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1

Thermo-diffusion MHD convection in enclosure using heat and mass lines visualization techniques

Kushawaha Durgesh, Yadav Sushil, Singh Dwesh K., Behera Pravata Kumar

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

63--76

EN

The numerical study of two-dimensional laminar thermo-diffusion natural convection in an exponentially heated and concentrated square enclosure of unit length in the presence of a uniform horizontal magnetic field is presented in this paper. The left and right vertical walls are assumed to have higher and lower temperatures and concentrations, respectively, and are governed by exponential functions, whereas the horizontal walls are assumed to be adiabatic and non-diffusive. The mathematical formulation of heat and mass functions has been completed, and heat and mass line contours have been drawn based on these functions to investigate the behavior of heat and mass in the cavity. The flow governing equations were solved using a finite difference method in conjunction with the Successive Over-Relaxation (SOR) technique and then converted to a vorticity-stream function form. A detailed comparison of isotherms with heatlines and isosolutes with masslines has been performed. Furthermore, the reduction for lower Rayleigh numbers Ra surpassing the reduction for higher values of Ra. The maximum reduction in overall heat and mass transfer has been observed for higher Hartmann (Ha = 8).

2

Więzy termomechaniczne w teorii gradientowej

Kubik J.

Roczniki Inżynierii Budowlanej

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2013

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z. 13

111--114

EN

The theory of thermomechanical constraints is used in the work in order to derive particular forms of equations for the theory of gradient thermodiffusion. The constraints are considered for the material medium in which the gradient components of the equations do not cause additional energy dissipation.

3

Odpowiedniość równań mikropolarnej teorii mieszanin i termodyfuzji

Kubik J.

Roczniki Inżynierii Budowlanej

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2012

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z. 12

21--24

EN

The mass and momentum balances in micropolar mixture theory and thermodiffusion is analysed. One can conclude that not only local and global mass balances should be fulfilled but also local and global inertia moment should be considered. As the consequence we should make similar generalization in reminding balance.

4

Wybrane problemy wzbogacania chromem stali S235JR i S355J2 przy zastosowaniu lasera CO₂

Górka A., Kocańda D.

Biuletyn Wojskowej Akademii Technicznej

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2012

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Vol. 61, nr 2

297-310

PL

Zagadnieniem rozważanym w pracy jest wzbogacanie warstwy wierzchniej stali kadłubowych S235JR i S355J2 chromem w celu zwiększenia ich odporności na działania korozyjne wody morskiej. Podjęto próbę wyjaśnienia mechanizmu tworzenia stabilnych struktur w głębi warstw dyfuzyjnych przy domieszkowaniu stali chromem. Jako modelowego aktywatora termodynamicznych zjawisk, zachodzących podczas tego procesu z zastosowaniem zmodyfikowanej metody zgrzewania tarciowego FSP (Friction Stir Processing), użyto wiązki promieniowania laserowego. Analizowano dyfuzyjny transport masy, który jest pierwotnym i dominującym zjawiskiem w przemianach fazowych zachodzących w mikroobszarach o znacznym gradiencie temperatury w procesie tworzenia cienkich warstw antykorozyjnych. Przedstawiono rozkłady stężenia chromu w warstwie dyfuzyjnej stali, wynikające ze wzorów analitycznych podanych w pracy i z własnych pomiarów doświadczalnych, otrzymanych przy wydłużonym i skróconym czasie nagrzewania wiązką laserową powłoki galwanicznego chromu osadzonej na podłożu stalowym.

EN

The paper deals with the problem of the S235JR and S355J2 hull steels enriching with chromium in order to improve their corrosive resistance against sea water. The attempts were undertaken to explain the mechanism of formation of durable layers inside the diffusion zone while the steel alloying process is providing. In the research to model the process of surface layer steel modification by means of modified friction stir processing (FSP), a continuous wave CO₂ laser beam was served as a model activator of thermodynamical phenomena appearing there. Mainly, diffusive mass transport was considered in the work as it being the original and basic phenomenon affected in the phase transformations. The theoretical and experimental concentration distributions of chromium inside the diffusive layers being diffused from galvanic coating of steel matrix were performed in the paper for short and long laser exposition time. Thanks determining in the work the conditions for collaboration of the concentration diffusion component as well the complex thermodiffusive mass transport, it is possible to design the structure and the thickness of diffusive layers in the matrix.

5

Model formowania warstwy dyfuzyjnej typu Fe-Cr w polu temperatury

Górka A., Kocańda D.

