# Phase-Transition Thermal Charging of a Channel-Shape Thermal Energy Storage Unit: Taguchi Optimization Approach and Copper Foam Inserts

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Model Description

_{h}and attached to the left side of the thermal energy storage unit. The other walls of the unit are well insulated, so they are assumed to permit zero heat flux. The thickness of the porous layer is l, and the distance between its center and the bottom wall is h. The local thermal equilibrium approach is used to model a copper foam filled by the NePCM composite. The matrix of the NePCM that is considered is capric acid, and the nanoparticles are copper. The properties of both materials are given in Table 1.

#### 2.2. Physical Model and Governing Equations

_{k}= ε inside the metal foam and 1 otherwise. The subscripts of “NeP,l” indicate the NePCM in the liquid state. The subscript j can be 1 for x or 2 for y; k is 1 for metal foam and 2 for clear flow. Both j and k are index parameters with no inherent physical meaning. The thermophysical properties are dynamic viscosity, μ, and density, ρ. Here f is a source term containing the body forces of the Darcy term, buoyancy forces, and a mushy zone control term. The porosity function and f are introduced as:

_{k}is permeability, and it is artificially large outside the foam and equal to K inside the foam. β is the thermal expansion coefficient, and g is the acceleration due to gravity. A

_{mush}is relatively large (~10

^{6}) compared to ϛ (~10

^{−3}). ξ(T) is the melt fraction. It is 1 for a fully liquid PCM and 0 when the PCM is fully solid. In the temperature range in which the PCM is expected to be partially melted, around the melting temperature, T

_{m}, ξ(T) is assumed to take a linear distribution. The permeability, melting fraction, and gravity acceleration are introduced by Equation (3c–e), respectively.

_{p}is the pore size, and ω is the pore density (pores per inch (PPI)).

_{p}, h

_{sf}, and λ are the heat capacity, latent heat of phase change, and thermal conductivity, respectively. The subscript “eff” indicates the effective property for foam and the material inside the pores (composite property). The effective properties are introduced in Equations (6a,b) and (7), where the subscripts “sm” and “s” denote the copper foam and solid PCM, respectively.

_{na}is the volumetric concentration of nanoparticles in the PCM. The Brinkman model (Equation (10c)) is utilized to estimate the dynamic viscosity [47].

#### 2.3. Characteristic Parameters

## 3. Solution Approach and Validation

#### 3.1. Numerical Method

^{−6}.

#### 3.2. Impact of Mesh Size

_{na}= 0.06.

#### 3.3. Validation

^{−7}m

^{2}(Equation (4a,b)).

^{5}and a Prandtl number Pr = 50 [54,55]. This case shows the melting of a PCM in a rectangular cavity when the heated wall is subject to a uniform hot temperature. The melting fronts are plotted in Figure 5 for all cases, which denotes fair proximity between all captured melting fronts. The melting front computed by Kashani et al. [56] has also been added for the sake of comparison.

## 4. Results and Discussion

_{na}≤ 0.08).

^{5}= 1024 experiments would be required, which is a prohibitive number. Taguchi’s method is used to reduce the number of required tests, and the L16 orthogonal array concept is utilized (Table 4). As can be seen in that table, only 16 tests with different combinations of the control factors are sufficient to define the influence of the factors, based on Taguchi’s algorithm.

_{MVF}

_{=1}, is the result that should be optimized. As the objective is to reduce t|

_{MVF}

_{=1}to achieve more efficient thermal storage, then the lower-the-better approach is appropriate. In addition, a linear regression equation is derived to create a simple predictive relationship between t|

_{MVF}

_{=1}and the various control factors:

_{MVF}

_{=1}(s) = −5100 − 10.0 w (mm) − 215.0 L (mm) + 177.50 h (mm) + 9600 ε − 11250 υ

_{na},

_{na}= 0.04, which gives melting time, t|

_{MVF}

_{=1}= 2025 s, while performing a test with optimal condition that gives t|

_{MVF}

_{=1}= 3291.6 s. This value of t|

_{MVF}

_{=1}is lower than that of all the 16 combinations performed earlier, indicating that the factors are likely to be optimized. The optimum case shows a 58% reduction in melting time, 100 × (7900−3291.6)/7900, compared to the design case of no. 9 of Table 4.

_{na}> w. This indicates that the properties of the porous layer, i.e., its height, porosity, and thickness, are the most significant parameters, followed by the volume fraction of the dispersed nanoparticles, while the thickness of the copper wall is the least significant parameter.

