Rough Set Analysis of the Heat Conduction in the Steam Pipe

2015 ◽  
Vol 750 ◽  
pp. 371-375
Author(s):  
Wan Li Zhong ◽  
Wei Wang ◽  
Jie Dong Lin ◽  
Ming Nie ◽  
Chang Hong Liu
Keyword(s):  
Heat Conduction ◽  
Rough Set ◽  
Heat Conduction Equation ◽  
Uncertain Parameters ◽  
Lower Approximation ◽  
Interval Equations ◽  
Steam Pipe ◽  
Analysis Of Stability ◽  
The Stability ◽  
Composite Pipes

During the analysis of stability heat conduction in the composite pipes, firstly, when the heat equation contained fuzzy and random uncertain parameters, interval equations of the heat conduction are presented in the rough set. Secondly, the error expecting of heat conduction equation is presented. Finally, with upper (lower) approximation in rough set, a new method of the rough set analysis to deal with the stability heat conduction is presented.

Instability of the Planar Freeze Front During Solidification of an Aqueous Binary Solution

1980 ◽  
Vol 102 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. G. O’Callaghan ◽  
E. G. Cravalho ◽  
C. E. Huggins
Keyword(s):  
Heat Transfer ◽  
Temperature Field ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Binary Solution ◽  
Concentration Field ◽  
Planar Interface ◽  
Solute Redistribution ◽  
The Stability ◽  
Planar Solidification

A model describing the heat transfer, solute redistribution, and interface stability during the planar solidification of an aqueous binary solution has been developed. The temperature field was calculated using a modified Ka´rma´n-Pohlhausen integral technique. With this technique, the averaged heat conduction equation was solved with the full set of boundary conditions using an assumed spatial variation of the temperature profile. The concentration field was solved for analytically. The stability of the planar freezing morphology was determined using the Mullins-Sekerka stability criterion, in which the time variation of a sinusoidal perturbation of the planar interface was calculated. Application of this criterion to the freezing of saline indicates that for any practical freezing rate the planar interface was unstable. This represents an indictment of the planar freezing model and indicates the tendency for aqueous solutions to freeze dendritically.

Difference Schemes with Nonlocal Boundary Conditions

2001 ◽  
Vol 1 (1) ◽  
pp. 62-71 ◽  
Author(s):  
Alexei V. Goolin ◽  
Nikolai I. Ionkin ◽  
Valentina A. Morozova
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Conduction Equation ◽  
The Stability ◽  
The One ◽  
Necessary And Sufficient

AbstractThe paper deals with the stability, with respect to initial data, of difference schemes that approximate the heat-conduction equation with constant coefficients and nonlocal boundary conditions. Some difference schemes are considered for the one-dimensional heat-conduction equation, the energy norm is constructed, and the necessary and sufficient stability conditions in this norm are established for explicit and weighted difference schemes.

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals FORMULATION OF THE HEAT CONDUCTION EQUATION FOR HETEROGENEOUS MEDIA WITH MULTIPLE SPATIAL SCALES USING REITERATED HOMOGENIZATION

2016 ◽  
Vol 15 (1) ◽  
pp. 96
Author(s):  
E. Iglesias-Rodríguez ◽  
M. E. Cruz ◽  
J. Bravo-Castillero ◽  
R. Guinovart-Díaz ◽  
R. Rodríguez-Ramos ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Small Parameter ◽  
Heat Conduction Equation ◽  
Heterogeneous Media ◽  
Spatial Scales ◽  
Conduction Equation ◽  
Small Scale ◽  
Multiple Spatial Scales ◽  
Effective Coefficients ◽  
Reiterated Homogenization

Heterogeneous media with multiple spatial scales are finding increased importance in engineering. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. The objective in this paper is to formulate the strong-form Fourier heat conduction equation for such media using the method of reiterated homogenization. The phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter ε. The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter ε . The technique leads to two pairs of local and homogenized equations, linked by effective coefficients. In this manner the medium behavior at the smallest scales is seen to affect the macroscale behavior, which is the main interest in engineering. To facilitate the physical understanding of the formulation, an analytical solution is obtained for the heat conduction equation in a functionally graded material (FGM). The approach presented here may serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.

scholarly journals A Monte Carlo Method of Solving Heat Conduction Problems

1980 ◽  
Vol 102 (1) ◽  
pp. 121-125 ◽  
Author(s):  
S. K. Fraley ◽  
T. J. Hoffman ◽  
P. N. Stevens
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Monte Carlo Method ◽  
Transport Equation ◽  
Heat Conduction Equation ◽  
Analytical Techniques ◽  
Conduction Equation ◽  
Geometric Complexity ◽  
New Approach ◽  
Temperature Distributions

A new approach in the use of Monte Carlo to solve heat conduction problems is developed using a transport equation approximation to the heat conduction equation. A variety of problems is analyzed with this method and their solutions are compared to those obtained with analytical techniques. This Monte Carlo approach appears to be limited to the calculation of temperatures at specific points rather than temperature distributions. The method is applicable to the solution of multimedia problems with no inherent limitations as to the geometric complexity of the problem.

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