Cengel Numerical

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Special Case: Incompressible FlowThe conservation of mass relations can be simplified even further when the

fluid is incompressible, which is usually the case for liquids. Canceling the

density from both sides of the general steady-flow relation gives

Steady, incompressible flow: (522)

For single-stream steady-flow systems it becomes

Steady, incompressible flow (single stream): (523)

It should always be kept in mind that there is no such thing as a conserva-

tion of volume principle. Therefore, the volume flow rates into and out of a

steady-flow device may be different. The volume flow rate at the outlet of

an air compressor is much less than that at the inlet even though the mass

flow rate of air through the compressor is constant (Fig. 511). This is due

to the higher density of air at the compressor exit. For steady flow of liq-

uids, however, the volume flow rates, as well as the mass flow rates, remain

constant since liquids are essentially incompressible (constant-density) sub-

stances. Water flow through the nozzle of a garden hose is an example ofthe latter case.

The conservation of mass principle is based on experimental observations

and requires every bit of mass to be accounted for during a process. If you

can balance your checkbook (by keeping track of deposits and withdrawals,

or by simply observing the conservation of money principle), you should

have no difficulty applying the conservation of mass principle to engineer-

ing systems.

EXAMPLE 51 Water Flow through a Garden Hose Nozzle

A garden hose attached with a nozzle is used to fill a 10-gal bucket. Theinner diameter of the hose is 2 cm, and it reduces to 0.8 cm at the nozzle

exit (Fig. 512). If it takes 50 s to fill the bucket with water, determine

(a) the volume and mass flow rates of water through the hose, and (b) the

average velocity of water at the nozzle exit.

SOLUTION A garden hose is used to fill a water bucket. The volume andmass flow rates of water and the exit velocity are to be determined.

Assumptions 1 Water is an incompressible substance. 2 Flow through the

hose is steady. 3 There is no waste of water by splashing.

Properties We take the density of water to be 1000 kg/m3 1 kg/L.

Analysis (a) Noting that 10 gal of water are discharged in 50 s, the volume

and mass flow rates of water are

(b) The cross-sectional area of the nozzle exit is

Ae pr2e p(0.4 cm)

2 0.5027 cm2 0.5027 104m2

m#

rV#

(1 kg/L)(0.757 L/s) 0.757 kg/s

V#

V

t

10 gal

50 sa3.7854 L

1 galb 0.757 L/s

V#

1 V#

2 V1A1 V2A2

ain

V#

aout

V# (m3/s)

178FLUID MECHANICS

m1 = 2 kg/s

Air

compressor

m2 = 2 kg/s

V2 = 0.8 m3/s

V1 = 1.4 m3/s

FIGURE 511

During a steady-flow process,

volume flow rates are not necessarily

conserved although mass flow

rates are.

Nozzle

BucketGardenhose

FIGURE 512

Schematic for Example 51.


2/24

The volume flow rate through the hose and the nozzle is constant. Then the

average velocity of water at the nozzle exit becomes

Discussion It can be shown that the average velocity in the hose is 2.4 m/s.

Therefore, the nozzle increases the water velocity by over six times.

EXAMPLE 52 Discharge of Water from a Tank

A 4-ft-high, 3-ft-diameter cylindrical water tank whose top is open to the

atmosphere is initially filled with water. Now the discharge plug near the bot-

tom of the tank is pulled out, and a water jet whose diameter is 0.5 in

streams out (Fig. 513). The average velocity of the jet is given by

V , where h is the height of water in the tank measured from the

center of the hole (a variable) and g is the gravitational acceleration. Deter-

mine how long it will take for the water level in the tank to drop to 2 ft from

the bottom.

SOLUTION The plug near the bottom of a water tank is pulled out. Thetime it will take for half of the water in the tank to empty is to be deter-

mined.

Assumptions 1 Water is an incompressible substance. 2 The distance

between the bottom of the tank and the center of the hole is negligible com-

pared to the total water height. 3 The gravitational acceleration is 32.2 ft/s2.

Analysis We take the volume occupied by water as the control volume. The

size of the control volume decreases in this case as the water level drops,

and thus this is a variable control volume. (We could also treat this as a

fixed control volume that consists of the interior volume of the tank by disre-

garding the air that replaces the space vacated by the water.) This is obvi-

ously an unsteady-flow problem since the properties (such as the amount ofmass) within the control volume change with time.

The conservation of mass relation for a control volume undergoing any

process is given in the rate form as

(1)

During this process no mass enters the control volume (m.

in 0), and the

mass flow rate of discharged water can be expressed as

(2)

where Ajet pD2jet/4 is the cross-sectional area of the jet, which is constant.

Noting that the density of water is constant, the mass of water in the tank at

any time is(3)

where Atank pD2tank/4 is the base area of the cylindrical tank. Substituting

Eqs. 2 and 3 into the mass balance relation (Eq. 1) gives

r22ghAjet d(rAtankh)

dt r22gh(pD2jet/4)

r(pD2tank/4) dh

dt

mCV rV rAtankh

m#

out (rVA)out r22ghAjet

m#

in m#

out dmCV

dt

12gh

Ve V

#

Ae

0.757 L/s

0.5027 104 m2a 1 m3

1000 Lb 15.1 m/s

179CHAPTER 5

Water

Air

0 Dtank

Djeth2

h0

h

FIGURE 51

Schematic for Example 52


3/24

Canceling the densities and other common terms and separating the vari-

ables give

Integrating from t 0 at which h h0 to t tat which h h2 gives

Substituting, the time of discharge is determined to be

Therefore, half of the tank will be emptied in 12.6 min after the discharge

hole is unplugged.

Discussion Using the same relation with h2 0 gives t 43.1 min for the

discharge of the entire amount of water in the tank. Therefore, emptying the

bottom half of the tank takes much longer than emptying the top half. This

is due to the decrease in the average discharge velocity of water with

decreasing h.

53 MECHANICAL ENERGY AND EFFICIENCY

Many fluid systems are designed to transport a fluid from one location to

another at a specified flow rate, velocity, and elevation difference, and the

system may generate mechanical work in a turbine or it may consume

mechanical work in a pump or fan during this process. These systems do

not involve the conversion of nuclear, chemical, or thermal energy to

mechanical energy. Also, they do not involve any heat transfer in any signif-

icant amount, and they operate essentially at constant temperature. Suchsystems can be analyzed conveniently by considering the mechanical forms

of energy only and the frictional effects that cause the mechanical energy to

be lost (i.e., to be converted to thermal energy that usually cannot be used

for any useful purpose).

The mechanical energy can be defined as the form of energy that can be

converted to mechanical work completely and directly by an ideal mechani-

cal device such as an ideal turbine. Kinetic and potential energies are the

familiar forms of mechanical energy. Thermal energy is not mechanical

energy, however, since it cannot be converted to work directly and com-

pletely (the second law of thermodynamics).

A pump transfers mechanical energy to a fluid by raising its pressure, and

a turbine extracts mechanical energy from a fluid by dropping its pressure.Therefore, the pressure of a flowing fluid is also associated with its mechan-

ical energy. In fact, the pressure unit Pa is equivalent to Pa N/m2

N m/m3 J/m3, which is energy per unit volume, and the product Pv or

its equivalent P/rhas the unit J/kg, which is energy per unit mass. Note that

pressure itself is not a form of energy. But a pressure force acting on a fluid

through a distance produces work, called flow work, in the amount of P/r

per unit mass. Flow work is expressed in terms of fluid properties, and it is

convenient to view it as part of the energy of a flowing fluid and call it flow

t24 ft 22 ft

232.2/2 ft/s2a3 12 in

0.5 inb2 757 s 12.6 min

t

0

dt D2tank

D2jet22g

h2

h0

dh

2h t

2h0 2h2

2g/2aDtankDjet

b2

dt D2tank

D2jet

dh

22gh

180FLUID MECHANICS


4/24

occurs in EGL and HGL whenever mechanical energy is removed from

the fluid (by a turbine, for example), as shown in Fig. 536.

