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- Question : 1P - A gas at 208C may be considered raref ed, deviating from the continuum concept, when it contains less than 1012 molecules per cubic millimeter. If Avogadro
- Question : 2P - Table A.6 lists the density of the standard atmosphere as a function of altitude. Use these values to estimate, crudely
- Question : 3P - For the triangular element in Fig. P1.3, show that a tilted free liquid surface, in contact with an atmosphere at pressure pa, must undergo shear stress and hence begin to ?ow. Hint: Account for the weight of the ? uid and show that a no-shear condition will cause horizontal forces to be out of balance.
- Question : 4P - Sand, and other granular materials, appear to ? ow; that is, you can pour them from a container or a hopper. There are whole textbooks on the
- Question : 5P - The mean free path of a gas, l, is def ned as the average distance traveled by molecules between collisions. A proposed formula for estimating l of an ideal gas is l 5 1.26 ? ?1RT What are the dimensions of the constant 1.26? Use the formula to estimate the mean free path of air at 208C and 7 kPa. Would you consider air raref ed at this condition?
- Question : 6P - Henri Darcy, a French engineer, proposed that the pressure drop Dp for ? ow at velocity V through a tube of length L could be correlated in the form
- Question : 7P - Convert the following inappropriate quantities into SI units: (a) 2.283 E7 U.S. gallons per day; (b) 4.5 furlongs per minute (racehorse speed); and (c) 72,800 avoirdupois ounces per acre.
- Question : 8P - Suppose we know little about the strength of materials but are told that the bending stress ? in a beam is proportional to the beam half-thickness y and also depends on the bending moment M and the beam area moment of inertia I. We also learn that, for the particular case M 5 2900 in ? lbf, y 5 1.5 in, and I 5 0.4 in4, the predicted stress is 75 MPa. Using this information and dimensional reasoning only, f nd, to three signifcant fgures, the only possible dimensionally homogeneous formula ? 5 y f(M, I ).
- Question : 9P - A hemispherical container, 26 inches in diameter, is flled with a liquid at 208C and weighed. The liquid weight is found to be 1617 ounces. (a) What is the density of the ?uid, in kg/m3? (b) What ? uid might this be? Assume standard gravity, g 5 9.807 m/s2.
- Question : 10P - The Stokes-Oseen formula [33] for drag force F on a sphere of diameter D in a ?uid stream of low velocity V, density ?, and viscosity ? is F 5 3??DV 1 9? 16 ?V2D2 Is this formula dimensionally homogeneous?
- Question : 11P - In English Engineering units, the specif c heat cp of air at room temperature is approximately 0.24 Btu/(lbm-8F). 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.
- Question : 12P - For low-speed (laminar) steady ? ow through a circular pipe, as shown in Fig. P1.12, the velocity u varies with radius and takes the form u 5 B
- Question : 13P - The effciency ? of a pump is def ned as the (dimensionless) ratio of the power developed by the ? ow to the power required to drive the pump: ? 5 Q
- Question : 14P - Figure P1.14 shows the ?ow of water over a dam. The volume ?ow 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 ?ow rate?
- Question : 15P - The height H that ?uid rises in a liquid barometer tube depends upon the liquid density ?, 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?
- Question : 16P - Algebraic equations such as Bernoulli
- Question : 17P - The Hazen-Williams hydraulics formula for volume rate of ?ow Q through a pipe of diameter D and length L is given by Q < 61.9 D2.63 a
- Question : 18P - For small particles at low velocities, the f rst term in the Stokes-Oseen drag law, Prob. 1.10, is dominant; hence, F < KV, where K is a constant. Suppose a particle of mass m is constrained to move horizontally from the initial position x 5 0 with initial velocity V0. Show (a) that its velocity will decrease exponentially with time and (b) that it will stop after traveling a distance x 5 mV0/K.
- Question : 19P - In his study of the circular hydraulic jump formed by a faucet ?owing into a sink, Watson [53] proposed a parameter combining volume ?ow rate Q, density ?, and viscosity ? of the ? uid, and depth h of the water in the sink. He claims that his grouping is dimensionless, with Q in the numerator. Can you verify this?
- Question : 20P - Books on porous media and atomization claim that the viscosity ? and surface tension Y of a ?uid 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 208C and a velocity of 3.5 cm/s. Note: You get extra credit if you know the name of this parameter.
