Heat Transfer During Phase Change of Paraffin Wax Stored in Spherical Shells

Hisham Ettouney*e-mail: [email protected]

Hisham El-Dessouky

Amani Al-Ali†

Department of Chemical Engineering,College of Engineering and Petroleum,

Kuwait University,P.O. Box 5969—Safat 13060—Kuwait

Heat Transfer During PhaseChange of Paraffin Wax Stored inSpherical ShellsThis study concerns experimental evaluation of heat transfer during energy storage andrelease for the phase change of paraffin wax in spherical shells. Measurements are madeusing air as the heat transfer fluid (HTF), copper spheres with diameters of 2, 3, 4, and6 cm. A detailed temperature field is obtained within the spheres using 10 thermocouplewires. Values of the air velocity and temperature used in the experiments are 4–10 m/sand 60–90°C, respectively. Measured times for melting and solidification varied over arange of 5–15 and 2–5 minutes, respectively. Calculations show that the Nusselt numberin the phase change material (PCM) during melting is one order of magnitude higherthan during solidification. Results indicate that the Nusselt number for melting has astrong dependence on the sphere diameter, lower dependence on the air temperature, anda negligible dependence on the air velocity. Variations in the Fourier number for meltingand solidification show similar trends. An increase in the Nusselt number for a largersphere diameter is attributed to increase in natural convection cells in the PCM inside thespheres. The larger volume allows for the free motion for the descent and rise of coolerand hotter molten wax. During the solidification process, the solid wax is evenly formedthrough the sphere, starting from the outer surface and moving inward. As the solidifica-tion proceeds, the melt volume decreases with a simultaneous decrease in the magnitudeof natural convection within the melt. The higher values of Fourier number for meltingindicate the consumption of a large part of the HTF energy in heating the molten waxrather than melting of the solid wax. �DOI: 10.1115/1.1850487�

1 IntroductionEnergy production and consumption is the main driver of all

urban and industrial activities. At present more than 80% of theworld energy is provided by the combustion of fossil fuels, whichincludes oil, gas, and coal. The combustion of fossil fuels gener-ates a wide range of air pollutants, which includes sulfur andnitrogen oxides, hydrocarbons, and soot. In the industry more thanone half of the input energy is discarded as waste heat in the fluegases or in the form of low-grade energy in cooling water �1�.Energy storage provides the means for the efficient managementof excess energy, sustainable energy sources �i.e., wind, solar,geothermal, or ocean thermal� and low-grade energy discardedfrom various industrial applications. The storage of excess energyis made during low demand and recovery is made during peakloads. The adoption of such systems implies savings in the expan-sion of power plants and the construction of new units to meet theincreasing load �2�. Energy storage combined with solar collectorsand photovoltaic have been developed over the years for homesand large buildings. Although these systems provide energy at ahigher cost than fossil fuels, their main advantage is their limitedimpact on the environment and sustainability of the energy source�3,4�. Hot and cold climates should be a major target for the use ofenergy storage systems, where energy consumption for indoor airconditioning or heating may exceed 50% of the peak load of elec-tric power generation during the summer or winter seasons. Iceformation in spherical capsules precedes overnight in large hold-ing tanks, which are then melted during the day for district cool-ing. Similar arrangements are found for district heating usingplate, shell and tube, and spherical capsules heat exchange units.

The majority of literature research on the energy storage system

is performed for the shell and tube configuration as well as theplate and frame system. A very limited number of studies arefound on the energy storage in spherical shells. The following areexamples of these studies:

i. Beasely et al. �5� evaluated the thermal response of apacked bed of spheres containing PCM’s. They developed phase-change models for both isothermal and nonisothermal melting.Their model takes into considerations axial thermal dispersion ef-fects. Also, they found good agreement between model predic-tions and measurements from a commercial size energy storageunit.

ii. Saitoh et al. �6� presented a detailed simulation study forphase change energy storage in spherical shells. The detailedmodel proved to be valuable in system design and performanceevaluation.

iii. Yagi and Akiyama �1� performed experimental and math-ematical modeling of high temperature latent heat energy storagesystem. It is concluded that metallic PCMs provide an almostuniform temperature distribution during phase change.

