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EN
As a combination of the traditional finite element method and boundary element method, the n-sided polygonal hybrid finite element method with fundamental solution kernels, named as HFS-FEM, is thoroughly studied in this work for two-dimensional heat conduction in fully anisotropic media. In this approach, the unknown temperature field within the polygon is represented by the linear combination of anisotropic fundamental solutions of problem to achieve the local satisfaction of the related governing equations, but not the specific boundary conditions and the continuity conditions across the element boundary. To tackle such a shortcoming, the frame temperature field is independently defined on the entire boundary of the polygonal element by means of the conventional one-dimensional shape function interpolation. Subsequently, by the hybrid functional with the assumed intra- and inter-element temperature fields, the stiffness equation can be obtained including the line integrals along the element boundary only, whose dimension is reduced by one compared to the domain integrals in the traditional finite elements. This means that the higher computing efficiency is expected. Moreover, any shaped polygonal elements can be constructed in a unified form with the same fundamental solution kernels, including convex and non-convex polygonal elements, to provide greater flexibility in meshing effort for complex geometries. Besides, the element boundary integrals endow the method higher versatility with a non-conforming mesh in the pre-processing stage of the analysis over the traditional FEM. No modification to the HFS-FEM formulation is needed for the non-conforming mesh and the element containing hanging nodes is treated normally as the one with more nodes. Finally, the accuracy, convergence, computing efficiency, stability of non-convex element, and straightforward treatment of non-conforming discretization are discussed for the present n-sided polygonal hybrid finite elements by a few applications in the context of anisotropic heat conduction.
2
Content available remote Modelowanie procesów wymiany ciepła w dzianinach futerkowych
PL
Ciepłochronność jest podstawowym parametrem determinującym praktyczne zastosowanie dzianin futerkowych. Analiza wymiany ciepła oraz metodyka określenia podstawowych parametrów struktury do optymalizacji konstrukcji z uwagi na wymagany poziom izolacyjności cieplnej nie zostały jeszcze dotychczas opisane dla dzianin futerkowych. Brak rozwiązań problemów ciepłochronności dla tego typu dzianin wskazuje na celowość podjęcia rozważanej tematyki. Celem prac prowadzonych w ramach dysertacji doktorskiej było opracowanie modelu przepływu ciepła, który może być wykorzystany do projektowania dzianin futerkowych o wymaganych właściwościach termofizycznych. Zaprezentowany model opisu zjawiska przepływu ciepła ma uzasadnienie praktyczne w odniesieniu do dzianin futerkowych. Przyjęta metoda pozwala uniknąć badań wielu parametrów, ograniczając je do minimum: udział objętościowy poszczególnych składników w stosunku do każdej warstwy oraz stosunek grubości poszczególnych warstw do grubości całego wyrobu. Do badań wykorzystuje się powszechnie dostępne urządzenia pomiarowe. Uzyskuje się ponadto możliwość symulacji eksperymentu, bez konieczności wytwarzania dzianiny futerkowej. Metoda ta pozwala na dowolne modelowanie warunków brzegowych i początkowych, co nie zawsze jest możliwe dla metod empirycznych ze względu na ograniczenia sprzętowe. Opracowany model obliczeniowy przepływu ciepła przez dzianiny futerkowe umożliwia uzyskanie rozkładu temperatury w tej konstrukcji oraz może stanowić punkt wyjścia doboru optymalnego struktury dla osiągnięcia 40 Anna Więzowska wymaganych właściwości wyrobów. Projektowanie dzianiny futerkowej o wymaganym poziomie izolacyjności cieplnej, z zastosowaniem przedstawionego modelu, może posłużyć do powstania rzeczywistego materiału o określonych właściwościach ciepłochronnych.
