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  1. 51 Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
  2. 51 Scientific Research of the Institute of Mathematics and Computer Science
  3. 28 Zeszyty Naukowe. Mechanika / Politechnika Opolska
  4. 12 Archives of Foundry Engineering
  5. 11 Archives of Foundry
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  1. 166 Majchrzak E.
  2. 41 Mochnacki B.
  3. 27 Kałuża G.
  4. 24 Freus K.
  5. 20 Mendakiewicz J.
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  1. 1 2020
  2. 1 2019
  3. 3 2018
  4. 3 2017
  5. 4 2016
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1
Content available 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
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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
Content available remote Second-order two-temperature model of heat transfer processes in a thin metal film subjected to an ultrashort laser pulse
Majchrzak E., Dziatkiewicz J.
Archives of Mechanics
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2019
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Vol. 71, nr 4-5
377--391
EN
Thermal processes in domain of thin metal film subjected to an ultrashort laser pulse are considered. A mathematical description of the process discussed is based on the system of four equations. Two of them describe the electrons and lattice temperature, while third and fourth equations represent the generalized Fourier law, it means the dependencies between the electrons (lattice) heat flux and the electrons (lattice) temperature gradient. In the generalized Fourier law the heat fluxes are delayed in relation to the temperature gradients which consequently causes the appearance of heat fluxes time derivatives in the appropriate equations. Depending on the order of the generalized Fourier law expansion into the Taylor series, the first- and the second-order model can be obtained. In contrast to the commonly used first-order model, here the second-order two-temperature model is proposed. The problem is solved using the implicit scheme of the finite difference method. The examples of computations are also presented. It turns out that for the low laser intensities the results obtained using the first- and the second-order models are very similar.
3
Content available Review of the selected problems connected with the thermal comfort analysis
Mochnacki B., Majchrzak E.
Zeszyty Naukowe Wyższej Szkoły Zarządzania Ochroną Pracy w Katowicach
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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.
4
Content available 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.
5
Content available 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
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2018
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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.
6
Content available Implicit scheme of the finite difference method for 1D dual-phase lag equation
Majchrzak E., Mochnacki B.
Journal of Applied Mathematics and Computational Mechanics
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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.
7
Content available remote Numerical model of thin metal film heating using the boundary element method
Majchrzak E., Mochnacki B.
Computer Methods in Materials Science
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2017
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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.
8
Content available remote Analysis of thermal processes occurring in heated multilayered metal films using the dual-phase lag model
Majchrzak E., Kałuża G.
Archives of Mechanics
|
2017
|
Vol. 69, nr 4-5
275--287
EN
A multilayered thin metal film subjected to an ultra-short laser pulse is considered. A mathematical description of the discussed process is based on the system of the dual-phase lag equations supplemented by appropriate boundary and initial conditions. Special attention is devoted to the ideal contact conditions at the interfaces between the layers, which in the case of the dual-phase lag model must be formulated in a different way than in the macroscopic Fourier model. To solve the problem the explicit scheme of the finite difference method is developed. In the final part of the paper the example of computations is shown.
9
Content available 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.
10
Content available 1D generalized dual-phase lag equation. Sensitivity analysis with respect to the porosity
Kałuża G., Majchrzak E., Turchan Ł.
Journal of Applied Mathematics and Computational Mechanics
|
2016
|
Vol. 15, nr 1
49--58
EN
Thermal processes occurring in the heated tissue are described by the 1D generalized dual-phase lag equation supplemented by appropriate boundary and initial conditions. Using the sensitivity analysis method, the additional problem connected with the porosity is formulated. Both problems are solved by means of the explicit scheme of the finite difference method. In this way it is possible to estimate the temperature changes due to the perturbation of porosity. In the final part of the paper, the example of computation is shown and the conclusions are formulated.
11
Content available 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.
12
Content available 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.
13
Content available Heat flux formulation for 1D dual-phase lag equation
Majchrzak E., Kałuża G.
