Heat Transfer Analysis of Functionally Graded Hollow Cylinders Subjected to an Axisymmetric Moving Boundary Heat Flux

The transient heat transfer analysis of functionally graded (FG) hollow cylinders subjected to a distributed heat flux with a moving front boundary on its inner surface is presented. The heat flux is assumed to be axisymmetric, and its front boundary moves along the axis of the cylinder. A method composed of the finite element and differential quadrature methods is employed to discretize the governing equations in the spatial domain. After demonstrating the convergence and accuracy of the method, the effects of different parameters on the temperature distribution and time history of the temperature at different points of FG cylinder are investigated.     

Inverse Thermo-Mechanical Analysis of a Functionally Graded Fin

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent base heat flux of a functionally graded fin from the knowledge of temperature measurements taken within the fin. Subsequently, the distributions of temperature and thermal stresses in the fin can be determined as well. It is assumed that no prior information is available on the functional form of the unknown base heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors and measurement location upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Inverse Transient Heat Conduction Problems of a Multilayered Functionally Graded Cylinder

Inverse transient heat conduction problems of a multilayered functionally graded (FG) cylinder are presented. The approach is based on measurement of temperature on the outer surface of the cylinder to estimate the heat flux and convection heat transfer coefficient on its inner surface. The non-Fourier heat transfer equation is employed to accurately formulate the problem. The conjugate gradient method (CGM) is used for the optimization procedure and the incremental differential quadrature method (DQM) is applied to solve the direct, sensitivity, and adjoint problems. The accuracy of the presented approach is examined by simulating the exact and noisy data through different examples. Good accuracy of the obtained results validates the presented approach.     

Estimation of Heat Flux and Thermal Stresses in Functionally Graded Hollow Circular Cylinders

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Thermal Residual Stresses in One-Directional Functionally Graded Plates Subjected to In-Plane Heat Flux

This study carries out the transient thermal residual stress analyses of functionally graded clamped plates for different in-plane material compositions and in-plane heat fluxes. The heat conduction and Navier equations representing the two-dimensional thermoelastic problem were discretized using the finite-difference method, and the set of linear equations were solved using the pseudo singular value method. Both in-plane temperature distributions and the heat transfer period were affected considerably by the compositional gradient. The type of in-plane heat flux had a minor effect on the temperature profile, but on the heat transfer period. The high stress levels appeared in the ceramic-rich regions. The normal and equivalent stresses exhibited a sharp change in the plates with ceramic-rich as well as metal-rich compositions, and the concentrated on a narrow ceramic layer. A smooth stress variation was achieved through the graded region with a balanced composition of ceramic and metal-phases, and the stress discontinuities disappeared. The in-plane shear stress was negligible. The equivalent stress exhibited a linear temporal variation for both constant and sinusoidal heat fluxes, but a nonlinear variation for the exponential heat flux. In case the heat flux is applied along the metal edge (metal-to-ceramic plate) instead of the ceramic edge, the displacement and stress components exhibited similar distributions to those of a ceramic-to-metal plate but in the opposite direction. As a result, the distribution of in-plane material composition affects only normal stress distributions, whereas the peak stress levels occur in the ceramic-rich regions. Since the normal stresses concentrate along a narrow ceramic layer for ceramic-rich or metal-rich compositions, a balanced in-plane material composition distribution of ceramic and metal would be useful to avoid probable local ceramic fracture or damage.     

Two-Dimensional Heat Conduction Problem in a Functionally Graded Plate with a Slanting Boundary to the Functional Gradation

This article presents analytical results for temperature in a functionally graded material plate (FGMP) with a slanting boundary to the functional gradation subjected to a partial heating. The heat conductivity is expressed in terms of a exponential function of the position. The general solution of the heat conduction equation for FGMP with a slanting boundary to the functional gradation is derived by use of the variable separation method and the analytical solution, which satisfies the boundary condition is obtained. Numerical calculations are carried out for ZrO₂/Ti-6Al-4V and ZrO₂/stainless (SUS304) functionally graded plates, when the ceramic surface is partially heated. Temperature and heat flux are graphically displayed for these two cases.     

