Statistical methods in finite element and its application in heat transfer

Abstract

Most of the time information The initial and boundary conditions are not precisely defined and cause errors in the output of the problem.

These initial values ??can be obtained through experiments and statistical methods. It is often not possible to study individual members of the society due to high cost, little time, or lack of sufficient facilities, so we select a sample of the society and generalize the result of the study to the entire society.

with statistics The information of the obtained results also has a percentage of error whose values ??are determined according to the statistical information of the data.

In this thesis, the statistical effect of the input information on the results obtained by using the finite element method is studied.

This method is used in all branches of science, but we study this method in heat transfer.

In two-dimensional heat transfer, the temperature is studied based on boundary conditions and coefficients of conduction and displacement, and the error percentage will be checked from the first order.

Finally, the applicability of this method will be proven.

Keywords:

finite element, heat transfer, numerical analysis, standard deviation, average, coefficient of variation, statistics

Chapter One: Introduction

For the accurate analysis of engineering problems, accurate input information and specific boundary conditions are needed, which in most cases, the initial information and boundary conditions are not precisely defined, and this information causes errors they bring that they cause errors in the output.

One of the methods of obtaining this basic information and boundary conditions is through experiments and statistical methods, in which the percentage of errors in the output can be calculated from the statistical information of the data. to investigate in solving heat transfer problems.

Here we solve the problem by entering different displacement and conductivity coefficients and different boundary conditions and study the results and get the error percentages.

1-1 research objectives

  Due to the advancement of technology in energy conversion and the creation of high heat energy in this process, the need to transfer the heat generated from the energy conversion environment is strongly felt.

    One of the methods of this heat transfer in small industries is a tool that increases the heat transfer by conduction and displacement methods, which are called vanes or fins. In order to study the rate of heat transfer from the fin, it is necessary to know the basic equations governing heat transfer, and partial differential equations will appear when these equations appear and using them, in order to solve the problem of these equations, one should take help from existing numerical methods, which is one of the strongest numerical solution methods in heat transfer problems is the finite element method. In order to do this, it is necessary to minimize the errors in the initial information and boundary conditions so that unreasonable results are not obtained. In order to better examine the results of the initial information and the boundary conditions, which in this type of problem are the conductive heat transfer coefficient and the displacement heat transfer coefficient and the boundary conditions, we enter statistically and from the laboratory method and compare the obtained results.In order to study the rate of heat transfer from the fin, it is necessary to know the basic equations governing heat transfer, and with the appearance of these equations and using them, partial differential equations will appear. In order to solve the problem of these equations, one should take help from existing numerical methods, which is one of the most powerful numerical solution methods in heat transfer problems, the finite element method, in this method, it works by completely removing the differential equations or simplifying them to ordinary equations that must be numerically stable. In order to do this, it is necessary to minimize the errors in the initial information and boundary conditions so that unreasonable results are not obtained. To better examine the results of the initial information and the boundary conditions, which in this type of problem are the conductive heat transfer coefficient and the displacement heat transfer coefficient and the boundary conditions of the problem, we enter statistically and from the laboratory method and compare the obtained results. The analysis is done and we relate the results with statistical information and boundary conditions. It is transferred to the object through conduction and is transferred to the environment through displacement. The analysis of such complex problems is very important from a practical point of view. With the increase or decrease of each of these problems, the output condition, which is the temperature, undergoes changes. Sometimes the excessive increase of each of them, for example, the heat transfer level when the heat transfer coefficient is large, will not only have an effect on increasing the amount of heat transfer, but due to the increase in conduction resistance, it will cause a decrease in heat transfer.

In this research, assuming four types of fins and four different states of the heat transfer coefficient, the displacement of a two-dimensional fin with the base temperature and with the condition that the end of the fin is insulated and the temperature differential is zero according to the length of the fin at the end of the fin, and we will analyze the problem according to the temperatures obtained.

1-3 research background

     Finite element method was developed in the early forties of the 20th century in solving complex problems of elasticity and analysis of structures in civil and aerospace engineering by dividing a continuous domain of matter into a series of sub-domains or smaller parts called elements. This solution method was quickly developed in heat transfer problems and because the first challenge of solving equations Differential is finding equations that approximate the original equations and are numerically stable. So, the use of laboratory data and the sorting of primary data and boundary conditions were analyzed statistically, and the first and second order error percentages were investigated.

Finite element statistical methods were first researched in Iran by Dr. Niazi and Dr. Kadivar in 1990 in the solids and heat branch, and one of the latest researches until now can be found in the research of Mr. Maisham Aliabadi in the fluid branch in the university. Hormozgan pointed around a rotating sphere and an airfoil. The computer program that is available in the finite element method gives the basic information, which here is the heat transfer and displacement coefficients, we get the answers for each of these basic information, then we check and analyze the outputs according to the changes in temperature and analyze the results.

1-5 summary of chapters

In the first chapter of this thesis, the goals, assumptions, background and research method are briefly stated, and in the second chapter, the theory of heat transfer, displacement and conduction and their governing equations are discussed briefly, and then numerical solution methods The characteristics of the element method have been investigated and this method has been expressed in statistical form, which is the problem solving method in this research, and the results and outputs have been examined in the next chapter, and the final chapter has been assigned to summarizing the research and suggestions.