Numerical study for enhancement of solidification of phase change materials using trapezoidal cavity

Powder Technology 268 (2014) 38–47

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Numerical study for enhancement of solidification of phase change materials using trapezoidal cavity R.K. Sharma a, P. Ganesan a,⁎, J.N. Sahu b,⁎, H.S.C. Metselaar a, T.M.I. Mahlia c a b c

Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia Petroleum and Chemical Engineering Programme Area, Faculty of Engineering, InstitutTeknologi Brunei (A Technology University), Tungku Gadong, P.O. Box 2909, Brunei Darussalam Department of Mechanical Engineering, Faculty of Engineering, UNITEN Putrajaya Campus, 43000 Kajang, Selangor, Malaysia

a r t i c l e

i n f o

Article history: Received 19 May 2014 Received in revised form 6 August 2014 Accepted 10 August 2014 Available online 17 August 2014 Keywords: Solidification Phase change materials (PCM) Nanofluid Trapezoidal cavity Computational fluid dynamics (CFD)

a b s t r a c t This paper reports the results of a numerical study on the heat transfer during process of conduction dominated solidification of copper–water nanofluid in isosceles trapezoidal cavity under controlled temperature and concentration gradients. The suspended nanoparticles have proven to increase the heat transfer rate substantially. The horizontal walls of the cavities are insulated while side walls are kept at constant but different temperatures. The total solidification time of pure fluid and nanofluid filled in the cavity is investigated for three different inclinations of side walls and the results are compared with square cavity. Results revealed that the total solidification time for pure fluid as well for nanofluid for all nano particle concentrations decreases. The effects of Grashof number (105–107) on the heat transfer phenomenon and solid–liquid interface are also numerically investigated and presented graphically. The enthalpy–porosity technique is used to trace the solid–liquid interface. Inclination angle can be used efficiently to control the solidification time. In addition, average Nusselt number along the hot wall for different angles, nanoparticles volume fractions, and Grashof number is presented graphically. The proposed predictions are very helpful in developing improved latent heat thermal energy storage for solar heat collector and for casting and mold design. © 2014 Elsevier B.V. All rights reserved.

1. Introduction A continuous increase in the gap between the energy demand and supply and the depletion of fossil fuels have received the attention of researchers in the last few decades. This increasing demand forces researchers to develop renewable energy sources. This demand and supply gap is efficiently bridged by employing a suitable energy storage system. As an example, solar energy is not available during the night and on cloudy days, so the heat stored in phase change materials (PCM) during sunny days can be used as and when needed during cloudy hours or in the night. Latent heat thermal energy storage (LHTES) devices are more attractive than sensible heat storage devices due to their high energy storage density and constant charging and discharging temperature. In general, these devices use a PCM such as pure water, paraffin wax, etc. Solid–liquid phase change problems, sometime also known as moving boundary or Stefan problems are encountered in many industrial applications such as casting and laser drilling, latent heat thermal energy storage, food and pharmaceutical processing, microelectronics, and protective clothing. Depending on ⁎ Corresponding authors. Tel.: +60 3 79675204/7670 (Office); fax: +60 3 79674579 (Office). E-mail addresses: [email protected] (P. Ganesan), [email protected] (J.N. Sahu).

http://dx.doi.org/10.1016/j.powtec.2014.08.009 0032-5910/© 2014 Elsevier B.V. All rights reserved.

the initial temperature of the material, they are categorized as oneregion, two-region or multiple region problems [1]. This phase change conversion (solid–liquid) absorbs latent heat during charging (melting) and releases it during solidification (discharging). Design and development of such devices have been greatly assisted by many experimental and numerical investigations. The low thermal conductivity of PCMs is the primary limitation in many engineering devices, however the dispersion of solid particles in the base fluid overcomes this limitation which can be done either at micro or nano levels. Dispersion of micrometer or millimeter sized particles may cause the clogging in the fluid flow and increase the pressure drop because of the rapid settling characteristics. Nano sized particles behave like liquid particle, show little or no pressure drop and flow with or without little chance of clogging in the pipes. A mixture of a base fluid and solid particles of nano size is called a nanofluid [2]. Khodadadi and Hosseinizadeh [3] reported that the latent heat decreases as the mass fraction of dispersed particles increases. They numerically analyzed the solidification of nanofluid (water + nano copper particles) in a square cavity and found that the speed of the solid/liquid interface increases as the time elapses for nanofluid of higher mass fractions. This resulted in considerable reduction in the total solidification time of the nanofluid. Encapsulation of PCMs is a promising way to store the latent heat energy due to numerous advantages such as large heat transfer area, reduction in PCMs interaction

R.K. Sharma et al. / Powder Technology 268 (2014) 38–47

Nomenclature A cp dp f g Gr h H He k kf keff ks L Lx Ly Lh M n NuL Nuave Pr Q S t T u v U, V x, y X, Y

Aspect ratio (L/H) Specific heat at constant pressure Nanoparticle diameter Liquid fraction Acceleration due to gravity Grashof number, gβfΔTH3/ϑ2f Specific enthalpy Height of the cavity Total enthalpy Thermal conductivity Fluid thermal conductivity Effective thermal conductivity Solid thermal conductivity Length of the cavity Length of square enclosure in the x direction Length of square enclosure in the y direction Latent heat of fusion Mushy zone constant Direction normal to the surface Local Nusselt number Average Nusselt number Prandtl number, μcp/k Total heat transfer from left wall Source term Time Temperature Velocity component along x-axis Velocity component along y-axis Dimensionless velocity components Cartesian coordinates Dimensionless coordinates

Greeks symbols β Volumetric expansion coefficient ∝ Thermal diffusivity ∅ Solid volume fraction ϑf Kinematic viscosity ρ Density μ Dynamic viscosity θ Inclination angle

