Proof of Lemma 7.3

13.25 Find the densityof a ¯uid at a depthhin terms of its density ₀ at the surface.

If a massmof ¯uid has volumeV₀at the surface, then it will have volumeV₀ Vat a depthh. The density at depthhis then

ˆ m

V0 V while 0ˆ m

V0

₀ˆ V0

V₀ Vˆ 1

1 …V=V₀† which gives

However, from Chapter 12, the bulk modulus is BˆP=…V=V₀† and so V=V₀ˆP=B. Making this substitution, we obtain

₀ˆ 1

1 P=B

If we assume thatis close to₀, then the pressure at depthhis approximately₀gh, and so

₀ˆ 1

1 …₀gh=B†

Supplementary Problems

13.26 A 60-kg performer balances on a cane. The end of the cane in contact with the ¯oor has an area of 0.92 cm². Find the pressure exerted on the ¯oor by the cane. (Neglect the weight of the cane.) Ans. 6.4 MPa 13.27 A certain town receives its water directly from a water tower. If the topof the water in the tower is 26.0 m

above the water faucet in a house, what should be the water pressure at the faucet? (Neglect the e€ects of other water users.) Ans. 255 kPa

13.28 At a height of 10 km (33 000 ft) above sea level, atmospheric pressure is about 210 mm of mercury. What is the resultant normal force on a 600 cm² window of an airplane ¯ying at this height? Assume the pressure inside the plane is 760 mm of mercury. The density of mercury is 13 600 kg/m³. Ans. 4.4 kN 13.29 A narrow tube is sealed onto a tank as shown in Fig. 13-7. The base of the tank has an area of 80 cm².

(a) Find the force on the bottom of the tank due to oil when the tank and capillary are ®lled with oil …ˆ0:72 g/cm³†to the heighth₁. (b) Repeat forh₂: Ans. (a) 11 N downward; (b) 20 N downward 13.30 Repeat Problem 13.29, but now ®nd the force on the top wall of the tank due to the oil. Ans. (a) 1.1 N

upward; (b) 9.6 N upward

13.31 Compute the pressure required for a water supply system that will raise water 50.0 m vertically.

Ans. 490 kPa

13.32 The area of a piston of a force pump is 8.0 cm². What force must be applied to the piston to raise oil …ˆ0:78 g/cm²† to a height of 6.0 m? Assume the upper end of the oil is open to the atmosphere.

Ans. 37 N

13.33 The diameter of the large piston of a hydraulic press is 20 cm, and the area of the small piston is 0.50 cm². If a force of 400 N is applied to the small piston, (a) what is the resulting force exerted on the large piston?

(b) What is the increase in pressure underneath the small piston? (c) Underneath the large piston? Ans. (a) 2:510⁵N; (b) 8.0 MPa; (c) 8.0 MPa

13.34 A metal cube, 2.00 cm on each side, has a density of 6600 kg/m³. Find its apparent mass when it is totally submerged in water. Ans. 44.8 g

13.35 A solid wooden cube, 30.0 cm on each edge, can be totally submerged in water if it is pushed downward with a force of 54.0 N. What is the density of the wood? Ans. 800 kg/m³

13.36 A metal object ``weighs'' 26.0 g in air and 21.48 g when totally immersed in water. What is the volume of the object? Its mass density? Ans. 4.55 cm³, 5:7210³kg/m³

13.37 A solid piece of aluminum…ˆ2:70 g/cm³) has a mass of 8.35 g when measured in air. If it is hung from a thread and submerged in a vat of oil …ˆ0:75 g/cm³), what will be the tension in the thread?

