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Chapter 8

Nanofluid Conductive Heat Transfer in Solidification Mechanism

Abstract

Latent heat thermal energy storage systems (LHTESS), which work based on energy storage and retrieval during solid–liquid phase change is used to establish balance between energy supply and demand. LHTESS stores and retrieves thermal energy during solid–liquid phase change, while in SHTESS phase change doesn’t occur during the energy storage and retrieval process. LHTESS has a lot of advantages in comparison to SHTESS; the most important one is storing a large amount of energy during phase change process, which makes the energy storage density in LHTESS much higher than SHTESS. Due to this property, LHTESS have a wide application in different cases, such as solar air dryer, HVAC systems, electronic chip cooling, and engine heat recovery. The main restriction for these systems is thermal conductivity weakness of common PCMs. In this chapter, the method of adding nanoparticles to pure PCM and making nanoenhanced phase change material (NEPCM) and using fin with suitable array are presented to accelerate solidification process. The numerical approach, which is used in this paper is standard Galerkin finite element method.

Keywords

nanoparticle

NEPCM

solidification

fin

LHTESS

standard Galerkin finite element method

8.1. Discharging process expedition of NEPCM in Y-shaped fin-assisted latent heat thermal energy storage system

8.1.1. Problem definition and Mathematical model

8.1.1.1. Problem definition

The main geometry of present study is a fin-assisted LHTESS, which HTF exists in the inner tube, and in the space between inner and outer tube, is filled by PCM. Y-shaped fin is connected to the outer side of HTF tube to improve heat transfer from HTF to PCM (Fig. 8.1) [1]. The two-dimensional solution domain is presented in Fig. 8.2. The angle between fin branches and axes is assumed to be β = π/16, 2π/16, 3π/16, 4π/16; fin length and thickness assumed to be equal to L = 2, 2.4, 2.8 cm and t = 0.5, 0.75, 1 mm, respectively. The constant temperature boundary condition equal to 240K is applied to inner tube and the initial temperature of liquid PCM is assumed equal to 278K. The PCM, nanoparticles, and fin properties are listed in Table 8.1.

Figure 8.2 Two-dimensional solution domain used in present study.

Table 8.1

The physical properties of water as PCM, copper as nanoparticles, and aluminum fin

Property	PCM	Nanoparticles	Fin
ρ (kg/m³)	997	8,954	2,700
C_p (J/kg K)	4179	385	902
k (W/mK)	0.6	400	200
L_f (J/kg)	335,000	—	—

The transient Governing equations of conduction-dominated solidification process are as follows:

$ρ C_{p} \frac{d T}{d t} = \nabla (k \nabla T) + L_{f} \frac{d S}{d t}$ $ρ C_{p} \frac{d T}{d t} = \nabla (k \nabla T) + L_{f} \frac{d S}{d t}$

(8.1)

$\begin{array}{l} S = 1 & T < T_{m} - T_{0} \\ S = 0 & T > T_{m} + T_{0} \\ S = \frac{(T_{m} + T_{0} / 2 - T)}{T_{0}} & T_{m} - T_{0} < T < T_{m} + T_{0} \end{array}$ $\begin{array}{l} S = 1 & T < T_{m} - T_{0} \\ S = 0 & T > T_{m} + T_{0} \\ S = \frac{(T_{m} + T_{0} / 2 - T)}{T_{0}} & T_{m} - T_{0} < T < T_{m} + T_{0} \end{array}$

(8.2)

When pure PCM is applied in LHTESS, pure PCM properties are used in governing equations but when copper nanoparticles are dispersed in PCM, the NEPCM properties should be obtained by using the following equations and be applied in governing equations [2]:

$ρ_{n f} = (1 - φ) ρ_{b f} + φ ρ_{p}$ $ρ_{n f} = (1 - φ) ρ_{b f} + φ ρ_{p}$

(8.3)

${(ρ C_{p})}_{n f} = (1 - φ) {(ρ C_{p})}_{b f} + φ {(ρ C_{p})}_{p}$ ${(ρ C_{p})}_{n f} = (1 - φ) {(ρ C_{p})}_{b f} + φ {(ρ C_{p})}_{p}$

(8.4)

${(ρ L_{f})}_{b f} = (1 - φ) {(ρ L)}_{b f}$ ${(ρ L_{f})}_{b f} = (1 - φ) {(ρ L)}_{b f}$

