Heat Diffusion Equation Research Articles

Effect of defects on the thermomagnetic instabilities of superconducting radio-frequency cavities with a multilayer structure

Abstract Vortex motion can lead to significant energy dissipation, resulting in hot spots and thermomagnetic instabilities that are detrimental to the application of superconductors. This paper presents a theoretical examination of thermomagnetic instabilities triggered by vortex motion within a Nb3Sn-I-Nb cavity featuring a multilayer structure. The investigation is conducted using Ginzburg–Landau theory in conjunction with the heat diffusion equation. The numerical simulations align well with experimental data from Nb3Sn superconducting cavities. Given that the performance of superconducting radio-frequency (SRF) cavities is highly sensitive to various defects, this study also considers the interaction between vortices and these defects. It reveals the impact of edge cracks on temperature rise and the quality factor. The findings indicate that edge cracks significantly reduce the threshold field for thermomagnetic instability in SRF cavities. The performance of SRF cavities is influenced not only by the RF field amplitude and frequency but also by the length and number of edge cracks. These results offer valuable insights for evaluating the performance of SRF cavities subjected to RF fields.

Implementasi Fast Fourier Transform dalam Penyelesaian Persamaan Difusi Panas Satu Dimensi

The Fast Fourier Transform (FFT) method for solving the 1-D heat diffusion equation offers an efficient approach for resolving partial differential equations (PDEs) with various time steps . FFT is used to transform the 1-D heat diffusion equation into the frequency domain and back to the time domain through inverse FFT. Using mathematical modeling with initial and Dirichlet boundary conditions, the numerical solutions produced by FFT are compared with analytical solutions. The accuracy of the method is validated using MAE and MSE calculated in Matlab. At several time intervals , the obtained MAE and MSE values indicate a good agreement between the numerical and analytical solutions, with very small errors. Numerical stability analysis confirms the reliability of the FFT method across various The variation in time step has a significant impact on the accuracy and stability of the solution. Smaller time steps improve accuracy and stability but require longer computation times. The optimal time step selected in this study is Increasing the number of discretization points also enhances accuracy but implies an increase in computational load and memory usage. The FFT method demonstrates good numerical consistency with increasing

A closed-form solution for thermally induced affine deformation in unbounded domains with a temporally accelerated anomalous thermal conductivity

Abstract One of the successful mathematical tools in the 21st century is the distributed-order fractional derivative, particularly, the bi-fractional diffusion equation of natural form (Chechkin, Gorenflo, and Sokolov, Phys. Rev. E 2002) and the bi-fractional diffusion equation of the modified form (Sokolov, Chechkin, and Klafter, Acta Phys. Pol. B 2004). The present study adopts the bi-fractional diffusion heat equation of the modified form that uses two Riemann-Liouville time-fractional derivatives with different fractional orders and the Riesz space-fractional derivative, and bears the property of variable thermal conductivity with temporal acceleration; i.e., the material thermal conduction transits from a low value at the small values of time ($t\to 0$) to a larger value at the long-time domain ($t\to \infty$). The corresponding fractional thermoelasticity theory, that uses the bi-fractional diffusion heat equation of the modified form, is exclusively considered in this work. For a quasi-static unbounded Hookean domain, exact solutions for the temperature, hydrostatic stress, and the displacement fields are derived in both short-time and long-time domains, in terms of the Fox \textit{H}-function. The exact solutions for the linearized problem are derived using the inversion of problem variables from the Laplace-Fourier space. Adding a zero initial condition on the normal stress of the unbounded space transforms the conventional Cauchy problem to a mixed initial-boundary value problem in order to derive a generalized form of the displacement. Additionally, thermal energy is not conserved in the presence of space fractality. The effect of such an accelerating transition in the thermal conduction on the displacement component is focused on, and it is found that the displacement inherits a similar transition behavior.

