Page 294 - Introduction to Computational Fluid Dynamics

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APPENDIX A 0 521 85326 5 May 20, 2005 13:6

Derivation of Transport Equations

A.1 Introduction

In the study of transport phenomena in moving fluids, the fundamental laws of
motion (conservation of mass and Newton’s second law) and energy (first law of
thermodynamics) are applied to an elemental fluid. Two approaches are possible:

1. a particle approach or
2. a continuum approach.

In the particle approach, the fluid is assumed to consist of particles (molecules,
atoms, etc.) and the laws are applied to study particle motion. Fluid motion is then
described by the statistically averaged motion of a group of particles. For most ap-
plications arising in engineering and the environment, however, this approach is too
1
cumbersome because the significant dimensions of the flow are considerably big-
ger than the mean-free-path length between molecules. In the continuum approach,
therefore, statistical averaging is assumed to have been already performed and the
fundamental laws are applied to portions of fluid (or control volumes) that contain
a large number of particles. The information lost in averaging must however be
recovered. This is done by invoking some further auxiliary laws and by empirical
specifications of transport properties such as viscosity µ, thermal conductivity k,
and mass diffusivity D. The transport properties are typically determined from ex-
periments. Notionally, the continuum approach is very attractive because one can
now speak of temperature, pressure, or velocity at a point and relate them to what
is measured by most practical instruments.
Guidance for deciding whether the particle or continuum approach is to be used
can be obtained from the Knudsen number Kn = l/L, wherel is the mean-free-path
length between molecules and L is a characteristic dimension (say, the radius of

1 This can be appreciated from Avogadro’s number, which specifies that, at normal temperature and
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pressure, a gas will contain 6.022 × 10 molecules per kmol. Thus in air, for example, there will
3
16
be 10 molecules/mm .
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