Maxwell Cattaneo Vernotte Research Articles

A unified mixed [formula omitted]-finite element framework for modeling laser pulse heating processes in the refined thermodynamics

In this paper, new multi-field variational formulations are derived for solving the following thermodynamic models: (i) ballistic-conductive system, (ii) the Guyer–Krumhansl heat conduction model and (iii) the Maxwell–Cattaneo–Vernotte model as some models of the extended irreversible thermodynamic, handling the temperature, the heat flux and the current density of heat flux as independent field variables. Based on these variational approaches as mathematical background, a family of mixed hp-version finite element methods, which is capable of reliably and efficiently modeling the temperature responses, is designed. The solutions provided by the constructed hp-FE framework are illustrated for the following two heat pulse experiments as benchmark problems: (1) sinusoid laser pulse heating process and (2) rectangular (step-like) laser pulse train. It is shown that stable, oscillation-free temperature response functions can be obtained not only for the ballistic-conductive system and the Maxwell–Cattaneo–Vernotte model but also for the under-diffuse and the over-diffuse parameter settings of the Guyer–Krumhansl heat conduction model.

Numerical analysis of the Maxwell-Cattaneo-Vernotte nonlinear model

In the literature, one can find numerous modifications of Fourier’s law, the first of which is the Maxwell–Cattaneo–Vernotte heat equation. Although this model has been known for decades and successfully used to model low-temperature damped heat wave propagation, its nonlinear properties are rarely investigated. This paper aims to present the functional relationship between the transport coefficients and the consequences of their temperature dependence, particularly focusing on thermal conductivity. These are the consequences of the second law of thermodynamics. Furthermore, we introduce a particular implicit numerical scheme in order to solve such nonlinear heat equations reliably, free from artificial numerical errors. We investigate the scheme’s stability, dissipation, and dispersion attributes. We demonstrate the effect of temperature-dependent thermal conductivity on two different initial-boundary value problems, including time-dependent boundaries and heterogeneous initial conditions, for which we discuss the nontrivial constraints following irreversible thermodynamics.

Exact solution of Maxwell–Cattaneo–Vernotte model: Diffusion versus second sound

In this work we study the Maxwell–Cattaneo–Vernotte model for heat transport by conduction. We present a methodology to find the exact solution of the model without decoupling the system, so we do not need the restriction (∂T/∂t)(x,0)=0. Furthermore, we found a critical value for the thermal relaxation time depending on the physical constants of the system, which allows us to classify heat transport as diffusion or second sound. Finally, we conducted computer simulations and demonstrated that the relaxation time may not be as small as expected for certain initial conditions.

Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables

Abstract By using a procedure of classical irreversible thermodynamics with internal variable (CIT-IV), some possible interactions among heat conduction and viscous-anelastic flows for rheological media are studied. In particular, we introduce as internal variables a second rank tensor ε α β ( 1 ) \varepsilon _{\alpha \beta }^{(1)} that is contribution to inelastic strain and a vector ξα that influences the thermal transport phenomena and we derive the phenomenological equations for these variables in the anisotropic and isotropic cases. The stress-strain equations, the general flows and the temperature equation in visco-anelastic processes are obtained and when the medium is isotropic, we obtain that the total heat flux J (q) can be split in two parts: a first contribution J (0), governed by Fourier law, and a second contribution J (1), obeying Maxwell-Cattaneo-Vernotte (M-C-V) equation.

Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating

The present study provides a theoretical estimate for the thermal stress distribution and the displacement vector inside a nano-thick infinite plate due to an exponentially temporal decaying boundary heating on the front surface of the elastic plate. The surface heating is in the form of a circular ring; therefore, the axisymmetric formulation is adopted. Three different hyperbolic models of thermal transport are considered: the Maxwell-Cattaneo-Vernotte (MCV), hyperbolic Dual-Phase-Lag (HDPL) and modified hyperbolic Dual-Phase-Lag (MHDPL), which coincides with the two-step model under certain constraints. A focus is directed to the main features of the corresponding hyperbolic thermoelastic models, e.g., finite-speed thermal waves, singular surfaces (wave fronts) and wave reflection on the rear surface of the plate. Explicit expressions for the thermal and mechanical wave speeds are derived and discussed. Exact solution for the temperature in the short-time domain is derived when the thermalization time on the front surface is very long. The temperature, hydrostatic stress and displacement vector are represented in the space-time domain, with concentrating attention on the thermal reflection phenomenon on the thermally insulated rear surface. We find that the mechanical wave speeds are approximately equal for the considered models, while the thermal wave speeds are entirely different such that the modified hyperbolic dual-phase-lag thermoelasticity has the faster thermal wave speed and the Lord-Shulman thermoelasticity has the slower thermal wave speed.

