General Thermal Boundary Conditions Research Articles

Analytical solution of temperature in laminated beams subjected to general thermal boundary conditions

Analytical solution of temperature in laminated beams subjected to general thermal boundary conditions

The Onset of Buoyancy and Surface Tension Driven Convection in a Ferrofluid Layer by Influence of General Boundary Conditions

This paper investigated the buoyancy and surface tension-driven ferro-thermal-convection (FTC) in a ferrofluid (FF) layer due to influence of general boundary conditions. The lower surface is rigid with insulating to temperature perturbations, while the upper surface is stress-free and subjected to general thermal boundary condition. The numerically Galerkin technique (GT) and analytically regular perturbation technique (RPT) are applied for solving the problem of eigenvalue. It is analyzed that increasing Biot number, decreases the magnetic and Marangoni number is to postponement the onset. Additionally, magnetization nonlinearity parameter has no effect on FTC in the non-existence of Biot number. The results under the limiting cases are found to be in good agreement with those available in the literature.

Analytical solution of micropolar functionally graded materials (FGM) hollow cylinder under thermal asymmetric load (r, θ)

AbstractThis paper focuses on the analysis of two‐dimensional steady‐state thermal stresses based on micropolar theory in hollow and solid cylinders which are made of functionally graded materials (FGM) under asymmetric loading condition (r, θ) with the general thermal and mechanical boundary conditions along the inside and outside surfaces. The heat conduction and Navier equations were solved using the separation of variables and complex Fourier series for micropolar theory. Finally, the exact numerical results for both micropolar and classical solutions are illustrated and discussed using two examples: In Example 1, based on the micropolar theory, when thermal tensions are exerted on the inside surface of the cylinder, some of the strain energy is used for the object rotation; therefore, considering this condition, the calculated values resulted from the micropolar theory are smaller than those of the classic theory. In Example 2, by exerting normal tension on the inside surface of the cylinder, the rotation created is very small; therefore, the results of the micropolar and classic theories are similar. In addition to FGM materials, this solution is applicable to minerals and organic matter multiphase fiber and particulate composites, soil, rock, concrete, and various granular materials [59].

Two-dimensional analytical solution of micropolar magneto-thermoelasticity FGM hollow cylinder under asymmetric load (r, θ)

This paper presents a two-dimensional analytical solution (r, θ) to study micropolar magneto-thermoelasticity for a hollow cylinder, made of FGMs, under steady-state conditions. The physical properties of materials are in the form of a power function and undergo changes in the direction of the radius. To solve the heat transfer equation and Navier equations, the complex Fourier series and the power-law functions are used. By solving the equations using the general thermal and mechanical asymmetric boundary conditions on the inner and outer surface of the cylinder, radial displacement, circumferential displacement, force stresses, coupling stresses, and micro-rotation are obtained. Numerical examples of Aluminum-epoxy composite are presented for the three theories of classical, micropolar, and micropolar magneto thermoelasticity. Results indicated that the inner and outer surface of the cylinder subject to asymmetric harmonic temperature and mechanical distribution show that the values of the micropolar magnet theory are less than those of the classical theory and more than the values of the micropolar theory. Where harmonic radial stresses are applied to the inner surface of the cylinder, the values of the classical theories correspond to those of the micropolar theory. The presence of a magnetic field makes differences between the values of the micropolar magnet theory and the values of the two classical and micropolar theories. The study was validated by examining an example of homogenous materials under ambient conditions.

Effect of General Thermal Boundary Conditions on the Dynamic and Buckling of Polymeric Hybrid Nanocomposite Beam with Variable Thickness

This paper deals with the effect of general thermal boundary conditions on the dynamic and buckling of polymeric hybrid nanocomposite beam. The Timoshenko beam with variable thickness is made of a polymer matrix reinforced with a combination of carbon nanotubes (CNT) and nanoclay (NC) particles. The thermal and mechanical properties of the hybrid nanocomposite beam are obtained by Halpin–Tsai micromechanical method. The governing equations are derived through using Hamilton's principle and then solved by using the differential quadrature method (DQM). In this work, for the first time, the general thermal boundary conditions are applied to hybrid nanocomposite beam with variable thickness. The numerical results of dynamic response are obtained for different thermal and mechanical boundary conditions. Besides the effects of the amount as well as shape factor of the nanofiller, and different types of thermal boundary conditions on the natural frequencies and critical buckling load are investigated. The results show that the dynamic response of the beam with uniform thickness is more affected by changes in the amount of nanoclay, while beam with non-uniform thickness is more sensitive to changes in the amount of CNT. In addition, thermal boundary conditions involving a constant heat flux on one side have different responses with the increase of CNT value.

