Problem 6.8
When water at 20C is in steady turbulent flow through an 8-cm-diameter pipe, the wall shear
stress is 72 Pa. What is the axial pressure gradient (
p/
x) if the pipe is (a) horizontal; and
(b) vertical with the flow up?
Solution 6.8
Equation (6.9b) applies in both cases, noting that
w is negative:
Problem 6.9
A light liquid (
≈ 950 kg/m3) flows at an average velocity of 10 m/s through a horizontal
smooth tube of diameter 5 cm. The fluid pressure is measured at 1-m intervals along the pipe, as
follows:
x, m:
0
1
2
3
4
5
6
p, kPa:
304
273
255
240
226
213
200
Estimate (a) the total head loss, in meters; (b) the wall shear stress in the fully developed section
of the pipe; and (c) the overall friction factor.
Solution 6.9
As sketched in Fig. 6.6 of the text, the pressure drops fast in the entrance region (31 kPa in the
first meter) and levels off to a linear decrease in the “fully developed” region (13 kPa/m for this
Problem 6.10
Water at 20C flows through an inclined 8-cm-diameter pipe. At sections A and B, pA = 186 kPa,
VA = 3.2 m/s, zA = 24.5 m, while pB = 260 kPa, VB = 3.2 m/s, and zB = 9.1 m. Which way is the
flow going? What is the head loss in meters?
Solution 6.10
Guess that the flow is from A to B and write the steady flow energy equation:
Problem 6.11
Water at 20C flows upward at 4 m/s in a 6-cm-diameter pipe. The pipe length between points
1 and 2 is 5 m, and point 2 is 3 m higher. A mercury manometer, connected between 1 and 2, has
a reading h = 135 mm, with p1 higher.
(a) What is the pressure change (p1 − p2)?
(b) What is the head loss, in meters?
(c) Is the manometer reading proportional to head loss? Explain. (d) What is the friction factor of
the flow?
Solution 6.11
A sketch of this situation is shown at right. By moving through the manometer, we obtain the
pressure change between points 1 and 2, which we compare with Eq. (6.9b):
Problem 6.12
A 5-mm-diameter capillary tube is used as a viscometer for oils. When the flow rate is
0.071 m3h, the measured pressure drop per unit length is 375 kPam. Estimate the viscosity of
the fluid. Is the flow laminar? Can you also estimate the density of the fluid?
Neglect minor losses.
Solution 6.12
NOTE: IN PROBLEMS 6.12 TO 6.99, MINOR LOSSES ARE NEGLECTED.
Assume laminar flow and use the pressure drop formula (6.12):
Problem 6.13
A soda straw is 20 cm long and 2 mm in diameter. It delivers cold cola, approximated as water at
10C, at a rate of 3 cm3s. (a) What is the head loss through the straw? What is the axial pressure
gradient
p
x if the flow is (b) vertically up or (c) horizontal? Can the human lung deliver this
much flow?
Neglect minor losses.
Solution 6.13
For water at 10C, take
= 1000 kgm3 and
= 1.307E−3 kgms. Check Re:
Problem 6.14
Water at 20C is to be siphoned through a tube 1 m long and 2 mm in diameter, as in Fig. P6.14.
Is there any height H for which the flow might not be laminar? What is the flow rate if H = 50 cm?
Neglect the tube curvature.
Neglect minor losses.
Solution 6.14
For water at 20C, take
= 998 kgm3 and
= 0.001 kgms. Write the steady flow energy
equation between points 1 and 2 above:
Problem 6.15
Professor Gordon Holloway and his students at the University of New Brunswick went to a fast-
food emporium and tried to drink chocolate shakes (
1200 kg/m3,
6 kg/ms) through fat
straws 8 mm in diameter and 30 cm long. (a) Verify that their human lungs, which can develop
approximately 3000 Pa of vacuum pressure, would be unable to drink the milkshake through the
vertical straw. (b) A student cut 15 cm from his straw and proceeded to drink happily. What rate
of milkshake flow was produced by this strategy?
Neglect minor losses.
Solution 6.15
(a) Assume the straw is barely inserted into the milkshake. Then the energy equation predicts
Problem 6.16
Fluid flows steadily, at volume rate Q, through a large pipe and then divides into two small
pipes, the larger of which has an inside diameter of 25 mm and carries three times the flow of the
smaller pipe. Both small pipes have the same length and pressure drop. If all flows are laminar,
estimate the diameter of the smaller pipe.
Neglect minor losses.
Solution 6.16
For laminar flow in a horizontal pipe, the volume flow is a simple formula,
Eq. (6.12):
Problem 6.17
A capillary viscometer measures the time required for a specified volume
of liquid to flow
through a small-bore glass tube, as in Fig. P6.17. This transit time is then correlated with fluid
viscosity. For the system shown, (a) derive an approximate formula for the time required,
assuming laminar flow with no entrance and exit losses. (b) If L = 12 cm, l = 2 cm,
= 8 cm3, and
the fluid is water at 20C, what capillary diameter D will result in a transit time t of 6 seconds?
Neglect minor losses.
Solution 6.17
(a) Assume no pressure drop and neglect velocity heads. The energy equation reduces to:
Problem 6.18
SAE 50W oil at 20ºC flows from one tank to another through a tube 160 cm long and 5 cm in
diameter. Estimate the flow rate in m3/hr if z1 = 2 m and z2 = 0.8 m.
Neglect minor losses.
Solution 6.18
From Table A.4 for SAE 50W oil, ρ = 902 kg/m3 and μ = 0.86 kg/m∙s. Write the energy
equation surface 1 and surface 2 to find the tube head loss, with minor losses neglected:
Problem 6.19
An oil (SG = 0.9) issues from the pipe in Fig. P6.19 at Q = 35 ft3/h. What is the kinematic
viscosity of the oil in ft3/s? Is the flow laminar?
