Study of the effective parameters on stress distribution around triangular hole in metallic plates subjected to uniform heat flux

On the basis of the steady-state two-dimensional theory of thermoelasticity, stress field around a triangular hole in an infinite isotropic plate is discussed. A metallic plate subjected to uniform heat flux and thermal-insulated condition along the hole boundary is assumed. The method used for this study is the expansion of Goodier and Florence's method. They used the complex variable method for stress analysis of infinite isotropic plates with an elliptical or circular hole. The rotation angle of the hole, bluntness, aspect ratio of hole size, and angle of heat flux are important parameters considered in this paper.     

Thermal stress analysis of infinite anisotropic plate with elliptical hole under uniform heat flux

Lekhnitskii’s complex variable method was developed to investigate the effect of uniform heat flux on perforated anisotropic plate with elliptical hole. The Cauchy’s integral formula was simplified by conformal mapping, and infinite area external to the hole was represented by the area outside the unit circle. In this article, Neumann boundary conditions and thermal-insulated condition along with the hole boundary were considered. Important parameters affecting stress distribution and displacement were those of rotation angle of hole, aspect ratio of hole size, and fiber angle. Results determined in this article were verified by finite element analysis.     

SOLUTION OF DISPLACEMENT BOUNDARY VALUE PROBLEM UNDER UNIFORM HEAT FLUX

The general solution of the displacement boundary value problem is obtained for an infinite plate with an arbitrary shaped hole under uniform heat flux in any direction. The complex stress functions, the dislocation method, and a rational mapping function are used and the closed solution is obtained. An infinite plate with a circular hole and a slit is analyzed under the condition of the constrained displacements. The singularity at the tip of the slit of the constrained displacement is investigated     

Analyses of Thermal Conduction and Stress Induced by Electric Current in an Infinite Thin Plate with an Elliptical Hole

Uniform electric current at infinity is applied to a thin infinite conductor with an elliptical hole disturbing the electric current, which gives rise to Joule heat, temperature increase and heat flux. Joule heat produces uniform and uneven temperature fields which in turn initiate thermal stress. These electrical current, Joule heat, temperature, heat flux and thermal stress analyses are carried out and their closed form solutions are obtained. The heat conduction problem is solved as a temperature boundary value problem. Figures of distribution of Joule heat, temperature, heat flux and stress are shown. A dislocation and a rotation terms for thermal stress analysis appear, which makes problem complex. Solutions of Joule heat, temperature, heat flux and thermal stress are nonlinear for the direction of electric current. For an infinite plate with a circular hole, stress components do not occur on the whole plate. As a special case, a crack problem is analyzed and intensities at the crack tip of each problem are investigated. Relations between melting temperature and electric current density, and between fracture toughness value and electric current density are investigated for some crack lengths for steel.     

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

Green's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.     

THERMAL STRESS PROBLEM FOR MIXED HEAT CONDUCTION BOUNDARY AROUND AN ARBITRARILY SHAPED HOLE WITH CRACK UNDER UNIFORM HEAT FLUX

This article deals with a thermal stress problem for thermal conduction around an arbitrarily shaped hole with a crack under uniform heat flux. Two cases for the hole edge and the crack faces are assumed: adiabatic and isothermal conditions or vice versa (isothermal and adiabatic). A closed-form solution is obtained using conformal mapping, dislocation functions, and the complex variable method. Results of temperature, heat flux, stress, and stress intensity factor are illustrated.     

Heat Transfer Characteristics of Laminar Flow in Internally Finned Tubes under Various Boundary Conditions

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NUMERICAL PREDICTION OF LAMINAR FORCED CONVECTION IN TRIANGULAR DUCTS WITH UNSTRUCTURED TRIANGULAR GRID METHOD

The flow and heat transfer characteristics of smooth triangular ducts with different apex angles of 15, 30, 60, and 90 under the fully developed laminar flow condition were predicted numerically using a finite volume method. The SIMPLE-like algorithm was employed together with an unstructured triangular grid method, where the grid was generated by a Delaunay method. The triangular grid was adopted instead of the traditional rectangular grid to fit better into the triangular cross section of the duct. Two kinds of boundary condition (uniform wall temperature and uniform wall heat flux) were considered. Comparison of the predictions with previous computational results indicated a very good agreement. Both the friction factor and Nusselt number (Nu) showed a strong dependence on apex angle of the triangular duct. When the apex angle was 60, the duct provided the highest steady-state forced convection from its inner surface to the airflow under the laminar flow condition.     