The pressure (gage) of a fluid is zero at locations where the HGL

intersects the fluid. The pressure in a flow section that lies above the HGL

is negative, and the pressure in a section that lies below the HGL is

positive (Fig. 537). Therefore, an accurate drawing of a piping system

and the HGL can be used to determine the regions where the pressure inthe pipe is negative (below the atmospheric pressure).

The last remark enables us to avoid situations in which the pressure drops

below the vapor pressure of the liquid (which causes cavitation, as dis-

cussed in Chap. 2). Proper consideration is necessary in the placement of a

liquid pump to ensure that the suction side pressure does not fall too low,

especially at elevated temperatures where vapor pressure is higher than it is

at low temperatures.

Now we examine Fig. 534 more closely. At point 0 (at the liquid surface),

EGL and HGL are even with the liquid surface since there is no flow there.

HGL decreases rapidly as the liquid accelerates into the pipe; however, EGL

decreases very slowly through the well-rounded pipe inlet. EGL declines con-tinually along the flow direction due to friction and other irreversible losses in

the flow. EGL cannot increase in the flow direction unless energy is supplied

to the fluid. HGL can rise or fall in the flow direction, but can never exceed

EGL. HGL rises in the diffuser section as the velocity decreases, and the sta-

tic pressure recovers somewhat; the total pressure does notrecover, however,

and EGL decreases through the diffuser. The difference between EGL and

HGL is V21/2g at point 1, and V22/2g at point 2. Since V1 V2, the difference

between the two grade lines is larger at point 1 than at point 2. The downward

slope of both grade lines is larger for the smaller diameter section of pipe

since the frictional head loss is greater. Finally, HGL decays to the liquid sur-

face at the outlet since the pressure there is atmospheric. However, EGL is

still higher than HGL by the amount V2

2/2g since V3

V2 at the outlet.

55 APPLICATIONS OF THEBERNOULLI EQUATION

In Section 54, we discussed the fundamental aspects of the Bernoulli equa-

tion. In this section, we demonstrate its use in a wide range of applications

through examples.

EXAMPLE 55 Spraying Water into the Air

Water is flowing from a hose attached to a water main at 400 kPa gage (Fig.538). A child places his thumb to cover most of the hose outlet, causing a

thin jet of high-speed water to emerge. If the hose is held upward, what is

the maximum height that the jet could achieve?

SOLUTION Water from a hose attached to the water main is sprayed intothe air. The maximum height the water jet can rise is to be determined.

Assumptions 1 The flow exiting into the air is steady, incompressible, and

irrotational (so that the Bernoulli equation is applicable). 2 The water pressure

194FLUID MECHANICS

PumpTurbine

EGL

HGL

Wpump Wturbine

FIGURE 536

A steep jump occurs in EGL and HGL

whenever mechanical energy is added

to the fluid by a pump, and a steep drop

occurs whenever mechanical energy is

removed from the fluid by a turbine.

Negative P

P= 0

P= 0HGL

Positive P

Positive P

FIGURE 537

The pressure (gage) of a fluid is zero

at locations where the HGL intersects

the fluid, and the pressure is negative

(vacuum) in a flow section that lies

above the HGL.


5/24

in the hose near the outlet is equal to the water main pressure. 3 The surface

tension effects are negligible. 4 The friction between the water and air is neg-

ligible. 5 The irreversibilities that may occur at the outlet of the hose due to

abrupt expansion are negligible.

Properties We take the density of water to be 1000 kg/m3.

Analysis This problem involves the conversion of flow, kinetic, and potential

energies to each other without involving any pumps, turbines, and wastefulcomponents with large frictional losses, and thus it is suitable for the use of

the Bernoulli equation. The water height will be maximum under the stated

assumptions. The velocity inside the hose is relatively low (V1 0) and wetake the hose outlet as the reference level (z1 0). At the top of the water

trajectory V2 0, and atmospheric pressure pertains. Then the Bernoulli

equation simplifies to

Solving for z2 and substituting,

Therefore, the water jet can rise as high as 40.8 m into the sky in this case.

Discussion The result obtained by the Bernoulli equation represents the

upper limit and should be interpreted accordingly. It tells us that the water

cannot possibly rise more than 40.8 m, and, in all likelihood, the rise will be

much less than 40.8 m due to irreversible losses that we neglected.

EXAMPLE 56 Water Discharge from a Large Tank

A large tank open to the atmosphere is filled with water to a height of 5 m

from the outlet tap (Fig. 539). A tap near the bottom of the tank is now

opened, and water flows out from the smooth and rounded outlet. Determine

the water velocity at the outlet.

SOLUTION A tap near the bottom of a tank is opened. The exit velocity ofwater from the tank is to be determined.

Assumptions 1 The flow is incompressible and irrotational (except very close

to the walls). 2 The water drains slowly enough that the flow can be approxi-

mated as steady (actually quasi-steady when the tank begins to drain).

Analysis This problem involves the conversion of flow, kinetic, and potential

energies to each other without involving any pumps, turbines, and wasteful

components with large frictional losses, and thus it is suitable for the use of

the Bernoulli equation. We take point 1 to be at the free surface of water so

that P1 Patm (open to the atmosphere), V1 0 (the tank is large relative tothe outlet), and z1 5 m and z2 0 (we take the reference level at the

center of the outlet). Also, P2 Patm (water discharges into the atmosphere).

Then the Bernoulli equation simplifies to

P1

rg

V21

2g z1

P2

rg

V22

2gz2 z1

V22

2g

40.8 m

z2 P1 Patm

rg

P1, gage

rg

400 kPa

(1000 kg/m3

)(9.81 m/s2

) a1000 N/m2

1 kPa b a1 kg m/s2

1 N b

P1

rg

V21

2g z1

P2

rg

V 22

2g z2

P1

rg

Patm

rg z2

195CHAPTER 5

Water jet

2

Hose

0

z

1

FIGURE 53


Water5m

1

2

V2

FIGURE 53


0 0 0Q Q

0 0Q


6/24

Solving for V2 and substituting,

The relation V is called the Toricelli equation.

Therefore, the water leaves the tank with an initial velocity of 9.9 m/s.

This is the same velocity that would manifest if a solid were dropped a dis-

tance of 5 m in the absence of air friction drag. (What would the velocity beif the tap were at the bottom of the tank instead of on the side?)

Discussion If the orifice were sharp-edged instead of rounded, then the flow

would be disturbed, and the velocity would be less than 9.9 m/s, especially

near the edges. Care must be exercised when attempting to apply the

Bernoulli equation to situations where abrupt expansions or contractions

occur since the friction and flow disturbance in such cases may not be neg-

ligible.

EXAMPLE 57 Siphoning Out Gasoline from a Fuel Tank

During a trip to the beach (Patm

1 atm

101.3 kPa), a car runs out ofgasoline, and it becomes necessary to siphon gas out of the car of a Good

Samaritan (Fig. 540). The siphon is a small-diameter hose, and to start the

siphon it is necessary to insert one siphon end in the full gas tank, fill the

hose with gasoline via suction, and then place the other end in a gas can

below the level of the gas tank. The difference in pressure between point 1

(at the free surface of the gasoline in the tank) and point 2 (at the outlet of

the tube) causes the liquid to flow from the higher to the lower elevation.

Point 2 is located 0.75 m below point 1 in this case, and point 3 is located

2 m above point 1. The siphon diameter is 4 mm, and frictional losses in

the siphon are to be disregarded. Determine (a) the minimum time to with-

draw 4 L of gasoline from the tank to the can and (b) the pressure at point

3. The density of gasoline is 750 kg/m3.

SOLUTION Gasoline is to be siphoned from a tank. The minimum time ittakes to withdraw 4 L of gasoline and the pressure at the highest point in

the system are to be determined.

Assumptions 1 The flow is steady and incompressible. 2 Even though the

Bernoulli equation is not valid through the pipe because of frictional losses,

we employ the Bernoulli equation anyway in order to obtain a best-case esti-

mate. 3 The change in the gasoline surface level inside the tank is negligible

compared to elevations z1 and z2 during the siphoning period.