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.

Lagrangian based thermal conduction

Based on the Lagrangian description of the dissipative oscillator, the Hamiltonian description of Fourier heat conduction is treated here. The method enables us to calculate the solution of thermal propagation involving the Maxwell-Cattaneo-Vernotte (MCV) telegrapher type also, in which both the initial and the boundary conditions are taken into account. The presented study offers a new kind of solution to certain partial differential equations.

A Case Study of Non-Fourier Heat Conduction Using Internal Variables and GENERIC

Abstract Applying simultaneously the methodology of non-equilibrium thermodynamics with internal variables (NET-IV) and the framework of General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), we demonstrate that, in heat conduction theories, entropy current multipliers can be interpreted as relaxed state variables. Fourier’s law and its various extensions—the Maxwell–Cattaneo–Vernotte, Guyer–Krumhansl, Jeffreys type, Ginzburg–Landau (Allen–Cahn) type and ballistic–diffusive heat conduction equations—are derived in both formulations. Along these lines, a comparison of NET-IV and GENERIC is also performed. Our results may pave the way for microscopic/multiscale understanding of beyond-Fourier heat conduction and open new ways for numerical simulations of heat conduction problems.

New perspectives for modelling ballistic-diffusive heat conduction

Among the three heat conduction modes (diffusive, second sound, and ballistic propagation), the ballistic one is the most difficult to model. In the present paper, we discuss its physical interpretations and show different alternatives to its modelling. We highlight two of them: a thermo-mechanical model in which the generalized heat equation—the so-called Maxwell–Cattaneo–Vernotte equation—is coupled to thermal expansion. The other one uses internal variables in the framework of non-equilibrium thermodynamics. For the first one, we developed a numerical solution and tested it on the heat pulse experiment performed by McNelly et al. on NaF samples. For the second one, we found an analytical solution that emphasizes the role of boundary conditions. This analytical method is used to validate an earlier developed numerical code.

Numerical treatment of nonlinear Fourier and Maxwell-Cattaneo-Vernotte heat transport equations

The second law of thermodynamics is a useful and universal tool to derive the generalizations of the Fourier’s law. In many cases, only linear relations are considered between the thermodynamic fluxes and forces, i.e., the conduction coefficients are independent of the temperature. In the present paper, we investigate a particular nonlinearity in which the thermal conductivity depends on the temperature linearly. Also, that assumption is extended to the relaxation time, which appears in the hyperbolic generalization of Fourier’s law, namely the Maxwell-Cattaneo-Vernotte (MCV) equation. Although such nonlinearity in the Fourier heat equation is well-known in the literature, its extension onto the MCV equation is rarely applied. Since these nonlinearities have significance from an experimental point of view, an efficient way is needed to solve the system of partial differential equations. In the following, we present a numerical method that is first developed for linear generalized heat equations. The related stability conditions are also discussed.

Evaluating the impact of transport inertia on the electrochemical response of lithium ion battery single particle models

The description of the transport mechanisms in operating Li ion battery cells is of key importance for a correct evaluation of their performance and for their optimization.In this work, we revise the Fickian approach for the description of the lithium transport in intercalation-type active materials. We adopt the Maxwell-Cattaneo-Vernotte (MCV) theory to capture the impact of lithium transport inertia on the electrochemical response of graphitic materials, taken here as an application example. We formalize this theory by means of an analytical mathematical expression which allows extracting the values of the lithium diffusion coefficient DMCV and the inertia characteristic time τ from potentiostatic intermittent titration technique (PITT) experiments. The implications of adopting the MCV theory in single particle models to calculate transient current response during the graphite lithiation are discussed (i) on the basis of the fitting of the calculations with in house PITT results and, (ii) by comparing the estimated diffusion coefficients with the ones resulting from the fitting using the classical Fickian approach.

Long-time behavior of quasilinear thermoelastic Kirchhoff–Love plates with second sound

We consider an initial–boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell–Cattaneo–Vernotte law giving rise to a ‘second sound’ effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev Hk-spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small – not in the full topology of our solution class, but in a lower topology corresponding to weak solutions we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.

