Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

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382 IIIb. Fluid Mechanics: Incompressible Viscous Flow

head

V = 0

-h

E- Since pressure in the pipe is now larger

than the tank pressure, flow starts to enter

the tank and the pressure wave reverses

course.

head

V = 0

-h

F- As the pressure wave travels towards

the valve, more stagnant water begins to

flow to the tank.

head

-h

t = 2L/c

G- At time 2L/c, pressure wave reaches the

closed valve and begins to reverse course

while water in the entire pipe is flowing to

the tank.

head

-h

V = 0

H- At the valve, head rises to –cV

/g and

the disturbance travels towards the tank.

head

V = 0

I- With water still flowing in the pipe to-

wards the valve, more water becomes stag-

nant in the pipe

head

V = 0

t = 3L/c

J- At time 3L/c, the wave front reaches

the tank and water in the entire pipe be-

comes stagnant.

head

V = 0

K- With the tank water being again at

higher pressure head than water in the pipe,

water flows towards the valve and so does

the pressure wave

head

V = 0

L- The closer the pressure wave gets to

the valve, the less stagnant water there is

next to the valve.

head

t = 4L/c

M- Finally, at time 4L/c, water in the entire

pipe reaches the closed valve and the con-

figuration is identical with case A.

Figure IIIb.7.5. Head versus pipe length for flow in a frictionless pipe with sudden valve

closure

Questions and Problems 383

QUESTIONS

− Does a boundary layer develop for the flow of inviscid fluids?

− Are the Navier-Stokes equations directly applicable to turbulent flow? What is

an eddy?

− What is the molecular viscosity?

− What are the two types of stresses in the boundary layer over a flat plate?

− Is it fair to say that the shear stress profile in pipe flow remains linear regardless

of the flow regime?

− What is the Hagen-Poiseuille flow?

− For the flow of a viscous fluids in a pipe, specify the location of the viscous

sub-layer.

− What is the friction velocity? Does it exist in an ideal flow?

− What is the Darcy formula? Define the difference between Darcy-Weisbach and

Hazen-Williams formulae.

− What is a valve flow coefficient?

− What is vena contracta? Define the discharge coefficient.

− What is the difference between a globe and a gate valve? For what applications

is each valve used?

− Specify the difference between waterhammer, rigid column, and quasi-steady

problems.

− What is an elastic analysis?

− Instant throttling of a valve disturbs the steady flow of a liquid in a pipe. At

what velocity does this disturbance propagate upstream?

− What factors affect the speed that a disturbance would propagate in a pipe flow?

− Consider the flow of water in two identical piping systems except for one pipe

being copper and the other plastic. In which system is the head rise higher fol-

lowing an instantaneous closure of a valve?

PROBLEMS

Sections 1 and 2

1. a) A viscous fluid at a specified flow rate enters a pipe of diameter d. The Rey-

nolds number indicates that the flow is laminar. The same fluid at the same mass

flow rate enters another pipe of diameter d/2 and flow becomes turbulent. What

flow property is the reason for the change in the Reynolds number?

b) Water at room temperature enters a pipe having I.D. = 4 in at a rate of 30

lbm/s. Determine the flow regime (i.e., whether the flow is laminar or turbulent).

2. The x-component of a flow velocity is given as V(x, y, z, t). Use the definition

of the time-averaged mean velocity

V =

(

)

tdV

to show that the time-

averaged fluctutating velocity (

′

) in the time domain of interest (

) is zero.

384 IIIb. Fluid Mechanics: Incompressible Viscous Flow

3. Find the minimum diameter of a smooth tube to ensure the flow of water at an

average velocity of 100 cm/s remains turbulent. Use v

water

= 1E-6 m

/s. [Ans.: 4

mm].

4. Show that the Re number for the flow of fluids in pipes and tubes (A =

/4)

is given as Re = 4



πµ

D). Find the maximum tube diameter to ensure the flow

of water at a rate of 100 kg/h remains turbulent. Use

water

= 1E-3 N·s/m

. [Ans.:

8.8 mm].

5. Air at 1 atm and 20 C flows through a smooth pipe. The centerline velocity

and the pipe diameter are 6 m/s and 15 cm, respectively. Find a) the wall shear

stress and b) the average velocity in the pipe.

6. Air flows over a sphere at 77 C. The diameter of the sphere is 10 cm. Find the

air velocity that results in the Reynolds number to reach the transition from the

laminar to turbulent flow if such transition takes place at a Reynolds number of

about 250,000. [Ans.: 52.3 m/s].

