Developments in Heat Transfer Part 11 potx

Schumann’s equations have been widely adopted in the analysis of thermocline heat storage utilizing solid filler material inside a tank.. To consider a non-uniform initial temperature di

Its dimensionless form is:

Fig 9 Comparison of dimensionless energy storage in the ‘washer’, normalized by the ideal

maximum energy change in the ‘washer’, due to different methods of solution

(Bi washer=h d( / 2) /i k s=3.0; /D eq d = i 6.0; t*=t/[( / 2) / ]d i 2 αs ; w c= 0.83442)

The parameter cluster of k s/(ρs s C HU) is a dimensionless term If it is sufficiently large, the

axial conduction term in Eq (27) may not be dropped off A basic effect of significant axial

heat conduction is that it will destroy the thermocline effect—a temperature gradient with

hot material being on top of cold This can lower the thermal storage performance in

general Therefore, to take into account the axial heat conduction effect, a similar correction

via the introduction of another factor to the modified heat transfer coefficient is proposed

This results in a new modified heat transfer coefficient of:

p

s washer c

For most thermal storage materials, such as rocks, molten salts, concrete, soil, and sands, the

value of k s/(ρs s C HU) is very small (in the order of 1 10× −6); while other terms in Eq.(27)

are in the order of 1.0 Therefore, the axial heat conduction effect in the thermal storage

material in Eq (27) is negligible

5.2.7 Application of the model to the storage system with PCM

For thermal storage with phase-change involved, the PCM can be enclosed in capsules to

form a packed bed as shown in Fig 2(a), or simply put in a storage tank that has heat

transfer tubes inside as shown in Fig 2(b) The governing equations discussed above are still

applicable to the heat transfer at locations where either phase change has not yet occurred or

has already been completed However, at locations undergoing phase change, the energy

equations must account for the melting or solidification process (Halawa & Saman, 2011;

Wu et al., 2011) The key feature in a melting or solidification process is that the temperature

of the material stays constant

Considering the energy balance for the thermal storage material:

where Γ is the fusion energy of the material, and Φ is the ratio of the liquid mass to the

total mass in the control volume of dz For melting, Φ increases from 0 to 1.0, while for

solidification it decreases from 1.0 to 0

Considering the invariant of the temperature of the material during a phase change process,

the energy balance equation for HTF is:

Equation (29) and (30) can be reduced to dimensionless equations by introducing the same

group of dimensionless parameters:

where a new dimensionless parameter ψ=(T H−T C L) /s Γ is introduced Since the phase

change temperature is known, Eqs (31) and (32) can be solved separately

5.3 Numerical methods and solution to governing equations

5.3.1 Solution for the case of no phase change

A number of analyses and solutions to the heat transfer governing equations of a working

fluid flowing through a packed-bed have been presented in the past (Schumann, 1929;

Shitzer & Levy, 1983; McMahan, 2006; Beasley, 1984; Zarty & Juddaimi, 1987) As the

pioneering work, Schumann (Schumann, 1929) 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 utilizing solid filler material

inside a tank His analysis and solutions were for the special case where there is a fixed fluid

temperature at the inlet to the storage system In most solar thermal storage applications this

may not be the actual situation To overcome this limitation, Shitzer and Levy (Shitzer &

Levy, 1983) employed Duhamel’s theorem on the basis of Schumann’s solution to consider a

transient inlet fluid temperature to the storage system The analysis of Schumann, and

Shitzer and Levy, however, still carry with them some limitations Their method does not

consider a non-uniform initial temperature distribution For a heat storage system,

particularly 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 vice versa Therefore, non-uniform and nonlinear temperature

distribution is typical for both charge and discharge processes To consider a non-uniform

initial temperature distribution and varying fluid temperature at the inlet in a heat storage

system, numerical methods have been deployed by researchers in the past

To avoid the long mathematical analysis necessary in analytical solutions, numerical

methods used to solve the Schumann equations were discussed in the literature by

McMahan (McMahan, 2006, 2007), and Pacheco et al (Pacheco et al., 2002), and

demonstrated in the TRNSYS software developed by Kolb and Hassani (Kolb & Hassani,