_{optimum}= 3291.6 s, was taken as the test time, and the melted volume fraction was measured at this time for each experiment. A volume fraction of melt that is lower than 1 indicates that full melting is not yet achieved, and the time required for full melting in the experiment is greater than t|

_{optimum}. It can be seen in Table 7 that in 10 out of 12 experiments, the PCM has not fully melted at t|

_{optimum}, which supports the supposition that the parameters that were calculated previously are indeed optimized. In the remaining two experiments, complete melting is fully achieved at t|

_{optimum}, when w was increased from 4 mm to 6 mm and when ε was increased from 0.8 to 0.85. Furthermore, the difference between the obtained values for melting time and t|

_{optimum}is lower when w is varied, which supports the rank defined based on Table 6. The variation of the foam layer’s location from 2 mm to 6 mm increased the MVF from 0.917 to 0.998, which shows an 8.8% improvement in the melting rate.

_{na}, on the flow patterns and the isothermal contours. In fact, as the porous layer’s location, size, and porosity are the same in all the cases, the geometry does not change when the volume fraction of the particles is varied. The variations of the MVF and the ES as a function of time are shown in Figure 18. It can be seen that the full melting is reached simultaneously for all the values of v

_{na}. Moreover, the values of ES show limited change with the variation of v

_{na}. In addition, compared to Figure 9, Figure 12, and Figure 15, Figure 18 shows that the impact of the nanoparticles’ concentration, v

_{na}, is limited on the values of MVF and ES, thus confirming the outcomes of Table 6, in which v

_{na}was shown to have a rank-four importance compared to the other control factors.

## 5. Conclusions

- In a cavity with side length, L = 40 mm, the optimal values obtained for the control factors are the following: w = 4 mm, l = 8 mm, h = 11 mm, ε = 0.8, and υ
_{na}= 0.04. By their decreasing order of influence, the control factors are ranked as: h > ε > l > υ_{na}> w. The variation of the design parameters could induce a 58% variation in the melting time. - When the left wall is heated, PCM starts melting, and convective flow takes place. The presence of the porous layer in the cavity improves heat transfer and contributes to PCM melting.
- The size and location of the porous layer affect the thermal behavior of the PCM. Increasing the layer size, l, enhances and accelerates heat transfer. The charging power increases with l. Just shifting the porous layer from 2 mm to 6 mm increased the melting rate by 8.8%.
- Moving the porous layer upwards (increasing h) hinders the convective effects in the bottom part of the cavity and lowers the contribution of the porous layer to PCM melting. The charging power decreases when h is raised.
- Reducing the porosity of the porous layer, ε, which is equivalent to a higher presence of the solid matrix and, consequently, a higher thermal conductivity, enhances heat transfer and PCM melting. The charging power increases as ε decreases.
- The width of the heated copper wall, w, has a very limited effect on the charging power.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Dependences of MVF as a function of time on grid size when L = 40 mm, w = 3 mm

_{,}l = 6 mm, h = 11 mm, ε = 0.85, and υ

_{na}= 0.06.

**Figure 4.**Comparison of the melting interface between the numerical results of the present study and published experimental results. (

**a**) shows the images of the melting field at various time steps, reported in [39]. (

**b**) shows the computed melting field of the present simulations. Adapted with permission from ref. [39]. Copyright 2019 Elsevier.

**Figure 6.**Signal to noise ratios of tested levels: the optimum levels are: w = 4, L = 4, h = 1, ε = 1, and υ

_{na}= 3.

**Figure 7.**Streamlines and interface of melting (white dashed line) for different control parameters at three times when L = 40 mm, w = 4 mm

_{,}h = 11 mm, ε = 0.80, and υ

_{na}= 0.04.

**Figure 8.**Isotherm lines for different control parameter at three times when L = 40 mm, w = 4 mm

_{,}h = 11 mm, ε = 0.80, and υ

_{na}= 0.04.

**Figure 9.**Variation of MVF (

**left**) and ES (

**right**) as a function of time for optimum porous layer thickness when L = 40 mm, w = 4 mm

_{,}h = 11 mm, ε = 0.80, and υ

_{na}= 0.04.

**Figure 10.**Streamlines and interface of melting (white dashed line) for different control parameter at three times when L = 40 mm, w = 4 mm, l = 8 mm, ε = 0.80, and υ

_{na}= 0.04.

**Figure 11.**Isotherm lines for different control parameter at three times when L = 40 mm, w = 4 mm, l = 8 mm, ε = 0.80, and υ

_{na}= 0.04.

**Figure 12.**Variation of MVF (

**left**) and ES (

**right**) as a function of time for the optimum height porous layer when L = 40 mm, w = 4 mm, l = 8 mm, ε = 0.80, and υ

_{na}= 0.04.

**Figure 13.**Streamlines and interface of melting (white dashed line) for different control parameter at three times when L = 40 mm, w = 4 mm, l = 8 mm, h = 11 mm, and υ

_{na}= 0.04.

