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Therefore, to achieve dimensional homogeneity, we somehow must combine bending
moment, whose dimensions are {ML2T–2}, with area moment of inertia, {I} = {L4}, and end up
with {ML–2T–2}. Well, it is clear that {I} contains neither mass {M} nor time {T} dimensions,
but the bending moment contains both mass and time and in exactly the combination we need,
{MT–2}. Thus it must be that
is proportional to M also. Now we have reduced the problem to:
2
22
M ML
yM fcn(I), or {L} {fcn(I)}, or: {fcn(I)}
LT T
= = =
4
{L }
−
Problem 1.9
A hemispherical container, 26 inches in diameter, is filled with a liquid at 20C and weighed.
The liquid weight is found to be 1617 ounces. (a) What is the density of the fluid, in kg/m3?
(b) What fluid might this be? Assume standard gravity, g = 9.807 m/s2.
Solution 1.9
First find the volume of the liquid in m3:
Hemisphere volume = 1
2(𝜋
6) 𝐷3 = 1
2(𝜋
6)(26 𝑖𝑛)3 = 4601 𝑖𝑛3÷(61024𝑖𝑛3
𝑚3)
= 0.0754𝑚3
Problem 1.10
The Stokes-Oseen formula [33] for drag force F on a sphere of diameter D in a fluid stream low
velocity V, density
, and viscosity
is:
22
9
F 3 DV V D
16
=+
Is this formula dimensionally homogeneous?
Solution 1.10
Write this formula in dimensional form, using Table 1-2:
Problem 1.11
In English Engineering units, the specific heat cp of air at room temperature is approximately
0.24 Btu/(lbm-F). When working with kinetic energy relations, it is more appropriate to express
cp as a velocity-squared per absolute degree. Give the numerical value, in this form, of cp for air
in (a) SI units, and (b) BG units.
Solution 1.11
From Appendix C, Conversion Factors, 1 Btu = 1055.056 J (or N-m) = 778.17 ft-lbf, and 1 lbm
= 0.4536 kg = (1/32.174) slug. Thus the conversions are:
Problem 1.12
For low-speed (laminar) steady flow through a circular pipe, as shown in Fig. P1.12, the velocity
u varies with radius and takes the form
( )
22
o
p
u B r r
=−
where
is the fluid viscosity and p the pressure drop from entrance to exit. What are the
dimensions of the constant B?
Figure P1.12
Solution 1.12
Using Table 1-2, write this equation in dimensional form:
22
22
{ p} L {M/LT } L
{u} {B} {r }, or: {B?} {L } {B?} ,
{ } T {M/LT} T
= = =
Problem 1.13
The efficiency
of a pump is defined as the (dimensionless) ratio of the power developed by the
fl ow to the power required to drive the pump:
Qp
Input Power
=
where Q is the volume flow rate of flow and
p the pressure rise produced by the pump. Suppose
that a certain pump develops a pressure rise of 35 lbf/in2 when its flow rate is 40 L/s. If the input
power is 16 hp, what is the efficiency?
Solution 1.13
The student should perhaps verify that Qp has units of power, so that
is a dimensionless ratio.
Then convert everything to consistent units, for example, BG:
Problem 1.14
Figure P1.14 shows the flow of water over a dam. The volume flow Q is known to depend only
on crest width B , acceleration of gravity g , and upstream water height H above the dam crest. It
is further known that Q is proportional to B . What is the form of the only possible dimensionally
homogeneous relation for this flow rate?
Solution 1.14
So far we know that Q = B fcn(H,g). Write this in dimensional form:
Problem 1.15
The height H that fluid rises in a liquid barometer tube depends upon the liquid density ρ, the
barometric pressure p, and the acceleration of gravity g. (a) Arrange these four variables into a
single dimensionless group. (b) Can you deduce (or guess) the numerical value of your group?
Solution 1.15
This is a problem in dimensional analysis, covered in detail in Chapter 5. Use the symbols for
dimensions suggested with Eq. (1.2): M for mass, L for length, T for time, F for force,
Problem 1.16
Algebraic equations such as Bernoulli’s relation, Eq. (1) of Example 1.3 (see below), are
dimensionally consistent, but what about differential equations? Consider, for example, the
boundary-layer x -momentum equation, first derived by Ludwig Prandtl in 1904:
x
u u p
u v g
x y x y
+ = − + +
where τ is the boundary-layer shear stress and g x is the component of gravity in the x direction.
Is this equation dimensionally consistent? Can you draw a general conclusion?
2
0
1
2
p p V gZ
= + +
(1)
Solution 1.16
This equation, like all theoretical partial differential equations in mechanics, is dimensionally
homogeneous. Test each term in sequence:
Problem 1.17
The Hazen-Williams hydraulics formula for volume rate of flow Q through a pipe of diameter D
and length L is given by
0.54
2.63 p
Q 61.9D L
where Δp is the pressure drop required to drive the flow.
