International Journal of Heat and Mass Transfer 54 (2011) 4923–4930

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Approximate analytical solutions for the solidification of PCMsin fin geometries using effective thermophysical properties

Thomas Bauer ⇑Institute of Technical Thermodynamics, German Aerospace Center (DLR), Pfaffenwaldring 38-40, 70569 Stuttgart, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 November 2010Received in revised form 28 June 2011Accepted 29 June 2011Available online 25 July 2011

Keywords:Phase changeConductionFinEffective propertiesMeltingSolidification

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.07.004

⇑ Tel.: +49 711 6862 681.E-mail address: thomas.bauer@dlr.de

Heat transfer enhancement of phase change materials (PCMs) is essential in order to achieve high chargeand discharge powers of latent heat storage systems. The utilisation of highly conductive fins is an effec-tive method to improve heat transfer. In the presented paper, solidification times of two fundamentalgeometries are examined by analytical modelling and numerical computation. These geometries arethe finned plane wall and a single tube which is radial-finned on the outside. The paper describes approx-imate analytical solutions based on the effective medium approach which include the boundary condi-tions, as well as material and geometric parameters.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Thermal energy storage systems have a high potential in termsof the utilization of the intermittent renewable sources and theimprovement of the energy efficiency. The three major types ofthermal energy storage are sensible, latent and thermo-chemical[1–3]. This paper considers latent heat storage using phase changematerials (PCM) as a storage medium. PCMs can be classified in dif-ferent ways. For example, PCMs can be classified as either organic(e.g. paraffins, fatty acids, sugar alcohols) or inorganic materials(e.g. water, salt hydrates, anhydrous salts). Most PCMs are basedon the solid–liquid phase change. Here, the process of meltingand solidification is utilized. Characteristic values of a PCM arethe phase change enthalpy and the phase change temperature.For the different temperature regions, material groups includelow (e.g. water/ice), medium (paraffins, fatty acids, sugar alcohols,salt hydrates) and high temperature PCMs (e.g. anhydrous salts).PCMs may be utilized in different applications. They can stabilisethe temperature, such as for the indoor temperature in buildings,or store and supply heat or cold in a small temperature interval[1–3]. It has been demonstrated, that in particular the utilizationof two-phase heat carriers (e.g. water/steam) is advantageous inorder to realise approximately isothermal storage systems [4].The capacity of the storage system is mainly determined by the la-tent heat value of the PCM. The charge and discharge power is

ll rights reserved.

dominated by the thickness of the PCM layer (or the distance ofthe heat exchanger surfaces) and the thermal diffusivity (or ther-mal conductivity) of the PCM. In many applications the lowthermal conductivity of the PCM presents a major challenge tothe design of storage systems with a high discharge power. Theutilization of a single heat exchanger, such as a tube register, isoften not economical. The additional usage of fins is one majorstorage design concept to reduce the tube register volume. Thefin volume should be as small as possible in order to enhance theeconomics of the storage system and to reach a high PCM volume(or large system capacity). In other words, an optimised utilisationof the fins is important. Here parameters such as the length, thick-ness, volume fraction and spacing of the fins can be adapted. Forthe fins, conductive materials such as graphite foil, aluminiumand copper can be used.

Literature on the modelling of solid–liquid phase change is vast.The heat transfer enhancement using fins within the PCM has beenalso examined by several authors. An overview of the literature isgiven by Ismail [5] and Zalba et al. [6]. Important modelling criteriainclude the geometry, the free convection in the melt and the fintype.

The major purpose of the presented work is to gain a betterunderstanding of the effective utilization of fins in latent heat stor-age design. In this work two geometries are discussed. The firstgeometry is a plane wall and the second geometry is a tube sur-rounded by the PCM-fin arrangement. As a boundary condition aconstant wall temperature is assumed. There are several character-istic parameters for this transient heat conduction problem with

Nomenclature

cp specific heat (J/kgK)CF cylinder factorFo Fourier number Fo = at/l2

k thermal conductivity (W/mK)l fin length (m)l/R dimensionless fin lengthL latent heat of the PCM (J/kg)m gradient of the fitted straight lineNEF non-equilibrium factorq heat flow (W)R tube radius (m)S shape factor (m)St Stefan number St = cp(Tm � Tw)/Lt time (s)T temperature (K)vf fin volume fraction vf = 1 � vs

vs PCM volume fraction vs = ws/(ws + wf)wf half width of the fin (m)ws half width of the PCM (m)ws/l dimensionless PCM width

x x-coordinate of the physical models (m)y y-coordinate of the physical models (m)

Greek symbolsa thermal diffusivity (m2/s) a = k/(qcp)e dimensionless fin factor e = vfkf/(vsks)q density (kg/m3)

Subscriptsaas approximate analytical solutioncyl cylinder geometryeff effective property of PCM and finf fini initialm meltingnum numerical computed Fluent results storage material (PCM)w wall or wall geometry

4924 T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930

phase change. They include the temperature boundary condition,the fin length, the tube radius, the volume fraction of fin andPCM, the spacing of the fins and the thermophysical properties ofPCM and fin. The motivation of the present work is the develop-ment of approximate analytical solutions for the solidification timeof PCMs including these parameters.

