Heat transfer enhancement of phase change materials by fins under simultaneous charging and discharging

Energy Conversion and Management 152 (2017) 136–156

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Heat transfer enhancement of phase change materials by fins under simultaneous charging and discharging

MARK

⁎

Mahmood Mastani Joybaria, Fariborz Haghighata, , Saeid Seddeghb, Abduljalil A. Al-Abidic a b c

Department of Building, Civil and Environmental Engineering, Concordia University, Montreal H3G 1M8, Canada School of Engineering & ICT, University of Tasmania, Hobart, TAS 7001, Australia Department of HVAC Engineering, Sana’a Community College, P.O. Box 5695, Sana’a, Yemen

A R T I C L E I N F O

A B S T R A C T

Keywords: Triplex tube heat exchanger Phase change material Natural convection Simultaneous charging and discharging Heat transfer enhancement Fin

Due to the inherent intermittency of renewable energy sources such as solar, latent heat thermal energy storage in phase change materials (PCMs) has received considerable attention. Among several techniques to enhance PCMs’ thermal conductivity, the majority of studies have focused on fin integration due to its simplicity, ease of manufacturing, and low cost. In this study, utilization of extended surfaces (by longitudinal fins) was investigated by development of a numerical model to study the performance of a triplex tube heat exchanger (TTHX) equipped with a PCM under simultaneous charging and discharging (SCD). Governing equations were developed and numerically solved using ANSYS Fluent v16.2. Three conventional fin geometries and six developed fin configurations were compared based on the temperature, liquid fraction, and natural convection behavior under both SCD and non-SCD conditions. The intensity of natural convection was investigated for different fins for the inside heating/outside cooling scenario based on the solid–liquid interface evolution over time. The results indicated that since the buoyancy forces induce upward melted PCM motion, the inner hot tube requires fins on its lower half, while the outer cold one should be extended from its upper half. It was concluded that the case with 3 hot tube fins and 1 cold tube fin is most compatible with natural convection and provides the best performance under SCD conditions.

1. Introduction In recent decades, latent heat storage in phase change materials (PCMs) received considerable attention. Advantages of PCMs include high energy storage density, almost constant temperature storage, chemical stability and economic efficiency [1]. Besides, PCMs are suitable for management of time mismatch between energy supply and demand [2,3]. Therefore, PCMs have found several applications, e.g. in solar systems [4], ventilation systems [5], refrigeration systems [6], net zero energy buildings [7], hot water tanks [8], etc. Four regimes hierarchically occur during melting of PCMs; (1) pure conduction, (2) combined conduction and natural convection, (3) natural convection dominated, and (4) shrinking solid [9,10]. In a horizontal cylindrical storage, at the onset through the early stages of melting, conduction is the dominant heat transfer mechanism. Later, natural convection develops and dominates by eroding the solid PCM at the upper positions, enhancing the melting rate [11–14]. On the other hand, during solidification, conduction is the dominant heat transfer

mechanism, which results in a cylindrical melting front shape around the heat transfer fluid (HTF) tube [15]. Therefore, unlike melting, the upper and lower halves of a horizontal [12] as well as a vertical [16] shell and tube heat exchanger had almost similar temperature profiles. The reason why conduction is identified as the dominant heat transfer mechanism in early stages of melting as well as the entire solidification is that regardless of the height of thermocouples, the recorded temperature and heat transfer were almost identical [17,18]. This shows that gravity did not affect the process; consequently, natural convection was almost negligible. During solidification, solid PCM layers are deposited on the tube wall over time and grow outward, which results in an increasing heat transfer resistance over time [11,12]. Therefore, recovering the stored heat from a PCM takes longer than its charging period due to the high thermal resistance of the PCM. Research in the area of phase change process of PCMs had a routine trend over the course of recent years. This trend includes investigation of a PCM under a one-time charging or one-time discharging process or considering consecutive charging and discharging periods. Therefore, it

Abbreviations: CFD, computational fluid dynamics; CHTF, cold heat transfer fluid; DHW, domestic hot water; HHTF, hot heat transfer fluid; HTF, heat transfer fluid; PCM, phase change material; SCD, simultaneous charging and discharging; SDHW, solar domestic hot water; TTHX, triplex tube heat exchanger ⁎ Corresponding author. E-mail address: [email protected] (F. Haghighat). http://dx.doi.org/10.1016/j.enconman.2017.09.018 Received 2 April 2017; Received in revised form 17 August 2017; Accepted 7 September 2017 0196-8904/ © 2017 Elsevier Ltd. All rights reserved.

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Nomenclature

C Cp D f h h H k Nu p Pr r Re S t T v

β γ ΔH ε λ μ ρ ηf

mushy zone parameter (kg·m−3·s−1) specific heat (J·kg−1·K−1) diameter (m) friction factor (–) sensible specific enthalpy (J·kg−1) convective heat transfer coefficient (W·m−2·K−1) total specific enthalpy (J·kg−1) thermal conductivity (W·m−1·K−1) Nusselt number (–) pressure (N·m−2) Prandtl number (–) radius (m) Reynolds number (–) source term time (s or min) temperature (K) velocity (m·s−1)

thermal expansion coefficient (K−1) liquid fraction (–) latent specific enthalpy (J·kg−1) small number to avoid division by zero, 0.001 (–) latent heat of fusion (J·kg−1) dynamic viscosity (kg·m−1·s−1) density (kg·m−3) fin effectiveness (–)

Subscripts

h i l m o ref s std

hydraulic inner; initial; corresponding liquidus middle outer reference solidus steady

Greek symbols

α

thermal diffusivity (m2·s−1)

assuming a symmetric 1D model and was solved by explicit finite difference approach for temperature and liquid fraction. Recently, the same authors experimentally compared the performance of the same system under natural as well as forced convective heat transfer mechanism from the shell to the air [24]. The results indicated that the latter caused lower melting rate where higher air velocities had longer transition time to shift the heat transfer mechanism from conduction to natural convection within the PCM. Therefore, the high thermal resistance of the solid PCM negatively affected the performance of the system. The most noticeable shortcoming of PCMs is their low thermal conductivity. For organic and salt hydrate PCMs, typical thermal conductivity values are in the range of 0.15–0.3 W/mK and 0.4–0.7 W/mK, respectively [25]. This results in high thermal resistance within PCMs during their operation, affecting their thermal response and efficiency [1]. Several solutions have been investigated in order to enhance the thermal conductivity of PCMs, which could be generally classified into the following major categories; introduction of high thermal conductivity additives/metal matrices, microencapsulation of PCMs, using multiple PCMs, increasing heat transfer surface area by utilization of extended surfaces (fins) or multiple tubes, etc. [13]. Advantages of fins include manufacturing and utilization simplicity as well as low cost [25]. Furthermore, they have been reported to be effective for organic PCMs [26]. Therefore, in this study, application of extended surfaces by fins is investigated. In a cylindrical geometry, fins could be positioned radially or longitudinally. The main reasons to use radial fins are ease of design and manufacturing as well as low cost [1]. However, several experimental results indicated superior performance for longitudinal fins since they did not suppress the natural convection during melting and also helped the solidification process [27]. It is important to consider the fact that application of fins should not interfere with or prevent the natural convection during melting [17]. For instance, a configuration with four perpendicular short longitudinal fins was found to intensify natural convection [28]. The same study also reported that longer fins were more effective than shorter ones in terms of shortening the phase change process. Therefore, there is a conflict between heat transfer enhancement by fins and effectiveness of natural convection within PCMs. For instance, due to the utilization of longitudinal fins, the melted PCM motion was limited to the space left between fins and the

