Heat Conduction Equation Research Articles

Heat Transfer Efficiency While Cooling with a Water Spray, Air-Assisted Water Spray and Water Jet Under Boiling and Single-Phase Forced Convection Conditions

The main purpose of this paper was to determine and compare the boundary conditions of heat transfer on the cooled surface of a cylindrical sensor made of Inconel 600 alloy while cooling with a water jet, water spray and air-assisted water spray under high-temperature conditions. The inverse method for the heat conduction equation was used to determine the boundary conditions. Experimental tests were carried out, including temperature measurements at several points inside the cylinder while cooling with all the tested systems from a temperature of 900 °C for three values of water pressure: 0.05 MPa, 0.1 MPa and 0.2 MPa. Temperature measurements were used as the input data to identify the heat transfer boundary conditions. The temperature field of the axially symmetric sensor was determined using the finite element method. The boundary conditions were determined as average values of the heat transfer coefficient and heat flux and local values of the heat transfer coefficient. A comparison of the amount of thermal energy dissipating from the sensor surface as a result of boiling and a forced single-phase convection is also presented in the paper. The highest uniformity of cooling was obtained during air-assisted water spray-cooling.

Two-phase heat conduction problems with Sturm-type boundary conditions

AbstractIn this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Heat Transfer Characteristics of Cryogenically Frozen Kulfi (A Dairy Dessert)

Kulfi was frozen from concentrated milk as well as from reconstituted dry mix with solids content of 60%. The convective heat transfer coefficients during freezing of kulfi by deep freeze and cryogenic methods were determined using one-dimensional transient heat conduction equation. The heat transfer coefficient for cryogenic freezing of kulfi from milk and dry mix concentrates was 210.57 W m-2K-1 and 216.84 W m-2K-1, respectively when compared to 8.33 W m-2K-1 for conventional deep-frozen kulfi. Freezing was achieved in 267 min in deep freeze method, while it reduced to just 185-190 s under cryogenic conditions. The freezing time was predicted using empirical equations, and the prediction accuracies were compared. Cryogenic freezing was nearly 87.66 times faster than deep freezing, and Pham’s model best-predicted the freezing time. Organoleptic evaluation using fuzzy-logic revealed that the order of ranking of quality attributes of kulfi was body and texture (highly important) > melting characteristics (highly important) > flavour and taste (highly important) > colour and appearance (important), and cryogenically frozen kulfi was statistically (p<0.05) superior to conventionally frozen kulfi.

ON THE HEAT CONDUCTION EQUATION AND THE HEAT DISSIPATION FUNCTION IN ANISOTROPIC REACTING FLUID MIXTURES WITH MAGNETIC RELAXATION

In some previous papers, within the framework of the thermody namics of irreversible processes with internal variables, a linear theory for magnetic relaxation phenomena in anisotropic mixtures, consisting of n reacting fluid components, was developed. In particular, assuming that the macroscopic magnetization m can be split in two irreversible parts m = m(0) + m(1) a generalized Snoek equation was derived. In this paper we derive for these reacting anisotropic mixtures the heat conduction equation. We show that the heat dissipation function is due to the chemical reactions, the magnetic relaxation, the electric conduc tion, the viscous, magnetic, temperature fields and the diffusion and the concentrations of the n fluid components. Also, the Snoek and De Groot special cases are studied. The obtained results find applications in nuclear resonance, in biology, in medicine and other fields, where different species of molecules have different magnetic susceptibilities and relaxation times and contribute to the total magnetization.

Crystallization and crystal morphology of polymers: A multiphase-field study

In this paper, we introduce a coarse-grained model of polymer crystallization using a multiphase-field approach. The model combines a multiphase-field method, Nakamura’s kinetic equation, and the equation of heat conduction for studying microstructural evolution of crystallization under isothermal and non-isothermal conditions. The multiphase-field method provides flexibility in adding any number of phases with different properties making the model effective in studying blends or composite materials. We apply our model to systems of neat PA6 and study the impact of initial distribution of crystalline grains and cooling rate on the morphology of the system. The relative crystallinity (conversion) curves show qualitative agreement with experimental data. We also investigate the impact of including carbon fibers on the crystallization and grain morphology. We observe a more homogeneous crystal morphology around fibers. This is associated with the higher initial volume fraction of crystal grains and higher heat conductivity of the fiber (compared to the polymer matrix). Additionally, we observe that the crystalline grains at the fiber surface grow perpendicular to the surface. This indicates that the vertical growth observed in experiments is merely due to geometrical constraints imposed by the fiber surface and neighbouring crystalline regions.

