theoretical methods to characterize thermal stratification

Theoretical methods to characterize thermal

stratification – A literature review

Report for the Ph.D. course: Thermal stratification in solar storage tanks at DTU,

Denmark 10/2007

Authors:

Cynthia Cruickshank, Ph.D. Candidate, Queen’s University, Canada

MSc. Michel Haller, Ph.D. Candidate, Graz University of Technology, Austria

1 INTRODUCTION 1

2 LITERATURE REVIEW OF STRATIFICATION INDICES 2

2.1 GRAPHICAL PRESENTATIONS 3 2.2 STRATIFICATION INDICES USING THE ENERGY APPROACH 5 2.2.1 FRACTION OF RECOVERABLE HEAT 5 2.2.2 THERMAL STORAGE EFFICIENCIES AND CHARGING/DISCHARGING EFFICIENCIES 5 2.2.3 DAVIDSON-ADAMS MIX NUMBER (1994) 5 2.3 STRATIFICATION INDICES USING THE EXERGY APPROACH 6 2.4 COMBINED METHOD 7 2.5 OTHER APPROACHES 9

3 EXAMPLE CALCULATION OF STRATIFICATION INDICES 9

3.1 EXAMPLE CASE 9 3.2 PANTHALOOKARAN STORAGE EVALUATION NUMBERS 10 3.3 DAVIDSON MIX NUMBER 11

4 DISCUSSION 11

5 CONCLUSIONS 13

6 LITERATURE STUDIED 14

APPENDIX A I

OTHER LITERATURE (NOT STUDIED) I

APPENDIX B I

TABLE OF REPORTS INTRODUCING AND/OR USING STRATIFICATION INDICES I

APPENDIX C I

DETAILED DESCRIPTION OF “OTHER METHODS” TO CHARACTERIZE STRATIFICATION I

APPENDIX D I

DETAILED CALCULATION OF EXAMPLES I

Cruickshank and Haller 2007 – Methods to characterize thermal stratification

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1 Introduction Solar energy has been identified as a promising source of environmentally friendly energy. However, as a time-dependent source of energy, solar energy is not always available when there is an energy demand. Therefore, to maximize the benefits of solar energy systems, effective energy storage devices are required. It has been shown that thermal stratification in the energy store may considerably increase system performance, especially for low flow solar heating systems (e.g. Lavan 1977, Phillips and Dave 1982, Hoerger and Phillips 1987, Hollands and Lightstone 1989, Cristofari et al. 2003). It is therefore desirable to have an index or “measure“ for the level of stratification that exists in a thermal energy storage, in order to be able to compare different tank designs, control strategies, etc., and to give an idea of the level of stratification compared to the ideal case.

A primary objective in creating a stratified thermal storage is to maintain the thermodynamic quality of energy so it can be extracted at the same temperature that it was stored at. The separation of the fluid into volumes with different temperatures in storage tanks is the key factor in achieving this objective. An obvious way to achieve this separation is to use two tanks with different temperature levels, but this adds to the cost of the storage system. Alternatively, a flexible diaphragm can be used to separate the two regions within a single tank. One other concept is referred to as a labyrinth tank in which the water is forced to flow through a maze separating hot and cold regions. These concepts tend be complicated and may be expensive or unreliable, and as such the concept of a single, naturally stratified storage tank has been widely used due to its simplicity and low cost. In a naturally stratified storage tank warm water floats on top of cold water due to its lower density (Zurigat and Ghajar 2002).

Figure 1 illustrates three thermal energy stores with equal amounts of energy stored, but different levels of stratification.

T

X

Hot Zone

Cold Zone

Thermocline

T

X

Hot Zone

Cold Zone

Thermocline

T

X

Uniform Temperature

(a) (b) (c)

Figure 1: Differing levels of stratification within a storage tank with the same amount of heat stored (a) top, highly stratified, (b) center, moderately stratified and (c) bottom, showing fully mixed, unstratified storage

Several processes destroy stratification during charging / discharging and standby of a store:

1. Mixing due to the kinetic energy of a water jet entering the tank

2. Water entering the store at a height with different temperature will move towards a position of equal temperature due to buoyancy effects. While moving towards its position, it mixes with surrounding water and carries surrounding water along. This process is called plume entrainment. The same effect may occur when water around an internal heat exchanger is cooled or heated.

3. Heat losses through upper tank walls will cause internal buoyancy driven movements of water, causing cold fluid to flow down along the walls, being replaced by fluid that will move upwards in the middle of the tank.

4. Thermal diffusivity of the water and heat transfer along the tank wall and internal components will tend to equalize temperatures within the tank

5. Unwanted circulation due to buoyancy effects in pipes connected to the store may cause a complete turnover of the tank volume.


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A good review of heat transfer and stratification in sensible heat storage systems is given in Chapter 6 of the book “Thermal Energy Storage” published in 2002 by Dincer and Rosen (Zurigat and Ghajar 2002). Different aspects of thermal stratification in liquid fluids are illustrated in Figure 2. This report will give a review of methods to characterize thermal stratification in sensible heat1 thermal energy stores based on a liquid fluid with a temperature dependent density. In most cases, these heat stores are water stores.

stratification (st.) of

liquid fluids

factors influencing st.

simulation of st. stores

methods to characterize st.

influence of st. on energy system performance

aspect ratio (height / diameter)

inlet and outlet pos.

heat exchanger design

dimensionless numbers: − Peclet − Reynolds − Prandtl − Richardson − Fourier − Froude

1D

2D

3D

graphical numerical figures

energy efficiency(first law)

exergy efficiency(second law)

others

simulations / emulations

analytical (e.g. Phillips coefficient)

Figure 2: different aspects of thermal stratification in water stores that can be found in literature, in this report, only the shaded area will be treated in this report

Care has to be taken not to confuse factors that influence stratification with factors that evaluate the goodness of stratification (stratification indices or stratification parameters) for a particular store after a particular charging and/or discharging procedure. The height to diameter ratio for example is a factor that influences stratification, however, it is certainly not a figure that gives any information about the actual stratification in a store, since there are a lot of other factors that influence the stratification like the height of inlets and outlets, the geometry of the inlet, the history of temperatures and mass flow for charging and discharging the store, etc. Also the dimensionless numbers of Peclet, Richardson, Froude etc. are parameters that influence thermal stratification rather than parameters to describe the actual stratification of a store (see Figure 2).

2 Literature review of stratification indices A wide range of literature is available about thermal stratification in water stores (e.g. Abdoly and Rapp 1982, Jaluria and Gupta 1982, Jordan and Furbo 2005, Andersen et al. 2007). Many authors have found different ways to present stratification or to calculate an indicator for the goodness of stratification in a store, and a lot of factors and figures have been found that influence stratification. Two fundamentally different ways of looking at stratification can be found: a density approach used by environmental scientists (Stefan and Gu 1992, Moretti and McLaughlin 1977), and a temperature approach used by thermal engineers (e.g. Sliwinski et al. 1978, Kandari 1990, Davidson et al. 1994).

1 Thermal energy may be stored by latent or sensible heat phenomena. Latent heat storage systems primarily use the energy absorbed or released during a

change in phase, without a change in temperature (isothermal). Sensible heat storages use a heat storage medium where the additional or removal of heat

results in a change in temperature.


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In this report, only approaches based on temperatures will be discussed, since energy stores and the thermodynamic quality or temperature of the energy stored are of interest in thermal engineering.

A common way of showing the actual thermal stratification of a store is to present a graph with the storage height on the x-axis and the temperature on the y-axis. Dimensional and non dimensional graphical presentations will be treated in section 2.1. Some authors calculate numerical figures of merit that show the degree of stratification of a store in order to be able to compare different temperature distributions and/or stores quickly. Some of these figures are based on the first law of thermodynamics (energy-approach). A review of these approaches will be given in section 2.2. Figures based on the second law of thermodynamics (exergy- or entropy approaches) are treated in section 2.3, and a recent approach combining energy and exergy approaches will be treated in section 2.4. Figures that can be calculated to show the goodness or degree of stratification of a store that are not based on energy or exergy approaches are treated briefly in section 2.5, and more extensively in Appendix C. Examples for the calculation of stratification indices will be given in section 3.