Biuletyn Wojskowej Akademii Technicznej

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2012

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Vol. 61, nr 2

233-244

PL

W artykule przedstawiono model tworzenia warstw dyfuzyjnych na przykładzie chromu, jako elementu nasycającego, i żelaza jako elementu nasycanego w polu laserowej inicjacji cieplnej z zastosowaniem lasera CO₂ emitującego wiązkę o normalnym rozkładzie gęstości mocy w trybie pracy ciągłej. Zamieszczone w pracy wyniki badań weryfikacyjnych dokumentują istnienie termodyfuzyjnego strumienia masy w strefie nagrzewanego materiału. Proces ten jest dominujący w pierwszym, wyodrębnionym w tym modelu etapie tworzenia warstwy, to jest w czasie nagrzewania materiału. Obserwowane efekty oddziaływania tego strumienia na strukturę w modyfikowanej warstwie uzależnione były w pierwszym rzędzie od czasu realizacji procesu oraz od parametrów cieplno-geometrycznych obszaru wpływu ciepła, takich jak: głębokość przetopienia i grubość zastosowanej powłoki chromu nałożonego galwanicznie. Przy spełnieniu odpowiednich warunków relacji pomiędzy wymienionymi parametrami, rozkłady stężenia chromu posiadały maksima w głębi warstw, a nie na powierzchni, jak ma to miejsce w konwencjonalnych procesach dyfuzyjnych realizowanych przy stałej temperaturze.

EN

In the paper, the model of formation of the Fe-Cr diffusion layer on a carbon steel substrate in the thermal field produced by continuous laser beam with normal power density distribution is presented. Here, the enclosed results of verification investigations prove that thermodifusional mass transport is a dominant phenomenon in the first stage of the diffusion layer formation process when the chromium galvanic surface layer is under heating. The effect of the thermodiffusion flux action depends on the heating period as well as on thermo-geometrical parameters of the heat affected zone such as melting depth and the thickness of the chemical chromium layer coating on the steel substrate. While the particular relations between the above mentioned parameters of heat affected zone are satisfied then the maximum values of chromium concentration distribution can be found in the depth of diffusion layer. It seems to be on contrary to the conventional diffusion process running under constant temperature whereas the maximum concentration of chromium metal is found on the surface of diffusion layer.

6

On the magnetorotatory thermosolutal convection (MRTC) of the Veronis type in the presence of Soret effect

Mohan H., Kumar P., Singh P.

Studia Geotechnica et Mechanica

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2009

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Vol. 31, nr 2

51-63

EN

In the present paper, we mathematically prove that the Soret-driven thermosolutal convection of the Veronis type under the simultaneous effect of uniform vertical rotation and magnetic field cannot manifest itself as oscillatory motions of growing amplitude if the thermosolutal Rayleigh number Rs, the Lewis number ζ, the Prandtl number δ and the magnetic Prandtl number δ 1 satisfy the inequalities

7

Analogie równań opisujących proces termodyfuzji w ośrodku ciągłym i mikropolarnym

Wieczorek B.

Roczniki Inżynierii Budowlanej

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2008

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z. 8

93--96

EN

In the paper are presented the formulation thermodiffusion problems in viscoelastic solid with multicomponent body. There is analysed the model of the continuous medium and the micropolar medium. The balance of mass, momentum, angular momentum, energy and entropy in a solid are proposed for each of the ingredients. Next, the balance equations are obtained for the body. The physical equations of the process are obtained assuming the thermodynamic potentials. Next, the final systems of equations are obtained which describe a character of mutual reaction of heat, diffusion and stress fields.

8

Sprzężenie przepływu cieplno-dyfuzyjnego w ujęciu numerycznym

Pilśniak J., Wieczorek B.

Roczniki Inżynierii Budowlanej

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2007

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z. 7

87--90

EN

The paper is gone on the scientific research which apply to phase transition in the thermodiffusion processes. They proceed under the influence of action external agents. There was presented results of phase transition solid body to liquid body. The process is described by the system equations, which allow for mutually coupled heat and diffusion flows. The numerical solution of these equations were obtained by finite difference method of numerical analysis. The computer program was written Visual C++. The program enables the simulation of the mutually process for different boundary and initial conditions and changeable source, which generate transitions. Moreover there was made an analysis of convergence of applied computational method.

9

Rozwiązanie fundamentalne zagadnienia quasi-statycznego stacjonarnej termodyfuzji w ośrodku sprężystym o właściwościach mikropolarnych

Wieczorek B.