Section 3

7. Water flows steadily in a horizontal and well-insulated pipe. Consider loca-

tions a and b along the length of the pipe and chose the correct answers for veloc-

ity (V), pressure (P), and temperature (T):

< V

, V

= V

, V

> V

, P

< P

, P

= P

, P

> P

, T

< T

, T

= T

, T

> T

8. Plot and compare the friction factors obtained from the Colebrook, Haaland,

Churchill, and McAdams correlations (Equations IIIb.3.5, IIIb,3.5-1, IIIb.3.5-2,

and IIIb.3.6) as a function of Reynolds number for a smooth pipe.

9. Compare the frictional pressure drop, due to skin friction, of air and water for a

smooth circular pipe of inside diameter D and length L. For this comparison,

consider the mass flow rate of air to be equal to the mass flow rate of water.

[Ans.: Using the McAdams’ correlation for f we find,

∆P

/∆P

= (

)

1.2

10. A large water tank is connected to a small nozzle located 5 ft below the water

surface in the tank (Figure a). The nozzle diameter at the exit is 1 in. Find the

flow rate through the nozzle. We now attach a diffuser to the nozzle (Figure b).

Will the flow rate a) increass, b) remain the same, or c) decrease?

Questions and Problems 385

(a) (b)

h h

11. An incompressible fluid flows steadily in a pipe of diameter d. This pipe is

connected to a pipe of larger diameter D by a sudden expansion. Use the

continuity and the momentum equations to derive a relation for the sudden

expansion loss coefficient, K

in terms of the dimensionless diameter

= d/D.

12. A pump, having a pump head of h

pump

delivers a liquid at a volumetric flow

rate of V



. Show that the power consumed by the pump is

()



= where

the liquid density. Find the pump head of a 10 hp pump, delivering water at a rate

of 400 GPM. [Ans. § 100 ft].

13. Find the water flow rate at an average temperature of 15 C in the shell-side of

a heat exchanger. Use L = 7.62 m, shell inside diameter D = 0.914 m, tube outside

diameter d = 3.81 cm, number of tubes N

tube

= 200, ∆P = 110.32 kPa, and total

loss coefficient K = 35. [Ans.: 0.492 m

/min].

14. Consider the flow of water at 70 F and a rate of 2.5 GPM in a wrought iron

pipe of ¾ in diameter pipe (I.D. = 0.824 in.) There are two 90-degree medium

sweep elbows, a swing check valve, and a fully open angle valve. Find the pres-

sure drop over 100 ft of this piping system. [Ans.: 0.83 psi].

15. A pump operating at 660 hp delivers water, from a large pressurized supply

tank to a large pressurized reservoir, at a rate of 150 ft

/s. Total length of the suc-

tion and the discharge piping is 1000 ft. Pressures in the supply and the reservoir

tanks are 30 psig and 15 psig, respectively. These pressures are maintained

throughout the pumping process. Water level heights with respect to the pump

centerline are 15 ft and 50 ft, respectively. Find the pipe diameter. Ignore varia-

tions in water level in both tanks. [Ans.: D

≈ 3.5 ft].

Pump

Water

Check

Valve

Gate

Valve

Suction

piping

Discharge

piping

Air

386 IIIb. Fluid Mechanics: Incompressible Viscous Flow

16. Consider the pumping system of Problem 15 but with both reservoirs at at-

mospheric pressure. Total length of the suction and the discharge piping, from the

supply to the discharge reservoir is L, having an inside diameter of D. Our goal is

to deliver the same flow rate to the same elevation but without using a pump.

Find the elevation of the bottom of the supply reservoir to provide sufficient grav-

ity head for this purpose. [Ans.: H = c

where c

= (fL/D + ΣK)(



/2gA

) and

= 1 – (f



/2gDA

)].

17. In problem 16 we investigated the effect of gravity head to provide the desired

flow rate without using a pump. In this problem we want to investigate the effect

of static pressure to provide the same flow rate without using a pump. For this

purpose, we maintain the same configuration of the piping system of Problem 15.

However, we maintain the discharge reservoir at atmospheric pressure. Find the

required pressure in the supply reservoir to deliver the desired flow rate. [Ans. P

– P

atm

= (fL/D + ΣK)(



/2gA

)].

18. A pump delivers water to a differential elevation of H

– H

= 40 ft. Diame-

ter of the suction and discharge piping is 5 in and total length of the piping is 450

ft. Pressure in the supply tank is 10 psig and in the receiving reservoir 5 psig.