Numerous authors have bracketed the degree of stratification that exists in a storage between two hypothetical temperature distribution limits consisting of a fully mixed storage at uniform temperature (i.e., zero stratification) and a perfectly stratified storage consisting of two uniform temperature regimes separated by an adiabatic boundary.

methods to characterize st.

graphical numerical figures based on temperature distribution

energy efficiency (first law)

numerical figures based on density distribution

dimensional

non dimensional

exergy efficiency(second law)

others

fraction of heat/cold recovered

thermal charging / discharging and storage efficiencies

MIX number (Davidson 1994)

outside tank balance

inside tank balance

combined approach (Panthalookaran 2007)

Wu and Bannerot 1987

Koldhekar 1981

Thermocline gradient methods

extraction efficiencies based on volumes

disturbed zone length

Moretti and McLaughlin 1977

Stefan and Gu 1992

Figure 3: Different methods to characterize thermal stratification in a water store. The shaded area contains the methods that will be treated in this report

2.1 Graphical presentations Graphs presenting stratification of a store may be dimensional, or non dimensional, as shown in Figure 4.

Non dimensional storage height

In the non dimensional presentation of the storage height *z , the actual height of a position in the storage z is divided by the storage height H . Thus, *z z H= , resulting in a value of 0 for the


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bottom of the tank, and a value of 1 for the top. Some authors define the dimensionless storage height the other way around, thus the top will be given a value of 0 and the bottom a value of 1.

Figure 4: Graphs showing dimensional (left) and non dimensional (right) presentation of stratification in a store (sources: Abdoly and Rapp 1982, Chan et al. 1983)

Non dimensional temperature

In the non dimensional presentation of temperature *T , the temperature difference between the actual value and the lower reference value minactualT T− is divided by the maximum possible temperature

difference of the system / process max minT T− . Thus, the lowest possible temperature will be given a value of 0, the highest possible temperature a value of 1:

* min

max min

actualT TTT T

−=

−

In the case where stratification was achieved by introducing a cold fluid of constant temperature into a hot tank of originally uniform temperature, minT will represent the cold fluid entering the tank, and maxT will represent the hot tank before the discharging process.

Non dimensional time

The non dimensional time is usually given for a charging or discharging process and defined in a way that time is equal to 0 at the beginning of the charging / discharging and 1 at the time the integrated heat capacity flow corresponds to the heat capacity of the tank. For a constant heat capacity and mass flow this corresponds to:

* P

P store

C t mtC m

⋅ ⋅=

⋅

where

PC specific heat of fluid, J/s t time, s m fluid mass flow, kg/s

storem mass of fluid in the store, kg


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2.2 Stratification indices using the energy approach Stratification indices using the energy approach are based on the first law of thermodynamics, usually calculating a fraction of energy that is lost / recovered after a certain charging / discharging or standby procedure.

2.2.1 Fraction of recoverable heat Abdoly and Rapp (1982) introduced a Parameter P representing recoverable heat, which only considers heat that has not been degraded more than 20% of its original temperature value towards ambient temperature. The fraction of recoverable heat from total heat F(t) can be calculated for any time t. Instead of temperature degradation towards ambient temperature, temperature degradation towards any useful temperature may be defined. This method was also used by Nelson (1999) who seperated it into a charging efficiency and a discharging efficiency.

2.2.2 Thermal storage efficiencies and charging/discharging efficiencies

Chan et al. (1983) analyzed thermal storage efficiencies for thermal storage devices used for building solar heating and cooling theoretically. They cite Duffie and Beckman (1975) and Hill et al. (1976) for the definition of a storage efficiency ( )1 tη . In order to determine this storage efficiency, a tank has to

be at a uniform (cold) temperature, iniT , at the beginning. Then the tank is charged with a constant

volume flow and the (hot) temperature , inT , and the mass weighted average temperature of the tank,

( )avT t , is monitored. Any mixing occurring during this charging process will be detected as soon as the water leaving the tank is not at the initial temperature anymore, and thus the energy stored in the tank (reflected by ( )av iniT t T− ) will be lower than if no mixing occurred (reflected by in iniT T− ):

( )( )

[ ]( )

1P av ini av ini

P in ini in ini

C V T t T T t Tt

C V T T T Tρ

ηρ

⎡ ⎤− −⎣ ⎦= =− −

where:

PC Vρ thermal mass of storage device, J/K t time, s

Chan et al. argued that the above definition does not always reflect the actual physical situation, and introduced a more physical version of this definition, the transient thermal efficiency:

( )( )

[ ]2P av ini

P in ini

C V T t Tt

C Qt T Tρ

ηρ

⎡ ⎤−⎣ ⎦=−

whereQ [m3/s] is the inlet volume flow rate

The two efficiencies 1η and 2η are interlinked by ( ) ( ) *2 1t t tη η= , where *t [-] is the dimensionless

filling time.

Similar approaches were used by Yee and Lai (2001), Yoo and Pak (1993), Tran et al. (1989), Wildin (1990), Hahne and Chen (1998), Mavros et al. (1994), Shah et al. (2005), Van Berkel et al. (1999), Bouhdjar and Harhad (2002), Bahnfleth and Song (2005).

2.2.3 Davidson-Adams MIX number (1994) The Davidson-Adams MIX number (Davidson et al. 1994) is calculated based on a “momentum of energy”. The momentum of energy M of a store is an integration of the energy density found in the store along its vertical axis, weighed with the dimensionless height along the vertical axis:


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1

N

i ii

M y E=

= ⋅∑

where:

iy distance of the center of node / volume i from the bottom of the tank, m

iE energy contained in node / volume i of the tank, J

This way, for the same amount of energy stored, the momentum of energy is higher the higher up in the store it is stored. In other words, a highly stratified store will have a high momentum of energy. The dimensionless height is taken at the middle of each node.

To get the dimensionless MIX-number, the difference of the momentum of energy between a perfectly stratified store (Mstr) and the actual store under investigation (Mexp) is divided by the difference of momentum of energy between a stratified store and a fully mixed store (Mmix). This number will be 0 if the store is fully stratified, and 1 if the store is fully mixed.

expstr

str mix

M MMIX

M M−

=−

It is important to note that the fully mixed store as defined by Davidson is NOT obtained by mixing a stratified store that has been obtained by a real charging process, but by using the same inlet temperature, different from the initial tank temperature, that caused stratification, and assuming that during the whole charging/discharging process the tank will always be fully mixed. Thus the amount of energy stored in the fully mixed store is smaller than the amount of energy stored in the stratified store if Tin > Tout, and larger if Tin < Tout. It is also important to note that both “theoretical” store temperature profiles, stratified as well as fully mixed, are calculated including heat losses to the surrounding with the heat loss coefficient that has been determined for the real store.

2.3 Stratification indices using the exergy approach Stratification indices using the exergy-approach are based on the second law of thermodynamics. Unlike the first law approaches, when calculating exergy or entropy generation, a reference “dead point” temperature has to be defined whose value will always be an arbitrary choice by the user based on whatever he/she thinks is a useful temperature for the energy system. Exergy approaches are especially useful when the energy stored will be used to produce work, since the exergy stored, and not the energy stored, is the thermodynamic limit of the work that can be produced. Since a better stratification is always going hand in hand with a higher exergy content, but not always with a higher energy content, exergy based figures may be used to give information about the goodness of stratification.

Moran and Keyhani (1982) studied peak load heat stores used in electricity production with turbine generators. Consequently, they were interested in the available energy for electricity production, which is a synonym for the exergy. In their work they divided the exergy introduced into a tank by a fluid flow into three fractions according to:

1 s out dε ε ε= + +

where sε is the fraction of the amount of exergy carried into the unit which is stored, outε is the

fraction of the exergy carried into the unit which exits the control volume with the flowing fluid, and dε is the fraction destroyed as a result of irreversibilities within the control volume. The measurements performed by Moran and Keyhani do not give an insight into what happens at any time within the storage tank. Thus, exergy losses only become evident at the time of discharging. A similar approach of exergy balance based on fluid flows entering and leaving the tank was used by Krane (1987) who defined an entropy generation number SN for a storage-removal cycle as the total availability (=exergy) destroyed during a cycle divided by the total availability of the cooled and heated (gas) streams that enter the storage unit.


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Rosengarten (1999) defined a non dimensionalized exergy looking at the actual distribution of temperatures in the store, and thus giving information about the goodness of stratification. Therefore, he also called it the stratification efficiency stη :

( ) ( ) ( )0

01 ln

Hdel

stdel mean del mean

T T dymc T T H T T T y

ξη⎛ ⎞

= = − ⋅⎜ ⎟⎜ ⎟− − ⎝ ⎠∫

where:

ξ exergy of stored liquid, J m mass, kg c heat capacity, J/kgK

delT temperature at which water is delivered (e.g. Temperature required for heating a room, useful temperature level), K

meanT mass weighted mean tank water temperature, K

T environmental temperature, K H tank height, m y vertical coordinate inside tank, m

A similar approach was later on used by Rosen (2001) and Shah and Furbo (2003).