EN
The fundamental function of the knitted fur fabrics, which are commonly used to manufacture the clothes and shoes, is to protect the human body against heat loss in low environmental temperature. Thus, the heat-insulating properties of fur fabrics are the basic criterion for their functional characteristics. The aim of this work is to determine the heat transfer model, which can be applied to design the knitted fur fabrics of the required thermophysical properties. The mathematical model of heat transport within knitted fur fabrics allows to obtain the temperature distribution in the structure and can be a starting point to optimize the structural shape in respect of the requested properties. The available literature does not introduce the heat transport problems in the knitted fur fabrics. A few works describe in general form some parameters of the structure whereas the corresponding standards are inaccessible. To explain the nature of heat transfer inside the knitted fur fabrics, the dissertation describes the basic phenomena of heat transfer. The heat conduction is defined more precisely as the dominant heat transport mechanism in textiles. The material properties influencing the heat flux density transferred in textile product are also described. The particular cases of heat conduction mechanisms in the single layer and the multi-layer structure have been analysed and next applied to determine the heat transfer in the homogenized fur fabric. The solution methodology of simple and complex heat transfer problems has been explained. The literature pertaining to the measurements of heat isolation in textiles has been reviewed in respect of the measurement methods, character of processes, measured parameters and field of their application vs. both structure of examined material as well as the layers arrangement. Assuming the complex structure of knitted fur fabrics (i.e. the multilayer arrangement, participation of glue in bottom layer, air inside the void spaces under inclined fibers), we have rejected the measurements methods influencing the material structure during the test. Heat transport within knitted fur fabrics is described by means of the heat conduction coefficient tested in steady conditions. The measured conduction coefficient is often of substitute nature and can additionally include the convective and radiative heat transport. The thermal properties of knitted fur fabrics are determined by means of the test device Tilmet 75. The preliminary investigations were conducted using the device Tilmet 75 for different knitted fabrics made of homogenous materials with the diversified thickness and surface mass. The fur fabrics were characterized using the same standard indexes which are tested for the knitted fabrics i.e. the thickness and surface mass. There are tested the heat conduction and heat permeability as well as determined the structure of knitted fabrics samples vs. the characteristics of thermal properties. The empirically determined heat conduction coefficient and heat resistances vs. basic structural parameters do not describe the influence of raw material in the knitted fur fabrics on the material heat characteristics. Both growing surface mass and growing fabric thickness does not determine unequivocally the gradation of these features in respect of thermal properties. According to the preliminary test results, it is necessary to change the factors determining the complex knitted fur fabrics in respect of the structure and raw material composition. The description of the knitted fur fabric can cause difficulties in heat transport correlations. Parameters of fur fabrics of the complex, space and multilayer structure are hard to determine and investigate using the standard test methods. The structure consists of the bottom layer and the fleece layer which are made of different raw materials: the yarn, band and glue as well as considerable volume fraction of air inside. Thus, the description of physical model is troublesome. Let us introduce the following assumptions concerning the fur fabric: (i) the same height of fleece layer; (ii) the uniform distribution of fibres density in cover layer; (iii) the uniform distribution of yarn, fibres and air in bottom layer; (iv) the void spaces between the yarn in bottom layer are filled by both air and glue; (v) the glue does not penetrate the cover layer. Under the above assumptions, the space 3D description can be simplified to the plane 2D problem introducing the homogenized particular layer of fabric. The structure is defined by the volume fraction and heat conduction coefficient of each layer. The heat conduction coefficient is determined using the rule of mixture which limits the domain of study of corresponding parameters to the volume fractions of particular component in every layer. The principal investigations were conducted for the knitted fur fabrics of the diversified both length of fleece and basis weight, subjected to the different finishing processes. The test methods applicable for the different textile materials were analysed in respect of the measurement characteristics i.e. applicable for thickness, density, mass related to materials / textile products and their particular layers. The complex structure of knitted fur fabrics can be characterized by the innovative, non-standard test methods as well as the standard methods, which are not usually applied for those materials. The thickness of knitted fur fabric tested for the various pressures strongly depends on the load applied. The presented model of heat transfer description is practically motivated for the knitted fur fabric. The adopted method can avoid the large number of tests of required parameters and restricts the analysis to the following cases: the volume fraction of particular component within each layer and thickness fraction of particular layer to the complete thickness of product. The commonly available test equipment is used during the tests. Additionally, the experiments are simulated which substitute the manufactured knitted fur fabric. This method allows to model 75 optionally the boundary and initial conditions which is not always applicable for empirical methods due to equipment limitations. Design of knitted fur fabric of the requested thermal isolation level, based on the model presented in this work, can help to create the real material of the prescribed heat-insulating properties. The next stage of the current investigations can be focused on determination of indexes characterizing the knitted fur fabrics in presented model vs. technological parameters necessary to manufacture the designed product.
