# Mathematical Model to Calculate Heat Transfer in Cylindrical Vessels with Temperature-Dependent Materials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

^{2}/s), k is thermal conductivity (W/mK), c

_{e}is specific heat capacity (J/kg K), ρ is density (kg/m

^{3}), t the time (s), and, finally x, y, z are the spatial coordinates. As previously indicated, the thermal conductivity, density, and specific heat capacity, and therefore the thermal diffusivity, will be expressed as a function of temperature.

^{2}K), A is the heat transfer surface area (m

^{2}), T

_{s}is the temperature of the solid surface (K), and finally, T

_{e}is the environmental temperature (K).

^{2}K

^{4}), respectively.

## 3. Network Model

## 4. Nondimensionalization Technique

_{i}is the initial temperature of the vessel, T

_{e}is room temperature, H is the height, and τ is the time at which the problem reaches a steady state. It should be noted that in the case of the radius, as shown in Figure 1, only the vessel structure without the fluid will be studied, R = r

_{1}− r

_{2}, since it is where the heating of the container with the air is taking place, although the solution obtained will also be valid in the event that the entire cylinder is solid, R = r

_{1}. On the other hand, it should also be indicated that the study is being carried out with symmetry, and therefore, the results obtained will also be valid for the complete cylinder. Finally, the cylinder is considered to be heating since T

_{e}> T

_{i}. In case it is cooling, T

_{i}> T

_{e}, the procedure to be followed would be the same.

_{1}represents the relationship of the temperature change with diffusion phenomena and the monomial π

_{2}the geometric relationship of the dimensions of the cylinder.

_{3}.

_{1}and π

_{3}, and it is interesting that it is only found in one of them, a new monomial can be obtained without this unknown by means of simple mathematical operations between both monomials.

_{4}monomial would have been ${\mathsf{\pi}}_{4}=\frac{\mathrm{h}\mathrm{D}}{\mathrm{k}}$, where D is the cylinder diameter, which is the well-known Nusselt number, Nu, which relates the heat transfer coefficient, h, and the material thermal conductivity, k [39]. This same monomial could have been deduced by nondimensionalization from the equivalence between Equations (6) and (8) on the boundary [26].

_{1}= Ψ (π

_{2,}π

_{4}).

_{1}, r

_{2}and H, will be the same for all materials. The speed at which the temperature increases for each of the vessels will depend on the thermal diffusivity, α, as expected, and on the relationship between the heat transfer coefficient, h, and the material thermal conductivity, k, that is, the Nusselt number. Finally, as previously indicated, the properties of the materials in this study depend on temperature, so this dependence should have been applied to the previous procedure. However, the relationships obtained between the properties of the materials would have been similar, being qualified by their dependence on temperature [21].

## 5. Material Properties and Model Validation

#### 5.1. Material Properties Depending on Temperature

#### 5.2. Model Validation

## 6. Results, Case Studies, and Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Network model of a volume element. (

**a**) Temperature, (

**b**) Density, (

**c**) Thermal conductivity, (

**d**) Specific heat capacity, and (

**e**) Thermal diffusivity.

**Figure 4.**Al 319 relation between material properties and temperature. (

**a**) Density, (

**b**) Thermal conductivity, and (

**c**) Specific heat.

**Figure 5.**Water relation between material properties and temperature. (

**a**) Density, (

**b**) Thermal conductivity, and (

**c**) Specific heat.

**Figure 6.**PET relation between material properties and temperature. (

**a**) Density and (

**b**) Specific heat.

**Figure 7.**PP relation between material properties and temperature. (

**a**) Density, (

**b**) Thermal conductivity, and (

**c**) Specific heat.

**Figure 8.**Comparison between experimental and simulated data for the Al 319 temperature at the bottle centre.

**Figure 9.**Comparison between experimental and simulated data for the PET temperature at the bottle centre.

**Figure 10.**Comparison between experimental and simulated data for the PP temperature at the bottle centre.

**Figure 11.**Distribution of temperatures for each of the materials at one hour and an ambient temperature of 30 °C. (

**a**) Al319, (

**b**) PET, and (

**c**) PP.

**Figure 12.**Distribution of temperatures for each of the materials at one hour and an ambient temperature of 40 °C. (

**a**) Al319, (

**b**) PET, and (

**c**) PP.

**Figure 13.**Distribution of temperatures for each of the materials at one hour and an ambient temperature of 50 °C. (

**a**) Al319, (

**b**) PET, and (

**c**) PP.

Material | Al319 | PET | PP |
---|---|---|---|

Length (cm) | 20.5 | 22.0 | 23.0 |

Radio (cm) | 3.50 | 2.75 | 3.50 |

Thickness (mm) | 3.05 | 1.80 | 2.20 |

Temperature (°C) | |||
---|---|---|---|

Time (Minutes) | Al319 | PET | PP |

0 | 54.1 | 66.7 | 55.7 |

15 | 53.6 | 58.7 | 50.7 |

30 | 53.2 | 52.5 | 46.7 |

45 | 52.5 | 48.0 | 43.3 |

60 | 52.2 | 43.6 | 40.3 |

Temperature at Reference Point (°C) | |||
---|---|---|---|

Room Temperature (°C) | Al319 | PET | PP |

30 | 23.72 | 27.77 | 27.58 |

40 | 27.43 | 35.59 | 35.14 |

50 | 31.13 | 43.48 | 42.69 |

60 | 34.72 | 51.43 | 50.22 |

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**MDPI and ACS Style**

Fernández-Gracía, M.; Sánchez-Pérez, J.F.; del Cerro, F.; Conesa, M. Mathematical Model to Calculate Heat Transfer in Cylindrical Vessels with Temperature-Dependent Materials. *Axioms* **2023**, *12*, 335.
https://doi.org/10.3390/axioms12040335

**AMA Style**

Fernández-Gracía M, Sánchez-Pérez JF, del Cerro F, Conesa M. Mathematical Model to Calculate Heat Transfer in Cylindrical Vessels with Temperature-Dependent Materials. *Axioms*. 2023; 12(4):335.
https://doi.org/10.3390/axioms12040335

**Chicago/Turabian Style**

Fernández-Gracía, Martina, Juan Francisco Sánchez-Pérez, Francisco del Cerro, and Manuel Conesa. 2023. "Mathematical Model to Calculate Heat Transfer in Cylindrical Vessels with Temperature-Dependent Materials" *Axioms* 12, no. 4: 335.
https://doi.org/10.3390/axioms12040335