Journal of Applied Mathematics and Computational Mechanics
|
2015
|
Vol. 14, nr 1
71--78
EN
The thin metal film subjected to the ultra-short laser pulse is analyzed. Heat transfer processes occurring in the domain considered are described by the dual-phase lag model in which the unknown is the heat flux, not, as usual, temperature. This approach is especially convenient in the case of Neumann boundary conditions, which are taken into account here. The mathematical model supplemented by initial conditions is solved using the explicit scheme of finite difference method. In the final part of the paper the examples of computations are shown and the conclusions are formulated.
14
Content available Analysis of ultrashort laser pulse interactions with metal films using a two-temperature model
Majchrzak E., Dziatkiewicz J.
Journal of Applied Mathematics and Computational Mechanics
|
2015
|
Vol. 14, nr 2
31--39
EN
In the paper the problem of thin metal film subjected to the action of the high laser fluence and the ultrashort pulse width is considered. The mathematical model consists of the equations describing the electrons and phonons temperatures and the relationships between the heat fluxes and temperature gradients of electrons and phonons. The problem is solved using the explicit scheme of the finite difference method with staggered grid. In the final part the results of computations and conclusions are presented.
15
Content available 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.
16
Content available remote Modeling of skin tissue heating using the generalized dual phase-lag equation
Majchrzak E., Turchan Ł., Dziatkiewicz J.
Archives of Mechanics
|
2015
|
Vol. 67, nr 6
417--437
EN
This paper concerns the numerical modeling of skin tissue heating. To describe the analyzed process the system of three generalized dual phase-lag equations corresponding to the successive layers of the skin: epidermis, dermis and sub-cutaneous region is applied. On the surfaces between the layers the ideal thermal contact is assumed, on the skin surface the Neumann condition describing the external heating of tissue can be accepted, and on the remaining surfaces the no-flux condition is taken into account. Initial temperature of the tissue and the blood is known. The problem is solved using the explicit scheme of finite difference method. In the final part of the paper the results of computations are shown.
17
Content available 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.
18
Content available Temperature field in burned and healthy tissue : sensitivity analysis with respect to the thermal parameters
Freus K., Freus S., Majchrzak E.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
Vol. 13, nr 3
47--58
EN
A non-homogeneous system being the composition of burn wound and healthy tissue is considered. The heat exchange between sub-domains and environment is described by the system of partial differential equations (the Pennes equations) supplemented by the assumed boundary conditions. Additional problems associated with sensitivity analysis with respect to thermal parameters occurring in the mathematical model are formulated. Both the basic problem and additional ones concerning the sensitivity with respect to selected parameters are solved using the boundary element method. In the final part of the paper the results of computations are shown.
19
Content available Simplified model of thermal interactions between environment, protective clothing and skin tissue
Mochnacki B., Majchrzak E., Duda M.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
Vol. 13, nr 4
91--96
EN
In the paper the simplified model of thermal processes proceeding in the domain of biological tissue secured with protective clothing is discussed. In particular, the simplification of the mathematical model consists in the omission of the real layer of fabric for which the transient temperature field is determined by the Fourier equation and the introduction in this place of the additional thermal resistance appearing in the boundary condition determining the heat exchange between tissue and environment. In this way both the mathematical model of the thermal processes in the system considered and also numerical realization are greatly simplified. To verify the effectiveness of the approach proposed, the solution of the basic problem and the simplified one have been solved (1D task) using the finite difference method and the results have been compared. It turned out that the results are close and from the practical point of view such simplification is fully acceptable.
20
Content available Numerical modeling of thermal processes in the living tissue domain secured with a layer of protective clothing
Mochnacki B., Majchrzak E., Duda M.
Journal of Applied Mathematics and Computational Mechanics
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2014
|
Vol. 13, nr 1
87--94
EN
In the paper the problem of thermal processes proceeding in the domain of biological tissue secured with protective clothing is discussed. In particular, the mathematical model of heat exchange corresponding to conditions of high temperature in the system environment - layer of protective clothing - air gap - skin tissue is formulated in the form of a certain boundary - initial problem. Next, the numerical algorithm based on the boundary element method is presented. In the final part of the paper the examples of numerical simulations are shown.
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