An experimental and numerical investigation of constrained melting heat transfer of a phase change material in a circumferentially finned spherical capsule for thermal energy storage

Applied Thermal Engineering 100 (2016) 1063–1075

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Applied Thermal Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / a p t h e r m e n g

Research Paper

An experimental and numerical investigation of constrained melting heat transfer of a phase change material in a circumferentially finned spherical capsule for thermal energy storage Li-Wu Fan a,b,*, Zi-Qin Zhu a,b, Sheng-Lan Xiao c, Min-Jie Liu a, Hai Lu d, Yi Zeng a,e, Zi-Tao Yu a,f, Ke-Fa Cen b a

Institute of Thermal Science and Power Systems, School of Energy Engineering, Zhejiang University, Hangzhou 310027, China State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China c Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong d Electric Power Research Institute, Yunnan Electric Power and Research Institute (Group), Kunming 650217, China e Department of Mechanical Engineering, Auburn University, Auburn, AL 36849, USA f Key Laboratory of Refrigeration and Cryogenic Technology of Zhejiang Province, Zhejiang University, Hangzhou 310027, China b

H I G H L I G H T S

• • • • •

Constrained melting heat transfer of a PCM in a spherical capsule is investigated. Heat transfer enhancement is attempted using a circumferentially positioned fin. The influence of the fin height is studied both experimentally and numerically. The melting duration time is shortened up to 30% at the highest fin height adopted. Enhanced heat transfer is attributed to both conduction and convection effects.

A R T I C L E

I N F O

Article history: Received 17 November 2015 Accepted 29 February 2016 Available online 7 March 2016 Keywords: Constrained melting Fin Heat transfer enhancement Phase change material Spherical capsule Thermal energy storage

A B S T R A C T

Constrained melting heat transfer of a phase change material (PCM) in a circumferentially finned spherical capsule was studied with application to latent heat thermal energy storage (TES). Attention was paid primarily to revealing the influence of fin height on melting heat transfer and TES performance of the PCM system. Visualized experiments were performed to observe the liquid–solid interface evolutions during melting, and to validate the numerical simulations that were conducted based on the enthalpy method. By means of measuring the instantaneous volume expansion upon melting, an indirect experimental method was proposed and implemented to acquire quantitatively the variations of melt fraction and heat transfer rate. Good consistency was observed between the experimental and numerical results. A combination of the two was able to offer an in-depth understanding on the fin effects by providing detailed knowledge on the interface evolutions and natural convective flow and heat transfer as well. It was shown that the TES performance is enhanced with increasing the fin height, and that the melting duration time is shortened up to nearly 30% at the highest fin height studied. The enhancement was attributed to the combined positive effects due to the presence of the fin, which are enhanced heat conduction by the extended heat transfer area and local natural convection induced in the vicinity of the fin. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Thermal energy storage (TES) is a means by which heat, in sensible, latent, or chemical forms, is temporarily buffered before it may be properly utilized [1]. TES units are deemed to be an effective, integrated part for improving the efficiency of thermal systems with renewable and sustainable sources featuring intermittent avail-

* Corresponding author. Tel.: +86 571 87952378; fax: +86 571 87952378. E-mail address: [email protected] (L.-W. Fan). http://dx.doi.org/10.1016/j.applthermaleng.2016.02.125 1359-4311/© 2016 Elsevier Ltd. All rights reserved.

ability and variable intensity. Among the various forms of TES, latent heat storage has been preferred due to its high energy storage density associated with solid–liquid phase transitions, i.e., melting and solidification, of phase change materials (PCM) [2]. Melting heat transfer inside enclosures have long been studied in relation to the endothermic process of encapsulated PCM for TES [3]. In a recent updated review [4], the presence of natural convection and its significant influence on melting heat transfer of PCM in containers with various shapes, including rectangular cavities, vertical/ horizontal cylinders, spherical capsules, and annular tubes, have been discussed.

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A representative shape with simplicity, spherical capsules have typically been utilized as the PCM encapsulation units in packed bed TES systems [5]. Experimental testing and numerical modeling on the thermal performance of packed bed TES tanks with spherical PCM containers have been carried out [6–14]. The heat transfer behaviors of an individual PCM-filled spherical capsule upon melting and/or solidification have also been investigated [15–24], with an emphasis on evaluating the rate of phase change inside the capsule as well as the external heat transfer coefficient between the capsule and the heat transfer fluid. In addition to the TES performance at system level, the phase change characteristics, especially the convection-dominated melting, and the associated heat transfer inside an individual spherical capsule have been studied extensively since the 1980s [25]. The melting process of a PCM inside an externally heated spherical capsule is rather complicated. If the unmelted solid bulk is not fixed in the capsule, it will migrate due to the density difference between the solid and liquid phases. Usually, close-contact melting will emerge at the bottom of the spherical capsule due to sinking of the solid PCM that is heavier (except for water). Such phenomenon has been classified as unconstrained melting in the literature [26]. The lubrication theory of viscous flow in thin films has been applied to solve analytically unconstrained melting heat transfer in spherical capsules by neglecting natural convection [27–30], followed by an extension to account for natural convection in the upper region above the unmelted solid PCM [31–33]. Recently, in order to better understand unconstrained melting in spherical capsules, visualized observations on the detailed evolutions of phase interface, convective flow, and heat transfer during melting have been performed, both numerically and experimentally [34–40]. In contrast, constrained melting may be encountered in case the unmelted solid PCM is held in position by a certain fixture mechanism [26]. Constrained melting heat transfer inside individual spherical capsules has been studied with an emphasis on the natural convection effect [41–43]. Unlike the situation of unconstrained melting, there is no close-contact melting in this scenario and the heat transfer in the lower region between the unmelted solid PCM and the capsule bottom is dominated by natural convection.

