problemas propuestos viscosidad - cengel munson white fox

COMPILADO DE PROBLEMAS DE VISCOSIDAD - 1 Problemas de Traslacin

R: F1=105,5N ; F2= 57,2N

R: F = 1,62N (Y = 6mm desde la pared en mvto)

Datos: Problemas de Rotacin

a) Discos

Nota: Los problemas 1.89 y 2.47 son semejantes.

R: T = 0,55 N.m , T=

CAPTULO 259

2-17I El anlisis de una hlice que opera en el agua a 70Fmuestra que la presin en las puntas de la misma cae hasta 0.1psia a altas velocidades. Determine si existe peligro de cavita-cin para esta hlice.2-18 Se usa una bomba para transportar agua hasta un depsi-to alto. Si la temperatura del agua es de 25C, determine la pre-sin ms baja que puede existir en la bomba sin cavitacin.

Energa y calores especficos

2-19C Cul es la diferencia entre las formas macroscpica ymicroscpica de la energa?2-20C Qu es energa total? Identifique las diferentes formasde energa que constituyen la energa total.2-21C Haga una lista de las formas de energa que contribu-yen a la energa interna de un sistema.2-22C Cmo estn interrelacionados el calor, la energa in-terna y la energa trmica?2-23C Qu es energa de flujo? Los fluidos en reposo tie-nen alguna energa de flujo?2-24C Qu comparacin existe entre las energas de un flui-do fluyente y uno en reposo? Nombre las formas especficas deenerga asociadas con cada caso.2-25C Con el empleo de calores especficos promedios, expli-que cmo se pueden determinar los cambios en la energa inter-na de los gases ideales y de las sustancias incompresibles.2-26C Con el empleo de calores especficos promedios, expli-que cmo se pueden determinar los cambios en la entalpa delos gases ideales y de las sustancias incompresibles.

Coeficiente de compresibilidad

2-27C Qu representa el coeficiente de compresibilidad deun fluido? Cul es su diferencia con la compresibilidad isotr-mica?2-28C Qu representa el coeficiente de expansin volumtri-ca de un fluido? Cul es su diferencia con el coeficiente decompresibilidad?2-29C Puede ser negativo el coeficiente de compresibilidadde un fluido? Qu se puede decir acerca del coeficiente deexpansin volumtrica?2-30 Se observa que la densidad de un gas ideal decrece en 10por ciento cuando se comprime en forma isotrmica de 10 atmhasta 11 atm. Determine el porcentaje de disminucin en la den-sidad del gas si se comprime en forma isotrmica de 100 atmhasta 101 atm.2-31 Con la definicin del coeficiente de expansin volum-trica y la expresin bgas ideal ! 1/T, demuestre que el porcentajede incremento en el volumen especfico de un gas ideal durantela expansin isobrica es igual al porcentaje de aumento en latemperatura absoluta.2-32 Se comprime en forma isotrmica agua a la presin de 1atm hasta una presin de 800 atm. Determine el incremento enla densidad del agua. Tome la compresibilidad isotrmica delagua como 4.80 " 10#5 atm#1.

2-33 Se calienta agua a 15C y una presin de 1 atm hasta100C, a presin constante. Con los datos del coeficiente deexpansin volumtrica, determine el cambio en la densidad delagua. Respuesta: #38.7 kg/m3

2-34 Se enfra lquido saturado de refrigerante-134a a 10Chasta 0C, a presin constante. Con los datos del coeficiente deexpansin volumtrica determine el cambio en la densidad delrefrigerante.2-35 Un tanque se llena por completo con agua lquida a20C. El material del tanque es tal que puede soportar tensincausada por una expansin volumtrica de 2 por ciento. Deter-mine la elevacin mxima en la temperatura admisible sinponer en peligro la seguridad.2-36 Repita el problema 2-35 para una expansin volumtricade 1 por ciento para el agua.2-37 La densidad del agua de mar en una superficie libredonde la presin es de 98 kPa es aproximadamente de 1 030kg/m3. Tome el mdulo de elasticidad de volumen del agua demar como 2.34 " 109 N/m2 y expresando la variacin de la pre-sin con la profundidad z como dP ! rg dz determine la densi-dad y la presin a una profundidad de 2 500 m. Descarte elefecto de la temperatura.

Viscosidad

2-38C Qu es viscosidad? Cul es la causa de su presenciaen los lquidos y en los gases? Tienen los lquidos una viscosi-dad dinmica ms elevada o los gases?2-39C Qu es un fluido newtoniano? Es el agua un fluidonewtoniano?2-40C Considere dos pequeas bolas de vidrio idnticas quese dejan caer en dos recipientes idnticos, uno lleno con agua yel otro con aceite. Cul de las dos bolas llegar primero al fon-do del recipiente? Por qu?2-41C Cmo vara la viscosidad dinmica de a) los lquidosy b) los gases con la temperatura?2-42C Cmo vara la viscosidad cinemtica de a) los lqui-dos y b) los gases con la temperatura?2-43 Se debe mover un bloque de 50 cm " 30 cm " 20 cmque pesa 150 N a una velocidad constante de 0.8 m/s sobre unasuperficie inclinada con un coeficiente de friccin de 0.27. a)Determine la fuerza F necesaria a aplicar en la direccin hori-zontal. b) Si se aplica una pelcula de aceite de 0.4 mm de espe-sor, con una viscosidad dinmica de 0.012 Pa ! s entre el bloquey la superficie inclinada, determine el porcentaje de reduccinen la fuerza necesaria.

150 N

F

= 0.8 m/s

30 cm50 cm

20

V

FIGURA P2-43

ENGEL 02C 2/22/06 4:41 AM Page 59

2-44 Considere el flujo de un fluido con viscosidad m por untubo circular. El perfil de velocidad en el tubo se expresa comou(r) ! umx(1 " rn/Rn), en donde umx es la velocidad mximade flujo, la cual se tiene en la lnea central; r es la distanciaradial desde la lnea central y u(r) es la velocidad de flujo encualquier posicin r. Desarrolle una relacin para la fuerza dearrastre ejercida sobre la pared del tubo por el fluido en ladireccin del flujo, por unidad de longitud del tubo.

60PROPIEDADES DE LOS FLUIDOS

aceite SAE 10W a 20C (m ! 0.1 Pa ! s), como se muestra enla figura P2-46. Si, especialmente en los lados, el espesor de lapelcula de aceite es de 1.2 mm, determine la potencia necesariapara mantener este movimiento. Determine tambin la reduc-cin en el consumo de potencia necesario cuando la temperatu-ra del aceite se eleva hasta 80C (m ! 0.0078 Pa ! s).2-47 El sistema de embrague que se muestra en la figura P2-47 se usa para transmitir par de torsin mediante una pelcula de aceite con m ! 0.38 N ! s/m2 que est entre dos discos idn-ticos de 30 cm de dimetro. Cuando la flecha impulsora gira a una velocidad de 1 450 rpm, se observa que la flecha impulsa-da gira a 1 398 rpm. Suponiendo un perfil lineal de velocidadpara la pelcula de aceite, determine el par de torsin transmi-tido.

2-45 Se jala horizontalmente de una placa plana delgada de20 cm # 20 cm a 1 m/s a travs de una capa de aceite de 3.6mm de espesor, que est entre dos placas, una estacionaria y laotra movindose a una velocidad constante de 0.3 m/s, como semuestra en la figura P2-45. La viscosidad dinmica del aceitees de 0.027 Pa ! s. Suponiendo que la velocidad en cada una delas capas de aceite vara en forma lineal, a) trace la grfica delperfil de velocidad y encuentre el lugar en donde la velocidaddel aceite es cero y b) determine la fuerza que se necesitaaplicar sobre la placa para mantener este movimiento.

r

R

umx

u(r) = umx(1 rn/Rn)

0

FIGURA P2-44

2-46 Un cuerpo en forma de cono cortado gira a velocidadangular constante de 200 rad/s en un recipiente lleno con

F

Pared fija

Pared en movimiento

= 1 m/sh1 = 1 mm

h2 = 2.6 mm w = 0.3 m/s

V

V

FIGURA P2-45

D = 12 cm

L = 12 cm

d = 4 cm

Caja

Aceite SAE 10W

r

z

FIGURA P2-46

2-48 Reconsidere el problema 2-47. Con el softwarede EES (o cualquier otro programa de este tipo),

investigue el efecto del espesor de la pelcula de aceite en elpar de torsin transmitido. Haga que el espesor de la pelculavare desde 0.1 mm hasta 10 mm. Trace la grfica de los re-sultados que obtenga y exprese sus conclusiones. 2-49 La viscosidad de algunos fluidos cambia cuando seaplica un fuerte campo elctrico en ellos. Este fenmeno seconoce como efecto electrorreolgico (ER) y los fluidos quemuestran un comportamiento de este tipo se conocen comofluidos ER. El modelo del plstico de Bingham para elesfuerzo cortante, el cual se expresa como t ! ty $ m(du/dy)se usa con amplitud para describir el comportamiento de losfluidos ER, debido a su sencillez. Una de las aplicacionesms promisorias de los fluidos ER es el embrague ER. Unembrague ER tpico de discos mltiples consta de varios dis-cos de acero igualmente espaciados de radio interior R1 yradio exterior R2, N de ellos sujetos a la flecha de entrada.La brecha h entre los discos paralelos se llena con un lquidoviscoso. a) Encuentre una relacin para el par de torsingenerado por el embrague cuando la flecha de salida estestacionaria y b) calcule el par de torsin para un embrague

30 cm

Flechaimpulsora

Flechaimpulsora

Aceite SAE 30W

3 mm

FIGURA P2-47

Casco

Flecha de entradaPlacas montadas sobre

la flecha de entradaCampo magntico variable

Flecha de salida

R2 R1

Placas montadassobre el casco

h = 1.2 mm

FIGURA P2-49

ENGEL 02C 2/22/06 4:41 AM Page 60

Problems 35

b

cb

2

V

V1

Figure P1.75

Figure P1.77

Figure P1.80

V0.1 mm gap

20

liquid of interest, the viscosity is given by Andrades equation (Eq.1.11) with and . By what percentage will the velocity increase as the liquid temperature isincreased from 40 !F to 100 !F? Assume all other factors remainconstant.