iv. The study by Watanabe et al. �7� includes a high rate charg-ing and discharging phase change energy storage system. This isachieved by the use of three types of PCMs with different meltingtemperatures. At the array top, shells of the high melting tempera-ture PCM where placed and are followed by the lower meltingtemperature shells.

v. The study by Baraba and Spiga �8� show that spherical cap-sules give the largest energy density and the most rapid chargeand release times, when compared against the slab or the cylin-drical geometry.

vi. A spherical capsule can have a sub-micron diameter, whichis the case micro-encapsulation in polymer coating �9�. Themicro-spheres can then be pressed to form pellets, cylinders, orspheres. Also, large capsules with a few centimeters diameter canbe used as an energy storage media. The large capsules can be

*Corresponding author.†Ministry of Electricity and Water—Kuwait.Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF ME-

CHANICAL ENGINEERS. Manuscript received by the ASME Solar Division July 9,2004; final revision August 3, 2004. Associate Editor: Jane Davidson.

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made of metal or thin plastic shells. Optimum sphere size dependson handling, energy density, and pressure drop between thespherical capsules.

Irrespective of the limited number of citations on phase changeenergy storage in spherical shells, there are a large similaritieswith the most common system, which is based on the double pipeconfiguration. The following is a brief evaluation for the majorfindings of these studies:

• Marshall and Dietsche �10� and Marshall �11� investigatedthe performance of a hybrid system of phase change energystorage and conventional water solar collector. Results showa comparable performance of the commercial and high puritywaxes. In addition, the performance of the hybrid system isshown to provide little improvement over the stand-alone so-lar collector water system.

• Ibrahim et al. �12� performed experimental measurements inenergy storage canisters at ground and space conditions. Theexperiments show the effects of natural convection and radia-tion upon variations in the gravitational pull. The analysiswas based on the study by Kerslake and Ibrahim �13�, whichincluded one- and two-dimensional models.

• Farid and Yacoub �14� presented a direct contact phasechange energy storage unit. Kerosene is used as the HTF,which is bubbled in a bed of salt hydrates. Higher efficiencywas obtained upon increase in the system size.

• Bansal and Buddhi �15� analyzed the performance of flatplate phase-change solar collectors. The study is motivatedby need for size reduction of the large size sensible heat solarcollectors.

• Hoogendoorn and Bart �16� studied the enhancement of theeffective thermal conductivity of paraffin wax using alumi-num matrix inserts. Enhancement factors up to 5 are reportedas the volumetric percentage of the inserts is increased to 2%.Similarly, Tong et al. �17� reported an order of magnitudeenhancement during melting and solidification upon insertingan aluminum matrix in the PCM.

• Choi and Kim �18� studied heat transfer characteristics forMgCl2•6H2O in a double pipe phase change energy storagesystem with air as the HTF. Smooth surface and finned heatexchange tubes are used in the experimental investigation.Results show that the heat transfer coefficient for finned tubesis 3.5 times larger than smooth tubes.

The above literature review can be summarized in the followingpoints:

• Innovations in the phase-change heat storage systems are es-sential to improve the system performance, especially, thecycle time, the specific heat transfer area, and the thermalefficiency.

• The use of finned heat exchanger tubes, a composite of metalfibers and PCM, or metal structure in the PCM, improves theheat transfer rates and as a result the melting and solidifica-tion time is drastically reduced.

• The use of more than one PCM with different melting pointsin the same energy storage unit increases the system outputpower by 30%.

This study focuses on experimental measurements and theanalysis of energy storage and release in spherical capsules. Thestudy is motivated by the fact that spheres provide the largestvolume per unit surface area. In addition, limited literature studiesare found on mechanism of heat transfer, analysis, and perfor-mance of spherical energy storage systems. This study will pro-vide the literature with valuable data on variations in the heattransfer coefficient and other system parameters.

2 Elements of Experimental SystemThe experimental system shown in Fig. 1 includes the

following:

• Spherical shells made of copper with diameters of 2, 3, 4, and6 cm.

• The wall thickness of the copper spheres is 1 mm for the 2cm sphere, 1.2 mm for the 3, 4, and 6 cm spheres.

• The copper spheres are machined in two halves with fittingrims.

• A heating element for the air stream with a total power of 2kW.

• PID temperature controller for the air heater with an accuracyof �1°C.

