Water Hydraulics - Mathematics and Statistics for the Environment - page 598

2 2

P
w

V P

w

V

AABB
+++ (22.18)

2 g 2 g

In Figure 22.4, as water passes through the constricted section of the pipe (section B), we know
from continuity of flow that the velocity at section B must be greater than the velocity at section
A, because of the smaller flow area at section B. This means that the velocity head in the system
increases as the water flows into the constricted section. However, the total energy must remain con-
stant. For this to occur, the pressure head, and therefore the pressure, must drop. In effect, pressure
energy is converted into kinetic energy in the constriction.
The fact that the pressure in the narrower pipe section (constriction) is less than the pressure in
the bigger section seems to defy common sense; however, it does follow logically from continuity
of flow and conservation of energy. The fact that there is a pressure difference allows measurement
of flow rate in the closed pipe.

■ EXAMPLE 22.13
Problem: In Figure 22.4, the diameter at section A is 8 in., and at section B it is 4 in. The flow rate
through the pipe is 3.0 cfs and the pressure at section A is 100 psi. What is the pressure in the con-
striction at section B?

Solution: Compute the flow area at each section, as follows:

2

π ( 0.666 ft )

2

A = = 0.349 ft (rounded)

A

4

) 2 π ( 0. 333 ft

2

A = = 0.087 ft (rounded)

B

4

From Q = A × V or V = Q / A , we get

3

3.0 ft /s
0.349 ft

V A = = 8.6 ft/s (rounded)

2

and

3

3.0 ft /s
0.087 ft

V B = = 34.5 ft/s (rounded)

2

22.5 CALCULATING MAJOR HEAD LOSS

Darcy, Weisbach, and others developed the first practical equation used to determine pipe friction
in about 1850. The equation or formula now known as the Darcy-Weisbach equation for circular
pipes is

2

LV
Dg

hf

f =

(22.19)

2

2
25

f = 8

fLQ
gD

h (22.20)

π