Enhancement of phase change heat transfer inside PCM encapsulations has long been sought for improving the energy storage/ retrieval rates in view of the undesirably low thermal conductivity of common PCM candidates. The introduction of highly-conductive extended surfaces, in the form of metal fins/foams/foils/beads for example [44], into TES units has been a straightforward, effective solution to this issue. For PCM contained in spherical capsules, thermal performance improvement has been attempted using copper plate fins [19] and stainless steel beads [21]. Although the efficacy of this approach has clearly been exhibited, there is, however, a lack of detailed analysis on melting/solidification heat transfer, either constrained or unconstrained, in spherical capsules in the presence of fins. In an effort to extend the existing knowledge, a combined experimental and numerical investigation was carried out to understand constrained melting heat transfer inside a spherical capsule that is internally circumferentially finned. The height of the fin was varied for a parametrical study on its influence on the convective flow and heat transfer during melting as well as on the overall TES performance. 2. Experimental work 2.1. Experimental setup and procedure A spherical melting facility was designed and constructed as schematically shown in Fig. 1. The facility consisted of a spherical capsule, two water tanks, a thermostat chamber, and other auxiliary units. Both the spherical capsule and water tanks were made of Plexiglas to allow for melting visualization. In order to mount the annularshaped straight fin made of aluminum (see Fig. 1), the spherical capsule was assembled from two hemispherical shells with flanges. The two shells were bolted together with flat silicone sealing rings being placed in between the fin and flanges. The inner diameter of the assembled spherical capsule was 100 mm, i.e., inner radius R = 50 mm, with a wall thickness of 2 mm. As clearly shown in the picture in Fig. 1, a straight Plexiglas tube, with an inner radius of Re = 6 mm, was attached to the upper hemispherical shell in such

Fig. 1. Experimental setup of the spherical melting facility with an annular fin.

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a way that its central line was positioned vertically and coincided with the central line of the spherical capsule. This tube served as a space for accommodating the volume expansion of the PCM upon melting, which was used as an indirect means to quantify the variation of melt fraction during melting through the scale attached to the tube, as shown in the inset in Fig. 1. The thickness of the fin was fixed at D = 2 mm, while the effective fin height H inside the spherical capsule was varied. Through the expansion tube, a thin wood stick of a radius of Rs = 1 mm was suspended in the capsule along its central line. This stick with a low thermal conductivity served as a fixture mechanism to hold the remaining solid PCM in position during melting. Several small and thin wood plates were attached to the stick at an equal spacing to help prevent the solid PCM from sinking down (see Fig. 1). By doing so, the constrained melting scenario was able to be realized. The spherical capsule was filled by a molten PCM while being submerged in a water tank that was connected to a constanttemperature circulation water bath maintained at a temperature much lower than the melting point (Tm) of the PCM. The filling of PCM was performed slowly by following a layer-by-layer manner to minimize void formation. At each layer, the molten PCM was allowed to solidify completely prior to pouring the next layer. The total volume of the solidified PCM in the capsule was precisely controlled so that the top surface of the PCM was aligned horizontally with the bottom of the expansion tube where zero on the scale was marked. Therefore, volume expansion of the PCM upon melting was able to be sensed by the rising height (He) in the expansion tube, as illustrated in Fig. 1. The water bath was then set at an appropriate initial temperature (T0) with the filled spherical capsule being held in it for a sufficiently long time, typically more than 12 h, to assure temperature uniformity within the PCM. In addition to the initial temperature control system, a second set of water bath and tank combination was adopted, which was maintained at a temperature (Th) higher than Tm to provide isothermal heating on the outside surface of the spherical capsule as a constant-temperature boundary condition for melting. A melting process was initiated by quickly transferring the filled spherical capsule from the low-temperature water tank to the hightemperature one. As shown in Fig. 1, the spherical capsule was suspended to submerge in the high-temperature water tank in such a way that the bottom of the expansion tube, i.e., zero on the scale, was aligned with the free water surface. As a consequence, the entire expansion tube was exposed to air for better reading of the scale. The high-temperature water tank was placed inside a thermostat chamber that was set to the same heating temperature Th. Therefore, by maintaining a nearly isothermal environment inside the chamber, the heat losses from the expansion tube were minimized. The compensating heating by the surrounding air was able to prevent the rising molten PCM in the expansion tube from local re-solidification. This also helped prevent heat losses from the hightemperature water tank to the ambient, and hence improve the temperature uniformity within the tank. The temperature uniformity was further improved by positioning two baffle plates in the vicinity of the inlet and outlet ports to minimize the local jet impingement effect. As shown in Fig. 1, a set of three type-T thermocouples (TCs) were mounted at different locations on the outside surface of the spherical capsule. Prior to use, the TCs were calibrated to possess an accuracy of ±0.5 °C. As represented by the three TCs, the temperature uniformity over the outside surface of the capsule was found to be within ±0.3 °C, which was even lower than the uncertainty of the TCs. In addition, the stability of the boundary temperature at various locations was found to be less than ±0.2 °C. In this work, a long-chain alkane, n-octadecane (C18H38), with a nominal melting point from 28 to 30 °C was adopted as the PCM. A set of important thermophysical properties of n-octadecane in both