- Question : 21P - Aeronautical engineers measure the pitching moment M0 of a wing and then write it in the following form for use in other cases: where V is the wing velocity, A the wing area, C the wing chord length, and ? the air density. What are the dimensions of the coeffcient ??
- Question : 22P - The Ekman number, Ek, arises in geophysical ?uid 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.
- Question : 23P - During World War II, Sir Geoffrey Taylor, a British ?uid 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.
- Question : 24P - Air, assumed to be an ideal gas with k 5 1.40, ?ows isentropically through a nozzle. At section 1, conditions are sea level standard (see Table A.6). At section 2, the temperature is 2508C. Estimate (a) the pressure, and (b) the density of the air at section 2.
- Question : 25P - On a summer day in Narragansett, Rhode Island, the air temperature is 748F and the barometric pressure is 14.5 lbf/in2. Estimate the air density in kg/m3.
- Question : 26P - When we in the United States say a car
- Question : 27P - For steam at a pressure of 45 atm, some values of temperature and specif c volume are as follows, from Ref. 23: T, 8F 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 approx i mate an ideal gas? If not, explain.
- Question : 28P - Wet atmospheric air at 100 percent relative humidity contains saturated water vapor and, by Dalton
- Question : 29P - A compressed-air tank holds 5 ft3 of air at 120 lbf/in2
- Question : 30P - Repeat Prob. 1.29 if the tank is f lled with compressed water instead of air. Why is the result thousands of times less than the result of 215,000 ft ? lbf in Prob. 1.29?
- Question : 31P - One cubic foot of argon gas at 108C and 1 atm is compressed isentropically to a pressure of 600 kPa. (a) What will be its new pressure and temperature? (b) If it is allowed to cool at this new volume back to 108C, what will be the fnal pressure?
- Question : 32P - A blimp is approximated by a prolate spheroid 90 m long and 30 m in diameter. Estimate the weight of 208C 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)?
- Question : 33P - A tank contains 9 kg of CO2 at 208C and 2.0 MPa. Estimate the volume of the tank, in m3.
- Question : 34P - Consider steam at the following state near the saturation line: (p1, T1) 5 (1.31 MPa, 2908C). 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.
- Question : 35P - In Table A.4, most common gases (air, nitrogen, oxygen, hydrogen) have a specif c 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?
- Question : 36P - Experimental data [55] for the density of n-pentane liquid for high pressures, at 508C, are listed as follows: Pressure, kPa 100 10,230 20,700 34,310 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 ? at 30 MPa.
- Question : 37P - A near-ideal gas has a molecular weight of 44 and a specifc heat c v 5 610 J/(kg ? K). What are (a) its specifc heat ratio, k, and (b) its speed of sound at 1008C?
- Question : 38P - In Fig. 1.7, if the ? uid is glycerin at 208C 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?
- Question : 39P - Knowing ? for air at 208C from Table 1.4, estimate its viscosity at 5008C by (a) the power law and (b) the Sutherland law. Also make an estimate from (c) Fig. 1.6. Compare with the accepted value of ? < 3.58 E-5 kg/m ? s.
- Question : 40P - Glycerin at 208C flls the space between a hollow sleeve of diameter 12 cm and a f xed coaxial solid rod of diameter 11.8 cm. The outer sleeve is rotated at 120 rev/min. Assuming no temperature change, estimate the torque required, in N
- Question : 41P - An aluminum cylinder weighing 30 N, 6 cm in diameter and 40 cm long, is falling concentrically through a long vertical sleeve of diameter 6.04 cm. The clearance is flled with SAE 50 oil at 208C. Estimate the terminal (zero acceleration) fall velocity. Neglect air drag and assume a linear velocity distribution in the oil. Hint: You are given diameters, not radii.
- Question : 42P - Helium at 208C has a viscosity of 1.97 E-5 kg/(m ? s). Use the data of Table A.4 to estimate the temperature, in 8C, at which helium
- Question : 43P - For the ? ow of gas between two parallel plates of Fig. 1.7, reanalyze for the case of slip ? ow at both walls. Use the simple slip condition, ?uwall 5 < (du/dy)wall, where < is the mean free path of the ? uid. Sketch the expected velocity prof le and f nd an expression for the shear stress at each wall.
