Analytical solution for transient response of counter flow heat exchanger with finite wall capacitance

The Laplace Transform Method is applied to solve the transient response of the counter flow heat exchanger with finite wall capacitance problem. The mathematical model is based on three local energy balance equations which are solved assuming that only the fluid 1 inlet condition is perturbed (step change). As any counter flow problem could be reduced to an adequate integral equation, collocation method is used for solving such equation in the presented case. Results are given for the outlet temperatures of both fluids and temperature distributions for both fluids and the wall as an explicit analytical formula.

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Analytische Lösung für das transiente Verhalten eines Gegenstrom-Wärmetauschers mit endlicher Wärmekapazität der Wand

Zusammenfassung Die Methode der Laplace-Transformation wird eingesetzt, um das Übertragungsverhalten des Gegenstrom-Wärmetauschers mit endlicher Wärmekapazität der Wand zu lösen. Das mathematische Modell fußt auf drei lokalen Energiebilanzgleichungen, welche unter der Voraussetzung gelöst werden, daß nur am Eintritt des Fluids 1 eine Störung (Sprung) auftritt. Da sich jedes Gegenstrom-Problem auf eine Integralgleichung zurückführen läßt, wird die Kollokationsmethode zur Lösung einer solchen, im vorliegenden Problem auftretenden Gleichung herangezogen. Die Ergebnisse in Form expliziter analytischer Ausdrücke betreffen die Austrittstemperaturen beider Fluide und die Wandtemperatur.

     

Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux

A fractional Cattaneo model is derived for studying the heat transfer in a finite slab irradiated by a short pulse laser. The analytical solutions for the fractional Cattaneo model, the classical Cattaneo-Vernotte model, and the Fourier model are obtained with finite Fourier and Laplace transforms. The effects of the fractional order parameter and the relaxation time on the temperature fields in the finite slab are investigated. The results show that the larger the fractional order parameter, the slower the thermal wave. Moreover, the higher the relaxation time, the slower the heat flux propagates. By comparing the fractional order Cattaneo model with the classical Cattaneo-Vernotte and Fourier models, it can be found that the heat flux predicted using the fractional Cattaneo model always transports from the high temperature to the low one, which is in accord with the second law of thermodynamics. However, the classical Cattaneo-Vernotte model shows that the unphysical heat flux sometimes transports from the low temperature to the high one.     

Analytical solution to a notch tip field in rubber-like materials under tension

Two kinds of elastic laws have been adopted by Gao, 1990 and Gao, 1997 to analyze the deformation fields near a crack tip. One of them contains the response to volume change and shape change; the other contains the response to extension and compression. In this paper the two kinds of constitutive relations are examined by typical large deformation, and the restrictions on constitutive parameters are discussed.     

Analytical solution of radiation sound pressure of double cylindrical shells in fluid medium

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The rate dependent response of a bistable chain at finite temperature

We study the rate dependent response of a bistable chain subjected to thermal fluctuations. The study is motivated by the fact that the behavior of this model system is prototypical to a wide range of nonlinear processes in materials physics, biology and chemistry. To account for the stochastic nature of the system response, we formulate a set of governing equations for the evolution of the probability density of meta-stable configurations. Based on this approach, we calculate the behavior for a wide range of parametric values, such as rate, temperature, overall stiffness, and number of elements in the chain. Our results suggest that fundamental characteristics of the response, such as average transition stress and hysteresis, can be captured by a simple law which folds the influence of all these factors into a single non-dimensional quantity. We also show that the applicability of analytical results previously obtained for single-well systems can be extended to systems having multiple wells by proper definition of rate and of the transition stress.     

ANALYTICAL SOLUTION OF RADIATED SOUND PRESSURE OF RING－STIFFENED CYLINDRICAL SHELLS IN FLUID MEDIUM

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Poromechanical response of a finite elastic cylinder under axisymmetric loading

Drained or undrained cylindrical specimens under axisymmetric loading are commonly used in laboratory testing of soils and rocks. Poroelastic cylindrical elements are also encountered in applications related to bioengineering and advanced materials. This paper presents an analytical solution for an axisymmetrically-loaded solid poroelastic cylinder of finite length with permeable (drained) or impermeable (undrained) hydraulic boundary conditions. The general solutions are derived by first applying Laplace transforms with respect to the time and then solving the resulting governing equations in terms of Fourier–Bessel series, which involve trigonometric and hyperbolic functions with respect to the z-coordinate and Bessel functions with respect to the r-coordinate. Several time-dependent boundary-value problems are solved to demonstrate the application of the general solution to practical situations. Accuracy of the numerical solution is confirmed by comparing with the existing solutions for the limiting cases of a finite elastic cylinder and a poroelastic cylinder under plane strain conditions. Selected numerical results are presented for different cylinder aspect ratios, loading and hydraulic boundary conditions to demonstrate the key features of the coupled poroelastic response.     