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Table 1 Measured thermophysical properties of n-octadecane. Property (unit)

Phase

Value

Density (kg/m3)

Solid Liquid Solid Liquid Solid Liquid Liquid Liquid – –

867 773 2.086 2.410 0.286 0.156 0.00352 0.000932 28.0 241.8

Specific heat capacity (kJ/kgK) Thermal conductivity (W/mK) Dynamic viscosity (kg/ms) Thermal expansion coefficient (1/K) Melting point (°C) Latent heat of fusion (kJ/kg)

liquid and solid phases over a temperature range from 10 to 60 °C were in-house measured due to the inaccessibility to some of these properties in available literature. The density in liquid phase (ρl) was measured by a densitometer, while the solid density (ρs) was determined by displacement technique based on the Archimedes’ principle. The thermal expansion coefficient (β) in liquid phase was then found via the variation of liquid density as a function of temperature. The specific heat capacity in both phases (cp,l and cp,s) was measured using a differential scanning calorimeter (DSC), by which the latent heat of fusion (L) and Tm were also determined. The thermal conductivity in both phases (kl and ks) was measured using the transient plane source (TPS) method with a customized sample holder for temperature control. In addition, determination of the viscosity in liquid phase (μ) was performed on a rotational viscometer with precise temperature control as well. The details regarding experimental setup, temperature control, and data reduction for DSC and TPS measurements can be referred to a recent work of the authors by which the n-eicosane-based composite PCM were characterized [45]. The measured thermophysical properties for n-octadecane are listed in Table 1. The reported properties were averaged over parallel tests on multiple specimens, with those being determined at 20 and 40 °C to represent the solid and liquid phases, respectively. The measured properties in the present work are in good agreement with those, if available, documented in Tan et al. [42] and Vélez et al. [46]. According to the measured Tm of n-octadecane (28.0 °C), which was identified as the onset of melting of the endothermic peaks on the DSC curves, the initial temperature was set to T0 = 27.0 °C, indicating an initial subcooling degree of 1.0 °C, while the heating boundary temperature was set to Th = 40 °C for all experiments. By fixing the spherical capsule size and the initial and boundary thermal conditions, only the effects of fin height were studied parametrically in this work. The fin height was varied to have a height-todiameter ratio H/R from 0 to 0.75. In order to assure reproducibility of the experimental results, repetitive experiments up to 3 times were conducted for each case. In each run, the transient evolution of the solid–liquid phase interface was recorded directly by taking a series of pictures on the visualized spherical capsule at representative time instants. The TC readings were logged by a data acquisition system at a frequency of 0.5 Hz, while the rising height of molten PCM in the expansion tube was also intermittently recorded in synchronization with photographing and temperature readings. 2.2. Data reduction and uncertainty analysis The transient melt fraction was able to be calculated directly by image processing of the snapshots on the liquid–solid interface, however the uncertainty associated with this method was deemed to be high as a result of the presence of the opaque flange structures along the equator as well as light refraction through the walls of the water tank and spherical capsule. Therefore, an indirect

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method was adopted for a better quantitative analysis of the melting processes. Having discussed briefly, the volume expansion upon melting was utilized to quantify the transient melt fraction by monitoring the rising height of molten PCM in the expansion tube. Using the notation system shown in Fig. 1, the instantaneous volume expansion (Ve) for a given rising height (He) upon melting was simply calculated by

Ve = π ( Re2 − r 2 ) He

(1)

The volume of the spherical capsule (Vc) was equal to the initial volume of the filled solid PCM, which was approximated by

Vc =

4 3 2 πR − π ⎡⎣ R 2 − ( R − H ) ⎤⎦ D − 2πr 2R 3

(2)

Table 2 Experimental uncertainties of the various thermophysical properties and quantities. Property/quantity (unit)

Symbol

Uncertainty (%)

Density (kg/m3) Specific heat capacity (kJ/kgK) Thermal conductivity (W/mK) Dynamic viscosity (kg/ms) Latent heat of fusion (kJ/kg) Volume expansion (m3) Melt faction by mass (–) Surface-averaged heat flux (W/m2) Overall heat transfer coefficient (W/m2K) Nusselt number (–) Stefan number (–) Grashof number (–) Fourier number (–)

ρ cp k μ L Ve f q″ h Nu Ste Gr Fo

0.03 0.89 1.03 0.06 0.58 2.25 2.31 9.80 8.92 9.88 1.09 0.09 0.52

leading to estimation of the total mass (M) of PCM by

M = ρsVc

(3)

