Heat Diffusion Equation Research Articles

Inverse problem on determining two kernels in integro-differential equation of heat flow

Inverse problem on determining two kernels in integro-differential equation of heat flow

Frequency-domain probe beam deflection method for measurement of thermal conductivity of materials on micron length scale.

Time-domain thermoreflectance and frequency-domain thermoreflectance (FDTR) have been widely used for non-contact measurement of anisotropic thermal conductivity of materials with high spatial resolution. However, the requirement of a high thermoreflectance coefficient restricts the choice of metal coating and laser wavelength. The accuracy of the measurement is often limited by the high sensitivity to the radii of the laser beams. We describe an alternative frequency-domain pump-probe technique based on probe beam deflection. The beam deflection is primarily caused by thermoelastic deformation of the sample surface, with a magnitude determined by the thermal expansion coefficient of the bulk material to measure. We derive an analytical solution to the coupled elasticity and heat diffusion equations for periodic heating of a multilayer sample with anisotropic elastic constants, thermal conductivity, and thermal expansion coefficients. In most cases, a simplified model can reliably describe the frequency dependence of the beam deflection signal without knowledge of the elastic constants and thermal expansion coefficients of the material. The magnitude of the probe beam deflection signal is larger than the maximum magnitude achievable by thermoreflectance detection of surface temperatures if the thermal expansion coefficient is greater than 5 × 10-6K-1. The uncertainty propagated from laser beam radii is smaller than that in FDTR when using a large beam offset. We find a nearly perfect matching of the measured signal and model prediction, and measure thermal conductivities within 6% of accepted values for materials spanning the range of polymers to gold, 0.1-300 W/(m K).

Guaranteed phase margin performance in closed-loop control of linear irrational systems

Recently stability of linear irrational systems has been formulated as a constraint satisfaction problem in (Malti, Rapaić & Turkulov, 2023) and a guaranteed solution has been obtained using algorithms from interval arithmetics. This paper extends the aforementioned result to compute controller parameters of irrational systems with prescribed phase margins (based on interval arithmetics) which is the main contribution of the paper. It is applied to control temperature of a one-dimensional heat diffusion equation at a given distance subject to a heating thermal flux applied to its boundary.

A symmetry theorem in two-phase heat conductors

<abstract><p>We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumptions that one medium is bounded and the interface is of class $ C^{2, \alpha} $, we show that if the interface is stationary isothermic, then it must be a sphere. The method of moving planes due to Serrin is directly utilized to prove the result.</p></abstract>

Global existence and finite time blow-up of solutions for a class of Dirichlet-to-Neumann operator heat flow equations with critical growth

Global existence and finite time blow-up of solutions for a class of Dirichlet-to-Neumann operator heat flow equations with critical growth

The method for the heat-diffusion equation in analytic number theory

This paper addresses the entire functions via theory of the tempered xi function. The zeros of the entire functions by the Fourier sine and cosine integrals are stud?ied in detail. The method for the heat-diffusion equation is proposed to structure the connection between Fourier analysis and heat-diffusion equation show the open problems in analytic number theory.

Indirect Thermographic Temperature Measurement of a Power-Rectifying Diode Die Based on a Heat Sink Thermogram

This article concerns the indirect thermographic measurement of the junction temperature of a D00-250-10 semiconductor diode. Herein, we show how the temperature of the semiconductor junction was estimated on the basis of the heat sink temperature. We discuss the methodology of selecting the points for thermographic measurement of the heat sink temperature and the diode case. The method of thermographic measurement of the heat sink temperature and the used measurement system are described. The simulation method used to obtain the temperature of the semiconductor diode junction on the basis of the thermographic measurement of the heat sink temperature, as well as the method of determining the emissivity and convection coefficients, is presented. In order to facilitate the understanding of the discussed issues, the construction of the diode and heat sink used, the heat flow equation and the finite element method are described. As a result of the work carried out, the point where the diode casing temperature is closest to the junction temperature was indicated, as well as which fragments of the heat sink should be observed in order to correctly estimate the temperature of the semiconductor junction. The indirect measurement of the semiconductor junction temperature was carried out for different values of the power dissipated in the junction.

