Manual de Ayudantias

Ayudant´ıas Resueltas de

Mec´ anica de Fluidos General

´ rdenas Zamorano Pablo Ca 2016

PCZ

Pre´ ambulo El objetivo de este documento es, por un lado, colaborar con el aprendizaje de los fundamentos de la Mec´ anica de los Fluidos a trav´es de una gu´ıa ordenada y comentada, y por otro, tener un respaldo acerca de los conocimientos m´ınimos que se espera que nuestras compa˜ neras y compa˜ neros tengan una vez aprobado un ramo introductorio sobre la Mec´ anica de los Fluidos. Este documento fue elaborado y redactado el a˜ no 2016 en la UTFSM de Valpara´ıso, Chile, luego de varios semestres poniendo a prueba y corrigiendo algunos problemas que ac´a se presentan. S´olo existe (y existir´ a) en formato digital a trav´es de los medios disponibles en el departamento de mec´anica de la universidad, sin embargo, los motivo a compartir y difundir este documento tanto como ustedes estimen conveniente. Los ejercicios ac´ a resueltos son, en general, una recopilaci´on de diversos libros (Cengel, Fox, White, etc.) y est´ an planteados de modo que sean una introducci´on a cada tema. Entonces, no son dif´ıciles de resolver y en ning´ un caso representan un problema real de un certamen/prueba/examen del ramo (exceptuando quiz´ as un par de casos). Agradecimientos especiales a mis profesores de Mec´anica de Fluidos, en especial a Alex Flores por permitir la continuidad en mi labor como ayudante, a mis compa˜ neros y compa˜ neras que asistieron a las ayudant´ıas y que semana a semana retroalimentaban indirectamente este documento, y a mis amigas y amigos por diversas correcciones en el formato, en especial a Sebasti´an Cueto por sus numerosas recomendaciones.

Pablo.

P´ agina 2

PCZ

´Indice Parte I: Fundamentos y An´ alisis Intgral Ayudant´ıa 1: Viscosidad e Hidrost´ atica . . . . . Ayudant´ıa 2: An´ alisis Integral. Conservaci´on de Ayudant´ıa 3: An´ alisis Integral. Conservaci´on de Ayudant´ıa 4: Repaso . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . Masa y Momentum Lineal . . Momentum Angular y Energ´ıa . . . . . . . . . . . . . . . . . .

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Parte II: An´ alisis Diferencial y Dimensional Ayudant´ıa 5: An´ alisis Diferencial. Cinem´atica y Conservaci´on de Masa . . . . . . . . Ayudant´ıa 6: An´ alisis Diferencial. Conservaci´on del Momentum Lineal. Ecuaci´ones de Ayudant´ıa 7: An´ alisis Diferencial. Flujo Ideal, Ecuaci´on de Euler y Bernoulli . . . . Ayudant´ıa 8: An´ alisis Dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 4 6 7 8

9 . . . . . . . . 9 Navier-Stokes 10 . . . . . . . . 11 . . . . . . . . 12

Parte III: Aplicaciones 14 Ayudant´ıa 9: Introducci´ on a Capa L´ımite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Ayudant´ıa 10: P´erdida de Carga en Tuber´ıas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Ayudant´ıa 11: Fundamentos de Turbom´ aquinas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Soluciones Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa Ayudant´ıa

1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11

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19 19 22 25 28 31 35 41 45 49 53 57

Anexo A: Operadores Diferenciales

61

Anexo B: Leyes de Conservaci´ on

64

P´ agina 3

the radius of the pipe, r is the radial distance from the center of the pipe, and umax is the maximum flow velocity, which occurs at the center. Obtain (a) a relation for the drag force applied by the fluid on a section of the pipe of length L and (b) the value PCZ of the drag force for water flow at 20°C with R 5 0.08 m, L 5 30 m, umax 5 3 m/s, and m 5 0.0010 kg/m?s.

