Dual Phase Lag Heat Conduction Research Articles

On the thermo-vibrational response of multi-layer viscoelastic skin tissue to laser irradiation

This paper investigates temperature and vibration responses of human multi-layer skin tissue to laser irradiation. Due to the viscoelastic property of skin, a standard linear solid model known as the Zener model with two relaxation times with dual phase-lag heat conduction model has been used to describe the temperature and displacement in a one-dimensional form. The obtained differential equations have been discretized in the weak form to balance thermal and vibrational energies. The extracted time-dependent ordinary differential equations have been solved numerically by integration over time. The investigations are based on the effect of four relaxation times of temperature, heat, strain, and stress with laser effect in the form of a single step or repeated pulses on the behavior of the skin. By examining their results, the effect of phase lags has also been discussed, showing that the heat phase lag has a complex impact on thermal response and depends on the type of laser radiation. The phase lag of temperature reduces the temperature change rate, and the strain phase lag spreads the displacement at later times, gives inertia to the system, and dampens the system's vibration.

Analysis of dual phase-lag heat conduction in the cornea subjected to laser irradiation

Analysis of dual phase-lag heat conduction in the cornea subjected to laser irradiation

Two dimensional MHD nanofluid flow analysis of fractional dual-phase-lag heat conduction between inclined cylinders with variable thickness

PurposeThis study aims to investigate the two-dimensional magnetohydrodynamic flow and heat transfer of a fractional Maxwell nanofluid between inclined cylinders with variable thickness. Considering the cylindrical coordinate system, the constitutive relation of the fractional viscoelastic fluid and the fractional dual-phase-lag (DPL) heat conduction model, the boundary layer governing equations are first formulated and derived.Design/methodology/approachThe newly developed finite difference scheme combined with the L1 algorithm is used to numerically solve nonlinear fractional differential equations. Furthermore, the effectiveness of the algorithm is verified by a numerical example.FindingsBased on numerical analysis, the effects of parameters on velocity and temperature are revealed. Specifically, the velocity decreases with the increase of the fractional derivative parameter α owing to memory characteristics. The temperature increase with the increase of fractional derivative parameter ß due to a decrease in thermal resistance. From a physical perspective, the phase lag of the heat flux vector and temperature gradients τq and τT exhibit opposite trends to the temperature. The ratio τT/τq plays an important role in controlling different heat conduction behaviors. Increasing the inclination angle θ, the types and volume fractions of nanoparticles Φ can increase velocity and temperature, respectively.Originality/valueFractional Maxwell nanofluid flows from a fixed-thickness pipe to an inclined variable-thickness pipe, and the fractional DPL heat conduction model based on materials is considered, which provides a basis for the safe and efficient transportation of high-viscosity and condensable fluids in industrial production.

Eigen value approach with memory dependant derivative on homogeneous isotropic infinitely extended rotating plate of a finite thickness in absence of heat source

Our present manuscript is an attempt to derive a model of generalized thermoelasticity with dual phase lag heat conduction by using the methodology of memory dependent derivative for a isotropic rotating plate subject to the prescribed boundary conditions with constant magnetic and electric intensities. Two integral transform such as Laplace transform for time variable and Fourier transform for space variable are employed to the governing equations to formulate vector-matrix differential equation which is then solved by eigenvalue approach methodology. The inversion of two integral transformations is carried out using suitable numerical techniques. Numerical computations for displacement, thermal strain and stress component, temperature distribution are evaluated and presented graphically under influences of different physical parameters.

Thermoelastic characteristics of moving viscoelastic nanobeams based on the nonlocal couple stress theory and dual-phase lag model

Thermal behavior of a moving viscoelastic nanobeam under the influence of periodic thermal load is considered in the framework of Kelvin-Voigt viscoelastic model with fractional operators. The equation of motion for axially moving nanobeam is modeled by employing the Eringen’s nonlocal elastic theory in conjunction with the couple stress hypothesis and the conventional Euler–Bernoulli beam model. The thermoelastic features is then established by employing the generalized dual phase-lag heat conduction model. After utilizing the Laplace transform, the thermomechanical equations are coupled and solved. The current results are validated by presenting numerical examples and comparing with previous solutions obtained by traditional theories in the literature. According to the provided numerical simulations, the deflection of the axially moving nanobeam as well as its temperature change reduce with the axial velocity and the influences of small scale and nonlocal parameters are also revealed and discussed.