*1.72 Use the value of the viscosity of water given in Table B.2at temperatures of 0, 20, 40, 60, 80, and to determine theconstants D and B which appear in Andrades equation 1Eq. 1.112.Calculate the value of the viscosity at and compare withthe value given in Table B.2. 1Hint: Rewrite the equation in theform

and plot ln versus From the slope and intercept of this curve,B and D can be obtained. If a nonlinear curve-fitting program isavailable, the constants can be obtained directly from Eq. 1.11 with-out rewriting the equation.21.73 For a certain liquid m " 7.1 # 10$5 lb # s/ft2 at 40 !F and m" 1.9 # 10$5 lb # s/ft2 at 150 !F. Make use of these data to deter-mine the constants D and B which appear in Andrades equation(Eq. 1.11). What would be the viscosity at 80 !F?1.74 For a parallel plate arrangement of the type shown inFig. 1.5 it is found that when the distance between plates is 2 mm,a shearing stress of 150 Pa develops at the upper plate when it ispulled at a velocity of 1 m/s. Determine the viscosity of the fluidbetween the plates. Express your answer in SI units.

1.75 Two flat plates are oriented parallel above a fixed lower plateas shown in Fig. P1.75. The top plate, located a distance b abovethe fixed plate, is pulled along with speed V. The other thin plateis located a distance cb, where 0 % c % 1, above the fixed plate.This plate moves with speed V1, which is determined by the vis-cous shear forces imposed on it by the fluids on its top and bot-tom. The fluid on the top is twice as viscous as that on the bot-tom. Plot the ratio V1/V as a function of c for 0 % c % 1.

1.76 There are many fluids that exhibit non-Newtonian behavior (see, for example, Video V1.6). For a given fluid the dis-tinction between Newtonian and non-Newtonian behavior is usu-ally based on measurements of shear stress and rate of shearingstrain. Assume that the viscosity of blood is to be determined by measurements of shear stress, !, and rate of shearing strain,du/dy, obtained from a small blood sample tested in a suitable vis-cometer. Based on the data given below, determine if the blood isa Newtonian or non-Newtonian fluid. Explain how you arrived atyour answer.

!(N/m2) 0.04 0.06 0.12 0.18 0.30 0.52 1.12 2.10du/dy ( ) 2.25 4.50 11.25 22.5 45.0 90.0 225 4501.77 The sled shown in Fig. P1.77 slides along on a thin hor-izontal layer of water between the ice and the runners. The horizon-tal force that the water puts on the runners is equal to 1.2 lb whenthe sleds speed is 50 ft/s. The total area of both runners in contactwith the water is , and the viscosity of the water is0.08 ft2

s$1

GO

1&T.m

ln m " 1B2 1T' ln D

50 C

100 C

B " 4000 RD " 5 # 10$7 lb # s&ft2Determine the thickness of the water layer un-

der the runners. Assume a linear velocity distribution in the water layer.

1.78 A 25-mm-diameter shaft is pulled through a cylindri-cal bearing as shown in Fig. P1.78. The lubricant that fills the 0.3-mm gap between the shaft and bearing is an oil having a kine-matic viscosity of and a specific gravity of 0.91.Determine the force P required to pull the shaft at a velocity of 3 m/s. Assume the velocity distribution in the gap is linear.

1.79 A piston having a diameter of 5.48 in. and a length of 9.50in. slides downward with a velocity V through a vertical pipe. Thedownward motion is resisted by an oil film between the piston andthe pipe wall. The film thickness is 0.002 in., and the cylinderweighs 0.5 lb. Estimate V if the oil viscosity is 0.016 lb # s/ft2. As-sume the velocity distribution in the gap is linear.

1.80 A 10-kg block slides down a smooth inclined sur-face as shown in Fig. P1.80. Determine the terminal velocity ofthe block if the 0.1-mm gap between the block and the surfacecontains SAE 30 oil at 60 F. Assume the velocity distributionin the gap is linear, and the area of the block in contact withthe oil is 0.1 m2.

1.81 A layer of water flows down an inclined fixed surfacewith the velocity profile shown in Fig. P1.81. Determine the mag-nitude and direction of the shearing stress that the water exerts onthe fixed surface for .U " 2 m&s and h " 0.1 m

8.0 # 10$4 m2&s

3.5 # 10$5 lb # s&ft2.

0.5 m

LubricantBearing

ShaftP

Figure P1.78

c01Introduction.qxd 2/13/12 3:53 PM Page 35

Chapter 11.2 1a2 MLT!1; 1b2 ML4 L!2; 1c2 MT!21.4 1a2 FLT!2; 1b2 FL!1T!1; 1c2 FL!3 T1.10 1a2 FL!1; 1b2 FL!3; FL1.12 LT!1; F0 L0 T0; LT!11.14 Yes1.16 1!2, !1!21.18 Dimensionless1.20 1a2 4.32 mm!s; 1b2 70.2 kg; 1c2 13.4 N; 1d2 22.3 m!s2; 1e2 1.12 N " s!m21.22 1a2 6.47 # 105 m2; 1b2 56.8 # 10!2 m3; 1c2 3.86 # 105 m; 1d2 5.90 # 104 W; 1e2 289 K1.30 1a2 0.240 mi3; 1b2 4.41 # 105 lb1.32 30.6 kg; 37.3 N1.34 1150 kg!m3; 11.3 kN!m31.36 0.0186 ft3; No1.38 0.9971 lb1.40 991.5 kg!m31.42 16.0 kN!m3; 1.63 # 103 kg!m3; 1.631.44 4.76 kg1.46 1a2 0.0214 kg!m3; 1b2 rMars!rearth $ 1.75%1.48 6.44 # 10!3 slugs!ft3; 0.622 lb1.50 98.7 psia1.52 668 lb1.56 0.6 N " s!m2; 1.3 # 10!2 lb " s!ft21.58 0.727 N " s!m21.60 23.7, 2.55 # 10!21.62 0.277 N " s!m21.64 2.05 # 10!5 N " s!m21.66 1a2 No; 1b2 Not correct1.68 1841.70 C $ 1.43 # 10!6 kg!1m " s " K1!22; S $ 107 K1.72 D $ 1.767 # 10!6 N " s!m2; B $ 1.870 # 103 K;

5.76 # 10!4 N " s!m21.74 0.300 N " s!m21.76 non-Newtonian1.78 286 N1.80 0.0883 m!s1.82 0.268 ft!s1.84 3.43 # 10!4 lb1.86 9.53 # 10!4 ft " lb1.88 1a2 12.7 ft2!rev; 1b2 4.73 # 10!3 lb " s!ft21.94 2.03 # 103 psi1.96 4.14 # 103 psi

1.98 1a2 343 m!s; 1b2 1010 m!s; 1c2 446 m!s1.100 104 psi 1gage21.102 2.88 kg!m31.104 4.25 # 10!3 slugs!ft3; 305 %F1.108 1.061.110 14.1%1.112 4.74 psi 1abs21.114 13 kPa 1abs21.116 5.81 kPa 1abs2; 0.842 psi 1abs21.118 0.060 N!m1.120 538 Pa1.122 97.9 Pa1.124 1a2 24.5 deg1.126 0.0614 in.; Yes1.128 1.80 # 10!2 ft1.130 7.49 mm

Chapter 22.2 59.2 kPa2.6 34.7 psi2.8 404 kPa2.10 1a2 p $ !Ev ln 11 ! r0gh!Ev2; 1b2 61.4 MPa; 1c2 60.6 MPa2.12 p $ Kh2!2 & '0h2.18 464 mm2.20 62.9%2.24 4,250 ft2.26 543 m2.28 60 kPa2.32 14.4 psia; 99.3 kPa 1abs22.34 1a2 18.2 ft; 1b2 8.73 psi; 21.7 psia2.36 0.317 ft2.38 6.28 ft2.40 !3.32 kPa2.42 1a2 4.00 ft; 1b2 2.08 ft2.44 1.45 ft2.46 1a2 26.9 kPa; 1b2 0.202 m2.48 1.55 slugs!ft32.50 94.9 kPa2.54 21.6 ft2.56 575 lb!ft22.58 0.100 m2.60 27.8%2.62 0.212 m

Answers toSelected Even-Numbered Homework Problems

ANS-1

BMAns.qxd 2/15/12 9:01 PM Page ANS-1*P1.55 The device in Fig. P1.54 is called a rotating disk viscometer[27]. Suppose that R ! 5 cm and h ! 1 mm. If the torquerequired to rotate the disk at 900 r/min is 0.537 N " m,what is the viscosity of the fluid? If the uncertainty in eachparameter (M, R, h, #) is $1 percent, what is the overalluncertainty in the viscosity?

