Heat Conduction Equation Research Articles

Unified model modal expansion method for full field reconstruction and inversion under mechanical and thermal loading

In this paper, a novel method termed UMM-T&S (Unified Model Modal reconstruction and inversion method for Temperature field and Structural field under limited sensors) is proposed. This method aims to enhance high-precision structural performance inversion and reconstruction under mechanical and thermal loading conditions, particularly in the context of structural health monitoring within the aviation and aerospace sectors. The approach begins by deriving the “temperature modal” solution of the heat conduction equation, drawing upon the modal expansion techniques used in multi-degree-of-freedom systems. This is followed by constructing a parameter mapping relationship within the unified model, enabling the solution of both the structural field (encompassing strain and displacement) and the temperature field through a single structural model modal expansion. Furthermore, the paper details a numerical validation using two different temperature fields and a thermal loading experimental verification on a stiffened plate under five distinct working conditions. The proposed UMM-T&S method yields lower errors (ranging between 10%-40%) and more stable performance across various complex working conditions, in comparison to existing methods. The analysis also delves into the sources of errors and their cumulative impacts.In conclusion, the innovative UMM-T&S method offers a swift, precise, and robust solution for the intricate task of reconstructing and inverting structural performance under combined mechanical and thermal stresses. The method's efficacy and potential are substantiated through experimental validation and comparative analysis against an established methodology, underscoring its capability to effectively tackle the challenges faced in structural monitoring under such demanding conditions.

A new method for solving multidimensional heat conduction problems

In this study, we propose an efficient boundary-type method named the half boundary method (HBM) for solving multidimensional heat conduction equations. Unlike conventional numerical methods, the core idea behind the HBM is to concentrate variables along half of the boundary and employ them to represent node variables within the solution domain. This approach streamlines the solving process by dealing with smaller-order matrices, substantially reducing storage requirements. We derived the fundamental equations of the HBM for solving multidimensional heat conduction differential equations. Furthermore, by introducing the Cayley–Hamilton formula, we optimize the program to enhance computational efficiency. Additionally, we explored heat conduction problems in solution domains differing from rectangular regions. Notably, we pioneer the application of the HBM to address three-dimensional (3-D) problems for the first time, ensuring its accuracy through rigorous validation.

Vibrations of skew cylindrical shells made of FG-GPLRCs under rapid surface heating

An analysis is performed in this research to examine the thermally induced vibrations in skew cylindrical shells. It is assumed that shell is made from a composite laminated material where each layer is reinforced with graphene platelets (GPLs). The amount of GPLs in the layers may be different which results in a piecewise functionally graded media. The one-dimensional transient heat conduction equation is established through the thickness of the shell. Heat conduction equation is solved using the finite difference method for each layer and continuity of temperature and heat flux is applied between the layers. Crank Nicolson algorithm is implemented to obtain the temperature profile at each time step and after that thermally induced force and moment are evaluated. The kinematics of the shell are estimated following the first order shear deformation shell theory. Also it should be noted that an oblique coordinates system in addition to orthogonal system is defined which make it easier to apply the boundary conditions. The definition of strain and kinetic energies are constructed and Following the Ritz method whose shape functions are estimated by Chebyshev polynomials, energies are provided in discrete form of the governing equations. Results of this study are provided to explore the effects of thermal and mechanical boundary conditions, geometry of the shell, weight fraction and distribution patterns of GPLs. It is highlighted that thermally induced vibrations indeed exist especially in thin class of shells.

An Inverse Problem for Estimating Spatially and Temporally Dependent Surface Heat Flux with Thermography Techniques

In this study, an inverse conjugate heat transfer problem is examined to estimate temporally and spatially the dependent unknown surface heat flux using thermography techniques with a thermal camera in a three-dimensional domain. Thermography techniques encompass a broad set of methods and procedures used for capturing and analyzing thermal data, while thermal cameras are specific tools used within those techniques to capture thermal images. In the present study, the interface conditions of the plate and air domains are obtained using perfect thermal contact conditions, and therefore we define the problem studied as an inverse conjugate heat transfer problem. Achieving the simultaneous solution of the continuity, Navier–Stokes, and energy equations within the air domain, alongside the heat conduction equation in the plate domain, presents a more intricate challenge compared to conventional inverse heat conduction problems. The validity of our inverse solutions was verified through numerical simulations, considering various inlet air velocities and plate thicknesses. Notably, it was found that due to the singularity of the gradient of the cost function at the final time point, the estimated results near the final time must be discarded, and exact measurements consistently produce accurate boundary heat fluxes under thin-plate conditions, with air velocity exhibiting no significant impact on the estimates. Additionally, an analysis of measurement errors and their influence on the inverse solutions was conducted. The numerical results conclusively demonstrated that the maximum error when estimating heat flux consistently remained below 3% and higher measurement noise resulted in the accuracy of the heat flux estimation decreasing. This underscores the inherent challenges associated with inverse problems and highlights the importance of obtaining accurate measurement data in the problem domain.

