Fluid Statics Examples 1u6w43

CIVE1400: Fluid Mechanics

Examples: Answers


Examples: Answers

b)

Pressure and Manometers

Ugh

p

1.1

103 u 9.81 u 0.4

What will be the (a) the gauge pressure and (b) the absolute pressure of water at depth 12m below the surface? Uwater = 1000 kg/m3, and p atmosphere = 101kN/m2. [117.72 kN/m2, 218.72 kN/m2]

c)

Z

a)

Ug Ugh

p

Ugh

p gauge

7.9 u 103 u 0.4

1000 u 9.81 u 12

Ugh

p

117.7 kN / m2 , ( kPa )

520 u 9.81 u 0.4

b)

218.7 kN / m2 , ( kPa )

patmosphere

1.2 At what depth below the surface of oil, relative density 0.8, will produce a pressure of 120 kN/m2? What depth of water is this equivalent to? [15.3m, 12.2m]

pabsolute

p gauge patmospheric 13.6 u 10 3 u 9.81 u 0.05 105 N / m2 , ( Pa ) 93.33 kN / m2 , ( kPa )

U JU water

1.5 What height would a water barometer need to be to measure atmospheric pressure? [>10m]

0.8 u 1000 kg / m 3 Ugh p Ug

1bar 1 u 105 N / m2

Ugh patmospheric

a)

h

120 u 10 3 15.29m of oil 800 u 9.81

patmosphere | 1bar 10

5

b) h

U 1000 kg / m3 h

2040 N / m2

1.4 A manometer connected to a pipe indicates a negative gauge pressure of 50mm of mercury. What is the absolute pressure in the pipe in Newtons per square metre is the atmospheric pressure is 1 bar? [93.3 kN/m2]

p gauge patmospheric (117 720 101) N / m2 , ( Pa )

p

3160 N / m 2

d)

117 720 N / m 2 , ( Pa )

pabsolute

3924 N / m 2

120 u 10 3 12.23 m of water 1000 u 9.81

h

1 u 105 N / m2

Ugh 105 . m of water 1019 1000 u 9.81 5 10 0.75 m of mercury (13.6 u 103 ) u 9.81

1.3 What would the pressure in kN/m2 be if the equivalent head is measured as 400mm of (a) mercury J=13.6 (b) water ( c) oil specific weight 7.9 kN/m3 (d) a liquid of density 520 kg/m3? [53.4 kN/m2, 3.92 kN/m2, 3.16 kN/m2, 2.04 kN/m2] a)

U JU water 13.6 u 1000 kg / m3 p

Ugh

13.6 u 103 u 9.81 u 0.4 Examples: Answers

53366 N / m2


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2


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1.6 An inclined manometer is required to measure an air pressure of 3mm of water to an accuracy of +/- 3%. The inclined arm is 8mm in diameter and the larger arm has a diameter of 24mm. The manometric fluid has density 740 kg/m3 and the scale may be read to +/- 0.5mm. What is the angle required to ensure the desired accuracy may be achieved? [12q 39’] p1


1.7 Determine the resultant force due to the water acting on the 1m by 2m rectangular area AB shown in the diagram below. [43 560 N, 2.37m from O] O

P

p2

diameter d

Examples: Answers

1.22m

1.0m 45° C

A

diameter D

x

er

le

d ea

2.0 m

R

2.0 m

z2

a Sc

D

B

Datum line z1

The magnitude of the resultant force on a submerged plane is: R = pressure at centroid u area of surface θ

R

Ugz A 1000 u 9.81 u 122 . 1 u 1 u 2

p1 p2 U man gh Volume moved from left to right

z2 A sin T 2

= z1 A1 z1 z1

SD

2

This acts at right angle to the surface through the centre of pressure.

xA2

Sc

2

z 2 Sd 4 sin T 4 z2 d 2 d2 x 2 2 sin T D D

x

Sd

2

Sc

d · ¸ D2 ¹

§ ©

2

Uman gx¨ sin T

Uwater gh

§ 008 2 · 0.74 u Uwater gx¨© sinT 00..024 2¸ ¹

G

d

For a rectangle I GG As the wall is vertical, Sc

The head being measured is 3% of 3mm = 0.003x0.03 = 0.00009m This 3% represents the smallest measurement possible on the manometer, 0.5mm = 0.0005m, giving 0.00009 sin T

T

I GG x Ax

G

. ) 0.74 x (sin T 01111

h

2nd moment of area about a line through O 1st moment of area about a line through O

b

d · ¸ D2 ¹

Uwater gh

I OO Ax

By the parallel axis theorem (which will be given in an exam), I oo I GG Ax 2 , where IGG is the 2nd moment of area about a line through the centroid and can be found in tables.

4

2

§ ©

Uman gx¨ sin T

p1 p2

43 556 N / m2

Uman g z1 z2

Sc

0.74 u 0.0005 (sin T 01111 . ) 0132 .