Homan (2003) defined an Entropy generation number SN as the ratio of entropy generated in

stratified charging / discharging Sσ over the entropy generated in fully mixed charging fσ .

He also calculated the minimum achievable entropy generation number for any charging process from a theoretical analysis as a value only dependent on the Peclet number (since Peclet relates advection to diffusion, and diffusion will always cause some mixing):

1 28SN

Peπ⎛ ⎞⎜ ⎟⎝ ⎠

2.4 Combined Method Panthalookaran et al. (2007) have presented a new method for the characterization of a stratified thermal energy store based on efficiency definitions that include terms for both the first law and second law of thermodynamics.

They define the term "energy response" to describe a total change in energy stored within a thermal energy storage during a charge or discharge cycle or during standby. A large energy response is desirable for the charge and discharge sequences, but not for the process of storing heat. Because of these conflicting requirements in a good thermal energy storage, the authors chose to define two independent efficiencies to characterize the charging/discharging process and the storing process, independently.

As a basis of comparison, Panthalookaran et al. defined an ideally stratified thermal storage and a fully mixed thermal storage with the same energy content as the real storage for any instant of time. To determine the characteristics of ideally stratified storage, the energy level of the system at time 2t is “reordered” such that storage contains only two temperature regimes, one at a uniform temperature equivalent to the highest temperature in the “real” storage, and the other at a uniform temperature corresponding to the lowest temperature in the “real” storage. At the same time, the temperature level of a uniformly mixed storage is calculated for the same energy level as the real storage. Then, the change in entropy as the storage goes from t1 to t2 can be calculated for all three cases as:


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mixed real stratifiedS S SΔ ≥ Δ ≥ Δ . From these values, the entropy generation number EGR for the real system as a measure for the second law efficiency can be calculated:

real stratifiedEG

mixed stratified

( )( )

S SR

S SΔ −Δ

=Δ −Δ

The entropy generation ratio is 0 if ideal stratification is achieved, and 1 if the tank is not stratified at all (fully mixed case).

In a similar way, the “energy response” RE can be calculated as a measure for the store’s energy response:

realR

ideal

EE

EΔ

=Δ

where real idealE EΔ ≤ Δ .

The term realEΔ represents the change in energy of a system within a specified time interval

( 2 1t t t−Δ = ) in the charging case and represents the energy extracted from the system in a specified

time interval for the discharging case. The term idealEΔ is calculated based on an ideal situation in which the maximum energy response is achieved. This is defined for the charging/discharging process as a piston flow adiabatically separated from the other parts of the system.

The entropy generation number EGR and the energy response number, RE , are then combined in the so-called storage evaluation numbers (SEN).

The storage evaluation number SEN1η is used for a charging or discharging process, where a maximum energy response is wanted, but at the same time as little as possible entropy generation:

ideal,cEG real stratifiedSEN1

R mixed stratified real,c1 100% 1 100%

ES SRE S S Eη

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

ΔΔ −Δ= − ⋅ = − ⋅ ⋅

Δ −Δ Δ

The storage evaluation number SEN2η is used for a storing process, where a minimum energy response (= heat loss) is wanted, and again at the same time as little as possible entropy generation (entropy generation = destruction of stratification):

real,sreal stratifiedRSEN2 EG

mixed stratified ideal,s1 100% 1 100%

ES SR E S S Eη

⎡ ⎤⎡ ⎤ ⎢ ⎥⎣ ⎦ ⎢ ⎥

⎣ ⎦

ΔΔ −Δ= − ⋅ ⋅ = − ⋅ ⋅

Δ −Δ Δ


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2.5 Other approaches Other approaches for the characterization of thermal stratification in energy storage tanks include:

− Phillips stratification coefficient which is a measure of the benefit achieved from stratification for a solar heating system (Phillips and Dave 1982)

− Lavan (1977) extraction efficiency which is a measure for the volume of the store that can be withdrawn before the temperature level has degraded to a certain amount.

− Thermocline gradient, thermocline gradient decay and thermocline gradient thickness. Sliwinski et al. (1978), Shyu et al. (1987/89), Bahnfleth and Song (2005)

− Stratification coefficients by Koldhekar (1981), Wu and Bannerot (1987), Ghaddar et al. (1989) and Kandari (1990)

A more detailed description of these methods can be found in Appendix C.

3 Example calculation of stratification indices

3.1 Example case A water filled storage tank, of volume 1 m3, is charged with a constant flow of hot water at 10 L per minute and 50°C during 50 minutes. The store is initially divided into four equal volumes of uniform temperature (see Figure 5). Calculations will be performed for one interval of time, spanning from 1t to

2t . The temperature distribution in the “real” storage tank at the initial time 1t and at the end of the charge is shown in Figure 5. Hypothetical temperature distributions for the stratified case, fully mixed cases and the ideal piston flow case are also shown.

45oC

45oC

40oC

30oC

0.25m3

0.25m3

0.25m3

0.25m3

45oC

30oC

0.66m3

0.33m3

40oC

1m3

real case E = Ereal

stratified case Estrat = Ereal

mixed case Emix = Ereal

(Panthalookaran)

time = t1

time = t2

50oC

50oC

40oC

30oC

0.25m3

0.25m3

0.25m3

0.25m3

„ideal“ piston flow caseEideal > Ereal

calculation

calculation

40oC

1m3

mixed case Emix ≠ Ereal (Davidson)

34.8°C

Figure 5: Illustration of a storage charge sequence during the time interval ( 2 1t t− ).

Note: The temperature distribution in the storage tanks is shown at the beginning of the charge interval (top) and at the end (lower left). Hypothetically charged storages of equivalent energy as the real tank at time t2 are shown below representing the ideally stratified and mixed cases. The lower right shows an ideally charged store with water at 50oC that perfectly stratified and pushed down the water in storage tank (plug flow) and a tank that was completely mixed during the whole charging period.


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3.2 Panthalookaran storage evaluation numbers The method proposed by Panthalookaran (2007) uses two storage evaluation numbers which are each based on the first law (energy response number ER) and second law (entropy generation number REG) of thermodynamics to characterize efficiency during storage and during charging/discharging.

To calculate the Entropy generation ratio REG, it is first necessary to determine the change in entropy change as the storage goes from its original state at t1 to the “real” state and the two hypothetical states “stratified” and “mixed” at t2:

2stratified

1 1ln 832.61p

n

ii

TS m c kJ

T=Δ = =∑

2mixed

1ln 836.89ptotal

TS c kJ

TmΔ = =

As anticipated, mixed real stratifiedS S SΔ ≥ Δ ≥ Δ . These values are then combined to calculate the

entropy generation ratio, EGR :

real stratifiedEG

mixed stratified

( ) (833.49 832.61 ) 0.206( ) (836.89 832.61 )

S S kJ kJRS S kJ kJΔ −Δ −= = =Δ −Δ −

This number is consistent with the expected value for a charging sequence where entropy generation is relatively low and stratification is maintained but at a level below the ideal case.

In a second step, the energy change of the real storage, realEΔ , is compared to the energy change

idealEΔ that an ideal storage would have undergone in the same charging procedure:

( )4

2 11

real ( ) ( ) 63000i p i ii

T t T tE m c kJ=

⋅ ⋅ − =Δ =∑

( )4

2 11

ideal ( ) ( ) 73500i p i ii

T t T tE m c kJ=

⋅ ⋅ −Δ = =∑

From these values, the “energy response” RE is calculated:

realR

ideal

63000 0.85773500

E kJEE kJ

=Δ

= =Δ

This value indicates that the achieved energy response corresponds to 86% of the energy response that an ideal storage would have achieved.

Finally, the so-called storage evaluation number (SEN) or charging efficiency can be determined:

EGSEN1

R

0.2061 100% 1 100% 75.96%0.857REη

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

= − ⋅ = − ⋅ =

In a similar fashion, SEN1η can be calculated for a discharge sequence and SEN2η for a storage process (stand-by).

2

1 1real ln 833.49 /

n

pii

TS m c kJ KT==Δ =∑


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3.3 Davidson MIX Number Considering the same example case as described above in Section 3.1, the MIX number according to Davidson et al. (1994) is calculated. For this purpose, the height of the tank is assumed to be 2 m, and a constant specific heat of water of 4.2 kJ/kgK is used.

