Analytical solution for transient temperature and thermal stresses within convective multilayer disks with time-dependent internal heat generation,Part I: Methodology

This work deals with the exact solution for asymmetric transient problem of heat conduction and accordingly thermal stresses within multilayer hollow or solid disks which lose heat by convection to the surrounding ambient. The combination of the separation of variables method (SVM) and Duhamel's theorem is applied to the heat conduction problem which provides a versatile technique. The temperature distribution is obtained by the SVM which concerns the heat conduction problem with time-independent internal heat generation. Applying Duhamel's theorem on the previous solution, temperature distribution with time-dependent internal heat generation can be achieved. Accordingly, assuming plane stress condition, radial and tangential stresses are obtained which are incorporated into the equivalent tensile stress formulation to calculate von Mises stress. The comprehensive methodology described here can be useful addition for many new emerging fields in which both transient and steady-state temperature distributions and thermal stresses for composite disks are important.     

GREEN'S FUNCTION APPROACH TO THREE-DIMENSIONAL HEAT CONDUCTION EQUATION OF FUNCTIONALLY GRADED MATERIALS

A Green's function approach based on the laminate theory is adopted to solve the three-dimensional heat conduction equation of functionally graded materials (FGMs) with one-directionally dependent properties. An approximate solution for each layer is substituted into the governing equation to yield an eigenvalue problem. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the three-dimensional transient temperature. The eigenvalues and the corresponding eigenfunctions are determined from the homogeneous boundary conditions at outer sides and from the continuous conditions of temperature and heat flux at the interfaces. A three-dimensional transient temperature solution with a source is formulated by the Green's function. Numerical calculations are carried out for an FGM plate, and the numerical results are shown in tables and figures.     

Transient analytical solution to heat conduction in composite circular cylinder

An analytical method leading to the solution of transient temperature filed in multi-dimensional composite circular cylinder is presented. The boundary condition is described as time-dependent temperature change. For such heat conduction problem, nearly all the published works need numerical schemes in computing eigenvalues or residues. In this paper, the proposed method involves no such numerical work. Application of ‘separation of variables’ is novel. The developed method represents an extension of the analytical approach derived for solving heat conduction in composite slab in Cartesian coordinates. Close-formed solution is provided and its agreement with numerical result is good which demonstrates a good accuracy of the developed solution form.     

Thermal buckling of temperature-dependent composite super elliptical plates reinforced with carbon nanotubes

In this work, the problem of thermal buckling of composite plates reinforced with carbon nanotubes (CNTs) is investigated. Distribution of CNTs as reinforcements through the thickness direction of the plate is assumed to be either uniform or functionally graded (FG). Properties of the reinforcement and matrix are both temperature dependent. Properties of the composite media are obtained according to a refined rule of mixture approach where the e?ciency parameters are introduced. The plate is in a super elliptical shape where the simple elliptical shape and rectangular shapes are obtained as especial cases. In these types of plates due to the round corners, stress concentration phenomenon is eliminated. Based on the Ritz method where the shape functions are of the polynomial type, the governing equations are obtained. These equations are solved using an iterative eigenvalue problem since the properties are temperature dependent. Numerical results are validated for the simple case of an isotropic plate. Novel numerical results are provided for plates reinforced with CNTs in different shapes, various volume fractions and different patterns of CNT distribution. It is shown that FG-X pattern of CNTs in matrix results in the maximum critical buckling temperature.     

Semi-analytical solution of the extended Graetz problem for combined electroosmotically and pressure-driven microchannel flows with step-change in wall temperature

Analytical solutions are obtained for the temperature and Nusselt number distribution in the thermal entrance region of a parallel plate microchannel under the combined action of pressure-driven and electroosmotic transport mechanisms, by taking into account the effects of viscous dissipation, Joule heating and axial conduction simultaneously, in the framework of an extended Graetz problem. Step changes in wall temperature are considered to represent physically conceivable thermal entrance conditions. The solution of the temperature distributions at the various channel sections essentially involves the determination of a set of eigenvalues and eigenfunctions corresponding to a Sturm Liouiville problem with non self-adjoint operators. The resultant eigenfunctions are non orthogonal in nature, and are obtained in the forms of hypergeometric functions. Parametric variations on the effects of the relative strengths of the pressure gradients and the electric field, ratio of the rate of heat generation to the rate of wall heat transfer, and the Peclet number are analyzed in details, in terms of their influences on the temperature field as well as the Nusselt number distribution.     

