Dual Phase Lag Model Research Articles

Thermoelastic damping in the size-dependent micro/nanobeam resonator with nonlocal dual-phase-lag heat conduction

Precisely calculating quality factor based on thermoelastic damping (TED) is of great importance for the design and optimization of micro/nano-resonators. With the consideration of size-dependent effects in the thermal and mechanical fields and the non-Fourier (NF) heat conduction effect, analytical expressions of TED in the micro/nanobeam resonator are proposed by adopting the theories of the modified-couple-stress (MCS) model and the nonlocal dual-phase-lag (DPL) model. TED models expressed in the series form and the explicit form are both developed basing on the energy-definition approach. The differences among TED models are discussed. In the simulation, silicon and gold, which are two representative materials with typical intrinsic length-scale quantities and phase-lag times, are selected for illustration. Additionally, the nonlocal thermal length-scale parameter refers to the mean-free-path (MFP) of energy carriers, and the MCS mechanical length-scale parameter is not constant but proportional to the beam thickness. TED results obtained by the classical model and the previous DPL model are also shown for comparison. The peak phenomena of TED spectra including the peak damping and the peak frequency are investigated according to the present simple model. Results reveal that TED in micro/nanobeam resonators is of dependence observably on the effect of nonlocal DPL-NF heat conduction associated with the thermal field, and the quality factor is improved by the MCS mechanical size-dependent effect.

Effect of Magnetic Field on Thermoelastic Micro-Elongated Solid with Diffusion under Dual-Phase-Lag Model

This study considers a thermoelastic micro-elongated solid which is being heated by a laser pulse in order to investigate the corresponding impact of thermal loading and magnetic field. A laser beam that is non-Gaussian in nature is used to heat the surface of the bounded plane with a pulse duration [Formula: see text][Formula: see text]ps. The dual-phase-lag (DPL) model is considered to develop a better understanding of the problem. Also, the problems are solved using normal mode analysis as well as it helps in attaining the appropriate expressions for micro-rotation, components of displacement, the force stress, couple stress, concentration field, the temperature distribution the scalar micro-elongated function, and the micro-stress. The results obtained from the thermal variations are validated by comparing them to Coupled Theory (CT) of thermo-elasticity, Lord and Shulman (L–S) theory, and the DPL model. The graphical illustration of these comparisons has been presented as it helps in exploring the impacts of the laser pulse at two different times and the magnetic field parameter. The investigation also helped deduce some specific cases of interest.

Mathematical modeling of short pulsed laser irradiation in the cornea: a dual phase lag model

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Finite element modeling of Non-Fourier heat transfer in a cancerous tissue with an injected fat layer during hyperthermia treatment.

Hyperthermia technique has received much attention over the last decade being less invasive among the others. Laser therapy is among the most commonly investigated types of ablative hyperthermia for treatment of cancer. In this method an external heat source provided by a laser fiber leads the cancerous tissue to the necrosis stage. For its simulation a cylindrical geometry of a breast tissue containing a tumor is acted upon by a Gaussian form of laser radiation. Then the feasibility of a fat layer injection around the tumor during the therapy is investigated numerically. In order to consider the finite speed of heat transfer, dual phase lag (DPL) model is implemented for prediction of the thermal results. The therapy is addressed with and without the presence of a fat layer around the breast tumor. Results show that the temperature in the tumor increases up to 15% by the injection of a fat layer. Also, the presence of a fat layer around the tumor shows that the irreversible ablation happens at a faster rate.

Thermoelastic behavior of skin tissue induced by laser irradiation based on the generalized dual-phase lag model.

This paper analyzes the thermoelastic responses of skin tissue during laser irradiation based on a generalized dual-phase-lag (DPL) model. The method of separation of variables is utilized to obtain the analytical solutions for thermal and mechanical responses. The influences of some crucial parameters on temperature, displacement and stress evolutions are discussed, including the phase lag of heat flux, the phase lag of temperature gradient and the phase lag of laser pulse, the coupling factor between tissue and blood, the porosity of tissue, the equivalent diameter of tissue and the diameter of blood vessels. The generalized DPL bio-heat transfer model predicts different results from those by the classical DPL model and Pennes model. The equivalent diameter of tissue affects the coupling factor between tissue and blood, while the diameter of blood vessels mainly affects the porosity of tissue.