Roczniki Inżynierii Budowlanej

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2007

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z. 7

129--134

EN

In a paper the problem of displacement evaluation coupled with the flow of mass and heat consideration is analysed in the elastic solid with micropolar structure. The problem of quasi-static and stationary process is described diffusion and conductivity equations and systems of motion equations for displacements and rotations. The system of equations has eight partial differential equations of the second order. The searched quantities in the problem are: the temperature T, the concentration C, the displacement field ui and rotation field ϕi. There is presented the method of the solution for the equations’ system. The transformation methods are used in the construction of fundamental solutions of the system of equations. A fundamental solution of that system is built on the basis of Fourier’s transformation, its properties and theory of distribution. On their basis are obtained the solution of the initial system. The fundamental solutions and the reciprocity theorem is applied to find the solution of the analyzed initial-boundary problem.

10

Zależności między współczynnikami sprzęgającymi w procesie termodyfuzji w ośrodku o własnościach mikropolarnych

Wieczorek B.

Roczniki Inżynierii Budowlanej

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2006

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z. 6

149--152

EN

The problem of quasi-statical convoluted elastic thermodiffusion in described with the system of eight partial differential equations of the second order, describing character of mutual reaction of the heat, diffusion and stress fields. There were different forms of these equations systems, resulted of the way of fields reactions. They were derived from the different thermodynamic formulations of problem and from the way of interaction of fields for the considered problem. The thermodynamic potentials and the dependences between them, also the constitutive equations were used to specify interdependence between the coefficients, which are in the system of these equations. If there are known the experimental constants for one of the formulation thermodiffusion than the relations allow to determine the constants for the others of the formulation process. It is very importance, because there isn’t possible to carry out an experiment and obtain the applicable conjugates.

11

O termodyfuzji w polu elektrycznym

Jędrzejczyk-Kubik J.

Roczniki Inżynierii Budowlanej

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2006

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z. 6

41--45

EN

An n-th component mixture placed in electric field is analysed. It is assumed that there exists a component which has a much greater density than those of the other components. Such an assumption validates the determination of the thermodiffusion equations in the electric field. In the final step, the constitutive equations for viscoelastic thermodiffusion concerning the electric field are formulated.

12

On the thermodiffusion equation for electrically charged matter

Piekarski S.

Journal of Technical Physics

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2005

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Vol. 46, no 2

83-95

EN

In this paper, we propose the thermodiffusion equation for electrically charged matter. It is a generalisation of the modified Fick law, which has been introduced in the paper: S. Piekarski, "On the modified Fick law and its potential applications" (J. Tech. Phys., 44, 2, 125-131, 2003). In the modified Fick law there exists a potential force acting on the diffusing matter which is described as the gradient of the potential U(x). Now we replace U(x) by U(x) - q[phi](x,t) where [phi] is the electrostatic potential, while q is the electric charge of a diffusing carrier. We write the corresponding field equations and consider the electromotive force induced by the temperature gradient. It is easy to compute this force under the assumption that the system is quasineutral. However, it seems that a more realistic model of the electromotive force induced by the temperature gradient should allow a deviation from quasineutrality. Both versions are compared and it is interesting whether the resulting expressions can be verified experimentally.

13

Global solution to the initial value problem for nonlinear system of equations of thermodiffusion without displacements

Łazuka J.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

557--570

EN

In the paper we shall present the proof of global-in-time solution to the initial value problem for nonlinear partial differential equations describing physical proc-cesses of thermodiffusion without displacements. Time decay of global solution will be also shown.

14

Thin dielectric electrodiffusion masks for Ag+ - Na+ ion exchange applications

Mrozek E., Mrozek P.

Journal of Technical Physics

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2003

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Vol. 44, no 3

289-294

EN

A new method of thin dielectric mask preparation for Ag+ - Na+ ion exchange applications was developed. Electrodiffusion bonding of Al-film to glass was used to obtain very thin dielectric Al2O3 layer in the Al-glass transition zone. After Al removal, the quality of the dielectric mask was tested in Ag pure thermal diffusion and electrodiffusion processes in molten AgNO3.

15

On the modified Fick law and its potential applications

Piekarski S.

