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Cattaneo-Vernotte bio-heat transfer equation. Identification of external heat flux and relaxation time in domain of heated skin tissue

Mochnacki Bohdan, Paruch Marek

Computer Assisted Methods in Engineering and Science

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2018

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Vol. 25, no. 2-3

71--80

EN

A cylindrical skin tissue domain subjected to an external heat flux is considered. Thermal processes in the domain considered are described by the Cattaneo-Vernotte equation supplemented by the appropriate boundary and initial conditions. The aim of considerations is the identification of external heat flux and relaxation time on the basis of ‘measured’ heating/cooling curves at the set of selected points located on the surface of the skin. The direct problem is solved using the implicit scheme of the Finite Difference Method (FDM), while at the stage of the inverse problem solution, the evolutionary algorithm is applied. In the final part of the paper the examples of computations are presented.

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Comparison of bio-heat transfer numerical models based on the Pennes and Cattaneo-Vernotte equations

Ciesielski M., Duda M., Mochnacki B.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

33--38

EN

The homogeneous soft tissue domain subjected to an external heat source is considered. Thermal processes in this domain are described using the well known Pennes equation and next the Cattaneo-Vernotte one. Within recent years the prevailing view is that the Cattaneo-Vernotte equation better describes the thermal processes proceeding in the biological tissue (it results from the specific internal tissue structure). Appearing in this equation the delay time of heat flux with respect to the temperature gradient (τq) is of the order of several seconds and the different values of τq are taken into account. At the stage of numerical modeling the finite difference method is used. In the final part of the paper, the examples of computations are shown.

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Cattaneo-Vernotte bioheat transfer equation. Stability conditions of numerical algorithm based on the explicit scheme of the finite difference method

Mochnacki B., Tuzikiewicz W.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

137-144

EN

The Cattaneo-Vernotte (CVE) equation is considered. This equation belongs to the group of hyperbolic PDE. Supplementing this equation by two additional terms corresponding to perfusion and metabolic heat sources one can apply the CVE as a mathematical model describing the heat transfer processes proceeding in domain of the soft tissue. Such an approach is recently often preferred substituting the classical Pennes model. At the stage of numerical computations the different numerical methods of the PDE solving can be used. In this paper the problems of stability conditions for the explicit scheme of the finite difference method (FDM) are discussed. The appropriate condition limiting the admissible time step have been found using the von Neumann analysis.

4

Cattaneo-Vernotte equation : identification of relaxation time using evolutionary algorithms

Mochnacki B., Paruch M.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 4

97--102

EN

The Cattaneo-Vernotte equation describing the heat conduction process in domain of solid body results from the generalization of the well - known Fourier law, in which the delay time’ (relaxation time τq) is introduced. The Cattaneo-Vernotte equation should be, among others, used in a case of microscale heat transfer analysis when the thermal processes are characterized by the extremely short duration (e.g. ultrafast laser pulse), the considerable temperature gradients and the very small dimensions (e.g. thin metal film). In the paper the problem of relaxation time identification is considered. In particular, the heat conduction process proceeding in domain of thin metal film subjected to a laser pulse is analyzed. The inverse problem solution is obtained using the evolutionary algorithms. The information concerning the time-dependent temperature distribution on the surface of metal film is assumed to be known. At the stage of numerical computations the finite difference method (FDM) is applied. In the final part of the paper the example of computations is shown.

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The finite difference method for fractional Cattaneo-Vernotte equation

Ciesielski M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2009

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Vol. 8, nr 1

13-18

EN

In this work a numerical solution of modified Cattaneo-Vernotte equation is presented. This equation is obtained by replacing the second order time derivative by the fractional derivative in Caputo sense. In order to solve the problem with classical boundary-initial conditions, the finite difference method is applied. In the final part of the paper the examples of computations are shown.

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Modelowanie rozkładu temperatury w piecu koksowniczym

Kardaś D., Polesek-Karczewska S., Mertas B.

Karbo

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2009

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Nr 3

151-155

PL

Analiza transportu ciepła w piecu koksowniczym ma istotne znaczenie z punktu widzenia modelowania procesu koksowania, gdyż jakość koksu oraz żywotność baterii koksowniczej zależy przede wszystkim od dynamiki zmian pola temperatury w warstwie wsadu węglowego. Koksowanie jest procesem, któremu towarzyszą nie tylko zjawiska chemiczne, jak piroliza czy reakcje pomiędzy składnikami uwolnionych gazów, ale i zjawiska fizyczne związane z przepływem gazów pirolitycznych, wymianą ciepła i przemianami fazowymi produktów procesu. Kompletny opis wymagałbyzatem zastosowania bardzo złożonych modeli. Prostszym podejściem do zagadnienia modelowania procesu koksowania jest opis wybranego zjawiska. W niniejszej pracy skupiono się na transporcie ciepła. W ogólności, w procesie koksowania udział biorą trzy mechanizmy wymiany ciepła - przewodzenie, konwekcja i promieniowanie. Do celów przedstawionej analizy przyjęto, że transport ciepła odbywa się jedynie na drodze przewodzenia. Zaproponowano uproszczony jednowymiarowy model transportu ciepła w komorze koksowniczej, oparty na konstytutywnym równaniu Cattaneo-Vernotte'a, wiążącym gęstość strumienia ciepła z gradientem temperatury. Warstwę węgla potraktowano jako ośrodek dwuskładnikowy typu ciało stałe-gaz, wprowadzając jego uśrednione (efektywne) własności termofizyczne na podstawie przyjętej porowatości. Postawione zagadnienie brzegowo-początkowe rozwiązano stosując metodę objętości kontrolnych dla równania energii oraz metodę różnic skończonych dla równania ewolucji strumienia. Uzyskane rezultaty wskazały na falowy charakter transportu ciepła w złożu węgla i są jakościowo zgodne z danymi pomiarowymi zamieszczonymi w literaturze.

EN

Due to the fact that coke quality, as well as the coke oven battery life depend mainly on the dynamics of temperature field changes in the coal layer during the coking process, the analysis of heat transfer problem in a coke oven plays a significant role in view of modelling the process. Coking is a process accompanied not only by chemical phenomena such as pyrolysis or reactions between the released gases components, but also by physical ones connected with flow of pyrolytic gases, heat transfer and phase transitions of process products. Thus, complete description would involve application of complex models. The simpler approach is to model coking process basing on the description of chosen phenomenon. Studies reported in the paper focus on the heat transfer problem. Generally, three mechanisms of heat exchange - conduction, convection and radiation - take part in the coking process. For the purpose of presented analysis it was assumed that heat is transferred only by conduction. The one-dimensional model of heat transfer in a coking chamber, based on the constitutive Cattaneo-Vernotte equation for the heat flux density, is proposed in the paper. Coal charge was treated as a two-component medium of type solid-gas and its averaged (effective) thermophysical properties, based on assumed porosity were introduced. Considered boundary-initial value problem was solved applying the Finite Volume Method for the energy balance equation and Finite Difference Method for heat evolution equation. Obtained results indicate on the wavy character of heat transport in a bed of coal and are in qualitative agreement with the measurement data presented in the literature.

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Application of the DRBEM for numerical solution of Cattaneo-Vernotte bioheat transfer equation

Majchrzak E., Kałuża G., Poteralska J.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2008

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Vol. 7, nr 1

111-120

EN

The 1D problem of heat transfer in the tissue subjected to action of external heat source is considered. This phenomenon is described by Cattaneo-Vernotte equation supplemented by adequate boundary and initial conditions. To solve the problem formulated the dual reciprocity boundary element method (DRBEM) is applied. In the final part of the paper the examples of computations are shown.