Properties The density of gasoline is given to be 750 kg/m3.

Analysis (a) We take point 1 to be at the free surface of gasoline in the

tank so that P1 Patm (open to the atmosphere), V1 0 (the tank is largerelative to the tube diameter), and z2 0 (point 2 is taken as the reference

level). Also, P2 Patm (gasoline discharges into the atmosphere). Then the

Bernoulli equation simplifies to


V2 22gz1 22(9.81 m/s2)(0.75 m) 3.84 m/s

P1

rg

V21

2g z1

P2

rg

V22

2gz2 z1

V22

2g

12gz

V2 22gz1 22(9.81 m/s2)(5 m) 9.9 m/s

196FLUID MECHANICS

0 0Q

0

z

z3

z1

z2

Gastank

0.75 m

2 m

Gasoline

siphoning

tube

Gas can

1

2

3

FIGURE 540

Schematic for Example 57.


7/24

The cross-sectional area of the tube and the flow rate of gasoline are

Then the time needed to siphon 4 L of gasoline becomes

(b) The pressure at point 3 can be determined by writing the Bernoulli equa-

tion between points 2 and 3. Noting that V2 V3 (conservation of mass), z2 0, and P2 Patm,

Solving for P3 and substituting,

Discussion The siphoning time is determined by neglecting frictional

effects, and thus this is the minimum timerequired. In reality, the time will

be longer than 53.1 s because of friction between the gasoline and the tube

surface. Also, the pressure at point 3 is below the atmospheric pressure. If

the elevation difference between points 1 and 3 is too high, the pressure at

point 3 may drop below the vapor pressure of gasoline at the gasoline tem-

perature, and some gasoline may evaporate (cavitate). The vapor then may

form a pocket at the top and halt the flow of gasoline.

EXAMPLE 58 Velocity Measurement by a Pitot Tube

A piezometer and a Pitot tube are tapped into a horizontal water pipe, as

shown in Fig. 541, to measure static and stagnation (static dynamic)

pressures. For the indicated water column heights, determine the velocity at

the center of the pipe.

SOLUTION The static and stagnation pressures in a horizontal pipe aremeasured. The velocity at the center of the pipe is to be determined.

Assumptions 1 The flow is steady and incompressible. 2 Points 1 and 2 are

close enough together that the irreversible energy loss between these two

points is negligible, and thus we can use the Bernoulli equation.

Analysis We take points 1 and 2 along the centerline of the pipe, with point1 directly under the piezometer and point 2 at the tip of the Pitot tube. This

is a steady flow with straight and parallel streamlines, and the gage pres-

sures at points 1 and 2 can be expressed as

P2 rg(h1 h2 h3)

P1 rg(h1 h2)

81.1 kPa

101.3 kPa

(750 kg/m

3

)(9.81 m/s

2

)(2.75 m)a1 N

1 kg m/s2b a1 kPa

1000 N/m2b

P3 Patm rgz3

P2

rg

V22

2gz2

P3

rg

V23

2gz3

Patm

rg

P3

rgz3

t VV

# 4 L0.0753 L/s

53.1 s

V#

V2A (3.84 m/s)(1.96 105m2) 7.53 105m3/s 0.0753 L/s

A pD2/4 p(5 103m)2/4 1.96 105m2

197CHAPTER 5

0

h3= 12 cm

h2= 7 cm

h1= 3 cm

Stagnation

point

Water1 2

V1

FIGURE 54Schematic for Example 5


8/24

Noting that point 2 is a stagnation point and thus V2 0 and z1 z2, the

application of the Bernoulli equation between points 1 and 2 gives

Substituting the P1 and P2 expressions gives


Discussion Note that to determine the flow velocity, all we need is to mea-

sure the height of the excess fluid column in the Pitot tube.

EXAMPLE 59 The Rise of the Ocean Due to a Hurricane

A hurricane is a tropical storm formed over the ocean by low atmospheric

pressures. As a hurricane approaches land, inordinate ocean swells (very

high tides) accompany the hurricane. A Class-5 hurricane features winds in

excess of 155 mph, although the wind velocity at the center eye is very

low.

Figure 542 depicts a hurricane hovering over the ocean swell below. The

atmospheric pressure 200 mi from the eye is 30.0 in Hg (at point 1, gener-

ally normal for the ocean) and the winds are calm. The hurricane atmo-

spheric pressure at the eye of the storm is 22.0 in Hg. Estimate the ocean

swell at (a) the eye of the hurricane at point 3 and (b) point 2, where the

wind velocity is 155 mph. Take the density of seawater and mercury to be

64 lbm/ft3 and 848 lbm/ft3, respectively, and the density of air at normal

sea-level temperature and pressure to be 0.076 lbm/ft3.

SOLUTION A hurricane is moving over the ocean. The amount of oceanswell at the eye and at active regions of the hurricane are to be determined.

Assumptions 1 The airflow within the hurricane is steady, incompressible,

and irrotational (so that the Bernoulli equation is applicable). (This is cer-

tainly a very questionable assumption for a highly turbulent flow, but it is jus-

tified in the solution.) 2 The effect of water drifted into the air is negligible.

Properties The densities of air at normal conditions, seawater, and mercury

are given to be 0.076 lbm/ft3, 64 lbm/ft3, and 848 lbm/ft3, respectively.

Analysis (a) Reduced atmospheric pressure over the water causes the water

to rise. Thus, decreased pressure at point 2 relative to point 1 causes the

ocean water to rise at point 2. The same is true at point 3, where the storm air

velocity is negligible. The pressure difference given in terms of the mercury

column height can be expressed in terms of the seawater column height by

P (rgh)Hg (rgh)sw hsw rHg

rswhHg

V1 22gh3 22(9.81 m/s2)(0.12 m) 1.53 m/s

V21

2g

P2 P1rg

rg(h1 h2 h3) rg(h1 h2)

rg h3

P1

rg

V21

2gz1

P2

rg

V22

2g z2

V21

2g

P2 P1

rg

198FLUID MECHANICS

0Q

Calm

ocean

level

Ocean

HurricaneEye

1

2

3

h1h2

A B

FIGURE 542

Schematic for Example 59. The

vertical scale is greatly exaggerated.


9/24

Then the pressure difference between points 1 and 3 in terms of the seawa-

ter column height becomes

which is equivalent to the storm surge at the eye of the hurricanesince thewind velocity there is negligible and there are no dynamic effects.

(b) To determine the additional rise of ocean water at point 2 due to the high

winds at that point, we write the Bernoulli equation between points A and B,

which are on top of the points 2 and 3, respectively. Noting that VB 0 (theeye region of the hurricane is relatively calm) and zA zB (both points are

on the same horizontal line), the Bernoulli equation simplifies to

Substituting,

where r is the density of air in the hurricane. Noting that the density of an

ideal gas at constant temperature is proportional to absolute pressure and

the density of air at the normal atmospheric pressure of 14.7 psia 30 inHg is 0.076 lbm/ft3, the density of air in the hurricane is

Using the relation developed above in part (a), the seawater column height

equivalent to 803 ft of air column height is determined to be

Therefore, the pressure at point 2 is 0.70 ft seawater column lower than the

pressure at point 3 due to the high wind velocities, causing the ocean to rise

an additional 0.70 ft. Then the total storm surge at point 2 becomes

h2 h1 hdynamic 8.83 0.70 9.53 ft

Discussion This problem involves highly turbulent flow and the intense

breakdown of the streamlines, and thus the applicability of the Bernoulli

equation in part (b) is questionable. Furthermore, the flow in the eye of the

storm is notirrotational, and the Bernoulli equation constant changes across

streamlines (see Chap. 10). The Bernoulli analysis can be thought of as the

limiting, ideal case, and shows that the rise of seawater due to high-velocity

winds cannot be more than 0.70 ft.

