FLUIDS

346

175 mm5 mm

Sprinkler 15o

40 rpm

Hose 10 L/min

Support

Figure 4.93: Two-arm sprinkler

4.78. The four-arm water sprinkler shown in Figure 4.94 has a radius-of-rotation of 0.7 m. The nozzles at the four outlets of the sprinkler each have an area of 30 mm2, and are oriented at an angle of of 40◦ to the circle-of-rotation. Consider the case where water is supplied to the sprinkler at a rate of 8 L/s. (a) What torque would be necessary to hold the sprinkler rotors stationary so that they do not rotate? (b) If the restraining torque is equal to zero, at what speed would the sprinkler rotate?

40o

Nozzle

0.7 m

Figure 4.94: Four-arm sprinkler

4.79. Consider the system shown in Figure 4.95 where a water sprinkler is used to generate electric power by rotating a shaft that is connected to an electric generator. Each arm of the sprinkler has a length of 10 cm, the nozzle at the end of each arm has a diameter of 1 cm, the total flow through the sprinkler is 8 L/s, and the rotational speed of the sprinkler rotor can be set by the sprinkler operator, using a device called a governor. (a) At what rotational speed, in rpm, would the maximum power be generated? (b) What is the maximum power that can be generated by the sprinkler-generator?

Nozzle

10 cm

Ou!low

Figure 4.95: Sprinkler for electricity generation

347

Section 4.6: Conservation of Energy

4.80. (a) Air flows at 0.05 kg/s through an insulated circular duct such that the (gage) pressure immediately downstream of a blower is 10 kPa. If the pressure at a downstream location within the duct is measured as 8 kPa, estimate the change in the temperature of the air that would be expected at the downstream location. You may neglect the impact of elevation change in your analysis. (b) If a blower were to be used to raise the pressure in the duct described in Part (a) by 2 kPa at a single location, estimate the power required to drive the blower if it has an efficiency of 90%.

4.81. An incompressible fluid flows with an average velocity of 1.2 m/s in a horizontal 300-mm diameter pipe. If the pressure drop between two cross-sections is 15 kPa, what is the rate at which energy is being lost between these two sections?

4.82. Water flows along the straight 200-mm diameter conduit shown in Figure 4.96, where the pressure at the lower-elevation section of the conduit is 400 kPa, and the pressure at the higher-elevation section is 360 kPa. The conduit makes an angle of 15◦ with the horizonal, and the distance between the gaged sections is 10 m. If the fluid is incompressible, what is the direction of flow, and what is the head loss between the two sections?

10 m 200 mm

15o

400 kPa

360 kPa

Flow

Conduit

Figure 4.96: Flow in an inclined conduit

4.83. A valve connects two pipes, where the upstream pipe has a diameter of 250 mm and the downstream pipe has a diameter of 400 mm. When water flows through the valve and the velocity in the upstream pipe is 2.0 m/s, the difference in pressure between the inflow pipe and the outflow pipe is measured as 50 kPa. If the inflow and outflow pipes are at the same elevation, what is the head loss through the valve? What is the power loss across the valve?

4.84. Calculate the rate of heat loss between the upstream and downstream sections in Figure 4.52. Assume that the temperature of the water does not change between the upstream and downstream sections.

4.85. Water flows at a velocity of 1.8 m/s in a 150-mm diameter pipeline as shown in Figure 4.97. At a particular section, the (gage) pressure is equal to 150 kPa, and downstream of this section the pipe elevation increases by 1.2 m and the water is discharged through a 50-mm nozzle. Estimate the head loss between the section where the pressure is 150 kPa and the section just downstream of the nozzle discharge.

348

1.2 m

150 mm

50 mm

1.8 m/s

g

Nozzle

Pipeline

150 kPa

Figure 4.97: Pipeline with nozzle discharge

4.86. Water at 20◦C flows through a pump in which the inflow pipe has a diameter of 150 mm and the outflow pipe has a diameter of 75 mm. The centerlines of the inflow and outflow pipes are aligned so that there is no difference in elevation. When the flow rate through the pump is 30 L/s, the inflow and outflow pressures are 150 kPa and 500 kPa, respectively, and measurements indicate that the temperature of the water increases by 0.1◦C between the inflow and the outflow. Estimate the rate at which energy is being delivered by the pump to the water. Approximately what percentage of the energy input by the pump goes towards raising the temperature?

4.87. Calculate the rate of energy loss in the reducer described in Problem 4.26.

4.88. A pump is attached to a fire hydrant as shown in Figure 4.98, where it is desired to produce a jet that has a maximum height of 40 m and delivers water at a rate of 100 L/s. At the attachment location, the diameter is 100 mm and the pressure is estimated as 150 kPa. If friction losses are neglected, estimate the head that must be added by the pump to achieve the desired objective. Approximately how much power will it take to run the pump?

P

p = 150 kPa

D = 100 mm 40 m

Pump

Hydrant 100 L/s

Jet

Figure 4.98: Pump attached to fire hydrant

4.89. Calculate the rate of energy loss in the junction described in Problem 4.35.

4.90. Water at 20◦C in a 10-m diameter, 2-m deep aboveground swimming pool is to be emptied by unplugging a 3-cm diameter, 25-m-long horizontal plastic drain pipe attached to the bottom of the pool. Determine the time it will take to empty the swimming pool completely assuming the entrance to the drain pipe is well-rounded with negligible loss. Take the friction factor of the pipe to be 0.022.

349

2 m

Swimming pool

10 m

3 cm

25 m

Drain pipe

Figure 4.99: Swimming pool

4.91. Water flows at a rate of 0.025 m3/s in a horizontal pipe whose diameter increases from 6 to 11 cm by an enlargement section. If the head loss across the enlargement section is 0.45 m and the kinetic energy correction factor at both the inlet and the outlet is 1.05, determine the pressure change.

4.92. The suction and discharge pipes to a pump both have diameters of 150 mm and are at the same elevation. Under a particular operating condition, the pump delivers 1600 L/min, the pressures in the suction and discharge lines are 30 kPa and 300 kPa, respectively, and the power consumption is 8 kW. Estimate the efficiency of the pump under this operating condition. Assume water at 20◦C.

4.93. Water is to be be pumped from a lake to irrigate a tree as shown in Figure 4.100. The head loss in the 50-cm diameter pipeline, hℓ [m], is a function of the flow rate, Q [m3/s], according to the relation

hℓ = 1.05Q 2

If the desired flow rate is 1 m3/s and the pump efficiency is assumed to be 80%. (a) What size pump (in terms of hp) would you select for the job? (b) If a nozzle of diameter 25 cm is attached to the end of the pipeline, will the same size pump be adequate and, if not, calculate the pump size that would be required in this case.

P

Lake

5 m

0.3 m

3 m

Ground Tree

Pump

Pipeline

Figure 4.100: Tree irrigation

350

4.94. Water is pumped from a lower reservoir to a higher reservoir using a pump that delivers 30 kW of power. The elevation difference between the upper and lower reservoir is 15 m, and the head loss in the pipeline is estimated as 6 m. Find the flow rate in the pipeline, the rate at which energy is lost from the system, and the fraction of input power that is lost.

4.95. Water is pumped from an open lower reservoir to a pressurized upper reservoir as shown in Figure 4.101. Under design conditions, the water level in the upper reservoir is 10 m above the water level in the lower reservoir, and it is desired to have a minimum delivery rate of 15 L/s from the lower to the upper reservoir. If a 5 kW pump is used and friction losses in the pipeline are neglected, what is the maximum allowable air pressure in the upper reservoir?

P

Air

Upper reservoir

Lower reservoir

Pump 10 m

Figure 4.101: Water delivery between reservoirs

4.96. A pump delivers water from a source reservoir to a terminal reservoir through a 200-mm diameter pipe as shown in Figure 4.102. The water-surface elevations in the reservoirs differ by 2 m, and the delivery pipe discharges at an elevation 5 m below the water-surface elevation of the source reservoir. Both the frictional head loss, hℓ, and the head added by the pump, hp, are functions of the flow rate, Q, between reservoirs, and these functional relationships are given by

hℓ = 3 V 2

2g , hp = 18(1− 100Q2)

where V is the velocity in the pipeline, hp is in m, and Q is in m3/s. (a) Estimate the flow rate between the reservoirs. (b) If the terminal reservoir is empty such that the pipeline discharges into the open atmosphere, what would be the flow rate under this condition?

2 mPump

200 mm

Source reservoir Terminal reservoir

5 m

P

Figure 4.102: Pump delivering water from a source to a terminal reservoir

4.97. A pump delivers water from a 125-mm diameter pipeline to a 75-mm pipeline. When the flow through the pump is 40 L/s, the inflow pressure is 150 kPa, the outflow pressure is 450 kPa, and the head loss within the pump is estimated as 18 m. Estimate the power required to drive the pump motor, and the efficiency of the pump in converting electrical energy to mechanical

351

energy. Assume the pump is 90% efficient in transforming electrical energy to the energy of the rotating impeller.

4.98. The inflow and outflow pipes to a turbine have diameters of 400 mm and 500 mm, respectively, and are located at approximately the same elevation. Under a particular operating condition, the flow rate of water through the turbine is 1 m3/s and the power extracted from the water by the turbine is 100 kW. Estimate the change in water pressure across the turbine. Assume water at 20◦C.

4.99. A hydroelectric facility has an upstream reservoir elevation of 110 m and a downstream river elevation of 85 m. The river can be taken as having a velocity of 1 m/s. The head loss, hℓ, through the tunnel leading to and from the turbine is given by

hℓ = 0.0826Q 2

where hℓ is in meters and the Q is the flow in m3/s. The efficiency of the turbine is 80%. Determine the flow through the turbine that is required to generate 1 MW of power.

4.100. Water flows in a 1-m diameter pipeline from a reservoir, through a turbine, and discharges freely as shown in Figure 4.103. Under design conditions, the water surface in the reservoir is 30 m above the discharge point. The head loss through the system can be estimated using the relation hℓ = 0.8V 2/2g, where V is the velocity in the pipeline. (a) If the flow rate through the system is 10 m3/s, estimate the power extracted by the turbine. (b) If the turbine were removed, what would be the flow rate through the system?

T Free discharge

Turbine30 m

1 m

Reservoir

Figure 4.103: Flow through turbine with free discharge

4.101. Water flows through a turbine as shown in Figure 4.104, where the pipeline upstream of the turbine has a diameter of 1.2 m and the downstream pipeline has a diameter of 1.7 m. The inflow section of the upstream pipeline is 4 m above the outflow section of the downstream pipeline. When the flow through the turbine is 6 m3/s, the pressure at the inflow section is 450 kPa, the pressure at the outflow section is −30 kPa, and the head loss between these two sections is estimated as 20 m. What is the power being delivered to the turbine?

352

T

6 m3/s

1.2 m

1.7 m

–30 kPa

4 m Turbine

450 kPa

Figure 4.104: Flow through a turbine

4.102. Angel falls in Venezuela is the second highest waterfall in the world, with a vertical drop of approximately 979 m. If adiabatic conditions were to exist along the entire length of the waterfall, estimate the difference in temperature in an element of water in the upstream pool and this same element of water within the downstream pool after the kinetic energy acquired during the fall has been dissipated.

4.103. A room has dimensions 10 m × 10 m × 3 m, and air from the room is vented to the outside through a 100-mm diameter vent. The head loss in this type of vent can be estimated as hℓ = 0.1V 2/2g, where V is the flow velocity through the vent. If the pressure inside the room is 2 kPa higher than the outside pressure, estimate the volume flow rate of air through the vent and the time that is takes to fully exchange the air in the room. Assume air at standard conditions.

4.104. Air flows from a large tank through a nozzle such that the change in enthalpy between the nozzle inflow an outflow sections is 200 kJ/kg. The flow through the nozzle is approximately adiabatic, and the inflow velocity is approximately equal to zero. Estimate the nozzle exit velocity.

4.105. Air flows through a turbomachine at a steady mass-flow rate of 1 kg/s. The turbomachine is well insulated to minimize heat exchange with the surroundings. The inflow and outflow ducts are of the same size, the temperature and pressure on the inflow side are approximately equal to standard atmospheric (sea-level) conditions, and the temperature and pressure on the discharge side are 150◦C and 600 kPa, respectively. The discharge velocity is 120 m/s. Estimate the power input to the turbomachine.

4.106. Air flows into a into a compressor through a large intake duct at a pressure of 101 kPa and a temperature of 25◦C. Air flows out of the compressor through a 300-mm diameter pipe at a pressure of 400 kPa and a temperature of 50◦C. The mass flow rate of air through the compressor is 10 kg/s and the power input to the compressor is 500 kW. Estimate the rate at which heat is transferred from the compressor.

4.107. A 0.8-m diameter fan is driven by a 1-kW motor. The fan pulls in air from a large room, and air leaves the fan at 10 m/s. Estimate the efficiency of the fan in converting electrical energy to the energy added to the air by the fan (i.e., the efficiency of the fan-motor system). Assume air under standard conditions.

4.108. A 200-L insulated storage tank contains compressed air at a pressure of 600 kPa and a temperature of 18◦C. A valve is opened to release air from the tank, and at the instant that

353

the valve is opened the mass flow rate of air out of the tank is 0.02 kg/s. Estimate the rate of change of temperature in the tank at the instant that the valve is opened. You may neglect the macroscopic kinetic energy of the outflow air.

Chapter 5

Differential Analysis

5.1 Introduction

Fundamental equations that govern the flow of fluids are the conservation equations of mass, mo- mentum, and energy. In the previous chapter, these equations were put in forms that are applicable to finite control volumes, and the finite-control-volume forms of the governing equations are par- ticularly useful when we are interested in the overall features of a flow, such as the mass flow rate into and out of a control volume, or the net force exerted on a structure in contact with a flowing fluid. Although the control-volume forms of the governing equations are exact, and useful in many cases, in some cases we are interested in the detailed spatial variations in fluid flow that are not resolved by the spatially coarse finite-control-volume formulation. In those cases where finer detail is required, it is preferable to utilize the governing flow equations in forms of differential equations that are applicable at each point in the flow field. This approach is called the differential analysis of fluid flow, and the differential equations that govern fluid flow can be derived from the equations for finite control volumes by taking the limit as the size of the control volume becomes infinitesimally small. In mathematical terms, utilization of the control-volume forms of the governing equations constitute an integral approach to the analysis of fluid flow, while utilization of the differential- equation forms of the governing equations constitute a differential approach to the analysis of fluid flow.

Applications of the material in this chapter. In many engineering applications, finite- control-volume analyses are sufficient to achieve the requisite objectives and the application of differential analysis is not required. For such cases, the material contained in this chapter is not essential to the analysis and no loss of accuracy is associated with using control-volume formula- tions. In those cases where detailed spatial and temporal variations in the flow field are of interest, solution of the flow equations in differential form is required. The material contained in this chap- ter is particularly useful in describing those few flow fields can be described exactly by analytic expressions. Also, in many commonly encountered flow fields, the velocity gradients and velocities are sufficiently small that viscous and compressibility effects can be neglected in the flow field. Under these circumstances, the fluid can be approximated by an ideal fluid, which is inviscid and incompressible, and analytic expressions for several commonly encountered ideal-fluid flows are covered in this chapter.

355

448

This form of the energy equation is applicable to Newtonian fluids in cases where radiation heat transfer and internal sources of heat are not present. Equation 5.311 can be applied to unsteady and compressible flows. In cases where the density (ρ), specific heat at constant volume (cv), viscosity (µ), and thermal conductivity (k) can be assumed constant, Equation 5.311 simplifies to

ρcv dT

dt = k∇2T + Φ (for ρ, cv, µ, and k constant) (5.312)

where the relations ∇ · v = 0 and du = cv dT have been applied in deriving Equation 5.312 from Equation 5.311. For fluids at rest, Φ = 0, and cv is replaced by cp in Equation 5.312. Also, note that cv ≈ cp in liquids. Numerical methods are usually used to obtain solutions to these equations for given boundary and initial conditions.

Applications of the differential energy equation. The differential form of the energy equa- tion should generally be used in cases where spatial variations in temperature affect the variables in the continuity and momentum equations. This is generally the case in compressible flows where the pressure and density are related to temperature by an equation of state. Additionally, the flows of incompressible fluids can be significantly influenced by temperature variations. For example, in free convection, the relationship between density and temperature is of central importance, and in lubrication flows the relationship between temperature and viscosity can significantly influence the flow.

Key Equations in Differential Analysis of Fluid Flows

The following list of equations are particularly useful in the solution of problems related to the differential analysis of fluid flows. If one is able to recognize these equations and recall their appropriate use, then the learning objectives of this chapter have been met to a significant degree. Derivations these equations, definitions of the variables, and detailed examples of usage can be found in the main text.

Kinematics

Velocity: V = ui+ vj+ wk

Acceleration: a = DV

Dt = ∂V

∂t + u

∂V

∂x + v

∂V

∂y + w

∂V

∂z

Angular velocity: ω = 1

2

( ∂w

∂y − ∂v ∂z

) i+

1

2

( ∂u

∂z − ∂w ∂x

) j+

1

2

( ∂v

∂x − ∂u ∂y

) k

ω = 1

2 (∇×V) = 1

2

∣∣∣∣∣∣∣∣∣

i j k

∂

∂x

∂

∂y

∂

∂z u v w

∣∣∣∣∣∣∣∣∣

Vorticity: ζ = ∇×V = 2ω

449

ζ =

( 1

r

∂vr ∂θ

− ∂vθ ∂z

) er +

( ∂vr ∂z

− ∂vz ∂r

) eθ +

1

r

( ∂(rvθ)

∂r − ∂vr

∂θ

) ez

Angular deformation: Ωxy = ∂v

∂x + ∂u

∂y

Ωxz = ∂w

∂x + ∂u

∂z

Ωyz = ∂w

∂y + ∂v

∂z

Volumetric dilatation rate: ∇ ·V

Conservation of Mass

Continuity equation: ∂ρ

∂t + ∂(ρu)

∂x + ∂(ρv)

∂y + ∂(ρw)

∂z = 0

∂ρ

∂t +

1

r

∂(rρvr)

∂r +

1

r

∂(ρvθ)

∂θ + ∂(ρvz)

∂z = 0

∂ρ

∂t +∇ · (ρV) = 0

1

ρ

Dρ

Dt +∇ ·V = 0

∇ ·V = 0 (incompressible fluid/flow)

∂(ρu)

∂x + ∂(ρv)

∂y + ∂(ρw)

∂z = 0 (steady flow)

∂u

∂x + ∂v

∂y + ∂w

∂z = 0 (incompressible flow)

1

r

∂(rvr)

∂r +

1

r

∂(vθ)

∂θ + ∂(vz)

∂z = 0 (incompressible flow)

Mach criterion: Ma = V

c < 0.3 (for approximate incompressibility)

Stream function: u = ∂ψ

∂y , v = −∂ψ

∂x

vr = 1

r

∂ψ

∂θ , vθ = −

∂ψ

∂r

vr = − 1

r

∂ψ

∂z , vz =

1

r

∂ψ

∂r

Flow between streamlines: q =

∫ ψ2

ψ1

dψ = ψ2 − ψ1

Conservation of Momentum

450

Momentum equation: ρg +∇ · σij = ρ [ ∂V

∂t + (V ·∇)V

]

ρgi +∇ · σij = ρ [ ∂vi ∂t

+V ·∇vi ]

Navier-Stokes equation: ρ

( ∂V

∂t +V ·∇V

) = −∇p+ ρg+ µ∇2V

ρ DV

Dt = −∇p+ ρg+ µ∇2V

ρ

( ∂Vi ∂t

+V ·∇Vi )

= −∇p+ ρgi + µ∇2Vi

(St) ∂V∗

∂t∗ +V∗ ·∇∗V∗ = −(Eu)∇p∗ +

( 1

Fr2

) g∗ +

( 1

Re

) ∇∗2V∗

V∗ ·∇∗V∗ = −(Eu)∇∗p∗ + (

1

Fr2

) g∗ +

( 1

Re

) ∇∗2V∗ (steady state)

(St) ∂V∗

∂t∗ +V∗ ·∇∗V∗ = −(Eu′)∇∗p′∗ +

( 1

Re

) ∇∗2V∗ (no free surface)

(St) ∂V∗

∂t∗ +V∗ ·∇∗V∗ = −(Eu)∇p∗ +

( 1

Fr2

) g∗ (inviscid flow)

Continuity equation: ∇∗ ·V∗ = 0 (incompressible flow)

Solutions of the Navier-Stokes Equation

Fixed plates (Hele-Shaw): p = −ρgz + ( ∂p

∂x

) x+ p0, u =

1

2µ

( ∂p

∂x

) (z2 − h2)

q = −2h 3

3µ

∂p

∂x , V =

h2

3µ

∂p

∂x , umax = −

h2

2µ

∂p

∂x =

3

2 V

Moving plate (Couette flow): u = U z

b +

1

2µ

( ∂p

∂x

) (z2 − bz), q = −

( ∂p

∂x

) b3

12µ +

Ub

2

Moving vertical plate: w = γ

2µ x2 − γh

µ x+W, q = Wh− γh

3

3µ , V = W − γh

2

3µ

Flow down inclined plane: u(y) = ρg sin θ

2µ (2h− y)y, q = ρgh

3 sin θ

3µ

Flow through circular tube: p = −ρg(r sin θ) + f(x), vx = 1

4µ

( ∂p

∂x

) (r2 −R2)

Q = − πD 4

128µ

( ∂p

∂x

) , V = − D

2

32µ

( ∂p

∂x

) , vmax = −

D2

16µ

( ∂p

∂x

)

451

Flow through annulus: vx = 1

4µ

( ∂p

∂x

)[ r2 −R2o +

R2i −R2o ln(Ro/Ri)

ln r

Ro

]

Q = − π 8µ

( ∂p

∂x

)[ R4o −R4i −

(R2o −R2i )2

ln(Ro/Ri)

]

V = − 1 8µ

( ∂p

∂x

)[ R2o +R

2 i −

R2o −R2i ln(Ro/Ri)

]

Flow between rotating cylinders: vθ = Ro/r − r/Ro

Ro/Ri −Ri/Ro Riω, T =

4πµR2i R 2 oLω

R2o −R2i

P = Tω = 4πµR2i R

2 oLω

2

R2o −R2i

Inviscid Flow

Navier-Stokes equation (inviscid):ρ

[ ∂V

∂t + (V ·∇)V

] = −∇p+ ρg

Euler equation: ρ DV

Dt = −∇p+ ρg

Pressure gradient (streamline): dp

ds +

1

2 ρ d(V 2)

ds + ρg

dz

ds = 0

Along streamline:

∫ dp

ρ +

V 2

2 + gz = constant,

p

ρ +

V 2

2 + gz = constant

In irrotational flow field: p

ρ +

V 2

2 + gz = constant,

p

ρ +

V 2

2 + gz = constant

Velocity potential: V = ∇φ, ∇2φ = 0

Bernoulli equation: p

ρ +

V 2

2 + gz = constant

Continuity equation: 1

r

∂

∂r

( r ∂φ

∂r

) +

1

r2 ∂2φ

∂θ2 + ∂2φ

∂z2 = 0

Stream function: ∇2ψ = 0

Streamline and stream function: ∂φ

∂x = ∂ψ

∂y ,

∂φ

∂y = −∂ψ

∂x

Plane potential flow: vr = 1

r

∂ψ

∂θ , vθ = −

∂ψ

∂r

Axisymmetric potential flow: vr = − 1

r

∂ψ

∂z , vz =

1

r

∂ψ

∂r , r

∂

∂r

( 1

r

∂ψ

∂r

) + ∂2ψ

∂z2 = 0

Fundamental and Composite Potential Flows

452

Uniform flow: φ = V x, φ = V (x cos θ + y sin θ)

ψ = V y, ψ = V (y cos θ − x sin θ)

Line source/sink flow: φ = q

2π ln r, φ =

q

2π ln √ (x− x0)2 + (y − y0)2

ψ = q

2π θ, ψ =

q

2π tan−1

( y − y0 x− x0

)

Line vortex flow: φ = Γ

2π θ, φ =

Γ

2π tan−1

( y − y0 x− x0

)

ψ = − Γ 2π

ln r, ψ = − Γ 2π

ln √ (x− x0)2 + (y − y0)2

Spiral flow: φ = q

2π ln r +

Γ

2π θ, ψ =

q

2π θ − Γ

2π ln r

Doublet: ψ = −K sin θ r

, φ = K cos θ

r

x2 +

( y +

K

2ψ0

)2 =

( K

2ψ0

)2 (streamline)

( x− K

2φ0

)2 + y2 =

( K

2φ0

)2 (equipotential)

Flow around half body: φ = V r cos θ + q

2π ln r, ψ = V r sin θ +

q

2π θ

b = q

2πV , rs =

b(π − θs) sin θs

, p0 + 1

2 ρV 2 = ps +

1

2 ρv2s

v2s = V 2

[ 1 +

sin 2θs π − θs

+ sin2 θs

(π − θs)2

]

Rankine oval: ψ = V r sin θ − q 2π

tan−1 ( 2ar sin θ

r2 − a2

)

φ = V r cos θ + q

2π (ln r1 − ln r2) ,

b

a =

√ q

πV a + 1

h

a =

1

2

[( h

a

)2 − 1 ] tan

[ 2

( πV a

q

) h

a

]

Vmax V

= 1 + ( q πV a

) 1 1 + h2/a2

Flow around circular cylinder: ψ = V r sin θ − K sin θ r

, ψ = V r

( 1− R

2

r2

)

φ = V r cos θ + K cos θ

r , φ = V r

( 1 +

R2

r2

)

453

ps = p0 + 1

2 ρV 2(1− 4 sin2 θ), Fx = FD = −

∫ 2π

0 ps cos θR dθ

Fy = FL = − ∫ 2π

0 ps sin θR dθ

Rotating cylinder: ψ = V r

( 1− R

2

r2

) sin θ − Γ

2π ln r, φ = V r

( 1 +

R2

r2

) cos θ +

Γ

2π θ

ps = p0 + 1

2 ρV 2

( 1− 4 sin2 θ + 2Γ sin θ

πRV − Γ

2

4π2R2V 2

)

FD = 0, FL = −ρV Γ

Turbulent Flow

Turbulent shear stress: τturb = −ρu′v′, τturb = η dū

dy

Eddy viscosity: η = −ρu′v′ ( dū

dy

)−1 = ρℓ2

dū

dy

Total shear stress: τ = µ dū

dy + η

dū

dy = ρ(ν + ϵ)

dū

dy

Turbulence intensity: I =

√ (u′)2

ū

Turbulent velocity: v(t) = v̄∆t + v ′

Conservation of Energy

Fourier’s law: q = −k∇T

Energy equation: ρ du

dt + p (∇ · v) = ∇ · (k∇T ) + Φ

ρcv dT

dt = k∇2T + Φ (for ρ, cv, µ, and k constant)

Problems

Section 5.2: Kinematics

5.1. Velocity measurements in a steady-state two dimensional incompressible flow field are taken at the four points shown in Figure 5.49. The x and y components of the velocity, designated

454

as u and v, respectively, are as follows:

Point u v

(m/s) (m/s)

1 3.23 0.81 2 4.18 0.52 3 5.85 0.36 4 3.51 –

(a) Estimate the x component of the fluid acceleration at Point 2. (b) The velocity v was inadvertently not measured at Point 4; estimate this missing value based on the available measurements and the continuity equation.

1 2 3

4

150 mm

150 mm 150 mm

x

y

Figure 5.49: Velocity measurements in a two-dimensional flow field

5.2. A steady two-dimensional flow field of a compressible fluid is described by the velocity com- ponents u = 10(x2+y2), and v = 10xy. At the coordinate location (x, y) = (3, 2), the density of the fluid is 1.7 kg/m3. Determine the rate of change of the density of the fluid as it passes through this point.

5.3. Consider a flow with the following steady-state velocity field:

V = 5xyi+ 2yzj+ 3xz2k

Determine the acceleration field. Determine the velocity and acceleration at (x, y, z) = (2,3,1).

5.4. The nozzle shown in Figure 5.50 is designed such that the velocity, v, at any point near the centerline of the nozzle under steady-state conditions can be estimated by

v = 8

1− x i m/s

where x is the distance in meters from the center of the nozzle entrance, measured along the centerline of the nozzle. The centerline velocities at the entrance and exit of the nozzle are 8 m/s and 20 m/s, respectively, and the length of the nozzle is 0.6 m. What is the acceleration of a fluid element at the center of the mid-section of the nozzle, where x = 0.3 m?

455

0.6 m

x 8 m/s 20 m/s

Nozzle

y

Figure 5.50: Flow through a nozzle

5.5. Prove Equation 5.8

5.6. The velocity field in a 2-D flow is given by

V = (1.1 + 3.0x+ 0.74y)i+ (0.9− 2.5x− 5.3y)j

Determine the vorticity field and assess the rotationality of the flow.

5.7. The velocity components in a two-dimensional velocity field in the xy plane are: u = x2 m/s, v = −2xy m/s, where x and y are in meters. Determine the rate of rotation of a fluid element about the point (0.5 m, 0.5 m). Indicate whether the rotation is in the clockwise or anticlockwise direction.

5.8. The velocity field, V [m/s] in the two-dimensional converging duct of width 3 m shown in Figure 5.51 is given by

V = (5 + 2x)i− 2yj where x [m] is the coordinate measured along the centerline of the duct, and y [m] is the coordinate measured normal to the centerline. Assess the rotationality of the flow within the duct.

x

y

5 m/s

Duct

Centerline

3 m

Figure 5.51: Flow through a duct

5.9. A two-dimensional velocity field is given by

V = 2(x2 + y2)i+ 3(x2 − y2)j

What is the rate of angular deformation at location (x, y) = (3,2). Identify any locations where the rate of angular deformation is equal to zero.

456

5.10. Consider the following steady, three-dimensional velocity field:

V = (3.0 + 2.0x− y)i+ (2.0x− 2.0y)j+ (0.5xy)k

Determine the angular velocity in the xy plane at (x, y) = (1,2).

5.11. A viscous fluid is contained between two parallel plates spaced 20 mm apart as shown in Figure 5.52. The bottom plate remains stationary, the top plate moves at 3 m/s, and the velocity of the fluid varies linearly between the bottom and top plate. (a) Determine the rate of angular rotation and the vorticity of a fluid element contained between the plates, and (b) determine the rate of angular deformation of the fluid element.

3 m/s

20 mm

Fixed plate

Moving plate

Fluid g

Figure 5.52: Flow between parallel plates

5.12. Measurements in a flow field indicate that the velocity components are u = 4 m/s and v = −3 m/s at a location where x = 1 m and y = 2 m. Express the given location in polar coordinates (r, θ), and determine the r and θ components of the velocity, which are commonly represented by vr and vθ.

5.13. Measurements in a flow field indicate that the velocity components are vr = 3 m/s and vθ = −2 m/s at a location where r = 3.5 m and θ = 50◦. Express the given location in Cartesian coordinates (x, y), and determine the x and y components of the velocity.

5.14. The velocity profile, u(r), in a circular conduit under laminar flow conditions is shown in Figure 5.53 and can be expressed analytically as:

u(r) = U0

[ 1− 4

( r D

)2]

where U0 is the centerline velocity, r is the distance from the centerline, and D is the diameter of the conduit. Determine functional expressions for the angular deformation of a fluid element as a function of location in the conduit.