Thermoelastic analysis of biological tissue during hyperthermia treatment for moving laser heating using fractional dual-phase-lag bioheat conduction

In this paper, a time-fractional dual-phase-lag (DPL) bio-thermoelastic model is presented to solve the thermoelastic response of a biological tissue during hyperthermia treatment by a moving laser heating. To improve the elimination rate of diseased tissue and reduce the side effects to surrounding normal tissue in hyperthermia, it is necessary to study the thermoelastic response of biological tissues under high-speed and high-intensity laser heating. The laser heating process causes a large instantaneous temperature difference and heat flow in the tissue, which leads to abnormal diffusion of heat conduction. The classical Fourier heat conduction model cannot describe this anomalous diffusion phenomenon due to infinite heat propagation velocity. In order to describe anomalous diffusion in tissue, the fractional derivative is introduced into the DPL heat conduction model. In order to solve the elastic deformation of tissue under heating, the biological fractional thermoelastic theory is established by combining the fractional DPL model with one-dimensional thermoelastic theory. The physical significance of the phase lags and fractional order is elucidated. The thermoelastic responses are compared with those based on existing models. Finally, the effects of blood perfusion rate and laser velocity on temperature distribution and thermal stress response during percutaneous laser thermal ablation are analyzed.

Numerical simulation and parameters estimation of the time fractional dual-phase-lag heat conduction in femtosecond laser heating

In this paper, considering the heat flux and the temperature gradient approximated by the fractional Taylor's series expansions with different orders in dual-phase-lag (DPL) model, we propose the time fractional DPL heat conduction model for thin gold films heating by femtosecond laser pulses. The solutions of the model are investigated analytically and numerically. Firstly, the semi-analytical solution is presented by the Laplace transform method and the Fourier cosine transform method. With the L1 approximation of the Caputo fractional derivative, the finite difference scheme is derived to describe the effects of femtosecond laser pulses on the temperature distributions of the gold films. The effectiveness of the numerical algorithm is examined by the comparison with the semi-analytical solution. And then based on the numerical algorithm and the experimental datasets of femtosecond laser pulses heating on gold films of various thicknesses, the estimations of the model parameters are considered by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO). Using the optimal values of the parameters estimation, we verify the consistencies between the semi-analytical solutions of the model and the datasets. It proves the accuracy of the time fractional DPL heat conduction model in characterizing the temperature distributions of thin gold films. Finally, the varieties of temperature distribution with time and position are considered and then the effects of the model parameters on the temperature distribution are also discussed.

Fractional order thermo-viscoelastic theory of biological tissue with dual phase lag heat conduction model

Viscoelasticity is an inherent property of the biological tissue and is often used as a diagnosis parameter in characterizing cancer. Recently, it has been found that fractional calculus agrees well with experimental results when describing the viscoelastic materials. In this study, the fractional linear thermo-viscoelastic theory is developed with considering the fractional relaxation effect of dual phase lag heat conduction model and viscoelastic constitutive relation simultaneously. The transient coupling thermal and mechanical behavior of biological tissue are investigated. The governing equations are solved by Laplace transformation. The effects of fractional parameter on the distributions of temperature, displacement and stress are obtained and illustrated graphically.

The biothermal analysis of a human eye subjected to exponentially decaying laser radiation under the dual phase-lag heat conduction law

In this paper, a biomathematical model of the human eye subjected to an exponentially decaying laser concerning the change in blood perfusion, porosity, evaporation rate, and ambient temperatures, has been constructed based on dual phase-lag (DPL) heat conduction law. The human eye has been divided into six layers. Appropriate boundary and interface continuity conditions have been considered. A separable function has been assumed, and twelve equations have been formulated in a matrix form. The solutions have been calculated by using MAPLE. The results have been shown in figures with different cases. The absolute temperature distribution based on various values of the power density of laser irradiation and relaxation times parameters have been discussed first. The effect of the porosity, evaporation rate, time, and ambient temperatures have also been discussed. The power density of laser irradiation, porosity, evaporation rate, time, and ambient temperatures have significant effects on the temperature passing through the human eye layers.