• Air fan with variable speed that provides a velocity range of1–12 m/s.

• A glass column with a diameter of 20 cm and height of 40cm, where the cold/hot air flow surrounds the PCM sphere.

• Data logger for storing temperature measurements of theinlet/outlet air and inside the spherical shell.

• Ten thermocouples are inserted inside the spheres. Seventhermocouples are inserted in the direction of air flow and theother three are inserted in a direction perpendicular to the airflow. The thermocouples are K type, Omega model TFAL-010, with an accuracy of �0.5°C and nominal diameter of0.25 mm. The ten thermocouples are placed inside a T-shape1 mm diameter plastic tube. The tube has equally spacedholes, where the thermocouple head penetrates through intothe PCM. Use of the plastic tube keeps the thermocouples inthe desired location.

• As is shown in Fig. 1, the seven thermocouples are placedalong the entire vertical axis of the sphere. The remainingthree thermocouples are placed along one half of the horizon-tal axis of the sphere. This selection is made to provide tem-perature readings along the vertical and horizontal axes of thesphere. These readings are used to determine the local heattransfer coefficient in the PCM near the bottom, top, and sideof the sphere. Also, the entire temperature field is used toconstruct the temperature contours as a function of radial andangular position.

• Three thermocouples to measure the temperature measure-ments above, below, and at the center of the PCM sphere. Thethermocouples are of the same type as those used inside thesphere.

• Velocity probe for measurements of the air velocity. Theprobe is AIRFLOW model number TA-2-30. The measuringrange is 0–30 m/s for a working temperature range of0–100°C.

The combined effect of the large weight for the PCM and themoderate velocity of the air did not cause any noticeable vibra-tions of the sphere. The sphere was suspended inside the glasscolumn by the thermocouple wires, which were placed inside thesphere and connected to the data logger, Fig. 1. A guiding rod isused to ensure that the sphere was located along the axis of theglass column. The physical properties of the PCM �commercialgrade paraffin wax� and HTF �air� are given in Table 1. As isshown, the melting temperature of the PCM is 52°C and the per-centage decrease in volume upon melting is close to 10%. Also,the percentage decrease in the specific heat in the PCM uponphase change from solid to liquid is approximately 30%.

3 System Evaluation and OperationPrior to system operation, the two parts of the spherical capsule

are filled with molten wax; simultaneously, the thermocouple tubeassembly is then placed along the horizontal and vertical axes ofone half, as shown in Fig. 1. The two parts are then assembledupon solidification of the wax. The top part of the spherical cap-sule has a small opening of less than 1 mm in diameter. This is toallow for free movement of the air during melting and solidifica-tion. This also would prevent wax bursting through the sphere rimduring solidification due to volume expansion. The experimentalprocedure includes the following steps:

358 Õ Vol. 127, AUGUST 2005 Transactions of the ASME


• The air fan and heater are adjusted to the desired flow rateand temperature.

• The spherical capsule is placed in the air stream.• The temperature field is logged until all thermocouple read-

ings within the sphere approach the air temperature, which ishigher than the wax melting temperature.

• The spherical capsule is then moved to an air duct that oper-ates at ambient temperature and the same air speed as in themelting mode. Cooling continues until all temperatureswithin the sphere approach the ambient air temperature.

• Frequent calibration of the thermocouples and data loggersystem is performed to ensure the reproducibility and accu-racy of measurements.

An additional experiment is performed to determine the effectof the thermocouple wires on the system thermal performance.This test involved the use of only one thermocouple, which isplaced at the center of the PCM sphere. The experiments are per-formed as a function of sphere diameter, air temperature, and airflow rate. The melting time for the case of a single thermocoupleis compared against those for the spheres with ten thermocouples.The following are the main features for this comparison:

• The percentage volume of the ten thermocouple assembly is0.6% for the 2 cm diameter sphere and decreases to 0.06%for the 6 cm diameter sphere. On the other hand, the percent-age volume for the single thermocouple varies from 0.02% to0.002%.

• Energy reduction upon the insertion of the ten thermocouplesis less than 1%; while energy reduction for the single ther-mocouple is less than 0.1%. Energy reduction is caused bythe replacement negligible portion of the wax with the ther-mocouple wires.