Non-Axisymmetric Thermal Stress Around a Circular Hole in a Functionally Graded Infinite Plate

This paper presents the non-axisymmetric two-dimensional problem of thermal stresses in a functionally graded plate with a circular hole based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the plate having radial arbitrary elastic properties is derived when it is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the stress around the circular hole in the functionally graded material plate can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

四边对流换热的内含热源各向异性矩形域稳态热传导解析

应用复级数方法给出的含热源各向异性矩形域稳态传导解析解。结合引入板角流角点条件,首次解析分析了四边与流体换热的含热源各向异性矩形域温度场。讨论了铺设角,各向异性程度,边界对流换热系数及跨宽对比温度场分布的影响。     

Heat and mass transfer in a cross-flow membrane-based enthalpy exchanger under naturally formed boundary conditions

Heat and mass transfer mechanisms in a cross-flow parallel plate membrane-based enthalpy exchanger for heat and moisture recovery from exhaust air streams are investigated. The flow is assumed laminar and hydrodynamically fully developed, but developing in thermal and concentration boundaries. Contrary to the traditional methods to assume a uniform temperature (concentration) or a uniform heat flux (mass flux) boundary condition, in this study, the real boundary conditions on the exchanger surfaces are obtained by the numerical solution of the coupled equations that govern the transfer of momentum, thermal energy, and moisture in the two cross-flow air streams and through the membrane. The naturally formed heat and mass boundary conditions are then used to calculate the local and mean Nusselt and Sherwood numbers along the cross-flow passages, in the developing region and thereafter. A comparison was made with those results under uniform temperature (concentration) and uniform heat flux (mass flux) boundary conditions, for rectangular ducts of various aspect ratios. An experiment is done to verify the prediction of outlet moisture content.     

GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.     

APPLICATION OF AN ALTERNATING METHOD TO THE THERMOELASTIC PROBLEMS WITH MULTIPLE CIRCULAR HOLES IN AN INFINITE DOMAIN

This paper presents a new efficient procedure to analyze the thermoelastic problems with multiple circular holes in two-dimensional infinite domain using an alternating method. To achieve this purpose, the analytical solutions including the temperature and associated thermal stress for the arbitrary heat flux across the single circular hole boundary in an infinite domain are first derived. Both the temperature and thermal stress fields in the thermal problems are simultaneously solved by these analytical solutions with the successive iterative superposition process. Compared with the solution of the conventional finite-element technique, the present method has been more accurate and has more advantages. Effects of the distributions and sizes of the holes on the stress concentration in the thermal problems also are evaluated in detail herein.     

Convective heat transfer in cross-corrugated triangular ducts under uniform heat flux boundary conditions

Cross-corrugated triangular ducts provide high heat mass transfer capabilities in membrane based air-to-air heat mass exchangers. The mixing effect would intensify the convective heat mass transfer coefficients on membrane surfaces. In this study, the fluid flow and convective heat transfer in a cross-corrugated triangular duct under uniform heat flux boundary condition is modeled and experimentally studied. A low Reynolds number k–ω (LKW) turbulence model is employed to account for the turbulence in the flow. Heat transfer experiments and high speed hot wire anemometry technology are used to validate the model. The transitional behavior of fluid flow in the duct is disclosed by velocity measurements and Fourier transforms. Correlations are provided for estimation of the pressure drop and the mean Nusselt numbers under uniform heat flux boundary conditions. The established correlations can be extended to estimate the convective mass transfer coefficients through heat mass analogy.     

GREEN'S FUNCTION FOR THERMAL STRESS MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ARBITRARY HOLE UNDER A POINT HEAT SOURCE

Green's function of a point heat source is derived for a mechanical mixed boundary value problem of an infinite plane with an arbitrary hole, for which zero-displacement and traction-free boundary conditions are prescribed to its boundary. As the thermal boundary condition on the hole, either an adiabatic or isothermal condition is considered. By employing the mapping technique and complex variable method, an explicit solution including a hypergeometrical function is obtained. Stress distributions are shown in illustrative examples for a square hole.     

