Chapter 6 Viscous Flow in Ducts - Simon Fraser University

[Pages:129]Chapter 6 ? Viscous Flow in Ducts

P6.1 An engineer claims that flow of SAE 30W oil, at 20?C, through a 5-cm-diameter smooth pipe at 1 million N/h, is laminar. Do you agree? A million newtons is a lot, so this sounds like an awfully high flow rate.

Solution: For SAE 30W oil at 20?C (Table A.3), take = 891 kg/m3 and = 0.29 kg/m-s. Convert the weight flow rate to volume flow rate in SI units:

Q = w = (1E6 N / h)(1/ 3600 h / s) = 0.0318 m3 = (0.05m)2 V , solve V = 16.2 m

g (891kg / m3 )(9.81m / s 2 )

s4

s

Calculate

Re D

=

VD

= (891kg / m3 )(16.2m / s)(0.05m) 0.29 kg / m - s

2500

(transitional)

This is not high, but not laminar. Ans. With careful inlet design, low disturbances, and a very smooth wall, it might still be laminar, but No, this is transitional, not definitely laminar.

6.2 Air at approximately 1 atm flows through a horizontal 4-cm-diameter pipe. (a) Find a formula for Qmax, the maximum volume flow for which the flow remains laminar, and plot Qmax versus temperature in the range 0?C T 500?C. (b) Is your plot linear? If not, explain.

Solution: (a) First convert the Reynolds number from a velocity form to a volume flow form:

V

=

Q ( /4)d 2

,

therefore

Red

=

Vd

=

4Q 2300 d

for laminar flow

Maximum laminar volume flow is given by

Qmax

=

2300 d 4

Ans. (a)

With d = 0.04 m = constant, get and for air from Table A-2 and plot Qmax versus T ?C:

Chapter 6 ? Viscous Flow in Ducts

435

Fig. P6.2

The curve is not quite linear because = / is not quite linear with T for air in this range. Ans. (b)

6.3 For a thin wing moving parallel to its chord line, transition to a turbulent boundary layer occurs at a "local" Reynolds number Rex, where x is the distance from the leading edge of the wing. The critical Reynolds number depends upon the intensity of turbulent fluctuations in the stream and equals 2.8E6 if the stream is very quiet. A semiempirical correlation for this case [Ref. 3 of Ch. 6] is

Re1x/c2rit

-1+ (1+ 13.25 2 )1/2 0.00392 2

where is the tunnel-turbulence intensity in percent. If V = 20 m/s in air at 20?C, use this formula to plot the transition position on the wing versus stream turbulence for between 0 and 2 percent. At what value of is xcrit decreased 50 percent from its value at = 0?

Solution: This problem is merely to illustrate the strong effect of stream turbulence on the transition point. For air at 20?C, take = 1.2 kg/m3 and = 1.8E-5 kg/ms. Compute Rex,crit from the correlation and plot xtr = Rex/[(20 m/s)] versus percent turbulence:

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Solutions Manual ? Fluid Mechanics, Fifth Edition

Fig. P6.3

The value of xcrit decreases by half (to 1.07 meters) at 0.42%. Ans.

6.4 For flow of SAE 30 oil through a 5-cm-diameter pipe, from Fig. A.1, for what flow rate in m3/h would we expect transition to turbulence at (a) 20?C and (b) 100?C?

Solution: For SAE 30 oil take = 891 kg/m3 and take = 0.29 kg/ms at 20?C (Table A.3)

and 0.01 kg/m-s at 100?C (Fig A.1). Write the critical Reynolds number in terms of flow

rate Q:

(a)

Recrit

= 2300 =

VD

= 4Q D

= 4(891 kg/m3)Q , (0.29 kg/ms)(0.05 m)

solve Q = 0.0293 m3 = 106 m3 Ans. (a)

s

h

(b)

Recrit

= 2300 =

VD

=

4Q D

=

4(891 kg/m3 )Q (0.010 kg/ms)(0.05

, m)

solve Q = 0.00101 m3 = 3.6 m3 Ans. (b)

s

h

Chapter 6 ? Viscous Flow in Ducts

437

6.5 In flow past a body or wall, early transition to turbulence can be induced by placing a trip wire on the wall across the flow, as in Fig. P6.5. If the trip wire in Fig. P6.5 is placed where the local velocity is U, it will trigger turbulence if Ud/ = 850, where d is the wire diameter [Ref. 3 of Ch. 6]. If the sphere diameter is 20 cm and transition is observed at ReD = 90,000, what is the diameter of the trip wire in mm?

