An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

Hsu, Heng-Pin; Tu, Te-Wen; Chang, Jer-Rong

doi:10.3390/axioms12050416

Cited by 3 publications

(11 citation statements)

References 30 publications

(47 reference statements)

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“…The temperatures at the four corners of the rectangular region can be functions of time. (2) The correctness of the solution in this study is verified by comparing the solutions of some cases using the proposed method with those of Young et al [8], the previous work [28], and Siddique [20]. To the best of the authors' knowledge, the other cases in this paper have never been presented in past studies.…”

Section: Introductionsupporting

confidence: 58%

“…The previous study [28] presented a method to derive an analytic solution to a 2D heat conduction problem with general Dirichlet boundary conditions. However, the heat generation was not considered in the model.…”

Section: The Solution Methodsmentioning

confidence: 99%

“…Zhou et al [27] proposed a polygonal boundary element method to solve the nonlinear heat conduction problems with space-dependent heat source and temperature-dependent thermal properties. The authors [28] derived a closed-form solution to the 2D heat conduction problem with the general Dirichlet boundary condition using the shifting function method with the eigenfunction expansion theorem. However, the temperature values of the boundary conditions and initial condition at the four corners of the rectangular region were restricted to zero, which limited the applicability of this study.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, a closed solution to the transient heat conduction problems in a rectangular cross-section of an infinite bar with nonhomogeneous space-time-dependent Dirichlet boundary conditions and heat sources was developed. This study addresses the limitations of the previous study [28]; that is, the values of the boundary conditions and initial condi-tions at the four corners of the rectangular area do not need to be limited to zero and become functions of time. The space-time-dependent heat sources are also considered in this study.…”

Section: Introductionmentioning

confidence: 99%

“…(1) The analytic solution to 2D heat conduction problems with the general Dirichlet boundary conditions using the shifting function method with the expansion theorem method was proposed in our previous study [28]. However, there were two restrictions, the temperature values at the four corners of the rectangular area should be zero, and the heat source was also set to zero.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

Hsu

Chang

Weng

et al. 2023

Axioms

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This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.

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Section: Introductionsupporting

confidence: 58%

Section: The Solution Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

Hsu

Chang

Weng

et al. 2023

Axioms

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The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

Eso,

Napirah,

Usman

et al. 2024

J. Phys.: Conf. Ser.

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In this paper, the two-dimensional conduction heat transfer equation on a square plate is analyzed using a finite difference method. We have developed both the forward time-centered space (FTCS) and Crank-Nicolson (CN) finite difference schemes for the two-dimensional heat equation, employing Taylor series. Subsequently, these schemes were employed to solve the governing equations. The primary objective of this study is to compare the efficiency of the two methods in solving the conduction heat transfer equation. This was accomplished by implementing Spreadsheet Excel instructions. The results are presented, highlighting a comparison between the exact and approximate solutions. Furthermore, to demonstrate the convergence of the numerical schemes, we estimated the error between the actual and approximate solutions for a specific numerical problem and presented the results graphically. The data utilized in this research included the thermal conductivity of the medium of the square plate concerning width, grid, and compliance with initial and boundary conditions. The findings indicate that the Crank-Nicolson method is more accurate than the forward time-centered space method, as it approaches the exact solution more effectively. Furthermore, this study confirms that the solution and simulation of the heat transfer equation on a square plate can be accurately performed using an Excel spreadsheet as well as other numerical software.

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Modeling Thermal Behavior in High-Power Semiconductor Devices Using the Modified Ohm’s Law

Kımuya

2024

Eurasian Journal of Science Engineering and Technology

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This paper addresses the challenge of thermal management in high-power semiconductor devices, where increasing power densities and complex operating environments demand more accurate thermal prediction methods. Traditional approaches often rely on simplified models that do not account for the crucial factor of temperature-dependent resistance variations. This limitation leads to inaccurate device temperature predictions, potentially compromising device reliability. This work proposes a novel approach for thermal management by introducing the first empirical application of a Modified Ohm’s Law. This modified law incorporates an exponential term to account for the non-linear relationship between temperature, current, and resistance. The paper demonstrates through simulations and empirical validation that the Modified Ohm’s Law offers a more accurate representation of thermal behavior compared to the standard version. This translates to more precise predictions of device temperature, especially during periods of rapid temperature changes. The validation process goes beyond simply establishing the Modified Ohm’s Law. It provides valuable insights into the thermal dynamics of the device, allowing for the refinement of simulation parameters used to assess various cooling strategies. These strategies include simulating different heat sink geometries and materials, modifying airflow rates over the device’s surface, and exploring the impact of Thermal Interface Materials (TIMs) between the device and the heat sink. By incorporating these elements, the simulations provide a more comprehensive picture of the device’s thermal behavior under various operating conditions and cooling configurations. Ultimately, this paper not only advances the theoretical understanding of thermal management but also offers practical benefits. Through enabling more accurate thermal predictions, the Modified Ohm’s Law model paves the way for informed decision-making in device design and optimization.

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An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

Cited by 3 publications

References 30 publications

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

Modeling Thermal Behavior in High-Power Semiconductor Devices Using the Modified Ohm’s Law

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