The instantaneous total volume occupied by the PCM, in both liquid and solid phases, was equal to the addition of the volume expansion to the volume of the spherical capsule. This relationship was used to estimate the instantaneous mass (m) of the molten PCM by

Ve + Vc =

m M −m + ρl ρs

(4)

Given the knowledge on the instantaneous mass of molten PCM, the instantaneous melt fraction (f) was then readily calculated by

f=

m ⎡ ρl ⎤ = Ve M ⎢⎣ ( ρs − ρl )Vc ⎥⎦

(5)

which is apparently a single-value function of the instantaneous rising height. The surface-averaged heat transfer coefficient (h) over the total interior surface (S), including both the spherical shell and the fin, was defined by

q′′ ΔQ = Th − Tm S (Th − Tm ) Δt

h=

(6)

where S was approximated by (7)

and ΔQ represents the total heat transferred into the PCM during a given time interval Δt, which was evaluated by calculating the instantaneous total heat (Q) over the entire duration since the onset of heating by

(M + m) cp,s (Tm − T0 ) mcp,l (Th − Tm ) 2

+

2

+ mL

(8)

The surface-averaged Nusselt (Nu) number was defined by

Nu =

hR kl

(9)

cp,l (Th − Tm ) L

g βρl2 (Th − Tm ) R 3 μ2

(11)

with g being the gravitational acceleration (g = 9.8 m/s2). Last, the dimensionless time, i.e., the Fourier (Fo) number, was defined by

Fo =

klt ρlcp,l R 2

(12)

It is noted that the thermophysical properties of the PCM in liquid phase were utilized in the definitions of these nondimensional numbers for the melting problem. The density and specific heat capacity of the PCM in both phases were utilized in the calculations, although they were assumed to be constant in each phase. The experimental uncertainties associated with data reduction of this indirect method were primarily related to the accuracy of the rising height readings, which had a resolution of 1 mm based on the scale. Other sources of the uncertainties included determination of the dimensions and thermophysical properties. Using the error propagation theory, the relative uncertainty of the measured volume expansion was determined approximately by 2 2 δ Ve ⎡⎛ δ He ⎞ ⎛ δR ⎞ ⎤ = ⎢⎜ + 4⎜ e ⎟ ⎥ ⎟ ⎝ Re ⎠ ⎦ Ve ⎣⎝ He ⎠

(13)

where the symbol δ in front of a quantity denotes the corresponding absolute uncertainty. The relative uncertainties for the rest of the reduced quantities were found in a similar way. The experimental uncertainties for the various thermophysical properties and reduced quantities are listed in Table 2, where the uncertainties for the properties were estimated by spreading of the measured results over parallel tests on multiple specimens. 3. Numerical simulation 3.1. Physical model and numerical solution

where the inner diameter R of the spherical capsule was chosen as the characteristic length. The definitions of other important dimensionless groupings that govern the melting problem follow. First, the Stefan (Ste) number that quantifies the intensity of driving force for melting was defined by

Ste =

Gr =

12

2 S = 4πR + 2π ⎡⎣ R 2 − ( R − H ) ⎤⎦ 2

Q=

Second, the Grashof (Gr) number that determines the intensity of natural convection in molten PCM during melting was defined by

(10)

To supplement the above-mentioned experimental work, a numerical study was undertaken that enables a detailed analysis on the natural convective flow pattern and temperature distribution during melting. The three-dimensional problem of constrained melting in a spherical capsule was reduced by only considering a semicircular region (see Fig. 2a), under the assumption of axisymmetric melting about the central line along the vertical di-

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Fig. 2. Schematic diagram showing (a) the semicircular computational domain and (b) a typical grid system with about 11,000 cells.

rection. Due to the consideration of constrained melting, the density difference between the liquid and solid phases of the PCM was neglected so that the relative migration of the remaining solid bulk was inhibited in the numerical simulations by setting a constant liquid density for both phases. The volume expansion was hence not captured in the numerical simulations. The density variation with temperature was only considered in the body force term using the Boussinesq approximation to take into account the buoyancy effect leading to natural convection during melting. As shown in Fig. 2a, the structural complexities associated with the flanges and expansion tube were simplified in the physical model. Similar to density, the other thermophysical properties were also assumed constant in the simulations and the values for the PCM in liquid phase were adopted, as listed in Table 1. The phase interface was assumed to be sharp and to always maintain at Tm during melting. It is noted that the case studied by Tan et al. [42] is equivalent to the unfinned baseline case in this work, because of the utilization of very close dimensions of the spherical capsule, the same PCM, and identical thermal conditions. The pertinent governing equations for conservation of mass, momentum, and thermal energy may be referred to as those given by Tan et al. [42] for simulating constrained melting in a spherical capsule in the absence of internal fins. The natural convective flow was assumed to be laminar. As mentioned, the PCM system was initially maintained at T0 = 27 °C (for t = 0), followed by a sudden increase of the boundary temperature Tw on the outside surface of the spherical shell, i.e., along the outer semicircle as shown in Fig. 2a, to a higher temperature Th = 40 °C for t > 0. Axisymmetric condition was applied along the central line of the spherical capsule, i.e., the vertical dashed line in Fig. 2a. All simulations were performed until a complete melting in the spherical capsule was obtained. Numerical solution to the governing equations was obtained using the fix-grid enthalpy method, as described by Tan et al. [42]. The simulations were performed by a commercial computational fluid dynamics (CFD) code ANSYS Fluent with the built-in functionality of melting/solidification modeling based on the enthalpy method. The equations were solved iteratively using the SIMPLE algorithm, while the PRESTO! scheme was adopted for pressure interpolation [42]. Spatial discretization of the governing equations was conducted using the second-order upwind scheme, whereas temporal discretization was implemented by second-order implicit time integration that is unconditionally stable. The convergence crite-