What are the dimensions of the constant 61.9? Can this equation be used with confidence for a
variety of liquids and gases?
Solution 1.17
Write out the dimensions of each side of the equation:
Problem 1.18*
For small particles at low velocities, the first term in Stokes-Oseen drag law, Prob. 1.10, is dominant,
hence
F ≈ KV, where K is a constant.
Solution 1.18*
Set up and solve the differential equation for forces in the x-direction:
Problem 1.19
In his study of the circular hydraulic jump formed by a faucet flowing into a sink, Watson [53]
proposes a parameter combining volume flow rate Q, density
and viscosity
of the fluid, and
depth h of the water in the sink. He claims that the grouping is dimensionless, with Q in the
numerator. Can you verify this?
Solution 1.19
Check the dimensions of these four variables, from Table 1.2:
Problem 1.20
Books on porous media and atomization claim that the viscosity
and surface tension
of a
fluid can be combined with a characteristic velocity U to form an important dimensionless
parameter. (a) Verify that this is so. (b) Evaluate this parameter for water at 20C and a velocity
of 3.5 cm/s. NOTE: You get extra credit if you know the name of this parameter.
Solution 1.20
Problem 1.21
Aeronautical engineers measure the pitching moment Mo of a wing and then write it in the
following form for use in other cases: 𝑀o= 𝛽 𝑉2 𝐴 𝐶 𝜌
where V is the wing velocity, A the wing area, C the wing chord length, and ρ the air density.
What are the dimensions of the coefficient β?
33
{ / }{ / }
{ } {1} dimensionless . Watson is correct.
{ / }{ }
= = =
QM L L T Ans
h M LT L
Solution 1.21
Write out the dimensions of each term in the formula:
Problem 1.22
The Ekman number, Ek, arises in geophysical fluid dynamics. It is a dimensionless parameter
combining seawater density
, a characteristic length L, seawater viscosity
, and the Coriolis
frequency
sin
, where is the rotation rate of the earth and
is the latitude angle. Determine
the correct form of Ek if the viscosity is in the numerator.
Solution 1.22
First list the dimensions of the various quantities:
Note that sin
is itself dimensionless, so the Coriolis frequency has the dimensions of .
Problem 1.23
During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to
estimate the energy released by an atomic bomb explosion. He assumed that the energy released, E, was
a function of blast wave radius R, air density
, and time t. Arrange these variables into a single
dimensionless group, which we may term the blast wave number.
Solution 1.23
These variables have the dimensions {E} = {ML2/T2}, {R} = {L}, {
} = {M/L3}, and
-3 -1 -1 -1
{ } {ML } ; { } {L} ; { } {ML T } ; { sin } {T }L
= = = =
Problem 1.24
Air, assumed to be an ideal gas with k = 1.40, flows isentropically through a nozzle. At section 1,
conditions are sea level standard (see Table A.6). At section 2, the temperature is –50C. Estimate
(a) the pressure, and (b) the density of the air at section 2.
Solution 1.24
From Table A.6, p1 = 101350 Pa, T1 = 288.16 K, and
1 = 1.2255 kg/m3. Convert to absolute
temperature, T2 = -50°C = 223.26 K. Then, for a perfect gas with constant k,
Problem 1.25
On a summer day in Narragansett, Rhode Island, the air temperature is 74ºF and the barometric
pressure is 14.5 lbf/in2. Estimate the air density in kg/m3.
Solution 1.25
This is a problem in handling awkward units. Even if we use the BG system, we have to convert.
But, since the problem calls for a metric result, better we should convert to SI units:
Problem 1.26
/( 1) 1.4/(1.4 1) 3.5
22
11
2
223.16
( ) (0.7744) 0.4087
288.16
Thus (0.4087)(101350 ) .( )
()
kk
pT
pT
p Pa Pa Ans a
−−
= = = =
==
41,400
When we in the United States say a car’s tire is filled “to 32 lb,” we mean that its internal
pressure is 32 lbf/in2 above the ambient atmosphere. If the tire is at sea level, has a volume of
3.0 ft 3 , and is at 75 F, estimate the total weight of air, in lbf, inside the tire.
Solution 1.26
Convert the temperature from 75F to 535R. Convert the pressure to psf:
2 2 2 2 2
p (32 lbf/in )(144 in /ft ) 2116 lbf/ft 4608 2116 6724 lbf/ft= + = +
Problem 1.27
For steam at a pressure of 45 atm, some values of temperature and specific volume are as
follows, from Ref. 23:
T, ºF
500
600
700
800
900
v, ft3/lbm
0.7014
0.8464
0.9653
1.074
1.177
Find an average value of the predicted gas constant R in m2/(s2∙K). Does this data reasonably
approximate an ideal gas? If not, explain.
