# Preliminary Design of a Space Habitat Thermally Controlled Using Phase Change Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

- The analysis of multiple coupled PCM modules is reduced to the study of a single PCM cell.
- Since the habitat would operate in reduced gravity and $L\ll R$, convective flows in (the liquid phase of) the PCM are neglected, and only the conductive transport of heat is considered through the cell.
- As a result of these two simplifications, the dynamics of the system are reduced to one dimension [24].
- The external and internal boundaries of the wall holding the PCM are considered to be perfectly conducting, i.e., only the PCM undergoing the phase change is analyzed.
- The habitat interior temperature matches the PCM temperature at $x=L$ and thus, this boundary is adiabatic for the analysis.
- The external boundary is subjected to cycles of illumination and eclipse (denoted by ‘i’ and ‘e’, respectively) with a period $\mathcal{T}$ so that

#### Numerical Model

## 3. Results

#### 3.1. Estimates Based on the Energy Balance

#### 3.2. Effect of the PCM Length and Eclipse Fraction

#### 3.3. Effect of the Thermo-Optical Relationship and Selection of Its Optimal Value

#### 3.4. Determining the Minimum PCM Length

#### 3.5. PCM Selection and Effect of Sinusoidal Solar Flux

## 4. Conclusions and Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of habitat concept and the associated 1-dimensional model. The main design parameters are the PCM length L, the absorptivity $\alpha $ and emissivity $\epsilon $ at the external radiated boundary, and the selected PCM. The solar cycle is modeled using ${\varphi}_{S}$ and either a step or a sinusoidal function that accounts for periods of illumination and eclipse. The colormap illustrates a typical temperature field in the liquid phase of the PCM.

**Figure 2.**Estimate of the optimal thermo-optical relationship, ${(\alpha /\epsilon )}_{\mathrm{opt}}$, as a function of ${\tau}_{i}$ for n-octadecane (black), n-heptadecane (red), and n-hexadecane (blue). The dashed vertical lines indicate values of ${\tau}_{i}$ used in the analysis below.

**Figure 3.**(

**a**,

**b**) Time–evolution of the temperature T at the external $x=0$ (red) and internal $x=L$ (blue curves) boundaries of the PCM for (

**a**) $L=22.5$ mm and (

**b**) 100 mm, and ${\tau}_{e}=33$%. Results are shown for n-octadecane, $\alpha =\epsilon =0.95$, and ${g}_{1}$. The horizontal dashed blue line indicates the mean temperature at the stationary state. Panel (c) illustrates the associated evolution of the liquid fraction, $\mathcal{L}$.

**Figure 4.**(

**a**) Maximum, minimum, and mean (labeled) steady values of T at the interior PCM boundary as a function of L. (

**b**) Mean stationary T at the interior PCM boundary as a function of ${\tau}_{e}$ for $L=100$ mm. Results are shown for n-octadecane.

**Figure 5.**Maximum, minimum and mean (labeled) steady T at the (

**a**) external and (

**b**) internal PCM boundaries as a function of the absorptivity–emissivity ratio, $(\alpha /\epsilon )$. The value of ${T}_{q}$, as derived in Section 2, is included in both panels.

**Figure 6.**Contours showing the time evolution of the temperature field along the PCM length for (

**a**) $(\alpha /\epsilon )=0.46$, (

**b**) 0.48 and (

**c**) 0.5 during 8 periods of the solar cycle. The S/L front position is marked using a solid black line. Results are shown for n-octadecane, $\epsilon =0.66,\phantom{\rule{0.166667em}{0ex}}{\tau}_{e}=25\%,\phantom{\rule{0.166667em}{0ex}}L=100\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ and ${g}_{1}$. Panel (b) corresponds to the optimal value ${(\alpha /\epsilon )}_{\mathrm{opt}}$.

**Figure 7.**Maximum liquid fraction, ${\mathcal{L}}_{\mathrm{max}}$, as a function of L for optimal $(\alpha ,\phantom{\rule{0.166667em}{0ex}}\epsilon )$ combinations. Each set of results fits to $\propto {L}^{-1}$, allowing to extrapolate the minimum length ${L}_{\mathrm{min}}$ for which a maximum liquid fraction of 100% is reached; these values are indicated with vertical lines.