The wind power of hurricanes is not the only cause of damage to coastal

areas. Ocean flooding and erosion from excessive tides is just as serious, as

are high waves generated by the storm turbulence and energy.

hdynamic rair

rswhair a0.056 lbm/f t

3

64 lbm/ft3b (803 ft) 0.70 ft

rair Pair

Patm airratm air a22 in Hg

30 in Hgb (0.076 lbm/ft3) 0.056 lbm/f t3

PB PArg

V2A2g

(155 mph)2

2(32.2 ft/s2)a1.4667 ft/s

1 mph b 2 803 ft

PA

rg

V2A

2g zA

PB

rg

V2B

2g zB

PB PA

rg

V2A

2g

h1 rHg

rswhHg a848 lbm/ft

3

64 lbm/ft3b3(30 22) in Hg4a 1 f t

12 inb 8.83 ft

199CHAPTER 5

0Q


10/24

EXAMPLE 510 Bernoulli Equation for Compressible Flow

Derive the Bernoulli equation when the compressibility effects are not negli-

gible for an ideal gas undergoing (a) an isothermal process and (b) an isen-

tropic process.

SOLUTION The Bernoulli equation for compressible flow is to be obtainedfor an ideal gas for isothermal and isentropic processes.

Assumptions 1 The flow is steady and frictional effects are negligible. 2 The

fluid is an ideal gas, so the relation P rRT is applicable. 3 The specific

heats are constant so that P/rk constant during an isentropic process.

Analysis (a) When the compressibility effects are significant and the flow

cannot be assumed to be incompressible, the Bernoulli equation is given by

Eq. 540 as

(1)

The compressibility effects can be properly accounted for by expressing r in

terms of pressure, and then performing the integration dP/r in Eq. 1. But

this requires a relation between Pand r for the process. For the isothermalexpansion or compression of an ideal gas, the integral in Eq. 1 can be per-

formed easily by noting that T constant and substituting r P/RT. It

gives

Substituting into Eq. 1 gives the desired relation,

Isothermal process: (2)

(b) A more practical case of compressible flow is the isentropic flow of ideal

gasesthrough equipment that involves high-speed fluid flow such as nozzles,

diffusers, and the passages between turbine blades. Isentropic (i.e.,reversible and adiabatic) flow is closely approximated by these devices, and

it is characterized by the relation P/rk C constant, where k is the spe-

cific heat ratio of the gas. Solving for r from P/rk Cgives r C1/kP1/k.

Performing the integration,

(3)

Substituting, the Bernoulli equation for steady, isentropic, compressible flow

of an ideal gas becomes

Isentropic flow: (4a)

or

(4b)

A common practical situation involves the acceleration of a gas from rest

(stagnation conditions at state 1) with negligible change in elevation. In

that case we have z1 z2 and V1 0. Noting that r P/RT for ideal

a kk 1

bP1r1

V212

gz1 a kk 1bP2r2

V222

gz2

a kk 1

bPR

V2

2 gz constant

dPr C1/kP1/k dP C1/k P1/k1

1/k 1

P1/k

rP1/k1

1/k 1a k

k 1bPr

RTlnP V2

2 gz constant

dPr dPP/RT RTln P

dPr V2

2 gz constant (along a streaml ine)

200FLUID MECHANICS


11/24

gases, P/rk constant for isentropic flow, and the Mach number is defined

as Ma V/cwhere is the local speed of sound for ideal gases,

Eq. 4b simplifies to

(4c)

where state 1 is the stagnation state and state 2 is any state along the flow.Discussion It can be shown that the results obtained using the compress-

ible and incompressible equations deviate no more than 2 percent when the

Mach number is less than 0.3. Therefore, the flow of an ideal gas can be

considered to be incompressible when Ma 0.3. For atmospheric air at

normal conditions, this corresponds to a flow speed of about 100 m/s or

360 km/h, which covers our range of interest.

56 GENERAL ENERGY EQUATION

One of the most fundamental laws in nature is the first law of thermody-

namics, also known as the conservation of energy principle, which pro-vides a sound basis for studying the relationships among the various formsof energy and energy interactions. It states that energy can be neither cre-ated nor destroyed during a process; it can only change forms. Therefore,every bit of energy must be accounted for during a process.

A rock falli ng off a cli ff , for example, picks up speed as a result of itspotential energy being converted to kinetic energy (Fig. 543). Experimen-tal data show that the decrease in potential energy equals the increase inki neti c energy when the air resistance is negli gibl e, thus confi rmi ng the con-servation of energy principle. The conservation of energy principle alsoforms the backbone of the diet industry: a person who has a greater energyinput (food) than energy output (exercise) will gain weight (store energy in

the form of fat), and a person who has a smaller energy i nput than outputwill lose weight. The change in the energy content of a system is equal tothe dif ference between the energy input and the energy output, and theconservation of energy principle for any system can be expressed simply asEin Eout E.

The transfer of any quantity (such as mass, momentum, and energy) isrecognizedat the boundaryas the quantity crosses the boundary. A quantityis said to enter a system if it crosses the boundary from the outside to theinside, and to exit the system if it moves in the reverse direction. A quantitythat moves from one location to another within a system is not consideredas a transferred quantity in an analysis since it does not enter or exit the sys-tem. Therefore, it is important to specif y the system and thus clearly iden-

tify its boundaries before an engineering analysis is performed.The energy content of a fixed quantity of mass (a closed system) can be

changed by two mechanisms:heat transfer Qandwork transfer W. Then theconservation of energy for a fixed quantity of mass can be expressed in rateform as (Fig. 544)

(549)Q#

net in W#

net in dEsys

dt or Q

#

net in W#

net in d

dtsys

re dV

P1P2

c1 ak 12 b M a22d

k/(k1)

c 1kRT

201CHAPTER 5

PE1= 10 m

K E1= 0

PE2= 7 km K E2= 3 k

z

FIGURE 54

Energy cannot be created o

destroyed duri ng a procesit can only change form

Wshaft, i n = 6 k

= 18 kJ

Qout

= 3 kJ

Qin

= 15 kJ

E= (15 3) + 6

FIGURE 54

The energy change of a systeduring a process is equal to the nwork and heat transfer between th

system and its surrounding


12/24

239CHAPTER 6

When the mass mof the control volume remains nearly constant, the fi rstterm of the Eq. 628 simply becomes mass times acceleration since

Therefore, the control volume in this case can be treated as a solid body, wi tha net force or thrust of

Thrust (629)

acting on the body. This approach can be used to determine the linear accel-eration of space vehicles when a rocket is fired (Fig. 619).

EXAMPLE 62 The Force to Hold a Deflector Elbow in Place

A reducing elbow is used to deflect water flow at a rate of 14 kg/s in a hori-

zontal pipe upward 30 while accelerating it (Fig. 620). The elbow dis-

charges water into the atmosphere. The cross-sectional area of the elbow is

113 cm2 at the inlet and 7 cm2 at the outlet. The elevation difference

between the centers of the outlet and the inlet is 30 cm. The weight of theelbow and the water in it is considered to be negligible. Determine (a) the

gage pressure at the center of the inlet of the elbow and (b) the anchoring

force needed to hold the elbow in place.

SOLUTION A reducing elbow deflects water upward and discharges it to theatmosphere. The pressure at the inlet of the elbow and the force needed to

hold the elbow in place are to be determined.

Assumptions 1 The flow is steady, and the frictional effects are negligible.

2 The weight of the elbow and the water in it is negligible. 3 The water is

discharged to the atmosphere, and thus the gage pressure at the outlet is

zero. 4 The flow is turbulent and fully developed at both the inlet and outlet

of the control volume, and we take the momentum-flux correction factor to

be b 1.03.Properties We take the density of water to be 1000 kg/m3.

