圆柱沿半径和轴向的热传导微分方程

The formal solutions of problems involving transient heat conduction in infinite internally bounded cylindrical solids may be obtained by the Laplace transform method. Asymptotic series representing the solutions for large values of time are given in terms of functions related to the derivatives of the reciprocal gamma function. The results are applied to the case of the internally bounded infinite cylindrical medium with, (a) the boundary held at constant temperature; (b) with constant heat flow over the boundary; and (c) with the ``radiation'' boundary condition. A problem in the flow of gas through a porous medium is considered in detail.

This research aims to develop a mathematical method for expressing the Laplace operator in cylindrical coordinates and applying it to solve heat conduction equations in various scenarios. The method commences by transforming Cartesian coordinates into cylindrical coordinates and identifying the necessary substitutions. The result is the expression of the Laplace operator in cylindrical coordinates, which is subsequently employed to address heat conduction equations within cylindrical coordinates. Various cases encompassing different initial and boundary conditions, as well as variations in the conduction coefficient over time, are meticulously considered. In each instance, precise mathematical solutions are determined and subjected to thorough analysis. This study carries substantial implications for comprehending heat transfer within cylindrical coordinate systems and finds relevance in a wide array of scientific and engineering contexts. The research's findings can be harnessed for technology development, heating system design, and heat transfer modeling across diverse applications, including mechanical engineering and materials science. Therefore, the research's contribution holds paramount significance in advancing our understanding of heat transfer within cylindrical coordinates and in devising more efficient and accurate solutions for an array of heat-related issues within the realms of science and engineering.

… developed to tackle transient heat problems for one-… heat conduction in composite cylindrical shells. As calculation methods and results exhibit differently for varied types of cylindrical …

… is to investigate the hyperbolic heat conduction subjected to heat flux boundary conditions. The governing equation is expressed in cylindrical coordinates. Equations are solved by …

… This paper presents a steady analytical solution for heat conduction in a cylindrical multilayer composite laminate in which the fiber direction may vary between layers. The analytical …

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

Abstract. According to the differential equations of heat conduction on cylindrical and spherical coordinate system, numerical solution of the discrete formula on cylindrical and spherical coordinate system with high accuracy were derived. Compared with the analytical solution, this discrete formula was verified with a high degree of accuracy. To make the complex dispersion coefficient of diffusion term θ more concrete in spherical coordinates, this paper derived the discretion coefficient of diffusion term θ by the first mean value theorem of integral. The accurate schemes provide a good reference for researchers whose work in solving the equation of heat conduction of three-dimensional cylindrical coordinates and spherical coordinates, and it will provide accurate numerical schemes and the theoretical basis for solving practical engineering problems.

… in the cylindrical … heat conduction equation in the cylindrical coordinate system. After that, the general solution is applied to establish analytical solutions for two practical heat conduction …

The heat conduction problem, in the presence of a change of state, was solved for the case of an indefinitely long cylindrical layer cavity. As boundary conditions, it is imposed that the internal surface of the cavity is maintained below the fusion temperature of the infilling substance and the external surface is kept above it. The solution, obtained in nondimensional variables, consists in two closed form heat conduction equation solutions for the solidified and liquid regions, which formally depend of the, at first, unknown position of the phase change front. The energy balance through the phase change front furnishes the equation for time dependence of the front position, which is numerically solved. Substitution of the front position for a particular instant in the heat conduction equation solutions gives the temperature distribution inside the cavity at that moment. The solution is illustrated with numerical examples.

Anisotropy is a prevailing property in most substances in the real world. The thermal conductivity characteristic of anisotropy must be determined for utilizing geothermal resources and assessing battery performances. Most core samples were primarily obtained by drilling and intended to be cylindrical in shape, with the cores resembling quantities of familiar batteries. Although Fourier’s law could be used to measure the axial thermal conductivity of square or cylindrical samples, there is still a need to develop a new method to measure the radial thermal conductivity of cylindrical samples and evaluate their anisotropy. Thus, we established a testing method for cylindrical samples using the theory of complex variable functions following the heat conduction equation and implemented a numerical simulation to determine the difference between this method and typical ones via a finite element model for various samples. Results show that the method could perfectly gauge the radial thermal conductivity of cylindrical samples with more powerful availability.

