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Article
Publication date: 25 June 2019

Krzysztof Grysa and Artur Maciag

The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion…

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Abstract

Purpose

The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion and then comparing field of temperature in a flat slab made of skin in the case when perfusion is taken into account, with the situation when a Fourier model is considered. When considering the process of heat conduction in the skin, one needs to take into account the average values of its thermal properties. When in biological bodies relaxation time is of the order of 20 s, the thermal wave propagation appears. The initial-boundary problems for Pennes’ model and SPL with perfusion model are considered to investigate the effect of the finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. As a reference model, the solution of the classic Fourier heat transfer equation for the considered problems is calculated. A heat flux has direction perpendicular to the surface of skin, considered as a flat slab. Therefore, the equations depend only on time and one spatial variable.

Design/methodology/approach

First of all the T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. Then, an approximate solutions of the problems are expressed in the form of a linear combination of the T-functions. The T-functions satisfy the equation modeling the problem under consideration. Therefore, approximating a solution of a problem with a linear combination of n T-functions one obtains a function that satisfies the equation. The unknown coefficients of the linear combination are obtained as a result of minimization of the functional that describes an inaccuracy of satisfying the initial and boundary conditions in a mean-square sense.

Findings

The sets of T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. An infinite set of these functions is a complete set of functions and stands for a base functions layout for the space of solutions for the equation used to generate them. Then, an approximate solutions of the initial-boundary problem have been found and compared to find out the effect of finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation.

Research limitations/implications

The methods used in the literature to find an approximate solution of any bioheat transfer problems are more complicated than the one used in the presented paper. However, it should be pointed out that there is some limitation concerning the T-function method, namely, the greater number of T-function is used, the greater condition number becomes. This limitation usually can be overcome using symbolic calculations or conducting calculations with a large number of significant digits.

Originality/value

The T-functions for the Pennes’ equation and for the SPL equation with perfusion have been reported in this paper for the first time. In the literature, the T-functions are known for other linear partial differential equations (e.g. harmonic functions for Laplace equation), but for the first time they have been derived for the two aforementioned equations. The results are discussed with respect to practical applications.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 6
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 1 March 2000

Artur Maciag and Mohammed Jehad Al‐Khatib

The inverse heat conduction problem with many internal responses is considered. After discretization with respect to time, the problem is described by a system of the Helmholtz…

240

Abstract

The inverse heat conduction problem with many internal responses is considered. After discretization with respect to time, the problem is described by a system of the Helmholtz equations in a recurrent form. An approximate solution of a heat conduction problem in an integral form is shown. Then, an approximate solution of an inverse heat conduction problem in a flat slab is presented for many internal temperature responses. Stability of the solution with respect to the internal response errors is investigated for two cases: when the integrals are calculated with the use of the average value theorem and when they are calculated numerically. Analysis of the norm of a matrix that is essential for the solution stability shows that in the case of three sensors the norm slightly changes with change of the middle sensor location. If more than three sensors are taken into consideration, the results practically will not change comparing to the case of three sensors. The internal temperature response errors are suppressed if the time step is greater than a certain critical value.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 10 no. 2
Type: Research Article
ISSN: 0961-5539

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