D and x

bd 3 12

z,

1 u 23 122 . 1 . 1 121 u 2 122 2.37 m from O

7.6$

[This is not the same as the answer given on the question sheet] Examples: Answers


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4


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1.8 Determine the resultant force due to the water acting on the 1.25m by 2.0m triangular area CD shown in the figure above. The apex of the triangle is at C. [43.5u103N, 2.821m from P]

2.1 Obtain an expression for the depth of the centre of pressure of a plane surface wholly submerged in a fluid and inclined at an angle to the free surface of the liquid. A horizontal circular pipe, 1.25m diameter, is closed by a butterfly disk which rotates about a horizontal axis through its centre. Determine the torque which would have to be applied to the disk spindle to keep the disk closed in a vertical position when there is a 3m head of fresh water above the axis. [1176 Nm]

G d

d/3

So you need to know the resultant force exerted on the disc by the water and the distance x of this force from the spindle.

2 10 . 2 cos 45 1943 . m. 3

Depth to centre of gravity is z R

The question asks what is the moment you have to apply to the spindle to keep the disc vertical i.e. to keep the valve shut?

bd 3 36

For a triangle I GG

Examples: Answers

Forces on submerged surfaces

b

G


We know that the water in the pipe is under a pressure of 3m head of water (to the spindle)

Ugz A . · § 2.0 u 125 1000 u 9.81 u 1943 . u¨ ¸ © 2.0 ¹ 23826 N / m

Distance from P is x

2.375

I oo Sc

z / cos 45 2.748m

F

I oo Ax I GG Ax 2 I GG x Ax 2.829m

3

h’

Distance from P to centre of pressure is Sc

h

2

x

Diagram of the forces on the disc valve, based on an imaginary water surface. h 3

3m , the depth to the centroid of the disc

h’ = depth to the centre of pressure (or line of action of the force)

. u2 125 2.748 . 2.748 36125

Calculate the force: F

Ugh A § 1.25· 1000 u 9.81 u 3 u S ¨ ¸ © 2 ¹

2

36.116 kN Calculate the line of action of the force, h’. h'

2nd moment of area about water surface 1st moment of area about water surface I oo Ah

By the parallel axis theorem 2nd moment of area about O (in the surface) I oo I GG Ah 2 where IGG is the 2nd moment of area about a line through the centroid of the disc and IGG = Sr4/4.

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5

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6


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h'

I GG Ah

Examples: Answers

If we take moments from the surface,

h

fd1 fd 2 fd 3

DR

Sr 4 3 4(Sr 2 )3 r2 3 12


D 3 f 12

3.0326m

f d1 d 2 d 3

d1 d 2 d 3

Taking the first beam, we can draw a pressure diagram for this, (ignoring what is below),

So the distance from the spindle to the line of action of the force is

x

h ' h

2H/3

3.0326 3 0.0326m

And the moment required to keep the gate shut is

moment

Fx

H

F=58860

36.116 u 0.0326 1176 . kN m

2.2 A dock gate is to be reinforced with three horizontal beams. If the water acts on one side only, to a depth of 6m, find the positions of the beams measured from the water surface so that each will carry an equal load. Give the load per meter. [58 860 N/m, 2.31m, 4.22m, 5.47m]

We know that the resultant force, F

H

First of all draw the pressure diagram, as below:

1 UgH 2 , so H 2

2 u 58860 1000 u 9.81

2F Ug

2F Ug

3.46 m

And the force acts at 2H/3, so this is the position of the 1st beam,

d1

2h/3

h

position of 1st beam f

R

f

d2

2 H 3

2.31m

Taking the second beam into consideration, we can draw the following pressure diagram,

d3

f

f

The resultant force per unit length of gate is the area of the pressure diagram. So the total resultant force is

R

1 Ugh 2 = 0.5 u 1000 u 9.81 u 62 2

F=2u58860

176580 N ( per m length)

R

H

176580 N ( per m length)

Examples: Answers

2F Ug

2 u (2 u 58860) 1000 u 9.81

(2 u 58860) u 3.27 depth to second beam d 2

The three beams should carry an equal load, so each beam carries the load f, where

R 3

f

4.9 m

The reaction force acts at 2H/3, so H=3.27m. Taking moments from the surface,

This is the resultant force exerted by the gate on the water.

f

d2

The reaction force is equal to the sum of the forces on each beam, so as before

Alternatively the resultant force is, R = Pressure at centroid u Area , (take width of gate as 1m to give force per m)

h Ug u h u 1 2

d1=2.31

2H/3

H

58860 u 2.31 58860 u d 2 4.22 m

For the third beam, from before we have,

58860 N

12 d1 d 2 d 3 depth to third beam d 3 12 2.31 4.22 5.47m


7

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2.3 The profile of a masonry dam is an arc of a circle, the arc having a radius of 30m and subtending an angle of 60q at the centre of curvature which lies in the water surface. Determine (a) the load on the dam in N/m length, (b) the position of the line of action to this pressure. [4.28 u 106 N/m length at depth 19.0m]


