Fractal and Fractional

19 pages, 1262 KiB

Open AccessArticle

Pore Structure Characterization and Fractal Characteristics of Tight Limestone Based on Low-Temperature Nitrogen Adsorption and Nuclear Magnetic Resonance

by Wei Lin, Xinli Zhao, Mingtao Li and Yan Zhuang

Fractal Fract. 2024, 8(7), 371; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070371 (registering DOI) - 25 Jun 2024

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Abstract

Pore structure characterization and fractal analysis have great significance for understanding and evaluating tight limestone reservoirs. In this work, the pore structure of tight limestone, low-temperature nitrogen adsorption (LTNA), and low-field nuclear magnetic resonance (NMR) are characterized, and the fractal dimension of the [...] Read more.

Pore structure characterization and fractal analysis have great significance for understanding and evaluating tight limestone reservoirs. In this work, the pore structure of tight limestone, low-temperature nitrogen adsorption (LTNA), and low-field nuclear magnetic resonance (NMR) are characterized, and the fractal dimension of the pore structure of tight limestone is discussed based on LTNA and NMR data. The results indicate that the pores of tight limestone have H3 and H4 types, the pore size distribution (PSD) of the H3 type is a wave distribution ranging from 2 to 10 nm, and the PSD of the H4 type is a unimodal distribution ranging from 2 to 10 nm. The transverse relaxation time (T₂) spectrum of tight limestone shows a single peak (DF), double peak (SF), and triple peak (TF), and the ranges for the T₂ spectra for micropores, mesopores, and macropores are 0.1 to 10 ms, 10 to 100 ms, and greater than 100 ms, respectively. The LTNA fractal dimension of tight limestone (D_L) ranges between 2.4446 and 2.7688, with an average of 2.5729, and the NMR fractal dimensions of micropores (D_NMR1), mesopores (D_NMR2), and macropores (D_NMR3) are distributed between 0.3744 and 1.1293, 2.4263 and 2.9395, and 2.6582 and 2.9989, respectively. Moreover, there is a negative correlation between D_L and average pore radius, a positive correlation between D_L and specific surface area, and a positive correlation between D_NMR2 and D_NMR3 and micropore content, while D_NMR2 and D_NMR3are negatively correlated with the content of mesopores and macropores. Full article

(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)

18 pages, 2169 KiB

Open AccessArticle

A Fractal Adsorption Model on Methane in Coal with Temperature Effect Dependence

by Fei Guo, Gaofeng Liu, Zhen Zhang, Runsheng Lv, Baoan Xian, Jia Lin, George Barakos and Ping Chang

Fractal Fract. 2024, 8(7), 370; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070370 (registering DOI) - 25 Jun 2024

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Abstract

The traditional Langmuir equation displays drawback in accurately characterizing the methane adsorption behavior in coal, due to it assuming the uniform surface of coal pores. Additionally, the decay law of gas adsorption capacity with an increasing coal reservoir temperature remains unknown. In this [...] Read more.

The traditional Langmuir equation displays drawback in accurately characterizing the methane adsorption behavior in coal, due to it assuming the uniform surface of coal pores. Additionally, the decay law of gas adsorption capacity with an increasing coal reservoir temperature remains unknown. In this study, the fractal adsorption model is proposed based on the fractal dimension (D_f) of coal pores and the attenuation coefficient (n) of the adsorption capacity. The principles and methods of this fractal adsorption model are deduced and summarized in detail. The results show that the pore structures of the two coal samples exhibit obvious fractal characteristics, with the values of fractal dimensions (D_f) being 2.6279 and 2.93. The values of adsorption capacity attenuation coefficients (n) are estimated as −0.006 and −0.004 by the adsorption experiments with different temperatures. The proposed fractal adsorption model presents a greater theoretical significance and higher accuracy than that of the Langmuir equation. The accuracy of the fractal adsorption model with temperature effect dependence is verified, establishing a prediction method for methane adsorption capacity in deep coal reservoirs. This study can serve as a theoretical foundation for coalbed methane exploration and development, as well as provide valuable insights for unconventional natural gas exploitation. Full article

(This article belongs to the Section Engineering)

33 pages, 742 KiB

Open AccessReview

A Brief Review of Fractional Calculus as a Tool for Applications in Physics: Adsorption Phenomena and Electrical Impedance in Complex Fluids

by Giovanni Barbero, Luiz. R. Evangelista, Rafael S. Zola, Ervin K. Lenzi and Antonio M. Scarfone

Fractal Fract. 2024, 8(7), 369; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070369 (registering DOI) - 25 Jun 2024

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Abstract

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, [...] Read more.