Ans. 0.059 N

13.38 A beaker contains oil of density 0.80 g/cm³. By means of a thread, a 1.6-cm cube of aluminum …ˆ2:70 g/cm³) is submerged in the oil. Find the tension in the thread. Ans. 0.076 N

13.39 A tank containing oil of spgrˆ0:80 rests on a scale and weighs 78.6 N. By means of a wire, a 6.0 cm cube of aluminum, spgrˆ2:70, is submerged in the oil. Find (a) the tension in the wire and (b) the scale reading if none of the oil over¯ows. Ans. (a) 4.0 N; (b) 80 N

13.40 Downward forces of 45.0 N and 15.0 N, respectively, are required to keep a plastic block totally immersed in water and in oil. If the volume of the block is 8000 cm³, ®nd the density of the oil. Ans. 620 kg/m³ 13.41 Determine the unbalanced force acting on an iron ball…rˆ1:5 cm,ˆ7:8 g/cm³) when just released while

totally immersed in (a) water and (b) mercury…ˆ13:6 g/cm³). What will be the initial acceleration of the ball in each case? Ans. (a) 0.94 N down, 8.6 m/s²down; (b) 0.80 N up, 7.3 m/s²up

Fig. 13-7

13.42 A 2.0 cm cube of metal is suspended by a thread attached to a scale. The cube appears to have a mass of 47.3 g when measured submerged in water. What will its mass appear to be when submerged in glycerin, spgrˆ1:26? (Hint: Findtoo.) Ans. 45 g

13.43 A balloon and its gondola have a total (empty) mass of 2:010²kg. When ®lled, the balloon contains 900 m³of helium at a density of 0.183 kg/m³. Find the added load, in addition to its own weight, that the balloon can lift. The density of air is 1.29 kg/m³. Ans. 7.8 kN

13.44 A certain piece of metal has a measured mass of 5.00 g in air, 3.00 g in water, and 3.24 g in benzene.

Determine the mass density of the metal and of the benzene. Ans. 2:5010³kg/m³, 880 kg/m³ 13.45 A spring, which may be either bronze (sp gr 8.8) or brass (sp gr 8.4), has a mass of 1.26 g when measured in

air and 1.11 g in water. Which is it? Ans. brass

13.46 What fraction of the volume of a piece of quartz…ˆ2:65 g/cm³) will be submerged when it is ¯oating in a container of mercury…ˆ13:6 g/cm³)? Ans. 0.195

13.47 A cube of wood ¯oating in water supports a 200-g mass resting on the center of its top face. When the mass is removed, the cube rises 2.00 cm. Determine the volume of the cube. Ans. 1:0010³cm³

13.48 A cork has a measured mass of 5.0 g in air. A sinker has a measured mass of 86 g in water. The cork is attached to the sinker and both together have a measured mass of 71 g when under water. What is the density of the cork? Ans. 2:510²kg=m³

13.49 A glass of water has a 10-cm³ice cube ¯oating in it. The glass is ®lled to the brim with cold water. By the time the ice cube has completely melted, how much water will have ¯owed out of the glass? The sp gr of ice is 0.92. Ans. none

13.50 A glass tube is bent into the form of a U. A 50.0 cm height of olive oil in one arm is found to balance 46.0 cm of water in the other. What is the density of the olive oil? Ans. 920 kg/m³

13.51 On a day when the pressure of the atmosphere is 1:00010⁵Pa, a chemist distills a liquid under slightly reduced pressure. The pressure within the distillation chamber is read by an oil-®lled manometer (density of oilˆ0:78 g/cm³). The di€erence in heights on the two sides of the manometer is 27 cm. What is the pressure in the distillation chamber? Ans. 98 kPa

Chapter 14

Fluids in Motion

FLUID FLOW OR DISCHARGE …J†: When a ¯uid that ®lls a pipe ¯ows through the pipe with an average speed v, the ¯ow or discharge J is

J ˆAv

where A is the cross-sectional area of the pipe. The units of J are m³/s in the SI and ft³/s in U.S.

customary units.

SometimesJ is called the rate of ¯owor thedischarge rate.

EQUATION OF CONTINUITY: Suppose an incompressible (constant-density) ¯uid ®lls a pipe and ¯ows through it. Suppose further that the cross-sectional area of the pipe is A₁ at one point and A₂ at another. Since the ¯ow through A₁ must equal the ¯ow through A₂, one has

J ˆA₁v₁ ˆA₂v₂ˆconstant

wherev₁ andv₂ are the average ¯uid speeds overA₁ andA₂, respectively.

THE SHEAR RATE of a ¯uid is the rate at which the shear strain within the ¯uid is changing.