• The melting and solidification time for the ten thermocouplesis less than the melting and solidification time for the singlethermocouple wire. Reduction varies from 5% to 12% upon adecrease in the sphere diameter and increase in the air tem-perature and air flow rate.

The diameter of the glass column is more than three times

Fig. 1 Single sphere energy storage system. Thermocouples number „1…, „7…, and „10… arelocated at radial positions equal to the inner sphere radius and at angular locations of À90 deg,0 deg, and 90 deg.

Table 1 Physical properties of PCM and HTF air properties arefor a temperature range of 25–100°C

Physical property Air �HTF�* Paraffin wax �PCM�

Melting temperature, °C ¯ 48.51Latent heat, kJ/kg ¯ 210Solid density, kg/m3

¯ 860Liquid density, kg/m3

¯ 780Vapor density, kg/m3 1.137 ¯

0.964Specific heat, kJ/kg °C 1.0048 2.9 �solid�

1.009 2.1 �liquid�Thermal conductivityW/m °C

2.49�10�2 0.24 �solid�

3.115�10�2 0.15 �liquid�Viscosity, kg/m s 1.9�10�5 0.205

2.15�10�5

Journal of Solar Energy Engineering AUGUST 2005, Vol. 127 Õ 359


larger than the diameter of the largest sphere. To assess the effectof the column diameter on the accuracy of the measurements, theair temperature is measured near the column wall and near thesurface of the spherical capsule. Measurements indicate variationsof less than �1°C. Similar measurements are made for the axialtemperature variations, where the air temperature is measured atone sphere diameter above and below the sphere. The axial tem-perature measurements indicate similar variations to the radialtemperature measurements, where the variations are below �1°C.This test indicates that the thermal load of the air stream is verylarge, when compared to losses to the surroundings and heating ofthe spherical capsule. Therefore, the assumption of isothermalconditions in the foregoing analysis is verified.

4 Error AnalysisTo estimate the uncertainties in the results presented in this

work, the approach described by Barford �19� was applied Theuncertainty in measurements is defined as the root sum square ofthe fixed error by the instrumentation and the random error ob-served during different measurements.

The measured experimental errors are �0.5°C for temperatures,�0.01 m/s for air velocity, 2.5�10�6 m for sphere inner and outerdiameter, and �1 s for time for melting or freezing. Accordingly,the resulting errors are �7.59%, and �6.65% in the calculatedNusselt and Fourier numbers.

5 Data AnalysisThe measurements, which include the temperature field inside

the PCM sphere and in the air stream and the air velocity, are usedto evaluate the heat transfer coefficient in the PCM inside thesphere during melting and freezing as well as the dimensionlesstime for melting and freezing. Data analysis is based on the fol-lowing equations:

• The heating load of the sphere

q�hh„Ao�Ta�Ts�… (1)

This equation is used to determine the heat transfer rate �q�from the surrounding moving air into the spherical shell. Thisequation requires the determination of the outside heat transferarea of the sphere (Ao) and measurements of the surrounding airtemperature (Ta) and the sphere outside temperature (Ts). Theheat transfer coefficient for the air moving around the sphere (hh)is obtained from Eq. �2�.

• The HTF heat transfer coefficient,

hh�„2��0.4 Reh0.5�0.06 Reh

0.3�Prh0.4…kh /�D0� (2)

The heat transfer coefficient on the air side is obtained from thecorrelation developed by Whitaker �20�. The correlation is validover the following ranges:

0.71�Prh�380

3.5�Reh�7.6�104

In the above equation, all properties are evaluated at Ta .

• The PCM heat transfer coefficient

hp�q/„Ai�Ti�Ti�1�… (3)

The heat transfer coefficient on the wax side is obtained fromthe heat flux equation. This requires measurements of the waxtemperature at two points in the wax near the inner wall of thesphere. In this case, it is assumed that the heat transfer rate fromthe sphere surface obtained from Eq. �1� remains constant and isequal to the heat transfer rate into the wax.

• The Fourier number correlation

Fo�a BibStc (4)

The Fourier number gives a measure of melting or solidificationtimes. It is proportional to the Biot and Stefan numbers in thePCM. The Biot number is the ratio of the heat transfer rate byconvection to conduction, while the Stefan number gives the ratioof the sensible heat to the latent heat.

