Axisymmetric Temperature Distribution Research Articles

Morton effect prediction with validation using a CFD based CNN for pad inlet temperatures

The Morton Effect (ME) is a thermal-fluid–structure interaction instability occurring in rotating machinery supported by hydrodynamic journal bearings. The mechanism of ME consists of a bowed rotor or mass imbalance induced shaft synchronous whirl vibration in the bearing, which causes local, asymmetric heating of the journal, which causes shaft bending, potentially leading to increasing vibration. This study presents an original ME simulation approach that includes a CFD (Computational Fluid Dynamics) bearing groove model, embedded in a deep learning algorithm for computational efficiency and non-expert usage. The groove model provides a 2D oil temperature distribution at the leading edge of the bearing pads, yielding a more accurate journal axisymmetric temperature distribution which is the source of the ME. The paper provides validation of the approach by test result correlation, and illustrates the effects of parameter variation and configuration variation, by examining various oil injection types. The approach may be used for correcting ME occurring in existing machinery, or for designing machinery to avoid the ME.

A comparison of mixture and separated-phase models of heat transfer in a stationary wet granular bed

Solvent removal by thermal drying of granular mixtures is encountered in many industrial operations. A mixture model that treats the granular solid and the liquid solvent along with any interstitial gas phase as a single phase, is a useful first approximation to estimate the time required to heat the system to the wet-bulb temperature at which the solvent starts to vaporize. An analytical solution for the transient, axisymmetric temperature distribution in a stationary wet granular mixture in a heated cylindrical vessel is developed as an approximation to the typical geometry of an industrial dryer . Besides being a good first approximation to obtain an order-of-magnitude estimate of the heating time, the analytical solution serves as an excellent verification case for numerical simulations. A parametric study to assess the effect of fill height and wall temperature on the heating time is performed. Before analyzing the mass transfer aspects of the drying phenomena, it is critical to understand the fundamentals of interphase heat transfer in multiphase mixtures that can be found in such assemblies. Therefore, a 0-D separated-phase (multiphase) heat transfer model is developed to account for the effect of interphase heat transfer in the prediction of heating time of wet granular mixtures in cylindrical vessels. The 0-D model is extended to 3-D simulations using a newly developed OpenFOAM solver with multiphase heat transfer capability. The same problem that was solved using the analytical solution is solved using the 0-D model, and in 3-D using the newly developed OpenFOAM solver, to study the effect of interphase heat transfer on the prediction of heating time. It was found that inclusion of interphase heat transfer in the heat diffusion equations has a significant impact on the predictions of heating time. Further analysis of the problem results in the identification of the non-dimensional thermal interphase transfer number that determines the conditions under which a mixture model can be used in place of a separated phase model in problems involving heat transfer.

Thermoelastic interactions on a thick granular plate with type II thermoelasticity under axisymmetric temperature distribution

This paper deals with the effects of type II thermoelasticity on wave propagation in a generalized thermoelastic granular medium. A two-dimensional thick granular plate with axisymmetric temperature distribution is considered under the generalized theory (with one relaxation time) and type II thermoelasticity. The lower and upper surfaces of the plate are assumed to be stress-free and subjected to a thermal condition. The analytical solution of the different fields like temperature, displacement, stresses, rotation, and couple stresses are obtained by the Laplace transform procedure along with the Hankel transform method. The short-time approximation and Boley’s theorem are used to analyze the fields like temperature and rotation vector in the physical domain. The numerical values are displayed graphically in the middle plane for different models, and comparison illustrates the problem. The study may be helpful in soil mechanics, geophysical-prospecting, seismology, and mining engineering.

2D PROBLEM FOR A SPHERE IN THE FRACTIONAL ORDER THEORY THERMOELASTICITY TO AXISYMMETRIC TEMPERATURE DISTRIBUTION

In the present article, we implement the fractional thermoelasticity theory to a 2D issue for a sphere whose surface is free from traction, subject to a provided axisymmetric temperature distribution of heat. The medium is supposed to be quiescent initially. A direct method is used to get a solution and the Laplace transform technique is used. Mathematical models for copper material are designed as a particular instance. Numerical results are computed with help of Mathcad software and graphically represented and the fractional-order parameter effect has been explained.

