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

Highlights

• Dual phase lag, Cattaneo-Vernotte and Fourier heat conduction models are compared.

• Modal analysis provides realistic values for lag times of DPL and CV models.

• Approximate series solutions converge in terms of an energy norm which is stronger than the maximum norm.

• Heat pulse progresses slower into specimen for DPL and CV models than for Fourier model.

• Vibrational analysis provides insight into series solutions of DPL and CV models.

Abstract

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.

Introduction

The invention of high-energy, short-pulse laser sources has led to significant development in the fields of materials processing and medical surgery. With these developments comes the requirement to understand, and being able to control the effects of short-pulse heating of material specimens. Heat transfer modelling is an important tool in obtaining such insight and control. Modelling essentially relies on the choice of mathematical model, the availability of model parameters (e.g. heat conductivity $k$ and thermal diffusivity $\alpha$ in the case of heat transfer) and knowledge of the mathematical techniques used to analyse these models.

The Fourier model (or diffusion equation) is a well-known and established mathematical model used to describe heat transfer, but it is considered to be inadequate when modelling heat transfer induced by short-pulse lasers, or alternatively, when the material specimen has small dimensions ([1,2]). This stems from the assumption in the derivation of the Fourier model that the heat flux vector and temperature gradient occur simultaneously. Several alternative models have been derived based on the concept of lagging responses in the heat flux and the temperature gradient. These models introduce two parameters ${\tau }_q$ and ${\tau }_{{}_T},$ known as the phase lags of the heat flux vector and the temperature gradient, respectively ([3]). The phase lags are however not reliably known and therefore pose a challenge when applying the models in specific situations. The alternative models include, but are not limited to the Cattaneo-Vernotte (CV) model, also known as the thermal wave model, and the dual-phase lag (DPL) model ([4], [5], [6], [7], [8], [9]). Two review papers [10] and [11] summarize the early progress in modelling the transmission of heat by thermal waves.

The main aim of this paper is to show that modal analysis of the DPL and CV heat conduction models provides a valid and effective approach for analysing and comparing the models. Simplified one-dimensional versions of the general three-dimensional models are used. This simplification is justified, in part, by the convention followed in international metrology to determine the thermal diffusivity $\alpha$ of materials relying on one-dimensional heat transfer modelling ([12], [13], [14]).

When comparing models, an option that provides interesting insights is to determine the predicted temperature distributions that result from the same input. In this paper, a pulse problem where a single heat pulse is applied to one endpoint of a specimen for a short time interval, is used for these comparisons. Of specific interest is the temporal and spatial behaviour of the pulse as it propagates through the specimen. The approach developed in [15] is well suited for such comparisons.

The well-posedness of the DPL and CV model problems is established using a general existence result in [16]. The weak formulations of both these models are special cases of an abstract hyperbolic problem in a Hilbert space setting. A formal series solution for the abstract problem is obtained, using a generalised separation of variables method. It is well-known that finite sums of modal solutions may be used as approximate solutions for the different models. However, the convergence in energy, as introduced in [15], means that the accuracy of these approximate solutions depends only on the accuracy of the partial sum approximations of the initial values. Consequently, an upper bound for the approximation error at any time $t>0$ is available.

There are several other advantages to using this modal approach. Firstly, the associated eigenvalue problem is the same for all three models, and consequently the functions that model the time behaviour do not depend on the spatial dimension of the problem, i.e. do not depend on the dimension of the eigenvalue problem. Conditions can be derived in terms of the physical parameters of the models for whether modes are underdamped or overdamped. This classification of modes is not typically used for heat conduction models, but borrowed from vibration analysis. It provides useful insight into the properties of the series solutions.

Furthermore, one of the important findings of this paper is that the modal solutions can be used to find justifiable values for the (dimensionless) model parameters: the lag times ${\tau }_q$ and ${\tau }_{{}_T},$ and the thermal diffusivity $\alpha$. Upper bounds for the lag times are obtained from the fact that, in some cases, the Fourier model yields physically realistic results after a sufficiently long time. This implies that in these cases, the solutions of the DPL and CV models should approach the solution of the Fourier model after some time.

The pulse problem is solved using an equivalent transformed problem with homogenised boundary conditions. The number of terms to use in the approximate solutions is obtained from the partial sum approximation of the initial value. But for the transformed problem the initial temperature distribution has a discontinuity at the boundary, due to the step function modelling of the boundary temperature pulse, and the approximation error in energy cannot be calculated. However, this initial temperature distribution can be replaced, with arbitrary small mean square error, by a piecewise differentiable function. The theory in [15] can then be applied.

The derivation of the DPL and CV heat conduction models is discussed in Section 2. In Section 3, relevant results pertaining to the existence and convergence of the formal series solutions are presented. The properties of the energy norm are discussed in this section. Expressions for the series solutions of the heat conduction models are derived in Section 4, and parameter values are obtained in Section 5. In Section 6, the initial temperature distribution for the pulse problem is discussed. A comparison of the pulse problem for the three models is presented in Section 7, followed by some concluding remarks in Section 8.