Subscripts ave Average c Cold eff Effective f Fluid h Hot nf Nanofluid o Reference value s Solid w Wall

Abbreviations CW Clockwise CCW Counter clockwise

with the external environment, and controlling the variation in volume change during phase change. The geometry of the enclosure (e.g. rectangular, cylindrical, and spherical) also plays an important

39

role in the heat transfer rate in fluid which has been extensively reviewed by Ostrach [4]. Most of the previous studies are concerned with the analysis of the conduction or natural convection heat transfer of PCMs in regular shapes like asquare/rectangular cavity [3,5–13], cylindrical cavity [14–22], and spherical cavity [23–31]. Gau and Viskanta [8] experimentally investigated the buoyancy driven flow in gallium and its effect on solid–liquid interface motion and heat transfer during melting and solidification in a rectangular cavity. They found that at the beginning of melting, the fluid motion was very slow and the interface was flat and parallel to the heated wall, which can be interpreted as conduction dominated heat transfer. As time progresses, buoyancy-driven flow increases and the solid–liquid interface no longer remains straight and appeared to be receding from top to bottom wall. Later, Brent et al. [32] proposed the enthalpy– porosity technique to trace the solid–liquid interface in melting and solidification phenomenon. They used this approach to reproduce the results of experimental work [8] using a 2D rectangular cavity, keeping the same boundary conditions and material used and found that their numerical predictions are in very good agreement with the experimental results. Recently Arasu and Majumdar [5] carried out a numerical investigation on the melting of a PCM (paraffin wax + Al2O3) in a square enclosure heated from the bottom side and from a vertical side with different volume fractions of nanoparticles (0.02 and 0.05). They reported that the effective thermal conductivity of a paraffin wax latent heat storage medium could be increased significantly by a smaller volumetric concentration of alumina particles in the paraffin wax in both cases. Darzi, Farhadi, and Sedighi [18] numerically simulated the melting process of n-eicosane in concentric and eccentric double pipe heat exchangers and studied the effect of the internal tube position on the melting rate. They found that the melting rate was same in the early stage of melting, and then decreased in the concentric model. Spherical geometry represents an interesting case for heat storage application, since spheres are often used in packed beds and other energy storage applications. Recently an experimental and numerical study [23,24] has been carried out for constrained melting of PCMs in a spherical shell. It was mentioned that the conductive heat transfer dominates during the early period of melting, but as the buoyancy-driven convection is strengthened, melting in the top region of the sphere is faster than in the bottom region. As discussed earlier, most of the solidification/melting studies at present are based on regular shapes and limited studies are carried out using irregular shapes. To the best of our knowledge, there is no solidification analysis of NEPCM filled in a differentially heated trapezoidal cavity. The trapezoidal cavity has received significant attention of researchers due to its applicability in various fields, for example in the moderate concentrating solar collector [33]. Solidification of a binary mixture in a trapezoidal cavity is of practical importance in the casting and mold design because of the common practice to make the ingot wall with a small slope to facilitate withdrawal of the casting [34]. One of the possible methods to improve the performance of LHTES devices is to enhance the heat transfer rate in the nanofluid by increasing the surface area normal to the direction of heat transfer which can be done using trapezoidal cavity. Duggiralaet al. [34] experimentally investigated the effect of initial concentration of ammonium chloride (NH4Cl) in the range of 0–19.8% and boundary temperatures in the range of − 30 to 0 °C on the solidification of binary alloy (NH4Cl + H2O) in a trapezoidal cavity. However, this study does not explicitly investigate the effect of the trapezoidal cavity on solidification rate; Moreover, this study is not based on NEPCM. Therefore, a detailed study is required to understand the true effect of using a trapezoidal cavity for the solidification of NEPCM especially since the presence of non-vertical walls in a trapezoidal cavity makes the conduction/convective flow analysis more difficult than that in a square and rectangular cavity.

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R.K. Sharma et al. / Powder Technology 268 (2014) 38–47

The purpose of the present work is to analyze the solidification of the water–Cu nanofluid filled in an isosceles trapezoidal cavity using the Ansys-Fluent CFD commercial package. The effects of nanoparticle volume fractions, inclination angles and temperature differences between hot and cold walls and the Grashof number on the solidification process will be presented. The solidification time of the nanofluid in the trapezoidal cavity and square cavity will be compared keeping the internal area of both cavities equal. The average Nusselt number along the hot wall calculated for Gr = 103–105 is also presented.

The viscosity of nanofluid is given by Brinkman [35]: μf

μ nf ¼

ð1−ϕÞ2:5

:

ð6Þ

The heat capacitance of the nanofluid and part of the Boussinesq are:   ρcp

    ¼ ð1−ϕÞ ρcp þ ϕ ρcp

nf

f

s

ðρβÞnf ¼ ð1−ϕÞðρβÞ f þ ϕðρβÞs :

ð7Þ ð8Þ

2. Methodology The latent heat of the nanofluid is evaluated using 2.1. Geometry

ðρLh Þnf ¼ ð1−ϕÞðρLh Þ f :

A two-dimensional (2D) trapezoidal cavity of 10 mm2 internal area, as shown in Fig. 1, is considered in this study. The length (L) of the cavity is 10 mm and the height (H) is varied in such a way that the internal area of the cavity remains constant at 10 mm2. Three types of trapezoidal cavities are formed based on three different angles such as θ = 2.72°, 5.42° and 7.69°. 2.2. Mathematical formulations

The thermal conductivity of the quiescent (subscript 0) nanofluid is given as [36]:   ks þ 2k f −2ϕ k f −ks knfo  : ð10Þ ¼ kf k þ 2k þ 2ϕ k −k s

Continuity ∂u ∂v þ ¼ 0; ∂x ∂y

ð1Þ

X-momentum

f

s


 ∂p 2 − þ μ nf ∇ u þ ðρβÞnf gx ðT−T C Þ þ Sx ; ∂x

ð2Þ

keff ¼ knf 0 þ kd ;