Because strain has no units, the SI unit for shear rate is s ¹.

THE VISCOSITY …† of a ¯uid is a measure of how large a shear stress is required to produce unit shear rate. Its unit is that of stress per unit shear rate, or Pas in the SI. Another SI unit is the Ns=m² (or kg/ms†, called the poiseuille (Pl): 1 Plˆ1 kg/msˆ1 Pas. Other units used are the poise (P), where 1 Pˆ0:1 , and the centipoise (cP), where 1 cPˆ10 ³ Pl. A viscous ¯uid, such as tar, has large :

POISEUILLE'S LAW: The ¯uid ¯ow through a cylindrical pipe of length L and cross-sectional radius R is given by

JˆR⁴…P_i P_o† 8L

whereP_i P_o is the pressure di€erence between the two ends of the pipe (input minus output).

THE WORK DONE BY A PISTON in forcing a volume V of ¯uid into a cylinder against an opposing pressure P is given by PV:

THE WORK DONE BY A PRESSURE P acting on a surface of area A as the surface moves through a distance x normal to the surface (thereby displacing a volume AxˆV) is

WorkˆPAxˆPV 157

Copyright 1997, 1989, 1979, 1961, 1942, 1940, 1939, 1936 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

BERNOULLI'S EQUATION for the steady ¯ow of a continuous stream of ¯uid: Consider two di€erent points along the stream path. Let point 1 be at a height h₁, and let v₁, ₁, and P₁ be the ¯uid-speed, density, and pressure at that point. Similarly de®ne h₂, v₂, ₂, and P₂ for point 2.

Then, provided the ¯uid is incompressible and has negligible viscosity, P₁‡¹₂v²₁‡h₁gˆP₂‡¹₂v²₂‡h₂g where₁ˆ₂ˆandgis the acceleration due to gravity.

TORRICELLI'S THEOREM: Suppose that a tank contains liquid and is open to the atmo-sphere at its top. If an ori®ce (opening) exists in the tank at a distance h below the topof the liquid, then the speed of out¯ow from the ori®ce is 

p2gh

, provided the liquid obeys Bernoulli's equation and the topof the liquid may be regarded as motionless.

THE REYNOLDS NUMBER …NR† is a dimensionless number that applies to a ¯uid of viscosity and density ¯owing with speed v through a pipe (or past an obstacle) with diameter D:

N_RˆvD

For systems of the same geometry, ¯ows will usually be similar provided their Reynolds numbers are close.Turbulent ¯owoccurs if N_Rfor the ¯ow exceeds about 2000 for pipes or about 10 for obstacles.

Solved Problems

14.1 Oil ¯ows through a pipe 8.0 cm in diameter, at an average speed of 4.0 m/s. What is the ¯owJin m³/s and m³/h?

JˆAvˆ…0:040 m†²…4:0 m=s† ˆ0:020 m³=s

ˆ …0:020 m³=s†…3600 s=h† ˆ72 m³=h

14.2 Exactly 250 mL of ¯uid ¯ows out of a tube whose inner diameter is 7.0 mm in a time of 41 s.

What is the average speed of the ¯uid in the tube?

FromJˆAv, since 1 mLˆ10 ⁶m³, vˆJ

Aˆ…25010 ⁶m³†=…41 s†

…0:003 5 m†² ˆ0:16 m=s

14.3 A 14 cm inner diameter (i.d.) water main furnishes water (through intermediate pipes) to a 1.00 cm i.d. faucet pipe. If the average speed in the faucet pipe is 3.0 cm/s, what will be the average speed it causes in the water main?

The two ¯ows are equal. From the continuity equation, we have JˆA₁v₁ˆA₂v₂

Letting 1 be the faucet and 2 be the water main, we have v₂ˆv₁A₁

A2ˆv₁r²₁

r²₂ˆ …3:0 cm=s† 1 14

2

ˆ0:015 cm=s

14.4 How much water will ¯ow in 30.0 s through 200 mm of capillary tube of 1.50 mm i.d., if the pressure di€erential across the tube is 5.00 cm of mercury? The viscosity of water is 0.801 cP and for mercury is 13 600 kg/m³.