EN
The aim of this work is to investigate, how in the adopted model of hydrodynamic lubrication of a conical slide bearing, the change of the heat flux value at the bearing shaft, affects bearing operating parameters. In this research, the authors use, the known from the literature, Reynolds type equation, describing the stationary hydrodynamic lubrication process of a conical slide bearing. The analytical, solutions, that determine the components of the lubricating oil velocity vector and the equation (analytical solution of energy equation) determining the threedimensional temperature distribution in the lubrication gap, was also adopted from previous works. In order to obtain numerical solutions, the Newton’s method was used, and the derivatives in the Reynolds type equation were approximated by the finite differences. An application of the method of subsequent approximations allowed considering the influence of temperature, pressure and shearing rate on the viscosity of lubricating oil. The considerations were performed by adopting the Reynolds condition of the hydrodynamic oil film. It was tested, how the assumed value of the heat flux on the bearing shaft surface affects the values of the obtained operating parameters, i.e. the transverse and longitudinal component of the load carrying capacity, friction force and coefficient of friction.
EN
Two-dimensional stationary problem of heat conduction and thermoelasticity for infinite elastic body containing periodic system of inclusions and cracks is considered. Solution of the problem is constructed using the method of singular integral equations (SIEs). The numerical solution of the system integral equations are obtained by the method of mechanical quadrature for a plate heated by a heat flow, containing periodic system elliptic inclusions and thermally insulated cracks. There are obtained graphic dependences of stress intensity factors (SIFs), which characterise the distribution of intensity of stresses at the tops of a crack, depending on the length of crack, elastic and thermoelastic characteristics inclusion, relative position of crack and inclusion.
EN
In this paper, the effect of a fractional order of time-derivatives occurring in fractional heat conduction models on the temperature distribution in a composite sphere is investigated. The research concerns heat conduction in a sphere consisting of a solid sphere and a spherical layer which are in perfect thermal contact. The solution of the problem with a classical Robin boundary condition and continuity conditions at the interface in an analytical form has been derived. The fractional heat conduction is governed by the heat conduction equation with the Caputo time-derivative, a Robin boundary condition and a heat flux continuity condition with the Riemann-Liouville derivative. The solution of the problem of non-local heat conduction by using the Laplace transform technique has been determined, and the temperature distribution in the sphere by using a method of numerical inversion of the Laplace transforms has been obtained.
EN
In this paper, a new formulation based on the method of fundamental solutions for two/three- -dimensional steady-state heat conduction problems involving internal curved line/surface heat sources is presented. Arbitrary shapes and non-uniform intensities of the curved heat sources can be modeled by an assemblage of several parts with quadratic variations. The presented mesh-free modeling does not require any internal points as in domain methods. Four numerical examples are studied to verify the validity and efficiency of the proposed method. Our analyses have shown that the presented mesh-free formulation is very efficient in comparison with conventional boundary or domain solution techniques.
EN
One of the main parameters affecting the hydrodynamic lubrication of slide bearings is the viscosity of lubricating oil. Many studies show, that significant changes in the viscosity of oil occur along with changes in its temperature. The influence on the temperature distribution in the lubrication gap of the slide bearing have a variety of factors, and one of them is the amount of heat exchanged between the lubricant and the environment. The temperature of the lubricating oil of operating bearing is usually higher than the ambient temperature. In addition to the convection, which occurs during the flow (heat exchange related to the oil supply and discharge system) some amount of heat is transferred to the bearing sleeve material (and also to the bearing shaft), and then it is conducted to sleeve outer surface. The amount of heat transferred through the bearing sleeve is mainly dependent on the difference of temperatures between inner and outer sleeve surfaces and also depend on the heat conduction coefficient of sleeve material. This article presents the results of modelling of the influence of amount of heat conducted through the bearing material, on the hydrodynamic lubrication of a conical slide bearing. The study concerned various values of the heat conduction coefficient of the bearing material to investigate its influence on the temperature values of lubricating oil, and thus, on its viscosity, on the distribution of hydrodynamic pressure and on the calculated values of bearing load carrying capacities and friction forces.