ria were set to be 10 −4 and 10 −6 , respectively, for continuity/ momentum and energy equations. 3.2. Independence test on grid and time step sizes Meshes were generated within the computational domain using quadrilateral cells. A typical grid system of a total number of about 11,000 cells is shown in Fig. 2b. Along the inner surface of the spherical shell and surrounding the fin, 12 layers of structured boundary layer cells with a growing factor of 1.02, as clearly demonstrated in the inset of Fig. 2b, were generated that are more densely packed than those in the rest area to have good resolution within the momentum and thermal boundary layers. As only pure heat conduction was expected to happen within the spherical shell and fin, very few layers of structured cells were deemed to be sufficient to render the temperature distributions. It is noted that a conjugate heat transfer problem was solved due to the involvement of the fin in the computational domain, where heat conduction through the fin was solved along with melting heat transfer in the PCM. Independence tests on both grid and time step sizes were performed using the transient melt fraction as an indicator for comparison. As shown in Fig. 3a, at a given physical time instant, the predicted melt fraction seems to be nearly independent of the number of cells when it increases to above 9000. Then, while fixing the total number of cells at about 10,000, the time step size was varied from 0.01 s to 0.1 s. The relative variation of the predicted melt fraction upon using the various time step sizes was found to be within only 1.5% (see Fig. 3b). In order to achieve a balance between computational accuracy and expenses, all simulations were performed with a total number of about 10,000 cells and a time step size of 0.05 s. 4. Results and discussion As mentioned, the emphasis of the present work was focused on the effect of fin height while the size of the spherical capsule, the PCM, and heating temperature being kept unvaried. Given the dimensions, thermal boundary conditions, and thermophysical properties of the PCM, the couple of dimensionless groupings that govern this problem were fixed at Gr = 6.27 × 105 and Ste = 0.127. The value of the Gr number justifies the assumption of laminar flow in the numerical simulations. The fin height was set so that 3 finned cases

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(a)

(b)

Fig. 3. Variation of the predicted melt fraction as a function of the (a) number of cells and (b) time step size.

were considered with H/R = 0.25, 0.5, and 0.75, in addition to the unfinned baseline case (H/R = 0). 4.1. Validation of numerical results by visualized melting processes As presented by Tan et al. [42], the numerical results of constrained melting in a spherical capsule using the CFD code with the built-in enthalpy-based model were validated against the experimental data using the variation of melt fraction as a means for comparison. Although a confidence had already been established on the numerical methodology and results in that work, a specific validation for the various finned cases was conducted in comparison to the visualized melting experiments. As shown in Fig. 4, the photographed and predicted evolutions of liquid–solid interface during melting are compared side by side at several representative time instants toward completion of melting. The images are arranged in an order that the fin height increases from left to right. In the right halves that show the numerical results, the blue and red regions represent the solid and molten PCM, respectively. Although the shape of the solid PCM was directly visible through the upper and lower hemispherical shells, the local interface was unable to be observed in the vicinity of the fin as a result of the presence of flanges. This issue confirms that the indirect method was necessary in order to perform a precise quantification on the melt process. In general, the agreement between the visualized and predicted interface evaluations is fairly good during the entire course of melting for all cases, no matter whether the capsule was unfinned or finned, thus clearly exhibiting validation of the numerical predictions. As shown in the first row corresponding to an elapsed time of 6 min, melting starts shortly after heating, due to the small subcooling degree, from the inside surface of the capsule as well as the surfaces of the fin, forming a thin layer of molten PCM with a nearly uniform thickness. This indicates that heat transfer in the liquid PCM is initially dominated by conduction. The thin layer surrounding the solid PCM keeps growing almost uniformly as melting proceeds, as seen from the second row of images (t = 9 min) in Fig. 4.