2006) Based on the regular finite-difference method, McMahan provided both explicit and

implicit discretized equations for the Schumann equations Whereas the explicit solution

method had serious stability issues, the implicit solution method encountered an additional

computational overhead, thus requiring a dramatic amount of computation time The

solution for the complete power plant with thermocline storage provided by the TRNSYS

model in Kolb’s work (Kolb & Hassani, 2006) cites the short time step requirement for the

differential equations of the thermocline as one major source of computer time

consumption To overcome the problems encountered in the explicit and implicit methods,

McMahan et al also proposed an infinite-NTU method (McMahan, 2006, 2007) This model

however is limited to the case in which the heat transfer of the fluid compared to the heat

storage in fluid is extremely large

The present study has approached the governing equations using a different numerical

method (Van Lew et al., 2011) The governing equations have been reduced to

dimensionless forms which allow for a universal application of the solution The

dimensionless hyperbolic type equations are solved numerically by the method of

characteristics This numerical method overcomes the numerical difficulties encountered in

McMahan’s work—explicit, implicit, and the restriction on infinite-NTU method (McMahan,

2006, 2007) The current model yields a direct solution to the discretized equations (with no

iterative computation needed) and completely eliminates any computational overhead A

grid-independent solution is obtained at a small number of nodes The method of

characteristics and the present numerical solution has proven to be a fast, efficient, and

accurate algorithm for the Schumann equations

The non-dimensional energy balance equations for the heat transfer fluid and filler material

can be solved numerically along the characteristics (Courant & Hilbert, 1962; Polyanin, 2002;

Ferziger, 1998) Equation (9) can be reduced along the characteristic t * =z * so that:

) (

1 Dt

D

f r r

τ

Similarly, Eq.(12) for the energy balance for the filler material is reposed along the

characteristic z *=constant so that:

) (

H dt

d

f r r

CR

τ

The solution for Eq (35) is very similar to that for Eq (33) but with the additional factor of

CR

H The term H CR is simply a fractional ratio of fluid heat capacitance to filler heat

capacitance Therefore, the equation for the solution of θr will react with a dampened speed

when compared to θf , as the filler material must have the capacity to store the energy

being delivered to it, or vice versa Finally, separating and integrating along the characteristic

for Eq.(35) results in:

f r r

There are now two characteristic equations bound to intersections of time and space A

discretized grid of points, laid over the time-space dimensions will have nodes at these

intersecting points A diagram of these points in a matrix is shown in Fig 10 In space, there

are i = 1, 2, ,M nodes broken up into step sizes of Δz * to span all of z * Similarly, in

time, there are j = 1, 2, ,N nodes broken up into time-steps of Δt * to span all of t *

Looking at a grid of the ϑ nodes, a clear picture of the solution can arise To demonstrate a

calculation of the solution we can look at a specific point in time, along z * where there are

two points, ϑ1 , 1 and ϑ2 , 1 These two points are the starting points of their respective

characteristic waves described by Eq (33) and (36) After the time Δt * there is a third point

2 , 2 1 , 1

* f r r

d

ϑ ϑ

ϑ

τ

The numerical integration of the right hand side is performed via the trapezoidal rule and

the solution is:

* f f r r r f

2 2

1 2 , 2 1 , 1 2 , 2 1 , 1 1

, 1 2 ,

τ θ

CR

d

ϑ ϑ

ϑ

τ

The numerical integration of the right hand side is also performed via the trapezoidal rule

and the solution is:

* f f r r r

CR r

2 2

H 2 , 2 2 , 1 2 , 2 2 , 1 1

, 2 2 ,

τ θ

θ , while θf and θr at grid points ϑ1 , 1 and ϑ2 , 1 are known

* CR r

r

* CR f

r

* r r

* f

r f

r

* CR r

* CR

r

* r

*

2

t H 1 2

t H

2

t 2

t 1

2

t H 1 2

t H

2

t 2

t 1

1 , 2 1

, 2

1 , 1 1

, 1

2 , 2

2 , 2

τ

Δ θ

τ

Δ θ

τ

Δ θ τ

Δ θ

θ

θ

τ

Δ τ

Δ

τ

Δ τ

Fig 10 Diagram of the solution matrix arising from the method of characteristics

From the grid matrix in Fig.10 it is seen that the temperatures of the filler and fluid at grids

1

,

ϑ are the initial conditions The temperatures of the fluid and filler at grid ϑ1 , 1 are the inlet conditions which vary with time The inlet temperature for the fluid versus time is given The filler temperature (as a function of time) at the inlet can be easily obtained using Eq.(12), for which the inlet fluid temperature is known Now, as the conditions at ϑ1 , 1, ϑ1 , 2, and ϑ2 , 1 are known, the temperatures of the rocks and fluid at ϑ2 , 2 will be easily calculated from Eq.(41)

Extending the above sample calculation to all points in the ϑ grid of time and space will give the entire matrix of solutions in time and space for both the rocks and fluid While the march of Δz * steps is limited to z * =1 the march of time Δt * has no limitation

The above numerical integrations used the trapezoidal rule; the error of such an implementation is not straightforwardly analyzed but the formal accuracy is on the order of

)

t

(

O Δ 2 for functions (Ferziger, 1998) such as those solved in this study

5.3.2 Solutions for the case with phase change

For the governing equations of the phase change case, the adopted convention of having the z-direction coordinate always follow the flow direction is preserved, such that for heat

charging, z=0 is for the top of a tank, and for heat discharging, z=0 is for the bottom of a

tank The two governing equations (Eq (31) and Eq.(32)) for the phase change process can

be discretized using finite control volume methodology:

The procedures for finding the solution of phase change problem are as follows:

1 Solve the non-phase-change governing equation analytically using Eq.(12) for the phase

change material for the inlet point

2 Monitor the temperature at each time step as given by Eq.(12), and see if the

temperature at a time step is greater than the fusion temperature, if yes, the solution for

that and subsequent time steps are to be solved using the phase change equation (Eq

(43)

3 For each time step solved using Eq (43), monitor the fusion ratio, Φ ; when it becomes

larger than 1.0 then the solution for that and subsequent time steps are to be solved

using the non-phase-change governing equation (Eq.(12)) for the remainder of the

required time

4 March a spatial step forward and repeat all of the above steps However, now in part (1)

of this procedure, Eq.(41) must be used to solve the temperatures of both the fluid and

PCM for time steps before phase change starts; and also in part (3) of this procedure

Eq.(41) should be used to solve the temperatures of the fluid and PCM for steps after

the phase change is over The repetition of parts (1) to (3) of this procedure is to be

continued until all the spatial steps are covered

6 Results from simulations and experimental tests

6.1 Numerical results for the temperature variation in a packed bed

The first analysis of the storage system was done on a single tank configuration of a chosen

geometry, using a filler and fluid with given thermodynamic properties The advantage of

having the governing equations reduced to their dimensionless form is that by finding the

values of two dimensionless parameters (τrand H CR) all the necessary information about

the problem is known The properties of the fluid and filler rocks, as well as the tank

dimensions, which determined τrand H CR for the example problem, are summarized in

Table 7

The numerical computation started from a discharge process assuming initial conditions of

an ideally charged tank with the fluid and rocks both having the same high temperature

throughout the entire tank, i.e θf =θs= After the heat discharge, the temperature 1

distribution in the tank is taken as the initial condition of the following charge process The

discharge and charge time were each set to 4 hours The fluid mass flow rate was

determined such that an empty (no filler) tank was sure to be filled by the fluid in 4 hours