is presumably assumed that the PCM is initially completely melted/ solidified and then it undergoes solidification/melting. This assumption ignores the fact that in real life applications, PCMs might go through simultaneous charging and discharging (SCD). In such scenarios, a PCM undergoes charging from one side, while it simultaneously faces discharging from the other. Specifically, SCD can occur when dealing with unpredictable occupant behavior or intermittent consumption. For instance, in the case of solar domestic hot water (SDHW) systems, water consumption might occur during sunshine hours; i.e. at the same time when the thermal storage is being charged. PCMs have been investigated in numerous studies under either charging, discharging, or consecutive charging and discharging process. However, SCD of PCMs has rarely been investigated in the literature. Liu et al. experimentally studied SCD for a heat pipe heat exchanger [19]. It was reported that the low PCM thermal conductivity prevented efficient performance of the system. Shabgard et al. used NaCl as the PCM in a heat pipe-assisted heat exchanger and numerically investigated three modes of charging only, SCD and discharging only [20]. They found negligible natural convection effect and that the PCM could serve as a capacitor, damping the input temporal variation and shifting the peak load. Using Cu-0.3Si as PCM, Sharifi et al. numerically investigated the same three modes in a heat pipe-assisted heat exchanger [21]. In that study, first, the PCM underwent charging until it reached a liquid fraction value of 0.5; then, an hour of SCD was imposed and finally discharging was carried out. In such a scenario, natural convection was found to be weakly effective compared to the high thermal conductivity of the PCM. Murray and Groulx experimentally investigated consecutive charging/discharging [17] as well as simultaneous charging/discharging [22] in a vertical cylindrical storage for SDHW. The PCM was continuously charged by the hot HTF, while the SCD was imposed intermittently to mimic a typical SDHW usage. In such a scenario, the final steady state was reported to be the fully melted and independent from the initial PCM condition. Moreover, it was found that during SCD, low thermal conductivity of the solid PCM controlled the heat exchange process and the natural convection was not beneficial in this regard. In another study, SCD of a PCM in a horizontal shell and tube heat exchanger (PCM at the shell side) was modeled and experimentally investigated [23]. An electric heater was mounted at the tube side, while air was blown over the shell side to cool down the structure. The numerical analysis was greatly simplified 137

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In summary, the main limitations of the abovementioned studies include:

shell wall [13]. An experimental study reported that at low Fourier numbers, the fin effectiveness was less than unity, which was attributed to the suppression of natural convection by fins [29]. In a recent study, an inflection point was reported during melting inside a longitudinally finned horizontal sleeve tube heat exchanger [28]. The point was attributed to the completion of melting at the upper half of the storage due to the buoyancy-induced motion. Beyond that point, conduction was found to replace natural convection to melt the rest of the PCM. Longitudinal fin configurations were investigated in several horizontal shell and tube heat exchangers with PCM at the shell side. Comparing the thermal behavior of a finned tube with two vertically and horizontally elliptic tube designs revealed that integration of fins was the best option to shorten the phase change process [30]. In one study, four fins of straight and angled designs were compared where the former included two vertical and two horizontal fins, while the latter was a rotation of the former by 45° [12]. It was found that during melting under low HTF temperatures, the angled configuration presented superior performance. This was due to the higher natural convection by the increased heat transfer surface area at the lower half of the storage. Another study found that the key to enhancing heat transfer during melting by fins is to allocate fins to the lower half of a horizontal shell and tube system [28]. Given this, it was proposed that the optimum fin configuration is three fins on the tube located at the bottom of the system where one fin is vertically positioned at the bottom and the two others are located at an angle between 60° to 90° on both sides of it. In a recent study, the number of finned tubes in a multiple tube heat exchanger was optimized for discharging process [31]. An optimum number of tubes and number of fins were reported beyond which a marginal reduction in discharging time was achieved. Khan et al. developed a novel shell and tube geometry with multiple finned tubes using commercial grade paraffin as PCM [32]. The numerical results indicated that increasing the number of tubes enhanced the heat transfer due to the increased heat transfer surface area. Another configuration which has received attention, due to its increased heat transfer surface area, is a triplex tube heat exchanger (TTHX). Basically, a TTHX is a concentric configuration of three tubes where the middle tube is filled with a PCM while the inner and outer ones carry HTFs. Similar to double pipe heat exchangers, a TTHX can benefit from fins. Nevertheless, advantages of a TTHX over double pipe heat exchangers include its increased heat transfer surface area, possibility of low temperature heating/cooling of a PCM, and faster melting/ solidification process [33]. Table 1 shows a summary of the studies with a PCM in TTHXs. In a recent series of studies, Al-Abidi et al. investigated a TTHX either for heat or cold energy storage (i.e. non-SCD condition) with/without fins using a 2D heat transfer model [13,33–37]. Their results indicated that the more the number of fins, the shorter was the phase change process. Similar results were reported for longer fins. However, the effect of fin thickness was found to be insignificant. It should be noted that despite the capability of TTHXs to provide SCD, the heat transfer enhancement by integration of fins in such systems has not been investigated under SCD.

• In the investigated configurations with SCD, the solid PCM pre• • •

vented heat transfer due to its higher thermal resistance. This could be tackled by means of longitudinal fins inside a TTHX due to the increased heat transfer surface area and more dominant natural convection, According to Table 1, the PCM was either charging or discharging and SCD in TTHXs has not been investigated so far (except in our earlier study without fins [40]), Natural convection suppression occurred in several earlier studies with fins due to ignoring the mechanism of buoyancy-driven upward melted PCM motion, Several finned configurations have been investigated with latent heat thermal energy storage in PCMs. Nevertheless, to the best of the authors’ knowledge, none of the studies focused on the SCD process.

In our earlier study [40], a numerical model was developed and validated to study the heat transfer mechanism of the phase change process under SCD in a horizontal TTHX. It was found that for both initially melted and solidified PCMs under internal heating/external cooling, the upward melted PCM motion solely affected the upper half of the domain. On the other hand, the lower half remained solid with high thermal resistance, preventing effective heat transfer. Therefore, in this study, application of finned tubes is investigated to enhance the heat transfer inside a TTHX. First, three conventional fin configurations for non-SCD applications in TTHXs (from the literature) are studied to investigate their effectiveness under SCD. Thereafter, six configurations are developed, which are then analyzed for both SCD and non-SCD conditions. Finally, the effect of fin geometry including their number, length, and thickness are investigated under SCD. 2. System description In this section, properties of the investigated PCM as well as the modeling approach are presented. 2.1. TTHX characteristics In this study, the United States standard copper tube sizing approach was used in which “Type M” tubes are recommended for water heating applications [41]. The heat exchanger (Fig. 1) had inner, middle and outer tubes with nominal diameters of 2, 5 and 8 in, respectively. Table 2 shows the dimensions of the tubes in the investigated TTHX. 2.2. PCM thermophysical properties A commercial PCM (RT31, Rubitherm Technologies GmbH) was used in this study. Table 3 shows the thermophysical properties of the

Table 1 Summery of the studies with PCM in triplex tube heat exchanger. Ref.