EN
Selective Catalytic Reduction (SCR) is well known method for reducing NOx emission in diesel engine exhaust gas. Urea-water solution (UWS) injected into hot stream decomposes due to thermolysis into ammonia and isocyanic acid which hydrolyses further into more ammonia and carbon dioxide. Resultant ammonia is the NOx reductor, producing water vapour and carbon dioxide from the reduction reaction. To provide sufficient NOx reduction efficiency, UWS needs to be properly atomized and mixed with exhaust gas. However, due to more and more restrictive emissions regulations provided by European Union and Close Coupled trend of aftertreatment systems in vehicles the design process is very complex and demanding. Computational Fluid Dynamics (CFD) simulations are integral part of product development, allowing save time and reduce costs of preparing prototypes for further tests. However, it is necessary to understand all the processes and problems connected with NOx reduction in SCR system. Strong turbulent flow of hot stream gas, urea-water solution spray injection, droplets interaction with wall, wallfilm generation are included. The objective of this work is to investigate the impact of heat transfer modelling inside mixing elements of SCR system on urea mixing uniformity and wallfilm deposit on the walls of the system. Simplified and more complex approach is compared with no heat transfer cases. All the simulations were conducted using AVL FIRETM software. Results showed that wall heat transfer might have an impact on mixing efficiency and wallfilm formulation. It is necessary to take into account the effect of mixing elements heat conduction in CFD simulations during the aftertreatment design process.
EN
In this paper, we consider the problem of locating coated inclusions in a 2D dimensional conductor material in order to obtain a suitable thermal environment. The mathematical model is described by elliptic partial differential equation with linear boundary condition, including heat transfer coefficient. A shape optimization problem is formulated by introducing a cost functional to solve the problem under consideration. The shape sensitivity analysis is rigorously performed for the problem by means of a Lagrangian formulation. The optimization problem is solved by means of gradient-based strategy and numerical experiments are carried out to demonstrate the feasibility of the approach.
PL
W artykule opisano metody obliczeniowe wykorzystywane do badania zjawiska przepływu ciepła. Do analizy użyte zostały wybrane kształty prętów i żeber wykonanych z materiałów o różnych właściwościach cieplnych. Stworzono model numeryczny przewodzenia ciepła analizowanych obiektów. Zbudowano stanowisko laboratoryjne do wizualizacji rozkładu temperaturyw prętach prostych oraz stożkowych z wykorzystaniem kamery termowizyjnej. Głównym celem prezentowanej pracy było sprawdzenie poprawności stosowania modelu analitycznego.Zbadano również strumienie ciepła przekazywane przez wybrane żebra. Przedstawiono wyniki różnych metod obliczeniowych oraz wyciągnięto z nich wnioski. Porównano wpływ kształtu oraz materiału prętów na wymianę ciepła i rozkłady temperatury.
EN
The article describes the calculation methods used to study the heat transfer. Selected shapes of pin fins and materials with different thermal properties were used for the analysis. Anumerical model of heat conduction of the analyzed objects was created. A laboratory set up was built to visualize the temperature distribution of the straight pin fins and pin fins of cone profile with blunt tip using the infrared camera. The main purpose of the presented chapter of the monograph was to check the right of using the analytical model. The results and conclusions of different calculation methods were presented. The shapes and materials of fins were also compared.
EN
The work deals with possibilities of using this specific material. It is focused on cast metal foams with a regular arrangement of internal cells and it refers to already used casting technologies – the production of metal foams with the aid of sand cores. Metal foams are used in many industries, such as: automotive, aerospace, construction, power engineering. They have unique propertiesand due to lower weight with sufficient strength and greater contact surface can be used, for example, for the conduction of heat. This article deals with the use of the metal foam as a heat exchanger. The efficiency of the heat exchanger depends on its shape and size and therefore the study is focused first on the optimization of the shape before the proper manufacture.
