pcm storage for solar dhw: an unfulfilled promise?

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Solar Energy 82 (2008) 861–869

PCM storage for solar DHW: An unfulfilled promise?

Ella Talmatsky, Abraham Kribus *,1

School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Received 5 August 2007; received in revised form 13 March 2008; accepted 10 April 2008Available online 8 May 2008

Communicated by: Associate Editor V. Wittwer

Abstract

Phase change materials (PCM) have been repeatedly proposed for use in solar domestic hot water (DHW) systems. PCM storage designshave been proposed, but no detailed evaluation has been made of the actual contribution of the PCM to the total heat storage under typicalend-use conditions. In this work annual simulations were done to compare the performance of a storage tank with PCM to a standard tankwithout PCM. A model was constructed to describe the heat storage tank with and without PCM, the collector, pump, controller and auxi-liary heater. Realistic environmental conditions and typical end-user requirements were imposed. Annual simulations were carried out fordifferent sites, load profiles, different PCM volume fractions, and different kinds of PCM. The results of all simulation scenarios indicatethat, contrary to expectations, the use of PCM in the storage tank does not yield a significant benefit in energy provided to the end-user.The main reason for this undesirable effect is found to be increased heat losses during nighttime due to reheating of the water by the PCM.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Thermal energy storage; Heat storage; Phase change materials; Domestic hot water

1. Introduction

Heat storage is common in energy systems, to compen-sate for a temporal mismatch of supply and demand ofheating or cooling. Heat storage for domestic hot water(DHW) systems is commonly done as sensible heat, by rais-ing the temperature of a stored volume of water. The use oflatent heat stored and released in the solid–liquid transitionof phase change materials (PCM), replacing or augmentingthe sensible heat storage, has been repeatedly proposedduring the last decades (Zalba et al., 2003). Different typesof materials were used, notably of the paraffin and salthydrate families, and recently combined with graphite asa method to increase heat transfer (Cabeza et al., 2002).Different methods to incorporate the PCM into theDHW system were proposed, including integration of thePCM into the solar collector (Rabin et al., 1995; Alva

0038-092X/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2008.04.003

* Corresponding author. Tel.: +972 3 6405924; fax: +972 3 6407334.E-mail address: [email protected] (A. Kribus).

1 Member ISES.

et al., 2006), and addition of PCM elements inside a stan-dard water storage tank (Esen et al., 1998; Mehling et al.,2003; Lee et al., 2006; Nallusamy et al., 2006).

Analysis of the performance of the PCM within theDHW system was usually addressed to several mainaspects: the question of heat transfer between the PCMand the surrounding water during charging and discharg-ing, the static behavior of a storage tank with PCM, toderive how long the storage can maintain a suitable temper-ature at the top layer in readiness for end-user withdrawaland the recovery of heat from the PCM after a complete dis-charge (Mehling et al., 2002; Cabeza et al., 2006). Very littlework was done on evaluating the long-term performance ofa DHW system with PCM, and the overall contributionthat the PCM will make to the energy delivered to theend-user under realistic operation conditions. A singlereport was found (Ibanez et al., 2006) that attempts toanswer this question, using a simulation with TRNSYS.This work indicates that for a single-family DHW systemin Lleida, Spain, the solar fraction has increased by up to8% due to addition of PCM in the storage tank. However,

mailto:[email protected]

Heater

Collector

TC,outIC

PCM

Nomenclature

A area [m2]C specific heat [kJ/kg K]h convection coefficient [W/m2 K]I incident solar flux [W/m2]m mass [kg]_m mass flow rate [kg/s]N number of layers in the tank modelQ thermal power [W]T temperature [�C]t time [h]U overall heat transfer coefficient [W/m2 K]X normalized temperature difference [m2 K/W]x position along tank axis [m]

Greek symbols

g efficiency

D element (of area, mass, etc.) belonging to a layerin the tank model

Sub/superscripts

a ambientC collectori layer indexin inletout outletl liquidM makeup water from gridP PCMs solidW waterU end-user

862 E. Talmatsky, A. Kribus / Solar Energy 82 (2008) 861–869

insufficient detail is given on the setup and boundary condi-tions of the simulation, so this result cannot be reproduced.Also, the result depends on many factors, including site con-ditions, load profile, type and amount of PCM, etc., andtherefore cannot be considered as a universal result.