[38] [39] [33] [34] [13] [37] [36] [35]

Study Type

Objective

Numerical

Experimental

✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

PCM

DHW heating by air source heat pump Waste heat recovery Energy supply and demand mismatch Energy supply and demand mismatch Energy supply and demand mismatch Energy supply and demand mismatch Energy supply and demand mismatch Liquid desiccant air conditioning

138

Paraffin 56# n-Hexacosane RT82 RT82 RT82 RT82 RT82 RT82

Process Melting

Solidification

✓ ✓

✓ ✓ ✓

✓ ✓ ✓ ✓ ✓

✓

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3.2. Modeling Since the problem was axisymmetric (line of symmetry was θ = 0 ), half of the annulus was modeled to save computational time. The governing equations were developed as follows: Continuity equation:

ri

∂ρ 1 ∂ 1 ∂ + (ρrvr ) + (ρvθ ) = 0 ∂t r ∂r r ∂θ

(1)

Momentum balance: Darcy’s law was used to account for the flow of the mushy PCM due to the natural convection. The law appears as a source term in the momentum balance [43]: Fig. 1. Schematic representation of the triplex tube heat exchanger.

C (1−γ )2 ui γ3 + ε

Si = Table 2 Dimensions of the tubes in the TTHX [41] Tube

Nominal diameter (in)

Inner radius (mm)

Thickness (mm)

Inner Middle Outer

2 5 8

25.51 62.32 98.89

1.47 2.77 4.32

where γ is the liquid fraction of the melted PCM, which is presented later. To avoid division by zero, ε is added to the denominator with the value of 0.001 [44]. The mushy zone parameter C in Equation (2) is a measure that indicates the steepness of reaching zero velocity during solidification. For most modeling approaches, values between 104 kg·m−3·s−1 and 107 kg·m−3·s−1 are recommended for the mushy zone parameter (C) [45]; therefore, in this study, 105 kg·m−3·s−1 was considered. The following equations were developed for the two velocity components:

Table 3 Thermophysical properties of RT31 [42] Property

Symbol

Value

Solidus temperature Liquidus temperature Solid density Liquid density Specific heat capacity Latent heat of fusion Thermal conductivity Thermal expansion coefficient Dynamic viscosity

Ts Tl ρs ρl Cp λ k β μ

300.15 K (27 °C) 306.15 K (33 °C) 880 kg/m3 760 kg/m3 2000 J/kg·K 170,000 J/kg 0.2 W/m·K 0.00076 1/K 0.002508 kg/m·s

(2)

∂ (ρvr ) ∂t

−

∂p ∂r

+

∂

+ μ ⎡ ∂r ⎣

∂ (ρvθ) ∂t

−

1 ∂ (ρrvr vr ) r ∂r

+

1 ∂p r ∂θ

(

1 ∂ r ∂r

1 ∂ (ρrvr vθ) r ∂r ∂

+ μ ⎡ ∂r ⎣

(

+

1 ∂ (ρvr vθ) = r ∂θ

)

(rvr ) +

+

1 ∂ r ∂r

1 ∂2vr 2 ∂vθ − ⎤ r 2 ∂θ 2 r 2 ∂θ ⎦

+ ρg cosθ−

C (1 − γ )2 γ3 + ε

vr

(3)

1 ∂ (ρvθ vθ) = r ∂θ

)

(rvθ ) +

1 ∂2vθ r 2 ∂θ 2

+

C (1 − γ )2 2 ∂vr ⎤−ρg sinθ− 3 vθ r 2 ∂θ ⎦ γ +ε

(4)

Boussinesq approximation was introduced to account for the natural convection process in the melted PCM [37,44]:

PCM, which filled the middle space in the TTHX as shown in Fig. 1.

ρ = ρl [1−β (T −Tl )]

(5)

where β is the thermal expansion coefficient and ρl is the liquid PCM density at its melting temperature Tl. Energy balance: Enthalpy method was adopted since it has been proven to prevent the difficulties of phase change front tracking of the normal energy balance equation [44]. Energy balance in terms of enthalpy variations is:

3. Numerical approach In this section, the modeling procedure and its assumptions as well as the considered initial and boundary conditions are presented. 3.1. Assumptions According to the physics of the problem as well as the common assumptions used in the literature, the following assumptions were made for the PCM phase change modeling:

∂ (ρh) 1 ∂ (ρrvr h) 1 ∂ (ρvθ h) 1 ∂ 2h k ⎡ 1 ∂ ⎛ ∂h ⎞ r + 2 2 ⎤−Sh = + + r r ∂θ Cp ⎢ r r r r ∂ ∂ ∂θ ⎥ ∂r ∂t ⎝ ⎠ ⎦ ⎣ (6)

• The flow of the liquid PCM was considered to be laminar and incompressible Newtonian fluid flow, • Viscous dissipation was neglected since large velocity gradients were not present in the study, • No-slip boundary condition was present, • At boundaries, heat transfer occurred with the corresponding HTF (Robin boundary condition), • Liquid fraction variation was assumed to change linearly with temperature, • Thermophysical properties of the PCM were phase-wise constant

where the source term is:

Sh =

∂ (ρΔH ) 1 ∂ (ρrvr ΔH ) 1 ∂ (ρvθ ΔH ) + + r r ∂θ ∂r ∂t

(7)

In these equations h and ΔH account for sensible and latent heat, respectively. Furthermore:

ΔH = γλ

(8)

where γ is the liquid fraction whose variation was assumed to be linear during the phase change process:

except for the density, which was evaluated based on Boussinesq approximation to account for the buoyancy forces.

γ=

139

T < Ts 0 ⎧ ⎪ T − Ts Ts < T < Tl ⎨ Tl − Ts ⎪1 T > Tl ⎩

(9)

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3.3. Initial and boundary conditions

Table 4 Grid size independency verification based on liquid fraction (time step size of 0.1 s).

According to the assumptions developed earlier and Fig. 1, the PCM was heated by the inner tube carrying hot heat transfer fluid (HHTF), while it was simultaneously cooled down by the cold heat transfer fluid (CHTF). Therefore, initial and boundary conditions for inside heating/ outside cooling mode were defined as:

t = 0 → T = Ti vr = 0 ⎧ ⎪ r = ri → vθ = 0 ⎨ ∂T ⎪−k ∂r = hHHTF (THHTF −T ) ⎩ vr = 0 ⎧ ⎪ r = rm → vθ = 0 ⎨ ∂T ⎪−k ∂r = hCHTF (T −TCHTF ) ⎩

2 3

(11)

where f, Re and Pr are the friction factor, Reynolds number and Prandtl number, respectively. Equation (11) is valid for Reynolds numbers up to 5 × 104. Reynolds number was calculated by:

ρuDh μ

(12)

where ρ and μ are two thermophysical properties (the density and dynamic viscosity) of the fluid, while u and Dh are two geometrical properties (the fluid velocity and hydraulic diameter). Once the Reynolds number is calculated, the friction factor (f) could be evaluated by [46]:

f = (1.58lnRe−3.28)−2

4800 s

9600 s

21,606 24,686 27,182

0.1171 0.1067 0.1059

0.1552 0.1452 0.1437

0.2056 0.2017 0.2004

0.2771 0.2862 0.2872

0.3517 0.3647 0.3655

0.385 0.3981 0.4004

3.6. Model validation In this study, SCD of a PCM is investigated; therefore, in order to validate the model, the results were compared with the experimental data from two earlier studies. Model validation was carried out for melting and solidification processes, separately.