Thermocapillary migration of a compound drop in an arbitrary viscous flow

The thermocapillary migration of a concentric compound drop in an arbitrary viscous flow under the consideration of negligible Reynolds number is investigated. The thermocapillary effect refers to the migration of a drop under the influence of a temperature gradient. The thermal and hydrodynamic problems are examined. The thermal field is governed by the heat conduction equation whereas the hydrodynamic fluid velocities are governed by the linearized Navier–Stokes equations. Presence of temperature gradient results in variation of the interfacial tension which is assumed to depend on temperature linearly. Variation of interfacial gradient leads to the coupling of the hydrodynamic problem with the thermal problem through the boundary condition. A complete general solution of Stokes equations is utilized to obtain closed-form expressions for the velocity vector and pressure. The hydrodynamic forces acting on the compound drop are obtained and expressed in terms of Fax́en’s law. Some important asymptotic limiting cases of hydrodynamic drag are also derived. The hydrodynamic drag for cases of uniform flow, shear flow, and heat source with the known ambient flow are derived and it is found that in the case of shear flow, the hydrodynamic drag is contributed only by the thermal component and the shear flow has no effect on it. The obtained results for drag and torque in the limiting cases are in agreement with the existing results in the literature. Furthermore, the migration velocity of the compound drop is obtained by equating the hydrodynamic drag force to zero. The results obtained for migration velocity are explained with the aid of graphs. The migration velocity is found to be a monotonic function of the Marangoni number and the radius of the innermost drop.

Local existence of compressible magnetohydrodynamic equations without initial compatibility conditions

In this paper, we study the initial‐boundary value problem of three‐dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions are established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term and strong nonlinear term . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

Anisotropic Fourier Heat Conduction and phonon Boltzmann transport equation based simulation of time domain thermo-reflectance experiments

The anisotropic Fourier Heat Conduction Equation (FHCE) and the multidimensional phonon Boltzmann Transport Equation (BTE) were solved numerically in cylindrical coordinates and in time domain to simulate a Time Domain Thermo-Reflectance (TDTR) experimental silicon/aluminum substrate/transducer setup. The out-of-phase response of the probe laser was predicted at various beam offset distances for a pump laser pulse frequency of 80 MHz and modulation frequency of 10 MHz and compared against experimental measurements for a silicon substrate. The isotropic FHCE was also solved for comparison. Results show that the isotropic FHCE with bulk thermal conductivity of 145 W/m/K significantly underpredicts the out-of-phase temperature difference, particularly at smaller beam offsets. With an isotropic thermal conductivity of 105 W/m/K, the computed results match experimental data at smaller beam offsets well, but overpredicts the experimental data at larger beam offsets. An almost-perfect match is obtained by using an anisotropic thermal conductivity wherein the radial (in-plane) thermal conductivity is set to 85 W/m/K and the axial (through-plane) conductivity is set to 130 W/m/K. The multidimensional frequency and polarization dependent phonon BTE is next solved. The BTE results for the out-of-phase temperature difference match experimental observations well at small and intermediate beam offsets, but overpredicts the experimental data at larger beam offsets. FHCE results are fitted to the BTE predictions, and the extracted (best fit) thermal conductivity is found to be 110 W/m/K.

Comparing firn temperature profile retrieval based on the firn densification model and microwave data over the Antarctica

Abstract. The firn temperature is a crucial parameter for understanding firn densification processes of the Antarctic Ice Sheet (AIS). Simulations with firn densification models (FDM) can be conceptualized as a function that relies on forcing data, comprising temperature and surface mass balance, together with tuning parameters determined based on measured depth-density profiles from different locations. The simulated firn temperature is obtained in the firn densification models by solving the one-dimensional heat conduction equation. Microwave satellite data on brightness temperature at different frequencies can also provide remote sensing monitoring of firn temperature variations across the AIS (i.e., the L-band up to 1500 meters). The firn temperature can be estimated by the microwave emission model and the regression method, but these two methods need more observations of temperature profiles for correction and validation. Therefore, we compiled a dataset with temperature profiles and temperature observations with depth around 10 meters. In this work, two methods were used to simulate/retrieve firn temperature across the Antarctic ice sheet. One method estimated the temperature profiles by solving the one-dimensional heat conduction equation driven by reanalyses and regional climate models, which are used in the simulation of FDMs. The other one established a relationship between the multi-frequency brightness temperature data from microwave remote sensing satellites and the firn temperature.