Photothermoelastic Investigation of Semiconductor Material Due to Distributed Loads

A dynamic mathematical model of photothermoelastic (semiconductor) medium is developed to analyze the deformation due to inclined loads. The governing equations for photothermoelastic with dual phase lag model are framed for two dimensional case and are further simplified by using potential function. Appropriate transforms w.r.t time (Laplace) and w.r.t space variables (Fourier) are employed on the resulting equations which convert the system of equations into differential equation. The problem is examined by deploying suitable mechanical boundary conditions. Specific types of distributed loads as uniformly distributed force and Linearly distributed force are taken to examine the utility of the model. The analytic expressions like displacements, stresses, temperature distribution and carrier density are obtained in the new domain (transformed).To recover the quantities in the physical domain, numerical inversion technique is employed. Numerical computed results with different angle of inclination vs distance are analyzed with and without dual phase lag theories of thermoelasticity in the form of visual representations. It is seen that physical field quantities are sensitive towards photothermoelastic and phase lag parameters.

Identification of the Thermal Constants of the DPL Heat Transfer Model of a Single Layer Porous Material

This paper deals with parameters’ identification of the Dual Phase Lag (DPL) thermal model of a 3D printed porous materials. The experiments were performed for two porous materials with different filling factors. The Laplace transform was applied for the heat transfer equation and together with different optimization methods it allowed to identify the thermal time constants of the DPL model. Several optimization methods were tested with known parameters in order to confirm the correctness of the parameters’ estimation.

Continuum Modeling Perspectives of Non-Fourier Heat Conduction in Biological Systems

Abstract The thermal modeling of biological systems is increasingly important in the development of more advanced and more precise techniques such as ultrasound surgery. One of the primary barriers is the complexity of biological materials: the geometrical, structural, and material properties vary in a wide range. In the present paper, we focus on the continuum modeling of heterogeneous materials of biological origin. There are numerous examples in the literature for non-Fourier thermal models. However, as we realized, they are associated with a few common misconceptions. Therefore, we first aim to clarify the basic concepts of non-Fourier thermal models. These concepts are demonstrated by revisiting two experiments from the literature in which the Cattaneo–Vernotte and the dual phase lag models are utilized. Our investigation revealed that these non-Fourier models are based on misinterpretations of the measured data, and the seeming deviation from Fourier’s law originates from the source terms and boundary conditions.

A Fibonacci Wavelet Method for Solving Dual-Phase-Lag Heat Transfer Model in Multi-Layer Skin Tissue during Hyperthermia Treatment

In this article, a novel wavelet collocation method based on Fibonacci wavelets is proposed to solve the dual-phase-lag (DPL) bioheat transfer model in multilayer skin tissues during hyperthermia treatment. Firstly, the Fibonacci polynomials and the corresponding wavelets along with their fundamental properties are briefly studied. Secondly, the operational matrices of integration for the Fibonacci wavelets are built by following the celebrated approach of Chen and Haiso. Thirdly, the proposed method is utilized to reduce the underlying DPL model into a system of algebraic equations, which has been solved using the Newton iteration method. Towards the culmination, the effect of different parameters including the tissue-wall temperature, time-lag due to heat flux, time-lag due to temperature gradient, blood perfusion, metabolic heat generation, heat loss due to diffusion of water, and boundary conditions of various kinds on multilayer skin tissues during hyperthermia treatment are briefly presented and all the outcomes are portrayed graphically.

Thermoelastic interactions in a functionally graded material with gravity and rotation under dual-phase-lag heat conduction

Present work is aimed to study the solution of a problem on thermoelastic interactions in a functionally graded material under dual-phase-lag model in the presence of rotation and gravity. By employing the normal mode technique, the analytical expressions for the displacement components, stresses and temperature field are obtained in the physical domain. These expressions are also calculated numerically and represented graphically in the presence and absence of gravity and rotation. Discussions have been made to highlight the effects of non-homogeneity, rotation and gravity on these physical fields. Moreover, the significant differences in the numerical results for three models known as dual-phase-lag (DPL) model, Lord–Shulman (L–S) model, and coupled thermoelastic (C-T) model are also observed for two different times. Some particular cases of interest have been inferred from the present study. Although various investigations do exist to observe the disturbances in a thermoelastic medium under the effects of different parameters, the work in its present form i.e., thermally induced vibrations in a non-homogeneous thermoelastic material with rotation and gravity under DPL model has not been studied till now.

Thermoelastic response of non-simple solid cylinder with variable properties under a continuous line heat source

This paper focuses on the thermoelastic response of a non-simple solid cylinder with variable thermal conductivity due to a continuous line heat source. The solid is interspersed with a uniform magnetic field and magneto-thermoelastic interactions are studied under the purview of the dual-phase lag model. Laplace transform technique is utilized to get the exact expressions for the displacement, the temperature, and the thermal stress distributions. It is important to delve into the influence of the continuous line heat source in the thermoelastic process. Comparisons are made with results between different models of thermoelasticity. Numerical results are also performed for appropriate material to clarify the results.