• The PCM Nusselt number correlation

Nu�a StbRacFod (5)

The Nusselt number for the PCM is finally correlated as afunction of the Stefan, Biot, and Fourier numbers.

6 Results and DiscussionExperimental measurements are made for the following condi-

tions:

i. Copper sphere with diameters of 2, 3, 4, and 6 cm.ii. Air velocity of 4, 6, 8, and 10 m/s.iii. Inlet air temperatures during melting of 60, 70, 80, and

90°C.iv. Inlet air temperature of 24°C during solidification.

An example for the transient temperature profile during freez-ing and melting is shown in Fig. 2. The data is for an air velocityof 10 m/s, air temperature of 60°C, and a sphere diameter of 3 cm.As is shown, the melting time is much larger than the solidifica-tion time. This is associated with nonsymmetric melting, whereasa molten layer is formed at the inner surface of the spherical shellthe un-molten solid sphere descends in the melt. Also, a cellularmotion is induced due to the temperature gradient within the melt.During melting a considerable part of the heat added to the sphereis consumed in superheating the melt instead of melting the re-maining solid. This is enhanced in part by the natural convectioncells within the sphere. As a result, hotter melt at the base of thesphere would rise upward and loses part of its heat to the sur-rounding cooler melt and solid wax. This should initiate a naturalconvection cell within the molten wax. On the other hand, thesolidification process is dominated by conduction, where solidlayers grow from the sphere surface towards the center. This re-duces the melt volume rapidly and diminishes the role of naturalconvection.

The transient of the temperature profile along the axis of thesphere in the same direction of the air flow is shown in Fig. 3. Asis shown, large temperature gradients are found throughout the

Fig. 2 Variation in the transient temperature profile inside thePCM for an inlet air velocity of 10 mÕs, inlet air temperature of60°C, and a sphere diameter of 3 cm



sphere at small times. As melting progresses and the volume ofthe melt increases, then, natural convection cells smoothes out thetemperature field. This is evident in the top and middle parts of thesphere. However, a larger temperature gradient exists in the lowerpart of the sphere. This part of the sphere faces the hot air stream.Also, the un-melted solid wax drops in that region. As a result andbecause of the consumption of the latent heat of fusion a largetemperature gradient exists in that region. As is shown in Fig. 3,once the temperature exceeds the melting temperature throughoutthe sphere then the sphere is moved into the cold air stream. Ananalysis of the data shown in Fig. 3 shows the following:

i. Heating of the sphere from ambient temperature �23°C� tothe point where the first melt drop is formed takes 2.67minutes.

ii. Complete melting requires an additional 7.33 minutes ofheating.

iii. Once solidification is initiated, it takes 4.67 minutes toachieve complete solidification.

iv. Reaching ambient temperature requires an additional 10minutes.

The temperature field inside the PCM sphere is shown in Fig. 4.The temperature field is constructed using the temperature read-ings from the ten thermocouples. This allows for graphing thetemperature field as a function of the radial and angular location.For example, thermocouple �1� shown in Fig. 1 is located at aradial position equal to the sphere radius �r� and an angular loca-tion of ��90°�. Similarly, thermocouples �7� and �10� are locatedat the same radial position and at angular locations of �0°� and�90°�, respectively. The figure includes four temperature fields attimes equal to 7.33, 15.33, 18.67, and 34.67 minutes. At a timeequal to 7.33 minutes, about 20% of the PCM is above the meltingtemperature. This amount is found in an annular region near theinner surface of the sphere. At a time equal to 15.33 minutes, theentire PCM material is above the melting temperature. At thiscondition, superheating of the melt occurs throughout the PCMsphere with relatively small temperature gradients because of thesmall difference between the melting temperature and the HTFtemperature. However, larger temperature gradients are found for

Fig. 3 Transient of temperature profile along the sphere axisfor a sphere diameter of 3 cm, air velocity of 10 mÕs, and airtemperature of 60°C during energy storage and air temperatureof 23°C during energy release

Fig. 4 Temperature profile inside the sphere during various modes of melting for air velocityof 2 mÕs, inlet air temperature of 120°C, and a sphere diameter of 2 cm



the cases where the HTF temperature is raised to 70, 80, or 90°C.Further heating of the sphere and at a time of 18.67 minutes, alarger portion of the PCM approaches the HTF temperature. Dur-ing the cooling cycle and after 34.67 minutes, the temperaturecontours within the sphere are formed at equal radial distances,where the highest temperature is found at the sphere center andthe lowest at the surface.

