Page 23 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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INTRODUCTION
certain region surrounding it, and its temperature is a measure of the temperature distribution
in that region. The numerical accuracy of these calculations depends strongly on the number
of designated nodal points, which control the number of elements generated. The accuracy
approaches an exact value as the mesh size (region size) approaches zero.
Further details on the numerical methods, for example, accuracy and error will be
discussed in later chapters.
1.8 Summary

In this chapter, the subject of heat transfer was introduced and various modes of heat trans-
port were discussed. The fundamentals of energy conservation principles and the application
of such principles to some selected problems were also presented. Finally, the general heat
conduction equations in multi-dimensions were derived and the appropriate boundary and
initial conditions were given. Although brief, we trust that this chapter gives the reader
the essential fundamental concepts involved in heat transfer in general and some detailed
understanding of conduction heat transfer in particular.

1.9 Exercise

Exercise 1.9.1 Extend the problem formulation of the plate subjected to a solar heat flux in
Section 1.4.1 for a square plate. Assume the bottom surface of the plate is insulated.
Exercise 1.9.2 Repeat the incandescent lamp problem of Section 1.4.2 but now assume that
the light source is within an enclosure (room). Assume that the enclosure is also participating
in conserving energy.

Exercise 1.9.3 Derive the energy balance equations for a rectangular fin of variable cross
section as shown in Figure 1.6. The fin is stationary and is attached to a hot heat source.
(Hint: This is similar to the problem given in Section (1.4.3), but without relative motion).
Exercise 1.9.4 Consider the respective control volumes in both cylindrical and spheri-
cal coordinates and derive the respective heat conduction equations. Verify these against
Equations 1.42 and 1.44.

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Exercise 1.9.5 The inner body temperature of a healthy person remains constant at 37 C,
while the temperature and humidity of the environment change. Explain, via heat transfer
mechanisms between the human body and the environment, how the human body keeps itself
cool in summer and warm in winter.
Exercise 1.9.6 Discuss the modes of heat transfer that determine the equilibrium temper-
ature of a space shuttle when it is in orbit. What happens when it re-enters the earth’s
atmosphere?

Exercise 1.9.7 A closed plastic container, used to serve coffee in a seminar room, is made of
two layers with an air gap placed between them. List all heat transfer processes associated