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Exercise 1.18. Suppose the atmospheric temperature varies according to: T = 15 − 0.001z, where
T is in degrees Celsius and height z is in meters. Is this atmosphere stable?
Solution 1.18. Compute the temperature gradient:
Exercise 1.19. A hemispherical bowl with inner radius r containing a liquid with density
ρ
is
inverted on a smooth flat surface. Gravity with acceleration g acts downward. Determine the
weight W of the bowl necessary to prevent the liquid from escaping. Consider two cases:
a) the pressure around the rim of the bowl where it meets the plate is atmospheric, and
b) the pressure at the highest point of the bowl’s interior is atmospheric.
c) Investigate which case applies via a simple experiment. Completely fill an ordinary soup bowl
with water and concentrically cover it with an ordinary dinner plate. While holding the bowl and
plate together, quickly invert the water-bowl-plate combination, set it on the level surface at the
bottom of a kitchen sink, and let go. Does the water escape? If no water escapes after release,
hold onto the bowl only and try to lift the water-bowl-plate combination a few centimeters off
the bottom of the sink. Does the plate remain in contact with the bowl? Do your answers to the
first two parts of this problem help explain your observations?
Solution 1.19. a) When the pressure around the inverted bowl's rim is atmospheric, the surface
does not support the weight of the liquid and any increase in elevation above the rim must
θ
=0
θ
=0
where p is the pressure inside the bowl and po is the pressure outside the bowl (atmospheric
pressure). Here
θ
is the polar angle (
θ
= 0 defines the positive z-axis). In this coordinate system z
= rcos
θ
, and the pressure inside and outside the bowl must match when z = 0, so p – po = –
ρ
gz =
Exercise 1.20. Consider the case of a pure gas planet where the hydrostatic law is:
dp dz =−
ρ
(z)Gm(z)z2
, where G is the gravitational constant and
m(z)=4
π ρ
(
ζ
)
ζ
2
o
z
∫d
ζ
is the
planetary mass up to distance z from the center of the planet. If the planetary gas is perfect with
gas constant R, determine
ρ
(z) and p(z) if this atmosphere is isothermal at temperature T. Are
these vertical profiles of
ρ
and p valid as z increases without bound?
Solution 1.20. Start with the given relationship for m(z), differentiate it with respect to z, and use
the perfect gas law, p =
ρ
RT to replace the
ρ
with p.
Matching exponents of z across the last equality produces: n – 4 = 2n – 5, and this requires n = 1.
For this value of n, the remainder of the equation is:
Exercise 1.21. Consider a gas atmosphere with pressure distribution
p(z)=p01−(2 /
π
)tan−1(z/H)
( )
where z is the vertical coordinate and H is a constant length scale.
a) Determine the vertical profile of
ρ
from (1.14)
b) Determine N2 from (1.35) as function of vertical distance, z.
c) Near the ground where z << H, this atmosphere is unstable, but it is stable at greater heights
where z >> H. Specify the value of z/H above which this atmosphere is stable.
Solution 1.21. a) Start from (1.14):
b) First determine d
ρ
/dz:
c) In the limit as
z H →0
, the terms inside the big parentheses go to –1/
γπ
, which is negative,
so N2 is negative, and this indicates an unstable atmosphere. In the limit as
z H → ∞
, both terms
inside the big parentheses become large, but the first one is larger so N2 is positive, and this
Exercise 1.22. Consider a heat-insulated enclosure that is separated into two compartments of
volumes V1 and V2, containing perfect gases with pressures and temperatures of p1 and p2, and T1
and T2, respectively. The compartments are separated by an impermeable membrane that
conducts heat (but not mass). Calculate the final steady-state temperature assuming each gas has
constant specific heats.
Solution 1.22. Since no work is done and no heat is transferred out of the enclosure, the final
Exercise 1.23. Consider the initial state of an enclosure with two compartments as described in
Exercise 1.22. At t = 0, the membrane is broken and the gases are mixed. Calculate the final
temperature.