The approach taken in this work is based on utilization of theeffective thermophysical properties of the PCM-fin sandwich struc-ture in order to describe the solidification time of the PCM. Variousauthors have utilized the effective medium approach, or volumeaveraging method, to describe transient heat conduction problemswith phase change in various physical structures (e.g. foams andporous media) [7–12]. However, the effective medium method isusually not applied to macroscopic PCM-fin structures. The majoradvantage of the method is that the complexity of the heat conduc-tion problem can be reduced considerably. The heat transfer pro-cesses in the two media, here fins and PCM, are not describedseparately. Instead, the heat transfer is described for one mediumon a continuum macroscopic level. The essential assumption ofthe effective medium approach is that both media are in local ther-mal equilibrium [13]. The presented paper examines the area ofvalidity of this assumption for structures with thin fins in thePCM. The paper describes a large number of numerical modellingruns with an exact description of the heat transfer in the PCMand fin. These numerical results are compared to analytical equa-tions using effective thermophysical properties of the fin andPCM. The paper focuses on critical parameters which need to bemet in order to achieve a good agreement between the exactnumerical model and the analytical model with effective proper-ties. Cases where the equilibrium is not fulfilled are another majorfocus of this paper. Here, approximate analytical equations for thesolidification time based on the effective medium solution aredeveloped.

2. Assumptions for transient heat conduction with phasechange

The numerical analysis outlined in this paper is based ontwo-dimensional physical models. The only mode of heat transferconsidered is heat conduction. In this work the quasistationaryapproximation is assumed. This means that the sensible heat canbe neglected compared to the latent heat. Hence, small Stefan num-

bers St 6 0.1 are considered. In the simulation one-dimensional heatconduction in the fins (thin fin approximation) and two-dimensionalconduction in the PCM is modelled. For the fin and PCM, isotropicand homogeneous material properties are assumed. Furtherassumptions are that the thermophysical properties are indepen-dent of temperature and that the PCM properties are equal in thesolid and liquid phase. Hence, density differences and void formationare not included. A clearly defined solid–liquid interface due to melt-ing of the PCM at a single temperature is assumed. In all simulations,the initial temperature of both media is at the melting temperature.The PCM at melting temperature is in the liquid phase (Ti = Tm).Hence, no temperature gradient exists in the PCM melt and freeconvection in the melt is assumed to be absent. The thermalconductivity ratio of fin and PCM is in the range 10 < kf/ks < 1000for the parametric study. For the reference case modelling this rangewas extended to 1.5 < kf/ks < 3000.

Numerical solutions have been obtained using a commercialsoftware package (Fluent Inc., Version 6.2.16). The finite volumemethod has been used for the transient modelling. The solidifica-tion model is based on the enthalpy-porosity method describedby Voller and Parkash [14] and Brent et al. [15]. The time step sizehas been adapted in order to achieve 100 to 500 time steps for theentire solidification process. The primitive rectangular mesh con-sisted of 280 to 1200 volume elements. The impact of the energyconvergence criterion on the solidification time has been found acritical parameter. This criterion was then decreased until nonoticeable change in the solidification time occurred. The numer-ical computer code for the phase change (Fluent) has been verifiedby several test cases. They included the melting of a slab of icewith a thickness of 10 cm and two-dimensional melting in a rect-angular body under imposed temperature. Analytical equations inliterature describe these cases [16]. In addition, a reference cylin-drical geometry with fins and PCM has been examined by anothercode. This internal two-dimensional verification of the Fluent codewith an own developed finite difference method showed a differ-ence of less than 3% for the melting time of the complete PCMvolume.

3. Effective thermophysical properties of PCM and fin

For the macroscopic analysis of heat conduction in heteroge-neous media, the use of effective, or local volume averaged,

Cooled isothermal wall

fins

symmetry

symmetry

fin

adia

batic

boun

dary

PCMCoo

led

Iost

herm

alw

all

l

ws wfx

y

Fig. 1. Definition of the plane wall geometry. The figure shows the schematicdiagrams of the storage setup with fins without PCM (top) and the two-dimensionalphysical model (bottom).

T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930 4925

material properties is a well established method [17]. There arevarious physical structures where this method is applied. Examplesinclude composites, filled foams and saturated powder fills. Thesestructures may not only be modelled for thermal energy storagebut also for other applications such as in hydroscience. The so-lid–liquid phase change in fills, capsules, foams and compositeswith effective material properties has been modelled by severalauthors. Weaver and Viskanta reported about an analytical andexperimental study on a PCM in a spherical bed. They used wateras a PCM and showed that for glass beads the heat transfer couldbe expressed by effective media properties. However, for highlyconductive aluminium beads they showed that the local thermalequilibrium becomes invalid [7]. Shiina and Inagaki examined cir-cular capsules filled with various PCMs, as well as metal foams forheat transfer enhancement. They compared experiments withmodelling results. The modelling assumed effective material prop-erties of the PCM-foam geometry [8]. Chow et al. modelled cap-sules with various heat transfer enhancement arrangements.They also included a metal-PCM composite (Ni–LiH mixture) witheffective thermophysical properties in their analysis [9]. Inaba andTu examined composites based on a paraffin and high densitypolyethylene as a supporting material. They characterised theseshape-stabilized PCM in terms of their thermophysical properties.They showed that the measured composite properties (density,specific heat and latent heat) can be calculated by effective proper-ties using the properties of each constituent [10]. Morisson mod-elled PCM-composites based on a porous graphite matrices andnitrate salts as a PCM. They applied the volume averaging proce-dure and assumed local thermal equilibrium. Their work aimedto identify a relationship between the effective (macroscopic)properties and the structure and properties of the single compo-nents in terms of the conductivity [11]. Alawadhi and Amon ana-lysed heat sinks composed of organic PCMs and conductive metalmatrices (aluminium, steel, titanium). For the numerical model-ling, they employed effective thermophysical properties of the heatsink. In their work, it was found that modelling by average materialproperties is a valid approach [12].

