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In the past few decades, numerous heat conduction models extending beyond Fourier's have been developed to account for large gradients, fast phenomena, wave propagation, and heterogeneous material structures typical of biological systems, superlattices, and thermal metamaterials. Navigating through these models has become challenging due to their varying thermodynamic backgrounds and potential compatibility issues. Furthermore, recent discoveries in the field of non-Fourier heat conduction have complicated the interpretation and utilization of specific non-Fourier heat equations, especially when designing materials for the new generation of thermal metamaterials. The situation is further compounded by the existence of numerous modeling strategies in the literature, each offering different interpretations of even the same heat equation. This complexity makes it increasingly difficult to gain a comprehensive understanding of this research field. Therefore, this review aims to facilitate the navigation of advanced heat equations beyond Fourier by discussing their properties and potential practical applications in the context of experiments. We begin with the simplest models and their fundamental principles, progressing toward more complex, coupled phenomena, such as ballistic heat conduction. We do not delve into the often intricate technical details of each thermodynamic framework or aim to compare each approach from a methodological perspective. Instead, we focus on reviewing models primarily from the Rational Extended Thermodynamics, Extended Irreversible Thermodynamics, and Non-Equilibrium Thermodynamics with Internal Variables frameworks. Additionally, we discuss relevant models from kinetic theory, fractional derivatives, thermomass, and phase lag approaches. We provide background information on these models to highlight their origins, any limitations they may have, and the corresponding stability conditions, if applicable. Furthermore, as the field of non-Fourier heat conduction has become quite segmented, this paper also seeks to establish a common foundation, promoting a comprehensive mutual understanding of the fundamentals of each model and the phenomena to which they can be applied. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Highlights • An analytical solution of Guyer-Krumhansl equation is presented for a particular case of boundary and initial conditions. • The convergence of the solution has been shown. • The characteristical solutions of Fourier and Maxwell–Cattaneo-Vernotte are reproduced. • The solution is validated by using a numerical code. Abstract The existence of non-Fourier heat conduction is known for a long time in small and low temperature systems. The deviation from Fourier’s law has been found at room temperature in heterogeneous materials like rocks and metal foams (Both et al., 2016; Ván et al., 2017). These experiments emphasized that the so-called Guyer-Krumhansl equation is adequate for modeling complex materials. In this paper an analytic solution of Guyer-Krumhansl equation is presented considering boundary conditions from laser flash experiment. The solutions are validated with the help of a numerical code (Kovács et al., 2015) developed for generalized heat equations. [ABSTRACT FROM AUTHOR]

The variety and complexity of heterogeneous materials in the engineering practice are continuously increasing, open-cell metal foams filled with phase change materials are typical examples. These are also having an impact on the recent developments in the energy industry. Earlier room temperature heat pulse experiments on macroscale foam samples showed non-Fourier over-diffusive behavior on a particular time scale. Since there is a need to investigate such complex structures on larger spatial scales and extend the one-dimensional analysis on two-, and three-dimensional settings, here we develop a two-dimensional analytical solution for the Guyer-Krumhansl and Jeffreys-type heat equations in cylindrical coordinates to investigate the transient thermal behavior of large bodies. We provide the steady-state and transient temperature and heat flux distributions for a space-dependent heat source. The solutions presented here will be helpful for the thermal characterization of complex materials and for the validation of numerical methods. [ABSTRACT FROM AUTHOR]

Highlights • Implicit numerical schemes are developed for the presented shifted field discretization. • The effectiveness of the scheme is compared for different parameter domains. • The scheme is validated using an analytical solution. • The presented numerical method is compared to a commercial software COMSOL. Abstract There are various situations where the classical Fourier’s law for heat conduction is not applicable, such as heat conduction in heterogeneous materials (Both et al., 2016; Ván et al., 2017) or for modeling low-temperature phenomena (Kovács and Ván, 2015, 2016, 2018). In such cases, heat flux is not directly proportional to temperature gradient, hence, the role – and both the analytical and numerical treatment – of boundary conditions becomes nontrivial. Here, we address this question for finite difference numerics via a shifted field approach. Based on this ground, implicit schemes are presented and compared to each other for the Guyer–Krumhansl generalized heat conduction equation, which successfully describes numerous beyond-Fourier experimental findings. The results are validated by an analytical solution, and are contrasted to finite element method outcomes obtained by COMSOL. [ABSTRACT FROM AUTHOR]

Heat conduction at low temperatures shows several effects that cannot be described by the Fourier law. In this paper, the performance of various theories is compared in case of wave-like and ballistic propagation of heat pulses in NaF. [ABSTRACT FROM AUTHOR]

• Finding solutions of the two-temperature model and investigating the behaviour of the solutions. • Comparing the mathematical and physical properties of the two-temperature and Guyer-Krumhansl models. • Testing the two-temperature model on heat pulse experiments. • Comparing the fitting properties of the two-temperature model to the Guyer-Krumhansl equation. The thermal behaviour is critical in numerous applications, affecting any device's lifetime and safe operations. Adding that the technological development resulted in the appearance of complex material structures, making the research on advanced thermal models even more important. Materials like 3D printed structures, foams, or with a biological origin often show non-Fourier thermal response. Modelling such a problem requires a thorough analysis and a deep understanding of non-Fourier models. The present paper focuses on two advanced models, called two-temperature and Guyer-Krumhansl equations. We compare their physical background and characteristic behaviour. Additionally, we perform the first experimental test of the two-temperature model. It is found that the solutions of the two-temperature model deviate from the ones predicted by the Guyer-Krumhansl equation. However, both approaches can model the experimental data under certain conditions and offer different insights about the observed heat conduction phenomenon. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• The thermodynamic compatibility of initial conditions is investigated. • Analytical solutions of the Guyer-Krumhansl equation is provided. • The maximum principle for the Guyer-Krumhansl equation is proved. • The results are supported by computational calculations. In the literature, one can find numerous heat conduction models with different backgrounds. In general, a heat equation beyond Fourier consists of additional time and spatial derivatives, which makes it more challenging to solve the mathematical model correctly. In the present paper, we revisit the famous Guyer-Krumhansl heat equation for which it has been reported earlier that can lead to negative temperatures by violating the maximum principle. We show that for proper initial and boundary conditions with thermodynamic compatibility, the maximum principle is fulfilled. Our work further emphasizes the importance of the thermodynamic origin of heat equations and their compatibility with the second law. Furthermore, we use two different approaches how to determine the initial state in a thermodynamically compatible way. Computational simulations support our results. [ABSTRACT FROM AUTHOR]