applied energy 73 (2002) 83–98 www.elsevier.com/locate/apenergy

Transient behaviour of a latent-heat thermal-energy store: numerical and experimental studies

Pedro D. Silva*, L.C. Goncalves, L. Pires

Department of Electromechanical Engineering, University of Beira Interior, 6200 Covilha, Portugal

Abstract

The phase change material (PCM) considered is a commercial paraffin wax. The mainenclosure that contains the wax has a rectangular geometry and is placed in a vertical posi-

tion. One of the walls is heated at constant rate during the charge process and the other wall iscooled by an airflow stream during the discharge. To describe the performance of the store aone–dimensional model based on the enthalpy method is developed. An apparent specific heat

of the PCM is determined using a differential scanning calorimeter (DSC). This parameterpermits the enthalpy of the PCM to be determined, and linearized for the numerical model.The results show that the simplified numerical model can be used to predict, with reasonable

accuracy the dynamic performance of this kind of heat exchange unit.# 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Thermal energy storage systems are important for the effective utilization ofenergy flows at low-temperature (<120 �C). The development of a latent heat ther-mal energy storage (LHTES) system has been dictated by the performance of avail-able materials and associated heat-exchangers [1,2].Suitable phase-change materials (PCMs) that may be used for LHTES appli-

cations are reviewed by Abhat [3]. Among them the paraffin waxes appear as goodPCMs because they are cheap, available for a large temperature range and also havereasonable latent-heat energy densities.Recently the operation of LHTES devices with a constant-rate heating mode, par-

ticulary for rectangular geometries since they allow a greater diversity of applications

Applied Energy 73 (2002) 83–98

www.elsevier.com/locate/apenergy

0306-2619/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0306-2619(02 )00060 -0

* Corresponding author. Fax: +351-275-320-820,.

E-mail address: [email protected] (P.D. Silva).

[4], has been discussed. In these investigations, we find experimental studies of theprocess of melting in the presence of natural convection in the melt [5,6]. The workof Zhang and Bejan [5] seems to be one of the first studies with this boundary con-dition. Those works also include numerical studies with aplications to thermal con-trol for the effective cooling of electronic equipment or for energy conservation inbuildings [7–9]. Also numerical/experimental studies for the direct utilization ofsolar energy [10] or for using electricity off-peak for domestic heating [11,12] havebeen developed.In the studies referred to above, the latent heat of the PCM is used in the

absorption, or to delay the absorption, of the heat flux that passes through theboundaries of the store.This paper is concerned with investigations about the transient behaviour of a

LHTES. The store consists in a vertical rectangular enclosure that contains a com-mercial paraffin wax, with one of the vertical walls subjected to constant-rate heat-ing and the other one to an airflow stream. Experimentally, the objective is to showthe importance of the heat-transfer mechanism inside the store during the melting(charge) process and the solidification (discharge) process, and to validate the modelresults. Numerically the objective is to develop a model based on the enthalpymethod in which the enthalpy-temperature relationship is linearized from differ-ential-scanning calorimeter (DSC) data of the PCM. This model is validated withthe results obtained from the experimental unit, and used to study the effects of thevariation of relevant parameters during the melting and solidification processes.

2. Experimental description

The thermal-energy store used in this study is shown in Fig. 1. The store consistsof two vertical, parallel, aluminium walls separated by a gap of 3 cm. These wallshave a thickness of 4 mm. The other walls that define the store volume are made ofPlexiglass, having a thickness of 1.5 cm to ensure a good thermal and mechanicalresistance and permiting visualization inside the store. The global dimensions of thePCM store are 813 mm and 623 mm for the total vertical and horizontal lengths.These dimensions are much greater than the aluminium-walls separation in order toreduce a possible perturbation of the melting/solidification front propagation. Oneof the aluminim walls—the warm wall—is heated at a constant rate by mean of anelectrical circuit made of NiCr wire. This circuit is bonded to the wall using a high-temperature chemical cement, so that one obtains a uniform heat-flux distribution.In order to have the minimum heat loss, a good insulation is applied, so this wall iscovered with a 15 cm layer of glass wool.The cooling process of the LHTES is provided by an laminar air-flow stream,

between the other aluminium wall—the cold wall—and a polyurethane wall of 15cm thickness. This air cooling channel has a width of 5 mm. The air inlet tempera-ture and the mass flow rate are maintained constant during the experimental pro-cesses. The uncertainty associated with the measured values of the air mass flow rateis estimated as 4%. The electric power dissipated was calculated from current and