Radiative-conductive heat transfer dynamics in dissipative dispersive anisotropic media

Abstract We develop a self-consistent theoretical formalism to model the dynamics of heat transfer in dissipative, dispersive, anisotropic nanoscale media, such as metamaterials. We employ our envelope dyadic Green’s function method to solve Maxwell’s macroscopic equations for the propagation of fluctuating electromagnetic fields in these media. We assume that the photonic radiative heat transfer mechanism in these media is complemented by dynamic phononic mechanisms of heat storage and conduction, accounting for effects of local heat generation. By employing the Poynting theorem and the fluctuation-dissipation theorem, we derive novel closed-form expressions for the radiative heat flux and the coupling term of photonic and phononic subsystems, which contains the heating rate and the radiative heat power contributions. We apply our formalism to the paraxial heat transfer in uniaxial media and present relevant closed-form expressions. By considering a Gaussian transverse temperature profile, we also obtain and solve a system of integro-differential heat diffusion equations to model the paraxial heat transfer in uniaxial reciprocal media. By applying the developed analytical model to radiative-conductive heat tranfer in nanolayered media constructed by layers of silica and germanium, we compute the temperature profiles for the three first orders of expansion and the total temperature profile as well. The results of this research can be of interest in areas of science and technology related to thermophotovoltaics, energy harvesting, radiative cooling, and thermal management at micro- and nanoscale.

Variations of heat equation on the half‐line via the Fokas method

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

Dynamics of Fourier's and Fick's laws on the convectively heated oscillatory sheet under Arrhenius kinetics: The finite-difference technique

The significance of chemical reaction with activation energy and convective boundary conditions on the fluid flow via an oscillatory stretchy surface in the presence of permeable media and radiation is analyzed in this study. This inspection presents Fourier and Fick's laws-based equations for heat, mass transport, and liquid flow through an oscillating stretchy sheet. Understanding these dynamics aids in the optimisation of catalytic reaction settings, where gradients greatly influence reaction rates in concentration and temperature. The governing differential equations of the current study are modelled and changed into their non-dimensional form by employing suitable similarity variables. The finite difference method (FDM) is also used to numerically solve the obtained dimensionless equations. The influence of many factors on the several profiles is portrayed with graphical representations. The outcome of the unsteadiness and porosity parameters on the velocity profile with time coordinate is depicted. The increase in the radiation parameter and Biot number upsurges the thermal profile. The temperature reduces as the unsteadiness parameter and temperature relaxation time parameter grow. The upsurge in the activation energy parameter intensifies the mass transport. The rise in concentration relaxation time parameter diminishes the concentration profile.

Thermal analysis of GaN HEMTs using nongray multi-speed phonon lattice Boltzmann method under Joule heating effect

Gallium Nitride (GaN) high electron mobility transistors (HEMTs) exhibit superior electrical properties for power and radio frequency applications, but performance is compromised by localized Joule heating, increasing channel temperatures. Precise thermal analysis during design is essential for optimizing device architecture and management strategies. Traditional methods like the Fourier heat diffusion equation (HDE) and the phonon lattice Boltzmann method (PLBM) with the D2Q8 scheme inadequately model phonon ballistic transport at high Knudsen numbers. This study introduces a nongray multi-speed PLBM integrated with the drift-diffusion model to analyze electro-thermal processes in GaN HEMTs. Validated through simulations of thermal conductivities in two-dimensional GaN thin films, the approach examines internal temperature rise in GaN HEMTs under different gate voltages, comparing results with HDE and gray BTE models, and emphasizing the need for non-Fourier effects in thermal analysis. It also evaluates the impact of GaN layer thickness on temperature distribution, providing a robust solution for thermal analysis in GaN HEMTs and other field-effect transistors.

Data-driven simulation-assisted-Physics learned AI (DPAI) for heat diffusion in large grain polycrystalline materials

In this paper, we propose Data-driven simulation-assisted Physics-learned Artificial Intelligence (DPAI), a deep-learning algorithm to simulate heat diffusion in large-grain polycrystalline materials. The DPAI model captures the spatio-temporal representation of heat diffusion in the material from input sequences from the training dataset. The training dataset consists of various temperature plots of polycrystalline materials taken from Finite Element (FE) simulations having varying numbers of grains oriented in random directions with a single-point heat source at the center. The arbitrary plane of the 3D microstructure of these materials is represented using 2D Voronoi tessellations. Voronoi configurations are used to model the geometry of the 2D Computer-Aided Design (CAD) model. Each cell of the Voronoi tessellation represents one grain of the microstructure. This CAD model is used as an input to the FE for solving heat diffusion equations. To model the near-realistic material anisotropy and accurately measure temperature differences at cell boundaries, a smaller mesh size is required in FE modeling, which takes considerable solver time. Therefore, the proposed Deep learning model significantly reduces the computational time while maintaining accuracy as compared to conventional numerical techniques. After training, the effectiveness of the trained DPAI model is examined by modeling larger domain problems involving a greater number of grains and varying material properties. The simulation result is qualitatively compared with the experiment. A scaled-up version of the microstructure is represented using Unidirectional Carbon Fiber laminate. The laminate is heated with a point heat source and the temperature plots are captured using Infrared Camera.

