Problem 5.69
A simple flow-measurement device for streams and channels is a notch, of angle
, cut into the
side of a dam, as shown in Fig. P5.69. The volume flow Q depends only on
, the acceleration of
gravity g, and the height
of the upstream water surface above the notch vertex. Tests of a model
notch, of angle
= 55, yield the following flow rate data:
, cm:
10
20
30
40
Q, m3/h:
8
47
126
263
(a) Find a dimensionless correlation for the data. (b) Use the model data to predict the flow rate
of a prototype notch, also of angle
= 55, when the upstream height
is 3.2 m.
Solution 5.69
(a) The appropriate functional relation is Q = fcn(
, g,
) and its dimensionless form is
Problem 5.70
A diamond-shaped body, of characteristic length 9 in, has the following measured drag forces when
placed in a wind tunnel at sea-level standard conditions:
V, ft/s: 30 38 48 56 61
F, lbf 1.25 1.95 3.02 4.05 4.81
Use these data to predict the drag force of a similar 15-in diamond placed at similar orientation in
20C water flowing at 2.2 m/s.
Solution 5.70
For sea-level air, take
= 0.00237 slug/ft3,
= 3.72E−7 slug/ft·s. For water at 20C, take
Problem 5.71
The pressure drop in a venturi meter (Fig. P3.128) varies only with the fluid density, pipe approach
velocity, and diameter ratio of the meter. A model venturi meter tested in water at 20C shows a 5–
kPa drop when the approach velocity is 4 m/s. A geometrically similar prototype meter is used to
measure gasoline at 20C and a flow rate of 9 m3/min. If the prototype pressure gage is most
accurate at 15 kPa, what should the upstream pipe diameter be?
Solution 5.71
Given p = fcn(
, V, d/D), then by dimensional analysis p/(
V2) = fcn(d/D). For water at
Problem 5.72
A one-twelfth-scale model of a large commercial aircraft is tested in a wind tunnel at 20C and
1 atm. The model chord length is 27 cm, and its wing area is 0.63 m2. Test results for the drag
of the model are as follows:
V, mi/h
50
75
100
125
Drag, N
15
32
53
80
In the spirit of Fig. 5.8, use this data to estimate the drag of the full-scale aircraft when flying at
550 mi/h, for the same angle of attack, at 32,800 ft standard altitude.
Solution 5.72
Compute the model drag coefficients and Reynolds numbers, plot them, and extrapolate in the
spirit of Fig. 5.8 of the text. For the first point, 50 mi/h = 22.35 m/s. At 20C and 1 atm,
Problem 5.73
The power P generated by a certain windmill design depends upon its diameter D, the air density
,
the wind velocity V, the rotation rate , and the number of blades n. (a) Write this relationship in
dimensionless form. A model windmill, of diameter 50 cm, develops 2.7 kW at sea level when
V = 40 m/s and when rotating at 4800 rev/min. (b) What power will be developed by a
geometrically and dynamically similar prototype, of diameter 5 m, in winds of 12 m/s at 2000 m
standard altitude? (c) What is the appropriate rotation rate of the prototype?
Solution 5.73
(a) For the function P = fcn(D,
, V, , n) the appropriate dimensions are {P} = {ML2T–3}, {D}
Problem 5.74
A one-tenth-scale model of a supersonic wing tested at 700 m/s in air at 20C and 1 atm shows a
pitching moment of 0.25 kN·m. If Reynolds-number effects are negligible, what will the pitching
moment of the prototype wing be flying at the same Mach number at 8-km standard altitude?
Solution 5.74
If Reynolds number is unimportant, then the dimensionless moment coefficient M(
V2L3)
Problem 5.75
According to the web site USGS Daily Water Data for the Nation, the mean flow rate in the New
River near Hinton, WV is 10,100 ft3/s. If the hydraulic model in Fig. 5.9 is to match this
condition with Froude number scaling, what is the proper model flow rate?