Find the water flow rate. The head developed by the pump is 65 ft and

ΣK = 5.

[Ans.: 630 GPM].

19. A piping system is to be designed to deliver 0.5 m

/s water from a lake to a

reservoir located at an elevation of 78 m from the pump centerline. The reservoir

is at atmospheric pressure. The pump centerline is 1 m above the surface of the

lake. The total pipe run from the lake to the reservior is 420 m. Both intake and

discharge pipings should have the same nominal pipe size of 12 in schedule 40.

The list of the fittings and valves of the piping system includes one globe valve,

two gate valves, two swing check valves, twelve elbows (90

), and two standard

Ts. Find the required pump head and pumping power. Assume a pump efficiency

of 70%. [Ans.: § 1.2 MW].

20. Find the pressure drop between locations a and b for flow in a pipe using the

following data:

Working fluid: Water

Pipe: Stainless steel, Schedule No. 120

Nominal pipe size (cm): 10

Flow rate (m

/min): 2.271

Temperature (C): 38

Length (m): 45.72

Solve for the following 3 cases: a) Z

= Z

=33.5 m, b) Z

= Z

+ 15 m, and c) Z

= Z

+ 46 m.

21. Find the head loss over 15 ft of 1 in pipe carrying oil (S.G. = 0.86 and

= 3

lbm/ft

⋅h) at 0.5 GPM. Is the flow laminar or turbulent? [Ans.: h

= 1.74 ft].

Questions and Problems 387

22. Head loss in 1000 ft of a 12 in cast-iron pipe, for water at 60 F, is 15 ft. Find

the flow rate,

ε/

D = 85E-4. [Ans.: V = 7.4 ft/s and volumetric flow rate = 5.8

/s].

23. Find the water flow rate at 15 C in a welded steel pipe of diameter 250 mm.

Head loss is 5m/500m.

24. Find the hydraulic diameter for the conduits shown in Figures (a), (b), (c) and

(d). For Figure (d), the rod pitch (the distance between the centerline of the

neighboring rods) is 2 inch and the sides of the square do not constitute solid

boundaries.

Flow

area

Flow

area

Flow

area

(a) (b) (c) (d)

Section 4

25. Find pressure drop for the sudden contraction and sudden expansion as shown

in the figure, d

= d

= 1 in and D

= D

= 5 in. a) Find pressure drop for the flow

of water at 65 F, 14.7 psia and a rate of 5000 lbm/h. b) Find pressure drop for air

at the same conditions as water.

26. In this problem our goal is to compare pressure drop for the flow of an in-

compressible fluid in a straight pipe to the pressure drop of a compressible fluid in

the same pipe and under the same conditions. For the incompressible case, con-

sider the flow of water at a rate of 5000 lbm/s in the pipe. Then compare the re-

sults with the flow of air. The pipe is smooth and has a diameter of 1 in and

length of 20 ft. For both cases, fluid enters the pipe at 14.7 psia and 100 F and

leaves the pipe at the same temperature.

27. Water enters a heated pipe at a rate of 5000 lbm/s, pressure of 14.7 psia and

temperature of 100 F. The pipe is smooth and has a diameter and length of 1 in

and 20 ft, respectively. Water temperature at the exit of the pipe is 200 F. Find

the total pressure drop from A to B.

BA B

Figure for Problems 27, 28, 29, and 50

388 IIIb. Fluid Mechanics: Incompressible Viscous Flow

28. Water enters a pipe at a rate of 5000 lbm/s, pressure of 14.7 psia and tempera-

ture of 200 F. The pipe is smooth and has diameter and length of 1 in and 20 ft,

respectively. A cold fluid flows over the pipe so that water temperature at exit is

100 F. Find total pressure drop from A to B.

29. Air enters a heated pipe at a rate of 2265 kg/s, pressure of 1 atm and tempera-

ture of 38 C. The pipe is smooth and has a diameter and length of 2.54 cm and 6

m, respectively. Air temperature at the exit of the pipe is 371 C. Find the total

pressure drop from A to B.

30. Air enters a pipe at a rate of 5000 lbm/s, pressure of 14.7 psia and temperature

of 700 F. The pipe is smooth and has a diameter and length of 1 in and 20 ft, re-

spectively. A cold fluid flows over the pipe so that air temperature at the exit is

100 F. Find the total pressure drop from A to B.

31. Water at 60 F flows is a tube with d = 1/2 in and L = 5 ft. Find the friction

pressure drop associated with the flow of water in the tube if water flows at a

velocity of a) 0.5 ft/s and b) 5 ft/s.