Examples: Answers

As this force act normal to the surface, it must act through the centre of radius of the dam wall. So the depth to the point where the force acts is, y = 30sin 39.31q=19m 2.4 The arch of a bridge over a stream is in the form of a semi-circle of radius 2m. the bridge width is 4m. Due to a flood the water level is now 1.25m above the crest of the arch. Calculate (a) the upward force on the underside of the arch, (b) the horizontal thrust on one half of the arch. [263.6 kN, 176.6 kN]

Draw the dam to help picture the geometry, R a

The bridge and water level can be drawn as: 60°

1.25m y

FR

h R

2m

Fh

a) The upward force on the arch = weight of (imaginary) water above the arch.

Rv

Fv h

30 sin 60 2598 . m

a

30 cos 60 150 . m

volume Rv

§ S 22 · . 2) u 4 ¨ (125 ¸ u4 2 ¹ © 1000 u 9.81 u 26.867

26.867 m 3

263568 . kN

b)

Calculate Fv = total weight of fluid above the curved surface (per m length)

The horizontal force on half of the arch, is equal to the force on the projection of the curved surface onto a vertical plane.

Ug (area of sector - area of triangle)

Fv

Ug u volume of water

. u 15· º 60 · § 2598 ª§ = 1000 u 9.81 u «¨ S 302 u ¸ ¨ ¸» ¹¼ 360¹ © 2 ¬© . kN / m 2711375

1.25

Calculate Fh = force on projection of curved surface onto a vertical plane

1 Ugh 2 2 05 . u 1000 u 9.81 u 2598 . 2

Fh

2.0

3310.681 kN / m Fh

The resultant, 2 v

2 h

F F

FR

2

3310.681 2711375 .

pressure at centroid u area

Ug125 . 1 u 2 u 4

2

176.58 kN

4279.27 kN / m acting at the angle tan T

T Examples: Answers

Fv Fh

0.819

39.32 $


9

2.5 The face of a dam is vertical to a depth of 7.5m below the water surface then slopes at 30q to the vertical. If the depth of water is 17m what is the resultant force per metre acting on the whole face? [1563.29 kN] Examples: Answers CIVE1400: Fluid Mechanics 10


Examples: Answers


Examples: Answers d1 d2

h2

d3

f1 F f2

f3

h1 60q

density of oil Uoil = 0.9Uwater = 900 kg/m3. Force per unit length, F = area under the graph = sum of the three areas = f1 + f2 + f3

x h2 = 17.0 m, so h1 = 17.0 - 7.5 = 9.5 . x = 9.5/tan 60 = 5.485 m.

f1

Vertical force = weight of water above the surface,

f2

Ug h2 u x 0.5h1 u x

Fv

f3

9810 u 7.5 u 5.485 0.5 u 9.5 u 5.485 659.123 kN / m

F

1 Ugh 2 2 0.5 u 1000 u 9.81 u 17 2

f 1d1 f 2 d 2 f 3d 3 2 15 2 . 165544 D 52974 u 2 79461 u (2 ) 33109 u (2 15 .) 3 2 3 D 2.347m ( from surface) 1153 . m ( from base of wall) DF

1417.545 kN / m The resultant force is

Fv2 Fh2

FR

(1000 u 9.81 u 15 . ) u 15 . u 3 33109 N 2 f 1 f 2 f 3 165544 N

To find the position of the resultant force F, we take moments from any point. We will take moments about the surface.

The horizontal force = force on the projection of the surface on to a vertical plane.

Fh

(900 u 9.81 u 2) u 2 u 3 52974 N 2 (900 u 9.81 u 2) u 15 . u 3 79461 N

659.1232 1417.5452

1563.29 kN / m And acts at the angle tan T

T

Fv Fh

0.465

24.94$

2.6 A tank with vertical sides is square in plan with 3m long sides. The tank contains oil of relative density 0.9 to a depth of 2.0m which is floating on water a depth of 1.5m. Calculate the force on the walls and the height of the centre of pressure from the bottom of the tank. [165.54 kN, 1.15m] Consider one wall of the tank. Draw the pressure diagram:

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12


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Examples: Answers 2 B

Application of the Bernoulli Equation

pB p A u u zA 1000 g 2g

3.1 In a vertical pipe carrying water, pressure gauges are inserted at points A and B where the pipe diameters are 0.15m and 0.075m respectively. The point B is 2.5m below A and when the flow rate down the pipe is 0.02 cumecs, the pressure at B is 14715 N/m2 greater than that at A. v2 where v is the velocity at A, Assuming the losses in the pipe between A and B can be expressed as k 2g find the value of k. If the gauges at A and B are replaced by tubes filled with water and connected to a U-tube containing mercury of relative density 13.6, give a sketch showing how the levels in the two limbs of the U-tube differ and calculate the value of this difference in metres. [k = 0.319, 0.0794m] dA = 0.2m