Analysis (a) We take the elbow as the control volume and designate the

inlet by 1 and the outlet by 2. We also take the x- and z-coordinates as

shown. The continuity equation for this one-inlet, one-outlet, steady-flow sys-

tem is m.1 m

.2 m

. 14 kg/s. Noting that m

. rAV, the inlet and outlet

velocities of water are

V2 m

#

rA2

14 kg/s

(1000 kg/m3)(7 10 4m2) 20.0 m/s

V1 m

#

rA1

14 kg/s

(1000 kg/m3)(0.0113 m2) 1.24 m/s

F

body mbodya

ain

bm#V

aout

bm#V

d(mV

)CV

dt mCV

dV

CV

dt (ma

)CV

= 2 m

V0= 2000 m/s

FIGURE 61

The thrust needed to lift the spacshuttle is generated by the rockeengines as a result of momentum

change of the fuel as it is acceleratefrom about zero to an exit speed o

about 2000 m/s after combustionNAS

FRz

FRx

Patm

30

30 cm

P1,gage

z

x

1

2

mV1

mV2

FIGURE 62



13/24

240FLUID MECHANICS

We use the Bernoulli equation (Chap. 5) as a first approximation to calculate

the pressure. In Chap. 8 we will learn how to account for frictional losses

along the walls. Taking the center of the inlet cross section as the reference

level (z1 0) and noting that P2 Patm, the Bernoulli equation for a stream-

line going through the center of the elbow is expressed as

(b) The momentum equation for steady one-dimensional flow is

We let the x- and z-components of the anchoring force of the elbow be FRxand FRz, and assume them to be in the positive direction. We also use gage

pressure since the atmospheric pressure acts on the entire control surface.

Then the momentum equations along the x-and z-axes become

Solving for FRx and FRz, and substituting the given values,

The negative result for FRx indicates that the assumed direction is wrong,

and it should be reversed. Therefore, FRx acts in the negative x-direction.

Discussion There is a nonzero pressure distribution along the inside walls of

the elbow, but since the control volume is outside the elbow, these pressures

do not appear in our analysis. The actual value of P1, gage will be higher than

that calculated here because of frictional and other irreversible losses in theelbow.

EXAMPLE 63 The Force to Hold a Reversing Elbow in Place

The deflector elbow in Example 62 is replaced by a reversing elbow such that

the fluid makes a 180 U-turn before it is discharged, as shown in Fig. 621.

FRz bm#V2sin u (1.03)(14 kg/s)(20 sin 30m/s)a 1 N

1 kg m/s2b 144 N

232 2285 2053 N

(202,200 N/m2)(0.0113 m2)

1.03(14 kg/s)3(20 cos 30 1.24) m/s4a1 N

1 kg m/s2b

FRx bm#(V2cos u V1) P1, gageA 1

FRz bm#V2sin u

FRx P1, gageA1 bm#V2cos u bm

#V1

a F

aout bm#

V

ain bm#

V

P1, gage 202.2 kN/m2

202.2 kPa (gage)

a(20 m/s)2 (1.24 m/s)22(9.81 m/s2)

0.3 0b a 1 kN1000 kg m/s2

bP1 Patm (1000 kg/m

3)(9.81 m/s2)

P1 P2 rgaV22 V

21

2g z2 z1b

P1

rgV2

12g

z1 P

2rg

V22

2g z2

FRzFRx

Patm

P1,gage

1

2

mV2

mV1

FIGURE 621

Schematic f or Example 63.


14/24

The elevation difference between the centers of the inlet and the exit sec-

tions is still 0.3 m. Determine the anchoring force needed to hold the elbow

in place.

SOLUTION The inlet and the outlet velocities and the pressure at the inletof the elbow remain the same, but the vertical component of the anchoring

force at the connection of the elbow to the pipe is zero in this case ( FRz 0)since there is no other force or momentum flux in the vertical direction (we

are neglecting the weight of the elbow and the water). The horizontal compo-

nent of the anchoring force is determined from the momentum equation

written in the x-direction. Noting that the outlet velocity is negative since it

is in the negative x-direction, we have

Solving for FRx and substituting the known values,

Therefore, the horizontal force on the flange is 2591 N acting in the nega-

tive x-direction (the elbow is trying to separate from the pipe). This force is

equivalent to the weight of about 260 kg mass, and thus the connectors

(such as bolts) used must be strong enough to withstand this force.

Discussion The reaction force in the x-direction is larger than that of Exam-

ple 62 since the walls turn the water over a much greater angle. If the

reversing elbow is replaced by a straight nozzle (like one used by firefighters)

such that water is discharged in the positive x-direction, the momentum

equation in the x-direction becomes

since both V1 and V2 are in the positive x-direction. This shows the impor-tance of using the correct sign (positive if in the positive direction and nega-

tive if in the opposite direction) for velocities and forces.

EXAMPLE 64 Water Jet Striking a Stationary Plate

Water is accelerated by a nozzle to an average speed of 20 m/s, and strikes

a stationary vertical plate at a rate of 10 kg/s with a normal velocity of

20 m/s (Fig. 622). After the strike, the water stream splatters off in all

directions in the plane of the plate. Determine the force needed to prevent

the plate from moving horizontally due to the water stream.

SOLUTION A water jet strikes a vertical stationary plate normally. The forceneeded to hold the plate in place is to be determined.

Assumptions 1 The flow of water at nozzle outlet is steady. 2 The water

splatters in directions normal to the approach direction of the water jet. 3

The water jet is exposed to the atmosphere, and thus the pressure of the

water jet and the splattered water leaving the control volume is atmospheric

pressure, which is disregarded since it acts on the entire system. 4 The ver-

tical forces and momentum fluxes are not considered since they have no

FRx P1, gageA1 bm#V2 bm

#V1 FRx bm

#(V2 V1) P1, gageA1

306 2285 2591 N

(1.03)(14 kg/s)3(20 1.24) m/s4a 1 N1 kg m/s2

b (202,200 N/m2)(0.0113 m2)FRx bm

#(V2 V1) P1, gageA1

FRx P1, gageA1 b2m#(V2) b1m

#V1 bm

#(V2 V1)

241CHAPTER 6

z

x

Patm

In

Out

V1

V2

1

FIGURE 62



15/24

effect on the horizontal reaction force. 5 The effect of the momentum-flux

correction factor is negligible, and thus b 1.Analysis We draw the control volume for this problem such that it contains

the entire plate and cuts through the water jet and the support bar normally.

The momentum equation for steady one-dimensional flow is given as

Writing it for this problem along the x-direction (without forgetting the nega-

tive sign for forces and velocities in the negative x-direction) and noting that

V1, x V1 and V2, x 0 gives

Substituting the given values,

Therefore, the support must apply a 200-N horizontal force (equivalent to

the weight of about a 20-kg mass) in the negative x-direction (the opposite

direction of the water jet) to hold the plate in place.Discussion The plate absorbs the full brunt of the momentum of the water

jet since the x-direction momentum at the outlet of the control volume is

zero. If the control volume were drawn instead along the interface between

the water and the plate, there would be additional (unknown) pressure forces

in the analysis. By cutting the control volume through the support, we avoid

having to deal with this additional complexity. This is an example of a wise

choice of control volume.

EXAMPLE 65 Power Generation and Wind Loading

of a Wind Turbine

A wind generator with a 30-ft-diameter blade span has a cut-in wind speed

(minimum speed for power generation) of 7 mph, at which velocity the tur-

bine generates 0.4 kW of electric power (Fig. 623). Determine (a) the effi-

ciency of the wind turbinegenerator unit and (b) the horizontal force exerted

by the wind on the supporting mast of the wind turbine. What is the effect of

doubling the wind velocity to 14 mph on power generation and the force

exerted? Assume the efficiency remains the same, and take the density of air

to be 0.076 lbm/ft3.

SOLUTION The power generation and loading of a wind turbine are to beanalyzed. The efficiency and the force exerted on the mast are to be deter-

mined, and the effects of doubling the wind velocity are to be investigated.Assumptions 1 The wind flow is steady and incompressible. 2 The efficiency

of the turbinegenerator is independent of wind speed. 3 The frictional

effects are negligible, and thus none of the incoming kinetic energy is con-

verted to thermal energy. 4 The average velocity of air through the wind tur-

bine is the same as the wind velocity (actually, it is considerably lesssee

the discussion that follows the example). 5 The wind flow is uniform and

thus the momentum-flux correction factor is b 1.