 ∂p 2 − þ μ nf ∇ v þ ðρβÞnf gy ðT−T C Þ þ Sy ; ∂y

nf

The empirically-determined constant C is obtained from the work of Wakao and Kaguei [38]. In Eqs. (2)–(4), S is the source terms which are given by: Sx ¼ Að f Þu; Sh ¼

ð3Þ

Energy equation 2 2  3  3 k þ k k þ k nfo d nfo d ∂T ∂T ∂T ∂ 6 ∂T 7 ∂ 6 ∂T 7 þu þv ¼ 4   5þ 4   5−Sh : ð4Þ ∂t ∂x ∂y ∂x ∂x ∂y ∂y ρcp ρcp nf

nf

S ¼ Að f Þv; h y i ∂ ðρLf Þnf

ð13Þ

∂t 2

fÞ causes the gradual change in velocity from a finite where Að f Þ ¼ Mðf1− 3 þε

value in the liquid to zero in the solid in the computational cells. Here ε = 0.001, a small computational constant used to avoid division by zero and M (105 in the current investigation) is the mushy zone constant reflecting the morphology of the melting/solidification. The sensible enthalpy h, and total enthalpy, He is written as: ZT

The density of the nanofluid is given by: ρnf ¼ ð1−ϕÞρ f þ ϕρs :

ð11Þ

where kd is the thermal conductivity enhancement term due to the thermal dispersion and is given as [37]:   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð12Þ kd ¼ C ρcp u2 þ v2 ϕdp :

Y-momentum ∂v ∂v ∂v 1 þu þv ¼ ∂t ∂x ∂y ρnf

f

The effective thermal conductivity of the nanofluid is:

The continuity, momentum considering the Boussinesq approximation, and energy equations for this study can be written as follows:

∂u ∂u ∂u 1 þu þv ¼ ∂t ∂x ∂y ρnf

ð9Þ

h ¼ hc þ

C p;nf dT

ð14Þ

Tc

ð5Þ

H e ¼ h þ f Lh

ð15Þ

where fLh is latent heat which varies between zero for solid to Lh for liquid, and f is given by:

f ¼

8 0 if > > < T−T

TbT s s

if T s bTbT l > T −T s > : l 1 if T NT l

ð16Þ

where Ts and Tl are the solidus and liquidus temperature respectively. The calculation of the Nusselt number is an effective way to represent the heat transfer rate of a nanofluid. A local Nusselt number (NuL) is calculated with the following equation:

Fig. 1. Sketch of the two dimensional trapezoidal cavity.

 knf H ∂T  NuL ¼  : k f ΔT ∂n w

ð17Þ

R.K. Sharma et al. / Powder Technology 268 (2014) 38–47

The initial and boundary conditions for the present investigation are as follows:

Table 1 Thermo-physical properties of the base fluid (water) and the Cu nanoparticles. Property

Copper nanoparticles

Base fluid (water)

ρ [kg = m3] μ [Pa s] cp[J/kg K] k [W/m K] β [1/K] L [J/kg] Pr Ste dp [m]

8954 – 383 400 1.67 × 10−5 – – – 10−9

997.1 8.9 × 10−4 4179 0.6 2.1 × 10−4 3.35 × 105 6.2 0.125 –

1 Si

ZS ð18Þ

NuL ds

At the left inclined wall; u ¼ v ¼ 0; T ¼ T h At the right inclined wall; u ¼ v ¼ 0; T ¼ T c ∂T At horizontal surfaces; u¼v¼ ¼0 ∂y

at x ¼ 0 at x ¼ L

and 0≤ y≤H and 0≤ y≤H

at y ¼ 0; H

and 0≤ x≤ L

0

where s is the distance along the inclined wall and Si is the total length of the inclined wall.

2.3. Boundary conditions The horizontal walls are assumed to be insulated, non-conducting, and impermeable to heat transfer. The above mathematical relations are identical to the work of Khanafer et al. [39] and Khodadadi and Hosseinizadeh [3], thus they can be used to benchmark our numerical model by comparing our findings with theirs. The left and right inclined walls are kept at constant temperatures of T h = 283.15 K and Tc = 273.15 K, respectively. The nanofluid in the cavity is considered to be Newtonian, laminar, and incompressible. Thermophysical properties of the nanofluid as shown in Table 1, are assumed to be constant, whereas the density variation in the buoyancy force is handled by the Boussinesq approximation. The nanoparticles are assumed to have a uniform shape and size (10 nm diameter). The left lower corner of the cavity is the origin of a coordinate system. Gravity acts in the negative y-direction, i.e., gx = 0 and gy = − g.

A total of 62 CFD simulations as shown in Table 2 were carried out for validation and to study the effect of various parameters, such as the inclination angle of a trapezoidal section, nanoparticle volume fraction, and wall temperature difference on the total solidification time of the nanofluid. Case 1 CFD model, which is based on a rectangular cavity of 8.89 cm × 6.35 cm filled with solid gallium, is used to investigate the propagation of the melt front of gallium. The obtained results were compared with that from the experimental data of Gau and Viskanta [8] and numerical predictions of Brent et al. [32] and Khodadadi and Hosseinizadeh [3]. Case 2 is based on the solidification of water filled in a trapezoidal cavity and is used to investigate the temperature distribution at a certain height of cavity. The results are compared with experimental data of Duggirala et al. [34]. Eight simulations, i.e., Cases 3–10 are based on a square cavity of 1 cm length, filled with nanofluid, and these cases are used to validate the U-velocity at the vertical mid plane and the temperature profiles at the horizontal mid plane for Gr = 104 and 105 against such results from Khanafer et al. [39]. The remaining cases are based on a trapezoidal cavity. Thirteen simulations, i.e., Cases 11–23, were used to investigate the effect of nanoparticle volume fraction, ϕ, inclination angle, θ, and temperature difference, ΔT on the total solidification time of the nanofluid in a trapezoidal cavity. Three simulation cases (24–26) are used to investigate the effect of Gr on the mode of heat transfer. Cases 27–62 (36 simulations) are used to calculate an average Nusselt number along the hot wall of the trapezoidal cavity for various values of Gr, θ, and ϕ. 2.5. Mesh independency test Structured and uniform square grid spacings for x and y-directions, as shown in Fig. 2, are used for all the numerical simulation cases

