Page 201 - 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 163

5.2 Fluid Transport 163

Taking a small length along the perforated pipe (see sketch) dx, then
L
2
h f = K Q x dx
L 0
Q x L - x
But =
Q 0 L
L - x
Q x = Q 0
L
2 L
Q 0 2
So h f = K (L - x) dx
2
L L 0
2 L
Q 0 2 2
h f = K 2 3(L - 2xL + x )dx4
L L 0
2 3 L
Q 0 2 2 x
h f = K cL x - x L + d
L 2 3 0
2 3
Q 0 3 3 L
h f = K 2 cL - L + d
L 3
L
2
h f = KQ 0
3
Hence, h f 1>3 (h f ) 0
(h f ) 0 3h f
That is, the head loss h f in a pipe is equal to three times the head loss h f in a perforated pipe.

5.2.2 Exponential Equation for Surface Resistance
Because of practical shortcomings of the Weisbach formula, engineers have resorted to
so-called exponential equations in flow calculations. Among them the Chezy formula is
the basic for all:
V = C1rs (5.18)
where v average velocity, ft/s; C coefficient; r hydraulic radius, which is defined as
the cross-section area divided by the wetted perimeter, ft; and s slope of water surface or
energy gradient:

r A>P w (5.19)
s h f >L (5.20)
2
2
where r hydraulic radius, ft or m; A cross-section area, ft or m ; and P w wetted
perimeter, ft or m; s slope of water surface, dimensionless; and L pipe length, ft or m.
The coefficient C can be obtained by using one of the following expressions:
Chezy expression: C (8g>f) 0.5 (5.21)
Manning expression: C (1.486>n)(r) 1>6 (5.22)
Bazin expression: C 157.6>(1 mr 0.5 ) (5.23)
Kutter expression:
0.5
C (41.65 0.00281>s 1.811>n)>[1 (n>r )(41.65 0.00281>s)] (5.24)