Page 38 - Introduction to Computational Fluid Dynamics

P1: IWV/ICD
CB908/Date
0 521 85326 5
0521853265c02

2 1D Heat Conduction May 25, 2005 10:49

2.1 Introduction

A wide variety of practical and interesting phenomena are governed by the 1D heat
conduction equation. Heat transfer through a composite slab, radial heat transfer
through a cylinder, and heat loss from a long and thin fin are typical examples. By
1D, we mean that the temperature is a function of only one space coordinate (say x
or r). This indeed is the case in steady-state problems. However, in unsteady state,
the temperature is also a function of time. Thus, although there are two relevant
independent variables (or dimensions), by convention, we refer to such problems as
1D unsteady-state problems. The extension dimensional thus always refers to the
number of relevant space coordinates.
The 1D heat conduction equation derived in the next section is equally applicable
tosomeoftheproblemsarisinginconvectiveheattransfer,indiffusionmasstransfer,
and in fluid mechanics, if the dependent and independent variables of the equation
are appropriately interpreted. In the last section of this chapter, therefore, problems
from these neighbouring fields will be introduced. Our overall objective in this
chapter is to develop a single computer program that is applicable to a wide variety
of 1D problems.

2.2 1D Conduction Equation

Consider the 1D domain shown in Figure 2.1, in which the temperature varies only
in the x direction although cross-sectional area A may vary with x.
The temperature over the cross section is thus assumed to be uniform. We shall
now invoke the first law of thermodynamics and apply it to a typical control volume
of length x. The law states that (Rate of energy in) − (Rate of energy out) + (Rate
of generation of energy) = (Rate of change of Internal energy), or

∂

Q x − Q x+ x + q A x = [ρ A xC T ] W, (2.1)
∂t
17