As natural convection starts to play its role, the liquid layer becomes asymmetric between the upper and lower regions since t = 18 min. Toward the end of melting, the asymmetry appears to be more pronounced as the PCM in the upper region melts much faster than that in the lower region. Although it was not directly visible, the concave-shaped interface at the bottom of the solid PCM (see the last two rows in Fig. 4), as a result of the natural convective jet impingement effect as identified by Tan et al. [42], was successfully predicted by the numerical simulations. The presence of the fin adds a heat conduction path into the PCM leading to greatly increased heat transfer area, which also strengthens the asymmetry of the liquid–solid interface shape due to the local heating effect above and underneath the fin. As increasing the fin height, the interface shapes in the upper and lower regions become more independently developed. It is observed in the last row of images in Fig. 4 (t = 63 min) that the solid PCM becomes smaller from left to right, thus indicating accelerated melting upon increase of the fin height. A number of differences between the visualized and predicted results were identified as follows. First, it is noted that small-scale sawtooth structures are present on the predicted liquid–solid interfaces, as opposed to the smooth interfaces observed experimentally, although the large-scale interface shapes are likely unaffected by such unreal small features. Second, approaching the end of melting, the predicted melting rate is slightly greater than the measured (see the last row in Fig. 4). Such discrepancy is consistent with the observation in Tan et al. [42]. Third, repeated longitudinal groove structures are observed leading to pumpkinshaped lower halves of the solid PCM. These localized wavy patterns are related to three-dimensional natural convective flow instability [42], which was simply neglected in the numerical simulations due to the axisymmetric simplification. The major sources for the aforementioned differences were attributed to (a) the difference between the real physical scenario and the simplified computational model, and (b) the utilization of constant thermophysical properties of the PCM in the numerical simulations, especially the constant density in both phases. The numerical simulations could be further improved by considering

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Fig. 4. Comparison between the visualized (left half) and predicted (right half) transient evolutions of liquid–solid interface during melting.

a three-dimensional model with temperature-dependent thermophysical properties of the PCM. Particularly, the density variation and the resultant volume expansion could be accounted for if a volume of fluid model is applied for the PCM system [35]. 4.2. Predicted evolutions of convective flow and temperature Having identified several hints on the presence of natural convection during melting, however the intensity and flow patterns of natural convection were unable to be detected directly through the visualized experiments. The numerical results were thus utilized to perform a quantitative analysis of the convective flow and heat transfer. As presented in Fig. 5, the streamlines in molten PCM and the contours of temperature distribution inside the capsule, at the same time instants as those in Fig. 4, are adopted to represent the convective flow and heat transfer characteristics, respectively. The arrangement of the images in Fig. 5 is identical to that in Fig. 4, where the color bar represents a temperature range between 27 and 40 °C, corresponding to the initial and heating boundary temperatures, respectively.

For the unfinned baseline case (H/R = 0), there are only very sparse streamlines in the thin liquid layer at the initial stage of melting (t = 6 min), indicating that heat conduction is the dominant mode of heat transfer at this moment. This is confirmed by the nearly concentric isotherms surrounding the nearly circular solid PCM. As pointed out by Khodadadi and Zhang [41], it is understood that in the absence of natural convection, solution to the pure heat conduction equation in the liquid layer is only a function of the radial position so that the isotherms present as concentric circles. At the same time instant, the presence of the fin with various heights seems not to affect the concentric isotherms in the upper and lower regions. The isotherms become parallel lines in the vicinity of the fin, which are slightly distorted in the corners above and underneath the fin where convective recirculation cells are present, as clearly seen from the streamlines. Apparently, the presence of the fin leads to formation of a heated-from-below configuration above it as well as a heated-from-above configuration underneath it. Such configurations are not only able to enhance heat conduction through the added heat transfer area, but also able to augment local natural convection effect.

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Fig. 5. The predicted transient evolutions of the flow (left half) and temperature fields (right half) during melting.

In particular, the heated-from-below configuration resembling the classical Rayleigh–Bénard problem is rather unstable to cause natural convection, as clearly indicated by the several recirculation cells above the highest fin studied (H/R = 0.75) at t = 9 min. With increasing the fin height, the streamlines become more densely packed along both the internal surface of the capsule and the liquid– solid interface, suggesting that natural convection in enhanced. As melting proceeds, natural convection becomes more intensive, with a great number of recirculation cells emerging at the bottom of the capsule when t = 18 min. As molten PCM occupies more area (at t = 36 min), the isotherms becomes stratified in the upper region and the small recirculation cells start merging together to form large ones. When the fin height is relatively large (H/R = 0.5 and 0.75 for example), the secondary stratification effect is clearly seen underneath the fin. Toward the end of melting (at t = 63 min), however, the secondary stratification disappears with the effect of the fin becoming almost negligible, because it now submerges in isothermal molten PCM. This ineffective performance of the fin may be improved by positioning it at a lower position or titling the finned capsule. In addition, the above-mentioned jet impingement effect from the bottom of the capsule is clearly present during such late stage of melting.

4.3. Measured and predicted variations of melt fraction Having shown qualitatively the comparison of transient melting rate through the visualized and predicted liquid–solid interface, a quantitative comparison of the melting rate among the cases with various fin heights was made by means of the variations of melt fraction. The definition of the instantaneous melt fraction by mass has been given previously in Eq. (5), by which the variations of melt fraction were calculated based on both the measured and predicted results, as presented in Fig. 6. The agreement between the experimental and numerical results is again revealed by the comparisons of the melt fraction variation for all cases, with the maximum discrepancy less than 20% being found during the intermediate stage of melting. However, the duration time for melting, which was defined as the elapsed time for f = 1 when the melting process ends, is found to be nearly consistent between the experimental and numerical results for each case. The melting duration time decreases, indicating accelerated melting process, with increasing the fin height. The measured melting duration times are 98, 86, 78, and 69 min for the cases of H/R = 0, 0.25, 0.5, and 0.75, respectively, whereas the predicted values are 94, 84, 75, and 67 min, respectively. The actual melting rate was slightly

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Fig. 6. Comparison of the measured and predicted variations of melt fraction during melting among the various fin heights.