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.

Heat waves interference regarding dual-phase-lag, hyperbolic and Fourier heat conduction in CNT reinforced composites under a thermal shock

We present a framework to investigate the effects of transient heat conduction on heatwave interference and the overshooting phenomenon in carbon nanotube (CNT)-reinforced composites. Material properties of the nanocomposite are temperature and volume dependent. Different kinds of heat conduction models such as Fourier, hyperbolic or single-phase-lag (SPL), and dual-phase-lag (DPL) are considered while the media is subjected to a heat pulse or periodic thermal shock. This paper attempts to add three contributions to the existing literature: (1) using the differential quadrature method (DQM) to solve for the first time, the nonlinear DPL heat conduction equations while the materials and properties are geometry- and temperature-dependent; (2) showing the effects of the volume fraction of CNTs and their distribution on the transient heat conduction and heatwave interference; (3) demonstrating heatwave interference and the high-temperature gradient as the origins of the overshooting phenomenon. The accuracy of the present solution is confirmed by comparing it with available results from the literature.

Semi-analytical solutions for different non-linear models of dual phase lag equation in living tissues

In this study, the heat transfer in a multilayer heated living tissue is investigated. A non-linear dual phase lag (DPL) heat conduction model is presented assuming temperature-dependent blood perfusion rate, heat conductivity and metabolic heat generation. Semi -analytical solutions for the non-linear model are obtained employing the Galerkin weighted residuals method. The effects of the variable thermophysical properties on the temperature distribution within the tissue are investigated through comparison of the results with the corresponding results for the constant thermophysical properties. The results show that the temperature dependence of the blood perfusion rate has a significant effect on the temperature distribution in the different layers of the tissue. As far as the tumor region is considered, 0.43 °C increase in its temperature is observed in comparison with that obtained assuming the constant blood perfusion rate at t = 1800 s. Moreover, increasing the temperature dependence of the thermal conductivity leads to the reduction in the temperature of the tissue except in the muscle layers. Considering the temperature-dependent metabolic heat generation increases the temperature of all of the layers compared to the corresponding temperatures obtained assuming the constant metabolic heat generation.

Reflection phenomena of waves through rotating elastic medium with micro-temperature effect

Abstract In this article, we analyzed the effect of variable thermal conductivity on reflected elastic waves. The waves are propagating through a thermoelastic medium rotating with some angular frequency. The concept of micro-temperature is also been considered, in which microelements of the medium contain a high temperature. A heat conduction phenomenon is encountered by dual phase-lag heat conduction model. P (or SV)-type wave is incident on the medium with some specific angle of incidence. After reflection from the surface incident, P-wave is converted into quasi longitudinal and quasi transverse waves and propagates back into the medium. Helmholtz’s potential function along with the harmonic wave solution is used to obtain the solution of the model. Analytically, we calculated the amplitude ratios and attenuation factor for each reflected wave against the angle of incidence. The obtained results are also represented graphically for different values of rotational frequency and variable thermal conductivity for a particular material.

Memory Response in a Magneto-Thermoelastic Rod with Moving Heat Source Based on Eringen’s Nonlocal Theory Under Dual-Phase Lag Heat Conduction

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of dual-phase lag (DPL) theory of thermoelasticity based on Eringen’s nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for temperature, displacement and stress within the rod is carried out and displayed graphically. The effect of moving heat source speed, time instance, memory-dependent derivative, magnetic-field and nonlocality on temperature, displacement and stress are studied.