*P1.56 The device in Fig. P1.56 is called a cone-plate viscometer[27]. The angle of the cone is very small, so that sin % !%, and the gap is filled with the test liquid. The torque Mto rotate the cone at a rate # is measured. Assuming a lin-ear velocity profile in the fluid film, derive an expressionfor fluid viscosity & as a function of (M, R, #, %).

& !'(8r

L04

Q)p'

Pipe end effects are neglected [27]. Suppose our capillaryhas r0 ! 2 mm and L ! 25 cm. The following flow rateand pressure drop data are obtained for a certain fluid:

Q, m3/h 0.36 0.72 1.08 1.44 1.80

)p, kPa 159 318 477 1274 1851

What is the viscosity of the fluid? Note: Only the first threepoints give the proper viscosity. What is peculiar about thelast two points, which were measured accurately?

P1.59 A solid cylinder of diameter D, length L, and density *sfalls due to gravity inside a tube of diameter D0. The clear-ance, D0 + D ,, D, is filled with fluid of density * andviscosity &. Neglect the air above and below the cylinder.Derive a formula for the terminal fall velocity of the cylin-der. Apply your formula to the case of a steel cylinder,D ! 2 cm, D0 ! 2.04 cm, L ! 15 cm, with a film of SAE30 oil at 20C.

P1.60 For Prob. 1.52 suppose that P ! 0.1 hp when V ! 6 ft/s,L ! 4.5 ft, b ! 22 in, and h ! 7/8 in. Estimate the vis-cosity of the oil, in kg/(m " s). If the uncertainty in eachparameter (P, L, b, h, V) is $1 percent, what is the over-all uncertainty in the viscosity?

*P1.61 An air-hockey puck has a mass of 50 g and is 9 cm in di-ameter. When placed on the air table, a 20C air film, of0.12-mm thickness, forms under the puck. The puck isstruck with an initial velocity of 10 m/s. Assuming a lin-ear velocity distribution in the air film, how long will ittake the puck to (a) slow down to 1 m/s and (b) stop com-pletely? Also, (c) how far along this extremely long tablewill the puck have traveled for condition (a)?

P1.62 The hydrogen bubbles which produced the velocity pro-files in Fig. 1.13 are quite small, D! 0.01 mm. If the hy-drogen-water interface is comparable to air-water and thewater temperature is 30C estimate the excess pressurewithin the bubble.

P1.63 Derive Eq. (1.37) by making a force balance on the fluidinterface in Fig. 1.9c.

P1.64 At 60C the surface tension of mercury and water is 0.47and 0.0662 N/m, respectively. What capillary heightchanges will occur in these two fluids when they are incontact with air in a clean glass tube of diameter 0.4 mm?

P1.65 The system in Fig. P1.65 is used to calculate the pressurep1 in the tank by measuring the 15-cm height of liquid inthe 1-mm-diameter tube. The fluid is at 60C (see Prob.1.64). Calculate the true fluid height in the tube and thepercent error due to capillarity if the fluid is (a) water and(b) mercury.

Problems 51

R R

Clearance h

Oil

P1.54

FluidR

*P1.57 For the cone-plate viscometer of Fig. P1.56, suppose thatR ! 6 cm and % ! 3. If the torque required to rotate thecone at 600 r/min is 0.157 N " m, what is the viscosity ofthe fluid? If the uncertainty in each parameter (M, R, #,%) is $1 percent, what is the overall uncertainty in the vis-cosity?

*P1.58 The laminar-pipe-flow example of Prob. 1.12 can be usedto design a capillary viscometer [27]. If Q is the volumeflow rate, L is the pipe length, and )p is the pressure dropfrom entrance to exit, the theory of Chap. 6 yields a for-mula for viscosity:

P1.56

32 Solutions Manual Fluid Mechanics, Fifth Edition

1.58 The laminar-pipe-flow example of Prob. 1.14 leads to a capillary viscometer [27], using the formula = ro4p/(8LQ). Given ro = 2 mm and L = 25 cm. The data are

Q, m3/hr: 0.36 0.72 1.08 1.44 1.80 p, kPa: 159 318 477 1274 1851

Estimate the fluid viscosity. What is wrong with the last two data points?

Solution: Apply our formula, with consistent units, to the first data point: 4 4 2o

3 2r p (0.002 m) (159000 N/m ) N sp 159 kPa: 0.040

8LQ 8(0.25 m)(0.36/3600 m /s) m = =

Do the same thing for all five data points:

p, kPa: 159 318 477 1274 1851 , Ns/m2: 0.040 0.040 0.040 0.080(?) 0.093(?) Ans.

The last two estimates, though measured properly, are incorrect. The Reynolds number of the capillary has risen above 2000 and the flow is turbulent, which requires a different formula.

1.59 A solid cylinder of diameter D, length L, density s falls due to gravity inside a tube of diameter Do. The clearance, o(D D) D, ! is filled with a film of viscous fluid (,). Derive a formula for terminal fall velocity and apply to SAE 30 oil at 20C for a steel cylinder with D = 2 cm, Do = 2.04 cm, and L = 15 cm. Neglect the effect of any air in the tube.

Solution: The geometry is similar to Prob. 1.47, only vertical instead of horizontal. At terminal velocity, the cylinder weight should equal the viscous drag:

2z z s

o

Va 0: F W Drag g D L DL,

4 (D D)/2 ! "= = + = +

# $% &

or: V .Ans= s ogD(D D)8

For the particular numerical case given, steel 7850 kg/m3. For SAE 30 oil at 20C, 0.29 kg/ms from Table 1.4. Then the formula predicts

3 2s o

terminalgD(D D) (7850 kg/m )(9.81 m/s )(0.02 m)(0.0204 0.02 m)V

8 8(0.29 kg/m s) .Ans

= =

0.265 m/s

32 Solutions Manual Fluid Mechanics, Fifth Edition

1.58 The laminar-pipe-flow example of Prob. 1.14 leads to a capillary viscometer [27], using the formula = ro4p/(8LQ). Given ro = 2 mm and L = 25 cm. The data are

Q, m3/hr: 0.36 0.72 1.08 1.44 1.80 p, kPa: 159 318 477 1274 1851

Estimate the fluid viscosity. What is wrong with the last two data points?

Solution: Apply our formula, with consistent units, to the first data point: 4 4 2o

3 2r p (0.002 m) (159000 N/m ) N sp 159 kPa: 0.040

8LQ 8(0.25 m)(0.36/3600 m /s) m = =

Do the same thing for all five data points:

p, kPa: 159 318 477 1274 1851 , Ns/m2: 0.040 0.040 0.040 0.080(?) 0.093(?) Ans.

The last two estimates, though measured properly, are incorrect. The Reynolds number of the capillary has risen above 2000 and the flow is turbulent, which requires a different formula.

1.59 A solid cylinder of diameter D, length L, density s falls due to gravity inside a tube of diameter Do. The clearance, o(D D) D, ! is filled with a film of viscous fluid (,). Derive a formula for terminal fall velocity and apply to SAE 30 oil at 20C for a steel cylinder with D = 2 cm, Do = 2.04 cm, and L = 15 cm. Neglect the effect of any air in the tube.

Solution: The geometry is similar to Prob. 1.47, only vertical instead of horizontal. At terminal velocity, the cylinder weight should equal the viscous drag:

2z z s

o

Va 0: F W Drag g D L DL,

4 (D D)/2 ! "= = + = +

# $% &

or: V .Ans= s ogD(D D)8

For the particular numerical case given, steel 7850 kg/m3. For SAE 30 oil at 20C, 0.29 kg/ms from Table 1.4. Then the formula predicts

3 2s o

terminalgD(D D) (7850 kg/m )(9.81 m/s )(0.02 m)(0.0204 0.02 m)V

8 8(0.29 kg/m s) .Ans

= =

0.265 m/s

Problems 37

1.88 One type of rotating cylinder viscometer, called a Stormerviscometer, uses a falling weight, !, to cause the cylinder to ro-tate with an angular velocity, v, as illustrated in Fig. P1.88. Forthis device the viscosity, m, of the liquid is related to ! and vthrough the equation ! ! Kmv, where K is a constant that depends only on the geometry (including the liquid depth) of theviscometer. The value of K is usually determined by using a cali-bration liquid (a liquid of known viscosity).(a) Some data for a particular Stormer viscometer, obtained usingglycerin at 20 "C as a calibration liquid, are given below. Plot val-ues of the weight as ordinates and values of the angular velocity asabscissae. Draw the best curve through the plotted points and de-termine K for the viscometer.

!(lb) 0.22 0.66 1.10 1.54 2.20v (rev/s) 0.53 1.59 2.79 3.83 5.49

(b) A liquid of unknown viscosity is placed in the same viscometerused in part (a), and the data given below are obtained. Determinethe viscosity of this liquid.