Dr U 0 x

u Velocity profile

Conduit

Figure 5.53: Flow in a circular conduit

457

5.15. Consider the velocity field, V, given by

V = 2xzi− 3yzj+ 4xyk m/s

where x, y, and z, are the coordinate locations in meters. Find the dilatation rate of the fluid at (x, y, z) = (2 m,3 m,1 m) and assess whether the fluid is being compressed or expanded.

Section 5.3: Conservation of Mass

5.16. A duct is being used to transport methane at a temperature of 25◦C. Estimate the maximum allowable speed in the duct for which the flow can be assumed incompressible.

5.17. Observations on the spatial variations in velocity within a fluid indicate that the velocity components can be estimated by: u = 0, v = z(z2 − 3y2), and w = y(3z2 − y2). Determine whether the fluid is likely to be incompressible.

5.18. The theoretical flow field surrounding a structure called a Rankine half-body is given by

vr = V cos θ + q

2π

1

r , vθ = −V sin θ

where V and q are parameters of the flow field. Show that this flow field applies to an incompressible fluid.

5.19. When a fluid flow approaches a solid surface as shown in Figure 5.54, the velocity of the fluid in contact with the solid surface must necessarily be equal to zero, and the viscosity of the fluid causes the velocity of the fluid away from the boundary to be retarded. Retardation of the fluid is significant within a region called the boundary layer, which can be taken to have a thickness, δ, that depends on the distance x from the front of the surface. Observations indicate that the x-component of the velocity, u, can be expressed in normalized form by

u

V = 2

(y δ

) − (y δ

)2 , where:

δ

x =

4.91

Re 1 2

, Re = V x

ν

where V is the free-stream velocity, and ν is the dynamic viscosity of the fluid. (a) Assuming the fluid is incompressible, determine an expression for the normalized y-component of the velocity, v/V , as a function of x, y, and δ. (b) Boundary layer analyses typically assume that v is negligible compared to u. Assess the justification of this assumption by finding the range of values of Re for which v/u ≤ 0.1 at the outer limit of the boundary layer.

x

y δ(x)

Boundary layer

Free stream

V

Solid surface

Velocity profile, u(x,y)

Figure 5.54: Flow in a boundary layer

458

5.20. In a two-dimensional incompressible flow field, the y component of the velocity, v is given by v = 2x2. The x component of the velocity, u(x, y), is unknown, but it is known that u(x, y) must satisfy the boundary condition that u(0, y) = 0. Determine u(x, y).

5.21. The velocity components in a three dimensional velocity field are given by u = ax2z, v = bxz2, and w = cxz2 + d, where a, b, c, and d are constants. Determine the relationship between the constants that would be required for the flow to be incompressible.

5.22. The x and z components of the velocity in a three-dimensional flow field are given by: u = ayz−bxy2 and w = 2az+bxz+cx2. If the fluid is incompressible, determine the y component of the velocity, v, as a function of x, y, and z.

5.23. A velocity field experienced by an incompressible fluid is given by

V = 5xy2i+ 2xzj+ xzf(x, y)k

where f(x, y) is an unknown function of x and y. Determine the required functional form of f(x, y).

5.24. Flow of an incompressible fluid is in the xy plane, and the y component of the velocity, v, is given by v = 3y. (a) Determine the required functional form of the velocity field. (b) Give a particular velocity field that satisfies the required functional form.

5.25. Two-dimensional unsteady flow of an incompressible fluid occurs in the xy plane such that the x-component of the velocity, u, is given by: u = 2xy2 + 3y. Determine the required functional form of the y-component of the velocity.

5.26. A three-dimensional flow field has velocity components that can be approximated by:

vr = r

2 sin θ, vθ = 3r cos θ, vz = 2z sin θ

(a) Assess whether this flow field can be classified as incompressible. (b) If the flow field can be classified as incompressible, does this mean that the fluid is incompressible?

5.27. Two-dimensional unsteady flow of an incompressible fluid occurs in the xy plane. When cylindrical polar coordinates are used, the θ component of the velocity is given by:

vθ = − C sin θ

r2

where C is a constant. Determine the required functional form of the r-component of the velocity.

5.28. Two-dimensional unsteady flow of an incompressible fluid occurs in the xy plane. When cylindrical polar coordinates are used, the r component of the velocity is given by:

vr = r 2 sin θ + r cos θ

Determine the required functional form of the θ-component of the velocity.

459

5.29. A two-dimensional steady flow in the rθ plane has a θ component of the velocity given by

vθ = 10

( 1 +

1

r2

) cos θ − 15

r

A boundary condition that must be satisfied by the flow field is vr(1, θ) = 0. Determine the r component of the velocity as a function of r and θ.

5.30. Air flows in a steady-state through a straight pipe as illustrated in Figure 5.55. Measurements at Section A indicate that the temperature is 15◦C, the pressure is 400 kPa absolute, and the velocity is 90 m/s. At Section B, which is 300 mm downstream of Section A, the velocity is measured as 94 m/s. Estimate the density of the air at sections A and B. The flow can be assumed to be one-dimensional in the x direction.

A B

300 mm

90 m/s 94 m/s

15oC

400 kPa Air

x

Conduit

Figure 5.55: Air flow through a pipe

5.31. Air flows through a 1-m long two-dimensional converging duct as shown in Figure 5.56. Under steady-state conditions, the inflow and outflow velocities are 50 m/s and 100 m/s, respectively, and the inflow and outflow densities are 1.3 kg/m3 and 0.8 kg/m3, respectively. Locations within the duct are identified relative to Cartesian coordinate axes with the origin at the point where the centerline of the duct intersects the inflow section. The x component of velocity, u, varies nonlinearly between the entrance and the exit according to the relation u = a+ bx2, and the density, ρ, varies linearly between the entrance and exit according to the relation ρ = c + dx, where a, b, c, and d are constants. It is known that the y component of the velocity is equal to zero along the centerline of the duct, but is nonzero elsewhere. Determine v as a function of x and y.

100 m/s50 m/s

1 m

x

y Centerline

Conduit

1.3 kg/m3 0.8 kg/m3

Figure 5.56: Two-dimensional flow through a converging duct

460

5.32. Standard air is contained within an insulated chamber with a fixed side and a moving side as shown in Figure 5.57. The moving side oscillates such that within a time interval of 0.1 seconds the velocity, u, in the direction of oscillation is given by

u = 10x

0.12 + 10t m/s

where x is the distance from the fixed side in meters, and t is the time since the beginning of the time interval in seconds. Within this time interval, the density, ρ, is spatially uniform within the chamber, but varies with time. The density of the air at the beginning of the time interval is 3 kg/m3. What is the air density within the chamber at the end of the time interval?

x

ρ(t)

P

u(x,t) Moving side

Fixed side Air chamber (insulated)

Figure 5.57: Expanding insulated air chamber

5.33. A two-dimensional velocity field is given by: V = 2xi− 2yj. Determine the location(s) of the stagnation points, and the expression for the stream function.

5.34. The two-dimensional velocity field of an incompressible fluid under a particular scenario is given by: u = 2y(2x + 1), v = −2y2 + x2. Determine the analytic expression for the stream function, and describe how you would use this stream function to plot the streamlines of the flow.

5.35. The stream function in a two-dimensional flow field is given by ψ = x2 − y2 m2/s, where x and y are the Cartesian coordinates in meters. Determine the rate of angular rotation of a fluid element located at (1 m, 1 m). How does this rate of rotation compare with the rate of rotation at other points in the flow field?

5.36. A fluid flows over a flat plate such that the velocity increases linearly with distance from the surface of the plate as shown in Figure 5.58. (a) Determine the stream function. (b) Use the stream function to calculate the volumetric flow rate of the fluid between the plate surface and a distance 2 m from the surface. (c) Determine distance beyond 2 m for the volumetric flow rate to be the same as that between the plate surface and 2 m.

461

z

x

2 m

15 m/s

Flat surface

Velocity profile

Flow

Figure 5.58: Flow adjacent to flat plate

5.37. The two-dimensional fluid-flow between two flat plates is illustrated in Figure 5.59. The velocity distribution, u(z), between the plates can be expressed as u(z) = a(h2 − z2), where a is a constant, z is the vertical coordinate measured from the mid-plane between the plates, and 2h is the distance from the bottom plate to the top plate. (a) Determine the stream function that can be used to describe the flow. (b) Determine the flow between the plates using the derived stream function. (c) Verify the result in Part (b) by integrating the velocity distribution over the interval between the plates.

x

yz

(Infinite) Flat plate

h

h

Velocity profile gu(z)

Figure 5.59: Flow between parallel plates

5.38. The stream function of a particular two-dimensional flow field is given by ψ = 0.1xy m2/s, where x and y are the Cartesian coordinates in meters. Determine the volumetric flow rate between the points (1 m, 1 m) and (2 m, 3 m).

5.39. The two-dimensional flow field of an incompressible fluid is described in radial coordinates as: vr = 1/r, vθ = 2/r2. Determine the analytic expression for the stream function.

5.40. A particular flow field experienced by an incompressible fluid is described in radial coordinates by the stream function: ψ(r, θ) = −5r sin θ+3θ m2/s, where r is in meters and θ is in radians. (a) Determine the r and θ components of the velocity vector as a function of r and θ. (b) At the point r = 1 m and θ = π/6 = 30◦, determine the r and θ components of the velocity, and the x and y components of the velocity, where x and y are the Cartesian coordinates.

5.41. A free vortex is a rotational flow in which there are to external force acting on the fluid. Streamlines of a particular free vortex is shown in Figure 5.60, where the tangential component of the velocity, vθ, is given by vθ = 3.5/r, where r is the distance from the center of the vortex. The radial component of the velocity is equal to zero. (a) Determine the stream function.

462

(b) Explain how you can deduce from the result in Part (a) that the streamlines are circles. (c) Use the stream function to estimate the volumetric flow rate between the streamlines at r = 1.2 m and r = 1.4 m.

Velocity profile

1.2 m

1.4 m Streamline

Figure 5.60: Flow in a free vortex

Section 5.4: Conservation of Momentum

5.42. A velocity field can be represented by the velocity components: u = 8z, v = 0, and w = 2x, where gravity has a magnitude g and acts in the negative z direction. Within the flow field, the viscosity is constant, and the pressure and density are equal to p0 and ρ0, respectively, at the location (x0, y0, z0). Use the Navier-Stokes equation of determine the pressure distribution in terms of the given parameters.

5.43. Observations of a fluid flow indicate that a velocity field can be represented by the velocity components: u = 8z, v = 6x, and w = 2x, where gravity has a magnitude g and acts in the negative z direction. Within the flow field, the viscosity is presumed constant. Assess whether this given velocity field satisfies the Navier-Stokes equation. If not, speculate on what may be the reason. If so, find an expression for the pressure distribution.

5.44. A liquid with a density ρ and viscosity µ flows between parallel plates as shown in Figure 5.61. The parallel plates are inclined at angle of θ with the horizontal, and the flow is driven by the force of gravity. Flow conditions are steady and do not change in the dimension perpendicular to the page. The x coordinate is measured along the incline, the z coordinate is measured perpendicular to the incline, and the magnitude of the acceleration due to gravity is g. Write the simplified form of the Navier-Stokes equation in Cartesian coordinates that applies to this scenario.

x

z

!

Flow Velocity profile

Top plate

Bo!om plate g

Figure 5.61: Flow between inclined parallel plates

463

5.45. A viscous liquid flows down an inclined plane as shown in Figure 5.62. The flow is two dimensional in the xy plane, and the x axis is oriented in the flow direction. Apply the Navier- Stokes equation to this problem and write the components of the Navier-Stokes equation in their most simplified forms.

!

h

Fluid-solid interface

Fluid-air interface

Velocity profile

x

y

g

u(y)

Figure 5.62: Flow down an inclined plane

Section 5.5: Solutions of the Navier-Stokes Equation

5.46. Consider the case of a viscous fluid flowing down an inclined plane as shown in Figure 5.62. The only non-zero component of the velocity is u(y), and there is no pressure gradient in the x direction. (a) Apply the Navier-Stokes equation in the x direction to determine the longitudinal velocity distribution u(y) and the volumetric flow rate down the incline. In your derivation, use the no-slip condition and the condition that the shear stress is negligible at the air-liquid interface. (b) If the fluid is SAE 30 oil at 20◦C, the depth of flow is 20 mm, and the angle of incline is 15◦, determine the volumetric flow rate.

5.47. SAE 50 oil at 20◦C flows between two horizontal parallel plates spaced a distance 5 mm apart. The plates are 2 m long and 1 m wide, and the oil flow is driven by a pressure differential between the inflow and outflow sections. The inflow pressure is 200 kPa gage and the outflow pressure is atmospheric. (a) Estimate the volumetric flow rate of the oil between the plates. (b) Estimate the average flow velocity between the plates. (c) Validate your results using a Reynolds number analysis.

5.48. A fluid flows freely between two vertical parallel plates that are of width W and spaced a distance h apart as shown in Figure 5.63. The force driving the flow is gravity, and it can be reasonably approximated that the pressure remains constant throughout the fluid. (a) Determine an expression for the volumetric flow rate of the fluid as a function of the fluid properties, W , and h. (b) If the fluid is SAE 30 oil at 20◦C, the spacing between plates is 5 mm, and the width of the plates is 1.5 m, determine the volumetric flow rate of the oil between the plates.

464

x

yz

h

g w(x)

Ver!cal plate

Figure 5.63: Flow between vertical parallel plates

5.49. Consider the steady flow of SAE 10 oil at 20◦C between flat plates spaced 10 mm apart. The average pressure between the plates is on the order of 100 kPa, the average velocity is on the order of 0.05 m/s. What simplified form of the Navier-Stokes equation would be applicable in this situation.

5.50. SAE 10 oil at 20◦C flows between parallel plates spaced 10 mm apart. The length of the plates in the flow direction is 1 m and the width of the plates is 80 cm. Find the volumetric flow rate between the plates and the force exerted by the fluid on each of the plates if a pressure gradient of 100 Pa/m is applied to drive the flow.

5.51. SAE 30 oil at 20◦C flows between 0.50-m wide parallel plates separated by 30 mm. The length of the top and bottom plates in the direction of flow is 2 m, the bottom plate is stationary, the top plate moves at 3 cm/s, an adverse pressure gradient of 700 Pa/m is applied between the plates, and the flow is laminar between the plates. Determine the flow rate between the plates and the force that must be applied to move the top plate.

5.52. Consider the case of a vertically oriented plate moving with a (vertical) velocity W through a fluid reservoir. Show that the distance, x0, from the plate surface to the location where the velocity in the fluid is equal to zero is given by

x0 h

= 1−

√

1− 2µW γh2

where h is the thickness of the fluid layer, and γ and µ are the specific weight and dynamic viscosity of the fluid, respectively.

5.53. SAE 50 oil at 20◦C is introduced uniformly at a volumetric flow rate of q per unit width at the top of a plane wall as shown in Figure 5.64. The oil ultimately attains a steady velocity profile and a uniform thickness, h. if the thickness of the oil layer is 5 mm, at what rate is oil being added at the top of the wall?

465

g

Oil Air

Wall

q

h

Figure 5.64: Oil flow down a wall

5.54. Show that the shear stress on the wall of a tube of diameter D in which laminar flow is occurring is given by

τwall = − D

4

( ∂p

∂x

)

where ∂p/∂x is the pressure gradient along the tube.

5.55. SAE 30 oil at 20◦C flows in a 10-m long 250-mm diameter pipe. The pipe is horizontal and under a particular flow condition the volumetric flow rate is 20 L/min. (a) Verify that Hagen- Poiseuille flow can be assumed in the pipe. (b) What is the difference in pressure between the entrance and exit of the pipe?

5.56. Gasoline at 20◦C is to be delivered in a 20-m long tube where the difference in pressure between the entrance and exit of the tube is 5 kPa. If laminar flow conditions are to be maintained in the tube, (a) what is the maximum tube diameter than can be used, and (b) what is the flow rate through the tube when using this maximum diameter?

5.57. Ethylene glycol at 20◦C is to be delivered between two pressurized reservoirs that have a pressure difference of 5 kPa. Delivery is to be through multiple tubes, with each tube having a length of 1.2 m and a diameter of 2 mm. If the desired volumetric flow rate between the reservoirs is 1.5 L/min, how many tubes are required?

5.58. A fluid flows under the influence of gravity in a vertical tube of diameter D as shown in Figure 5.65. It can be reasonably approximated that the pressure remains constant throughout the fluid. (a) Determine an expression for the volumetric flow rate as a function of the fluid properties and D. (b) If the fluid is SAE 30 oil at 20◦C, and the diameter of the tube is 50 mm, determine the volumetric flow rate in the tube.

466

x

yz

D

g w(r)

Ver!cal tube

r

Figure 5.65: Flow in a vertical tube

5.59. A fluid flows through a glass annulus with an inner diameter of 5 mm and an outer diameter of 10 mm. The fluid has a density of 930 kg/m3 and a dynamic viscosity of 0.200 Pa·s. If a pressure gradient of −100 Pa/m is applied, what flow rate is expected through the annulus? How would you expect this flow rate to change if the annulus were made of steel rather than glass?

5.60. An incompressible fluid is contained between two cylinders as shown in Figure 5.66. The radii of the inner and outer cylinders are Ri and Ro, respectively, and the viscosity of the fluid is µ. The fluid flow is caused by both a pressure gradient in the flow direction and by moving the inner cylinder at a velocity V in the direction of flow. (a) Determine an expression for the steady-state velocity distribution in the annular region as a function of r, Ri, Ro, µ, and the pressure gradient. (b) If the fluid is SAE 10 oil at 20◦C, the inner and outer radii are 10 mm and 20 mm, respectively, the pressure gradient is −80 Pa/m, and the inner cylinder moves at 50 mm/s, determine the fluid velocity midway between the inner and outer cylinders.

x

r R o

Annulus

R i

g

V

Figure 5.66: Flow between cylinders

5.61. A liquid with a density ρ and viscosity µ is contained in the annular space between two cylinders as shown in Figure 5.67. The inner cylinder rotates at a constant angular speed of ω, the outer cylinder is stationary, flow conditions in the fluid are at steady state and uniform in the z dimension (perpendicular to the page), and gravity acts in the z direction. Write the simplified form of the Navier-Stokes equation in cylindrical coordinates.

467

ω

Inner cylinderOuter cylinder

Fluid

Velocity profile

r "

Figure 5.67: Flow in the annular space between cylinders

5.62. A viscous liquid is contained between two cylinders as shown in Figure 5.68. The inner cylinder has a radius Ri, the outer cylinder has a radius Ro, and the outer cylinder rotates at an angular speed ω. Use the θ-component of the Navier-Stokes equation along with the appropriate inner and outer boundary conditions to determine the velocity distribution in the fluid contained between the two cylinders.

ω

R i

R o

Inner cylinder

(fixed)

Outer cylinder

(rota!ng) r

Fluid

Figure 5.68: Fluid between cylinders

5.63. A lubricant is located in the annular region between a 1.0-m long rotating shaft of diameter 40 mm and an outer cylinder of diameter 50 mm. The density and viscosity of the lubricant are 900 kg/m3 and 0.35 Pa·s, respectively. What is the power required to rotate the shaft at 50 rpm? Is the flow of the lubricant stable?

Section 5.6: Inviscid Flow

5.64. A two-dimensional velocity field in the xy plane is described by the following velocity com- ponents

u = 10x

x2 + y2 v =

10y

x2 + y2

represent a possible incompressible flow? If so, find the pressure gradient ∇p assuming a frictionless flow with negligible body forces.

5.65. A two-dimensional velocity field in the xz plane is described by the velocity components u = 2(x2− z2)− 6x m/s and w = 4xz− 6z m/s, where x and z are the Cartesian coordinates in meters. The gravity force acts in the negative z direction, and the fluid has a density of 998 kg/m3. Determine the pressure gradients in the x and z directions at the point x = 2 m, z = 1 m.

468

5.66. A two-dimensional velocity field in the xy plane is described by the velocity components u = −3x+ 9y2 m/s and v = 3y m/s, where x and y are the Cartesian coordinates in meters. The gravity force acts in the negative z direction, and the fluid has a density of 1.2 kg/m3. Determine ∇p at the point x = 2 m, z = 1 m.

5.67. A two-dimensional velocity field in the rθ plane is described by the velocity components vr = −6/r m/s and vθ = 3/r m/s, where r and θ are cylindrical coordinates in meters and radians, respectively. The gravity force acts in the negative z direction, and the fluid has a density of 1.20 kg/m3. Calculate the pressure gradients in the r, θ, and z directions at r = 2 m and θ = π/4 rad.

5.68. A fundamental two-dimensional flow field that is used as a building block for construction more complex ideal-fluid flows is that of a doublet, which has a flow field described by the following velocity components

vr = − K cos θ

r2 , vθ = −

K sin θ

r2

where K is a constant. The gravity force acts in the negative z direction, the acceleration due to gravity is g, and the density of the fluid is ρ. (a) Determine a functional expression for the pressure gradient, ∇p, in terms of r, θ z, ρ, and g. (b) If K = 3 m3/s, ρ = 1000 kg/m3, and g = 9.81 m2/s, what is the pressure gradient at r = 2 m, θ = π/4 rad, z = 5 m?

5.69. A two-dimensional velocity field in the xz plane is described by the velocity components u = 5(x + z) m/s and w = 5(x − z) m/s, where x and z are the Cartesian coordinates in meters. The fluid has a density of 998.2 kg/m3. Determine the pressure gradient in the streamline direction at the point x = 2 m, z = 1 m.

5.70. A two-dimensional velocity field is described by the velocity components u = 4x−10 m/s and v = 6 − 4y m/s, where x and y are the Cartesian coordinates in meters. The gravity force is in the negative z direction, and the density of the fluid is equal to 1.5 kg/m3. (a) Show that the flow field is incompressible and irrotational. (c) Estimate the difference in pressure between the points (1 m, 1 m) and (2 m, 2 m).

5.71. A flow field in which the radial component of the velocity is everywhere equal to zero, and the tangential component of the velocity depends only on the radial distance, r, from a center point is called a vortex. Consider the vortex illustrated in Figure 5.69, where Point P is the center of the vortex, Point 1 is on one streamline, and Point 2 is on another streamline. For which one of the following velocity fields can the Bernoulli equation be applied between Points 1 and 2: vθ = ar or vθ = a/r, where a is any constant?

1

2

P

r

Streamline

Figure 5.69: Streamlines in a vortex

469

5.72. The hydrofoil shown in Figure 5.70 was tested and the minimum pressure on the surface of the foil was found to be 70 kPa when the foil was submerged 1.80 m and towed at a speed of 8 m/s. At the same depth, at what speed will cavitation first occur? Assume irrotational flow for both cases and a water temperature of 10◦C.

Water @ 10oC1.83 m

V

Hydrofoil

Figure 5.70: Flow around a hydrofoil

5.73. A fluid is that is approximately inviscid and incompressible flows through a 3-m long hori- zontal diffuser with a cross-sectional area, A, that increases according to the relation A(x) = 0.1(1 + 0.5x2) m2, where x is the distance from the entrance of the diffuser in meters. The fluid has a density of 1100 kg/m3, and under a particular steady-state condition the pressure and velocity of the fluid at the entrance to the conduit are 240 kPa and 2 m/s, respectively. (a) Determine an expression for the pressure in the diffuser as a function of x. (b) Determine an expression for the pressure gradient in the diffuser as a function of x.

5.74. The stream function of a flow field is given as ψ = x2 − y2. (a) Is the flow irrotational throughout the flow field? (b) If the flow is irrotational, determine the potential function of the flow field.

5.75. The velocity components in a two-dimensional flow field are u = x+ 2 and v = 3− y. (a) Is the flow irrotational throughout the flow field? (b) If the flow is irrotational, determine the potential function of the flow field.

5.76. Consider a two-dimensional flow described by the following stream function:

ψ = 2r sin θ − 4θ m2/s

where r is in meters, θ is in radians. The density of the fluid is 1000 kg/m3. (a) If possible, find the potential function for this flow. (b) Find the difference in pressure between two points with (x, y) locations given by (1 m,1 m) and (3 m,2 m).

5.77. The stream function of two irrotational flows are given by

ψ1 = 15x 2y − 5y3, ψ2 = −2(x− y)

(a) What would be the stream function of the flow field generated by combining these two flows? (b) Show that the combined flow field is irrotational.

470

5.78. The stream function, ψ, for a particular flow field is given by

ψ = 4x+ 3y

(a) Describe the velocity field, (b) if possible, find the potential function for the velocity field, and (c) determine the velocity at (x, y) = (1,2).

5.79. An incompressible and irrotational flow field has a potential function, φ, that can be described by the relation

φ = y6 − 15x2y4 + 15x4y2 − x6

Determine an expression for the stream function that can be used in plotting the streamlines.

5.80. An incompressible and irrotational flow field has a potential function, φ, that can be described by the relation

φ = r cos θ + ln r

Determine the corresponding expression for the stream function in polar coordinates.

Section 5.7: Fundamental and Composite Potential Flows

5.81. Consider the line sink embedded in a solid boundary as shown in Figure 5.71. The sink withdraws fluid at a rate of 3 L/s per m length. The fluid is inviscid and the flow is at steady state. Determine: (a) an analytic expression for the velocity field, and (b) analytic expressions for the stream and potential functions.

Streamline

Sink Solid boundary

160o

Figure 5.71: Sink embedded in a solid boundary

5.82. A two-dimensional flow field is generated in the xy plane by injecting a liquid uniformly at a rate of 2 m3/s over the 3-m depth as shown in Figure 5.72. In the absence of the injection, the liquid has a uniform velocity of 0.015 m/s. If the origin of the coordinate axes is located on the injection line, determine the (x, y) coordinates where the fluid velocity is equal to zero (i.e., the location of the stagnation point).

3 m1.5 m/s

x

y

z

2 m3/s

Figure 5.72: Injection into a two-dimensional flow field

471

5.83. The streamlines resulting from the superposition of a sink and uniform flow are illustrated in Figure 5.73. This composite flow is commonly used by groundwater hydrologists to represent the flow field surrounding a pumped well (sink) in a regional flow (uniform flow). Locations in the flow field are referenced to coordinate axes centered at the sink location, the magnitude of the uniform flow is V , and the strength of the sink is q. (a) Determine the coordinate location of the stagnation point, P, in terms of V and q. (b) A portion of the upstream flow is intercepted by the sink, and a portion of the upstream flow bypasses the sink. Determine the width, W , of the upstream flow that is intercepted by the sink. (c) If a pumped well extracts 70 L/s over a vertical interval of 5 m in an aquifer that has a regional velocity of 1 m/day, determine the distance of the stagnation point from the well, and the width of the regional flow intercepted by the well.

Streamline

Sink

Uniform flow P Stagna!on point

x

y

W

Figure 5.73: Sink in uniform flow

5.84. A two-dimensional velocity field has the velocity components: u = −2x and v = 2y. (a) Verify that this flow field can be described by a potential function. (b) Determine the circulation, Γ, around a closed path of straight lines connecting the following points: (1,1), (2,1), (2,4), (1,4), and (1,1). Explain the significance of your result.

5.85. A flow field closely approximates that of a free line vortex with a vortex strength Γ. The pressure far from the vortex center is p∞, and the density of the fluid is ρ. (a) Determine the pressure at any distance r from the vortex center in terms of r, p∞, ρ, and Γ. (b) If the fluid is air with a density of 1.225 kg/m3, the pressure far from the influence of the vortex is 101 kPa, and the strength of the vortex is 3000 m2/s, determine the radial distance from the vortex center to where the pressure is 10% less than the undisturbed pressure of 101 kPa.

5.86. The velocity field in a hurricane can be approximated by a forced vortex between the center and a radial distance of R from the center, and as a free vortex beyond the radial distance R. In such an approximation, the θ component of the velocity can be approximated by

vθ =

⎧ ⎨

⎩

rω, 0 ≤ r ≤ R Γ 2πr

, r ≥ R

where ω is the rotational speed of the forced vortex, Γ is the circulation of the free vortex, and R is the match point of the two velocity distributions. The radial component of the velocity,

472

vr, is equal to zero. (a) Explain why R is where the wind speed is a maximum. (b) If the undisturbed pressure outside of the hurricane is p0 and the maximum wind speed is Vmax, determine an expression for the pressure at the match point in terms of the given variables and the relevant properties of the air. (c) For a intense Category 4 hurricane, the maximum velocity is 251 km/h (156 mph), the match-point radius is 15 km (9.3 mi), and conditions outside the hurricane are standard sea-level conditions. Determine the angular speed, ω, and the circulation, Γ, of the hurricane.

5.87. A liquid forms a free vortex around a drain as shown in Figure 5.74. If the depth of liquid at a distance r0 from the drain is h0, determine an expression for the depth, h, of the liquid as a function of the distance r from the drain. Your final result should be in terms of the strength Γ of the vortex. Assume steady flow.

z

r

h 0 h(r)

r 0

r

Drain

Figure 5.74: Vortex surrounding a drain

5.88. A two-dimensional flow field can be approximated by the superposition of two line sources and a line vortex in the xy plane. The first line source is located at the point (1 m, 1 m) and has a strength of 2 m2/s, the second line source is located at the point (2 m, 2 m) and has a strength of 3 m2/s, and the line vortex is located at the point (3 m, 3 m) and has a strength of 2.5 m2/s. Determine the x and y components of the velocity at the point (4 m, 4 m).

5.89. Consider a flow field that can be approximated as a spiral free vortex, with sink strength q and vortex strength Γ. If locations within the flow field are specified in polar coordinates r and θ, express the pressure distribution within a spiral free vortex in terms of r, θ, q, and Γ.

5.90. The air-flow velocity is measured 35 m from the center of a tornado as shown in Figure 5.75. The measured velocity has a magnitude of 110 km/h and is directed at an angle of 60◦ south of west. Assuming that the tornado can be represented as a spiral free vortex, find the equa- tion of the streamline that passes through the measurement location.

473

35 m

x

y

40o

110 km/h

60o

P

North

East

Figure 5.75: Measured wind velocity

5.91. Consider a tornado that is approximated as a spiral free vortex. In a particular tornado, the circulation is estimated as 8000 m2/s, and at a distance of 50 m from the center of the tornado the pressure is measured as 2 kPa below the pressure outside the tornado. Conditions outside the tornado are approximately equal to standard atmospheric conditions at sea level. (a) Estimate the sink strength of the tornado. (b) If a circle of radius 50 m is drawn around the center of the tornado, at what angle do the velocity vectors cross this circle?

5.92. A flow field is generated by two doublets, each of strength 5 m3/s. If one doublet is located at (x, y) = (−1 m, 0 m) and the other doublet is located at (x, y) = (1 m, 0 m), determine the resulting flow velocity at (x, y) = (2 m, 2 m).

