Session 12: Heat and Mass Diffusion and Analytical Methods

Chairs:

Helcio Rangel Barreto Orlande
Programa de Engenharia Mecanica, COPPE/UFRJ

Marco Túlio Mena Barreto de Vilhena
Departamento de Engenharia Nuclear UFRGS








s12p01

A FRACTIONAL DIFFUSION EQUATION FOR CALCULATING THERMAL PROPERTIES OF THIN FILMS FROM SURFACE TRANSIENT THERMOREFLECTANCE MEASUREMENTS

Vladimir V. Kulish - [email protected]
School of Mechanical & Production Engineering
Nanyang Technological University
50 Nanyang Ave., Singapore 639798

José L. Lage - [email protected]
Pavel L. Komarov - [email protected]
Peter E. Raad - [email protected]
Mechanical Engineering Department
Southern Methodist University
Dallas, Texas, 75275-0337

The thermoreflectance method, suitable for determining the thermal properties of thin films, consists of measuring changes in the reflectivity of a thin film under pulsed laser
heating, and relating these changes to corresponding temperature variations. Analytical or numerical solutions of the diffusion problem are then used to determine the thermal property of the material following an iterative matching process between the analytical and the experimental results. The existing analytical or numerical solutions, valid when the laser energy is absorbed at the surface or when it is absorbed volumetrically by the thin film, allow for the determination of only one thermal property (thermal conductivity or diffusivity), with the other one assumed to be equal to the bulk material property. A complete solution to the non-homogeneous diffusion equation with surface and volumetric heating, found using fractional calculus and presented in a semi-derivative form, provides the means to determine the two thermal properties of thin films (thermal conductivity and diffusivity) concomitantly. This can be achieved by utilising a secondary laser for heating the material. The solution component for surface heating is validated by comparison with experimental data for a GaAs bulk sample using the classical thermoreflectance method.

Keywords: thermoreflectance, thin films, fractional calculus
 
 



s12p02

BENCHMARK SOLUTIONS TO STEADY NATURAL CONVECTION IN AN INCLINED ENCLOSURE THROUGH INTEGRAL TRANSFORM METHOD

Jesús Salvador Pérez Guerrero - [email protected]
Marco Aurélio Leal - [email protected]
Comissão Nacional de Energia Nuclear, Coordenação de Rejeitos Radioativos - COREJ
Rua General Severiano, 90 - Botafogo - Rio de Janeiro - RJ - 22294.900- Brazil

The Generalized Integral Transform Technique (GITT) is employed in the solution of two-dimensional laminar natural convection within inclined enclosures filled by air (Prandtl number of 0.71), subjected to differentially heated walls and insulated horizontal surfaces. The hybrid nature of the GITT approach allows for the establishment of reference results in the solution of non-linear partial differential systems, as the coupled set of heat and fluid flow equations that govern the steady natural convection problem under consideration. The aim of the present work is to provide reference results to steady-state natural convection in square cavities with Rayleigh numbers equal to 10 4 and 10 5 for several different inclination angles, a = 40 0 , 60 0 , 120 0 and 140 0 . Numerical values of the mean Nusselt numbers and streamfunctions are presented to all examined situations.

Keywords: Natural convection, Integral transform, Hybrid method
 
 



s12p03

ANALYTICAL SOLUTION OF THE ONE DIMENSIONAL TRANSIENT POLLUTANT PROBLEM WITH VARIABLE EDDY DIFFUSIVITY COEFFICIENT

Sérgio Wortmann - [email protected]
Angela B. D. Moura - [email protected]
Marco T. M. B. Vilhena - [email protected]
PROMEC - Universidade Federal do Rio Grande do Sul
Sarmento Leite, 425, 3 º andar - 90035-972 - Porto Alegre, RS,Brasil

The objective of this work consists of presenting an analytical solution, being used the Generalized Integral Transform Technique (GITT) and Laplace Transform, for the vertical dispersion of polutants in a stable atmospheric boundary layer, with a diffusion coefficient derived from the local similarity and of the statistical diffusion theories. Numerical simulations are reported.

Keywords: GITT, Analitical Solution, Atmospheric Dispersion, Laplace Transform.
 