Solution 5.75
For Froude scaling, the volume flow rate is a blend of velocity and length terms:
Problem 5.76*
A 2-ft-long model of a ship is tested in a freshwater tow tank. The measured drag may be split into
“friction” drag (Reynolds scaling) and “wave” drag (Froude scaling). The model data are as follows:
Tow speed, ft/s:
0.8
1.6
2.4
3.2
4.0
4.8
Friction drag, lbf:
0.016
0.057
0.122
0.208
0.315
0.441
Wave drag, lbf:
0.002
0.021
0.083
0.253
0.509
0.697
The prototype ship is 150 ft long. Estimate its total drag when cruising at 15 kn in seawater at 20C.
Solution 5.76
For fresh water at 20C, take
= 1.94 slug/ft3,
= 2.09E−5 slug/fts. Then evaluate the
Problem 5.77
A dam 75 ft wide, with a nominal flow rate of 260 ft3/s, is to be studied with a scale model 3 ft
wide, using Froude scaling. (a) What is the expected flow rate for the model? (b) What is the
danger of only using Froude scaling for this test? (c) Derive a formula for a force on the model
as compared to a force on the prototype.
Solution 5.77
The length scale ratio is
/ 75 / 3 25.
pm
L L ft ft==
(a) The velocity scales as the Froude number,
and flow rate scales as velocity times area:
Problem 5.78
A prototype spillway has a characteristic velocity of 3 m/s and a characteristic length of 10 m. A
small model is constructed by using Froude scaling. What is the minimum scale ratio of the model
which will ensure that its minimum Weber number is 100? Both flows use water at 20C.
Solution 5.78
For water at 20C,
= 998 kg/m3 and Y = 0.073 N/m, for both model and prototype.
Problem 5.79
An East Coast estuary has a tidal period of 12.42 h (the semidiurnal lunar tide) and tidal currents of
approximately 80 cm/s. If a one-five-hundredth-scale model is constructed with tides driven by a
pump and storage apparatus, what should the period of the model tides be and what model current
speeds are expected?
Solution 5.79
Problem 5.80
A prototype ship is 35 m long and designed to cruise at 11 m/s (about 21 kn). Its drag is to be
simulated by a 1-m-long model pulled in a tow tank. For Froude scaling find (a) the tow speed,
(b) the ratio of prototype to model drag, and (c) the ratio of prototype to model power.
Solution 5.80
Problem 5.81
An airplane, of overall length 55 ft, is designed to fly at 680 m/s at 8000-m standard altitude. A one-
thirtieth-scale model is to be tested in a pressurized helium wind tunnel at 20C. What is the
appropriate tunnel pressure in atm? Even at this (high) pressure, exact dynamic similarity is not
achieved. Why?
Solution 5.81
For air at 8000–m standard altitude (Table A–6), take
= 0.525 kg/m3,
Problem 5.82
A one-fiftieth scale model of a military airplane is tested at 1020 m/s in a wind tunnel at sea-
level conditions. The model wing area is 180 cm2. The angle of attack is 3°. If the measured
model lift is 860 N, what is the prototype lift, using Mach number scaling, when it flies at
10,000 m standard altitude under dynamically similar conditions? Note: Be careful with the area
scaling.
Solution 5.82
At sea-level,
= 1.2255 kg/m3 and T = 288 K. Compute the speed of sound and Mach number
for the model:
Problem 5.83
A one-fortieth–scale model of a ship’s propeller is tested in a tow tank at 1200 r/min and exhibits a
power output of 1.4 ft·lbf/s. According to Froude scaling laws, what should the revolutions per
minute and horsepower output of the prototype propeller be under dynamically similar conditions?
Solution 5.83
Given
= 1/40, use Froude scaling laws:
Problem 5.84
A prototype ocean-platform piling is expected to encounter currents of 150 cm/s and waves of 12-s
period and 3-m height. If a one-fifteenth-scale model is tested in a wave channel, what current
speed, wave period, and wave height should be encountered by the model?