!(lb) 0.04 0.11 0.22 0.33 0.44v (rev/s) 0.72 1.89 3.73 5.44 7.42

Section 1.7 Compressibility of Fluids1.92 Obtain a photograph/image of a situation in which the com-pressibility of a fluid is important. Print this photo and write a briefparagraph that describes the situation involved.1.93 A sound wave is observed to travel through a liquidwith a speed of 1500 m/s. The specific gravity of the liquid is 1.5.Determine the bulk modulus for this fluid.1.94 A rigid-walled cubical container is completely filled with wa-ter at 40 "F and sealed. The water is then heated to 100 "F. Deter-mine the pressure that develops in the container when the waterreaches this higher temperature. Assume that the volume of thecontainer remains constant and the value of the bulk modulus ofthe water remains constant and equal to 300,000 psi.1.95 In a test to determine the bulk modulus of a liquid it wasfound that as the absolute pressure was changed from 15 to 3000psi the volume decreased from 10.240 to 10.138 in.3 Determine thebulk modulus for this liquid.1.96 Estimate the increase in pressure (in psi) required todecrease a unit volume of mercury by 0.1%.1.97 A volume of water is contained in a rigid con-tainer. Estimate the change in the volume of the water when a pis-ton applies a pressure of 35 MPa.1.98 Determine the speed of sound at 20 C in (a) air, (b) helium,and (c) natural gas (methane). Express your answer in m/s.1.99 Calculate the speed of sound in m/s for (a) gasoline, (b) mer-cury, and (c) seawater. 1.100 Air is enclosed by a rigid cylinder containing a pis-ton. A pressure gage attached to the cylinder indicates an initialreading of 25 psi. Determine the reading on the gage when the pis-ton has compressed the air to one-third its original volume. Assume the compression process to be isothermal and the local atmospheric pressure to be 14.7 psi.1.101 Repeat Problem 1.100 if the compression process takes placewithout friction and without heat transfer (isentropic process).1.102 Carbon dioxide at and 300 kPa absolute pres-sure expands isothermally to an absolute pressure of 165 kPa. Determine the final density of the gas.1.103 Oxygen at 30 "C and 300 kPa absolute pressure expandsisothermally to an absolute pressure of 120 kPa. Determine the final density of the gas.1.104 Natural gas at and standard atmospheric pressureof 14.7 psi (abs) is compressed isentropically to a new absolute pres-sure of 70 psi. Determine the final density and temperature of thegas.1.105 Compare the isentropic bulk modulus of air at 101 kPa 1abs2with that of water at the same pressure.*1.106 Develop a computer program for calculating the final gagepressure of gas when the initial gage pressure, initial and final vol-umes, atmospheric pressure, and the type of process 1isothermal orisentropic2 are specified. Use BG units. Check your programagainst the results obtained for Problem 1.100.1.107 Often the assumption is made that the flow of a certain fluidcan be considered as incompressible flow if the density of the fluidchanges by less than 2%. If air is flowing through a tube such thatthe air pressure at one section is 9.0 psi and at a downstream sec-tion it is 8.6 psi at the same temperature, do you think that thisflow could be considered an incompressible flow? Support youranswer with the necessary calculations. Assume standard atmos-pheric pressure.

70 F

30 C

1-m3

Rotating plate

0.1-in. gap

Torque

Figure P1.89

LiquidFixed outercylinder

Weight

! Rotatinginner

cylinder

Figure P1.88

1.89 A 12-in.-diameter circular plate is placed over a fixedbottom plate with a 0.1-in. gap between the two plates filled withglycerin as shown in Fig. P1.89. Determine the torque required torotate the circular plate slowly at 2 rpm. Assume that the velocitydistribution in the gap is linear and that the shear stress on the edgeof the rotating plate is negligible.

1.90 Vehicle shock absorbers damp out oscillations caused byroad roughness. Describe how a temperature change may affect theoperation of a shock absorber.1.91 Some measurements on a blood sample at

indicate a shearing stress of 0.52 for a corre-sponding rate of shearing strain of . Determine the apparentviscosity of the blood and compare it with the viscosity of water at the same temperature.

200 s#1N$m2198.6 F2 37 C






r

R

umx

u(r) = umx(1 rn/Rn)

0

FIGURA P2-44


F

Pared fija

Pared en movimiento

= 1 m/sh1 = 1 mm

h2 = 2.6 mm w = 0.3 m/s

V

V

FIGURA P2-45

D = 12 cm

L = 12 cm

d = 4 cm

Caja

Aceite SAE 10W

r

z

FIGURA P2-46



30 cm

Flechaimpulsora

Flechaimpulsora

Aceite SAE 30W

3 mm

FIGURA P2-47

Casco



Flecha de salida

R2 R1


h = 1.2 mm

FIGURA P2-49

ENGEL 02C 2/22/06 4:41 AM Page 60

Chapter 2 Properties of Fluids

PROPRIETARY MATERIAL

2-47 Solution A clutch system is used to transmit torque through an oil film between two identical disks. For specified rotational speeds, the transmitted torque is to be determined.

Assumptions 1 The thickness of the oil film is uniform. 2 The rotational speeds of the disks remain constant.

Properties The absolute viscosity of oil is given to be = 0.38 Ns/m2.

SAE 30W oil

Driving shaft

Driven shaft

3 mm 30 cm

Analysis The disks are rotting in the same direction at different angular speeds of 1 and of 2 . Therefore, we can assume one of the disks to be stationary and the other to be rotating at an angular speed of 21 . The velocity gradient anywhere in the oil of film thickness h is V /h where rV )( 21 = is the tangential velocity. Then the wall shear stress anywhere on the surface of the faster disk at a distance r from the axis of rotation can be expressed as

h

rhV

drdu

w)( 21 ===

Then the shear force acting on a differential area dA on the surface and the torque generation associated with it can be expressed as

h

1r 2r

drrh

rdAdF w )2(

)( 21 ==

drrh

drrh

rrdFd 321

221 )(2)2(

)(T

=== Integrating,

hDr

hdrr

h

D

r

D

r 32)(

4)(2)(2

T4

212/

0

42132/

0

21 =====

Noting that 2 n = , the relative angular speed is ( ) ( ) ( )1 2 1 2 1 min2 2 rad/rev 1450 1398 rev/min 5.445 rad/s60 sn n

= = = ,

Substituting, the torque transmitted is determined to be

mN 0.55 ==m) 003.0(32

m) 30.0(/s) 445.5)(s/mN 38.0(T42

Discussion Note that the torque transmitted is proportional to the fourth power of disk diameter, and is inversely proportional to the thickness of the oil film.

. 2006 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

2-21

b) Otras Configuraciones Geomtricas

whose properties are ! " 0.003 m2/s and SG " 0.88. Es-timate the force required to pull the shaft at a steady ve-locity of 0.4 m/s.

P1.48 A thin plate is separated from two fixed plates by very vis-cous liquids #1 and #2, respectively, as in Fig. P1.48. Theplate spacings h1 and h2 are unequal, as shown. The con-tact area is A between the center plate and each fluid.(a) Assuming a linear velocity distribution in each fluid,derive the force F required to pull the plate at velocity V.(b) Is there a necessary relation between the two viscosi-ties, #1 and #2?

sured torque is 0.293 N $ m, what is the fluid viscosity?Suppose that the uncertainties of the experiment are as fol-lows: L (%0.5 mm), M (%0.003 N $ m), & (%1 percent),and ri or ro (%0.02 mm). What is the uncertainty in themeasured viscosity?

P1.52 The belt in Fig. P1.52 moves at a steady velocity V andskims the top of a tank of oil of viscosity #, as shown. As-suming a linear velocity profile in the oil, develop a sim-ple formula for the required belt-drive power P as a func-tion 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 oilat 20C, with L " 2 m, b " 60 cm, and h " 3 cm?

50 Chapter 1 Introduction

Liquid film ofthickness h

W

V Block contactarea A

P1.45

P1.48

h1

h2

1

2

F, V

P1.49 The shaft in Prob. 1.47 is now fixed axially and rotated in-side the sleeve at 1500 r/min. Estimate (a) the torque (N $ m) and (b) the power (kW) required to rotate the shaft.

P1.50 An amazing number of commercial and laboratory deviceshave been developed to measure the viscosity of fluids, asdescribed in Ref. 27. The concentric rotating shaft of Prob.1.49 is an example of a rotational viscometer. Let the in-ner and outer cylinders have radii ri and ro, respectively,with total sleeve length L. Let the rotational rate be &(rad/s) and the applied torque be M. Derive a theoreticalrelation for the viscosity of the clearance fluid, #, in termsof these parameters.

P1.51 Use the theory of Prob. 1.50 (or derive an ad hoc expres-sion if you like) for a shaft 8 cm long, rotating at 1200r/min, with ri " 2.00 cm and ro " 2.05 cm. If the mea-

LV

Oil, depth h

Moving belt, width b

P1.52

*P1.53 A solid cone of angle 2', base r0, and density (c is rotat-ing with initial angular velocity )0 inside a conical seat,as shown in Fig. P1.53. The clearance h is filled with oilof viscosity #. Neglecting air drag, derive an analytical ex-pression for the cones angular velocity )(t) if there is noapplied torque.

Oil

h

Baseradius r0

(t)

2

P1.53

*P1.54 A disk of radius R rotates at an angular velocity & insidea disk-shaped container filled with oil of viscosity #, asshown in Fig. P1.54. Assuming a linear velocity profileand neglecting shear stress on the outer disk edges, derivea formula for the viscous torque on the disk.