5.93. Air flows at 10 m/s towards a structure whose side can be approximated by a Rankine half body as shown in Figure 5.76. The geometry of the structure is referenced to Cartesian axes that are centered at the “toe” of the structure. A point of interest on the surface of the structure has coordinates (4 m, 4 m). (a) Determine the coordinates of Point C that represents the location of the imaginary source used in constructing the Rankine half body. (b) What is the equation of the surface of the structure in terms of radial coordinates centered at C? (c) Determine the equation of the stream function that can be used to generate the streamline of air flow over the structure.

x (m)

y (m)

0 2 4 6 8 10

2

4 (4,4)

P

10 m/s Rankine half body

r

!C

Figure 5.76: Air flow over a Rankine half body

474

5.94. Water flowing with a velocity of 0.4 m/s and a depth of 3 m in a channel encounters a hump whose shape can be approximated as a Rankine half body as shown in Figure 5.77. The flow in the channel can be approximated as irrotational and two dimensional in the xz plane. (a) Estimate the velocity and gage pressure at the Point P that is 3 m downstream and 1.5 m above the toe of the hump. (b) Compare the velocity and gage pressure at Point P calculated in Part (a) with the velocity and gage pressure at Point P that would be obtained if the flow were assumed to be normal to the flow section at P. Comment on any discrepancies

3 m0.4 m/s P

1.5 m

3 m

Hump

x

z

Flow sec!on

Figure 5.77: Water flow over a Rankine half body

5.95. Air flows flows towards a steep hill at 60 km/h, where the shape of the hill can be approxi- mated as a Rankine half body of height 100 m as illustrated in Figure 5.78. The air is at a temperature of 20◦C and atmospheric pressure at the base of the hill is 101 kPa. Estimate the maximum velocity and the minimum pressure on the face of the hill. Neglect any change in temperature as the air flows up the hill.

60 km/h

Hill

100 m

Figure 5.78: Air flow over a hill

5.96. Water at 20◦C flows around a submerged structure that has the shape of a Rankine half body with a width of 3 m. The absolute pressure in the water upstream of the structure is 130 kPa. Determine the maximum allowable approach velocity such that cavitation does not occur on the surface of the structure.

5.97. Air flows towards a surface with the shape of a Rankine half body as shown in Figure 5.79. The point P is the source location that determines the shape of the half body. Pressure taps at points 1 and 2 measure pressures of 207 kPa and 205 kPa, respectively. Estimate the velocity of the air flowing towards the surface.

475

x

y

45o

1

2

P

Rankine half body

Air flow b

Figure 5.79: Air flow towards Rankine half body

5.98. Fluid flows around an elliptical body that is 8 m long and 7 m wide. The fluid approaches the ellipse along its major at a speed of 1 m/s. Assuming that the flow around the ellipse can be approximated as the flow around a Rankine oval, (a) estimate the parameters describing the Rankine-oval flow field, and (b) estimate the maximum velocity on the surface of the Rankine oval.

5.99. Water at 20◦C flows towards a structure that has the shape of a Rankine oval as shown in Figure 5.80. The structure has a length of 3 m and a width of 1.5 m. The absolute pressure in the water approaching the structure is equal to 125 kPa. Estimate the approach velocity, V , of the water that will initiate cavitation on the structure.

Rankine oval

1.5 m

3 m

x

y

P

V

125 kPa

Figure 5.80: Water flow towards a Rankine oval

5.100. A storage bunker has the shape of a half-cylinder of radius 5 m as shown in Figure 5.81. The bunker structural supports are to be designed to withstand the lift force generated by a 96-km/h (60-mph) wind when both the static atmospheric pressure and the air pressure inside the bunker are 101 kPa, and the air temperature is 15◦C. What is the lift force on the bunker?

476

5 m

Storage bunker96 km/h

101 kPa

101 kPa

Figure 5.81: Air flow over a storage bunker

5.101. Air under conditions of standard temperature and pressure flows at 15 m/s towards a vertical cylindrical flagpole with a diameter of 0.5 m. (a) Estimate the maximum and minimum speed of the air adjacent to the flagpole. (b) Estimate the maximum and minimum pressure deviation from standard pressure by the air on the flagpole.

5.102. Water at 20◦C flows towards a 3-m diameter cylindrical column as shown in Figure 5.82. Pressure measurements at point P indicate that the pressure at that point is approximately 12.5 kPa less than the pressure in the flow approaching the column. (a) Estimate the approach velocity. (b) Determine the doublet strength that would be used with the potential theory to simulate the flow field.

1 m x

y

45o

P Cylinder

Flow

Figure 5.82: Water flow towards a cylinder

5.103. Consider the flow towards the cylinder as shown in Figure 5.82. If the approach velocity is equal to 5 m/s, estimate the rate of rotation of the cylinder that would make the point P a stagnation point. What direction of rotation would be required, clockwise or anticlockwise?

5.104. Air flows towards a rotating cylindrical cable at 100 km/h, where the diameter of the cable is 150 mm and the length of the cable is 10 m. Force measurements indicate that the pressure lift on the rotating cable is 1.00 kN when the temperature of the air is 20◦C and atmospheric pressure is 101 kPa. If the flow is approximated as potential flow past a rotating cylinder,

477

(a) what is the value of the circulation parameter needed to simulate the flow? (b) Are there stagnation points on the cable? (c) What is the maximum pressure at any point on the cable?

Chapter 6

Dimensional Analysis and Similitude

6.1 Introduction

The primary objective of dimensional analysis is to transform functional relationships between dimensional variables into functional relationships between nondimensional groups. The advantage of using dimensional analysis is that there is generally a lesser number dimensionless groups than variables, thereby facilitating experimentation to determine empirical relationships between the variables. Similitude is the study of the relationship between behavior of similar systems, where the behavior of one system can be inferred from the behavior of another (similar) system. The rules of similitude can be derived using dimensional analysis, thereby facilitating the design of experiments to study the behavior of full-scale systems using laboratory-scale models. Such model studies and similitude applications are particularly important in the design of large and expensive structures, where safety and economic concerns require increased certainty in how the structure will perform once it is constructed. Although the focus of this book is on the applications of dimensional analysis in the field of fluid mechanics, dimensional analysis is a mathematical technique that is applicable in any field where interrelated variables can be identified.

6.2 Dimensions in Equations

A dimension is a measure of a physical quantity, and the seven fundamental dimensions in nature are mass, length, time, temperature, electric current, amount of light, and amount of matter. In contrast to a dimension, a unit is a particular convention used to measure a dimension, for example centimeters, meters, and kilometers are all units of length. Equations that do not depend on the units used, provided that the units are consistent, are called dimensionally homogeneous equations. For example, suppose that the SI system of units is used in a dimensionally homogeneous equation, such that length, mass, and time are expressed in terms of meters, kilograms, and seconds. Since the equation is dimensionally homogeneous, any calculated quantity will have length, mass, and time units that are in meters, kilograms, and seconds. If the units in the same equation were changed to another system of units, such as U.S. Customary units, using feet, slugs, and seconds, then the magnitude of the calculated quantity would be the same as before, with the only differences being the numerical value and the system of units in which the calculated quantity is represented. Dimensionally homogeneous equations have the property that if they involve sums (or differences)

479

508

Example 6.12 A 10-m diameter wind turbine is to be designed for an area where the characteristic wind speed is 40 km/h. The turbine rotor will be restricted to rotating at 8 rpm, and the turbine is expected to deliver 550 kW under design conditions. For the turbine shape being considered, viscous effects are negligible for Reynolds numbers greater than 105. A 1:20 scale model of the wind turbine is tested in a wind tunnel with an air speed that is the same as in the prototype. (a) What rotational speed should be used in the model turbine? (b) What is the power output expected in the model? Assume standard air at sea level.

Solution. From the given data: Lp = 10 m, Vp = Vm = 40 km/h = 11.11 m/s, ωp = 8 rpm, Pp = 550 kW, Lr = 20, and Recrit = 10

5. For standard air at sea level, ν = 1.461× 10−5 m2/s. A dimensional analysis of the relevant variables will yield the following functional relationship:

P 1 2ρV

3L2 = f

( ωL V

, V L ν

) → P1

2ρV 3L2

= f1 (St,Re)

Check the Reynolds numbers in the prototype and model to determine whether viscous effects can be neglected:

Rep = VpLp ν

= (11.11)(10) 1.461× 10−5 = 7.60× 10

6, Rem = VmLm ν

= (11.11)(10/20) 1.461× 10−5 = 3.80× 10

5

Since Rep > 10 5 and Rem > 10

5, viscous effects can be neglected in the model and prototype. The relevant functional relationship becomes

P 1 2ρV

3L2 = f2 (St)

Therefore, the model tests should be designed in accordance with Strouhal-number similarity.

(a) Strouhal number similarity requires that [ ωL V

]

p

=

[ ωL V

]

m

→ [ (8)(10) (40)

]

p

=

[ ω(10/20)

40

]

m

→ ωm = 160 rpm

Therefore, the rotor speed in the model should be set at 100 rpm.

(b) If Strouhal number similarity is enforced, then [

P 1 2ρV

3L2

]

p

=

[ P

1 2ρV

3L2

]

m

→ [

550 (40)3(10)2

]

p

=

[ P

(40)3(10/20)2

]

m

→ Pm = 1.38 kW

Therefore, the power output from the model is expected to be 1.38 kW.

Key Equations for Dimensional Analysis and Similitude

The following list of equations are particularly useful in the solution of problems related to dimen- sional analysis and similitude. If one is able to recognize these equations and recall their appropriate use, then the learning objectives of this chapter have been met to a significant degree. Derivations these equations, definitions of the variables, and detailed examples of usage can be found in the main text.

Modeling and Similitude

509

Dimensional analysis: Π0 = f(Π1,Π2, . . . ,ΠN )

Similitude: (Π0)p = (Π0)m

Reynolds and Mach similarity: νm νp

=

( cm cp

)( Lm Lp

)

Reynolds and Froude similarity: νm νp

=

( Lm Lp

) 3 2

Problems

Section 6.2: Dimensions in Equations

6.1. It is known from engineering analysis that the gage pressure, p, at a height z above the water surface is given by

p = −γ ( 1 + 0.24

Q2

gD5

) z (6.25)

where γ is the specific weight the water, Q is the volumetric flow rate (L3/T), and D is the diameter of the pipe. Show that Equation 6.25 is dimensionally homogeneous.

6.2. When a liquid of depth h flows down an inclined plane that makes an angle θ with the horizontal, the velocity u in the flow direction at a distance y from the bottom of the channel can be estimated by the relation

u(y) = ρg

µ sin θ

( hy − y

2

2

) (6.26)

where ρ and µ are the density and viscosity of the liquid, and g is the gravity constant. Determine whether Equation 6.26 is dimensionally homogeneous.

6.3. The displacement x [L] of a vibrating particle and a function of time t [T] can sometimes be described by the equation

d2x

dt2 + a

dx

dt + bx = c

What are the dimensions of a, b, and c required for this equation to be dimensionally homo- geneous?

6.4. If a heated object of mass m is placed in a liquid that is maintained at a temperature Tℓ, the temperature, T , of the object as a function of time, t, can be estimated using the relation

dT

dt =

h

mc (T − Tℓ)

where c is the specific heat of the object, and h is the heat transfer coefficient. It a typical application, the units of the variables are as follows: T (◦C), t (s), m (kg), c (kJ/kg·◦C), and Tℓ (◦C). What units should h be expressed in?

510

6.5. Consider the energy equation given in the form

p1 γ

+ V 21 2g

+ z1 = p2 γ

+ V 22 2g

+ z2 + hf (6.27)

State the SI unit of energy, and state the SI unit(s) of the terms (not the individual variables) in Equation 6.27. Explain why each of the terms in Equation 6.27 represents energy, and express Equation 6.27 as a nondimensional equation.

6.6. The Reynolds number, Re, of the flow at any cross-section of a closed conduit is defined by the relation

Re = ρV Dh

µ

where ρ is the density of the fluid, V is the average velocity, Dh is the hydraulic diameter, and µ is the kinematic viscosity. (a) Determine the dimensions of Re and explain how the dimensions of Re would change if U.S. Customary units were used instead of SI units? (b) Determine the Reynolds number at a section of pipe with a hydraulic diameter of 150 mm and where methane is flowing at a velocity of 30 m/s. The temperature of the methane is 20◦C and the pressure of the methane is standard sea-level pressure of 101 kPa.

6.7. The Froude number, Fr, at any cross-section of an open channel is defined by the relation

Fr = V√ gD

where V is the average velocity, g is the acceleration due to gravity, and Dh is the hydraulic depth. The hydraulic depth is defined as A/T , where A is the flow area, and T is to top width of the flow area. (a) Show that Fr is dimensionless. (b) Determine the Froude number in a trapezoidal channel of bottom width 2 m, side slopes 3:1 (H:V), and in which water is flowing with an average velocity of 0.3 m/s and a depth of 1.2 m.

6.8. One-dimensional flow in a horizontal conduit is illustrated in Figure 6.6, where the velocity profile is symmetric about the centerline of the conduit and varies with time. The differential equation describing the velocity profile, u(r, t), is given by

ρ ∂u

∂t = µ

( 1

r

∂u

∂r + ∂2u

∂r2

)

where ρ is the density of the fluid, t is time, µ is the viscosity of the fluid, and r is the distance from the conduit centerline. Consider the case where R is the radius of the conduit, and the velocity fluctuates such that the maximum velocity is U and the temporal frequency of the oscillation is ω. Using R, ω, and U , as reference variables, express the governing equation in nondimensional form.

511

R r

ConduitVelocity profile

u(r,t)

Figure 6.6: One-dimensional flow in a conduit

6.9. A two-dimensional flow field in the xy plane has velocity components u and v in the x and y directions, respectively, and the density, ρ, of the fluid varies in both space (x, y) and time, t. The continuity equation for this two-dimensional compressible flow is given by

∂ρ

∂t +

∂

∂x (ρu) +

∂

∂y (ρv) = 0 (6.28)

The velocity, length, time, and density scales in the flow field are V , L, T , and ρ0, respectively. It is common practice to represent the time scale, T , in terms of a frequency scale, ω, where ω = 2π/T ; in such cases the time scale becomes ω−1. (a) Express Equation 6.28 in normalized form. Use the Strouhal number, St, defined as St = ωL/V , in the final expression. (b) Give the asymptotic form of the normalized continuity equation as the Strouhal number becomes small, and state a physical scenario in which the Strouhal number might become negligibly small.

6.10. The momentum equation that describes the motion of a nonviscous incompressible fluid in a three-dimensional (xyz) flow field can be expressed in the form

ρ

[ ∂V

∂t + u

∂V

∂x + v

∂V

∂y + w

∂V

∂z

] = −∇p− ρgk (6.29)

where ρ is the density of the fluid, V is the velocity vector with components u, v, and w, p is the pressure, and g is the gravity constant. Consider the case in which the only relevant scales are the length scale, L, and the velocity scale, V . Note that time can be normalized by L/V and pressure can be normalized by ρV 2. Express Equation 6.29 in normalized form, using the Froude number, Fr, defined as Fr = V/

√ gL, in the final expression. What happens

to the effect of gravity on the flow as the Froude number becomes large?

6.11. In a particular two-dimensional flow field of an incompressible fluid in the xz plane, the z component of the momentum equation is given by

ρu ∂w

∂x = µ

( ∂2w

∂x2 + ∂2w

∂z2

) − ρg (6.30)

where u and w are the x and z components of the velocity, respectively, ρ and µ are the density and viscosity of the fluid, respectively, and g is the gravity constant. The relevant scales are the length scale, L, and the velocity scale, V . Express Equation 6.30 in normalized form, using the Reynolds number, Re, defined as Re = ρV L/µ, and the Froude number, Fr, defined as Fr = V/

√ gL, in the final expression. What is the asymptotic form of the governing

equation as the Reynolds number becomes large?

512

6.12. The temperature distribution, T (x, y), within a two-dimensional flow field of an incompress- ible and non-viscous fluid is governed by the following (energy) equation

ρcp

[ u ∂T

∂x + v

∂T

∂y

] = k

( ∂2T

∂x2 + ∂2T

∂y2

) (6.31)

where ρ and cp are the density and specific heat of the fluid, respectively, u and v are the velocity components in the x and y directions, respectively, and k is the thermal conductivity of the fluid. The relevant scales are the length scale, L, the velocity scale, V , and the temperature scale, T0. Express Equation 6.31 in normalized form, using the Prandtl number, Pr, defined as Pr = µcp/k, and the Reynolds number, Re, defined as Re = ρV L/µ, where µ is the dynamic viscosity of the fluid.

Section 6.3: Dimensional Analysis

6.13. The velocity distribution, V (r), in a pipe with fully developed laminar flow is given by

V = ∆p

4µL

( D2

4 − r2

)

where ∆p is the pressure drop over a distance L, µ is the dynamic viscosity, D is the diameter of the pipe, and r is the distance from the pipe centerline. Derive this functional relationship using the Buckingham pi theorem.

6.14. Under laminar flow conditions, the average velocity, V , in a pipe of diameter D depends on the pressure gradient in the direction of flow, ∂p/∂x, and the dynamic viscosity of the fluid. Use dimensional analysis to determine the functional relationship between V and the influencing variables.

6.15. The force F on a reducer is related to the flow conditions and pipe geometry by

F = p1A1 − p2A2 − ρA1V1(V2 − V1)

where p1 and p2 are the pressures upstream and downstream of the reducer, respectively; A1 and A2 are the corresponding flow areas; and V1 and V2 are the velocities. Derive this functional relationship using the Buckingham pi theorem.

6.16. When a liquid flows at a high velocity through a pipe, the sudden closure of a valve can cause the generation of pressure waves that can be of sufficient magnitude to cause damage to the pipe. This phenomenon is called water hammer. It is known that the maximum water hammer pressure, pmax, depends of the pipe diameter, D, the velocity of flow in the pipe prior to valve closure, V , the density of the fluid prior to valve closure, ρ, and the bulk modulus of elasticity of the liquid, Ev. Use dimensional analysis to determine the relationship between pmax and the influencing variables.

6.17. A pump impeller of diameter D contains a mass m of fluid and rotates at an angular velocity of ω. Use dimensional analysis to obtain a functional expression for the centrifugal force F on the fluid in terms of D, m, and ω.

513

6.18. The energy per unit mass, e, added by a pump of a given shape depends on the pump size, D, flow rate, Q, speed of the rotor, ω, density of the fluid, ρ, and dynamic viscosity of the fluid, µ. This functional relation can be stated as:

e = f(D,Q,ω, ρ, µ)

Express this as a relationship between dimensionless groups. What is gained by express- ing the pump performance as an empirical relationship between dimensionless groups versus expressing the pump performance as a relationship between the given dimensional variables?

6.19. The power required to drive a rotary hydraulic machine typically depends on the speed of rotation, the size of the rotor, and the density and viscosity of the working fluid. Determine the relationship between the power requirement and the influencing variables, expressed in terms on nondimensional groups. Identify any named conventional nondimensional groups.

6.20. An orifice plate is a flat plate with a central opening that is sometimes used to measure the flow rate in a pipe. The pressure drop across an orifice plate can be assumed to be a function of the diameter of the opening in the plate, the diameter of the pipe, the velocity of flow in the pipe, and the density and viscosity of the fluid. Use dimensional analysis to determine the functional relationship between the pressure drop across the plate and the influencing variables. Express this functional relationship in both a dimensional and nondimensional form. Identify any named conventional dimensionless groups that appear in the nondimensional relationship.

6.21. When freewheeling, the angular velocity, Ω, of a windmill is assumed to be a function of the windmill diameter, D, the wind velocity, V , the air density, ρ, air viscosity, µ, the windmill height, H, as compared to the atmospheric boundary layer, L, and the number of blades, N :

Ω = f

( D,V, ρ,

H

L ,N

)

If viscosity effects are negligible, find the appropriate pi groups and rewrite the function in dimensionless form. Identify any conventional non-dimensional groups.

6.22. The air flow in large wind tunnels are driven by large fans that exert a thrust force on their supporting structure. The thrust force, T , on a supporting structure can be expected to be a function of the density and viscosity of the air, the size of the fan, and that rate at which the fan rotates. Determine a non-dimensional functional expression that relates the thrust generated by a fan to the influencing variables.

6.23. A model differential equation for chemical reaction dynamics in a plug reactor is as follows:

u ∂C

∂x = D

∂2C

∂x2 − kC − ∂C

∂t

where u is the velocity, D is a diffusion coefficient, k is a reaction rate, x is distance along the reactor, and C is the (dimensionless) concentration for a given chemical in the reactor. Determine the appropriate dimensions of D and k. Using a characteristic length scale L, average velocity V , and background concentration C0 as parameters, rewrite this equation in dimensionless form and comment on any pi groups appearing.

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6.24. A fluid of density ρ and viscosity µ flows with a velocity V towards a rectangular plate of width W , height H, and thickness T . The fluid flow is in the plane normal to the plane of the rectangular plate and makes an angle θ with the normal to the plate. Determine the functional relationship between dimensionless groups that would be appropriate for studying the relationship between the drag force on the plate and the given independent variables.

6.25. For the flow condition shown in Figure 6.7, the velocity, u, at any distance y from a fixed bottom plate depends on the velocity V of the top plate, the spacing h between the plates, the viscosity µ of the fluid, and the pressure gradient dp/dx. (a) Use the Buckingham pi theorem to obtain a functional relationship between dimensionless groups. (b) A theoretical analysis postulates that the actual relationship between the variables in this problem is give by

u = V y

h +

1

2µ

dp

dx (y2 − hy)

Nondimensionalize this equation and show that the resulting non-dimensional equation vali- dates your analysis in Part (a).

h u

y

Sta!onary plate

Moving plate V

Pressure gradient = dp dx

Figure 6.7: Flow between parallel plates

6.26. A cable of length L and diameter D is strung tightly between two poles. A fluid of density ρ and viscosity µ flows at a velocity V past the cable producing a deflection δ. The modulus of elasticity of cable material is E, and the cable is sufficiently long that the geometry of the end poles do not affect the cable deflection. Determine the functional relationship between dimensionless groups that would be appropriate for studying the relationship between the cable deflection and the given independent variables.

6.27. Wind blowing at a steady velocity V towards vertical column of diameter, D, generates veloc- ity fluctuations behind the column that have a frequency ω. The frequency of the generated velocity fluctuations can also depend on the density and viscosity of the air. Determine a nondimensional functional relationship that relates the frequency of the velocity fluctuations to the influencing variables. Identify any named conventional dimensionless groups.

6.28. Standard air blows at 160 km/h past a 300-mm diameter flagpole during an intense hurricane. (a) Estimate the magnitude of the viscous force relative to the inertial force of the wind. (b) Estimate the magnitude of the gravity force relative to the inertial force of the wind. (c) Partially based on the results in Parts (a) and (b), estimate the relative importance of fluid viscosity and gravity in determining the force of the wind on the flagpole.

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6.29. Air flowing past an object typically exerts a drag force on the object. For given shape of the object, the drag force is commonly assumed to depend on the speed at which the air approaches the object, the size of the object, and the density and viscosity of the air. At sufficiently his velocities, the drag force asymptotically becomes independent of the viscosity of the fluid. Use dimensional analysis to express the asymptotic functional behavior of the drag force, and state the functional relationship between the drag force and the influencing variables under the asymptotic condition.

6.30. A sphere of diameter D that falls in a fluid ultimately attains a constant velocity when the drag force exerted by the fluid is equal to the net weight of the sphere in the fluid. The net weight of a sphere can be represented by ∆γV, where ∆γ is the difference between the specific weight of the sphere and the specific weight of the fluid, and V is the volume of the sphere. In cases where the sphere falls slowly in a viscous fluid, it is known that the drag force on the sphere depends only on the size of the sphere, D, the velocity of the sphere, V , and the viscosity of the fluid, µ. Use dimensional analysis to determine the functional relationship between the terminal velocity and the other relevant variables.

6.31. When an object of a given shape falls very slowly in a fluid, the steady-state velocity of the object depends only on the size of the object and the density and viscosity of the fluid. Use dimensional analysis to determine the functional relationship between the steady-state velocity and the influencing variables.

6.32. The force, F , on a satellite in the earth’s upper atmosphere depends on the mean path length of molecules, λ, the density, ρ, the diameter of the body, D, and the molecular speed, c. Express this functional relationship in terms of nondimensional groups.

6.33. Surface waves are generated by wind blowing over large water bodies. It is postulated that a wind of speed V blowing over a length of water, F , called a “fetch”, generates waves of height H in a body of water of depth h and density ρw. Other important variables in this relationship are expected to be the density of the air, ρair, and the acceleration due to gravity, g. Using h, ρw and V , as repeating variables, determine an appropriate relationship between nondimensional groups that could be used to concisely express the relationship between the variables.

6.34. The propagation speed, V , of small ripples in a shallow depth, h, of a liquid depends on the surface tension, σ, at the air-liquid interface, the density of the liquid, ρ, and the gravity constant, g. Determine the relationship between these variables in terms of dimensionless groups. If possible, also express this relationship in terms of named conventional dimensionless groups.

6.35. Observations on the International Space Station indicate that if a small spherical liquid droplet suspended in space is deformed slightly, then its surface will oscillate with a frequency that depends on the diameter of the droplet, and the density and viscosity of the liquid. Use dimensional analysis to estimate the functional relationship between the oscillation frequency of the droplet surface and the influencing variables.

6.36. A 8-mm deep film of SAE 30 oil flows steadily down an inclined plane at a velocity of 5 cm/s. (a) Estimate the magnitude of the viscous force relative to the inertial force. (b) Estimate

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the magnitude of the gravity force relative to the inertial force. (b) Estimate the magnitude of the surface-tension force relative to the inertial force. (d) Partially based on the results in Parts (a),(b), and (c), estimate the relative importance of fluid viscosity, gravity, and surface tension on the flow rate of the oil.

6.37. The height of rise, h, of a fluid in a capillary tube depends on the specific weight, γ, the surface tension, σ, of the fluid, and the diameter, D, of the capillary tube. In other words,

h = f(γ,σ, D)

Express this relationship in terms of dimensionless groups. How does your result compare with the theoretical expression

h = 4σ

γD

6.38. The characteristic diameter of the liquid droplets delivered into the air from a spray bottle depend on the diameter of the jet in the spray bottle, the velocity of the spray jet, and the density, viscosity, and surface tension of the liquid. Determine a functional relationship, in terms of nondimensional groups, that relates the characteristic diameter of the liquid droplets to the influencing variables. Identify and named conventional dimensionless groups involved in this relationship.

6.39. It is well known that tiny objects can be supported on the surface of a liquid by surface-tension forces. For an object of a particular shape, the weight of an object that can be supported is a function of the size of the object, the surface tension and density of the fluid, and the gravity constant. Determine a nondimensional functional relationship that relates the weight of the supported object to the influencing variables.

6.40. An orifice in the side of an open storage tank discharges the stored liquid at an average velocity of V when the depth of liquid above the orifice is h. The liquid has a density and viscosity of ρ and µ, respectively, and the acceleration due to gravity is g. (a) Show by dimensional reasoning that if viscous effects are negligible, then the density of the fluid does not influence the average discharge velocity. (b) If viscous effects are negligible, use dimensional analysis to determine that functional relationship between V and the influencing variables.

6.41. An orifice of diameter D in the side of an open storage tank discharges the liquid at an average velocity of V when the depth of liquid above the orifice is h. The liquid has a density and viscosity of ρ and µ, respectively, and the acceleration due to gravity is g. If experiments are to be done to determine the relationship between these variables, identify the nondimensional groups that should be used in analysing the experimental data.

6.42. The flow over a sharp-crested weir, Q, is commonly expressed in the form

Q = f

( We,Re,

H

Hw

) b √ gH

3 2 (6.32)

where We is the Weber number defined by

We = ρV 2H

σ

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and V is the flow velocity over the weir, σ is the surface tension of the fluid, ρ is the fluid density, H is the depth of fluid over the crest of the weir, and Re is the Reynolds number defined by

Re = ρV H

µ

where µ is the viscosity of the fluid, Hw is the height of the weir, and b is the length of the weir crest. Derive Equation 6.32 using dimensional analysis.

6.43. A hydraulic jump is a phenomenon that is associated with high-velocity flow of a liquid in an open channel in which the flow depth of the liquid suddenly changes from a lower depth, h1, to a higher depth depth, h2. When the flow is occurring in a horizontal rectangular channel, the relationship between h1 and h2 depends only on the velocity of flow, V1, where the flow depth is h1, and the gravity constant, g. Use dimensional analysis to determine the functional relationship between h2 and the influencing variables, expressed in terms of dimensionless groups. Identify any named conventional dimensionless groups that occur in this relationship.

6.44. A ship of length L moves at a cruising speed of V in a calm ocean where the density and viscosity of the seawater are ρ and µ, respectively. Synoptic experimental measurements of the drag force on the ship and the influencing variables are available. Determine the dimensionless groups that should be used to organize these data in order to determine an empirical relation between the drag force and the influencing variables. Why is it important to include gravity as a variable?

6.45. The thrust, T , generated by a ship’s propeller is generally thought to be a function of the diameter, D, forward speed, V , and rate of rotation, ω, of the propeller, the density, ρ, and viscosity, µ of the liquid in which the propeller is submerged, the pressure, p, at the level of the propeller, and gravity, g. Determine a nondimensional functional relationship between the thrust, T , and its influencing variables. Identify an named conventional dimensionless groups that appear in this relationship.

6.46. The lift force on an airfoil is known to depend on the forward speed of the airfoil, the angle that the airfoil makes with the forward direction, the length of the airfoil, the thickness of the airfoil, and the density of the air. Determine a nondimensional functional relationship that relates the normalized lift force to the influencing variables.

6.47. The drag force on a supersonic aircraft of a particular shape depends on its size, its speed, the density of the air, and the sonic speed in the air. Determine a nondimensional func- tional relationship between the drag force and its influencing variables. Identify any named conventional dimensionless groups.

6.48. The length, Lw, of the wake behind a particular model aircraft depends on the speed of the aircraft, the size of the aircraft, L, and the density and viscosity of the ambient air. Develop a nondimensional functional relationship that relates the size of the wake to its influencing variables.

6.49. The power, P , required to drive the propeller on a a turboprop aircraft is thought to be a function of the diameter, D, forward speed, V , and rate of rotation, ω, of the propeller,

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the density, ρ, and viscosity, µ of the air, and the sonic speed, c, in the air. Determine a nondimensional functional relationship between the power, P , and its influencing variables. Identify an named conventional dimensionless groups that appear in this relationship.

Section 6.4: Dimensionless Groups as Force Ratios

6.50. Show that the ratio of the inertial force to the pressure force is measured by the Euler number.

6.51. Show that the ratio of the inertial force to the surface tension force is measured by the Weber number.

6.52. Show that the ratio of the inertial force to the compressibility force is measured by the Cauchy number.

6.53. An “atomizer” is a common term that describes a spray nozzle that produces small droplets of a liquid that is drawn into the nozzle by the low pressure that exists there. Consider such a nozzle of diameter D that generates droplets of diameter d when the velocity at the nozzle exit is V . The relevant liquid properties are the density, ρ, the viscosity, µ, and the surface tension, σ. With the objective of predicting the size of the droplets generated by an atomizer, express the relationship between the relevant variables in nondimensional form where, to the extent possible, dimensionless groups representing force ratios are used with each of the fluid properties.

