Page 836 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix D: Fluid Mechanics—Reviews

of Selected Topics

Fluid mechanics is pertinent in the design of nearly all unit additive effect of this pressure profile, i.e., the integration of
processes. Topics of particular interest include drag, pipe flow the infinitesimal forces over the whole surface, causes a
(incompressible and compressible), flow distribution by mani- resultant force opposite to the plate velocity vector.
folds, porous media flow, flow measurement, turbulence, and The calculation of the drag of a body is given by
dispersion. These areas are reviewed here to provide a ready
reference for key equations and associated coefficients. In the v 2
F D ¼ C D r A (D:1)
case of compressible flow, the logic is outlined as well as the 2
final equations. The review is intended to provide reference
for frequently sought equations, but may not be readily in which
accessible. The equations may also be useful for spreadsheet F D is the drag force on body (N)
applications. C D is the coefficient of drag (dimensionless)
A is the area of body projection perpendicular to the
2
direction of motion (m )
D.1 FLUID DRAG
v o is the velocity of body relative to fluid (m=s)
2
Drag is a fundamental notion of fluid mechanics and it per- r is the density of fluid (kg=m )
meates many kinds of problems. Examples include sedimen-
tation, sediment transport, headloss in pipes, pump design, Equation D.1 is applicable to both friction drag and form drag.
airfoil design, design of mixers, energy dissipation, etc. Prob- The coefficient, C D , is quite different in each case.
lems of particular interest in unit operations include fall
velocity of particles, paddle wheel design in flocculation,
D.1.2 FLOW REGIME
mixing, etc.
The drag coefficient depends upon the shape of the body and
the Reynolds number, i.e., R,defined,
D.1.1 DRAG EQUATIONS
Drag occurs when a fluid in motion contacts a surface of a rvD
(D:2)
‘‘body,’’ e.g., an impeller, a pipe wall, an airfoil, an automo- R ¼ m
bile, a solid particle settling in water, a gas bubble rising in
water, ad infinitum. The two types of drag are friction drag in which
and form drag. Both occur in most situations and the total R is the Reynolds number (dimensionless)
drag on a ‘‘body’’ is the sum of the two. v is the velocity of body relative to fluid (m=s)
The friction drag is due to boundary layer shear. Headloss D is the characteristic length (m)
2
in a pipe, calculated by the Darcy–Weisbach equation is m is the dynamic viscosity (N s=m )
caused by friction drag. The resistance of an airfoil is due
mostly to friction drag. The other kind of drag is form drag. The length term, D, in Equation D.2 is selected to characterize
The ‘‘lift’’ on an airfoil is due to form drag. A section of an the system, e.g., pipe diameter, impeller diameter, etc. At low
axial flow impeller is really an ‘‘airfoil’’ and the associated Reynolds numbers, i.e., R < 10, the flow regime is laminar.
pumping is due to lift. The pressure profile caused by flow As R increases, the flow becomes transitional, and then as R
around a body causes form drag (a ‘‘streamline’’ shape of an increases further, the regime is fully turbulent. The specific
object reduces form drag). The shape of the body determines Reynolds number at each transition depends upon the kind of
the pressure profile. shape.
A flat rectangular plate provides a means to illustrate the
two types of drag. If the plate is oriented with its surface
D.1.3 DRAG COEFFICIENTS
parallel to the flow, the drag on the plate is caused by friction.
If the plate is oriented with its surface perpendicular to the As noted in Section D.1.2, drag coefficients depend on (1) the
flow, the drag is caused by separation effects, called form form (or shape) of the body and (2) the Reynolds number, R,
drag. A pressure profile around the plate would show a nearly associated with the velocity of the fluid. The drag coefficient,
uniform positive pressure on the upstream face and a nearly C D , must be determined by experiment, using a wind tunnel, a
uniform negative pressure on the downstream face. The towing tank, channel, or any other means to measure the drag

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