Download 1 An empty straight pipe of length L and diameter d is connected to a

An empty straight pipe of length L and diameter d is connected to a water source having pressure Po. At time t = 0,
a valve is opened allowing water to flow in the pipe. Assuming a horizontal pipe, and the only viscous losses are
those due to friction, develop an equation for the time required for the water to exit the pipe. You may assume the
pipe discharges to atmosphere and the friction factor is independent of time.
Note: Friction only operates over that portion of the pipe containing water.
SOLUTION
Mass conservation on the liquid in the pipe gives
Ax
dx
 vAx
dt
where x is the distance the fluid has traveled down the pipe. Since the cross sectional area and the density are
constant
dx
v
dt
To relate the amount of fluid in the pipe to the fluid velocity, write Bernoulli’s equation from the reservoir (water
source) to the end of the water column
Po  Patm 
v 2
2

fx v 2
d 2
or
fx  v 2

Po  Patm  1  
d  2

Solving for velocity
2( Po  Patm ) / 
fx 

1  
d 

v
and substituting for velocity in the continuity equation
dx

dt
2( Po  Patm ) / 
fx 

1  
d 

or
1/ 2
fx 

1  
d 

dx

dt
2( Po  Patm )

which may be directly integrated as follows. Let
1
u 1
fx
f
 du  dx
d
d
such that
u1 / 2 du 
2( Po  Patm )
f
d

dt
Integrate from time t  0 when x  0 , to time t when x  L
1

fL
d 1/ 2
u
du 
1
1
2 3/ 2
u
3
1
2 
fL 
1 

3 
d 
fL
d
3/ 2

f
d
f
d
2( Po  Patm )

 dt
0
2( Po  Patm )
 f
 1 
 d
t

t
2( Po  Patm )

t
which may be solved directly for t.
2