Arbitrary Temperature Distribution Research Articles

Diffusion through composite media

The diffusion equation for the temperature distribution in each of k sections of a composite with internal heat generation and either heating, cooling or perfect thermal contact at the k − 1 interfaces is solved. The composite consists of k discrete plates, cylinders or spheres each of different material. The k sections have an arbitrary initial temperature distribution and the media are exchanging heat at the external boundaries through two different, though constant, arbitrary film coefficients with two different time dependent surroundings. The solution is obtained by using a unique dependent variable substitution which gives a new partial differential equation with homogeneous external boundary conditions. The solution for this derived partial differential equation is then obtained by using the Vodicka type of orthogonality relationship. The solution, thus obtained, gives the temperature distribution in any of the k plates, cylinders or spheres for any position x and time t for the most general type of linear boundary, internal and initial conditions.

On the gravitational potential energy of vertically stratified water columns

An expression for the gravitational potential energy of a vertically stratified water column is obtained for arbitrary temperature distribution. This expression is minimized under the condition that heat content and mass remain unaltered, and the resultant temperature profile compared with those experimentally obtained. A hitherto neglected term in the expression of the potential energy involves atmospheric pressure, and may imply differences in the manners of diffusion and heat and salt.

The effect of forced flow on heat transfer from a disc rotating near a stator

This paper describes the application of the Spalding-Patankar numerical integration procedure to the case of heat transfer from an air-cooled disc rotating close to a stationary casing. Results are calculated to show the effect of frictional heating, arbitrary disc temperature distributions and non-unity Prandtl numbers on heat transfer, and these effects are shown to be qualitatively similar to those predicted by Dorfman for a free disc. Mean Nusselt numbers, calculated for a range of flow parameters, tend to the empirical correlation of Kapinos at high Reynolds numbers but diverge at lower Reynolds numbers. The latter divergence is consistent with the weak dependence of Nusselt number on rotational speed noted by Mitchell and Kreith at low Reynolds numbers.

Analysis of heat conduction in bridge slab

An analytical procedure is presented for the heat transfer in a bridge slab of finite dimensions and arbitrary initial temperature distribution. The thermal conductivity and surface conductance may be different on the top and bottom surfaces of the slab. The ambient temperature and heat input into the slab may vary both spacewise and timewise.

Transient heat conduction in non-homogeneous spherical systems

A comprehensive study encompassing a general analytical development and an archival presentation of results is made for transient heat conduction in thermally coupled spherical regions. The system consists of a sphere and its surrounding environment, each region having different thermal properties and different initial temperatures. The general closed form solution, which accommodates an arbitrary inital temperature distribution in the sphere, is specialized to apply to a number of interesting problems. Among these, the situation in which the sphere and the surroundings are initially at different uniform temperatures constitutes a basic problem in the theory of heat conduction. Correspondingly, a comprehensive presentation of transient temperature histories is made for various locations in the sphere and in the surroundings, with relevant thermal property ratios serving as parameters. Characteristics such as the duration of the transient period and the penetration depth of the temperature field into the surroundings are illuminated. Another interesting situation is that in which the thermal conductivity of the sphere is much greater than that of the surroundings, so that the sphere temperature may be regarded as being spatially uniform. In addition to a presentation of temperature histories, the conditions are identified under which the assumed spatial uniformity of the sphere temperature is valid. For the case of a metallic sphere situated in a gaseous environment, it is demonstrated that the transient response can be represented by a single universal curve.

Heat transfer in composite media subject to distributed sources, and time-dependent discrete sources and surroundings

The equation for the temperature distribution in each of k sections of a composite is solved. The composite consists of k discrete plates, cylinders of spheres, each of different material solidly joined at their k- 1 interfaces. The composite media have distributed sources, and also have k- 1 discrete sources at the interfaces. The composite media additionally have an arbitrary initial temperature distribution, and are exchanging heat at their mutual external boundaries with two different, arbitrary time-dependent surroundings through two different arbitrary constant film coefficients. The solution is obtained by means of a substitution that reduces the problem to that of solving a partial differential equation with homogeneous external boundary conditions. The solution is finally developed by means of a Vodicka type of orthogonality relationship.