Dual-phase-lag heat conduction in the composites by introducing a new application of DQM

The first application of the differential quadrature method (DQM) in solving the nonlinear dual-phase-lag (DPL) heat conduction equation is demonstrated here. To show the effect of DPL parameters, the temperature response of the medium was obtained from Fourier’s low, hyperbolic heat conduction and hyperbolic type DPL heat conduction model were compared. Furthermore, the transient temperature and resultant heat flux distributions have been found for various types of dynamic thermal loading. We show whether thermal waves exist or not in hyperbolic type DPL heat conduction by considering the time lag parameter in the microstructural interactions of fast transient heat conduction. Also, overshooting which is one of the hyperbolic heat conduction is investigated here. The numerical solution at each time level depends on the solutions at its previous levels. This means the temperature and heat flux obtained at the time nth are the initial conditions for the time (n + 1)th. After demonstrating the convergence and accuracy of the method, the effects of different parameters on the temperature and heat flux distribution of the medium are studied.

Dual-phase-lag heat conduction in FG carbon nanotube reinforced polymer composites

The numerical simulation of non-Fourier, dual-phase-lag (DPL) heat conduction in carbon nanotube (CNT) reinforced composites has been performed by developing the differential quadrature method (DQM) application. Although using the non-Fourier heat conduction has become popular, most of the simulations are conducted over simple geometries due to numerical restrictions. DQM copes with this problem easily even with complicated boundary conditions. The proposed method discretizes the time domain in the form of blocks. Each block contains several time levels, and each block needs to solve separately by using the output of one block as an initial condition of the next block. The effect of volume fraction of CNT on the temperature and heat flux profile is investigated, and all parameters are considered temperature dependent. To show the influence of time lag parameters, the Fourier, hyperbolic heat conduction and DPL heat conduction models were compared. Furthermore, various types of dynamic thermal loading have been examined to obtain the transient temperature distributions. We show the presence of heat wave in DPL heat conduction by considering the time lag parameter in the microstructural interactions of fast transient heat conduction. It is found that considering the mechanical properties of the CNTs dependent to the temperature is crucial to obtain accurate results.

A Closed Form Solution of Dual-Phase Lag Heat Conduction Problem With Time Periodic Boundary Conditions

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

On the Increase in Signal Depth due to High-Order Effects in Micro- and Nanosized Deformable Conductors

With regard to the transmission of a thermomechanical signal on extremely short temporal and spatial scales, which represents an issue particularly important dealing with micro- and nanosized electromechanical systems, it is well known that the so-called non-Fourier effects become not negligible. In addition, it has to be considered that the interaction among multiple energy carriers has, as a direct consequence, the involvement of high-order terms in the time-differential formulation of the dual-phase lag heat conduction constitutive equation linking the heat flux vector with the temperature variation gradient. Accepting that the deformations caused by the temperature variations are small enough to be modeled under the assumptions typical of the linear thermoelasticity, in the present article we take into account the highest Taylor expansion orders able to guarantee (under appropriate assumptions) stability conditions, thermodynamic consistency, and at the same time the existence of an influence domain of the external data linked to the energy transmission as thermal waves. To this aim, a cylindrical domain filled by an anisotropic and inhomogeneous thermoelastic material is investigated, although the results obtained will be independent from the considered geometry: for such a reason, we will be able to consider as illustrative examples some simulations referred to single-layer graphene and to show how the expansion orders selected strongly influence the domain of influence depth.

Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.

Electro-thermal modeling for InxGa1-xN/GaN based quantum well heterostructures

The joint effect of the heat transfer and the electronic properties in the InGaN/GaN based quantum well (QW) heterostructures has been investigated theoretically and numerically. One-dimensional Schrödinger equation solver coupled with Poisson equation solver and Dual-phase-lagging (DPL) heat conduction solver has been developed. The numerical results suggest that the DPL heat conduction equations capture the microscale responses caused by the phonon-electron interaction. Both effects of the polarization charge and conduction band offset between the InGaN/GaN interfaces lead to the creation of the two-dimensional electron gas (2DEG) on the lower interface of the QW. It is found that the 2DEG density at the triangular quantum well increases with increasing Indium (In) composition. This increase is the same for the conduction band offset and the electron density. As a consequence, an increase of the heat dissipation and the temperature is observed at the lower interface of the quantum well.