C1.1 Sometimes we can develop equations and solve practi- cal problems by knowing nothing more than the dimen- sions of the key parameters in the problem. For exam- ple, consider the heat loss through a window in a building. Window efficiency is rated in terms of “R value,” which has units of (ft2 h F)/Btu. A certain manufacturer advertises a double-pane window with an R value of 2.5. The same company produces a triple- pane window with an R value of 3.4. In either case the window dimensions are 3 ft by 5 ft. On a given winter day, the temperature difference between the inside and outside of the building is 45F.
(a) Develop an equation for the amount of heat lost in a given time period t, through a window of area A, with Rvalue R, and temperature difference T. How much heat (in Btu) is lost through the double-pane window in one 24-h period?
(b) How much heat (in Btu) is lost through the triple-pane window in one 24-h period?
(c) Suppose the building is heated with propane gas, which costs $3.25 per gallon. The propane burner is 80 percent efficient. Propane has approxi- mately 90,000 Btu of available energy per gallon.
In that same 24-h period, how much money would a homeowner save per window by installing triple- pane rather than double-pane windows?
(d) Finally, suppose the homeowner buys 20 such triple-pane windows for the house. A typical winter has the equivalent of about 120 heating days at a temperature difference of 45F. Each triple-pane
window costs $85 more than the double-pane window. Ignoring interest and inflation, how many years will it take the homeowner to make up the additional cost of the triple-pane windows from heating bill savings?
C1.2 When a person ice skates, the surface of the ice actually melts beneath the blades, so that he or she skates on a thin sheet of water between the blade and the ice.
(a) Find an expression for total friction force on the bottom of the blade as a function of skater velocity V, blade length L, water thickness (between the blade and the ice) h, water viscosity , and blade width W.
(b) Suppose an ice skater of total mass m is skating along at a constant speed of V0when she suddenly stands stiff with her skates pointed directly forward, allowing herself to coast to a stop. Neglecting fric- tion due to air resistance, how far will she travel before she comes to a stop? (Remember, she is coasting on two skate blades.) Give your answer for the total distance traveled, x, as a function of V0, m, L, h, , and W.
(c) Find x for the case where V04.0 m/s, m100 kg, L30 cm, W5.0 mm, and h0.10 mm. Do you think our assumption of negligible air resistance is a good one?
C1.3 Two thin flat plates, tilted at an angle , are placed in a tank of liquid of known surface tension and contact angle , as shown in Fig. C1.3. At the free surface of the liquid in the tank, the two plates are a distance L apart FE1.6 The only possible dimensionless group that combines
velocity V, body size L, fluid density , and surface tension coefficient is
(a) L/V, (b) VL2/, (c) V2/L, (d) LV2/, (e) LV2/
FE1.7 Two parallel plates, one moving at 4 m/s and the other fixed, are separated by a 5-mm-thick layer of oil of specific gravity 0.80 and kinematic viscosity 1.25 E-4 m2/s. What is the average shear stress in the oil?
(a) 80 Pa, (b) 100 Pa, (c) 125 Pa, (d) 160 Pa, (e) 200 Pa
FE1.8 Carbon dioxide has a specific heat ratio of 1.30 and a gas constant of 189 J/(kg C). If its temperature rises from 20 to 45C, what is its internal energy rise?
(a) 12.6 kJ/kg, (b) 15.8 kJ/kg, (c) 17.6 kJ/kg, (d) 20.5 kJ/kg, (e) 25.1 kJ/kg
FE1.9 A certain water flow at 20C has a critical cavitation number, where bubbles form, Ca0.25, where Ca 2(papvap)/V2. If pa1 atm and the vapor pressure is 0.34 pounds per square inch absolute (psia), for what water velocity will bubbles form?
(a) 12 mi/h, (b) 28 mi/h, (c) 36 mi/h, (d) 55 mi/h, (e) 63 mi/h
FE1.10 Example 1.10 gave an analysis that predicted that the viscous moment on a rotating disk M R4/(2h).
If the uncertainty of each of the four variables (, , R, h) is 1.0 percent, what is the estimated overall uncertainty of the moment M?
(a) 4.0 percent (b) 4.4 percent (c) 5.0 percent (d) 6.0 percent (e) 7.0 percent
and have width b into the page. The liquid rises a distance hbetween the plates, as shown.
(a) What is the total upward (z-directed) force, due to sur- face tension, acting on the liquid column between the plates?
(b) If the liquid density is , find an expression for surface tension in terms of the other variables.
C1.3
C1.4 Oil of viscosity and density drains steadily down the side of a tall, wide vertical plate, as shown in Fig. C1.4.
In the region shown, fully developed conditions exist; that is, the velocity profile shape and the film thickness are independent of distance z along the plate. The vertical velocity w becomes a function only of x, and the shear resistance from the atmosphere is negligible.
C1.4
(a) Sketch the approximate shape of the velocity profile w(x), considering the boundary conditions at the wall and at the film surface.
z
␦
x g Oil film
Air Plate
h
z
g
L
(b) Suppose film thickness , and the slope of the veloc- ity profile at the wall, (dw/dx)wall, are measured by a laser Doppler anemometer (to be discussed in Chap.
