Page 27 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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SOME BASIC DISCRETE SYSTEMS
importance of the finite element method finds a place here, that is, finite element techniques,
in conjunction with the digital computer, have enabled the numerical idealization and
solution of continuous systems in a systematic manner. This in effect has made possible
the practical extension and application of classical procedures to very complex engineering
systems.
We deal here with some basic discrete, or lumped-parameter systems, that is, systems
with a finite number of degrees of freedom. The steps in the analysis of a discrete system
are as follows:
Step 1: Idealization of system: System is idealized as an assembly of elements
Step 2: Element characteristics: The characteristics of each element, or component, is found
in terms of the primitive variables
Step 3: Assembly: A set of simultaneous equations is formed via assembly of element
characteristics for the unknown state variables
Step 4: Solution of equations: The simultaneous equations are solved to determine all the
primitive variables on a selected number of points.
We consider in the following sections some heat transfer and fluid flow problems.
The same procedure can be extended to structural, electrical and other problems, and the
interested reader is referred to other finite element books listed at the end of this chapter.

2.2 Steady State Problems

2.2.1 Heat flow in a composite slab

Consider the heat flow through a composite slab under steady state conditions as shown in
Figure 2.1. The problem is similar to that of a roof slab subjected to solar flux on the left-
2
hand face. This is subjected to a constant flux q W/m and the right-hand face is subjected
to a convection environment. We are interested in determining the temperatures T 1 , T 2 and
T 3 at nodes 1, 2 and 3 respectively.
The steady state heat conduction equation for a one-dimensional slab with a constant
thermal conductivity is given by Equation 1.41, that is,
2
d T
= 0 (2.1)
dx 2
Integration of the above equation yields the following temperature gradient and tem-
perature distribution:
dT
= a (2.2)
dx
and
T = ax + b (2.3)

Consider a homogeneous slab of thickness L with the following boundary conditions
(in one dimension):
At x = 0,T = T 1 and At x = L, T = T 2 (2.4)