An Exact Analytical Solution for Heat Conduction in a Functionally Graded Conical Shell

In this study, an exact analytical solution for the heat conduction problem in a truncated conical shell is presented. The cone is made of functionally graded materials and it is considered that the material properties vary according to power-law functions. The general thermal boundary conditions are applied to cover a wide variety of actual applications. The results are successfully validated. Two examples, which are tried to mimic practical conditions, are studied using the derived solution, and a parametric study is done to shed light on the problem. The outcomes of this research provide useful information for understanding the nature of heat transfer behavior in the specific geometry of a cone. Regarding the specific applications of conical shells, the results can be used in the prefabrication process of these shells and tailoring the design parameter of functionally graded materials.

WITHDRAWN: Linear stability analysis of Benard-Marangoni ferroconvection in presence of vertical throughflow

The stability analysis of throughflow effect on Bénard-Marangoni ferroconvection is investigated hypothetically. The bottom of the fluid layer is taken as rigid with temperature fixed, while the top of the fluid layer is assumed as free and, on the boundary, surface tension effect depending on temperature is taken as non-deformable and subjected to general thermal boundary condition. The Regular perturbation method is applied to solve the problem analytically. The results reveal that the direction of the throughflow doesn’t have any control on the characteristics of stability. Further, Peclet number Q is to delay the ferroconvection. The effect in increasing magnetic number Rm and Prandtl number Pr is to accelerate the convection. It is noticed that M 3 , representing non-linearity of fluid magnetization doesn’t affect the Bénard-Marangoni ferroconvection.

Two-dimensional analytical solution for temperature distribution in FG hollow spheres: General thermal boundary conditions

Abstract The paper aimed at obtaining an analytical solution for the steady-state heat transfer in a hollow sphere made of functionally graded material. Two-dimensional distribution of temperature is considered to be in both radial and peripheral directions and the conductivity coefficients in both directions are a function of radius. The general boundary conditions for the interior and exterior surface of the sphere are considered so that the resulting solution can be extended to treat a wide range of functional cases. The obtained solutions are in the form of Bessel and Legendre functions. Unknown solution coefficients are achieved by applying boundary conditions and orthogonal Legendre functions' relations. The paper additionally provides results for two practical test cases to assess the robustness of the achieved solution. The influence of material constants and conductivity ratio are investigated in order to shed a light on material selection. The results confirm that the achieved general exact solution is able to adequately calculate the distribution of temperature. The validated findings of the current paper could be considered as a clue for tailoring of functionally graded spheres, like spherical vessels, based on the actual thermal boundary conditions in the manufacture process.

On 2D asymmetric heat conduction in functionally graded cylindrical segments: A general exact solution

The paper has dealt with an exact analytical solution of steady-state heat conduction for the special case of a functionally graded (FG) cylindrical sector. In this regard, Fourier theory is utilized to develop the steady-state temperature field. The material properties according to the power-law function are considered to vary in radial and circumferential directions and in both directions, the most general thermal boundary conditions are applied. Adequate verification of the solution is demonstrated. The correctness of the exact analytical solution in terms of industrial examples is examined by solving a heat conduction problem for a cylindrical segment subjected to a combination of boundary conditions. Furthermore, the effects of various parameters such as material constant, geometry, and thermal conductivity ratio on the temperature distribution are explored. The findings are advantageous to comprehend the flexibility of the two-dimensional FG materials for designing process and optimizing configurations under multi-functional requirements such as intelligent control applications.