84 P.D. Silva et al. / Applied Energy 73 (2002) 83–98

voltage measurements, and the uncertainty associated is estimated to be 2%. Theheat flux to the store was obtained from the electric power dissipated divided by thewarm-wall surface area. Temperature measurements inside the store were taken by athree-level temperature probe (see Fig. 1). The distances between each level and thebottom of the store are 16.5, 37.5, and 58.5 cm. The thermojunctions at each level,are positioned at 6, 12, 18 and 24 mm from the warm wall and located in the verticalmid-plane of the store. Temperature measurements are obtained through thermo-couples located in each aluminium wall at the same probe levels. Thermocouples(type T) of copper and constantan wires of 0.2 mm diameter connected to a data-acquisition system were used. Temperature measurements were taken with an accu-racy �0.4 �C.The store was filled with commercial paraffin wax. A small air gap was left at the top

of the store to accommodate the variation of volume associated with the melting/solidi-fication cycles. The thermophysical properties of the wax are given in Table 1. In thistable, the thermal conductivity, the absolute viscosity and the volumetric coefficient of

Table 1

Thermophysical properties of the PCM

Melting temperature 52.1 �C

Latent heat 243.5 kJ/kg

Density at 15 �C 809.5 kg/m3

Density at 70 �C 771.0 kg/m3

Thermal conductivity 0.15 W/mK

Absolut viscosity 5.5�10�3 kg/ms

Volumetric coefficient of expansion 7.9�10�4 K�1

Fig. 1. Schematic diagram of the experimental apparatus.

P.D. Silva et al. / Applied Energy 73 (2002) 83–98 85

expansion are estimated at the melting temperature from data available for otherparaffin waxes [14]. The specific heat as a function of temperature and the latent heatwere obtained through experiments using a differential scanning calorimeter. Thevalues must be considered to have an uncertainty of �3%.

3. Mathematical formulation

The following assumptions will be made to establish the mathematical modeldescribing the physical phenomena under consideration: one-dimensional conduc-tion controls the phase-change process within the PCM; the fluid flow (of air) is fullydeveloped with temperature-independent properties and its capacitance is ignored;thermal radiation between the wall and the fluid is neglected. Based on theseassumptions, energy balances for the PCM [13] and for the circulating fluid [15,12]yield the following equations:

@H

@t¼

@

@xk@T

@x

� �ð1Þ

m:fcf

dTf

dy¼

AhcL

T� w�Tf

� �ð2Þ

Here, T and k are the temperature and thermal conductivity of the PCM, m:f; cf;Tf

and hc are the mass flow rate, specific heat, bulk temperature and convection coeffi-cient of the fluid, respectively. The flow direction is given by y, and L;A;T� w are thelength, surface area and mean temperature of the wall in contact with the fluid,respectively. In Eq. (1), H represents the specific enthalpy and can be related to thetemperature field by a linear relation [16,17]:

T ¼ A Tð ÞHþ B Tð Þ ð3Þ

where

A Tð Þ ¼1

�cð4Þ

B Tð Þ ¼ Tref�Href

�cð5Þ

and

pc ¼1

T� Tref

ðTTref

�c �ð Þd� ð6Þ


Tref and Href represent reference values for the temperature and enthalpy. Theparameters A and B in Eqs. (4) and (5) are determined using the PCM experimentalvalues for density, r, and specific heat, c.The initial condition and the boundary conditions for the Eqs. (1) and (2) are:

t ¼ 0 ! T ¼ Tf ¼ Tfi

x ¼ 0 ! �k@T

@x¼

qe in charge process

0 in discharge process

�

x ¼ ‘ ! �k@T

@x¼ qc

y ¼ 0 ! Tf ¼ Tfi

ð7Þ

where qe represents a heat flux obtained from the electric power dissipated per unitarea, and ‘ is the PCM thickness. The rate at which heat is transferred by convec-tion, per unit area, from the cold wall to the air flow is given by:

qc ¼ hc T�w � T� f

� �¼

m:fcfA

Tfo � Tf ið Þ ð8Þ

Tfo and Tfi are the fluid temperature at the outlet and at the inlet, respectively. AsEq. (2) has an exact solution, it can be obtained as an averaged bulk air temperatureover the channel length:

T� f � T� w

Tfi � T� w¼1� e�NTU

NTUð9Þ

where NTU=Ahc/m:fcf represents the number of thermal units. Taking into account

the heat convection effects in the melting process, an equivalent thermal con-ductivity can be used for design purposes [18]. This kind of procedure has been usedrecently in several investigations [11,19,20]. In this work, the correlation proposedby Farid and Kanzawa [21] is applied:

keqkl

¼ C1Ra0:25�

‘

� �ð10Þ

where kl, Ra, and d are the liquid thermal conductivity, the Rayleigh number andthe melt-layer thickness. The constant C1 will be determined by the experimentalresults. For the air-flow convection coefficient, a correlation for the Nusselt numberfor laminar fully-developed flow between two walls is used with the boundary con-ditions of uniform temperature at one wall and the other wall insulated [22]. In thissituation, Nu2hc‘c/kf=4.86, where ‘ indicates the air-channel width.


4. Numerical scheme

The differential Eq. (1) was discretized using the enthalpy formulation, andapplying a time-explicit scheme for any internal-node results [23]:

Hnþ1j ¼ Hn

j þ�tn�xj

qnj�1

2� qn

j�12

� �ð11Þ

where

qj�12¼ �

Tnj � Tn

j�1

Rj�12

; Rj�12¼

�xj�12kj�1

þ�xj2kj

ð12Þ

are the discrete heat flux and discrete thermal resistance. To ensure the numericalstability of the scheme, the time increment is limited to:

�t41

2

�x2min

maxð13Þ

where �x represents the control volume length, and a the thermal difusivity of thePCM. To describe the true enthalpy-temperature relationship, more than one linearrelation is needed [see Eq. (3)]. For the numerical model, using the state equationstogether with the nodal enthalpy updating at each time step, avoids the missing outof the enthalpy effects related to the phase-change process [24].

5. Results and discussion

The results shown Fig. 2 were obtained from experimental tests using samples ofparaffin wax in a DSC. The graph shows two phase transitions: a solid/solid transi-tion (31–38 �C) and a solid/liquid transition (48–56 �C). These results can be iden-tified in Fig. 3. The temperature value of 52 �C can be considered as the referencevalue for the solid/liquid transition which is in good agreement with the data of theparaffin-wax supplier (see Table 1). However it must pointed out that the phasechange takes place over a temperature range of the latent heat delivered, aproxi-mately 25% is related to the solid/solid transition.The experimental results presented in Figs. 4–6 were obtained for an electric

power of 245 W and for an imposed heat flux of 480 W/m2. The mass flow rate andthe temperature of the cooling air were kept constant with values of 5.488�10�3 kg/sand 19.3 �C, respectively. Experimental tests for the charge and discharge wereconducted during a period of 24 h. For the charge process, the end time was con-sidered to occur when the cold-wall temperature at the lower level reached the valueof 54 �C. This means that the paraffin wax was totally liquid, as confirmed byvisual inspection. Time for the charge was about 8.4 h. For the charge process, theresults of the warm-wall temperature at each level are shown in Fig. 4. It must be


pointed out that there is a period of time during which the temperature varies uni-formly, which indicates the domination of the conduction mode. The thermal stra-tification that leads to the thermal gradient at the wall appears as a result of naturalconvection inside the PCM. At this stage, the temperature remains near constantand steady during the phase-change process where solid and liquid phases existtogether. This situation creates a thermal protection effect at the wall. As soon asthe solid phase disappears, the wall temperature increases and the thermal protec-tion ceases. Experimentally it was verified that thermal stratification increases with

Fig. 2. Apparent specific heat of the wax.

