scholarly journals Steady-state heat transfer analysis in a spherical domain revisited

2021 ◽  
Vol 347 ◽  
pp. 00006
Author(s):  
Jat du Toit ◽  
Christiaan Pretorius
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Numerical Solution ◽  
Heat Transfer Analysis ◽  
Finite Element Code ◽  
Dimensionless Numbers ◽  
Galerkin Finite Element ◽  
Spherical Domain ◽  
Steady State Heat Transfer ◽  
The One

The paper discusses the numerical solution of the one-dimensional radially axi-symmetric non-linear second-order differential equation to model the conduction and radiation transfer through a spherical domain as a result of an exothermic heat source. The equation is transformed to a non-dimensional form. The dimensionless numbers emanating from the transformation represent the effect of the reaction rate, reaction type, activation energy, radiation and the convection on the temperature. The non-dimensional differential equation for the temperature distribution was previously solved using the Runge-Kutta-Fehlberg method coupled with a Shooting technique. In this paper the solution of the non-dimensional differential equation using an iterative Galerkin finite element method approach employing the Picard method is described. The commercial finite element code Comsol is also employed to solve the non-dimensional differential equation. The current study was motivated by inconsistencies that were observed in the previous results that were presented. Although the assumed underlying physics is used to evaluate the results, the study focuses purely on the numerical solution of the non-dimensional differential equation. The results obtained by the Galerkin finite element code and Comsol were found to be in exact agreement and also exhibit no inconsistencies.

scholarly journals Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 469 ◽  
Author(s):  
Azhar Iqbal ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Spline Method ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
B Spline

This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 ,   I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement.

scholarly journals APPLICATION OF GALERKIN FINITE ELEMENT METHOD IN THE SOLUTION OF 3D DIFFUSION IN SOLIDS

2009 ◽  
Vol 8 (2) ◽  
pp. 79 ◽  
Author(s):  
E. C. Romão ◽  
M. D. De Campos ◽  
J. A. Martins ◽  
L. F. M. De Moura
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Heat Diffusion ◽  
Average Error ◽  
Galerkin Finite Element Method ◽  
Diffusion In Solids ◽  
Galerkin Finite Element ◽  
L2 Norm ◽  
Element Method

This paper presents the numerical solution by the Galerkin Finite Element Method, on the three-dimensional Laplace and Helmholtz equations, which represent the heat diffusion in solids. For the two applications proposed, the analytical solutions found in the literature review were used in comparison with the numerical solution. The results analysis was made based on the the L2 Norm (average error throughout the domain) and L¥ Norm (maximum error in the entire domain). The two application results, one of the Laplace equation and the Helmholtz equation, are presented and discussed in order to to test the efficiency of the method.

Overpressure Fragility Evaluation of a Mark I Drywell Using Thermal-Mechanical Finite Element Analysis

2014 ◽  
Author(s):  
Mohamed M. Talaat ◽  
David K. Nakaki ◽  
Kyle S. Douglas ◽  
Philip S. Hashimoto ◽  
Yahya Y. Bayraktarli
Keyword(s):  
Finite Element ◽  
Stress Analysis ◽  
Fe Analysis ◽  
Heat Transfer Analysis ◽  
Fe Model ◽  
Head Region ◽  
Element Analysis ◽  
Differential Movement ◽  
Steady State Heat ◽  
Steady State Heat Transfer

The overpressure fragility of a Mark I boiling water reactor drywell was performed by detailed finite element (FE) analysis. The drywell overpressure capacity is controlled by the onset of leakage in the bolted head flange connection once separation exceeds the capacity of the silicone rubber O-ring seals. The FE analysis was conducted at 6 discrete accident temperatures, ranging from 150 to 425°C. The overpressure evaluation used an axisymmetric model of the drywell head region for computational efficiency, and verified it by comparing to results from one FE model which used 3D solid elements. The mechanical properties of the steel materials were defined as temperature-dependent linear-elastic. The median overpressure capacity at each temperature was determined using a 2-step thermal-stress analysis procedure. First, a steady-state heat transfer analysis was conducted to map out the temperature distribution in the drywell wall, which is exposed to the accident temperature on the inside and ambient temperature on the outside. Second, a quasi-static multi-step stress analysis was performed. The vertical differential movement between the flange surfaces was monitored and compared to the O-ring rebound capacity to define the pressure at the onset of leakage. After leakage occurred, the relationship between leakage area and increased pressure was recorded. The evaluation predicted the median overpressure capacity and the lognormal standard deviation for uncertainty in O-ring rebound capacities, bolt preload, and model sophistication, in addition to the median pressure-leak area relationship.

Sign in / Sign up

Export Citation Format

Share Document