- Question : 44P - One type of viscometer is simply a long capillary tube. A commercial device is shown in Prob. C1.10. One measures the volume ?ow rate Q and the pressure drop Dp and, of course, the radius and length of the tube. The theoretical formula, which will be discussed in Chap. 6, is
- Question : 45P - A block of weight W slides down an inclined plane while lubricated by a thin f lm of oil, as in Fig. P1.45. The flm contact area is A and its thickness is h. Assuming a linear velocity distribution in the f lm, derive an expression for the
- Question : 46P - A simple and popular model for two nonnewtonian ?uids in Fig. 1.8a is the power-law: ? < C adu dy bn where C and n are constants f t to the ? uid [16]. From Fig. 1.8a, deduce the values of the exponent n for which the ? uid is (a) newtonian, (b) dilatant, and (c) pseudoplastic. Consider the specif c model constant C 5 0.4 N ? sn/m2, with the ? uid being sheared between two parallel plates as in Fig. 1.7. If the shear stress in the ?uid is 1200 Pa, fnd the velocity V of the upper plate for the cases (d) n 5 1.0, (e) n 5 1.2, and (f) n 5 0.8.
- Question : 47P - Data for the apparent viscosity of average human blood, at normal body temperature of 378C, varies with shear strain rate, as shown in the following table. Strain rate, s21 1 10 100 1000 Apparent viscosity, 0.011 0.009 0.006 0.004 kg/(m
- Question : 48P - A thin plate is separated from two fxed plates by very viscous liquids ?1 and ?2, respectively, as in Fig. P1.48. The plate spacings h1 and h2 are unequal, as shown. The contact area is A between the center plate and each ?uid. (a) Assuming a linear velocity distribution in each ?uid, derive the force F required to pull the plate at velocity V. (b) Is there a necessary relation between the two viscosities, ?1 and ?2?
- Question : 49P - An amazing number of commercial and laboratory devices have been developed to measure ? uid viscosity, as described in Refs. 29 and 49. Consider a concentric shaft, fxed axially and rotated inside the sleeve. Let the inner and outer cylinders have radii ri and ro, respectively, with total sleeve length L. Let the rotational rate be V (rad/s) and the applied torque be M. Using these parameters, derive a theoretical relation for the viscosity ? of the ?uid between the cylinders.
- Question : 50P - A simple viscometer measures the time t for a solid sphere to fall a distance L through a test ? uid of density ?. The ? uid viscosity ? is then given by ? < Wnett 3?DL if t $ 2?DL ? where D is the sphere diameter and Wnet is the sphere net weight in the ?uid. (a) Prove that both of these formulas are dimensionally homogeneous. (b) Suppose that a 2.5 mm diameter aluminum sphere (density 2700 kg/m3) falls in an oil of density 875 kg/m3. If the time to fall 50 cm is 32 s, estimate the oil viscosity and verify that the inequality is valid.
- Question : 51P - An approximation for the boundary-layer shape in Figs. 1.5b and P1.51 is the formula u( y) < U sina? 2? yb, 0 # y # ? where U is the stream velocity far from the wall and ? is the boundary layer thickness, as in Fig. P1.51. If the ?uid is helium at 208C and 1 atm, and if U 5 10.8 m/s and ? 5 3 cm, use the formula to (a) estimate the wall shear stress ?w in Pa, and (b) fnd the position in the boundary layer where ? is one-half of ? w.
- Question : 52P - The belt in Fig. P1.52 moves at a steady velocity V and skims the top of a tank of oil of viscosity ?, as shown. Assuming a linear velocity prof le in the oil, develop a simple formula for the required belt-drive power P as a function of (h, L, V, b, ?). What belt-drive power P, in watts, is required if the belt moves at 2.5 m/s over SAE 30W oil at 208C, with L 5 2 m, b 5 60 cm, and h 5 3 cm?
- Question : 53P - A solid cone of angle 2?, base r0, and density ?c is rotating with initial angular velocity ?0 inside a conical seat, as shown in Fig. P1.53. The clearance h is f lled with oil of viscosity ?. Neglecting air drag, derive an analytical expression for the cone
- Question : 54P - A disk of radius R rotates at an angular velocity V inside a disk-shaped container f lled with oil of viscosity ?, as shown in Fig. P1.54. Assuming a linear velocity profle and neglecting shear stress on the outer disk edges, derive a formula for the viscous torque on the disk.
- Question : 55P - A block of weight W is being pulled over a table by another weight Wo, as shown in Fig. P1.55. Find an algebraic formula for the steady velocity U of the block if it slides on an oil f lm of thickness h and viscosity ?. The block bottom area A is in contact with the oil. Neglect the cord weight and the pulley friction. Assume a linear velocity profle in the oil flm.