With the current configuration, after five discharge and charge cycles the results of all subsequent discharge processes were identical—likewise for the charge processes It is therefore assumed that the solution is then independent of the first-initial condition The data presented in the following portions of this section are the results from the cyclic discharge and charge processes after 5 cycles

Table 7 Dimensions and parameters of a thermocline tank (Van Lew et al., 2011)

Fig 11 Dimensionless fluid temperature profile in the tank for every 0.5 hours

Shown in Fig 11 are the temperature profiles in the tank during a discharge process, in which cold fluid enters into the tank from bottom of the tank The location of z =* 0 is at the bottom of a tank for a discharge process The temperature profile evolves as discharging proceeds, showing the heat wave propagation and the high temperature fluid moving out of

the storage tank The fluid temperature at the exit (z =* 1) of the tank gradually decreases after 3 hours of discharge At the end of the discharge process, the temperature distribution along the tank is shown in Fig 12 At this time the fluid and rock temperatures, θf and

s

θ respectively (θsis denoted by θrwhen the filler material is rock), in the region with z*below 0.7 are almost zero, which means that the heat in the rocks in this region has been completely extracted by the passing fluid In the region from z =* 0.7 to z =* 1.0the temperature of the fluid and rock gradually becomes higher, which indicates that some heat has remained in the tank

Fig 12 Dimensionless temperature distribution in the tank after time t * = 4 of discharge

(Here θr is used to denote θs, as rocks are used as the storage material in the example)

Fig 13 Dimensionless temperature distribution in the tank after time t * = 4 of charge

(Here θr is used to denote θs, as rocks are used as the storage material in the example)

A heat charge process exhibits a similar heat wave propagation scenario The temperature for the filler and fluid along the flow direction is shown in Fig 13 after a 4 hour charging process During a charge process, fluid flows into the tank from the top, where z* is set as zero It is seen that for the bottom region (z from 0.7 to 1.0) the temperatures of the fluid *and rocks decrease significantly A slight temperature difference between heat transfer fluid and rocks also exists in this region

Fig 14 Dimensionless temperature histories of the exit fluid at z * = 1 for charge and

discharge processes

The next plots of interest are the variation of θfat z = as dimensionless time progresses * 1for a charging or discharging process Figure 14 shows the behavior of θfat the outlet during both charge and discharge cycles For the charge cycle, θf begins to increase when

all of the initially cold fluid has been ejected from the thermocline tank For the present thermocline tank, the fluid that first entered the tank at the start of the cycle has moved completely through the tank at t =* 1, which also indicates that the initially-existing cold fluid of the tank has been ejected from the tank Similarly, during the discharge cycle, after the initially-existing hot fluid in the tank has been ejected, the cold fluid that first entered the tank from the bottom at the start of the cycle has moved completely through the tank at

* 1

t = At t =* 2.5, or t=2.5 hours, the fluid temperature θf starts to drop This is because the energy from the rock bed has been significantly depleted and incoming cold fluid no longer can be heated to θf= 1 by the time it exits the storage tank

The above numerical results agree with the expected scenario as described in section 4 To validate the above numerical method, analytical solutions were obtained using a Laplace Transform method by the current authors (Karaki, et al, 2010), which were only possible for cases with a constant inlet fluid temperature and a simple initial temperature profile Results compared in Fig 15 are obtained under the same operational conditions—starting from a

fully charged initial state and run for 5 iterations of cyclic discharge and charge processes The fluid temperature distribution along the tank (z*=0 for bottom of the tank) from numerical results agrees with the analytical results very well This comparison essentially proves the effectiveness and reliability of the numerical method developed in the present study

Fig 15 Comparison of numerical and analytical results of the temperature distribution in

the tank after time t * = 4 of a discharge (Here θris used to denote θs, as rocks are used as the storage material in the example)