We shall make use of Poiseuille's Law with

P_i P_oˆghˆ …13 600 kg=m³†…9:81 m=s²†…0:050 0 m† ˆ6660 N=m² ˆ …0:801 cP† 10 ³kg=ms

cP

ˆ8:0110 ⁴kg=ms and

Thus, we have

Jˆr⁴…Pi Po†

8L ˆ …7:510 ⁴m†⁴…6660 N=m²†

8…8:0110 ⁴kg=ms†…0:200 m†ˆ5:210 ⁶m³=sˆ5:2 mL=s In 30.0 s, the quantity that would ¯ow out of the tube is (5.2 mL/s)(30 s)ˆ1:610²mL.

14.5 An artery in a certain person has been reduced to half its original inside diameter by deposits on the inner artery wall. By what factor will the blood ¯ow through the artery be reduced if the pressure di€erential across the artery has remained unchanged?

From Poiseuille's Law,J/r⁴. Therefore, Jfinal

Joriginalˆ rfinal

roriginal

₄

ˆ 1 2

4

ˆ0:062 5

14.6 Under the same pressure di€erential, compare the ¯ow of water through a pipe to the ¯ow of SAE No. 10 oil. for water is 0.801 cP; for the oil is 200 cP.

From Poiseuille's Law,J/1=. Therefore, Jwater

J_oil ˆ 200 cP 0:801 cPˆ250

so the ¯ow of water is 250 times as large as that of the oil under the same pressure di€erential.

14.7 Calculate the power output of the heart if, in each heartbeat, it pumps 75 mL of blood at an average pressure of 100 mmHg. Assume 65 heartbeats per minute.

The work done by the heart isPV. In one minute,Vˆ …65†…7510 ⁶m³†. Also Pˆ …100 mmHg†1:0110⁵Pa

760 mmHg ˆ1:3310⁴Pa Powerˆwork

time ˆ…1:3310⁴Pa†‰…65†…7510 ⁶m³†Š

60 s ˆ1:1 W

so

14.8 What volume of water will escape per minute from an open-top tank through an opening 3.0 cm in diameter that is 5.0 m below the water level in the tank? (See Fig. 14-1.)

We can use Bernoulli's equation, with 1 representing the top level and 2 the ori®ce. ThenP₁ˆP₂and h1ˆ5:0 m,h2ˆ0.

P1‡¹₂v²1‡h1gˆP2‡¹₂v²2‡h2g

12v²₁‡h₁gˆ¹₂v²₂‡h₂g

If the tank is large,v1can be approximated as zero. Then, solving forv2, we obtain Torricelli's equation:

v₂ˆ 

2g…h₁ h₂†

p ˆ 

2…9:81 m=s²†…5:0 m†

q

ˆ9:9 m=s and the ¯ow is given by

Jˆv₂A₂ˆ …9:9 m=s†…1:510 ²m†²ˆ7:010 ³m³=sˆ0:42 m³=min

14.9 A water tank springs a leak at position 2 in Fig. 14-2, where the water pressure is 500 kPa. What is the velocity of escape of the water through the hole?

We use Bernoulli's equation withP₁ P₂ˆ5:0010⁵N/m²,h₁ˆh₂, and the approximationv₁ˆ0.

Then

…P1 P2† ‡ …h1 h2†gˆ¹₂v²2

v₂ˆ



2…P1 P2† s

ˆ



2…5:0010⁵N=m²† 1000 kg=m³ s

ˆ31:6 m=s whence

14.10 Water ¯ows at the rate of 30 mL/s through an opening at the bottom of a tank in which the water is 4.0 m deep. Calculate the rate of escape of the water if an added pressure of 50 kPa is applied to the topof the water.

Fig. 14-1

Fig. 14-2

From Bernoulli's equation in the case wherev1is essentially zero, …P1 P2† ‡ …h1 h2†gˆ¹₂v²2

We can write this twice, before the pressure is added and after.