Solution 1.23. No heat is transferred out of the enclosure and the work done by either gas is
delivered to the other so the total energy is unchanged. First consider the energy of either gas at
temperature T, and pressure P in a container of volume V. The energy E of this gas will be:
However, the perfect gas law implies: p1f(V1+V2) = n1RuTf, and p2f(V1+V2) = n2RuTf where n1 and
n2 are the mole numbers of gases "1" and "2", and Ru is the universal gas constant. The mole
numbers are obtained from:
Exercise 1.24. A heavy piston of weight W is dropped onto a thermally insulated cylinder of
cross-sectional area A containing a perfect gas of constant specific heats, and initially having the
external pressure p1, temperature T1, and volume V1. After some oscillations, the piston reaches
an equilibrium position L meters below the equilibrium position of a weightless piston. Find L. Is
there an entropy increase?
Solution 1.24. From the first law of thermodynamics, with Q = 0, ΔE = Work = (W + p1A)L. For
a total mass m of a perfect gas with constant specific heats, E = mcvT so ΔE = E2 – E1 = mcv(T2 –
The initial length Lo of the column of gas is V1/A, so this final answer can be written:
L
Lo
=W p1A
γ
1+W p1A
( )
.
Exercise 1.25. Starting from 295 K and atmospheric pressure, what is the final pressure of an
isentropic compression of air that raises the temperature 1, 10, and 100 K.
Solution 1.25. Start from the first equation of (1.32), and solve for the pressure:
Exercise 1.26. Compute the speed of sound in air at –40°C (very cold winter temperature), at
+45°C (very hot summer temperature), at 400°C (automobile exhaust temperature), and 2000°C
(nominal hydrocarbon adiabatic flame temperature)
Solution 1.26. The speed of sound c of a perfect gas is given by (1.33):
Exercise 1.27. The oscillation frequency Ω of a simple pendulum depends on the acceleration of
gravity g, and the length L of the pendulum.
a) Using dimensional analysis, determine single dimensionless group involving Ω, g and L.
b) Perform an experiment to see if the dimensionless group is constant. Using a piece of string
slightly longer than 2 m and any small heavy object, attach the object to one end of the piece of
string with tape or a knot. Mark distances of 0.25, 0.5, 1.0 and 2.0 m on the string from the
center of gravity of the object. Hold the string at the marked locations, stand in front of a clock
with a second hand or second readout, and count the number (N) of pendulum oscillations in 20
seconds to determine Ω = N/(20 s) in Hz. Evaluate the dimensionless group for these four
lengths.
c) Based only on the results of parts a) and b), what pendulum frequency do you predict when L
= 1.0 m but g is 16.6 m/s? How confident should you be of this prediction?
Solution 1.27. a) Construct the parameter & units matrix using Ω, g, and L.
b) The following table lists the results of the simple pendulum experiments:
L (m) N Ω (s–1) Ω(L/g)1/2
0.25 20 1.00 0.160
0.50 14.5 0.725 0.164
Exercise 1.28. The spectrum of wind waves, S(
ω
), on the surface of the deep sea may depend on
the wave frequency
ω
, gravity g, the wind speed U, and the fetch distance F (the distance from
the upwind shore over which the wind blows with constant velocity).
a) Using dimensional analysis, determine how S(
ω
) must depend on the other parameters.
b) It is observed that the mean-square wave amplitude,
η
2=S(
ω
)d
ω
0
∞
∫
, is proportional to F.
Use this fact to revise the result of part a).
c) How must
η
2 depend on U and g?
Solution 1.28. a) Construct the parameter & units matrix using S,
ω
, g, U, and F. The units of S
are (length2)(time).
S
ω
g U F
where the final equality follows from changing the integration variable to
γ
=
ω
F/U. From this
equation, the only way that
η
2 can be proportional to F is for the undetermined function be
Exercise 1.29. One military technology for clearing a path through a minefield is to deploy a
powerfully exploding cable across the minefield that, when detonated, creates a large trench
through which soldiers and vehicles may safely travel. If the expanding cylindrical blast wave
from such a line-explosive has radius R at time t after detonation, use dimensional analysis to
determine how R and the blast wave speed dR/dt must depend on t, r = air density, and E´ =
energy released per unit length of exploding cable.