(8.5)

In present study, thermal conductivity of NEPCM is evaluated by the following equation based on Maxwell [2] work:

$\frac{k_{n f}}{k_{b f}} = \frac{k_{p} + 2 k_{b f} - 2 φ (k_{b f} - k_{p})}{k_{p} + 2 k_{b f} + 2 φ (k_{b f} - k_{p})}$ $\frac{k_{n f}}{k_{b f}} = \frac{k_{p} + 2 k_{b f} - 2 φ (k_{b f} - k_{p})}{k_{p} + 2 k_{b f} + 2 φ (k_{b f} - k_{p})}$

(8.6)

where φ, k_p, and k_bf are nanoparticle volume fraction, nanoparticle and base fluid thermal conductivity, respectively.

Total energy density released during discharging process is calculated from the following equation:

$E_{d e n s i t y} = \frac{ρ}{V} \int (C_{p} T + (1 - s) L_{f}) d V$ $E_{d e n s i t y} = \frac{ρ}{V} \int (C_{p} T + (1 - s) L_{f}) d V$

(8.7)

8.1.1.2. Numerical method

Standard Galerkin finite element method with cubic interpolation over triangles is implemented to solve the present phase change problem. Nodal values are placed on corners and sides of the grid cells. The Galerkin equations are formed by symbolic analysis, which substitutes definitions, segregates dependencies on variables, applies integration by parts, integrates over cells, and ultimately differentiates the resulting system with respect to system variables to form the coupling matrix. Equations are solved simultaneously by an iterative method. For nonlinear systems Newton–Raphson iteration process with backtracking is used. For time dependent problems, such as solidification problem, an implicit backward difference method for integration in time is used. Variables are approximated by quadratic polynomials in time, and the time step is controlled to keep the cubic term smaller than the required value of error. The residual Galerkin integral over a patch of cells surrounding each mesh node is minimized by the finite element equations. Then the residuals in each cell independently are analyzed as a measure of compliance, and subdivide each cell in which the required error tolerance is exceeded. Any cell, thus, split can be remerged whenever the cell error drops to the splitting tolerance. Adaptive grid refinement is used to simulate solidification process in present work. When the initial mesh generation is performed, code estimates the error, and refines mesh in order to reach to the desired accuracy. In unsteady problems, this procedure also has to be applied to the initial values of the variables in order to refine the mesh where rapid change in variables occurs (Fig. 8.3). Comparison between the present code based on Galerkin finite element method and experimental results obtained by Ismaeil et al. [3] indicates good agreement, which validates the present code (Fig. 8.4). Moreover, it proves that ignoring natural convection in numerical simulation of solidification phenomenon leads to results close to reality.

Figure 8.3 Adaptive grid refinement procedure for β = 3π/16, L = 2.4 cm, t = 1 mm.

Figure 8.4 Comparison between solidification front in fin-assisted LHTESS in present study and experimental work by Ismail et al. [3].

8.1.2. Effects of active parameters

8.1.2.1. Applying Y-shaped fin to LHTESS

In this section the effect of adding Y-shaped fins to LHTESS containing pure PCM during solidification process and expedition of energy retrieval will be investigated. For LHTESS without fin, solidification process begins only in the region adjacent to the inner tube containing HTF, but by attaching fins with high thermal conductivity to the inner tube, solidification also begins in the regions adjacent to the fins surfaces, and as a result, solidification rate will be enhanced. In order to achieve this purpose, Y-shaped fin array as can be seen from Figs. 8.1 and 8.2 for different values of branch angle, thickness and length will be applied to the system. In Table 8.2 the mentioned geometry parameters are listed.

Table 8.2

Geometry parameters of Y-shaped fins

Parameter	Different values used in simulation
β	$π / 16, 2 π / 16,3 π / 16,4 π / 16$ $π / 16, 2 π / 16,3 π / 16,4 π / 16$
L	2, 2.4, 2.8 cm
t	0.5, 0.75, 1 mm

