Nonlinear Heat Conduction Research Articles

Unified modeling and kinetic analysis of the near-cathode region and hot cathode in atmospheric-pressure arc discharges

The near-cathode region plays a crucial role in exploring the transport characteristics of the transition from arc column to the hot cathode in atmospheric-pressure arc discharges because of the existing non-equilibrium phenomena. A one-dimensional unified model, including the near-cathode region and the cathode body, is developed for an argon arc discharge with the tungsten cathode at atmospheric pressure in this paper. The electrostatic model coupled with an external circuit in the near-cathode region is solved based on the implicit particle-in-cell coupled Monte Carlo collision method without any assumptions of thermal or ionization equilibrium or quasi-neutrality. A detailed description of the arc plasma–cathode and cathode–gas interactions is obtained by calculating the nonlinear heat conduction equation in the cathode. It is shown that the space-charge sheath strongly affects particle transport in the near-cathode region and energy transport from arc plasma to the thermionic cathode. The total current density has significant effects on the kinetic characteristics of arc plasma by feedback-like mechanisms. The Joule heating by the external circuit and charged particles deposited into the cathode are dominating mechanisms of energy transfer from the near-cathode region to the cathode, while energy loss by radiation is more significant compared with natural convection.

Simultaneous Bayesian estimation of the temperature-dependent thermal properties of a metal slab using a three-dimensional transient experimental approach

This study presents an experimental technique to simultaneously estimate the thermal conductivity (k) and the specific heat (cp) as functions of temperature of AISI 304 stainless steel. The thermal model was based on the transient nonlinear heat conduction across a three-dimensional slab. This enabled one better assess the heat diffusion effects, increasing the sensitivity, and addressing a more realistic case study. Bayesian statistics evaluated the agreement between the numerical model and the measurements to infer constant k and cp for intervals of 10 °C. Temperature dependence was determined via a linear fit to these piecewise estimates. The direct problem was solved in COMSOL Multiphysics, obtaining the temperature field from known initial and boundary conditions. D-optimality-based sensitivity analysis was used to determine the best location to collect the measurements so that data from a single thermocouple were sufficient to identify these thermal properties. This sensitive temperature information was provided by a low-cost experiment at room temperature. The inverse estimation quality was shown by comparing the outcomes with data from literature and by assessing the 95% confidence bounds, which had deviations below 3.6% for both properties. The simultaneous estimation accuracy was further investigated using uncertainty analysis, which indicated a value close to 3.5%.

Application of agent-based approach for heat conduction processes simulation

Heat transfer processes are the basis of the most technological processes of the energy sector. Thus, the development of modern approaches for computer simulation and visualization of the phenomena of thermal energy transfer in various objects are of great importance and is relevant. Classical models in the form of differential equations of various types describe processes in continuous space and time. So, it is difficult to apply classical models to study nonlinear phenomena, and processes in inhomogeneous media in the presence of discontinuous solutions at the boundaries. In these cases, simplifying assumptions are used, thus, the adequacy of the models is reduced. It is of great interest to apply fundamentally different approaches to describe transfer processes, which include discrete dynamic models. The purpose of this project is to study the possibilities to apply discrete approaches to simulate nonlinear heat transfer processes under conditions of material inhomogeneity and the presence of volume sources of variable power. The paper studies the possibilities to apply the agent-based approach to simulate models of complex systems. This approach allows us to consider a continuum as a set of interacting elements (agents). The behavior of the elements is completely described by local dependencies. At the same time, the laws of functioning of individual elements are accepted as deterministic and they correspond to the fundamental principles of the theory of heat transfer. The possibility to apply a discrete approach for simulating the process of heat transfer by the molecular mechanism has been studied. The general methodology to develop an agent-based deterministic model is described. Its applicability to describe quasi-linear and nonlinear heat conduction processes is considered. The examples of simulation of combustion processes complicated by exothermic and endothermic effects are considered. The advantages and disadvantages of the proposed method are indicated. The results of the study have shown that discrete agent models are a good alternative to classical continuum approaches to study heat transfer processes in inhomogeneous media. The results obtained do not contradict modern approaches to the description of thermal processes. It has also been found that the simulation algorithms used in the agent-based approach are quite universal and easily adapt to changes under the conditions of problem setting. The analysis of the results makes it possible to recommend a discrete agent-based approach to develop simulation models of complex technological processes and systems.

