Wyniki wyszukiwania - BazTech

1

Integro-differential form of the first-order dual phase lag heat transfer equation and its numerical solution using the Control Volume Method

Ciesielski M., Mochnacki B., Majchrzak E.

Archives of Mechanics

|

2020

|

Vol. 72, nr 5

415--444

EN

The start point of the dual phase lag equation (DPLE) formulation is the generalized Fourier law in which two positive constants (the relaxation and thermalization times) appear. This type of equation can be used (among others) to describe the heat conduction processes proceeding in micro-scale. Depending on the number of components in the development of the generalized Fourier law into a power series, one can obtain both the first-order DPLE and the second-order one. In this paper the first-order dual phase lag equation is considered. The primary objective of this research is the transformation of DPLE differential form to the integro-differential one supplemented by the appropriate boundary-initial conditions. The obtained form of the differential equation is much simpler and more convenient at the stage of numerical computations – the numerical algorithm based on the three-time-level scheme reduces to the two-time-level one. To find the numerical solution, the Control Volume Method is used (the heating of thin metal film subjected to a laser beam is considered). The choice of the numerical method was not accidental. The method has a simple physical interpretation ensuring the preservation of the local and global energy balances. To our knowledge, it has not been used so far in this type of tasks. In the final part of the paper the examples of numerical simulations are presented and the conclusions are formulated.

2

Review of the selected problems connected with the thermal comfort analysis

Mochnacki B., Majchrzak E.

Zeszyty Naukowe Wyższej Szkoły Zarządzania Ochroną Pracy w Katowicach

|

2018

|

Nr 1(14)

117--126

EN

Thermal comfort according to the ISO 7730 standards is defined in the following way: it is the situation of the mind condition expressing satisfaction with the thermal situation in man’s environment (thermal neutrality). This definition is of the verbal character and specialists put a lot of effort to express it in the language of mathematics and physics. The work will present the bases of the numerical approach to thermal comfort and particularly the PMV (Predicted Mean Vote) parameter determining comfort conditions in the seven-level scale (or derogations from these conditions). The present work is of the overview character and the appropriate equations leading to determining PMV were taken from literature. The authors’ own contribution is a computer programme in the Delphi language supporting calculating this indicator.

PL

Komfort cieplny wg standardów ISO 7730 [4] definiowany jest następująco: jest to sytuacja stanu umysłu wyrażająca satysfakcję z termicznej sytuacji w otoczeniu człowieka (neutralność cieplna). Ta definicja ma charakter werbalny i specjaliści włożyli wiele wysiłku, aby wyrazić ją językiem matematyki i fizyki. W pracy przedstawione zostaną podstawy liczbowego podejścia do problemu komfortu cieplnego, a w szczególności parametru PMV (Predicted Mean Vote) determinującego w siedmio-stopniowej skali warunki komfortu (lub odstępstwa od tych warunków). Prezentowana praca ma charakter przeglądowy i odpowiednie równania prowadzące do wyznaczenia PMV zaczerpnięto z literatury. Udziałem własnym autorów jest program komputerowy w języku Delphi wspomagający obliczenia tego wskaźnika.

3

Implicit scheme of the finite difference method for the second-order dual phase lag equation

Majchrzak E., Mochnacki B.

Journal of Theoretical and Applied Mechanics

|

2018

|

Vol. 56 nr 2

393--402

EN

The second-order dual phase lag equation (DPLE) as a mathematical model of the microscale heat transfer is considered. It is known that the starting point determining the final form of this equation is the generalized Fourier law in which two positive constants (the relaxation and thermalization times) appear. Depending on the order of the generalized Fourier law expansion into the Taylor series, different forms of the DPLE can be obtained. As an example of the problem described by the second-order DPLE equation, thermal processes proceeding in the domain of a thin metal film subjected to a laser pulse are considered. The numerical algorithm is based on an implicit scheme of the finite difference method. At the stage of numerical modeling, the first, second and mixed order of the dual phase lag equation are considered. In the final part of the paper, examples of different solutions are presented and conclusions are formulated.