Dimensional analysis of laboratory data:

6.54. The performance of the expansion shown in Figure 6.8 was studied in the laboratory to deter- mine the relationship between the pressure rise, ∆p, across the expansion and the diameter ratio D1/D2, where D1 and D2 are the upstream and downstream diameters, respectively. (a) Assuming that the flow is fully turbulent, perform a dimensional analysis to determine the functional relationship between nondimensional groups to be investigated in the laboratory. (b) A laboratory investigation of the expansion is performed using a fixed upstream diameter of 150 mm and a constant flow rate of 20 L/s. The downstream diameter was varied and the corresponding increases in pressure between the upstream and downstream sections were measured. The results are as follows:

D2 (mm) 750 375 250 188

∆p (Pa) 81.8 174.6 275.2 326.5

Viscous effects were negligible for all flow conditions. Plot the measured data in nondimen- sional form, and derive an empirical relationship the can be used to estimate the change in pressure for a given diameter ratio.

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p 1

D 2

D 1

Flow

p 2

V 1

Figure 6.8: Performance of an expansion

6.55. The performance of the sudden contraction shown in Figure 6.9 was studied in the laboratory. The goal of the study was to determine the head-loss coefficient, K, of the contraction, where the head loss, hℓ, in the contraction can be expressed as

hℓ = K V 22 2g

where V2 is the velocity in the section downstream of the contraction. The laboratory study was conducted by fixing the diameter, D2, of the downstream section at 150 mm, using water at 20◦C as the fluid, and fixing the flow rate at 42.41 L/s. The diameter, D1, of the upstream section was varied between experiments, and the corresponding pressure differences, p1 − p2, between the upstream and downstream sections were measured. The measured laboratory data were as follows:

D2 p1 − p2 D2 p1 − p2 (mm) (Pa) (mm) (Pa)

150 0 330 1630 165 796 375 1651 180 1287 450 1631 210 1637 600 1569 240 1688 750 1547 270 1652 1500 1523 300 1660 − −

(a) Perform a dimensional analysis on the variables and present the laboratory results in an appropriate nondimensional form that relates the pressure change to the characteristics of the contraction, fluid, and flow. Plot the laboratory results in dimensionless form. (b) Use the results of the laboratory study to plot the head loss coefficient, K, as a function of the diameter ratio D1/D2. Estimate an empirical equation that relates K to D1/D2.

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D 2

D 1

p 1

p 2

Flow V 2

Figure 6.9: Performance of sudden contraction

Section 6.6: Modeling and Similitude

6.56. A rectangular duct is to be used to transport a very viscous fluid and you have been com- missioned to perform laboratory experiments to determine the shear stress on the walls of the duct. Identify the relevant variables, and use dimensional analysis to express a functional relationship that could be used as a basis for designing your experiments. Explain the benefit of performing a dimensional analysis and identify any conventional non-dimensional groups that occur in your functional relationship. For a given fluid, what equation would be used to relate the fluid velocity in the model to the fluid velocity in the prototype in order to achieve dynamic similarity between the model and the prototype?

6.57. Show that if viscous forces are dominant in a prototype, Reynolds-number similarity is used in a model study, and the same fluid is used in the model and prototype, then the force on a particular component of the model will always be equal to the force on the corresponding component of the prototype.

6.58. The shear stress, τ0, exerted on a pipeline of diameter, D, by a fluid of density, ρ, and viscosity, µ, moving with a velocity, V , is given by the following dimensionless relation

τ0 ρV 2

= f

( ρV D

µ , ϵ

D

)

where ϵ is the height of the roughness elements on the boundary of the pipe. In a prototype pipeline to be modeled, the velocity of flow is 2 m/s, the diameter of the pipeline is 3 m, and the height of the roughness elements are 2 mm. If a model of the pipeline is to be constructed based on Reynolds number similarity (since viscous force is important), a model scale of 1:20, and the same fluid is used in both the model and prototype, what velocity and roughness height should be used in the model? If the shear stress on the pipe boundary in the model is measured at 2.25 kPa, then what is the shear stress in the prototype?

6.59. The performance of a valve used to regulate the flow of water in a pipeline is to be tested using a 13 -scale model. At the prototype scale, the characteristic dimension of the valve is 200 mm and the design flow rate through the valve is 300 L/s. What scaling law should be used to design the model test? What is the model flow rate corresponding to the design flow rate in the prototype?

6.60. Kerosene at 20◦C flows through a section of 150-mm diameter pipeline with a velocity of 3 m/s. The pressure loss in the section of pipeline is to be studied using a scale model that

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has a diameter of 25 mm, and water at 20◦C is to be used in the model study. What water- velocity should be used in the model? If a pressure loss of 10 kPa is measured in the model, what would be the corresponding pressure drop in the actual pipeline section?

6.61. Water flows out of a fire hose at 30 L/s, where the hose has a diameter of 15 cm and the nozzle at the end of the hose has an exit diameter of 7 cm. (a) What force must a fireman exert to hold the nozzle in place? (b) If a scale model of the nozzle is constructed to have the same Froude number, with a model flow rate of 0.01 L/s, what must be the dimension of the model nozzle? What restraining force would you expect in the model nozzle?

6.62. Water with a kinematic viscosity of 10−6 m2/s flows through a 10 cm pipe. What would the velocity of water have to be for the water flow to be dynamically similar to oil (ν = 10−5

m2/s) flowing through the same pipe at a velocity of 2 m/s? What is meant by “dynamic similarity”?

6.63. The speed of sound, c, in a fluid that is contained within a conduit is given by

c =

√√√√√ Ev/ρ

1 +

( EvD

eE

) (6.33)

where Ev is the bulk modulus of elasticity of the fluid, ρ is the density of the fluid, D is the diameter of the conduit, e is the thickness of the conduit wall, and E is the modulus of elasticity of the pipe-wall material. (a) Express Equation 6.33 as a relationship between nondimensional groups. (b) Perform a dimensional analysis on the variables in Equation 6.33 to determine a functional relationship between nondimensional groups. Identify any commonly used dimensionless groups. (c) Discuss whether your result in Part (b) is confirmed by your result in Part (a). (d) A model of a steel pipe is to be constructed so as to study the propagation of a pressure wave. The scale model is to have a diameter that is 1/20 the diameter of the prototype, and the model will use benzene at 20◦C instead of water at 20◦C in the prototype. If the pipe-wall thickness in the prototype is 20 mm, what pipe-wall thickness would you use in the model to yield dynamic similarity between pressure-wave propagation in the model and in the prototype?

6.64. A performance of a prototype water pump is to be tested using a scale model. The prototype will have an impeller of diameter 300 mm, and when delivering 1000 L/s the difference in water pressure between the inlet and the outlet of the pump is expected to be 340 kPa. The impeller diameter in the model is to be 60 mm, and the model tests are to use water at the same temperature as used in the prototype. (a) What dimensionless functional relationship would describe the performance of the pump? Express this relationship in terms of conventional dimensionless groups. (b) What flow rate should be used in the model in order to achieve dynamic similarity? (c) What would be the expected pressure rise between the inlet and the outlet in the model?

6.65. A prototype pump increases the pressure by 500 kPa when the mass flow rate through the pump is 600 kg/s. The performance of this pump is to be studied using a 1:8 scale model. (a) State the nondimensional relationship between the relevant variables that forms the basis

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of the model study. (b) If water at the same temperature is used in both the prototype and the model, what mass flow rate should be used in the model? (c) What pressure is expected to be added by the model pump?

6.66. A prototype water pump with an impeller diameter of 500 mm is designed to operate at a rotational speed of 900 rpm, and at this speed the pump delivers a volumetric flow rate of 2 m3/s. The temperature of the water in the prototype pump is 20◦C. The performance of a 1:5 scale model of the pump is tested using standard air as the fluid. The model pump has a rotational speed of 1800 rpm, and power required to drive the model pump is 100 W. The Reynolds number in the prototype and the model are both sufficiently high that Reynolds similarity is not a requirement. (a) Determine the volumetric flow rate of air in model that corresponds to the design volumetric flow rate of water in the prototype. (b) Determine the power requirement of the prototype that corresponds to the measured power requirement in the model.

6.67. A manufacturer makes several fans in different sizes, but all have the same shape (i.e, they are geometrically similar). Such a series of models is called a homologous series. Two fans in the series are to be operated under dynamically similar conditions. The first fan has a diameter of 250 mm, a rotational speed of 2400 rpm, and produces an air-flow rate of 0.8 m3/s. The second fan is to have a rotational speed of 1600 rpm, and produce a flow rate of 4 m3/s. What diameter should be selected for the second fan?

6.68. A prototype turbine is to be designed to deliver a power of 300 kW when the flow rate through the turbine is 2 m3/s. The performance of this turbine is to be studied using a 1:10 scale model. (a) State the nondimensional relationship between the relevant variables that forms the basis of this investigation, and identify the scaling relationship to be used in the model study. (b) If water at the same temperature is used in both the prototype and the model, what flow rate should be used in the model? (c) What power is expected to be produced by the model?

6.69. A new boomerang is being proposed that has a characteristic size of 200 mm, travels at an average velocity of 20 m/s, and rotates at 170 rpm. The performance of a model of this boomerang is to be tested in a wind tunnel, where the model is to be three times the size of the prototype and Reynolds number and Strouhal similarity laws are to be used in designing the experiments. What wind speed and boomerang-rotation rates should be used in the laboratory tests?

6.70. In open-channel flow in a microchannel, both gravity and surface tension forces are important. A 120 -scale model is to be constructed to study microchannel flow, where the fluid used in the model is to have the same surface tension as the fluid in the prototype. What similarity laws should be used in designing the model study? What is the ratio of the density of the fluid in the model to the density of the fluid in the prototype in order to achieve dynamic similarity between the model and the prototype?

6.71. A 1:15 model of a spillway is tested under a particular upstream condition. The measured velocity and flow rate over the model spillway are 0.8 m/s and 0.1 m3/s, respectively. What are the corresponding velocity and flow rate in the actual spillway?

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6.72. A hydraulic model is to be constructed to study the performance of a spillway. The prototype spillway has a length of 30 m and has a design overflow rate of 150 m3/s. The model scale is to be 1:20. (a) What scaling law should be used in designing the model study? (b) What flow rate should be used in the model? (c) If the force on a component of the spillway is measured in the model as 100 N, what is the force on the corresponding component in the prototype spillway?

6.73. A sluice gate is to be installed to control the flow in a coastal canal. The flow through the gate, Q, depends on the upstream water depth, h1, the downstream water depth, h2, the gate opening, s, the width of the gate, b, and the acceleration due to gravity, g. Use dimensional analysis to determine a functional relationship between nondimensional groups that can be used to guide laboratory model experiments on the gate. Identify the physical significance of each nondimensional group. A 17 scale model having a width of 1 m is constructed in the laboratory, and the following operating condition is observed in the model:

h1 h2 s Q (m) (m) (m) (m3/s)

0.57 0.50 0.16 0.14

Find the corresponding operating condition in the prototype. How would you justify neglect- ing the viscosity of the water in your analysis?

6.74. Water flows at 30 m3/s in a 14-m wide rectangular channel at a depth of 2 meters. A 1:20 scale model of the channel is to be built in the laboratory. (a) What flow rate should be used in the model to achieve Reynolds similarity? (b) What flow rate should be used in the model to achieve Froude similarity? (c) Find the viscosity of the model fluid required to achieve both Reynolds and Froude similarity. Does such a fluid exist? (d) Explain the circumstances under which a Froude model would be preferable to a Reynolds model; and (e) A force of 1.5 N is measured on a hydraulic structure placed in a Froude model. What is the corresponding force in the prototype?

6.75. A simple flow measurement device for streams and channels is a notch, of angle α, cut into the side of a dam as shown in Figure 6.10. The volume flow rate, Q, depends only on α, the acceleration of gravity, g, the height, δ, of the upstream water surface above the notch vertex. Tests of a model notch, of angle α = 55◦, yield the following flow rate data:

δ (cm) 10 20 30 40 Q (m3/h) 8 47 126 263

(a) Find a dimensionless correlation for the data. (b) Use the model data to predict the flow rate of a prototype notch, also of angle α = 55◦, when the upstream height δ is 3.2 m.

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α δ

Notch

Figure 6.10: Flow over a V-notch weir

6.76. Consider the case where a particle of diameter d and density ρs rests on the bottom of a open channel where a liquid with density ρ and viscosity µ flows over the particle. The particle begins to move when the fluid velocity is Vc, and, in this analysis, gravity, g, is a relevant variable. (a) Determine a nondimensional grouping of the given variables that measures the ratio of the fluid force tending to move the particle to the weight of the particle. Take into account the fact that the particle does not weigh the same when submerged in a liquid as it does when not submerged. (b) Using this result, perform a dimensional analysis on the relevant variables to determine a functional relationship between nondimensional groups that can be used to analyze measurements in a scale model to measure the critical fluid velocity, Vc, to move a particle of a given size, d. In your dimensional analysis, incorporate the fact that ρs and g can be combined into the single variable: (ρs− ρ)g. (c) If a scale model is to be built that is twice the size of the actual system, what dimensionless group(s) would you make the same in the model and the prototype? If model measurements using a 3 mm diameter particle show a critical velocity of 25 cm/s, what is the corresponding critical velocity and particle-size in the prototype?

6.77. The performance of a low-flying aircraft at a speed of 150 km/h is to be investigated using a model in a wind tunnel. Standard air is representative flying conditions, and the wind tunnel will also use standard air. If the maximum air speed that can be achieved in the wind tunnel is 100 m/s, what scale ratio corresponds to the smallest size model aircraft that can be used in the study?

6.78. A supersonic aircraft is has a design speed of 1500 km/h when operating at an elevation of 14 km. If the aircraft were to operate at an elevation of 10 km, what speed would cause the same compressibility effects as when the aircraft is operating at the higher altitude?

6.79. A light aircraft cruises at 400 km/h at an elevation of 4 km and the performance of this aircraft is to be studied in a wind tunnel using a 1:15 scale model. If standard air is to be used in the wind tunnel tests, what air speed should be used in the model study? Comment on the practicality of using the calculated air speed in the model.

6.80. A small plane is designed to cruise at 360 km/h at an elevation of 1 km, where the typical atmospheric pressure is 90 kPa. The drag characteristics of the aircraft are to be studied in a pressurized wind tunnel, where the air in the wind tunnel is at the same temperature as the air in the prototype. The model is to be at 1:8 scale and the air speed in the wind tunnel is to be 300 km/h. (a) What scaling law should be used in designing the model study? (b)

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What pressure should be used in the wind tunnel? (c) If the drag on a component of the model aircraft is measured as 10 N, what is the drag on the corresponding component in the prototype?

6.81. A model of a new airplane is tested in a wind tunnel. Measurements taken during the model tests indicate that when the approach air speed is 950 km/h the stagnation pressure on the nose of the airplane is 50 kPa higher than the free-stream pressure. The temperature of the air in the wind tunnel is 25◦C. The prototype airplane is designed to cruise at an altitude of 10.6 km. What are the air speed and the relative stagnation pressure in the prototype that correspond to the model scenario?

6.82. Model aircraft are sometimes tested in a water tunnel instead of a wind tunnel. For a 1:5 scale model of an aircraft, estimate the ratio of the required air speed in a wind tunnel to the required water speed in a water tunnel. For the wind-tunnel tests assume standard air, and for the water-tunnel tests assume water at 20◦C.

6.83. The performance of an aircraft is to be tested using a scale model. The condition of interest in the prototype is at a speed of 200 km/h. Ambient conditions in the prototype can be taken as standard air at sea level. A 1:10 scale model is to be used in the study. (a) If the performance of the aircraft is to be tested in a wind tunnel, determine the required air speed in the wind tunnel, which will also use standard air at sea level. Comment on the practicality of using the required speed. (b) If the performance of the aircraft is tested in a water tunnel, determine the required speed of the water. Assume water at 20◦C and comment on the practicality of the required speed.

6.84. The lift force generated by a new wing (airfoil) design is to be tested in a pressurized wind tunnel. Under design conditions, the aircraft will have a speed of 180 km/h and operate at an elevation where the temperature is 10◦C and the absolute pressure is 92.5 kPa. The model airfoil will have a scale of 1:8, and the temperature of the air in the wind tunnel will be maintained at 10◦C. Determine the air speed and pressure that should be maintained in the wind tunnel in order to achieve dynamic similarity and minimize compressibility effects.

6.85. The performance of a new blimp is to be tested using a scale model in a wind tunnel. The full-scale blimp is designed to move at 54 km/h in standard air. If a 1:12 model is used in a wind tunnel with air at 15◦C and an air speed of 80 m/s, what must be the air pressure in the wind tunnel to achieve dynamic similarity? If dynamic similarity is achieved, and a drag force of 200 N is measured in the model, what is the corresponding drag force in the prototype?

6.86. When a fluid flows around a cylinder, under some circumstances transient oscillating vortices are formed behind the cylinder as illustrated in Figure 6.11. The path of these vortices is called the Von Kármán vortex street. Consider the case in which a fluid of density ρ and viscosity µ approaches a cylinder of diameter D and generates vortices with a frequency ω. A model is to be constructed to study this phenomenon in the laboratory. (a) What scaling law should be used to design the model? (b) In a prototype, standard air flows past a cylinder of diameter 300 mm at a velocity of 70 km/h. If the model cylinder is to have a diameter of 50 mm, and the model test is to be conducted in a water tunnel using water at 20◦C,

526

what approach velocity should be used in the model? (c) If a shedding frequency of 32 Hz is observed in the model, what shedding frequency is expected in the prototype?

Cylinder Vortex street Streamline

DV

Figure 6.11: Vortex street behind cylinder

6.87. A 1:60 scale model of a ship is used in a water tank to simulate a ship speed of 10 m/s. What should be the model speed? If a towing force of 10 N is measured in the model, what force is expected on the prototype? Neglect viscous effects.

6.88. A 1:50 scale model of a ship is to be tested to determine the wave drag. The geometry and surface properties of the ship are such that viscous drag is negligible. In the model test, the model is moved at a velocity of 2 m/s and the measure drag on the model is 20 N. Water at the same temperature is used in both the model and the prototype. What are the corresponding velocity and drag force in the prototype?

6.89. The performance of a ship is to be tested using a 1:50 scale model. In this particular ship, under normal operating conditions both gravity and viscous effects have a significant effect on the drag. Estimate the dynamic viscosity of the fluid that must be used in the model test to achieve dynamic similarity with the prototype. Assume seawater at 20◦C for the prototype.

6.90. A boat of length L is designed to move through water at a velocity V while generating bow waves of height H. To study the range of wave heights that will be generated by the boat, a scale model of the boat is constructed and tested in a hydraulics laboratory. Viscous effects might be important at the laboratory scale. (a) What nondimensional relationship will you use to design your laboratory study? (b) How will you use this relationship? (c) If your model scale is 1:10, the model length is 60 cm, and you measure a wave height of 5 cm when the model boat is moving at 30 cm/s, what are the corresponding conditions in the full-scale model? (d) How does the ratio of wave height to boat length in the model compare with the corresponding ratio in the prototype?

6.91. A prototype ship is 35 m long and designed to cruise at 11 m/s (about 21 knots). Its drag is to be simulated by a 1-m long model pulled in a tow tank. For Froude scaling, determine the tow speed, the ratio of prototype to model drag, and the ratio of prototype to model power.

6.92. The performance of a new hydrofoil is tested in a laboratory open (water) channel using a 1:15 scale model. The geometry of the hydrofoil is sufficiently aerodynamic that viscous effects have a negligible effect on the flow around the hydrofoil. Laboratory measurements show that when the speed of the model hydrofoil is 10 m/s, then the force on a critical component of the model hydrofoil is 10 N. The water properties in the model and prototype can be assumed to be approximately the same. Determine the speed and force in the prototype hydrofoil corresponding to the model measurements.

527

6.93. A 1:15 scale model of a hydrofoil is to be tested using Froude number similarity. Under design conditions, the prototype hydrofoil will operate in freshwater at 10◦C, the free-stream pressure at the hydrofoil level is 150 kPa, and the hydrofoil has a cruising speed of 100 km/h. In order to duplicate the prototype cavitation conditions in the model test, the cavitation number must be the same in both the model and the prototype. To facilitate achieving complete similarity, the temperature of the water used in the model tests is 50◦C. Determine the free- stream pressure at the level of the model hydrofoil that is required to achieve similarity in cavitation conditions.

6.94. A particular submarine is capable of cruising at 20 km/h at both the surface of the ocean and far below the surface of the ocean. Tests of both of these conditions are to be performed using a 1:40 scale model of the submarine. Determine the required model speed and the ratio of prototype drag to model drag for the following model conditions: (a) a model of cruising on the surface, and (b) a model of cruising far below the surface. Assume that the prototype ocean-water properties can be duplicated in the model tests.

6.95. The performance of a new submarine propeller is to be tested using a 1:15 scale model. Under design conditions, the prototype propeller is expected to rotate at 1500 rpm. When the submarine is near the surface, gravitational effects are much more important that viscous effects, and when the submarine is far below the surface, viscous effects are much more important than gravity effects. Determine the theoretical model rotational speed that should be used to simulate each of these conditions. Comment on the practicality of the model rotational speeds.

6.96. A particular torpedo is designed to travel underwater at a velocity of 50 km/h, and the performance of this torpedo is to be studied using a 1:8 scale model. Options suggested for the model tests are to use standard air in a wind tunnel, or to use water at 20◦C in a water tunnel. Determine the required test speed for each option and suggest what option is more feasible.

6.97. The performance of a torpedo is to be tested using a scale model in a wind tunnel. The full-scale torpedo is designed to move at 30 m/s in water at 20◦C. If a 1:4 model is used in a wind tunnel with air at 20◦C and an air speed of 110 m/s, what must be the air pressure in the wind tunnel to achieve dynamic similarity? If dynamic similarity is achieved, and a drag force of 600 N is measured in the model, what is the corresponding drag force in the prototype?

6.98. The aerodynamic drag on a car is to be predicted at a speed of 60.0 mi/h at an air temperature of 25◦C. Automotive engineers build a one-fourth scale model of the car to test in a wind tunnel. Determine the required air speed in the wind tunnel. If the aerodynamic drag on the model is measured as 36.5 lbf, estimate the drag at full-scale.

6.99. A small wind tunnel has a test section that is 20 in. × 20 in. in cross section, is 4.0 ft long, and can sustain an air speed of 160 ft/s. A model of an 18-wheeler truck is to be built to study the aerodynamic drag on the truck. A full-size tractor-trailer truck is 52 ft long, 8.33 ft wide, and 12 ft high. Both the air in the wind tunnel and the air flowing over the prototype is at 80◦F and at atmospheric pressure. What is the largest scale model that can be accommodated

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in the wind tunnel and still stay within the rule of thumb guidelines for blockage? What are the dimensions of the model truck in inches? What is the maximum model truck Reynolds number that can be simulated in the wind tunnel. Can Reynolds number independence be achieved? What is meant by Reynolds number independence?

6.100. The drag force on a truck at a speed of 88 km/h is to be measured in a wind tunnel using a 1:10 scale model. Standard air is to be used as a basis for assessment in both the prototype and the model. (a) What air speed should be used in the model tests? (b) Determine whether compressibility will be a complicating factor in the model tests.

6.101. A prototype vehicle has a design speed of 88 km/h at an ambient air temperature of 20◦C. The drag on this vehicle is to be investigated using a 1:8 scale model in a wind tunnel. The temperature of the air in the wind tunnel is 5◦C. Both the model and prototype air are at the same pressure. (a) Estimate the air speed at which the model tests should be run in the wind tunnel. (b) Determine whether compressibility will preclude dynamic similarity at the wind speed calculated in Part (a).

Chapter 7

Flow in Closed Conduits

7.1 Introduction

A flow in which a fluid completely fills a conduit is classified as a flow in a closed conduit. Gasses will generally fill the conduit in which they are being transported. In contrast to gasses, liquids only undergo closed-conduit flow when there is no free surface, in other words, only when the liquid completely fills the conduit. The cross sections of closed conduits can be of any shape or size, and the conduits can be made of any solid material. For the most part, the same general equations can be used to describe the flow of liquids and gasses in closed conduits, with the only difference being the fluid properties. In cases where the closed conduit has a circular cross section, the conduit is commonly referred to as a pipe, and in cases where the cross section is not circular the conduit is commonly referred to as a duct, particularly when the fluid being transported is a gas. Small-diameter pipes are commonly referred to as tubes.

Laminar versus turbulent flow. Flows in closed conduits can be laminar or turbulent, with each type of flow having sufficient distinguishing features that it is worthwhile to consider them separately. The type of flow (laminar or turbulent) is usually determined by the Reynolds number, Re, which, for pipes of circular cross section, is generally defined by

Re = V D

ν (7.1)

where V is the average flow velocity, D is the pipe diameter, and ν is the kinematic viscosity of the fluid being transported by the pipe. Laminar flows in pipes usually occur for Re < 2000 and turbulent flows usually occur for Re > 4000, with either laminar or turbulent flows usually occurring in the range 2000 < Re < 4000. It is also common to cite the range of laminar flow as Re < 2300 instead of Re < 2000, with the former condition more widely used in the United States and the latter condition more widely used in the United Kingdom. In order to accommodate both conditions, it is safe to assume that laminar flow generally occurs for Re < 2000. In exceptional and highly controlled cases, laminar flows have been maintained up to Re = 100,000. Turbulent flows commonly occur is applications involving the flow of water in large-diameter pipes, such as in water-distribution networks and household plumbing systems. Laminar flows commonly occur in applications involving the transport of viscous fluids in small-diameter pipes, such as in hydraulic and lubrication systems.

529

587

Hydro-Pneumatic Systems. Hydro-pneumatic systems use pressurized air and stored water contained in the same tank. A pump supplies water to the tank, with the water being contained in an expandable bladder surrounded by compressed air. As the volume of water stored in the bladder changes, the volume of air in the pressurized tank adjusts, thereby maintaining the water pressure in the pipes that are connected to the bladder.

Key Equations for Flow in Closed Conduits

The following list of equations are particularly useful in the solution of problems related to flow in closed conduits. If one is able to recognize these equations and recall their appropriate use, then the learning objectives of this chapter have been met to a significant degree. Derivations these equations, definitions of the variables, and detailed examples of usage can be found in the main text.

Steady Incompressible Flow

Steady-state energy equation:

( p1 γ

+ v21 2g

+ z1

) − ( p2 γ

+ v22 2g

+ z2

) = hℓ (= hf)

Energy equation, constant diameter:

( p1 γ

+ z1

) − ( p2 γ

+ z2

) = hf

Head loss due to friction, momentum: hf = 2τL

rγ =

4τ0L

Dγ = constant

Shear stress distribution: τ = τ0 r

D

Dimensional analysis: τ0

1 8ρV

2 = f

( Re,

ϵ

D

) ,

τ0 1 2ρV

2 = fB

( Re,

ϵ

D

)

Darcy-Weisbach equation: hf = fL

D

V 2

2g

Pressure gradient: dp

dx = −γ

( hf L

+ sin θ

)

Entrance length: Le D

=

⎧ ⎨

⎩ 0.058Re (laminar flow)

4.4Re 1 6 (turbulent flow)

Hydraulic radius: Rh = A

P

Hydraulic diameter: Dh = 4 A

P

Head loss, noncircular conduits: hf = τ0L

γRh , hf =

4τ0L

γDh , hf =

fL

Dh

V 2

2g

Friction Effects in Laminar Flow

588

Velocity distribution: u(r) = hfγ

16µL (D2 − 4r2), u(r) = 2V

( 1− r

2

R2

)

Centerline velocity: Vc = hfγ

16µL D2

Average velocity: V = hfγ

32µL D2

Volumetric flow rate: Q = πhfγ

128µL D4

Friction factor, laminar flow: f = 64

Re

Friction Effects in Turbulent Flow

Darcy-Weisbach equation: hf = fL

D

V 2

2g

Thickness of viscous sublayer: δv = 5 ν

u∗

Friction factor, turbulent flow: 1√ f =

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

−2 log (

2.51

Re √ f

) , for smooth pipe

−2 log ( ks/D

3.7

) , for rough pipe

1√ f = −2 log

( ks/D

3.7 +

2.51

Re √ f

) (Colebrook equation)

f = 0.25

[ log

( ks/D

3.7 +

5.74

Re0.9

)]2 (Swamee-Jain equation)

Limit of fully turbulent regime: 1√ f =

Re

200(D/ks)

Velocity distribution: v(r) =

[ (1 + 1.326

√ f)− 2.04

√ f log

( R

R− r

)] V

v(r) = V0 ( 1− r

R

) 1 n

Energy correction factor: α = 1 + 2.7f

Momentum correction factor: β = 1 + 0.98f

Practical Applications

Flow rate (Darcy-Weisbach): Q = −0.965D2 √

gDhf L

ln

( ks/D

3.7 +

1.784ν

D √ gDhf/L

)

589

Pipe diameter: D = 0.66

[ k1.25s

( LQ2

ghf

)4.75 + νQ9.4

( L

ghf

)5.2]0.04

Hazen-Williams formula: V = 0.849CHR 0.63S0.54f

Head loss, Hazen-Williams formula: hf = 6.82 L

D1.17

( V

CH

)1.85

Manning formula: V = 1

n R

2 3S

1 2 f

Head loss (Manning formula): hf = 6.35 n2LV 2

D 4 3

Local head loss: hlocal = K V 2

2g

Head-loss coefficient (general): K = − ∆p1 2ρV

2

Head-loss coefficient (sudden contraction): Ksc =

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

0.42

( 1− D

2 2

D21

) , for D2/D1 ≤ 0.76

( 1− D

2 2

D21

)2 , for D2/D1 > 0.76

Head-loss coefficient (sudden expansion): Kse =

( 1− D

2 1

D22

)2

Head-loss coefficient (gradual expansion): Kge = 2.61 sin

( θ

2

)( 1− D

2 1

D22

)2 +

fgeLge Dge

, for θ ≤ 45◦

Equivalent length: Leq = K

f D

Energy equation: E0 − NP∑

i=1

fiLi Di

V 2i 2g

− NL∑

j=1

Kj V 2j 2g

+ hp − ht = E1

Water Hammer

Water-hammer pressure: ∆p = ρ0cV

Speed of pressure wave: c =

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

√ Ev ρ0 , (rigid walls)

√ Ev/ρ0

1 + (EvD/eEp) , (flexible walls)

Critical valve closure time: tc = 2L

c

Pipe Networks

590

Continuity equation:

NP(j)∑

i=1

Qij − Fj = 0, j = 1, . . . ,NJ

Energy equation: h2 = h1 − ( fL

D + ∑

Km

) Q|Q| 2gA2

+ Q

|Q|hp (nodal method)

NP(i)∑

j=1

(hL,ij − hp,ij) = 0, i = 1, . . . ,NL (loop method)

Head-loss parameters: r = fL

2gA2D , n = 2

Hardy Cross equation: ∆Qi = −

∑NP(i) j=1

rijQj |Qj |n−1

∑NP(i) j=1

nrij |Qj |n−1

Building Water-Supply Systems

Energy equation: p0 γ

− [ p1 γ

+ V 21 2g

+ ∆z

] = ∑

hf + ∑

hlocal

Head loss: hf =

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

0.230

( fL

D5

) Q2, Darcy–Weisbach equation

85.4

( L

C1.85H D 4.87

) Q1.85, Hazen–Williams equation

Problems

Section 7.1: Introduction

7.1. (a) If water at 20◦C flows with a velocity of 2 m/s in a pipeline, what is the maximum diameter for which the flow is likely to be laminar, and what is the minimum diameter for which the flow is likely to be turbulent? (b) If standard air flows with a velocity of 2 m/s in a pipeline, what is the maximum diameter for which the flow is likely to be laminar, and what is the minimum diameter for which the flow is likely to be turbulent? (c) What is the maximum air velocity for which the flow is likely to be laminar in a 1-mm diameter pipeline? (c) What is the minimum air velocity for which the flow is likely to be turbulent in a 1-mm pipeline conduit?

