Page 193 - Fair, Geyer, and Okun's Water and wastewater engineering : water supply and wastewater removal

JWCL344_ch05_154-193.qxd 8/2/10 9:44 PM Page 155

5.1 Fluid Mechanics, Hydraulics, and Water Transmission 155
Fluid flow may be steady or unsteady, uniform or nonuniform. Steady flow occurs at
any point if the velocity of successive fluid particles is the same at successive instants, that
is, if the fluid velocity is constant with time. The uniform flow occurs when the velocity of
successive fluid particles does not change with distance. This book introduces mainly the
fundamentals of the steady and uniform flows involving water and wastewater.
In steady flow, the mass of fluid passing any and all sections in a stream of fluid per
unit of time is the same. For incompressible fluids, such as water and wastewater, the
following principle of conservation of mass governs:
A v A v Constant Q (5.4)
2 2
1 1
2
2
where A and A are, respectively, the cross-sectional areas (ft or m ) at Station 1 and
2
1
Station 2; V and V are, respectively, the average velocity of the stream (ft/s or m/s) at
2
1
3
3
Section 1 and Section 2; Q is the flow rate (ft /s or m /s).
The Bernoulli equation results from application of the principle of conservation of en-
ergy to fluid flow, and is written between two points in a hydraulic system:
H H H H H B (5.5a)
a
A
e
l
H H H (H H ) H h f (5.5b)
a
B
A
e
l
B
2
H P > v >2g Z A (5.6a)
A
A
A
2
H P > v >2g Z B (5.6b)
B
B
B
where
H A energy at Section A
H B energy at Section B
H a energy added
H l energy lost
H e energy extracted. (The units of H A , H B , H a , H l , and H e are feet or meters of the fluid.)
2
2
2
P A , P B pressures at Sections A and B (lb/in. or kN/m [1,000 Newton/m ])
3
3
specific weight of water (lb/ft or kN/m )
2
2
g acceleration due to gravity (32.2 ft/s or 9.81 m/s )
V A , V B velocities at Sections A and B (ft/s or m/s)
Z A , Z B heights of stream tube above any assumed datum plane at Sections A and B (ft or m)
h f head loss (ft or m).
It is important to determine the magnitude, direction, and sense of the hydraulic force
exerted by fluids in order to design the constraints of the structures satisfactorily. The force
2
2
P (lb or kg) exerted by a fluid on a plane area A (ft or m ) is equal to the product of the
f
3
3
specific weight, (lb/ft or kN/m ), of the liquid, the depth (ft and m) of the center of grav-
2
2
ity, h of the area, and the area A (ft or m ). The engineering equation is
cg
P ( h ) A I A (5.7)
f
cg
p
where I is the intensity of pressure at the center of gravity of the area. The line of the
p
action of the force passes through the center of pressure, which can be determined by the
following equation:
y [I >y A] y cg (5.8)
cp
cg
cg
where
y cg moment of inertia of the area about its center of gravity axis
y cp distance of the center of pressure measured along the plane from an axis located at the
intersection of the plane and the water surface, extended if necessary
y cg distance the center of gravity measured along the plane from an axis located at the inter-
section of the plane and the liquid surface, extended if necessary.