The described literature shows that the length scale of the en-tire geometry needs to be large compared to the PCM-enhance-ment structure in order to apply the volume averagingprocedure. The effective density, heat capacity and latent heatcan be obtained by the simple averaging method (Eqs. (1)–(3)).

qeff ¼ vsqs þ v f qf ð1Þ

cp;eff ¼qs

qeffvscp;s þ

qf

qeffv f cp;f ð2Þ

Leff ¼qs

qeffvsL ð3Þ

Heat conduction through heterogeneous media depends not only onthe conductivity and volume of each material but also on the phys-ical structure of the phases. Important aspects are the interconnec-tion of the conductive structure (or the continuity of the conductivematerial), contact resistances and the thermal conductivity ratio ofthe enhancement structure and the PCM [7,17]. For the effectivethermal conductivity, a large number of models have been devel-oped. Examples are geometries such as beds of spherical conductiveparticles and highly interconnected foams. The present work aimsfor a high effective thermal conductivity with a small volume frac-tion of highly conductive fin material (1 to 10 Vol%). This require-ment is met by the parallel arrangement of fins and PCM in theheat flow direction. Eq. (4) defines the effective conductivity ofthe parallel arrangement assuming one-dimensional steady-stateconductive heat transfer [17].

keff ¼ v sks þ v f kf ð4Þ

4. Plane wall geometry

4.1. Analytical solution without fins

Analytical solutions for the non-linear problems of melting andsolidification date back to the 19th century [16,18]. The solidifica-tion time of a PCM slab (plane wall) can be expressed by Eq. (5).Important assumptions are that the liquid PCM is at the meltingtemperature Tm, the PCM is confined in a space of a finite thickness0 < x < l, for times t > 0 the boundary surface at x = 0 is maintainedat a constant temperature Tw below Tm and the sensible heat is ne-glected compared to the latent heat. Further assumptions havebeen already defined in Section 2.

tw;s ¼Lqs l2

2ksðTm � TwÞð5Þ

4.2. Assumptions of the physical model with fins and PCM

It is generally accepted that conduction heat transfer betweenwall and PCM is significantly improved by additional fins withinthe PCM volume. Heat transfer phenomena of PCMs with fins havebeen studied by several authors. A good overview of the numericaland experimental work is given by Lamberg and Siren [19]. Thework on approximate analytical solutions is generally limited.Lamberg developed approximate analytical solutions for the planewall geometry with finite fins and PCM. The approximate analyti-cal method by Lamberg is based on a separate treatment of heattransfer from the end wall and the fin. The method requires an ana-lytical description of the one-dimensional temperature distribu-tion in the fin. The described approximate analytical methodrequires several equations to be solved in order to calculate thetime for complete solidification of the PCM [19].

The present paper aims for an alternative approximate analyti-cal solution for the solidification time of the PCMs in fin geome-tries. Fig. 1 at the top shows the considered plane wall geometry.Due to symmetry conditions, a simple two-dimensional physicalmodel can be defined (Fig. 1 at the bottom). In the physical modelthere are four boundary conditions. The heat extraction wall on the

02468

1012141618202224262830

0 10 20 30 40 50 60 70 80 90 100 110 120Dimensionless fin factor ε = vfkf / (vsks)

t num

/ t w

,eff

ws/l=0.5; St=0.1ws/l=0.5; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.1; St=0.1ws/l=0.1; St=0.01

Fig. 2. Plane wall geometry: normalised modelling results depending on the finfactor e.

1.1

1.15

1.2

1.25

1.3

/ t w

,aas

ws/l=0.5; St=0.1ws/l=0.5; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.1; St=0.1ws/l=0.1; St=0.01

4926 T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930

left hand side is isothermal and the thickness of the isothermalwall is assumed zero. For the wall on the right hand side, the adi-abatic condition is assumed. For the other walls (top and bottom)symmetry conditions can be applied. For the simulation, dimen-sionless geometric parameters are defined. The dimensionlessPCM width is varied up to ws/l 6 0.5. This assumption results inlong fins compared to the fin spacing. Hence, heat transfer intothe PCM is dominated by indirect heat transfer from the wall viathe fin to the PCM. For the storage application, a small volume frac-tion of the fin vf 6 0.1 compared to the total fin-PCM volume isstudied.