7.2. Air at 80◦C and standard pressure is to be carried in a pipeline with a weight-flow-rate of 0.5 N/s. For what range of pipeline diameters will the flow be laminar?

7.3. Methane at 10◦C and 600 kPa (absolute) is to be transported via pipeline at a flow rate of 0.1 N/s. (a) Calculate the density of methane under these conditions and perform an online

591

search to determine the dynamic viscosity of methane at the given temperature and pressure. (b) For what range of pipe diameters will the flow be turbulent?

7.4. Carbon dioxide at 10◦C and 300 kPa (absolute) is to be transported via a 50-mm diameter pipeline. (a) Calculate the density of carbon dioxide under these conditions and perform an online search to determine the dynamic viscosity of carbon dioxide at the given temperature and pressure. (b) For what range of flow rates will the flow be laminar?

Section 7.2: Steady Incompressible Flow

7.5. A fluid flows at a rate of 6 L/min in a straight 25-mm diameter pipe. The fluid has a density of 918 kg/m3, a viscosity of 440 mPa·s, and pressure measurements at two locations 10 m apart show an upstream pressure of 400 kPa and a downstream pressure of 340 kPa. The downstream section is 1.70 m higher than the upstream section. (a) Estimate the average shear stress on the pipe walls. (b) Estimate the friction factor for the flow. (c) Estimate the distance from the pipe entrance required to establish fully developed flow. (d) Would your answers to Part (a) and Part (b) be different if the flow between the upstream and downstream section is not fully developed?

7.6. A 150-mm diameter duct admits air from a chamber in which the air temperature is 80◦C. (a) Determine the minimum volumetric inflow rate required to ensure that the flow in the duct is turbulent. (b) At the limiting condition determined in Part (a), what is the hydrodynamic entrance length required for fully developed flow in the duct?

7.7. A fluid with a specific gravity and kinematic viscosity of 0.87 and 1.20× 10−4 m2/s, respec- tively, flows in a straight 125-mm diameter pipe that is 80 m long and inclined (upward) and an angle of 8◦ to the horizontal. If the pressure at the downstream (higher-elevation) end of the pipe is 150 kPa and the maximum allowable shear stress on the pipe boundary is 180 Pa, what is the maximum allowable pressure at the upstream end of the pipe?

7.8. Standard air from a large room enters a 225-mm duct at a flow rate of 150 L/s. Estimate the distance from the duct entrance to where the flow is fully developed.

Section 7.3: Friction Effects in Laminar Flow

7.9. A liquid is forced through a small horizontal tube in order to determine its viscosity. The diameter of the tube is 0.6 mm, the length of the tube is 1.1 m, and when a pressure difference of 0.8 MPa is applied across the tube the flow rate is 900 mm3/s. Estimate the viscosity of the liquid.

7.10. SAE 30 oil at 20◦C flows with an average velocity of 0.2 m/s in a 25-mm diameter pipe that is inclined at 10◦ to the horizontal. (a) Determine the maximum velocity and the volume flow rate in the pipeline. (b) What longitudinal pressure gradient is causing this flow? (c) What is the shear stress on the pipe wall?

7.11. Consider the case of laminar flow through a pipeline of diameter D, where hf is the head loss between two sections a distance L apart. (a) Explain how you would use pressure gages and a tape measure to measure hf . (b) For a given fluid and for measured values of hf and L,

592

determine (analytically) the relationship between the percentage error in the volumetric flow rate, Q, and the percentage error in estimating D.

7.12. A fluid with a density and viscosity of 850 kg/m3 and 0.5 Pa·s, respectively, flows in a straight inclined pipe of diameter 50 mm at a rate of 3 L/min. Verify the assumption of laminar flow. Two sections are of interest that are spaced 15 m apart. (a) What is the head loss between the two sections? (b) What pipe inclination would cause the pressure at the two sections to be the same? (c) What is the average shear stress on the pipe boundary between the two sections?

7.13. For the general case of laminar flow in a pipe of diameter D, at what distance from the pipe centerline is the velocity equal to the average velocity? Give your answer a fraction of the pipe diameter.

7.14. Engine oil (SAE 50) at 20◦C flows in a 1-cm diameter pipe. The pressure at one location in the pipe is measured as 30 kPa, and at a location 20 m downstream the pipe elevation is 2 m higher and the measured pressure is 5 kPa. If the dynamic viscosity of the oil is 0.86 Pa·s and the density is 902 kg/m3, determine the following: (a) the average velocity in the pipe; (b) the velocity distribution in the pipe; (c) the Reynolds number, and verify that the flow is laminar according to the Reynolds number criterion; and (d) the friction factor in the Darcy-Weisbach equation for head loss.

7.15. Water at 20◦C flows in a 2-mm pipe that is inclined downward at an angle of 5◦. (a) What is the maximum allowable pressure gradient along the pipe so as to maintain laminar flow? (b) What is the volumetric flow rate in the pipe under the limiting laminar-flow condition?

7.16. Blood at 37◦C flows in a 50-mm diameter tube inclined at 80◦ to the horizontal. If the pressure gradient along the tube is measured as −15 kPa/m, estimate the volumetric flow rate in the tube.

Section 7.4: Friction Effects in Turbulent Flow

7.17. A fluid is to be carried in a 150-mm diameter copper pipe such that the Reynolds number of the flow is equal to 5× 104. The pipe is highly polished and it can be assumed that ks ≈ 0. Using the given data, estimate the thickness of the laminar boundary layer adjacent to the wall of the pipe. Is the flow turbulent? Is the flow fully turbulent?

7.18. A fluid with a specific gravity and kinematic viscosity of 0.87 and 1.20× 10−4 m2/s, respec- tively, flows in a straight 50-mm diameter PVC pipe. The roughness height of the PVC pipe is sufficiently small that when the flow is turbulent it can be assumed that the flow is hy- draulically smooth. If the maximum allowable shear stress on the pipe boundary is 150 Pa, what is the volumetric flow rate in the pipe?

7.19. Water at 20◦C is to be transported in a 200-mm pipe at a flow rate of 50 L/s. It is desired to use pipes with sufficiently smooth boundaries that the flow will be in the hydraulically smooth regime. What range of equivalent sand roughnesses can be used to achieve this objective?

7.20. Water is flowing in a horizontal 200-mm diameter pipe at a rate of 0.06 m3/s, and the pressures at sections 100 m apart are equal to 500 kPa at the upstream section and 400 kPa at the downstream section. Estimate the average shear stress on the pipe and the friction factor, f .

593

7.21. Water at 20◦C flows through a 100-mm diameter steel pipe at a rate of 20 L/s. The equivalent sand roughness of the pipe is estimated as 0.3 mm. (a) Estimate the friction factor of the flow. (b) Estimate the thickness of the laminar layer on the pipe boundary. (c) Compare the thickness of the laminar layer to the sand roughness to estimate whether the flow is in the smooth-pipe, transition, or rough-pipe regime.

7.22. Standard air flows with a velocity of 14 m/s in a 5-mm copper tube. (a) Confirm that the flow is turbulent and calculate the head loss per unit length of tube. (b) If great care is taken to maintain laminar flow conditions in the tube, determine the head loss per unit length under the laminar flow condition, and the percentage change in head loss per unit length that occurs when the flow transitions from the laminar to the turbulent flow condition.

7.23. Water at 20◦C flows at a velocity of 2 m/s in a 250-mm diameter horizontal ductile-iron pipe. (a) Compare the friction factors derived from the Moody diagram, the Colebrook equation, and the Swamee–Jain equation. State whether the flow is fully turbulent or not. (b) Estimate the change in pressure over 100 m of pipeline. (c) How would the friction factor and pressure change be affected if the pipe were not horizontal but 1 m lower at the downstream section?

7.24. A straight pipe has a diameter of 25 mm, a roughness height of 0.1 mm, and is inclined upward at an angle of 10◦. The water temperature is 20◦C. (a) If the pressure at a given section of the pipe is to be maintained at 550 kPa, determine the pressure at a section 100 m downstream for flows of 2 L/min, and 20 L/min (two answers are required here). (b) For a flow of 20 L/min, compare the value of the friction factor calculated using the Colebrook equation with the friction factor calculated using the Swamee–Jain equation; does your result support Swamee–Jain’s assertion that the results are within 1% of each other?

7.25. Show that the Colebrook equation can be written in the (slightly) more convenient form:

f = 0.25

{ log

[( ks/D

3.7

) +

2.51

Re √ f

]}2

Why is this equation termed “(slightly) more convenient”?

7.26. If you had your choice of estimating the friction factor either from the Moody diagram or from the Colebrook equation, which one would you pick? Explain your reasons.

7.27. A frequently cited empirical equation for determining the friction factor in pipes under tur- bulent flow conditions is the Haaland equation, which is given by

1√ f = −1.8 log

[( ks/D

3.7

)1.11 +

6.9

Re

]

Consider flows in which ks/D = 1×10−4 and Re is in the range of 104–108. (a) Determine the maximum percentage error that can be introduced by using the Haaland equation instead of the Colebrook equation. (b) Compare the maximum percentage error determined in Part (a) with that of the Swamee–Jain equation for the same range of ks/D and Re. (c) State which equation would be preferable for the given ranges of parameters.

594

7.28. Estimate the kinetic energy correction factor, α, for turbulent pipe flow with a velocity distribution given by

v(r) = V0

[ 1−

( r R

)2]

where v(r) is the velocity at a distance r from the centerline of the pipe, V0 is the centerline velocity, and R is the radius of the pipe.

7.29. The velocity profile, v(r), for turbulent flow in smooth pipes is sometimes estimated by the seventh-root law, originally proposed by Blasius (1913):

v(r) = V0 ( 1− r

R

) 1 7

where V0 is the maximum (centerline) velocity and R is the radius of the pipe. Estimate the energy and momentum correction factors corresponding to the seventh-root law.

7.30. Show that the kinetic energy correction factor, α, corresponding to the power-law velocity profile is given by Equation 7.51. Use this result to confirm your answer to Problem 7.29.

7.31. The logarithmic velocity distribution and the power-law velocity distribution are both used to describe turbulent flow in pipes, and these equations are given by Equations 7.46 and 7.49, respectively. The parameter accounting for pipe friction in the logarithmic equation is the friction factor, f , and the parameter accounting for pipe friction in the power-law equation is the exponent n. (a) Derive the relationship between f and n when both velocity distributions give the same average velocity and the same maximum velocity. (b) If f = 0.02, what is the corresponding value of n obtained using the relationship derived in Part (a)? (c) For f = 0.02, plot and compare the normalized logarithmic and power-law velocity distributions. Note that in the normalized distributions the velocity is normalized by the maximum velocity and the radial distance is normalized by the radius of the conduit.

Section 7.5: Practical Applications

7.32. Water at 20◦C is pumped at a rate of 15 L/s into a reservoir as shown in Figure 7.17. Between the pump and the reservoir is 150 m of 100-mm diameter PVC pipe, and the water surface elevation in the reservoir is 8 m above the centerline of the inflow pipe. The roughness height in the PVC pipe can be assumed to ne negligibly small. Estimate the gage pressure on the downstream side of the pump.

8 m

P

150 m

100 mm

15 L/s

Pump Reservoir

Figure 7.17: Pumped flow into a reservoir

595

7.33. Crude oil at 20◦C is transported at a rate of 500 L/s through a 750-mm diameter steel pipe that has an estimated equivalent sand roughness of 1.5 mm. The pipeline is 3 km long and the downstream end of the pipeline is at an elevation 1.5 m higher than the elevation at the beginning of the pipeline. (a) What change in pressure is to be expected over the length of the pipeline. (b) At what rate is energy being consumed to overcome friction? (c) If a smooth lining such is installed in the pipe such that the roughness height is reduced by 70%, what would be the percentage change in the quantities calculated in Parts (a) and (b)?

7.34. An oil pipeline has a diameter of 1200 mm, an estimated roughness height of 0.15 mm, and an upward slope of 5%. This pipeline is to be used to transport crude oil that has an average density of 860 kg/m3 and an average viscosity of 7.2 mPa·s. The design flow rate of oil is 3 m3/s, and pumps are to be placed at fixed intervals to maintain the flow rate. If the maximum allowable pressure in the pipeline is 8 MPa and the minimum allowable pressure is 350 kPa, estimate the required spacing between pumps.

7.35. Water leaves a treatment plant in a 500-mm diameter ductile-iron pipeline at a pressure of 600 kPa and at a flow rate of 0.50 m3/s. If the elevation of the pipeline at the treatment plant is 120 m, estimate the pressure in the pipeline 1 km downstream where the elevation is 100 m. Assess whether the pressure in the pipeline would be sufficient to serve the top floor of a 10-story building (approximately 30 m high).

7.36. Water flows at a rate of 100 L/s in a 100-m section of 200-mm diameter ductile-iron pipeline, where the end of the pipeline has an elevation that is 0.8 m higher than the beginning of the pipeline. Under the given flow condition, the pressure at the end of the pipeline is measured to be 90 kPa less than the pressure at the beginning of the pipeline. (a) Estimate the friction factor of the pipeline. (b) Assess the condition of the pipeline in terms of likely deterioration of the interior surface of the pipeline.

7.37. Methane at a temperature of 5◦C and a pressure of 500 kPa (absolute) flows in a 50-mm diameter duct at the rate of 5 L/s. The duct is inclined at 10◦ to the horizontal, and the pressure drop over a 20-m segment of the duct is measured as 300 Pa. An literature search for the properties of methane at the given temperature and pressure indicates that the dynamic viscosity of the gas is 0.01054 mPa·s under these conditions. Estimate the friction factor and equivalent sand roughness of the duct.

7.38. SAE 30 oil at 20◦C flows in the 3-cm diameter pipe shown in Figure 7.18, which slopes at 37◦. For the pressure measurements shown, determine the flow rate in m3/h.

596

180 kPa

500 kPa 15 m

20 m 37o

Pipe

Figure 7.18: Flow in pipe

7.39. A 25-mm diameter galvanized iron service pipe is connected to a water main in which the pressure is 400 kPa. If the length of the service pipe to a faucet is 20 m and the faucet is 2.0 m above the main, estimate the flow rate when the faucet is fully open.

7.40. A galvanized iron service pipe from a water main is required to deliver 300 L/s during a fire. If the length of the service pipe is 40 m and the head loss in the pipe is not to exceed 45 m, calculate the minimum pipe diameter that can be used. Use the Colebrook equation in your calculations.

7.41. Repeat Problem 7.40 using the Swamee–Jain approximation (Equation 7.59).

7.42. Air at standard pressure and at a temperature of 20◦C flows at a rate of 60 L/s through a 20- m long duct that has a circular cross section. The duct material has an estimated roughness height of 0.5 mm. Design limitations require that the pressure change in the air as it passes through the duct be less than 1.5 kPa. What is the minimum acceptable diameter of the duct?

7.43. Standard air is to be moved between two chambers where the difference in pressure is 350 kPa. The material to be used in the connecting conduit has an estimated roughness height of 1.5 mm and a length of 100 m. If the desired air transfer rate is 500 L/s, what duct diameter should be used? Use the Colebrook equation in your analysis.

7.44. Methane is delivered to a 200 mm distribution line at a rate of 50 N/s and at a temperature and pressure of 10◦C and 450 kPa, respectively. At the given temperature and pressure conditions (at the entrance to the pipeline), the dynamic viscosity of methane is estimated as 0.01073 mPa·s. The equivalent sand roughness of the pipeline is estimated as 1 mm. Elevation changes along the distribution pipeline are negligible, and the temperature remains approximately constant along the entire length of the pipeline. (a) Assuming incompressible flow, estimate the change in pressure per kilometer of pipeline. (b) If the total length of the pipeline is 15 km, compare the densities at the beginning and end of the pipeline, and reassess the assumption of incompressible flow.

7.45. Water flows at 5 m3/s in a 1 m × 2 m rectangular concrete pipe. Calculate the head loss over a length of 100 m.

597

7.46. Water flows at 10 m3/s in a 2 m × 2 m square reinforced-concrete pipe. If the pipe is laid on a (downward) slope of 0.002, what is the change in pressure in the pipe over a distance of 500 m?

7.47. Air flows at 10 m/s in a 400 mm × 400 mm square duct that has an estimated equivalent sand roughness of 0.003 mm. The temperature of the air is 70◦C and the air is at standard atmospheric pressure. (a) Determine the head loss per unit length in the square duct. (b) If the square duct is changed to a circular duct made of the same material, what diameter should be used for the circular duct so that the head loss per unit length remains unchanged.

7.48. A fan circulates standard air at 200 L/s in a horizontal 200 mm× 400 mm duct. The length of the duct is 60 m and the equivalent sand roughness of the duct material is estimated as 0.5 mm. What is the rate of energy loss in the air that can be attributed to friction between the moving air and the (stationary) duct. Is this loss rate approximately equal to the power that must be supplied by the fan to move the air through the duct?

7.49. Standard air flows at a rate of 500 L/s in an existing duct that is 250 mm wide and 500 mm high. The estimated equivalent sand roughness of the duct material is 0.1 mm. Ductwork renovations call for the replacement of the existing duct by another rectangular duct having the same cross-sectional area and the same length, but a different aspect ratio. The aspect ratio is defined as the height divided by the width. Aspect ratios in the range of 0.1–3 are being considered for the replacement duct. (a) Plot the ratio of the head loss in the replacement duct to the head loss in the existing duct as a function of the aspect ratio. (b) What aspect ratio in the replacement duct would give the least head loss?

7.50. Derive the Hazen–Williams head-loss relation, Equation 7.62, starting from Equation 7.60.

7.51. Compare the Hazen–Williams formula for head loss with the Darcy–Weisbach equation for head loss to determine the expression for the friction factor that is assumed in the Hazen– Williams formula. Based on your result, identify the type of flow condition (rough, smooth, or transition) incorporated in the Hazen–Williams formula.

7.52. Derive the Manning head-loss relation, Equation 7.65.

7.53. Compare the Manning formula for head loss with the Darcy–Weisbach equation for head loss to determine the expression for the friction factor that is assumed in the Manning for- mula. Based on your result, identify the type of flow condition (rough, smooth, or transition) incorporated in the Manning formula.

7.54. Determine the relationship between the Hazen–Williams roughness coefficient and the Man- ning roughness coefficient.

7.55. Given a choice between using the Darcy–Weisbach, Hazen–Williams, and Manning equations to estimate the friction losses in a pipeline, which equation would you choose? Why?

7.56. Water flows at a velocity of 2 m/s in a 300-mm new ductile-iron pipe. Estimate the head loss over 500 m using: (a) the Hazen–Williams formula, (b) the Manning formula, and (c) the Darcy–Weisbach equation. (d) Compare your results. (e) Calculate the Hazen–Williams roughness coefficient that should be used to obtain the same head loss as the Darcy–Weisbach

598

equation. (f) Calculate the Manning coefficient that should be used to obtain the same head loss as the Darcy–Weisbach equation.

7.57. Water leaves a reservoir at 0.06 m3/s through a 200-mm riveted steel pipeline that protrudes into the reservoir and then immediately turns a 90◦ bend with a local (minor) loss coefficient equal to 0.3. Estimate the length of pipeline required for the friction losses to account for 90% of the total losses, which includes both friction losses and so-called “minor losses.” Would it be fair to say that for pipe lengths shorter than the length calculated in this problem, the word “minor” should not be used?

7.58. Floodwater from a residential neighbourhood is discharged to a river through a 200 m long 100-mm diameter pipe that has an estimated roughness height of 0.5 mm. The discharge end of the pipe is open to the atmosphere, and is at an elevation that is 1.2 m below the head of the pipe. Appurtenances within the pipe combine to give a total local loss coefficient of 8.7. Under design conditions, the pressure at the head of the pipe is 300 kPa. Estimate the discharge through the pipe under design conditions. Assume water at 20◦C.

7.59. Water at 20◦C is to be delivered at a rate of 100 L/s through a 200 m long pipeline with an estimated roughness height of 0.1 mm and an estimated local-loss coefficient of 3.4. The downstream end of the pipe has an elevation that is 1.8 m higher than the upstream end of the pipe. The pressure at the head of the pipeline can be maintained at 450 kPa using a pump, and the minimum allowable pressure at the downstream end of the pump is 200 kPa. What is the minimum diameter pipe that can be used?

7.60. The 1-km long tunnel shown in Figure 7.19 is to be sized such that water can be moved from the upper reservoir to the lower reservoir at a rate of 2 m3/s when all control gates are fully open. The tunnel will be lined with a material that has an estimated roughness height of 1.5 mm, and it is expected that the water will be at approximately 20◦C. (a) What tunnel diameter is required to achieve the desired flow rate if local losses are neglected? (b) If local losses are lumped into a single loss coefficient equal to 15.3, what tunnel diameter would be required? What can you conclude about the effect of local losses in this system?

23 m TunnelUpper reservoir

Lower reservoir

L= 1 km

Figure 7.19: Flow through a tunnel

7.61. The large-capacity faucet is to be connected to a water main by 20-mm diameter copper tubing as shown in Figure 7.20. The copper tubing has an estimated equivalent sand roughness of 0.003 mm, and local head losses are created by the corporation stop (C-stop), three bends, and the faucet itself. The faucet exit has a diameter of 12 mm. What is minimum pressure in the water main required to obtain a desired flow rate of 50 L/min when the faucet is wide open?

599

Bend

Bend

Bend

Bend

Faucet

C-stop

Pipe

3.5 m

2 m

5 m

3 m

4 m

Fi!ng K

C-stop 9

Bend(s) 1.2

Faucet 10

Water main

Figure 7.20: Faucet connected to water main

7.62. Standard air flows through a 200-mm diameter duct at a rate of 100 L/s. The roughness height of the duct material is estimated as 1 mm. The air-duct system contains a three bends each with a head loss coefficient of 1.0, and a single valve with a head loss coefficient of 5.0. (a) Estimate the length of duct that will have the same head loss as each bend. (b) Estimate the length of duct that will have the same head loss as the valve. (c) If the total length of the duct system is 20 m, estimate what percentage of the system head loss is caused by the bends and valve combined.

7.63. Air at a temperature and pressure of 40◦C and 150 kPa, respectively, flows in a 100-mm diameter duct at a volumetric flow rate of 10 L/s. A fixture is inserted in the duct that causes the pressure to decrease by 20 Pa. What is the head-loss coefficient of the fixture?

7.64. Water at 20◦C flows at 6 L/s through a 75 mm diameter pipe, and a fixture is installed such that the diameter is suddenly reduced to 50 mm. (a) Estimate the head loss due to the sudden contraction. (b) What pressure change is expected at the contraction? (c) What would be the pressure change if the local head loss is not taken into account?

7.65. A 41-mm diameter Schedule 40 galvanized steel pipe is connected to a water main as shown in Figure 7.21. The pressure in the water main is 350 kPa, the flow in the pipeline is regulated by a globe valve, and flow exits the pipeline through a nozzle with an exit diameter of 30 mm. A globe valve is to be selected such that a water fountain as high as 5 m can be generated. The 90◦ bend between the globe valve and the pipe exit is estimated to have a head-loss coefficient equal to 1.2, the nozzle is estimated to have a loss coefficient of 0.05, and the galvanized steel pipe is estimated to have an equivalent sand roughness of 0.1 mm. Estimate the maximum allowable head-loss coefficient for the open globe valve. Assume water at 20◦C.

Water main Globe valve

5 m

15 m 3 m

1 m

Fountain

Pipe

Figure 7.21: Water fountain

600

7.66. A small playground is to be served by a galvanized-iron supply line connected to the water main. The pressure in the water main is 400 kPa, and the length of the supply line to the playground spigot is 50 m. It is estimated that the equivalent sand roughness of the supply line is 0.1 mm, and the sum of the local head loss coefficients of the pipeline appurtenances is equal to 30 when the spigot is fully open. If a maximum flow rate of 100 L/min is to be delivered to the playground, what is the minimum pipe diameter that should be used? Use the Colebrook equation. Assume water at 15◦C.

7.67. The storage tank shown in Figure 7.22 is to be used to supply a community such that the pressure at Point B is always in the range of 330–440 kPa. The water-demand of the com- munity is expected to vary in the range of 0.3–1 m3/s. The bottom of the storage tank in 42 m above ground level and the supply pipe is 1.5 m below ground level. The length of steel pipe between the bottom of the storage tank (Point A) and Point B is 1 km, the diameter of the pipe is 600 mm, and the pipe contains five 90◦ bends. The equivalent sand roughness of the pipe is estimated as 1 mm, and the head loss coefficient of each bend is estimated as 0.8. An auxiliary pumping system will ensure that the tank never becomes empty. Determine the range of water depth (h) in the tank that will meet the design pressure requirements at B. At the minimum flow rate what will be the minimum pressure? Assume water at 20◦C.

Storage tank

42 m

Supply pipe

A

Flow 1.5 m

h

B

Figure 7.22: Storage tank for water supply

7.68. The effect of a nozzle on the flow rate out of a garden hose is to be investigated. The 25-m long 19-mm diameter hose is connected to a spigot as shown in Figure 7.23, where the spigot maintains a (gage) pressure of 200 kPa and the hose exit is 0.9 m above the spigot. The equivalent sand roughness of the hose material is estimated as 0.1 mm. The nozzle has an exit diameter of 10 mm, and is estimated to have a head-loss coefficient of 0.05 based on the exit velocity. The water is at 15◦C. (a) What is the exit velocity and volumetric flow rate out of the hose without the nozzle? (b) What is the exit velocity and volumetric flow rate out of the hose with the nozzle attached? (c) Identify a benefit of using a nozzle on a hose.

601

0.9 m Spigot

Nozzle

Hose

Wall

Figure 7.23: Hose discharging water from a spigot

7.69. A pump creates a water pressure of 1 MPa in the supply line to a 65-mm diameter 300-m long hose with a roughness height of 0.5 mm. The end of the hose is at an elevation 2.8 m higher than the beginning of the hose. It is desired to place a nozzle at the end of the hose so as to create a jet of water that rises 30 m into the air. If the local head-loss coefficient of the nozzle is 0.04, what nozzle diameter should be used? Assume water at 20◦C.

7.70. The pressurized water source shown in Figure 7.24 is connected to a 200-mm diameter 1-km long pipeline that discharges into the atmosphere at an elevation 7 m below the connection point. When the pressure at the water source is 140 kPa, the pipeline discharges 60 L/s. A new connection midway along the existing pipeline is being contemplated such that the new connection will deliver 20 L/s while the outlet of the existing pipeline continues to deliver 60 L/s. What is the required pressure at the water source to accomplish this design objective? Assume water at 20◦C.

60 L/s

20 L/s

1 2

3

40 kPa

7 m

Pressurized water source

Figure 7.24: Existing pipeline with new connection

7.71. Water enters and leaves a pump in pipelines of the same diameter and approximately the same elevation. If the pressure on the inlet side of the pump is 30 kPa and a pressure of 500 kPa is desired for the water leaving the pump, what is the head that must be added by the pump, and what is the power delivered to the fluid?

7.72. Water at 20◦C is pumped from a lower reservoir to an upper reservoir as shown in Figure 7.25. The riveted steel pipeline has a length of 1 km and a diameter of 500 mm, and the difference between the water-surface elevations in the upper and lower reservoirs is 20 m. A antiquated gate valve controls the flow in the pipeline and when it is fully open it has a head loss coefficient of 45. The currently installed pump delivers 250 kW of power to the water when it is turned on, and produces a flow of 500 L/s when the gate valve is fully open. (a)

602

Estimate the equivalent sand roughness of the pipeline. (b) If the gate valve is replaced by a new one such that the head loss coefficient is reduced by 70%, what pumping power is needed to generate the same flow rate?

Lower reservoir

Upper reservoir

P

20 m

Riveted steel pipeline

Gate valve

Figure 7.25: Pumping from a lower to an upper reservoir

7.73. Water is pumped from a supply reservoir to a ductile-iron water-transmission line, as shown in Figure 7.26. The high point of the transmission line is at point A, 1 km downstream of the supply reservoir, and the low point of the transmission line is at point B, 1 km downstream of A. If the flow rate through the pipeline is 1 m3/s, the diameter of the pipe is 750 mm, and the pressure at A is to be 350 kPa, then: (a) estimate the head that must be added by the pump; (b) estimate the power supplied by the pump; and (c) calculate the water pressure at B.

Elev. 7 m

Pump

Elev. 10 m

Transmission line

A

B Elev. 4 m

Supply reservoir PP

Figure 7.26: Water pumped into transmission line

7.74. Water is to be pumped at at rate of 0.7 m3/s from a supply reservoir into a 600-mm steel water-supply pipeline as shown in Figure 7.27. The equivalent sand roughness of the steel pipe is estimated as 1 mm. The length of the pipeline from the reservoir to Point A is 1 km, from A to B is 1.5 km, and from B to C is 1.8 km. Point A is at the lowest elevation of the pipeline, Point B is at the highest elevation, and Point C is at the delivery point to the distribution network where the pressure is 400 kPa under the given flow condition. (a) Estimate the range of pressures in the pipeline downstream of the pump? (b) If the maximum pressure in the pipeline is to be no more than 600 kPa, what is the minimum diameter that should be used for the pipeline? Assume water at 20◦C.

603

p=400 kPa

P

5 m

Source reservoir

Water-supply pipeline

10 m

600 mm

A

B

12 m C

Figure 7.27: Water delivery from a supply reservoir

7.75. Water is to be pumped into a storage reservoir through a 230-m long galvanized pipeline that has an estimated roughness height of 0.2 mm. Under design conditions, the water level in the reservoir is 18 m above the centerline of the pump discharge. When the pump is operating, the desired water pressure on the discharge side of the pump is 400 kPa, and the desired flow rate is 50 L/s. What pipe diameter is required to attain these operating conditions? Assume water at 20◦C.

7.76. A pump is being used to fill a storage tank with water as shown in Figure 7.28. The bottom of the storage tank is 4 m above the center of the discharge line of the pump, and the depth of water in the tank is represented by h. The plan area of the tank is 10 m2, and the pipeline between the pump and the tank has a diameter of 75 mm, a length of 65 m, and an estimated roughness height of 0.5 mm. The pressure at the discharge side of the pump is maintained at 550 kPa. Estimate the time it takes to fill the tank from a depth of 0.5 m to a depth of 3 m. Assume fully turbulent flow and water at 20◦C.