According to Figs. 8.5 and 8.6 it is obvious that for all values of fin length and thickness, the best performance of LHTESS during solidification occurs when the branch angle is β = 3π/16 among the investigated values. When the value of β is small, branches of Y-shaped fin are close to the axes, therefore in the region between two branches in each quadrant of solution domain, the amount of PCM without fin is too high and because in this region, thermal penetration depth is not enhanced by fins, solidification rate is too low, which increases full solidification time. Similarly when the value of fin branch angle is too high, in region between two branches in each quadrant of solution domain, penetration depth is enhanced but in the regions between branches and axes, solidification rate is too low. So, it is obvious that the best choice for β is a value between π/16·4π/16, which in the values discussed in this paper, β = 3π/16 demonstrates the lowest full solidification time and the most uniform solidification process, which is obvious from related contour and diagrams. As can be observed in Figs. 8.5 and 8.7 full solidification time for three values of fin thickness are presented. These diagrams indicate that by increasing fin thickness from the value of 0.5 to 0.75 mm, solidification rate enhancement is considerable but by changing the value to 1 mm, solidification rate enhancement is not significant. Moreover, since increasing the fin thickness value leads to increasing fin volume in LHTESS and decreasing the volume of employed PCM and as a result, maximum energy storage capacity will decrease, so the best value of fin thickness according to reducing full solidification time and avoiding the reduction of maximum energy storage capacity is t = 0.75 mm.

Figure 8.5 Full solidification time for different values of fin geometry parameters, (A) L = 2 cm, (B) L = 2.4 cm, (C) L = 2.8 cm.

Figure 8.6 Effect of fin branch angle on temperature (left side) and solid fraction (right side) contours. L = 2.8 cm, t = 0.75 mm.

Figure 8.7 Effect of fin branch thickness on temperature (left side) and solid fraction (right side) contours β = 3π/16 [Rad], L = 2.8 cm.

According to Figs. 8.5 and 8.8, by increasing in fin length for all values of branch angles and thickness, full solidification time is decreased considerably and because of this significant enhancement, decrease in value of maximum energy storage capacity can be ignored. The main reason is the augmentation of penetration depth due to increase in fin length. So, the best choice for fin length among discussed values is L = 2.8 cm. Therefore, between all discussed cases, the best choice for geometry parameters is β = π/16, t = 0.75 mm, L = 2.8 cm. From solid fraction contour, this case shows the fastest and most uniform solidification process.

Figure 8.8 Effect of fin length temperature (left side) and solid fraction (right side) contours. β = 3π/16 [Rad], t = 0.75 mm.

According to Fig. 8.6, the temperature distribution and solid fraction contours has been illustrated to study the effect of fin branch angle on solidification process in LHTESS. As can be seen, in all time steps, average temperature for the case of β = 3π/16 is lower in comparison to the other cases, and the value of solid fraction is higher, which means that for the mentioned branch angle, solidification rate is higher. According to Fig. 8.7, the temperature distribution and solid fraction contours has been illustrated to study the effect of fin thickness on solidification process in LHTESS. As can be seen, in all time steps, average temperature for the case of t = 1 mm is lower and solid fraction value is higher, but the enhancement difference for the case of t = 1 mm in comparison to t = 0.75 mm is insignificant. According to Fig. 8.8, the temperature distribution and solid fraction contours has been illustrated to study the effect of fin length on solidification process in LHTESS. As can be observed, in all time steps, by increasing in fin length, average temperature is lower and the value of solid fraction in higher. It can be observed that when the solidification front moves away a little from fin tips, phase change process is not enhanced by fins anymore. By employing longer fins, the region affected by them will be wider and heat transfer enhancement will be more considerable.

8.1.2.2. Adding nanoparticles to LHTESS

The effect of adding copper particles as high thermal conductivity nanoparticles, to water as PCM and making NEPCM, on LHTESS without Y-shaped fins performance during solidification process is illustrated in Table 8.3 and Fig. 8.9; it can be observed that adding nanoparticles to LHTESS has moderately considerable effect on solidification process, this is because in conduction-dominated phase change procedures, adding nanoparticles has much more effect in comparison to natural convection-dominated procedures [4], therefore since in solidification process, conduction is predominant heat transfer mechanism, increasing of nanoparticles volume fraction shows rather significant enhancement in process rate.

Table 8.3

Effect of nanoparticles volume fraction on full solidification time and improvement

Nanoparticle volume fraction (φ)	Full solidification time (s)	Increment in rate of solidification (%)
0.00	13,000	—
0.025	11,847	8.8
0.050	10,800	16.9

Figure 8.9 Effect of nanoparticles volume fraction on solidification front position

According to Fig. 8.9, it can be observed that at the beginning of the process the enhancement of solidification rate by adding nanoparticles is insignificant, but as time progresses, the enhancement of the process rate because of nanoparticles dispersion augments. Due to the enhancement in the discharging rate by adding 5% copper nanoparticles, this volume fraction value can be reported as the best choice for nanoparticles volume fraction in this system.