4

Application of numerical methods for solving the non-Fourier equations. A review of our own and collaborators’ works

Majchrzak E., Mochnacki B.

Journal of Applied Mathematics and Computational Mechanics

|

2018

|

Vol. 17, nr 2

43--50

EN

Thermal processes occuring in the solid bodies are, as a rule, described by the well-known Fourier equation (or the system of these equations) supplemented by the appropriate boundary and initial conditions. Such a mathematical model is sufficiently exact to describe the heat transfer processes in the macro scale for the typical materials. It turned out that the energy equation based on the Fourier law has the limitations and it should not be used in the case of the microscale heat transfer and also in the case of materials with a special inner structure (e.g. biological tissue). The better approximation of the real thermal processes assure the modifications of the energy equation, in particular the models in which the so-called lag times are introduced. The article presented is devoted to the numerical aspects of solving these types of equations (in the scope of the microscale heat transfer). The results published by the other authors can be found in the references posted in the works cited below.

5

Implicit scheme of the finite difference method for 1D dual-phase lag equation

Majchrzak E., Mochnacki B.

Journal of Applied Mathematics and Computational Mechanics

|

2017

|

Vol. 16, nr 3

37--46

EN

The 1D dual-phase lag equation (DPLE) is solved using the implicit FDM scheme. The dual phase lag equation is the hyperbolic PDE and contains a second order time derivative and higher order mixed derivative in both time and space. The DPLE results from the generalization of the well known Fourier law in which the delay times are taken into account. So, in the equation discussed, two positive parameters appear. They correspond to the relaxation time τq and the thermalization time τ T. The DPLE finds, among others, the application as the mathematical description of the thermal processes proceeding in the micro-scale. In the paper, the numerical solution of DPLE based on the implicit scheme of the FDM is presented. The authors show that a such an approach in the case of DPLE leads to the unconditionally stable differential scheme.

6

Numerical model of thin metal film heating using the boundary element method

Majchrzak E., Mochnacki B.

Computer Methods in Materials Science

|

2017

|

Vol. 17, No. 1

12--17

EN

The subject of the paper is connected with the microscale heat transfer proceeding in the metal domain. In particular, the heating process of thin metal film subjected to an external heat flux is analysed. Thermal processes in the domain considered are described by the dual-phase lag equation (DPLE) supplemented by the appropriate boundary and initial conditions. At the stage of numerical modeling the variant of the boundary element method called the BEM using discretization in time is applied. So far, this method has not been used fo the hyperbolic equations describing the microscale heat transfer. In the final part the example of computations is shown.

PL

Temat pracy jest związany z mikroskalowym przepływem ciepła zachodzącym w ultracienkich warstwach metalowych. W szczególności rozpatruje się nagrzewanie warstwy poddanej działaniu zewnętrznego strumienia ciepła o zadanej wydajności. Procesy cieplne zachodzące w rozpatrywanym obszarze opisano wykorzystując równanie z dwoma czasami opóźnień uzupełnione odpowiednimi warunkami brzegowo-początkowymi. Na etapie obliczeń numerycznych wykorzystano tzw. kombinowany wariant metody elementów brzegowych. Jak dotąd, metoda ta nie była używana do przybliżonego rozwiązywania hiperbolicznych równań różniczkowych cząstkowych opisujących przepływ ciepła w mikroskali. W końcowej części pracy pokazano wyniki obliczeń numerycznych.

7

Numerical analysis of thermal processes in the system protective clothing – biological tissue subjected to an external heat flux

Mochnacki B., Duda M.

Journal of Applied Mathematics and Computational Mechanics

|

2016

|

Vol. 15, nr 2

85--93

EN

The non-homogeneous fragment of biological tissue is considered. Its shape roughly corresponds to the fragment of cross-section of the upper or lower limb. The tissue domain is protected by a layer of protective clothing. The purpose of numerical computations is to examine the effectiveness of the clothing insulation layer on the action of the external heat fluxes of differing intensity. Thermal processes in the tissue domain are described by the system of the Pennes equations. This system is supplemented by the appropriate boundary-initial conditions and the energy equations determining the transient temperature field in the fabric and air gap sub-domains (the air gap is treated as a solid body). At the stage of numerical computations, the program MSC. Marc has been used. In the final part of the paper, the examples of numerical simulations and also the conclusions are presented.