32. Water at 100 F is pumped at a rate of 125 gpm to a reservoir 50 ft higher. The

pipe length is 200 ft and the pipe inside diameter, I.D. = 2.067 in. The fittings

include 3 standard 90-degree elbows, two 45-degree elbows, and an open gate

valve. Pump efficiency is 70%. Find pump horsepower for a)

= 0 ft, b)

0.0008 ft.

33. A shell-and-tube heat exchanger has 40 tubes of I.D. = 1 in and L = 10 ft. Hot

stream enters the shell side and air at 14.7 psia and 200 F at a rate of 1000 lbm/h

enters the tubes. Find the total tube-side pressure drop. The diameter of both inlet

and outlet plena is 0.75 ft. [Ans.: 2 lbf/ft

34. Fully developed air flows inside the ¼ in tubes of an air-cooled heat exchang-

er at a rate of 1.5 lbm/h. The conditions at the inlet of the tube are atmospheric

pressure and 60 F. After being heated in the 2 ft long tube, air leaves at 780 F.

Find the pressure drop in the heat exchanger.

35. Oil flows in a pipe at a rate of 4000 GPM. The head loss over a length of

10,000 ft of the pipe having a surface roughness of 0.0018 in, is 75 ft. Find the

pipe diameter. Assume oil

= 0.36 ft

/h. [Ans. 16.85 in].

36. The number of tubes of a shell and tube heat exchanger N is given but we do

not know the tube diameter, d. To determine the tube diameter, a pump is used to

circulate flow inside the tubes. The flow rate and the pressure drop are then care-

fully measured. Use the specified data and find the tube inside diameter.

Data: N = 40, L

tube

= 3 m, ∆P = 34.5 kPa, V



= 31.55 lit/s, T = 20 C.

[Ans.: § 1.6 cm].

Questions and Problems 389

Water entering shell (tube bundle)

Water entering

tubes

Water exiting shell (tube bundle)

Water exiting

tubes

37. Use the definition of the flow coefficient C

and substitute the appropriate

equivalent units in the Darcy equation to show that the valve resistance or loss co-

efficient in terms of C

is given as K = 891d

C where the pipe diameter, d is in

inches.

38. Find a) the loss coefficient of a valve installed on a 8 in schedule 40 steel pipe

and b) the equivalent pipe length . The flow is fully turbulent and the valve has a

discharge coefficient of 2000. [Ans.: 0.9 and 43 ft].

39. Find K, C

, and L

for a 3 in Schedule 40 fully open lift check valve. [Ans.:

= 153 ft].

40. An incompressible viscous fluid flows in a pipe of length L

and flow area A

which is connected to a pipe of flow area A

and length L

as shown in the figure.

Write the momentum equation for both sections of the pipe and add them together

to obtain an equation for the combined piping system. Find the key assumption

that you have to make to obtain the results listed in Table IIIb.4.3-1. [Ans.: P

j-1

j+1

where P

j-1

and P

j+1

are pressures right before and right after the sudden

enlargement, respectively].

j+1

j-1

20 cm

40 cm

25 m

10 m

40 m

Problem 40 Problem 41

41. Air flows at 60 C in a duct as shown in the figure at a rate of 100 m

/min.

The duct divides into two branches, A and B, having lengths of 15 m and 40 m,

respectively. The depth of the duct remains constant at 50 cm. Branch A has a

width of 40 cm and branch B has a constant width of 20 cm. The discharge of

both branches is into the same outlet. The 90 elbows are well rounded with each

having a loss coefficient of 0.25. For the smooth surface of the ducts use a friction

factor of 0.022 and find flow of air in each branch.

42. In a conceptual nuclear reactor design, we want to divide the core into two

zones. Zone one contains N

bundles surrounded by zone two with N

bundles.

The design criteria requires the total power generated in zone one to be equal to

the total power produced in zone two. Additionally, the temperature rise in zone

390 IIIb. Fluid Mechanics: Incompressible Viscous Flow

one must also be equal to the temperature rise in zone two. To meet these design

criteria, it is necessary to reduce the flow rate in zone two by adding orifice blocks

to the inlet of the fuel bundles located in zone two. Each orifice block has five

orifices. Use the given data to find the diameter of the orifcing (D). Assume

smooth surfaces and negligible pressure losses in all parts of the fuel assemblies

other than the fuel bundle and the orifice block. Data:

∆P

Core

= 0.745 MPa,

Core



= 17.5E6 kg/h, N

= 65, N

= 85, L = 4 m,

= 40 cm,

water

= 800 kg/m

water

= 2E-4 N·s/m

, K

= 0.5, K

= 1.0, and all five channels in the orifice block

have equal diameters. [Ans.: 2 cm].