- Question : 56P - The device in Fig. P1.56 is called a cone-plate viscometer [29]. The angle of the cone is very small, so that sin ? < ?, and the gap is f lled with the test liquid. The torque M to rotate the cone at a rate V is measured. Assuming a linear velocity profle in the ?uid flm, derive an expression for ? uid viscosity ? as a function of (M, R, V, ?).
- Question : 57P - Extend the steady ? ow between a f xed lower plate and a moving upper plate, from Fig. 1.7, to the case of two immiscible liquids between the plates, as in Fig. P1.57. (a) Sketch the expected no-slip velocity distribution u(y) between the plates. (b) Find an analytic expression for the velocity U at the interface between the two liquid layers. (c) What is the result of (b) if the viscosities and layer thicknesses are equal?
- Question : 58P - The laminar pipe ? ow example of Prob. 1.12 can be used to design a capillary viscometer [29]. If Q is the volume ?ow rate, L is the pipe length, and Dp is the pressure drop from entrance to exit, the theory of Chap. 6 yields a formula for viscosity: ? 5 ?r4 0
- Question : 59P - A solid cylinder of diameter D, length L, and density ?s falls due to gravity inside a tube of diameter D0. The clearance, D0 2 D ,, D, is f lled with ? uid of density ? and viscosity ?. Neglect the air above and below the cylinder. Derive a formula for the terminal fall velocity of the cylinder. Apply your formula to the case of a steel cylinder, D 5 2 cm, D0 5 2.04 cm, L 5 15 cm, with a flm of SAE 30 oil at 208C.
- Question : 60P - Pipelines are cleaned by pushing through them a closeftting cylinder called a pig. The name comes from the squealing noise it makes sliding along. Reference 50 describes a new nontoxic pig, driven by compressed air, for cleaning cosmetic and beverage pipes. Suppose the pig diameter is 5-15/16 in and its length 26 in. It cleans a 6-in-diameter pipe at a speed of 1.2 m/s. If the clearance is flled with glycerin at 208C, what pressure difference, in pascals, is needed to drive the pig? Assume a linear velocity prof le in the oil and neglect air drag.
- Question : 61P - An air-hockey puck has a mass of 50 g and is 9 cm in diameter. When placed on the air table, a 208C air flm, of 0.12-mm thickness, forms under the puck. The puck is struck with an initial velocity of 10 m/s. Assuming a linear velocity distribution in the air f lm, how long will it take the puck to (a) slow down to 1 m/s and (b) stop completely? Also, (c) how far along this extremely long table will the puck have traveled for condition (a)?
- Question : 62P - The hydrogen bubbles that produced the velocity profles in Fig. 1.15 are quite small, D < 0.01 mm. If the hydrogen
- Question : 63P - Derive Eq. (1.33) by making a force balance on the ?uid interface in Fig. 1.11c.
- Question : 64P - Pressure in a water container can be measured by an open vertical tube
- Question : 65P - The system in Fig. P1.65 is used to calculate the pressure p1 in the tank by measuring the 15-cm height of liquid in the 1-mm-diameter tube. The ? uid is at 608C. Calculate the true ? uid height in the tube and the percentage error due to capillarity if the ? uid is (a) water or (b) mercury.
- Question : 66P - A thin wire ring, 3 cm in diameter, is lifted from a water surface at 208C. 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?
- Question : 67P - A vertical concentric annulus, with outer radius ro and inner radius ri, is lowered into a ? uid of surface tension Y and contact angle ? , 908. Derive an expression for the capillary rise h in the annular gap if the gap is very narrow.
- Question : 68P - Make an analysis of the shape h(x) of the water
- Question : 69P - A solid cylindrical needle of diameter d, length L, and density ?n may ? oat in liquid of surface tension Y. Neglect buoyancy and assume a contact angle of 08. Derive a formula for the maximum diameter d max able to ? oat in the liquid. Calculate dmax for a steel needle (SG 5 7.84) in water at 208C.
- Question : 70P - Derive an expression for the capillary height change h for a ?uid of surface tension Y and contact angle ? between two vertical parallel plates a distance W apart, as in Fig. P1.70. What will h be for water at 208C if W 5 0.5 mm?
- Question : 71P - 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. Assuming an isothermal process, derive an expression for fnding D3 as a function of D1, D2, patm, and Y.