(13)

Finally, the convective heat transfer coefficient (h ) is:

Nuk D

2400 s

Prior to the analysis, grid size as well as time step size independency was checked for the numerical procedure. Depending on the case under investigation, three different grid resolutions were examined to check the independency from the liquid fraction. Also, three different time steps of 0.05 s, 0.1 s and 0.2 s were checked. Tables 4 and 5 show the liquid fraction values for three grid sizes and time steps for Case 10 as an example. The liquid fraction values were compared over 160 min of SCD. Since increasing the grid resolution from 24,686 cells to 27,182 cells did not significantly affect the liquid fraction, the former was selected to reduce the computational time. Therefore, for Case 10, a grid size of 24,686 cells with a time step size of 0.1 s was selected for the simulation in order to achieve the momentum and energy equation convergence criteria of 10−3 and 10−5, respectively. The final grid resolution used for each case is shown in Table 6.

f 2

h =

1200 s

3.5. Model grid and time step independency

( ) (Re−1000) Pr Nu = 1 + 12.7 ( ) (Pr −1)

Re =

600 s

(10)

where the convective heat transfer coefficient (h ) values were determined by an empirical equation for Nusselt number (Nu). For a turbulent forced internal flow [46]:

1 f 2 2

300 s

sketched using the DesignModeler environment. Then, ANSYS Meshing generated the mesh. Thereafter, the simulation was setup in ANSYS Fluent environment. To solve the equations, pressure-based solver was used since it is the only option in the software for solving melting/ solidification processes [48]. Semi-implicit pressure-linked equation (SIMPLE) method was used for pressure-velocity coupling. For spatial discretization of the pressure and momentum, PRESTO (pressure staggering option) and QUICK schemes were adopted, respectively.

∂v

⎧ ∂θr = 0 ⎪ θ = 0, π → vθ = 0 ⎨ ∂T ⎪ ∂θ = 0 ⎩

Number of cells

(14)

3.6.1. Melting process Experimental data from an earlier study [37] was used for validation of the melting process. Validation was carried out by simulating the melting process of the paraffin RT82 (Rubitherm Technologies GmbH) with phase change temperature range of 77–85 °C. The experimental apparatus was a PCM-equipped horizontal triplex tube heat exchanger with internal and external fins to enhance the heat transfer through the PCM. Water with the temperature of 90 °C flowed inside the inner and outer tubes as the HTF to charge the PCM. Fig. 2 shows the comparison of numerical and experimental average temperatures. The numerical results followed the experimental ones but almost overestimated the average temperature. Nevertheless, there is an acceptable agreement with the experimental data, and the model can be used to numerically study melting in the PCM-equipped TTHX system.

Flow average temperatures of 313.15 K (40 °C) and 293.15 K (20 °C) were assumed for the HHTF and CHTF, respectively. The PCM was assumed to be initially fully melted/solidified at 313.15/293.15 K (40/ 20 °C) depending on the case under investigation. 3.4. Simulation tool There is different commercial software for numerical analysis of the melting/solidification process. However, according to a recent review [47], ANSYS Fluent is the most widely utilized software. Therefore, the numerical study was performed using ANSYS Fluent v16.2. The solidification and melting models were solved for the laminar flow with Navier-Stokes equations using the enthalpy-porosity technique. The liquid fraction was computed during each iteration at each cell in the domain based on enthalpy balance. The mushy zone was modeled as a pseudo porous medium whose porosity decreases from 1 to 0 as the material solidifies. Once the material in a cell reaches the fully solidified state, the porosity reaches zero; consequently, the velocity drops to zero. A step-by-step approach was taken in ANSYS Workbench to simulate the process. First, the 2D symmetric half of the annulus was

Table 5 Time step size independency verification based on liquid fraction (24,686 cells).

140

Time step size

300 s

600 s

1200 s

2400 s

4800 s

9600 s

0.05 s 0.1 s 0.2 s

0.1025 0.1027 0.1028

0.1395 0.1402 0.1407

0.1944 0.1957 0.1965

0.2749 0.2782 0.2794

0.3511 0.3617 0.3517

0.3833 0.3921 0.3974

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both sides heat transfer and (3) SCD only. Fig. 5 shows schematic representations of the scenarios, where the blue and red colors represent temperatures equal to the CHTF and HHTF, respectively. Furthermore, the dashed lines represent adiabatic boundary conditions. In order to analyze the effect of fin length, four lengths of 10, 20, 30 and 33 mm were selected (see Table 9). As shown in the table, the effect of fin length was examined for Case 10. Finally, the effect of fin thickness was examined for Case 14 for thicknesses of 0.5, 1, 2 and 4 mm (see Table 10). The reason to select these cases was that they benefit from the maximum number of fins, both internal and external, simplifying the comparison among the cases with different lengths and thicknesses.

Table 6 Grid resolution used for each case. Case Case Case Case Case Case

1 2 3 4 5

Grid size

Case

12,216 20,139 25,901 25,834 14,515

Case Case Case Case Case

Grid size 6 7 8 9 10

15,716 17,390 19,278 22,975 24,686

3.6.2. Solidification process In order to validate the model for the solidification process, experimental data from a recent study [49] was utilized. The process was simulated for the paraffin RT60 (Rubitherm technologies, GmbH) whose phase change temperature range was 55–61 °C. The melted PCM was sitting at the shell side of a vertical shell-and-tube heat exchanger with water flowing as the HTF at 10 °C. For validation purposes, the temperature variation in a specific thermocouple (T8 for Cylinder D in [49]) was compared with the predicted temperature at the same point obtained from the numerical model. Fig. 3 shows that the numerical results followed the experimental data. It should be noted that a source of deviation from the experimental data is the fact that the temperature recorded by any thermocouple is an indicator of a part of the PCM around it. Therefore, it cannot measure the temperature at an exact point. Nevertheless, the numerical results are in a reasonable agreement with the experimental data.

4. Results and discussion Under SCD conditions, a fully melted or fully solidified state is not reached at the end of the phase change process. Instead, a steady state is reached when neither the melting from the heat source nor the solidification from the heat sink can prevail the other one. Therefore, the simulations were run until achieving the steady state, i.e. the moment when changes would no longer occur in the solid-liquid interface position. In this section, first, conventional fin geometries are analyzed under SCD for both initially solid and initially liquid PCM conditions. Thereafter, the developed fin geometries, which are compatible with natural convection, are also examined under SCD for the same initial conditions. Then, the performance of all configurations is investigated under non-SCD conditions. Finally, the proper geometrical fin design for each application (SCD or non-SCD) is identified.