EN
The one-dimensional time-fractional heat conduction equation with heat absorption (heat release) proportional to temperature is considered. The Caputo time-fractional derivative is utilyzed. The fundamental solutions to the Cauchy and source problems are obtained using the Laplace transform with respect to time and the exponential Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically.
PL
Rozpatrywanie zagadnień początkowo-brzegowych dla przewodników warstwowych w ramach klasycznej teorii przewodnictwa cieplnego jest skomplikowane, ponieważ zagadnienia te są opisane przez równania różniczkowe o zmiennych i silnie oscylujących współczynnikach. W związku z tym poszukuje się modeli uproszczonych. W pracy porównano rozwiązania otrzymane w ramach dwóch uśrednionych modeli przewodnictwa cieplnego w periodycznych kompozytach warstwowych: modelu tolerancyjnego i jego wariantu asymptotycznego. Rozwiązania te otrzymano metodą różnic skończonych. Stwierdzono, że korzystanie z rozwiązań modelu asymptotycznego nie zawsze jest możliwe ze względu na zbyt dużą różnicę w wynikach względem modelu tolerancyjnego.
EN
Analysis of initial-boundary conditions for multilayered conductors within the framework of the classical theory of heat conduction is complex because these problems are described by differential equations with variable and highly oscillating coefficients. Due to this, simplified models are being sought. In this paper, the solutions obtained within the framework of two averaged models of heat conduction in multilayered materials have been compared: the tolerance model and its asymptotic version. These solutions have been obtained with use of the finite differences method. It has been stated that application of the asymptotic model is not always possible due to too large difference in results if compared to the tolerance model.
14
Content available remote Izolacje techniczne w obiektach rewitalizowanych
PL
W artykule omówiono kwestie prawne i przesłanki techniczne doboru i montażu izolacji technicznych w obiektach rewitalizowanych.
EN
The article further presents the legal aspects and technical criteria for selection and installation of engineering insulation at revitalized buildings.
EN
The purpose of this review article is to summarize observations accumulated over the years on director alignment phenomena in nematic and cholesteric liquid crystals by molecular dynamics simulation of molecular model systems and by experiment on real systems. The main focus is on the alignment angle between the director and external dissipative fields such as velocity gradients in various flow geometries and temperature gradients doing irreversible work on the system. A general observation is that the director attains an orientation relative to the field where the energy dissipation rate is minimal in the steady state. In the case of planar elongational flow, it can be proven by using symmetry arguments that the energy dissipation rate must be either maximal or minimal and simulations have shown that is minimal. In planar Couette flow both simulations and experiments imply that the energy dissipation rate is minimal in the steady state. Finally, in the case of heat conduction, symmetry arguments imply that the energy dissipation rate must be either minimal or maximal and simulations and experiments indicate that it is minimal. All these observations can be explained by applying a recently proven theorem according to which the energy dissipation rate is minimal in the steady state in the linear regime at low fields.
EN
An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.
EN
The article presents the results of an experimental study of energy-resource saving technologies of formation of massive amorphous structure. Considered are the methods of mathematical modeling and optimization of process production of massive amorphous structures, which can reduce experimental studies and material resources to create a highly efficient production of amorphous alloys. The results of physical experiments are compared with the results of the calculation. Received results can be used to analyze the physical regularities and justified choice of technological modes of formation of amorphous structures.
EN
The paper presents the analytic-numerical hybrid method using, among others, the Taylor transformation, thanks to which the solution of the Stefan problem is replaced by the solution of a nonlinear system of equations.
EN
In this paper an analytical solution of the time-fractional heat conduction problem in a spherical coordinate system is presented. The considerations deal the two-dimensional problem in multilayer spherical bodies including a hollow sphere, hemisphere and spherical wedge. The mathematical Robin conditions are assumed. The solution is a sum of time-dependent function satisfied homogenous boundary conditions and of a solution of the steady-state problem. Numerical example shows the temperature distributions in the hemisphere for various order of time-derivative.
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.
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