8

Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method

Majchrzak E., Mochnacki B.

Journal of Applied Mathematics and Computational Mechanics

|

2016

|

Vol. 15, nr 3

89--96

EN

The dual-phase lag equation (DPLE) is considered. This equation belongs to the group of hyperbolic PDE, contains a second order time derivative and higher order mixed derivative in both time and space. From the engineer’s point of view, the DPLE results from the generalized form of the Fourier law. It is applied as a mathematical model of thermal processes proceeding in the micro-scale and also in the case of bio-heat transfer problem analysis. At the stage of numerical computations the different approximate methods of the PDE solving can be used. In this paper, the authors present the considerations concerning the stability conditions of the explicit scheme of finite difference method (FDM). The appropriate conditions have been found using the von Neumann analysis. In the final part of the paper, the results of testing computations are shown.

9

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

|

2016

|

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.

10

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

|

2016

|

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.

11

Sensitivity of transient temperature field in domain of forearm insulated by protective clothing with respect to perturbations of external boundary heat flux

Mochnacki B., Ciesielski M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2016

|

Vol. 64, nr 3

591--598

EN

The problem discussed in the paper is numerical modeling of thermal processes in the domain of biological tissue secured by a layer of protective clothing being in thermal contact with the environment. The cross-section of the forearm (2D problem) is treated as non-homogeneous domain in which the sub-domains of skin tissue, fat, muscle and bone are distinguished. The air gap between skin tissue and protective clothing is taken into account. The process of external heating is determined by Robin boundary condition and sensitivity analysis with respect to the perturbations of heat transfer coefficient and ambient temperature is also discussed. Both the basic boundary-initial problem and the sensitivity problems are solved by means of control volume method using Voronoi polygons.

12

Estimation of Cast Iron Substitute Thermal Capacity Using the Experimental Data

Majchrzak E., Mendakiewicz J., Mochnacki B.

Archives of Metallurgy and Materials

|

2016

|

Vol. 61, iss. 1

307--314

EN

In the paper the problem of the cast iron substitute thermal capacity estimation is discussed. This parameter appears when the macroscopic mathematical model of alloys solidification bases on the one domain method (fixed domain approach). In the case of cast iron the form of function describing the course of temperature-dependent thermal capacity is quite complex. Using the experimental data, in particular the measured cooling, heating curves at the set of points selected in the casting – mould domain the identification problem has been solved using the gradient methods. The results presented concern the gray iron 3.21% C and 1.9% Si.

13

Modeling of Melting and Resolidification in Domain of Metal Film Subjected to a Laser Pulse

Majchrzak E., Mochnacki B.

Archives of Foundry Engineering

|

2016

|

Vol. 16, iss. 1

41--44

EN

Thermal processes in domain of thin metal film subjected to a strong laser pulse are discussed. The heating of domain considered causes the melting and next (after the end of beam impact) the resolidification of metal superficial layer. The laser action (a time dependent bell-type function) is taken into account by the introduction of internal heat source in the energy equation describing the heat transfer in domain of metal film. Taking into account the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered, the mathematical model of the process is based on the dual phase lag equation supplemented by the suitable boundary-initial conditions. To model the phase transitions the artificial mushy zone is introduced. At the stage of numerical modeling the Control Volume Method is used. The examples of computations are also presented.

14

Dual Phase Lag Model of Melting Process in Domain of Metal Film Subjected to an External Heat Flux

Mochnacki B., Ciesielski M.

Archives of Foundry Engineering

|

2016

|

Vol. 16, iss. 4

85--90

EN

Heating process in the domain of thin metal film subjected to a strong laser pulse are discussed. The mathematical model of the process considered is based on the dual-phase-lag equation (DPLE) which results from the generalized form of the Fourier law. This approach is, first of all, used in the case of micro-scale heat transfer problems (the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered). The external heating (a laser action) is substituted by the introduction of internal heat source to the DPLE. To model the melting process in domain of pure metal (chromium) the approach basing on the artificial mushy zone introduction is used and the main goal of investigation is the verification of influence of the artificial mushy zone ‘width’ on the results of melting modeling. At the stage of numerical modeling the author’s version of the Control Volume Method is used. In the final part of the paper the examples of computations and conclusions are presented.