Table 2 Simulation cases. Sim.⁎ cases

Inclination angle, θ°

Material

ΔT (°C)

Gr

Remarks

1 (square cavity)

0

Solid gallium

9.7

–

2 (trapezoidal cavity)

–

Pure water

–

–

3–10 (square cavity)

0

0, 0.1, and 0.2 Cu–H2O nanofluid

–

11–13 (trapezoidal cavity)

2.72

0, 0.1, and 0.2 Cu–H2O nanofluid

10

104 105 –

14–16 (trapezoidal cavity)

0.2 Cu–H2O nanofluid

10

–

0.2 Cu–H2O nanofluid

10

–

To study the phase front propagation in trapezoidal cavity

20–23 (trapezoidal cavity)

2.72 5.42 7.69 2.72 5.42 7.69 2.72

Used to validate the phase front of Gau and Viskanta [8], Brent et al. [32], and Khodadadi and Hosseinizadeh [3] Used to validate the temperature profile of solidifying water in the cavity [34] Used to validate the temperature profile and U-velocity results of Khanafer et al. [39] To study the effect of nanoparticle volume fraction on solidification of nanofluid To study the effect of inclination angles on solidification of nanofluid

0.2 Cu–H2O nanofluid

–

To study the effects of temperature difference on the solidification

24–26 (trapezoidal cavity)

2.72

0.2 Cu–H2O nanofluid

10 20 30 55 110 –

To study the effect of conduction or convection dominated heat transfer

27–62 (trapezoidal cavity)

2.72 5.42 7.69

0, 0.1, and 0.2 Cu–H2O nanofluid

–

105 106 107 103 104 105

17–19 (trapezoidal cavity)

⁎ Simulation.

:

2.4. Simulation cases

An average Nusselt number is calculated by integrating the local Nusselt number, using the following equation: Nuave ¼

41

To compute the values of average Nusselt number

42

R.K. Sharma et al. / Powder Technology 268 (2014) 38–47

Numerical simulations of solidification begin with the steady state natural convection within the trapezoidal cavity filled with nanofluid (Cu–H2O). At t = 0 s, the left wall of the cavity was kept at 10 °C higher than that of the solidification point of water (273.15 K) and the right wall was held at the solidification temperature. The nanofluid was maintained at uniform temperature 273.15 K throughout the cavity at t = 0 s. Then, the temperatures of both inclined walls are lowered by an equal amount (10 °C). Consequently, solidification of nanofluid begins at the right wall and the solid–liquid interface travels towards the left until the complete solidification of the nanofluid within cavity. This phenomenon is discussed below. 3. Results and discussion

Fig. 2. Meshed trapezoidal cavity.

reported here. The grid independence study is carried out from coarse to fine grid using five different uniform grids 50 × 50, 80 × 80, 120 × 120, 150 × 150, and 200 × 200. The complete solidification time for all grids was numerically computed and shown in Table 3. Between the grid size of 50 × 50 and 120 × 120, the fluctuation in results is as high as 4% but increasing the grid size beyond 120 × 120 reduces the variation substantially down (to 1%), so the grid size of 120 × 120 is considered in the present study with the finer grid on the side walls.

2.6. Numerical methods The SIMPLE method within the commercial CFD package ANSYS Fluent [40] is used to solve the governing equations (1)–(4). The enthalpy–porosity approach [32] is used for the phase change region inside the PCM, by which the porosity in each cell is set equal to the liquid fraction in that cell. The QUICK differencing scheme was used for solving the momentum and energy equations, whereas the pressure staggering option (PRESTO) scheme was used for pressure correction equations. In the enthalpy method, the solution is based on a fixed grid and governing equations are modified such that they are valid for both phases. Also, the mushy zone constant (M) is set to 105 [kg/m3s]. The time step size used for all the simulations in this study is 0.5 s and the number of iterations for each time step is 800. The underrelaxation factor of all the components, such as velocity components, pressure correction, thermal energy, etc., is kept at 0.3. Convergence criteria are set at 10− 6 for continuity and momentum and 10− 8 for thermal energy.

In Section 3.1, our CFD predictions of the melt front of the gallium in a square cavity is compared with the experimental data of Gau and Viskanta [8] and the numerical predictions by Brent et al. [32] and Khodadadi and Hosseinizadeh [3]. The U and V-velocities for Gr = 104 and 105 are compared with the numerical results of Khanafer et al. [39] to benchmark natural convection for the case of nanofluid. The present calculated values of average Nusselt number along the hot wall of a square cavity are compared with those available in the literature. Section 3.2 presents the CFD results of the effect of the nanoparticle volume fraction of solidification. While Sections 3.3, and 3.4 present the CFD results of the effect of inclination angles, θ, and few sets of temperature difference between hot and cold walls, ΔT on the solidification time. In addition, average values of the Nusselt number, Nuave calculated for various values of Grbf, ϕ, and θ along the hot wall of trapezoidal cavities are presented in Section 3.5. 3.1. Validation of the model The results of the Case 1 CFD model are compared with those of an experimental study of Gau and Viskanta [8], numerical predictions by Brent et al. [32] and Khodadadi and Hosseinizadeh [3] of the melting of solid gallium in a two-dimensional rectangular cavity (height, Ly = 6.35 cm; width, Lx = 8.9 cm) in Fig. 3. The horizontal walls are adiabatic and the vertical walls have different but constant temperatures. A uniform internal temperature at the melting point of gallium, 302.95 K is set to the cavity. At t = 0 s, the temperature of the left vertical wall is suddenly raised to a prescribed temperature above the melting point, resulting in melting of the gallium. The values of the governing dimensionless numbers and aspect ratio are listed in Table 4. The melt front is plotted at different times, i.e., 2, 6, 10 and 17 min, in Fig. 3. It is evident from this figure

Table 3 Mesh dependency test. Sr. no.