From the temperature distribution shown in Figure 5 for the “real” case, the momentum of energy Mexp is calculated:

4

exp1

181125i ii

M y E kJ m=

= ⋅ = ⋅∑

The ideally stratified storage at time t2, as defined by Davidson et al. corresponds to the ideal piston flow case shown in Figure 5. Therefore, Mstr is:

196875strM kJ m= ⋅

To determine the value of Mmix at t2, it must be assumed that the store is instantly mixed at time t1 and remains mixed during the whole charging process until time t2. The temperature of a fully mixed tank with the same energy content as the one shown in Figure 5 is 25 °C, and the temperature at the end of the mixing process can be calculated as:

( ) ( )( )2 1 34.8storetQ Vin inT t T T T t e C−= − − = °

Thus:

146160mixM kJ m= ⋅

To get the dimensionless MIX-number, the difference of the momentum of energy between a perfectly stratified store and the actual store under investigation is divided by the difference of momentum of energy between a stratified store and a fully mixed store:

exp 196,875 181,125 0.31196,875 146,160

str

str mix

M MMIX

M M− −

= = =− −

This means that the charging process lead to a mixing of the store of 31%, (or maintained 69% stratification). In a typical application, the MIX number would be calculated at each time step in the charge or discharge process.

4 Discussion A large amount of different methods can be found in literature to describe stratification in a thermal energy storage tank. Graphs have been widely used and they give a good and complete visualization of the current state of stratification of a single storage tank or the change of stratification with time within one tank. Graphs usually show temperature curves over time or over tank height, either in a dimensional or a non dimensional way. Especially for the comparison of charging / discharging procedure with different flow rates, but also for other cases, non dimensional graphs may be easier to interpret since they put the values into the context of the experimental boundary conditions.

For comparison of a large number of different tanks or experiments and to give an idea of how close the current stratification is to an ideal case, it is useful to have single numerical values that characterize the degree of stratification or mixing.

Solar engineers are interested in thermal stratification because a higher energy efficiency can be achieved with a better stratification. Therefore it is quite natural to base figures of merit or stratification


12/16

parameters on the first and/or second law of thermodynamics. Most authors using these approaches compare a tested store with a hypothetical store of equal size that shows an ideal stratified behavior and a hypothetical store that is fully mixed. The definition of these hypothetical stores may differ from one author/method to another.

By using the first law of thermodynamics (energy approach), heat losses of a tank can be described and charging and discharging efficiencies can be calculated. With a pure energy approach it is usually not possible to detect destratification or mixing that occurs during a standby period unless the tank is discharged to a certain point. This is usually the time when the center of the expected thermocline is about to be discharged. A discharge efficiency taken at the dimensionless time 1 (i.e., when the thermocline is expected to be discharged halfway) is certainly a useful number. When showing the discharge efficiency over time, however, care has to be taken that the graph is not misinterpreted. For example, the “inefficiency” or mixing appears to occur at the end of the discharge, (i.e., when the thermocline leaves the tank), however, the mixing of the thermocline region most likely occured at the beginning of the discharge, when the thermocline was close to the inlet of the cold water entering the tank (see Figure 6).

90%

92%

94%

96%

98%

100%

0 0.2 0.4 0.6 0.8 1Dimmensionless time

Mixing ocurred here, ...

...but it shows here!

90%

92%

94%

96%

98%

100%

0 0.2 0.4 0.6 0.8 1Dimmensionless time

Mixing ocurred here, ...Mixing ocurred here, ...

...but it shows here!...but it shows here!

Figure 6: Graph showing discharging efficiency over dimensionless time.

Note: Although mixing occurred most likely in the beginning of the test, it shows only at the end of the test when the mixed region / thermocline is leaving the tank

The MIX number proposed by Davidson (1994) is not only based on energy, but also on the tank height at which the energy is stored at. By calculating a “momentum of energy”, information regarding destratification can be obtained during standby and without having to discharge the store. An additional advantage of the MIX number is that destratification or mixing can be shown at the precise time it occurs, given that the time constants of the measurement equipment are small enough.

For heat stores that are used in systems where the heat is used to produce work, it is quite natural to use an exergy approach instead of an energy approach, since the upper limit of work that can be produced is given by the exergy, and not the energy content of a store. In order to use the exergy approach it is necessary to define a reference “dead point” temperature, which is usually the heat sink temperature of the process used to produce work. Since exergy always gives more esteem to energy stored at higher temperatures than it does to the same amount of energy stored at a lower temperature2, exergy approaches may also be used to characterize stratification. Furthermore, heat losses from a store, as well as temperature differences across a heat exchanger, result in exergy losses. However, in a system that is not used to produce work, the definition of the reference temperature will be more of an arbitrary choice by the user than in a store used to produce work. Problems may arise with the exergy approach if store temperatures drop below the reference temperature. In this case, a pure exergy analysis would assume that it is now possible to use the store as a heat sink and the reference temperature as a heat source. As a result, exergy would be positive, not reflecting that the store is less useful for a heating system than a store at dead-point temperature with no exergy at all.

A combined energy and exergy approach has been proposed by Panthalookaran et al. (2007). This method combines the advantages of both approaches. Since it includes the same energy approach described above for the discharging efficiency, this method will also show efficiencies at the end of a

2 at least as long as this temperature is above the reference „dead point“ temperature.


13/16

complete discharge. The entropy or exergy approach included in this method however will show the mixing at the time it occurs.

Other parameters proposed to describe thermal stratification are the thermocline gradient, thermocline gradient decay, thermocline gradient thickness or undisturbed percentage of tank volume. These methods can be applied to describe the results of a specific charging or discharging procedure, but it is difficult to apply them after a complex charging or discharging history that may result in more than one thermocline in the tank. Furthermore, these methods do not give any information about the energy level as such or about the temperature distribution outside the center of the thermocline. These methods may be useful in certain cases, but they are limited in application.

5 Conclusions Many methods to characterize thermal stratification in water stores have been proposed and used in literature. For thermal energy engineering, methods based on temperature distributions or thermal energy and exergy in the tank are certainly more useful than methods based on density distributions that are often used to describe stratification in lakes by environmental scientists. Both dimensional and non dimensional graphs are a good means of illustrating stratification within a tank at a certain time or for several times during a charging process. In order to compare different tank designs or charging / discharging procedures with each other, it is useful to have a single numerical value in terms of a stratification or mixing efficiency.

Methods to calculate stratification parameters or indices may be separated into those based on measurements at the tank inlet and outlet only (outside tank measurements), and those that need measurements to be performed within the tank. Outside tank measurements (e.g. used to calculate charging and discharging efficiencies) do not show the decay of a thermocline at the time it occurs, but only at the time the thermocline is being discharged. Although this is a considerable limitation of these methods, they are valuable in cases where inside tank measurements are not possible because of technical and/or financial limitations (e.g. testing of a large number of products sold on the market). For scientific purpose, however, it is always advantageous to measure inside tank temperatures. Methods to characterize stratification based on inside tank measurements are the MIX number, the stratification parameter of Wu and Bannerot, or methods that include exergy analysis of different tank levels including the recently proposed method of Panthalookaran. Figures based on these methods are able to show destratification at the precise time it occurs. As a result, the processes that cause mixing or destratification can be identified more easily.