The Heat Transfer in Skin Tissues Under The General Two-Temperature Three-Phase-Lag Model of Heat Conduction with a Comparative Study

In this paper, a new and more general model of heat conduction that depends on the drift velocity due to thermomass motion assumption will be established and will be applied to the skin tissue. Four different heat conduction models will be incorporated into a unified equation of heat conduction: the Pennes, Vernotte-Cattaneo, dual-phase-lag of Tzou, and the general two-temperature three-phase-lag of Youssef. The governing partial differential equations of the general two-temperature three-phase-lag model of bioheat conduction will be implemented and solved directly in the domain of the Laplace transformation. The numerical solutions of the Laplace transform will be calculated by executing the Tzou iteration formula. The ramp-type heat on the surface of the skin tissue will be considered as thermal loading. The conductive and dynamic temperature increment reactions have been studied and discussed with different values of ramp-time heat, characteristic length, and drift velocity parameters. The novelty of this work is to introduce some comparisons of the four under-studied bioheat conduction models and show the differences between them in the figures. The numerical results show that the ramp-time heat, drift velocity, and characteristic length parameters have major impacts on the increment of both dynamical and conductive temperature distributions.

Development of mathematical models of heat conductivity for modern electronic devices with elements containing foreign inclusions

This paper considers the heat conduction process for an isotropic medium containing a foreign half-through inclusion and heated by a locally concentrated heat flow. Linear and non-linear mathematical models for determining the temperature field have been built to establish the temperature regimes for the effective operation of electronic devices. The coefficient of thermal conductivity of a non-uniform structure is represented as a whole, using asymmetric unit functions, which automatically provides the conditions of ideal thermal contact on the surfaces of materials. This results in solving one heat conduction equation with discontinuous and singular coefficients. A linearizing function was introduced to linearize the nonlinear boundary value problem. Analytical-numerical solutions of linear and nonlinear boundary-value problems have been obtained in a closed form. A linear temperature dependence of the coefficient of thermal conductivity of structural materials was chosen for a heat-sensitive medium. As a result, an analytical-numerical solution was derived, which determines the temperature distribution in this medium. On this basis, a numerical experiment was performed, the results of which are graphically displayed and confirm the adequacy of the constructed mathematical models to a real physical process. The materials of the plate and inclusion are silicon and silver. The results for these materials based on the linear and non-linear model differ by 7 %. Their slight difference is explained by the fact that the values of the temperature coefficient of thermal conductivity are small. The models built make it possible to analyze the given environments in terms of their thermal resistance. As a result, it becomes possible to improve it, and protect structures from overheating, which could lead to the failure of individual nodes and their elements and the entire electronic device

Deep thermal state on the southern margin of the Zhangzhou Basin based on the electrical conductivity model

Exploring the internal spatial and thermal structure of the Zhangzhou Basin is of great scientific significance in understanding the properties of the deep heat sources and the heating mechanism of hot springs in this region. This study estimates the temperature distribution within the upper mantle of the Basin's southern margin using the Arrhenius equation and Hashin-Shtrikman bounds based on a two-dimensional crust-mantle electrical resistivity model. We also employ a layered simulation technique to calculate the crustal temperature distribution using a one-dimensional steady-state heat conduction equation, constrained by the upper mantle's top and ground surface temperatures. This approach displays the characteristics of the longitudinal variations and horizontal inhomogeneities in crust-mantle temperature. Additionally, we estimate the heat flow values within the study area. Our findings reveal that: (i) the upper mantle (at depths of 30 - 50 km) exhibits a temperature range of 700 - 1100 °C, with the presence of local Moho and upper mantle uplifts; (ii) the crustal temperature spans from 21 - 900 °C, with a diminishing influence of the upper mantle uplift area on crustal temperature at shallower depths; (iii) the surface heat flow values derived from our simulations range between 87 and 100 mW/m2, averaging at 93.23 mW/m2; (iv) the exploration of dry heat rock in this region is likely to reach a depth of at least 6 km. These results suggest that the genesis of hot springs in the study area is not solely influenced by the heat energy extracted from large-area granitic surrounding rocks during a long transport process, but is also considerably affected by local deep thermal anomalous bodies and deep-large faults.

