Homework Set #5:
Due date: 10/20/06, by 11:00 am.
Problem #1
:
A fluid with viscosity 18.3 cp and density 1.32 g/cm3 is flowing in a long horizontal tube of
radius 1.05 in (2.67 cm). For what pressure gradient will the flow become turbulent?
Answer:
The minimum value of
Re
4 /(
)
wD
π
μ
=
needed to produce turbulent flow in a long, smooth
tube is about 2100. Poiseuille’s law holds until this critical Re value, giving
()
4
0
critical
Re<Re
8
L
PP
R
w
L
πρ
−
=
Hence, the pressure gradient needed to initiate the laminar-turbulent transition is
2
5
44
3
88
4
Re
2100
1.1 10 Pa/km
4
critical
dp
w
D
dz
R
R
R
μμ
πρπ
ρ
==
≈
≈
Problem #2:
Water is flowing through a long, straight, level run of smooth 6.00 in inner diameter pipe, at
temperature 68
0
F. The pressure gradient along the length of the pipe is 1 psi/mile.
(a)
Determine the wall shear stress
τ
0
in Pa?
(b)
Assume the flow to be turbulent and determine the radial distance from the pipe wall at
which
, using Fig. 5.5-3 in the textbook (second
edition), also available in the lecture note. Pay attention at the definition of dimensionless
variables.
,max
0.0
0.2
uu
=
, 0.1,
, 0.4, 0.7, 0.85, 1.0
/
zz
(c)
Plot the complete velocity profile,
versus r/R.
,max
/
(d)
Is the assumption of turbulent flow justified?
Answer:
(a) Wall shear stress
τ
0
( )
0
0
0.1633 Pa
2
L
pp
R
L
τ
−
(b) The radial distances from the pipe wall at which
,max
0.0, 0.1, 0.2, 0.4, 0.7, 0.85, 1.0
/
=
Physical properties of water at temperature 68
0
F
ρ
= 0.9992 10
3
kg/m
3
,
μ
= 1.0019 10
-3
Pa-s,
ν
=1.0037 10
-6
m
2
/s