Heat transport during drop impact onto a heated wall covered with an electrospun nanofiber mat: The influence of wall superheat, impact velocity, and mat thickness

Nanofiber surface coating is a promising method for the enhancement of heat transfer during spray cooling. In the present work, the drop dynamics as well as local and overall heat transfer during single drop impact onto a heated wall covered with a polyacrylonitrile (PAN) nanofiber mat are investigated to obtain insight into the mechanisms governing the heat transport enhancement. The influence of wall superheat, drop impact velocity, and mat thickness on the hydrodynamics and heat transfer from the heated wall to the fluid is studied. The experiments were conducted inside a temperature-controlled test cell with a pure vapor atmosphere maintained with refrigerant FC−72 (perfluorohexane). The temperature field at the solid–fluid interface was observed with a high-speed infrared camera, and the heat flux field was derived by solving a three-dimensional transient heat conduction equation within the substrate. The presence of the nanofiber mat on the heater surface suppresses the drop receding phase due to the pinning of the contact line at the end of the spreading phase. Two different heat transfer scenarios are observed depending on the wall superheat and drop impact velocity: scenario (I), in which the liquid drop completely penetrated through the porous nanofiber mat and made contact with the heater surface; and scenario (II) in which the vapor produced inside the pores of the nanofiber mat prevented the liquid drop from touching the heater surface. In scenario (I), an up 450% heat flow enhancement during the sessile drop evaporation stage has been observed. In scenario (II), a 80% heat flow enhancement at this stage has been registered.

Experimental‐Assisted Approach to Develop a Numerical Model for Simulating the Reaction Propagation in Reactive Multilayers

One outstanding feature of self‐propagating reactions is their ability to release heat of reaction over both temporal and spatial scales, enabling the sustained progression of the reaction after a local ignition. They propagate in the form of a continuous reaction front through the mixture of the starting materials. Previous research on reactive materials has predominantly focused on unraveling the microstructure property relationships influencing released energy in reacting multilayers. This involved considering coupled differential equations, including the heat conduction equation and Fick's law. In this study, the introduction of a purely thermal numerical macroscale model is made, incorporating two states of material properties that differentiate between the thermal characteristics before and after phase formation. The homogenization of material properties before the phase formation is accomplished through the consideration of directional‐temperature‐dependent thermal conductivity and temperature‐dependent‐specific heat capacity. The energy‐release function is derived using experimental data for the reaction velocity depending on bilayer thickness. This model allows for the exploration of reaction motion and temperature profiles, achieving qualitative conformity with experimental measurements for freestanding foil, and necessitating reasonable computational effort.

Thermal-magnetic coupling inhomogeneous temperature field analysis of cylindrical eddy current brakes

With the increasing requirements for safety and reliability, it is now necessary to analyze the thermal characteristics to the same extent as the electromagnetic design for eddy current brakes(ECB), because the working characteristic will be greatly influenced by temperature. In this paper, the transient thermal-magnetic coupling analysis of permanent magnet cylindrical eddy current brake, which is used as the recoil mechanism in artillery, is presented. First, the structure and working principle of this ECB are described in detail. Second, the analytical electromagnetic model is established by multilayer theory to derive the heat source of this ECB. Thirdly, the thermal-magnetic coupling model is established to calculate the dynamic inhomogeneous temperature field of the ECB. The heat conduction equation is solved in different regions with inhomogeneous internal heat source by Finite Difference method(FDM) in two-dimensional cylindrical coordinate. Finally, the validity of the proposed model is verified by commercial Finite Element software. The thermal characteristic and its effects on braking force is analyzed during working. The maximum value of temperature approximately rises by 16℃ in each firing. The braking force will decline by 8% considering temperature.