Some researchers have stated that adding PCM to thestorage tank is aimed to compensate for losses at the topsurface and to reheat the top layer after a period of dis-charge (Mehling et al., 2003). These effects would improvethe availability of hot water to the end-user. However, theauxiliary heater that is present in practically every DHWsystem can also accomplish these functions. The more sig-nificant criterion for evaluating the addition of PCM iswhether the presence of the PCM will lead to an overallannual increase in the solar fraction, i.e., a better use ofthe collected solar energy, and a reduction in the requiredamount of auxiliary energy.

The goal of the work reported here is to evaluate the fea-sibility of using PCM in the storage tank of a DHW sys-tem, based on the criterion of annual performance. If theaddition of PCM to the storage tank leads to a clear andconsistent advantage in energy delivered to the end-user,then the added cost of the PCM may be justified. However,if the energy advantage is not clear, then the use of PCMmay be undesirable. We compare the annual performanceof a DHW system with PCM in the storage tank, to thatof a similar system without PCM, under the same condi-tions. The comparison is repeated for a wide range of con-ditions and assumptions, to ascertain the validity androbustness of the results.

Tank

Pump

TC,in mC.

Fig. 1. Layout of the DHW system.

2. Simulation model

A mathematical model of a complete solar heating sys-tem was developed in MATLAB environment. This systemincludes a storage tank, solar collector, pump, controller

and auxiliary heater (Fig. 1). The model assumes a forcedcirculation system with a pump rather than a thermo-siphon natural circulation system. This is preferred in orderto obtain a well-defined flow rate through the system,rather than relying on a naturally generated flow that is dif-ficult to simulate.

The model of the mass and energy transport in the stor-age tank assumes that the problem is one-dimensional,with temperature variations and flow in the vertical direc-tion only, based on the model used in TRNSYS. The stor-age volume is divided into N fully mixed horizontal layers.Each layer contains water and optionally PCM. The PCMis arranged in small cylindrical containers spaced apart ineach layer, rather than in a single unit, in order to increasethe surface area for convection. The temperature of thePCM in each layer can be different from the water temper-ature in the same layer. The PCM is of a composite typewith high thermal conductivity, and therefore the lumpedsystem assumption (the PCM is represented by a singletemperature) can be used. This assumption was validatedby estimation of the Biot number, which was in the range

Table 1Thermo-physical properties model parameters for two composite PCM:NaOAc�3H2O-graphite (SAT-G) and RT42-graphite (RT42-G)

SAT-G RT42-G

hsf kJ/kg 173 139.7Ts �C 57.31 37.0Tf �C 60.75 47.0Cs kJ/kg K 4.02 1.57Cf kJ/kg K 3.68 2.0k W/m K 5.0 5.0

Table 2Storage tank and solar collector data

E. Talmatsky, A. Kribus / Solar Energy 82 (2008) 861–869 863

of 0.05 to 0.16 for all cases considered. No PCM is presentin the top and bottom layers, to avoid direct heat loss fromthe PCM to the environment. Water inlets and outlets aredefined at the top and bottom layers.

The energy balance equations for the ith layer, for thewater and for the PCM element (if present in the givenlayer), are:

DmWi CW dT W

i

dt¼X

j

QWi;j ð1Þ

DmPi CP dT P

i

dt¼X

j

QPi;j ð2Þ

Collector area 2.67 m2

Collector gC0 0.735Collector gC1 4.6

Maximum flow rate 0.0225 kg/sTank volume 0.15 m3

Tank heat loss coefficient 0.57–1.13 W/m2 K

Heat storage capacity (20–70 �C)

Water only 31.5 MJ6% SAT-G 33.9 MJ6% RT42-G 31.4 MJ25% SAT-G 41.6 MJ25% RT42-G 31.2 MJ

Qi,j represent all possible heat transfer paths j, by con-duction and convection to neighboring layers, loss to theenvironment through the tank walls, exchange betweenthe water and PCM within the same layer and to neighbor-ing layers, and advection from the inlet and outlet streamsfor the relevant layers. The detailed models for each ofthese terms are given in (Talmatsky, 2007), and a summaryis presented in the Appendix.