Solution 5.84
Given
= 1/15, apply straight Froude scaling (Fig. 5.6b) to these results:
Problem 5.85*
As shown in Ex. 5.3, pump performance data can be non-dimensionalized. Problem P5.61 gave
typical dimensionless data for centrifugal pump “head”, H = p/
g, as follows:
where Q is the volume flow rate, n the rotation rate in r/s, and D the impeller diameter. This type
of correlation allows one to compute H when (
, Q, D) are known. (a) Show how to rearrange
these pi groups so that one can size the pump, that is, compute D directly when (Q, H, n) are
known. (b) Make a crude but effective plot of your new function. (c) Apply part (b) to the
following example: When H = 37 m, Q = 0.14 m3/s, and n = 35 r/s. Find the pump diameter for
this condition.
Solution 5.85*
(a) We have to eliminate D from one or the other of the two parameters. The writer chose to
remove D from the left side. The new parameter will be
2
2 2 3
6.0 120()
g H Q
n D nD
−
Problem 5.86
Solve Prob. P5.49 for glycerin at 20°C, using the modified sphere-drag plot of Fig. 5.11.
Solution 5.86
Recall this problem is identical to Prob. 5.85 above except that the fluid is glycerin, with
Problem 5.87
In Prob. P5.61 it would be difficult to solve for because it appears in all three of the
dimensionless pump coefficients. Suppose that, in Prob. 5.61, is unknown but D = 12 cm and
Q = 25 m3/h. The fluid is gasoline at 20°C. Rescale the coefficients, using the data of Prob.
P5.61, to make a plot of dimensionless power versus dimensionless rotation speed. Enter this
plot to find the maximum rotation speed for which the power will not exceed 300 W.
Solution 5.87
For gasoline,
= 680 kg/m3 and
= 2.92E−4 kg/m·s. We can eliminate from the power
coefficient for a new type of coefficient:
Problem 5.88
Modify Prob. 5.61 as follows: Let = 32 r/s and Q = 24 m3/h for a geometrically similar pump.
What is the maximum diameter if the power is not to exceed 340 W? Solve this problem by
rescaling the data of Fig. P5.61 to make a plot of dimensionless power versus dimensionless
diameter. Enter this plot directly to find the desired diameter.
Solution 5.88
We can eliminate D from the power coefficient for an alternate coefficient:
Problem 5.89
Wall friction
w, for turbulent flow at velocity U in a pipe of diameter D, was correlated, in 1911,
with a dimensionless correlation by Ludwig Prandtl’s student H. Blasius:
2 1/ 4
0.632
( / )
w
U UD
Suppose that (
U,
,
w) were all known and it was desired to find the unknown velocity U.
Rearrange and rewrite the formula so that U can be immediately calculated.
Solution 5.89
The easiest path the writer can see is to get rid of U2 on the left hand side by multiplying both
sides by the Reynolds number squared:
Problem 5.90
Knowing that p is proportional to L, rescale the data of Example 5.10 to plot dimensionless p
versus dimensionless viscosity. Use this plot to find the viscosity required in the first row of data in
Example 5.10 if the pressure drop is increased to 10 kPa for the same flow rate, length, and density.
Solution 5.90
Recall that Example 5.10, where p/L = fcn(
, V,
, D), led to the correlation
Problem 5.91*
The traditional “Moody–type” pipe friction correlation in Chap. 6 is of the form
where D is the pipe diameter, L the pipe length, and
the wall roughness.
Note that pipe average velocity V is used on both sides. This form is meant to find p when V is
known.
(a) Suppose that p is known and we wish to find V. Rearrange the above function so that V is
isolated on the left-hand side. Use the following data, for
/D = 0.005, to make a plot of your
new function, with your velocity parameter as the ordinate of the plot.
f
0.0356
0.0316
0.0308
0.0305
0.0304
VD/
15,000
75,000
250,000
900,000
3,330,000
(b) Use your plot to determine V, in m/s, for the following pipe flow: D = 5 cm,
= 0.025 cm,
L = 10 m, for water flow at 20C and 1 atm. The pressure drop p is 110 kPa.
Solution 5.91
We can eliminate V from the left side by multiplying by Re2. Then rearrange:
2
2( , )
p D VD
f fcn
V L D
==
8.01E6
1.78E8
1.92E9
2.47E10
3.31E11