Fig. P6.5

Solution:

For the same U and ,

Red

=

Ud

=

850;

ReD

=

UD

=

90000,

or

d

=

D Red ReD

=

(200

mm)

850 90000

1.9

mm

P6.6 For flow of a uniform stream parallel to a sharp flat plate, transition to a turbulent boundary layer on the plate may occur at Rex = Ux/ 1E6, where U is the approach velocity and x is distance along the plate. If U = 2.5 m/s, determine the distance x for the following fluids at 20?C and 1 atm: (a) hydrogen; (b) air; (c) gasoline; (d) water; (e) mercury; and (f) glycerin.

Solution: We are to calculate x = (Rex)()/(U) = (1E6)()/[ (2.5m/s)]. Make a table:

FLUID

? kg/m3

- kg/m-s

x - meters

Hydrogen

0.00839

9.05E-5

43.

Air

1.205

1.80E-5

6.0

Gasoline

680

2.92E-4

0.17

Water

998

0.0010

0.40

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Solutions Manual ? Fluid Mechanics, Fifth Edition

Mercury

13,550

1.56E-3

0.046

Glycerin

1260

1.49

470.

Clearly there are vast differences between fluid properties and their effects on flows.

6.7 Cola, approximated as pure water at 20?C, is to fill an 8-oz container (1 U.S. gal = 128 fl oz) through a 5-mm-diameter tube. Estimate the minimum filling time if the tube flow is to remain laminar. For what cola (water) temperature would this minimum time be 1 min?

Solution: For cola "water", take = 998 kg/m3 and = 0.001 kg/ms. Convert 8 fluid ounces = (8/128)(231 in3) 2.37E-4 m3. Then, if we assume transition at Re = 2300,

Recrit

=

2300

=

VD

=

4Q D

,

or:

Qcrit

= 2300 (0.001)(0.005) 9.05E-6 4(998)

m3 s

Then tfill = /Q = 2.37E-4/9.05E-6 26 s Ans. (a)

(b) We fill in exactly one minute if Qcrit = 2.37E-4/60 = 3.94E-6 m3/s. Then

Qcrit

= 3.94E-6

m3 s

=

2300 D 4

if water 4.36E-7 m2 /s

From Table A-1, this kinematic viscosity occurs at T 66?C Ans. (b)

6.8 When water at 20?C ( = 998 kg/m3, = 0.001 kg/ms) flows through an 8-cmdiameter pipe, the wall shear stress is 72 Pa. What is the axial pressure gradient ( p/ x)

if the pipe is (a) horizontal; and (b) vertical with the flow up? Solution: Equation (6.9b) applies in both cases, noting that w is negative:

(a) Horizontal: dp = 2w = 2(-72 Pa) = -3600 Pa Ans. (a)

dx R 0.04 m

m

(b) Vertical,up:

dp = 2w

-

g

dz

1

=

-3600

-

998(9.81)

=

- 13,

400

Pa

Ans. (b)

dx R

dx

m

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439

6.9 A light liquid ( = 950 kg/m3) flows at an average velocity of 10 m/s through a horizontal smooth tube of diameter 5 cm. The fluid pressure is measured at 1-m intervals along the pipe, as follows:

x, m:

0 1 2 3 456

p, kPa: 304 273 255 240 226 213 200

Estimate (a) the total head loss, in meters; (b) the wall shear stress in the fully developed section of the pipe; and (c) the overall friction factor.

Solution: As sketched in Fig. 6.6 of the text, the pressure drops fast in the entrance region (31 kPa in the first meter) and levels off to a linear decrease in the "fully developed" region (13 kPa/m for this data). (a) The overall head loss, for z = 0, is defined by Eq. (6.8) of the text:

hf

=

p g

=

304,000 - 200,000 Pa (950 kg/m3 )(9.81 m/s2 )

= 11.2 m

Ans. (a)

(b) The wall shear stress in the fully-developed region is defined by Eq. (6.9b):

| p

L fully

developed =

13000 Pa 1 m

=

4 w d

=

4w , 0.05 m

solve for w = 163 Pa

Ans. (b)

(c) The overall friction factor is defined by Eq. (6.10) of the text:

foverall

=

h f , overall

d L

2g V2

=

(11.2

m)

0.05 m 6 m

2(9.81 m/s2 ) (10 m/s)2

=

0.0182

Ans. (c)

NOTE: The fully-developed friction factor is only 0.0137.