The variation of the PCM properties during the phasechange process was modeled by a variable specific heat,with values taken from measurements for several types ofmaterials (Mehling, 2005). The specific heat is modeledby a piecewise linear function, as shown in Fig. 2, with fivefree parameters that are found by matching to the experi-mentally available data. The model parameters and otherthermo-physical properties for two types of compositePCM are given in Table 1.

The solar collector model is based on linear variation ofthe efficiency with the normalized temperature difference:

gC ¼ gC0 � gC1 � X : ð3Þ

Solid

Liquid

Temperature

Specific Heat

Cl

Cs

Cmax

TlTs

Fig. 2. Model of variable specific heat of the PCM as a function oftemperature: the gray area includes the phase change enthalpy.

The constants gC0, gC1 were taken from the performancedata of real collectors (Table 2). The normalized tempera-ture difference is defined as:

X ¼12ðT C;in þ T C;outÞ � T a

IC: ð4Þ

The collector energy balance provides another equationfor the collector efficiency:

gCICAC ¼ _mCCðT C;out � T C;inÞ: ð5ÞEqs. (3) and (5) are solved simultaneously to find the

collector water outlet temperature.The pump controller activates the pump when the col-

lector water temperature exceeds 34 �C, or when the differ-ence between inlet and outlet of the collector exceeds 4 �C.These conditions are typical of control schemes used byDHW system manufacturers in Israel.

In many DHW systems, an auxiliary electrical heatingelement is installed inside the storage tank. This may leadto activation of the auxiliary heater in some situationswhen solar heat is available and there is no need for auxil-iary heat. In this work the auxiliary heater is placed on theoutlet pipe outside the storage tank (Fig. 1). The auxiliaryheater is equipped with a thermostat that measures thetemperature of the water delivered to the end-user: whenthis temperature falls below 40 �C, the heater is activatedat an average power sufficient to bring the temperatureback to 40 �C. This setup enables a clear separationbetween solar-derived heat and auxiliary heat, and defini-tion of the minimum amount of backup electricity that isneeded to satisfy the user’s requirements.

360

240

120

0

Flow

Rat

e [k

g/h]

Time [h]6 12 18 240

Statistical profileStep profile

Fig. 3. End-user daily load profile: mass flow rate of water provided to theend-user at 40 �C.

Table 3Summary of annual results: gain is the increase in solar fraction of systemwith PCM relative to the same system without PCM

Case Site % PCM Load PCM Gain (%)

A Tel Aviv 6 Stepwise SAT-G �0.05B Tel Aviv 25 Stepwise SAT-G �0.05C Tel Aviv 25 Statistical SAT-G +0.7D Munich 25 Statistical SAT-G �0.6E Tel Aviv 25 Stepwise RT42-G �0.1F Tel Aviv 25 Statistical RT42-G �0.3


Environmental data for a typical meteorological year(hourly insolation and ambient temperature) were obtainedusing METEONORM software for two different sites: TelAviv, Israel (representing warm climate and good insola-tion) and Munich, Germany (representing cold climateand low annual insolation).

Two user load profiles were used, as shown in Fig. 3. Asimple stepwise profile that concentrates hot water con-sumption during three periods each day, and a more elabo-rate statistical model that accounts also for variations inconsumption during weekends and holidays (Jordan andVajen, 2000). For both profiles, the overall daily consump-tion was normalized to 180 l/day of water at 40 �C. Thewater from the tank was diluted with cold water when itstemperature was higher, and the auxiliary heater was turnedon when the water temperature was lower than 40 �C.

All component models were implemented in MATLABand the overall system model was integrated in time foran entire year. Several validation cases were run to ensurethe correctness of the simulation (Talmatsky, 2007).