*P1.55 The device in Fig. P1.54 is called a rotating disk viscometer[27]. Suppose that R ! 5 cm and h ! 1 mm. If the torquerequired to rotate the disk at 900 r/min is 0.537 N " m,what is the viscosity of the fluid? If the uncertainty in eachparameter (M, R, h, #) is $1 percent, what is the overalluncertainty in the viscosity?


& !'(8r

L04

Q)p'


Q, m3/h 0.36 0.72 1.08 1.44 1.80

)p, kPa 159 318 477 1274 1851









Problems 51

R R

Clearance h

Oil

P1.54

FluidR



P1.56

1.68 h ! ("/#g)1/2 cot $1.70 h ! 2" cos $/(#gW)1.72 z! 4800 m1.74 Cavitation occurs for both (a) and (b)1.76 z! 7500 m1.78 (a) 25C; (b) 4C1.80 x2y % y3/3 ! constant1.82 y ! x tan $ & constant1.84 x ! x0{ln (y/y0) & ln2 (y/y0)}

Chapter 22.2 'xy !%289 lb/ft2, (AA !%577 lb/ft22.4 x ! Const e%2Cz/B2.6 (a) 30.3 ft; (b) 30.0 in; (c) 10.35 m; (d) 13,100 mm2.8 DALR ! 9.77C/km2.10 10,500 Pa2.12 8.0 cm2.14 74,450 Pa with air; 75,420 Pa without air2.16 (a) 21,526 cm3; (b) 137 kPa2.18 1.562.20 14 lbf2.22 0.94 cm2.24 psealevel! 117 kPa, mexact ! 5.3 E18 kg2.26 (a) 2580 m; (b) 5410 m2.28 4400 ) 400 ft2.30 101,100 Pa2.32 22.6 cm2.34 *p ! *h[+water(1 & d2/D2) % +oil(1 % d2/D2)]2.36 252.38 (a) p1,gage ! (#m % #a)gh % (#t % #a)gH2.40 21.3 cm2.42 pA % pB ! (#2 % #1)gh2.44 (a) 171 lb/ft2; (b) 392 lb/ft2; manometer reads friction

loss

Chapter 11.2 1.3 E44 molecules1.4 1.63 slug/ft3, 839 kg/m31.6 (a) {L2/T2}; (b) {M/T}1.8 '! 1.00 My/I1.10 Yes, all terms are {ML/T2}1.12 {B} ! {L%1}1.14 Q ! Const B g1/2H3/21.16 All terms are {ML%2T%2}1.18 V ! V0e%mt/K1.20 zmax ! 64.2 m at t ! 3.36 s1.22 (a) %0.372U,2 /R; (b) x !%1.291 R1.24 e ! 221,000 J/kg1.26 Wair ! 0.71 lbf1.28 #wet ! 1.10 kg/m3, #dry ! 1.13 kg/m31.30 W1-2 ! 21 ft - lbf1.32 (a) 76 kN; (b) 501 kN1.34 1300 atm1.36 (a) BN2O ! 1.33 E5 Pa; (b) Bwater ! 2.13 E9 Pa1.38 ( ! 1380 Pa, ReL ! 281.40 A ! 0.0016 kg/(m - s), B ! 1903 K1.42 ./.200K ! (T K/200 K)0.681.44 Data 50 percent higher; Andrade fit varies )50 percent1.46 V! 15 m/s1.48 F! (.1/h1 & .2/h2)AV1.50 . !M(ro % ri)/(2/0ri3L)1.52 P! 73 W1.54 M! /.0R4/h1.56 . ! 3M sin $/(2/0R3)1.58 . ! 0.040 kg/(m - s), last 2 points are turbulent flow1.60 . ! 0.88 ) 0.023 kg/(m - s)1.62 28,500 Pa1.64 (a) %0.023 m; (b) & 0.069 m1.66 F ! 0.014 N806

Answers to Selected Problems





r

R

umx

u(r) = umx(1 rn/Rn)

0

FIGURA P2-44


F

Pared fija

Pared en movimiento

= 1 m/sh1 = 1 mm

h2 = 2.6 mm w = 0.3 m/s

V

V

FIGURA P2-45

D = 12 cm

L = 12 cm

d = 4 cm

Caja

Aceite SAE 10W

r

z

FIGURA P2-46



30 cm

Flechaimpulsora

Flechaimpulsora

Aceite SAE 30W

3 mm

FIGURA P2-47

Casco



Flecha de salida

R2 R1


h = 1.2 mm

FIGURA P2-49

ENGEL 02C 2/22/06 4:41 AM Page 60

ER con N ! 11 para R1 ! 50 mm, R2 ! 200 mm, y n.

! 2 400rpm, si el fluido es SAE 10, con m ! 0.1 Pa ! s, ty ! 2.5 kPa, yh ! 1.2 mm. Respuesta: b) 2 060 N ! m2-50 La viscosidad de algunos fluidos, llamados fluidos mag-netorreolgicos (MR), cambia cuando se aplica un campo mag-ntico. Esos fluidos contienen partculas magnetizables con ta-mao del orden de micras, suspendidas en un lquido portadorapropiado y son adecuados para usarse en embragues hidruli-cos controlables. Vase la figura P2-49. Los fluidos MR puedentener viscosidades mucho ms altas que los ER y, a menudo,muestran un comportamiento de adelgazamiento al corte, en elcual la viscosidad del fluido disminuye conforme aumenta lafuerza cortante aplicada. Este comportamiento tambin se cono-ce como seudoplstico y se puede representar con xito me-diante el modelo constitutivo de Herschel-Bulkley, expresadocomo t ! ty " K(du/dy)m. Aqu t es el esfuerzo cortante apli-cado, ty es el esfuerzo en el punto de fluencia, K es el ndice deconsistencia y m es el ndice de potencia. Para un fluido deHerschel-Bulkley con ty ! 900 Pa, K ! 58 Pa ! sm, y m !0.82, a) encuentre una relacin para el par de torsin transmiti-do por un embrague MR, para N platos sujetos a la flecha deentrada cuando sta se encuentra girando a una velocidad angu-lar de v mientras que la flecha de salida est estacionaria y b)calcule el par de torsin transmitido por un embrague de ese ti-po, con N ! 11 platos, para R1 ! 50 mm, R2 ! 200 mm, n

.

! 2 400 rpm, y h ! 1.2 mm.2-51 Se va a medir la viscosidad de un fluido con un vis-cosmetro construido de dos cilindros concntricos de 75 cm delargo. El dimetro exterior del cilindro interior es de 15 cm y labrecha entre los dos cilindros es de 0.12 cm. Se hace girar elcilindro interior a 200 rpm y se mide que el par de torsin es de0.8 N ! m. Determine la viscosidad del fluido.

2-54 Repita el problema 2-53 para umx ! 5 m/s. Respuesta:b) 0.942 N

Tensin superficial y efecto de capilaridad2-55C Qu es la tensin superficial? Qu la causa? Porqu la tensin superficial tambin recibe el nombre de energasuperficial?2-56C Considere una pompa de jabn. La presin dentro de lapompa es ms alta o ms baja que la del exterior?2-57C Qu es el efecto de capilaridad? Qu lo causa? C-mo lo afecta el ngulo de contacto?2-58C Se inserta un tubo de dimetro pequeo en un lquidocuyo ngulo de contacto es 110. El nivel del lquido en el tuboascender o descender? Explique.2-59C El efecto de capilaridad es mayor en los tubos de di-metro pequeo o en los de dimetro grande?2-60I Se introduce un tubo cuyo dimetro es de 0.03 pulgadasen queroseno a 68F. El ngulo de contacto del queroseno conuna superficie de vidrio es de 26. Determine el ascenso por ca-pilaridad del queroseno en el tubo. Respuesta: 0.65 pulgadas

CAPTULO 261

2-52I Se va a medir la viscosidad de un fluido con un vis-cosmetro construido con dos cilindros concntricos de 3 piesde largo. El dimetro interior del cilindro exterior mide 6 pul-gadas y la brecha entre los dos cilindros es de 0.05 pulgadas. Sehace girar el cilindro interior a 250 rpm y se mide que el par detorsin es de 1.2 lbf ! ft. Determine la viscosidad del fluido.Respuesta: 0.000648 lb ! s/ft2

2-53 En las regiones alejadas de la entrada, el flujo de un flui-do por un tubo circular es unidimensional y el perfil de veloci-dad para el flujo laminar se expresa como u(r) ! umx(1 #r2/R2), donde R es el radio del tubo, r es la distancia radial des-

0.12 cmFluido

200 rpm

Cilindroestacionario

FIGURA P2-51

r R umx

umx( )1 r2R2

o

FIGURA P2-53

2-61 Se introduce un tubo de dimetro de 1.9 mm en un lqui-do desconocido cuya densidad es de 960 kg/m3 y se observaque el lquido asciende 5 mm en el tubo y forma un ngulo decontacto de 15. Determine la tensin superficial del lquido.

h

0.03 pulg

Queroseno

FIGURA P2-60E

de el centro de ese tubo y umx es la velocidad mxima de flujo,la cual se tiene en el centro. Obtenga a) una relacin para lafuerza de resistencia al movimiento aplicada por el fluido enuna seccin del tubo de longitud L y b) el valor de la fuerza deresistencia al movimiento para flujo de agua a 20C, con R !0.08 m, L ! 15 m, umx ! 3 m/s, y m ! 0.0010 kg/m ! s.