3.7. Analysis procedure Table 7 shows the considered conventional fin configurations. Case 1 was the base case with no fins. Case 2 had two vertical external (over cold tube) and two horizontal internal (over hot tube) fins, while Cases 3 and 4 had 4 fins instead. It should be noted that the fin configuration of Case 4 was a rotation of that of Case 3 by 45°. In order to reduce the natural convection suppression by fins in the horizontal thermal storage system, fins should be allocated on the lower and upper halves of the hot inner tube and cold outer tube, respectively. Based on this, six fin geometries have been developed as tabulated in Table 8. Since the outer tube has a larger heat transfer surface area compared to that of the inner one, initially internal fins were considered only (Cases 5–7). Then, Cases 8–10 were developed with fixed 3 internal fins of Case 7 but different external ones. Fig. 4 shows schematic representations of all cases. In order to conduct a comprehensive evaluation of both developed and conventional fin configurations, three different scenarios were investigated; (1) non-SCD with single side heat transfer, (2) non-SCD with

4.1. Analysis of conventional fin configurations under SCD conditions In earlier studies, the two fluids (HTFs) that flowed inside the inner and outer tubes of a TTHX had the same temperature and were used either to charge the storage or discharge it. Therefore, the PCM faced either heating or cooling from its both sides. However, as presented earlier, the PCM might undergo SCD conditions. In this section, the performance of conventional fin configurations is presented under SCD. According to our earlier study (without fins) [40], the final PCM condition under SCD was found to be different depending on the initial PCM status, i.e. initially solid or liquid. Therefore, in this section, the results are presented for both cases. 4.1.1. Initially solid Fig. 6 shows the liquid fraction (left) and temperature (right) contours for such conditions. Since the PCM was initially solid, the heat

PCM average Temperature (°C)

100

Fig. 2. Comparison of the numerical and experimental average temperatures for melting.

90 80 70 60 50 40 30 20

Numerical

10 0

Experimental 0

10

20

30

40

50

60

Time (min) 141

70

80

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100

Fig. 3. Comparison of the numerical and experimental temperatures for solidification.

Numerical Experimental

90

Temperature (ºC)

80 70 60 50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (hour) the number of fins, the higher was the amount of melted PCM after a certain duration. Interestingly, comparing Cases 3 and 4 it was concluded that the angled inner fin configuration (Case 4) was greatly different from the non-angled one (Case 3). This was due to the presence of internal fins on the lower half of the system where natural convection was developed above the fins, which would otherwise remain solid for a long period of time during the phase change process (see Case 1). Besides, the angled configuration created a region at the lower half close to the inner tube where the PCM remained melted, enhancing the heat transfer to the solid PCM.

Table 7 Characteristics of conventional fin configurations. Case

No. of internal fins

No. of external fins

Fin length (mm)

Fin thickness (mm)

1 2 3 4

0 2 4 4

0 2 4 4

– 30 30 30

– 1 1 1

Table 8 Characteristics of the developed configurations. Case

No. of internal fins

No. of external fins

Fin length (mm)

Fin thickness (mm)

5 6 7 8 9 10

1 2 3 3 3 3

0 0 0 1 3 5

30 30 30 30 30 30

1 1 1 1 1 1

4.1.2. Initially liquid Fig. 7 shows the liquid fraction (left) and temperature (right) contours of Cases 1–4 for initially liquid PCM. As the PCM was already liquid at the beginning of the phase change process, the natural convection was established sooner than the cases with initially solid PCM. Over time, solid PCM was deposited on the cold external tube. However, due to the continuous buoyancy-driven hot liquid PCM motion at the upper half of the system, the solid deposition was greater on the lower half. 4.1.3. Comparison Comparing the contours in Figs. 6 and 7 after 150 min of SCD, it was concluded that integration of fins enhanced the heat transfer. Moreover, for both initial conditions, the solid-liquid interfaces after

transfer began by melting the solid PCM around the hot tube. The figure shows that the integration of fins enhanced the heat transfer within the solid PCM; however, the magnitude of impact was different. The more

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

Case 9

Case 10

Fig. 4. Schematic representation of the investigated cases.

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Scenario 1

Scenario 2

Scenario 3

Fig. 5. Schematic representation of the considered scenarios.

Initially solid

Initially liquid

could be identified by the slope of the curves in Fig. 8 where Cases 2 and 4 had almost similar rates for initially solid PCM, whereas Case 3 followed by Case 4 had the highest rate for initially melted PCM. However, according to the table, Case 3 has an effectiveness value less than unity, which means that addition of the fins undesirably resulted in less thermal energy storage in the PCM. In addition, Case 2 had the highest steady liquid fraction value among the cases.

Table 9 Investigated cases to analyze the effect of fin length. Case

No. of internal fins

No. of external fins

Fin length (mm)

Fin thickness (mm)

11 12 13 14

3 3 3 3

5 5 5 5

10 20 30 33

1 1 1 1

4.2. Analysis of the developed configurations under SCD conditions In the previous section, conventional fin configurations, which are normally used to charge and discharge PCMs consecutively were investigated under SCD. In this section, the developed configurations, which are compatible with natural convection, are investigated.

Table 10 Investigated cases to analyze the effect of fin thickness. Case

No. of internal fins

No. of external fins

Fin length (mm)

Fin thickness (mm)

15 16 17 18

3 3 3 3

5 5 5 5

33 33 33 33

0.5 1 2 4

4.2.1. Initially solid Fig. 9 shows the liquid fraction (left) as well as temperature (right) contours for Cases 5–10. Comparing Case 5 with Case 1 (in Fig. 6), it was found that the addition of only one fin at the bottom of the internal tube enhanced the heat transfer to the lower half of the system. Instead of one vertical fin at the lower half (Case 5), Case 6 benefited from two fins with an angle of 45° from the gravity axis. According to the contours, not only the PCM above the fins was heated up by the natural convective motion, but also the PCM resting at the region between the two fins (at the lower half of the system) was melted, unlike Case 5. This shows that for internal heating at least two angled fins at the lower half are required. Case 7 was basically a combination of Cases 5 and 6 where three fins were added to the internal tube. Compared to Case 6, the vertical fin in Case 7 obviously melted the PCM at further vertical positions close to the bottom of the system. This in turn eased the heat transfer to the external cold tube through a thinner solid layer. Overall, Case 7 was found to be the best option among the cases with internal fins. Therefore, the rest of the cases were developed on this basis. Case 8 was similar to Case 7 except that it had one vertical external fin at the top of the storage. As shown in the figure, the fin enabled heat transfer to the heat sink by 30 min (for Case 8). However, for Case 7 this type of heat transfer was achieved after 30 min. Therefore, the addition of this external fin resulted in faster response of the system under SCD. The same explanation applies to Cases 9 and 10 and they enabled more rapid response of the system. However, due to the addition of the fins, a continuous melted PCM motion was created at the right and left sides of the system, which confirmed continuous melting and solidification of the PCM at that region due to SCD.