Neglect minor losses.
Solution 6.19
Apply steady-flow energy:
Problem 6.20
The oil tanks in Tinyland are only 160 cm high, and they discharge to the Tinyland oil truck
through a smooth tube 4 mm in diameter and 55 cm long. The tube exit is open to the
atmosphere and 145 cm below the tank surface. The fluid is medium fuel oil,
= 850 kg/m3 and
= 0.11 kg/m-s. Estimate the oil flow rate in cm3/h.
Neglect minor losses.
Solution 6.20
The steady flow energy equation, with 1 at the tank surface and 2 the exit, gives
Problem 6.21
In Tinyland, houses are less than a foot high! The rainfall is laminar! The drainpipe in Fig. P6.21
is only 2 mm in diameter. (a) When the gutter is full, what is the rate of draining? (b) The gutter
is designed for a sudden rainstorm of up to 5 mm per hour. For this condition, what is the
maximum roof area that can be drained successfully? (c) What is Red?
Neglect minor losses.
Solution 6.21
If the velocity at the gutter surface is neglected, the energy equation reduces to
Problem 6.22
A steady push on the piston in Fig. P6.22 causes a flow rate Q = 0.15 cm3/s through the needle.
The fluid has
= 900 kg/m3 and
= 0.002 kg/(ms). What force F is required to maintain the
flow?
Neglect minor losses.
Solution 6.22
Determine the velocity of exit from the needle and then apply the steady-flow energy equation:
12
Q 0.15 306 cm/s
A( /4)(0.025)
V
= = =
Problem 6.23
SAE 10 oil at 20C flows in a vertical pipe of diameter 2.5 cm. It is found that the pressure is
constant throughout the fluid. What is the oil flow rate in m3/h? Is the flow up or down?
Neglect minor losses.
Solution 6.23
For SAE 10 oil, take
= 870 kg/m3 and
= 0.104 kg/ms. Write the energy equation between
point 1 upstream and point 2 downstream:
Problem 6.24
Two tanks of water at 20C are connected by a capillary tube 4 mm in diameter and 3.5 m long.
The surface of tank 1 is 30 cm higher than the surface of tank 2. (a) Estimate the flow rate in
m3/h. Is the flow laminar? (b) For what tube diameter will Red be 500?
Neglect minor losses.
Solution 6.24
For water, take
= 998 kg/m3 and
= 0.001 kg/ms. (a) Both tank surfaces are at atmospheric
pressure and have negligible velocity. The energy equation, when neglecting minor losses,
reduces to:
Problem 6.25
For the configuration shown in Fig. P6.25, the fluid is ethyl alcohol at 20C, and the tanks are
very wide. Find the flow rate that occurs, in m3/h. Is the flow laminar?
Neglect minor losses.
Solution 6.25
For ethanol, take
= 789 kg/m3 and
= 0.0012 kg/ms. Write the energy equation from upper
free surface (1) to lower free surface (2):
Problem 6.26
Two oil tanks are connected by two 9-m-long pipes, as in Fig. P6.26. Pipe 1 is 5 cm in diameter
and is 6 m higher than pipe 2. It is found that the flow rate in pipe 2 is twice as large as the flow
in pipe 1. (a) What is the diameter of pipe 2? (b) Are both pipe flows laminar? (c) What is the
flow rate in pipe 2 (m3/s)?
Neglect minor losses.
Solution 6.26
(a) If we know the flows are laminar, and (L,
,
) are constant, then Q D4:
Problem 6.27*
Let us attack Prob. 6.25 in symbolic fashion, using Fig. P6.27. All parameters are constant except
the upper tank depth Z(t). Find an expression for the flow rate Q(t) as a function of Z(t). Set up a
differential equation, and solve for the time t0 to drain the upper tank completely. Assume quasi-
steady laminar flow.
Neglect minor losses.
Problem 6.25
For the configuration shown in Fig. P6.25, the fluid is ethyl alcohol at 20C, and the tanks are
very wide. Find the flow rate that occurs, in m3/h. Is the flow laminar? Neglect minor losses.
Solution 6.27
The energy equation of Prob. 6.25, using symbols only, is combined with a control-volume mass
balance for the tank to give the basic differential equation for Z(t):
Problem 6.28
For straightening and smoothing an airflow in a 50-cm-diameter duct, the duct is packed with a
“honeycomb” of thin straws of length 30 cm and diameter 4 mm, as in Fig. P6.28. The inlet flow
is air at 110 kPa and 20C, moving at an average velocity of 6 m/s. Estimate the pressure drop
across the honeycomb.
Neglect minor losses.
Solution 6.28
For air at 20C, take
1.8E−5 kg/ms and
= 1.31 kg/m3. There would be approximately
Problem 6.29
SAE 30W oil at 20C flows through a straight pipe 25 m long, with diameter 4 cm. The average
velocity is 2 m/s. (a) Is the flow laminar? Calculate (b) the pressure drop; and (c) the power
required. (d) If the pipe diameter is doubled, for the same average velocity, by what percent does
the required power increase?
Neglect minor losses.
Solution 6.29
For SAE 30W oil at 20C, Table A.3,
= 891 kg/m3, and
= 0.29 kg/m-s. (a) We have enough
information to calculate the Reynolds number:
Problem 6.30
SAE 10 oil at 20C flows through the 4-cm-diameter vertical pipe of Fig. P6.30. For the mercury
manometer reading h = 42 cm shown, (a) calculate the volume flow rate in m3/h, and (b) state
the direction of flow.
Neglect minor losses.
Solution 6.30
For SAE 10 oil, take
= 870 kg/m3 and
= 0.104 kg/ms. The pressure at the lower point (1) is