the inner and alge function

Ayudant´ıa 1: Mec´ anica de Fluidos General Mi´ ercoles 17 de Agosto, 2016

Contenidos

2 u max 1 – r 2

(

r R o

R

)

umax

Ley de Viscosidad de Newton, Hidrost´atica

Problema 1

FIGURE P2–88

72 PROPERTIES OF FLUIDS

2–136 In a water distri Un eje con di´ ametro D = 80 [mm] y de largo L = 400 [mm], Bearing can be 2–89 Repeat Prob. 2–88 for umax 5 7 m/s. Answer: (b) 2.64 N as low as 1.4 p como se muestra en la figura, es empujado con velocidad h2 water allowed in the pipi h1 Viscous oil, μ y (a) 50°F (b) 77°F (c) 2–90 A frustum-shaped body is rotating at a constant anguconstante U = 5 [m/s] por un rodamiento con di´ametro Shaft x 2–137 lar speed of 200 rad/s in a container filled with SAE 10W The thermal ene variable. El espacio entre el eje y el rodamiento, el cual D U (a)the Sensible energy (b) L oil at 20°C (m 5 0.100 Pa?s), as shown in Fig. P2–90. If (c) Sensible 1 latent ener var´ıa desde h1 = 1,2 [mm] hasta h2 = 0,4 [mm] es llenado thickness of the oil film on all sides is 1.2 mm, determine 2–138 The difference b L por un lubricante newtoniano cuya viscosidad din´amicathe es power required to maintain this motion. Also determine stationary fluid per unit m FIGURE P2–127 Enthalpy (b) Flow e 0,10 [Pa·s]. Determinar la fuerza requerida para mantener the reduction in the required power input when the oil (a) tem(d ) Kinetic energy (e) perature rises to 80°C (m 5 0.0078 Pa?s). el movimiento axial del eje. A 2–128 Reconsider Prob. 2–127. The shaft now rotates with 2–139 The 2–92 pressure of a constant angular speed of n 5 1450 rpm in a bearing with m/s Th ov 1200 kPa by a4pump. variable diameter. The clearance between shaft and bearing, by 0.15°C. The density 20°C. A which varies from h1 5 1.2 mm to h2 5 0.4 mm, is filled with heat is cp 5 4.18 kJ/kg?° film, as a Newtonian lubricant whose dynamic viscosity is 0.1 Pa?s. during this process is s Cengel 03.qxd 2/22/06 5:26 AM Page 111 Case Determine the torque required to maintain the motion. upper pl (a) 1100 kJ/kg (b) 0.63 Problema 2 (e) 4.2 kJ/kg velocity 2–129 A 10-cm-diameter cylindrical shaft rotates inside a 40-cm-long 10.3-cm diameter bearing. The space between the 2–140 The coefficient Un cuerpo con forma de cono truncado gira con rapidez an- shaft and the bearing is completely filled with SAE 10Wvisoil oil whose pressible substance is D = 12 cmtemperature is 0.300 N?s/m2. cosity at anticipated operating (a) 0 (b) 0.5 (c) 1 ( gular constante de 200 [rad/s] en un contenedor lleno con Determine the power required to overcome friction when the 2–141 The pressure of z aceite SAE 10W a 20 [◦ C] (µ = 0, 100 [Pa·s]), como se mues- shaftLrotates at a speed of (a) 600 rpm and (b) 1200 rpm. = 12 cm be raised to 210 atm to tra en la figura. Si el espesor de la capa de aceite en todos 2–130 Some rocks or bricks contain small air pockets in coefficient of compressib d =structure. 4 cm Assuming the air spaces them and have a spongy (a) 209 atm (b) 20,900 r of 0.006 mm, deterlos lados es de 1,2 [mm], determine la potencia requerida form columns of an average diameter (e) 210,000 atm mine how high water can rise in such a material. Take the 90 la reChapter Fluid Staticsinterface in that material to 2–142 When a liquid para mantener este movimiento. Tambi´en, determine surface tension of the air–water h =y5 mm w bloques vencerán la abrupt flow restriction (s altura del del lodo a N/m. la cual a) los fricción ducci´ on en la potencia requerida cuando la temperatura be 0.085 compressed. The resultin empezarán a resbalar y b) los bloques se voltearán. strike the pipe surfaces, aceite aumenta a 80 [◦ C] (µ = 0, 0078 [Pa·s]). Fundamentals of Engineering (FE) Exam Problems