A semi-analytic solution of a wellbore in a non-isothermal low-permeability porous medium under non-hydrostatic stresses

The transient stress, displacement, pore pressure and temperature fields around a wellbore in a thermo-poro-elastic (THM) medium subject to non-hydrostatic remote stresses are analyzed under non-isothermal plane-strain conditions. The linear THM model proposed by Coussy (1989) is adopted in the analysis with a focus on thermal effects in low-permeability saturated rocks, characterized by a latent heat associated with local changes of fluid mass content. Non-dimensionalized parameters are identified by reformulating the fully-coupled governing equations and boundary conditions. The wellbore problem is simplified by decomposing it into axisymmetric and deviatoric loading cases. The corresponding analytical solutions are obtained in Laplace space. The inverse Laplace transforms are performed numerically to find the time-dependent distributions of field variables in the rock mass around the wellbore. These numerical results show that although the pore pressure diffusion has little influence on temperature and stress, temperature changes can strongly affect the pore pressure and stress around the wellbore. The temperature change can lead to changes in near-well stresses and the resulting significant change in wellbore breakdown pressure illustrates the importance of considering the THM coupling.     

Analytical solution for temperature field in thin film initially heated by a short-pulse laser source

Analytical solution for electron and lattice temperature distribution in the solid initially heated by a laser short-pulse is presented. Strained parameters method is introduced when formulating electron and lattice temperature distributions. Laser short pulse heating of gold film is simulated numerically and temperature data at the end of the heating pulse are adopted as initial condition to the governing equations of energy transport for analytical solutions. This enables to solve the governing equations of energy analytically in the cooling period. It is found that electron temperature decays sharply while lattice site temperature increases gradually in the surface regions during the cooling cycle. As the depth from the surface increases change in both temperatures become gradual.     

Two-temperature generalized thermoelasticity under ramp-type heating by finite element method

In this work, a general finite element model is proposed to analyze transient phenomena in thermoelastic half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory (Youssef in IMA J. Appl. Math. 71(3):383–390, 2006). A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. A finite element scheme is presented for the high accuracy numerical purpose. The numerical solutions of the non-dimensional governing partial differential equations of the problem have been shown graphically and some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating and the parameter of two-temperature.     

Thermo-mechanical response of tension panels under intense rapid heating

The temperature dependence of the fracture strength of 7075-T6 aluminum alloy was determined under rapid heating (0.02–.50 sec) conditions by exposing thin-section specimens to intense surface irradiation while under constant tensile load. When combined with a numerical thermal analysis and an appropriate limit analysis, these data enabled accurate prediction of the heating time required to produce ductile fracture in spot irradiated tension panels. The transient heat transfer model employed in the computations incorporated material removal due to melting, temperature-dependent material properties, and convection/radiation losses.     

Analytical solution of the contact problem for a system of bodies under collective wear

The contact problem is considered for a system of bodies subject to wear on a common base. The deformation properties of the bodies and the base are described by the Winkler model. The problem is reduced to a system of ordinary differential equations and an integral equation of hereditary type with difference kernel. The solution of the problem is constructed with the use of the Laplace transform. The asymptotics of the solution at large times is studied. The continuity conditions for the contact of each of the bodies with the base are derived.     

Analytical solution of a non-smooth freezing front

Short-time analytical solution has been obtained for a non-smooth freezing front which develops when the temperature is prescribed over a finite length on the surface of a melt occupying a two-dimensional half space. An interesting feature of the solution is that at the ends of the finite length over which the solidification initiates, removable singularities occur which tend to limits which are consistent with the physics of the problem. The spreads along the surface have been obtained in a novel way from the solution of an auxiliary problem. The spread along the surface can have unusual growth rates. Some inverse problems have been discussed.     

Analytical solution for a reinforcement layer bonded to an elliptic hole under a remote uniform load

A general solution to a reinforced elliptic hole embedded in an infinite matrix subjected to a remote uniform load is provided in this paper. Investigations on the present elasticity problem are rather tedious due to the presence of material inhomogeneities and complex geometric configurations. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacement and stresses in a reinforcement 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. The results show that there exists an optimum design of a reinforcement layer such that both the magnitude of stress concentration and the interfacial stresses could be fairly reduced.