Example 11-17 Viscous Laminar Flow in a Long, Narrow Pipe
Water at 20°C moves in laminar flow at an average flow speed of 0.800 m>s through a pipe of radius 0.500 mm = 5.00 *
1024 m. The pipe is 2.00 m long. Determine what the pressure difference must be between the two ends of the pipe.
Set Up
We saw in part (a) of Example 11-16 that the flow is
laminar for water at this temperature moving through
such a pipe at this speed. Because the pipe is long and narrow, viscosity is important and laminar flow is an example
of Hagen-Poiseuille flow. (The density and viscosity are
given in Example 11-16.) Our goal is to find the pressure
difference p between the ends of the pipe, and we use the
Hagen-Poiseuille equation, Equation 11-30.
Solve
In Example 11-16 we calculated the volume flow rate.
Solve the Hagen-Poiseuille equation for the pressure
­difference p and substitute values.
Hagen-Poiseuille equation
for laminar flow of a
R
viscous fluid:
Q =
pR4
p
8hL
(11-30)
L
∆p = ?
Volume flow rate:
Q = 6.28 * 10-7 m3 >s
p =
=
8hLQ
pR4
811.002 * 10-3 Pas2 12.00 m2 16.28 * 10-7 m3 >s2
= 5.13 * 104 Pa
p 15.00 * 10-4 m2 4
Reflect
Even though the viscosity of water is low, the pressure
required to sustain the flow is substantial (about half an
atmosphere) because the pipe is so narrow and long.
p = 15.13 * 104 Pa2 a
= 0.506 atm
1 atm
b
1.01325 * 105 Pa