 



s12p05

CONDUÇÃO DE CALOR HIPERBÓLICA: FORMULAÇÃO E SIMULAÇÃO

Fernando P. Duda
Universidade Federal do Rio de Janeiro, COPPE
Programa de Engenharia Mecânica, C.P. 68503, 21945-970 - Rio de Janeiro, RJ, Brasil
Luiz C. G. Pimentel
Universidade Federal do Rio de Janeiro
Departamento de Meteorologia, RJ, Brasil
Jesús S. Perez Guerrero
Comissão Nacional de Energia Nuclear CNEN/COREJ, RJ, Brasil

Resumo: Neste trabalho consideraremos aspectos relacionados com a formulação do modelo de Cattaneo visando a simulação do problema de condução hiperbólica.  Obteremos a solução analítica para o problema de condução em uma placa finita, usando a Técnica da Transformada Integral e o Mathematica.  A solução encontrada apresenta comportamento não realista do ponto de vista físico, o que pode estar relacionado com o fato do modelo de Cattaneo utilizado ser inconsistente com a segunda lei da termodinâmica.  Finalmente, mostraremos como o arcabouço da termomecânica do contínuo pode ser utilizado para tornar o modelo de Cattaneo consistente com a segunda lei, resultando numa forma especial para a equaçaão constitutiva para a energia interna.

Palavras-chave: condução de calor hiperbólica, termomecânica do contínuo, transformada integral
 
 



s12p07

SOURCE TERM ESTIMATION IN NONLINEAR HEAT CONDUCTION PROBLEMS WITH ALIFANOV'S ITERATIVE REGULARIZATION METHOD

Jiazheng Wang - [email protected]
Jian Su- [email protected]
Antônio J. Silva Neto - [email protected], [email protected]
Instituto Politécnico, Universidade do Estado do Rio de Janeiro, CP 97282, CEP 28601-970, Nova Friburgo, RJ, Brasil.
Programa de Engenharia Nuclear - COPPE - Universidade Federal do Rio de Janeiro, CP 68509, CEP 21945-970, Rio de Janeiro, RJ, Brasil.

In this work we consider the inverse problem of heat sources intensity estimation, with spatial and timewise dependency, in nonlinear heat conduction problems. The formulation and solution of the inverse problem with Alifanov's iterative regularization method is presented. A comparison is done between the results obtained with the non-linear formulation and those resulting from an approximation with a linear formulation.

Keywords: Inverse problems, Nonlinear heat conduction, Conjugate gradient, Adjoint problem.
 
 



s12p13

HYBRID SOLUTION FOR TRANSIENT CONVECTION IN CHANNELS WITH AXIAL DIFFUSION VIA INTEGRAL TRANSFORMS AND TRANSIENT FILTERING

Romberg Rodrigues Gondim - [email protected]
Universidade Federal da Paraíba - CT/DTM, Campus I, Laboratório de Energia Solar, Caixa Postal 5115 - CEP 58051-970 - João Pessoa, PB, Brasil.

Renato Machado Cotta - [email protected]
Mechanical Engineering Department, EE/COPPE/UFRJ - Universidade Federal do Rio de Janeiro, Caixa Postal 68503 - CEP 21945-970, Rio de Janeiro - RJ.

Transient forced convection within planar channels is solved for a thermally developing laminar flow situation, considering the presence of axial diffusion in the fluid energy equation. The analysis employs the generalized integral transform technique combined with a transient filtering solution, aimed at enhancing the convergence behavior of the associated eigenfunction expansions, and allowing for the inspection of the Peclet number influence on the final converged solution. The numerical results so obtained are critically compared against available results in the open literature for the infinite Peclet number situation. Finally, a computational cost analysis is performed among different possible filtering strategies for the same problem.

Keywords: Transient convection, Integral transform, Axial
 
 



s12p14

HIGHER ORDER LUMPED ANALYSIS OF TRANSIENT HEAT TRANSFER IN A NUCLEAR FUEL ROD

Jian Su - [email protected]
Nuclear Engineering Department, COPPE/UFRJ, CP 68509, Rio de Janeiro, 21945-970, Brazil
Renato M. Cotta - [email protected]
Mechanical Engineering Department, COPPE/UFRJ
CP 68503, Rio de Janeiro, 21945-970, Brazil

Abstract. Transient heat transfer in a nuclear fuel rod is analyzed by an improved lumped parameter approach.  The circunferential symmetry is assumed with the heat transfer through the gap modelled by a heat transfer coefficient.  Higher order (H1,1) Hermite approximation for integration is used to obtain the average temperatures in the radial direction.  A significant improvement over the classical lumped parameter formulation has been achieved.  The proposed fuel rod heat conduction model can be used in stability analysis of BWR, simplified model of PWR or real-time simulator of nuclear power plants.