150 min were almost the same for Cases 3 and 4. However, our earlier study without fins revealed that the final solid-liquid interface position depended on the initial PCM condition; i.e. solid or liquid (compare the contours for Case 1 in Figs. 6 and 7). This difference was due to the fact that fins enhanced heat transfer, especially through the highly resistance solid PCM. For Case 1, which lacked this enhancement, the low thermal conductivity of the PCM prevented effective heat transfer between the heat source and heat sink. Three key parameters should be considered to evaluate the performance of a thermal storage under SCD: (1) the steady state time (or stabilization time), which is the time required to achieve the steady state condition under SCD, (2) the steady liquid fraction value, which represents how much of the storage desirably remains charged at the steady conditions, and (3) fin effectiveness (ηf), which is the ratio of the heat stored in the PCM with fins to that without fins. Fig. 8 shows the liquid fraction variation of the cases for initially solid (left) and liquid (right) PCMs, respectively. Moreover, Table 11 shows the abovementioned key parameters for the conventional fin geometries under SCD. The steady state was assumed to be reached when the difference between two consecutive liquid fraction values was lower than 10−3, which indicated the time beyond which negligible heat was stored within the PCM while heat continuously transferred through the storage from the heat source to heat sink. The table shows that integration of more fins affected the final steady state liquid fraction in a way that there was no significant difference for the two initial conditions of solid and liquid. Moreover, Case 4 was the fastest one to reach the steady state for both initially solid and liquid PCMs. The rate of phase change

4.2.2. Initially liquid The liquid fraction and temperature contours of the initially liquid PCM under SCD are shown at left and right sides of Fig. 10, 143

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Case 1

Case 2

Case 3

Case 4

5 min

30 min

60 min

90 min

120 min

150 min Fig. 6. Liquid fraction (left) and temperature (right) contours for the conventional fin configurations with initially solid PCM under SCD.

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Case 1

Case 2

Case 3

Case 4

5 min

30 min

60 min

90 min

120 min

150 min Fig. 7. Liquid fraction (left) and temperature (right) contours for the conventional fin configurations with initially liquid PCM under SCD.

utilization of two internal fins at the lower half. Nevertheless, comparing Cases 5 and 6 it was found that the solidified PCM layer located at the bottom of the storage was undesirably thicker for Case 6. Therefore, a vertical internal fin should be integrated and Case 7 showed superior performance to Cases 5 and 6. Addition of external fins in Cases 8–10 solidified more PCM at the upper half, which enabled continuous melting and solidification of the PCM at the sides due to

respectively. According to the figure, a solid PCM layer was deposited by time over the external wall. Comparing Case 5 with Case 1 (in Fig. 7) it was found that the sole vertical internal fin kept the PCM melted at the lower half of the system. This together with the conduction through the highly resistant solid PCM resulted in easier heat transfer to the heat sink at the lower half of the system as compared to Case 1. In Case 6, a bigger portion of the system remained liquid after 150 min due to the 145

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0.6

1

0.5

0.9

Liquid fraction

Liquid fraction

M.M. Joybari et al.

0.4 0.3 0.2

Case 1 Case 2 Case 3 Case 4

0.1 0 0

60

120

180

240

Case 1 Case 2 Case 3 Case 4

0.8 0.7 0.6 0.5 0.4

300

Time (min)

0

60

120

180

240

300

Time (min)

Fig. 8. Liquid fraction variation of initially solid (left) and initially liquid (right) PCM with conventional fin configurations under SCD.

4.3.1. One side heat transfer For this scenario, the storage was facing either heating from the inner tube or cooling from the outer one under initially solid and liquid conditions, respectively. Therefore, one side heat transfer was investigated and the other tube was assumed to be insulated. Figs. 11 and 12 show the liquid fraction contours of the conventional and developed fin geometries under either internal heating (left) or external cooling (right) conditions, respectively. As the figures show, for internal heating (left contours), the upward melted PCM motion greatly affected the upper half of the system for all cases. Therefore, allocating more fins to the lower half (e.g. Cases 6–10), enhanced the heat transfer and shortened the melting process. However, such allocation in Cases 6 and 7 was not helpful for the solidification process (see right contours). Integration of only one external fin in Case 8 greatly enhanced the heat transfer within the solid PCM. Overall, Cases 9 and 10 benefited from more external fins and had shorter solidification process.

Table 11 Steady state time and liquid fraction for the conventional fin configurations under SCD conditions. Case

Case Case Case Case

Initially solid

1 2 3 4

Initially liquid

tstd (min)

γstd

ηf

tstd (min)

γstd

ηf

94 83 84 79

0.3101 0.5398 0.4553 0.5051

1.00 1.74 1.47 1.63

76 58 83 54

0.5855 0.6792 0.4940 0.5907

1.00 1.16 0.84 1.01

SCD.

4.2.3. Comparison Table 12 shows the steady state liquid fraction values of the developed cases as well as the time required to achieve it. Comparing cases with internal fins only (Cases 5–7), it was found that Case 7 enabled the highest storage within an acceptable stabilization time compared to other cases. For the rest of the cases with external fins, although Case 8 had the highest steady liquid fraction, Case 10 had almost half an hour faster stabilization time for the initially solid PCM. This could be a requirement for some equipment where rapid response of the system is essential. In such cases, a compromise of the steady liquid fraction value as well as fin effectiveness might be the solution. It should be noted that the stabilization time for initially solid PCMs was longer than liquid ones. This was due to the initiation, domination, establishment and finally stabilization of natural convection. In other words, the melted PCM around the inner tube started to move upward as much as possible until the process was stabilized due to SCD. However, for initially liquid PCMs, since the natural convection was already established, it stabilized sooner under SCD.

4.3.2. Both sides heat transfer Fig. 13 shows the liquid fraction contours of the conventional cases under melting (left) and solidification (right) from both sides of the storage. The same contours for the developed fin configurations are shown in Fig. 14. As the figures show, for melting under both sides heat transfer, the higher heat transfer surface area (compared to one side heat transfer) together with the buoyancy-driven motion significantly shortened the phase change process. Therefore, all cases achieved the fully melted condition within 90 min. On the other hand, although the heat transfer surface area attributed to shortening the solidification period, due to the lack of natural convection, the process was longer than melting. Interestingly, addition of more fins created some liquid cells surrounded by solid PCM for Cases 3, 4, 9 and 10 after 60 min. This finding underscores the significance of fins to enhance solidification. These figures, similar to the figures for one side heat transfer (Figs. 11 and 12), show how fast melting is compared to solidification thanks to the natural convection. This reveals that the bottleneck is to find a suitable geometry that accelerates solidification by enhancing the conductive heat transfer through the PCM.

4.3. Analysis of the configurations under non-SCD conditions It is worth pointing out that it is not desirable to always run thermal storage systems under SCD since in such cases the intermediate PCM could be eliminated to enable direct heat transfer between the HTFs. Nevertheless, as mentioned earlier, there are some scenarios when such systems might be simultaneously charged and discharged. So far in this study, the heat transfer enhancement by fins under SCD process has been investigated. Since the developed fin configurations are consistent with natural convection, they could be beneficial for non-SCD conditions as well. In this section, their performance is investigated and compared with that of the conventional configurations.

4.3.3. Comparison Unlike SCD, the cases investigated in this section could achieve a fully solid or liquid condition at the end of the phase transition process. Therefore, the evaluation of the performance was based on the time required to reach such condition (see Table 13). It should be noted that the simulations were run for 14 h (840 min) of the phase change process and if a fully melted or solidified state was not achieved, the final liquid fraction value was reported in parentheses. The reason is since renewable energy sources are inherently intermittent, they are not 146

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Case 5

Case 6

Case 7

Case 8

Case 9

Case 10

5 min

30 min

60 min

90 min

120 min

150 min Fig. 9. Liquid fraction (left) and temperature (right) contours for the developed fin configurations with initially solid PCM under SCD.