Thermal conductivity changes of photo-elastic semiconductor excited in gravitational field with hydrostatic initial stress and internal heat source

In this work, the effect of hydrostatic initial stress is studied with the exited semiconductor material. Electrons are exited at the free surface of the elastic semiconductor medium using the photo-thermal-elastic theory during the transport (diffusion) process. The thermal memories are due to the dual-phase-lag (DPL) relaxation times of the heat conduction equation with a novel model. In this model, the thermal conductivity is chosen as a linear dependent function of the thermal temperature impact which is variable. The effect of gravitational field is taken into account during the two-dimensional (2D) elastic deformation under the impact of an internal heat source. The Kirchhoff transformation mapping is used. The harmonic wave (normal mode) method is applied for the 2D governing equations to obtain the main physical fields analytically. Some thermal and mechanical forces are applied with other elastic-plasma conditions at the free surface of the semiconductor material to get the complete solutions. Some comparisons used a novel parameters which depend on the DPL model and the differences in the thermal conductivity parameters and they are illustrated graphically and discussed.

Analysis of plane wave propagation under the purview of three phase lag theory of thermoelasticity with non-local effect

Present study characterizes the propagation of harmonic plane waves in Eringen's non-local thermoelastic medium. Three phase lag (TPL) theory of thermoelasticity has been applied to account for the interactions between elastic and thermal fields. Two sets of coupled longitudinal waves (elastic and thermal) and one independent vertically shear wave have been achieved. Longitudinal waves are found to be dispersive and experience attenuation. All these waves are found to be influenced by the elastic non-local parameter. Further shear-type wave obtains a critical frequency while the coupled longitudinal waves may face critical frequencies conditionally. High and low frequency asymptotes of physical fields have been calculated. By considering an illustrative example, numerical results have been computed to explore the physics of the problem. The results for the TPL model have been compared with those for the dual phase lag theory. The study reveals that the distribution of phase velocities and attenuation coefficients for the TPL model are significantly different from those of the dual phase lag models with respect to the frequency as well as non-local elastic parameter. TPL theory is found to be capable in predicting better results as compare to the dual phase lag theory of thermoelasticity.

Response of deflection and thermal moment of Timoshenko microbeams considering modified couple stress theory and dual-phase-lag heat conduction model

The present work focuses on the vibration analysis of Timoshenko microbeams considering the effects of the modified couple stress theory (MCST) and the dual-phase-lag (DPL) heat conduction model. The explicit expressions for equations of motion and corresponding boundary conditions of microbeam are constructed using the Hamilton’s principle and DPL heat conduction theory. For the simply supported microbeam, the deflection and thermal moment are obtained subjected to a constant impulsive force per unit length. A comparison of results between classical theory and the MCST under DPL model is specifically taken into consideration. The obtained results are also compared to the existing results for Lord-Shulman (LS) and Fourier heat conduction models. Besides, the influences of some parameters, such as the aspect ratios of length to thickness and material length-scale parameter to thickness on deflection and thermal moment of the microbeam, are analyzed in a detailed manner. The effects of phase-lag parameters associated with the DPL heat conduction model on deflection and thermal moment are also discussed. For a short time range, the present results show a good agreement with the corresponding results under LS and Fourier heat conduction models. But for a large time range, the present results show a lower rate of energy dissipation in comparison to LS and Fourier heat conduction models. In addition, the size-dependency of microbeam under the DPL model has been observed which plays an important role on the response of deflection and thermal moment. It is also revealed that there is a significant influence of phase-lags on the variation of microbeam’s deflection and thermal moment.

Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dual-phase lags

The aim of this article is to investigate the thermal and mechanical vibration properties of functionally graded microbeams. The governing system of equations is formulated on the basis of classical Euler–Bernoulli beam model incorporating the generalized dual-phase lag model of thermoelasticity rather than the conventional steady-state Fourier heat conduction. In our analysis, it is also assumed that the mechanical and thermal properties such as modulus of elasticity, density and coefficient of thermal conductivity change through the thickness by an exponential law distribution, with the exception of Poisson’s ratio. The effects of system parameters on different field quantities have been depicted for initial stress, material gradient index and pulses duration . The numerical results are compared with benchmark findings, for the sake of verification. Presented results demonstrate the considerable effects of temperature and the material gradients on the dynamic behavior of microscopic structures. Finally, a comparison is conducted with the results discussed in the literature to justify the quality of the present technique.

Temperature dependent thermal conductivity during photothermal excitation process of semiconductor medium with an internal heat source in gravitational field

In this work, a novel model of semiconducting elastic material is studied during a process of photothermal excitation. The dual phase lag (DPL) for the heat conduction equation is introduced in the context of the variable of the thermal conductivity. The thermal conductivity can be chosen as a linear function of temperature (depend on temperature). The generalized thermoelasticy theory is investigated when an internal heat source is steady with a constant speed in the context of the gravitational field. The harmonic wave technique is used in two-dimensional deformations to obtain the considered physical fields. Some thermal loading subjected to thermal shock and other mechanical forces at the outer free surface of a semi-infinite elastic medium (semiconductor as silicon) are applied. The considered physical fields were illustrated graphically. The influences of some several variables are obtained which based on the DPL (thermal memories) model and the variable thermal conductivity. The results obtained are investigated and are depicted graphically.