Keywords: Transient Heat Conduction, Lumped Parameter Analysis, Hermite Approximation, Nuclear Reactor Thermohydraulics
 
 



s12p15

AN ANALYTICAL APPROXIMATED SOLUTION FOR THE NONLINEAR COUPLED CONDUCTIVE-RADIATIVE H EAT TRANSFER PROBLEM

Rubem Mário Figueiró Vargas - [email protected]
PUCRS - FEN G - DEQ
Porto Alegre - RS - Av Ipiranga 6681 CEP 90619.900 -  Prédio 3 0 Bloco 6 Sala 216

Marco Tullio de Vilhena - [email protected]
Jacques Brancher
UFRGS - PROMEC

In this work is proposed an analytica l solution for the nonlinear coupled conductive-radia tive hea t p roblem. The solutio n is o b ta ined usin g the decomposition method. This method is a powerful tool to solve nonlinear p roblems. Numerical results a re a ttained a s for iso tropic med ium as anisotropic med ium. These ones a re compared with results encountered in the literature. An analyses is made comparing numerical and analytica l solutions.
 
 



s12p16

SOLUTION OF THE TRANSIENT HEAT CONDUCTION EQUATION IN MULTICOMPONENT PLATES

Ivanilda B. Aseka - [email protected]
Universidade Federal de Santa Maria, Departamento de Matem_atica
Universidade Federal do Rio Grande do Sul, Programa de Pos-Graduação em Minas, Metalurgica e de Materiais - PPGEM
Osvaldo Aranha, 99/613 - 90035-190, Porto Alegre, RS, Brasil

Marco T. Vilhena - [email protected]
Universidade Federal do Rio Grande do Sul, Instituto de Matemática PPGEM

Paulo O. Beyer - [email protected]
Marcus V. A. Bianchi - [email protected]
Universidade Federal do Rio Grande do Sul, Departamento de Engenharia Mecânica
Cx. P. 17819 - 90035-972 - Porto Alegre, RS, Brasil

In this work a semi-analytic method is presented for the solution of a two- dimensional transient heat conduction problems in a multicomponent plates. That method consists of the application of the nodal method combined with the Laplace transform technique, which allows to _nd expressions for the average temperatures and for the temperature in the boundary.

Keywords: semi-analytic, transient conduction, plates multicomponent
 
 



s12p18

SOLUÇÃO SEMI-ANALÍTICA DA EQUAÇÃO DE TRANSFERÊNCIA RADIATIVA NÃO-LINEAR

José V. P. de Oliveira
Universidade Federal de Santa Maria, Departamento de Matemática
Av. Roraima, S/N - CCNE - 97105-900 - Santa Maria, RS, Brasil.

Augusto V. Cardona - [email protected]
Pontifícia Universidade Católica do Rio Grande do Sul, Faculdade de Matemática
Av. Ipiranga, 6681 - Prédio 15 - 90619-900 - Porto Alegre, RS, Brasil.

Marco T. M. B. de Vilhena - [email protected]
Universidade Federal do Rio Grande do Sul, PPGEM
Av. Osvaldo Aranha, 99 - 6 o andar - 90046-900 - Porto Alegre, RS, Brasil.

Ricardo Barros - [email protected]
Universidade do Estado do Rio de Janeiro, Instituto Politécnico - IPRJ
Caixa Postal 97282 - 28630-050 - Nova Friburgo, RJ, Brasil.

Analytical Solution of the Non-Linear Radiative Transfer Equation. In this paper, we describe a new approach to solve the radiative transfer problem, combining the LTSN  method  and Spectral method. We use the essence of the spectral methods, where the intensity of radiation is expanded in time in a truncated series of Laguerre polynomials yielding a set of stationary one-dimensional transport problems, that we solve using the LTSN method. The material temperature and the intensity of radiation are determined by an iteration from the initial temperature. Numerical results and comparisons with the results found in the literature are also presented.