… heat conduction term in the 2D model, we derived another 1-D model the 1D-radial model, which considers only radial heat conduction. … a 2-D radial and axial model, reducing solution …

A steady state thermal conductivity measuring setup based on the comparative radial heat flow method is presented. The setup consists of a pair of coaxial cylinders as its main components, with test fluid placed in the annular space between these cylinders with water tight cover plates at the top and bottom of the cylinders. Experiment involves heating the coil at the concentric-center of the inner cylinder; steady state data are acquired for the calculation of the thermal conductivity. Thermal conductivity is calculated by comparing the radial heat flow between the cylinders and the test fluid (comparative method). Thermal conductivity of water, glycerol, and ethylene glycol was measured for varying temperatures and is in good agreement with the published thermal conductivity values in literature.

Enhanced conduction heat transfer via sinusoidal oscillatory flow through circular tubes connecting two fluid reservoirs maintained at different temperatures is examined. Using a multiscale expansion technique for the solution of the governing partial differential equation, the spatial and temporal temperature variation within the tubes is determined for low values of α2 Pr, where α is the Womersley number and Pr the fluid Prandtl number. From this result, a calculation of the effective thermal diffusivity is made and used to determine the enhanced conduction heat transfer from the high- to low-temperature reservoir. The enhanced heat transfer produced by the oscillations is shown to be proportional to the square of the oscillation amplitude and a function of the Prandtl number, the frequency, and the tube radius. Values of effective conductivity in liquid metals three orders of magnitude greater than the normal heat conductivity are shown to be readily obtainable under typical experimental conditions.

This paper presents both analytical and numerical approaches for solving the non-homogeneous heat conduction equation in a cylindrical domain with time-dependent polynomial and harmonic boundary conditions. The analytical solution is constructed using the superposition principle and Fourier-Bessel series expansion, with rigorous proofs of convergence in the \(L_2\) norm and asymptotic stability. For numerical implementation, we develop a Bessel Collocation Method that achieves spectral accuracy with moderate computational cost. Theoretical error estimates are derived, showing exponential convergence in space and second-order accuracy in time. Numerical experiments confirm the theoretical predictions, with absolute errors decreasing from \(10^{-2}\) to \(10^{-15}\) as the number of basis functions increases. The results demonstrate that the proposed framework provides reliable solutions for heat transfer problems with complex boundary regimes in circular geometries.

… of coordinate transformation, the diffusion term in the r direction of the heat conduction equation in a cylindrical coordinate … The object in this paper is a two-dimensional cylindrical heat …

… the angular derivative term appearing in the radiative transfer equation in spherical or cylindrical coordinates. This term is generally approximated by a finite differencing scheme [18]. …

… equations of the materials in cylindrical coordinate system … heat conduction has been established for a cylindrical … non-Fourier heat conduction in the cylindrical coordinate system are …

… on the time-fractional heat conduction equation are considered in cylindrical coordinates. … The solutions to the fractional heat conduction equation under the Dirichlet boundary condition …

Abstract Finding analytical solutions for the unsteady conduction heat transfer equation is generally very difficult due to the intrinsic complexity of the problem. In most cases, the equation has no analytical solutions. A numerical approach that surmounts these difficulties consists of the finite difference technique. The alternate directions finite-difference technique especially suits to the two-dimension heat conduction. The paper considers a hollow cylinder consisting of a conduction heat transfer medium, and the two-direction unsteady heat propagation within this medium is studied by means of the alternate directions method. The approach simulates the heat conduction process during a charge-discharge cycle of a single borehole heat storage.