ENGEL 02C 2/22/06 4:41 AM Page 61



2-49 Solution A multi-disk Electro-rheological ER clutch is considered. The ER fluid has a shear stress that is expressed as )( dyduy += . A relationship for the torque transmitted by the clutch is to be obtained, and the numerical value of the torque is to be calculated.

Assumptions 1 The thickness of the oil layer between the disks is constant. 2 The Bingham plastic model for shear stress expressed as )( dyduy += is valid.

Properties The constants in shear stress relation are given to be = 0.1 Pas and y = 2.5 kPa.

Output shaft

Input shaft

Plates mounted on shellPlates mounted on input shaft

Shell

Variable magnetic field

R2 R1

h = 1.2 mm Analysis (a) The velocity gradient anywhere in the oil of film thickness h is V/h where V = r is the tangential velocity relative to plates mounted on the shell. Then the wall shear stress anywhere on the surface of a plate mounted on the input shaft at a distance r from the axis of rotation is expressed as

hr

hV

drdu

yyyw +=+=+=

Then the shear force acting on a differential area dA on the surface of a disk and the torque generation associated with it are expressed as

drrhrdAdF yw )2(

+==

drhrrdrr

hrrrdFd yy

+=

+==

322)2(T

Integrating,

+=

+=

+=

== )(4)(324322T 41423132

4332

2

1

2

1

RRh

RRhrrdr

hrr y

R

Rryy

R

Rr

This is the torque transmitted by one surface of a plate mounted on the input shaft. Then the torque transmitted by both surfaces of N plates attached to input shaft in the clutch becomes

+= )(

4)(

34T 41

42

31

32 RRh

RRN y

(b) Noting that rad/s3.251 rad/min080,15) rev/min2400(22 ==== n and substituting,

mN 2060 =

+= ])m 05.0(m) 20.0[(

m) 0012.0(4/s) 3.251)(s/mN 1.0(])m 05.0(m) 20.0[(

3N/m 2500)11)(4(T 44

233

2

Discussion Can you think of some other potential applications for this kind of fluid?


2-23

*P1.55 The device in Fig. P1.54 is called a rotating disk viscometer[27]. Suppose that R ! 5 cm and h ! 1 mm. If the torquerequired to rotate the disk at 900 r/min is 0.537 N " m,what is the viscosity of the fluid? If the uncertainty in eachparameter (M, R, h, #) is $1 percent, what is the overalluncertainty in the viscosity?


& !'(8r

L04

Q)p'


Q, m3/h 0.36 0.72 1.08 1.44 1.80

)p, kPa 159 318 477 1274 1851









Problems 51

R R

Clearance h

Oil

P1.54

FluidR



P1.56

1.68 h ! ("/#g)1/2 cot $1.70 h ! 2" cos $/(#gW)1.72 z! 4800 m1.74 Cavitation occurs for both (a) and (b)1.76 z! 7500 m1.78 (a) 25C; (b) 4C1.80 x2y % y3/3 ! constant1.82 y ! x tan $ & constant1.84 x ! x0{ln (y/y0) & ln2 (y/y0)}

Chapter 22.2 'xy !%289 lb/ft2, (AA !%577 lb/ft22.4 x ! Const e%2Cz/B2.6 (a) 30.3 ft; (b) 30.0 in; (c) 10.35 m; (d) 13,100 mm2.8 DALR ! 9.77C/km2.10 10,500 Pa2.12 8.0 cm2.14 74,450 Pa with air; 75,420 Pa without air2.16 (a) 21,526 cm3; (b) 137 kPa2.18 1.562.20 14 lbf2.22 0.94 cm2.24 psealevel! 117 kPa, mexact ! 5.3 E18 kg2.26 (a) 2580 m; (b) 5410 m2.28 4400 ) 400 ft2.30 101,100 Pa2.32 22.6 cm2.34 *p ! *h[+water(1 & d2/D2) % +oil(1 % d2/D2)]2.36 252.38 (a) p1,gage ! (#m % #a)gh % (#t % #a)gH2.40 21.3 cm2.42 pA % pB ! (#2 % #1)gh2.44 (a) 171 lb/ft2; (b) 392 lb/ft2; manometer reads friction

loss

Chapter 11.2 1.3 E44 molecules1.4 1.63 slug/ft3, 839 kg/m31.6 (a) {L2/T2}; (b) {M/T}1.8 '! 1.00 My/I1.10 Yes, all terms are {ML/T2}1.12 {B} ! {L%1}1.14 Q ! Const B g1/2H3/21.16 All terms are {ML%2T%2}1.18 V ! V0e%mt/K1.20 zmax ! 64.2 m at t ! 3.36 s1.22 (a) %0.372U,2 /R; (b) x !%1.291 R1.24 e ! 221,000 J/kg1.26 Wair ! 0.71 lbf1.28 #wet ! 1.10 kg/m3, #dry ! 1.13 kg/m31.30 W1-2 ! 21 ft - lbf1.32 (a) 76 kN; (b) 501 kN1.34 1300 atm1.36 (a) BN2O ! 1.33 E5 Pa; (b) Bwater ! 2.13 E9 Pa1.38 ( ! 1380 Pa, ReL ! 281.40 A ! 0.0016 kg/(m - s), B ! 1903 K1.42 ./.200K ! (T K/200 K)0.681.44 Data 50 percent higher; Andrade fit varies )50 percent1.46 V! 15 m/s1.48 F! (.1/h1 & .2/h2)AV1.50 . !M(ro % ri)/(2/0ri3L)1.52 P! 73 W1.54 M! /.0R4/h1.56 . ! 3M sin $/(2/0R3)1.58 . ! 0.040 kg/(m - s), last 2 points are turbulent flow1.60 . ! 0.88 ) 0.023 kg/(m - s)1.62 28,500 Pa1.64 (a) %0.023 m; (b) & 0.069 m1.66 F ! 0.014 N806

Answers to Selected Problems

ER con N ! 11 para R1 ! 50 mm, R2 ! 200 mm, y n.

! 2 400rpm, si el fluido es SAE 10, con m ! 0.1 Pa ! s, ty ! 2.5 kPa, yh ! 1.2 mm. Respuesta: b) 2 060 N ! m2-50 La viscosidad de algunos fluidos, llamados fluidos mag-netorreolgicos (MR), cambia cuando se aplica un campo mag-ntico. Esos fluidos contienen partculas magnetizables con ta-mao del orden de micras, suspendidas en un lquido portadorapropiado y son adecuados para usarse en embragues hidruli-cos controlables. Vase la figura P2-49. Los fluidos MR puedentener viscosidades mucho ms altas que los ER y, a menudo,muestran un comportamiento de adelgazamiento al corte, en elcual la viscosidad del fluido disminuye conforme aumenta lafuerza cortante aplicada. Este comportamiento tambin se cono-ce como seudoplstico y se puede representar con xito me-diante el modelo constitutivo de Herschel-Bulkley, expresadocomo t ! ty " K(du/dy)m. Aqu t es el esfuerzo cortante apli-cado, ty es el esfuerzo en el punto de fluencia, K es el ndice deconsistencia y m es el ndice de potencia. Para un fluido deHerschel-Bulkley con ty ! 900 Pa, K ! 58 Pa ! sm, y m !0.82, a) encuentre una relacin para el par de torsin transmiti-do por un embrague MR, para N platos sujetos a la flecha deentrada cuando sta se encuentra girando a una velocidad angu-lar de v mientras que la flecha de salida est estacionaria y b)calcule el par de torsin transmitido por un embrague de ese ti-po, con N ! 11 platos, para R1 ! 50 mm, R2 ! 200 mm, n

.

! 2 400 rpm, y h ! 1.2 mm.2-51 Se va a medir la viscosidad de un fluido con un vis-cosmetro construido de dos cilindros concntricos de 75 cm delargo. El dimetro exterior del cilindro interior es de 15 cm y labrecha entre los dos cilindros es de 0.12 cm. Se hace girar elcilindro interior a 200 rpm y se mide que el par de torsin es de0.8 N ! m. Determine la viscosidad del fluido.