Based on the results shown in Fig 15, the temperature distribution along z* at the end of a charge is nonlinear This distribution will be the initial condition for the next discharge cycle Similarly a discharge process will result in a nonlinear temperature distribution, which will be the initial condition for the next charge It is evident that the analytical solutions developed by Schumann (Schumann, 1929) could not handle this type of situation

Fig 16 Comparison of dimensionless temperature distributions in the tank after time t* = 4

of discharge for different numbers of discretized nodes

Another special comparison was made to demonstrate the efficiency of the method of

characteristics at solving the dimensionless form of the governing equations Shown in Fig

16 are the temperature profiles at t =* 4 obtained by using different numbers of nodes (20,

100, and 1000) for z* The high level of accuracy of the current numerical method, even with

only 20 nodes, demonstrates the accuracy and stability of the method with minimal

computing time

6.2 Comparison of modeling results with experimental data from literature

6.2.1 Temperature variations in charge processes

The authors have conducted experimental tests (Karaki at al., 2011) The test conditions for a

given heat charge process are listed in Table 8, which also shows the dimensionless

parameters As shown in Fig 17, at the initial time the thermocline tank has a uniform

temperature equal to room temperature The temperature readings from the thermocouples

at the top of the tank provide the inlet fluid temperatures in a charge process

Tank Length 0.65(m) Initial temperature 21.9 (oC)

Tank inner diameter 0.241 (m) High temperature 79.82 (oC)

Rock nominal diameter 0.01 (m) Low temperature 21.9 (oC)

Density of rocks 2632.8 kg/m3 Π =c t c/( / )H U 7.2511

Time (s) 0 856 1713 2570 3427 4284

Dimensionless time t* 0 1.45 2.9 4.35 5.80 7.25

Table 8 Conditions of a heat charging test (HTF is Xceltherm 600 by Radco Industries)

0 10 20 30 40 50 60 70 80 90

Fig 17 Temperatures in the center of the tank along the height of 65 cm for a charging

process (Thermocouples were set every 5cm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

The thermal storage performance test results of a thermocline tank reported in the literature (Pacheco et al., 2002) was also referenced to validate the current modeling work The experimental tests used eutectic molten salt (NaNO3-KNO3, 50% by 50%) as the heat transfer fluid and quartzite rocks and silica sands as the filler material Thermocouples in the test apparatus were imbedded in the packed-bed Temperatures in the tank at different height locations were recorded during a two-hour heat discharge process after the tank was charged for the same length of time The storage tank dimensions, packed-bed porosity, and properties of the fluid and filler material are listed in Table 9

Using the modeling of section 5, the heat charge followed by heat discharge was simulated

In Fig 19, the predicted temperatures at several height locations of the tank at different time instances during the discharge process were compared to the experimental data reported (Pacheco et al., 2002) The trend of temperature curves from the modeling prediction and the experiment is quite consistent Considering the uncertainties in the experimental test and the properties of materials considered, the agreement between the experimental data and the modeling prediction is quite satisfactory This comparasion firmly validates the current modeling and its numerical solution method

Tank Length measured 0.65(m) Initial temperature 129.0 (oC)

Tank inner diameter 0.241 (m) High temperature 129.0 (oC)

Rock nominal diameter 0.01 (m) Low temperature 56.0 (oC)

Density of rocks 2632.8 kg/m3 Π =d t d/( / )H U 8.0596

Fig 19 Comparison of modeling predicted results with experimental data from reference

(Pacheco et al., 2002)

6.2.2 Temperature variations in a discharge process

A heat discharge experiment was conducted under the conditions listed in Table 10, which also includes the dimensionless parameters Shown in Fig 20 is the temperature variation versus time at different locations in the tank along the tank height The thermocouples at the

top (z*=1.0) of the tank measure the temperature of the discharged fluid The degradation of

discharged fluid temperature is clearly shown in the figure

0 20 40 60 80 100

Fig 20 Temperatures in the center of the tank along the height of 65 cm for a discharging process (Thermocouples were set every 5cm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