Bending of circular sandwich plates due to asymmetric temperature distribution

I. Introduction T HERMOELASTIC behavior of sandwich plates has been considered by many investigators. However, most of these investigations pertain to the analysis of rectangular plates. Thermal deflection of circular sandwich plates was investigated by Bruun 1 and Huang and Ebcioglu,2 but with the restriction of rotational symmetry. Nonrotationally symmetrical bending of circular sandwich plates was considered by the author3; however, no thermal effect was included in the analysis. In this paper the thermoelastic equations for the bending of circular sandwich plates subjected to arbitrary temperature distribution are derived by means of the principle of stationary complementary energy. To illustrate the use of these equations an example, a clamped circular sandwich plate subjected to arbitrary temperature distribution which is expressible in the form of a double infinite series, is presented. The assumptions made in the presented analysis are as follows. 1) Two faces are isotropic but may be different in material, thickness, and temperature. The Poisson's ratios of the faces are the same. Bending rigidities of the individual faces may be neglected. 2) The core is considered to be isotropic and is assumed to take only the transverse shear stress. 3) Small deflection theory is considered.

Laminar natural-convective heat transfer from the outer surface of a vertical cylinder

Laminar natural convection along the outer surface of a vertical cylinder is compared with that along a vertical flat plate on heat transfer. For any Prandtl number and for arbitrary vertical temperature or heat flux distribution at the cylinder surface, local heat-transfer coefficients are represented non-dimensionally by the following approximate formulae. When surface temperature distributions are given, (Nu x) c = (Nu x) p + 0.435 x r w , x r w ≲ 0.7 (Nu) x) p , and when surface heat flux distributions are given. (Nu x) c = (Nu x) p + 0.345 x r w , x r w ≲ 0.7 (Nu) x) p , where ( Nu x ) c and ( Nu x ) p are the local Nusselt numbers for a cylinder and a flat plate respectively, x the vertical distance from the leading edge, and r w the radius of the cylinder.

General solutions to a class of unsteady heat conduction problems in a rectangular parallelepiped

General expressions are presented for the three-dimensional and unsteady heating of a finite solid rectangular parallelepiped under the influence of an arbitrary volume heat source and an arbitrary initial temperature distribution when convective type of time-dependent boundary conditions are prescribed on the six plane surfaces. These expressions given in various forms and not available hitherto, contain the solution of numerous special problems of technological importance. Corresponding general expressions for the rectangle and for the slab are deduced as limiting cases. The particular problem treated recently by Cobble is shown to be a very special case of the results derived here for the parallelepiped.

Unsteady Temperature Distribution in a Sphere Subjected to Time-Dependent Surface Heat Flux and Internal Heat Source

General expressions are derived for the three-dimensional unsteady heating of a solid sphere under the influence of a variable internal heat source and an arbitrary initial temperature distribution when the entire surface is subjected to a variable heat flux. Both the volume heat source and the surface heat flux are prescribed in terms of arbitrary functions of space and time. Various forms of solutions are noted. These expressions, not available hitherto, contain the solutions of many special problems of technological importance. The results presented in two recent studies on the arbitrary heating of spheres are shown to be special cases of the general sphere problem treated here and the errors appearing in these studies are corrected. Other special cases are also discussed.

Heat transfer in heat-generating fluid with flat velocity profile and arbitrary fluid inlet and circumferential wall temperature distributions

This paper presents a theoretical analysis of the steady state temperature distribution in a heat-generating fluid flowing in a circular tube with a flat velocity profile, i.e., ‘plug flow’. The fluid has a volumetric heat generation rate depending linearly on local temperature, and an arbitrary inlet temperature distribution. The temperature distribution at the tube wall varies circumferentially but is constant longitudinally.

Unsteady heat conduction in finite hollow circular cylinders under time-dependent boundary conditions of the second kind

A complete analytical study is presented for the determination of the three-dimensional unsteady temperature distributions in a finite hollow circular cylinder under the influence of a variable internal heat source and an arbitrary initial temperature distribution when the entire surface is subjected to variable heat flux rates. The volume heat source as well as the surface flux rates are prescribed in terms of arbitrary functions of space and time. General expressions are derived and their various alternative forms are noted. Corresponding general expressions for the two-dimensional and the one-dimensional heat flow problems are systematically deduced as limiting cases.

Hotspot Flexure of Plate on Circular Support

The equations for bending of an infinite plate resting on a circular support due to a hotspot are derived. Bending due to any arbitrary temperature distribution on faces of the plate can be obtained by integrating these equations. Finite circular plates with clamped or simply supported edges can be treated as special cases.

Strain energy expression for large deformations of isotropic elastic shells subjected to arbitrary temperature distribution

A general strain energy expression for large deflections of thin isotropic elastic shells is derived, including the effects of heating. With this expression, the thermal buckling and post-buckling behavior of heated shells may be treated. The results are specialized for circular cylindrical shells, for spherical shells and for right-circular conical shells. Important applications of thetheory presented occur with high-speed aircraft and rocket structures and with nuclear reactor structures.