Journal of Technical Physics

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2003

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Vol. 44, no 2

125-131

EN

It can be observed that the standard Einstein-Smoluchowski equation for isothermal processes can be written in the two equivalent forms which, however, become distinct when the temperature becomes spatially varying. Therefore, we have two different equations which could potentially describe the flux of diffusing particles subjected to the potential external forces and variable temperature. The first is the standard Einstein-Smoluchowski equation, while the second is its modification containing an additional term proportional to the temperature gradient. In order to fix the terminology, the above mentioned modification of standard Einstein-Smoluchowski equation shall be called '^the modified Fick law". In this paper, we shall discuss the origin of this equation and we shall try to make a preliminary discussion of its solutions and potential applications to thermodiffusion. We propose two models: one exploiting Onsager's thermodynamics and the second one independent from it. Elementary solutions of both models are identical in the limit of small concentrations.

16

On diffusion and thermodiffusion in a gravity field

Piekarski S.

Journal of Technical Physics

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2003

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Vol. 44, no 3

329-337

EN

The modified Fick law has been introduced in the paper: S. Piekarski, On the modified Fick law and its potential applications (J. Tech. Phys., 44, 2, 125-131, 2003). The first purpose of the present paper is to give a simpler "derivation" of the modified Fick law than that originally presented. Namely, it can be introduced as a particular case of the Wojnar equation (R. Wojnar, "Nonlinear heat equation and thermodiffusion", p. 296, Eq. (2.1), Rep. on Math. Phys., No 1/2, Vol. 46, 2000). The third equation discussed is the Einstein-Smoluchowski equation. All equations mentioned above, contain a potential for the force, acting on the diffusing matter. The important fact is that, for isothermal conditions, all the equations mentioned above are identical. In this article we try to describe the effects of the constant gravitational field on the diffusion processes. We propose an explicit form of the potential for which the equilibrium concentration profiles are consistent with the experimental results. We compare shortly our results with two models of the diffusion processes in a gravitational field described in Huang's book [8] as well as with other opinions concerning this subject. As an example of technological application of the presented theory, the diffusion of the atomic hydrogen in fluid iron is proposed.

17

Twierdzenie o istnieniu i jednoznaczności rozwiązania zagadnienia początkowego dla nieliniowych równań termodyfuzji w przestrzeni trójwymiarowej

Szymaniec A.

Biuletyn Wojskowej Akademii Technicznej

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2003

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Vol. 52, nr 7

101-109

PL

W pracy udowodniono twierdzenie o istnieniu i jednoznaczności lokalnego w czasie rozwiązania zagadnienia Cauchy'ego dla nieliniowego hiperboliczno-parabolicznego układu równań różniczkowych cząstkowych opisujących trójwymiarowy ośrodek termodyfuzyjny.

EN

Theorem of the existence and uniqueness local (in time) solution the Cauchy problem for nonlinear hyperbolic-parabolic partial differential equations describing the process of the thermodiffusion in 3D space is proved.

18

Naprężenia termiczne i dyfuzyjne w ciele o transwersalnej izotropii

Łupieżowiec M.

Zeszyty Naukowe. Budownictwo / Politechnika Śląska

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2000

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z. 89

143-152

PL

W pracy analizuje się stan naprężenia spowodowany przepływem ciepła i masy w ciele o tranwersalnej izotropii. Wykorzystując sformułowanie wariacyjne i program ANSYS oblicza się naprężenia w walcu poddanym działaniu sprzężonego pola termodyfuzyjnego.

EN

In the paper the state of stress caused by the thermal and mass flux in a body of transverse isotropy has been analysed. Making use of variational formulation and ANSYS program, the stresses in a cylinder subjected to the operation of the coupled thermo-diffusion field have been computed.

19

Termodyfuzja w periodycznie warstwowych sprężystych kompozytach

Kaczyński A., Matysiak S. J.

Vibrations in Physical Systems

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2000

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Vol. 19

148--149

20

Rozwiązania fundamentalne zagadnienia quasi-statystycznego stacjonarnej termofuzji lepkosprężystej

Kubik J., Wieczorek B.

Zeszyty Naukowe. Budownictwo / Politechnika Śląska

|

1999

|

z. 86

131-143

PL

W opracowaniu analizowane jest zadanie początkowo-brzegowe dotyczące wyznaczenia stanu przemieszczenia w ośrodku lepkosprężystym z uwzględnieniem przepływów masy i ciepła, opisanych rówaniami dyfuzji i przewodnictwa cieplnego. Rozpatruje się quasi-statystyczne zagadnienia szczególne termodyfuzji lepkosprężystej, w których pomija się wpływy niestacjonarności na rozwiązania zadań brzegowych.

EN

There is analysed an initial-boundary problem of the displacement evaluation in the viscoelastic body with the flow of mass and heat consideration, described with diffusion and conducivity equations.