OPTIMIZATION OF MATERIAL COMPOSITION OF FUNCTIONALLY GRADED PLATE FOR THERMAL STRESS RELAXATION USING A GENETIC ALGORITHM

In this article, a genetic algorithm is applied to an optimization problem of material composition for a plate of step-formed functionally graded materials. The step-formed functionally graded plate is analyzed as a laminated composite plate made of numerous layers with homogeneous and different isotropic material properties. First, the onedimensional transient temperature distribution for a laminated plate is analyzed theoret ically. In addition, the thermal stress components for such an infinitely long plate are formulated under the mechanical condition of being traction-free. As a numerical example, a plate composed of zirconium oxide and titanium alloy is considered. In addition, for the optimization problem of minimizing the thermal stress distribution, the numerical calculations are made using a genetic algorithm without supposing a distribution function of material composition and the optimal material composition of each layer is determined taking into account the effect of the temperature dependency of material properties. Furthermore, the results obtained when a distribution function isn't specified and the results found when a distribution function is specified are compared.     

Solution of the complex eigenvalue problem for transient laminar forced convection inside tubes

Two solutions are presented for the complex eigenvalue problem arising in laminar transient forced convection inside tubes. The “method of matched asymptotic expansions” is applied to the complex eigenvalue problem yielding a closed form algebraic expression for the complex eigenvalues and two asymptotic expansions, one valid near the tube centerline and another valid near the tube wall. These expansions are combined yielding a single uniformly valid composite asymptotic expansion for the complex eigenfunctions. The present asymptotic solution is accurate for all normalized complex eigenfunctions except for those corresponding to the smaller eigenvalues. Another solution of the complex eigenvalue problem, which is accurate for the smaller eigenvalues, is presented which is based on the finite Bessel transform method. This latter solution is utilized for checking the accuracy of the closed form asymptotic solutions for the eigenvalues and the eigenfunctions of the complex eigenvalue problem.     

Experimental verification of thermal remote sensing method for recovering temperature distribution in glass

The thermal remote sensing method for recovering the temperature distribution in glass from spectral emission data is examined experimentally. An analytical model is formulated and the desired temperature distribution is obtained using an optimization scheme which determines the temperature profile in the form of a polynomial or a set of discrete points. In order to evaluate the accuracy and validity of the thermal remote sensing method, the recovered temperatures are compared with independent measurements using surface thermocouples and a Mach-Zehnder interferometer. Experimental results are reported for fused silica (Corning Code 7940) glass samples using a Perkin-Elmer spectrometer to measure the spectral radiant energy emerging from the layer of glass. Opaque (high and low emittance) boundary conditions at the heated surface of the glass were considered. Temperatures in the range from 500 to about 900 K were examined. Spectral emission data between 3.3 and 4.8 μm were used in recovering the temperature distribution in the glass samples. The results showed that the recovered and interferometrically measured temperature profiles agreed well, with the maximum deviation never exceeding approximately 2 per cent.     

The thermal problem of friction during braking for a three-element tribosystem with a composite pad

The analytical solution to a thermal problem of friction during braking with constant retardation for a three-element system (a foundation/strip/semi-space) is obtained. The solution allows to find the evolution and distribution of transient temperature in the caliper/pad/disk tribosystem. Unlike known solutions for three-element tribosystem, this one is obtained on the assumption that material of the pad (strip) is the periodic composite. The every unit cell of the composite contains four sub-cells with rectangular cross-section and with different thermo-physical properties. It is assumed, that intensity of the heat generation on the contact surface is equal to power of friction and through this surface the heat transfer takes place. The influence of the geometrical dimensions and thermo-physical properties of composite sub-cells on the maximum temperature in the system has been investigated.     

Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

Analysis of laminar dissipative flow and heat transfer in a porous saturated circular tube with constant wall heat flux

A theoretical study of the thermal development of forced convection was performed using a circular tube filled with a saturated porous medium, with constant wall heat flux, and with the effect of viscous dissipation. The solution was obtained using the method of separation of variables. The Sturm–Liouville system was solved for the eigenvalues. Ordinary differential equations for the eigenfunctions were solved numerically by the fourth-order Runge–Kutta method. Results show that, in the presence of the viscous dissipation, both the level and distribution of temperature are altered remarkably, even for small values of the Brinkman number, Br, which is the ratio of heat generation caused by viscous dissipation to the value of heat flux at the wall. The value of the local Nusselt number, Nu, is demonstrably independent of Br, unlike the situation in which the wall temperature remains constant.     

STRESS DISTRIBUTION IN A DISSIMILAR MATERIALS JOINT FOR COMPLEX SINGULAR EIGENVALUES UNDER THERMAL LOADING

The problem of a joint of dissimilar materials with arbitrary edge angles exposed to thermal loading is treated. Emphasis is placed on the investigation of the stress distribution around the singular point for complex singular eigenvalues. For this case, an analytical form of the stress is obtained. Some examples are given to show the stress distribution around the singular point and the oscillation of the stress. The range of the stress oscillations is extremely small.     

One‐dimensional transient heat conduction in a bilayer finite slab: A case study in the printing process

In this work, a typical case of heat distribution is examined during a paper printing process, based on one‐dimensional transient heat conduction in two‐layer finite slabs with an insulated free surface, and a constant temperature free surface. Analytical solutions were obtained in non‐dimensional form. Various examples of applying these solutions are presented. The accuracy of the solutions, with respect to time, is analyzed considering the eigenvalues of their infinite solutions. It is observed that the larger the number of eigenvalues in consideration, the better the accuracy of the solutions. The model related to a two‐layer slab describes the simplified case in which all heat transfer occurs only by conduction. The solutions obtained are finally compared with the solutions for heat conduction in two semi‐infinite solids. The comparison between the two solutions shows that results are in good agreement only during short time scales. The heat distribution study is expected to be helpful in knowing the effectiveness of various mediums to be used as the reciever during the printing process; however, there is scope for development of more robust models.     

Buckling of Functionally Graded Cylindrical Shells Under Combined Thermal and Compressive Loads

This article focuses on analytical solutions for bifurcation buckling of FGM cylindrical shells under thermal and compressive loads. A new solution methodology is established based on Hamilton's principle. The fundamental problem is subsequently transformed into the solutions of symplectic eigenvalues and eigenvectors, respectively. Then, by applying a unidirectional Galerkin method, imperfection sensitivity of an imperfect FGM cylindrical shell is discussed in detail. The solutions reveal that boundary conditions, volume fraction exponent, FGM properties, and temperature rise distribution significantly influence the buckling behavior. Critical stresses are reduced greatly due to the existence of initial geometric imperfections.     

OPTIMIZATION CONTROL OF COMPOSITE LAMINATES FOR MAXIMUM THERMAL BUCKLING AND MINIMUM VIBRATIONAL RESPONSE

The problem of maximizing the thermal buckling and minimizing the vibrational response of composite laminates is solved using optimal design and active control procedures. The problem is formulated based on a first-order shear deformation laminate theory with various cases of boundary conditions. The design objective is to maximize thermal buckling using ply thickness and the fiber orientation angle as design variables. The active control objective is to minimize the laminate vibrational response with the minimum possible expenditure of control energy. The vibrational response is expressed in terms of the total elastic energy of the laminate and a penalty functional of closed-loop control force. Liapunov-Bellman theory is used to obtain solutions for controlled deflections and optimal control force. Comparative examples are given for angle-ply antisymmetric laminates subjected to a uniform temperature distribution. A general representation for the design variables is presented such that the ply thickness is a function of the number of layers. Some of the obtained numerical results are compared with their counterparts in the literature.     