3

3.66

The gate shown is hinged at H. The gate is 3andmreflect wide 3 along the pi

The specific gravity of a fluid is specified to be 0.82. produce a familiar 3-74 problema 3-73 para bloques de concreto consound normalRepita to 2–131 theel plane of the diagram. Calculate the force The specific volume of this fluid is (a) Condensation (b) C FIGURE 3 3 3 un ancho de 0.4 m. (b) 0.00122 /kg (c) 0.0082 m /kg m /kg the (d ) Compression (e) W required at(a)A0.00100 to hold gate mclosed.

3-75

(e) 820 m3/kg (dFIGURE ) 82 m3/kgP2–90

2–143 The density of a 2–132 The specific gravity of mercury is 13.6. The specific stant pressure2–93 when itsA coefficient of volume exp of mercury is 2–91 weight A rotating viscometer consists of two concentric cal tube 3 3 3 3 21 1.5 m (a) 1.36 kN/m (b) 9.81 kN/m (c) 106 kN/m (d ) 133 kN/m (a) 0.01 K (b) 0.005 K H inner cyliner of radius Ri and an cylinders—a film thic (e) 13,600stationary kN/m3 2–144 Water is compres

Una compuerta de 4 m de largo con forma de un cuarto de círculo de radio 3 m y de peso despreciaProblema 3 ble está articulada alrededor de su borde Fsuperior A, como se rotating at angular outer ofideal inside radius and La compuerta mostrada est´ a pivoteada en H. Esta com2–133 An gas flows in at 20°C. The density of velocity stant TheL, in ocompuerta muestra encylinder la figura P3.75. Laa Rpipe controla el temperature. flujoD de 3 and its molar mass is 44 kg/kmol. The therate) gas is v 1.9. kg/m and the isothermal co In the tiny gap between the two cylinders (rotation the cylin o puerta mide 3 [m] de largo normal al plano de agua la figura. 21 . The f sobre pressure el reborde está comprimida unatmreof the gasen is B, donde 4.8por 3 1025 A is the fluid whose viscosity (m)3ismto be measured. The length q 3 t S (a) 1000 kg/mas (a) 7 kPa (b) 72 kPa (c) 105 kPa (d ) 460 kPa (e) 4630 kPa Calcular la magnitud de la fuerza F para mantener la comWater sorte. Determine la fuerza mínima necesaria del resorte para (b) 1 (d ) 1003.5 kg/m3 (e) 9 30° 2–134 A gas mixture consists of 3 kmol oxygen, 2 kmol nitrogen, and 0.5 kmol water vapor. The total pressure of the gas mixture is 100 kPa. The partial pressure of water vapor in this gas mixture is (a) P3.66 5 kPa (b) 9.1 kPa (c) 10 kPa (d ) 22.7 kPa (e) 100 kPa

puerta cerrada.

Problema 4

2–145 The speed of a s atmospheric air at 240°C (a) 35 .9 (b) 0.85 (c)

2–146 The dynamic vis 1.83 3 1025 kg/m?s. The k 2–135 Liquid water vaporizes into water vapor as it flows A long, square wooden is pivoted along one edge. (a) 0.525 3 1025 m2/s in the piping of a boiler. If block the temperature of water in the (c) 1.47 1025 m2/s is 180°C, the vapor pressure of theimmersed water in the pipein is water The block pipe is in equilibrium when to3the un cuarto (a) 1002 kPa (b) 180 kPa (c) 101.3 kPa (d ) 18 kPa (e) 100 kPa (e) 0.380 3 1025 m2/s

3.67

Una compuerta de 4 [m] de largo con forma de depth shown. Evaluate the specific gravity of the wood, if de c´ırculo de radio 3 [m] y de peso despreciable est´a artifriction in the pivot is negligible. culada alrededor de su borde superior A, como se muestra 037-074_cengel_ch02.indd 68 L Air en la figura. La compuerta controla el flujo de agua sobre el reborde en B, donde est´ a comprimida por un resorte. Deterd = 0.5 72m mine la fuerza m´ınima necesaria del resorte para mantener037-074_cengel_ch02.indd Wood L = 1.0 m cerrada la compuerta cuando el nivel del agua se eleva hasta A en el borde superior de la compuerta Water