14/16

6 Literature studied Abdoly M.A., Rapp D. (1982) Theoretical and experimental studies of stratified thermocline storage of hot water; Energy Conversion and Management 22 (3), pp. 275-285

Alizadeh S. (1999) An experimental and numerical study of thermal stratification in a horizontal cylindrical solar storage tank; Solar Energy 66 (6), pp. 409-421

Al-Najem N.M., Al-Marafie A., Ezuddin K.Y. (1993) Analytical and experimental investigation of thermal stratification in storage tanks, International Journal of Energy Research 17, pp. 77-88

Andersen E., Furbo S., Fan J. (2007) Multilayer fabric stratification pipes for solar tanks, Solar Energy, In Press, Corrected Proof, Available online 21 February 2007

Bahnfleth W.P., Song J. (2005) Constant flow rate charging characteristics of a full-scale stratified chilled water storage tank with double-ring slotted pipe diffusers; Applied Thermal Engineering 25, pp. 3067–3082

Bejan A. (1978) Two thermodynamic optima in the design and operation of thermal energy storage systems. Journal of Heat Transfer 100, pp. 708–712

Bouhdjar A., Harhad A. (2002) Numerical analysis of transient mixed convection flow in storage tank: influence of fluid properties and aspect ratios on stratification. Renewable Energy 25, pp. 555–567

Cabeza L.F., Castell A., Medrano M. (2006) Project Report: Dimensionless parameters used to characterize water tank stratification – A technical report of Subtask D, A report of IEA-SHC Task 32 Advanced storage concepts for solar and low energy buildings, Version 1: November 2006

Chan A.M.C., Smereka P.S. and Giusti D. (1983) A Numerical Study of transient mixed convection flows in a thermal storage tank, ASME Journal of Solar Energy Engineering 105, pp. 246-253

Cole R.L., Bellinger F.O. (1982) Thermally stratified tanks, ASHRAE Transactions 88, Part 2(1), pp. 1005-1017

Cristofari C., Notton G., Poggi P., Louche A. (2003) Influence of the flow rate and the tank stratification degree on the performances of a solar flat-plate collector. International Journal of Thermal Sciences 42 (5), pp. 455-469

Dave, R.N. (1991) Effect of the Circulation Number on the Stratification Coefficient for Solar Heating-Systems; Journal of Solar Energy Engineering-Transactions of the ASME 113, pp. 250-256

Davidson, J.H., Adams, D.A., Miller, J.A. (1994) A coefficient to characterize mixing in solar water storage tanks, Journal of Solar Energy Engineering 116, pp. 94-99

Dincer I., Rosen, M. (2002) Thermal Energy Storage, Wiley

Ghaddar N.K., Al-Marafie A.M., Al-Kandari A. (1989) Numerical simulation of stratification behaviour in thermal storage tanks; Applied Energy 32 (3), pp. 225-239

Ghajar A.J., Zurigat Y.H. (1991). Numerical study of the effect of inlet geometry on stratification in thermal energy storage, Numerical Heat Transfer 19, pp. 65-83

Hahne E., Chen Y. (1998). Numerical study of flow and heat transfer characteristics in hotwater stores; Solar Energy 64, pp. 9-18

Hoerger C.R.B., Phillips,W.F. (1987) The Stratification Coefficient Approach for Predicting the Performance of Solar Air Heating-Systems Utilizing Packed-Bed Heat-Storage; Journal of Solar Energy Engineering-Transactions of the ASME 109, pp. 179-184

Hollands K. G. T., Lightstone M. F. (1989) A review of low-flow, stratified-tank solar water heating systems; Solar Energy 43 (2), pp. 97-105

Homan K.O. (2003) Internal entropy generation limits for direct sensible thermal storage. Journal of Energy Resources Technology, Transactions of ASME 125, pp. 85–93

Huhn R. (2007) Beitrag zur thermodynamischen Analyse und Bewertung von Wasserwärmespeichern in Energieumwandlungsketten, Technische Universität Dresden, Germany, Doctoral Thesis (German)

Jaluria Y., Gupta S.K. (1982) Decay of thermal stratification in a water body for solar energy storage; Solar Energy 28 (2), pp. 137-143

Jaluria Y., O'Mara B. T. (1989) Thermal field in a water body for solar energy storage and extraction due to a buoyant two-dimensional surface water jet; Solar Energy 43 (3), pp. 129-138;

Johannes K., Fraisse G., Achard G., Rusaouën G. (2005) Comparison of solar water tank storage modelling solutions; Solar Energy 79 (2), pp 216-218

Jordan U., Furbo S. (2005) Thermal stratification in small solar domestic storage tanks caused by draw-offs; Solar Energy 78 (2), pp. 291-300


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Jordan U., Vajen K. (2001) Influence Of The DHW Load Profile On The Fractional Energy Savings: A Case Study Of A Solar Combi-System With TRNSYS Simulations; Solar Energy 69, Supplement 6, pp. 197-208

Kandari, A.M. (1990) Thermal stratification in hot storage-tanks; Applied Energy 35 (4), pp. 299-315

Kenjo L., Inard C., Caccavelli D. (2007) Experimental and numerical study of thermal stratification in a mantle tank of a solar domestic hot water system; Applied Thermal Engineering, 27 (11-12), pp. 1986-1995

Kleinbach E.M., Beckman W.A., Klein S.A. (1993) Performance study of onedimensional models for stratified thermal storage tanks, Solar Energy 50, pp. 155-166

Knudsen S., Furbo S. (2004) Thermal stratification in vertical mantle heat-exchangers with application to solar domestic hot-water systems; Applied Energy 78 (3), pp. 257-272

Krane R.J. (1987) A second law analysis of the optimum design and operation of thermal energy storage systems; International Journal of Heat Mass Transfer 30, pp. 43–57

Krane R.J., Krane M.J. (1992) The optimum design of stratified thermal energy storage systems––part II: completion of the analytical model, presentation and interpretation of the results. Journal of Energy Resources and Technology 114/205, pp. 204–208

Lavan Z., Thompson J. (1977) Experimental study of thermally stratified hot water storage tanks; Solar Energy, 19(5), pp. 519-524

Liu W., Davidson J.H., Kulacki F.A. (2004) Thermal characterization of prototypical IKCS systems with immersed heat exchangers, Proceedings of ISEC 2004, July 11-14 2004, Portland, Oregon

Lund P.D. (1988) Effect of storage thermal behavior in seasonal storage solar heating systems; Solar Energy 40 (3), pp. 249-258

Madhlopa A., Mgawi R., Taulo J. (2006) Experimental study of temperature stratification in an integrated collector–storage solar water heater with two horizontal tanks; Solar Energy 80 (8), pp. 989-1002

Mather D.W., Hollands K.G.T., Wright J.L. (2002) Single- and multi-tank energy storage for solar heating systems: fundamentals; Solar Energy 73 (1), pp. 3-13

Mavros P., Belessiotis V., Haralambopoulos D. (1994) Stratified energy storage vessels: Characterization of performance and modeling of mixing behavior; Solar Energy 52 (4), pp. 327-336

Mawire A., McPherson M. (2007) Experimental characterisation of a thermal energy storage system using temperature and power controlled charging; Renewable Energy, In Press, Corrected Proof, Available online 12 June 2007

Mehling H., Cabeza L.F., Hippeli S., Hiebler S. (2003) PCM-module to improve hot water heat stores with stratification; Renewable Energy 28 (5), pp. 699-711

Mishra R.S. (1991) Thermal stratification in thermosyphonic solar water heating systems; Energy Conversion and Management 31 (5), pp. 425-430

Misra R.S. (1993) Evaluation of Thermal Stratification in Thermosiphonic Solar Water Heating-Systems; Energy Conversion and Management 34 (5), pp. 347-361

Misra R.S. (1994) Thermal Stratification with Thermosiphon Effects in Solar Water-Heating Systems; Energy Conversion and Management 34 (3), pp. 193-203

Moran M.J., Keyhani V. (1982) Second law analysis of thermal energy storage systems. In: Proceedings of the Seventh International Heat Transfer Conference, Munich, vol. 6, pp. 473–478

Moretti P.M., McLaughlin D.K. (1977) Hydraulic modeling of mixing in stratified lakes; Journal of the Hydraulics Division 103, pp. 367–380

Morrison G.L., Nasr A., Behnia M., Rosengarten G. (1998) Analysis of horizontal mantle heat exchangers in solar water heating systems; Solar Energy 64 (1-3), pp. 19-31

N. Tran, Kreider J.F., Brothers P. (1989) Field measurements of chilled water storage thermal performance, ASHRAE Transactions 95 (1), pp. 1106–1112

Nelson J.E.B., Balakrishnan A.R., Murthy S.S. (1999) Experiments on stratified chilled water tanks, International Journal of Refrigeration 22, pp. 216-234

Panthalookaran V., Heidemann W., Müller-Steinhagen H. (2007) A new method of characterization for stratified thermal energy stores; Solar Energy 81 (8), pp. 1043-1054

Phillips W.F. (1981) Effects of Stratification on the Performance of Solar Air Heating-Systems; Solar Energy 26, pp. 287-295

Phillips W.F., Dave R.N. (1982) Effects of stratification on the performance of liquid-based solar heating systems; Solar Energy 29 (2), pp. 111-120


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Rosen M.A. (2001.) The exergy of stratified thermal energy storages; Solar Energy 71, pp. 173–185

Rosengarten G. (1999) A second law approach to characterising thermally stratified hot water storage with application to solar water heaters; Journal of Solar Energy Engineering-Transactions of the ASME 121, pp. 194-200