Electro-thermal dynamics in superconducting generators: An In-Depth investigation of AC losses and cooling techniques for a 10 MW rotor coil system

In the pursuit of optimizing superconducting generator systems, a comprehensive understanding of their electro-thermal behaviors is crucial. This study builds upon previous research on an 8-pole, 10 MW superconducting generator by Inanir et al. (2022, Journal of Superconductivity and Novel Magnetism, 35(11), 3189) and focuses on an in-depth electro-thermal analysis of the rotor coil system. The emphasis is on assessing the AC losses experienced during transitory current changes. By simultaneously solving Ampere's equation and the heat conduction equation under specific boundary conditions, the research provides a detailed insight into the interaction between electrical and thermal dynamics within the rotor coils. The paper highlights the importance of temperature dynamics in various cooling and current conditions. The results demonstrate that achieving an optimal balance in cooling, particularly with the use of liquid nitrogen, is critical for efficient operation. This study not only advances the understanding of superconducting generators but also highlights potential strategies for improving their reliability and efficiency.

Constitutive modeling of thermo-chemical decomposition and thermo-mechanical deformation, coupled with transient heat conduction, in ablative matrix composite

Several thermal protection systems employ sacrificial composite layer that undergoes thermo-chemical decomposition in high-temperature environment. This results in the pyrolysis gas formation (endothermic reaction) that gets trapped inside the voids generated in the ablative matrix phase. These trapped gases apply pore pressure on the structure, along with the mechanical loading, thus significantly influencing the structure failure. A novel thermo-chemical (TC) decomposition and thermo-mechanical (TM) deformation-based coupled multi-physics formulation, applicable to ablative composite systems, is thus presented. A novel shrinkage expression, due to ablative matrix decomposition, is derived. The TC + TM coupled formulation is converted to stress update process, and its results are validated against the available experimental data. The proposed formulation is also converted to boundary value problem employing non-linear finite element framework (NL-FEM). The Jacobian matrices, for one- and two-dimensional cases, are systematically derived, and the proposed NL-FEM formulation is successfully verified against several benchmark problems.The transient heat conduction equation is finally coupled with the proposed TC + TM formulation (one-way coupling) thus enabling the analysis of more realistic situations where the constant heating rate assumption is not valid. The coupled formulation is finally implemented for several test cases and it is demonstrated that, it has a significant influence on pore pressure and porosity evolution (through pore volumetric strain) within the ablative matrix phase.

Numerical search for the effective thermal conductivity of cracked media

Spacecraft parts accumulate damage during operation and defects that are invariably present even in new designs may grow. This leads to changes in the behavior of individual parts of the space vehicle and, consequently, to the risk of fracture. A more accurate assessment of spacecraft safety requires internal defects to be included in the material models under consideration. One of the main hazardous effects on space objects is multiple temperature heating and cooling due to periodic action of solar rays. This paper presents a study of thermal conduction of media containing cracks. It is carried out with the help of a technique developed by the authors to determine the effective thermal conductivity of materials and based on approximate numerical solution of the steady-state thermal conduction problem for a three-dimensional medium with cracks by the boundary element method. This technique allows to obtain the distribution of the temperature field and heat flux density at any point of the body under consideration, as well as to calculate the effective parameters of materials with high accuracy at relatively low calculation time using ordinary personal computers of average power. The basis of the numerical method presented in this paper is the decomposition of the desired solution into a series of some pre-calculated analytical solutions of the heat conduction equations. The dependence of the effective thermal conductivity on the density of thermally insulated cracks was considered. The formula of this dependence is proposed. Verification of the proposed methodology was carried out by comparing the numerical results of a number of problems with the results of other authors.