Transient Thermal Stress Analysis of Orthotropic Functionally Graded Materials with a Crack

The transient thermal fracture problem of a crack (perpendicular to the gradient direction) in a graded orthotropic strip is investigated. Most of the materials properties are assumed to vary as an exponential function of thickness direction. The transient two-dimensional temperature problem is analyzed by the methods of Laplace and Fourier transformations. A system of singular integral equations are obtained and solved numerically. Numerical results are figured out to show the variation of the temperature on the crack faces and extended line and stress intensity factors for different material parameters with dimensionless time.     

Numerical Simulation of Non-Fourier Temperature Behavior of Functionally Graded Materials in a Slab with Surface Radiation

Functionally graded materials (FGMs) are used in many applications that presumably produce the wave nature of thermal energy transport. This study investigates the hyperbolic and parabolic heat conduction problem for a solid slab made of FGM numerically. A constant heat flux is considered at both sides of the slab, and boundaries dissipate heat by radiation into an ambient. An exponential space-dependent function of volume fraction is considered. MacCormack's explicit predictor-corrector scheme is used to solve the nonlinear equation in order to handle discontinuities at the wave front quite satisfactorily with small oscillations. Results are compared to the results obtained with the assumption of constant and linear spatial variation of volume fraction function. Further effects of different nondimensional numbers on the temperature distribution is sought. Numerical results are validated by the analytical solution of a special case that shows excellent agreement.     

Transient Thermoelastic and Piezothermoelastic Problems of Functionally Graded Materials

This paper is concerned with the theoretical treatment of transient thermoelastic problems involving functionally graded thick plate, laminated composite strip with an interlayer of functionally graded material, and functionally graded hollow cylinder, and transient piezothermoelastic problems involving functionally graded piezoelectric cylindrical panel. The thermal, thermoelastic and piezoelectric constants of the functionally graded materials are expressed as power functions of the radial coordinate variable or exponential functions of the thickness coordinate variable. The exact solutions for the two-dimensional temperature change in a transient state, and thermoelastic or piezothermoelastic response under the state of plane strain are presented herein. Some numerical results are shown in figures.     

Dual-Phase-Lag Heat Conduction in Finite Slabs With Arbitrary Time-Dependent Boundary Heat Flux

Applying a constant or transient heat flux on a plane slab is a common technique in microelectronics technology and material processing, including laser patterning, micromachining, and laser surface treatment processes. Although Fourier's law is typically very precise for evaluating temperatures in solids, a number of experimental observations suggest the existence of non-Fourier transient conduction in these applications. Since the dual-phase-lag (DPL) model of heat conduction can be compatible with the hypothesis of local equilibrium thermodynamics (as shown here), the effects of temperature gradient relaxation time on the non-Fourier hyperbolic conduction in a finite slab subjected to an arbitrary time-dependent surface heat flux is examined by this model. The combination of diffusion- and wave-like features in heat conduction process is properly monitored by the DPL model for two types of heat flow regimes, namely, gradient precedence and flux precedence. The results indicate considerable deviations between the predictions of these regimes.     

Thermal Buckling Analysis of Functionally Graded Thin Circular Plate Made of Saturated Porous Materials

This study presents the buckling analysis of thermal loaded solid circular plate made of porous material. It is assumed that the material properties of the porous plate vary across the thickness. The edge of the plate is clamped and the plate is assumed to be geometrically perfect. The geometrical nonlinearities are considered in the Love–Kirchhoff hypothesis sense. Equilibrium and stability equations, derived through the variational formulation, are used to determine the prebuckling temperatures and critical buckling temperatures. The equations are based on the Sanders non-linear strain-displacement relation.The porous plate is assumed of the form where pores are saturated with fluid. Also, the effect of pores distribution and thermal distribution on the critical buckling temperature is investigated.     

