Page 22 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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forward. Equation 1.47 can be rewritten with the direction cosines of the outward nor-
mals as
∂T ∂T ∂T INTRODUCTION
˜
k x l + k y ˜ m + k z ˜ n = C on qf (1.50)
∂x ∂y ∂z
Similarly, Equation 1.48 can be rewritten as
∂T ∂T ∂T
˜
k x l + k y ˜ m + k z ˜ n = h(T − T a ) on qc (1.51)
∂x ∂y ∂z
where l, ˜m and ˜n are the direction cosines of the appropriate outward surface normals.
˜
In many industrial applications, for example, wire drawing, crystal growth, continuous
casting, and so on, the material will have a motion in space, and this motion may be
restricted to one direction, as in the example (Section 1.4.3) cited previously. The general
energy equation for heat conduction, taking into account the spatial motion of the body is
given by

∂ ∂T ∂ ∂T ∂ ∂T ∂T ∂T ∂T ∂T
k x + k y + k z + G = ρc p + u + v + w
∂x ∂x ∂y ∂y ∂z ∂z ∂t ∂x ∂y ∂z
(1.52)
where u, v and w are the components of the velocity in the three directions, x, y and z
respectively.
The governing equations for convection heat transfer are very similar to the above and
will be discussed in Chapter 7.

1.7 Solution Methodology

Although a number of analytical solutions for conduction heat transfer problems are avail-
able (Carslaw and Jaeger 1959; Ozisik 1968), in many practical situations, the geometry
and the boundary conditions are so complex that an analytical solution is not possible.
Even if one could develop analytical relations for such complicated cases, these will
invariably involve complex series solutions and would thus be practically difficult to imple-
ment. In such situations, conduction heat transfer problems do need a numerical solution.
Some commonly employed numerical methods are the Finite Difference (Ozisik and Czisik
1994), Finite Volume (Patankar 1980), Finite Element and Boundary Elements (Ibanez and
Power 2002) techniques. This text will address the issues related to the Finite Element
Method (FEM) only (Comini et al. 1994; Huang and Usmani 1994; Lewis et al. 1996;
Reddy and Gartling 2000).
In contrast to an analytical solution that allows for the temperature determination at any
point in the medium, a numerical solution enables the determination of temperature only
at discrete points. The first step in any numerical analysis must therefore be to select these
points. This is done by dividing the region of interest into a number of smaller regions.
These regions are bounded by points. These reference points are termed nodal points and
their assembly results in a grid or mesh. It is important to note that each node represents a