The temperature distribution along the height in the tank at different times is shown in Fig

21 At the end of the discharge, the temperature on top of the tank (at z*=1) becomes low, which means that stored energy has been discharged Using the measured initial temperature distribution and the inlet fluid temperature (at z*=1) together with the properties listed in Table 10, numerical simulation results were obtained and are compared with the test results in Fig 21 The real time and the dimensionless time are listed in Table

10 Again, the agreement between the experimental data and the modeling simulation is satisfactory

6.3 Correlation of energy delivery effectiveness to dimensionless parameters

Based on the above discussion and the dimensionless governing equations obtained in section 5.2.3, the energy delivery effectiveness, η, is a function of four dimensionless parameters, Πc/Π , d Π , d τr, and H CR Solutions of the dimensionless governing equations for energy charge and discharge allow us to develop a database so that a series of charts and diagrams for η= Πf( c/Π Πd, d, ,τr H CR) may be prepared for reference by engineers in the design of thermocline storage tanks Illustrated in Fig 22 is a configuration

of a group of database charts for a givenΠ , in which multiple graphs, each with a specific d

r

τ , may be provided In each graph, multiple curves, each with a given HCR, for the energy storage effectiveness η versus Πc/Π are provided To build a large database, more dgraphs in the same configuration can be provided covering a large range of Π values d

Figure 23 shows four charts of η versus Πc/Π at d Π =4.0 Each chart is for a specific d τrwith multiple curves for different H CR All the data of energy delivery effectiveness were obtained based on several cyclic operations of the energy charge and discharge, and the results are consequently independent of the number of cycles More charts with wide range

of Π , d τr, andH CRmay be developed However, in actual design practice, the ranges of

(c) τr=0.1

(d) τr=0.2Fig 23 Multiple graphs from modeling results for energy storage effectiveness versus /

Π Π at Π =4.0 d

Observing the above four graphs one can easily draw the following conclusions:

1 The energy delivery effectiveness never reaches 1.0 if Πc/Π <d 1.0 This proves that only for an ideal thermocline storage tank can η=1.0 at Πc/Π =d 1.0

2 With the decrease of τr, η will increase For example, at a ratio of Πc/Π =d 1.5and

0.25

CR

H = , the energy delivery effectiveness approaches 1.0 when τrchanges from 0.2

to 0.01 This is because a decrease of τr is due to an increase in the volume of the storage tank

3 It is understood that a small HCR corresponds to the case where (ρC)s is relatively large when compared to (ρC)f, and therefore the energy storage capability is improved, and

η can approach 1.0 easier On the other hand, when the void fraction in a packed bed approaches 1.0, it will makeH CR→ ∞ , and the thermal storage effectiveness can approach that of an ideal case However, in most practical applications, a low void fraction in a thermocline tank is required for the purpose of using less heat transfer fluid, and therefore a smaller H CR value is practical and preferable

4 For cases where η could never approach 1.0, even at large Πc/Π values, it is dobviously attributable to the fact that the storage tank is too small, and reselection of a larger storage tank is needed

Designers for a thermal storage system often need to calibrate or confirm that a given storage tank can satisfy an energy delivery requirement Under such a circumstance, the dimensions of the storage tank and the power plant operational conditions are known, which means the values of τr, Π , and d H CR are essentially given One can easily check whether over a range of values of Πc/Π the energy delivery effectiveness ηd can approach 1.0

7 Procedures of sizing and design of a thermal storage system

The required operational conditions of the power plant dictate the size of the thermocline storage tank The relevant operational conditions include: electrical power, thermal efficiency

of the power plant, the extended period of operation based on stored thermal energy, the required high temperature of the heat transfer fluid from the storage tank, and the low temperature of the fluid returned from the power plant, the specific choice of heat transfer fluid and thermal storage material, 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:

1 Select a minimum required volume for a thermocline tank using Eqs (1) and (3)

2 Choose a radius, R, and the corresponding height, H, from the minimum volume decided in step (1) Using these dimensions, the parameters—Π , d τr, H CR for a thermocline tank with filler material, can be evaluated, where Π is determined based d

on the required operational time

3 Look up the design charts (such as those in Fig 23) and see if an energy delivery effectiveness of 1.0 can be achieved Often the energy delivery effectiveness will not approach 1.0 for the first trial design This is because the first trial uses a minimum volume However, with the results from the first trial one can predict the required height or volume of the tank necessary to decrease τrand Π in the same proportion d

A couple of trail iterations may be needed to eventually satisfy the criterion of ηclosing to 1.0

If the energy delivery effectiveness from step (1) cannot approach 1.0 even if a large Πc/Π d

is chosen, one actually has two ways to improve the effectiveness during the second trial These are to decrease H CR, or decrease both τrand Π in the same proportion However, d

CR

H is determined by properties of the fluid and filler material, which has very limited

options, and therefore a decrease of τr and Π is more practical The decrease of d τr can be done by an increase of the height of a storage tank This means that to achieve an effectiveness of 1.0 one has to increase the size of the storage tank When the height of tank

is increased, Π decreases accordingly since it depends on the height of the tank d

Occasionally, a calibration analysis requires a designer to find a proper time period of energy charge that can satisfy the needed operation time of a power plant The known parameters will be the tank volume, τr, as well as H CR at a required operation period of Π dThe first step of the calibration should be the examination of the criterion given in Eq (3), from which a minimum tank volume can be chosen If the minimum tank volume is satisfied, the second step of calibration will be to find a proper Πc/Π that can make the denergy delivery effectiveness approach 1.0 Graphs including curves at the required H CR

and the given τr and Π must be looked up Conclusions can be easily made depending on dwhether the energy delivery effectiveness can approach 1.0 for a particular value of Πc/Π dTwo practical examples of thermocline thermal storage tank design are provided to help industrial designers practice the design procedures proposed in this work Readers can easily repeat the design procedures while using their specific material parameters and operational conditions

7.1 Design example 1—a system as shown in Fig 2(a)

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 Therminol® VP-1 The power plant requires high and low fluid temperatures of 390 °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 The required time period of energy discharge is 4 hours, the storage tank diameter is chosen to

be 8 m The rock diameter is 4 cm

The solution is discussed as follows:

1 Making use of Eqs (1) to (3) and the above given details on the power plant, as well as the properties of Therminol® VP-1, we can find a necessary mass flow rate of 25.34 kg/m3 and an ideal tank height of 9.59 m The minimum volume of the storage tank can

be determined from Eq (3) Here, the ideal volume is used in the first trial of the design Using Eqs (19) and (20) we find the modified heat transfer coefficient to be 32.05 W/(m2 K) With this information, the values of H CR, Π , and d τr are found to be 0.451, 3.03, and 0.0227, respectively Given in Fig 24 is a chart for Π =3 and d τr=0.0227 at various values of H CR and Πc/Π It is seen that on the curve of d H CR=0.45, there is

no time ratio Πc/Π that allows the energy delivery effectiveness to be close to 1.0 dTherefore, the ideal volume chosen will not satisfy the energy storage need

2 One option to provide the ability to store and deliver more energy and approach an effectiveness of 1.0 is to increase the height of the storage tank When the height is increased to 12 m, the values of Π and d τr changed to 2.42 and 0.0181, respectively Figure 25 gives the chart for Π =2.42 and d τr=0.0181 It is seen on the curve of H CR= 0.45 that at the time period ratio, Πc/Π =1.2, the energy delivery effectiveness dapproaches 0.99 This should be an acceptable design

In this example, compared to an ideal thermal storage tank, the rock-packed-bed thermocline tank uses about 40.0% of the heat transfer fluid To avoid the temperature degradation of the heat transfer fluid in a required discharge time period, a 20% longer charging time than discharging time is always applied in every charge and discharge cycle