8.1.2.3. Comparison between adding nanoparticles and applying Y-shaped fins

Adding nanoparticles decreases maximum energy storage capacity because of decreasing the value of heat capacity and latent heat, while applying fin in LHTESS decreases maximum energy storage capacity because of decreasing in PCM mass. But both of these enhancement methods enhance solidification rate. In order to compare these techniques from the viewpoint of both parameters of solidification rate and maximum energy storage capacity, by using Eq. (8.7), Figs. 8.10 and 8.11 have been produced.

Figure 8.10 Solid fraction during discharging process.

Figure 8.11 Total energy released during discharging process.

According to Fig. 8.10 it is obvious that in Y-shaped fin-assisted LHTESS, discharging rate is significantly higher than the case of nanoparticles dispersion and as a result, full solidification time is significantly lower. Moreover, according to Fig. 8.11, comparison between the maximum energy storage capacity in the case of adding nanoparticles to the system and adding Y-shaped fin, indicates that by employing fin in the system, maximum energy storage capacity, which is equal to total energy at the beginning of discharging process, is almost the same as the LHTESS containing pure PCM without fin, and the LHTESS containing NEPCM, but the discharging rate enhancement is significantly higher, therefore adding fin to LHTESS is a better enhancement technique in comparison to nanoparticle dispersion from the viewpoint of either discharging rate or maximum energy storage capacity.

8.2. Snowflake-shaped fin for expediting discharging process in latent heat thermal energy storage system containing nanoenhanced phase change material

8.2.1. Problem statement

The main geometry of the present study is a Snowflake shaped fin-assisted LHTESS, which has been illustrated in Fig. 8.12 [5]. HTF flows in the inner tube, and the space between inner tube and storage container shell, which is square shaped [6], is filled with PCM. The Snowflake-shaped fin is connected to the outer side of the HTF tube in order to enhance thermal penetration depth into the space filled with PCM. The two-dimensional solution domain is illustrated in Fig. 8.13. The fin geometry parameters are listed in Table 8.4. Constant temperature equal to 240K is applied to the inner tube and the liquid PCM initial temperature is assumed equal to 285K. In Fig. 8.14, solidification front for different values of fin branch angles and length have been illustrated at the same time step, in order to study the behavior of solidification process at different values of fin branch length. Three different values for each branch length have been investigated, from this figure it can be inferred that for all values of branches direction, thermal penetration depth increases and solidification rate has strictly increasing behavior. Therefore, fin branches length has been chosen long enough and constant, which for longer branches, they will interfere or exceed of the calculation domain, for different values of branches directions. This choice reduces geometry parameters and simplifies the analysis of the other parameters, which have unknown effect on the performance of LHTESS. Fin thickness has been changed in order to keep constant the cross section area of the fin. This choice has been made because among the three geometry parameters in the present study including β, L, and w, the effect of fin thickness on solid–liquid phase change is not significant [3], therefore its effect on solidification process has not been investigated. On the other hand, if the volume of the employed fin varies for different cases, the value of energy storage capacity will be different, so the value of fin volume, in two-dimensional simulation cross section of the fin system, has to be the same in order to study the effect of fin geometry parameters on solidification rate, at the same value of energy storage capacity. To do so, fin length and direction is changed to study their effect on the system performance, and fin thickness, which has negligible effect on the process, has been changed in order to keep constant the cross section area of the fin and energy storage capacity.

Figure 8.12 Three-dimensional view of Snowflake shaped fin-assisted LHTESS.

Figure 8.13 Solution domain and geometry parameters.

Table 8.4

Geometry parameters of Snowflake-shaped fin structure

Geometry parameters	Values
β₁	[2π/12–5π/12] [Rad]
β₂	[2π/12–5π/12] [Rad]
X₁	0.1L, 0.2L
X₂	0.6L, 0.7L

Figure 8.14 Phase change front at t = 1000 s, for Snowflake shaped fin-assisted LHTESS with β₁ = β₂ = 2π/12, 4π/12 and 4π/12.

