# An Optimum Enthalpy Approach for Melting and Solidification with Volume Change

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Numerical Solution

#### 3.1. Optimum Approach

#### 3.2. Optimum Density Approach

#### 3.3. Source Based Method

#### 3.4. Test Cases

#### 3.4.1. One-Dimensional Solidification

#### 3.4.2. Two-Dimensional Melting with Convection

#### 3.5. Validation Case

#### 3.6. Mesh Influence for the Convection Test Case

## 4. Results and Discussion

#### 4.1. One-Dimensional Solidification

#### 4.2. Two-Dimensional Melting of Octadecane with Natural Convection

#### 4.3. Validation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${c}_{p}$ | heat capacity |

f | function |

h | enthalpy |

$\overrightarrow{g},g$ | gravitational acceleration |

k | heat transfer coefficient |

l | length |

m | mass |

$\overrightarrow{n}$ | normalized gradient |

p | pressure |

t | time |

$\overrightarrow{u}$ | velocity vector |

x | coordinate |

y | coordinate |

$\overrightarrow{A}$ | Darcy term |

A | area |

D | Darcy constant |

E | identity matrix |

L | latent heat of fusion |

T | temperature |

V | volume |

$\alpha $ | phase fraction |

$\beta $ | volumetric coefficient of thermal expansion |

$\gamma $ | abbreviation |

$\u03f5$ | small numerical constant |

$\lambda $ | thermal conductivity |

$\eta $ | viscosity |

$\rho $ | density |

$\tau $ | stress tensor |

$\omega $ | underrelaxation factor |

c | cold |

g | global |

h | hot |

i | index |

$ini$ | initial |

k | index |

l | liquid |

m | index |

o | old |

$ref$ | reference |

$res$ | residuum |

s | solid |

y | y component |

L | liquidus |

S | solidus |

T | transpose |

## Appendix A

#### Appendix A.1. Derivation of the Pressure Equation

#### Appendix A.2. PISO-Algorithm

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**Figure 1.**Flow chart of the numerical solution process inside a time step. In the following, iterations needed to solve the linearized energy equation are called linear iterations and iterations for the nonlinear energy equation are called nonlinear T-h iterations.

**Figure 3.**State of the simulation domain at $t>0\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. The left side is heated to ${T}_{h}$, while the right side is kept at ${T}_{c}$, which is the same as the initial temperature ${T}_{ini}$. The PCM is partially molten and the influence of convection is visible.

**Figure 4.**State of the extended simulation domain at $t>0\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. The copper and the aluminum plate are not simulated, instead a constant temperature is prescribed.

**Figure 5.**Relative difference in the global liquid fraction between the coarse/medium and the fine mesh over time.

**Figure 6.**Temperature profile of the one dimensional test case after four time units. The kink corresponds to the position of the phase boundary.

**Figure 9.**Velocity at x = 2.5. At the beginning, u is greater than 0.1. The maximum velocity is 0.51.

**Figure 10.**Number of nonlinear T-h iterations for the two different approaches and four different time steps [0.5,1.0,2.0,4.0]. The missing bar indicates a solver crash.

**Figure 11.**(

**Left**) nonlinear T-h iterations over time; and (

**Right**) linear iterations over time (max Co = 1.0/dt = $1.0\phantom{\rule{0.166667em}{0ex}}\mathrm{s})$.

**Figure 12.**Convergence history for the two different approaches. The graph shown here refers to t = 20 min (max Co = 1.0/dt = $1.0\phantom{\rule{0.166667em}{0ex}}\mathrm{s}).$

**Figure 13.**(

**Left**) Numerical phase fronts with adiabatic top and bottom boundary; and (

**Right**) experimental phase and numerical phase fronts with the extended simulation domain. In both figures, the lines refer to 1, 2, 3 and 4 h (left to right).

**Figure 14.**Experimentally (

**left**); and numerically (

**right**) obtained velocity field inside the PCM after 4 h.

**Figure 15.**Temperature profile in the PCM, the pipe walls and the insulation after: 1 h (

**left**); and 3 h (

**right**). The black lines were inserted to guide to eye and mark the pipe walls. The melting temperature is white.

Property | Unit | Value |
---|---|---|

Melting temperature | K | 301.15 [25] |

Melting Range (${T}_{L}$-${T}_{S}$) | K | 0.05 |

Latent heat of fusion | kJ/kg | 243.68 [26] |

Density (solid) | kg/m^{3} | 867.00 |

Density (liquid) | kg/m^{3} | 775.60 [26] |

Volumetric thermal expansion coefficient | 1/K | 8.36 × ${10}^{-4}$ [26] |

Specific heat capacity (solid) | kJ/(kg K) | 1.90 [26] |

Specific heat capacity (liquid) | kJ/(kg K) | 2.24 [26] |

Thermal conductivity (solid) | W/(m K) | 0.32 [26] |

Thermal conductivity (liquid) | W/(m K) | 0.15 [26] |

Dynamic viscosity ($T=309.15\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) | mPas | 3.75 |

Darcy constant | kg/(m^{3}s) | ${10}^{10}$ |

Property | Unit | Value |
---|---|---|

Density PC | kg/m^{3} | 1200.0 |

Density PS | kg/m^{3} | 40.0 |

Specific heat capacity PC | kJ/(kg K) | 1170.0 |

Specific heat capacity PS | kJ/(kg K) | 1500.0 |

Thermal conductivity PC | W/(m K) | 0.21 |

Thermal conductivity PS | W/(m K) | 0.04 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Faden, M.; König-Haagen, A.; Brüggemann, D.
An Optimum Enthalpy Approach for Melting and Solidification with Volume Change. *Energies* **2019**, *12*, 868.
https://doi.org/10.3390/en12050868

**AMA Style**

Faden M, König-Haagen A, Brüggemann D.
An Optimum Enthalpy Approach for Melting and Solidification with Volume Change. *Energies*. 2019; 12(5):868.
https://doi.org/10.3390/en12050868

**Chicago/Turabian Style**

Faden, Moritz, Andreas König-Haagen, and Dieter Brüggemann.
2019. "An Optimum Enthalpy Approach for Melting and Solidification with Volume Change" *Energies* 12, no. 5: 868.
https://doi.org/10.3390/en12050868