Comparing the dual phase lag, Cattaneo-Vernotte and Fourier heat conduction models using modal analysis

This paper deals with phase lag (or time-lagged) heat conduction models: the Cattaneo-Vernotte (or thermal wave) model and the dual phase lag model. The main aim is to show that modal analysis of these second order partial differential equations provides a valid and effective approach for analysing and comparing the models. It is known that reliable values for the phase lags of the heat flux and the temperature gradient are not readily available. The modal solutions are used to determine a range of realistic values for these lag times. Furthermore, it is shown that using partial sums of the series solutions for calculating approximate solutions is an efficient procedure. These approximate solutions converge in terms of a so-called energy norm which is stronger than the maximum norm. A model problem where a single heat pulse is applied to a specimen, is used for comparing these models with the Fourier (or parabolic) heat conduction model.

Dual-phase lag model of heat transfer between blood vessel and biological tissue.

A single blood vessel surrounded by the biological tissue with a tumor is considered. The influence of the heating technique (e.g. ultrasound, microwave, etc.) is described by setting a fixed temperature for the tumor which is higher than the blood and tissue temperature. The temperature distribution for the blood sub-domain is described by the energy equation written in the dual-phase lag convention, the temperature distribution in the biological tissue with a tumor is described also by the dual-phase lag equation. The boundary condition on the contact surface between blood vessel and biological tissue and the Neumann condition are also formulated using the extended Fourier law. So far in the literature, the temperature distribution in a blood vessel has been described by the classical energy equation. It is not clear whether the Fourier's law applies to highly heated tissues in which a significant thermal blood vessel is distinguished, therefore, taking into account the heterogeneous inner structure of the blood, the dual-phase lag equation is proposed for this sub-domain. The problem is solved by means of the implicit scheme of the finite difference method. The computations were performed for various values of delay times, which were taken from the available literature, and the influence of these values on the obtained temperature distributions was discussed.

Numerical analysis of non-Fourier thermal response of lung tissue based on experimental data with application in laser therapy

Background and objectiveThe thermal therapy is a minimally invasive technique used as an alternative approach to conventional cancer treatments. There is an increasing concern about the accuracy of the thermal simulation during the process of tumor ablation. This study is aimed at investigating the effect of finite speed of heat propagation in the biological lung tissue, experimentally and numerically. MethodsIn the experimental study, a boundary heat flux is applied to the lung tissue specimens and the temperature variation is measured during a transient heat transfer procedure. In the numerical study, a code is developed based on the finite volume method to solve the classical bio-heat transfer, the Cattaneo and Vernotte, and the Dual-phase-lag (DPL) equations. The thermal response of tissue during the experiments is compared with the predictions of the three heat transfer models. ResultsIt is found that the trend of temperature variation by the DPL model resembles the experimental results. The experimental observation in parallel with the numerical results reveals that the accumulated thermal energy diffuses to the surrounding tissue with a slower rate in comparison with the conventional bio-heat transfer model. The DPL model is implemented to study the temperature elevation in the laser irradiation to lung tissue in the presence of gold nanoparticles (GNPs). It is concluded that the extent of the necrotic tumoral region and the area of the damaged healthy tissue are reduced, when the non-Fourier heat transfer is taken into account. ConclusionsResults show that considering the phase lags is crucial in planning for an effective thermal treatment, in which the cancerous tissue is ablated and the surrounding tissues are preserved from irreversible thermal damage.

A numerical approach to non-Fourier heat transfer in liver tumor during laser irradiation

Thermal therapy is a type of cancer treatment that uses heat to kill cancer cells, but it also may harm healthy tissue. Numerical simulations can help to accurately analyze the thermal damage of the tissue during heat exposure. The target of this study is to investigate the effect of time lags on the thermal response of the biological tissue during laser irradiation to the tumoral tissue. The classical Fourier, single phase lag (SPL) and dual phase lag (DPL) models of bio-heat transfer are implemented and compared. The numerical solution based on the finite volume method (FVM) is applied to solve the bio-heat transfer equations. Beer-Lambert’s law is applied to determine the heat source distribution caused by the laser irradiation. The thermal damage caused by the laser exposure for the three models is discussed. Results show that the DPL model predicts a significantly different thermal damage from the classical Fourier and the SPL models. It is observed that the DPL model predicts the maximum temperature 4.1 ˚C and 5.7 ˚C less than the Fourier and the SPL models, respectively. The deviation between the maximum temperatures obtained by the three models can be attributed to the finite speed of thermal wave propagation in the non-Fourier models.