Jan Adam Kołodziej

Analytical approximations of the shape factors for conductive heat flow in circular and regular polygonal cross-sections

The heat flux in steady heat conduction through cylinders whose cross-section has an inner or an outer contour in the form of a regular polygon or a circle is considered. To calculate the shape factor the temperature field is determined. Three cases are considered: (a) temperature field for hollow prismatic cylinders bounded by isothermal inner circles and outer regular polygons, (b) temperature field for hollow prismatic cylinders bounded by isothermal inner regular polygons and outer circles, (c) temperature field for hollow prismatic cylinders bounded by isothermal inner and outer regular polygons. The boundary collocation method in the least squares sense for solving appropriate boundary value problems is used. By means of nonlinear approximation (Marquardt method), for the three considered geometry cases, the simple analytical formulas for the shape factors are proposed.

Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem

Abstract The paper considers the determination of heat sources in unsteady 2-D heat conduction problem. The determination of the strength of a heat source is achieved by using the boundary condition, initial condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the θ -method with the method of fundamental solution and radial basis functions is proposed. Due to ill conditioning of the inverse transient heat conduction problem the Tikhonov regularization method based on SVD decomposition was used. In order to determine the optimum value of the regularization parameter the L-curve criterion was used. For testing purposes of the proposed algorithm the 2-D inverse boundary-initial-value problems in square region Ω with the known analytical solutions are considered. The numerical results show that the proposed method is easy to implement and pretty accurate. Moreover the accuracy of the results does not depend on the value of the θ parameter and is greater in the case of the identification of the temperature field than in the case of the identification of the heat sources function.

Permeability Tensor for Heterogeneous Porous Medium of Fibre Type

A theoretical model which allows us to determine the permeability of a fibrous porous medium is proposed. Fibres are assumed to be parallel and nonuniform in space and material with a low volume fraction of fibres is considered. The model includes two geometric parameters: the diameter of fibres and the diameter of caverns or fissures inside the bundle of fibres. The tensor of permeability of the porous medium is determined based upon a generalized cell model. The components of permeability tensor depend on two parameters which are determined using experimental data and least-squares approximation. The influence of the geometric parameters on components of permeability tensor is discussed.

The determination of heat sources in two dimensional inverse steady heat problems by means of the method of fundamental solutions

This article deals with the inverse determination of heat sources in steady 2-D heat conduction problems. The problem is described by Poissons equation in which the right-hand side function is unknown. The identification of the strength of a heat source is given by using the boundary conditions for the temperature and heat flux at chosen points on the boundary. For the solution of the inverse problem of identification of the heat source, the method of fundamental solutions with radial basis functions and the Tikhonov regularization technique is proposed. Accurate results have been obtained for six test problems where the analytical solutions are available.

Analysis of Carreau fluid flow between corrugated plates

The paper deals with a problem of Carreau fluid flow between corrugated plates. A meshless numerical procedure for solving nonlinear governing equation is constructed using the Picard iteration method in combination with the method of fundamental solutions and the radial basis functions. The dimensionless average velocity and the product of friction factor and Reynolds number are calculated for different values of the fluid model and parameters of the considered region.

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

Abstract The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.

Fluid flow and heat transfer of a power-law fluid in an internally finned tube with different fin lengths

In the paper an analysis of fluid flow and heat transfer of a power-law fluid in an internally finned tube with different fin length is conducted. Nonlinear momentum equation of a power-law fluid flow and nonlinear energy equation are solved using the Picard iteration method. Then on each iteration step the solution of inhomogeneous equation consists of two parts: the general solution and the particular solution. Firstly the particular solution is obtained by interpolation of the inhomogeneous term by means of the radial basis functions and monomials. Then the general solution is obtained using the method of fundamental solutions and by fulfilling boundary conditions.

Solution of Cauchy problem to stationary heat conduction equation by modified method of elementary balances with interpolation of the solution in physical plane

This article presents a solution of stationary direct and inverse problems (Cauchy problem) of cooling a circular ring with the modified method of elementary balances. The idea of the method itself relies on the division of the considered range into elements and interpolation of the solution within the elements with the help of linear combination of base functions. In the vicinity of every mesh node, the control regions are created in which the energy is balanced. The regions include or interpenetrate with each other. The numerical calculation has been carried out with the use of a quadrilateral mesh with four nodes and a triangular mesh with six nodes. The presented solution to the inverse problem with randomly perturbed boundary conditions of the flux and temperature at the outer boundary of the ring with maximal error reaching up to 10% gave very good results. The solution of the inverse problem has been obtained in the sense of singular value decomposition algorithm.

Motion equations of the dynamic theory of consolidation from the point of view of the theory of transient pipe flows

An elastic fluid-saturated porous medium is modeled as a bundle of parallel cylindrical tubes aligned in a direction parallel to the fluid movement. The pore space is filled with viscous compressible liquid. A cell model and the theory of transient pipe flow are used to derive one-dimensional governing equations in such media. All macroscopic constants in these equations are defined by the individual material constants of the fluid and solid. The interaction force includes an additional term not found in Biots theory.

Comparison of different approaches in the Trefftz method for analysis of fluid flow between regular bundles of cylindrical fibres

In the paper three different approaches for the Trefftz method are compared in analysis of the fluid flow between regular bundles of cylindrical fibres. The approximate solution is a linear combination of such trial functions which fulfil exactly the governing equations. The trial functions can be defined in the Cartesian coordinate system (the first approach), in the cylindrical coordinate system (and can fulfil also some boundary conditions - the second approach) or be defined as a fundamental solutions (the third approach - the method of fundamental solutions).The average velocity and the product of the friction factor and the Reynolds number ƒ ⋅ Re are compared for selected parameters of a considered region.