Shah L.J., Andersen E., Furbo S. (2005) Theoretical and experimental investigations of inlet stratifiers for solar storage tanks, Applied Thermal Engineering 25 (14-15), pp. 2086-2099

Shah L.J., Furbo S. (2003) Entrance effects in solar storage tanks; Solar Energy 75 (4), pp. 337-348

Shin M.-S., Kim H.S., Jang D.-S., Lee S.-N., Lee Y.-S., Yoon H.-G. (2004) Numerical and experimental study on the design of a stratified thermal storage system, Applied Thermal Engineering 24, pp. 17–27

Shyu R. J., Lin J.Y. and Fang L. J. (1989) Thermal analysis of stratified storage tanks; ASME Journal of Solar Energy Engineering 111, pp. 54–61

Shyu R.J., Hsieh C.K. (1987) Unsteady natural convection in enclosure with stratified medium; Journal of Solar Energy Engineering 109, pp. 127–133

Sliwinski B.J., Mech A.R., Shih T.S. (1978) Stratification in thermal storage during charging; Proceedings of the 6th International Heat Transfer Conference, Toronto, vol. 4, pp. 149-154

Stefan H.G., Gu, R. (1992) Efficiency of jet-mixing of temperature stratified water; Journal of Environmental Engineering, ASCE 118, pp. 363–379

Van Berkel J., Rindt C.C.M., van Steenhoven A. (1999) Modelling of two-layer stratified stores, Solar Energy 67 (1-3), pp. 65-78

Wildin M.W. (1990) Diffuser design for naturally stratified thermal storage; ASHRAE Transactions 96 (1), pp. 1094–1102

Wu L., Bannerot R. B. (1987) Experimental Study of the Effect of Water Extraction on Thermal Stratification in Storage; Proceedings of the 1987 ASME-JSME-JSES Solar Energy Conference, Honolulu, Vol. 1, pp. 445-451

Yee C.K., Lai F.C. (2001) Effects of a porous manifold on thermal stratification in a liquid storage tank; Solar Energy 71(4), pp. 241-254

Yoo H., Pak E. (1993) Theoretical model of the charging process for stratified thermal storage tank; Solar Energy 51 (6), pp. 513–519

Zachár A., Farkas I., Szlivka F. (2003) Numerical analyses of the impact of plates for thermal stratification inside a storage tank with upper and lower inlet flows; Solar Energy 74 (4), pp. 287-302

Zurigat Y.H., Ghajar A.J. (2002) Chapter 6: Heat Transfer and Stratification in Sensible Heat Storage, in Dincer I. and Rosen, M., Thermal energy Storage, Wiley

Zurigat Y.H., Maloney K.J., Ghajar A.J. (1989) A comparison study of one-Dimensional models for stratified thermal storage tanks. ASME Journal of Solar Energy Engineering 111, pp. 205–210

Cruickshank and Haller 2007 – Methods to characterize thermal stratification Appendix A

Appendix A Other Literature (not studied) Adams D.E. (1993) Design of a Flexible Stratification Manifold for Solar Water Heating Systems. Master Thesis, Colorado State University, Fort Collins, CO

Bahnfleth W., Musser A. (1998) Thermal performance of a full-scale stratified chilled water thermal storage tank, ASHRAE Transactions 104 (2), pp. 377–388

Banfleth W., Musser A. (1998) Thermal performance of a full-scale stratified chilled water thermal storage tank, ASHRAE Transactions 104 (2), pp. 377–388.

Bejan A. (1982) Entropy Generation through Heat and Fluid Flow. John Wiley & Sons, New York

Bejan A. (1996) Entropy Generation Minimization. CRS Press, New York.

Bouhdjar A., Bekhelifa A., Harhad, A. (1997) Numerical study of transient mixed convection in a cylinder cavity, Numerical Heat Transfer, Part A, Vol. 31, pp. 305-324

Brumleve T.D. (1974) Sensible Heat Storage in Liquids, Sandia Labs Report, SLL-73-0263

Davidson J.H., Adams D.A. (1994). Fabric stratification manifolds for solar water heating, Journal of Solar Energy Engineering 130, pp. 130-136

Furbo S. (2005) Class notes for PhD Course Thermal Stratification in Solar Storage Tanks, DTU

Ismail K.A.R., Leal J.F.B., Zanardi M.A. (1997) Models of liquid storage tanks, International Journal of Energy Research 22, pp. 805-815

Jordan U., Furbo S. (2003) Thermal stratification in small solar domestic storage tanks caused by draw-offs. Proceedings of ISES 2003 Solar World Congress, Göteborg, Sweden

Koldhekar S. M. (1981) Temperature stratification in hot water solar thermal storage tanks, American Institute of Aeronautics and Astronautics, Aerospace Sciences Meeting, 19th, St. Louis, Mo., Jan 12-15 1981, 10 p.

Lin E.I., Sha W.T., Michaels A.I. (1979) On thermal energy storage efficiency and the use of COMMIX-SA for its evaluation and enhancement, Proceedings of Solar Energy Storage Options, Vo. I, San Antonio, Texas, 1979, pp. 501-513

McCarthy (1990) Effects of Various Load Profiles on Solar Storage Tank Stratification Parameters, University of Arizona Master’s Thesis

Moran M.J. (1989) Availability Analysis: A Guide to Efficient Energy Use. ASME, New York

Musser A., Bahnfleth W. (1998) Evolution of temperature distributions in a full-scale stratified chilled water storage tank, ASHRAE Transactions 104 (1), pp. 55–67

Oppel F. J., Ghajar, A. J., Moretti P. M. (1986) Computer simulation of stratified heat storage; Applied Energy 23, 205

Ramsayer R.M. (2001) Numerische Untersuchung der Strömungs – und Wärmetransportvorgänge bei der thermischen Beladung eines Warmwasserspeichers, Student report, Institut für Thermodynamik und Wärmetechnik, Universität Stuttgart, Germany

Tran N., Kreider J.F., Brothers P. (1988) Final Report field measurement of chilled water storage system performance. Joint center for energy management, University of Colorado, Boulder

Phillips, W. F. (1981) Effects of stratification on the performance of solar air heating system. Solar Energy 26, 175

Wildin M.W., Truman C.R. (1985) Evaluation of stratified chilled-water storage techniques, EPRI EM 4325, Vols. 1 and 2, RP 2036-5

Zurigat Y.H., Liche P.R., Ghajar A.J. (1990) Influence of the inlet geometry on mixing in thermocline thermal energy storage, International Journal of Heat and Mass Transfer 34 (1), pp. 115-125

Cruickshank and Haller 2007 – Methods to characterize thermal stratification Appendix B

Appendix B Table of reports introducing and/or using stratification indices Table 1: Authors that cite (C) / use (U)/ introduce (I) parameters for stratification (left column) of other authors that defined them (upper header)

This Author is citing (C), using (U) or introduced (I)

cited authers / methods La

van

1977

Sl

iwin

ski 1

978

Phill

ips

1982

Ab

doly

198

2 W

u an

d B

ann.

198

7 Sh

yu 1

987

Kran

e 19

87

Tran

198

7/89

H

oerg

er 1

987

Kand

ari 1

990

Wild

in 1

990

Dav

e 19

91

Yoo

1993

M

avro

s 19

94

Dav

idso

n 19

94

Van

Berk

el 1

997/

99

Hah

ne 1

998

Ros

enga

rten

1999

N

elso

n 19

99

Ros

en 2

001

Bouh

djar

200

2 Za

char

200

3 Sh

ah 2

003

Hom

an 2

003

Liu

2004

Sh

ah 2

005

Cab

eza

2006

An

ders

en 2

007

Pant

halo

okar

an 2

007

Lavan Extraction efficiency (1977)

I C C

Sliwinski temperature gradient of thermocline (1978)

I C

Koldhekar stratification number (1981)

C C

Philips stratification coefficient (1982)

I U U C C C

Moran and Keyhani (1982, 1989)

C

Abdoly Extraction efficiency (1982)

I U U1

Chan storage efficiency (1983) C U

Wildin efficiency (1985) C C

Wu-Bannerot stratification factor (1987)

I C U C C

Shyu decay rate of stratification (1987/1989)

I C

Krane entropy generation number and second law eff. (1987)

I C C

Tran figure of merit (1987/1989)

I U C C

Kandari (1990) extraction efficiency

I C

Wildin figure of merit (1990) U

Yoo charging process efficiency (1993)

I U C U* C* U C* C* U*

Davidson-Adams MIX (1994) I C C C U U C

Bouhdjar (1997) delete? (if replaced by Chan...)