Inclination angle

Mesh size

Total solidification time (s)

Error

1. 2. 3. 4. 5.

2.72° 2.72° 2.72° 2.72° 2.72°

50 80 120 160 200

1292 1304 1360 1378 1392

– 0.92% 4.118% 1.306% 1.006%

× × × × ×

50 80 120 160 200

Fig. 3. Progress of the melting phase front with time: Comparison among the prediction of Khodadadi and Hosseinizadeh [3] and Brent et al. [32], experimental data of Gau and Viskanta [8], and present work.

R.K. Sharma et al. / Powder Technology 268 (2014) 38–47 Table 4 Parameters used in the validation of melting of gallium. AR Ra Pr Ste

Aspect ratio Rayleigh number Prandtl number Stefan number

0.714 3 × 105 0.021 0.040

that a reasonably good agreement exists between the computed and experimental melt front positions with a small discrepancy between the measured and calculated results. This discrepancy between the predicted phase front of the present CFD model and the experimental results may be due to the sub-cooling of approximately 2 °C in the solid [41]. In addition, at some times, e.g., 2 min, 6 min and 10 min, our results are relatively much closer to that of the experiment than the numerical prediction of Khodadadi and Hosseinizadeh [3]. To benchmark our model for solidification of water based PCMs, the results of the numerical study (Case 2) have been compared with the experimental data as reported by Duggirala et al. [34] and presented in Fig. 4. To validate the current numerical model, a 2-D trapezoidal cavity with a bottom and top widths of 65 mm and 165 mm respectively and a height of 130 mm was used. This cavity was completely filled with pure water and the initial temperature was kept at 0 °C. Top and bottom walls were made insulated and both inclined walls were kept at − 30 °C. As soon as the simulation was started, solidification began immediately. The temperature profile at a distance of 37 mm from the bottom wall and t = 72 min was plotted and compared with the experimental data available in literature. Numerical prediction is in good agreement with the experimental data. The results from Cases 3–10 CFD models, which are based on a square cavity of 1 cm, have been checked against numerical results reported by Khanafer et al. [39]. Fig. 5(a) and (b) shows the U-velocity distribution at the vertical mid plane for Gr = 104 and 105, respectively based on nanoparticle volume fractions of 0, 0.1, and 0.2. Fig. 5(c) shows the temperature distribution at the horizontal mid plane of the differential heated square cavity of height 1 cm filled with nanofluid of nanoparticle volume fraction 0.2, for Gr = 104 and 105. In general, for all of the nanoparticle volume fractions, the well-established trend of the horizontal velocity is seen exhibiting an accelerated flow near the top and bottom horizontal walls and a weak flow near the center of the cavity which shows the nanofluid behaves like a base fluid (water) near the center. Overall, the CFD results for all nanoparticle volume fractions are close to those from literature [39]. In particular, the results for the water (ϕ = 0) for both Grashof numbers have an excellent agreement. However, small discrepancies can be seen for ϕ = 0.1 and ϕ = 0.2 especially at the peaks. Fig. 5(c) shows the

Fig. 4. Comparison of the temperature profile at y = 37 mm in the pure water between the experimental data of Duggirala et al. [34] and present work.

43

temperature distribution at the horizontal mid plane for Gr = 104 and 105 based on nanoparticle volume fraction of 0.2. The temperature distribution of the CFD models is consistent with that of Khanafer et al. [39] for both Gr. All the above comparisons were used to validate our CFD models. Table 5 shows the comparison of variation of average Nusselt number (Nuavg) given in Eq. (18) with the Grashof number along the hot wall to validate the present results with those reported by Khanafer et al. [39], Ho et al. [42], and Das and Ohal [43] for a square cavity filled with pure water. A good agreement is obtained between the present and benchmark solutions.

Fig. 5. Comparison of the U velocity along the vertical mid plane of the cavity between present work and those of Khanafer et al. [39] for (a) Gr = 104, and (b) Gr = 105, and (c) comparison of temperature on the mid plane for ϕ = 0.2.

44

R.K. Sharma et al. / Powder Technology 268 (2014) 38–47

Table 5 Comparison of the average Nusselt number for square cavity where Pr = 6.2 (water) for ϕ = 0.0. Literature

Grbf = 103

Grbf = 104

Grbf = 105

Present Khanafer et al. [39] Ho et al. [42] Das and Ohal [43]

1.9377 1.9806 Not applicable 2.3097

4.1093 4.0653 3.9494 4.6512

8.2521 8.3444 7.8258 9.3662

3.2. Effect of nanoparticle volume fraction on solidification time

Fig. 7. Effects of inclination angles on solidification time of nanofluid within a trapezoidal cavity at ΔT = 10 °C and ϕ = 0.2.

a) 10

Y (mm)

Progress of Phase Front

5

0

1200

900

600

300

100

10

Time(s)

b) 10

Y (mm)