Thermal Postbuckling Analysis of Functionally Graded Beams

Thermal buckling and postbuckling analysis of functionally graded (FG) beams is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. For discretizing the governing equations and the related boundary conditions differential quadrature method (DQM) as a simple and computationally efficient numerical tool is used. Based on displacement control method, a direct iterative method is employed to obtain thermal postbuckling behavior of FG beams with different boundary conditions and geometrical parameters.     

An Evaluation of Heat Transfer Surface Materials Used in Fouling Applications

Fouling is a very important and complex problem that extends into many fields, including natural, chemical, medical, and industrial processes. Fouling of a surface takes place as a result of the complex reactions that cause deposits to form on process surfaces. A number of parameters influence fouling development, including flow velocity, surface temperature, surface material/finish, surface geometry and fluid properties. Fouling is a transient process that begins with a clean process surface and progresses until the surface no longer can be used effectively. The event sequence of the fouling process appears in general to be universal, beginning when fluid comes into contact with a process surface. During the induction period, the conditioning film forms with heat transfer efficiencies not changing significantly. Conditioning film development is followed by a rapid accumulation of deposit growth. It is during this growth phase that the heat transfer across the process surface starts to dramatically change. Finally, a pseudo steady-state period takes place when accumulation is almost constant. Deposit accumulation causes efficiencies to significantly decrease, and a complete surface cleaning may be required. Conclusions and observations regarding the materials/surfaces that are commonly used in designs where fouling may be a concern are presented here. Comparisons of fouling rate and deposit thickness are given for several materials.     

Thermoelastic Analysis of Functionally Graded Rotating Disks with Variable Thickness Involving Non-Uniform Heat Source

The research aims to investigate thermoelastic behavior of functionally graded rotating disks with variable thickness involving a non-uniform heat source. We assume material properties and thickness of rotating disks to vary in the radial direction. Axisymmetric thermal loads including non-uniform heat source, heat flux, and temperature boundary conditions are considered. To conduct corresponding simulations, two user subroutines are edited and incorporated into the commercial finite-element code ABAQUS. For verification, analytical formulations are derived and solved uniquely by symbolic calculations using the computing software Mathematica. The developed finite-element technique is then verified with very good agreement between results by ABAQUS and Mathematica.     

A High Order Control Volume Finite Element Method for Transient Heat Conduction Analysis of Multilayer Functionally Graded Materials with Mixed Grids

This paper describes a new two-dimensional(2-D) control volume finite element method(CV-FEM) for transient heat conduction in multilayer functionally graded materials(FGMs). To deal with the mixed-grid problem, 9-node quadrilateral grids and 6-node triangular grids are used. The unknown temperature and material properties are stored at the node. By using quadratic triangular grids and quadratic quadrilateral grids, the present method offers greater geometric flexibility and the potential for hig...     

Study on Laminar Natural Convection Heat Transfer from a Hemisphere with Uniform Heat Flux Surface

By employing the modified model based on Bejan et al., laminar natural convection heat transfer from a hemisphere with uniform heat flux surface has been numerically investigated. Extensive results of two different surface boundary conditions are obtained for a wide range of Grashof numbers(10 ≤ Gr ≤ 10~7) and Prandlt number of 0.72. The characteristics of heat transfer and fluid flow are analyzed in terms of isotherm contours and streamline patterns, radial and tangential velocities, dimensionless temperature profiles, local friction and pressure drag coefficients, as well as local and average Nusselt numbers. Meanwhile, the effects of Grashof number and adiabatic surface on flow motion and heat transfer have been studied. No recirculation zone or flow separation generates over the top of the hemisphere compared to the isothermal sphere. Owing to the curvature effect, the maximum values of local friction and pressure drag coefficients appear at the corner point B. Comparisons with the previous results are also reported in detail. All the results are in good agreement with the numerical data. Moreover, both local and average Nusselt numbers show a positive dependence on Grashof number. The values of the non-adiabatic case are smaller than that of the adiabatic case due to the preheating effect. Finally, two precise and general correlations of average Nusselt number varying with Grashof numbers have been presented, which can provide an effective prediction for the heat transfer rate in engineering applications, and offer academic values for the future research.     