…P₁ P₂†_before‡ …h₁ h₂†gˆ¹₂…v²₂†_before …P₁ P₂†_before‡510⁴N=m²‡ …h₁ h₂†gˆ¹₂…v²₂†_after If the opening and the top of the tank are originally at atmospheric pressure, then

…P₁ P₂†_beforeˆ0 and division of the second equation by the ®rst gives

…v²₂†_after

…v²₂†_beforeˆ510⁴N=m²‡ …h₁ h₂†g …h₁ h₂†g

…h1 h2†gˆ …4:0 m†…1000 kg=m³†…9:81 m=s²† ˆ3:910⁴N=m² But

…v₂†_after …v2†_beforeˆ



8:910⁴N=m² 3:910⁴N=m² s

ˆ1:51 Therefore;

SinceJˆAv, this can be written as J_after

J_beforeˆ1:51 or J_afterˆ …30 mL=s†…1:51† ˆ45 mL=s

14.11 How much work W is done by a pump in raising 5.00 m³ of water 20.0 m and forcing it into a main at a gauge pressure of 150 kPa?

Wˆ(work to raise water)‡(work to push it in)ˆmgh‡PV

W ˆ …5:00 m³†…1000 kg=m³†…9:81 m=s²†…20:0 m† ‡ …1:5010⁵N=m²†…5:00 m³† ˆ1:7310⁶J

14.12 A horizontal pipe has a constriction in it, as shown in Fig. 14-3. At point 1 the diameter is 6.0 cm, while at point 2 it is only 2.0 cm. At point 1,v₁ˆ2:0 m/s andP₁ˆ180 kPa. Calculatev₂andP₂: We have two unknowns and will need two equations. Using Bernoulli's equation withh₁ˆh₂, we have

P₁‡¹₂v²₁ˆP₂‡¹₂v²₂ or P₁‡¹₂…v²₁ v²₂† ˆP₂

Furthermore,v₁ˆ2:0 m/s, and the equation of continuity tells us that v₂ˆv₁A1

A₂ˆ …2:0 m=s† r1

r₂

2

ˆ …2:0 m=s†…9:0† ˆ18 m=s Fig. 14-3

Substituting then gives

1:8010⁵N/m²‡¹₂…1000 kg/m³†‰…2:0 m/s)² …18 m/s)²Š ˆP2

from whichP₂ˆ0:2010⁵N/m²ˆ20 kPa.

14.13 What must be the gauge pressure in a large ®re hose if the nozzle is to shoot water straight upward to a height of 30.0 m?

To rise to a heighth, a projectile must have an initial speed 

p2gh

. (We obtain this by equating¹₂mv²0to mgh.) We can ®nd this speed in terms of the di€erence between the pressures inside and outside the hose (the gauge pressure) by writing Bernoulli's equation for points just inside and outside the nozzle:

P_in‡¹₂v²_in‡h_ingˆP_out‡¹₂v²_out‡h_outg Herehouthinand because the hose is largevin0, therefore

Pin Poutˆ¹₂v²out

Substitution of 

p2gh

forv_outgives

P_in P_outˆghˆ …1000 kg=m³†…9:81 m=s²†…30:0 m† ˆ294 kPa How could you obtain this latter equation directly from Torricelli's Theorem?

14.14 At what rate does water ¯ow from a 0.80 cm i.d. faucet if the water pressure is 200 kPa?

We use Bernoulli's equation for points just inside and outside the faucet:

Pin‡¹₂v²in‡hingˆPout‡¹₂v²out‡houtg Takingh_outˆh_inandP_in P_outˆ200 kPa, we have

v²_out v²_inˆ …20010³Pa†2

Assumingv²_inv²_out, we solve to obtainv_outˆ20 m/s. The ¯ow rate is then

JˆvAˆ …20 m=s†…r²† ˆ …20 m=s†…†…0:1610 ⁴m²† ˆ1:010 ³m³=s

14.15 The pipe shown in Fig. 14-4 has a diameter of 16 cm at section 1 and 10 cm at section 2. At section 1 the pressure is 200 kPa. Point 2 is 6.0 m higher than point 1. When oil of density 800 kg/m³ ¯ows at a rate of 0.030 m³/s, ®nd the pressure at point 2 if viscous e€ects are negligible.