6.10 Water at 20?C ( = 998 kg/m3) flows through an inclined 8-cm-diameter pipe. At sections A and B, pA = 186 kPa, VA = 3.2 m/s, zA = 24.5 m, while pBB = 260 kPa, VBB = 3.2 m/s, and zB = 9.1 m. Which way is the flow going? What is the head loss?

Solution: Guess that the flow is from A to B and write the steady flow energy equation:

pA g

+

VA2 2g

+

zA

=

pB g

+

VB2 2g

+

zB

+

hf

,

or:

186000 9790

+

24.5

=

260000 9790

+

9.1 +

hf

,

or: 43.50 = 35.66 + hf , solve: hf = +7.84 m Yes, flow is from A to B. Ans. (a, b)

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Solutions Manual ? Fluid Mechanics, Fifth Edition

6.11 Water at 20?C flows upward at 4 m/s in a 6-cm-diameter pipe. The pipe length between points 1 and 2 is 5 m, and point 2 is 3 m higher. A mercury manometer, connected between 1 and 2, has a reading h = 135 mm, with p1 higher. (a) What is the pressure change (p1 - p2)? (b) What is the head loss, in meters? (c) Is the manometer reading proportional to head loss? Explain. (d) What is the friction factor of the flow?

Solution: A sketch of this situation is shown at right. By moving through the manometer, we obtain the pressure change between points 1 and 2, which we compare with Eq. (6.9b):

p1 + wh - mh - wz = p2,

or:

p1 - p2 = 133100 - 9790

N m3

(0.135

m)

+

9790

N m3

(3

m)

= 16650 + 29370 = 46,000 Pa Ans.(a)

From Eq. (6.9b),

hf

=

p - z = w

46000 Pa 9790 N/m3

-3

m = 4.7 - 3.0 = 1.7 m

Ans. (b)

The friction factor is

f

= hf

d 2g L V2

=

(1.7

m)

0.06 m 5 m

2(9.81 m/s2 (4 m/s)2

)

=

0.025

Ans. (d)

By comparing the manometer relation to the head-loss relation above, we find that:

hf

=

( m - w ) h w

and thus head loss is proportional to manometer reading.

Ans. (c)

NOTE: IN PROBLEMS 6.12 TO 6.99, MINOR LOSSES ARE NEGLECTED.

6.12 A 5-mm-diameter capillary tube is used as a viscometer for oils. When the flow rate is 0.071 m3/h, the measured pressure drop per unit length is 375 kPa/m. Estimate the viscosity of the fluid. Is the flow laminar? Can you also estimate the density of the fluid?

Solution: Assume laminar flow and use the pressure drop formula (6.12):

p L

=?

8Q R4

,

or:

375000

Pa m

=?

8(0.071/3600) (0.0025)4

,

solve 0.292 kg m s

Ans.

Chapter 6 ? Viscous Flow in Ducts

441

Guessing

oil 900

kg m3

,

check

Re =

4Q d

=

4(900)(0.071/3600) (0.292)(0.005)

16

OK, laminar

Ans.

It is not possible to find density from this data, laminar pipe flow is independent of density.

6.13 A soda straw is 20 cm long and 2 mm in diameter. It delivers cold cola, approximated as water at 10?C, at a rate of 3 cm3/s. (a) What is the head loss through the straw? What is the axial pressure gradient p/x if the flow is (b) vertically up or (c) horizontal? Can the human lung deliver this much flow?

Solution: For water at 10?C, take = 1000 kg/m3 and = 1.307E-3 kg/ms. Check Re:

Re = 4Q = 4(1000)(3E-6 m3/s) = 1460 (OK, laminar flow) d (1.307E-3)(0.002)

Then, from Eq. (6.12),

h f

=

128LQ gd4

=

128(1.307E-3)(0.2)(3E-6) (1000)(9.81)(0.002)4

0.204

m

Ans. (a)

If the straw is horizontal, then the pressure gradient is simply due to the head loss:

p L

|horiz =

ghf L

= 1000(9.81)(0.204 m) 9980 0.2 m

Pa m

Ans. (c)

If the straw is vertical, with flow up, the head loss and elevation change add together:

p L

|vertical

=

g(hf + L

z)

=

1000(9.81)(0.204 0.2

+

0.2)

19800

Pa m

Ans. (b)

The human lung can certainly deliver case (c) and strong lungs can develop case (b) also.

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