2 A

15 0.065 . 2.5 1045 . k

k

2 A

u 2g

0.065k 0.319

Part ii)

p xxL p xxR p xxL

A

Uw gz B p B Um gR p Uw gz A Uw gR p p A p xxR

U w gz B p B

Um gR p U w gz A U w gR p p A

pB p A

U w g z A z B gR P Um U w

14715 1000 u 9.81 u 2.5 9.81R p 13600 1000 Rp

dB = 0.2m

3.2

B

A Venturimeter with an entrance diameter of 0.3m and a throat diameter of 0.2m is used to measure the volume of gas flowing through a pipe. The discharge coefficient of the meter is 0.96. Assuming the specific weight of the gas to be constant at 19.62 N/m3, calculate the volume flowing when the pressure difference between the entrance and the throat is measured as 0.06m on a water U-tube manometer. [0.816 m3/s]

Rp

Part i) dA

0.15m

pB p A hf

dB

0.079 m

0.075m

Q

d2 = 0.2m

0.02 m 3 / s

d1 = 0.3m

14715 N / m 2

kv 2 2g

Taking the datum at B, the Bernoulli equation becomes: p A u 2A z Ug 2 g A zA

Z2

p B u B2 u 2A zB k Ug 2 g 2g

2.5

zB

Z1

Rp

h

0

By continuity: Q = uAAA = uBAB uA

0.02 / S 0.075 2

uB

0.02 / S 0.0375 2

1132 . m/ s 4.527 m / s

giving

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14


Examples: Answers

What we know from the question:

0.96

d1

0.3m

d2

0.2 m

Examples: Answers

3.3 A Venturimeter is used for measuring flow of water along a pipe. The diameter of the Venturi throat is two fifths the diameter of the pipe. The inlet and throat are connected by water filled tubes to a mercury U-tube manometer. The velocity of flow along the pipe is found to be 2.5 H m/s, where H is the manometer reading in metres of mercury. Determine the loss of head between inlet and throat of the Venturi when H is 0.49m. (Relative density of mercury is 13.6). [0.23m of water]

U g g 19.62 N / m 2 Cd


Calculate Q.

u1

Q / 0.0707

u2

Q / 0.0314

For the manometer: p1 U g gz p1 p 2

p 2 U g g z 2 R p Uw gR p 19.62 z 2 z1 587.423

(1)

For the Venturimeter

p1 u12 z Ug g 2g 1 p1 p 2

p2 u 22 z Ug g 2g 2

Z2

19.62 z 2 z1 0.803u 22

Z1

(2 )

H

h

Combining (1) and (2) 0.803u 22 u 2 ideal Qideal Q

587.423 27.047 m / s § 0.2 · 27.047 u S ¨ ¸ © 2 ¹

Cd Qidea

2

0.96 u 0.85

For the manometer:

0.85m 3 / s

p1 Uw gz1 0.816m 3 / s

p1 p2

p2 Uw g z2 H Um gH Uw gz2 Uw gH Um gH Uw gz1

(1)

For the Venturimeter

p1 u2 1 z1 Uw g 2 g p1 p2

p2 u2 2 z2 Losses Uw g 2 g

Uw u22 2

Uw gz2

Uw u12 2

Uw gz1 LUw g

( 2)

Combining (1) and (2)

p1 u12 z Uw g 2 g 1 LU w g

p2 u 22 z Losses Uw g 2 g 2 Hg Um Uw

Uw 2

u

2 2

u12

(3)

but at 1. From the question u1 u1 A1 175 . uS

Examples: Answers


15

Examples: Answers

d2 4 u2

2.5 H

175 . m/ s

u 2 A2 § 2d · u2 S ¨ ¸ © 10 ¹

2

10.937 m / s CIVE1400: Fluid Mechanics

16


Examples: Answers


Examples: Answers

Substitute in (3)

Losses

L

2 3

u 2g 5.99 m / s

183 .