Solution 1.29. Follow example 1.10 and construct the parameter & units matrix using R, t, r, and
E´. The units of E´ are energy/length. Construct the parameter & units matrix.
Exercise 1.30. One of the triumphs of classical thermodynamics for a simple compressible
substance was the identification of entropy s as a state variable along with pressure p, density
ρ
,
and temperature T. Interestingly, this identification foreshadowed the existence of quantum
physics because of the requirement that it must be possible to state all physically meaningful
laws in dimensionless form. To see this foreshadowing, consider an entropic equation of state for
a system of N elements each having mass m.
a) Determine which thermodynamic variables amongst s, p,
ρ
, and T can be made dimensionless
using N, m, and the non-quantum mechanical physical constants kB = Boltzmann’s constant and c
= speed of light. What do these results imply about an entropic equation of state in any of the
following forms: s = s(p,
ρ
), s = s(
ρ
,T), or s = s(T,p)?
b) Repeat part a) including
= Planck’s constant (the fundamental constant of quantum
physics). Can an entropic equation of state be stated in dimensionless form without
?
Solution 1.30. a) Select each thermodynamic variable in turn, and see if it can be made
dimensionless using m, kB, and c. Here N is dimensionless and it is the only extensive variable so
it need not be considered when seeking the dimensionless forms of the intensive (per unit mass)
thermodynamic variables. The dimensions of the remaining parameters and constants are:
s p
ρ
T m kB c
M 0 1 1 0 1 1 0
Exercise 1.31. The natural variables of the system enthalpy H are the system entropy S and the
pressure p, which leads to an equation of state in the form: H = H(S, p, N), where N is the
number of system elements.
a) After creating ratios of extensive variables, use exponent algebra to independently render H/N,
S/N, and p dimensionless using m = the mass of a system element, and the fundamental constants
kB = Boltzmann’s constant,
= Planck’s constant, and c = speed of light.
b) Simplify the result of part a) for non-relativistic elements by eliminating c.
c) Based on the property relationship (1.24), determine the specific volume =
υ
=1
ρ
=∂h∂p
( )
s
from the result of part b).
d) Use the result of part c) and (1.25) to show that the sound speed in this case is
5p3
ρ
, and
compare this result to that for a monotonic perfect gas.
Solution 1.31. a) Here H and S are extensive variables so they must be proportional to N. So
parameter & units matrix can be constructed as:
H/N S/N p m kB
c
M 1 1 1 0 1 1 0
b) To eliminate c, extract the second dimensionless group from the argument of Θ, raise it to the
–2/5 power, and multiply on the left side with the altered group involving H:
where H/Nm = h, and s = S/Nm has been used for the second-to-last equality.
c) Use the final equality of part b), and perform the indicated differentiation:
d) Use the final result of part c) get an equation for the pressure,
p=2
m8 3
2
5Θs
mkB
"
#
$%
&
'
(
)
*+
,
-
5 3
ρ
5 3
, and
Exercise 1.32. A gas of noninteracting particles of mass m at temperature T has density
ρ
, and
internal energy per unit volume
ε
.
a) Using dimensional analysis, determine how
ε
must depend on
ρ
, T, and m. In your formulation
use kB = Boltzmann’s constant,
= Plank’s constant, and c = speed of light to include possible
quantum and relativistic effects.
b) Consider the limit of slow-moving particles without quantum effects by requiring c and
to
drop out of your dimensionless formulation. How does
ε
depend on
ρ
and T? What type of gas
follows this thermodynamic law?
c) Consider the limit of massless particles (i.e., photons) by requiring m and
ρ
to drop out of
your dimensionless formulation of part a). How does
ε
depend on T in this case? What is the
name of this radiation law?
