# 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

- Bejan, A. Convection Heat Transfer; John Wiley & Sons: Hoboken, NJ, USA, 1995; p. 623. [Google Scholar]
- Bejan, A.; Kraus, A. Heat Transfer Handbook; John Wiley & Sons: Hoboken, NJ, USA, 2003; Available online: https://books.google.com/books?hl=es&lr=&id=d4cgNG_IUq8C&oi=fnd&pg=PP11&ots=28zTda2t1B&sig=6yXJBvtqgJxIlD72No6ehnMMHYw (accessed on 2 January 2022).
- Jordan, J.L.; Casem, D.T.; Bradley, J.M.; Dwivedi, A.K.; Brown, E.N.; Jordan, C.W. Mechanical Properties of Low Density Polyethylene. J. Dyn. Behav. Mater.
**2016**, 2, 411–420. [Google Scholar] [CrossRef] [Green Version] - Zenkour, A.M.; Abbas, I.A. A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties. Int. J. Mech. Sci.
**2014**, 84, 54–60. [Google Scholar] [CrossRef] - Ding, S.; Wu, C.P. Optimization of material composition to minimize the thermal stresses induced in FGM plates with temperature-dependent material properties. Int. J. Mech. Mater. Des.
**2018**, 14, 527–549. [Google Scholar] [CrossRef] - Haopeng, S.; Kunkun, X.; Cunfa, G. Temperature, thermal flux and thermal stress distribution around an elliptic cavity with temperature-dependent material properties. Int. J. Solids Struct.
**2021**, 216, 136–144. [Google Scholar] [CrossRef] - Demirbas, M.D. Thermal stress analysis of functionally graded plates with temperature-dependent material properties using theory of elasticity. Compos. B Eng.
**2017**, 131, 100–124. [Google Scholar] [CrossRef] - Noda, N. Thermal stresses in materials with temperature-dependent properties. Appl. Mech. Rev.
**1991**, 44, 383–397. [Google Scholar] [CrossRef] - del Cerro Velázquez, F.; Gómez-Lopera, S.A.; Alhama, F. A powerful and versatile educational software to simulate transient heat transfer processes in simple fins. Comput. Appl. Eng. Educ.
**2008**, 16, 72–82. [Google Scholar] [CrossRef] - Mark, J.E. Physical Properties of Polymers Handbook, 2nd ed.; Springer Science+Business Media, LLC: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Ozawa, S.; Morohoshi, K.; Hibiya, T. Influence of oxygen partial pressure on surface tension of molten type 304 and 316 stainless steels measured by oscillating droplet method using electromagnetic levitation. ISIJ Int.
**2014**, 54, 2097–2103. [Google Scholar] [CrossRef] [Green Version] - Fukuyama, H.; Higashi, H.; Yamano, H. Thermophysical Properties of Molten Stainless Steel Containing 5 mass % B4C. Nucl. Technol.
**2019**, 205, 1154–1163. [Google Scholar] [CrossRef] [Green Version] - Pichler, P.; Leitner, T.; Kaschnitz, E.; Rattenberger, J.; Pottlacher, G. Surface Tension and Thermal Conductivity of NIST SRM 1155a (AISI 316L Stainless Steel). Int. J. Thermophys.
**2022**, 43, 66. [Google Scholar] [CrossRef] - Plebanski, T. Recommended Reference Materials for Realization of Physicochemical Properties. Pure Appl. Chem.
**2007**, 52, 2392–2404. [Google Scholar] [CrossRef] [Green Version] - Daubert, T.E.; Danner, R.P. Physical and thermodynamic properties of pure chemicals: Data compilation. Choice Rev. Online
**1990**, 27, 3319. [Google Scholar] [CrossRef] - Coker, A.K. Ludwig’s Applied Process Design for Chemical and Petrochemical Plants, 4th ed.; Gulf Professional Publishing: Houston, TX, USA, 2010; Volume 2. [Google Scholar] [CrossRef]
- Alhama, F.; Campo, A. Electric network representation of the unsteady cooling of a lumped body by nonlinear heat transfer modes. J. Heat Transfer.
**2002**, 124, 988–991. [Google Scholar] [CrossRef] - Alhama, F.; González-Fernández, C.F. Network simulation method for solving phase-change heat transfer problems with variable thermal properties. Heat Mass Transfer/Waerme- und Stoffuebertragung
**2002**, 38, 327–335. [Google Scholar] [CrossRef] - Sánchez-Pérez, J.F.; Marín, F.; Morales, J.L.; Cánovas, M.; Alhama, F. Modeling and simulation of different and representative engineering problems using network simulation method. PLoS ONE
**2018**, 13, e0193828. [Google Scholar] [CrossRef] [Green Version] - Perez, J.F.S.; Conesa, M.; Alhama, I. Solving ordinary differential equations by electrical analogy: A multidisciplinary teaching tool. Eur. J. Phys.
**2016**, 37, 065703. [Google Scholar] [CrossRef] - Sánchez-Pérez, J.F.; Alhama, I. Universal curves for the solution of chlorides penetration in reinforced concrete, water-saturated structures with bound chloride. Commun. Nonlinear. Sci. Numer. Simul.
**2020**, 84, 105201. [Google Scholar] [CrossRef] - Sánchez-Pérez, J.F.; Alhama, F.; Moreno, J.A.; Cánovas, M. Study of main parameters affecting pitting corrosion in a basic medium using the network method. Results Phys.