FIGURA P3-75

Pivot, O

P3.67

mantener cerrada la compuerta agua se ele3.68 A solid concrete dam is tocuando be builteltonivel hold del back a depth A en el ease borde de lathe compuerta. va hasta D of water. For ofsuperior construction walls of the dam must P´ agina 4

be planar. Your asks you consider the 4following 3-76 Repita el supervisor problema 3-75 paratoun radio de m para la dam cross-sections: a rectangle, a right triangle with the Respuesta: 314 kN compuerta.

3

P2.92 *P2.93 In Fig. P2.93, a one-quadrant spherical shell of radius R is submerged in liquid of specific weight  and depth h  R. PCZ Find an analytic expression for the resultant hydrostatic force, and its line of action, on the shell surface. z

Problema 5

P2.95

P2.96 Curved pan lar to the b paper, estim on the pane

ρ, γ h

Un cuadrante de un cascar´ on esf´erico de radio R es sumergido en un l´ıquido con peso espec´ıfico γ y profundidad h > R. Encuentre, planteando la integral respectiva, la fuerza hidrost´ atica sobre el cascar´ on en la direcci´ on vertical.

R R

z

R

R

x

P2.96

P2.93

P´ agina 5

x

u

2h

4.106 Air at standard conditions flows along a flat plate. The

undisturbed freestream speed is U0 5 20 m/s. At L 5 0.4 m downstreamwhi29346_ch03_138-227.qxd from the leading edge of the plate,PCZ the boundary10/28/09 17:10 Page 200 Debd 208:MHDQ176:whi29346:0073529346:w layer thickness is δ 5 2 mm. The velocity profile at this Ayudant´ıa 2: Mec´ anica de isFluidos General location approximated as u/U0 5 y/δ. Calculate the horP4.101 Mi´ e rcoles 24 de Agosto, 2016 component of force per forunit width 200 Chapter 3 Integral Relations a Control Volume required to hold the 158 Chapter 4 Basic Equations in Integral Formizontal for a Control Volume An incompressible fluid flows steadily in the entrance plate stationary. number is 0.90. For one-dimensional flow, calculate 4.39 Water (a)enters two-dimensional, 4.44 A cyli d= the massaflow; (b) p ; (c) V ; and (d) thesquare change in channel of (3) of a circular tube of radius R 5 75elbow mm.isThe flow rate entropy 1 mm, andsystem 2. with (e) Howuniform you explain the to produce constant width, 5 75.5 velocity, The into in its bo 4.92 A 30 reducing shown. The fluid is water. 4.98 A nozzle forhsplitter abetween spray isdo designed a hole 4.107 A sharp-edged plate inserted partU.way a flat Contenidos entropy change? 3  makesofawater. 90 bend that distorts thethe flownozzle to produce thevelocity components force must be provided by flat channel radial sheet The sheet leaves at V2 5 the flow rate uniform U1ofat thethat entrance. .1 m /s. Find theEvaluate P3.29 In elementary compressible flow theory (Chap. 9), comstream 10 ofm/s, flowing water produces theexit, flow  airprofile the rate of c 5 2The shown. the linear velocity shown at the with vpattern pressed will exhaust from a small hole in a tank at 1.5 max the adjacent pipes to keep the elbow from moving. covers 180 of arc, and has thickness t 5 mm. (2) ocity distribution atTTR a section is (Masa ydownstream Momentum Lineal) the v mass flow rate C, m/s. where  is the air density in m˙ 7.5 P3.32 v . Evaluate , if U 5 min min 4.45 A tan Analyze the situation to evaluate θ as a function of α, where nozzle discharge radius is R 5 50 mm. The water supply pipe the tank and C is a constant. If  is the initial density  r 2 Elbow mass, M = 10 kg in a tank of volume , derive a formula for the density P3.33 In some wind tunnels valve is sectio op u the test 5 150 kPa is 35 mm in diameter and the inlet pressure is p 0 # α , 0.5. Evaluate force toformula hold1 theoutsplitter plate changethe (t) after the hole isneeded opened.yApply your fluid and provide a thin viscou ¼ 12 through an Internal volume, V = 0.006 m3 tothe the following case: a spherical tank of diameter 50 spray cm, (abs). Evaluate axial force exerted by the nozzle on test section wall in Fig. P3.33 c Problema 1 umax R with initial pressure 300force kPa and temperature 100°C, the 5-mm passing thrm in place.the(Neglect any friction between water diameterstream each per square V Q = 0.11 m3/s max coupling. and a hole whose initial exhaust rate is 0.01 kg/s. suction velocity through each hole Find thePlot time required for theθtank density R by functions test-section entrance theconssplitter plate.) both and as ofpressure α.velocity isisV V to drop xVmin e the maximum velocity downstream section. Entra agua aat unthe canal cuadrado, bidimensional deand ancho 50 percent. the air in th 1 U1 ρ = 750 kg/m3