**Figure 14.**Isotherm lines for different control parameter at three times when L = 40 mm, w = 4 mm, l = 8 mm, h = 11 mm, and υ

_{na}= 0.04.

**Figure 15.**Variation of MVF (

**left**) and ES (

**right**) as a function of time for optimum porosity when L = 40 mm, w = 4 mm, l = 8 mm, h = 11 mm, and υ

_{na}= 0.04.

**Figure 16.**Streamlines and interface of melting (white dashed line) for different control parameter at three times when L = 40 mm, l = 8 mm, h = 11 mm, ε = 0.80, and w = 4 mm.

**Figure 17.**Isotherm lines for different control parameter at three times when L = 40 mm, l = 8 mm, h = 11 mm, ε = 0.80, and w = 4 mm.

**Figure 18.**Variation of MVF (

**left**) and ES (

**right**) of copper nano-additives as a function of time for optimum wall thickness of Copper when L = 40 mm, l = 8 mm, h = 11 mm, ε = 0.80, and w = 4 mm.

Properties | Capric Acid | Copper |
---|---|---|

Density (kg m^{−3}) | Solid: 1018 Liquid: 888 | 8933 |

Kinematic viscosity (m^{2} s^{−1}) | 3 × 10^{−6} | N/A |

Thermal expansion coefficient (K^{−1}) | 1 × 10^{−3} | 1.67 × 10^{−5} |

Thermal conductivity (Wm^{−1} K^{−1}) | Solid: 0.372 Liquid: 0.153 | 401 |

Latent heat (kJ kg^{−1}) | 152.7 | N/A |

Phase change temperature (°C) | 32 | N/A |

Specific heat (kJ kg^{−1} K^{−1}) | Solid: 1.9 Liquid: 2.4 | 0.385 |

**Table 2.**Details of uniform grid check cases in which specified conditions are L = 40 mm, w = 3 mm, l = 6 mm, h = 11 mm, ε = 0.85, and υ

_{na}= 0.06.

Cases | Mesh Size in Wall | Mesh Size in PCM | $\mathbf{MVF}\left(\right)open="|">{}_{\mathit{t}\mathbf{=}\mathbf{2000}\mathit{s}}$ | Computational Time |
---|---|---|---|---|

Case I | 4 × 75 | 75 × 75 | 0.7281 | 14 h 19 min 15 s |

Case II | 5 × 100 | 100 × 100 | 0.7273 | 9 h 45 min 20 s |

* Case III | 6 × 125 | 125 × 125 | 0.7327 | 10 h 13 min 36 s |

Case IV | 7 × 150 | 150 × 150 | 0.7369 | 10 h 14 min 17 s |

Case V | 8 × 175 | 175 × 175 | 0.7401 | 12 h 33 min 11 s |

Factors | Description | Level 1 | Level 2 | Level 3 | Level 4 |
---|---|---|---|---|---|

A | w / mm (Copper wall thickness) | 1 | 2 | 3 | 4 |

B | l / mm (Porous layer thickness) | 2 | 4 | 6 | 8 |

C | h / mm (Porous layer height) | 11 | 17 | 23 | 29 |

D | ε (Porosity) | 0.80 | 0.85 | 0.90 | 0.95 |

E | υ_{na}(Nanoparticles volume fraction) | 0.00 | 0.02 | 0.04 | 0.06 |

Case No. | Control Parameters | MVF = 1 | ||||||
---|---|---|---|---|---|---|---|---|