Analysis of Thermal Nonequilibrium Inverse Heat Transfer in a Porous Channel

This article is concerned with the one-dimensional transient inverse heat conduction between two parallel plates filled with a porous medium under a nonthermal equilibrium condition between the solid and the fluid phases. The estimation of transient heat flux using the conjugate gradient method (CGM) along with the differential adjoint equations has been carried out under nonthermal equilibrium conditions between two phases. Derivation of the adjoint differential equations in the case of nonthermal equilibrium and calculation of gradient function from coupled adjoint equations are presented in detail here. The transient wall heat flux imposed on the porous boundary is estimated using the aforementioned method, and results show that sensor locations and existing error in the measured data have important effects on the calculated heat flux. Nonetheless, accurate heat flux estimation is quite achievable.     

Thermal Buckling Analysis of Perforated Functionally Graded Plates

In this research, the buckling behavior of functionally graded (FG) plates under thermal loading is investigated based on finite element analysis. It is assumed the plate is subjected to a uniform temperature rise across plate thickness. First-order shear deformation theory (FSDT) is utilized for developing the solution method. By using an appropriately designed mesh structure for a perforated plate, the critical thermal buckling temperature is obtained by numerical solution of the problem based on finite element method (FEM). The FG plate is perforated by multiple cutouts. The number of cutouts is assumed one, two, four, or six. Also different geometrical shapes of cutouts including triangle, square, rhombus, pentagon, hexagon, and circle are considered. The influence of the number of cutouts and their geometrical shapes on thermal buckling response is investigated. The effects of the number of sides of cutouts from three (triangle) to infinity (circle) are discussed. Two different boundary conditions are taken into account. Also the influences of the distance between the cutouts and the orientation of cutouts on critical buckling temperature are studied. In addition, the effects of the orientation of ellipse cutouts are studied. Some remarkable conclusions are gained that can be useful in practical applications.     

Nonaxisymmetric Thermomechanical Analysis of Functionally Graded Hollow Cylinders

Analytical solutions for nonaxisymmetric, thermomechanical response of functionally graded hollow cylinders are obtained in this article. The hollow cylinders are assumed to be subjected to nonaxisymmetric mechanical and transient thermal loads. Properties of functionally graded material are considered as temperature-independent and continuously varying in radial direction. Employing complex Fourier series and Laplace transform techniques, analytical solutions of time-dependent temperature and thermomechanical stresses are obtained. Numerical values of temperature and stresses of a FGM hollow cylinder under assumed thermomechanical loads are presented in graphical form.     

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

A novel multistep method based on the nonuniform rational basis spline curves is developed for the solution of a system of nonlinear differential equations. Efficiency of the presented method in terms of convergence and accuracy is demonstrated through the two-dimensional nonlinear transient heat transfer analysis of multilayered functionally graded truncated cone subjected to an internal thermal load with radiative–convective boundary condition at outer surface. In order to discretize the governing equations in the spatial domain, a layerwise-differential quadrature method is implemented. This transforms the partial differential equations into a system of nonlinear ordinary differential equations in the temporal domain, which is then solved using the presented technique. Comparative studies between predictions of the presented multistep technique and various existing numerical methods demonstrate superior computational efficiency of the method with low computational cost. High efficiency of this method indicated that the presented multistep approach can be successfully employed in various nonlinear problems. Finally, the effects of different parameters on the transient temperature of the truncated cone are studied. It is expected that the presented multistep method will be applied on various practical engineering problems in future studies.