FromJˆv₁A₁ˆv₂A₂we have v₁ˆ J

A₁ˆ 0:030 m³=s

…8:010 ²m†²ˆ1:49 m=s v2ˆ J

A₂ˆ 0:030 m³=s

…5:010 ²m†²ˆ3:82 m=s We can now use Bernoulli's equation:

P1‡¹₂v²1‡g…h1 h2† ˆP2‡¹₂v²2

SettingP₁ˆ2:0010⁵N=m²;h₂ h₁ˆ6 m andˆ800 kg=m³gives

P₂ˆ2:0010⁵N=m³‡¹₂…800 kg=m³†‰…1:49 m=s†² …3:82 m=s†²Š …800 kg=m³†…9:81 m=s²†…6:0 m†

ˆ1:4810⁵ N=m²ˆ1:510⁵kPa.

14.16 A venturi meter equipped with a di€erential mercury manometer is shown in Fig. 14-5. At the inlet, point 1, the diameter is 12 cm, while at the throat, point 2, the diameter is 6.0 cm. What is the ¯owJ of water through the meter if the mercury manometer reading is 22 cm? The density of mercury is 13.6 g/cm³:

From the manometer reading we obtain

P₁ P₂ˆghˆ …13 600 kg=m³†…9:81 m=s²†…0:22 m† ˆ2:9310⁴N=m²

SinceJˆv1A1ˆv2A2, we havev1ˆJ=A1andv2ˆJ=A2. Using Bernoulli's equation withh1 h2ˆ0 gives

…P₁ P₂† ‡¹₂…v²₁ v²₂† ˆ0 2:9310⁴N=m²‡¹₂…1000 kg=m³† 1

A²₁ 1 A²₂

J²ˆ0 where

A1ˆr²1ˆ…0:060†²m²ˆ0:011 31 m² and A2ˆr²2ˆ…0:030†²m²ˆ0:002 8 m² Substitution then givesJˆ0:022 m³/s.

14.17 A wind tunnel is to be used with a 20 cm high model car to approximately reproduce the situation in which a 550 cm high car is moving at 15 m/s. What should be the wind speed in the tunnel? Is the ¯ow likely to be turbulent?

We want the Reynolds numberN_Rto be the same in both cases, so that the situations will be similar.

That is, we want

N_Rˆ vD

tunnelˆ vD

air

Bothandare the same in the two cases, so we have v_tD_tˆv_aD_a from which v_tˆv_aDa

D_t ˆ …15 m=s†…550=20† ˆ0:41 km=s

To investigate turbulence, we evaluateNR usingˆ1:29 kg/m³andˆ1:810 ⁵Pas for air. We ®nd that N_Rˆ5:910⁶, a value far in excess of that required for turbulent ¯ow. The ¯ow will obviously be turbulent.

Fig. 14-4 Fig. 14-5

Supplementary Problems

14.18 Oil ¯ows through a 4.0 cm i.d. pipe at an average speed of 2.5 m/s. Find the ¯ow in m³/s and cm³/s. Ans. 3:110 ³m³=sˆ3:110³cm³=s

14.19 Compute the average speed of water in a pipe having an i.d. of 5.0 cm and delivering 2.5 m³of water per hour. Ans. 0.35 m/s

14.20 The speed of glycerin ¯owing in a 5.0 cm i.d. pipe is 0.54 m/s. Find the ¯uid's speed in a 3.0 cm i.d. pipe that connects with it, both pipes ¯owing full. Ans. 1.5 m/s

14.21 How long will it take for 500 mL of water to ¯ow through a 15 cm long, 3.0 mm i.d. pipe, if the pressure di€erential across the pipe is 4.0 kPa? The viscosity of water is 0.80 cP. Ans. 7.5 s

14.22 A molten plastic ¯ows out of a tube that is 8.0 cm long at a rate of 13 cm³/min when the pressure di€erential between the two ends of the tube is 18 cm of mercury. Find the viscosity of the plastic. The i.d. of the tube is 1.30 mm. The density of mercury is 13.6 g/cm³. Ans. 0.097 kg/msˆ97 cP