A closed-form solution for axisymmetric conduction in a finite functionally graded cylinder

An exact general analytical solution is derived for heat conduction problem in an axisymmetric cylinder made of functionally graded material whose thermal conductivity differs in two directions. It is assumed that the thermal conductivity coefficients in radial and longitudinal directions are the power functions of radius. This is accomplished by taking advantage of Sturm-Liouville theory to get a suitable Fourier transformation. The general inhomogeneous thermal boundary conditions are adopted in both radial and longitudinal directions. The obtained solution is adequately verified against available analytical results. Also, two illustrative cases are considered to guarantee the solution's accuracy from a practical perspective. Particularly, the importance of the material constants on temperature distribution in both radial and longitudinal directions is uncovered. The results show that the present exact solution satisfies well the energy equation and arbitrary complex boundary conditions.

Cascaded lattice Boltzmann method for incompressible thermal flows with heat sources and general thermal boundary conditions

Cascaded or central-moment-based lattice Boltzmann method (CLBM) is a relatively recent development in the LBM community, which has better numerical stability and naturally achieves better Galilean invariance for a specified lattice compared with the classical single-relation-time (SRT) LBM. Recently, CLBM has been extended to simulate thermal flows based on the double-distribution-function (DDF) approach [L. Fei et al., Int. J. Heat Mass Transfer 120, 624 (2018)]. In this work, CLBM is further extended to simulate thermal flows involving complex thermal boundary conditions and/or a heat source. Particularly, a discrete source term in the central-moment space is proposed to include a heat source, and a general bounce-back scheme is employed to implement thermal boundary conditions. The numerical results for several canonical problems are in good agreement with the analytical solutions and/or numerical results in the literature, which verifies the present CLBM implementation for thermal flows.

Solution for Equation of Two-Dimensional Transient Heat Conduction in Functionally Graded Material Hollow Sphere With Piezoelectric Internal and External Layers

In this paper, an exact solution for the equation of two-dimensional transient heat conduction in a hollow sphere made of functionally graded material (FGM) and piezoelectric layers is developed. Transient temperature distribution, as a function of radial and circumferential directions and time with general thermal boundary conditions on the inside and outside surfaces, is analytically obtained for different layers, using the method of separation of variables and Legendre series. The results are the sum of transient and steady-state solutions that depend upon the initial condition for temperature and heat source, respectively. The FGM properties are assumed to depend on the variable r and they are expressed as power functions of r.

A unified method for stresses in FGM sphere with exponentially-varying properties

Using the Complementary Functions Method (CFM), a general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of functionally graded material (FGM) is presented. The mechanical properties are assumed to obey the exponential variations in the radial direction, and the Poisson's ratio is assumed to be constant, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. In the present paper, a semi-analytical iterative technique, one of the most efficient unified method, is employed to solve the heat conduction equation and the Navier equation. For different values of inhomogeneity constant, distributions of radial displacement, radial stress, circumferential stress, and effective stress, as a function of radial direction, are obtained. Various material models from the literature are used and corresponding temperature distributions and stress distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases and virtually exact results are obtained.

Entropy generation analysis of laminar flow of a nanofluid in a circular tube immersed in an isothermal external fluid

This paper is an analytical study of entropy generation in the laminar flow of nanofluids in a circular tube. The tube is immersed in an isothermal external fluid – which is the most general thermal boundary condition but has not been studied in much detail in literature. Two nanofluids, namely – water–Al2O3 and ethylene glycol–Al2O3 have been chosen for this study. The effects of the external Biot number, non-dimensional temperature difference and volume fraction on the entropy generation characteristics of the flow have been shown through graphs and the physical reasoning behind the observed trends has been discussed threadbare. It is shown that the addition of nanoparticles is beneficial only at smaller Reynolds number and for less viscous base fluids. Most importantly, it is proved that the entropy generated in the case of a tube immersed in an isothermal external fluid is bounded by those for uniform heat flux and uniform wall temperature boundary conditions.

