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1Fundamentals of Heat and Mass Transfer, 6e
Fundamentals of Heat and Mass
Transfer, 6th edition
Presented： 杜文静E-mail: [email protected]
Telephone: 88399596-2511
2Fundamentals of Heat and Mass Transfer, 6e
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)
3Fundamentals of Heat and Mass Transfer, 6e
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
4Fundamentals of Heat and Mass Transfer, 6e
4.2 Method of separation of variables
Dimensionless excess temp无量纲过余温度
separation of variables
= 2
Boundary conditions
Two ordinary differential equation
常微分方程
Partial differential equation偏微分方程
5Fundamentals of Heat and Mass Transfer, 6e
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.
6Fundamentals of Heat and Mass Transfer, 6e
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
Temp difference between 2 prescribed boundaries21T
7Fundamentals of Heat and Mass Transfer, 6e
Sh
ap
e fa
cto
r for s
ele
cte
d s
yste
m in
2D
or 3
D s
yste
ms
8Fundamentals of Heat and Mass Transfer, 6e
Sh
ap
e fa
cto
r for s
ele
cte
d s
yste
m in
2D
or 3
D s
yste
ms
9Fundamentals of Heat and Mass Transfer, 6e
4.3 Conduction shape factor and
dimensionless conduction heat rate
• Shape factor may also be defined in 1-D geometries
plane
cylindrical
spherical
Shape factor method is applicable for heat rate
calculations between 2 described temperature surfaces
LAS /
)/ln(/2 12 rrLS
)/(4 1221 rrrrS
10Fundamentals of Heat and Mass Transfer, 6e
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
11Fundamentals of Heat and Mass Transfer, 6e
12Fundamentals of Heat and Mass Transfer, 6e
Shape factor in 2D case
13Fundamentals of Heat and Mass Transfer, 6e
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
• Nodes节点nodal network,
grid, mesh 网格(m, n)
02
2
2
2
y
T
x
T
14Fundamentals of Heat and Mass Transfer, 6e
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）
15Fundamentals of Heat and Mass Transfer, 6e
4.4 Finite-difference form of heat
equation
• 2D steady state, without Eg , Taylor series method :
02
2
2
2
y
T
x
T
16Fundamentals of Heat and Mass Transfer, 6e
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!!!
17Fundamentals of Heat and Mass Transfer, 6e
4.4The energy balance method
•Convenient to formulate the energy balance by assuming all heat flow is into the node.
18Fundamentals of Heat and Mass Transfer, 6e
4.4The energy balance method
19Fundamentals of Heat and Mass Transfer, 6e
If h or q equal to 0,
then ??
20Fundamentals of Heat and Mass Transfer, 6e
Example 4.2 p218
21Fundamentals of Heat and Mass Transfer, 6e
4.4The energy balance method
• Thermal resistance application
With contact resistance
22Fundamentals of Heat and Mass Transfer, 6e
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
23Fundamentals of Heat and Mass Transfer, 6e
4.5 Solving the finite-difference
equations
• The matrix inversion method
24Fundamentals of Heat and Mass Transfer, 6e
4.5 Solving the finite-difference
equations
• Gauss-seidel iteration procedure
• 1.reorder equations (diagonally dominant主对角线占优)
• 2. rewrite equations
• 3. assume initial value （初始值， k=0）
• 4. calculation (k=1)
• 5. iteration
• 6. termination
naaaa 1131211 ,,
25Fundamentals of Heat and Mass Transfer, 6e
4.5 Solving the finite-difference
equations
Precautions：注意
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
26Fundamentals of Heat and Mass Transfer, 6e
Example 4.3 p224
27Fundamentals of Heat and Mass Transfer, 6e
28Fundamentals of Heat and Mass Transfer, 6e
4.6 Summary
2-D steady state conduction cases
• Exact solution
• Graphical solution Approximate
• Numerical solution Approximate
• Finite difference method (energy balance
method)
29Fundamentals of Heat and Mass Transfer, 6e
Exercises in class
• 4.35 finite-difference equation
• 4.38 contact resistance
• 4.40 composite materials
• 4.41
• 4.43
30Fundamentals of Heat and Mass Transfer, 6e
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