15

Simulations of thermal processes in tooth proceeding during cold pulp vitality testing

Ciesielski M., Mochnacki B., Siedlecki J.

Acta of Bioengineering and Biomechanics

|

2016

|

Vol. 18, no. 3

33--41

EN

Purpose: This paper deals with the mathematical modeling of the thermal processes occurring in the tooth, during a very brief contact (a few seconds) with a very cold liquid on a part of the tooth crown. In this way one can simulate a heat transfer in tooth proceeding during a dental diagnostic test - pulp vitality testing. The impact of rapid ambient thermal changes acting on the tooth can cause toothache. Methods: The mathematical model: a system of partial differential equations with initialboundary conditions (the axially-symmetrical problem) and their numerical solutions using the control volume method is discussed. Results: Simulation results of the kinetics of the temperature changes inside the tooth are presented. The example of the control volume mesh (using the Voronoi polygons) well describing the shape of a molar tooth is given. Conclusions: The simulation results (the temperature distribution in the tooth at any moment of the simulation time and the kinetics of temperature variation at the points of the considered tooth domain) can help dentists in the selection of an appropriate method of treatment.

16

3D model of thermal interactions between human forearm and environment

Duda M., Mochnacki B.

Journal of Applied Mathematics and Computational Mechanics

|

2015

|

Vol. 14, nr 3

17--23

EN

Thermal processes in the domain of a human forearm are considered. The external surface of forearm is in the direct thermal contact with the environment. The steady state problem is considered. From the mathematical point of view, the task is described by the system of the Poisson-type equations, the boundary conditions given on the contact surface between tissue sub-domains, the boundary conditions determining heat transfer between blood vessels and tissue and the boundary conditions on the external surface of the system. The non-homogeneous forearm domain is reconstructed as accurately as possible (3D task). At the stage of numerical modelling, the finite element method has been used. In the final part of the paper the example of computations is presented.

17

Micro-scale heat transfer : algorithm basing on the Control Volume Method and the identification of relaxation and thermalization times using the search method

Mochnacki B., Ciesielski M.

Computer Methods in Materials Science

|

2015

|

Vol. 15, No.2

353--361

EN

The thermal processes proceeding in micro-domains can be described, among others, using the dual phase lag model (DPLM). According to the newest opinions the DPLM constitutes a very good description of the real heat transfer processes proceeding in the micro-scale, in particular on account of extremely short duration, extreme temperature gradients and the very small geometrical dimensions of domain considered. The base of DPLM formulation is a generalized form of Fourier law in which two times τq, τγ appear (the relaxation time and thermalization one, respectively). The numerical solution of the problem discussed bases on the author’s version of the Control Volume Method adapted to resolve the hyperbolic partial differential equations. The example illustrating the method application concerns the estimation of τq and τγusing the algorithm basing on the search method and the thin metal film subjected to the laser pulse is considered.

PL

Procesy cieplne zachodzące w mikro-obszarach mogą być opisane między innymi za pomocą modelu matematycznego z dwoma czasami opóźnień (DPLM). Według najnowszych opinii, model DPLM stanowi bardzo dobry opis rzeczywistych procesów przepływu ciepła w mikroskali, w szczególności ze względu na ekstremalnie krótki czas ich trwania, ekstremalne gradienty temperatury i bardzo małe wymiary geometryczne rozważanego obszaru . Podstawą formułowania DPLM jest uogólnienie prawa Fouriera, w którym występują dwa czasy opóźnień τq i τγ (odpowiednio-czas relaksacji i termalizacji). Numeryczne rozwiązanie omówionego zagadnienia opiera się na autorskiej wersji Metody Bilansów Elementarnych dostosowanej do rozwiązywania hiperbolicznych równań różniczkowych cząstkowych. Przykład ilustrujący zastosowanie metody dotyczy oszacowania czasów τq i τγ za pomocą algorytmu opartego na metodzie przeszukiwania, oraz rozpatrywana jest cienka folia metalowa poddawana działaniu impulsu laserowego.

