Problem 1.62
The hydrogen bubbles that produced the velocity profiles in Fig. 1.15 are quite small,
D ≈ 0.01 mm. If the hydrogen– water interface is comparable to air–water and the water
temperature is 30°C, estimate the excess pressure within the bubble.
Solution 1.62
At 30C the surface tension from Table A-1 is 0.0712 N/m. For a droplet or bubble with one
Problem 1.63
Derive Eq. (1.37) by making a force balance on the fluid interface in Fig. 1.11c.
11
12
()p R R
−−
=  +
(1.37)
Solution 1.63
The surface tension forces YdL1 and YdL2 have a slight vertical component. Thus summation of
forces in the vertical gives the result
Problem 1.64
Pressure in a water container can be measured by an open vertical tube—see Fig. P2.11 for a
sketch. If the expected water rise is about 20 cm, what tube diameter is needed to ensure that the
error due to capillarity will be less than 3 percent?
Figure P2.11
Solution 1.64
For water on glass, take ϒ = 0.073 N/m and θ = 0º. We want the capillary rise h to be less than
Problem 1.65
The system in Fig. P1.65 is used to estimate the pressure p1 in the tank by measuring the 15-cm
height of liquid in the 1–mm-diameter tube. The fluid is at 60C. Calculate the true fluid height
in the tube and the percent error due to capillarity if the fluid is (a) water; or
(b) mercury.
Figure P1.65
Solution 1.65
This is a somewhat more realistic variation of Ex. 1.9. Use values from that example for contact
angle
:
Problem 1.66
A thin wire ring, 3 cm in diameter, is lifted from a water surface at 20C. Neglecting the wire
weight, what is the force required to lift the ring? Is this a good way to measure surface tension?
Should the wire be made of any particular material?
Solution 1.66
In the literature this ring-pull device is called a DuNouy Tensiometer. The forces are very
Problem 1.67
A vertical concentric annulus, with outer radius ro and inner radius ri, is lowered into fluid of
surface tension Y and contact angle
90. Derive an expression for the capillary rise h in the
annular gap if the gap is very narrow.
Solution 1.67
For the figure above, the force balance on the annular fluid is
Problem 1.68*
Make an analysis of the shape
(x) of the water-air interface near a plane wall, as in Fig. P1.68,
assuming that the slope is small, R−1 d2
/dx2. Also assume that the pressure difference across
the interface is balanced by the specific weight and the interface height, p
g
. The boundary
conditions are a wetting contact angle
at x = 0 and a horizontal surface at
= 0 as x → .
What is the maximum height h at the wall?
Figure P1.68
Solution 1.68
This is a two-dimensional surface-tension problem, with single curvature. The surface tension rise is
balanced by the weight of the film. Therefore the differential equation is
Problem 1.69
A solid cylindrical needle of diameter d , length L , and density ρn may float in liquid of surface
tension Y . Neglect buoyancy and assume a contact angle of 0 . Derive a formula for the
maximum diameter dmax able to float in the liquid. Calculate dmax for a steel needle (SG = 7.84)
in water at 20 C.
Solution 1.69
The needle “dents” the surface downward and the surface tension forces are upward, as shown. If
these tensions are nearly vertical, a vertical force balance gives:
Problem 1.70
Derive an expression for the capillary height change h for a fluid of surface tension ϒ and
contact angle θ between two vertical parallel plates a distance W apart, as in Fig. P1.70. What
will h be for water at 20C if W = 0.5 mm?
Fig. P1.70
Solution 1.70
With b the width of the plates into the paper, the capillary forces on each wall together balance
the weight of water held above the reservoir free surface:
Problem 1.71*
A soap bubble of diameter D1 coalesces with another bubble of diameter D2 to form a single
bubble D3 with the same amount of air. For an isothermal process, derive an expression for
finding D3 as a function of D1, D2, patm, and surface tension ϒ.
Solution 1.71*
The masses remain the same for an isothermal process of an ideal gas:
Problem 1.72
Early mountaineers boiled water to estimate their altitude. If they reach the top and find that
water boils at 84C, approximately how high is the mountain?
Solution 1.72
From Table A-5 at 84C, vapor pressure pv 55.4 kPa. We may use this value to interpolate in the
Problem 1.73
A small submersible moves at velocity V in fresh water at 20C at a 2-m depth, where ambient
pressure is 131 kPa. Its critical cavitation number is Ca = 0.25. At what velocity will cavitation
bubbles begin to form on the body? Will the body cavitate if V = 30 m/s and the water is cold
(5C)?
Solution 1.73
From Table A-5 at 20C read pv = 2.337 kPa. By definition,
Problem 1.74
Oil, with a vapor pressure of 20 kPa, is delivered through a pipeline by equally-spaced pumps,
each of which increases the oil pressure by 1.3 MPa. Friction losses in the pipe are 150 Pa per
meter of pipe. What is the maximum possible pump spacing to avoid cavitation of the oil?
Solution 1.74
The absolute maximum length L occurs when the pump inlet pressure is slightly greater than
20 kPa. The pump increases this by 1.3 MPa and friction drops the pressure over a distance L
Problem 1.75
An airplane flies at 555 mi/h. At what altitude in the standard atmosphere will the airplane’s
Mach number be exactly 0.8?
Solution 1.75
First convert V = 555 mi/h x 0.44704 = 248.1 m/s. Then the speed of sound is
Problem 1.76
Derive a formula for the bulk modulus of an ideal gas, with constant specific heats, and calculate
it for steam at 300ºC and 200 kPa. Compare your result to the steam tables.
Solution 1.76
If an ideal gas is isentropic, p = Cρk, where k = cp/cv. Evaluate the bulk modulus:
Problem 1.77
Assume that the n-pentane data of Prob. P1.36 represents isentropic conditions. Estimate the
value of the speed of sound at a pressure of 30 MPa. [Hint: The data approximately fit Eq. (1.19)
with B = 260 and n = 11.]
Solution 1.77
Problem 1.78
Sir Isaac Newton measured the speed of sound by timing the difference between seeing a
cannon’s puff of smoke and hearing its boom. If the cannon is on a mountain 5.2 mi away,
estimate the air temperature in degrees Celsius if the time difference is ( a ) 24.2 s and
( b ) 25.1 s.
Solution 1.78
Cannon booms are finite (shock) waves and travel slightly faster than sound waves, but what the
heck, assume it’s close enough to sound speed:
Problem 1.79
From Table A.3, the density of glycerin at standard conditions is about 1260 kg/m3. At a very
high pressure of 8000 lb/in2, its density increases to approximately 1275 kg/m3. Use this data to
estimate the speed of sound of glycerin, in ft/s.
Solution 1.79
For a liquid, we simplify Eq. (1.38) to a pressure-density ratio, without knowing if the process is
isentropic or not. This should give satisfactory accuracy:
Problem 1.80
In Problem P1.24, for the given data, the air velocity at section 2 is 1180 ft/s. What is the Mach
number at that section?