P

h

4 m

Pump

Storage tank

Flow

550 kPa

Figure 7.28: Filling a storage tank using a pump

7.77. A pipeline is to be run from a water-treatment plant to a major suburban development 3 km away. The average daily demand for water at the development is 0.0175 m3/s, and the peak demand is 0.578 m3/s. Determine the required diameter of ductile-iron pipe such that the flow velocity during peak demand is 2.5 m/s. Round the pipe diameter upward to the nearest 25 mm (i.e., 25 mm, 50 mm, 75 mm, . . .). The water pressure at the development is to be at least 340 kPa during average demand conditions, and 140 kPa during peak demand. If the water at the treatment plant is stored in a ground-level reservoir where the level of the water is 10.00 m NGVD and the ground elevation at the suburban development is 8.80 m NGVD, determine the pump power (in kilowatts) that must be available to meet both the average daily and peak demands.

604

7.78. Water is pumped at a rate of 240 L/min from an open well as shown in Figure 7.29. Ground- water in the well enters the vertical 50-mm diameter intake pipe at a depth of 15 m from the suction side of the pump. The equivalent sand roughness of the intake pipe is estimated as 0.5 mm. The distance h represents the difference in elevation between the suction side of the pump and the water level in the well. Atmospheric pressure is 101.3 kPa and the temperature of the groundwater is 15◦C. (a) For the given pumping rate and intake characteristics, what is the maximum allowable value of h so as to prevent the occurrence of cavitation on the suction side of the pump? (b) Estimate the maximum allowable value of h regardless of the pumping rate. (c) What general statement can you make about the maximum elevation of pumps relative to their source reservoirs?

P Flow

h Water table

Open well

15 m Intake pipe

Figure 7.29: Pumping from a well

7.79. Water exits a reservoir through a 125-m long 5-cm diameter horizontal cast-iron pipe as shown in Figure 7.30. The pipe entrance is sharp-edged, the water flows through a turbine, and the discharge is to the atmosphere. If the flow rate is 4 L/s, what power is extracted by the turbine? Describe a practical situation in which you might encounter this type of problem.

40 m

T

Turbine Open globe valve

125 m

Figure 7.30: Flow out of a reservoir

7.80. Water flows from an upper reservoir to a lower reservoir through a 150-m long 350-mm diame- ter tunnel that feeds a turbine as shown in Figure 7.31. The tunnel has an estimated roughness height of 1 mm and the turbine is expected to extract energy from the water at a minimum rate 60 kW. The entrance and exit loss coefficients are 0.8 and 1.0, respectively. What is the minimum flow rate through the tunnel that is required to accomplish this objective?

605

T

45 mTunnel

Turbine

Upper reservoir

Lower reservoir

Figure 7.31: Flow between two reservoirs

Section 7.6: Water Hammer

7.81. Plot the water-hammer pressure versus time at the midpoint of the pipeline shown in Figure 7.10. Assume that the valve is closed instantaneously.

7.82. Water flows in a 100-m long pipe at 3 m/s. If the water temperature is 20◦C, determine the minimum valve-closure time to avoid creating water-hammer pressures. What is the maximum water-hammer pressure that can occur? How is this pressure affected if the water temperature drops to 10◦C?

7.83. Based on your result in Problem 7.82, do you think that water hammer can be a serious problem in household plumbing?

7.84. Water at 20◦C flows in a 150-m long 50-mm diameter ductile iron pipe at 4 m/s. The thickness of the pipe wall is 1.5 mm and the modulus of elasticity of ductile iron is 1.655×105 MN/m2. What is the maximum water-hammer pressure that can occur?

7.85. The pipeline described in Problem 7.84 is replaced by 50-mm diameter PVC pipe with a wall thickness of 2 mm and a modulus of elasticity of 1.7× 104 MN/m2. How will this affect the maximum water-hammer pressure?

Section 7.7: Pipe Networks

7.86. A water-supply pipeline is to deliver 0.6 m3/s from a water treatment plant through a 1000 mm diameter 3.2-km long ductile iron pipeline. The water surface elevation in the source storage reservoir is at 10.06 m NGVD and the water is to be pumped from the storage reservoir into the delivery pipe. The pump is located 1.5 m below the water surface in the source reservoir, and the end of the delivery pipe is at elevation 11.52 m NGVD. (a) Estimate the pump requirement in kW so that the pressure at the delivery point is 350 kPa. (b) What is the pressure in the pipeline immediately downstream of the pump? (c) Recalculate the answer to Part (a) if the pipeline is divided into two 600-mm diameter 3.2-km long ductile- iron pipelines; the pipes are separated just downstream of the pump and rejoined at the downstream location.

7.87. Reservoirs A, B, and C are connected as shown in Figure 7.32. The water elevations in reservoirs A, B, and C are 100 m, 80 m, and 60 m, respectively. The three pipes connecting

606

the reservoirs meet at the junction J, with pipe AJ being 900 m long, BJ 800 m long, CJ 700 m long, and the diameter of all pipes equal to 850 mm. If all pipes are made of ductile iron and the water temperature is 20◦C, find the flow into or out of each reservoir.

Elev.

100 m

A Elev.

80 m

B

Elev.

60 mC

L AJ = 900 m

L BJ

= 800 m

L JC = 700 m

J

Figure 7.32: Connected reservoirs

7.88. A 3-km long 150-mm diameter existing riveted steel pipeline connects two reservoirs as shown in Figure 7.33. The existing pipeline extends from locations A to C and has an estimated roughness height of 1 mm. To increase the flow rate between reservoirs, it is proposed to add a new 200 mm diameter pipeline such that the new pipeline is connected to the existing pipeline at location B that is 1.5 km from A. The new pipeline will have a length of 1.6 km and an estimated roughness height of 0.5 mm. Under design conditions, the difference in elevation between the water surfaces in the two reservoirs is 8 m. For this analysis, you may assume that the flows is all pipes are fully turbulent, and that local losses are negligible compared to friction losses. (a) Compare the existing and new flows between the reservoirs. (b) What percentage change in flow will occur in the existing pipe? (c) Using the calculated flows in the new system, review your assumption of fully turbulent flow and state how you would modify your analysis based on your assessment results. Assume water at 20◦C.

8 m

A

B

C D

Exis!ng 150-mm pipeline

New 200-mm pipeline

Figure 7.33: Existing and modified pipeline system

7.89. The water main shown in Figure 7.34 diverges a Point A and serves locations at Points B, C, and D. The lengths and the diameters of the pipeline segments are tabulated in Figure 7.34, along with the elevations of Points A, B, C, and D. All pipes are made of ductile iron with an estimated roughness height of 0.2 mm. Under normal operating conditions, the synoptic pressures at the key points in the system are tabulated in Figure 7.34. A 1.8-km long 150-mm diameter ductile iron pipe is being considered for installation that will connect Points B and C. (a) Estimate the flows QB, QC, and QD for the existing system. (b) Estimate the flows QB, QC, and QD with the new pipeline connecting B to C in operation. Assume water at 20◦C. Do not assume that the flow is fully turbulent.

607

A

B

C

D

Q B

Q C

Q D

Exis!ng

Planned Line Length Diameter (km) (mm)

AB 3.0 300

AC 2.0 200

AD 2.5 250

(BC) 1.8 (150)

Point Eleva!on Pressure

(m) (kPa)

A 3.40 450

B 2.80 330

C 3.20 300

D 2.60 360

Figure 7.34: Flows in pipeline system

7.90. The water-supply network shown in Figure 7.35 has constant-head elevated storage tanks at A and B, with inflows and withdrawals at C and D. The network is on flat terrain, and the pipeline characteristics are as follows:

Pipe L D

(km) (mm)

AD 1.0 400 BC 0.8 300 BD 1.2 350 AC 0.7 250

If all pipes are made of ductile iron, calculate the inflows/outflows from the storage tanks. Assume that the flows in all pipes are fully turbulent.

0.2 m3/s

Reservoir

Elev. 25 m

0.2 m3/s

Reservoir

Elev. 20 m

A C

D B

Figure 7.35: Water-supply network

7.91. Water is handled by a system of pipes as shown in Figure 7.36 and the pipe characteristics

608

are as follows:

Pipe Length Diameter

f (m) (m)

AC, BC 100 0.50 0.0055 CD 300 0.75 0.0050 DE 500 0.30 0.0060 DF 400 0.25 0.0060 DG 500 0.30 0.0060

The elevation of outlets E, F, and G is 100 m above the elevation of inlets A and B. All outlets and inlets are at atmospheric pressure. If the mean velocity in the pipes AC and BC is 2.5 m/s, calculate the flow rate through the pump P, the pressure difference across the pump, and the power consumed by the pump. Take the pump efficiency as 76%.

Flow Flow

Flow

Flow

E

F

G

A

C D

BFlow

P

Figure 7.36: Pipe system

7.92. Consider the pipe network shown in Figure 7.37. The Hardy Cross method can be used to calculate the pressure distribution in the system, where the friction loss, hf , is estimated using the equation

hf = rQ n

and all pipes are made of ductile iron. What value of r and n would you use for each pipe in the system? The pipeline characteristics are as follows:

L D Pipe (m) (mm)

AB 1000 300 BC 750 325 CD 800 200 DE 700 250 EF 900 300 FA 900 250 BE 950 350

You can assume that the flow in each pipe is hydraulically rough.

609

A C

0.5 m3/s

0.5 m3/s

F D

B

E

Figure 7.37: Two-loop pipe network

7.93. A portion of a municipal water-distribution network is shown in Figure 7.38, where all pipes are made of ductile iron and have diameters of 300 mm. Use the Hardy Cross method to find the flow rate in each pipe. If the pressure at point P is 500 kPa and the distribution network is on flat terrain, determine the water pressures at each pipe intersection.

150 m 150 m

100 m

100 m

100 m

100 m

0.1 m3/s

0.07 m3/s

0.06 m3/s

0.05 m3/s

0.06 m3/s

150 m 150 m

150 m 150 m

P

Figure 7.38: Four-loop pipe network

7.94. Water is delivered at a rate of 48,000 m3/d at node C of the distribution system shown in Figure 7.39, and the water is withdrawn at a rate of 8000 m3/d at each of the nodes. All pipes are made of steel with roughness heights of 0.05 mm and are of diameter 1120 mm. The pipe lengths are as follows: CD = 5 km, DE = 12 km, EF = 9 km, FC = 6 km, FG = 7 km, GH = 8 km, and HC = 10 km. Determine the flow distribution in the pipes.

48,000 m3/d

8000 m3/d 8000 m3/d

8000 m3/d8000 m3/d8000 m3/d

8000 m3/d

C DH

G F E

Figure 7.39: Water-supply system

610

Section 7.8: Building Water-Supply Systems

7.95. Water in a household plumbing system originates at the neighborhood water main where the pressure is 480 kPa, the velocity is 5 m/s, and the elevation is 2.44 m. A 19-mm (3/4-in.) copper service line supplies water to a two-story residence where the faucet in the master bedroom is 40 m (of pipe) away from the main and at an elevation of 7.62 m. If the sum of the minor-loss coefficients is 3.5, estimate the maximum (open faucet) flow. Is this flow rate typical of a bathroom faucet? How would this flow be affected by the operation of other faucets in the house?

7.96. A 50-mm diameter pipeline made of copper-tubing is connected directly to the water main as shown in Figure 7.40.

1 2

3 4

5 6

Globe valve

Ou!low

Water main

(p = 400 kPa)

15 m

7

PVC pipe

Figure 7.40: Pipeline

The pressure in the water main at point 1 is 400 kPa and the faucet of interest is at point 6, which is 15.0 m vertically above the water-main connection. The length of pipeline from the water main to the faucet is 30.0 m, and the pipeline has the following characteristics:

Location Type of Fixture

1 sharp-edged entrance 2, 3, 5 90◦ elbow 4 tee 6 globe valve

The is no flow in the line from point 4 to point 7. According to the manufacturer of the faucet (= globe valve) the local loss coefficient is equal to 4.0 when the faucet is fully open. In this case the copper tubing is so smooth that the roughness height can be taken as zero. Estimate the flow rate in the pipeline when the faucet is fully open.

7.97. The top floor of an office building is 40 m above street level and is to be supplied with wa- ter from a municipal pipeline buried 1.5 m below street level. The water pressure in the municipal pipeline is 450 kPa, the sum of the local loss coefficients in the building pipes is 10.0, and the flow is to be delivered to the top floor at 20 L/s through a 150-mm di- ameter PVC pipe. The length of the pipeline in the building is 60 m, the water temper- ature is 20◦C, and the water pressure on the top floor must be at least 150 kPa. Will a

611

booster pump be required for the building? If so, what power must be supplied by the pump?

7.98. Design flow rates in household plumbing are based on the number and types of plumbing fixtures as measured by “fixture units.” The Hunter curve is used to relate fixture units to flow rate. For a particular office building, the total fixture units is determined to be 120 and the corresponding flow rate is 4.67 L/s. The pressure at the water main is estimated to be 380 kPa, and the maximum velocity in the plumbing is not to exceed 2.4 m/s, copper lines are to be used, the length of the line to the most remote fixture is 110 m, the minimum allowable pressure in the pipe (after accounting for head losses) is 240 kPa, and the elevation difference between the curb and the most remote fixture in the building is 3 m. Copper pipe is available in diameters starting at 12.5 mm (1/2 in.) and increasing in increments of 6.25 mm (1/4 in.). Determine the minimum diameter that could be used for the plumbing line. Neglect local losses.

7.99. Design the hot-water pipes for the building given in Example 7.17

7.100. Water is to be delivered from a public water supply to a two-story building. Under design conditions, each floor of the building is to be simultaneously supplied with 200 L/min. The pipes in the building plumbing system are to be made of galvanized iron. The length of pipe from the public water supply to the delivery point on the first floor is 20 m, the length of pipe from the delivery point on the first floor to the delivery point on the second floor is 5 m, the water delivery point on the first floor is 2 m above the water main connection, and the delivery point on the second floor is 3 m above the delivery point on the first floor. If the water pressure at the water main (meter) is 380 kPa, what is the minimum diameter pipe in the building plumbing system to ensure that the pressure is at least 240 kPa on the second floor? Neglect minor losses and consider pipe diameters in increments of 1/4 in., with the smallest allowable diameter being 1/2 in. For the selected diameter under design conditions, what is the water pressure on the first floor?

Chapter 8

Turbomachines

8.1 Introduction

Fluid machines can be broadly categorized as either turbomachines and positive-displacement ma- chines. Turbomachines are devices that either add or extract energy from a fluid by means of a rotating element. The prefix “turbo” generally means “spinning”, and so a turbomachine is literally a “spinning machine”. Machines with rotating elements are also called rotodynamic machines. In contrast to turbomachines, positive-displacement machines move fluids by forcing a fluid into and out of a chamber. The main focus of this chapter is on turbomachines, which are widely used in engineering applications. Turbomachines that add mechanical energy to flowing fluids are generi- cally classified as turbopumps, while turbomachines that convert energy in a flowing fluid into the mechanical energy are classified as turbines. A change in pressure is usually the dominant com- ponent of the change in mechanical energy between the inlet and outlet of a turbomachine; with pressure generally increasing across pumps and decreasing across turbines. In common usage, the term “pump” is usually applied to machines that move liquids, while machines that move gasses are variously called fans, blowers, and compressors; depending on their application. Typically, if the pressure rise is very small a gas pump is called a fan, if the pressure rise in on the order of one atmosphere the gas pump is called a blower, and for pressure rises much greater than one atmosphere the gas pump is called a compressor. The primary use of compressors is to increase the pressure of a gas, while the primary use of fans and blowers is to move a gas. Turbines also have a variety of uses, for example, hydraulic turbines are used to extract energy from flowing water in hydroelectric facilities, wind turbines and windmills are used to extract energy from wind, and gas turbines and steam turbines are used to extract energy from various flowing gasses.

Characteristics of turbomachines. All turbomachines include one or more rotating elements, called rotors, between the inflow and outflow section of the turbomachine. In the case of turbop- umps, the rotor is commonly called an impeller and is powered by an external motor, and the impeller does work on the fluid. In the case of turbines, the rotor is commonly called a runner, the fluid does work on the runner which transmits mechanical energy to an external device connected to a generator. The flow of a fluid towards the rotor of a turbine is typically controlled by fixed or adjustable vanes or blades. For both turbopumps and turbines, the assembly of rotor and vanes (for turbines) is usually contained within a casing, which is sometimes called a housing. Turbomachines can be classified by the direction of fluid flow relative to the rotor, with the three classifications

613

672

Homologous points (pump): Q2 = Q1

( ω2 ω1

) and h2 = h1

( ω2 ω1

)2

Fans

Affinity laws (fans):

[ Q

ωD3

]

1

=

[ Q

ωD3

]

2

→

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[ ∆p

ρω2D2

]

1

=

[ ∆p

ρω2D2

]

2

[ P

ρω3D5

]

1

=

[ P

ρω3D5

]

2

η1 = η2

Specific speed, dimensionless (fan): ns = ωQ

1 2 ρ

3 4

(∆p) 3 4

Impulse Turbines

Shaft torque: Tshaft = ṁR(V2θ − V1θ)

Shaft power: Ẇshaft = ṁU(Vj − U)(cosβ − 1)

Bucket speed, optimum: U = 12Vj

Shaft power, maximum: Ẇshaft ∣∣∣ max

= 14ṁV 2 j (cosβ − 1)

Efficiency, wheel: ηw = |Ẇshaft| 1 2ṁV

2 j

, ηw,theory = 2U(Vj − U)(1− cosβ)

V 2j

Efficiency, turbine: ηt = |Ẇshaft| γQhe

, ηt ∣∣∣ max

= 12C 2 v(1− cosβ)

Head available to runner: ht = he − kj V 2j 2g

− k v 2 2

2g − V

2 2

2g

Power, theoretical: Pt = γQht

Efficiency, hydraulic: ηt = γQht γQhe

= ht he

Efficiency, overall: η = Tω

γQhe

Reaction Turbines

Shaft torque: Tshaft = ṁ (r2V2θ − r1V1θ)

Shaft power: Ẇshaft = Tshaftω = ṁω (r2V2θ − r1V1θ)

Efficiency, hydraulic: ηh = ω (r1V1θ − r2V2θ)

gha

673

Efficiency, overall: ηo = P

ṁght

Head extracted by turbine: ht = H − hℓ − htd − V 2

2g

Turbine performance curve: ght ω2D2

= Φ1

( Q

ωD3

) ,

Ẇshaft ρω3D5

= Φ2

( Q

ωD3

) , η = Φ3

( Q

ωD3

)

Affinity laws:

[ Q

ωD3

]

1

=

[ Q

ωD3

]

2

→

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[ ght ω2D2

]

1

=

[ ght ω2D2

]

2

[ Ẇshaft ρω3D5

]

1

=

[ Ẇshaft ρω3D5

]

2

η1 = η2

Scale effect: 1− η2 1− η1

=

( D1 D2

) 1 5

, 0.95− η2 0.95− η1

=

( Q1 Q2

)0.32

Specific speed, dimensionless: ns = ωQ

1 2

(ght) 3 4

, n′s = ω √ Ẇshaft/ρ

(ght) 5 4

Specific speed, dimensional: N ′s = ne

√ BHP

h 5 4 t

Diameter of rotating element, optimal: D = 2φ∗

√ 2ght ω

Synchronous speed: 60Ω

no. of pairs of poles

Net positive suction head, definition: NPSH = pd γ

+ V 2d 2g

− pv γ

Net positive suction head, minimum: NPSHmin = σcht

Maximum discharge height: zmax = patm − pv

γ − hℓ −NPSHmin

Hydropower generation potential: P = γQhtη

Problems

Section 8.2: Mechanics of Turbomachines

8.1. Water at 20◦C flows through the pump impeller shown in Figure 8.33, where the inner and

674

outer diameters of the impeller are 160 mm and 400 mm, respectively, and height of the pump vanes is 55 mm. The impeller rotates at 3600 rpm. Under a particular operating condition, water enters in a direction normal to the inflow surface, the velocity normal to the outflow surface, Vr, is equal to 20 m/s, and the magnitude of the absolute velocity, V, at the outflow surface is 35 m/s. Estimate the shaft power required to drive the pump at this operating condition.

ω 400 mm

Pump impeller

Vane

V

160 mm

Ou!low surfaceVr

Inflow surface

Figure 8.33: Flow through a pump impeller I

8.2. A centrifugal blower has a 900-mm diameter impeller with a width of 40 mm that rotates at a rate of 1725 rpm. When the flow rate through the impeller is 0.20 m3/s, the inflow velocity is normal to the inflow surface and the outflow velocity makes an angle of 35◦ with the radial direction as shown in Figure 8.34. Assuming that the blower is 100% efficient, estimate the energy per unit mass added to the air as it passes through the blower.

ω V

35o

Ou!low surface

Impeller

Figure 8.34: Flow through a blower

8.3. A water-pump impeller has radial vanes as shown in Figure 8.35. The inner and outer diameters of the impeller are 50 mm and 125 mm, respectively. When the rotational speed of the impeller is 3000 rpm, the pump delivers 10 L/s, and the speed of the water relative to the vane at the exit is equal to 3 m/s. Determine (a) the width of the impeller at the outflow surface, and (b) the power required to drive the pump. Assume water at 20◦C.

675

ω 125 mm50 mm

Pump impeller Ou!low surface

Vane

Inflow surface

Figure 8.35: Flow through a pump impeller II

8.4. A pump impeller has an outer diameter of 350 mm, a blade height of 30 mm, and rotates at 800 rpm. At a particular operating condition, the inflow is normal to the inflow surface, and the (absolute) outflow velocity is 38 m/s and makes an angle of 40◦ with the normal to the outflow surface, as shown in Figure 8.36. Estimate the shaft power required to drive the pump under this condition. Assume water at 20◦C.

ω

Pump impeller

V

40o

Ou!low surface

Figure 8.36: Flow through a pump impeller III

8.5. The pump impeller shown in Figure 8.37 has a diameter of 320 mm, a blade height of 30 mm, an exit blade angle of 42◦, and rotates at 800 rpm. Water at 20◦C flows through the pump at a flow rate of 75 L/s and this flow enters the impeller in a direction normal to the inflow surface. Estimate the required shaft power corresponding to the design flow rate.

ω 325 mm

Pump impeller

Vane

U

WV

42o

Figure 8.37: Flow through a pump impeller IV

676

8.6. Consider the general case of flow through a pump impeller as illustrated in Figure 8.38. (a) Develop a general expression for the power input to the pump in terms of the inner and outer impeller radii, r1 and r2, the inner and outer blade widths, b1 and b2, inner and outer blade angles, β1 and β2, rotational speed, ω, volumetric flow rate, Q, and density of the fluid, ρ. (b) Estimate the power input and head added to the pump for r1 = 200 mm, r2 = 600 mm, b1 = 60 mm, b2 = 40 mm, β1 = 70◦, β2 = 75◦, ω = 700 rpm, and Q = 0.90 m3/s. Assume the liquid is water at 20◦C.

! 1

! 2

Ou!low surfaceInflow surface

Impeller

Ou!low blade angle

Inflow blade angle

Blade

r 1

r 2

ω

Figure 8.38: Flow through a pump impeller V

8.7. A pump impeller similar to the one shown in Figure 8.38 has inner and outer radii of 150 mm and 300 mm, respectively, inner and outer blade widths of 45 mm and 40 mm, respectively, and inner and outer blade angles of 25◦ and 10◦, respectively. The rate of angular rotation is 600 rpm. The liquid being pumped is water at 20◦C. (a) Estimate the volumetric flow rate required for the entrance velocity to be normal to the inflow surface. (b) Estimate the power added by the pump for the flow rate calculated in Part (a).

8.8. A pump impeller has inner and outer radii of 100 mm and 300 mm, respectively, inner and outer blade widths of 40 mm and 35 mm, respectively, and inner and outer blade angles of 50◦ and 60◦, respectively. The flow rate is 250 L/s. The liquid being pumped is water at 20◦C. (a) Estimate the angular speed required for the entrance velocity to be normal to the inflow surface. (b) Estimate the power added by the pump for the angular speed calculated in Part (a).

8.9. A pump impeller has inner and outer radii of 150 mm and 400 mm, respectively, inner and outer blade widths of 50 mm and 45 mm, respectively, and the outer blade angle is 70◦. The flow rate is 500 L/s and the rotational speed is 600 rpm. The liquid being pumped is water at 20◦C. (a) Estimate the inner blade angle required for the entrance velocity to be normal to the inflow surface. (b) Estimate the power added by the pump for the angular speed calculated in Part (a).

8.10. A pump impeller is to be designed to have an inner and outer radius of 125 mm and 250 mm, respectively, and an inner and outer blade width of 60 mm and 50 mm, respectively. The impeller is to have a rotational speed of 1725 rpm, and is to deliver a volumetric flow rate of 0.3 m3/s against a head of 13.8 m. The inflow velocity is desired to be normal to the inflow surface. (a) Design the blade angles at the inflow and outflow surfaces. (b) If the liquid to be pumped is water at 20◦C, what is the power requirement of the pump?

677

8.11. A pump delivers 100 L/s of water when the rotational speed of the impeller is 600 rpm. The inflow velocity is normal to the inflow surface, the outflow velocity relative to the rotating blade is 5 m/s, and the blade angle at the outflow surface is 90◦. The motor consumes 8 kW of power, and it is estimated that the pump motor has an efficiency of 70%. Estimate the radius and width of the outflow surface. Assume water at 20◦C.

8.12. A centrifugal pump uses 8 kW of power and has an efficiency of 80%. The pump delivers gasoline at a rate of 4 L/s. Estimate the maximum change in pressure between the inlet ant outlet of the pump. How would the estimated pressure change be different if a liquid with increased density were used?

8.13. A pump unit housed within a pump station has a rotational speed of 2400 rpm and has an efficiency of 73%. The pump unit is driven by a motor that has an efficiency of 82% and delivers a shaft torque of 8 N·m. The discharge side of the pump station is at an elevation 2.4 m higher than the suction side of the station. Measurements taken when the pump is delivering 6 L/s show a suction-side pressure and velocity of 90 kPa (gage) and 2 m/s, respectively, and the velocity on the discharge side is 5 m/s. Estimate the power delivered to the motor, and the pressure on the discharge side of the pump station. Assume water at 20◦C.

8.14. The blade of a turbine is illustrated in Figure 8.39, where the relative inflow velocity makes an angle of 70◦ with the inflow surface, and the relative outflow velocity makes an angle of 60◦ with the outflow surface. Relative velocities are measured relative to the blades mounted on the rotor. The inflow surface has a radius of 1.75 m, the outflow surface has a radius of 1.05 m, and the height of the rotor is 0.5 m. Under design conditions, the flow rate through the turbine is 35 m3/s and the rotational speed is 150 rpm. Estimate the shaft power extracted by the turbine. Assume water at 20◦C.

1.05 m

1.75 m

150 rpm

W

70o

60o Ou!low surface

Figure 8.39: Flow through a turbine I

8.15. Water at 20◦C flows through a turbine at a rate of 400 L/s and the velocities on the inflow and outflow surfaces of the rotor are shown in Figure 8.40. The absolute velocity and the velocity relative to the rotating blade are represented by V and W, respectively, and the subscripts “1” and “2” indicate conditions at the inflow and outflow surfaces, respectively. On the inflow

678

surface, the incoming flow makes an angle of 25◦ with the tangent to the surface, and it is desired that the velocity relative to the rotor make an angle of 55◦ to the inflow surface so as to match the rotor angle. On the outflow surface, it is assumed that the velocity relative to the moving blade will make an angle of 50◦ with the outflow surface so that it matches the inner blade angle. The inflow and outflow surfaces have radii if 800 mm and 400 mm, respectively, and the rotor rotates at 130 rpm. (a) Determine the required height of the rotor. (b) Estimate the shaft power generated by the turbine.

Inflow surface

Ou!low surface

V 1

W 1

W 2

V 2

25o

55o

50o

!

Figure 8.40: Flow through a turbine II

Section 8.3: Hydraulic Pumps and Pumped Systems

8.16. A proposed pump design has an impeller with inner and outer radii of 125 mm and 225 mm, respectively, inflow- and outflow-area widths of 50 mm and 45 mm, respectively, and the blade angles at the inflow and outflow surfaces of 35◦ and 15◦, respectively. The impeller rotates at 1140 rpm, and the estimated efficiency of the pump is 85%. Estimate (a) the optimal flow rate through the pump, (b) the head added by the pump at the optimal flow rate, and (c) the estimated power requirement to run the pump. Assume water at 20◦C.

8.17. A homologous series of centrifugal pumps is driven by 1200-rpm motor. For a 500-mm size, the best efficiency of 81% occurs when the flow rate is 250 L/s and the total dynamic head is 63.7 m. What is the best-efficiency operating point for a 250-mm size? Estimate the efficiency?

8.18. Pump manufacturers typically present the performance characteristics of their pumps in di- mensional form as shown in Figure 8.41, where the flow rate is in m3/h and the head added by the pump is in m. The efficiencies in % at various operational states are represented by dashed lines on the performance curve. The performance curves shown in Figure 8.41 are for homologous pumps with a motor speed of 3500 rpm and impeller sizes of 178 mm, 165 mm, 152 mm, 140 mm, and 127 mm. (a) Plot on a single graph the nondimensional performance curves for each of the pumps. (b) Compare the nondimensional performance curves and as- sess whether these pumps are really from a homologous series. (c) Identify the maximum efficiency and specific speed for each impeller size.

679

H e

a d

, h

p (

m )

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30 35

Flow rate, Q (m3/h)

2 5

%

3 5

%

4 0

%

4 5

%

5 0

%

5 3

% 5

5 %

5 7

%

59%

59.5%

57%

55% 53%

50%

45%

178 mm

165 mm

152 mm

140 mm 127 mm

40

Figure 8.41: Performance curves of homologous pump series

8.19. Water at 15◦C is pumped at a rate of 20 L/s using a 5 kW pump. If the efficiency of the pump is 80%, what is the head added to the pump?

8.20. A Model X pump with an impeller diameter of 300 mm and rotational speed of 1800 rpm has its maximum efficiency at a flow rate of 300 L/s, at which point the head added by the pump is 65 m and the brake horsepower is 240 kW. A geometrically similar Model X pump with an impeller size of 250 mm is driven by a 1400 rpm motor. The working fluid is water at 20◦C. (a) Determine the flow coefficient, head coefficient, and power coefficient of the 300-mm pump when it is operating at its most efficient point. (b) What is the flow rate, head added, and brake horsepower of the 250 mm pump when it is operated at its most efficient point?

8.21. A pump with a rotary speed of 1725 rpm delivers 25 L/s at its most efficient operating point. Under this condition, the inflow velocity is normal to the inflow surface of the impeller, and the component of the velocity normal to the outflow surface of the impeller is 4 m/s. The width of the impeller at the outflow surface is 15 mm, and the blade angle at the outflow surface is 50◦. (a) Estimate the head added by the pump. (b) Use the affinity laws to estimate the added head and flow rate delivered by the pump when the rotational speed is changed to 114 0 rpm.

8.22. A pump manufacturer makes three homologous series of pumps with three different specific speeds, and specifies the performance of each series by nondimensional functional relation- ships. An engineer identifies the required homologous series, which has a best-efficiency point with a flow coefficient of 0.035, a head coefficient of 0.14, and a power coefficient of 0.006. For a specified impeller diameter of 600 mm and a rated rotational speed of 1140 rpm, determine the discharge, head, brake horsepower, and efficiency of the requisite pump. Assume water at 20◦C.