Fig. 7. The correlated variations of melt fraction during melting for the various fin heights.

overestimated by the numerical simulations, with the possible reasons having been discussed before. The relative shortening of the melting duration time for the finned cases of H/R = 0.25, 0.5, and 0.75 with respect to the unfinned baseline case is found, both experimentally and numerically, to be approximately 11%, 20%, and 29%, respectively. That is, by simply adding a circumferential thin straight fin with a height of H/R = 0.75, a nearly 30% enhancement in the TES performance of such PCM system was able to achieve. Having discussed in terms of the observed convective flow and heat transfer during melting, the enhanced performance was mainly attributed to the combined contribution of enhanced heat conduction through the extended heat transfer surfaces and emerging local natural convection phenomena in the vicinity of the fin. The measured instantaneous melt fraction was further analyzed as a function of the dimensionless time, i.e., the Fo number as defined in Eq. (12). As suggested by Kamkari and Shokouhmand [47], a correlation for the measured melt fraction was proposed in terms of a modified dimensionless time involving the dimensionless fin height, which is given by

tions of the Nu number is plotted for all cases in Fig. 8. In general, the Nu number is infinitely large at the initial time instant and then drops quickly to a value of around 20 within a time period of 2–3 min for all cases, followed by a sudden increase to a local peak value around 35, corresponding to the onset of natural convection. In the presence of the fin, the onset of melting occurs earlier than that for the unfinned baseline case. As melting proceeds, a number of secondary peaks with relative smaller magnitudes are found due to the emergence of local natural convection phenomena in the vicinity of the fin, as having been shown in Fig. 5. The Nu number then gradually decreases with a small slope toward the end of melting. Despite a fairly good agreement in the trend between the experimental and numerical results, the predicted Nu number is shown to overestimate slightly up to the intermediate stage of melting, whereas it becomes underestimated to some extent during the late stage. The decreasing slope of the predicted results is therefore larger than the measured. As compared to the numerical results showing local small fluctuations, the measured variations are somewhat flatten out as a result of the relatively low temporal resolution of data acquisition that was limited by manual recording of the rising height for calculating the volume expansion. As shown by the comparison among Figs. 8a through d, the Nu number seems to only slightly vary upon increasing the fin height, suggesting that the fin effects were difficult to identify from the almost consistent surfaceaveraged Nu number variations when the fin area was involved in the calculations, as given by Eq. (9). If the Nu number were defined based on the surface area of the capsule, then considerable discrepancy would be able to observe for the various cases. Similar to the correlation for melt fraction, a correlation for the Nu number was also sought as a function of the modified Fo number involving the dimensionless fin height to the power of 0.6. As shown in Fig. 9, the measured Nu number variations may be correlated to

⎡ Fo (1 + H R )0.6 ⎤ f = 1.6262 − 1.6166 exp ⎢ − ⎥ 0.2003 ⎦ ⎣

(14)

where both the exponent and coefficients were determined by curve fitting. The measured variations of melt fraction almost collapse to the single curve corresponding to the above correlation, as shown in Fig. 7. The coefficient of determination for curve fitting is as high as 0.9989 with very small relative deviations being within only ±3%. Since the Gr and Ste numbers were kept constant in this work, they are absent in the correlation while the transient melt fraction for the finned cases simply varies with the scaled Fo number with a coefficient of the dimensionless fin height to the power of 0.6, suggesting clearly the exponential fin height dependence on the overall TES performance. 4.4. Measured variations of surface-averaged Nusselt number Having discussed the overall TES performance by means of the melt fraction, an analysis of heat transfer during constrained melting was conducted using the surface-averaged Nu number, as defined in Eq. (9). Comparison of the measured and predicted transient varia-

Nu (1 + H R )

0.2

⎡ Fo (1 + H R )0.6 ⎤ = 21.758 − 1.746 exp ⎢ ⎥ 0.087 ⎣ ⎦

(15)

where the Nu number is scaled by the dimensionless fin height to the power of 0.2. Due to the presence of significant fluctuations during the initial stage of melting, this correlation seems to be only valid for the intermediate to late stage with the modified Fo number being greater than 0.05. Despite a relatively low coefficient of determination of 0.9331 for all data points, the relative deviations for

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Fig. 8. Comparison of the measured and predicted variations of the surface-averaged Nu number during melting among the various fin heights.

the valid range are found to be within ±10% that are fairly good for engineering applications. As mentioned, the scaling of the Nu number in terms of the Gr and Ste numbers that were both fixed was unavailable.

4.5. Analysis of fin effectiveness and beyond In addition to the previous identification on the fin effects on the melting heat transfer and TES performance, the fin effectiveness was

Fig. 9. The correlated variations of the surface-averaged Nu number during melting for the various fin heights.