Three-dimensional temperature profile in a dome-shaped habitat structure on the moon

This paper presents the methodology to determine the temperature profile at the surface and through the wall thickness of a monolithic hemispherical dome-shaped habitat structure built on the lunar surface. The three-dimensional thermodynamic equation of heat diffusion is used to determine the temperature profiles using the explicit finite difference scheme. The direct solar radiation and lunar albedo are taken as the heat sources on the habitat structure whereas non-blackbody radiation and habitat albedo were taken as heat sinks. In this study, all the applicable modes of heat transfer – radiation in the external wall surface, convection in the internal wall surface (interior habitat will have air for human missions), and conduction in the structural wall body – have been considered. This temperature analysis study also takes into consideration of the self-shadowing effect of the habitat structure itself. The temperatures at the surface and through the wall thickness of the monolithic dome habitat structure were determined for the habitat structure made of sintered lunar regolith simulant, and for the purpose of comparison, regular terrestrial concrete, for three complete lunar day-and-night cycles. While the maximum temperature of around 390 K was observed at the apex of the sintered lunar regolith simulant dome during the lunar noon, the minimum temperature observed was around 213.5 K during the night time on the external surface of the habitat wall. The corresponding results for the lunar habitat structure made of terrestrial concrete were 364 K and 222.5 K, respectively. At the inner surface of the 40 cm thick habitat structure (in contact with interior habitat environment), the temperature fluctuation (difference between maximum and minimum temperatures) can be up to 30 K for each material type. The results from this study should help to determine the structural stresses and deflections caused by such lunar environment temperature fluctuations and the heat regulation for habitability in the future habitat structure on the Moon. Moreover, the methodology presented in this study should be applicable to conduct parametric studies to evaluate the effect of different parameters on the structural habitat temperature results.

A comparison of a commercial hydrodynamics TCAD solver and Fermi kinetics transport convergence for GaN HEMTs

Various simulations of a GaN HEMT are used to study the behaviors of two different energy-transport models: the Fermi kinetics transport model and a hydrodynamics transport model as it is implemented in the device simulator Sentaurus from Synopsys. The electron transport and heat flow equations of the respective solvers are described in detail. The differences in the description of electron flux and the discretization methods are highlighted. Next, the transport models are applied to the same simulated device structure using identical meshes, boundary conditions, and material parameters. Static simulations show the numerical convergence of Fermi kinetics to be consistently quadratic or faster, whereas the hydrodynamic model is often sub-quadratic. Further comparisons of large-signal transient simulations reveal the hydrodynamic model produces certain anomalous electron ensemble behaviors within the transistor structure. The fundamentally different electron dynamics produced by the two models suggest an underlying cause for their different numerical convergence characteristics.

Thermal Bridge Modeling According to Time-Varying Indoor Temperature for Dynamic Building Energy Simulation Using System Identification

It is not easy to dynamically analyze thermal bridges that require multi-dimensional analysis in building energy simulations, which are mostly one-dimensional platforms. To solve this problem, many studies have been conducted and, recently, a study was conducted to model a thermal bridge based on the data obtained by approaching this in a similar way to steady-state analysis, showing high accuracy. This was an early-stage study, which is only applicable when the indoor temperature is constant. By extending the study, a thermal bridge model that can be applied even when the indoor temperature changes over time in building energy simulations is proposed and validated. Since the governing equation, the heat diffusion equation, is linear, the key idea is to create and apply two thermal bridge transfer function models by expressing the heat flow that enters the room as a linear combination of the transfer function for indoor temperature and the transfer function for outdoor temperature. For the proposed thermal bridge model, the NRMSE of the model itself showed a high accuracy of 0.001, and in the verification through annual simulation using the model, the NRMSE showed an accuracy of 0.1.

Thermomagnetic instability and accompanied stress intensity factor jumps in type-II superconducting bulks under various magnetization processes

For type-II superconducting bulks used as trapped-field magnets, the thermomagnetic instability, manifested as flux jumps and temperature spikes, frequently takes place, resulting in a large amount of energy dissipation in a short time and further the crack problem due to electromagnetic and thermal stresses. In this paper, based on the magnetic and heat diffusion equations and fracture theory, we develop a thermal-magnetic-mechanical coupling model to analyze the flux-jump and fracture behaviors in bulk samples of BiSrCaCuO under various magnetization processes. This model has an important advantage that the simulation domain can be restricted to the sample itself, without having to consider the air region around it, and its reliability is verified by the existing experimental and numerical results. The effects of the sample size, the ambient temperature, and the sweep rate, direction, and uniformity of the external magnetic field on the flux jumps, and Mode I and Mode II stress intensity factors are fully analyzed under different cooling conditions. It is found that as ambient temperature or field inclined angle increases or field sweep rate decreases, the first flux-jump field presents a trend of monotonically increasing for zero-field-cooling magnetization but it has an opposite trend for field-cooling magnetization. The flux jump can lead to the jump of temperature, electromagnetic force, and stress intensity factor. In addition, the sensitivity of flux-jump and fracture behaviors to different parameters and the influence of flux jump on the demagnetization behavior under crossed magnetic fields are discussed. We also find the levitation force jumping phenomenon when the bulk sample is magnetized in a nonuniform magnetic field. From the results obtained, we provide some general guidelines on how the system parameters of superconducting bulk magnets could be chosen to improve the thermal-magnetic-mechanical stability.