3. Results

3.1. Annual simulations

The basic data values used in the simulation are given inTable 2. Several simulations were performed, with variouscombinations of system design parameters, differentboundary conditions, and variations of model assump-tions, to examine the sensitivity of the results. These varia-tions included:

� Sites: Tel Aviv, Israel and Munich, Germany.� Load profiles: stepwise and statistical.� PCM: composite sodium acetate trihydrate mixed with

graphite particles (SAT-G) and RT42 paraffin in agraphite matrix (RT42-G).� Amount of PCM: 6%–25%, corresponding to an

increase in heat storage capacity relative to that of wateralone, over the temperature swing of 20–70 �C, between32% (for 25% SAT-G) and close to zero (for RT42-G),as shown in Table 2.

� Water inlet (makeup water): fixed temperature (20 �C)and variable temperature (ambient).� Internal convection coefficient (PCM to water): fixed,

and variable with flow conditions; values found fromcorrelations (see Appendix) in the range 50–163 W/m2 K.� Tank heat loss coefficient in the range 0.57–1.13 W/

m2 K.

For each set of parameters, the overall annual thermalenergy delivered to the end-user was divided into energyoriginating from the sun vs. energy from the auxiliary hea-ter. The solar contribution was compared between two sys-tem variants, defined with and without PCM in the storagetank, but otherwise identical. A summary of results for sixcases with different parameters is shown in Table 3.

The most striking result seen from Table 3 is that formost of the cases studied, the gain in solar energy to theend-user due to the addition of PCM is negative. Thismeans that less solar energy is delivered to the end-useras a result of installing PCM in the storage tank. A secondobservation is that the differences are always small, lessthan 1%, and are therefore not significant when consideringthe uncertainties of the model and the numerical calcula-tion. The gain has been normalized with respect to theoverall annual amount of energy provided to the consumer,which is a constant reference value for all cases in thisstudy.

The sensitivity of the results to various assumptionsmade in the model was also considered. Additional caseswere simulated in an extensive investigation (Talmatsky,2007) with variations in model parameters, where the base-line for comparison was case C of Table 3. Varied param-eters included: the tank heat loss coefficient, the convectionheat transfer between the PCM and the surrounding water,the PCM properties and the inlet temperature of the coldmakeup water. The results of the sensitivity studies aregiven in Table 4. In each of these cases, the details of thesimulation, such as the temperature history of each layerin the tank etc., were different. However, the end result –the comparison of annual solar energy delivered to theend-user from the system with PCM vs. the system withoutPCM – was the same: the differences remained very small,less than 1%, which is insignificant compared to the uncer-tainty of the simulation.

Table 4Sensitivity study, relative to case C as reference: gain is the increase insolar fraction of system with PCM relative to the same system withoutPCM

Parameter Originalvalues

Change Value Gain(%)

Reference case – – – +0.7Tank heat loss

coefficient1.133 Decrease

50%0.567 +0.6

Cold water inlettemperature

20 �C Variable Ambient +0.4

PCM convectioncoefficient

163 W/m2 K

Variable Correlation +0.8

PCM specific heat Table 1 Increase15%

– +1.0

0

200

400

600

800

Jan Mar May Jul Sep NovMonth

Sola

r Ene

rgy

to U

ser [

MJ]

awith PCM without PCM

b

0

200

400

600

800

Jan Mar May Jul Sep NovMonth

Auxi

liary

Ene

rgy

[MJ]

with PCM without PCM

Fig. 4. Monthly results for case C: (a) solar energy to end-user and (b)auxiliary energy to end-user.


Additional details of the annual results for the best case(C as defined in Table 3) are shown in Table 5. The differ-ences between the variants with and without PCM areinsignificant, not only for the overall energy, but for quan-tities such as the heat losses from the tank, and the collec-tor efficiency.

The results for case C, broken into monthly sums, areshown in Fig. 4. It can be seen that also on a monthly basis,the amount of solar energy delivered to the end-user isalmost identical between the systems with and withoutPCM. As a result, the amount of needed backup energyfrom the auxiliary heater is also almost the same. Theannual integral for this case does show an advantage forthe system with PCM but it is less than 1% of the totalenergy delivered to the end-user. This is insignificant rela-tive to the uncertainty in any simulation and relative tothe year-to-year variation in available solar energy. Thisinsignificant difference is typical also of the other casesinvestigated in this work.

