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A numeric simulation of transient thermal field in a transistorradiator system and its verification by measurements

Gołębiewski J., Łukjaniuk A., Bycul R. P.

Przegląd Elektrotechniczny

2013

R. 89, nr 11

169-174

A physical and mathematical model of heat transfer in a system of a power transistor and forced air stream cooled radiator was presented. An important part of the model was the estimation of the heat transfer coefficient for forced convection and radiation. Transient thermal field was determined with the use of the finite element method by using a professional computer program NISA v. 16. The results were verified by doing measurements. Thermoelectric sensors of the T type, an NI 9211 module and an NI Signal Express computer program were utilized during the measuring. The relative difference between the results of the computations and the measurements did not exceed 15,5% in the transient state and it dropped to 6% in the steady state.

W pracy przedstawiono fizyczny i matematyczny model propagacji ciepła w układzie tranzystora mocy i radiatora chłodzonego wentylatorem. Istotnym elementem modelu było oszacowanie współczynnika przejmowania ciepła z układu w warunkach wymuszonej konwekcji i promieniowania. Nieustalone pole termiczne wyznaczono metodą elementów skończonych za pomocą profesjonalnego programu NISA v.16. Wyniki numerycznej symulacji zweryfikowano na drodze pomiarowej. W tym celu wykorzystano termoelektryczne czujniki typu T, moduł NI 9211 oraz oprogramowanie NI Signal Express. Okazało się, że względna różnica między wynikami obliczeń i pomiarów nie przekroczyła w stanie przejściowym 15,5% i w stanie ustalonym zmalała do 6%.

Transient thermal analysis of functionally graded shallow shells by the MLPG

Sladek J., Sladek V., Solek P.

Computer Methods in Materials Science

2009

Vol. 9, No. 2

171-177

In recent years the demand for construction of huge and lightweight shell and spatial structures is increasing. To minimize the weight of shell structures a layered profile of the shell is utilized frequently. In such a case a delaminating of individual layers may occur due to a jump change of the material properties. To remove this phenomenon the functionally graded materials (FGMs) has been introduced recently. FGMs are multi-phase materials with a pre-determined property profile, where the phase volume fractions are varying gradually in space. This results in continuously nonhomogenous material properties at the macroscopic structural scale. Often, these spatial gradients in the material behaviour render FGMs as superior to conventional composites because of their continuously graded structures and properties. FGMs may exhibit isotropic or anisotropic material properties, depending on the processing technique and the practical engineering requirements. Many linear bending studies are focused only to a lateral pressure load with assumption of uniformly distributed temperature in the whole shell. However, in shells with FGM properties the role of thermal loading is more imperative. Therefore, it is interesting to analyze shells under a general thermal load. Literature sources on this subject are poor and they are mostly restricted to analyses of plates. Due to the high mathematical complexity of the boundary or initial-boundary value problems, analytical approaches for FGMs are restricted to simple geometry and boundary conditions. The choice of an appropriate mathematical model together with a consistent computational method is important for such kind of structures. Most significant advances in shell analyses have been made using the finite element method (FEM). It is well known that numerical results by standard displacement-based type shell element are over stiff with yielding the shear locking phenomena in thin shells. Locking problems arise due to inconsistencies in discrete representations of the transverse shear energy and the membrane energy. The boundary element method (BEM) has emerged as an alternative numerical method to solve plate and shell problems. It is a very powerful computational method if a fundamental solution is available for considered problem. However, the fundamental solution for a thick orthotropic shell wit continuously varying material properties is not available according to the best of the author?s knowledge. Meshless approaches for solution of problems of continuum mechanics have attracted much attention during the past decade. One of the most rapidly developed meshfree methods is the Meshless Local Petrov-Galerkin method (MLPG). The solution of the uncoupled problem in the present paper is split into two tasks. In the first task the temperature distribution in the shell has to be obtained by solving the diffusion equation. The temperature distribution in shell has to be analyzed as 3-D problem. The MLPG is applied to transient heat conduction equations in a continuously nonhomogeneous solid. The Laplace transform technique is used to eliminate the time variable. Several quasi-static boundary value problems must be solved for various values of the Laplace-transform parameter. The Stehfest?s inversion method is applied to obtain the time-dependent solution. In the second task, the set of governing differential equations for Reissner-Mindlin shell bending theory with Duhamel-Neumann constitutive equations is solved. Since thermal changes in solids are relatively slow with respect to elastic wave velocity, the inertial terms in Reissner-Mindlin governing equations are not considered. The problem is considered as quasi-static with time dependent thermal forces. The MLPG method is applied again to solution of that problem with the meshless Moving Least-Squares (MLS) approximation of primary field variables. The nodal points are spread freely in the analyzed domain and on its boundary. The essential boundary conditions are satisfied by collocation of approximated fields at nodes with prescribed values. In other nodes, the governing PDEs are considered on subdomains around these nodes in the local weak-form with using unit test functions. The resulting local integral equations are discretized within the assumed approximation of field variables. Numerical results for simply supported and clamped square shells are presented to illustrate the efficiency of the present computational method.

Tematem niniejszej pracy jest wykorzystanie lokalnej beziastkowej metody Petrova-Galerkina (MLPG) do problemu odkształceń termicznych powłok Reissnera-Mindlina. W studium wykorzystano model funkcjonalnych materiałów gradientowych z założeniem ciągłej zmiany własności na grubości elementu powłokowego. Forma słaba równań występujących w teorii Reissnera-Mindlina została przeniesiona na zbiór równań całkowych rozwiązywanych w obszarze zdefiniowanych poddomen. Cylindryczne poddomeny losowo otaczają wygenerowane punkty węzłowe. W rozwiązaniu wykorzystano beziatkową aproksymację metody Moving Least-Squares (MLS).