Onset of Bénard–Marangoni ferroconvection with a convective surface boundary condition: The effects of cubic temperature profile and MFD viscosity

The combined effects of basic cubic temperature profiles and magnetic field dependent (MFD) viscosity on the onset of Bénard-Marangoni convection in a ferrofluid layer are studied. The lower boundary is rigid-isothermal, while the upper free boundary open to the atmosphere is flat and subject to a general thermal boundary condition. The Galerkin technique is employed to extract the critical stability parameters numerically. The results indicate that the basic cubic temperature profiles have a profound influence on the stability characteristics of the system and can be effectively used to either suppress or augment the onset of Bénard–Marangoni ferroconvection. Besides, increasing the magnetic Rayleigh number and the nonlinearity of magnetization hastens, while an increase in the Biot number and MFD viscosity parameter delays the onset of Bénard–Marangoni ferroconvection. The existing results in the literature are obtained as particular cases from the present study.

Deformation and Stresses Analysis in FG Rotating Hollow Disk and Cylinder Subjected to Thermal and Mechanical Load

In this paper, a new solution is presented for one-dimensional steady-state mechanical and thermal stresses in a FG rotating hollow disk and cylinder. The material properties for FG are expressed as nonlinear exponential functions through the radius and Poisson’s ratio is taken to be constant. The temperature distribution is derived from first law of thermodynamics by solving energy equation, with a general thermal and mechanical boundary conditions on the inside and outside surfaces. Heat conduction and Navier equations are solved analytically by choosing elliptic cylinder coordinates system and the results are shown for displacement and stress components along the radial direction.

Thermomechanical Analysis in FG Rotating Hollow Disk

In this paper, mechanical and thermal stresses of rotating hollow disks composed of functionally graded materials (FGMs) is presented. The material properties for FG are expressed as nonlinear exponential functions through the radius of disk and Poisson’s ratio is taken to be constant. The temperature distribution is derived from first law thermodynamics by solving energy equation, general thermal and mechanical boundary conditions are assumed on the inside and outside surfaces of the disk. Heat conduction and Navier equations of a FGM disk are expressed in elliptic cylinder coordinates system and solved analytically. The results are shown for displacement and stresses components along the radial direction.

Radially Symmetric Steady State Thermal and Mechanical Stresses of a Poro FGM Hollow Sphere

A general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of porous functionally graded material (FGPM) is presented. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. The material properties, except Poisson's ratio, are assumed to vary along the radius r according to a power law function. The analytical solution of the heat conduction equation and the Navier equation lead to the temperature profile, radial displacement, radial stress, and hoop stress as a function of radial direction.

Two-Dimensional Stresses in a Hollow FG Sphere with Heat Source

This work studied the theoretical solution for axisymmetric steady-state mechanical and thermal stresses in hollow functionally graded spheres with respect to heat source. The material properties of the FG sphere change continuously across the thickness direction according to the power functions of radial direction. The steady-state temperature, displacements, and stresses are derived due to the general mechanical and thermal boundary conditions as function of radial and circumferential directions. The temperature and Navier equations are solved analytically, using Taylor and Legendre series. With increasing the power law indices the temperature distribution due to heat source is decreased. Circumferential stress and radial displacement due to heat source are decreased as the power law index increases.

Analytical and numerical analysis for the FGM thick sphere under combined pressure and temperature loading

Thermo-mechanical analysis of functionally graded hollow sphere subjected to mechanical loads and one-dimensional steady-state thermal stresses is carried out in this study. The material properties are assumed to vary non-linearly in the radial direction, and the Poisson’s ratio is assumed constant. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. In the analysis presented here, the effect of non-homogeneity in FGM thick sphere was implemented by choosing a dimensionless parameter, named βi (i = 1, . . . , 3), which could be assigned an arbitrary value affecting the stresses in the sphere. It is observed that the solution process for βi(i = 3) = −1 are different from those obtained for other values of βi (i = 1, . . . , 3). Cases of pressure, temperature, and combined loadings were considered separately. It is concluded that by changing the value of βi (i = 1 . . . 3), the properties of FGM can be so modified that the lowest stress levels are reached. The present results agree well with existing results. Using FEM simulations, the analytical findings for FGM spheres under the influence of internal pressure and temperature gradient were compared to finite element results.