scholarly journals Heat Conduction in Hollow Cylinders with Radiation

1964 ◽  
Vol 14 (2) ◽  
pp. 159-164 ◽  
Author(s):  
E. Marchi ◽  
G. Zgrablich
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Integral Transformation ◽  
Finite Integral Transformation ◽  
Finite Integral ◽  
Hollow Cylinders ◽  
Radiation Type

AbstractA new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.

scholarly journals Analytic Solutions for Heat Conduction in Functionally Graded Circular Hollow Cylinders with Time-Dependent Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Sen-Yung Lee ◽  
Chih-Cheng Huang
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Integral Transformation ◽  
Transient Heat Conduction ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Time Dependent ◽  
Power Functions ◽  
Hollow Cylinders

An analytic solution method, without integral transformation, is developed to find the exact solutions for transient heat conduction in functionally graded (FG) circular hollow cylinders with time-dependent boundary conditions. By introducing suitable shifting functions, the governing second-order regular singular differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. The exact solution of the system with thermal conductivity and specific heat in power functions with different orders is developed. Finally, limiting studies and numerical analyses are given to illustrate the efficiency and the accuracy of the analysis.

scholarly journals New finite integral transform for the Laplace equation in an arbitrary domain

2020 ◽  
Vol 3 ◽  
pp. 115-124
Author(s):  
M.G. Berdnyk ◽  
◽  
Keyword(s):  
Analytical Solution ◽  
Laplace Equation ◽  
Integral Transformation ◽  
Boundary Value ◽  
Inverse Transformation ◽  
Maximum Deviation ◽  
Arbitrary Domain ◽  
First Order ◽  
Finite Integral Transformation ◽  
Finite Integral

Reliability, survivability, as well as the optimal operating mode of operation of the supercomputer will depend on the architecture and efficiency of the cooling system of the hot components of the supercomputer. That is why the number of problems, of great theoretical and practical interest, is the problem of studying the temperature fields arising in elements of arbitrary configuration, cooling a supercomputer. To solve this class of heat conduction problems, the method of finite integral transformations turned out to be the most convenient. This article is the first to construct a new finite integral transformation for the Laplace equation in an arbitrary domain bounded by several closed piecewise-smooth contours. An inverse transformation formula is given. Finding the core of the constructed new finite integral transformation by the finite element method in the Galerkin form for simplex first-order elements reduces to solving a system of algebraic equations. To test the operability of the new integral transformation, calculations were carried out of solutions of the boundary value problem for the Laplace equation obtained using the developed new integral transformation and the well-known analytical solution. The results of comparison the calculations of the solution of the Laplace equation are presented. In the case of a square with a side length equal to one and on one side of the square, the temperature is unity, and on the other, the temperature is zero, with a well-known analytical solution and a solution obtained using the new integral transformation. These results were obtained for 228 simplex first-order elements and 135 nodes. The maximum deviation modulo of these solutions is 0,096, the mathematical expectation of deviations is 0,009, and the variance of the type is 0,001. The developed integral transformation makes it possible to obtain a solution to complex boundary value problems of mathematical physics.

Adjoint Based Heat Conduction Optimization of Struts Parameters Within Hollow Blade

2021 ◽  
Author(s):  
Yougang Ruan ◽  
Zhenping Feng
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Inverse Design ◽  
Ansys Fluent ◽  
Average Temperature ◽  
Hollow Blade ◽  
Target Values ◽  
Solid Region ◽  
Continuous Adjoint

Abstract In gas turbine, the interaction between hot gas mainstream and blade solid region becomes more and more obvious as the turbine inlet temperature increases, thus heat conduction within the blade solid regions should be taken into consideration in optimization design process. In this paper, an adjoint-based optimization method for heat conduction problems in the solid region was built based on ANSYS Fluent and OpenFOAM Solver. The continuous adjoint equation and the corresponding boundary conditions for three typical conduction boundary conditions were derived in detail. To validate the correctness of this method, inverse design problems within the hollow cylinder and hollow blade were calculated, respectively. Inner shape inverse design of the hollow cylinder and the blade thickness inverse design were performed, and the target values were found successfully. Adjoint gradients were compared with finite-difference method or theoretical results. Then a Conjugate Heat Transfer (CHT) calculation was performed using ANSYS Fluent software, and the numerical methods were validated against the experimental results. An optimization of the struts place and thickness within hollow blade for average temperature was performed based on the CHT calculation results. Average temperature within the solid region of the optimized blade decreased 11.1K as compared to the original case.

scholarly journals INTEGRAL TRANSFORMATION IN A PARTIALLY BOUNDED REGION WITH A RADIAL THERMAL FLOW

2018 ◽  
Vol 13 (6) ◽  
pp. 89-96
Author(s):  
E. M. Kartashov
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Green's Functions ◽  
Green’S Functions ◽  
Integral Transformation ◽  
Bounded Region ◽  
Integral Transformations ◽  
Functional Relations ◽  
Special Cases ◽  
Unsteady Heat Conduction