0.49 u 9.8113600 1000 1000 / 2 10.937 2 1.75 2

u3

9.81 u 1000

Q u 3 A3

0.233m 3.4 Water is discharging from a tank through a convergent-divergent mouthpiece. The exit from the tank is rounded so that losses there may be neglected and the minimum diameter is 0.05m. If the head in the tank above the centre-line of the mouthpiece is 1.83m. a) What is the discharge? b) What must be the diameter at the exit if the absolute pressure at the minimum area is to be 2.44m of water? c) What would the discharge be if the divergent part of the mouth piece were removed. (Assume atmospheric pressure is 10m of water). [0.0752m, 0.0266m3/s, 0.0118m3/s]

0.02665 5.99 u S d3

If the mouth piece has been removed, p1 p1

Ug

z1 u2

d 32 4

0.0752m

p2 p2

Ug

u 22 2g

2 gz1

5.99 m / s

0.052 Q 5.99S 4

0.0118 m 3 / s

3.5 A closed tank has an orifice 0.025m diameter in one of its vertical sides. The tank contains oil to a depth of 0.61m above the centre of the orifice and the pressure in the air space above the oil is maintained at 13780 N/m2 above atmospheric. Determine the discharge from the orifice. (Coefficient of discharge of the orifice is 0.61, relative density of oil is 0.9). [0.00195 m3/s]

h

2

3 P = 13780 kN/m2

From the question:

d2

0.05m p2

minimum pressure p1

10m

Ug

0.66m

2.44 m

Ug

oil

p3

Ug do = 0.025m

Apply Bernoulli: p1 u12 z Ug 2 g 1

p2 u22 z Ug 2 g 2

p3 u32 z Ug 2 g 3

From the question

If we take the datum through the orifice:

z1

183 . m

z2

z3

u1

0

negligible

Between 1 and 2

Uo

900 0.61

u2 Q

u2 A2

Apply Bernoulli,

10 183 . 2.44

Between 1 and 3 p1 Examples: Answers

0.9

Cd u 22 2g 1357 . m/s § 0.05· 1357 . u S¨ ¸ © 2 ¹

Uo Uw

V

p1 u12 z Ug 2 g 1

p2 u22 z Ug 2 g 2

2

0.02665 m 3 / s

Take atmospheric pressure as 0,

p3 CIVE1400: Fluid Mechanics

17

Examples: Answers


18


Examples: Answers

13780 0.61 Uo g


Examples: Answers

2 2

u 2g

p1

u 2g

Ug

p1 p 2 u12 hf Ug Ug 2 g

u 22 2g

Ug

u2

6.53 m / s

Q

§ 0.025· 0.61 u 6.53 u S ¨ ¸ © 2 ¹

2

0.00195 m 3 / s

2 1

25

3.6 The discharge coefficient of a Venturimeter was found to be constant for rates of flow exceeding a certain value. Show that for this condition the loss of head due to friction in the convergent parts of the meter can be expressed as KQ2 m where K is a constant and Q is the rate of flow in cumecs. Obtain the value of K if the inlet and throat diameter of the Venturimeter are 0.102m and 0.05m respectively and the discharge coefficient is 0.96. [K=1060]

u hf 2g

u 22 3.77 2 2.5 2g 2g u 2 21.346 m / s Q u 2 A2 0.0667 d2

3.7 A Venturimeter is to fitted in a horizontal pipe of 0.15m diameter to measure a flow of water which may be anything up to 240m3/hour. The pressure head at the inlet for this flow is 18m above atmospheric and the pressure head at the throat must not be lower than 7m below atmospheric. Between the inlet and the throat there is an estimated frictional loss of 10% of the difference in pressure head between these points. Calculate the minimum allowable diameter for the throat. [0.063m]

p2

2 2

21.346 u S

d 22 4

0.063m

3.8 A Venturimeter of throat diameter 0.076m is fitted in a 0.152m diameter vertical pipe in which liquid of relative density 0.8 flows downwards. Pressure gauges are fitted to the inlet and to the throat sections. The throat being 0.914m below the inlet. Taking the coefficient of the meter as 0.97 find the discharge a) when the pressure gauges read the same b)when the inlet gauge reads 15170 N/m2 higher than the throat gauge. [0.0192m3/s, 0.034m3/s]

d1 = 0.15m

d1 = 0.152m

d2

From the question:

d1 = 0.076m

d1 u1 p1

Ug

. m 015 Q / A 3.77 m / s

240 m 3 / hr

Q p2

18m

Ug

0.667 m3 / s

7 m

Friction loss, from the question: hf

0.1

p

1

p2

Ug

From the question: d1 d2

Apply Bernoulli:

0.152 m 0.076m

A1 A2

0.01814 m 0.00454 m

800 kg / m 3 Cd 0.97

U Apply Bernoulli: Examples: Answers


19

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20


Examples: Answers

p1

Ug a) p1

2 1

p2

2 2

Ug

u12 z 2g 1

u 22 z 2g 2

4.1 A reservoir is circular in plan and the sides slope at an angle of tan-1(1/5) to the horizontal. When the reservoir is full the diameter of the water surface is 50m. Discharge from the reservoir takes place through a pipe of diameter 0.65m, the outlet being 4m below top water level. Determine the time for the water level to fall 2m assuming the discharge to be 0.75a 2 gH cumecs where a is the cross sectional area of the pipe in m2 and H is the head of water above the outlet in m. [1325 seconds]

p2

By continuity: Q u2 u12 0.914 2g

Examples: Answers

Tank emptying

u z 2g 2

u z 2g 1


u1 A1 u 2 A2 A1 u1 4 u1 A2

50m

16u12 2g

r H

Q

0.914 u 2 u 9.81 15 Cd A1 u1

Q

. 0.96 u 0.01814 u 10934

u1

. 10934 m/ s

x

0.019 m 3 / s

1

b)