8.2.2. Numerical results

In this section, the reason of employing Snowflake-shaped fin structure in square-shaped LHTESS will be investigated. Solid fraction–temperature contour plots for LHTESS containing pure PCM without fin are illustrated in Fig. 8.15. The contour plots indicate that in square-shaped LHTESS, solidification process at the corners happens too slowly and it increases the full process time. As shown in Fig. 8.15, at 15,000 s, if the LHTESS shape had been cylindrical, the solidification process would have been completed by this time step, but for square shape LHTESS the remaining PCM at the corners slowdowns the process significantly, when full solidification is achieved after 22, 500 s from the beginning of the process. This issue shows the requirement of increasing thermal penetration depth at the corners of the square-shaped LHTESS, which in this paper is carried out by employing fin with an innovative array in order to increase the solidification rate of PCM. In order to increase the thermal penetration depth at the corners of the square-shaped LHTESS, we will use an innovative fin configuration, which we have named Snowflake-shaped fin structure as illustrated in Fig. 8.13. In this structure, each fin has four branches; the main question is that the bigger branch should be near the cold wall or at the right side after the smaller branch. To answer this, a secondary problem is considered in order to justify the reason of special Snowflake crystal structure from the viewpoint of heat transfer analysis. In this problem, conduction heat transfer through the Snowflake-shaped fin with thermal conductivity equal to 200 W/mK, with constant temperature equal to 273K in one side, zero heat flux at the end side, surrounding temperature equal to 298K, and convection heat transfer coefficient equal to 15 W/m²K have been simulated. It should be noted that the boundary condition at the surfaces is applied by convection, and this is not relevant to the natural convection effect on solidification process of the main problem of this paper, To investigate the performance of fin branches, efficiency is measured as indicated in Eq. (8.8) [7]. Where q_f is the actual heat transfer from the fin surface area, h is the convection heat transfer coefficient, A_f is the fin surface area, and θ_b is the difference between fin base temperature and ambient temperature.

Figure 8.15 Solid fraction (left side) and temperature (right side) contour plots at three different time steps during solidification process of LHTESS without fins.

$η \equiv \frac{q_{f}}{q_{\max}} = \frac{q_{f}}{h A_{f} θ_{b}}$ $η \equiv \frac{q_{f}}{q_{\max}} = \frac{q_{f}}{h A_{f} θ_{b}}$

(8.8)

In Fig. 8.16A–B the efficiency of two branches, when the bigger branch is in left side—near the cold wall—or right side, has been illustrated. Comparison between these two figures indicates that, when the bigger branch is at left, the efficiency values of the two branches are close to each other for all values of branches distance from the end side. This will be used to achieve the desired uniform solidification process in LHTESS. But when the bigger branch locates at right side, for all values of distance from the end side, its efficiency will be lower and since bigger branch gets more volume in the LHTESS instead of PCM in comparison with smaller branch, the lower efficiency of bigger branch is not reasonable in this application. The other result that can be inferred from Fig. 8.6 is that the closer the two branches locates to the cold wall, the higher the efficiency of both branches.

Figure 8.16 Fin branches efficiency when the bigger branch is in left position—near the cold wall (A) or in the right (B) versus the distance of branches from the end side.

It can be claimed that the mysteriousness of Snowflake crystal structure (Fig. 8.17) can be justified by heat transfer reasons, which in the aforementioned sentences, it has been indicated that this structure, with smaller branch near the end side, causes more uniform temperature distribution on the branches in comparison with other arrays, which is Snowflakes in nature, the reason of this structure is to prevent the Snowflakes from melting and destruction. This concept can be applied to fin structure to achieve more uniform temperature distribution on the fin surface to improve heat transfer rate between the fin surface area and surroundings. Snowflake-shaped fin structure is firstly proposed in this work and has not been investigated before. It is obvious that by adding branches to the main fin, the ability of fin to widespread heat into the PCM will be improved. But the configuration of branches and the effect of geometry parameters have to be investigated. As mentioned earlier, Snowflake structure is able to achieve uniform temperature distribution on its surface, and this feature can be used to achieve uniform solidification by immersing this fin structure into the LHTESS. Therefore, solidification process simulation in Snowflake shaped fin-assisted LHTESS has been carried out for different values of geometry parameters to obtain the best fin structure. In Fig. 8.18, full solidification time for different geometry parameters is illustrated. From Fig. 8.18, it can be observed that, changing the direction of smaller branch (β₂) doesn’t have considerable effect on the solidification rate of PCM, but changing the bigger branch direction (β₁) has a significant effect on the process rate. According to Fig. 8.18, the best case among the investigated cases of geometry parameters is X₁ = 0.1L, X₂ = 0.6L, β₁ = 45°, β₂ = 75°.