These results raise the question: why does the PCM notproduce a significant advantage relative to the case withoutPCM? On an intuitive basis, the presence of the PCMshould lower the average storage tank temperature due tothe higher thermal capacitance. Therefore, heat loss tothe environment should be lower on average, and the netenergy to the end-user should be higher. The system withPCM should also return colder water to the collector, lead-ing to increased collector efficiency. Both effects shouldlead to higher overall solar fraction. So the next step is tocheck if these intuitive statements are correct for the sys-tems under consideration.

Table 5Results of the annual simulation for case C (25% vol. of SAT-G PCM,Tel Aviv, statistical load profile)

With PCM Without PCM Gain (%)

Solar energy to consumer [MJ] 6844 6790 0.7Electrical backup required [MJ] 784 838 �0.7Tank losses [MJ] 2354 2381 �0.35Collector efficiency [%] 45.6 45.5 0.1

3.2. Storage tank losses

The examination of the heat loss from the storage tankproceeds by considering the detailed temperature historyduring several days during the year. Fig. 5(a) shows theevolution of the water temperature of the top layer duringa selected period of 48 h (January 30–31). During the after-noon, the effect of the PCM is clearly visible: when thewater reaches the phase change temperature, the watertemperature for the system with PCM remains nearly con-stant, while the temperature of the same layer in the systemwithout PCM continues to increase. The losses to the envi-ronment due to this layer are therefore lower for the systemwith PCM, as hypothesized above. However, the trendchanges during the night hours. After about 6 PM, thereheat effect of the PCM is clearly visible: the temperatureof the water, reduced by the evening withdrawal by theend-user, increases again due to release of heat stored inthe PCM. This effect does not exist in the system withoutPCM and the water temperature remains low. Duringnighttime, the losses to the environment from the systemwith PCM are therefore higher, the opposite behavior rel-ative to the afternoon observation. Therefore, over a 24-hperiod, these two effects approximately cancel each other,leaving a negligible net difference between the two systems.

A more representative measure of the losses to the envi-ronment should include the behavior of all layers in the

0

25

50

75

100a

0 12 24 36 48Time [hr]

Tem

pera

ture

[C]

With PCMWithout PCM

Ambient

b

0

25

50

75

100

0 12 24 36 48Time [hr]

Tem

pera

ture

[C]

With PCM

Without PCM

Ambient

Fig. 5. Temperature history during January 30–31, for systems with andwithout PCM: (a) temperature of top layer and (b) area-weighted averagetemperature.

0

25

50

75

100a

0 12 24 36 48

Time [hr]

Tem

pera

ture

[C]

With PCM

Without PCM

Ambient

b

0 12 24 36 48

Time [hr]

0

25

50

75

100

Tem

pera

ture

[C] With PCM

Without PCM

Ambient

Fig. 6. Temperature history during April 1–2, for systems with andwithout PCM: (a) temperature of top layer and (b) area-weighted averagetemperature.

Table 6Comparison of daily performance indicators (heat loss from the tank tothe environment, collector efficiency) for selected days during the year

With PCM Without PCM

Tank heat loss (MJ)

January 30–31 13.09 13.12April 1–2 15.73 15.69

Collector efficiency (%)

January 30–31 42.14 41.51April 1–2 40.36 39.85


storage tank, rather than the top layer alone. Fig. 5(b)shows an area-weighted average temperature: the tempera-ture of each layer in the tank was weighed with the areathat this layer exposes to the environment, giving a higherweight to the top and bottom layers. The trend for thisaverage temperature is the same: during the afternoon,the system with PCM has lower losses, but during the nightit has higher losses due to the reheat effect.

A similar behavior can be observed at other periods dur-ing the year, for example April 1–2, as shown in Fig. 6. Thesame effects can be seen for the top layer and for an area-weighted average of layer temperatures: lower tempera-tures and lower losses for the system with PCM duringthe afternoon, but higher temperature and higher lossesduring nighttime and early morning, due to the reheat ofthe water from the energy stored in the PCM. Therefore,the net effect over time shows no clear preference for thesystem with PCM, as we have seen above in the annualand monthly energy results.

The total daily losses to the environment are shown inTable 6 for the two periods discussed above. The differ-ences between the system with PCM and the one withoutPCM are very small, as discussed qualitatively above.