- Question : 72P - Early mountaineers boiled water to estimate their altitude. If they reach the top and fnd that water boils at 848C, approximately how high is the mountain?
- Question : 73P - A small submersible moves at velocity V, in fresh water at 208C, at a 2-m depth, where ambient pressure is 131 kPa. Its critical cavitation number is known to be C a 5 0.25. At what velocity will cavitation bubbles begin to form on the body? Will the body cavitate if V 5 30 m/s and the water is cold (58C)?
- Question : 74P - 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?
- Question : 75P - dard atmosphere will the airplane
- Question : 76P - Derive a formula for the bulk modulus of an ideal gas, with constant specifc heats, and calculate it for steam at 3008C and 200 kPa. Compare your result to the steam tables.
- Question : 77P - 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 f t Eq. (1.19) with B 5 260 and n 5 11.]
- Question : 78P - Sir Isaac Newton measured the speed of sound by timing the difference between seeing a cannon
- Question : 79P - 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.
- Question : 80P - 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?
- Question : 81P - Use Eq. (1.39) to f nd and sketch the streamlines of the following ?ow f eld:
- Question : 82P - A velocity f eld is given by u 5 V cos?, v 5 V sin?, and w 5 0, where V and ? are constants. Derive a formula for the streamlines of this ?ow.
- Question : 83P - Use Eq. (1.39) to f nd and sketch the streamlines of the following ?ow f eld: u 5 K(x2 2 y2); v 5 22Kxy; w 5 0, where K is a constant. Hint: This is a frst-order exact differential equation.
- Question : 84P - In the early 1900s, the British chemist Sir Cyril Hinshelwood quipped that ? uid dynamics study was divided into
- Question : 85P - Do some reading and report to the class on the life and achievements, especially vis-
- Question : 86P - A right circular cylinder volume ? is to be calculated from the measured base radius R and height H. If the uncertainty in R is 2 percent and the uncertainty in H is 3 percent, estimate the overall uncertainty in the calculated volume. Hint: Read Appendix E.
- Question : 1FOEEP - The absolute viscosity ? of a fluid is primarily a function of (a) Density, (b) Temperature, (c) Pressure, (d) Velocity, (e) Surface tension
- Question : 2FOEEP - Carbon dioxide, at 208C and 1 atm, is compressed isentropically to 4 atm. Assume CO2 is an ideal gas. The fnal temperature would be (a) 1308C, (b) 1628C, (c) 1718C, (d) 2378C, (e) 3138C
- Question : 3FOEEP - Helium has a molecular weight of 4.003. What is the weight of 2 m3 of helium at 1 atm and 208C? (a) 3.3 N, (b) 6.5 N, (c) 11.8 N, (d) 23.5 N, (e) 94.2 N
- Question : 4FOEEP - An oil has a kinematic viscosity of 1.25 E-4 m2/s and a specifc gravity of 0.80. What is its dynamic (absolute) viscosity in kg/(m ? s)? (a) 0.08, (b) 0.10, (c) 0.125, (d) 1.0, (e) 1.25
- Question : 5FOEEP - Consider a soap bubble of diameter 3 mm. If the surface tension coeffcient is 0.072 N/m and external pressure is 0 Pa gage, what is the bubble
- Question : 6FOEEP - The only possible dimensionless group that combines velocity V, body size L, ?uid density ?, and surface tension coeffcient ? is (a) L??/V, (b) ?VL2/?, (c) ??V2/L, (d) ?LV2/?, (e) ?LV2/?
- Question : 7FOEEP - Two parallel plates, one moving at 4 m/s and the other fxed, are separated by a 5-mm-thick layer of oil of specifc 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
- Question : 8FOEEP - Carbon dioxide has a specif c heat ratio of 1.30 and a gas constant of 189 J/(kg ? 8C). If its temperature rises from 20 to 458C, 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
- Question : 1CP - A certain water ? ow at 208C has a critical cavitation number, where bubbles form, Ca < 0.25, where Ca 5 2(pa2 pvap)/?V2. If pa 5 1 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
- Question : 2CP - Example 1.10 gave an analysis that predicted that the viscous moment on a rotating disk M 5 ??VR4/(2h). If the uncertainty of each of the four variables (?, V, 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
- Question : 3CP - Sometimes we can develop equations and solve practical problems by knowing nothing more than the dimensions of the key parameters in the problem. For example, consider the heat loss through a window in a building. Window effciency is rated in terms of
- Question : 4CP - 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 V0 when she suddenly stands stiff with her skates pointed directly forward, allowing herself to coast to a stop. Neglecting friction 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 V0 5 4.0 m/s, m 5 100 kg, L 5 30 cm, W 5 5.0 mm, and h 5 0.10 mm. Do you think our assumption of negligible air resistance is a good one?