Figure 8.17 Snowflake crystal structure.

Figure 8.18 Full solidification time for different values of geometry parameters of Snowflake shaped fin-assisted LHTESS.

8.2.2.1. Effect of branches direction on solidification rate

In Fig. 8.19, the effect of changing bigger branch direction on solid fraction of PCM during solidification process, for the best values for the position of branches among the investigated cases, which are X₁ = 0.1L and X₂ = 0.6L, has been illustrated. According to this figure, changing the bigger branch direction has a rather considerable effect on the solidification rate of PCM. For all values of β₂, when the direction of bigger branch is β₁ = 5π/12, solidification rate, which is equal to the slope of solid fraction diagram versus time, is the lowest, and among the other values of second branch direction, β₁ = 3π/12 has the highest rate. The reason that can be explained here is that, in the bigger branch direction, two parameters should be considered, the first one is the increasing thermal penetration depth in the direction of the corners of the quarter of the solution domain which are named in Fig. 8.13 as A and B. The second one is keeping the optimized distance from the small branch in order to keep the amount of PCM in the space between branches in optimized value. Therefore, the best value of bigger branch direction according to both of aforementioned parameters is β₁ = 3π/12. Also, it can be observed that, the behavior of the solid fraction rate doesn’t change significantly by changing in smaller branch direction.

Figure 8.19 The effect of changing bigger branch direction on solid fraction of PCM during solidification process for X₁ = 0.1L and X₂ = 0.6L.

8.2.2.2. The effect of changing the distance between the branches on solidification rate

In Fig. 8.20, the effect of changing the distance between the branches on solid fraction of PCM during discharging process is illustrated. It can be observed that the best case for the branches distance from cold wall is X₁ = 0.1L, X₂ = 0.6L, which is the closest case to the cold wall for both branches. As mentioned before, when the branches are closer to the cold wall, both branches efficiency values are higher. Therefore, we have chosen X₁ so close to the cold wall and inevitably chosen X₂ equal to 0.6L in order to increase the thermal penetration depth at the corners of LHTESS. In Figs. 8.21 and 8.22, solid fraction and temperature contour plots are illustrated for LHTESS with Snowflake-shaped fin with best branch structure among the investigated cases and simple longitudinal fin with the same value of cross section area, in three time steps, which are 200, 1500, and 3200 s after the beginning of solidification process. It should be mentioned that 3200 s is the time that full solidification is achieved for the case of Snowflake shaped fin-assisted LHTESS with the best branches array, and these time steps are used in longitudinal fin-assisted LHTESS in order to simplify the comparison. Comparison between these cases and LHTESS without fin indicates that solidification rate in Snowflake shaped fin-assisted LHTESS is so higher than other cases. Although, longitudinal fin has the best direction, which increases thermal penetration depth at the corners of square-shaped LHTESS, solidification rate in Snowflake shaped fin-assisted LHTESS is higher approximately 36% than the case of longitudinal fin-assisted LHTESS.

Figure 8.20 The effect of changing the distance between the branches on the solid fraction of PCM during discharging process in Snowflake shaped fin-assisted LHTESS.

Figure 8.21 Solid fraction (left side) and temperature (right side) contour plots at three different time steps during solidification process of Snowflake shaped fin-assisted LHTESS.

Figure 8.22 Solid fraction (left side) and temperature (right side) contour plots at three different time steps during solidification process of longitudinal fin-assisted LHTESS.

8.2.2.3. Performance enhancement of discharging process in LHTESS by adding fins

In Fig. 8.23, average temperature over the whole domain during discharging process has been illustrated in order to analyze the efficiency of adding fin to the LHTESS. The average temperature has been calculated using the following equation:

Figure 8.23 Average temperature variations over computational domain during solidification process.