4.4. Proper geometry selection for SCD and non-SCD conditions

normally available for long periods of time. For instance, maximum solar energy availability would not exceed 14 h per day. According to Table 13, the process was much longer for Case 1 compared to the others due to the low thermal conductivity of the PCM as well as lack of fins. Addition of conventional fins greatly improved the melting process, while for the solidification process the magnitude of impact was lower. Overall, due to the absence of natural convection, solidification was longer than melting. It is interesting to point out that the final liquid fraction value during solidification for Case 2 under one side heat transfer was higher than Case 1. This was due to the fact that the horizontal internal fins suppressed the downward cold PCM motion and hot PCM zones were created under the internal fins.

In order to identify the best fin geometry for a specific application, all of the findings are compared in this section. The evaluation criteria differ based on the application of the storage; therefore, it is not possible to simply report one of the configurations as the best among others. In this section, the most suitable geometry for three specific scenarios, which TTHXs are capable of offering, are analyzed; (1) nonSCD with single side heat transfer, (2) non-SCD with both sides heat transfer and (3) SCD only.

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Case 5

Case 6

Case 7

Case 8

Case 9

Case 10

5 min

30 min

60 min

90 min

120 min

150 min Fig. 10. Liquid fraction (left) and temperature (right) contours for the developed fin configurations with initially liquid PCM under SCD.

4.4.1. Scenario 1: Non-SCD with one side heat transfer The storage in this scenario faces either internal charging or external discharging. An example for this scenario is peak load shaving of an air conditioning with latent heat thermal energy storage. In this case, a TTHX can be used with CHTF from the air conditioning system, which discharges the PCM during the night so as to be used later on to cool down the HHTF during daytime. The HHTF is then used inside a building for cooling applications. Keep in mind that, in such scenario, the HTFs have separate flow paths where the internal and external tubes are used for HHTF and CHTF, respectively. Therefore, the PCM is either internally charged or externally discharged. According to Fig. 15, Case 4 as well as Cases 6–10 achieved a fully melted condition within 14 h of the phase change process, where the

Table 12 Steady state time and liquid fraction for the developed fin configurations under SCD conditions. Case

Case Case Case Case Case Case

Initially solid

5 6 7 8 9 10

Initially liquid

tstd (min)

γstd

ηf

tstd (min)

γstd

ηf

92 120 119 114 100 82

0.3147 0.5585 0.5603 0.5388 0.4713 0.3637

1.00 1.77 1.78 1.71 1.50 1.16

69 43 46 46 54 69

0.6069 0.7089 0.7164 0.7136 0.6154 0.4552

1.00 1.17 1.18 1.18 1.01 0.75

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Case 1

Case 2

Case 3

Case 4

5 min

30 min

60 min

90 min

120 min

150 min Fig. 11. Liquid fraction contours for the conventional fin configurations under internal melting (left) or external solidification (right).

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Case 5

Case 6

Case 7

Case 8

Case 9

Case 10

5 min

30 min

60 min

90 min

120 min

150 min Fig. 12. Liquid fraction contours for the developed fin configurations under internal melting (left) or external solidification (right).

Fig. 16 shows the time required to achieve complete phase change for the cases. It should be noted that Case 1 did not achieve complete solidification after 14 h of heat transfer; hence; it is excluded from the figure. Interestingly, addition of only one vertical internal fin (Case 5) shortened the solidification time (compared to Case 1). Overall, melting was faster than solidification for both conventional and developed cases. Cases 3, 4 and 10 had almost similar performance but Case 3 was the best option to accomplish melting and solidification sooner than all other cases.

line shows the time required to achieve it. Although the times were close, Case 10 had a superior performance among others. For the solidification process, none of the configurations reached a fully solidified condition within 14 h. Nevertheless, Cases 3 and 10 desirably had the lowest final liquid fraction value. In summary, Case 10 is recommended as the proper geometry for this scenario. 4.4.2. Scenario 2: Non-SCD with both sides heat transfer In this scenario, the storage faces either charging or discharging from its both sides. An example of this storage is the same as the one for one side heat transfer except that the HTFs do not have separate flow paths; therefore, both the internal and external tubes are used for heat transfer to/from the PCM, consecutively.

4.4.3. Scenario 3: SCD only Under SCD, the PCM faces internal charging, while it is simultaneously discharged externally. An example of this scenario is a thermal 150

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Case 1

Case 2

Case 3

Case 4

5 min

30 min

60 min

90 min

120 min

150 min Fig. 13. Liquid fraction contours for the conventional fin configurations under melting (left) or solidification (right) from both sides.

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Case 5

Case 6

Case 7

Case 8

Case 9

Case 10

5 min

30 min

60 min

90 min

120 min

150 min Fig. 14. Liquid fraction contours for the developed fin configurations under melting (left) or solidification (right) from both sides.

the integration of an external fin. Overall, the final decision should be made based on the requirements of the storage, where either the fast stabilization or higher liquid fraction value would prevail the other. Assuming the necessity of the highest steady liquid fraction value, Case 8 had the best performance for both initially solid and liquid conditions.

storage where heat is stored inside the PCM by an internal supply HHTF from a solar collector so as to be extracted simultaneously for SDHW (i.e. the CHTF) heating. Fig. 17 shows the steady state liquid fraction value and its corresponding required time for both initially solid and liquid storages. Overall, it is desirable to achieve a higher steady state liquid fraction value since in such condition the objective of the storage, which is storing higher amounts of heat, is met. Accordingly, for initially solid conditions, Cases 2, 4 and 10 had the shortest steady state time. However, Case 2 had a higher steady liquid fraction value. For initially liquid PCM, Cases 4 and 6–8 had the shortest steady state time. However, Cases 7 and 8 had the highest steady liquid fraction value. As mentioned earlier, Case 8 enabled faster response of the system due to

5. Analysis of the effective parameters In this section, the effect of fin numbers, length and thickness are investigated for the developed configurations under SCD conditions.

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5.2. The effect of fin length

Table 13 Comparison of phase change time (only if accomplished by 14 h) or final liquid fraction value (after 14 h, shown in parentheses) for all cases under non-SCD conditions.

One side Melting Case Case Case Case Case Case Case Case Case Case

The effect of fin length was investigated for the fin configuration of Case 10 (see Table 9). Fig. 20 shows this effect for initially liquid (left) and solid (right) PCMs. In order to explain the effect of fin length under SCD, two key points should be considered. First, the impact of fin length on enhancement of conductive heat transfer from the corresponding surface to the PCM and second, the suppression of natural convection by adding longer fins. The overall outcome is dependent upon these opposite points. Case 11 had the shortest fin length (i.e. 10 mm) and the results showed that its steady liquid fraction value tended to be closer to the initial one. This means that for the initially solid PCM, its steady liquid fraction was the smallest compared to the rest of the cases, while it had the highest value under initially liquid conditions. Therefore, when the fin length was too short, the heat struggled to transfer within the PCM. Addition of another 10 mm to the fin length, Case 12 showed a better performance than Case 11. In addition to its enhanced steady liquid fraction value, the slope of the graphs revealed that the rate of heat transfer was also improved, especially for the initially liquid PCM. Regardless of the initial condition, elongating the fins for another 10 mm (Case 13) did not show a significant impact. This is a difference between SCD and non-SCD since earlier studies reported that under non-SCD conditions, longer fins shortened the phase change process [28]. For the initially solid PCM, Case 13 showed a sudden raise in the liquid fraction value at about 120 min. This was due to the fact that enough melted PCM was formed to pass through the space left between the HHTF tube and the fins over the CHTF tube. For shorter fin lengths, the gap was wider; hence, the melted PCM was flowing more freely. Interestingly, Case 14 also lacked such a raise, which was due to the very small gap size, greatly limiting the motion of the melted PCM. Overall, natural convection was suppressed by the fins over the CHTF tube for Cases 13 and 14.