Impact of variable thermal conductivity and diffusivity on two temperature fractional thermodiffusive elastic half space with dual phase lags

The motive of current research is to investigate the impact of variable conductivity and diffusivity on the elasto-thermo-diffusive interactions in a 2D half-space under the two temperature fractional dual phase lag model. At the bounding surface of half-space, thermal and concentration loadings are applied. Variable diffusivity and conductivity are assumed to be linearly dependent on concentration and thermodynamic temperature, respectively. Kirchhoff's transformations are taken into consideration to convert the non-linear heat conduction and diffusion equations into a linear form. Then, reduced governing equations are solved by employing the transform technique. The solutions of different physical quantities are presented in a compact form. Numerical simulation is carried out for copper material. A numerical inversion process is adopted to invert the solutions in the original domain. Consideration of two temperature, dual phase lag fractional thermoelasticity to study the transient responses of half space with specific type of loadings augment the novelty of the present work. The results for conductive and thermodynamic temperature, normal stress and mass concentration are plotted to illustrate the impact of variable diffusivity and conductivity, fractional and two temperature parameters. These parameters are observed to have varying degrees of influence on the physical fields.

Surface transfer coefficients estimation for heat conduction problem using the Bayesian framework

This work deals with an inverse two-dimensional nonlinear heat conduction problem to determine the top and lateral surface transfer coefficients. For this, the Bayesian framework with the Markov Chain Monte Carlo algorithm is used to determine the posterior distribution of unknown parameters. To handle the computational burden, a lumped one-dimensional model is proposed. The lumped model approximations are considered within the parameter estimation procedure thanks to the Approximation Error Model. The experiments are carried out for several configurations of chamber ventilator speed. Experimental observations are obtained through a complete measurement uncertainty propagation. By solving the inverse problem, accurate probability distributions are determined. Additional investigations are performed to demonstrate the reliability of the lumped model, in terms of accuracy and computational gains.

Estimation of the local convective heat transfer coefficients of low frequency two-phase pulsating impingement jets using the IHCP

ABSTRACT In this paper, a numerical-experimental investigation was performed for estimation of the local convective heat transfer coefficients of a two-phase pulsating heated jet impinging a cold disk-shaped mass. By solving the energy equation, temperature history of thermocouples was obtained and then using the “temperatures measured” at locations of the sensors and the “temperatures calculated” from the numerical solution of heat equation and then minimizing the sum of squared errors by the conjugate gradient method (CGM), the convective heat transfer coefficients are estimated using the nonlinear inverse heat conduction procedure (IHCP). To compare the heat transfer characteristics of the pulsating, steady, bubbly, and liquid-only jets, the heat transfer coefficients were estimated with respect to the square pulse frequency produced by a solenoid valve in the range of and the gas volume fraction in the range of . In the case of single-phase jet, results show that flow pulsation increases Nusselt number up to at the stagnation point and for farther radial distances as compared to the steady jet. Also, for steady jet, increasing Reynolds number resulted in a increase in the stagnation point Nusselt number and a increase at farther radial distances.

A new multi-objective optimization approach for process parameters optimization during numerical simulation of quenching steel parts

The paper presents the numerical simulation of quenching cylindrical steel sample immersed in three different quenchants: water, 5 % aquatensid solution, and isorapid oil. The quenching process starts from the initial temperature of the cylinder at 850 °C and moves through the air until it reaches the quenching bath. The quenchant is held at constant temperature of 40 °C. The cylinder is made of carefully selected steel which does not change its structure during quenching and heating. Cylindrical samples were manufactured in three different dimensions (R, H), (mm): (12.5 × 100), (25 × 150) and (37.5 × 225), so that four measuring points were installed in each sample. Each measuring point consists of thermocouple installed beneath the cylinder surface, capable of measuring the temperature every half second. Based on the experiment, the numerical simulation is recognized as transient and nonlinear two-dimensional heat conduction problem consisting of the two main tasks: direct heat transfer problem (DHTP) and inverse heat transfer problem (IHTP). The paper proposes a new multi-objective optimization approach for the estimation of heat transfer coefficients during the numerical simulation of quenching cylindrical steel sample. The proposed approach gained better results and less convergence time compared to the results from the literature. The paper includes methods, algorithms and the source code for the calculation of the temperature fields in time and heat transfer coefficient estimation of the IHTP. The simulation software has been implemented in C# programming language and can be found at http://github.com/bhrnjica/quenching_simulation.