2

2

2

0

g

con velocidad El canal30° te the pressuretante droph,that would exist in theU. channel iftiene un codo en 90◦ 1uniforme V2 lineal que se muesp1 =flujo, 200 kPa (abs) 2 distorsiona el produciendo el perfil friction at theque walls could be neglected. 2 A = 0.0182 m

1 tra en la figura. Considerando Vmax 2VkPa . Calcular Vmin en min p2 ==120 (abs) R A2 = 0.0081 m2 funci´ on de U.

r

P4.92

z

incompressible flow of ai ~.steady Watertlow (a) V0, (b)nozzle V2, and (c) Vf , in m/s. and strike

V

p1 standing point-down, P3.30 A hollow conical container, is2 1.2 m x high and has a total included cone angle of 80 . It is being filled with water from a hose at 50 gallons per minute. How long will it take to fill the cone? R wedgeP3.31 Water A bellows may be modeled as a deforming shaped volume as in Fig. P3.31. The check valve on the h U ft/s If b is the left (pleated) end is closed during the80 stroke. bellows width into the paper, derive an expression for outlet mass flow m˙ 0 as a function of stroke (t).

αh

V

-

θ

4.46

V

Df = 2.2 m Vf

P4.39, 4.80, 4.98

Uniform suctio

initialflo 6-61.The Water totalnozzle tank v diameter V2 whichthe is originally initial r

cart4.47 when itA L =attain 4rec m

Thickness, t Prob. L (..-57

u

Air e

originally at of res velocity Test section = 0.8 th m 3 seconds with aDasfter veloc

P3.33 near Mon Consider the steady adiabatic flow of air through a 4.40 Viscous liquid from a circular tank, D 5 300 in diamα hmmP3.34 P4.98 2 A rocket motorthrough is operating s the long straight pipe with 0.05 m cross-sectional area. At the eter, drains through a long of that radius R 5 50 mm. 6-58. The car h is used to circular scoop uptube water is lying in aFig. P3.34. The products of comb information  4.99 A small round object is tested in a 0.75-m diameter lO exhaust nozzle approximate a perfe Determine the force needed to p ull the trough at the tracks. inlet, the air is at 200 kPa (gage), 60 C, and has a velocity of The velocityhprofile at the tube discharge is 1 2 weight of 28. For the given conditio Splitter No Un150 fuelle como unaand cu˜ n a deformable. Consicar forward at constant ,. for each of thesections three cases. wind tunnel. The pressurevelocity is uniform 1 and m/s.puede At theser exit,modelado the air is at 80 kPa has a velocity  r 2 across U1 of 300 m/s. Calculate the axial force of the air on the pipe. Flo The scoop has a cross-sectional area of 0 θ30 (t) A and them density . The upstream pressure is mm H O (gage), the 2 u 5 umax 1 2 2 derando que la v´ alvula est´ a cerrada durante la carrera para echar Liquid oxygen: θ (t) 3 1 RV d