In this study, an analytical solution is proposed for the problem of transient anisotropic conductive heat transfer in composite cylindrical shells. The composite shells are considered to have directional heat transfer properties, which is due to the existence of fibers which can be winded in any direction. The composite shells usually show high conductivity in the direction parallel to fiber direction and low conductivity in other two orthogonal directions. To solve the heat transfer partial differential equation, finite Fourier transform and separation of variables method are used. The present solution is used to find the temperature distribution in a composite cylindrical vessel for which the composite material is graphite/epoxy and the vessel is prone to an external heat flux and also ambient flow. The analytical solution is verified perfectly by the data obtained from a second-order finite difference solution. The solution is used to investigate the effects of values of fiber angle and material conductivity coefficients on temperature distribution of the composite cylindrical vessel. The results show the important role of fiber angle values on the temperature distribution of vessel.

THE Green's Function of this paper is the temperature v at P (r, 0) at time t due to an instantaneous unit line source at P'(r', 0') at time zero, the solid being then at zero temperature, and …

… problems of heat conduction which is … cylindrical bodies with projecting longitudinal tins. Problems of this sort are met with in detailed analyses of heat conduction in finned tubes of heat …

… to cylindrical coordinates were also given by the writer (7). Taking as variables the three dimensionless quantities F,, Xr> and Nur, Equation … convection at the cylindrical surface and the …

… shape function in the rectangular coordinate systemI is applied to the present study in cylindrical and spherical coordinate systems, it is obvious that the solution form of equation (9) will …

… in a cylindrical tube. In this study, the analytical solutions in cylindrical and spherical coordinates are discussed by the method of relativistic transformation of coordinates and method of …

… in heat conduction for a wide class of geometrical shapes for which the heat equation cannot … It frequently happens, however, that, even when the equation of transient heat conduction …

… problem on the transfer of heat in a cylindrical duct of radius R′. We assume that a … coordinates and velocities, we use the Williams equation [14] which, in a cylindrical coordinate …

… the heating/cooling rate @T=@t plays a significant role in governing stability [1–7] in the prediction of the local heat flux… , transient heat equation in cylindrical coordinates is given by [14] …

… heat conduction problems in the cylindrical coordinate system using a hybrid Green’s function method. The major difficulty encountered in the numerical solutions of hyperbolic heat …

There are many applications for problems involving thermal conduction in two-dimensional (2D) cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green's functions where the principle of superposition is employed in combining a one-dimensional (1D) cylindrical case with a 1D Cartesian case to provide a temperature solution for a 2D cylindrical. Green's functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

In this paper, a spectral collocation method (SCM) is developed to solve transient conduction and radiation heat transfer problems in infinitely long annular cylinders with participating …

… of transient heat conduction in cylindrical multilayer composite laminates. This solution is valid for the most generalized linear boundary conditions consisting of the conduction, … transient …

The purpose of this study was to formulate and solve the problem of transient combined conduction and radiation in a grey absorbing, emitting, and scattering medium of cylindrical geometry. The medium is bounded by grey diffuse surfaces at known temperatures. The problem was solved by the Galerkin finite element method using linear interpolating functions. The transient terms were handled using the Crank-Nicolson scheme with the time steps chosen to avoid temperature fluctuations at early times. The results were compared with exact solutions for the extreme cases, i.e., pure conduction and radiative equilibrium. Results are also compared with an approximate solution for the case of combined conduction and radiation. The effects of various parameters on the solutions are presented in tabular as well as graphical forms. The results obtained compare well with the exact solutions. The Galerkin finite element technique is well suited to this problem because of the ease with which integrals with variable limits can be handled.