Initial modelling showed that important parameters are thedimensionless PCM width ws/l, the volume fraction of the fin vf

and the PCM vs = 1 � vf, the conductivity ratio of fin to PCM kf/ks

and the Stefan number St. Table 1 shows the total range of theparameters presented in this paper. Nine geometries were mod-elled (3 � 3). Four ratios of the thermal conductivity of the fin toPCM and two Stefan numbers lead to eight parameters (4 � 2).All combinations resulted in a total of 72 (8 � 9) simulations.

4.3. Numerical modelling results with fins and PCM

Numerically computed results showed that the fin and PCMmaterial can be described by a single material with effective prop-erties (Eqs. (1)–(4)), if the fin is sufficiently thin and the dimen-sionless PCM width ws/l is sufficiently small. Eq. (6) defines thesolidification time of a PCM-fin arrangement with effective proper-ties. Eq. (6) differs in terms of the effective properties compared toEq. (5).

tw;eff ¼Leff qeff l

2

2keff ðTm � TwÞð6Þ

As an important factor, the dimensionless fin factor e was identified(Eq. (7)). The fin factor can be considered as a one-dimensional stea-dy-state heat flow ratio, where S is the shape factor. The numeratorexpresses the flow in the fin which increases proportionally withthe thickness wf (or the volume fraction vf) and the thermal conduc-tivity of the fin kf. Similarly, the denominator defines the heat flowwithin the PCM without phase change.

e ¼qf

qs¼ kf Sf DT

ksSsDT¼ wf kf

wsks¼ v f kf

vsksð7Þ

For the analysis, the 72 numerical modelling results (defined in Ta-ble 1) have been normalised. The computed solidification timetw,num has been divided by the solidification time of the effectivemedium tw,eff according to Eq. (6). Fig. 2 shows the values tw,num/tw,eff

as a function of the fin factor e.For fin factors e close to zero, numerical modelling results show

that the normalised solidification time tw,num/tw,eff is approximately1 (Fig. 2). In other words, it can be said that a good agreement be-tween the numerical computed time tw,num and the analyticallytime with effective properties tw,eff is achieved. For larger valuesof the fin factor e, results show that the normalised time tw,num/tw,eff

increases in good approximation linearly with the fin factor e. Thenormalised computational results also show almost identicalvalues for different Stefan numbers (St = 0.1 and St = 0.01). This

Table 1Modelling parameters and their range in Fluent for the plane wall geometry.

Geometry parameters Material/boundary parameters

ws/l vf kf/ks St

0.1 0.01 10 0.010.3 0.05 100 0.10.5 0.1 300

1000

means that, due to the normalisation, results are approximatelyindependent of the Stefan number in the considered range. Ananalysis of the numerical results showed that the gradient m ofthe line is equal to (ws/l)2 (Fig. 2). Hence, cases with large fin factore can be also described analytically. From the results shown inFig. 2, the approximate analytical solution tw,aas was defined (Eq.(8)). Fig. 3 shows the deviations of the computed solutions tw,num

and the newly defined analytical solution tw,aas. The results showa deviation of less than ±10% for all cases. For most cases, it canbe seen that the deviation is around or below 5%. In this work,the term non-equilibrium factor (NEF) has been defined. The NEFdescribes the additional factor which is not included in the effec-tive medium solution tw,eff.

tw;aas ¼ws

l

� �2 v f kf

v sks|ffl{zffl}e

þ1

2664

3775tw;eff ¼ NEFw � tw;eff ð8Þ

Additionally, the validity of the Eq. (8) was tested by the assessmentof a reference case. Table 2 defines the values of the reference case.A single parameter has been set to a minimum or maximum valueand each numerical solution has been compared with the result ofEq. (8). The parameter values varied in a wide range. Resultsshowed that the accuracy of the solution is not affected by the abso-lute size of the geometry, if constant values for ws/l and vf are as-sumed. The l2 dependency is in accordance with the definition ofthe Fourier number Fo = at/l2. For all cases defined in Table 2 themaximum deviation of the computed solidification time tw,num fromthe approximate analytical solution tw,aas was below ±5%. These re-sults show that Eq. (8) is valid for the range of thermophysical prop-erties and boundary conditions relevant to PCM-storage design.

0.9

0.95

1

1.05

0 10 20 30 40 50 60 70 80 90 100 110 120Dimensionless fin factor ε = vfkf / (vsks)

t num

Fig. 3. Plane wall geometry: deviation of approximate analytical and numericalresults.

Table 2Reference case and single parameter analysis for the plane wall geometry.

Parameter Symbol Unit Min. value Reference value Max. value

Geometryla Fin length m 0.001 1 1000ws/l Dim. PCM width 1 0.1 0.3 0.5vf Fin vol. fraction 1 0.01 0.05 0.1

Thermophysical propertieskf Fin conductivity W/mK 1 150 500qf Fin density kg/m3 100 1000 20000cp,f Fin capacity J/kgK 200 1200 3000ks PCM conductivity W/mK 0.05 0.5 100qs PCM density kg/m3 100 2000 20000cp,s PCM capacity J/kgK 200 1500 2000b

L PCM latent heat J/kg 75000 b 100000 500000

Boundary conditionTm � Tw Temp. gradient K 0.1 5 6.67 b

a Values ws/l and vf were kept constant.b Limited by the assumption St 6 0.1 for tw,eff in Eq. (6).

