Page 166 - Mechanical Engineers' Handbook (Volume 4)

1 Conduction Heat Transfer 155

1.3 Two-Dimensional Steady-State Heat Conduction
Two-dimensional heat transfer in an isotropic, homogeneous material with no internal heat
2
2
generation requires solution of the heat-diffusion equation of the form T/ X T/ y
2
0, referred to as the Laplace equation. For certain geometries and a limited number of fairly
simple combinations of boundary conditions, exact solutions can be obtained analytically.
However, for anything but simple geometries or for simple geometries with complicated
boundary conditions, development of an appropriate analytical solution can be difficult and
other methods are usually employed. Among these are solution procedures involving the use
of graphical or numerical approaches. In the first of these, the rate of heat transfer between
two isotherms, T and T , is expressed in terms of the conduction shape factor, defined by
2
1
q kS(T T )
2
1
Table 9 illustrates the shape factor for a number of common geometric configurations. By
combining these shape factors, the heat-transfer characteristics for a wide variety of geo-
metric configurations can be obtained.
Prior to the development of high-speed digital computers, shape factor and analytical
methods were the most prevalent methods utilized for evaluating steady-state and transient
conduction problems. However, more recently, solution procedures for problems involving
complicated geometries or boundary conditions utilize the finite difference method (FDM).
Using this approach, the solid object is divided into a number of distinct or discrete regions,
referred to as nodes, each with a specified boundary condition. An energy balance is then
written for each nodal region and these equations are solved simultaneously. For interior
nodes in a two-dimensional system with no internal heat generation, the energy equation
takes the form of the Laplace equation discussed earlier. However, because the system is
characterized in terms of a nodal network, a finite difference approximation must be used.
This approximation is derived by substituting the following equation for the x-direction rate
of change expression

T T m 1,n T m 1,n 2T m,n
2
x 2 ( x) 2
m,n
and for the y-direction rate of change expression

T T m,n 1 T m,n 1 T m,n
2
y 2 ( y) 2
m,n
Assuming x y and substituting into the Laplace equation and results in the following
expression
T T T 4T 0
T m,n 1 m,n 1 m 1,n m 1,n m,n
which reduces the exact difference to an approximate algebraic expression.
Combining this temperature difference with Fourier’s law yields an expression for each
internal node

˙ q x y 1
T m,n 1 T m,n 1 T m 1,n T m 1,n 4T m,n 0
k
Similar equations for other geometries (i.e., corners) and boundary conditions (i.e., convec-
tion) and combinations of the two are listed in Table 10. These equations must then be
solved using some form of matrix inversion technique, Gauss-Seidel iteration method or
other method for solving large numbers of simultaneous equations.