6). Find an expression for the viscosity of the oil as a function of , , (dw/dx)wall, and the gravitational acceleration g. Note that, for the coordinate system given, both w and (dw/dx)wallare negative.
C1.5 Viscosity can be measured by flow through a thin-bore or capillary tube if the flow rate is low. For length L, (small) diameter DVL, pressure drop p, and (low) volume flow rate Q, the formula for viscosity is D4 p/(CLQ), where C is a constant.
(a) Verify that C is dimensionless. The following data are for water flowing through a 2-mm-diameter tube which is 1 meter long. The pressure drop is held constant at p5 kPa.
T, C 10.0 40.0 70.0
Q, L/min 0.091 0.179 0.292
(b) Using proper SI units, determine an average value of Cby accounting for the variation with temperature of the viscosity of water.
C1.6 The rotating-cylinder viscometer in Fig. C1.6 shears the fluid in a narrow clearance r, as shown. Assuming a linear velocity distribution in the gaps, if the driving torque M is measured, find an expression for by (a) neglecting and (b) including the bottom friction.
C1.6
C1.7 SAE 10W oil at 20C flows past a flat surface, as in Fig. 1.6b. The velocity profile u(y) is measured, with the following results:
y, m 0.0 0.003 0.006 0.009 0.012 0.015
u, m/s 0.0 1.99 3.94 5.75 7.29 8.46
L
Ω R
Viscous fluid
Solid cylinder
Δr << R
Using your best interpolating skills, estimate the shear stress in the oil (a) at the wall and (b) at y15 mm.
C1.8 A mechanical device that uses the rotating cylinder of Fig. C1.6 is the Stormer viscometer [29]. Instead of being driven at constant , a cord is wrapped around the shaft and attached to a falling weight W. The time t to turn the shaft a given number of revolutions (usually five) is measured and correlated with viscosity. The formula is
where A and B are constants that are determined by cali- brating the device with a known fluid. Here are calibra- tion data for a Stormer viscometer tested in glycerol, using a weight of 50 N:
, kg/m-s 0.23 0.34 0.57 0.84 1.15
t, sec 15 23 38 56 77
(a) Find reasonable values of A and B to fit this calibra- tion data. Hint: The data are not very sensitive to the value of B.
(b) A more viscous fluid is tested with a 100 N weight and the measured time is 44 s. Estimate the viscosity of this fluid.
C1.9 The lever in Fig. C1.9 has a weight W at one end and is tied to a cylinder at the left end. The cylinder has negli- gible weight and buoyancy and slides upward through a film of heavy oil of viscosity . (a) If there is no acceler- ation (uniform lever rotation), derive a formula for the rate of fall V2of the weight. Neglect the lever weight. Assume a linear velocity profile in the oil film. (b) Estimate the fall velocity of the weight if W 20 N, L1 75 cm, L2 50 cm, D 10 cm, L 22 cm, R 1 mm, and the oil is glycerin at 20C.
C1.9
C1.10 A popular gravity-driven instrument is the Cannon- Ubbelohde viscometer, shown in Fig. C1.10. The test
L1 L2
V1
V2?
Cylinder, diameter D, length L, in an oil film of thickness ⌬R.
pivot
W
t A
WB
liquid is drawn up above the bulb on the right side and allowed to drain by gravity through the capillary tube below the bulb. The time t for the meniscus to pass from upper to lower timing marks is recorded. The kinematic viscosity is computed by the simple formula:
Ct
where C is a calibration constant. For in the range of 100–500 mm2/s, the recommended constant is C 0.50 mm2/s2, with an accuracy less than 0.5 percent.
(a) What liquids from Table A.3 are in this viscosity range? (b) Is the calibration formula dimensionally con- sistent? (c) What system properties might the constant C depend upon? (d) What problem in this chapter hints at a formula for estimating the viscosity?
C1.11 Mott [Ref. 49, p. 38] discusses a simple falling-ball vis- cometer, which we can analyze later in Chapter 7. A small ball of diameter D and density bfalls though a tube of test liquid (, ). The fall velocity V is calculated by the time to fall a measured distance. The formula for calcu- lating the viscosity of the fluid is
This result is limited by the requirement that the Reynolds number (VD/) be less than 1.0. Suppose a steel ball (SG 7.87) of diameter 2.2 mm falls in SAE 25W oil (SG 0.88) at 20C. The measured fall velocity is 8.4 cm/s.
(a) What is the viscosity of the oil, in kg/m-s? (b) Is the Reynolds number small enough for a valid estimate?
C1.12 A solid aluminum disk (SG 2.7) is 2 in in diameter and 3/16 in thick. It slides steadily down a 14incline that is coated with a castor oil (SG 0.96) film one hundredth of an inch thick. The steady slide velocity is 2 cm/s. Using Figure A.1 and a linear oil velocity profile assumption, estimate the temperature of the castor oil.
(b)gD2 18 V
Upper timing mark Lower timing mark Capillary tube C1.10 The Cannon-Ubbe-
lohde viscometer. [Cour- tesy of Cannon Instrument Company.]