Analysis of Thermal Isotropy of Parallelepiped Shape of Bronze Samples for Mechanical Engineering Applications

Objective: The need to find out whether the specimens are ‘thermally thin’ or not is the basis for applying the law of conservation of energy or the Fourier differential equation of heat conduction in finding the time dependence of the specimen temperature. The aim of this study is to find out the thermal field distribution in a parallelepiped shaped bronze specimen and to verify the Bio number. Methods: A number of external variables make it difficult to maintain a monotonic change in sample temperature during the ‘heating’ stage. These factors led to the use of the cooling method to study tiny samples with low inertia of the thermal process. We used the cooling method to experimentally verify the isotropy of the thermal field in the sample and to study the kinetics of cooling relative to the sample axes over a wide range of temperatures. Results: It was found out that in natural air cooling the main mechanisms are conductive, convective and radiative heat transfer. The contribution of thermal radiation to the cooling process is noticeable at high temperatures. It was found that the typical cooling time increases with sequential radiative, conductive and convective heat transfer. By extending the temperature study area to, for the first time, we were able to observe each component of the heat transfer process independently. The values of the typical cooling time on all axes correspond to the experimental error bounds. It was found that the temperature value of the sample is independent of the coordinates and depends only on time. In this case, the law of conservation of energy can be used to explain the dependence of body temperature on cooling time. Conclusion: It is shown that the sample under study is ‘thermally thin’, i.e. the thermal field is constant in all directions. In this situation, the thermal balance equation should be applied rather than the Fourier differential equation of heat conduction, since the temperature gradient inside the body is practically zero. The results obtained are of great importance for the study of cooling processes of metallic products in thermal power engineering, heat engineering and thermophysics.

Effect of vibration frequency on tribological characteristics of vibration-assisted micro-nano machining

Experimental findings regarding the influence of vibration frequency on the machining outcomes of vibration-assisted machining (VAM) have been inconsistent. This work aims to address this issue through both experimental and theoretical analyses. The results indicate that the circuit's frequency response is the primary source of these discrepancies. When the vibration frequency is less than half the cutoff frequency, the friction force reaches a minimum while the wear volume peaks. The solution of the heat conduction equation demonstrated that the acoustic softening effect drives the softening mechanism in nanoscale VAM. Molecular dynamics (MD) simulations further elucidate the damage mechanisms during the wear process, showing that high-frequency vibrations contribute to achieving high-precision machining.

Cauchy Type Nonlinear Inverse Problem in a Two-Layer Cylindrical Area

Various types of protective coatings, such as ceramics, are used to increase the efficiency of metal parts that are exposed to high heat loads. Maintaining mechanical properties and acceptable thermal stresses requires temperature control for components such as turbine blades, gun barrels or combustion chambers. This paper presents a new method for controlling thermal parameters by solving a non-stationary Cauchy problem for the heat conduction equation. The solution is presented in a cylindrical two-layer area considering the variation of the thermophysical parameters of the metal and ceramic. The non-linear equation was solved by successive approximation method using differential quotients for the space and time variable. In order to regularize the ill-conditioned inverse problem, a quasi-regularization method was applied by performing an energy balance for the ceramic. Numerical calculations were carried out for different thicknesses of the ceramic layer. The temperature, heat flux density and heat transfer coefficient for the protective layer were calculated. For the same values of the heat transfer coefficient for 0.1 and 0.3 mm thick protective layers, the difference in permissible gas temperature differs by 10.5 °C.

Determination of the Temperature on the Inner Wall of the Piston Machine Body by Solving the Inverse Problem

In the study, the solution of the inverse heat conduction problem for the cross-section of the piston machine body is presented. The internal surface of the body is unevenly thermally loaded. In its upper part, where the heat flux on the wall is the highest, there is a cooling channel. Based on the temperature values at the measurement points inside the body, unknown temperature values on the inner wall surface were determined. Calculations were performed for time-averaged boundary conditions on the side of hot gases (inner surface of the body), the assumed temperature of the fluid, and the heat transfer coefficient in the cooling channel and on the external surface of the piston machine. A new method was used to solve the problem, in which the solution is sought in the form of a linear combination of basic functions that identically fulfill the heat conduction equation. The stability of the inverse problem was achieved using Tikhonov regularization.