FR bm#V

1 (1)(10 kg/s)(20 m/s)a 1 N1 kg m/s2

b 200 N

FR 0 bm#V

1

a F

aout bm#V

ain bm#V

242FLUID MECHANICS

1 2

Patm

Patm

mV2

mV1

FR

Streamline

x

FIGURE 623

Schematic f or Example 65.


16/24

Properties The density of air is given to be 0.076 lbm/ft3.

Analysis Kinetic energy is a mechanical form of energy, and thus it can be

converted to work entirely. Therefore, the power potential of the wind is

proportional to its kinetic energy, which is V2/2 per unit mass, and thus

the maximum power is m.V2/2 for a given mass flow rate:

Therefore, the available power to the wind turbine is 1.225 kW at the windvelocity of 7 mph. Then the turbinegenerator efficiency becomes

(or 32.7 )

(b) The frictional effects are assumed to be negligible, and thus the portion

of incoming kinetic energy not converted to electric power leaves the wind

turbine as outgoing kinetic energy. Noting that the mass flow rate remains

constant, the exit velocity is determined to be

or

We draw a control volume around the wind turbine such that the wind is

normal to the control surface at the inlet and the outlet and the entire con-

trol surface is at atmospheric pressure. The momentum equation for steady

one-dimensional flow is given as

Writing it along the x-direction and noting that b 1, V1, x V1, and V2, xV2 give

Substituting the known values gives

31.5 lbf

FR m#(V2 V1) (551.7 l bm/s)(8.43 10.27 ft/s) a 1 lbf

32.2 lbm ft/s2b

FR m# V2 m# V1 m# (V2 V1)

a F

aout

bm#V

ain

bm#V

V2 V121 hwind turbine (10.27 ft /s)21 0.327 8.43 ft/s

m#ke2 m

#ke1(1 hwind turbine) m

# V22

2 m

# V21

2(1 hwind turbine)

hwind turbine W

#

act

W#

max

0.4 kW

1.225 kW 0.327

1.225 kW

(551.7 lbm/s)(10.27 ft/s)2

2 a 1 lbf

32.2 lbm ft/s2b a 1 kW

737.56 lbf ft/sb

W#

max m#ke1 m

# V21

2

m#

r1V1A1 r1V1pD2

4 (0.076 lbm/ft3)(10.27 ft/s)

p(30 ft)2

4 551.7 lbm/s

V1 (7 mph)a1.4667 ft/s1 mph b 10.27 ft/s

243CHAPTER 6


17/24

The negative sign indicates that the reaction force acts in the negative x-

direction, as expected. Then the force exerted by the wind on the mast

becomes Fmast FR 31.5 lbf.

The power generated is proportional to V3 since the mass flow rate is pro-

portional to V and the kinetic energy to V2. Therefore, doubling the wind

velocity to 14 mph will increase the power generation by a factor of 23 8

to 0.4 8 3.2 kW. The force exerted by the wind on the support mast isproportional to V2. Therefore, doubling the wind velocity to 14 mph will

increase the wind force by a factor of 22 4 to 31.5 4 126 lbf.

Discussion To gain more insight into the operation of devices with propellers

or turbines such as helicopters, wind turbines, hydraulic turbines, and turbo-

fan engines, we reconsider the wind turbine and draw two streamlines, as

shown in Fig. 624. (In the case of power-consuming devices such as a fan

and a helicopter, the streamlines converge rather than diverge since the exit

velocity will be higher and thus the exit area will be lower.) The upper and

lower streamlines can be considered to form an imaginary duct for the

flow of air through the turbine. Sections 1 and 2 are sufficiently far from the

turbine so that P1 P2 Patm. The momentum equation for this large con-

trol volume between sections 1 and 2 was obtained to be

(1)

The smaller control volume between sections 3 and 4 encloses the turbine,

and A3 A4 A and V3 V4 since it is so slim. The turbine is a device

that causes a pressure change, and thus the pressures P3 and P4 are differ-

ent. The momentum equation applied to the smaller control volume gives

(2)

The Bernoulli equation is not applicable between sections 1 and 2 since the

path crosses a turbine, but it is applicable separately between sections 1

and 3 and sections 4 and 2:

Adding these two equations and noting that z1 z2 z3 z4, V3 V4,

and P1 P2 Patm gives

(3)

Substituting m.

rAV3 into Eq. 1 and then combining it with Eqs. 2 and 3

gives

(4)

Thus we conclude that the average velocity of a fluid through a turbine is the

arithmetic average of the upstream and downstream velocities. Of course, the

validity of this result is limited by the applicability of the Bernoulli equation.

Now back to the wind turbine. The velocity through the turbine can be

expressed as V3 V1(1 a), where a 1 since V3 V1. Combining this

expression with Eq. 4 gives V2 V1(1 2a). Also, the mass flow rate

through the turbine becomes m.

rAV3 rAV1(1 a). When the frictional

V3 V1 V2

2

V22 V21

2

P4 P3

r

P1

rg

V21

2g z1

P3

rg

V23

2g z3

and

P4

rg

V24

2g z4

P2

rg

V22

2g z2

FR P3 A P4 A 0 FR (P4 P3)A

FR m# (V2 V1)

244FLUID MECHANICS

1 3 4 2

Patm

Patm

V2V1

A

V3 =V4

Streamline

Streamline

Windturbine

FIGURE 624

The large and small control volumesfor the analysis of a wind turbinebounded by upper and lowerstreamlines.


18/24

effects and losses are neglected, the power generated by a wind turbine is

simply the difference between the incoming and the outgoing kinetic energies:

Dividing this by the available power of the wind W.max m.V21/2 gives the effi-

ciency of the wind turbine in terms of a,

The value of a that maximizes the efficiency is determined by setting the

derivative of hwind turbine with respect to aequal to zero and solving for a. It

gives a 1/3. Substituting this value into the efficiency relation just pre-

sented gives hwind turbine 16/27 0.593, which is the upper limit for the

efficiency of wind turbines and propellers. This is known as the Betz limit.

The efficiency of actual wind turbines is about half of this ideal value.

EXAMPLE 66 Repositioning of a Satellite

An orbiting satellite has a mass of msat 5000 kg and is traveling at a con-

stant velocity of V0. To alter its orbit, an attached rocket discharges mf 100 kg of gases from the reaction of solid fuel at a velocity Vf 3000 m/s

relative to the satellite in a direction opposite to V0 (Fig. 625). The fuel

discharge rate is constant for 2 s. Determine (a) the acceleration of the

satellite during this 2-s period, (b) the change of velocity of the satellite dur-

ing this time period, and (c) the thrust exerted on the satellite.

SOLUTION The rocket of a satellite is fired in the opposite direction tomotion. The acceleration, the velocity change, and the thrust are to be

determined.

Assumptions 1 The flow of combustion gases is steady and one-dimensional

during the firing period. 2 There are no external forces acting on the satel-

lite, and the effect of the pressure force at the nozzle exit is negligible.

3 The mass of discharged fuel is negligible relative to the mass of the satel-

lite, and thus the satellite may be treated as a solid body with a constant

mass. 4 The nozzle is well-designed such that the effect of the momentum-

flux correction factor is negligible, and thus b 1.Analysis (a) We choose a reference frame in which the control volume

moves with the satellite. Then the velocities of fluid streams become simply

their velocities relative to the moving body. We take the direction of motion

of the satellite as the positive direction along the x-axis. There are no exter-nal forces acting on the satellite and its mass is nearly constant. Therefore,

the satellite can be treated as a solid body with constant mass, and the

momentum equation in this case is simply Eq. 628,

0 d(mV

)CV

dt a

out

bm#V

ain

bm#V

msat

dV

sat

dt m

#fV

f

hwind turbine W

#

W#

max

2rAV31a(1 a)

2

(rAV1)V21/2

2rAV31a(1 a)2

W#

m#(ke1 ke2)

m#(V21 V

22)

2

rAV1(1 a)3V21 V21(1 2a)242

245CHAPTER 6

V0

Vf Satellitemsat

xCS

FIGURE 62



19/24

Noting that the motion is on a straight line and the discharged gases move

in the negative x-direction, we can write the momentum equation using mag-

nitudes as

Substituting, the acceleration of the satellite during the first 2 s is deter-

mined to be

(b) Knowing acceleration, which is constant, the velocity change of the satel-

lite during the first 2 s is determined from the definition of acceleration asat dVsat/dtto be

(c) The thrust exerted on the satellite is, from Eq. 629,

Discussion Note that if this satellite were attached somewhere, it would

exert a force of 150 kN (equivalent to the weight of 15 tons of mass) to its

support. This can be verified by taking the satellite as the system and apply-

ing the momentum equation.