5

p1 p 2 p1 p2 Ug 15170 Ug 55.8577 Q

15170 From the question:

u 22 u12 0.914 2g Q 2 220.432 5511 . 2 2g

Q

H = 4m

a = S(0.65/2)2 = 0.33m2

0.75a 2 gh 10963 . h

0.914

In time Gt the level in the reservoir falls Gh, so

Q 2 220.432 5511 . 2

Q Gt

0.035 m 3 / s

A Gh A Gh Q

Gt

Integrating give the total time for levels to fall from h1 to h2. T

³

h2

h1

A dh Q

As the surface area changes with height, we must express A in of h. A = Sr2 But r varies with h. It varies linearly from the surface at H = 4m, r = 25m, at a gradient of tan-1 = 1/5. r = x + 5h 25 = x + 5(4) x=5 so

A = S( 5 + 5h )2 = ( 25S + 25Sh2 + 50Sh )

Substituting in the integral equation gives Examples: Answers


21

Examples: Answers


22


T

³

h2

Examples: Answers 2

25S 25Sh 50Sh 10963 h . 2 h2 1 h 2h

h1

25S ³h1 10963 . . ³ 71641

1

h2

h1

. ³ 71641

h

h2

h h2 h


And we can write an equation for the discharge in of the surface height change:

dh

Q Gt

Gt

dh 2h h

T

h

T

³

h2

h1

A dh Q h2 A

168 . ³

2 4 º2 ª . «2h 1/2 h 53/ 2 h 3/ 2 » 71641 5 3 ¼ h1 ¬ From the question, h1 = 4m

A Gh A Gh Q

Integrating give the total time for levels to fall from h1 to h2.

dh

h 1/2 h 3/ 2 2h 1/2 dh

h1

Examples: Answers

(1)

dh

a) For the first 1m depth, A = 8 x 32 = 256, whatever the h.

h2 = 2m, so

So, for the first period of time:

4 2 4 2 ª§ · § ·º 71641 . «¨ 2 u 4 1/2 u 4 53/ 2 u 4 3/ 2 ¸ ¨ 2 u 2 1/2 u 2 53/ 2 u 2 3/ 2 ¸ » ¹ © ¹¼ 3 5 3 5 ¬© . >4 12.8 10.667 2.828 2.263 3.77 @ 71641

T

168 . ³

h2

256 h

h1

dh

>

430.08 h1 h2

. >27.467 8.862@ 71641 1333 sec

>

@

430.08 2.6 16 .

@

299 sec

4.2 A rectangular swimming pool is 1m deep at one end and increases uniformly in depth to 2.6m at the other end. The pool is 8m wide and 32m long and is emptied through an orifice of area 0.224m2, at the lowest point in the side of the deep end. Taking Cd for the orifice as 0.6, find, from first principles, a) the time for the depth to fall by 1m b) the time to empty the pool completely. [299 second, 662 seconds]

b) now we need to find out how long it will take to empty the rest. We need the area A, in of h. A 8L L 32 h 1.6 A 160h

32.0m

So

1.0m 2.6m

T

L

168 . ³

h2

h

dh

2 h 3/2 h2 3/2 3 1 2 3/ 2 3/ 2 268.9 16 . 0 3 362.67 sec

>

>

The question tell us ao = 0.224m2, Cd = 0.6 Apply Bernoulli from the tank surface to the vena contracta at the orifice:

p1 u12 z Ug 2 g 1

160h

h1

268.9

p1 = p2 and u1 = 0. u 2

h

h1

p2 u22 z Ug 2 g 2

@

@

Total time for emptying is, T = 363 + 299 = 662 sec

2 gh

We need Q in of the height h measured above the orifice.