The vertical walls of the square cavity were inclined to form a trapezoidal cavity in such a way that the internal area remains constant for all inclination angles. Solidification was studied for five different angles, θ = 1.14°, 2.26°, 2.72°, 5.42°, and 7.69°. For the first two values of the inclination angle, the reduction in total solidification time of NEPCM was less than 5% when compared with the total solidification time in a square cavity. For θ = 2.72°, the reduction in solidification time is almost 8% and then it further increases with inclination. In this study, the results obtained from first two inclination angles are not presented in this study due to their insignificant reduction in solidification time and the last three cases of trapezoidal cavity are discussed in the following sections. The dependency of the solidification time on the dispersed copper nanoparticle volume fractions (ϕ = 0, 0.1, and 0.2), in the base fluid (H2O) filled in a trapezoidal cavity of θ = 2.72° is investigated in Cases 11–13 CFD models. Note that, the internal area (10 mm2) of the trapezoidal cavity is kept equal to the square cavity used by Khodadadi and Hosseinizadeh [3]. The solidification time for Cases 11–13 and those from the literature are presented in Fig. 6. The increase of the nanoparticle volume fractions decreases the solidification time of the nanofluids for all the cases. For example, the total solidification time of the trapezoidal cavity (θ = 2.72°) for ϕ = 0 and 0.2 is about 2700 s and 1300 s, respectively. In general, the last 30% of the fluids needs about a half of the total solidification time before it completely solidifies and this is true for all cases. A possible reason for the reduction in solidification time with dispersion of solid nanoparticles is that, when there is a higher heat transfer rate of NEPCM, the crystal grows rapidly. It also may be due to the fact that less energy per unit mass of nanofluid is required to solidify it because of the lower latent heat of fusion. Now, compare the trapezoidal cavity with the square cavity. The nanofluid requires less time to solidify completely in the trapezoidal cavity than in the square cavity. The solidification time in the trapezoidal cavity of θ = 2.72° filled with pure water is almost 10% less than that of water filled in a square cavity, while this time is approximately 5% and 8% less for trapezoidal cavity of θ = 2.72° filled with nanoparticle

Progress of Phase Front

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Fig. 6. Effect of nanoparticle volume fraction on solidification time of nanofluid within the square and trapezoidal cavity of θ = 2.72°.

Fig. 8. Solid–liquid interface position at different time during solidification process for temperature difference of 10 °C and ϕ = 0.2. (a) θ = 2.72°, (b) θ = 5.42° and (c) θ = 7.69°.

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a)

b)

Fig. 9. Velocity field for pure water (ϕ = 0), and temperature difference ΔT = 10 °C (a) at t = 200 s and (b) for θ = 7.69°.

volume fractions of 0.1 and 0.2 respectively. This enhancement of the heat transfer may be due to a higher surface area normal to the direction of heat transfer in the trapezoidal cavity than that in the square one. In other words, the length of the side walls of the trapezoidal cavity is higher than that of the side vertical walls of the square cavity. 3.3. Effect of inclination angles Fig. 7 shows the effect of the inclination angle of the side walls, θ, on the solidification time for Cases 14–16, which are based on θ = 2.72°, 5.42° and 7.69°, ϕ = 0.2 and the temperature difference of 10 K. For comparison, the total solidification time of the nanofluid (ϕ = 0.2) under the same temperature difference in a square cavity is included in the figure. Note that the internal area of the trapezoidal cavities of these cases and that of the square cavity was kept constant and equal to 10 mm2. The solidification time of the nanofluid within the

trapezoidal cavities decreases with the increase of θ. The total solidification time for the trapezoidal cavity of θ = 7.69° filled with nanofluid (ϕ = 0.2) is approximately 140 s (11.3%) less than that for the square cavity. This may be because of the surface area normal to the direction of heat transfer increases with an increase of θ, resulting in the enhanced heat transfer rate. The heat transfer phenomenon for solidification consists of both conduction and convection energy transfer. To understand this phenomenon in the solidification of the nanofluid, the propagation of the solid–liquid interface for Cases 17, 18 and 19 are shown in Fig. 8(a), (b) and (c), respectively at 10 s, 100 s, 300 s, 600 s, 900 s and 1200 s. In general, the higher the inclination angle of side walls, the quicker the phase front moves. For example, at 900 s, the phase front has reached near the extreme left corner of the cavity for the case of θ = 7.69° (Fig. 8c). However, this is not the same for the case of θ = 2.72° (Fig. 8a) where the phase front is yet to reach the left inclined side wall. The figure also shows the apparent 10

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X (mm) Fig. 10. Effect of temperature difference on the solidification of nanofluid of nanoparticle volume fraction 0.2 inside cavity of θ = 2.72°.

Fig. 11. Effect of temperature difference on the solid–liquid interface at 100 s during solidification of nanofluid of nanoparticle volume fraction 0.2 inside cavity of θ = 2.72°.

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a) Gr 105 Gr 106 Gr 107

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moves towards the left, both vortices shrink in covered space and their strength decays. 3.4. Effects of temperature difference and Grashof number

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X(mm) Fig. 12. Solid–liquid interface position for ϕ = 0.2 and θ = 2.72° at (a) 100 s, and (b) 300 s.

difference in phase velocities for all inclination angles and the solid– liquid interface is parallel to the cold wall which means the heat transfer in all these cases is consistently conduction dominated. The effect of natural convection on the heat transfer rate is discussed in Section 3.4. Velocity vectors at a certain time during the solidification of pure water (ϕ = 0) are shown in Fig. 9. Fig. 9(a) shows the velocity vectors at 200 s for three inclination angles (θ = 2.72°, 5.42°, and 7.69°). The figure shows that for a particular ΔT, the strength of circulation diminishes with increase in θ. This is due to the decrease in vertical velocity with increase in θ. Instantaneous velocity vectors during the solidification of pure water filled in a cavity with θ = 7.69° are presented in Fig. 9(b). In the early time step of 50 s, two vortices nearly equal in size formed in CW and CCW directions. As the solidification progresses this dual vortex structure persists but because the solid–liquid interface