Keywords: Radiative transfer, Transient transport problem, Spectral method, LTSN method.
 
 



s12p19

SOLUCÃO ANALITICA DE UM MODELO TRIDIMENSIONAL DE DISPERSAO DE CONTAMINANTES NA CAMADA LIMITE PLANET ARIA CONVECTIVA: COEFICIENTES DE DIFUSAO DEPENDENTES DA DISTANCIA DA FONTE E DA ALTURA

Angela B. D. Moura - [email protected]
Marco T M. B. Vilhena - [email protected]
Universidade Federal do Rio Grande do Sul , PROMEC
Sarmento Leite 425/ 3 andar- Cx. P. 17819 - 90035.972 - Porto Alegre, RS, Brasil

Gervasio Degrazia - [email protected]
Universidade Federal de Santa Maria, Departamento de Fisica.

Cyntia Segatto- [email protected]
Universidade Federal do Rio Grande do Sul, Depto. de Matematica.

O objetivo deste trabalho consiste na determinaçao de uma soluçao analitica para o problema de difusao e adveçao tridimensional estacionario que representa a dispersao de contaminantes em uma camada limite planetaria. Este modelo e valido para a dispersao de um contaminante passivo na camada limite atmosferica, emitido a partir de uma fonte pontual contnua, e sujeito a situaçoes de turbulencia homogenea e com velocidade de vento medio uniformes, ou seja, para abandonos elevados que ocorrem em condiçoes de estabilidade intermediaria e na ausencia de fortes empuxos. A soluçao e alcançada com o uso do metodo da Transformada Integral Generalizada e s~ ao analiticas no sentido de que nenhuma aproximaçao foi feita ao longo de sua derivaçao. Simulaçoes e comparaçoes com resultados experimentais disponveis na literatura sao apresentadas. Nestas simulaçoes os coeficientes de dispersao foram considerados dependentes da distancia da fonte.

Palavras-chave: soluçao anaitica, GITT, dispersao atmosferica
 
 



s12p23

AN INVERSE MASS TRANFER PROBLEM IN GAS-LIQUID ADSORPTION SEPARATION PROCESS IN COLUMNS OF BUBBLE AND FOAM

Jader Lugon Junior 1 - [email protected]
Antônio J. Silva Neto 2 - [email protected]
Instituto Politécnico - Universidade do Estado do Rio de Janeiro, CP 97282, CEP 28601-970, Nova Friburgo, RJ, Brasil.

In this work is done the estimation of parameters related to the separation process of bio-molecules by adsorption in gas-liquid interfaces in bubble and foam columns. The results obtained with the Levenberg-Marquardt method and real experiment data of the concentration of the Bovine Serum Albumin (BSA) are presented.

Keywords: Inverse problems, Gas-liquid adsorption, Fractionation, Bubble columns.
 
 



s12p24

DISCRETE-ORDINATES SOLUTIONS TO SOME CLASSICAL FLOW PROBLEMS IN THE RAREFIED GAS DYNAMICS

M. Camargoy
P. Rodriguesy
Programa de Pos - Graduaçao em Engenharia Mecanica
Universidade Federal do Rio Grande do Sul
90050{170 Porto Alegre, RS, Brasil,
E-mail: [email protected], [email protected]

L. B. Barichellozy
zInstituto de Matem_atica
Universidade Federal do Rio Grande do Sul
91509{900 Porto Alegre, RS, Brasil
E-mail: [email protected]

A recently developed version of the discrete-ordinates method is used to solve in a uni_ed manner, for plane and cylindrical geometry, some classical ow problems
based on the Bhatnagar, Gross and Krook model in the theory of rare_ed-gas dynam- ics. In particular, the thermal-creep problem for the case of a semi-in_nite medium and the Poiseuille- ow problem, for a wide range of the Knudsen number, are solved. Ana- lytical solutions for the discrete-ordinates problem are obtained based on a \half-range" quadrature scheme which results in simpli_ed eigenvalue problems. Numerical results are presented to show that the solutions are specially accurate and easy to implement.

Key Words: Rare_ed Gas Dynamics, BGK Model, Discrete-Ordinates