A mathematical theory is developed for constructing integral transformations in a partially bounded region with a radial heat flow - a massive body bounded from the inside by a cylindrical cavity. Constructed: an integral transformation, the image of the operator on the right side of the equation of unsteady heat conduction, the inversion formula for the image of the desired function. The proposed approach favorably differs from the classical theory of differential equations of mathematical physics for the construction of generalized integral transformations based on the eigenfunctions of the corresponding singular Sturm-Liouville problems. The developed method is based on the operational solution of the initial boundary problems of unsteady heat conduction with an initial function of a general form L2(r0,∞) belonging to the r > r0 region and homogeneous boundary conditions and is associated with the calculation of the Riemann-Mellin contour integrals from images containing various combinations of modified Bessel functions. At the same time, for the above-mentioned region, the method of Green's functions was developed by constructing integral representations of analytical solutions of the first, second and third boundary value problems through inhomogeneities in the initial formulation of the problem (boundary conditions, source function in the initial equation). Mathematical models for finding the corresponding Green's functions are formulated, and functional relations of all three Green functions included in the presented integral formula are written out with the help of the developed theory of integral transformations. The functional relations constructed in the article can be used when considering numerous special cases of practical thermal physics. The specific possible applications of the presented results in many areas of science and technology are given.

On a heat flow problem in a hollow circular cylinder

1968 ◽  
Vol 64 (1) ◽  
pp. 193-202
Author(s):  
Nuretti̇n Y. Ölçer
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Eigenvalue Problems ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Frequency Equation ◽  
Hollow Circular Cylinder ◽  
New Ideas ◽  
Finite Integral

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

New Analytical Solutions of Buckling Problem of Rotationally-Restrained Rectangular Thin Plates

2019 ◽  
Vol 11 (10) ◽  
pp. 1950101 ◽  
Author(s):  
Salamat Ullah ◽  
Jinghui Zhang ◽  
Yang Zhong
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
Integral Transformation ◽  
Solution Procedure ◽  
Thin Plates ◽  
Algebraic Equations ◽  
Transform Method ◽  
Linear Algebraic Equations ◽  
Title Problem ◽  
Finite Integral

A double finite sine integral transform method is employed to analyze the buckling problem of rectangular thin plate with rotationally-restrained boundary condition. The method provides more reasonable and theoretical procedure than conventional inverse/semi-inverse methods through eliminating the need to preselect the deflection function. Unlike the methods based on Fourier series, the finite integral transform directly solves the governing equation, which automatically involves the boundary conditions. In the solution procedure, after performing integral transformation the title problem is converted to solve a fully regular infinite system of linear algebraic equations with the unknowns determined by satisfying associated boundary conditions. Then, through some mathematical manipulation the analytical buckling solution is elegantly achieved in a straightforward procedure. Various edge flexibilities are investigated through selecting the rotational fixity factor, including simply supported and clamped edges as limiting situations. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparing with the existing literature as well as the finite element method using (ABAQUS) software.

scholarly journals Time-Fractional Fourier Law in a finite hollow cylinder under Gaussian-distributed heat flux

2018 ◽  
Vol 180 ◽  
pp. 02008 ◽  
Author(s):  
Slawomir Blasiak
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Hollow Cylinder ◽  
Integral Transformation ◽  
Temperature Fields ◽  
Radial Coordinate ◽  
Finite Hollow Cylinder ◽  
The Laplace Transform ◽  
Transformation Methods ◽  
Fractional Heat Conduction

This paper presents the solution of the theoretical model of heat conduction based on timefractional Fourier equation for a finite hollow cylinder treated with heat flux on one of the front surfaces. A derivative of fractional order in the Caputo sense was applied to record the temperature derivative in time. The distributions of temperature fields in the hollow cylinder were determined with the use of Fourier-Bessel series, as surface functions of two variables (r, θ) . The distributions of temperature fields were determined using analytical methods and applying integral transformation methods. The Laplace transform with reference to time, the Fourier finite cosine transform with reference to axial coordinate z and Marchi-Zgrablich transform for radial coordinate r. The fractional heat conduction equation was analysed for 0 < α ≤ 2

scholarly journals NON-STATIONARY AXISYMMETRIC PROBLEM OF INVERSE PIEZOEFFECT FOR CIRCULAR BIMORPHOUS PLATE OF STEPPED VARIABLE THICKNESS AND RIGIDITY

2017 ◽  
Vol 19 (6) ◽  
pp. 133-140
Author(s):  
D.A. Shlyakhin
Keyword(s):  
Variable Thickness ◽  
Arbitrary Function ◽  
Axisymmetric Problem ◽  
Elastic System ◽  
Integral Transformation ◽  
Variable Rigidity ◽  
Closed Solution ◽  
Finite Integral Transformation ◽  
Finite Integral ◽  
Deformed State

Non-stationary axisymmetric problem for a thin circular bimorphous plate under the action on the end surfaces of electric potential, which is an arbitrary function of time is viewed. On the basis of the theory of Tymoshenko by method of finite integral transformation a new closed solution for the viewed electricity and elastic system of stepped variable rigidity and thickness is built. The obtained calculated correlations allow to explore the frequency characteristics and stress-deformed state of bimorphous elements.

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