T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930 4927

From the numerical modelling it can be concluded that thereare two major cases in terms of the non-equilibrium factor NEF.For NEF �1 the shortest solidification time for a given volume frac-tion and heat conductivity of the fin and PCM is achieved. In thiscase, the effective medium approach is valid. The solidificationtime can be calculated by the solution of a single material witheffective properties of the PCM-fin laminate teff. The analysis ofthe temperature profiles showed that, along the x-axis, the temper-ature differences in y-direction between fin and PCM are small(Fig. 4 top). Hence, it can be concluded that the fin and PCM are lo-cally (in y-direction) in thermal equilibrium.

Values NEF > 1 result in longer solidification times compared tothe solidification time of the effective medium teff (Eq. (8)). In otherwords, in these cases the effective medium approach can not beapplied. Instead the longer solidification time can be describedby the additional non-equilibrium factor defined in this work. ForNEF > 1, there is a lower temperature throughout the fin, or astronger cooling throughout the fin, compared to cases whereNEF � 1. The analysis of the temperature profiles showed that,along the x-axis, the temperature differences in y-direction be-tween fin and PCM are large (Fig. 4 bottom). Hence, it can be con-cluded that the fin and PCM are locally not in thermal equilibrium.The dimensionless PCM width ws/l, or the fin spacing, has beenidentified as the major design parameter to reduce the NEF to-wards 1. The adaption of this parameter allows for an optimalusage of the fin volume. For example, two thin fins with half ofthe thickness wf can replace a single fin. In this example the vol-ume fractions of the fin and PCM remain constant. Also, the solid-ification time of the effective medium teff is not altered. There is,

ε = 0.101

vf = 0.01

kf/ks = 10

ε = 111.1

vf = 0.1

kf/ks = 1000

Fig. 4. Plane wall geometry: example of the temperature profile in the solid phasefor a small fin factor e (top) and a large e value (bottom). For both cases constantvalues are St = 0.1 and ws/l = 0.3. The temperature scale is plotted on the left handside in Kelvin.

however, a significant reduction of the factor NEF due to the re-duced spacing of the fin (wf) and PCM (ws). Eq. (8) shows the qua-dratic dependency of the dimensionless PCM width ws/l within thefactor NEF.

4.4. Comparison of analytical solution with experimental literatureresults

In order to verify the approximate analytical solution (Eq. (8)),this solution has been compared to experimental data of themelting time reported in the literature [20]. The consideration ofmelting, rather than solidification, requires the adaption of thetemperature indices of Eq. (6). In the experiments, fins made ofaluminium and n-Octadecane with a melting temperature of28 �C as a PCM have been used. Variable parameters of the exper-iments were the wall temperature and the fin thickness. Thepublication defines the geometry, the material properties andboundary conditions exactly. For the comparison, the reportedmaterial properties have been utilized. The density values of thesolid and liquid range have been averaged. Constant geometricparameters of the experiment were l = 127 mm andws + wf = 15.875 mm. Results of the normalized melting time havebeen also reported. Table 3 shows the experimental results of themelting time tw,Henze and the calculated results by the approximateanalytical solution tw,aas. It needs to be considered that severalexperimental aspects are not included in the approximate analyti-cal solution. They include the aspects such as the free convection inthe melt, contact resistance between wall and fin and the deviationbetween initial and melting temperature (in the model 0 K and inthe experiment 1 K). Also, in the experiment the Stefan numberand the fin volume fraction vf were large compared to the rangeexamined in the analytical modelling. Although the modelling sim-plifications will lead to some errors, it can be seen that a goodagreement between the experimental and analytical modellingvalues are obtained (Table 3). The following section discusses themore complex hollow cylinder geometry using the same method-ology as for the plane wall.

Table 3Comparison of approx. analytical equation and experimental values by Henze.

Tw � Tm

[K]wf

[mm]ws/l vf St tw,Henze

[s]tw,aas

[s]Error(%)

20 3.175 0.1 0.2 0.22 7469 7086 +510 3.175 0.1 0.2 0.11 15170 14173 +710 1.5875 0.1125 0.1 0.11 21005 21308 �1

4928 T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930

5. Hollow cylinder geometry

5.1. Analytical solution without fins

An analytical solution for the solidification time of a PCMaround a cooled tube without fins (hollow cylinder of PCM) is givenby Baehr [21]. Eq. (9) defines this analytical solution as the productof the solution of the plane wall tw,s and the geometric cylinder fac-tor CF (Eq. (10)). For this solution, the assumptions are similar as inSection 4 (plane wall geometry). The solid PCM is at the meltingtemperature Tm, the PCM is confined in a space of a finite thicknessR < x < R + l, for times t > 0 the boundary surface at x = R is main-tained at a constant temperature Tw below Tm and the sensible heatis neglected compared to latent heat. Further assumptions havebeen defined in Section 2. The geometric cylinder factor CF de-pends solely on the dimensionless fin length l/R (Eq. (10)). Fig. 5shows this function up to a maximum value of l/R = 100. Highervalues are usually irrelevant for PCM-storage engineering.

tcyl;s ¼ tw;sCF ð9Þ

CF ¼ lnlRþ 1

� �Rlþ 1

� �2

� 12þ R

l

� �ð10Þ

wswf

PCMx

y

5.2. Assumptions of the physical model with fins and PCM

There are different configurations of finned tubes. The fins canbe located within the tube or connected to the outer circumfer-ence. The orientation and shape of the fin is also characteristic.Simple geometries are axial-finned and radial-finned tubes. Ismailand Zalba give an overview of experimental and numerical studiesof finned tubes [5,6,22]. Heinisch examined analytically a tubewith radial-fins on the outside by the Laplace method. The meltingand solidification time was calculated by means of numericalmethods and no explicit analytical solution for the phase changetime has been developed [23].