2-54 Repita el problema 2-53 para umx ! 5 m/s. Respuesta:b) 0.942 N

Tensin superficial y efecto de capilaridad2-55C Qu es la tensin superficial? Qu la causa? Porqu la tensin superficial tambin recibe el nombre de energasuperficial?2-56C Considere una pompa de jabn. La presin dentro de lapompa es ms alta o ms baja que la del exterior?2-57C Qu es el efecto de capilaridad? Qu lo causa? C-mo lo afecta el ngulo de contacto?2-58C Se inserta un tubo de dimetro pequeo en un lquidocuyo ngulo de contacto es 110. El nivel del lquido en el tuboascender o descender? Explique.2-59C El efecto de capilaridad es mayor en los tubos de di-metro pequeo o en los de dimetro grande?2-60I Se introduce un tubo cuyo dimetro es de 0.03 pulgadasen queroseno a 68F. El ngulo de contacto del queroseno conuna superficie de vidrio es de 26. Determine el ascenso por ca-pilaridad del queroseno en el tubo. Respuesta: 0.65 pulgadas

CAPTULO 261

2-52I Se va a medir la viscosidad de un fluido con un vis-cosmetro construido con dos cilindros concntricos de 3 piesde largo. El dimetro interior del cilindro exterior mide 6 pul-gadas y la brecha entre los dos cilindros es de 0.05 pulgadas. Sehace girar el cilindro interior a 250 rpm y se mide que el par detorsin es de 1.2 lbf ! ft. Determine la viscosidad del fluido.Respuesta: 0.000648 lb ! s/ft2

2-53 En las regiones alejadas de la entrada, el flujo de un flui-do por un tubo circular es unidimensional y el perfil de veloci-dad para el flujo laminar se expresa como u(r) ! umx(1 #r2/R2), donde R es el radio del tubo, r es la distancia radial des-

0.12 cmFluido

200 rpm

Cilindroestacionario

FIGURA P2-51

r R umx

umx( )1 r2R2

o

FIGURA P2-53

2-61 Se introduce un tubo de dimetro de 1.9 mm en un lqui-do desconocido cuya densidad es de 960 kg/m3 y se observaque el lquido asciende 5 mm en el tubo y forma un ngulo decontacto de 15. Determine la tensin superficial del lquido.

h

0.03 pulg

Queroseno

FIGURA P2-60E

de el centro de ese tubo y umx es la velocidad mxima de flujo,la cual se tiene en el centro. Obtenga a) una relacin para lafuerza de resistencia al movimiento aplicada por el fluido enuna seccin del tubo de longitud L y b) el valor de la fuerza deresistencia al movimiento para flujo de agua a 20C, con R !0.08 m, L ! 15 m, umx ! 3 m/s, y m ! 0.0010 kg/m ! s.

ENGEL 02C 2/22/06 4:41 AM Page 612.68 The viscometer of Problem 2.67 is used to measure theapparent viscosity of a fluid. The data below are obtained.What kind of non-Newtonian fluid is this? Find the values ofk and n used in Eqs. 2.16 and 2.17 in defining the apparentviscosity of a fluid. (Assume is 0.5 degrees.) Predictthe viscosity at 90 and 100 rpm.

Speed (rpm) 10 20 30 40 50 60 70 80

(N ! s/m2) 0.121 0.139 0.153 0.159 0.172 0.172 0.183 0.185

2.69 An insulation company is examining a new material forextruding into cavities. The experimental data is given belowfor the speed U of the upper plate, which is separated from afixed lower plate by a 1-mm-thick sample of the material,when a given shear stress is applied. Determine the type ofmaterial. If a replacement material with a minimum yieldstress of 250 Pa is needed, what viscosity will the materialneed to have the same behavior as the current material at ashear stress of 450 Pa?

(Pa) 50 100 150 163 171 170 202 246 349 444U (m/s) 0 0 0 0.005 0.01 0.025 0.05 0.1 0.2 0.3

2.70 A viscometer is used to measure the viscosity of apatients blood. The deformation rate (shear rate)"shearstress data is shown below. Plot the apparent viscosity versusdeformation rate. Find the value of k and n in Eq. 2.17, andfrom this examine the aphorism Blood is thicker thanwater.

du/dy (s21) 5 10 25 50 100 200 300 400

(Pa) 0.0457 0.119 0.241 0.375 0.634 1.06 1.46 1.78

2.71 A viscous clutch is to be made from a pair of closelyspaced parallel disks enclosing a thin layer of viscous liquid.Develop algebraic expressions for the torque and the powertransmitted by the disk pair, in terms of liquid viscosity, ,disk radius, R, disk spacing, a, and the angular speeds: i ofthe input disk and o of the output disk. Also developexpressions for the slip ratio, s5/i, in terms of i andthe torque transmitted. Determine the efficiency, , in termsof the slip ratio.

R

a

i o

R a

b

H

P2.71 P2.72

2.72 A concentric-cylinder viscometer is shown. Viscoustorque is produced by the annular gap around the inner

cylinder. Additional viscous torque is produced by the flatbottom of the inner cylinder as it rotates above the flatbottom of the stationary outer cylinder. Obtain an algebraicexpression for the viscous torque due to flow in the annulargap of width a. Obtain an algebraic expression for the viscoustorque due to flow in the bottom clearance gap of height b.Prepare a plot showing the ratio, b/a, required to hold thebottom torque to 1 percent or less of the annulus torque,versus the other geometric variables. What are the designimplications? What modifications to the design can yourecommend?

2.73 A viscometer is built from a conical pointed shaft thatturns in a conical bearing, as shown. The gap between shaftand bearing is filled with a sample of the test oil. Obtain analgebraic expression for the viscosity of the oil as a func-tion of viscometer geometry (H, a, and ), turning speed ,and applied torque T. For the data given, find by referring toFigure A.2 in Appendix A, the type of oil for which theapplied torque is 0.325 N !m. The oil is at 20#C. Hint: Firstobtain an expression for the shear stress on the surface of theconical shaft as a function of z.

a = 0.2 mm

= 75 rev/s

= 30

r

zH = 25 mm

P2.73

2.74 Design a concentric-cylinder viscometer to measure theviscosity of a liquid similar to water. The goal is to achieve ameasurement accuracy of 61 percent. Specify the config-uration and dimensions of the viscometer. Indicate whatmeasured parameter will be used to infer the viscosity of theliquid sample.

2.75 A spherical thrust bearing is shown. The gap betweenthe spherical member and the housing is of constant width h.Obtain and plot an algebraic expression for the nondimen-sional torque on the spherical member, as a function ofangle .

h

Oil film (viscosity, )

R

P2.75

52 Chapter 2 Fundamental Concepts

2.68 The viscometer of Problem 2.67 is used to measure theapparent viscosity of a fluid. The data below are obtained.What kind of non-Newtonian fluid is this? Find the values ofk and n used in Eqs. 2.16 and 2.17 in defining the apparentviscosity of a fluid. (Assume is 0.5 degrees.) Predictthe viscosity at 90 and 100 rpm.

Speed (rpm) 10 20 30 40 50 60 70 80

(N ! s/m2) 0.121 0.139 0.153 0.159 0.172 0.172 0.183 0.185


(Pa) 50 100 150 163 171 170 202 246 349 444U (m/s) 0 0 0 0.005 0.01 0.025 0.05 0.1 0.2 0.3


du/dy (s21) 5 10 25 50 100 200 300 400

(Pa) 0.0457 0.119 0.241 0.375 0.634 1.06 1.46 1.78


R

a

i o

R a

b

H

P2.71 P2.72




a = 0.2 mm

= 75 rev/s

= 30

r

zH = 25 mm

P2.73



h


R

P2.75


2.68 The viscometer of Problem 2.67 is used to measure theapparent viscosity of a fluid. The data below are obtained.What kind of non-Newtonian fluid is this? Find the values ofk and n used in Eqs. 2.16 and 2.17 in defining the apparentviscosity of a fluid. (Assume is 0.5 degrees.) Predictthe viscosity at 90 and 100 rpm.

Speed (rpm) 10 20 30 40 50 60 70 80

(N ! s/m2) 0.121 0.139 0.153 0.159 0.172 0.172 0.183 0.185


(Pa) 50 100 150 163 171 170 202 246 349 444U (m/s) 0 0 0 0.005 0.01 0.025 0.05 0.1 0.2 0.3


du/dy (s21) 5 10 25 50 100 200 300 400

(Pa) 0.0457 0.119 0.241 0.375 0.634 1.06 1.46 1.78


R

a

i o

R a

b

H

P2.71 P2.72




a = 0.2 mm

= 75 rev/s

= 30

r

zH = 25 mm

P2.73



h


R

P2.75


COMPILADO DE PROBLEMAS DE VISCOSIDAD - 3

Nota: El fluido cubre las reas superior, inferior y lateral del cono truncado. El espesor de la pelcula de aceite es mismo en cada una de las reas anteriores. R: P1 = 270 W ; P2 = 249W Problemas de Perfiles de Velocidad

R:

Ecuaciones diferenciales (Variables separables)

Datos: I = !!"!; !"#" = !!!!"#" R: = ! 52

R: a) 121s b) c)472m





r

R

umx

u(r) = umx(1 rn/Rn)

0

FIGURA P2-44


F

Pared fija

Pared en movimiento

= 1 m/sh1 = 1 mm

h2 = 2.6 mm w = 0.3 m/s

V

V

FIGURA P2-45

D = 12 cm

L = 12 cm

d = 4 cm

Caja

Aceite SAE 10W

r

z

FIGURA P2-46



30 cm

Flechaimpulsora

Flechaimpulsora

Aceite SAE 30W

3 mm

FIGURA P2-47

Casco



Flecha de salida

R2 R1


h = 1.2 mm

FIGURA P2-49

ENGEL 02C 2/22/06 4:41 AM Page 60



2-44 Solution The velocity profile of a fluid flowing though a circular pipe is given. The friction drag force exerted on the pipe by the fluid in the flow direction per unit length of the pipe is to be determined.

Assumptions The viscosity of the fluid is constant.