Thermal Spreading Resistance of Near Surface Localized Heating-Induced Size Effects in Semiconductors

Abstract Localized heating is encountered in various scenarios, including the operation of transistors, light-emitting diodes, and some thermal spectroscopy techniques. When localized heating occurs on a scale comparable to the mean free path of the dominant energy carriers, additional thermal resistance is observed due to ballistic effects. The main objective of this study is to find a relation between this resistance, problem geometry, and material thermal properties in situations involving localized heating. Models based on the solution of the Fourier heat diffusion equation and the gray phonon Boltzmann Transport Equation are solved simultaneously to calculate the additional thermal resistances that arise from localized heating. Subsequently, the results are analyzed to derive the desired relationship. It is noted that in the context of localized heating resistance, the effects of geometrical variables are non-linear and substantial, particularly when the Knudsen numbers for the boundary and heat source exceed certain thresholds. Specifically, when the Knudsen number for the heat source becomes comparable to 1 localized heating resistance is observed. However, when the Knudsen number based on heat source height and width surpasses 8 and 20, respectively the heat source behaves akin to a point source, no longer significantly affecting the localized heating resistance. At this juncture, the maximum resistance limit is reached.

Numerical simulation on magnetohydrodynamics convection in annulus inclined elliptical enclosure

ABSTRACT The present work numerically demonstrates the uniform magnetohydrodynamics Newtonian, laminar natural convection in elliptical cold enclosure with inner hot circular body considering the influence of its vertical movement on the fluid flow and heat transfer. The gap area between the elliptical enclosure and the inner body had been filled by the Al2O3-water nanofluid in the upper layer, whereas lower layer has been filled by the porous medium that has been saturated by an identical nanofluid. The local thermal equilibrium model had been implemented to model the nanofluid and porous media. Additionally, Darcy–Brinkman model considered in the representation of the porous media. The three governing equations of heat and fluid flow like energy and momentum of fluid, in addition to the continuity equation, had been solved numerically utilising finite element formulation. The parameters under investigation are the Rayleigh number value 10 3 ≤ Ra ≤ 10 6 , Darcy number 10 − 5 ≤ Da ≤ 10 − 1 , and Hartmann number 0 ≤ Ha ≤ 60 . Additionally, two geometrical parameters had been selected, which are the three different locations of inner cylinder (top, middle, bottom), as well as the four different values of the enclosure’s orientation angle γ = 0 ∘ , γ = 90 ∘ , γ = 180 ∘ , γ = 270 ∘ . The results have been presented to reflect the influence of the abovementioned parameters on isotherms and streamlines, besides Nusselt’s number. It has been proved that to improve the heat transfer rate, it is better to locate inner body in bottom region. The Nusselt number increases by 17.44% when it is moved from the top to the bottom. Additionally, the Nusselt number along the elliptical enclosure attached to the nanofluid layer when rotating the elliptical body from γ = 0 ∘ into γ = 180 ∘ leads to lower the Nusselt number by 32.3372%. This result is exactly inverse considering Nusselt number along nanofluid–porous layer which proved that increasing the orientation angle from γ = 0 ∘ to γ = 180 ∘ contributed to enhancing Nusselt number by 24.7009%.

Triple Mixed Integral Transformation and Applications for Initial-Boundary Value Problems

In this paper, we establish a new integral transformation method for solving the initial-boundary value problems of the partial differential equations. The method is a combined expression mainly based on the triple integral transformation of Laplace and Sumudu. We study the basic properties of the triple Laplace-Sumudu transform, and give the triple Laplace-Sumudu transforms of some basic functions. And as applications, we solve some heat flow equations and wave equations with initial-boundary value conditions by using the method. The results show that the triple Laplace-Sumudu transform is more efficient and useful to deal with these problems.