A Localized Method of Fundamental Solution for Numerical Simulation of Nonlinear Heat Conduction

In this study, an efficient localized method of fundamental solution (LMFS) is applied to nonlinear heat conduction with mixed boundary conditions. Since the thermal conductivity is temperature-dependent, the Kirchhoff transformation is used to transform the nonlinear partial differential equations (PDEs) into Laplace equations with nonlinear boundary conditions. Then the LMFS is applied to the governing equation, and the nonlinear equations are treated by the fictitious time integration method (FTIM). Both 2D and 3D numerical examples are proposed to verify the effectiveness of the LMFS.

Mathematical Simulation of Forest Fuel Pyrolysis and Crown Forest Fire Impact for Forest Fire Danger and Risk Assessment

In order to predict and assess the danger from crown forest fires, it is necessary to study the thermal degradation of different forest fuels in a high-temperature environment. In this paper, the main characteristics of pyrolysis accompanied by moisture evaporation in a foliage sample of angiosperms (birch) were investigated within conditions typical for a crown forest fire. The heat and mass transfer in the forest fuel element is described by the system of non-stationary non-linear heat conduction equations with corresponding initial and boundary conditions. The considered problem is solved within the framework of the three-dimensional statement by the finite difference method. The locally one-dimensional method was used to solve three-dimensional equations for heat conduction. The simple iteration method was applied to solve nonlinear effects caused by the forest fuel pyrolysis and moisture evaporation. The fourth kind of boundary conditions are applicable at the interface between the sub-areas. Software implementation of the mathematical model is performed in the high-level programming language Delphi in the RAD Studio software. The characteristic changes in the sample temperature field and the phase composition under high-temperature exposure from a forest fire are presented. The induction period of the thermal decomposition of dry organic matter in the sample was determined. Recommendations are made about key features of accounting for the pyrolysis and evaporation processes when predicting forest fire danger. The research results can be used in the development and improvement of systems for predicting forest fire danger and environmental consequences of the forest fires.

Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (4th-CASAM-N): II. Application to a Nonlinear Heat Conduction Paradigm Model

This work illustrates the application of the fourth-order comprehensive sensitivity analysis methodology for nonlinear systems (abbreviated as “4th-CASAM-N”), which enables the efficient computation of exactly determined 1st-, 2nd-, 3rd-, and 4th-order functional derivatives of results produced by computational models with respect to the model’s parameters. Results produced by computational models are called model “responses” and the respective functional derivatives are called “sensitivities” (with respect) to model parameters. The qualifier “comprehensive” indicates that the 4th-CASAM-N methodology enables the exact and efficient computation not only of response sensitivities with respect to customary model parameters (including computational input data, correlations, initial and/or boundary conditions) but also with respect to imprecisely known material boundaries, as would be caused by manufacturing tolerances. The 4th-CASAM-N enables the hitherto very difficult, if not intractable, exact computation of all of the 1st-, 2nd-, 3rd-, and 4th-order response sensitivities for large-scale systems involving many parameters, as usually encountered in practice. A paradigm model that describes nonlinear heat conduction through a material has been chosen to illustrate the application of the 4th-CASAM-N methodology, as this model enables the derivation of tractable closed-form analytical expressions of representative 1st-, 2nd-, 3rd-, and 4th-order response sensitivities while largely avoiding side-tracking algebraic manipulations. The responses chosen for this paradigm model include not only physically measurable quantities but also a synthetic response designed to illustrate the enormous possible reduction in the number of computation when using the 4th-CASAM-N (rather than other methods) for computing response sensitivities.

Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (4th-CASAM-N): I. Mathematical Framework

This work presents the fourth-order comprehensive sensitivity analysis methodology for nonlinear systems (abbreviated as “4th-CASAM-N”) for exactly and efficiently computing the first-, second-, third-, and fourth-order functional derivatives (customarily called “sensitivities”) of physical system responses (i.e., “system performance parameters”) to the system’s (or model) parameters. The qualifier “comprehensive” indicates that the 4th-CASAM-N methodology enables the exact and efficient computation not only of response sensitivities with respect to the customary model parameters (including computational input data, correlations, initial and/or boundary conditions) but also with respect to imprecisely known material boundaries, caused by manufacturing tolerances, of the system under consideration. The 4th-CASAM-N methodology presented in this work enables the hitherto very difficult, if not intractable, exact computation of all of the first-, second-, third-, and fourth-order response sensitivities for large-scale systems involving many parameters, as usually encountered in practice. Notably, the implementation of the 4th-CASAM-N requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. The application of the principles underlying the 4th-CASAM-N to an illustrative paradigm nonlinear heat conduction model will be presented in an accompanying work.

On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with degenerate heat conductivity

We consider the one dimensional compressible Navier-Stokes system with constant viscosity and the nonlinear heat conductivity being proportional to a positive power of the temperature which may be degenerate. This problem is imposed on the stress-free boundary condition, which reveals the motion of a viscous heat-conducting perfect polytropic gas with adiabatic ends putting into a vacuum. We prove that the solution of one dimensional compressible Navier-Stokes system with the stress-free boundary condition shares the same large-time behavior as the case of constant heat conductivity.

Glasdi: Parametric Physics-Informed Greedy Latent Space Dynamics Identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on-the-fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, an efficient k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 66 to 4,417x speed-up with 1 to 5% relative errors.

Dynamic thermal buckling of spherical porous shells

The aim of present work is to address nonlinear dynamic thermal buckling of shallow spherical functionally graded porous shells subjected to transient thermal loading using the first order shear deformation theory (FSDT). A power-law distribution as well as cosine-type porosity distribution are used to model the variation of constituents through the shell thickness. Thermomechanical properties are assumed to be temperature dependent. Using Crank–Nicolson time marching scheme, an iterative procedure is employed to solve nonlinear transient heat conduction equation. For thermal boundary conditions, the outer surface of shells is kept at a reference temperature, while the inner surface experiences a sudden temperature rise. Geometrical type of nonlinearity in the sense of von-Karman is taken into account. The highly coupled nonlinear governing equations of motion are extracted by constructing the appropriate weak form and also using multi-term Ritz–Chebyshev method. The resulting ODEs are then reduced to a system of nonlinear algebraic equations by employing the well-known Newmark family of time integration schemes. The latter equations are solved by means of Newton–Raphson iteration procedure. Budiansky criterion is used to recognize critical parameters of dynamic instability of shells due to applied thermal shocks. Some comparison studies are conducted in order to verify the accuracy of results of the present work. Moreover, various parametric studies are performed to assess the influence of involved parameters.

Application of two-sided approximations method to solution of first boundary value problem for one-dimensional nonlinear heat conductivity equation

The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes.

Existence of a weak solution to a nonlinear induction hardening problem with Leblond–Devaux model for a steel workpiece

In the present paper, we investigate a mathematical model for an induction hardening process of a steel workpiece. We describe the electromagnetic process by the eddy current equations, which are coupled with the energy balance in the workpiece via the heat equation and the kinetic phase transition. The nonlinearities of the system are caused by a nonlinear relation between the magnetic induction and the magnetic field, together with a nonlinear heat conduction and Joule heating. We propose a time discretization for the variational system, and perform some a priori estimates for discrete solutions. By aid of Rothe’s method and the theory of monotone operators, we show the convergence of the proposed scheme and the existence of a weak solution to the variational problems.