The variation of the liquid fraction against the solidification time of Cases 20–23, which are based on ϕ = 0.2 at five different temperature differences, ΔT = 10°, 20°, 30°, 55° and 110° in the trapezoidal cavity with inclination angle, θ = 2.72° is shown in Fig. 10. The increasing temperature difference consistently decreases the solidification time. Similar to the low temperature difference, the phase front moves parallel to the cold wall for large temperature difference of 110 °C as well, which signifies that conduction still dominates the heat transfer phenomenon as shown in Fig. 11. To study the effect of natural convection on heat transfer, the Gr is varied from 105 to 107 (Cases 24–26) and the results are shown in Fig. 12. The figure shows the solid–liquid interface at two different times (100 s and 300 s) during solidification for all three Grashof numbers. For Gr ≤ 105 the solid–liquid interface is almost parallel to the cold wall and moves towards hot wall with uniform velocity until complete solidification from which conduction dominated heat flow can be inferred. As the Gr increases (106, and 107), buoyancy force increases within the liquid which causes the interface to deflect and the rapid solidification appears in the lower half of the cavity compared to upper half which indicates the natural convection dominated heat flow. 3.5. Effect of Grashof number on Nusselt number Fig. 13 shows the variation of the average Nusselt number along the hot wall for Cases 27–62 where Grashof number (Gr) is varied from 103 to 105, cavity inclination angles (θ) from 0° to 7.69° and nanoparticle volume fractions (ϕ) from 0 to 0.2. Pr = 6.2 is used in all these cases. The Nusselt number depends strongly on the inclination angle and it increases with increasing angle. In general, the Nusselt number increases with an increase of the nanoparticle volume fraction and this occurs for all Grashof numbers because the thermal conductivity of nanofluid enhances with the void fraction. For low Grashof number (b 103 and 104), the increase in the Nusselt number versus the angles is small because of the dominant conductive heat transfer but as we increase Grashof number (≥ 105), the buoyancy-driven heat transfer dominates and a higher difference in Nusselt number versus the angle can be seen. It should be noted that the trend in the figure for the average Nusselt number along the hot wall against the inclination angle would be downward if the Nusselt number is based on the effective thermal conductivity keff. 4. Conclusions The solidification process of the nano-enhanced phase change materials in the isosceles trapezoidal cavities has been studied using CFD. Various nanoparticle volume concentrations, inclination angles, and Grashof numbers have been considered to study their effects on the heat transfer. This study has the following conclusions:

Fig. 13. Effect of Grashof number, nanoparticle volume fraction and inclination angle on average Nusselt number for Pr = 6.2.

• The heat transfer performance of NEPCM is significantly enhanced with the use of a trapezoidal cavity when compared to a square cavity having the same internal area. • For the temperature difference of 10 °C, and a small inclination angle (2.72°) in the side walls of square cavity decreasing the solidification time of nanofluid of 20% nanoparticle volume fraction by approximately 7% while increasing the inclination angle to 5.42° and 7.69°, it reduces by 11% and 11.5% respectively. In short, the solidification time reduces with the increase of the inclination angle. • NEPCM as used in the current study shows great ability to store/release the thermal energy in comparison to the conventional PCMs. Increasing nanoparticle volume fraction decreases the solidification time.

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• The heat transfer rate greatly depends on the temperature difference and increasing temperature difference increases the heat transfer rate for all values of nanoparticle volume fraction and inclination angles but, the mode of heat transfer remains conduction. • The mode of heat transfer depends on the value of the Grashof number. For low values of Gr (≤105), conduction is the main heat transfer mechanism while for higher values of Gr (N105), heat transfer takes place majorly by convection. Acknowledgment This research is financially supported by the University of Malaya, Ministry of Higher Education High Impact Research (UM.C/HIR/ MOHE/ENG/21) and University Malaya Research Grant (UMRG: RG121/11AET), Kuala Lumpur, Malaysia. References [1] Amir Faghri, T. Zhang, Transport Phenomena in Multiphase Systems, Elsevier Inc., London, UK, 2006. [2] S.U.S. Choi, J.A. Eastman, Enhancing Thermal Conductivity of Fluids with Nanoparticles, 1995. [3] J.M. Khodadadi, S.F. Hosseinizadeh, Nanoparticle-enhanced phase change materials (NEPCM) with great potential for improved thermal energy storage, Int. Commun. Heat Mass Transfer 34 (2007) 534–543. [4] S. Ostrach, Natural convection in enclosures, J. Heat Transf. 110 (1988) 1175–1190. [5] A.V. Arasu, A.S. Mujumdar, Numerical study on melting of paraffin wax with Al2O3 in a square enclosure, Int. Commun. Heat Mass Transfer 39 (2012) 8–16. [6] M. Pinelli, S. Piva, Solid/liquid phase change in presence of natural convection: a thermal energy storage case study, J. Energy Resour. Technol. 125 (2003) 190–198. [7] S. Wu, H. Wang, S. Xiao, D. Zhu, numerical simulation on thermal energy storage behavior of Cu/paraffin nanofluids PCMs, Procedia Eng. 31 (2012) 240–244. [8] C. Gau, R. Viskanta, Melting and solidification of a pure metal on a vertical wall, J. Heat Transf. 108 (1986) 174–181. [9] C. Gau, R. Viskanta, Melting and solidification of a metal system in a rectangular cavity, Int. J. Heat Mass Transf. 27 (1984) 113–123. [10] U. Stritih, An experimental study of enhanced heat transfer in rectangular PCM thermal storage, Int. J. Heat Mass Transf. 47 (2004) 2841–2847. [11] X.-Q. Wang, C. Yap, A.S. Mujumdar, A parametric study of phase change material (PCM)-based heat sinks, Int. J. Therm. Sci. 47 (2008) 1055–1068. [12] D.D. Ganji, A. Malvandi, Natural convection of nanofluids inside a vertical enclosure in the presence of a uniform magnetic field, Powder Technol. 263 (2014) 50–57. [13] M. Sheikholeslami, M. Gorji-Bandpy, S.M. Seyyedi, D.D. Ganji, H.B. Rokni, S. Soleimani, Application of LBM in simulation of natural convection in a nanofluid filled square cavity with curve boundaries, Powder Technol. 247 (2013) 87–94. [14] D.S. Riley, F.T. Smith, G. Poots, The inward solidification of spheres and circular cylinders, Int. J. Heat Mass Transf. 17 (1974) 1507–1516. [15] K.A.R. Ismail, C.L.F. Alves, M.S. Modesto, Numerical and experimental study on the solidification of PCM around a vertical axially finned isothermal cylinder, Appl. Therm. Eng. 21 (2001) 53–77. [16] K. Kim, H. Kim, C. Park, J. Peck, C. Kang, Experimental study on melting/solidification of adjoining two different PCMs in a concentric vertical cylinder, Proceedings of the KSME, 2011, pp. 130–135. [17] F. Agyenim, P. Eames, M. Smyth, Experimental study on the melting and solidification behaviour of a medium temperature phase change storage material (Erythritol) system augmented with fins to power a LiBr/H2O absorption cooling system, Renew. Energy 36 (2011) 108–117. [18] A.R. Darzi, M. Farhadi, K. Sedighi, Numerical study of melting inside concentric and eccentric horizontal annulus, Appl. Math. Model. 36 (2012) 4080–4086.