New fractional model for 2 dimensional half space problem with in the theory of generalized thermoelastic diffusion

AbstractIn this manuscript, the generalized thermoelastic diffusion theory is modified by using fractional derivatives. The revised equation is employed to study a two‐dimensional problem of the axisymmetric temperature distribution during a half space with a permeating substance in touch with the bounding plane. The surface of the half space is taken into account to be traction‐free and subject to an axisymmetric temperature distribution. Laplace and Hankel transformation techniques are used. The analytical solution within the transform domain is obtained by an immediate method. By employing a numerical method supported the Fourier expansion technique, the inverse of the double transforms is often obtained. Numerical results for the temperature, displacement, stress, concentration, and chemical potential are administered and represented graphically. The figures show some comparisons to estimate the impact of fractional parameter on all research areas.

2D hereditary thermoelastic application of a thick plate under axisymmetric temperature distribution

In this work, we present a new theory of generalized fractional hereditary thermoelasticity associated with Mittag–Leffler relaxation function. We analyze a two‐dimensional problem of a thick plate of hereditary thermoelastic. The lower and upper surfaces of the plate are assumed to be traction free and subjected to a given axisymmetric temperature distribution. Direct approach together with Laplace and Hankel transform techniques is employed to obtain the solution in the transformed domain. Hankel transforms are inverted analytically while a numerical method is used for the Laplace transform inversion. The distributions of different fields like temperature, displacement, and stresses have been computed and shown graphically.

Thermoelastic interactions on thick granular plate with three-phase-lag model under axisymmetric temperature distribution

The theory of thermoelasticity with three-phase-lag is used to study the phase lag effects on wave propagation in granular medium. A two-dimensional thick granular plate with axisymmetric temperature distribution is considered under the type III and three-phase-lag thermoelasticity. The plate’s lower and upper surfaces are assumed to be stress free and subjected to a thermal condition. The analytical solution of the different fields like temperature, displacement, stresses, rotation, and couple stresses is obtained by the Laplace together with the Hankel transform method. The characteristic features of the underlying theory are analyzed by the small-time approximation method with Boley’s theorem and by comparing these results with their counterparts in other generalized thermoelasticity theories (GN III and Lord Shulman theory). The numerical values are evaluated and displayed graphically in the middle plane for different models, and comparison is made.

Effect of Atangana–Baleanu fractional derivative on a two-dimensional thermoviscoelastic problem for solid sphere under axisymmetric distribution

A two-dimensional fractional thermoviscoelastic problem for solid sphere under axisymmetric temperature distribution is presented. The solution of the problem is obtained in the Laplace transform domain by using a direct approach. The surface of the sphere is taken to be traction-free and subjected to the given axisymmetric temperature distribution. The inversion of the Laplace transform is carried out using the inversion formulas of the transform together with Fourier expansion techniques. Numerical results are represented graphically and discussed.

Two dimensional spherical regions problem in the context of the theory of generalized thermoelastic diffusion

In this work a spherical thermoelastic region problem with a permeating substance in contact of the bounding plane is considered in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The general solution is obtained in the Laplace transform domain by using a direct approach without the use of potential functions. The resulting formulation is used to solve problem of a solid sphere. The surface is taken to be traction free, subjected to a given axisymmetric temperature distribution and the chemical potential also assumed to be a known function of time. The inversion of the Laplace transform is carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical methods are used to accelerate the convergence of the resulting series to obtain the temperature, displacement, concentration, stress distributions as well as the chemical potential in the physical domain. Numerical results are represented graphically and discussed.

The effect of fractional thermoelasticity on two-dimensional problems in spherical regions under axisymmetric distributions

Two-dimensional axisymmetric problems are considered within the context of the fractional order thermoelasticity theory. The general solution is obtained in the Laplace transform domain by using a direct approach without the use of potential functions. The resulting formulation is used to solve two problems of a solid sphere and of an infinite space with a spherical cavity. The surface in each case is taken to be traction free and subjected to a given axisymmetric temperature distribution. The inversion of the Laplace transforms is carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical methods are used to accelerate the convergence of the resulting series to obtain the temperature, displacement, and stress distributions in the physical domain. Numerical results are represented graphically and discussed. Some comparisons are shown in figures to estimate the effect of the fractional order parameter on all studied fields.

