This chapter provides a comprehensive exploration of numerical methods for solving steady and unsteady heat and mass transfer problems. It covers various numerical techniques employed to solve partial differential equations, enabling accurate and efficient solutions for complex engineering and scie...
Forsberg Heat Transfer
Chapter 5
Numerical Methods
(Steady and Unsteady)
Finite Difference Method
In general, the temperature in a body is a function
of location and time. In rectangular coordinates,
T = T (x , y , z , t ). The parameters x , y , z , t are continuous
variables over their respective ranges. For the finite
difference method, the parameters are not continuous.
They are only defined at specific locations or times:
nodes for the spatial parameters and time steps for
the time parameter.
The first step in a finite-difference analysis is to
determine the grid and nodal points for the object.
1
, Nodes for a Large Wall
The wall is "L " wide. The surfaces temperatures are known,
and we want to determine interior temperatures. We decide
to have 11 nodes, equally-spaced, with two of the nodes at
total length L
the surfaces. x = =
number of divisions 10
Nodes for a Rectangular Plate
The cross-hatched nodes show the extent of Node 1, a corner node;
Node 7, an interior node; and Node 15, an edge node. This 16-node
grid could be replaced with the 28-node square grid on the next slide.
2
, Nodes for a Rectangular Plate (Cont’d)
With the 28-node grid, we will get better information about
the temperature distribution in the plate than we would have
gotten with the 16-node grid.
Nodes for a Circular Plate
There are 5 nodes, with r = ro /4. Node 1 has a circular area
of radius r/2. Nodes 2, 3, and 4 have ring areas of r width.
Boundary Node 5 has a ring area of r / 2 width. Node 4
is cross-hatched in the figure.
3
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