FEATURES OF EXCITATION OF LINEAR AND NONLINEAR THERMAL WAVES IN DIELECTRIC FILMS ON A SUBSTRATE UNDER IRRADIATION WITH A HARMONICALLY MODULATED ION BEAM

A mathematical model has been formulated for the problem of generating linear and nonlinear thermal waves in dielectric films attached to a substrate and in the air by means of a harmonically frequency-modulated ion beam. To solve the formulated problem, a system of nonlinear heat conduction equations is used for two layers of the sample and the substrate. The temperature dependence of the thermophysical quantities of both parts of the sample and, as well as the heat transfer coefficient and emissivity degree, is taken in linear form using thermal coefficients. Temperature disturbances are presented as the sum of stationary and oscillatory components, and the oscillatory part as the sum of linear and nonlinear. In turn, the nonlinear part consists of the sum of oscillations at the fundamental and second harmonics. Due to the fact that the temporary change in the heat source due to the absorption of the ion flow has a harmonic form, the time dependence of the temperature fluctuation in the original equations is also presented in a harmonic form. By solving the boundary value problem, an explicit form of expression was obtained for all parts of the oscillatory component of the temperature disturbance. It was discovered that the frequency dependence of the amplitude of the linear component of the excited thermal wave in the irradiated layer, while for the non-irradiated layer and substrate

Fluid-solid-ablation coupled calculation of hypersonic vehicle based on rbf grid deformation method

The ablative phenomenon of the thermal protection system happens when the hypersonic vehicle suffers serious challenges of aerothermal due to high Mach number. To determine whether the thermal protection system is destroyed, the fluid-solid-ablation loosely coupled method for Carbon-Carbon composite materials is proposed to calculate the deformation of ablation and acquire the deformed shape. Navier-Stokes equation, heat conduction equation, and linear ablation model are applied to calculate the fluid field, the solid field, and the ablation receding amount. As one of the most widely applicable dynamic mesh methods, the Radial Basis Functions (RBF) dynamic method is used to get the deformed solid mesh. The ablation of a missile is computed and analyzed. It could be found that the surface temperature of the head increases during flight and the ablation of the surface results in the change of the vehicle’s shape. The ablative layer thickness of the thermal protection system should be greater than the max ablative deformation, reaching 4.4 mm of the missile. The proposed fluid-solid coupling method could effectively predict the ablation deformation of the vehicle and guide the design of hypersonic vehicle thermal protection systems.

New thermal solver for mitigating surface temperature instability in laser-induced heating

An accepted approach to computing laser-induced peak surface temperature is to employ the enthalpy formulation of the transient heat conduction equation [Grigoropoulos et al., Adv. Heat Transfer 28, 75–144 (1996); Sawyer et al., J. Laser Appl. 29, 022212 (2017)]. This approach is generally implemented using an explicit numerical scheme to solve the thermal transport equation. While it offers the advantage of modeling the solid-melt phase transition automatically, the approach results in instability-like behavior in the computed surface temperature. When laser-induced ablation becomes significant, the heating rate in the surface cell becomes unrealistically large. This results in spikes in the computed peak surface temperature due to large errors in calculating the heating rate. In this paper, we present a new approach, which we refer to as the Moving Frame Solver, that employs a moving-coordinate frame of reference, located at the receding evaporating surface. We also use an analytical representation for the phase transition region of the enthalpy-temperature relationship. The Moving Frame Solver combined with an implicit scheme leads to a stable solution without surface temperature, pressure, or velocity spikes. In other words, any instability in these computed parameters due to use of an explicit scheme (such as Dufort–Frankel) has been eliminated. Details of the new thermal solver and example calculations are presented. Numerical experiments suggest that the surface cell size needs to be small, ∼0.1 μm, to obtain a highly accurate solution with a typical metal such as aluminum. Using the Moving Frame Solver with a refined grid near the surface, but coarse elsewhere, enables accurate and stable surface temperature computation.

Development of Heat Conduction Equation using a Heat Propagation Model on ERK Solar Dryer Plates

Development of Heat Conduction Equation using a Heat Propagation Model on ERK Solar Dryer Plates

Оценка возможности использования пакета COMSOL для моделирования теплогидравлических процессов в реакторах ВВЭР