Analysis The wall shear stress is determined from its definition to be

Run

Rnru

Rr

drdu

drdu

Rrn

n

Rrn

n

Rrw

max1

maxmax 1 ==

==

=

==

u(r) = umax(1-rn/Rn)

R r

0

umax

Note that the quantity du /dr is negative in pipe flow, and the negative sign is added to the w relation for pipes to make shear stress in the positive (flow) direction a positive quantity. (Or, du /dr = - du /dy since y = R r). Then the friction drag force exerted by the fluid on the inner surface of the pipe becomes

LunLRRun

AF ww maxmax 2)2( ===

Therefore, the drag force per unit length of the pipe is

max2/ unLF = . Discussion Note that the drag force acting on the pipe in this case is independent of the pipe diameter.


2-18


1.82 A thin layer of glycerin flows down an inclined, wide platewith the velocity distribution shown in Fig. P1.82. For h ! 0.3 in.and a ! 20", determine the surface velocity, U. Note that for equi-librium, the component of weight acting parallel to the plate sur-face must be balanced by the shearing force developed along theplate surface. In your analysis assume a unit plate width.

1.85 The space between two 6-in.-long concentric cylindersis filled with glycerin The in-ner cylinder has a radius of 3 in. and the gap width between cylin-ders is 0.1 in. Determine the torque and the power required to rotatethe inner cylinder at The outer cylinder is fixed.Assume the velocity distribution in the gap to be linear.

1.86 A pivot bearing used on the shaft of an electrical in-strument is shown in Fig. P1.86. An oil with a viscosity of ! !0.010 lb . s/ft2 fills the 0.001-in. gap between the rotating shaft andthe stationary base. Determine the frictional torque on the shaftwhen it rotates at 5000 rpm.

1.87 The viscosity of liquids can be measured through the use of arotating cylinder viscometer of the type illustrated in Fig. P1.87. Inthis device the outer cylinder is fixed and the inner cylinder is rotatedwith an angular velocity, The torque required to develop ismeasured and the viscosity is calculated from these two measurements.(a) Develop an equation relating and Neglect endeffects and assume the velocity distribution in the gap is linear.(b) The following torque-angular velocity data were obtained with arotating cylinder viscometer of the type discussed in part (a).Torque 13.1 26.0 39.5 52.7 64.9 78.6Angular

velocity 1.0 2.0 3.0 4.0 5.0 6.0For this viscometer and Make use of these data and a standard curve-fitting program to de-termine the viscosity of the liquid contained in the viscometer.

/ ! 5.00 in.Ri ! 2.45 in.,Ro ! 2.50 in.,1rad#s2

1ft # lb2Ri.m, v, t, /, Ro,

"tv.

180 rev#min.

1viscosity ! 8.5 $ 10%3 lb # s#ft22.u__U

y__h

y2__h22 =

Uh

y u

Figure P1.81

u__U

y__h

y2__h22 =

Uyu

h

Figure P1.82

*1.83 Standard air flows past a flat surface, and velocitymeasurements near the surface indicate the following distribution:y 1ft2 0.005 0.01 0.02 0.04 0.06 0.08u 0.74 1.51 3.03 6.37 10.21 14.43The coordinate y is measured normal to the surface and u is thevelocity parallel to the surface. (a) Assume the velocity distribu-tion is of the form

and use a standard curve-fitting technique to determine the con-stants and (b) Make use of the results of part 1a2 to determinethe magnitude of the shearing stress at the wall and at

1.84 A new computer drive is proposed to have a disc, asshown in Fig. P1.84. The disc is to rotate at 10,000 rpm, and thereader head is to be positioned 0.0005 in. above the surface of thedisc. Estimate the shearing force on the reader head as a result ofthe air between the disc and the head.

y ! 0.05 ft.1y ! 02C2.C1

u ! C1y & C2y3

1ft#s2

Figure P1.87

Liquid

Fixedouter

cylinder!

"

Rotatinginner

cylinder

RiRo

Figure P1.86

0.2 in.

0.001 in.

5000 rpm

30

= 0.010 lb s/ft2

Figure P1.84

10,000 rpm0.0005 in.

Rotating disc2 in.

0.2-in.dia.Stationary reader head


Chapter 11.2 1a2 MLT!1; 1b2 ML4 L!2; 1c2 MT!21.4 1a2 FLT!2; 1b2 FL!1T!1; 1c2 FL!3 T1.10 1a2 FL!1; 1b2 FL!3; FL1.12 LT!1; F0 L0 T0; LT!11.14 Yes1.16 1!2, !1!21.18 Dimensionless1.20 1a2 4.32 mm!s; 1b2 70.2 kg; 1c2 13.4 N; 1d2 22.3 m!s2; 1e2 1.12 N " s!m21.22 1a2 6.47 # 105 m2; 1b2 56.8 # 10!2 m3; 1c2 3.86 # 105 m; 1d2 5.90 # 104 W; 1e2 289 K1.30 1a2 0.240 mi3; 1b2 4.41 # 105 lb1.32 30.6 kg; 37.3 N1.34 1150 kg!m3; 11.3 kN!m31.36 0.0186 ft3; No1.38 0.9971 lb1.40 991.5 kg!m31.42 16.0 kN!m3; 1.63 # 103 kg!m3; 1.631.44 4.76 kg1.46 1a2 0.0214 kg!m3; 1b2 rMars!rearth $ 1.75%1.48 6.44 # 10!3 slugs!ft3; 0.622 lb1.50 98.7 psia1.52 668 lb1.56 0.6 N " s!m2; 1.3 # 10!2 lb " s!ft21.58 0.727 N " s!m21.60 23.7, 2.55 # 10!21.62 0.277 N " s!m21.64 2.05 # 10!5 N " s!m21.66 1a2 No; 1b2 Not correct1.68 1841.70 C $ 1.43 # 10!6 kg!1m " s " K1!22; S $ 107 K1.72 D $ 1.767 # 10!6 N " s!m2; B $ 1.870 # 103 K;

5.76 # 10!4 N " s!m21.74 0.300 N " s!m21.76 non-Newtonian1.78 286 N1.80 0.0883 m!s1.82 0.268 ft!s1.84 3.43 # 10!4 lb1.86 9.53 # 10!4 ft " lb1.88 1a2 12.7 ft2!rev; 1b2 4.73 # 10!3 lb " s!ft21.94 2.03 # 103 psi1.96 4.14 # 103 psi

1.98 1a2 343 m!s; 1b2 1010 m!s; 1c2 446 m!s1.100 104 psi 1gage21.102 2.88 kg!m31.104 4.25 # 10!3 slugs!ft3; 305 %F1.108 1.061.110 14.1%1.112 4.74 psi 1abs21.114 13 kPa 1abs21.116 5.81 kPa 1abs2; 0.842 psi 1abs21.118 0.060 N!m1.120 538 Pa1.122 97.9 Pa1.124 1a2 24.5 deg1.126 0.0614 in.; Yes1.128 1.80 # 10!2 ft1.130 7.49 mm

Chapter 22.2 59.2 kPa2.6 34.7 psi2.8 404 kPa2.10 1a2 p $ !Ev ln 11 ! r0gh!Ev2; 1b2 61.4 MPa; 1c2 60.6 MPa2.12 p $ Kh2!2 & '0h2.18 464 mm2.20 62.9%2.24 4,250 ft2.26 543 m2.28 60 kPa2.32 14.4 psia; 99.3 kPa 1abs22.34 1a2 18.2 ft; 1b2 8.73 psi; 21.7 psia2.36 0.317 ft2.38 6.28 ft2.40 !3.32 kPa2.42 1a2 4.00 ft; 1b2 2.08 ft2.44 1.45 ft2.46 1a2 26.9 kPa; 1b2 0.202 m2.48 1.55 slugs!ft32.50 94.9 kPa2.54 21.6 ft2.56 575 lb!ft22.58 0.100 m2.60 27.8%2.62 0.212 m

Answers toSelected Even-Numbered Homework Problems

ANS-1

BMAns.qxd 2/15/12 9:01 PM Page ANS-1

whose properties are ! " 0.003 m2/s and SG " 0.88. Es-timate the force required to pull the shaft at a steady ve-locity of 0.4 m/s.

P1.48 A thin plate is separated from two fixed plates by very vis-cous liquids #1 and #2, respectively, as in Fig. P1.48. Theplate spacings h1 and h2 are unequal, as shown. The con-tact area is A between the center plate and each fluid.(a) Assuming a linear velocity distribution in each fluid,derive the force F required to pull the plate at velocity V.(b) Is there a necessary relation between the two viscosi-ties, #1 and #2?

sured torque is 0.293 N $ m, what is the fluid viscosity?Suppose that the uncertainties of the experiment are as fol-lows: L (%0.5 mm), M (%0.003 N $ m), & (%1 percent),and ri or ro (%0.02 mm). What is the uncertainty in themeasured viscosity?

P1.52 The belt in Fig. P1.52 moves at a steady velocity V andskims the top of a tank of oil of viscosity #, as shown. As-suming a linear velocity profile in the oil, develop a sim-ple formula for the required belt-drive power P as a func-tion 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 oilat 20C, with L " 2 m, b " 60 cm, and h " 3 cm?


Liquid film ofthickness h

W

V Block contactarea A

P1.45

P1.48

h1

h2

1

2

F, V

P1.49 The shaft in Prob. 1.47 is now

problemas propuestos viscosidad - cengel munson white fox

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