Phase change duration for non-SCD (min)

1 2 3 4 5 6 7 8 9 10

> 840 > 840 > 840 362 > 840 378 332 336 335 327

(0.8504) (0.8617) (0.8900) (0.8841)

Both sides Solidification

Melting

Solidification

> 840 > 840 > 840 > 840 > 840 > 840 > 840 > 840 > 840 > 840

75 58 34 37 75 49 48 48 46 36

> 840 (0.1039) 269 122 127 569 407 403 375 195 130

(0.2619) (0.3457) (0.1598) (0.3017) (0.2833) (0.2904) (0.3542) (0.3162) (0.2245) (0.1726)

5.1. The effect of fin numbers Fig. 18 shows the liquid fraction variation for cases with internal (left) and external (right) fins for initially solid PCM. Since internal fins were attached to the hot tube whereas the cold ones were integrated to the cold tube, their effect on the phase change process was completely different. As the figure shows, addition of only one internal fin (Case 5) did not enhance the final liquid fraction value as much as other cases with at least two angled internal fins. On the other hand, integration of external fins reduced the final liquid fraction value indicating higher extraction from the storage. Addition of only one external fin (Case 8) did not drop the final liquid fraction value as much as other cases. For initially liquid PCM, Fig. 19 shows the liquid fraction variation of cases with internal (left) and external (right) fins. Case 5 had the lowest final liquid fraction value among cases with internal fins. This was due to the fact that adding only one vertical fin was not effective enough to melt the PCM at the lower half of the system. Cases 6 and 7 had almost similar performance, which was because of adding at least two internal fins. For Cases 8–10, more external fins reduced the final liquid fraction values. The mechanism for the effect of fin numbers was explained in Section 3.2; thus, it is not repeated in this section.

Liquid fraction

Melting - Liquid fraction

5.3. The effect of fin thickness Earlier studies reported that the effect of fin thickness on the heat transfer enhancement was negligible; however, its effect was only investigated under non-SCD conditions. Fig. 21 shows the effect of fin thickness on the liquid fraction of initially solid (left) and liquid (right) PCMs under SCD. For the initially solid PCM (left), due to the simultaneous presence of conduction and natural convection, there was a very small difference between the cases. On the other hand, for the

Solidification - Liquid fraction

Melting - Time

1

500

0.9

450

0.8

400

0.7

350

0.6

300

0.5

250

0.4

200

0.3

150

0.2

100

0.1

50 0

0 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 153

Fig. 15. Phase change time (only if accomplished by 14 h) or final liquid fraction value (after 14 h) for the cases under non-SCD with one side heat transfer.

Time (min)

Case

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600 Melting

Fig. 16. Phase change time for the cases under non-SCD with both sides heat transfer.

Solidification

500

Time (min)

400

300

200

100

0

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

Case 9 Case 10

Initially solid - Liquid fraction

Initially liquid - Liquid fraction

Initially solid - Time

Initially liquid - Time

Fig. 17. Steady state time and liquid fraction for the cases under SCD.

0.8

140

0.7

120 100

Time (min)

0.5 80 0.4 60 0.3 40

0.2

20

0.1 0

0 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10

0.7

0.7

0.6

0.6

0.5

0.5

Liquid fraction

Liquid fraction

Liquid fraction

0.6

0.4 0.3 0.2

Case 5 Case 6 Case 7 Case 8

0.1 0 0

60

120

180

240

300

0.4 0.3 0.2

Case 7 Case 8 Case 9 Case 10

0.1 0

360

Time (min)

0

60

120

180

240

300

360

Time (min)

Fig. 18. Temporal liquid fraction variation for cases with different internal (left) and external (right) fin number (see Table 8) for initially solid PCM.

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1

0.8

Case 7 Case 8 Case 9 Case 10

0.9

Liquid fraction

0.9

Liquid fraction

1

Case 5 Case 6 Case 7 Case 8

0.7 0.6 0.5

0.8 0.7 0.6 0.5

0.4

0.4 0

60

120

180

240

300

360

0

60

Time (min)

120

180

240

300

360

Time (min)

0.6

1

0.5

0.9

Liquid fraction

Liquid fraction

Fig. 19. Temporal liquid fraction variation for cases with different internal (left) and external (right) fin number (see Table 8) for initially liquid PCM.

0.4 0.3 0.2 Case 11 Case 12 Case 13 Case 14

0.1 0 0

60

120

180

240

300

Case 11 Case 12 Case 13 Case 14

0.8 0.7 0.6 0.5 0.4

360

0

60

Time (min)

120

180

240

300

360

Time (min)

0.6

1

0.5

0.9

Liquid fraction

Liquid fraction

Fig. 20. Temporal liquid fraction variation for cases with different fin length (see Table 9) for initially solid (left) and liquid (right) PCMs.

0.4 0.3 0.2

Case 15 Case 16 Case 17 Case 18

0.1

Case 15 Case 16 Case 17 Case 18

0.8 0.7 0.6 0.5

0

0.4 0

60

120

180

240

300

360

0

Time (min)

60

120

180

240

300

360

Time (min)

Fig. 21. Temporal liquid fraction variation for cases with different fin thickness (see Table 10) for initially solid (left) and liquid (right) PCMs.

investigated the effect of fin number, length, and thickness on the phase change process of a TTHX under SCD conditions. Moreover, some fin configurations were developed and compared with the conventional ones under both SCD and non-SCD conditions. The most important findings were:

initially melted PCM (right), the lines could hardly be distinguished and almost coincided. Based on these results, it was concluded that regardless of heat transfer mechanism, the fin thickness had no significant effect on the heat transfer rate under SCD.

• For systems with SCD applications, the recommended fin config-

6. Concluding remarks

uration includes one external (at 90°) and three internal fins (at

This paper presented the results of a numerical study, which 155

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• • •

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225°, 270° and 315°). For systems with non-SCD applications having both sides heat transfer, the recommended fin configuration includes four internal fins (at 0°, 90°, 180°, 270°) and four external fins (at 45°, 135°, 225° and 315°). For systems with non-SCD applications having one side heat transfer, the recommended fin configuration includes five external fins (at 0°, 45°, 90°, 135° and 180°) at the upper half and three internal fins (at 225°, 270° and 315°) at the lower half of the system. Under SCD conditions, the effect of fin thickness was negligible. However, their number and length enhanced the heat transfer as long as the fins did not suppress the natural convection.

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