The parity between the systems is therefore a result of adaily cycle that compensates the daytime advantage ofthe PCM with the nighttime disadvantage due to reheating.

3.3. Collector efficiency

The second statement made above claimed that thewater entering the collector in the system with PCM mightbe at lower temperature, and therefore the collector effi-ciency may be better in the presence of PCM. This aspect


is now investigated by observing the temperature history ofthe water in the bottom layer of the storage tank, where theoutlet to the collector is installed. This temperature historyfor the periods of January 30–31 and April 1–2 is shown inFig. 7.

Considering only the gray areas in Fig. 7, indicatingperiods when the collector is active, we see again the sameconflicting behavior. In early morning, the temperature ofthe water supplied to the collector from the system withPCM is higher, due to the nighttime reheating effect. Thiseffect exists even when the bottom of the tank does not con-tain any PCM, since thermal conduction through the waterhas ample time during the night to spread heat over theentire volume of the tank. Therefore, the collector effi-ciency during the morning hours is lower for the systemwith PCM. In the afternoon, this trend is reversed, andthe water supplied to the collector from the system withPCM is cooler relative to the system without PCM, andthe collector efficiency should be better for the system withPCM. Over a full day we have, therefore, two conflictingeffects that cancel each other, leaving only a small differ-ence in daily average collector efficiency. A quantitativevalidation of this observation is given in Table 6.

b

0 12 24 36 48Time [hr]

0

25

50

75

100

Tem

pera

ture

[C] With PCM

Without PCM

Ambient

0

25

50

75

100a

0 12 24 36 48Time [hr]

Tem

pera

ture

[C]

With PCM

Without PCM

Ambient

Fig. 7. Temperature of the water in the bottom layer of the storage tank,for systems with and without PCM: (a) January 30–31 and (b) April 1–2.Gray areas indicate sunlight periods when the collector is active.

4. Discussion

The conclusion from the analysis above is somewhatsurprising. The prevailing impression in the extensive exist-ing literature is that the addition of PCM in the storagetank should be beneficial, and the only perceived objectionis the high cost of the materials. This common notion,obviously, has served as motivation for the investigationthat we report here. However, our conclusion is thatregardless of cost, the addition of PCM provides no ener-getic advantage to the end-user, and in some conditions,it may actually be detrimental. We have also analyzed thereason for this unexpected result, and found that thereheating of the water by the PCM during nighttime isresponsible for increased losses to the environment, in anamount that is sufficient to cancel gains made during theday.

The results shown above include several cases with dif-ferent types and amounts of PCM, different sites (solarradiation and ambient temperature boundary conditions),and different behaviors of the end-user. The results for eachcombination of these parameters are similar: no energeticadvantage of the system with PCM relative to the same sys-tem without PCM. The sensitivity of the results to variousmodel assumptions was also investigated, with the sameresult: very small performance differences between thetanks with and without PCM. This indicates that the resultis robust across a wide range of application scenarios, not acoincidence due to an unfortunate choice of some specificset of boundary conditions.

It should be noted that the results here are calculatedusing a simplified model, and the performance of real sys-tems sometimes may diverge from the model predictions.Nevertheless, the sensitivity studies and validation casesthat were run with the model indicate that the model is con-sistent and we may attribute reasonable confidence to theresults.

Another aspect of this comparison is the fact that solarwater heating systems operate within a wide range of tem-peratures: between ambient temperature and up to 80 �C.A PCM has much larger heat storage capacity relative towater over a narrow temperature range, close to its meltingtemperature. When operating in a single-phase far from itsmelting point the advantage of the latent heat is diluted bythe large amount of energy stored as sensible heat. More-over, the single-phase specific heat of many phase changematerials (such as the RT42 used in the current study) issignificantly lower than that of water. In these cases thePCM can be actually an inferior heat storage medium rel-ative to water, when used over a wide temperature range.Since the water in the storage tank undergoes a wide rangeof temperatures every day, as shown in our simulations, thesensible heat effect may be a contributing factor to the lackof advantage of the PCM.