Q

Cd a o u 2

Cd a o 2 gh

0.6 u 0.224 u 2 u 9.81 h 0.595 h Examples: Answers


23

Examples: Answers


24


Examples: Answers

CIVE1400: Fluid Mechanics Q Gt

4.3 A vertical cylindrical tank 2m diameter has, at the bottom, a 0.05m diameter sharp edged orifice for which the discharge coefficient is 0.6. a) If water enters the tank at a constant rate of 0.0095 cumecs find the depth of water above the orifice when the level in the tank becomes stable. b) Find the time for the level to fall from 3m to 1m above the orifice when the inflow is turned off. c) If water now runs into the tank at 0.02 cumecs, the orifice remaining open, find the rate of rise in water level when the level has reached a depth of 1.7m above the orifice. [a) 3.314m, b) 881 seconds, c) 0.252m/min]

Examples: Answers

A Gh A Gh Q

Gt

Integrating between h1 and h2, to give the time to change surface level T

³

h2

h1

A dh Q

. ³ 6018

h2

h1

h 1/ 2 dh

1203.6>h 1/ 2 @h

h2

Q = 0.0095 m3/s

1

1203.6>h21/ 2 h11/ 2 @

h1 = 3 and h2 = 1 so T = 881 sec c) Qin changed to Qin = 0.02 m3/s

h

From (1) we have Qout i.e. do = 0.005m

Q

Q

p2 u22 z Ug 2 g 2

u

2 gh .

We need Q in of the height h measured above the orifice. Cd a o u2

Cd a o 2 gh

§ 0.05· 0.6S ¨ ¸ © 2 ¹

Qin Qout

0.02 0.0068 0.0132 m 3 / s

Au Q 0.0132 A § S 22 · ¨ ¸ © 4 ¹

0.0042 m / s 0.252 m / min

4.4 A horizontal boiler shell (i.e. a horizontal cylinder) 2m diameter and 10m long is half full of water. Find the time of emptying the shell through a short vertical pipe, diameter 0.08m, attached to the bottom of the shell. Take the coefficient of discharge to be 0.8. [1370 seconds]

With the datum the bottom of the cylinder, z1 = h, z2 = 0

Qout

0.0068 m3 / s

As Q = Area x Velocity, the rate of rise in surface is

Apply Bernoulli from the water surface (1) to the orifice (2),

p1 = p2 and u1 = 0. u 2

0.00522 17 .

The rate of increase in volume is:

From the question: Qin = 0.0095 m3/s, do=0.05m, Cd =0.6

p1 u12 z Ug 2 g 1

Qout

0.00522 h . The question asks for the rate of surface rise when h = 1.7m.

2

d = 2m

2 u 9.81 h

0.00522 h

(1)

32m

For the level in the tank to remain constant: inflow = out flow Qin = Qout 0.0095 0.00522 h h 3.314 m

do = 0.08 m

(b) Write the equation for the discharge in of the surface height change:

From the question W = 10m, D = 10m do = 0.08m Cd = 0.8 Examples: Answers


25

Examples: Answers


26


Examples: Answers


Apply Bernoulli from the water surface (1) to the orifice (2),

p1 u12 z Ug 2 g 1 p1 = p2 and u1 = 0. u 2

1123.6³ 1123.6³

h2

>

2 u 9.81 h

Integrating between h1 and h2, to give the time to change surface level h1

h2 h1

4.5 Two cylinders standing upright contain liquid and are connected by a submerged orifice. The diameters of the cylinders are 1.75m and 1.0m and of the orifice, 0.08m. The difference in levels of the liquid is initially 1.35m. Find how long it will take for this difference to be reduced to 0.66m if the coefficient of discharge for the orifice is 0.605. (Work from first principles.) [30.7 seconds]

A Gh A Gh Q

³

@

749.07>2.828 1@ 1369.6 sec

Write the equation for the discharge in of the surface height change:

T

2 h dh

§ 2· 3/ 2 1123.6¨ ¸ 2 h © 3¹

2

0.0178 h

h2

2h h 2 dh h

h2

h1

Cd a o 2 gh

§ 0.08 · 0.8S ¨ ¸ © 2 ¹

Gt

2h h 2 dh h

h2

h1

We need Q in of the height h measured above the orifice.

Q Gt

20 2h h 2 dh . 01078 h h1

2 gh .

Cd a o u2

h2

h1

1123.6³

With the datum the bottom of the cylinder, z1 = h, z2 = 0

Qout

³

T

p2 u22 z Ug 2 g 2

Examples: Answers

d1 = 1.75m

d2 = 1.0m

A dh Q

But we need A in of h h = 1.35m

2.0m

1.0m a L h do = 0.108m

. Surface area A = 10L, so need L in of h 12 a

§ L· a2 ¨ ¸ © 2¹

S¨

do

0.08m,

2

2.4m2 ao

(1 h)

12

§ L· (1 h) 2 ¨ ¸ © 2¹

L

2h h

2

§ 0.08· ¸ © 2 ¹

S¨

A2

§ 1· © 2¹

2

S¨ ¸

0.785m2

2

0.00503m2

Cd

0.605

by continuity,

2

A1Gh1

2

defining, h = h1 - h2

2

Gh

A 20 2h h

A2Gh2

QGt

(1)