14.23 In a horizontal pipe system, a pipe (i.d. 4.0 mm) that is 20 cm long connects in line to a pipe (i.d. 5.0 mm) that is 30 cm long. When a viscous ¯uid is being pushed through the pipes at a steady rate, what is the ratio of the pressure di€erence across the 20-cm pipe to that across the 30-cm pipe? Ans. 1.6

14.24 A hypodermic needle of length 3.0 cm and i.d. 0.45 mm is used to draw blood…ˆ4:0 mPl). Assuming the pressure di€erential across the needle is 80 cmHg, how long does it take to draw 15 mL? Ans. 17 s 14.25 In a blood transfusion, blood ¯ows from a bottle at atmospheric pressure into a patient's vein in which the

pressure is 20 mmHg higher than atmospheric. The bottle is 95 cm higher than the vein, and the needle into the vein has a length of 3.0 cm and an i.d. of 0.45 mm. How much blood ¯ows into the vein each minute?

For blood,ˆ0:004 0 Pas andˆ1005 kg/m³. Ans. 3.4 cm³

14.26 How much work does the piston in a hydraulic system do during one 2.0-cm stroke if the end area of the piston is 0.75 cm²and the pressure in the hydraulic ¯uid is 50 kPa? Ans. 75 mJ

14.27 A large open tank of nonviscous liquid springs a leak 4.5 m below the top of the liquid. What is the theoretical velocity of out¯ow from the hole? If the area of the hole is 0.25 cm², how much liquid would escape in exactly 1 minute? Ans. 9.4 m/s, 0.014 1 m³

14.28 Find the ¯ow in liters/s of a nonviscous liquid through an opening 0.50 cm²in area and 2.5 m below the level of the liquid in an open tank. Ans. 0.35 liter/s

14.29 Calculate the theoretical velocity of e‚ux of water from an aperture that is 8.0 m below the surface of water in a large tank, if an added pressure of 140 kPa is applied to the surface of the water. Ans. 21 m/s 14.30 What horsepower is required to force 8.0 m³ of water per minute into a water main at a pressure of

220 kPa? Ans. 39 hp

14.31 A pump lifts water at the rate of 9.0 liters/s from a lake through a 5.0 cm i.d. pipe and discharges it into the air at a point 16 m above the level of the water in the lake. What are the theoretical (a) velocity of the water at the point of discharge and (b) power delivered by the pump. Ans. (a) 4.6 m/s; (b) 2.0 hp

14.32 Water ¯ows steadily through a horizontal pipe of varying cross-section. At one place the pressure is 130 kPa and the speed is 0.60 m/s. Determine the pressure at another place in the same pipe where the speed is 9.0 m/s. Ans. 90 kPa.

14.33 A pipe of varying inner diameter carries water. At point 1 the diameter is 20 cm and the pressure is 130 kPa.

At point 2, which is 4.0 m higher than point 1, the diameter is 30 cm. If the ¯ow is 0.080 m³/s, what is the pressure at the second point? Ans. 93 kPa

14.34 Fuel oil of density 820 kg/m³ ¯ows through a venturi meter having a throat diameter of 4.0 cm and an entrance diameter of 8.0 cm. The pressure drop between entrance and throat is 16 cm of mercury. Find the

¯ow. The density of mercury is 13 600 kg/m³. Ans. 9:310 ³m³/s

14.35 Find the maximum amount of water that can ¯ow through a 3.0 cm i.d. pipe per minute without turbulence.

Take the maximum Reynolds number for nonturbulent ¯ow to be 2000. For water at 208C, ˆ1:010 ³Pas. Ans. 0.002 8 m³

14.36 How fast can a raindrop…rˆ1:5 mm) fall through air if the ¯ow around it is to be close to turbulent, i.e., for NRclose to 10? For air,ˆ1:810 ⁵Pas andˆ1:29 kg/m³. Ans. 4.6 cm/s

In document Distributed Control and Computing: Optimal Estimation, Error Correcting Codes, and Interactive Protocols (Page 167-170)