An Efficient Replacement for Schrödinger's Partial Differential Equation

The classical finite difference method for solving time-dependent partial differential equations has become quite tedious and requires the use of off-the-shelf algorithms such as those in MATLAB. The treatment by finite difference method then solution of the n resulting first order algebraic equations is quite difficult since the underlying matrix is singular. We propose an alternative revolutionary statistical technique using Bmatrix chains, which are a product of the Cairo techniques. This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE. He completely neglects partial differential equations as if they never existed. The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.

Ginzburg–Landau simulations of three-terminal operation of a superconducting nanowire cryotron

Abstract Superconducting nanowire cryotrons (nTrons) are expected to be used as interfaces for super-high-performance hybrid devices in which superconductor and semiconductor circuits are combined. However, nTrons are still under development, and diverse analyses of these devices are needed. Accordingly, we have developed a numerical technique to simulate the three-terminal operation of an nTron by using the finite element method to solve the time-dependent Ginzburg–Landau (TDGL) equation and the heat-diffusion equation. Simulations using this technique offer understanding of the dynamics of the order parameter, the thermal behavior, and the characteristics of three-terminal operation, and the TDGL model reproduces qualitatively the results of nTron experiments conducted at the Massachusetts Institute of Technology. In addition, we investigated how some geometric and physical parameters (the design elements) affect the operation characteristics. The TDGL model has far fewer free parameters compared with the lumped-element electrothermal model commonly used for simulating superconducting devices. Furthermore, the TDGL model provides time-dependent visual information about the superconducting state and the normal state, thereby offering insights into the relationship between nTron geometry and three-terminal operation. These simulation results offer a route to nTron optimization and the development of nTron applications.

Simulation of Temperature Field in Steel Billets during Reheating in Pusher-Type Furnace by Meshless Method

Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. Inside each of the billets, the heat diffusion equation is solved on a two-dimensional central slice of the billet. The diffusion equation is solved in a strong form by the Local Radial Basis Function Collocation Method (LRBFCM) with explicit time-stepping. The ray tracing procedure solves the radiation, where the view factors are computed with the Monte Carlo method. The changing number of billets in the furnace at the start and the end of the loading and unloading of the furnace is considered. A sensitivity study on billets’ temperature evolution is performed as a function of a different number of rays used in the Monte Carlo method, different stopping times of the billets in the furnace, and different spacing between the billets. The temperature field simulation is also essential for automatically optimizing the furnace’s productivity, energy consumption, and the billet’s quality. For the first time, the LRBFCM is successfully demonstrated for solving such a complex industrial problem.

Heat Kernel of Networks with Long-Range Interactions

The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of “information” across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with thek-path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization.

Temperature eigenfunction basis for accelerated transverse mode instability simulation.

This work presents a model for the simulation of transverse mode instability (TMI) in rare earth doped optical fiber amplifiers. The model evaluates the internal temperature of a fiber using a superposition of a finite number of thermal eigenmodes. This simplification greatly enhances the speed of calculation with negligible impact on calculation accuracy. This new method is described and quantitatively compared to an older model that uses standard, spatially resolved FDTD to integrate the heat diffusion equation. When tested over a range of spatial and temporal resolutions, this model reduces runtime by a factor of ∼13.9 on average relative to identical simulations using the spatially resolved model.

Study of the photothermal response of a multilayer structure doped with VO2@Au nanoshells

In this paper, we demonstrate a theoretical study of a multiphysics problem to solve for the photothermal response of a one-dimensional multilayer structure containing a layer doped with VO2@Au nanoshells. The VO2@Au nanoshell consists of a gold (Au) shell and a core of the phase change material vanadium dioxide (VO2) where the VO2 core transitions from a semiconductor state to a conductor state at the critical temperature of 68 °C. This behaviour results in thermal induced optical tunability through this reversible phase change of the VO2, due to the temperature dependent optical and thermal properties. The presence of the VO2 core, functioning as an ultra-fast and reversible optical phase-change material, leads to the emergence of photothermal induced bistability. The layer doped with the VO2@Au nanoshell is approximated as an effective medium using the Maxwell-Garnett Theory to enable an analytical solution. In this study, the optical response of the multilayer structure is obtained using the Transfer Matrix Method, while the thermal response for both stationary and transient states is solved using the Green’s Function Method and Kirchhoff’s Transformation. These equations are interconnected through the heat source term in the heat diffusion equations, representing the local heat generation induced by the continuous-wave laser applied to the structure. Our findings indicate that at the wavelengths of 658 nm and 747 nm, there are two distinct photothermal responses arising from the phase change of the VO2 core. At these wavelengths, the absorption of light increases and decreases, respectively, because of the VO2 phase change. This analytical method not only offers a thorough exploration of the fundamentals of induced photothermal responses in multilayer structures but also holds considerable potential for various applications, including solar cells, photothermal therapy, and nanothermal sensors.

