 2006. Advanced Engineering Thermodynamics. John Wiley & Sons Inc., New Jersey. Brosseau, D., Kelton, J.W., Ray, D., Edgar, M., Chisman, K., Emms, B., 2005. Testing of thermocline filler materials and molten-salt heat transfer fluids for thermal energy storage systems in parabolic trough power plants. Journal of Solar Energy Engineering – Transactions of ASME 127, 109–116. Gabbrielli, R., Zamparelli, C., 2009. Optimal design of a molten salt thermal storage tank for parabolic trough solar power plants. Journal of Solar Energy Engineering 131, 041001. Gil, Antoni, Medrano, Marc, Martorell, Ingrid, Lazaro, Ana, Dolado, Pablo, Zalba, Belen, Cabeza, Luisa F., 2010. State of the art on high temperature thermal energy storage for power generation. Part 1— Concepts, materials and modellization. Renewable and Sustainable Energy Reviews 14 (1), 31–55. Hasnain, S.M., 1998. Review on sustainable thermal energy storage technologies. Part I: Heat storage materials and techniques. Energy Conversion and Management 39, 1127–1138. Herrmann, U., Kearney, D.W., 2002. Survey of thermal energy storage for parabolic trough power plants. Journal of Solar Energy Engineering 124 (2), 145–152.

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