Books on porous media and atomization claim that the viscosity and surface tension Y of

A gas at 20 8 C may be considered rarefi ed, deviating from the continuum concept, when it contains less than 10 12 molecules per cubic millimeter. If Avogadros number is 6.023 E23 molecules per mole, what absolute pressure (in Pa) for air does this represent?

Table A.6 lists the density of the standard atmosphere as a function of altitude. Use these values to estimate, crudely say, within a factor of 2the number of molecules of air in the entire atmosphere of the earth.

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 fl ow. Hint: Account for the weight of the fl uid and show that a no-shear condition will cause horizontal forces to be out of balance.

Sand, and other granular materials, appear to fl ow; that is, you can pour them from a container or a hopper. There are whole textbooks on the transport of granular materials [54]. Therefore, is sand a fl uid ? Explain.

The mean free path of a gas, l , is defi ned as the average distance traveled by molecules between collisions. A proposed formula for estimating l of an ideal gas is

Henri Darcy, a French engineer, proposed that the pressure drop Dp for fl ow at velocity V through a tube of length L could be correlated in the form p 5 LV2 If Darcys formulation is consistent, what are the dimensions of the coeffi cient ?

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.

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 in 4 , the predicted stress is 75 MPa. Using this information and dimensional reasoning only, fi nd, to three signifi cant fi gures, the only possible dimensionally homogeneous formula 5 y f ( M , I ).

A hemispherical container, 26 inches in diameter, is fi lled with a liquid at 20 8 C and weighed. The liquid weight is found to be 1617 ounces. ( a ) What is the density of the fl uid, in kg/m 3 ? ( b ) What fl uid might this be? Assume standard gravity, g 5 9.807 m/s 2

The Stokes-Oseen formula [33] for drag force F on a sphere of diameter D in a fl uid stream of low velocity V , density , and viscosity is F 5 3DV 1 9 16V2 D2 Is this formula dimensionally homogeneous?

In English Engineering units, the specifi c heat cp of air at room temperature is approximately 0.24 Btu/(lbm- 8 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.

For low-speed (laminar) steady fl ow through a circular pipe, as shown in Fig. P1.12, the velocity u varies with radius and takes the form u 5 B p (r0 2 2 r2 ) where is the fl uid viscosity and Dp is the pressure drop from entrance to exit. What are the dimensions of the constant B ? r = 0 r u (r) Pipe wall r

The effi ciency of a pump is defi ned as the (dimensionless) ratio of the power developed by the fl ow to the power required to drive the pump: 5 Qp input power where Q is the volume rate of fl ow and Dp is the pressure rise produced by the pump. Suppose that a certain pump develops a pressure rise of 35 lbf/in 2 when its fl ow rate is 40 L/s. If the input power is 16 hp, what is the effi ciency?

Figure P1.14 shows the fl ow of water over a dam. The volume fl 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 fl ow rate?

The height H that fl 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?

Algebraic equations such as Bernoullis relation, Eq. (1) of Example 1.3, are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer x -momentum equation, fi rst derived by Ludwig Prandtl in 1904: u 0u 0x 1 0u 0y 5 20p 0x 1 gx 1 0 0y where is the boundary-layer shear stress and gx is the component of gravity in the x direction. Is this equation dimensionally consistent? Can you draw a general conclusion?

The Hazen-Williams hydraulics formula for volume rate of fl ow Q through a pipe of diameter D and length L is given by Q < 61.9 D2.63 a p L b 0.54 where Dp is the pressure drop required to drive the fl ow. What are the dimensions of the constant 61.9? Can this formula be used with confi dence for various liquids and gases?

For small particles at low velocities, the fi 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

In his study of the circular hydraulic jump formed by a faucet fl owing into a sink, Watson [53] proposed a parameter combining volume fl ow rate Q , density , and viscosity of the fl 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?

Books on porous media and atomization claim that the viscosity and surface tension Y of a fl 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 20 8 C and a velocity of 3.5 cm/s. Note: You get extra credit if you know the name of this parameter

Aeronautical engineers measure the pitching moment M0 of a wing and then write it in the following form for use in other cases: M0 5 V2 AC Problems 47 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 coeffi cient ?

The Ekman number, Ek, arises in geophysical fl 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.