- Question : 5PC - Two thin ? at plates, tilted at an angle a, are placed in a tank of liquid of known surface tension Y 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 and have width b into the page. The liquid rises a distance h between the plates, as shown. (a) What is the total upward (z-directed) force, due to surface tension, acting on the liquid column between the plates? (b) If the liquid density is ?, fnd an expression for surface tension Y in terms of the other variables.54 Chapter 1 Introduction C1.3 h z g L ? ? ? ? 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 prof le shape and the f lm thickness d 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 z g x Oil flm Air Plate ? (a) Sketch the approximate shape of the velocity profle w(x), considering the boundary conditions at the wall and at the flm surface. (b) Suppose flm thickness ?, and the slope of the velocity profle 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)wall are negative. C1.5 Viscosity can be measured by ? ow through a thin-bore or capillary tube if the ? ow rate is low. For length L, (small) diameter D L, pressure drop Dp, and (low) volume ?ow rate Q, the formula for viscosity is ? 5 D4Dp/(CLQ), where C is a constant. (a) Verify that C is dimensionless. The following data are for water ? owing through a 2-mm-diameter tube which is 1 meter long. The pressure drop is held constant at Dp 5 5 kPa. T, 8C 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 C by accounting for the variation with temperature of the viscosity of water.
- Question : 6CP - The rotating-cylinder viscometer in Fig. C1.6 shears the ?uid in a narrow clearance Dr, as shown. Assuming a linear velocity distribution in the gaps, if the driving torque M is measured, fnd an expression for ? by (a) neglecting and (b) including the bottom friction.
- Question : 7CP - Make an analytical study of the transient behavior of the sliding block in Prob. 1.45. (a) Solve for V(t) if the block starts from rest, V 5 0 at t 5 0. (b) Calculate the time t1 when the block has reached 98 percent of its terminal velocity.
- Question : 8CP - 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 f ve) is measured and correlated with viscosity. The formula is t < A? W 2 B where A and B are constants that are determined by calibrating the device with a known ? uid. Here are calibration 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 f t this calibration data. Hint: The data are not very sensitive to the value of B. (b) A more viscous ? uid is tested with a 100 N weight and the measured time is 44 s. Estimate the viscosity of this ?uid.
- Question : 9CP - 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 negligible weight and buoyancy and slides upward through a flm of heavy oil of viscosity ?. (a) If there is no acceleration (uniform lever rotation), derive a formula for the rate of fall V2 of the weight. Neglect the lever weight. Assume a linear velocity prof le in the oil flm. (b) Estimate the fall velocity of the weight if W 5 20 N, L1 5 75 cm, L2 5 50 cm, D 5 10 cm, L 5 22 cm, DR 5 1 mm, and the oil is glycerin at 208C. L1 L2 V1 V2? Cylinder, diameter D, length L, in an oil flm of thickness ?R.
- Question : 10CP - A popular gravity-driven instrument is the CannonUbbelohde viscometer, shown in Fig. C1.10. The test 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: v 5 Ct where C is a calibration constant. For ? in the range of 100
- Question : 11CP - Mott [Ref. 49, p. 38] discusses a simple falling-ball viscometer, which we can analyze later in Chap. 7. A small ball of diameter D and density ?b falls through a tube of test liquid (?, ?). The fall velocity V is calculated by the time to fall a measured distance. The formula for calculating the viscosity of the ?uid is ? 5 (?b 2 ?)gD2 18 V This result is limited by the requirement that the Reynolds number (?VD/?) be less than 1.0. Suppose a steel ball (SG 5 7.87) of diameter 2.2 mm falls in SAE 25W oil (SG 5 0.88) at 208C. 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?
- Question : 12CP - A solid aluminum disk (SG 5 2.7) is 2 in in diameter and 3/16 in thick. It slides steadily down a 148 incline that is coated with a castor oil (SG 5 0.96) flm one hundredth of an inch thick. The steady slide velocity is 2 cm/s. Using Figure A.1 and a linear oil velocity profle assumption, estimate the temperature of the castor oil.

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