Page 39 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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SOME BASIC DISCRETE SYSTEMS
iterative procedure is necessary. An initial guess of the unknown temperature values is also
essential to start any iterative procedure.
In this example, if the time terms are neglected, we can recover the steady state for-
mulation. However, the time-dependent load terms are necessary to carry out any form of
transient analysis. In practice, the reduction of an appropriate discrete system that contains
all the important characteristics of the actual physical system is usually not straightforward.
In general, a different discrete model should be chosen for a transient response prediction
than that chosen for a steady state analysis.
The time-derivative terms used in the above formulation have to be approximated in
order to obtain a temperature distribution. As discussed in later chapters, approximations
such as backward Euler, central difference, and so on, may well be employed.

2.4 Summary

In this chapter, we have discussed some basic discrete system analyses. It is important to
reiterate here that this chapter gives only a brief discussion of the system analysis. We
believe that the material provided in this chapter is sufficient to give the reader a starting
point. It should be noted that the system analysis is straightforward and works for many
simple heat transfer problems. However, for complex continuum problems, a standard
discretization of the governing equations and solution methodology is essential. We will
discuss these problems in detail in the following chapters.

2.5 Exercise

Exercise 2.5.1 Use the system analysis procedure described in this chapter and construct
the discrete system for heat conduction through the composite wall shown in Figure 2.9.
Also, from the following data, calculate the temperature distribution in the composite wall.
2
2
2
Areas: A 1 = 2.0 m ,A 2 = 1.0 m and A 3 = 1.0 m .
Thermal conductivity: k 1 = 2.00 W/mK, k 2 = 2.5 W/mK and k 3 = 1.5 W/mK.
2
Heat transfer coefficient: h = 0.1 W/m K
◦
Atmospheric temperature: T a = 30 C
◦
Temperature at the left face of wall: T 1 = 75.0 C.
Exercise 2.5.2 The cross section of an insulated pipe carrying a hot fluid is shown in
Figure 2.10. The inner and outer radii of the pipe are r 1 and r 2 respectively. The thickness
of the insulating material is r 2 − r 3 . Assume appropriate conditions and form the discrete
system equations.

Exercise 2.5.3 The pipe network used to circulate hot water in a domestic central heating
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arrangement is shown in Figure 2.11. The flow rate at the entrance is Qm /s. Neglecting
any loss of mass, construct a system of simultaneous equations to calculate the pressure
distribution at selected points using a discrete system analysis. Assume laminar flow occurs
in the system.