Fundamentals of Heat and Mass Transfer, 6 th edition

Transcription

1 Fundamentals of Heat and Mass Transfer, 6 th edition Presented: 杜文静 wjdu@sdu.edu.cn Telephone:

2 Chapter 4 two-dimensional steady state conduction Contents and objectives Two-dimensional steady state conduction (temperature distribution & heat rate) Exact solutions & Approximate solutions Numerical methods (finite-difference, finite element, boundary element) 2

3 4.1 Alternative approaches Lines of constant temperature (isotherms 等温线 ) Heat flow lines( 热流线 ): represent the direction of heat flux vector no heat conduction across heat flow lines Two things to do in conduction cases 1 T (x, y) Heat diffusion equation 2 heat flux (x, y) Fourier s law Methods: Analytical separation of variables Exact solution Graphical approximate solution, see supplementary material 4S.1 Numerical approximate solution 3

4 4.2 Method of separation of variables Dimensionless excess temp 无量纲过余温度 Partial differential equation 偏微分方程 Boundary conditions separation of variables 2 = Two ordinary differential equation 常微分方程 4

5 4.2 Method of separation of variables The general solution Exact solution: for a simple 2-D steady conduction case An exact solution for other geometry and boundary conditions are presented in specialized books on conduction heat transfer. 5

6 4.3 Conduction shape factor and dimensionless conduction heat rate In some 2 D or 3D conduction systems: shape factor S 形状因子法 T 1 2 Temp difference between 2 prescribed boundaries 6

7 Shape factor for selected system in 2D or 3D systems 7

8 Shape factor for selected system in 2D or 3D systems 8

9 4.3 Conduction shape factor and dimensionless conduction heat rate Shape factor may also be defined in 1-D geometries plane cylindrical spherical S A/ L S 2 L / ln( r 2 / r1 ) S rr /( r ) r1 Shape factor method is applicable for heat rate calculations between 2 described temperature surfaces 9

10 4.3 Conduction shape factor and dimensionless conduction heat rate For the infinite cases: Dimensionless conduction heat rate, 无量纲导热速率 T1: object temp T2: infinite media temp Lc: Characteristic length 特征长度 As: the surface area of the object 10

11 11

12 Shape factor in 2D case 12

13 4.4 Finite-difference equations Numerical calculation applicable for more boundary and geometry conditions, also applicable for 3 D cases Finite difference, finite element, boundary element Control equation 2 T 2 x 2 T 0 2 y Nodes 节点 nodal network, grid, mesh 网格 (m, n) 13

14 4.4 Finite-difference equations Nodes discrimination equation 离散化 numerical calculation Temperature discrimination equation Taylor series ( 均匀 网格 ), Energy balance ( 非均匀网格 ) 14

15 4.4 Finite-difference form of heat equation 2D steady state, without E g, Taylor series method : 2 T 2 x 2 T 2 y 0 15

16 4.4The energy balance method Applicable for many different phenomena: with or without heat sources, asymmetric grid size Finite-difference equation can be obtained by apply conservation of energy to a node For node conduction rate, both the Fourier s law and thermal resistance method are applicable 热平衡法!!! 16

17 4.4The energy balance method Convenient to formulate the energy balance by assuming all heat flow is into the node. 17

18 4.4The energy balance method 18

19 If h or q equal to 0, then?? 19

20 Example 4.2 p218 20

21 4.4The energy balance method Thermal resistance application With contact resistance 21

22 4.5 Solving the finite-difference equations The matrix inversion method( 矩阵转置法, 矩阵求逆法 ) direct method Gauss-seidel iteration( 高斯 - 赛德尔迭代法 ) iteration method 22

23 4.5 Solving the finite-difference equations The matrix inversion method 23

24 4.5 Solving the finite-difference equations Gauss-seidel iteration procedure 1.reorder equations (diagonally dominant 主对角线占优 ) a11 a12, a13, a1 n 2. rewrite equations 3. assume initial value ( 初始值, k=0) 4. calculation (k=1) 5. iteration 6. termination 24

25 4.5 Solving the finite-difference Precautions: 注意 equations 1.Finite differential heat equation An approximate solutions check solutions by the energy balance equation Grid studies( 网格验证 ) Grid refinement gird-independent results 25

26 Example 4.3 p224 26

27 27

28 4.6 Summary 2-D steady state conduction cases Exact solution Graphical solution Approximate Numerical solution Approximate Finite difference method (energy balance method) 28

29 Exercises in class 4.35 finite-difference equation 4.38 contact resistance 4.40 composite materials

30 Homework Assignment 4.23 shape factors with thermal circuit 4.32 hint: boundary conditions change 4.39 hint :a correct control volume should be defined first 4.44 hint: symmetrical condition 4.49 hint: fin heat rate=base conduction rate 4.51(a): solving the finite-difference equation 30