Thermophysical modeling software packages are widely used for modeling various thermohydraulic processes including modeling nuclear reactor fuel element. Verification of such calculations is possible only if we compare the results obtained in different packages (cross-verification) due to the extreme complexity or impossibility to carry out analytical calculations or experiments. It seems relevant to carry out cross-verification of calculations of WWER reactor fuel elements in COMSOL Multiphysics software package using another popular ANSYS software package. Models of fuel elements for the analysis of thermohydraulic processes are developed using thermophysical modeling packages ANSYS Fluent and COMSOL Multiphysics for cross-verification of calculation results in the latter. Modeling of temperature fields is carried out based on the heat conduction equation with an internal heat source. The authors have considered several formulations of problems: with fuel element cladding and without it and the central hole in the fuel. Models of fuel elements of the WWER-1000 reactor have been designed for three formulations of problems with different designs in ANSYS Fluent and COMSOL Multiphysics software packages. Temperature fields have been determined using the finite element method. The results of the calculations of the fuel core maximum temperature are presented and compared for two software packages. The maximum relative deviation of the temperatures obtained is 0,97 %. A comparison of the temperature calculations results obtained using models developed in two packages shows that the relative deviation of the values corresponds to the error of the measuring instruments. It confirms that COMSOL Multiphysics can be successfully used to simulate processes in WWER reactors along with ANSYS Fluent.

Overview of the Current State of Physical-Mathematical Modelling of Solid Fuel Combustion Processes

This study provides a comprehensive introduction to the combustion of solid propellants. We conduct a thorough review of existing physical-mathematical models related to solid fuel combustion processes. Additionally, we explore the heat propagation within the solid phase, examine the physical chemical processes in the k–phase, and investigate the fundamental solution of the heat conduction equation in a multiphase (k–phase) medium. The heat conduction equation plays a crucial role in governing the temperature distribution within a given material, and a profound understanding of its fundamental solution is essential for various engineering and scientific applications. In this research, we expand our analysis to a multiphase scenario, considering different phases with distinct thermal properties coexisting.

Comment on “Numerical study with OpenFOAM on heat conduction problems in heterogeneous media” by K. Zhang, C.-A. Wang, J.-Y. Tan, International Journal of Heat and Mass Transfer, Vol. 124 (2018) 1156–1162

In the paper “Numerical study with OpenFOAM on heat conduction problems in heterogeneous media” by K. Zhang, C.-A. Wang, J.-Y. Tan, International Journal of Heat and Mass Transfer, Vol. 124 (2018) 1156–1162 the authors modify laplacianFoam from open-source CFD package OpenFOAM® to solve heat conduction problems in heterogeneous media by changing the diffusion coefficient from a constant value to a scalar field. Their paper clearly demonstrates a fundamental, overlooked, and common mistake in rearranging the heat conduction equation, which cannot be replaced with the temperature diffusion equation. In this comment, we provide clear evidence through mathematical derivations and numerical examples calculated in OpenFOAM®.

BEM Modeling for Stress Sensitivity of Nonlocal Thermo-Elasto-Plastic Damage Problems

The main objective of this paper is to propose a new boundary element method (BEM) modeling for stress sensitivity of nonlocal thermo-elasto-plastic damage problems. The numerical solution of the heat conduction equation subjected to a non-local condition is described using a boundary element model. The total amount of heat energy contained inside the solid under consideration is specified by the non-local condition. The procedure of solving the heat equation will reveal an unknown control function that governs the temperature on a specific region of the solid’s boundary. The initial stress BEM for structures with strain-softening damage is employed in a boundary element program with iterations in each load increment to develop a plasticity model with yield limit deterioration. To avoid the difficulties associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators. To validate the physical correctness and efficiency of the suggested formulation, a numerical case is solved.

A study on modeling of boiling heat transfer in core debris bed of SFR

In case of a hypothetical severe accident in a Sodium-cooled Fast Reactor (SFR), coolability of the debris bed in the post-accident phase plays a vital role in mitigating the accident and ensuring the structural integrity of the reactor vessel. Few numerical studies are reported in literature, in which the boiling heat transfer in debris bed is expressed as equivalent heat conduction using similarity law between heat conduction and two-phase heat transfer. However, these studies assumed steady state mass conservation for the boiling zone and neglected the gravity force. Hence, a detailed study has been carried out for various particle sizes and porosities of SFR debris to investigate the influence of above considerations. The effect of gravity on debris bed coolability is studied using steady state model of Lipinski, which showed that gravity has a non-negligible effect, for particle size of 0.3 mm and porosity of 0.5. However, the gravitation force was found to have a negligible effect in dryout heat flux estimation for the bottom cooled configuration. A transient numerical model is developed for simulating the boiling phenomena in debris beds and validated with the published experimental results. The assumption of steady state mass conservation is verified by carrying out transient analysis, which indicated early prediction of the dryout inception. For time dependent heat generation case, the unsteady mass conservation predicted higher DHF compared to constant heat generation.