Fuel rod

Fuel

bundle

(zone 1)

Fuel

bundle

(zone 2)

Section a-a

43. A thin-plate orifice is used to measure water flow rate. The water temperature

is 27 C and a mass flow rate of 50 lit/s is measured by the orifice. The pipe di-

ameter is 26 cm and the orifice diameter is 7 cm. Find the non-rcoverable pres-

sure drop. [Ans.: 21 m, C

= 0.6, K = 2.5].

44. We want to measure the flow rate of water at 20 C by a thin plate orifice. The

pipe diameter carrying the water is 20 cm and the thin-plate orifice dimater is 5

cm. A pressure drop of 70 kPa is measured between the taps. Find the flow rate

and the throat velocity. [Ans.: 23 lit/s, 11.6 m/s].

45. Obtaining the characteristics of an equivalent pipe (L

, A

, V

, D

, and K

) to

represent a compound piping (i.e. pipes connected in series or in parallel) depends

on the imposed constraint. In this problem, you are asked to develop a table simi-

lar to Table IIIb.4.3.

a) First, for several pipes connected in series, find the characteristics of an equiva-

lent pipe using the following constraints L

= ΣL

, (∆P

skin

)

= (Σ∆P

skin

)

, and

(

∆P

fittings-valves

)

= (Σ∆P

fitting-valves

)

b) Next, consider several pipes connected in parallel. Find the characteristics of

an equivalent pipe using the following constraints A

= ΣA

, (∆P

skin

)

= (∆P

skin

)

and (

∆P

fittings-valves

)

= (Σ∆P

fitting-valves

)

[Ans.:

Pipes in series Pipes in parallel

¦¦

iiE

LmLmL



flow

: V

Questions and Problems 391

EEE

ALI =

¦¦

iiE

LmLmI



()

iiEE

/KK

()

−

iiEE

KAA

46. For part b of Problem 35, find the flow rate in a parallel flow branch. Assume

all branches consist of smooth pips and use the friction factor as given by Equa-

tion IIIb.3.6.

[Ans. V])()[()()(V

8.1/1

8.1/8.4

8.1/1

8.1/8.4



−

47. Water enters a system of parallel piping at a rate of 500 GPM. The pipes

length and inside diameter are L

= 10 ft, D

= 2.5 in, L

= 20 ft, D

= 3.0 in, L

= 5

ft, D

= 2 in, and L

= 25 ft, D

= 3.5 in. All pipes are smooth and loss coefficients

are negligible. Find the diameter of the equivalent pipe. [Ans.: 6.06 in].

48. The riser of a spray system consists of 3 pipe segments connected in series.

The data for these three pipes are shown below. If we want to represent this sys-

tem with only one pipe, what diameter should we choose? The flow rate, density,

and viscosity are 1250 GPM, 62.4 lbm/ft

, and 0.7E–3, respectively.

Pipe LD K

No. (ft) (in) (-)

1 110.00 8.00 15.0

2 120.00 7.50 1.50

3 100.00 7.00 3.00

[Ans.:

Flow In Serial Pipes

Pipe

No.

(ft)

(in)

(ft

)

(ft/s)

(-)



GPM

× 1E–6

(-)

∆P

(psi)

1 110 8.00 0.349 7.98 15.0 1250 0.474 0.01348 7.38

2 120 7.50 0.307 9.07 1.50 1250 0.506 0.01331 2.25

3 100 7.00 0.267 10.43 3.00 1250 0.542 0.01312 3.84

Data For The Equivalent Pipe Representing The Compound Piping System

L (ft) D (in) A (ft

) V (ft/s) K(-) V



(GPM) I (ft

-1

) ∆P(psi)

331.87 7.34 0.307 9.07 17.08 1250.00 1080.4 13.50].

49. Water flows at a rate of 200 GPM into a piping system connected in series.

Length, diameter, and loss coefficient of each pipe are given below. For

= 62.4

lbm/ft

and

= 0.7E–3 lbm ft/s find total pressure drop and the diameter of an

equivalent pipe representing this system.