Kuo-Chi Liu

Numerical Analysis for Dual-Phase-Lag Heat Conduction in Layered Films

ABSTRACT The heat conduction induced by a pulsed volumetric source in a two-layered film is investigated with the dual-phase-lag model. An analytical method and a numerical scheme are used to solve the problem. The present numerical scheme is a hybrid application of the Laplace transform method and a control-volume formulation in conjunction with hyperbolic shape functions. Because of the difference in the relaxation times, τ q and τ T , between two films, the interfacial boundary condition introduces complexity and causes some mathematical difficulties for solving the present problem directly in the temperature domain. To show the efficiency and accuracy of the present numerical scheme, comparison between the present numerical results, the analytical solution, and solutions in the literature is made. Theoretical insight to the dual-phase-lag heat conduction in two-layered films is given.

Solution of an Inverse Heat Conduction Problem in a Bi-Layered Spherical Tissue

This work attempts to estimate the phase lag times of a tissue based on the dual-phase-lag model from the experimental data. The inverse dual-phase-lag bioheat transfer problem in the bilayered spherical tissue is studied. The difference between two layers in the thermophysical parameters, geometry effects, and measurement errors of the input data make it hard to be solved. To solve the present problem, a hybrid scheme based on the Laplace transform, change of variables, and the least-squares scheme is proposed. In order to evidence the validity and accuracy of the estimated results, the comparison of the history of temperature increase between the calculated results and the experimental data is made for various measurement locations. The effect of measurement location on the estimated results is also investigated.

Numerical analysis for the hyperbolic heat conduction problem under a pulsed surface disturbance

A hybrid application of the Laplace transform method and a control-volume formulation in conjunction with the hyperbolic shape functions is applied to investigate the hyperbolic diffusion problems with the pulsed boundary conditions in various coordinate systems. The Laplace transform method is used to remove the time-dependent terms in the governing differential equations and the boundary conditions, and then the transformed equations are discretized by the control volume scheme. The primary difficulty in dealing with the present problem is the suppression of numerical oscillations in the vicinity of sharp discontinuities. The results show that the numerical results agree well with the analytic solution and do not exhibit numerical oscillations in the vicinity of the jump discontinuity. The present method also can solve the problems with the singular point.

Analysis of non-Fickian diffusion problems in a composite medium

The one-dimensional non-Fickian diffusion problems in a two-layered composite medium for finite and semi-infinite geometry are analyzed by using a hybrid application of the Laplace transform technique and control-volume method in conjunction with the hyperbolic shape functions, where the effect of the potential field is taken into account. The Laplace transform method used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control volume scheme. To evidence the accuracy of the present numerical method, a comparison between the present numerical results and analytical solution is made for the constant potential gradient. Results show that the present numerical results are accurate for various values of the potential gradient, relaxation time ratio, and diffusion coefficient ratio. It can be found that these values play an important role in the present problem. An interesting finding is that when the mass wave encounters an interface of the dissimilar materials, a portion of the wave is reflected and the rest is transmitted. The speed of propagation can change owing to the penetration of the mass wave into the region of the different material. The wave nature is significant only for short times and quickly dissipates with time.

Numerical solutions for the hyperbolic heat conduction problems in a layered solid cylinder with radiation surface

The numerical scheme based on the hybrid application of the Laplace transform method and a control-volume formulation in conjunction with the hyperbolic shape functions is applied to investigate the hyperbolic heat conduction problems in an infinitely long layered solid cylinder with the nonlinear boundary conditions. In order to perform the Laplace transform method, the nonlinear boundary conditions are linearized with Taylors series expansion. Results show that the present numerical scheme can overcome the mathematical difficulties induced by the nonlinear boundary conditions, the geometry, the composite interface and the singular point, and has stability and reliability for such problems. Effects of the thermal properties of two dissimilar materials on heat transfer are also discussed.