18

Numerical Model Of Binary Alloys Solidification Basing On The One Domain Approach And The Simple Macrosegregation Models

Majchrzak E., Mochnacki B., Mendakiewicz J.

Archives of Metallurgy and Materials

|

2015

|

Vol. 60, iss. 3

2431--2435

EN

In the paper the thermal processes proceeding in the domain of solidifying binary alloy are considered. The mathematical model of solidification and cooling processes bases on the one domain method (or fixed domain method). In such a model the parameter called a substitute thermal capacity (STC) appears. At the stage of STC construction the macrosegregation process described by the lever arm rule or the Scheil model is taken into account. In this way one obtains the formulas determining the course of STC resulting from the certain physical considerations and this approach seems to be closer to the real course of thermal processes proceeding in domain of solidifying alloy. In the final part the examples of numerical solutions basing on the finite difference method are presented.

PL

W pracy rozpatruje się procesy cieplne zachodzące w obszarze krzepnącego i stygnącego stopu dwuskładnikowego. Model matematyczny tych procesów bazuje na podejściu nazywanym metodą jednego obszaru. W modelach tego typu pojawia się parametr nazywany zastępczą pojemnością cieplną. Na etapie jej definiowania autorzy uwzględnili proste modele makrosegregacji wynikające z reguły dźwigni i znanego modelu Scheila. Otrzymane zależności determinujące przebiegi pojemności zastępczej na podstawie pewnych rozważań fizycznych wydają się lepiej przybliżać rzeczywisty przebieg procesów cieplnych zachodzących w obszarze krzepnącego stopu. W końcowej części pracy pokazano wyniki rozwiązań numerycznych uzyskanych przy wykorzystaniu metody różnic skończonych.

19

The BEM-FDM model of thermal processes proceeding in the domain of the human finger

Majchrzak E., Mochnacki B., Tarasek D., Dziewoński M.

Acta of Bioengineering and Biomechanics

|

2015

|

Vol. 17, no. 4

85--96

EN

Purpose: The problem of the numerical modeling of thermal processes proceeding in the non-homogeneous domain of the human finger is discussed. The domain considered constitutes the assembling of soft and bone tissues and the system of supplying blood vessels (arteries and veins). The mathematical description of the process analyzed corresponds to the so-called vascular models. Methods: At the stage of numerical modeling the algorithm being the composition of the boundary element method (BEM) and the finite difference method (FDM) is applied. Results: The algorithm presented allows one to determine the steady state temperature field in the finger domain in natural convection conditions. To verify the effectiveness and exactness of the method of the problem solution, the thermal imaging measurements of the finger surface temperature have been done. Conclusions: The compatibility of numerical and experimental results (the natural convection conditions) has proved to be quite satisfactory. It is possible to use the algorithm proposed for the modeling of thermal processes proceeding in the conditions of low or high ambient temperatures and the big values of heat transfer coefficients. The impact of protective clothing on the temperature field in the domain of the finger can also be analyzed.

20

Application of the Control Volume Method using the Voronoi polygons for numerical modeling of bio-heat transfer processes

Ciesielski M., Mochnacki B.

Journal of Theoretical and Applied Mechanics

|

2014

|

Vol. 52 nr 4

927--935

EN

In the paper, the problem concerning the numerical modeling of thermal processes in the domain of a biological tissue being in thermal contact with the environment is discussed. The changing ambient temperature causes that the non-steady heat transfer process is con- sidered. The cross-section of the forearm (2D problem) is treated as a non-homogeneous domain in which the sub-domains of skin tissue, fat, muscle and bone are distinguished. From the mathematical point of view, the boundary-initial problem described by the system of energy equations (the Pennes equations), boundary conditions on the external surface of the system, boundary conditions on the surfaces limiting the successive sub-domains and the initial condition is analyzed. At the stage of numerical computations, the Control Volume Method using the Voronoi polygons is applied. In the final part of the paper, examples of computations are shown.