During World War II, Sir Geoffrey Taylor, a British fl 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

Air, assumed to be an ideal gas with k 5 1.40, fl ows isentropically through a nozzle. At section 1, conditions are sea level standard (see Table A.6). At section 2, the temperature is 2 50 8 C. Estimate ( a ) the pressure, and ( b ) the density of the air at section 2.

On a summer day in Narragansett, Rhode Island, the air temperature is 74 8 F and the barometric pressure is 14.5 lbf/in 2 . Estimate the air density in kg/m 3

When we in the United States say a cars tire is fi lled to 32 lb, we mean that its internal pressure is 32 lbf/in 2 above the ambient atmosphere. If the tire is at sea level, has a volume of 3.0 ft 3 , and is at 75 8 F, estimate the total weight of air, in lbf, inside the tire.

For steam at a pressure of 45 atm, some values of temperature and specifi c volume are as follows, from Ref. 23: T , 8 F 500 600 700 800 900 v , ft 3 /lbm 0.7014 0.8464 0.9653 1.074 1.177 Find an average value of the predicted gas constant R in m 2 /(s 2 K). Does this data reasonably approx i mate an ideal gas? If not, explain.

Wet atmospheric air at 100 percent relative humidity contains saturated water vapor and, by Daltons law of partial pressures, patm 5 pdry air 1 pwater vapor Suppose this wet atmosphere is at 40 8 C and 1 atm. Calculate the density of this 100 percent humid air, and compare it with the density of dry air at the same conditions

A compressed-air tank holds 5 ft 3 of air at 120 lbf/in 2 gage, that is, above atmospheric pressure. Estimate the energy, in ft-lbf, required to compress this air from the atmosphere, assuming an ideal isothermal process.

Repeat Prob. 1.29 if the tank is fi 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?

One cubic foot of argon gas at 10 8 C 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 10 8 C, what will be the fi nal pressure?

A blimp is approximated by a prolate spheroid 90 m long and 30 m in diameter. Estimate the weight of 20 8 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)?

A tank contains 9 kg of CO 2 at 20 8 C and 2.0 MPa. Estimate the volume of the tank, in m 3 .

Consider steam at the following state near the saturation line: ( p1 , T1 ) 5 (1.31 MPa, 290 8 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.

In Table A.4, most common gases (air, nitrogen, oxygen, hydrogen) have a specifi c heat ratio k < 1.40. Why do argon and helium have such high values? Why does NH 3 have such a low value? What is the lowest k for any gas that you know of?

Experimental data [55] for the density of n-pentane liquid for high pressures, at 50 8 C, are listed as follows:

( a ) Fit this data to reasonably accurate values of B and n from Eq. (1.19). ( b ) Evaluate at 30 MPa. P1.37 A near-ideal gas has a molecular weight of 44 and a specifi c heat cv 5 610 J/(kg K). What are ( a ) its specifi c heat ratio, k , and ( b ) its speed of sound at 100 8 C?

In Fig. 1.7, if the fl uid is glycerin at 20 8 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?

Knowing for air at 20 8 C from Table 1.4, estimate its viscosity at 500 8 C 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.

Glycerin at 20 8 C fi lls the space between a hollow sleeve of diameter 12 cm and a fi 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 m per meter of rod length, to hold the inner rod fi xed

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 fi lled with SAE 50 oil at 20 8 C. 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

Helium at 20 8 C has a viscosity of 1.97 E-5 kg/(m s). Use the data of Table A.4 to estimate the temperature, in 8 C, at which heliums viscosity will double.

For the fl ow of gas between two parallel plates of Fig. 1.7, reanalyze for the case of slip fl ow at both walls. Use the simple slip condition, uwall 5 < ( du/dy ) wall , where < is the mean free path of the fl uid. Sketch the expected velocity profi le and fi nd an expression for the shear stress at each wall

One type of viscometer is simply a long capillary tube. A commercial device is shown in Prob. C1.10. One measures the volume fl 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 p < 8QL/(R4 ) . For a capillary of diameter 4 mm and length 10 inches, the test fl uid fl ows at 0.9 m 3 /h when the pressure drop is 58 lbf/in 2 . Find the predicted viscosity in kg/m s.