**Figure 8.**Estimate of ${(\alpha /\epsilon )}_{\mathrm{opt}}$ as a function of ${\tau}_{i}$ for n-octadecane (black), n-heptadecane (red), and n-hexadecane (blue); see Equation (19). Results are calculated for the mean solar flux of ${g}_{2}$: ${\overline{\varphi}}_{S}=(2/\pi )\phantom{\rule{0.166667em}{0ex}}{\varphi}_{S}$.

**Table 1.**Thermo-physical properties of n-octadecane (nC18), n-heptadecane (nC17) and n-hexadecane (nC16); reproduced from Refs. [12,24]. The subscripts $\u2018l\u2019,\u2018s\u2019$ refer to the liquid and solid phases of the PCM, respectively. The value ${(\alpha /\epsilon )}_{\mathrm{min}}$, as derived in Section 2, is included for its relevance in the analysis.

Thermo-Physical Property | nC18 | nC17 | nC16 |
---|---|---|---|

Melting temperature, ${T}_{M}$ (${}^{\circ}$C) | 28.0 | 21.4 | 18.0 |

Liquid density, ${\rho}_{l}$ (kg m${}^{-3}$) | 780 | 772 | 765 |

Solid density, ${\rho}_{s}$ (kg m${}^{-3}$) | 865 | 772 | 765 |

Latent heat, ${c}_{L}$ (kJ kg${}^{-1}$) | 243 | 165 | 237 |

Liquid specific heat capacity, ${c}_{p\phantom{\rule{0.166667em}{0ex}}l}$ (J kg${}^{-1}$ K${}^{-1}$) | 2196 | 2300 | 2220 |

Solid specific heat capacity, ${c}_{p\phantom{\rule{0.166667em}{0ex}}s}$ (J kg${}^{-1}$ K${}^{-1}$) | 1934 | 1840 | 1950 |

Liquid thermal conductivity, ${k}_{l}$ (W m${}^{-1}$ K${}^{-1}$) | 0.148 | 0.146 | 0.145 |

Solid thermal conductivity, ${k}_{s}$ (W m${}^{-1}$ K${}^{-1}$) | 0.358 | 0.200 | 0.330 |

${(\alpha /\epsilon )}_{\mathrm{min}}$ [–] | 0.341 | 0.312 | 0.298 |

**Table 2.**Mesh convergence test for numerical simulations based on the average error E in the liquid fraction $\mathcal{L}$, as defined by Equation (15), and computational cost, measured by the simulation time. Results are shown for n-octadecane, $\epsilon =0.66,\phantom{\rule{0.166667em}{0ex}}(\alpha /\epsilon )=0.48,\phantom{\rule{0.166667em}{0ex}}{\tau}_{e}=25\phantom{\rule{0.166667em}{0ex}}\%,\phantom{\rule{0.166667em}{0ex}}L=100$ mm and ${g}_{1}$. The selected combination of parameters are marked in bold.

Parameters | #0 | #1 | #2 | #3 | #4 |
---|---|---|---|---|---|

$\Delta t$ (s) | $\mathbf{1}/\mathbf{2}$ | 1/2 | 5 | 1/2 | 1/4 |

$\mathcal{S}$ (mm) | $\mathbf{1}/\mathbf{3}$ | 2 | 1/3 | 1/6 | 1/6 |

E (%) | $\mathbf{3}.\mathbf{47}\times {\mathbf{10}}^{-\mathbf{4}}$ | $1.96\times {10}^{-1}$ | $2.29\times {10}^{-2}$ | $3.60\times {10}^{-3}$ | — |

Sim. time (min) | 137 | 94 | 14 | 213 | 275 |

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**MDPI and ACS Style**

Borshchak Kachalov, A.; Salgado Sánchez, P.; Martínez, U.; Ezquerro, J.M.
Preliminary Design of a Space Habitat Thermally Controlled Using Phase Change Materials. *Thermo* **2023**, *3*, 232-247.
https://doi.org/10.3390/thermo3020014

**AMA Style**

Borshchak Kachalov A, Salgado Sánchez P, Martínez U, Ezquerro JM.
Preliminary Design of a Space Habitat Thermally Controlled Using Phase Change Materials. *Thermo*. 2023; 3(2):232-247.
https://doi.org/10.3390/thermo3020014

**Chicago/Turabian Style**

Borshchak Kachalov, A., P. Salgado Sánchez, U. Martínez, and J. M. Ezquerro.
2023. "Preliminary Design of a Space Habitat Thermally Controlled Using Phase Change Materials" *Thermo* 3, no. 2: 232-247.
https://doi.org/10.3390/thermo3020014