The two-dimensional conduction equation is solved for a hollow cylinder subjected to a series of heat flux pulses on the inner boundary. The periodic heat flux is represented by an exponentially decreasing pulse with a spatial distribution of peak magnitude. The analytical techniques and representation of the boundary conditions apply to different situations involving pulsating boundary conditions. An application to the gun barrel heating problem is given. Calculated bore surface and internal temperature histories are in good agreement with experimental data. During the actual firing time in rapidly-firing guns, results show that external cooling is generally ineffective for controlling barrel bore surface temperature.

… We present the transient solution of a cylindrical heat source … of the present solution and the purely conductive model. … In Section 4, we present the MICS steady-state and transient …

… In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are …

… The processes of the transient coupled radiative-conductive heat transfer … transient thermal responses are investigated. It is found that, due to the radiation within medium, the transient …

… conduction analysis in cylindrical coordinates (r, ϕ, z) based on the DPL theory are obtained. Consider a radially graded FG cylindrical … The initial temperature of the cylindrical structure …

… Most recently, the steady and transient problems of heat transfer in a … transient heat transfer in a conducting, absorbing and emitting medium bounded by two infinite coaxial cylindrical …

… Fourier heat conduction problem in an FGM cylindrical shell … to analyze transient and steady Fourier heat conduction in a … related to hyperbolic heat conduction in FGM structures. Fang …

… to a cylindrical geometry. The objective of this study was to derive more general closed-form solutions for the transient heat conduction … Compared to the work in cylindrical coordinates …

Abstract The article aims to determine and experimentally validate the transient temperature fields in a cylindrical component. The applied in-house algorithm, which is based on the inverse heat conduction method, is solved by means of the finite volume method. On the basis of temperature measurements on the outer surface of the element, the algorithm enables thermomechanical analysis of the element. In order to perform experimental validation of the algorithm, the existing laboratory set-up was modernized. During the experiment, the thick-walled steam manifold was heated up rapidly with dry saturated steam. To reconstruct the unsteady temperature distribution within the element, 19 thermocouples were used and 3 thermocouples were inserted inside the wall of the element. The 3 thermocouple locations correspond with nodal positions in the numerical algorithm, thus allowing validation of the unstable temperature fields. The analysis of the time-step selection and of the influence of the accidental temperature measurement error on the accuracy of the solution have also been analyzed. Moreover, the algorithm has been validated based on the analytical method. The algorithm does not require knowledge of the boundary conditions on the element’s barely accessible internal surface – a highly valued feature in industrial applications.

… 2D transient heat conduction in a fiber-reinforced cylindrical composites. ► The effect of fibers' angle on the transient … The solution is valid for fiber-reinforced cylindrical composites with …

… (7) AND (8) were derived from the numerical solution of the heat conduction equation based … a finite cylinder or two dimensional rectangle. According to Eq. (7) and (8), only two design …

An approximate analytical solution to the transient heat-conduction problem in a large composite region with an internal cylindrical source is presented. The generalized orthogonal expansion technique is utilized in deriving the solution. Such problems are encountered in the design or simulation of the ground-coupled heat exchangers used in ground-coupled heat pumps. Solutions are presented for the nondimensional temperature as a function of the ratios of the thermal conductivities and thermal diffusivities of the two materials in the layers. To verify the correctness of the solution, comparisons are made between the two-layer composite solution and the classical homogeneous cylindrical source solution and a finite difference solution.

… on heat diffusion. We present a numerical simulation model for the study of three-dimensional heat diffusion through cylindrical … two-dimensional analytical solutions that are known for …

… Cylindrical medium The cylindrical domains studied in this section are an infinite cylinder and an infinite cylindrical ring bounded by black or gray walls [see Fig. 3(a) and (b)]. The …

… field and load in the Fourier series in terms of the circumferential coordinate, θ. The … Fourier’s heat conduction law. For a cylindrical geometry, we express ∇, κ and q in the cylindrical …

… The program is based on a direct statement of Fourier's law … For any Specific system of coordinates one needs only to … In the present study, only the cylindrical part of the heat flux …

… the cylindrical coordinate system. A singular integral equation is derived by applying the Fourier … far from its source, the classical Fourier law of heat conduction has been widely used in …

… for Poisson equation in cylindrical coordinates. The scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by …

Abstract The distribution of electromagnetic fields, forces and temperatures induced by a three- phase axially-symmetric system of electric current in a conducting cylinder of finite length has been calculated. An original method was used to calculate the mean values of electromagnetic forces. The magnetohydrodynamic (MHD) flow of viscous incompressible liquid and distribution of temperature are obtained by the finite-difference method, using monotonic finite difference schemes.