EXAMPLE 67 Net Force on a Flange

Water flows at a rate of 18.5 gal/min through a flanged faucet with a par-

tially closed gate valve spigot (Fig. 626). The inner diameter of the pipe at

the location of the flange is 0.780 in ( 0.0650 ft), and the pressure at

that location is measured to be 13.0 psig. The total weight of the faucet

assembly plus the water within it is 12.8 lbf. Calculate the net force on the

flange.

SOLUTION Water flow through a flanged faucet is considered. The net forceacting on the flange is to be calculated.

Assumptions 1 The flow is steady and incompressible. 2 The flow at the

inlet and at the outlet is turbulent and fully developed so that the momentum-

flux correction factor is about 1.03. 3 The pipe diameter at the outlet of the

faucet is the same as that at the flange.

Properties The density of water at room temperature is 62.3 lbm/ft3.

Analysis We choose the faucet and its immediate surroundings as the con-

trol volume, as shown in Fig. 626 along with all the forces acting on it.

These forces include the weight of the water and the weight of the faucet

assembly, the gage pressure force at the inlet to the control volume, and the

Fsat 0 m#

f(Vf) (100/2 kg/s)(3000 m/s)a 1 kN1000 kg m/s2

b 150 kN

dVsat asat dt Vsat asatt (30 m/s2)(2 s) 60 m/s

asat dVsat

dt

mf/t

msatVf

(100 kg)/(2 s)

5000 kg(3000 m/s) 30 m/s2

msatdVsat

dt m

#fVf

dVsat

dt

m#

f

msatVf

mf/t

msatVf

246FLUID MECHANICS

Wfaucet

Wwater

P1,gage

CV

Out

Spigot

Flange

x

z

In

FR

FIGURE 626

Control volume for Example 67 wi thall forces shown; gage pressure is usedfor convenience.


20/24

net force of the flange on the control volume, which we call F

R. We use gage

pressure for convenience since the gage pressure on the rest of the control

surface is zero (atmospheric pressure). Note that the pressure through the

outlet of the control volume is also atmospheric since we are assuming

incompressible flow; hence, the gage pressure is also zero through the outlet.

We now apply the control volume conservation laws. Conservation of mass

is trivial here since there is only one inlet and one outlet; namely, the massflow rate into the control volume is equal to the mass flow rate out of the

control volume. Also, the outflow and inflow average velocities are identical

since the inner diameter is constant and the water is incompressible, and

are determined to be

Also,

Next we apply the momentum equation for steady flow,

We let the x- and z-components of the force acting on the flange be FRx and

FRz, and assume them to be in the positive directions. The magnitude of the

velocity in the x-direction is V1 at the inlet, but zero at the outlet. The

magnitude of the velocity in the z-direction is zero at the inlet, but V2 at

the outlet. Also, the weight of the faucet assembly and the water within it

acts in the z-direction as a body force. No pressure or viscous forces act on

the chosen control volume in the z-direction.

The momentum equations along the x- and z-directions become

Solving for FRx and FRz, and substituting the given values,

Then the net force of the flange on the control volume can be expressed in

vector form as

F

R FRxi

FRzk

7.20 i

11.8 k

lbf

(2.568 lbm/s)(12.42 f t/s)a 1 lbf32.2 lbm ft/s2

b 12.8 lbf 11.8 lbfFRz m

#V2 Wfaucetwater

7.20 lbf

(2.568 lbm/s)(12.42 f t/s)a 1 lbf32.2 lbm ft/s2

b (13 lbf/in2)p(0.780 in)24

FRx m#V1 P1, gageA1

FRz Wfaucet Wwater m#(V2) 0

FRx P1, gageA1 0 m#(V1)

a F

aout

bm#V

ain

bm#V

m#

rV#

(62.3 lbm/f t3)(18.5 gal/mi n)a0.1337 ft31 gal

b a1 min60 s

b 2.568 lbm/s

V2 V1 V V

#

A c

V#

pD2/4

18.5 gal/mi n

p(0.065 ft)2/4a0.1337 ft3

1 gal b a1 min

60 sb 12.42 ft/s

247CHAPTER 6


21/24

From Newtons third law, the force the faucet assembly exerts on the flange

is the negative of F

R,

Discussion The faucet assembly pulls to the right and down; this agrees

with our intuition. Namely, the water exerts a high pressure at the inlet, butthe outlet pressure is atmospheric. In addition, the momentum of the water

at the inlet in the x-direction is lost in the turn, causing an additional force

to the right on the pipe walls. The faucet assembly weighs much more than

the momentum effect of the water, so we expect the force to be downward.

Note that labeling forces such as faucet on flange clarifies the direction of

the force.

65

REVIEW OF ROTATIONAL MOTIONAND ANGULAR MOMENTUM

The motion of a rigid body can be considered to be the combination of thetranslational motion of its center of mass and rotational motion about itscenter of mass. The translational motion can be analyzed using the linearmomentum equation, Eq. 616. Now w e discuss the rotational motionamotion during which all points in the body move in circles about the axis ofrotation. Rotational motion is described with angular quantities such as theangular distance u, angular velocity v, and angular acceleration a.

The amount of rotation of a point in a body is expressed in terms of theangleu swept by a line of length r that connects the point to the axis of rota-tion and is perpendicular to the axis. The angle u is expressed in radians (rad),which is the arc length corresponding to u on a circle of unit radius. Notingthat the circumference of a circle of radius r is 2pr, the angular distance trav-eled by any point in a rigid body during a complete rotation is 2p rad. Thephysical distance traveled by a point along its circular path is l ur, where ris the normal distance of the point from the axis of rotation and u is the angu-lar distance in rad. Note that 1 rad corresponds to 360/(2p) 57.3.

Angular velocity v is the angular distance traveled per unit time, andangular acceleration a is the rate of change of angular velocity. They areexpressed as (Fig. 627),

(630)

or

(631)

whereV is the linear velocity and at is the linear acceleration in the tangen-tial direction for a point located at a distance r from the axis of rotation.Note that v anda are the same for all points of a rotating rigi d body, but Vandat are not (they are proportional to r).

V rv and at ra

v du

dt

d(l/r)

dt

1

r

dl

dt

V

r and a dv

dt

d2u

dt2

1

r

dV

dt

at

r

F

faucet on fl ange F

R 7.20

11.8k

lbf

248FLUID MECHANICS

v

v= =

du

dt

u

r

r

V= rv

V

r

FIGURE 627

The relations between angulardistance u, angular velocity v,and linear velocity V.


22/24

EXAMPLE 81 Flow Rates in Horizontal and Inclined Pipes

Oil at 20C (r 888 kg/m3 and m 0.800 kg/m s) is flowing steadily

through a 5-cm-diameter 40-m-long pipe (Fig. 817). The pressure at the

pipe inlet and outlet are measured to be 745 and 97 kPa, respectively.

Determine the flow rate of oil through the pipe assuming the pipe is (a) hor-

izontal, (b) inclined 15 upward, (c) inclined 15 downward. Also verify thatthe flow through the pipe is laminar.

SOLUTION The pressure readings at the inlet and outlet of a pipe are given.The flow rates are to be determined for three different orientations, and the

flow is to be shown to be laminar.