Matrix Method of Analysis of Ship Structure (II)

In this paper studies are made on the application of displacement method to the analysis of two-different types of problems in two dimensional theory of elasticity.The first study is to develope the computer programming for stress analysis of mixed boundary value problems which are frequently encountered in engineering practice and yet reliable method of analysis is not well established. Taking a perforated square plate streched in one direction by a certain prescribed edge displacements as an example, stress distribution around a hole was calculated. Comparative study of the results obtained with that of previous study in which the plate is streched by uniformly distributed edge loads showed significant difference on the stress concentration factor and stress distribution.Computer programming is also developed for analysis of thermal stress problem under arbitrary temperature distribution. Thermal stress distribution of a rectangular plate under a given temperature distribution was also studied and good agreement was confirmed with the results of previous investigation. Study on the application of this method to the initial stress problem such as residual stress problems due to welding or heat treatment is underway.

平板の非定常空力加熱

Abstract The problems of the aerodynamic heating through the turbulent boundary layer of a thin flat plate is discussed in view of the unsteady temperature distribution along the surface. The fundamental system of equations for a turbulent boundary layer of a compressible fluid consists of the conservation equations for mass, momentum and energy, the equation of state and the equations describing the transport properties of air. The solutions depend upon the boundary conditions on the surface of a plate. For the temperature distribution in the boundary layer, either the adiabatic-wall assumption or the equitemperature-wall condition has usually been postulated, simply for the convenience of analysis. But strictly speaking, the temperature is hardly regarded as uniform. On the other hand, the wall temperature should be determined by solving the equations of heat conduction in the plate which is under the influence of aerodynamic heating through the boundary layer. However, because the rate of aerodynamic heating depends on the boundary layer flow, the heating rate itself turns out to be dependent on the wall temperature. This means that the boundary layer over a plate and the heat conduction in a plate must be solved simultaneously. Since we are interested in the temperature distribution over the wall, the problem will be reduced to that of determining the surface temperature by introducing the heat transfer rate through the boundary layer with arbitrary temperature distribution in the equation of heat conduction. Using FERRARI's formula for the aerodynamic heat of the turbulent boundary layer, we obtained numerical results for a thin steel plate of 10 mm thick. The theoretical unsteady temperature distributions are compared with the experimental results.

Theory of thermal natural frequency variations in disks

The paper discusses natural frequency variations caused by thermal membrane stresses. In particular induced thermal membrane stresses can increase the fundamental frequency of an important class of circular disk elements. This class includes grinding wheels, turbine disks and circular saws. The mathematical formulation for determination of stiffness variations arising from an arbitrary non-uniform temperature distribution is presented. The possible application of induced thermal stresses as a method of frequency control and optimization is discussed and illustrated by example.

Thermoelastic stresses and deformations in reactor fuel plates

The thermoelastic stresses and displacements in a reactor fuel plate are determined for arbitrary temperature distributions and normal surface tractions which do not vary across the width of the plate and are symmetrical about the mid-plane of the thickness. It is assumed that cross sections perpendicular to the width dimension are in a state of plane strain. The exact solution involving no thin plate approximations is obtained in part A through the use of the Papkovich stress functions and Fourier series methods. The determination of the coefficients of the resulting series for the stresses and displacements requires the numerical solution of infinite sets of simultaneous linear algebraic equations. In part B are derived two approximate solutions based on small thickness-to-length ratios; quantitative estimates of the errors involved in these approximate solutions are obtained by comparison with the exact solution of part A.

Quasi-analytic solution of the diffusion equation

An approximate method for solving the partial differential equation of conduction is developed that is continuous in time and discrete in space. The general solution is presented for a slab having five nodes and time dependent boundaries and having an arbitrary initial temperature distribution. The corresponding case, for a four node and a three node slab, is given in Appendix I. A typical example for a slab is solved using this quasi-analytic method, as well as by the traditional finite difference method. Both approximate solutions of the example problems are compared graphically with the exact solution.

Thermal stresses in rectangular plates

A method of determining the thermal stresses in a flat rectangular isotropic plate of constant thickness with arbitrary temperature distribution in the plane of the plate and with no variation in temperature through the thickness is presented. The thermal stress have been obtained in terms of Fourier series and integrals that satisfy the differential equation and the boundary conditions. Several examples have been presented to show the application of the method.