Unsteady heat conduction in two-dimensional two slab-shaped regions. Exact closed-form solution and results

The unsteady heat conduction analysis for multi-directional piecewise-homogeneous bodies is generally held to be complex and demanding, possibly explaining why practical guidelines for thermal field calculation are few and far between. The proposed solution method represents an extension of the new, ‘natural’ analytic approach derived in companion papers for solving one-dimensional multi-layer problems of time-dependent heat conduction. As the approach is new, it is presented in full, together with the complete temperature double-series solution prepared for computer implementation. By setting thermal diffusivity ratio unitary and assuming a uniform distribution of initial temperature, it emerges that, all other things being equal, the transient thermal response can be expressed as the product of two, separated, one-directional solutions, one across the layers and the other along the composite slab. The formulation deals properly with thermal conductivity ratios of all magnitudes. An efficient and accurate procedure of computing eigenvalues is given. Graphical and numerical output is presented and discussed.     

Integral transform solution of Luikov''s equations for heat and mass transfer in capillary porous media

The Luikov system of equations for coupled heat and mass transfer within capillary porous bodies is analytically handled through application of the generalized integral transform technique. The problem of temperature and moisture distribution during contact drying of a moist porous sheet is considered to illustrate the development of the present approach. The classical coupled auxiliary problem with the related complex eigenvalues is completely avoided and, instead, two decoupled eigenvalue problems for temperature and moisture are chosen, which are of the conventional Sturm-Liouville type. A set of benchmark results is generated and critically compared with previously reported approximate solutions.     

Transverse temperature distribution and heat generation rate in composite superconductors subjected to constant thermal disturbance

Analytical solution of one-dimensional, transient heat conduction with a distributed heat source is obtained to predict the transverse temperature distribution and heat generation rate per unit volume of the composite superconductor. The solution indicates that temperature distribution and heat generation rate depend on three dimensionless parameters: the dimensionless external disturbance w₀, the dimensionless interface temperature θ¹, and the dimensionless parameter φ that is dependent on the thickness and the thermal conductivity of the superconductor. Results of transient and steady-state solutions are presented. It is shown that the heat generation rate per unit volume of the composite, Q/Qc, is directly proportional to the current in the stabilizer.     

Transient conjugated forced convection in ducts with periodically varying inlet temperature

Transient forced convection for slug flow inside parallel-plate channels and circular ducts including conjugation to the walls is solved analytically and exactly for periodic variation of the inlet temperature. The periodic solution to the problem involved eigenfunctions and eigenvalues of a complex eigenvalue problem. The complex eigenvalue problem is solved by modifying the recently advanced Count Method, and benchmark results are presented for the eigenvalues in tabular form. The amplitude and phase lag of oscillations with respect to the conditions at the inlet are determined for the wall temperature, fluid bulk temperature and heat flux. The results for the cases of both parallel-plate channels and circular ducts are presented in the graphical form as a function of the axial position for different values of the parameters signifying the rate of energy storage in the walls. The effects of walls on damping the amplitude and altering the phase of temperature and heat flux oscillations along the duct are investigated.     

Inverse Boundary Design Problems in Enclosures With Non-Gray Media

In this work, the inverse analysis is applied to radiative heat transfer boundary design problems with non-gray media. The objective of the inverse problem is to find the power of the heaters on the heater surface that produces the desired output, that is, temperature and heat flux distribution over the design surface. The inverse problem is formulated as an optimization problem for minimization of an objective function, which is defined by the sum of the squared difference between estimated and desired heat flux distributions over the design surface. The non-gray optimization problem is solved using the conjugate gradient method, which is a gradient-based optimization method. The spectral line weighted-sum-of-gray-gases model (SLW) is used to account for non-gray gas radiation properties. The radiative transfer equation is solved by the discrete ordinates method combined with two models for simulation of non-gray media. Enclosures with diffuse and gray walls are considered. Radiation is assumed the dominant mode of heat transfer. Example problems including homogeneous/nonhomogeneous, isothermal/nonisothermal media are considered. The results obtained using the SLW model and the gray model are compared.