Solution 1.32. a) Construct the parameter & units matrix noting that kB and T must go together
since they are the only parameters that involve temperature units.
ε
ρ
kBT m
c
b) Dropping
means dropping Π3. Eliminating c means combining Π1 and Π2 to create a new
Exercise 1.33. A compression wave in a long gas-filled constant-area duct propagates to the left
at speed U. To the left of the wave, the gas is quiescent with uniform density
ρ
1 and uniform
pressure p1. To the right of the wave, the gas has uniform density
ρ
2 (>
ρ
1) and uniform pressure
is p2 (> p1). Ignore the effects of viscosity in this problem. Formulate a dimensionless scaling law
for U in terms of the pressures and densities.
Solution 1.33. a) Construct the parameter & units matrix:
U
ρ
1
ρ
2 p1 p2
M 0 1 1 1 1
U!
p1,
ρ
1!
u1 = 0!
p2,
ρ
2!
Exercise 1.34. Many flying and swimming animals – as well as human-engineered vehicles –
rely on some type of repetitive motion for propulsion through air or water. For this problem,
assume the average travel speed U, depends on the repetition frequency f, the characteristic
length scale of the animal or vehicle L, the acceleration of gravity g, the density of the animal or
vehicle
ρ
o, the density of the fluid
ρ
, and the viscosity of the fluid
µ
.
a) Formulate a dimensionless scaling law for U involving all the other parameters.
b) Simplify your answer for a) for turbulent flow where
µ
is no longer a parameter.
c) Fish and animals that swim at or near a water surface generate waves that move and propagate
because of gravity, so g clearly plays a role in determining U. However, if fluctuations in the
propulsive thrust are small, then f may not be important. Thus, eliminate f from your answer for
b) while retaining L, and determine how U depends on L. Are successful competitive human
swimmers likely to be shorter or taller than the average person?
d) When the propulsive fluctuations of a surface swimmer are large, the characteristic length
scale may be U/f instead of L. Therefore, drop L from your answer for b). In this case, will higher
speeds be achieved at lower or higher frequencies?
e) While traveling submerged, fish, marine mammals, and submarines are usually neutrally
buoyant (
ρ
o ≈
ρ
) or very nearly so. Thus, simplify your answer for b) so that g drops out. For this
situation, how does the speed U depend on the repetition frequency f?
f) Although fully submerged, aircraft and birds are far from neutrally buoyant in air, so their
travel speed is predominately set by balancing lift and weight. Ignoring frequency and viscosity,
use the remaining parameters to construct dimensionally accurate surrogates for lift and weight
to determine how U depends on
ρ
o/
ρ
, L, and g.
Solution 1.34. a) Construct the parameter & units matrix
U f L g
ρ
o
ρ
µ
M 0 0 0 0 1 1 1
c) When f is no longer a parameter, then
U=gL ⋅
ψ
3
ρ
o
ρ
( )
, so that U is proportional to
L
.
This scaling suggests that taller swimmers have an advantage over shorter ones. [Human
swimmers best approach the necessary conditions for this part of this problem while doing
freestyle (crawl) or backstroke where the arms (and legs) are used for propulsion in an
alternating (instead of simultaneous) fashion. Interestingly, this length advantage also applies to
ships and sailboats. Aircraft carriers are the longest and fastest (non-planing) ships in any Navy,
and historically the longer boat typically won the America’s Cup races under the 12-meter rule.
Exercise 1.35. The acoustic power W generated by a large industrial blower depends on its
volume flow rate Q, the pressure rise ΔP it works against, the air density
ρ
, and the speed of
sound c. If hired as an acoustic consultant to quiet this blower by changing its operating
conditions, what is your first suggestion?
Solution 1.35. The boundary condition and material parameters are: Q,
ρ
, ΔP, and c. The
solution parameter is W. Create the parameter matrix:
W Q ΔP
ρ
c
––––––––––––––––––––––––––––
Mass: 1 0 1 1 0