$T_{a v e} = \frac{\int T d A}{\int d A}$ $T_{a v e} = \frac{\int T d A}{\int d A}$

(8.9)

According to temperature contours illustrated in Figs. 8.15, 8.21, and 8.22 and average temperature over the computational domain in Fig. 8.23, it can be observed that the average temperature for the Snowflake shaped fin-assisted LHTESS is the lowest, which means that the whole computational domain temperature is closer to the cold wall and the fin system is more efficient in enhancing penetration depth into the PCM. Also in Fig. 8.23, full solidification time for LHTESS without fin, with simple longitudinal fin, and with Snowflake-shaped fin is illustrated, comparison between them indicates that adding fins to LHTESS is a highly efficient technique for expediting the discharging process in LHTESS, and the Snowflake fin structure has the best performance among the investigated cases, although the simple longitudinal fin has the best direction which improves thermal penetration depth at the corners of the square-shaped LHTESS. Moreover, the comparison between fin-assisted and without fin LHTESS indicates that employing fin of any configuration is an efficient method for increasing the solidification rate of PCM in LHTESS without changing the thermophysical properties of PCM.

8.2.2.4. Optimization of the Snowflake-shaped fin configuration

The optimization method, which has been employed in present study, is RSM. This approach is consisted of mathematical methods, which are appropriate for optimization studies. It is so efficient in problems where several design parameters affect the response of the system. In the optimization procedure by RSM, the responses that have been obtained by a limited number of cases properly chosen in the design space are characterized [8]. The main purpose of optimization procedure in this paper is to find the best configuration of Snowflake-shaped fin based on solidification expedition. The reason of employing this array in square container is to cover the space filled with PCM, in order to reduce thermal resistance as possible. The value of thermal resistance has an indirect relationship with thermal conductivity of PCM and direct relationship with the space between fins [7]. Therefore, thermal resistance of liquid PCM, due to its low thermal conductivity, is so high that it slows down the energy retrieval process of LHTESS. It is obvious that by immersing fin into the system, thermal properties of PCM don’t change, and thermal resistance is just controlled by the distance between fin branches. If there is no fin in the system, thermal resistance will be so high because the distance between the cold wall and the liquid PCM far from cold wall is so high that leads to lower discharging rate. By applying fin into the system, due to its high thermal conductivity, solidification procedure begins not only in the region adjacent to the cold wall, but also in the regions adjacent to the fin system, therefore it resembles the case that the distance between the cold wall and PCM decreases. This distance can be optimized to obtain minimum thermal resistance and more discharging rate.

8.2.2.4.1. Effect of bigger branch direction on solidification rate

From Fig. 8.18, it can be observed that the effect of changing bigger branch direction on solidification rate of PCM is significant. With the increase in the bigger branch direction, first the full solidification rate decreases and then increases. The reason of this behavior is that in the bigger branch direction investigation, two parameters should be considered; the first one is the increase of thermal penetration depth toward the corners of the solution domain, which are marked by (B) in Fig. 8.13. The second one is to keep the optimized distance from the smaller branch to achieve optimized mass distribution in the space between branches during solidification process. Therefore, the best value of bigger branch direction according to both of the aforementioned parameters is β₁ = 3.69π/12.

8.2.2.4.2. Effect of smaller branch direction on solidification rate

According to Fig. 8.18, with the increase in smaller branch direction, full solidification time decreases, because at the corners of the domain, which are marked by (A) in Fig. 8.13, thermal penetration depth is increased by the main fin branch as illustrated in Fig. 8.13, therefore the need for the increase of thermal penetration depth by the second branch at the corners is insignificant. But with the increase in second branch direction, mass distribution in the space between the bigger and smaller branches is controlled and optimized. According to the mentioned points, the optimized smaller branch direction is β₂ = 5π/12. It should be noted that the existence of smaller branch is necessary because the distance between cold wall and the corners of container is so high that causes high thermal resistance. Therefore, this branch covers the space filled with PCM at the corners and lowers thermal resistance and expedites energy retrieval of LHTESS.

8.2.2.4.3. Effect of bigger branch distance from cold wall on solidification rate

With the increase in bigger branch distance from cold wall, full solidification time doesn’t have absolute ascending or descending trend; first it decreases and then increases (Fig. 8.24). As the bigger branch is closer to the cold wall and therefore the average temperature on its surface is closer to the cold wall temperature, thus the bigger branch efficiency is higher according to Eq. 8.9, which leads to heat transfer enhancement and higher solidification rate. On the other hand, if the value of X₁ is higher, the distance between bigger and smaller branch is lower and mass distribution in the space between branches is more uniform. Based on the aforementioned reasons, the optimized value of bigger branch distance from cold wall is a value between the biggest and smallest value of X₁, which is X₁ = 0.16L.