Effects of the experimental parameters on the temperature tran-sients are shown in Figs. 5–7. Figure 5 shows variations in thetemperature transients for thermocouple �1� as a function of theHTF temperature. As is shown, increase in the HTF temperaturereduces the melting time. Also, the increased superheating effectsof the PCM at higher HTF temperatures results in longer solidifi-cation times. It should be noted that the decrease in the meltingtime levels off as the HTF temperature increases.

The effects of increasing the sphere diameter on the tempera-ture transients are shown in Fig. 6. As is shown, slight differencesare found between the temperature profiles of the 2 cm and 3 cmspheres. On the other hand, a wide deviation is found for the 6 cmsphere; where the melting time is almost double that for the smallspheres. The solidification times increases with the increase of thesphere diameter, however, this increase is not as dramatic as forthe case of melting.

Effects of the HTF velocity are shown in Fig. 7. As is shownthis parameter has a limited effect on the system performance. Theincrease in the air velocity reduces the air resistance to heat trans-

fer. However, the magnitude of this resistance remains muchlarger than that of the sphere wall or the PCM material.

Results for the specific melting and solidification times areshown in Figs. 8 and 9. The specific melting or solidification timesare defined as the time required for melting or solidifying a unitmass of the PCM. The specific melting time shows that the high-est time is obtained for the smallest sphere diameter. The specificmelting time decreases with the increase in the sphere diameter.This is because for the small sphere diameter the heat transferprocess is dominated by conduction; however, natural convectioneffects are more pronounced for the larger diameter spheres. Thisenhances the heat transfer process and reduces the specific melt-ing time. Variations in the specific solidification time are shown inFig. 9. As is shown the largest specific time is obtained for the 2cm sphere diameter. A further decrease occurs for the 3 cm spherediameter and then the specific solidification time starts to increasefor the 4 and 6 cm diameter spheres. The decrease and then de-

Fig. 5 Effect of the HTF temperature on temperature tran-sients at thermocouple „1… during melting and solidification fora sphere diameter of 2 cm and air velocity of 10 mÕs

Fig. 6 Effect of the sphere diameter on the temperature tran-sients at thermocouple „1… during melting and solidification forHTF temperature of 90°C and velocity of 10 mÕs

Fig. 7 Effect of the air velocity on the temperature transientsat thermocouple „1… during melting and solidification

Fig. 8 Variation in the measured specific melting time asa function of the sphere diameter, air temperature, and airvelocity



crease in the specific solidification time is a trade-off between thechange in specific heat transfer area and the increase in the ther-mal resistance within the PCM.

7 Correlations of Fourier and Nusselt NumberThe experimental data is used to determine the melting and

solidification times. Also, the data is used to calculate the PCMheat transfer coefficient. The data is then correlated as a functionof other operating condition expressed in dimensionless form. Thefollowing is a summary of these correlations:

• Fourier number for melting

Fm��0.199�St��0.56�Bi��0.212� (6)

which is valid over the following parameter range:

0.08�St�0.38

1.89�Bi�6

60�Th�90°C

The coefficient of determination (R2) for the above correlation is0.93.

• Fourier number for solidification

FS��0.344�St�0.62�Bi��2.01� (7)


0.08�St�0.38

1.89�Bi�6

23�Th�43°C

The coefficient of determination (R2) for the above correlation is0.91.

• Nusselt number for melting

Num��0.022�F ��0.68�St��0.62�Ra�0.22� (8)


0.007�F�0.2

0.08�St�0.38

50,000�Ra�35,000,000

60�Th�90°C

The coefficient of determination for (R2) the above correlation is0.91.

• Nusselt number for solidification

NuS��0.023�F ��0.36�St�0.28�Ra�0.21� (9)


0.003�F�0.04

0.08�St�0.38

1,100,000�Ra�28,000,000

23�Th�43°C

The coefficient of determination for (R2), the above correlation is0.94.

