1、Chapter 3,College of Nuclear Science and Technology,Steady-State Conduction Multiple Dimensions,1,Two dimensions(common),Three dimensions,Chapter 3,College of Nuclear Science and Technology,2,Introduction,For two-dimensional steady state,the Laplace equation applies.,It becomes harder to solve the t
2、wo-dimensional steady state equations than one-dimensional ones.,How to get the solution?,Chapter 3,College of Nuclear Science and Technology,3,Solution methods,Analytical method,Numerical method,Graphical analysis,Analog method,Chapter 3,College of Nuclear Science and Technology,4,Summary,Solution
3、principle of conduction,One Dimension?,No,Shape factor method?,No,Analytical method?,Complex,Numerical techniques,Chapter 3,College of Nuclear Science and Technology,Thank you!,5,Chapter 3,College of Nuclear Science and Technology,Analytical method,6,solving the equation directly by mathematical too
4、ls,such as Fouries series.,Limits,Adapt to easy conditions only,Example1,Definition,One important method is the method of separation of variables,Back up,Chapter 3,College of Nuclear Science and Technology,7,Graphical analysis,Definition,Using the graph of temperature and heat flow distribution to g
5、et the solution-sketching the curvilinear squares.,Example2,Chapter 3,College of Nuclear Science and Technology,8,The conduction shape factor,Definition,In a two-dimensional system where only two temperature limits are limits are involved.Define a conduction shape factor S,Table 3-1,Back up,Chapter
6、3,College of Nuclear Science and Technology,9,Numerical method,Definition,Use the set of finite and discrete points to replace the physical field where time and space are continuous.Then solve the equation including these discrete points to get the unknown physical parameters.,One important method i
7、s finite-difference techniques,Chapter 3,College of Nuclear Science and Technology,10,Finite-difference techniques,Finite differences are used to approximate differential increments in the temperature and space coordinates. The smaller we choose these finite increments,the more closely the true temp
8、erature distribution will be approximated.,Chapter 3,College of Nuclear Science and Technology,11,Finite-difference techniques,Consider a two-dimensional body which is to be divided into equal increments in both x and y directions,as shown in Fig 3-5.,Chapter 3,College of Nuclear Science and Technol
9、ogy,12,Finite-difference techniques,Chapter 3,College of Nuclear Science and Technology,13,Finite-difference techniques,For the body with heat generation,For a square grid in which x= y,Then,Chapter 3,College of Nuclear Science and Technology,14,Finite-difference techniques,For convection boundary c
10、ondition,the result is,Chapter 3,College of Nuclear Science and Technology,15,Finite-difference techniques,Solution techniques,Matrix method(using matrix and software to get solution; may be complex),Gauss-Seidel iteration,Chapter 3,College of Nuclear Science and Technology,16,Gauss-Seidel iteration
11、,When the number of nodes is very large,an iterative techniques may frequently yield a more efficient solution to the nodal equation than a direct matrix.,From the front page,we know,Chapter 3,College of Nuclear Science and Technology,17,Gauss-Seidel iteration,Procedure,Chapter 3,College of Nuclear
12、Science and Technology,18,Gauss-Seidel iteration,Obviously, the smaller the value of ,the greater calculation time required to obtain the desired result and the accurate the result will be.,Back up,Chapter 3,College of Nuclear Science and Technology,Analog method,19,Steady-state conduction in a homo
13、geneous material of constant resistivity is analogous to steady heat conduction in a body of similar geometric shape. So we can simulate temperature fields by electrical fields.,Definition,Back up,They both satisfy the Laplace equation.,Chapter 3,College of Nuclear Science and Technology,20,Example1
14、,Considering that the two space coordinates x and y are independent ,use Fourier equations.,Chapter 3,College of Nuclear Science and Technology,21,So the total heat flow is the resultant of qx and qy.,Example1,If the temperature distribution in the matarial is known,we may easily establish the heat flow.,Back up,Chapter 3,College of Nuclear Science and Technology,22,Example2,Chapter 3,College of Nuclear Science and Technology,23,Example2,Chapter 3,College of Nuclear Science and Technology,24,Example2,Back up,Chapter 3,College of Nuclear Science and Technology,25,Back up,