Nonlinear heat-transport equation beyond Fourier law: application to heat-wave propagation in isotropic thin layers

By means of a nonlinear generalization of the Maxwell–Cattaneo–Vernotte equation, on theoretical grounds we investigate how nonlinear effects may influence the propagation of heat waves in isotropic thin layers which are not laterally isolated from the external environment. A comparison with the approach of the Thermomass Theory is made as well.

On heat equation in the framework of classic irreversible thermodynamics with internal variables

In this paper, we show that, using a procedure of classical irreversible thermodynamics (CIT) with internal variables, it is possible to describe the relaxation of thermal phenomena, obtaining some well known results of extended irreversible thermodynamics (EIT). In particular, we introduce as internal variables a vector and a second rank tensor, that influence the thermal transport phenomena, and we derive in the anisotropic and isotropic case, the phenomenological equations for these variables. In the case, in which the medium is isotropic, it is obtained that the total heat flux can be split in two parts: a first contribution [Formula: see text], governed by Fourier law, and a second contribution [Formula: see text], obeying Maxwell–Cattaneo–Vernotte (MCV) equation, in which a relaxation time is present. The obtained results may have applications in describing the thermal behavior in nanosystems (semiconductors, nanotubes[Formula: see text]), where the phenomena are fast and there are high-frequency thermal waves.

Generalized heat conduction in heat pulse experiments

A novel equation of heat conduction is derived with the help of a generalized entropy current and internal variables. The obtained system of constitutive relations is compatible with the momentum series expansion of the kinetic theory. The well known Fourier, Maxwell–Cattaneo–Vernotte, Guyer–Krumhansl, Jeffreys-type, and Cahn–Hilliard type equations are derived as special cases.Some remarkable properties of solutions of the general equation are demonstrated with heat pulse initial and boundary conditions. A simple numerical method is developed and its stability is proved. Apparent faster than Fourier pulse propagation is calculated in the over-diffusion regime.

A Continuum Approach to Thermomass Theory

A continuum approach to the thermomass theory for nonlinear heat transport is developed and its compatibility with the general framework of continuum thermodynamics is investigated. The heat flux is supposed to depend on the absolute temperature together with a vectorial internal variable, which is proportional to the drift velocity of the heat carriers. A generalized heat-transport equation, which is capable to bring Fourier, Maxwell–Cattaneo–Vernotte and thermomass-theory equations as special cases is derived. Propagation of heat waves along a nonequilibrium steady state is analyzed as well.

Thermal performance analysis of arbitrary-profile fins with non-fourier heat conduction behavior

The thermal performance of variable cross-section fins is considered using the Maxwell-Cattaneo- Vernotte (MCV) heat conduction model. Four different fins, namely rectangular, triangular, convex, and concave fins, with a periodic thermal condition are examined. The governing equations are hyperbolic and are solved numerically using an implicit finite difference method. In the MCV model, the thermal wave propagates with a finite speed, and hence sharp discontinuities appear in the temperature profiles. In this study, temperature profiles at various times, heat transfer rates, and thermal efficiencies of Fourier and non-Fourier fins are presented. In addition, the effect of relaxation time is considered. The results show that the effects of cross-sectional area and relaxation time are considerable on the thermal performance of various non-Fourier fins. To validate our findings, the results for non-Fourier fins with constant cross-sectional area obtained from this study are compared to those of other numerical solutions. This comparison confirms the correctness of the current results.

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell–Cattaneo–Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An h-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.

Modified Robertson projection operator method and irreversible thermodynamics

A modified Robertson projection operator is used to obtain the evolution equation for the projected distribution function as well as various macroscopic evolution equations. These evolution equations, although not as yet proved to be consistent with the thermodynamic laws, form a basis for a formal theory of macroscopic processes. On introducing some approximations to the projected propagator we construct an approximate, thermodynamically consistent evolution equation (kinetic equation) and develop a thermodynamically consistent theory of irreversible processes with the approximate evolution equation so obtained. The mathematical structure of the theory of irreversible processes thus formulated can be brought into harmony with the theory based on the Boltzmann equation and the generalized Boltzmann equation, which was previously developed and reported in the literature. The Maxwell–Cattaneo–Vernotte type constitutive equations are derived on making a number of approximations on the exact evolution equations for the stress and the heat flux.