… of the heat conduction equation are presented for the arbitrary heating of a finite solid body of … of a circular solid cylinder of fmite length, general expressions are derived for the unsteady …

. In this paper, the effect of the fractional order of the Caputo time-derivative occurring in heat conduction models on the temperature distribution in a finite cylinder consisting of an inner solid cylinder and an outer concentric layer is investigated. The inner cylinder (core) and the cylindrical layer are in perfect thermal contact. The Robin boundary condition on the outer surface and the Neumann conditions on the ends of the cylinder are assumed. An internal heat source is represented in the mathematical model by taking into account in the heat conduction equation of a function which depends on the space and time variable. An analytical solution of the problem is derived in the form of the double series of eigenfunctions. Numerical examples are presented.

Steady periodic heating is an important experimental technique for measurement of thermal properties. In these methods the thermal properties are deduced from a systematic comparison between the data (such as temperature) and a detailed thermal model. This paper addresses steady-periodic heat transfer on cylindrical geometries with application to thermal-property measurements. The method of Green's functions is used to provide a comprehensive collection of exact analytical expressions for temperature in cylinders. Five kinds of boundary conditions are treated for one-, two-, and three-dimensional geometries. For some geometries an alternate form of the Green's function is given, which can be used for improvement of series convergence and for checking purposes to produce highly accurate numerical values. Numerical examples are given.

… In Section 4, we deduce the estimate for finite cylinders DT from the infinite cylinder case. For technical reasons, the results in Section 4 will only be established on smooth domains. Of …

… of the heat equation, as the wave like behaviour can be better evidenced in a simple geometry. Studies of periodic conduction in applied fields also exist for some particular case, like …

… value problems in the conduction of heat in composite circular cylinders is of considerable … hollow metallic cylinders of finite length are encountered under diverse heating and boundary…

… An infinite series solution was developed for a finite cylinder insulated at … paper the steady state solution is given for a finite cylinder maintained at z = L. As pointed out below equation …

… The finite cylinder is the most common of all the nuclear fuel configurations. In the highly … in which the thermal stresses in finite homogeneous cylinders vary on changing the length to …

Purpose – The purpose of this paper is to depict the effect of time and thermal and diffusion phase-lags due to axisymmetric heat supply in a ring. The problem is discussed within the context of dual-phase-lag heat transfer and dual-phase-lag diffusion models. The upper and lower surfaces of the ring are traction free and subjected to an axisymmetric heat supply. Design/methodology/approach – The solution is found by using Laplace and Hankel transform technique and a direct approach without the use of potential functions. The analytical expressions of displacements, stresses and chemical potential, temperature and mass concentration are computed in transformed domain. Numerical inversion technique has been applied to obtain the results in the physical domain. Numerically simulated results are depicted graphically. The effect of time and diffusion and thermal phase-lags are shown on the various components. Some particular cases of result are also deduced from the present investigation. Findings – It is observed that change in time changes the behaviour of deformations of the various components of stresses, displacements, chemical potential function, temperature change and mass concentration. The authors find that for t=0.2, trends are oscillatory in all the cases whereas for t=0.1, trends are quite different. A sound impact of diffusion and thermal phase-lags on the various quantities is observed. A lot of difference in the trends of single phase lag and dual phase lag is observed. The use of diffusion phase-lags in the equation of mass diffusion gives a more realistic model of thermoelastic diffusion media as it allows a delayed response between the relative mass flux vector and the potential gradient. Originality/value – This problem is totally new because dual phase lag is applied in heat conduction and diffusion equation while considering the problem of plate in axisymmetric heat supply.