8.23. A pump is to be chosen from a homologous series which has a best-efficiency point with a flow coefficient of 0.035, a head coefficient of 0.14, and a power coefficient of 0.006. The desired

680

pump is to have a best-efficiency flow rate of 200 L/s and a motor with a speed of 850 rpm. What is the impeller size of the required pump? Assume water at 20◦C.

8.24. An existing pump has a 400-mm diameter impeller and under maximum-efficiency conditions the efficiency of the pump is 85%, the flow rate is 200 L/s, the head added is 24 m, and the power consumed by the pump is 50 kW. The pump is to be refurbished by replacing the existing impeller with a geometrically similar impeller that has a diameter of 350 mm. (a) Neglecting scale effects, estimate the flow rate, head added, and power consumption of the pump with the new impeller when the pump is operating under-maximum-efficiency conditions. (b) Quantify the scale effect on the efficiency of the pump and assess whether the scale effect is expected to be significant.

8.25. A homologous series of centrifugal pumps are driven by 2400-rpm motors. For a 400-mm size within this series, the manufacturer claims that the best efficiency of 85% occurs when the flow rate is 500 L/s and the head added by the pump is 89.5 m. What would be the best-efficiency operating point for a 300-mm size within this homologous series, and estimate the corresponding efficiency.

8.26. A 1:4 scale model of a water pump is operated at a speed of 4500 rpm. At its best efficiency point, the efficiency of the model pump is 84%, and the model delivers a flow rate of 0.7 m3/s with an added head of 4.9 m. If the full-scale pump has a rotational speed of 120 rpm, what is the flow rate and head delivered by the full-scale pump? What is the power requirement of the full scale pump? Assume water at 20◦C.

8.27. Affinity laws are typically used to identify homologous values of flow rate, head, and power between geometrically similar pumps. Develop an affinity law for relating homologous values of torque.

8.28. A manufacturer tests a 1:10 scale model of a pump in the laboratory. The model pump has an impeller diameter of 200 mm and a rotational speed of 3450 rpm, and when the head across the pump is 40 m the pump delivers a flow rate of 10 L/s at an efficiency of 84%. The prototype pump is to develop the same head a the scale model, however, because of its increased size, the prototype pump is expected to have an efficiency of 90%. (a) What is the power supplied to the model pump? (b) What is the rotational speed, flow rate, and power supplied to the prototype pump under homologous conditions? Assume water at 20◦C.

8.29. A pump has an impeller diameter of 250 mm and a rotational speed of 1200 rpm. At the best-efficiency operating point, the pump adds a head of 8.2 m at a flow rate of 20 L/s. What is the specific speed of the pump? What type of pump is this likely to be?

8.30. What is the constant that can be used to convert the specific speed in SI units (Equation 8.39) to the specific speed in U.S. Customary units (Equation 8.40)?

8.31. Express the specific speed of a pump in terms of the head coefficient and flow coefficient at the best-efficiency point.

8.32. A pump is required to deliver 500 L/s from a lake to a storage reservoir. Application of the energy equation to the pipeline system shows that the head that must be added by the pump

681

is 80 m. If the pump motor is to have a rotational speed of 1800 rpm, what is the specific speed of the required pump? What type of pump is required?

8.33. The head a flow rate to be delivered by a particular pump installation are to be changed, however, the existing pump is to be retained with a new motor installed. The existing pump has a specific speed of 4.2. Under the new operating conditions, the pump is required to deliver 300 L/s with an added head of 10 m. What rotational speed should the new motor have? What type of pump is this?

8.34. What is the highest synchronous speed for a motor driving a pump?

8.35. A pump is to be selected such that it operates at or near its most efficient state when delivering 100 L/s with an added head of 50 m. A manufacturer has five models of homologous pumps, with specific speeds of 0.95, 0.81, 0.61, 0.36, and 0.25. Motors with any of the standard rated speeds can be provided with these pumps. Identify the best choice of specific speed and the rotational speed of the motor that should be used to drive the pump.

8.36. A pump has an impeller diameter of 450 mm, and at its most efficient operating point it delivers water at a flow rate of 650 L/s with an added head of 9.5 m. The specific speed of the pump is 1.5, and the motor delivers power to the shaft at a rate of 80 kW. (a) Estimate the shutoff head of the pump. (b) Estimate the efficiency of the pump at its best operating point. Assume water at 20◦C.

8.37. A pump with an impeller size of 250 mm is operated at a rotational speed of 1200 rpm. A review of the pump performance data shows that the pump has a shutoff head of 8.2 m and at the best-efficiency operating point the pump adds a head of 6.5 m at a flow rate of 20 L/s. (a) Use the performance data to fit a parabolic performance curve of the form hp = a− bQ2, where hp is the head added by the pump in m, Q is the flow rate in L/s, and a and b are constants. (b) If the rotational speed of the pump impeller is increased to 1800 rpm, estimate the parabolic performance curve at the adjusted speed.

8.38. A prototype water pump has a specific speed of 1.2, and when operating at its most efficient state delivers 5 L/s with an added head of 10 m. Operation of the pump is to be tested using a 15 -scale model with various test fluids that have dynamic viscosities in the range of 5–10 times that of water. It is desired that viscous effects be accurately accounted for in the model. What range of rotational speeds, flow rates, and heads will be required in model testing? Assess whether accurately accounting for viscous effects is realistic.

8.39. Pressure measurements are taken in the inflow and outflow pipes of a pump as shown in Figure 8.42. When the flow rate through the pump is 10 L/s, the measured (gage) pressure on the inflow side is −20 kPa, and the measured (gage) pressure on the outflow side is 100 kPa. The measurement section on the outflow side is 0.83 m above the measurement section of the inflow side. The diameters of the inflow and outflow pipes are 150 mm and 75 mm, respectively. The pumped fluid is water at 10◦C, and the head loss in the pipes between the measurement sections an the pump can be assumed negligible. Estimate the head added by the pump.

682

P

150 mm

75 mm 100 kPa

-20 kPa

0.83 m

10 L/s

Pump

Figure 8.42: Flow through a pump

8.40. A centrifugal pump is to be used to pump water at a rate of 50 L/s from a sump to a storage reservoir, where the water surface in the storage reservoir is 25 m higher than the water surface in the sump. The suction and discharge line from the pump have a total length of 48 m, a diameter of 125 mm, and an estimated roughness height of 0.5 mm. The sum of the local-loss coefficients in the pipeline is equal to 4, the specific speed of the pump is 0.8, and the manometric efficiency of the pump is 80%. In the homologous series from which the pump is selected, the impeller blades are forward curved and make an angle of 55◦ with the outflow surface, the height of the impeller is equal to 12.5% of the diameter, and the blades cover 7% of the outer area of the impeller. Inflow to the impeller is normal to the inflow surface. What size impeller is should be selected for this application? Assume water at 20◦C.

8.41. The performance characteristics of a pump are determined using the system shown in Figure 8.43, where simultaneous measurements are taken of the flow rate, Q, the pressures at the inflow and outflow pipes sections, p1 and p2, respectively, and the power consumption, P , of the pump. These measurements are as follows:

Q (L/s) 1.26 2.52 3.79 5.05 6.31 7.57 8.83

p2 − p1 (kPa) 671 665 653 629 593 557 498

P (kW) 3.54 4.22 4.97 5.60 6.27 7.20 7.77

The outflow pipe section has an elevation 0.38 m higher than the inflow section, and the temperature of the water used in the test is 20◦C. (a) Plot the performance curve of the pump, showing hp versus Q and η versus Q, where hp is the head added by the pump and η is the efficiency. Preferably, both of these relationships should be shown on the same graph. (b) What is the flow rate through the pump when it is operating at maximum efficiency, and what is the maximum efficiency of the pump?

683

Δz

p 1

p 2

Flow

Pump

Figure 8.43: Pump performance test

8.42. A pump is required to deliver 150 L/s (±10%) through a 300-mm diameter PVC pipe from a well to a reservoir. The water level in the well is 1.5 m below the ground surface and the water surface in the reservoir is 2 m above the ground surface. The delivery pipe is 300 m long, and local losses can be neglected. A pump manufacturer suggests using a pump with a performance curve given by

hp = 6− 6.67× 10−5Q2

where hp is in meters and Q in L/s. Is this pump adequate?

8.43. A pump with the performance curve shown in Figure 8.44 is being considered to pump water from a lower reservoir to a higher reservoir where the difference in water surface elevations is 9.0 m. The conduit is 100 m long, 100 mm in diameter, and has negligible roughness. Estimate the maximum flow rate that can be achieved if this pump is used. At what efficiency would the pump operate? Assess the desirability of using this pump.

0

4

8

12

600 1200 18000

Flow (L/min)

To ta

l d yn

am ic

h e

ad (

m )

20

40

60

0

Effi ci

e n

cy (

% )

head

efficiency

Figure 8.44: Pump performance curve

8.44. Water is pumped from a lower tank to a higher tank having a difference in elevation of 10 m. The piping system consists of 200-mm ductile iron pipe with a length of 2 km and minor losses equal to 6.2 times the velocity head. The pump characteristics are shown in Table 8.3. (a) Determine the expected rate of flow through the system, assuming fully-turbulent flow; and (b) what is the power of the motor required to drive the pump?

684

Table 8.3: Pump Characteristics

Discharge (L/s) 0 10 20 30 40 50

Total head (m) 25 23.2 20.8 17.0 12.4 7.3

Efficiency (%) – 45 65 71 65 45

8.45. The pumped-storage system illustrated in Figure 8.45 is designed to exchange 2 m3/s through a 1220-mm diameter 3.2-km long ductile-iron pipeline lined with bitumen. The elevation difference between the water surfaces in the upper and lower reservoirs is 61 m, and the pump/turbine is to operate for 8 hours during the day as a turbine (to generate electricity) and 8 hours during the night as a pump (to return the water to the upper reservoir). The pump efficiency is 85%, the turbine efficiency is 90%, the cost of pumping is $0.06/kWh, and the revenue from turbine operations (selling electricity) is $0.12/kWh. Determine the annual profit from this operation. Neglect the effect of storage on reservoir elevations. If the pump performance curve is given by hp = 80 − 3.5Q2 where hp is in meters and Q is in m3/s, estimate the change in profit when the elevation difference is 65 m. For an elevation difference of 65 m, assume that the flow is fully turbulent.

P/T

61 m

Pump/Turbine

Pipeline

Upper reservoir

Lower reservoir

Figure 8.45: Pumped storage system

8.46. A pump is to be selected to deliver water from a well to a treatment plant through a 300- m long pipeline. The temperature of the water is 20◦C, the average elevation of the water surface in the well is 5 m below the ground surface, the pump is 50 cm above the ground surface, and the water surface in the receiving reservoir at the water-treatment plant is 4 m above the ground surface. The delivery pipe is made of ductile iron (ks = 0.26 mm) with a diameter of 800 mm. If the selected pump has a performance curve of hp = 12−0.1Q2, where Q is in m3/s and hp is in m, then what is the flow rate through the system? Calculate the specific speed of the required pump (in U.S. Customary units), and state what type of pump will be required when the speed of the pump motor is 1200 rpm. Neglect local losses.

8.47. A pump is to be used to withdraw water from a reservoir at a rate of 1500 L/s. When operating at this flow rate, the head loss in the suction pipe is estimated to be 2.3 m, and the pump specifications give the required net positive suction head as 2.9 m. Standard sea- level atmospheric conditions are expected at the site, and under worst-case conditions the temperature of the water in the reservoir is 25◦C. What is the maximum allowable elevation of the suction side of the pump above the reservoir water surface?

685

8.48. Under design conditions a water pump is expected to deliver 500 L/s with an added head of 75 m. The diameter of the suction pipe is 550 mm. Under worst-case conditions, the temperature of the water is expected to be 80◦C and atmospheric pressure is 96 kPa. The manufacturer states that the cavitation parameter of the pump is equal to 0.12. A pressure gage is installed on the suction side of the pump to detect conditions when cavitation is likely to occur. What will be the pressure reading when cavitation is likely to occur?

8.49. Water at 20◦C is to be pumped out of a reservoir at a rate of 20 L/s through a vertical 150-mm diameter ductile-iron pipeline with an estimated roughness height of 0.3 mm. Appurtenances installed in the intake pipe are expected to contribute to a total local head loss coefficient of 12. The pump being considered for installation has a required net positive suction head of 5.5 m. (a) What is the maximum elevation of the pump relative to the water surface in the reservoir? (b) If the total head loss coefficient can be reduced to 1.2 m, how much higher could the pump be placed? Assume standard atmospheric conditions.

8.50. Tests on a pump under standard atmospheric conditions show that when water at 20◦C is pumped at 60 L/s and the head added by the pump is 40 m, cavitation occurs when the pressure head plus velocity head on the suction side of the pump is 3.9 m. (a) Determine the required net positive suction head and the cavitation number of the pump. (b) If this same pump is operated on a mountain under the same flow rate and added head condition, but the temperature of the water is 5◦C and the atmospheric pressure is 90 kPa, by how much must the elevation of the pump above the sump reservoir be reduced compared with the test condition? Assume that the friction loss in the suction pipe remains approximately the same and that the sump reservoir is open to the atmosphere in both cases.

8.51. A pump with an impeller diameter of 225 mm and a rotational speed of 1725 rpm is found to induce cavitation when pumping water at a temperature of 80◦C. Under these conditions, the volumetric flow rate is 50 L/s, the pressure on the suction side of the pump is 80 kPa, and the corresponding velocity in the suction pipe is 5 m/s. (a) What is the net positive suction head of the pump? (b) If a geometrically similar pump has a rotational speed of 1140 rpm and an impeller diameter of 675 mm, what is the net positive suction head than is expected in the larger pump?

8.52. Water is to be pumped out of a well and stored in an above-ground reservoir. The water surface in the well is 3 m below the ground surface, water is to be pumped through a 100-m long 50-mm diameter galvanized iron line and exit 19.3 m above the ground, and water is to be delivered to the upper reservoir at a rate of at least 370 L/min when the reservoir is empty. The sum of the local loss coefficients in the system is 1.8. A local pump salesman suggests a pump model with performance curves shown in Figure 8.46. Determine if this pump will meet the demands of the project and, if so, the pump size that is required. What is the maximum height that the pump can be placed above ground?

686

60

50

40

30

30

30

20

10 10

10

10

N P

SH in

fe et

N P

SH in

m et

er s

To ta

l h ea

d in

m et

er s

To ta

l h ea

d in

fe et

0 0 0

0

0 0

5

5

20

20

20

20

40

40352515

40

60

60

80

80

100

100

120

120

140

140

160

160

NPSH R EQ

180

200

180

220

240

260

40 %

45 %

45% 50

%

50%

53 %

53%

55 %

55%

57 %

57 %

59 %

59 %

35 %

70

Series 80-SC Bell & Gossett

7.0 in.

Capacity in cubic meters/hr

Capacity in U.S. gallons per minute

25 %

59.5%

6.5 in.

6.0 in.

5.5 in.

5.0 in.

3500 rpm 1½ × 1½ × 7B

Figure 8.46: Pump performance curves

8.53. Consider the 6.5-in. (165-mm) pump with the performance curve shown in Figure 8.46. The motor on this pump is changed from one with a rotational speed of 3500 rpm to one with a rotational speed of 2500 rpm. (a) Plot the performance curve of the modified pump, showing both the head added by the pump and the efficiency of the pump as a function of the flow rate. Use the following units in your plot: flow rate in L/s, head in m, and efficiency in %. (b) What are the flow rate, head added, and efficiency of the modified pump at its best efficiency point? (c) Calculate and compare the specific speeds of the existing and modified pumps and use these results to infer the type of pump.

8.54. A pump is located 2 m below a reservoir as shown in Figure 8.47. The piping between the pump and the reservoir consists of 3 m of 150 mm ductile iron pipe (DIP). It is estimated that the DIP has an equivalent sand roughness of 0.25 mm. The valves and bends between the source reservoir and the pump are estimated to have a total local loss coefficient of 50. Under design conditions, the water in the reservoir has a temperature of 25◦C and the atmospheric pressure is 101.3 kPa. The pump specifications require that the pump have a minimum net positive suction head of 4 m. What is the maximum allowable flow rate through the system so as to avoid cavitation?

687

P

2 m

3 m of 150 mm DIP

Pump

Flow

Reservoir

Figure 8.47: Flow in a pumped system

8.55. A pump lifts water through a 100-mm diameter ductile-iron pipe from a lower to an upper reservoir (Figure 8.48). If the difference in elevation between the reservoir surfaces is 10 m, and the performance curve of the 2400-rpm pump is given by

hp = 15− 0.1Q2

where hp is in meters and Q in L/s, then estimate the flow rate through the system. If the pump manufacturer gives the required net positive suction head under these operating conditions as 1.5 m, what is the maximum height above the lower reservoir that the pump can be placed and maintain the same operating conditions?

10 m

100 m

Lower reservoir 1 m

Upper reservoir

P

3 m

Figure 8.48: Water pumped from lower to upper reservoir

8.56. Water is being pumped from reservoir A to reservoir F through a 30-m long PVC pipe of diameter 150 mm (see Figure 8.49). There is an open gate valve located at C; 90◦ bends (threaded) located at B, D, and E; and the pump performance curve is given by

hp = 20− 4713Q2

where hp is the head added by the pump in meters and Q is the flow rate in m3/s. The specific speed of the pump (in U.S. Customary units) is 3000. Assuming that the flow is turbulent (in the smooth, rough, or transition range) and the temperature of the water is 20◦C, (a) write the energy equation between the upper and lower reservoirs, accounting for entrance, exit, and local losses between A and F; (b) calculate the flow rate and velocity in

688

the pipe; (c) if the required net positive suction head at the pump operating point is 3.0 m, assess the potential for cavitation in the pump (for this analysis you may assume that the head loss in the pipe is negligible between the intake and the pump); and (d) use the affinity laws to estimate the pump performance curve when the motor on the pump is changed from 800 rpm to 1600 rpm.

P

3 m

FE

A

B C

D 10 m

Figure 8.49: Water-delivery system

8.57. The performance curve of a Goulds Model 3656 irrigation pump is shown in Figure 8.50. The pump is to be used to deliver water at 20◦C from a pond to the center of a large field located 100 m from the pond. The desired pump is expected to deliver a maximum flow rate of 380 L/min through 107 m of 6-cm steel pipe having an equivalent sand roughness of 0.01 mm. The water level in the pond during the irrigation season is 10.00 m and the ground elevation of the field is 15.00 m. (a) Which of the pumps shown in Figure 8.50 would you select for the job? (b) Using the efficiency of the pump under operating conditions, calculate the size of the motor in kilowatts that must be used to drive the pump. (c) What is the maximum elevation above the pond that the pump could be located?

689

0

10

15

5

Model 3656 / Size 2×2-5

RPM 3450 Curve CN346R00 IMP. DWG. NO. 119-86

To ta

l d yn

am ic

h ea

d

0 10 20 30 40 m 3/h

Capacity

0

10

20

30

40

50

NOTE: Not recommended for operation beyond printed N-Q curve.

0 20 40 60 80 100 120 140 160 180

ft

4 in.

30% 43% 50%

NPSHR - ft 5 ft

6 ft

7 ft 9 ft

16 ft

55%

55% 50%

43%

60%

60%

64%

64%

gpm

m

3.75 in.

3.5 in.

3.25 in.

3.0 in.

Figure 8.50: Goulds Model 3656 pump curves

8.58. If the performance curve of a certain pump model is given by

hp = 30− 0.05Q2

where hp is in meters and Q is in L/s, what is the performance curve of a pump system containing n of these pumps in series? What is the performance curve of a pump system containing n of these pumps in parallel?

8.59. A pump is placed in a pipe system in which the energy equation (system curve) is given by

hp = 15 + 0.03Q 2

where hp is the head added by the pump in meters and Q is the flow rate through the system in L/s. The performance curve of the pump is

hp = 20− 0.08Q2

What is the flow rate through the system? If the pump is replaced by two identical pumps in parallel, what would be the flow rate in the system? If the pump is replaced by two identical pumps in series, what would be the flow rate in the system?

8.60. A wastewater pump station is required to handle a design flow rate of 1000 L/s, and the pump station must provide an added head of 9 m. The maximum feasible size of each pump/motor unit has a power demand of 35 kW, and the overall efficiency of these pump/motor units is 62%. The specific speed of each unit is around 1.5. (a) If the pump units are to be placed in

690

parallel to accommodate the design flow, how many pump units are required? (b) What motor speed would be required for optimal performance of each unit? Assume that the properties of the wastewater are approximately equal to those of water at 20◦C.

8.61. Nine pump units are placed in parallel at a pump station. Each unit has a power demand of 40 kW, and adds 35 m of head. The overall efficiency of each unit is 60%. The liquid being pumped is water at 20◦C. When all units are operating, what is the flow rate delivered by the pump station?

8.62. Water is to be pumped from a river to an elevated reservoir through a 250-mm diameter pipe that has an equivalent sand roughness of 0.1 mm. The water surface in the reservoir is 70 m above the water surface in the river, and the length of pipeline is 5 km. Initially, a flow rate between the river and the reservoir of 35 L/s will be required, however, it is expected that the required flow rate will double to 70 L/s at some time in the future. The chief engineer on the project has proposed that a pump from a homologous series with performance characteristics shown in Figure 8.51 be used to deliver the initial flow rate of 35 L/s, and that the pump station be designed such that additional pump(s) can be added in parallel to handle the increased flow when it becomes necessary. (a) Determine the required pump size to deliver 35 L/s, (b) what would be the efficiency of the pump when it is delivering 35 L/s, (c) what is the maximum height that this pump can be placed above the water surface of the river, and (d) how many pumps would be needed to deliver 70 L/s. Clearly explain your calculations. You can assume that the flow in the pipeline will generally be fully turbulent, and for Part b you can assume that the entrance loss and friction head loss in the suction pipe is negligible.

0 10 20 30 40 50 60

Flow rate (L/s)

20

40

60

80

100

120

140

H e

a d

( m

)

10 20 30 40 50 60

4 8

0

N P

SH r (

m )

150 mm

175 mm

200 mm

225 mm

250 mm 50 60 65 68 71

73 73

73 71

68

Flow rate (L/s)

Efficiency (%)

250 mm 150 mm

Figure 8.51: Pump performance characteristics

8.63. A 20-km long 1120-mm diameter steel pipe with an estimated roughness height of 0.05 mm is to deliver water from a water-supply reservoir through a system of parallel pumps as shown in Figure 8.52. The intake at A and the exit from the pump system at B both are at an elevation of 5 m, the water level in the reservoir is 2 m above the intake at A, and the elevation at the delivery point, C, is 15 m. There is negligible head loss due to friction between A and B, and

691

the performance curve for each pump in the system is given by

hp = 65− 7.6× 10−8Q2

where hp is the head added by the pump in m and Q is the flow through the pump in m3/d. When the system is delivering 48,000 m3/d at C, the required pressure at C is 448 kPa. (a) Determine how many pumps are required. (b) Assuming that the flow is fully turbulent, what is the actual flow rate at C when using the number of pumps determined in Part (a)? The temperature of the water is 20◦C.

P

P

P

P

A B

C

Figure 8.52: Parallel-pump system

8.64. The water-supply system shown in Figure 8.53 is to be constructed such that water is delivered from a reservoir at A to two communities located at C and D. The pipe lengths, diameters, and demand flow rates are as follows:

Length Diameter Flow Line (km) (mm) (L/s)

AB 1.05 200 27 BC 2.80 150 12 BD 2.50 150 15

The water-surface elevation of the supply reservoir is 3.00 m, and the elevations of the delivery pipes at C and D are 2.00 m and 5.00 m, respectively. Under the given demand conditions, it is desired to have water pressures of at least 350 kPa at C and D. All pipes are to be made of ductile iron. (a) Determine the minimum power that must be delivered by the pump. (b) A pump manufacturer will be able to match the minimum-power operating condition by providing several “micro-pumps” in series where each micro-pump has a performance performance given by

hp = 0.455D − 4000Q2

where hp is the head added in m, D is the size of the micro-pump in cm, and Q is the flow in m3/s. The manufacturer can deliver any size, D, in the range of 40–50 cm. How many micro-pumps will be needed and of what size?

692

Elev. 3.00 m A

B P

C

D

Elev. 2.00 m p ≥ 350 kPa

p ≥ 350 kPa Elev. 5.00 m

Figure 8.53: Water-supply system

8.65. The performance curve for a variable-speed pump operating at 600 rpm is given by

hp = 6− 0.05Q2

where hp is the head added by the pump in m and Q is the flow rate in m3/min. This pump is installed in a system where energy considerations require a system curve given by

hp = 3 + 0.042Q 2

Find the flow rate in the system when the pump is operating at 600 rpm and compare this with the flow rate when the pump is operating at 1200 rpm.

8.66. A large centrifugal fan generates a flow of 7 m3/s with a motor speed of 1140 rpm. A smaller geometrically similar fan has a motor speed of 1725 rpm, operates at the same efficiency as the larger fan, and generates the same pressure increase. What flow rate is generated by the smaller fan?

Section 8.5: Hydraulic Turbines and Hydropower

8.67. A single-jet Pelton-wheel hydropower plant is to be operated such that the shaft power de- veloped by the Pelton wheel is equal to 15 MW when the head just upstream of the nozzle is equal to 1600 m. The Pelton wheel is constructed such that is has a diameter of 4 m, a deflection angle of 170◦, and rotates at a controlled speed of 600 rpm. Determine the appro- priate nozzle diameter to be used in the project. Assume water at 20◦C and a nozzle loss coefficient of 0.03.

8.68. An existing Pelton wheel has a diameter of 3 m and is controlled to rotate at 150 rpm. The buckets on the Pelton wheel have a deflection angle of 170◦. The power output from the Pelton wheel is controlled by adjusting the jet nozzle diameter. If the desired power is 850 kW, what is the required nozzle diameter for the most efficient operation of the Pelton wheel? Assume water at 20◦C.

8.69. A Pelton wheel has a diameter of 2.7 m and buckets with a deflection angle of 167◦. The nozzle and water-supply characteristics are fixed such that, under design conditions, the incident jet has a diameter of 150 mm and a velocity of 12 m/s. What is the maximum power that can be extracted by this Pelton wheel? What is the rotational speed of the wheel under the maximum-power condition? Assume water at 20◦C.

693

8.70. A 3-m diameter Pelton wheel is driven by a single 200-mm diameter jet, and the pressure and velocity in the pipeline just upstream of the jet is 3 MPa and 5 m/s, respectively. The nozzle loss coefficient is estimated as 0.03. The buckets on the Pelton wheel have a deflection angle of 160◦. Estimate the following: (a) the maximum power that can be generated by the Pelton wheel, (b) the rotational speed at the maximum-power state, (c) the runaway rotational speed, (d) the torque on the wheel shaft at the maximum-power state, and (e) the torque on the wheel shaft when the wheel is held stationary. Assume water at 20◦C.

8.71. A manufacturer cites the wheel efficiency of an impulse turbine as 85% when the turbine is operating at its maximum-efficiency state. Estimate the blade angle of the buckets mounted on the turbine.

8.72. A Pelton wheel has an average radius of 1.95 m, and its buckets have a deflection angle of 160◦. According to the manufacturer, the Pelton wheel has a wheel efficiency of 85% under optimal operating conditions. In a particular installation, the turbine is driven by a jet with a diameter of 150 mm and a velocity of 95 m/s. (a) Estimate the optimal rotation rate of the turbine. (b) Estimate the maximum power that can be extracted by the turbine. Assume water at 20◦C.

8.73. Show that if the effective head on a Pelton wheel is he, the velocity coefficient of the nozzle is Cv, and the bucket speed of the wheel is U , then the theoretical maximum efficiency is attained by the Pelton when when

U = 12Cv √

2ghe

8.74. The peripheral velocity factor of a Pelton wheel, φ, is defined as φ = U/ √ 2ghe, where U is

the bucket speed, and he is the effective head on the turbine. In Problem 8.73 it is shown that the efficiency of a Pelton wheel is maximized when φ = 12Cv, where Cv is the velocity coefficient of the nozzle. Show that the maximum efficiency of a Pelton wheel turbine, ηt, under these conditions is given by

ηt = 1 2C

2 v(1− cosβ)

where β is the bucket angle.

8.75. The loss in the nozzle of a Pelton wheel is sometimes represented by a local head loss coef- ficient, kj, and is sometimes represented by a nozzle velocity coefficient, Cv. Show that the relationship between Cv and kj is given by

Cv =

√ 1

1 + kj

8.76. A Pelton wheel has a diameter of 4.2 m and a bucket angle of 165◦. The effective head at the nozzle is 524 m, and the nozzle is set such that the velocity coefficient is 0.94 and the diameter of the jet is 125 mm. Estimate the optimal flow rate, rotational speed of the wheel, and power output. Assume water at 20◦C.

694

8.77. A Pelton wheel is to be designed to harness the available hydropower from a site where the effective head on the turbine will be 150 m and the reliable flow rate through the turbine will be 5 m3/s. The wheel is to have a rotational speed of 440 rpm, the bucket angle will be 165◦, and the nozzle is expected to have a velocity coefficient of 0.92. (a) What diameter wheel would maximize the efficiency of the turbine? (b) What power can be expected from the turbine? (c) Assess whether a different type of turbine should be considered for this site. Assume water at 20◦C.

8.78. The Pelton wheel shown in Figure 8.54 is driven by two jets, where each jet has a diameter of 30 mm and a velocity of 40 m/s. The buckets mounted on the Pelton wheel each deflect their incident jets by 160◦. The diametric distance between the centers of the buckets is 1 m, and it can be assumed that the reaction forces on the each of the buckets act at the center of the bucket. (a) What torque is required to hold the Pelton wheel stationary? (b) If the Pelton wheel is allowed to “freewheel” such that the shaft torque is negligible, at what speed will the Pelton wheel rotate? (c) If a governor controls the rotational speed at 120 rpm, what power can be generated by the Pelton wheel? Assume water at 20◦C.

160o

Flow

Flow

Pelton wheel

Deflected jet

1 m

Figure 8.54: Two-jet Pelton wheel

8.79. The impulse-turbine system shown in Figure 8.55 uses water at 20◦C from a reservoir with a water-surface-elevation that is 80 m above the elevation of the nozzle. The length of the 300-mm delivery pipe is 600 m, and the delivery pipe has an estimated roughness of 0.5 mm. The nozzle at the end of the pipe is estimated to have a nozzle-loss coefficient of 0.05, and the buckets on the impulse turbine have a deflection angle of 160◦. (a) Identify the optimum nozzle diameter, and state the corresponding power that can be extracted by the turbine. (b) Determine the optimum nozzle diameter and corresponding power derived by assuming fully turbulent flow in the supply pipe, and compare your result with that obtained in Part (a). Comment on the appropriateness of assuming fully turbulent flow.

695

Reservoir

80 m Impulse turbine Delivery pipe

600 m

Nozzle Jet

Figure 8.55: Impulse-turbine system

8.80. At a Pelton-wheel installation the water-surface elevation of the supply reservoir is 85 m above the nozzle; the delivery line has a diameter of 0.60 m, a length of 300 m, and a roughness height of 8 mm. The discharge nozzle has a diameter of 50 mm and a head-loss coefficient of 0.8. The bucket friction loss coefficient is 0.5, the velocity of water relative to the bucket at the exit from the bucket is 2 m/s, and the absolute velocity of the water leaving the bucket is 6 m/s. Determine the power that could be derived from the system and the hydraulic efficiency of the turbine.