Optimizing analytical research on the current leads characteristics to the superconducting electromagnets

In the paper is proposed new analytical approach to the optimization of the characteristics of the current leads to the superconducting electromagnets, basing on the analytical solution of the heat diffusion equation in stationary as well as dynamic, one dimensional case. Then temperature distribution in copper current leads to superconducting electromagnets through which flows current has been determined for monolithic lead with thermal intercept. The values of the low temperature losses, i.e. heat leak flowing through the cold end of lead at FEL-s type construction were determined as the function of current, while minimum of this leakage was found for various leads geometries. This new analytical approach allows to follow, in more visible form than advanced numerical calculations, the influence of physical parameters on the current leads operation.

Mesh-free technique for enhancement of the lung CT image

Lung cancer is abnormal cell disease and a leading cause of mortality for both men and women according to Global cancer statistics 2020 (GLOBACON). Pulmonary nodules appear to be one of the first signs of lung cancer which needs to be recognized for early diagnosis. CT images are distorted due to various noises and poor resolution. Image pre-processing plays an important role to get rid of noises and improve image quality. Enhancement of lung CT image facilitates to determine the sign of early stage of lung cancer in terms of tumor size, malignancy, rounded hematoma. A partial differential equation based heat diffusion equation is used in this work for enhancement of lung CT-scan images. Mesh-less based radial basis function technique is used to solve the heat equation. Proposed work employed on randomly selected sample images of two lung dataset named as The Cancer Genome Atlas Lung Adenocarcinoma(TCGA-LUAD) and QIN. The performance of the proposed lung enhancement technique is quantified by various parameters namely, Structure Similarity Index Measure (SSIM), Peak Signal-to-Noise Ratio (PSNR), and Contract Improvement Index (CII). The observational investigations demonstrated that the proposed technique is computationally more efficient as compared to FDM, Median, and CLAHE.

Heat conduction in general relativity

We study the problem of heat conduction in general relativity by using Carter’s variational formulation. We write the creation rates of the entropy and the particle as combinations of the vorticities of temperature and chemical potential. We pay attention to the fact that there are two additional degrees of freedom in choosing the relativistic analog of Cattaneo equation for the parts binormal to the caloric and the number flows. Including the contributions from the binormal parts, we find a new heat-flow equations and discover their dynamical role in thermodynamic systems. The benefit of introducing the binormal parts is that it allows room for a physical ansatz for describing the whole evolution of the thermodynamic system. Taking advantage of this platform, we propose a proper ansatz that deals with the binormal contributions starting from the physical properties of thermal equilibrium systems. We also consider the stability of a thermodynamic system in a flat background. We find that new ‘Klein’ modes exist in addition to the known ones. We also find that the stability requirement is less stringent than those in the literature.

Computational Fluid Dynamics Approach for Oscillating and Interacting Convective Flows

The oscillation and collective behavior of convective flows is studied by a computational fluid dynamics approach. More specifically, the rising dynamics of heated fluid columns is simulated in gravitational field using a simplified 2D geometry. The numerical method uses the FEniCS package for solving the coupled Navier–Stokes and heat-diffusion equations. For the flow of a single heated fluid column, the effect of the inflow yield and the nozzle diameter is studied. In agreement with the experiments, for a constant nozzle diameter the oscillation frequency increases approximately linearly as a function of the the flow rate, while for a constant flow rate the frequency decreases as a power law with the increased nozzle diameter. For the collective behavior of two nearby flows, we find a counter-phase synchronization and a decreasing trend of the common oscillation frequency with the distance between the jets. These results are in agreement with the experiments, and our computational study also suggests that the phenomenon is present on largely different length-scales.