Analytical solution for magneto-thermo-elastic responses of an annular functionally graded sandwich disk by considering internal heat generation and convective boundary condition

In this paper, an exact solution for stress distribution of a rotating sandwich disk subjected to magnetic, thermal fields, and convection heat transfer with the consideration of internal heat generation is presented. The sandwich disk is composed of three different layers in which the central layer is made of functionally graded material, whereas the inner and outer layers are made from pure metal and ceramic materials, respectively. All mechanical and thermal properties of the central layer are assumed to obey a power-law form in the radial direction with different non-homogeneity constants. Axisymmetric temperature distributions in each layer are obtained by solving Fourier heat conduction equation. By substituting the constitutive equations in equilibrium equation, a second-order differential equation in terms of radial displacement for each layer is derived by considering centrifugal force and Lorentz magnetic force obtained from Maxwell’s relations. Thereafter, the governing equations are analytically solved and radial displacement, radial and circumferential stresses for the current disk are obtained. Numerical simulations reveal the effects of internal heat generation, convection heat transfer, magnetic field, material properties, and angular velocity on the magneto-thermo-elastic response of the functionally graded sandwich disk. Also, a comparison study between metal, functionally graded, ceramic, and functionally graded sandwich disks is performed.

A theoretical model for delayed hydride cracking velocity considering the temperature history and temperature gradients

Delayed hydride cracking (DHC) threatens the safety of Zircaloy components. The DHC velocity in the stage of stable crack growth needs to be studied. Based on the recent research, with a new idea the theoretical model is developed for the DHC velocity considering the temperature history and temperature gradient. The stress field ahead of the crack tip is calculated based on the fracture mechanics theory of second-order estimate of plastic zone size, and the redistributed stresses are statically equivalent with the stresses obtained from Linear Elastic Fracture Mechanics. It is confirmed that a simple linear function of the equivalent axisymmetric temperature distribution can be effectively and easily calibrated for predictions of the DHC velocity at different temperature gradients along the cracking direction. The developed model is validated to be effective because the predicted velocities agree well with some experimental results for the cases with and without temperature gradients, whether the test temperature is approached by heating or cooling. The theoretical results indicate that (1) with a higher positive temperature gradient along the cracking direction, the DHC velocity increases and a higher crack arrest temperature will occur; while a negative temperature gradient will remarkably depress DHC process; (2) The total hydrogen concentration in the hydrogen source has an important effect on the DHC velocity, and the crack arrest temperature can be evidently lowered for the cases with a small quantity of hydrogen atoms there;(3) At higher temperatures, negative flux contribution from the hydrogen concentration gradient in the hydrogen source is found to be the main mechanism of crack arrest.

Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source

This paper is concerned with fractional order thermoelastic response due to a heat source whose magnitude varies periodically with time within the context of generalized thermoelasticity with one relaxation time. Traction free boundary conditions are considered and the thick circular plate is subjected to a given axisymmetric temperature distribution. Integral transform technique is used to derive the solution in the transformed domain. Laplace transforms are inverted using a numerical scheme. Mathematical model is prepared for Copper material and results for temperature, displacement and stress distributions are computed and represented graphically.

A 2D problem of thermoelasticity without energy dissipation for a sphere subjected to axisymmetric temperature distribution

ABSTRACTWe solve a 2D problem for a sphere in the theory of thermoelasticity without energy dissipation WED. The sphere’s surface is taken to be traction free and subjected to an axisymmetric temperature distribution that is harmonic in time. A direct approach is used to solve the problem. We consider two different cases of the boundary conditions. The collocation method was used in one of them where we could not obtain the exact solution. Numerical results are represented graphically and discussed.

A detailed comparative study on responses of four heat conduction models for an axisymmetric problem of coupled thermoelastic interactions inside a thick plate

The objective of the present work is to analyze the thermoelastic interactions inside an infinitely extended thick plate due to an axisymmetric temperature distribution applied at the lower and upper surfaces of the plate under recent heat conduction models, namely Green-Naghdi-I model, Green-Naghdi-II model, dual phase-lag model and Green-Lindsay model. In order to investigate the problem under all these four heat conduction models simultaneously, we consider the basic governing equations under all these models and formulate our problem in a unified way. The potential function concept along with Laplace and Hankel transform techniques has been used to solve the problem. Inversion of Hankel transform has been carried out analytically to obtain the solution in Laplace transform domain. The short-time approximation technique is employed to invert the solutions obtained in Laplace transform domain and an appropriate analytical approach has been used to analyze the wave propagation and discontinuities of different wave fields. In addition, a numerical method has been used to invert the Laplace transform directly in order to find out the distributions of all the physical fields, like stress, temperature and displacement in the middle plane of the plate. Results are analyzed to make a comparative analysis of the predictions of Green-Naghdi-II model with the predictions by dual phase-lag thermoelastic model and other models. The special findings and differences among the predictions by four models have been highlighted.