Fig. 3. Enthalpy of the wax.


the imposed heat flux, as the effects of natural convection are greater. This situationinfluences the fusion-front propagation velocity and reduces the period of thermal-protection effect. It is expected that for the same imposed heat flux, the paraffin waxthickness will be crucial to define the thermal protection period.The cumulative enthalpy for each level is shown in Fig. 5 and represents the time

integral value of the enthalpy of the wax, obtained with both the experimental tem-peratures and the function H(T) (see Fig. 3). These experimental data points give usthe time instant when the phase transition solid/liquid finishes at each level. Also,

Fig. 4. Warm-wall temperature evolution.

Fig. 5. Time integral plot of the enthalpy inside the store.


the results give us the rate the thermal energy is stored (i.e. cumulative enthalpy). Assoon as the charge process ends, the temperature differences between the levelsdecrease and the heat-transfer mechanism becomes a conduction-dominated pro-cess.The evolution of the mean temperature values for the cold and warm walls and of

the air going out is presented in Fig. 6, for the charging and discharging processes.The experimental values are matched by the numerical results obtained from thecomputer simulations. The value adopted for the constant C1 [see Eq. (10)] was 0.18.For the thermal conductivity of the solid wax, the value 0.17 W/mK was employed.This value was adopted to achieve better overlapping of the numerical and experi-mental results [14,25]. The lack of experimental data and the importance of thermalconduction during the discharge process and the initial period of charge are thereasons for the methodology adopted. To describe the behaviour enthalpy versustemperature in the numerical model, five linear relationships similar to Eq. (3) wereused.The results presented in Fig. 7, were obtained for the following conditions:

imposed heat-flux 622 W/m2; air mass flow rate 6.265�10�3 kg/s; air inlet tempera-ture 19 �C. The experiments were conducted in order that the end of the dischargewas considered to occur when the warm-wall temperature at the upper level reached31 �C. The charge time for this experiment was about 6 h. It can be concluded thatthere is good agreement between the numerical and experimental results. Thenumerical model gives a good response for the thermal behaviour of a LHTES.Figs. 8–10 show some of the results obtained with the numerical model to illus-

trate the effects of varying different parameters in the charge and discharge pro-cesses. For this numerical study, the end of the charge time was considered to occurwhen the cold-wall temperature Tec reached 60

�C and the end of the discharge time

Fig. 6. Experimental and numerical results for the walls and air temperature (qe=480 W/m2).


when the warm wall temperature Ted reached 31�C. The air inlet temperature was

considered as the standard air temperature of 21 �C. Fig. 8 shows the effect of theimposed heat flux on the charge time for the same PCM thickness. As expected, thetime for charging increases with the PCM thickness increasing and the heat fluxdecreasing. However, there is a value for the imposed heat flux (i.e., a minimumvalue) for which no charge process occurs. The minimum value for the heat flux isindependent of the PCM thickness (see Fig. 8), and is clearly dependent upon theNTU value (see Fig. 9).

Fig. 7. Experimental and numerical results for the walls and air temperature (qe=622 W/m2).

Fig. 8. Effect of heat flux and PCM thickness upon the melting time.


The estimation of the minimum values for the heat flux can be obtained from thefollowing equation for the steady-state condition:

qmine ¼

hcNTU

Tec � Tfið Þ 1� e�NTU� �

ð14Þ

and this expression has two limits:

qmine ¼ hc Tec � Tfið Þ; NTU ! 0 ð15Þ

qmine ¼

m:C f

ATec � Tfið Þ; NTU ! 1 ð16Þ

Fig. 9. Effects of heat flux and NTU value upon the melting time.

Fig. 10. Effects of PCM thickness and NTU value upon its solidification time.


For values of the imposed heat flux greater than qemin, as defined by Eq. (15), the

charge time is almost independent of the NTU, so the store always charges (seeFig. 9). Also the numerical results show that the discharge time is not affected by theimposed heat-flux. The duration of the discharge process depends only on thethickness of the PCM and the NTU value (see Fig. 10).The information obtained for different case studies by the numerical model (see

[26] for details), permits the selection of dimensionless parameters to be presented ascharge and discharge characteristics curves of the LHTES. For the charge process,the non-dimensional heat flux becomes:

$ ¼qeqmine

ð17Þ

The inclusion of the minimum heat flux comes naturally because the charge pro-cess begins only when the imposed heat flux its greater than a minimum value. Forthe non-dimensional charge time, one must consider the energy stored in the paraffinwax through the latent heat, L, and the sensible energy stored in the two walls istaken in to account by the specific heat, c, and the mass m of the wall. In this way,the expression for the dimensionless time is:

¼tcqe

�L‘þ 2mc

A

� �wTec � Tfið Þ

ð18Þ

The non-dimensional representation of the charge time for the considered case studiesis given in Fig. 11. Mathematically, we can define three correlations in the form:


¼ 2:651$�0:9829 1:24$4 1:5

¼ 2:305$�0:5905 1:54$4 2:0

¼ 1:692� 7:958� 10�2$ 2:04$4 5:0 ð19Þ

The relative errors associated with these correlations are less than, respectively, 5,7, and 12% of the estimated values.For the discharge process as the imposed heat flux does not affect the discharge

period, the parameter qmine is introduced. This has the advantage of taking into

account the NTU value. The thickness of the PCM is a relevant parameter and isincluded, through the Biot number, with the thermal resistance of the fluid and thePCM, being defined as:

Bi ¼hc‘

ksð20Þ

The thermal conductivity included in the previous equation corresponds to thesolid phase of the PCM. So, the non-dimensional parameter can be represented inthe form:

' ¼qmine

qmine

� �NTU ! 0

BPi ð21Þ

In this equation, the exponent of the Biot number is a consequence of the non-linearity of the relationship between the discharge time and the PCM thickness, andits value is determined from an analysis of several numerical experiments. InsertingEqs. (14) and (15) into Eq. (21), it follows that:

' ¼1� e�NTU

NTUBPi ð22Þ

For the definition of a dimensionless discharge time, we proceed in an analogousway to that used to obtain Eq. (18). The energy stored in the walls is now taken intoaccount between the-end-charge and end-discharge temperatures.The dimensionless time becomes:

� ¼tdq

mine

�L‘þ 2mc

A

� �wTec � Tedð Þ

ð23Þ

The graphical representation is given in Fig. 12 for several numerical experimentsand can be represented mathemathically by the following expression:

� ¼ 1:647þ 1:028' ð24Þ


This correlation gives an estimation of the discharge time with a relative error ofless than 4%. The value obtained for the Biot number exponent p was 0.72.The charge and discharge correlations are valid for the following ranges of the

PCM thickness and NTU parameter:

0:014 ‘4 0:05

0:54NTU4 2:0 ð25Þ

The correlations developed predict in a satisfactory way the charge and dischargetimes, and so can be used in the design of this kind of thermal-energy storage system.

6. Conclusions

The dynamic thermal behaviour of a latent heat thermal energy store is presented.The experimental results obtained show the importance of each transfer mechanism,and it can be concluded that the heat conduction controls the initial charge periodand almost all the discharge process. As a result of convection effects, it can beconcluded that thermal stratification inside the store and a vertical temperature-gradient in the heated wall take place. When the solid and liquid phases coexist, thevalue of the temperature at the heated wall remains quasi-constant, showing, duringthis period, a thermal protection effect. The numerical model is validated withexperimental data obtained for several melting/solidification cycles. The resultsshow that the numerical model can predict well the dynamic thermal response of thiskind of heat-exchange unit. The numerical study shows that the melting time isdominated by the imposed heat flux and PCM thickness. The effect of the NTUvalue is important for the melting time just for values of the imposed heat flux close


to a minimum value defined by Eq. (15). It can also be concluded that the imposedheat-flux is irrelevant for the solidification time. This depends only on the NTUvalue and the PCM thickness. Finally we can conclude that the correlations devel-oped can be used for the rapid estimation of the charge and discharge times and socan be useful in the design of this kind of latent-heat thermal-energy store.

References

[1] Goncalves LCC. Theoretical predictions and experimental performance of packed-beds of encapsu-

lated phase-change materials. PhD thesis, Cranfield Institute of Technology, 1984.

[2] Goncalves LCC, Probert SD. Thermal-energy storage: dynamic performance characteristics of cans

each containing a phase-change material, assembled as a packed-bed. Applied Energy 1993;45:117–

55.

[3] Abhat A. Low-temperature latent heat thermal energy storage: heat storage materials. Solar Energy

1983;30(4):313–32.