**2019**, 12, 1015–1025. [Google Scholar] [CrossRef] - García-Ros, G.; Alhama, I.; Morales, J.L. Numerical simulation of nonlinear consolidation problems by models based on the network method. Appl. Math. Model
**2019**, 69, 604–620. [Google Scholar] [CrossRef] - Morales, N.G.; Sánchez-Pérez, J.F.; Nicolás, J.A.M.; Killinger, A. Modelling of alumina splat solidification on preheated steel substrate using the network simulation method. Mathematics
**2020**, 8, 1568. [Google Scholar] [CrossRef] - Baranovskii, E.S. Optimal Boundary Control of the Boussinesq Approximation for Polymeric Fluids. J. Optim. Theory Appl.
**2021**, 189, 623–645. [Google Scholar] [CrossRef] - Nigri, M.R.; Pedrosa-Filho, J.J.; Gama, R.M.S. An exact solution for the heat transfer process in infinite cylindrical fins with any temperature-dependent thermal conductivity. Therm. Sci. Eng. Prog.
**2022**, 32, 101333. [Google Scholar] [CrossRef] - Guart, A.; Bono-Blay, F.; Borrell, A.; Lacorte, S. Effect of bottling and storage on the migration of plastic constituents in Spanish bottled waters. Food Chem.
**2014**, 156, 73–80. [Google Scholar] [CrossRef] [PubMed] - Loyo-Rosales, J.E.; Rosales-Rivera, G.C.; Lynch, A.M.; Rice, C.P.; Torrents, A. Migration of Nonylphenol from Plastic Containers to Water and a Milk Surrogate. J. Agric. Food Chem.
**2004**, 52, 2016–2020. [Google Scholar] [CrossRef] - Hao, L.; Xu, F.; Chen, Q.; Wei, M.; Chen, L.; Min, Y. A thermal-electrical analogy transient model of district heating pipelines for integrated analysis of thermal and power systems. Appl. Therm. Eng.
**2018**, 139, 213–221. [Google Scholar] [CrossRef] - Kreith, F.; Manglik, R.M.; Bohn, M.S. Principles of Heat Transfer, SI Edition, 7th ed.; West Publishing Company: Eagan, MN, USA, 1999; Volume 2. [Google Scholar]
- Zheng, W.; Zhu, J.; Luo, Q. Distributed Dispatch of Integrated Electricity-Heat Systems with Variable Mass Flow. IEEE Trans Smart Grid
**2022**, 1. [Google Scholar] [CrossRef] - Holger, V.; Atkinson, G.; Nenzi, P.; Warning, D. Software “NgSpice”. Available online: https://ngspice.sourceforge.io/index.html (accessed on 2 January 2022).
- Nagel, L.W. SPICE2: A Computer Program to Simulate Semiconductor Circuits; University of California: Berkeley, CA, USA, 1975. [Google Scholar]
- Gear, C.W. The automatic integration of ordinary differential equations. Commun. ACM
**1971**, 14, 176–179. [Google Scholar] [CrossRef] - Nagel, L.W.; Pederson, D.O. SPICE (Simulation Program with Integrated Circuit Emphasis); EECS Department: Berkeley, CA, USA, 1973. [Google Scholar]
- Pérez, J.F.S.; Conesa, M.; Alhama, I.; Alhama, F.; Cánovas, M. Searching fundamental information in ordinary differential equations. Nondimensionalization technique. PLoS ONE
**2017**, 12, e0185477. [Google Scholar] [CrossRef] [Green Version] - Sánchez-Pérez, J.F.; Conesa, M.; Alhama, I.; Cánovas, M. Study of Lotka–Volterra Biological or Chemical Oscillator Problem Using the Normalization Technique: Prediction of Time and Concentrations. Mathematics
**2020**, 8, 1324. [Google Scholar] [CrossRef] - Conesa, M.; Pérez, J.F.S.; Alhama, I.; Alhama, F. On the nondimensionalization of coupled, non-linear ordinary differential equations. Nonlinear. Dyn.
**2016**, 84, 91–105. [Google Scholar] [CrossRef] - Madrid, C.; Alhama, F. Análisis Dimensional Discriminado en Mecánica de Fluidos y Transferencia de Calor; Editorial Reverté: Barcelona, Spain, 2012. [Google Scholar]
- Valencia, J.J.; Quested, P.N. Thermophysical Properties. ASM Handb. Cast.
**2008**, 15, 468–481. [Google Scholar] [CrossRef] - Mark, J.E. Physical Properties of Polymers Handbook. In Physical Properties of Polymers Handbook; Springer: New York, NY, USA, 2007. [Google Scholar] [CrossRef] [Green Version]
- Jagga, S.; Vanapalli, S. Cool-down time of a polypropylene vial quenched in liquid nitrogen. Int. Commun. Heat Mass. Transf.
**2020**, 118, 104821. [Google Scholar] [CrossRef] - Kalaprasad, G.; Pradeep, P.; Mathew, G.; Pavithran, C.; Thomas, S. Thermal conductivity and thermal diffusivity analyses of low-density polyethylene composites reinforced with sisal, glass and intimately mixed sisal/glass fibres. Compos. Sci. Technol.
**2000**, 60, 2967–2977. [Google Scholar] [CrossRef] - Toyo’oka, T.; Oshige, Y. Determination of alkylphenols in mineral water contained in PET bottles by liquid chromatography with coulometric detection. Anal. Sci.
**2000**, 16, 1070–1076. [Google Scholar] [CrossRef] [Green Version]

**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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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