2D Axisymmetric Problem for a Sphere with Heat Sources in the Theory of Generalized Thermoviscoelasticity

A solid sphere composed of a thermoviscoelastic material is subjected to an axisymmetric temperature distribution on its surface that is traction free. A distributed heat source acts inside an inner sphere. The problem is solved analytically and numerically. The solutions are represented graphically for different cases.

Thermoelastic-induced vibrations on an elliptical disk with internal heat sources

ABSTRACTIn this study, integral transform technique is used to investigate the thermally induced vibration of an elliptical disk. The axisymmetric temperature distribution in the disk is determined by conductivity equation and the corresponding initial and boundary conditions using an extended integral transform technique. The problem of thermally induced vibration of the disk with both ends clamped extremes is solved by developing an integral transform for double Laplace differential equation. The thermal moment is derived on the basis of temperature field, whereas maximum normal stresses are derived based on resultant bending moments per unit width. The results are obtained in series form in terms of Mathieu functions, and numerical results are shown in figures.

Mathematical modelling of the transient response of pipeline

Steam pipelines applied in power units operate at high pressures and temperatures. In addition, to stress from the pipeline pressure also arise high thermal stresses in transient states such as start-up, shutdown or a load change of the power unit. Time-varying stresses are often the cause of the occurrence of fatigue cracks since the plastic deformations appear at the stress concentration regions. To determine the transient temperature of the steam along the steam flow path and axisymmetric temperature distribution in the pipeline wall, a numerical model of pipeline heating was proposed. To determine the transient temperature of the steam and pipeline wall the finite volume method (FVM) was used Writing the energy conservation equations for control areas around all the nodes gives a system of ordinary differential equations with respect to time. The system of ordinary differential equations of the first order was solved by the Runge-Kutta method of the fourth order to give the time-temperature changes at the nodes lying in the area of the wall and steam. The steam pressure distribution along pipeline was determined from the solution of the momentum conservation equation. Based on the calculated temperature distribution, thermal stresses were determined. The friction factor was calculated using the correlations of Churchill and Haaland, which were proposed for pipes with a rough inner surface. To assess the accuracy of the proposed model, numerical calculations were also performed for the thin-walled pipe, and the results were compared to the exact analytical solution. Comparison of the results shows that the accuracy of the proposed model of pipeline heating is very satisfactory. The paper presents examples of the determination of the transient temperature of the steam and the wall.

An investigation on coupled thermoelastic interactions in a thick plate due to axi-symmetric temperature distribution under an exact heat conduction with a delay

Present work is concerned with a recently proposed heat conduction model: an exact heat conduction model with a single delay term. The main purpose of this paper is to examine the effects of the single phase-lag parameter/delay term on wave propagation in an infinitely extended thick plate due to axisymmetric temperature distribution applied in the lower and upper surfaces of the plate. The problem is formulated by using the new heat conduction model in such a way that other existing models can be extracted as special cases. Potential function concept including the Laplace and Hankel Transform method is used to find the solution in the transformed domain. Inversion of Hankel Transform is done analytically to obtain the solution in Laplace transform domain. By using a suitable analytical approach, the complete analysis on the wave propagation and discontinuities of different wave fields are found out. Furthermore, a numerical method for the Laplace inversion is applied to obtain the distributions of different physical fields like, temperature, displacement and stresses in the middle plane of the plate. Analysis of the results obtained under new model along with a comparison of the respective results obtained in other models is presented in detailed way. The findings in the present work are believed to bring out some lights concerning the new heat conduction model that involves a single delay parameter.

Efficient and robust method for simultaneous reconstruction of the temperature distribution and radiative properties in absorbing, emitting, and scattering media

A rapid computational method called generalized sourced multi-flux method (GSMFM) was developed to simulate outgoing radiative intensities in arbitrary directions at the boundary surfaces of absorbing, emitting, and scattering media which were served as input for the inverse analysis. A hybrid least-square QR decomposition–stochastic particle swarm optimization (LSQR–SPSO) algorithm based on the forward GSMFM solution was developed to simultaneously reconstruct multi-dimensional temperature distribution and absorption and scattering coefficients of the cylindrical participating media. The retrieval results for axisymmetric temperature distribution and non-axisymmetric temperature distribution indicated that the temperature distribution and scattering and absorption coefficients could be retrieved accurately using the LSQR–SPSO algorithm even with noisy data. Moreover, the influences of extinction coefficient and scattering albedo on the accuracy of the estimation were investigated, and the results suggested that the reconstruction accuracy decreased with the increase of extinction coefficient and the scattering albedo. Finally, a non-contact measurement platform of flame temperature field based on the light field imaging was set up to validate the reconstruction model experimentally.