8.81. An Francis-turbine unit rotating at 900 rpm is to be designed to deliver a head of 180 m when operating under design conditions. Based on experience with similar units, the overall efficiency of the turbine unit is expected to be 82% and the hydraulic efficiency is expected to be 90%. The guide vanes oriented to direct the inflow at an angle of 30◦ to the inflow surface, and the vanes in the runner are at to be at an angle of 60◦ to the inflow surface as shown in Figure 8.56. The height of the runner is 20% of the diameter of the inflow surface and the outflow velocity is normal to the outflow surface of the runner. What outer diameter of the turbine runner is required?

Inflow surface

V

U

W

Inflow from guide vane

Tip rota!on

Rela!ve velocity in runner

30o

60o

Figure 8.56: Turbine inflow velocities

8.82. A Francis turbine has an overall efficiency of 92% and a hydraulic efficiency of 95%. The turbine runner has an outer diameter of 2.5 m, a height of 0.35 m, and rotates at 300 rpm. The turbine is located 4 m above the discharge reservoir surface, and the head loss in the draft tube can be assumed negligible. Under design conditions, the flow rate through the turbine is 20 m3/s, and the velocity and pressure at the entrance to the scroll case are 10 m/s and 2447 kPa, respectively. The flow exits the runner in a direction normal to the outflow surface. (a) Estimate the power output of the turbine unit. (b) What is the specific speed of the turbine? (c) Estimate the guide vane and runner blade angles. Assume water at 20◦C.

696

8.83. The inflow surface of a runner in a reaction turbine has a diameter and height of 500 mm and 60 mm, respectively, and the outflow surface has a diameter and height of 350 mm and 80 mm, respectively. On both the inflow and outflow surfaces the blades occupy 6% of the flow area. Under design conditions, the inflow and outflow velocity components are shown in Figure 8.57, where V, U, and W are the absolute velocity, tip velocity, and velocity relative to the moving runner, respectively. The guide vane directs the inflow velocity at an angle of 24◦ to the inflow surface, the runner blade is at an angle of 80◦ to the inflow surface, and it is desired that the flow in the runner be aligned with the runner blade. The runner blade makes an angle of 30◦ with the outflow surface. It is estimated that the head extracted by the turbine is 65 m, of which 10% is lost due to hydraulic friction within the turbine, and 5% is lost due to mechanical friction in the turbine system. (a) At what speed should the turbine be operated so that shock losses are minimized, which means that the flow in the runner is aligned with the runner blade on inflow. (b) What is the power output of the turbine? Assume water at 20◦C.

V 1

U 1

W 1

W 2

V 2

U 2

80o

24o

30o

Inflow surface

Ou!low surface

Figure 8.57: Turbine flow velocities

8.84. Flow enters a Francis turbine at an angle of 20◦ to the inflow surface as shown in Figure 8.58. The runner has an outer diameter of 1600 mm, an inner diameter of 900 mm, and the width of the inflow surface is 80 mm. When the runner rotates at 120 rpm, the flow rate through the turbine is 2.9 m3/s, and the outflow velocity is normal to the outflow surface. Estimate the power produced by the turbine. Assume water at 20◦C.

ω 1600 mm

Turbine runner

V

900 mm

Inflow surface

Ou!low surface

20o

Figure 8.58: Flow through a turbine

697

8.85. A hydropower plant utilizes 8 Francis turbines. The change in head across the power plant is 250 m, and the flow through each turbine unit is 12 m3/s. The estimated efficiency of each turbine unit is 95%, and the efficiency of the generator and supporting power delivery systems is 91%. Estimate the power generating capacity of the hydropower facility. Assume water at 20◦C.

8.86. A hydropower installation with reaction-type (Francis) turbines is to be located where the downstream water-surface elevation is 100 m below the water-surface elevation in the reservoir. The 2.0-m-diameter concrete-lined penstock is 500 m long and has an estimated roughness height of 15 mm. When the flow rate through the system is 20 m3/s, the combined head loss in the turbine and draft tube is 5.0 m, and the average velocity in the tailrace is 0.80 m/s. Estimate the power that can be extracted from the system.

8.87. A large hydropower facility uses several Francis turbines in its power plant. Each turbine has a rated power output of 50 MW at a head of 70 m, and has a maximum efficiency of 90%. The generator in each unit has a rotational speed of 120 rpm. (a) Estimate the specific speed of each unit and verify that a Francis turbine is the best type of turbine to use. (b) Estimate the optimal flow rate through each turbine unit. Assume water at 20◦C.

8.88. A Francis turbine with a runner diameter of 600 mm has been working well at a particular hydropower installation for many years, and an analysis of available performance data indicate that the efficiency of this turbine when it is operating at its best-efficiency point is 85%. It is planned to use a geometrically similar turbine with a runner diameter of 2500 mm at another site. Estimate the maximum efficiency that can be expected from the larger unit?

8.89. A hydropower facility that is under design is to required accommodate a design flow rate of 44 m3/s, and under this condition the head that can be extracted by the turbine is 40 m. The turbine under consideration has a shaft rotation rate of 150 rpm, and can generate a shaft power 12 MW when operated at maximum efficiency. (a) What is the efficiency of the turbine under consideration? (b) What type of turbine is being considered? (c) If the head on the turbine during operation is reduced to 18 m and the turbine is operated at maximum efficiency, what shaft power could be extracted by the turbine? Assume water at 20◦C.

8.90. A site being considered for hydropower development has an available head of 786 m and a reliable flow rate of 3.2 m3/s. A generator operating at a speed of 520 rpm is feasible, and the expected efficiency of a turbine unit at this site is 90%. What type of turbine should be considered for use?

8.91. A hydropower facility is reported to have several turbine units, each with a rated flow rate of 9 m3/s at a head of 30 m. The rotational speed of the runner in each turbine is 130 rpm and the efficiency of each unit is 90%. Estimate the type and size of each unit.

8.92. A hydroelectric power plant is reported to have three turbine units, with each unit having a power output of 500 kW, a runner size of 850 mm, and a rotation speed of 210 rpm. When the head on the power plant is 10 m, the flow through each turbine unit is 7 m3/s. What type of turbine unit is likely being used at this site? Assume water at 20◦C.

698

8.93. A proposed turbine is being designed to generate a power of 25 MW, with a generator rota- tional speed of 100 rpm, and an available head of 20 m. A model of the turbine is to be tested in the laboratory, where the available head is 5 m, the power handling capacity is 40 kW, and it is expected that the model turbine will have an efficiency of 90%. What length scale, rotational speed, and flow rate should be used in the model tests? Assume water at 20◦C.

8.94. A Francis turbine that is operating successfully at a particular hydropower facility has a runner of diameter 2.00 m, a rotational speed of 115 rpm, and generates 170 MW of power when the head across the turbine is 68 m and the flow rate through the turbine is 300 m3/s. It is anticipated that these optimal conditions can be scaled up at a new hydropower facility, which will use a turbine with the same rotational speed, but the head across the new turbine will be 95 m. Estimate the runner size, flow rate, power delivered, and efficiency expected at the unscaled facility. Assume water at 20◦C.

8.95. The performance of a turbine is being studied using a 15 -scale model. The prototype (full- scale) turbine operates at a design head of 35 m when the flow rate through the turbine is 64.1 m3/s and the angular speed of the runner is 600 rpm. The model is to be tested at a head of 12 m. (a) What should be the angular speed and flow rate in the model to achieve similarity with the prototype? (b) If the shaft power generated in the model is measured as 110 kW and it is assumed that the efficiency in the prototype is 5% better than the efficiency in the model, estimate the power that is generated in the prototype under design conditions. (c) What is the specific speed of the turbine and what should be its type? Assume water at 20◦C.

8.96. A particular site in a river valley has an available head for hydropower generation of 30 m. Turbines used at such sites by the local power authority typically have a rotational speed of 120 rpm and efficiencies of approximately 90%. It is desired to generate 20 MW of power at this site. (a) What type of turbine is required? (b) Approximately what flow rate through the turbine would generate the required power at the most efficient operating condition? Assume water at 20◦C.

8.97. A Francis turbine is located at an elevation of 2000 m above sea level. The discharge from the turbine unit is 3 m above the tailwater pool, the water flowing through the turbine is at a temperature of 15◦C, and the critical cavitation parameter of the turbine is given by the manufacturer as 0.2. The head loss in the draft tube is negligible. Estimate the maximum head that can be extracted by the turbine without the occurrence of cavitation.

8.98. A proposed hydropower plant is to be located at a site with an available head of 8 m, and it is desired to obtain a (shaft) power of 32 MW from this site. The axial-flow turbine units under consideration operate at an angular speed of 140 rpm, have a specific speed of 4.9, and have an estimated efficiency of 85%. (a) How may of these units are required? (b) What total flow rate must be available to generate the desired power? Assume water at 20◦C.

8.99. (a) What is the typical efficiency associated with a hydropower installation? (b) If a potential site for hydropower generation has an average available head of 300 m and an average flow rate of 0.8 m3/s, estimate the amount of hydropower that could reasonably be expected from this site.

699

8.100. A hydroelectric project is to be developed where the 90-percentile flow rate is 1630 m3/s, and storage to a height of 15 m above the downstream stage is required to meet all the demands. Estimate the maximum installed capacity that would be appropriate for these conditions.

Appendix A

Units and Conversion Factors

A.1 Units

The Système International d’Unités (International System of Units or SI system) was adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960 and is now used by almost the entire world. In the SI system, all quantities are expressed in terms of seven base (fundamental) units. These base units and their standard abbreviations are as follows:

Meter (m): Distance light travels in a vacuum during 1/299 792 458 of a second.∗

Kilogram (kg): Mass of a cylinder of platinum–iridium alloy kept in Paris.

Second (s): Duration of 9 192 631 770 cycles of the radiation corresponding to the transition between two hyper fine levels of the ground state of the cesium-133 atom.

Ampere (A): Magnitude of the current that, when flowing through each of two long parallel wires of negligible cross section separated by one meter in a vacuum, results in a force between the two wires of 2× 10−7 newtons for each meter of length.

Kelvin (K): Defined in the thermodynamic scale by assigning 273.16 K to the triple point of water (freezing point, 273.16 K = 0◦C).

Candela (cd): Luminous intensity of 1/600 000 of a square meter of a radiating cavity at the temperature of freezing platinum (2042 K).

Mole (mol): Amount of substance which contains as many specified entities (molecules, atoms, ions, electrons, photons, etc.) as there are atoms in exactly 0.012 kg of carbon-12.

In addition to the seven base units of the SI system, there are two supplementary SI units: the radian and the steradian. The radian (rad) is defined as the angle at the center of a circle subtended by an arc equal in length to the radius, and the steradian (sr) is defined as the solid angle with its vertex at the center of a sphere that is subtended by an area of the spherical surface equal to the radius squared. The SI units should not be confused with the now obsolete metric units, which were developed in Napoleonic France approximately 200 years ago. The primary difference

∗“Meter” is the accepted spelling in the United States of America; the rest of the world uses the spelling “Metre.”

1015

1016

between metric and SI units is that the former uses centimeters and grams to measure length and mass, while these quantities are measured in meters and kilograms in SI units. The United States is gradually moving toward the use of SI units; however there is still widespread use of the “English” system of units, which are more appropriately referred to as “U.S. Customary” units.

In addition to the seven SI base units, there are several derived units that are given special names. These derived units are used for convenience rather than necessity, and are listed in Table A.1.

Table A.1: SI Derived Units

In terms of Unit name Quantity Symbol base units becquerel Activity of a radionuclide Bq s−1

coulomb Quantity of electricity, electric charge C A·s degree Celsius Celsius temperature ◦C K farad Capacitance F C·V−1 gray Absorbed dose of ionizing radiation Gy J·kg−1 henry Inductance H Wb·A−1 hertz Frequency Hz s−1

joule Energy, work, quantity of heat J N·m lumen Luminous flux lm cd·sr lux Illuminance lx lm·m−2 newton Force N kg·m·s−2 ohm Electric resistance Ω V·A−1 pascal Pressure, stress Pa N·m−2 siemens∗ Conductance S A·V−1 sievert Dose equivalent of ionizing radiation Sv J·kg−1 telsa Magnetic flux density T Wb·m−2 volt Electric potential, potential difference V W·A−1 watt Power, radiant flux W J·s−1 weber Magnetic flux Wb V·s

∗The siemens was previously called the mho.

When units are named after people, like the newton (N), joule (J), and pascal (Pa), they are capitalized when abbreviated but not capitalized when spelled out. The abbreviation capital L for liter is a special case, used to avoid confusion with one (1). In the SI system the unit of absolute temperature is the degree kelvin, which is abbreviated K without the degree symbol. The units of second, minute, hour, day, and year are correctly abbreviated as s, min, h, d, and y.

In using prefixes with SI units, multiples of 103 are preferred in engineering usage, with other multiples such as cm avoided if possible. It is conventional practice to separate sequences of digits into groups of three by spaces rather than commas.

A.2 Conversion Factors

In most cases, application of unit conversion factors results in converted numbers that have more significant digits than the original numbers. In these cases, the converted number should be rounded

1017

off such that rounding error is consistent with the rounding error of the converted number.

Example 1.1 The height of a water-control structure is reported as 19.3 feet. Convert this dimension to meters.

Solution. The conversion factor is given in Table A.2 as 1 foot = 0.3048 meters, hence,

19.3 ft = 19.3× 0.3048 = 5.88264 m

Since 19.3 ft could have resulted in rounding any number between 19.25 ft and 19.35 ft, the maximum possible

rounding error is ±0.05/19.3 = ±0.26%. Similarly, rounding 5.88264 m to 5.88 m gives a maximum rounding error of ±0.005/5.88 = ±0.085%, and rounding to 5.9 m gives an error of ±0.05/5.9 = ±0.85%. Hence, accuracy is lost by taking 19.3 ft as 5.9 m, while 5.88 m is more accurate than indicated by 19.3 ft. It is usually prudent not to discard

accuracy, so take 19.3 ft = 5.88 m.

A good rule of thumb is that the converted number should have the same number of significant digits as the original number, assuming that the conversion factor is always more accurate than the original number.

1018

Table A.2: Multiplicative Factors for Unit Conversion

Quantity Convert From Convert To Multiply By

Area ac ha 0.404687 mi2 km2 2.59000

Energy Btu J 1054.350264 ft·lb J 1.355818 cal J 4.184∗

Energy/Area ly† kJ/m2 41.84∗

Flow rate cfs m3/s 0.02831685 gpm L/s 0.06309 mgd† m3/s 0.04381

m3/d 3785.412 L/s 43.81

Force lbf N 4.4482216152605∗

Length ft m 0.3048∗

in. m 0.0254∗

mi (U.S. statute) km 1.609344∗

mi (U.S. nautical) km 1.852000∗

yd m 0.9144∗

Mass g kg 0.001∗

lbm kg 0.45359237∗

slug kg 14.59390

Permeability darcy m2 0.987 × 10−12

Power hp W 745.69987

Pressure atm kPa 101.325∗

bar kPa 100.000∗

mm Hg (at 0◦C) kPa 0.133322 psi kPa 6.894757 torr kPa 0.133322

Speed knot m/s 0.514444444 mph m/s 0.44704 mph km/h 1.609344

Viscosity (dynamic) cp Pa·s 0.001∗

Viscosity (kinematic) cs m2/s 10−6∗

Volume gal (U.S.) L 3.785411784∗

gal (imperial) 4.5461

Weight ton (U.S. short) metric ton‡ 0.90718486 ton (British long) metric ton 1.1060470

∗Exact conversion, †ly ≡ langley, mgd ≡ million gallons per day, ‡1 metric ton = 1000 kgf.

Appendix B

Fluid Properties

B.1 Water

Table B.1: Physical Properties of Water at Standard Sea-Level Pressure

Temp Density Dynamic Heat of Saturation Surface Bulk Expansion Viscosity Vaporization Vapor Pressure Tension Modulus Coefficient

(◦C) (kg/m3) (mPa·s) (MJ/kg) (kPa) (mN/m) (106 kPa) (10−3 K−1)

0 999.8 1.781 2.499 0.611 75.6 2.02 −0.07 5 1000.0 1.518 2.487 0.872 74.9 2.06 0.160 10 999.7 1.307 2.476 1.227 74.2 2.10 0.088 15 999.1 1.139 2.464 1.704 73.5 2.14 0.151 20 998.2 1.002 2.452 2.337 72.8 2.18 0.207 25 997.0 0.890 2.440 3.167 72.0 2.22 0.257 30 995.7 0.798 2.428 4.243 71.2 2.25 0.303 40 992.2 0.653 2.405 7.378 69.6 2.28 0.385 50 988.0 0.547 2.381 12.340 67.9 2.29 0.457 60 983.2 0.466 2.356 19.926 66.2 2.28 0.523 70 977.8 0.404 2.332 31.169 64.4 2.25 0.585 80 971.8 0.354 2.307 47.367 62.6 2.20 0.643 90 965.3 0.315 2.282 70.113 60.8 2.14 0.665 100 958.4 0.282 2.256 101.325 58.9 2.07 0.752

The properties given in Table B.1 are for pure water. Pure water seldom exists in nature, where the density of water can be significantly influenced by salinity, temperature, and possibly other properties through an equation of state. The general dependence of water density on temperature has been found to be approximately parabolic, with a maximum at 4◦C. However, the temperature corresponding to the maximum density of water changes with increasing salinity, decreasing to about 0◦C for highly saline systems. To a first-order approximation, density is linearly dependent on salinity over much of the normal range of interest.

Notes:

1019

1020

a. The kinematic viscosity, ν, of water at 20◦C is approximately equal to 1.004× 10−6 m2/s.

1021

B.2 Air

Table B.2: Physical Properties of Air at Standard Sea-Level Atmospheric Pressure

Temperature Density Dynamic Specific Heat Speed of Viscosity cp k = cp/cv Sound

(◦C) (kg/m3) (µPa·s) (J/kg·K) (–) (m/s)

−40 1.514 15.18 1002 1.401 306.2 −20 1.394 16.22 1005 1.401 319.1

0 1.292 17.29 1006 1.401 331.4 5 1.269 17.23 1006 1.401 334.4

10 1.246 17.72 1006 1.401 337.4 15 1.225 17.96 1007 1.401 340.4 20 1.204 18.21 1007 1.401 343.3 25 1.184 18.44 1007 1.401 346.3 30 1.164 18.68 1007 1.400 349.1 40 1.127 19.15 1007 1.400 354.7 50 1.092 19.61 1007 1.400 360.3 60 1.059 20.06 1007 1.399 365.7 70 1.028 20.51 1007 1.399 371.2 80 0.9994 20.95 1008 1.399 376.6 90 0.9718 21.38 1008 1.398 381.7 100 0.9458 21.81 1009 1.397 386.9 200 0.7459 25.77 1023 1.390 434.5 300 0.6158 29.34 1044 1.379 476.3 400 0.5243 32.61 1069 1.368 514.1 500 0.4565 35.63 1093 1.357 548.8

1000 0.2772 48.26 1184 1.321 694.8

Notes:

a. Standard air is air at 15◦C at standard sea-level pressure (= 101.325 kPa).

b. The specific gas constant, R, for standard air is approximately equal to 287.058 J/kg·K, based on a molar mass of 28.965 g/mol.

c. The kinematic viscosity, ν, of standard air is approximately equal to 1.461× 10−5 m2/s.

1022

B.3 The Standard Atmosphere

Table B.3: Physical Properties of the Standard Atmosphere†

Height above Temperature Absolute Density Absolute Speed Gravity ground pressure viscosity of sound (km) (◦C) (kPa) (kg/m3) (µPa·s) (m/s) (m/s2)

0 15.00 101.325 1.2250 17.894 340.294 9.80665 1 8.50 89.876 1.1117 17.579 336.43 9.8036 2 2.00 79.501 1.0066 17.260 332.53 9.8005 3 −4.49 70.121 0.90925 16.938 328.58 9.7974 4 −10.98 61.660 0.81935 16.612 324.59 9.7943 5 −17.47 54.048 0.73643 16.282 320.55 9.7912 6 −23.96 47.217 0.66011 15.949 316.45 9.7882 7 −30.45 41.11 0.5900 15.61 312.27 9.785 8 −36.94 35.651 0.52579 15.27 308.11 9.782 9 −43.42 30.80 0.4671 14.93 303.79 9.779 10 −49.90 26.499 0.41351 14.58 299.53 9.776 11 −56.50 22.632 0.3639 14.22 295.07 9.773 12 −56.50 19.330 0.3108 14.22 295.07 9.770 13 −56.50 16.510 0.2655 14.22 295.07 9.767 14 −56.50 14.102 0.2268 14.22 295.07 9.774 15 −56.50 12.11 0.1948 14.22 295.07 9.761 20 −56.50 5.529 0.08891 14.22 295.07 9.745 25 −51.60 2.549 0.04008 14.48 298.46 9.730 30 −46.64 1.197 0.01841 14.75 301.80 9.715 40 −22.80 0.287 0.003996 16.01 317.63 9.684 50 −2.50 0.07978 0.001027 17.04 329.80 9.654 60 −26.13 0.02196 0.0003097 15.84 314.07 9.624 70 −53.57 0.0052 0.00008283 14.38 295.61 9.594 80 −74.51 0.0011 0.00001846 13.21 281.12 9.564

†Latest version, last revised in 1976.

1023

B.4 Common Liquids

Table B.4: Physical Properties of Common Liquids at 20◦C and at Standard Sea-Level Atmospheric Pressure

Liquid Density Absolute Surface Saturation Modulus of Specific Viscosity Tension Vapor Pressure Elasticity Heat

(kg/m3) (mPa·s) (mN/m) (kPa) (MPa) (J/kg·K)

Ammonia 608 0.220 21.3 910.0 1820 − Benzene 876 0.65 29 10.0 1030 1720 Blood @ 37◦C 1060 3.5 58 − − 3600 Carbon tetrachloride 1590 0.958 26.9 13.0 1310 − Ethyl alcohol 789 1.19 22.8 5.9 1060 − Ethylene glycol 1117 21.4 48.4 0.012 − − Freon 12 1327 0.262 − − 795 − Gasoline 680 0.29 − 55.2 − 2100 Glycerin 1260 1500 63.3 0.000014 4520 2100 Kerosene 808 1.92 25 3.20 − 2000 Mercury 13550 1.56 510 0.00017 26200 139.4 Methanol 791 0.598 22.5 13.4 830 − Oil (Crude) 856 7.2 30 − − − SAE 10 Oil 918 82 37 − − − SAE 30 Oil 918 440 36 − − − SAE 30 Oil @ 15.6◦C 912 380 36 − − − SAE 50 Oil 902 860 − − − − Water (Fresh) 998 1.00 73 2.34 2171 4187 Water (Sea) @20◦C 1023 1.07 73.5 2.34 2300 3933

@10◦C 1025 1.39 74.8 1.20 − 4007

1024

B.5 Common Gasses

Table B.5: Physical Properties of Common Gasses at 20◦C and at Standard Sea-Level Atmospheric Pressure

Gas Chemical Molar Density Absolute Specific heat formula mass viscosity cp cv

(g/mol) (kg/m3) (µPa·s) (J/kg·K) (J/kg·K)

Air − 28.96 1.205 18.0 1003 716 Argon Ar 39.944 1.66 22.4 519 311 Carbon Dioxide CO2 44.01 1.84 14.8 858 670 Carbon Monoxide CO 28.01 1.16 18.2 1040 743 Chlorine Cl2 70.91 2.95 10.3 − − Helium He 4.003 0.166 19.7 5220 3143 Hydrogen H2 2.016 0.0839 9.0 14450 10330 Methane CH4 16.04 0.668 13.4 2250 1730 Nitrogen N2 28.02 1.16 17.6 1040 743 Nitric Oxide NO 30.01 1.23 19.0 − − Nitrous Oxide N2O 44.02 1.82 14.5 − − Oxygen O2 32.00 1.33 20.0 909 649 Propane† @ 0◦C C3H8 44.10 2.010 − − − Water Vapor H2O 18.02 0.747 10.1 1862 1400

†Propane has a saturation vapor pressure of 853.16 kPa (= 123.8 lb/in2) at 21.1◦C.

Notes:

a. The universal gas constant, Ru, is approximately equal to 8.31446 J/mol·K.

b. The gas constant, R, for each gas is given by R = Ru/M , where M is the molar mass.

1025

B.6 Nitrogen

Table B.6: Physical Properties of Nitrogen at Standard Sea-Level Atmospheric Pressure

Temperature Density Specific Dynamic Heat, cp Viscosity

(◦C) (kg/m3) (J/kg·K) (µPa·s)

−50 1.5299 957.3 13.90 0 1.2498 1035 16.40 50 1.0564 1042 18.74 100 0.9149 1041 20.94 150 0.8068 1043 23.00 200 0.7215 1050 24.94 300 0.5956 1070 28.49 400 0.5072 1095 31.66 500 0.4416 1120 34.51 1000 0.2681 1213 45.94 1500 0.1925 1266 55.62 2000 0.1502 1297 64.26

1026

Appendix C

Properties of Areas and Volumes

C.1 Areas

Table C.1: Geometric Properties of Areas

Moments of Shape Illustration Area Inertia

Rectangle C

a 2

a 2

b 2

b 2

ba

Ixc = 1 12ba

3

Iyc = 1 12ab

3

Ixyc = 0

Circle C

R

πR2 Ixc = Iyc =

πR4

4 Ixyc = 0

Semicircle C 4R

3ϖ RR

1 2πR

2

Ixc = 0.1098R4

Iyc = 0.3927R4

Ixyc = 0

1027

1028

Table C.1 continued

Moments of Shape Illustration Area Inertia

Quarter circle C

4R

3ϖ

4R

3ϖ

R

1 4πR

2 Ixc = Iyc = 0.05488R4

Ixyc = −0.01647R4

Ellipse C

a/2 a/2

b/2

b/2

1 4πab

Ixc = 1 64πab

3

Iyc = 1 64πba

3

Semiellipse C

a/2 a/2

b/2 2b

3ϖ

1 8πab

Ixc = 1

128πab 3

Iyc = 1

128πba 3

Triangle C

b

(b+h)/3

h

d

h

3

1 2bh

Ixc = 1 36ba

3

Ixyc = 1 72bh

2(b− 2d)

C.2 Properties of Circles and Spheres

C.2.1 Circles

There are several additional properties of a circle, beyond those shown in Section C.1 that are useful in many applications. These additional properties are shown in Figure C.1.

1029

s

R d

h

c

A 1

! R

A 2

!

Figure C.1: Dimensions and subareas of a circle

The basic parameters used to characterize a circle and a part of a circle are the radius, R, and the central angle, θ, where

θ = 2 cos−1 ( d

R

)

Using R and θ as parameters, and taking R in units of length and θ in units of radians, the following quantities can be calculated:

arc length: s = Rθ, chord length: c = 2R sin θ

2

height: h = R

( 1− cos θ

2

) , segment area: A1 =

R2

2 (θ − sin θ))

sector area: A2 = 1 2R

2θ

C.2.2 Spheres

The properties of a sphere that are most commonly of interest are the volume and the surface area. For a sphere of radius R and diameter D (= 2R), the volume, V , and surface area, A, are given by

V = 43πR 3 = 16πD

3, A = 4πR2 = πD2

1030

C.3 Volumes

Table C.2: Geometric Properties of Volumes

Shape Illustration Volume

Truncated cone

D 2

D 1

h 1 12πh(D

2 1 +D

2 2 +D1D2)

Appendix D

Pipe Specifications

D.1 PVC Pipe

Pipe dimensions of interest to engineers are usually the diameter and wall thickness. The pipe diameter is typically specified by the nominal pipe size and the wall thickness is usually specified by the schedule. Nominal pipe sizes are typically given in either inches or millimeters, and represent rounded approximations to the inside diameter of the pipe. The schedule of a pipe is a number that approximates the value of the expression 1000P/S, where P is the service pressure and S is the allowable stress. Higher schedule numbers correspond to thicker pipes, and schedule numbers in common use are 5, 5S, 10, 10S, 20, 20S, 30, 40, 40S, 60, 80, 80S, 100, 120, 140, and 160. The schedule numbers followed by the letter “S” are primarily intended for use with stainless steel pipe (ASME B36.19M).

Schedule 5 Schedule 10 Schedule 40 Schedule 80 Nominal Outside Wall Inside Wall Inside Wall Inside Wall Inside pipe size diameter thickness diameter thickness diameter thickness diameter thickness diameter (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm)

50 60 1.7 57 2.8 55 3.9 53 5.5 49 80 90 2.1 85 3.0 83 5.5 78 7.6 74

100 114 2.1 110 3.0 108 6.0 102 8.6 97 125 141 2.8 136 3.4 134 6.6 128 9.5 122 150 168 2.8 163 3.4 161 7.1 154 11.0 146

Source: Fetter (1999).

D.2 Ductile-Iron Pipe

Ductile-iron pipe is manufactured in diameters in the range of 100–1200 mm (4–48 in.), and for diameters in the range of 100–500 mm (4–20 in.) standard commercial sizes are available in 50-mm (2-in.) increments, while for diameters in the range of 600–1200 mm (24–48 in.) the size increments are 150 mm (6 in.). The standard lengths of ductile-iron pipe are 5.5 m (18 ft) and 6.1 m (20 ft).

1031

1032

Schedule 5 Schedule 10 Schedule 40 Schedule 80 Nominal Outside Wall Inside Wall Inside Wall Inside Wall Inside pipe size diameter thickness diameter thickness diameter thickness diameter thickness diameter (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.)

2 2.375 0.065 2.245 0.109 2.157 0.154 2.067 0.218 1.939 3 3.500 0.083 3.334 0.120 3.260 0.216 3.068 0.300 2.900 4 4.500 0.083 4.334 0.120 4.260 0.237 4.026 0.337 3.826 5 5.563 0.109 5.345 0.134 5.295 0.258 5.047 0.375 4.813 6 6.625 0.109 6.407 0.134 6.357 0.280 6.065 0.432 5.761

Source: Fetter (1999).

D.3 Concrete Pipe

D.4 Physical Properties of Common Pipe Materials

Young’s modulus, E

Material (GPa) Poisson’s ratio

Concrete 14–30 0.10–0.15 Concrete (reinforced) 30–60 — Ductile iron 165–172 0.28–0.30 PVC 2.4–3.5 0.45–0.46 Steel 200–207 0.30

Section E.4 Physical Properties of Common Pipe Materials 1033

Table D.1: Commercially Available Sizes of Concrete Pipe

Non-reinforced pipe Reinforced pipe

Diameter Diameter Diameter Diameter (mm) (in.) (mm) (in.)

100 4 — — 150 6 — — 205 8 — — 255 10 — — 305 12 305 12 380 15 380 15 455 18 455 18 535 21 535 21 610 24 610 24 685 27 685 27 760 30 760 30 840 33 840 33 915 36 915 36 — — 1065 42 — — 1220 48 — — 1370 54 — — 1525 60 — — 1675 66 — — 1830 72 — — 1980 78 — — 2135 84 — — 2285 90 — — 2440 96 — — 2590 102 — — 2745 108