Beyond Newton's law of cooling in evaluating magnetic hyperthermia performance: a device-independent procedure.

Accurate knowledge of the heating performance of magnetic nanoparticles (MNPs) under AC magnetic fields is critical for the development of hyperthermia-mediated applications. Usually reported in terms of the specific loss power (SLP) obtained from the temperature variation (ΔT) vs. time (t) curve, such an estimate is subjected to a huge uncertainty. Thus, very different SLP values are reported for the same particles when measured on different equipment/in different laboratories. This lack of control clearly hampers the further development of nanoparticle-mediated heat-triggered technologies. Here, we report a device-independent approach to calculate the SLP value of a suspension of magnetic nanoparticles: the SLP is obtained from the analysis of the peak at the AC magnetic field on/off switch of the ΔT(time) curve. The measurement procedure, which itself constitutes a change of paradigm within the field, is based on the heat diffusion equation, which is still valid when the assumptions of Newton's law of cooling are not applicable, as (i) it corresponds to the ideal scenario in which the temperature profiles of the system during heating and cooling are the same; and (ii) it diminishes the role of coexistence of various heat dissipation channels. Such an approach is supported by theoretical and computational calculations to increase the reliability and reproducibility of SLP determination. Furthermore, the new methodological approach is experimentally confirmed, by magnetic hyperthermia experiments performed using 3 different devices located in 3 different laboratories. Furthermore, the application of this peak analysis method (PAM) to a rapid succession of stimulus on/off switches which results in a zigzag-like ΔT(t), which we term the zigzag protocol, allows evaluation of possible variations of the SLP values with time or temperature.

Application of Implicit One-step Third Derivative Method for the Solution of Temperature Gradient in a Fish Chunk Placed in a Smoking Kiln

This paper described numerical solution of temperature gradient in a fish chunk placed in a smoking kiln, using an implicit one-step third-derivative numerical method which was developed by introducing more derivative properties into the conventional linear multistep method. The heat transferred from the furnace generated by the biomass through the metal plate to the drying chamber by conduction, to the fish chunk in the smoking kiln was evaluated by the Fourier first heat flow equation. Comparison of the numerical result with the exact solution showed that the implicit one–step third- derivative method was accurate and thus recommended for solving Engineering problems resolving in differential equations.

SIMPLIFIED HYDRAULIC CALCULATION OF THE CRYOGENIC PIPELINE OF EXTERNAL FACILITIES FOR FILLING LIQUEFIED NATURAL GAS

Natural gas is a mineral that is the most environmentally friendly energy fossil raw material. In recent decades, proven natural gas reserves have equaled oil reserves in terms of their indicators and, moreover, continue to grow actively. In this regard, the issue of transportation and storage of natural gas becomes relevant. One actively developing area has become the industry of natural gas liquefaction, transportation and distribution.The paper discusses the issues of shipment of liquefied natural gas (LNG) to gas tankers during its large-tonnage production. The authors propose the use of remote filling systems (VSN) for LNG in a similar way to similar devices for filling oil into tankers. However, the main feature of the VSN LNG is the cryogenic pipeline from the onshore facilities to the VSN. In this regard, it is necessary to make a simplified mathematical model of a cryogenic pipeline taking into account heat exchange with the environment and heat release during energy dissipation due to viscous friction forces.In the course of numerical simulation, the dependence of the maximum pumping distance was obtained for a given difference between the temperature of LNG on shore facilities and the maximum required temperature of LNG loading into a gas carrier. The influence of each component of the heat flow equation — heat exchange with the environment and LNG heating due to viscous friction forces was also evaluated.As an example, a section with a length of 10 km was selected and a simulation of the temperature distribution, as well as the required pressure on shore structures necessary to ensure a given loading speed of the gas carrier and the temperature of the supplied LNG was carried out.