Application of inverse heat transfer to fatigue fracture

Cyclic actuation tends to cause self-heating in the material as the structure experiences fatigue, where the movement and coalescence of defects lead to crack formation, propagation, and eventual fracture. This study explores the modeling aspects of the self-heating phenomenon using a one-dimensional inverse heat problem to analyze heat generation and dissipation to quantify the plastic work rates needed for predicting fatigue life. The approach is based on analyzing the surface thermography obtained using an infrared camera and numerically solving an inverse Fourier heat conduction equation. Formulation of the inverse problem via constrained optimization and method of solution for analyzing the fatigue behavior of CS 1018 flat dog bone specimens subjected to fully reversed bending fatigue are presented. The proposed model demonstrates superior accuracy in predicting the location of maximum heat generation and identifying potential fracture zones compared to traditional methods.

An innovative subdivision collocation algorithm for heat conduction equation with non-uniform thermal diffusivity.

This paper presents a subdivision collocation algorithm for numerically solving the heat conduction equation with non-uniform thermal diffusivity, considering both initial and boundary conditions. The algorithm involves transforming the differential form of the heat conduction equation into a system of equations and discretizing the time variable using the finite difference formula. The numerical solution of the system of heat conduction equations is then obtained. The feasibility of the algorithm is verified through theoretical and numerical analyses. Additionally, numerical and graphical representations of the obtained numerical solutions are provided, along with a comparison to existing methods. The results demonstrate that our proposed method outperforms the existing methods in terms of accuracy.

Coupled Thermo-Electric-Elastic Piezoelectric Vibration Energy Harvester With Axial Movement: Modeling, Verification, and Analysis

Abstract In the field of rail transport and aerospace field, vibration energy harvesting is inevitably subjected to coupled excitations, including train wheel–track interaction induced friction heat and forced vibration, periodic thermal radiation, and vibration excitation. This paper investigates a coupled thermo-electric-elastic piezoelectric vibration energy harvester with axial movement under external heat flux and mechanical force load. The coupled forced vibration equation, coupled electric equation, and coupled thermoelastic heat conduction equation are derived and solved by Green's function theory. To analyze the effect of excitations on the response characteristics, the decoupled method is utilized to solve the coupled multi-field equations and obtain the displacement, electric, and temperature distribution closed-form solutions. The displacement coupling effect induced temperature distribution and the thermo-electric coupling effect triggered displacement are respectively decoupled and analyzed. The obtained closed-form temperature distribution and displacement solutions are verified by the finite element method. To further verify the obtained solutions, a numerical method is conducted by decoupling the coupled multi-field equations and comparing them with prior solutions. Additionally, the different height-to-length ratios, axially moving speeds, and external force load are analyzed in detail. The results indicate that the displacement, temperature distribution, and output voltage vary with external conditions due to the coupled multi-field effect. Overall, this work investigates the thermo-electric-elastic coupling effect on the axially moving piezoelectric energy harvesting, which is beneficial to promote theoretical investigations of the coupled multi-field energy harvesting system and accelerate the practical applications in the aerospace field.

Effects of OpenCL-Based Parallelization Methods on Explicit Numerical Methods to Solve the Heat Equation

In recent years, the need for high-performance computing solutions has increased due to the growing complexity of computational tasks. The use of parallel processing techniques has become essential to address this demand. In this study, an Open Computing Language (OpenCL)-based parallelization algorithm is implemented for the Constant Neighbors (CNe) and CNe with Predictor–Corrector (CpC) numerical methods, which are recently developed explicit and stable numerical algorithms to solve the heat conduction equation. The CPU time and error rate performance of these two methods are compared with the sequential implementation and Euler’s explicit method. The results demonstrate that the parallel version’s CPU time remains nearly constant under the examined circumstances, regardless of the number of spatial mesh points. This leads to a remarkable speed advantage over the sequential version for larger data point counts. Furthermore, the impact of the number of timesteps on the crossover point where the parallel version becomes faster than the sequential one is investigated.