The work presented here considers PCM-storage systems basedon a register made of radial-finned tubes. The tubes contain theheat carrier in order to charge and discharge the system. In orderto gain an understanding of the entire system, a single tube whichis radial-finned on the outside is examined in detail. The wall tem-perature along the tube can be approximately isothermal, if a two-phase heat carrier is utilized. An example of such carrier fluid ispressurised water/steam. Here, the condensation and evaporationprocess is utilised [4].

As for the plane wall geometry (Section 4), the effective med-ium method in the hollow cylinder geometry with fins and PCMhas been applied. The aim has also been the development of anapproximate analytical solution for the solidification time of the

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50 60 70 80 90 100

Dimensionless fin length l/R

Cyl

inde

r fa

ctor

CF

Fig. 5. Geometric cylinder factor CF depending on the dimensionless fin length l/R.

PCMs. Fig. 6 at the top shows the considered geometry and atthe bottom the two-dimensional physical model including axialsymmetry conditions. In the physical model there are four bound-ary conditions. The tube for heat extraction is isothermal and thethickness of the wall is assumed zero. In the physical model thisis the left hand wall. The circumference area is adiabatic, whichcorresponds to the right hand wall in the physical model. For theother two walls symmetry conditions can be applied. In the phys-ical model these two walls are at the top and bottom. For the sim-ulation, dimensionless geometric parameters have been defined.The dimensionless PCM width varies up to ws/l 6 0.5. As for theplane wall geometry, this assumption results in long fins comparedto the fin spacing. Hence, heat transfer is dominated by indirecttransfer from the wall via the fin to the PCM. For the storage appli-cation, a small volume fraction of the fin vf 6 0.1 compared to thetotal fin-PCM volume has been assumed.

Table 4 shows the numerical modelling parameters and theirrange. In total 27 geometries have been modelled (3 � 3 � 3). Fourratios of the thermal conductivity of the fin to PCM and two Stefannumbers resulted in eight parameters (4 � 2). In total 216 (8 � 27)simulations have been carried out.

5.3. Numerical modelling results with fins and PCM

As for the geometry of the plane wall, the numerical modellingresults of the cylindrical geometry have been normalised. Thenumerical solidification time has been divided by the solidificationtime of the effective medium according to Eq. (11).

tcyl;eff ¼ tw;eff CF ð11Þ

These normalised results are plotted as a function of the fin factor ein Fig. 7. The definition of the fin factor (Eq. (7)) holds not only forthe plane wall geometry, but also for the cylinder geometry. Fig. 7presents three graphs with different l/R ratios.

fin

lR

Fig. 6. Definition of the cylinder geometry with radial fins. The figure shows theschematic diagrams of the storage setup with fins (top) and the two-dimensionalphysical model (bottom).

Table 4Modelling parameters and their range in Fluent for the cylinder geometry.

Geometry parameters Material/boundary parameters

ws/l l/R vf kf/ks St

0.1 1 0.01 10 0.010.3 10 0.05 100 0.10.5 100 0.1 300

1000

l/R = 1

0

2

4

6

8

10

12

14

16

18

20

22

24

t num

/ t c

yl,e

ff

ws/l=0.5; St=0.1ws/l=0.5; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.1; St=0.1ws/l=0.1; St=0.01

l/R = 10

0

2

4

6

8

10

12

14

16

18

20

22

24

t num

/ t c

yl,e

ff

ws/l=0.5; St=0.1ws/l=0.5; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.1; St=0.1ws/l=0.1; St=0.01

l/R = 100

0

2

4

6

8

10

12

14

16

18

20

22

24

0 10 20 30 40 50 60 70 80 90 100 110 120Dimensionless fin factor ε = vfkf / (vsks)

t num

/ t c

yl,e

ff

ws/l=0.5; St=0.1ws/l=0.5; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.1; St=0.1ws/l=0.1; St=0.01

Fig. 7. Hollow cylinder geometry: normalised modelling results depending on thefin factor e.

m = 0.7692 (L/R)-0.2498

m = 0.7407 (L/R)-0.2492

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80 90 100 110

Dimensionless fin length l/R

Gra

dien

t m

St=0.1St=0.01

Fig. 9. Hollow cylinder geometry: impact of dimensionless fin length l/R on thegradient m.

l/R = 1

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Dimensionless fin factor ε = vfkf / (vsks)

t num

/ t c

yl,a

as

ws/l=0.1; St=0.1ws/l=0.1; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.5; St=0.1ws/l=0.5; St=0.01

l/R = 10

1

1.05

1.1

1.15

1.2

1.25

1.3

t num

/ t c

yl,a

as

ws/l=0.1; St=0.1ws/l=0.1; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.5; St=0.1ws/l=0.5; St=0.01

T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930 4929

These results presented bear several similarities to the planewall results. They agree in terms of the following characteristics:

� tnum is approximately tcyl,eff for fin factors e close to zero.� tnum/tcyl,eff increases linearly with the fin factor e.� tnum/tcyl,eff is virtually independent of the Stefan number, for

0.01 6 St 0.1.

