Heat source problem of thermoelasticity in an elliptic plate with thermal bending moments

The principal aim of this article is to investigate the thermoelastic problems on an elliptical plate in which interior heat sources are generated within the solid, with compounded effect due to sectional heating and boundary conditions of the Dirichlet type. The analysis is based on the small-deflection theory of the elliptical plate and performed in the elliptical coordinate system. In addition, the intensities of bending moments, twisting moments, etc., are formulated involving the Mathieu and modified functions and their derivatives. The analytical solution for the thermal stress components is obtained in terms of resultant forces and resultant moments.     

Hygrothermoelastic response in the bending analysis of elliptic plate due to hygrothermal loading

Abstract

This study investigates the theoretical outline to couple both classical Fourier’s and Fick’s laws to frame a new model of two-temperature hygrothermoelastic diffusion theory for a non-simple rigid material. Based on hygrothermoelasticity method, a system of linearly coupled partial differential equations for the thermal and moisture diffusion for the case of a non-simple medium is established. The transient response using the decoupled technique of a multilayered elliptic plate perpendicular to the axial axis, subjected to hygrothermal loading is considered, to derive closed-form expressions for temperature, moisture, deflection, bending moments, and hygrothermal stresses. The solutions to the governing coupled equations and its boundary conditions are solved by employing a new integral transform technique. The small deflection equation is found and utilized to preserve the intensities of bending moments and stresses, involving the Mathieu functions and its derivatives. Moreover, the elliptical region can be degenerated into a circular part by applying limitations. Numerical results of the transient response of hygrothermoelastic fields are established graphically for the better understanding the underlying elliptic structure, improved understanding of its relationship to circular profile, and better estimates of the effect of the associated hygrothermoelastic responses.     

An analytical solution of the potential velocity field induced by a growing bubble from a plate orifice

An analytical solution is derived with the mirror image method of the velocity field of an inviscid liquid induced by a growing bubble from a plate orifice. The flow is assumed potential, and the bubble shape is idealised as spherical. In deriving the motion equation, the spherical image of a point source, which is a combination of a point source and a line source, is proved approximate to a double source. This approximation enables continuation of the effectiveness of mirror image method to the case studied in this paper. The derived velocity potential equation is verified for the boundary conditions on the bubble surface and the orifice plate. The streamlines of the velocity field are presented and compared with experimental results in the literature.     

Dynamic response of an infinite medium with a spherical cavity on temperature-dependent properties subjected to a thermal shock under fractional-order theory of thermoelasticity

In this work, we consider the problem for an infinite medium with a spherical cavity on temperature-dependent properties subjected to a stress shock and thermal shock under the fractional-order theory of generalized thermoelasticity. The modulus of elasticity and the coe?cient of thermal conductivity are taken as linear function of temperature. The governing equations for the problem are formulated and then solved by Laplace transform together with its numerical inversion. The nondimensional temperature, displacement, radial stress, and hoop stress are obtained and illustrated graphically. In the calculation, the emphasis is focused on investigating the effect of temperature-dependent properties on the variations of the considered variables. The graphical results indicate that the temperature-dependent modulus of elasticity plays a significant role on all the physical quantities.     

An exact analytical solution for two-dimensional,unsteady, multilayer heat conduction in spherical coordinates

Analytical series solution is proposed for the transient boundary-value problem of multilayer heat conduction in r–θ spherical coordinates. Spatially non-uniform, but time-independent, volumetric heat sources may exist in the concentric layers. Proposed solution is valid for any combination of homogenous boundary conditions of the first or second kind in the θ -direction. However, inhomogeneous boundary conditions of the first, second or third kind may be applied at the inner and outer radial boundaries of the concentric layers. It is noted that the proposed solution is “free” from imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of radial eigenvalues on those in the θ-direction. Solution is shown to be relatively simple for the most common spherical geometries?(multilayer) hemisphere and full sphere. An illustrative problem of heat conduction in a three-layer hemisphere is solved. Results along with the isotherms are shown graphically and discussed.     

Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds

This article is aimed at determining the thermoelastic displacement, stress, and temperature in a thick circular plate of finite thickness and infinite extent whose lower and upper surfaces are traction free, subjected to a given axisymmetric temperature distribution. The problem is formulated in the context of fractional order thermoelasticity theory with finite wave speeds. Integral transform technique is used to obtain the general solution in Laplace transform domain. Inversion of the Laplace transforms is done using a numerical scheme. A mathematical model is prepared for a copper material plate. Thermoelastic stresses, temperature and displacement are shown graphically and the effects of fractional-order parameters are discussed.     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate

The present article is concerned with analysis of large deflection of a heated thin annular sector plate with clamped edges under transient temperature distribution using Berger’s approximate methods. The prescribed surface temperature is at the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the Laplace transform as well as the classical method have been used for the solution of heat conduction equation. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using resultant bending moment and resultant forces per unit length. The calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the numerical results for temperature, deflection, resultant bending moments, and thermal stresses have been illustrated by graphs.     

Integral transform solution for hyperbolic heat conduction in a finite slab

An analytical integral transformation of the thermal wave propagation problem in a finite slab is obtained through the generalized integral transform technique (GITT). The use of the GITT approach in the analysis of the hyperbolic heat conduction equation leads to a coupled system of second order ordinary differential equations in the time variable. The resulting transformed ODE system is then numerically solved by Gear's method for stiff initial value problems. Numerical results are presented for the local and average temperatures with different Biot numbers and dimensionless thermal relaxation times, permitting a critical evaluation of the technique performance. A comparison is also performed with previously reported results in the literature for special cases and with those produced through the application of the Laplace transform method (LTM), and the finite volume-Gear method (FVGM).     

An exact solution for surface temperature in down grinding

A model has previously been developed for heat transfer in down grinding. A numerical solution algorithm was used to solve the system of equations. In this paper, an exact solution is found for this set of equations. The effect of the location of heat generation (i.e., at wear flats or shear planes) is explored for three typical grinding conditions: conventional grinding with aluminum oxide abrasives, creep feed grinding with aluminum oxide abrasives, and conventional grinding with cubic boron nitride (CBN) abrasives. It is found that during grinding with CBN, there is a strong effect of the assumed location of heat generation.     

Role of heat source/sink in transient free convective flow through a vertical cylinder filled with a permeable medium: An analytical approach

In this study, an investigation is performed to analyze the impact of the heat source/sink parameter on the laminar transient free convective flow through a vertical cylinder filled with a permeable medium. The governing nondimensional PDEs of the mathematical model along with their appropriate initial and boundary conditions are solved analytically by incorporating the Laplace transform scheme. Moreover, we explored the impact of emerging physical parameters of the considering model in the presence of the source/sink on the velocity profiles by graphs and tables. It is found that the velocity profile has a increasing tendency with enhancement in the numerical values of the time, which finally attains its steady-state solution in the presence of heat source/sink. Moreover, the Prandtl number, sink parameter, and viscosity ratio parameter lead to a decrease in the velocity profiles, whereas the reverse phenomenon occurs with the Darcy number and source parameter. Finally, the numerical values of the Nusselt number, skin friction, and mass flux are given in the tabular forms. The main result obtained in this paper is that the velocity is higher in the case of the source parameter, whereas an opposite behavior is observed in the case of the sink parameter.     

An analytical solution for a Stefan problem with variable latent heat

Governing equations for a one-phase Stefan melting problem with variable latent heat are presented. It is shown that these equations model the movement of the shoreline in a sedimentary basin. An analytical solution for the sedimentation rate and shoreline movement--based on a similarity variable--shows a square root dependence of shoreline position with time.     

Significance of time-dependent magnetohydrodynamic transient free convective flow in vertical annuli: An analytical approach with the finite Hankel transform

An analytical study is accomplished by using the Finite Hankel Transform Technique for the variation of the temperature and the velocity profile with several nondimensional parameters. In this problem, an electrically conducting fluid has been considered in the vertical concentric annulus, with a perpendicular radial magnetic field. Furthermore, a closed and exact type of expression for the velocity and the temperature are received in the form of Bessel functions of both kinds (first and second). The impact of emerging parameters in this model, such as time, Hartmann number, Prandtl number, and the annular gap between the cylinders, is discussed through graphs, whereas the numerical values of the skin friction, mass flux, and the Nusselt number are given in the tabular form. As a consequence of this, it is detected that the velocity and temperature distributions are increasing continuously with an enhanced time scale. Eventually, it gains its steady state very quickly. Moreover, the impact of the Prandtl number and the Hartmann number leads to a decrease in the velocity profiles.     