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thin straight fin was taken into account as a straightforward example to reveal the fin effects on constrained melting in a spherical capsule. In addition to the fin height, the effects of other key parameters that have not yet been considered, e.g., the Gr and Ste numbers and the inclination angle of the fin relative to gravity are of great interest in future studies. 5. Conclusions Constrained melting heat transfer of a PCM in a spherical capsule was studied, both experimentally and numerically, toward TES applications. In order to enhance heat transfer inside the capsule, a circumferential thin straight fin was introduced along the equator of the capsule. With fixing the other geometric and thermal conditions, the fin height was varied so that its effects on the constrained melting heat transfer as well as TES performance in the spherical capsule was investigated parametrically. Based on the preceding data analysis and discussion, some concluding remarks and recommendations for future work are drawn as follows: Fig. 10. The variation of measured fin effectiveness as a function of the finned-tounfinned ratio of heat transfer area.

1) The enthalpy-based numerical simulations can successfully predict the evolutions of the liquid–solid interface during melting, as validated by the visualized experiments. The proposed indirect method can be used to quantify the variations of melt fraction and heat transfer rate by means of monitoring the instantaneous volume expansion upon melting. By combining the experimental and numerical methods, an indepth understanding of the melting process can be made available with detailed knowledge on the interface development as well as natural convective flow and heat transfer. 2) The presence of the fin is able to both enhance heat conduction by extending heat transfer area as well as inducing local natural convection emerging from above and underneath the fin. The combination of both positive effects leads to enhanced TES performance. Under the cases studies, the TES performance is improved monotonously with increasing the fin height. The melting duration time is able to be shortened by a factor up to nearly 30% at the highest fin height, corresponding to a fin effectiveness of 1.39. A fin optimization problem may be posed if other constraints are applied for practical applications. 3) The proposed correlations can be utilized for predicting both the melt fraction and Nu number in terms of the dimensionless fin height, with relative deviations being within ±3% and ±10%, respectively. The effects of Gr and Ste number are absent in the correlations that are of great interest in future studies by varying the size of the capsule, thermal conditions, and selection of PCM as well. The effect of the inclination angle of the fin relative to gravity is also of significance, in view of the fact that the finned spherical capsules may be randomly positioned in practical applications.

also evaluated following the classical fin analysis. The fin effectiveness E serves as a measure for the improvement of heat transfer rate in the presence of fin relative to the unfinned case, which may be calculated by [47]

E=

qf′′Sf qnf ′′ Snf

(16)

where the quantity in the angle brackets represents the time-averaged value over the entire duration of melting. In Fig. 10, the measured fin effectiveness for the various finned cases is plotted as a function of the extended heat transfer area ratio A that is defined by

A=

Sf 3 1 ⎛ H ⎞ = − ⎜1− ⎟ Snf 2 2 ⎝ R⎠

2

(17)

As expected, the fin effectiveness raises upon increasing the heat transfer area ratio. As shown by the curve fitting in Fig. 10, the rise in fin effectiveness is found to nearly follow an exponential trend, with the growth being up to 1.39 at the highest fin height studied. The exponential trend implies that the fin effectiveness could be further increased by the adoption of a higher fin. For the spherical geometry under consideration, however, the maximum fin height is achieved when H/R = 1. In this limit case, the spherical capsule is actually divided into two closed parts with the fin serving as the bottom and top plates, respectively, for the upper and lower hemispherical cavities. At the standing point of thermal performance, the more the fin area added, the higher the fin effectiveness achieved. In practical TES applications, however, an optimization problem may be posed to pursue the maximum fin effectiveness under certain constraints. For instance, the utilization of a fin with larger area will increase the cost on material as well as decrease the effective TES capacity. Even with a given total mass of the fin material, the type and shape of the fin could be optimized to possibly achieve maximum enhancement on TES performance. Having the fin efficiency concept in mind, the present aluminum fin material may be replaced by copper for its higher thermal conductivity. However, trade-off considerations are required for practical applications to approach a balance among the thermal conductivity, density, material and machining costs, and chemical compatibility as well. Admittedly, to address these issues is out of the scope of the present work. Due to its geometric simplicity, the circumferential

Acknowledgements This work was supported financially by the National Natural Science Foundation of China (NSFC) under Grant No. 51276159, the China Postdoctoral Science Foundation (CPSF) under Grant Nos. 2012M511362 and 2013T60589, and the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars, Ministry of Education of China. L-W. Fan would like to thank a start-up fund granted by the “100 Talents Program” of Zhejiang University. Nomenclature A cp

Ratio of heat transfer area Specific heat capacity (kJ/kgK)

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D E f g h H k L m M q″ Q r R S t T V

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Fin thickness (m) Fin effectiveness Melt fraction by mass Gravitational acceleration (m/s2) Heat transfer coefficient (W/m2K) Fin height (m) Thermal conductivity (W/mK) Latent heat of fusion (kJ/kg) Mass of molten PCM (kg) Total mass of PCM (kg) Heat flux (W/m2) Total heat (J) Radius of stick (m) Radius of spherical capsule (m) Heat transfer surface area (m2) Time (min) Temperature (°C) Volume (m3)

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