Dynamic analysis of cracks under thermal shock considering thermoelasticity without energy dissipation

This article reports a study of a cracked finite isotropic medium under nonclassic thermal shock based on thermoelasticity without energy dissipation. The time history of stress intensity factors as well as the temperature distribution around the crack tip is analyzed thoroughly. The fully coupled governing equations are discretized in the space by employing the extended finite-element method. The Newmark method is used as the time integration scheme to solve discretized equations. The stress intensity factors, which are extracted using the interaction integral method, are compared with other theories of thermoelasticity. The results of a cracked plate under temperature shock demonstrate that the stress intensity factors based on thermoelasticity without energy dissipation are significantly greater than those based on classic and Lord–Shulman models, whereas the peaks of stress intensity factors under heat flux shock are nearly equal for various theories of thermoelasticity. Furthermore, a mobile cold region is created along slanted crack in the temperature distribution, in which the temperature is less than the applied thermal boundary condition.     

AXIALLY SYMMETRIC THERMAL STRESS OF A PENNY-SHAPED CRACK UNDER GENERAL HEAT FLUX

This paper presents a solution of the steady-state thermal stress and displacement in an infinite isotropic elastic solid containing a penny-shaped crack whose surfaces are exposed to the general mechanical loading and nonsymmetric heat flux. That is, the heat flux applied to both surfaces of the crack is not the same. The problem is solved using Hankel transforms and the Abel operator of the second kind. Using limiting values of the stress, displacement, and temperature fields at the crack plane in terms of the stress, displacement, temperature, and heat flux discontinuities and boundary conditions of the crack, the problem is reduced to that of solving Abel integral equations that admit closed-form solutions. For a special case of heat flux conditions, stresses at a general point of the medium have been computed and presented graphically.     

Possible transformations for natural convection on a vertical flat plate embedded in porous media with prescribed wall heat flux

This study derived the transformation of boundary layer equations for two-dimensional steady natural convection on a vertical wall embedded in porous media. Three different kinds of thermal boundary conditions are prescribed for wall heat flux: uniform distribution, power law variation, and exponential variation. The flow pattern contains three subregions based on the distance along the flat plate. When inertia resistance is ignored, similarity solution exists in case wall heat flux is in linear distribution. The known relationships of uniform wall temperature relative to wall heat flux variation and uniform wall heat flux relative to wall temperature variation in both cases of the pure fluid flow and the pure porous flow can all be obtained in the present transformation.     

Stress Analysis of an Infinite Plate with a Coated Elliptic Hole Under a Remote Uniform Heat Flow

A thermoelastic solution to a coated elliptic hole embedded in an infinite matrix subjected to a remote uniform heat flow is provided in this article. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the temperature and stresses in the coated layer and the matrix are derived explicitly in a series form. Some numerical results are provided to investigate the effects of the material combinations and geometric configurations on the interfacial stresses. It is found that a coated layer has a strong effect on thermal stresses of the problem with an elliptic hole embedded in an infinite plate.     

Effects of functional surface on performance of a micro heat pipe

The effect of a functional surface with the axial ladder contact angle distribution on the thermal performance of a triangular micro heat pipe has been analyzed based on a one-dimensional steady-state model. Compared with the traditional micro heat pipe (MHP) with a uniform contact angle distribution on its surface, the simulation results show that a MHP with a functional surface can remove a greater amount of heat under the same condition. The increase in thermal performance is more obvious with the increase in the ladder difference of the contact angles between the adjacent sections of the MHP. The increased thermal performance associated with the functional surface can be attributed to the increase of the liquid capillary force as well as the no obvious increase of the liquid shearing force provided by the functional surface, which also brings about the increase in condensate mass flow rate through the adiabatic section–evaporation section interface. It is also found that for the traditional MHP with uniform contact angle surface, there is an optimal contact angle leading to the maximum heat input. The deviation of the optimal value will decrease the capillary force and thermal performance of the MHP.