An analytical solution for air dehumidification by liquid desiccant in a packed column

This paper presents a new analytical solution of heat and mass transfer processes in a packed bed liquid desiccant dehumidifier based on the equilibrium humidity on the interface is assumed to be constant. In order to maintain the partial pressure difference on the interface, a high liquid desiccant flow rate is often applied in the practical absorber. Therefore, for a narrow range of operating conditions for practical dehumidification process, we can assume that the equilibrium humidity ratio on the interface is constant. The assumption of constant humidity ratio is applied in this paper for derivation of the analytical solution. The model and the analytical solution predictions were compared against a reliable set of experimental data available in the literature, with very good agreement. According to the Lewis definition in this present study, the Lewis number obtains 0.9. The effects of variables such as air and liquid desiccant flow rate, air temperature and humidity, desiccant temperature and concentration have been investigated on the condensation rate. The results present that design variables such as desiccant concentration, desiccant temperature, air flow rate, and air humidity ratio have the greatest impact on the performance of the dehumidifier. The liquid flow rate and the air temperature have not a significant effect. Furthermore, the effects of air and liquid desiccant flow rate have been reported on the humidity effectiveness of the column.     

Integral transform methods for inverse problem of heat conduction with known boundary of a thin rectangular object and its stresses

The three-dimensional inverse transient thermoelastic problem for a thin rectangular object is considered within the context of the theory of generalized thermoelasticity. The upper surface of the rectangular object occupying the space D: a≤x≤a; b≤y≤b; 0≤z≤h; with the known boundary conditions. Laplace and Finite Marchi-Fasulo transform techniques are used to determine the unknown temperature, temperature distribution, displacement and thermal stresses on upper plane surface of a thin rectangular object. The distributions of the considered physical variables are obtained and represented graphically.     

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

In this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.     

Fundamental solution for a line source of heat in the fractional order theory of thermoelasticity using the new Caputo definition

Abstract

The new Caputo Fabrizio fractional differential operator is used to investigate a problem in the fractional order theory of thermoelasticity. The problem concerns an infinite elastic space under the effect of a continuous line source of heat. The problem is solved using asymptotic expansions valid for short times. Laplace and Hankel transforms are used to solve the problem. A brief study to the nature of propagation of waves is introduced. Graphical results are presented and discussed.     

An analytical and experimental analysis of a very fast thermal transient

According to some international standards, some products, developed for use under heavy thermal conditions, have to be tested by subjecting them for a short time to a particular heating and cooling thermal stress to allow them an acceptable future operative life. It is possible to obtain these fast thermal gradients in confined environments, called climatic chambers where the air is heated by an electrical resistance and is cooled with a finned evaporator which is linked to a vapour compression system subjected to a particular control system of the refrigerating power. In particular, in this paper the air and object tested thermal transients are studied from an analytical and experimental point of view. The study of the mathematical model is realized assuming simplified hypotheses about the air, the object and the air cooled evaporator temperature. The most complex circumstances are related to a very fast temperature decrease because under this working condition the mathematical model is characterized by a nonlinear differential system. The nonlinear term is represented by the refrigerating power that varies in a definite range with the evaporator temperature according to a sinusoid trend. For this power a suitable analytical expression, derived by the control system performance and by the compressor characteristic, has been found. The analytical–experimental comparison during a cooling thermal stress of typical products subjected to international standard tests as the electronic boards, has been carried out showing acceptable results. The model presented is useful to foresee the climatic chamber performances in the presence of a specific refrigerating power trend; this is the start‐point for the design of the vapour compression plant and its control system. Copyright © 2001 John Wiley & Sons, Ltd.     

An axisymmetric crack in a functionally graded thermal barrier coating bonded to a homogeneous elastic substrate under transient thermal loading

Abstract

In this paper, the fracture problem of an axisymmetric crack in a functionally graded thermal barrier coating (FGTBC) bonded to a homogeneous substrate is considered. The problem is solved for the laminate that is suddenly heated from the upper surface of the FGTBC. The bottom surface of the homogeneous substrate is maintained at the initial temperature. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. By using both the Laplace and Hankel transforms, the thermo-mechanical fracture problem is reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The stress intensity factors of the crack are computed and presented as functions of the normalized time for various values of the nonhomogeneous and geometric parameters.