Thermodynamically consistent generalized hydrodynamic theory of thermal conduction and integral equations of thermal conductivity of simple fluids in electromagnetic fields

We have reviewed thermodynamically consistent evolution equations of fluids in an invited review article in the McGill University bicentennial memorial issue [B. C. Eu, Can. J. Chem. (in press) (2021)]. In this article, we apply the generalized hydrodynamic evolution equations of heat fluxes to derive integral equations for thermal conductivity coefficients of fluids in electromagnetic fields, which generalize the linear response theory formula of Green and Kubo to nonlinear heat conduction phenomena. The constitutive equation for the temperature field derived is compared with the temperature evolution equation obtained by Guo et al. [Physica D 342, 24 (2017)] on the basis of extended irreversible thermodynamics. They are considerably different.

MESHLESS SIMULATION OF HEAT CONDUCTION IN FUNCTIONALLY GRADED MATERIALS SUBJECTED TO TEMPERATURE-DEPENDENT HEAT SOURCES

In this paper, an efficient meshless local B-spline based finite difference (FD) method for analysis of functionally graded materials (FGMs) subjected to temperature-dependent heat sources is presented. The favourable properties of B-spline basis functions in having arbitrary degree for resolution of solution, partition of unity and the Kronecker delta properties are combined with high accuracy and low computational effort of differential quadrature method in approximating shape functions and their derivatives. In this study, the FGMs are assumed to have temperature-dependent materials properties that vary as a function of radial distance. The homogenized properties are evaluated with power-law mixture rule. The nonlinearities from material properties and heat source terms are handled by the predictor-corrector method along with the Crank-Nicolson scheme for time integration. Case of nonlinear 2D heat conduction in FGM due to temperature dependentheat sources is examined. The method is shown to be accurate and efficient for complex thermal analysis of FGMs taking into account temperature dependency of material properties and heat sources.

A general finite element method: Extension of variational analysis for nonlinear heat conduction with temperature-dependent properties and boundary conditions, and its implementation as local refinement

In simulation of heat conduction with temperature-independent physical properties and boundary conditions (BCs), Galerkin residual analysis and variational analysis yield equivalent finite element method (FEM), the conventional FEM. However, if the properties and BCs are temperature-dependent, it is discovered that their derivatives further induce nonlinearity of FEM which consequently generates divergence between the two analyses. A general FEM, extension of variational analysis, is derived as general form of conventional FEM modeling nonlinear heat conduction. Numerical examples demonstrate that the general FEM produces results with considerably higher accuracy and stability and also possesses higher performances on conforming with both two analyses. Since general FEM degenerates to conventional FEM if derivatives are of small-amplitude or zero and its direct implementation to the entire domain is costive, general FEM is alternatively utilized as local refinement of governing equation only to points with significant derivatives. The strategy of local refinement is optimized to enhance efficiency.

Mathematical Simulation of Forest Fuel Pyrolysis in One-Dimensional Statement Taking into Account Soot Formation

Pyrolysis (thermal decomposition) is considered as the most important stage of a forest fire before direct forest fuel ignition. This process is accompanied by soot particle formation. Such particles have a negative impact on public health in the vicinity of forest fires. The purpose of this article was to investigate the heat and mass transfer process occurring in a typical forest fuel element (birch leaf). The pyrolysis and soot formation processes were taken into account, and various forest fire scenarios were considered. Computational experiments were carried out using the high-level programming language Delphi. Heat and mass transfer processes were described by nonlinear non-stationary differential heat conduction equations with corresponding initial and boundary conditions. The differential equations were solved by the finite difference method. Nonlinearity was resolved using a simple iteration. The main results of the research were (1) physical and mathematical models proposed for modeling forest fuel pyrolysis, taking into account soot formation and conditions corresponding to various forest fires; (2) a computer program coded in the high-level programming language Delphi; (3) the obtained temperature distributions over leaf thickness; (4) volume fractions obtained for various components dependent on time and space coordinates. The qualitative analysis of the dependencies showed that the temperature distributions in the birch leaf structure are similar for all forest fire types and differ only in absolute value. The intensity of the soot formation process directly depends on the forest fire type. The presented results should be useful in predicting and assessing forest fire danger, including near the facilities of the Russian Railways.