Modeling of Thermal Traces Using Fractional Order, a Discrete, Memory-Efficient Model
Abstract
:1. Introduction
- Recalling of some elementary ideas and definitions from fractional calculus.
- The CFE-based method of solution of the FO state equation and its adaptation to solving of a scalar, initial problem.
- The use of the proposed model for modeling of thermal traces on flat wooden surfaces.
- Experimental verification of results.
2. Preliminaries
2.1. Elementary Ideas
2.2. The CFE Approximation
3. The Considered Heat System
4. Main Results
4.1. The Free, Scalar, IO State Equation
4.2. The Discrete, Free, Scalar, FO State Equation Using CFE Approximation
5. Experimental Verification of Results
6. Discussion of Results
- The FO model is more accurate than the IO model in points (1)–(3). In point (4), the IO model is more accurate.
- The sense of use of the FO model, where the initial condition is given as a function, not as a single point is well illustrated by result for point (1). At this point, the use of the initial function allows to correct prediction of the behavior of the temperature in contrast to IO model, where the prediction was incorrect.
- For the IO model, values of time constant are significantly different for each point.
- For the FO model, the values of the parameters are very close for all the tested points of the modeled temperature field. This allows the conclusion that the proposed, simplified model correctly describes the real thermal process under consideration.
7. Final Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Order L | ||
---|---|---|
L = 1 | ||
L = 3 | ||
L = 5 | ||
No. | x | y | ||
---|---|---|---|---|
1 | 50 | 50 | 17,959.12 | 0.0286 |
2 | 165 | 100 | 1721.28 | 0.0114 |
3 | 200 | 125 | 3486.51 | 0.0067 |
4 | 250 | 125 | 6455.36 | 0.0078 |
No. | x | y | a | |||
---|---|---|---|---|---|---|
1 | 50 | 50 | 0.4898 | 24.0387 | 1.1037 | 0.0072 |
2 | 165 | 100 | 0.4859 | 24.1231 | 1.1042 | 0.0102 |
3 | 200 | 125 | 0.4893 | 24.0492 | 1.1044 | 0.0061 |
4 | 250 | 125 | 0.4889 | 24.0538 | 1.1050 | 0.0102 |
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Oprzędkiewicz, K.; Rosół, M.; Mitkowski, W. Modeling of Thermal Traces Using Fractional Order, a Discrete, Memory-Efficient Model. Energies 2022, 15, 2257. https://doi.org/10.3390/en15062257
Oprzędkiewicz K, Rosół M, Mitkowski W. Modeling of Thermal Traces Using Fractional Order, a Discrete, Memory-Efficient Model. Energies. 2022; 15(6):2257. https://doi.org/10.3390/en15062257
Chicago/Turabian StyleOprzędkiewicz, Krzysztof, Maciej Rosół, and Wojciech Mitkowski. 2022. "Modeling of Thermal Traces Using Fractional Order, a Discrete, Memory-Efficient Model" Energies 15, no. 6: 2257. https://doi.org/10.3390/en15062257
APA StyleOprzędkiewicz, K., Rosół, M., & Mitkowski, W. (2022). Modeling of Thermal Traces Using Fractional Order, a Discrete, Memory-Efficient Model. Energies, 15(6), 2257. https://doi.org/10.3390/en15062257