0

2

4

6

8

10

12

14

16

18

20

22

24

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28Dimensionless fin factor ε (ws/l)²

t num

/ t c

yl,e

ff

l/R=1; St=0.1l/R=1; St=0.01l/R=10; St=0.1l/R=10; St=0.01l/R=100; St=0.1l/R=100; St=0.01

m = 0.7666m = 0.7381

m = 0.4357

m = 0.4202

m = 0.2426

m = 0.2343

Fig. 8. Hollow cylinder geometry: normalised modelling results depending one�(ws/l)2.

Following the methodology of the plane wall geometry, the cor-relation (ws/l)2 has been utilised. Fig. 8 shows the same numericalmodelling results as presented in Fig. 7 with the factor e�(ws/l)2 inthe abscissa. As can be seen, the three plots shown in Fig. 7 can bereduced to one plot (Fig. 8). Fig. 8 shows three major linear slopes

0.9

0.95

Dimensionless fin factor ε = vfkf / (vsks)

l/R = 100

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 10 20 30 40 50 60 70 80 90 100 110 120

Dimensionless fin factor ε = vfkf / (vsks)

t num

/ t c

yl,a

as

ws/l=0.1; St=0.1ws/l=0.1; St=0.01ws/l=0.3; St=0.1ws/l=0.3; St=0.01ws/l=0.5; St=0.1ws/l=0.5; St=0.01

Fig. 10. Hollow cylinder geometry: deviation of approximate analytical andnumerical results.

4930 T. Bauer / International Journal of Heat and Mass Transfer 54 (2011) 4923–4930

with different gradients m. The gradients depend primarily on thedimensionless fin length l/R. An increase of the ratio l/R results innumerical solution tnum closer to the analytical solution tcyl,eff.

Fig. 9 shows the gradients m, defined in Eq. (12), versus thedimensionless fin length l/R. In order to describe the correlation,a power function with two parameters has been fitted. The powerfunction has been adapted in order to achieve a close agreementbetween the computed numerical results and the approximateanalytical solution. Also, for simplicity round numbers have beenselected.

Eq. (12) defines the complete approximate analytical solutionfor the solidification time of the PCM in the finned tube geometry.The major parameters are the dimensionless fin length l/R, thedimensionless PCM width ws/l, the dimensionless fin factor e andthe solidification time of the effective PCM-fin medium tcyl,eff. Thefin factor e contains the volume fraction and the thermal conduc-tivity of the PCM and fin.

tcyl;aas ¼ 0:8ð lRÞ

14|fflfflfflffl{zfflfflfflffl}

m

ws

l

� �2 v f kf

vsks|ffl{zffl}e

þ1

2664

3775 tw;eff � CF|fflfflfflfflffl{zfflfflfflfflffl}

tcyl;eff

¼ NEFcyl � tcyl;eff ð12Þ

In order to evaluate the accuracy of the approximate analyticalsolution tcyl,aas, all 216 numerically computed results tnum have beendivided by tcyl,aas. The results can be seen in Fig. 10. For l/R = 100 andin particular for small fin factors e the deviations were large, but be-low 25% (Fig. 10 at the bottom). For the l/R ratios of 1 and 10, Fig. 10shows that all cases have a deviation of around or below 5%.

6. Summary and conclusions

Approximate analytical solutions based on the effective med-ium properties of the PCM-fin laminate have been developed.These solutions can predict the solidification time of the PCM intwo geometries. These are the finned plane wall and a single tubewhich is radial-finned on the outside. The effective medium ap-proach has been typically applied to structures which are smallcompared to the physical size of the modelled region. Examplesare foams and powder fills. In the presented paper the approachhas been transferred to macroscopic fins with PCMs and the valid-ity of the effective medium approach has been examined. Thedeveloped approximate analytical solutions are based on the prod-uct of two terms. On the one hand, this is the solidification time ofthe effective medium teff which can be utilized if equilibrium con-ditions are met. On the other hand the additional non-equilibriumfactor NEF has been included in this work in order to calculate thesolidification time if the effective medium approach is not valid.The major assumption of the approximate analytical solution in-clude the following: an isothermal wall, small Stefan numbers(0.01 6 St 0.1), physical long fins (ws/l 6 0.5), an ideal contact offin and wall, no free convection in the melt and no heat loss tothe surrounding. For the plane wall case, the solution has been con-firmed by experimental results presented in the literature.