Solution 1.27
If ideal, the calculated gas constant would, from Table A.4, be about 461 m2/(s2∙K). Try this for
the first value, at T = 500ºF = 960ºR. Change to SI units, using the inside front cover
conversions:
Problem 1.28
Wet atmospheric air at 100 percent relative humidity contains saturated water vapor and, by
Dalton’s law of partial pressures,
Solution 1.28
Change T from 40C to 313 K. Dalton’s law of partial pressures is
aw
tot air water a w
mm
p 1 atm p p R T R T
= = + = +
Problem 1.29
A compressed-air tank holds 5 ft3 of air at 120 lbf/in2 (gage). Estimate the energy in ft-lbf
required to compress this air from the atmosphere assuming an ideal isothermal process.
Solution 1.29
Integrate the work of compression, assuming an ideal gas:
Problem 1.30
Repeat Prob. 1.29 if the tank is filled with compressed water rather than air. Why is the result
thousands of times less than the result of 215,000 ftlbf in Prob. 1.29?
Solution 1.30
First evaluate the density change of water. At 1 atm,
o 1.94 slug/ft3.
At 120 psi(gage) = 134.7 psia, the density would rise slightly according to Eq. (1.22):
Problem 1.31
One cubic foot of argon gas at 10C and 1 atm is compressed isentropically to a new pressure of
600 kPa. (a) What will be its new pressure and temperature? (b) If allowed to cool, at this new
volume, back to 10C, what will be the final pressure?
Solution 1.31
This is an exercise in having students recall their thermodynamics. From Table A.4, for argon
gas, R = 208 m2/(s2-K) and k = 1.67. Note T1 = 283K. First compute the initial density:
Problem 1.32
A blimp is approximated by a prolate spheroid 90 m long and 30 m in diameter. Estimate the
weight of 20°C gas within the blimp for (a) helium at 1.1 atm; and (b) air at 1.0 atm. What might
the difference between these two values represent (see Chap. 2)?
Solution 1.32
Find a handbook. The volume of a prolate spheroid is, for our data,
2 2 3
22
LR (90 m)(15 m) 42412 m
33
= =
Problem 1.33
A tank contains 9 kg of CO2 at 20ºC and 2.0 MPa. Estimate the volume of the tank, in m3.
Solution 1.33
All we have to do is find the density. For CO2, from Table A.4, R = 189 m2/(s2∙K). Then
Problem 1.34
Consider steam at the following state near the saturation line: (p1, T1) = (1.31 MPa, 290°C).
Calculate and compare, for an ideal gas (Table A.4) and the steam tables (a) the density
1; and
(b) the density
2 if the steam expands isentropically to a new pressure of 414 kPa. Discuss your
results.
Solution 1.34
From Table A.4, for steam, k 1.33, and R 461 m2/(s2K). Convert T1 = 563 K. Then,
Problem 1.35
In Table A.4, most common gases (air, nitrogen, oxygen, hydrogen) have a specific heat ratio
k ≈ 1.40. Why do argon and helium have such high values? Why does NH3 have such a low
value? What is the lowest k for any gas that you know of?
Solution 1.35
In elementary kinetic theory of gases [21], k is related to the number of “degrees of freedom” of
the gas: k 1 + 2/N, where N is the number of different modes of translation, rotation, and
vibration possible for the gas molecule.
Problem 1.36
Experimental data [55] for the density of n-pentane liquid for high pressures, at 50ºC, are listed
as follows:
Pressure, kPa
100
10230
20700
34310
Density, kg/m3
586.3
604.1
617.8
632.8
(a) Fit this data to reasonably accurate values of B and n from Eq. (1.19).
(b) Evaluate ρ at 30 MPa.
Solution 1.36
Eq. (1.19) is p/po ≈ (B+1)(ρ/ρo)n – B. The first column is po = 100 kPa and ρo = 586.3 kg/m3.
(a) The writer found it easiest to guess n, say, n = 7 for water, and then solve for B from the data:
Problem 1.37
A near-ideal gas has a molecular weight of 44 and a specific heat of cv = 610 J/(kgK). What are
(a) its specific heat ratio, k, and (b) its speed of sound at 100°C?
Solution 1.37
Problem 1.38
In Fig. P1.7, if the fluid is glycerin at 20°C and the width between plates is 6 mm, what shear
stress (in Pa) is required to move the upper plate at 5.5 m/s? What is the Reynolds number if L is
taken to be the distance between plates?
Solution 1.38
a) For glycerin at 20°C, from Table 1.4,
1.5 N · s/m2. The shear stress is found from Eq. (1) of
Ex. 1.8:
Problem 1.39