C

Van Berkel cycle thermal efficiency (1997/1999)

I C

Rosengarten stratification efficiency (1999)

I C C

Nelson Mixing coefficient (1999)

I

Rosen Exergy (2001) I C C

Shah entropy and exergy efficiency (2003)

I

Homan Entropy gen. Number (2003)

I U

Panthalookaran storage eval. num. (2007)

I

1) Zachar et al. cite Van Berkel for a fraction of recoverable heat η90 that corresponds to the extraction efficiency introduced by Abdoly and Rapp, just with a limit of 10% degradation allowed instead of 20% as originally proposed by Abdoly and Rapp

* Yoo and Pak charging process efficiency is synonyme to Hahne charging/discharging coefficient. This is why authors citing Hahne have been attributed to cite (indirectly) Yoo and Pak.

Cruickshank and Haller 2007 – Methods to characterize thermal stratification Appendix C

Appendix C Detailed description of “other methods” to characterize stratification Phillips stratification coefficient (1981)

Already in 1981, a stratification coefficient for solar air heating systems with pebble bed stores was introduced (Phillips 1981). The method was then further developed and applied to liquid based heating systems and water storage tanks (Phillips and Dave 1982). This Phillips stratification coefficient is fundamentally different from other approaches, since it is actually not a figure that gives information about the goodness of stratification at a certain time, (e.g., after a certain charging / discharging procedure), but rather a figure telling us how much the solar gain is increased due to stratification in the store compared to a solar heating system with a fully mixed tank.

Based on his studies of the effect of stratification on solar air heating systems (W.F. Phillips, Solar Energy 26, 175 (1981)), Phillips presents a theoretical model which predicts the effects of solar tank stratification on the instantaneous performance of a liquid-based solar heating system. He uses a stratification coefficient KS, which is defined as the ratio of the actual useful energy gain to the energy gain that would be achieved if there were no thermal stratification in the storage tank. Thus,

SK is not only dependent on the heat store, but also on the whole solar heating “system” (collector field, hydraulics etc.):

( )( )

R e s R l iS

R e s R l s

F q F U T TK

F q F U T Tαα

∞

∞

− −=

− −

where:

RF collector heat removal factor, dimensionless

eα effective transmittance absorption product, dimensionless

sq solar flux on collector aperture, Wm-2

lU overall heat transfer coefficient between the receiver and the ambient, Wm-2K-1

iT temperature of the liquid leaving storage, K

sT mean storage temperature, K

T∞ Tinfinite ambient temperature, K

For his prediction of SK , Phillips needs a dimensionless parameter called the mixing number M, which is the ratio of conduction to convection in the storage tank:

S

S

A kMm Cp H

⋅≡

⋅ ⋅

where:

SA cross sectional area of the storage tank, m2

k thermal conductivity of the storage fluid, W/(mK)

m mass flow rate of the fluid through the tank, kg/s

Cp specific heat of the fluid, J/(kgK)

SH vertical height of storage tank, m

A closer look at M reveals that it is nothing else than the inverse of the Peclet number:


1MPe

= , since H V CpPe

kρ⋅ ⋅ ⋅

= and Sm A V ρ= ⋅

where:

ρ density of the fluid, kg/m3

V velocity of the fluid, m/s

Lavan extraction efficiency

One of the earliest approaches of defining a storage evaluation parameter that is influenced by charging / discharging stratification behaviour was the one by Lavan (1977). He defined an extraction efficiency not in terms of energy effeciency, but in terms of the volume of the store that can be withdrawn before the temperature level has degraded to a certain amount:

stopQtV

η ≡

where:

stopt time at which the initial inlet-exit temperature difference has dropped to some pre-assigned value (Lavan chose 10% of original value)

Q volume flow rate

V volume of the store

Thermocline gradient, gradient decay and thickness

Very early approaches also used the thermocline gradient dT/dz in the region of its maximum to give a measure of storage stratification. Sliwinski (1978) evaluated the maximum thermocline gradient as dT/dz between two points of storage height z that have a gradient of 10% less than the maximum. The measured gradient was then non-dimensionalized by dividing by:(Tin-Tini)/L, where L is the vertical distance between inlet and outlet.

Later on, Shyu et al. (1987/1989) defined a decay rate of stratification by the ratio tS of the centerline temperature gradient at time t in the initially stratified region to the temperature gradient at the same place at the initial time:

0

C

tt

C

dTdzS

dTdz

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

where ( )0CdT dz is the mean of the centerline temperature gradient in the initially stratified region,

and ( )C tdT dz is the mean of the temperature gradient in the same region at time t.

Bahnfleth (2005) mentiones that Musser and Bahnfleth (1998) defined the thermocline to be the region between endpoints defined by limiting values of the dimensionless temperature of a charge cycle. The thermocline thickness is defined by discarding equal fractions of the total difference on each end of the temperature distribution. So, if 0.15 is discarded, the thermocline encompasses the region from H = 0.15 to 0.85 and contains 70% of the overall temperature change.


Koldhekar (1981) according to Davidson 1994

According to Davidson (1994), Satish M. Koldhekar proposed a definition for the characterization of stratification as follows:

( )( )

avgactual actual

ideal avg ideal

TTT T

ξ Δ= ⋅Δ

which is the product of the actual temperature difference between water at the top and bottom of the storage tank to an “ideal” temperature difference and the ratio of the actual average temperature of the tank to the average temperature calculated assuming a linear vertical temperature profile. The “ideal” temperature difference is defined as the temperature of boiling water minus that of the cold supply water. Davidson criticized that this approach ignores the possibility that a tank of uniform hot temperature could result from an ideal tank in which no water displaces colder water without mixing.

Wu and Bannerot stratification coefficient (1987)

Wu and Bannerot define a stratification coefficient as follows:

( )2

i i avg

itotal

m T TST

m−

=∑

where:

im mass of section / node i of store, kg

iT temperature of section / node i of store, K

avgT mass weighted average temperature of store, K

totalm total mass of store, kg

In addition to this, an instantaneous energy extraction efficiency is defined:

( )( )

out makeup

avg makeup

T TT T

β−

=−

outT temperature of hot water leaving the tank (draw off), K

makeupT temperature of the makeup (tap) water, K

Liu et al. (2004) analyzed stratification in an integral collector storage solar system with an immersed heat exchanger. She based his stratification factor on the stratification factor ST by Wu and Bannerot, which she slightly modified to account for the height of tube bundles within his storage collector.

Ghaddar tank thermal turbulent efficiency (1989)

Ghaddar introduces a “tank’s thermal turbulent efficiency” Tη :

( )( )1 100T D Lx x Lη = − − ⋅

Where

Lx laminar model disturbed zone length

Dx disturbed zone length with eddy conductivity factor


Tη = 1, if no mixing because of the inlet jet occurs (only thermal diffusion causes mixing in this case). Unfortunately, Ghaddar does not tell us how the disturbed zone length shall be determined from field measurements.

Percentage ratio of non disturbed volume

Kandari (1990) defines an extraction efficiency Dη which is the percentage ratio of the non-disturbed volume to the total tank volume

( )1 100D D TV Vη = − ⋅

where TV and DV are the total and the disturbed volume of the tank respectively. DV is defined as the

tank volume covered within the temperature range of 1 0.05t t+ ⋅ and 2 0.05t t− ⋅ , 1t and 2t being the

lower and upper temperatures of the tank, and 2 1t t t= −

Cruickshank and Haller 2007 – Methods to characterize thermal stratification Appendix D

Appendix D Detailed calculation of examples The Panthalookaran method uses two storage evaluation numbers which are based on the first law (energy response number ER) and second law (entropy generation number REG) of thermodynamics to characterize efficiency during storage and during charging/discharging.

To calculate the Entropy generation ratio REG, it is first necessary to determine the change in entropy change as the storage goes from its original state at t1 to the “real” state at t2:

Panthalookaran storage evaluation numbers

33

2real

1 1 1

real

real

ln

4*0.25 *1000 *4.2 *

45 273 45 273 40 273 30 273* ln ln ln ln40 273 30 273 20 273 10 273

833.49 /

p

n n

i i ii i

TS m s m c

T

kg kJS m kgKm

S kJ K

= =

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

Δ = Δ =

Δ =

+ + + ++ + ++ + + +

Δ =

∑ ∑

This value is then compared to the change in entropy that would have occurred if the tank had perfectly stratified at time 2t or perfectly mixed at 2t , (see Figure 5). The calculations for each are shown below and correspond to the temperature distribution determined for the ideal cases at an equivalent energy level as the real storage at 2t .