The presented approximate analytical solutions can contributeto a better understanding of the transient heat conduction mecha-nisms with phase change in finned geometries. The analyticalmodels are applicable to a wide range of geometries and typicalutilised PCMs (e.g. water/ice, paraffins, salt hydrates and anhy-drous salts). The modelling was based on the solidification processbut is also applicable to melting, if the free convection in the meltis neglected. The work identified different characteristic dimen-sionless parameters and their impact on the solidification time.These dimensionless parameters correspond to design parametersin a PCM storage system, such as spacing, thickness and number of

fins, as well as the distance of the finned tubes. Hence, the devel-oped approximate analytical solutions can give guidance in thepre-design and optimisation of finned latent heat storage systemsbased on tube registers. In addition, the paper shows that the vol-ume averaging method is a useful approach not only for transientheat conduction problems in local thermal equilibrium, but also itcan act as a basis for non-equilibrium conditions.

Acknowledgements

The author wish to thank the German Federal Ministry of Edu-cation and Research (BMBF) for the financial support given to theLWS-Net project (Contract 03SF0307A-G). The author wishes to ex-tend also special thanks to Thomas Meyer for the large number ofnumerical simulations and the data evaluation during his diplomathesis at the University of Stuttgart.

References

[1] F. Dinter, M. Geyer, R. Tamme, Thermal Energy Storage for CommercialApplications: A Feasibility Study on Economic Storage Systems, Springer,Berlin, 1991.

[2] G. Beckman, P.V. Gilli, Thermal Energy Storage, Springer, New York, 1984.[3] H. Mehling, L.F. Cabeza, Heat and Cold Storage with PCM – An Up To Date

Introduction into Basics and Applications, Springer, Berlin, 2008.[4] R. Tamme, T. Bauer, J. Buschle, D. Laing, H. Müller-Steinhagen, W.-D.

Steinmann, Latent heat storage above 120�C for applications in the industrialprocess heat sector and solar power generation, Int. J. Energy Res. 32 (2008)264–271.

[5] K. Ismail, Chapter 8: Heat transfer with phase change in simple and complexgeometries, in: I. Dincer, M.A. Rosen (Eds.), Thermal Energy Storage, Systemsand Applications, Wiley, 2002.

[6] B. Zalba, J.M. Marin, L.F. Cabeza, H. Mehling, Review on thermal energy storagewith phase change: materials, heat transfer analysis and applications, Appl.Therm. Eng. 23 (2003) 251–283.

[7] J. Weaver, R. Viskanta, Freezing of liquid-saturated porous media, Trans. ASMEJ. Heat Transfer 108 (1986) 654–659.

[8] Y. Shiina, T. Inagaki, Study on the efficiency of effective thermal conductivitieson melting characteristics of latent heat storage capsules, Int. J. Heat MassTransfer 48 (2005) 373–383.

[9] L. Chow, J.K. Zhong, J.E. Beam, Thermal conductivity enhancement for phasechange storage media, Int. Commun. Heat Mass Transfer 23 (1996) 91–100.

[10] H. Inaba, P. Tu, Evaluation of thermophysical characteristics on shape-stabilized paraffin as a solid–liquid phase change material, Heat MassTransfer 32 (1997) 307–312.

[11] V. Morisson, E. Palomo Del Barrio, M. Rady, Heat transfer modelling withingraphite/PCM composite materials for high temperature energy storage, Proc.of the Tenth International Conference on Thermal Energy Storage, May 31 –June 2, 2006.

[12] E. Alawadhi, C. Amon, PCM thermal control unit for portable electronicdevices: experimental and numerical studies, IEEE Trans. Compon. Packag.Tech. 26 (1) (2003) 116–125.

[13] S. Whitaker, The Method of Volume Averaging, in the Series Theory andApplications of Transport in Porous Media, Vol. 13, Kluwer AcademicPublishers., Dordrecht, 1999.

[14] V. Voller, C. Parkash, A fixed grid numerical modelling methodology forconvection-diffusion mushy region phase-change problems, Int. J. Heat MassTransfer 30 (1987) 1709–1719.

[15] A. Brent, V.R. Voller, K.J. Reid, Enthalpy-porosity technique for modelingconvection–diffusion phase-change: application to the melting of a puremetal, Numer. Heat Transfer 13 (1988) 297–318.

[16] V. Alexiades, A.D. Solomon, Mathematical Modelling of Melting and FreezingProcesses, Hemisphere Publishing Corporation, Washigton DC, 1993.

[17] M. Kaviany, Principles of Heat Transfer in Porous Media, Springer, New York,1991.

[18] L. Yao, J. Prusa, Melting and freezing, Adv. Heat Transfer 19 (1989) 1–95.[19] P. Lamberg, K. Siren, Approximate analytical model for solidification in a finite

PCM storage with internal fins, Appl. Math. Model. 27 (2003) 491–513.[20] R. Henze, J. Humphrey, Enhanced heat conduction in phase change thermal

energy storage devices, Int. J. Heat Mass Transfer 24 (1981) 459–474.[21] H. Baehr, K. Stephan, Heat and Mass Transfer (in German: Wärme- und

Stoffübertragung), Fifth ed., Springer, Berlin, 2006.[22] K. Ismail, C. Alves, M. Modesto, Numerical and experimental study on the

solidification of PCM around a vertical axially finned isothermal cylinder, Appl.Therm. Eng. 21 (2001) 53–77.

[23] M. Heinisch, Analytical treatment of the melting and solidification of latentheat store materials in finned tube geometries (in German), doctoral thesis,University of Stuttgart, 1987.