… Governing equations The problem considered here is axisymmetric in nature and the governing equation for a single layer, in cylindrical coordinates, is given by:(1) ∂ 2 T ∂r 2 + 1 r ∂T …

… An analytical solution is derived for the axisymmetric thermo-elastic problem of multilayered material with anisotropic thermal diffusivity due to a buried heat source. By applying the …

… A simple lattice Boltzmann equation (LBE) model for axisymmetric thermal flow is proposed in this paper. The flow field is solved by a quasi-two-dimensional nine-speed (D2Q9) LBE, …

… diffusion for a half-space under thermal shock. This work is aimed at studying the thermoelastic diffusion interactions in a half-space under axisymmetric … of linear diffusion equation. …

… diffusion of heat and chemical species in axisymmetric flows. The boundary layer equations … The resulting similarity equations are of the same form as for the purely thermal point source …

… of axisymmetric heat supply on the phenomena of diffusion in … of generalized thermoelastic diffusion with one relaxation … free and subjected to an axisymmetric heat supply. The solution …

The effect of the mixed convection with anisotropic thermal diffusion on the bubble breakdown inside a cylindrical cavity with a rotating top and the stationary bottom is investigated in this article. The lattice Boltzmann multiple relaxation time axisymmetric method with the D2Q9 model is used. The three distribution functions, one for axial and radial components of the velocity field, second for an azimuthal component of the velocity field, and third for the temperature field, are used. The code is validated for the fluid flow inside a lid-driven cylindrical cavity and for hot rotating-top-lid cylindrical cavity. The results are compared with the benchmark data. The effect of mixed convection on the Bodewadt boundary layer thickness and the temperature boundary layer thickness is investigated. The simulations are perfermed for various Reynolds number (Re) from of 990 and 2494, the Richardson number (Ri in 0.01 and 1, and Rayleigh number (Ra from 9801 and 6.22 × 10 6). This study concludes that the Bodewadt boundary layer thickness ( δ B ) is ∝ to Ri and the temperature boundary layer thickness ( δ T c ) is ∝ to Ri for 0.01 ≤ Ri ≤ 0.1. Further, the δ B ≈ δ T c for isotropic thermal diffusion. For an anisotropic thermal diffusion, the δB remains constant and δ T c increases at Ri = 0.01 with an increase in the ratio of thermal diffusivity values (n). The bubble breakdown vanishes with an increase in Ri at a constant Re. The main application of this study is in the mixing/blending processes with convection inside cylindrical shaped reactors.

… is needed to recover the macro diffusion-convection equation for temperature filed. In … axisymmetric thermal flow. And natural convection in a vertical annulus with anisotropic diffusion …

… While the emphasis of this article is on axisymmetric geometries, it should be pointed out that the element boundary condition equations have been written in a form such that the …

… Having established the flow field, we now consider the axisymmetric energy equation … Equation (3), a convective diffusion equation for fl(p,v), is elliptic and lends itself to numerical …

… axisymmetric, or an object may subject to an axisymmetric energy source. Kim et al. (1990) found the analytical solution of HHC equation in the axisymmetric … of thermal diffusion (kD º …

… In the "gas phase," the distribution of the volume concentration Φg of magnetic particles obeys the diffusion equation and, for the problem in question, is described by Eq. (3), whereas in …

… -transformed thermal device and demonstrated its transient thermal cloaking behavior. To … thermal diffusion equation and used the anisotropic thermal diffusivity (instead of the thermal …

… It has been shown that the solutions of the linearized set of thermal diffusion equations can describe the motion of a liquid at extreme points of hydrodynamic fields. A generalization of …