A block of weight W slides down an inclined plane while lubricated by a thin fi lm of oil, as in Fig. P1.45. The fi lm contact area is A and its thickness is h . Assuming a linear velocity distribution in the fi lm, derive an expression for the terminal (zero-acceleration) velocity V of the block. Find the terminal velocity of the block if the block mass is 6 kg, A 5 35 cm 2 , 5 15 8 , and the fi lm is 1-mm-thick SAE 30 oil at 20 8 C. P1.45 L

6 A simple and popular model for two nonnewtonian fl uids in Fig. 1.8 a is the power-law : < C a du dy b n where C and n are constants fi t to the fl uid [16]. From Fig.1.8 a , deduce the values of the exponent n for which the fl uid is ( a ) newtonian, ( b ) dilatant, and ( c ) pseudoplastic. Consider the specifi c model constant C 5 0.4 N s n /m 2 , with the fl uid being sheared between two parallel plates as in Fig.1.7. If the shear stress in the fl uid is 1200 Pa, fi nd 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.

Data for the apparent viscosity of average human blood, at normal body temperature of 37 8 C, varies with shear strain rate, as shown in the following table. Strain rate, s 21 1 10 100 1000 Apparent viscosity, 0.011 0.009 0.006 0.004 kg/(m s) ( a ) Is blood a nonnewtonian fl uid? ( b ) If so, what type of fl uid is it? ( c ) How do these viscosities compare with plain water at 37 8 C?

A thin plate is separated from two fi xed 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 fl uid. ( a ) Assuming a linear velocity distribution in each fl 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 ?

An amazing number of commercial and laboratory devices have been developed to measure fl uid viscosity, as described in Refs. 29 and 49. Consider a concentric shaft, fi xed 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 Problems 49 applied torque be M. Using these parameters, derive a theoretical relation for the viscosity of the fl uid between the cylinders.

A simple viscometer measures the time t for a solid sphere to fall a distance L through a test fl uid of density . The fl uid viscosity is then given by < Wnett 3DL if t $ 2DL where D is the sphere diameter and Wnet is the sphere net weight in the fl uid. ( a ) Prove that both of these formulas are dimensionally homogeneous. ( b ) Suppose that a 2.5 mm diameter aluminum sphere (density 2700 kg/m 3 ) falls in an oil of density 875 kg/m 3 . If the time to fall 50 cm is 32 s, estimate the oil viscosity and verify that the inequality is valid.

An approximation for the boundary-layer shape in Figs. 1.5 b and P1.51 is the formula u( y) < U sina y 2 b, 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 fl uid is helium at 20 8 C 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 ) fi nd the position in the boundary layer where is one-half of w . P1.51 U y 0 u(y) y

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 profi 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 20 8 C, with L 5 2 m, b 5 60 cm, and h 5 3 cm?

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 fi lled with oil of viscosity . Neglecting air drag, derive an analytical expression for the cones angular velocity ( t ) if there is no applied torque. Oil h Base radius r0 (t) 2

A disk of radius R rotates at an angular velocity V inside a disk-shaped container fi lled with oil of viscosity , as shown in Fig. P1.54. Assuming a linear velocity profi le and neglecting shear stress on the outer disk edges, derive a formula for the viscous torque on the disk. R R Clearance h Oil P1.54

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 fi 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 profi le in the oil fi lm. P1.55 U W

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 fi lled with the test liquid. The torque M to rotate the cone at a rate V is measured. Assuming a linear velocity profi le in the fl uid fi lm, derive an expression for fl uid viscosity as a function of ( M , R , V , ). Fluid R

Extend the steady fl ow between a fi 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. h2 y x h1 1 2 V Fixed(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?

The laminar pipe fl ow example of Prob. 1.12 can be used to design a capillary viscometer [29]. If Q is the volume fl 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 r 4 0p 8LQ Pipe end effects are neglected [29]. Suppose our capillary has r0 5 2 mm and L 5 25 cm. The following fl ow rate and pressure drop data are obtained for a certain fl uid: Q , m 3 /h 0.36 0.72 1.08 1.44 1.80 Dp , kPa 159 318 477 1274 1851 What is the viscosity of the fl uid? Note: Only the fi rst three points give the proper viscosity. What is peculiar about the last two points, which were measured accurately?

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 fi lled with fl 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 fi lm of SAE 30 oil at 20 8 C.

0 Pipelines are cleaned by pushing through them a closefi tting 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 fi lled with glycerin at 20 8 C, what pressure difference, in pascals, is needed to drive the pig? Assume a linear velocity profi le in the oil and neglect air drag.