[4] Fukusako S, Yamada M. Recent advances in research on melting heat transfer problems. Proc of the

Tenth International Heat transfer Conference 1994;1:313–31.

[5] Zhang Y, Bejan A. Melting in an enclosure heated at constant rate. Int Journal of Heat and Mass

Transfer 1989;32(6):1063–76.

[6] Zhang Y, Chen Z, Wang Q, Wu Q. Melting in an enclosure with discrete heating at a constant rate.

Experimental Thermal and Fluid Science 1993;6:196–201.

[7] Sasaguchi K, Ishihara A, Zhang H. Numerical study on utilization of melting of phase change

material for cooling of a heated surface at a constant rate. Numerical Heat Transfer, Part A 1996;29:

19–31.

[8] Ho CJ, Chu CH. Numerical simulation of heat penetration through a vertical rectangular phase-

change material/air composite cell. Int Journal of Heat and Mass Transfer 9;39:3165–73.

[9] Pal D, Joshi Y. Application of phase-change materials for passive thermal control of plastic quad flat

packages: a computacional study. Numerical Heat Transfer, Part A 1996;30:19–34.

[10] Stritih U, Novak P. Numerical and experimental investigations of pcm solar wall. Proc of the

Eurotherm seminar 1996;49:95–100.

[11] Farid MM, Husian RM. An electrical storage heater using the phase-change method of heat storage.

Energy Convers Mgmt 1990;30(3):219–30.

[12] Laouadi A, Lacroix M. Thermal performance of a latent-heat energy storage ventilated panel for

electric load management. Int Journal of Heat and Mass Transfer 1999;42:275–86.

[13] Crank J. Free and moving boundary problems. Oxford University Press, Oxford, 1994.

[14] Humphries WR, Griggs EI. A design handbook for phase change thermal control and energy-storage

devices. NASA Technical Paper 1977;1074.

[15] Schmidt FW, Willmott AJ. Thermal energy storage and regeneration. Hemisphere Publishing Cor-

poration, New York, 1981.

[16] Voller VR, Swaminathan CR, Thomas BG. Fixed-grid techniques for phase change problems: a

review. Int Journal for Numerical Methods in Engineering 1990;30:875–98.

[17] Bellecci C, Conti M. Phase change thermal storage: transient behaviour analysis of a solar receiver/

storage module using the enthalpy method. Int Journal of Heat and Mass Transfer 1993;36:2157–63.

[18] Bonacini C, Comini G, Fasano A, Primicerio M. Numerical solution of phase–change problems. Int

Journal of Heat and Mass Transfer 1973;16:1825–32.

[19] Hirata T, Nishida K. An analysis of heat transfer using equivalent thermal conductivity of liquid

phase during melting inside an isothermally heated horizontal cylinder. Int Journal of Heat and

Mass Transfer 1989;32(9):1663–70.

[20] Lacroix M. Study of the heat transfer behaviour of a latent heat thermal energy storage unit with a

finned tube. Int Journal of Heat and Mass Transfer 1993;36(8):2083–92.

[21] Farid MM, Kanzawa A. Thermal performance of a heat storage module using pcms with different

melting temperatures: mathematical modelling. Journal of Solar Energy Engineering 1989;111:152–7.


[22] Shah RK, Bhatti MS. Laminar convective heat transfer in ducts. In: Kakac S, Shah RK, Aung EW,

editors. Handbook of single-phase convective heat-transfer. John Wiley & Sons, New York, 1987

[Chapter 3].

[23] Alexiades V, Solomon AD. Mathematical modeling of melting and freezing processes. Hemisphere

Publishing Corporation, New York, 1993.

[24] Swaminathan CR, Voller VR. On the enthalpy method. Int J Num Meth Heat Fluid Flow 1993;3:

233–44.

[25] Farid MM, Kim Y, Kanzawa A. Thermal performance of a heat-storage module using pcms with

different melting temperature: experimental. Journal of Solar Energy Engineering 1990;112:125–31.

[26] Silva PD. Experimental and numerical modelling of latent heat thermal energy storage systems. PhD

thesis, Beira Interior University, 2001 [in Portuguese].


applied energy 73 (2002) 83–98 www.elsevier.com/locate/apenergy

Documents