Assumptions 1 The flow is steady and incompressible. 2 The entrance

effects are negligible, and thus the flow is fully developed. 3 The pipe

involves no components such as bends, valves, and connectors. 4 The piping

section involves no work devices such as a pump or a turbine.

Properties The density and dynamic viscosity of oil are given to be r

888 kg/m3 and m 0.800 kg/m s, respectively.

Analysis The pressure drop across the pipe and the pipe cross-sectional

area are

(a) The flow rate for all three cases can be determined from Eq. 834,

where u is the angle the pipe makes with the horizontal. For the horizontal

case, u 0 and thus sin u 0. Therefore,

0.00311 m3/s

V#

horiz

PpD4

128mL

(648 k Pa)p(0.05 m)4

128(0.800 kg/m s)(40 m)a1000 N/m2

1 kPa b a1 kg m/s2

1 N b

V#

(P rgL sin u)pD4

128mL

A c pD2/4 p(0.05 m)2/4 0.001963 m2

P P1 P2 745 97 648 kPa

333CHAPTER 8

(b) For uphill flow with an inclination of 15, we have u 15, and

(c) For downhill flow with an inclination of 15, we have u 15, and

0.00354 m3/s

[648,000 Pa (888 kg/m3)(9.81 m/s2)(40 m)sin (15)]p(0.05 m)4

128(0.800 kg/m s)(40 m) a1 kg m/s2

1 Pa m2b

V#

downhill (P rgL sin u)pD4

128mL

0.00267 m3/s

[648,000 Pa (888 kg/m3)(9.81 m/s2)(40 m)sin 15]p(0.05 m)4

128(0.800 kg/m s)(40 m) a1 kg m/s2

1 Pa m2b

V#

uphill (P rgL sin u)pD4

128mL

+15

15

Horizontal

FIGURE 81



23/24

The flow rate is the highest for the downhill flow case, as expected. The

average fluid velocity and the Reynolds number in this case are

which is much less than 2300. Therefore, the flow is laminar for all three

cases and the analysis is valid.

Discussion Note that the flow is driven by the combined effect of pressure

difference and gravity. As can be seen from the flow rates we calculated,

gravity opposes uphill flow, but enhances downhill flow. Gravity has no effect

on the flow rate in the horizontal case. Downhill flow can occur even in the

absence of an applied pressure difference. For the case of P1 P2 97 kPa

(i.e., no applied pressure difference), the pressure throughout the entire pipe

would remain constant at 97 Pa, and the fluid would flow through the pipe at

a rate of 0.00043 m3/s under the influence of gravity. The flow rate increases

as the tilt angle of the pipe from the horizontal is increased in the negative

direction and would reach its maximum value when the pipe is vertical.

EXAMPLE 82 Pressure Drop and Head Loss in a Pipe

Water at 40F (r 62.42 lbm/ft3 and m 1.038 103 lbm/ft s) is

flowing through a 0.12-in- ( 0.010 ft) diameter 30-ft-long horizontal pipe

steadily at an average velocity of 3.0 ft/s (Fig. 818). Determine (a) the head

loss, (b) the pressure drop, and (c) the pumping power requirement to over-

come this pressure drop.

SOLUTION The average flow velocity in a pipe is given. The head loss, thepressure drop, and the pumping power are to be determined.

Assumptions 1 The flow is steady and incompressible. 2 The entrance

effects are negligible, and thus the flow is fully developed. 3 The pipe

involves no components such as bends, valves, and connectors.

Properties The density and dynamic viscosity of water are given to be r

62.42 lbm/ft3 and m 1.038 103 lbm/ft s, respectively.

Analysis (a) First we need to determine the flow regime. The Reynolds num-

ber is

which is less than 2300. Therefore, the flow is laminar. Then the friction

factor and the head loss become

(b) Noting that the pipe is horizontal and its diameter is constant, the pres-

sure drop in the pipe is due entirely to the frictional losses and is equivalent

to the pressure loss,

hL fL

D

V2avg

2g 0.0355

30 ft

0.01 ft

(3 ft/ s)2

2(32.2 ft/s2) 14.9 ft

f64Re

64

1803 0.0355

Re rVavgD

m

(62.42 lbm/ft3)(3 ft/s)(0.01 ft)

1.038 10 3lbm/ft s 1803

Re

rVavgD

m

(888 kg/m3)(1.80 m/s)(0.05 m)

0.800 kg/m s 100

Vavg V

#

A c

0.00354 m3/s

0.001963 m2 1.80 m/s

334FLUID MECHANICS

3.0 ft/s

30 ft

0.12 in

FIGURE 818Schematic f or Example 82.


24/24

(c) The volume flow rate and the pumping power requirements are

Therefore, power input in the amount of 0.30 W is needed to overcome the

frictional losses in the flow due to viscosity.

Discussion The pressure rise provided by a pump is often listed by a pump

manufacturer in units of head (Chap. 14). Thus, the pump in this flow needs to

provide 14.9 ft of water head in order to overcome the irreversible head loss.

85

TURBULENT FLOW IN PIPESM ost f lows encountered in engineering practice are turbulent, and thus i t isimportant to understand how turbulence affects wall shear stress. However,turbulent f low is a complex mechanism dominated by f luctuations, anddespite tremendous amounts of work done in thi s area by researchers, thetheory of turbulent f low remains largely undeveloped. Therefore, we mustrely on experiments and the empirical or semi-empirical correlations devel-oped for various situations.

Turbulent flow is characterized by random and rapid fluctuations ofswi rli ng regions of f luid, call ed eddies, throughout the flow. These fluctua-tions provide an additional mechanism for momentum and energy transfer.In laminar f low, fl uid parti cles fl ow in an orderly manner along pathli nes,

and momentum and energy are transferred across streamlines by moleculardif fusion. In turbulent fl ow, the swi rl ing eddies transport mass, momentum,and energy to other regions of flow much more rapidly than molecular dif-fusion, greatly enhancing mass, momentum, and heat transfer. A s a result,turbulent f low is associated with much higher values of fr icti on, heat trans-fer, and mass transfer coeffi cients (Fig. 819).

Even when the average f low is steady, the eddy moti on in turbulent fl owcauses signif icant fluctuations in the values of veloci ty, temperature, pres-sure, and even density (in compressible f low). Figure 820 shows the varia-tion of the instantaneous velocity component uwith time at a specified loca-tion, as can be measured wi th a hot-wi re anemometer probe or othersensitive device. We observe that the instantaneous values of the velocity

f luctuate about an average value, which suggests that the veloci ty can beexpressed as the sum of an average value uand a fluctuating component u,

(835)

This is also the case for other properties such as the velocity component vin the y-direction, and thus v v v,P P

P, and T T

T. The

average value of a property at some location is determined by averaging itover a time interval that is sufficiently large so that the time average levelsoff to a constant. Therefore, the time average of f luctuating components is

u u u

W#

pump V#

P (0.000236 ft3/s)(929 lbf /f t2)a 1 W0.737 lbf ft/s

b 0.30 WV

#

Vavg A c Vavg(pD2

/4) (3 ft/s)[p(0.01 ft)2

/4] 0.000236 f t3

/s

929 lbf/ft2 6.45 psi

P PL fL

D

rV2avg

2 0.0355

30 ft

0.01 ft

(62.42 lbm/ft3)(3 ft/s)2

2 a 1 lbf

32.2 lbm ft/s2b

335CHAPTER 8

(a) Before turbulence

2 2 2 2 2

55

7

12

7

12

7

12

7

12

7

12

5 5 5

(b) Af ter turbulence

12 2 5 7 5

122

7

2

7

5

1

2

12

7

5

12

5 7 2

FIGURE 81

The intense mixi ng in turbulent fl o

brings fluid particles at differenmomentums into close contact anthus enhances momentum t ransfe

u

uu

Time,

FIGURE 82

Fluctuations of the velocitcomponentuwith time at a specifie

location in turbulent f low

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