Figure 8.24 Time surfaces for different geometry parameters of Snowflake-shaped fin

8.2.2.4.4. Effect of smaller branch distance from cold wall on solidification rate

If smaller branch is close to the corners of the solution domain, far from cold wall, thermal penetration depth at the corners will be enhanced, but on the other hand, it should be closer to the cold wall in order to control the mass distribution in the space between smaller and bigger branches [9]. The interaction between these two factors makes the optimized value of smaller branch distance from cold wall X₂ = 0.66L.

In Figs. 8.25 and 8.26, solid-temperature contours are illustrated for LHTESS with simple longitudinal fin and with Snowflake-shaped fin with optimized structure in three time steps including 200, 1500, and 2800 s after the beginning of solidification process. It should be mentioned that 2800 s is the time that full solidification is achieved for the case of optimized Snowflake shaped fin-assisted LHTESS, and these time steps are used in longitudinal fin-assisted LHTESS in order to simplify the comparison. Comparison between these cases and LHTESS without fin indicates that solidification rate in Snowflake shaped fin-assisted LHTESS is significantly higher than other cases. It should be noted that the investigated longitudinal fin has the same cross section area as the optimized Snowflake-shaped fin, in order to compare these systems from the viewpoint of solidification rate, in a constant value of energy storage capacity. Although longitudinal fin has the best direction, which increases thermal penetration depth at the corners of square-shaped LHTESS; solidification rate in Snowflake shaped fin-assisted LHTESS is higher approximately 49% than the case of longitudinal fin-assisted LHTESS.

Figure 8.25 Solid-temperature contours at three different times during solidification process of Snowflake shaped fin-assisted LHTESS.

Figure 8.26 Solid–temperature contours at three different time steps during solidification process of longitudinal fin-assisted LHTESS.

8.2.2.5. Performance enhancement of discharging process in LHTESS by adding various structures of fin

According to temperature contour plots and average temperature over the solution domain in Fig. 8.27, it can be observed that the average temperature for the Snowflake shaped fin-assisted LHTESS in the lowest and it has the most uniform temperature distribution, which means that the solution domain temperature is closer to the cold wall and the fin system is more efficient in enhancing penetration depth into the PCM. Also in Fig. 8.27, full solidification time for LHTESS without fin, with simple longitudinal fin, and with Snowflake-shaped fin is illustrated, comparison between these cases indicates that adding fin of any structure to LHTESS is a highly efficient technique for expediting the discharging process in LHTESS without changing the physical properties of PCM, and the Snowflake-shaped fin configuration has the best performance among the investigated cases, although the simple longitudinal fin has the best direction which improves thermal penetration depth at the corners of the square-shaped LHTESS.

Figure 8.27 Average temperature variations over the solution domain during solidification process.

8.2.2.6. Comparison between the enhancement techniques

In Fig. 8.28, total energy released during discharging process of LHTESS has been illustrated in order to investigate two parameters. The first parameter is maximum energy storage capacity, which is equal to the sum of sensible and latent heat, and the second one is full solidification time. The reason that can be explained as the importance of these factors in present paper is that in the case of adding fins to the systems, energy storage capacity reduces because less amount of PCM is used in the system. But for the case of nanoparticles dispersion, energy storage capacity decreases because of a decrease in latent heat of fusion and heat capacity, which can be observed according to Eqs. (8.4) and (8.5). The effect of nanoparticles dispersion and making NEPCM, and adding fin to LHTESS has been investigated considering both of the aforementioned parameters. According to Fig. 8.28, full solidification time for Snowflake-shaped fin-assisted LHTESS is the lowest; moreover, maximum energy storage density is almost the same for all of these cases. Therefore, adding Snowflake-shaped fin enhances the solidification rate considerably without decreasing maximum energy storage capacity significantly, which verifies the efficiency of adding Snowflake-shaped fin configuration to LHTESS. Finite element methods has been applied in various other filed of physics in recent years [10–20].

Figure 8.28 Total energy released during the discharging process in LHTESS containing PCM without a fin, PCM with Snowflake-shaped fin, and NEPCM with φ = 2.5, 5.0%.

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Table of Contents for Chapter 8: Nanofluid Conductive Heat Transfer in Solidification Mechanism

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