An air-hockey puck has a mass of 50 g and is 9 cm in diameter. When placed on the air table, a 20 8 C air fi lm, 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 fi 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 )?

The hydrogen bubbles that produced the velocity profi les in Fig. 1.15 are quite small, D < 0.01 mm. If the hydrogen water interface is comparable to airwater and the water temperature is 30 8 C, estimate the excess pressure within the bubble.

Derive Eq. (1.33) by making a force balance on the fl uid interface in Fig. 1.11 c .

Pressure in a water container can be measured by an open vertical tubesee 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?

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 fl uid is at 60 8 C. Calculate the true fl uid height in the tube and the percentage error due to capillarity if the fl uid is ( a ) water or ( b ) mercury. P1.65 15 cm p

A thin wire ring, 3 cm in diameter, is lifted from a water surface at 20 8 C. 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?

A vertical concentric annulus, with outer radius ro and inner radius ri , is lowered into a fl uid of surface tension Y and contact angle , 90 8 . Derive an expression for the capillary rise h in the annular gap if the gap is very narrow

A vertical concentric annulus, with outer radius ro and inner radius ri , is lowered into a fl uid of surface tension Y and contact angle , 90 8 . Derive an expression for the capillary rise h in the annular gap if the gap is very narrow

A solid cylindrical needle of diameter d , length L , and density n may fl oat in liquid of surface tension Y . Neglect buoyancy and assume a contact angle of 0 8 . Derive a formula for the maximum diameter dmax able to fl oat in the liquid. Calculate dmax for a steel needle (SG 5 7.84) in water at 20 8 C.

Derive an expression for the capillary height change h for a fl 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 20 8 C if W 5 0.5 mm? P1.70

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 fi nding D3 as a function of D1 , D2 , patm , and Y .

Early mountaineers boiled water to estimate their altitude. If they reach the top and fi nd that water boils at 84 8 C, approximately how high is the mountain?

A small submersible moves at velocity V , in fresh water at 20 8 C, at a 2-m depth, where ambient pressure is 131 kPa. Its critical cavitation number is known to be Ca 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 (5 8 C)?

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?

An airplane fl ies at 555 mi/h. At what altitude in the standard atmosphere will the airplanes Mach number be exactly 0.8?

Derive a formula for the bulk modulus of an ideal gas, with constant specifi c heats, and calculate it for steam at 300 8 C and 200 kPa. Compare your result to the steam tables.

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 fi t Eq. (1.19) with B 5 260 and n 5 11.]

Sir Isaac Newton measured the speed of sound by timing the difference between seeing a cannons 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.

From Table A.3, the density of glycerin at standard conditions is about 1260 kg/m 3 . At a very high pressure of 8000 lb/in 2 , its density increases to approximately 1275 kg/m 3 . Use this data to estimate the speed of sound of glycerin, in ft/s.

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?

Use Eq. (1.39) to fi nd and sketch the streamlines of the following fl ow fi eld: u 5 Kx; v 5 2Ky; w 5 0, where K is a constant.

A velocity fi 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 fl ow.

Use Eq. (1.39) to fi nd and sketch the streamlines of the following fl ow fi eld: u 5 K(x2 2 y2 ); v 5 22Kxy; w 5 0, where K is a constant. Hint: This is a fi rst-order exact differential equation.

In the early 1900s, the British chemist Sir Cyril Hinshelwood quipped that fl uid dynamics study was divided into workers who observed things they could not explain and workers who explained things they could not observe. To what historic situation was he referring?

Do some reading and report to the class on the life and achievements, especially vis--vis fl uid mechanics, of ( a ) Evangelista Torricelli (16081647) ( b ) Henri de Pitot (16951771) ( c ) Antoine Chzy (17181798) ( d ) Gotthilf Heinrich Ludwig Hagen (17971884) ( e ) Julius Weisbach (18061871) ( f ) George Gabriel Stokes (18191903) ( g ) Moritz Weber (18711951) ( h ) Theodor von Krmn (18811963) ( i ) Paul Richard Heinrich Blasius (18831970) ( j ) Ludwig Prandtl (18751953) ( k ) Osborne Reynolds (18421912) ( l ) John William Strutt, Lord Rayleigh (18421919) ( m ) Daniel Bernoulli (17001782) ( n ) Leonhard Euler (17071783)

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.