A | B | C | D | E | t|_{MVF}_{=1} / s | * CP / J m^{−1} s^{−1} | S/N Ratio | |

w / mm | l / mm | h / mm | ε | υ_{na} | ||||

1 | 1 | 2 | 11 | 0.8 | 0 | 4100 | 131.5572 | −72.2557 |

2 | 1 | 4 | 17 | 0.85 | 0.02 | 4700 | 113.2655 | −73.4420 |

3 | 1 | 6 | 23 | 0.90 | 0.04 | 5800 | 91.0539 | −75.2686 |

4 | 1 | 8 | 29 | 0.95 | 0.06 | 6700 | 78.6821 | −76.5215 |

5 | 2 | 2 | 17 | 0.90 | 0.06 | 5600 | 93.6736 | −74.9638 |

6 | 2 | 4 | 11 | 0.95 | 0.04 | 4400 | 122.7105 | −72.8691 |

7 | 2 | 6 | 29 | 0.8 | 0.02 | 6300 | 80.5411 | −75.9868 |

8 | 2 | 8 | 23 | 0.85 | 0 | 5600 | 95.4117 | −74.9638 |

9 | 3 | 2 | 23 | 0.95 | 0.02 | 7900 | 68.6977 | −77.9525 |

10 | 3 | 4 | 29 | 0.90 | 0 | 7800 | 68.6409 | −77.8419 |

11 | 3 | 6 | 11 | 0.85 | 0.06 | 3300 | 159.8793 | −70.3703 |

12 | 3 | 8 | 17 | 0.8 | 0.04 | 3400 | 153.9024 | −70.6296 |

13 | 4 | 2 | 29 | 0.85 | 0.04 | 6900 | 73.5752 | −76.7770 |

14 | 4 | 4 | 23 | 0.8 | 0.06 | 5100 | 98.5855 | −74.1514 |

15 | 4 | 6 | 17 | 0.95 | 0 | 5600 | 98.7377 | −74.9638 |

16 | 4 | 8 | 11 | 0.90 | 0.02 | 3400 | 159.6877 | −70.6296 |

Optimum Factors | Optimum Melting Time at MVF = 1 | |||||
---|---|---|---|---|---|---|

W | L | H | ε | υ_{na} | Taguchi Prediction | Tested Case |

4 mm | 8 mm | 11 mm | 0.80 | 0.04 | 2025s | 3291.6 |

w / mm | l / mm | h / mm | ε | υ_{na} | |
---|---|---|---|---|---|

Level 1 | −74.37 | −75.49 | −71.53 | −73.26 | −75.01 |

Level 2 | −74.70 | −74.58 | −73.50 | −73.89 | −74.50 |

Level 3 | −74.20 | −74.15 | −75.58 | −74.68 | −73.89 |

Level 4 | −74.13 | −73.19 | −76.78 | −75.58 | −74.00 |

δ | 0.57 | 2.30 | 5.25 | 2.32 | 1.12 |

Rank | 5 | 3 | 1 | 2 | 4 |

**Table 7.**Further analysis around the optimum case (w = 4 mm, l = 8 mm, h = 11 mm, ε = 0.8, and υ

_{na}= 0.04).

Case No. | Parameter | Control Parameters | at t = 3291.6 s | |||||
---|---|---|---|---|---|---|---|---|

A | B | C | D | E | MVF | ^{*}CP / J m^{−1} s^{−1} | ||

w / mm | l / mm | h / mm | ε | υ_{na} | ||||

1 | l | 4 | 2 | 11 | 0.80 | 0.04 | 0.9171 | 144.6119 |

2 | 4 | 4 | 11 | 0.80 | 0.04 | 0.9845 | 156.9111 | |

3 | 4 | 6 | 11 | 0.80 | 0.04 | 0.9976 | 160.5965 | |

4 | h | 4 | 8 | 17 | 0.80 | 0.04 | 0.9998 | 158.3983 |

5 | 4 | 8 | 23 | 0.80 | 0.04 | 0.9317 | 144.8368 | |

6 | 4 | 8 | 29 | 0.80 | 0.04 | 0.8469 | 130.0390 | |

7 | ε | 4 | 8 | 11 | 0.85 | 0.04 | 1.0000 | 163.0727 |

8 | 4 | 8 | 11 | 0.9 | 0.04 | 0.9982 | 162.9335 | |

9 | 4 | 8 | 11 | 0.95 | 0.04 | 0.9635 | 157.2007 | |

10 | υ_{na} | 4 | 8 | 11 | 0.80 | 0.0 | 0.9856 | 163.2366 |

11 | 4 | 8 | 11 | 0.80 | 0.02 | 0.9961 | 163.2134 | |

12 | 4 | 8 | 11 | 0.80 | 0.06 | 1.0000 | 160.8724 |

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**MDPI and ACS Style**

Ghalambaz, M.; Mehryan, S.A.M.; Hajjar, A.; Younis, O.; Sheremet, M.A.; Pour, M.S.; Hulme-Smith, C.
Phase-Transition Thermal Charging of a Channel-Shape Thermal Energy Storage Unit: Taguchi Optimization Approach and Copper Foam Inserts. *Molecules* **2021**, *26*, 1235.
https://doi.org/10.3390/molecules26051235

**AMA Style**

Ghalambaz M, Mehryan SAM, Hajjar A, Younis O, Sheremet MA, Pour MS, Hulme-Smith C.
Phase-Transition Thermal Charging of a Channel-Shape Thermal Energy Storage Unit: Taguchi Optimization Approach and Copper Foam Inserts. *Molecules*. 2021; 26(5):1235.
https://doi.org/10.3390/molecules26051235

**Chicago/Turabian Style**

Ghalambaz, Mohammad, Seyed Abdollah Mansouri Mehryan, Ahmad Hajjar, Obai Younis, Mikhail A. Sheremet, Mohsen Saffari Pour, and Christopher Hulme-Smith.
2021. "Phase-Transition Thermal Charging of a Channel-Shape Thermal Energy Storage Unit: Taguchi Optimization Approach and Copper Foam Inserts" *Molecules* 26, no. 5: 1235.
https://doi.org/10.3390/molecules26051235