Soliton crystals in optical Kerr microresonators in the presence of thermo-optic effects

The influence of thermo-optic effects on shape profiles of soliton crystals in optical Kerr microresonators is investigated. The study rests on a model that consists of the Lugiato–Lefever equation, coupled to the one-dimensional heat diffusion equation with a source term proportional to the average power of the optical field. Using appropriate variable changes the model equations are transformed into a set of coupled first-order nonlinear ordinary differential equations. These equations are solved numerically with emphasis on the influence of thermo-optic effects on the amplitude and instantaneous frequency of the optical field, as well as on the temperature profile in the microresonator cavity. It is found that thermo-optic effects do not prevent soliton crystals from forming in optical Kerr microresonators, however, a strong thermal detuning will decrease the soliton-crystal amplitude. The model predicts a temperature profile in the microresonator cavity which is insensitive to the specific spatio-temporal profile of the soliton crystal propagating in the microresonator, a feature peculiar to the model.

A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into a continuous partial differential equation (PDE). We show that this continuation can be performed for both linear and nonlinear systems, including multidimensional, space-varying, and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing the evolution of a continuous-state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples, including transport equations, Kuramoto equations, and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original solution to Hilbert’s sixth problem. Finally, we leverage our derivation of Euler equations to solve a control problem multiagent systems, by designing a nonlinear control algorithm for robot formation based on its continuous approximation.

Numerical studies on conjugate convection from discretely heated electronic board

PurposeThis paper deals with numerical studies into combined conduction, convection and radiation from a heated vertical electronic board are provided here.Design/methodology/approachHere three inbuilt heaters with decrease in their heights were placed in the vertical electronic board. With respect to the non-heat portions, two configurations were studied. The first considers the non-heat portions to be adiabatic, while in the second, they are non-adiabatic. The heat that is produced in three heaters is conducted along the board and is dissipated either from the heater portions alone or from the whole board by convection and radiation. Air is considered as working medium, while the equations of heat transfer and flow of fluid are handled without boundary layer approximations. These equations were further solved using finite volume method with Gauss–Seidel iteration method.FindingsResults of various comparative studies were discussed to bring out the relevance of thermal conductivity, modified Richardson number and surface emissivity on different heat transfer and flow results concerning this problem.Originality/valueThe optimum values of surface emissivity, thermal conductivity and modified Richardson number have also been notionally explored.

Errors Incurred in Local Convective Heat Transfer Coefficients Obtained through Transient One-Dimensional Semi-Infinite Conduction Modeling: A Computational Heat Transfer Study

In typical turbulent flow problems, detailed heat transfer coefficient (h) maps obtained through short-duration experiments are based on inverse heat transfer methods that take the wall temperatures measured via liquid crystals or infrared thermography as input, and an error minimization routine is adopted to determine the best value of h that satisfies the wall temperature temporal evolution under a certain change in fluid temperature. A common practice involves modeling the solid as a one-dimensional semi-infinite medium by selecting the solid material that has low thermal conductivity and low thermal diffusivity. However, in certain flow scenarios, the neglection of the lateral heat diffusion may lead to significant errors in the deduced h values. It is imperative to understand the reasons behind large errors that may be incurred by using the 1D heat conduction assumption in order to accurately determine high-resolution h maps for better heat exchanger designs in a wide range of thermal management applications. This paper presents a computational heat transfer study on different jet impingement scenarios to demonstrate the errors incurred in the determination of h when calculated under the assumption of one-dimensional (1-d) heat conduction into a solid. To this end, three different cases are studied: (a) single jet, (b) array jet (theoretical distribution), (c) array jet (experimental distribution), along with three different mainstream temperature evolution profiles representing step change, moderately fast transient and slow transient nature of flow driving the heat transfer in the solid. A known distribution of heat transfer coefficient (“true h”) for each of the three cases is considered, and three-dimensional transient heat diffusion equations were solved to populate temperatures of each node in the solid at every time step. It is found that stagnation zones’ h1d calculations were lower than the “true h” while the low heat transfer zones exhibited significantly higher h1d compared to the “true h”. For the array jet (experimental distribution) case, it was observed that errors can be as high as 10% in certain low heat transfer zones. Different data reduction procedures, configurations, and conditions explored in this study indicate that a suitable balance can be achieved if shorter time durations in transient experiments are used as a reference for tracking in h1d calculations to keep the deviations from the “true h” low.

The regularized free fall II. Homology computation via heat flow

In [BOV20] Barutello, Ortega, and Verzini introduced a non-local functional which regularizes the free fall. This functional has a critical point at infinity and therefore does not satisfy the Palais-Smale condition. In this article we study the $L^2$ gradient flow which gives rise to a non-local heat flow. We construct a rich cascade Morse complex which has one generator in each degree $k\ge 1$. Calculation reveals a rather poor Morse homology having just one generator. In particular, there must be a wealth of solutions of the heat flow equation. These can be interpreted as solutions of the Schr\"odinger equation after a Wick rotation.