It should be noted that dependence of Fourier number duringmelting is inversely proportional to Stefan and Biot numbers. Thisimplies that the increase in Stefan or Biot numbers reduces theFourier number of the times for melting or solidification. A higherStefan number implies increase in the sensible heat thermal loadin comparison with the latent heat thermal load. Similarly, anincrease in the Biot number implies an increase in heat transfer byconvection versus that by conduction. Variations in the Nusseltand Fourier number as a function of the sphere diameter, air ve-locity, and air temperature are shown in Figs. 10–13. The datashows an increase in the Nusselt number upon an increase in thesphere diameter and air temperature. A lower dependence on theair velocity is found for both Nusselt and Fourier numbers. As isshown in Figs. 12 and 13 the melting Fourier number decreasesupon increase in the air temperature and opposite is found for thesolidification process.

8 ConclusionsThe study provides the literature with a detailed picture for the

temperature field during melting and solidification inside thesphere of phase change material. This achieved by the use of tenthermocouples. Previous literature studies are limited to the reportof one or two temperature readings inside the sphere. The mea-surements are made as a function of the sphere diameter, air tem-perature, and air velocity. Results show a higher melting time andhigher heat transfer coefficient during the melting process. This iscaused by the difference in the melting and solidification mecha-

Fig. 9 Variation in the measured specific solidification time asa function of the sphere diameter, air temperature, and airvelocity

Fig. 10 Variation in the melting Nusselt number as a functionair temperature, air velocity, and sphere diameter. Each dataset includes four air velocities.



nisms. During melting, the molten wax has hotter and coolerzones. As a result, convection cells are initiated within the melt.Also, the remaining solid wax descends within the melt. Thisresults in increased mixing within the melt wax and enhanced heattransfer rates. The natural convection role is enhanced upon anincrease in the sphere diameter and the air temperature. On theother hand, during solidification the wax layers are formed in-

ward, starting at the sphere walls. As solidification progresses, themelt volume becomes smaller and the role of natural convectiondiminishes rapidly.

AcknowledgmentThis research was supported by the Research Administration of

Kuwait University, Project No. ELC-012.

Nomenclature

A � Sphere heat transfer area, m2

Bi � PCM Biot number, dimensionless, Bi�hpD0 /kpCp � The specific heat at constant pressure, kJ/kg °CD � Sphere diameter, m

F0* � Fourier melting number, dimensionless,F0*�(�ptm)/D0

2

Fo � Fourier number, dimensionless, Fo�(�pt)/D02

g � Gravitational acceleration, m/s2

h � Heat transfer coefficient, kW/m2 °Ck � Thermal conductivity, kW/m °C

M � Mass flow rate, kg/sNup � PCM Nusselt number, Nup�hpD0 /kpPrh � HTF Prandtl number, Prh�Cph�h /khRap � PCM Rayleigh number, Rap�g�pCpp�pD0

2T/�pkpReh � HTF Reynolds number, Reh��hVhD/�h

St � Stefan number, St�CppT/pT � Temperature, °CU � The overall heat transfer coefficient, kW/m2 °C

Greek Symbols

�p � PCM thermal diffusivity, �p�kp /(�pCpp), m2/s�p � PCM expansion coefficient, �p��p /(�pTp)� � dynamic viscosity, kg/m s

p � latent heat of the PCM, kJ/kg� � density, kg/m3

T � difference of HTF and PCM temperatures, T�Th�Tm , °C

Tp � PCM temperature difference, °C�p � PCM density difference over a temperature difference

of Tp , kg/m3

Subscripts

i � inside of the PCM sphere or inlet airh � heat transfer fluidm � melting pointo � outside of the PCM sphere or outlet airp � phase change materialw � wall

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Fig. 11 Variation in the solidification Nusselt number as afunction air temperature, air velocity, and sphere diameter.Each data set includes four air velocities.

Fig. 12 Variation in the melting Fourier number as a functionair temperature, air velocity, and sphere diameter. Each dataset includes four air velocities.

Fig. 13 Variation in the solidification Fourier number as afunction air temperature, air velocity, and sphere diameter.Each data set includes four air velocities.



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Documents

Heat Transfer During Phase Change of Paraffin Wax Stored in Spherical Shells