Ideally Stratified Storage

3 3

3 33

3

2stratified

1 1 1

stratified

ln

3 45 273 3 45 273*ln *ln12 40 273 12 30 273

2 45 273 1 30 2734*1000 *4.2 * *ln *ln12 20 273 12 20 2733 30 273*ln

12 10 2

p

n n

i i ii i

TS m s m c

T

m m

kg kJS m mkgKm

m

= =

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Δ = Δ =

+ ++ ++ +

+ +Δ = + + ++ +

+++

∑ ∑

stratified

73

832.61 /S kJ K

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

Δ =


Ideally Mixed Storage

33

2mixed

1

mixed

mixed

ln

1 *1000 *4.2 *

40 273 40 273 40 273 40 273* ln ln ln ln40 273 30 273 20 273 10 273

836.89 /

ptotal total totalT

S s cT

kg kJS m kgKm

S kJ K

m m

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

Δ = Δ =

Δ =

+ + + ++ + ++ + + +

Δ =

As anticipated, mixed real stratified

836.89 / 833.49 / 832.61 /

S S S

kJ K kJ K kJ K

Δ ≥ Δ ≥ Δ

≥ ≥.

These values are then combined to arrive the entropy generation ratio, EGR , i.e.,

real stratifiedEG

mixed stratified

( ) (833.49 / 832.61 / ) 0.206( ) (836.89 / 832.61 / )

S S kJ K kJ KRS S kJ K kJ KΔ −Δ −= = =Δ −Δ −

This number is consistent with the expected value for a charging sequence where entropy generation is relatively low and stratification is maintained but at a level below the ideal case.

The second step of the analysis requires that the “energy response”, RE , be calculated from

realR

ideal

EE E

Δ= Δ

where realEΔ represents the change in energy of a system within a specified time interval (i.e.,

2 1t t− ) for the “real” storage, as calculated from the change in temperature distribution of the storage over the time interval i.e.,

( )4

2 11

real

real

( ) ( )

250 *4.2 *(45 40) 250 *4.2 *(45 30)

250 *4.2 *(40 20) 250 *4.2 *(30 10)

63000

i p i ii

o oo o

o oo o

T t T tE

E

E

m c

kJ kJkg C kg Ckg C kg CkJ kJkg C kg Ckg C kg C

kJ

=

⋅ ⋅ −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Δ

Δ

Δ

=

− + − +=+ − + −

=

∑

As previously described, the term idealEΔ is calculated based on an ideal situation in which the maximum energy response is achieved. This is defined for the charging process as a piston flow adiabatically separated from the other parts of the system. Assuming the storage was charged with 500 L of water at 50oC during the time interval (i.e., 2 1t t− ), then the ideal temperature distribution in the storage would be as shown in Figure 5.


And the hypothetical change in energy would be

( )4

2 11

ideal

ideal

( ) ( )

250 *4.2 *(50 40) 250 *4.2 *(50 30)

250 *4.2 *(40 20) 250 *4.2 *(30 10)

73500

i p i ii

o oo o

o oo o

T t T tE

E

E

m c

kJ kJkg C kg Ckg C kg CkJ kJkg C kg Ckg C kg C

kJ

=

⋅ ⋅ −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Δ

Δ

Δ

=

− + − +=+ − + −

=

∑

Based on these values the “energy response”, RE , be calculated

realR

ideal

63000 0.857173500

E kJEE kJ

=Δ

= =Δ

This value is indicates that the achieved energy response corresponds to 86% of the energy response that an ideal storage would have achieved.

Finally, using the derived values of the EGR and RE terms, the so-called storage evaluation number (SEN) or efficiencies as can be determined for the charge sequence, i.e.,

EGSEN1

R

SEN1

SEN1

1 100%

0.2061 100%0.857

75.96%

REη

η

η

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

= − ⋅

= − ⋅

=

These results show that a high (SEN) value was achieved for this charge sequence.

In a similar fashion, SEN1η can be calculated for a discharge sequence and SEN2η for a storage process (stand-by).

Example of the MIX Number calculation (Davidson et al., 1994) for the same case

Considering the same example case as described in Section 3.1, it is possible to calculate the MIX Number according to Davidson et al. The same initial and final tank temperatures are assumed as given in Figure 5. To define the MIX number it is also necessary to define the height of the tank considered. For simplicity the tank is given a volume of 1 m3 (i.e.,1000 L), and a height of 2 m. It is further assumed that 1 L of water has a mass of 1 kg and that the specific heat of water is constant at 4.2 kJ/kgK. As described in Section 2.2.3., the Mix number is calculated as

expstr

str mix

M MMIX

M M−

=−

where M is the “the momentum of energy” given by


1

N

i ii

M y E=

= ⋅∑ , where yi is the distance from the bottom of the storage tank to a temperature node

located at center of a defined uniform temperature layer.

As given in the previous example, the storage will be divided into 4 equal volume (and mass) layers, each at a uniform temperature. The temperature of the layers at the start and end of the charge interval will be assumed to be those shown in Figure 5.

In performing the calculation, it is necessary to initially calculate Mexp (or Mactual as Davidson et al. refer to in their paper). If the tank is divided into 4 equal segments and has a height of 2 m then y1 to y4 are 0.25, 0.75, 1.25, and 1.75m, respectively.

Therefore the value of Mexp at the final condition (Time = t2) is given as

4

1

exp

exp

,

0.25 *250 *4.2 *(45) 0.75 *250 *4.2 *(45)

1.25 *250 *4.2 *(40) 1.75 *250 *4.2 *(30)

181125

i ii

i i p i

o oo o

o oo o

y

where E m C T and therefore

M

M

M E

kJ kJm kg C m kg Ckg C kg CkJ kJm kg C m kg Ckg C kg C

kJ m

=

⋅

= ⋅ ⋅

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

=

+ +=+ +

= ⋅

∑

As a basis of comparison, the values of M are calculated for an ideally stratified and fully mixed storage, (i.e., Mstr and Mmix) assuming they both were at the temperature profile shown at the start of the charge sequence (i.e., at Time = t1 ).

In the case of the ideally stratified storage at time t2, and assuming the storage is charged with 500 L of water at 50oC that does not mix during the charge interval, the tank temperature profile for each of the 4 layers will be 50oC , 50oC , 40oC, 30oC from the top to the bottom. Performing a similar calculation as above but with these assumed temperatures, the value of Mstr is

0.25 *250 *4.2 *(30) 0.75 *250 *4.2 *(40)

1.25 *250 *4.2 *(50) 1.75 *250 *4.2 *(50)

196875

oo o

stro o

o o

str

M

M


kJ m

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

° + +=+ +

= ⋅

To determine the value of Mmix at t2, it is necessary to determine the equivalent mixed temperature at the end of the charge sequence from assuming that the charge energy was added to an equivalent fully mixed tank at the start of the charge sequence, (i.e., at time = t1 ). For the initial conditions given for time t1, the tank temperature distribution was 40 oC, 30 oC, 20 oC, 10oC for each of the equal volume layers. Therefore the equivalent initial mixed temperature would be 25oC. The energy added to the tank is given by the addition of 500 L of water at 50 oC and the removal or 500 L at 25 oC. The mixed temperature at t2 is then given by

( ) ( )( )2 1 34.8storetQ Vin inT t T T T t e C−= − − = °


Thus:

34.8 34.8

34.8 34.8

0.25 *250 *4.2 *( ) 0.75 *250 *4.2 *( )

1.25 *250 *4.2 *( ) 1.75 *250 *4.2 *( )

146160

o oo o

mixo o

o o

mix

M

M


kJ m

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

+ +=+ +

= ⋅

To get the dimensionless MIX-number, the difference of the momentum of energy between a perfectly stratified store and the actual store under investigation is divided by the difference of momentum of energy between a stratified store and a fully mixed store:

exp 196,875 181,125 0.31196,875 146,160

str

str mix

M MMIX

M M− −

= = =− −

This number will be 0 if the store is fully stratified and 1 if the store is fully mixed. In a typical application of these indices, the MIX number would be calculated at each time step in the charge or discharge process, ensuring that the time-step chosen is sufficiently short to tenure that the assumption of constant temperature over the interval holds.

theoretical methods to characterize thermal stratification

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