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems. Full article

(This article belongs to the Special Issue Anomalous Diffusion and Relaxations in Liquid Crystals: Fractal and Fractional Behavior)

30 pages, 3304 KiB

Open AccessArticle

Robust Speed Control of Permanent Magnet Synchronous Motor Drive System Using Sliding-Mode Disturbance Observer-Based Variable-Gain Fractional-Order Super-Twisting Sliding-Mode Control

by Ameen Ullah, Jianfei Pan, Safeer Ullah and Zhang Zhang

Fractal Fract. 2024, 8(7), 368; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070368 - 24 Jun 2024

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Abstract

This paper proposes a novel nonlinear speed control method for permanent magnet synchronous motors that enhances their robustness and tracking performance. This technique integrates a sliding-mode disturbance observer and variable-gain fractional-order super-twisting sliding-mode control within a vector-control framework. The proposed control scheme employs [...] Read more.

This paper proposes a novel nonlinear speed control method for permanent magnet synchronous motors that enhances their robustness and tracking performance. This technique integrates a sliding-mode disturbance observer and variable-gain fractional-order super-twisting sliding-mode control within a vector-control framework. The proposed control scheme employs a sliding-mode control method to mitigate chattering and improve dynamics by implementing fractional-order theory with a variable-gain super-twisting sliding manifold design while regulating the speed of the considered motor system. The aforementioned observer is suggested to enhance the control accuracy by estimating and compensating for the lumped disturbances. The proposed methodology demonstrates its superiority over other control schemes such as traditional sliding-mode control, super-twisting sliding-mode control, and the proposed technique. MATLAB/Simulink simulations and real-time implementation validate its performance, showing its potential as a reliable and efficient control approach for the system under study in practical applications. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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15 pages, 333 KiB

Open AccessArticle

Existence of Mild Solutions to Delay Diffusion Equations with Hilfer Fractional Derivative

by Yuhang Jin, Wenchang He, Luyao Wang and Jia Mu

Fractal Fract. 2024, 8(7), 367; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070367 - 23 Jun 2024

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Abstract

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, [...] Read more.

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions. Full article

(This article belongs to the Special Issue Fractional Order Systems with Time Delay: Theory, Stability Analysis and Applications)

19 pages, 320 KiB

Open AccessArticle

A Qualitative Analysis of a Non-Linear Coupled System under Two Types of Fractional Derivatives along with Mixed Boundary Conditions

by Abdelkader Amara, Mohammed El-Hadi Mezabia, Brahim Tellab, Khaled Zennir, Keltoum Bouhali and Loay Alkhalifa

Fractal Fract. 2024, 8(7), 366; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070366 - 22 Jun 2024

Viewed by 200

Abstract

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. [...] Read more.

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications. Full article

(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications)

18 pages, 6627 KiB

Open AccessArticle

The Regulation of Superconducting Magnetic Energy Storages with a Neural-Tuned Fractional Order PID Controller Based on Brain Emotional Learning

by Ashkan Safari, Hoda Sorouri and Arman Oshnoei

Fractal Fract. 2024, 8(7), 365; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8070365 - 21 Jun 2024

Viewed by 273

Abstract

Intelligent control methodologies and artificial intelligence (AI) are essential components for the efficient management of energy storage modern systems, specifically those utilizing superconducting magnetic energy storage (SMES). Through the implementation of AI algorithms, SMES units are able to optimize their operations in real [...] Read more.

Intelligent control methodologies and artificial intelligence (AI) are essential components for the efficient management of energy storage modern systems, specifically those utilizing superconducting magnetic energy storage (SMES). Through the implementation of AI algorithms, SMES units are able to optimize their operations in real time, thereby maximizing energy efficiency. To have a more advanced understanding of this issue, DynamoMan is presented in this paper. For use with SMES systems, DynamoMan, an Artificial Neural Network (ANN)-tuned Fractional Order PID Brain Emotional Learning-Based Intelligent Controller (ANN-FOPID-BELBIC), has been developed. ANN tuning is employed to optimize the key settings of the reward/penalty generator of a BELBIC, which are important for its overall efficacy. Following this, DynamoMan is integrated into the SMES control system and compared to scenarios in which a BELBIC, PID, PI, and P are utilized. The findings indicate that DynamoMan performs considerably better than other models, demonstrating robust and control attributes alongside a considerably reduced period of settling time, especially when incorporated with the power grid. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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Fractal Fract., Volume 8, Issue 7 (July 2024) – 7 articles

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