Does this mean that PCM should not be used as heatstorage medium in domestic hot water applications? No,this statement is too far-reaching. The limited conclusion


that we can draw from the current study is that the simpleaddition of PCM inside a standard storage tank is not agood idea. However, the use of PCM in some other design,a different geometry, or a different method of interactionwith the collector and the end-user, might still be beneficial.So the hope of using PCM and reaping a substantial benefitis still alive. But we should focus our efforts to find theinnovative design that would make the envisioned potentialof PCM come true.

Acknowledgments

Funding for this work was provided by the Israel Mini-stry of National Infrastructure. The authors are gratefulfor the valuable data provided by H. Mehling (ZAE Bay-ern, Germany) and by E. Shilton (Rand Industries, Israel).

Appendix

The energy balance equations for the water elementsand the PCM elements in the storage tank includes contri-butions that can be classified into paths describing conduc-tion, convection between the water and the PCM, internaland external (inlets and outlets) advection and loss to theenvironment.

Conduction

The contribution of conduction between water in layer i

and water in one of the neighboring layers (i ± 1) has theform:

kW DAW�Wi;i�1 ðT W

i�1 � T Wi Þ=Dxi ðA1Þ

DAW�W is the contact area of the water element i to theneighboring water element.

The contribution of conduction between the PCM inlayer i and PCM in one of the neighboring layers (i ± 1)has the form:

kPDAP�Pi;i�1ðT P

i�1 � T Pi Þ=Dxi ðA2Þ

AW�W is the contact area of PCM element i to the neigh-boring PCM element, if it exists in the relevant layer.

Obviously the same terms appear in the balance equa-tions for layers i ± 1, with the opposite sign.

Convection

Convection occurs between the water and the PCM ele-ment in the same layer, and possibly between the water anda PCM element in a neighboring layer, in case that thelayer does not contain PCM, or its PCM has a differentgeometric configuration relative to the neighboring layer.The contributions of the two types to the energy balanceof the water are:

hi;iDAW�Pi;i ðT P

i � T Wi Þ

hi;i�1DAW�Pi;i�1 ðT P

i�1 � T Wi Þ

ðA3Þ

Obviously the same contributions appear in the balanceequations for the PCM elements in layers i and i ± 1, withthe opposite sign.

The value of the convection coefficient h in Eq. (A3)was estimated during times when the fluid is stationary(no flow through the tank) using standard correlationsfor vertical and horizontal surfaces (Incropera and Dewitt,2002), and during times with net flow, using correlationsfor parallel flow along cylinders (El-Wakil, 1971; Esenet al., 1998). The results of these correlations vary, but asensitivity analysis (Talmatsky, 2007) has shown that theannual energy results are not sensitive to changes in thevalue of the convection coefficient within the range foundaccording to the various correlations.

Internal and external advection

In the following equations, the notation ()+ will be usedto signify that the expression in parenthesis keeps its valueif it is zero or positive, but is set to zero if it is negative.

The contribution of convection to the energy balance ofthe top layer (i = 1), including the effect of water inlet fromthe collector and water outlet to the end-user, is:

_mCCW ðT C;out � T W1 Þ þ ð _mU � _mCÞþCW ðT W

2 � T W1 Þ: ðA4Þ

The contribution of convection to the energy balance ofan intermediate layer (1 < i < N) is:

ð _mU � _mCÞþCW ðT Wiþ1 � T W

i Þ þ ð _mC � _mU ÞþCW ðT Wi�1 � T W

i Þ:ðA5Þ

The contribution of convection to the energy balance ofthe bottom layer (i = N), including the effect of water outletto the collector and inlet of makeup water from the exter-nal water grid, is:

_mU CW ðT M � T WN Þ þ ð _mC � _mU ÞþCW ðT W

N�1 � T WN Þ: ðA6Þ

Losses

The heat loss from each layer to the environment has theform:

UaDAiðT Wi � T aÞ: ðA7Þ

Ua is the overall loss coefficient, taking into account thetank insulation and external convection to the ambient air.DA is the relevant heat transfer area to the ambient, includ-ing the bottom and top bases of the cylindrical tank for thetwo relevant layers.

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pcm storage for solar dhw: an unfulfilled promise?

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