47

[19] G.R. Solomon, R. Velraj, Analysis of the heat transfer mechanisms during energy storage in a Phase Change Material filled vertical finned cylindrical unit for free cooling application, Energy Convers. Manag. 75 (2013) 466–473. [20] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid, Powder Technol. 254 (2014) 82–93. [21] M. Sheikholeslami, D.D. Ganji, Three dimensional heat and mass transfer in a rotating system using nanofluid, Powder Technol. 253 (2014) 789–796. [22] M. Hatami, M. Sheikholeslami, D.D. Ganji, Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method, Powder Technol. 253 (2014) 769–779. [23] E. Assis, L. Katsman, G. Ziskind, R. Letan, Numerical and experimental study of melting in a spherical shell, Int. J. Heat Mass Transf. 50 (2007) 1790–1804. [24] F.L. Tan, S.F. Hosseinizadeh, J.M. Khodadadi, L. Fan, Experimental and computational study of constrained melting of phase change materials (PCM) inside a spherical capsule, Int. J. Heat Mass Transf. 52 (2009) 3464–3472. [25] F.L. Tan, Constrained and unconstrained melting inside a sphere, Int. Commun. Heat Mass Transfer 35 (2008) 466–475. [26] L. Bilir, Z. İlken, Total solidification time of a liquid phase change material enclosed in cylindrical/spherical containers, Appl. Therm. Eng. 25 (2005) 1488–1502. [27] K.A.R. Ismail, J.R. Henrı́quez, Solidification of pcm inside a spherical capsule, Energy Convers. Manag. 41 (2000) 173–187. [28] S.F. Hosseinizadeh, A.A.R. Darzi, F.L. Tan, Numerical investigations of unconstrained melting of nano-enhanced phase change material (NEPCM) inside a spherical container, Int. J. Therm. Sci. 51 (2012) 77–83. [29] P. Chandrasekaran, M. Cheralathan, V. Kumaresan, R. Velraj, Enhanced heat transfer characteristics of water based copper oxide nanofluid PCM (phase change material) in a spherical capsule during solidification for energy efficient cool thermal storage system, Energy 72 (2014) 636–642. [30] S.F. Hosseinizadeh, A.A. Rabienataj Darzi, F.L. Tan, J.M. Khodadadi, Unconstrained melting inside a sphere, Int. J. Therm. Sci. 63 (2013) 55–64. [31] W. Zhao, A.F. Elmozughi, A. Oztekin, S. Neti, Heat transfer analysis of encapsulated phase change material for thermal energy storage, Int. J. Heat Mass Transf. 63 (2013) 323–335. [32] A.D. Brent, V.R. Voller, K.J. Reid, Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal, Numer. Heat Transfer 13 (1988) 297–318. [33] H. Saleh, R. Roslan, I. Hashim, Natural convection heat transfer in a nanofluid-filled trapezoidal enclosure, Int. J. Heat Mass Transf. 54 (2011) 194–201. [34] R.K. Duggirala, C.X. Lin, C. Ghenai, Investigation of double-diffusive convection during the solidification of a binary mixture (NH4Cl–H2O) in a trapezoidal cavity, Exp. Fluids 40 (2006) 918–927. [35] H.C. Brinkman, The Viscosity of concentrated suspensions and solutions the viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [36] F.J. Wasp, Solid–Liquid Slurry Pipeline Transportation, Trans. Tech., Berlin, 1977. [37] A. Amiri, K. Vafai, Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media, Int. J. Heat Mass Transf. 37 (1994) 939–954. [38] N. Wakao, S. Kaguei, Heat and Mass Transfer in Packed Beds, Gordon and Breach Science Publishers, New York, 1982. 175–205. [39] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639–3653. [40] A.F. 13.0, User's Guide, ANSYS Inc., USA, 2011. [41] Z.-X. Gong, S. Devahastin, A.S. Mujumdar, Enhanced heat transfer in free convectiondominated melting in a rectangular cavity with an isothermal vertical wall, Appl. Therm. Eng. 19 (1999) 1237–1251. [42] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transf. 51 (2008) 4506–4516. [43] M.K. Das, P.S. Ohal, Natural convection heat transfer augmentation in a partially heated and partially cooled square cavity utilizing nanofluids, Int. J. Numer. Methods Heat Fluid Flow 19 (2009) 411–431.