Study of Hyperbolic Heat Conduction Problem in the Film and Substrate Composite with the Interface Resistance

The hybrid application of the Laplace transform technique and the control-volume method in conjunction with the hyperbolic shape functions is employed to analyze the hyperbolic heat conduction problem in the film and substrate composite under a pulsed volumetric source adjacent to the exterior film surface. The model of the convective boundary condition is applied to specify the interface thermal resistance. The major difficulty encountered in the present problem is the numerical oscillations in the vicinity of the jump discontinuity. In order to evidence the accuracy of the present numerical method, a comparison between the present numerical results and the analytical solution is made. The results show that the present numerical method can suppress the numerical oscillations and the present numerical results agree well with the analytical solution. The propagation behavior of the thermal wave in the film and substrate composite depends on the thermal property ratios of two dissimilar materials and the interface thermal resistance. The interface thermal resistance restricts the energy transmission across the interface of the film and substrate composite and alters the strength of the reflected and transmitted waves.

Temperature Prediction for Tumor Hyperthermia with the Behavior of Thermal Wave

In magnetic nanoparticle hyperthermia for cancer treatment, controlling the heat distribution and temperature elevations is an immense challenge in clinical applications. It is expected for treatment quality to understand the heat transport occurring in biological tissue. The non-Fourier thermal behavior in biological tissue has been experimentally observed. This work uses the thermal wave model to predict the temperature excess occurring in a two-layer concentric spherical tissue with the heat source of Gaussian distribution. The solutions to the hyperbolic bio-heat equation with the space-dependent source term in the spherical coordinate system are presented. The influences of relaxation time, blood perfusion rate, and heating strength on the thermal response in tumor and normal tissue are discussed.

Study of hyperbolic heat conduction problem in IC chip

The hyperbolic heat conduction equation is applied to investigate the rapid transient heat conduction in IC chip instead of the classical Fourier equation. The present study applies an analytical method and a hybrid numerical scheme of the Laplace transform technique and the control volume method in conjunction with the hyperbolic shape functions to investigate the present problem. For the numerical scheme, the Laplace transform method is used to remove the time-dependent terms in the governing differential equation and the boundary conditions, and then the transformed equations are discretized by the control volume scheme. The analytical solution is obtained from a simple mathematical analysis. In order to evidence the reliability and accuracy of the present results, a comparison between the present numerical results and the analytical solution is made. The results show that the present numerical results do not exhibit numerical oscillations in the vicinity of the jump discontinuity and agree well with the analytic solution. It can also be observed that the time derivative of the heat generation and the relaxation time play the important role in the present problem and the non-Fourier effects are significant only for short times.

Effect of the potential field on non-Fickian diffusion problems in a sphere

Abstract The present study applies a hybrid numerical method to investigate the effect of a potential field on one-dimensional non-Fickian diffusion problems in a sphere. This hybrid numerical scheme involves the Laplace transform technique and the control volume method in conjunction with the suitable hyperbolic shape functions. The Laplace transform method is used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control volume scheme. It is worth noting that the boundary condition at r=0 should be carefully established for the present problems to determine an accurate numerical result. To evidence the accuracy of the present numerical method, a comparison of the mass concentration distribution between the present numerical results and the analytic solutions is made for the potential gradient dV/dr=0. The results show that the present numerical results agree well with the analytic solutions and do not exhibit numerical oscillations in the vicinity of the jump discontinuity for various potential values. The important findings are that dV/dr has a great effect on the mass concentration distribution, and the strength of the jump discontinuity can decrease with increasing the value of the dimensionless potential gradient.

Stability Analysis on Viscous Magnetic Fluid Film Flowing Down Along a Vertical Cylinder

The paper investigates the hydromagnetic stability theory of a thin electrically conductive fluid film flowing down along the outside surface of a vertical cylinder. The long-wave perturbation method is employed to solve for generalized kinematic equations with free film interface. The normal mode approach is used to compute the stability solution for the film flow. The modeling results display that the degree of instability in the film flow is further intensified by the lateral curvature of cylinder. This is somewhat different from that of the planar flow. It is also observed that by increasing the effect of the magnetic field and increasing the radius of the cylinder the film flow can become relatively more stable as traveling down along the vertical cylinder.