Gh1 Gh2

Substituting this in (1) to eliminate Gh2

Substitute this into the integral term, Examples: Answers

. · § 175 ¸ © 2 ¹

A1

2


27

Examples: Answers


28


Examples: Answers

A1Gh1

Gh1 A1

A2Gh A1 A2

A2 (Gh1 Gh) A2Gh A1 A2

A2Gh1 A2Gh

Q Gt

Gt

QGt

Examples: Answers

A Gh A Gh Q

Integrating between h1 and h2, to give the time to change surface level

( 2)

T

From the Bernoulli equation we can derive this expression for discharge through the submerged orifice:

Q


Cd ao 2 gh

So

A dh Q 60000 h2 1 dh 0.678 ³h1 h 3/ 2 ³

h2

h1

2 u 8849558 . >h 1/ 2 @h

h2

A Gh A1 2 A1 A2

Gt

1

Cd a o 2 gh Gt

From the question T = 3600 sec and h1 = 0.6m 3600 17699115 . >h21/ 2 0.6 1/ 2 @

A1 A2 1 Gh A1 A2 Cd ao 2 g h

h2

0.5815m

Total depth = 3.4 + 0.58 = 3.98m

Integrating T

A

A1 A2 A2 Cd a o 2 g

³

A

2 A1 A2 A2 Cd a o 2 g

1

1

h2

h1

1 dh h h2 h1

2 u 2.4 u 0.785

2.4 0.785 u 0.605 u 0.00503 2 u 9.81

0.8124 11619 .

30.7 sec 4.6 A rectangular reservoir with vertical walls has a plan area of 60000m2. Discharge from the reservoir take place over a rectangular weir. The flow characteristics of the weir is Q = 0.678 H3/2 cumecs where H is the depth of water above the weir crest. The sill of the weir is 3.4m above the bottom of the reservoir. Starting with a depth of water of 4m in the reservoir and no inflow, what will be the depth of water after one hour? [3.98m]

From the question A = 60 000 m2, Q = 0.678 h

3/2

Write the equation for the discharge in of the surface height change:

Examples: Answers


29

Examples: Answers


30


Examples: Answers


Examples: Answers

5.2 Derive an expression for the discharge over a sharp crested rectangular weir. A sharp edged weir is to be constructed across a stream in which the normal flow is 200 litres/sec. If the maximum flow likely to occur in the stream is 5 times the normal flow then determine the length of weir necessary to limit the rise in water level to 38.4cm above that for normal flow. Cd=0.61. [1.24m]

Notches and weirs

5.1 Deduce an expression for the discharge of water over a right-angled sharp edged V-notch, given that the coefficient of discharge is 0.61. A rectangular tank 16m by 6m has the same notch in one of its short vertical sides. Determine the time taken for the head, measured from the bottom of the notch, to fall from 15cm to 7.5cm. [1399 seconds]

From your notes you can derive: From your notes you can derive:

Q

From the question: Q1 = 0.2 m3/s,

For this weir the equation simplifies to Q 144 . H

Gt

Q2 = 1.0 m /s, So we have two situations:

A Gh A Gh Q

0.2 10 .

A dh Q 16 u 6 h2 1 ³h1 h5/2 dh . 144 2 h2 u 66.67>h 3/ 2 @h 1 3 ³

h2

44.44>0.075

3/ 2

015 .

2 C b 2 g x 3/ 2 1801 . bx 3/ 2 3 d 2 3/ 2 3/ 2 C b 2 g x 0.384 1801 . b x 0.384 3 d

(1) ( 2)

From (1) we get an expression for b in of x

h1

0111 . x 3/ 2

b

Substituting this in (2) gives, § x 0.384 · 10 . 1801 . u 0111 . ¨ ¸ ¹ © x

h1 = 0.15m, h2 = 0.075m T

h2 = x + 0.384

where x is the height above the weir at normal flow.


T

h1 = x

3

5/ 2

Write the equation for the discharge in of the surface height change: Q Gt

2 C b 2 gh 3/ 2 3 d

Q

8 T 2 g H 5/ 2 C tan 15 d 2

3 / 2

52 / 3

@

x

1399 sec

3/ 2

§ x 0.384 · ¨ ¸ © ¹ x 01996 m .

So the weir breadth is b

0111 . 01996 .

3/ 2

124 . m

Examples: Answers


31

Examples: Answers


32


Examples: Answers

8 T 2 g H 5/ 2 C tan 15 d 2

Q Gt

Gt

T = 90q

T

Cd KH 5/ 2 is in the notes.

h1 = 0.08m

h2 = 0.07m

T = 43.5sec

So

T

Q = 2.36 Cd h5/2

Gt

A Gh A Gh Q


A dh Q h2 10 1 dh 2.36Cd ³h1 h 5/2 2 4.23 3/2 0.08 u >h @0.07 3 Cd 2.82 435 . >0.07 3/2 0.083/2 @ Cd Cd 0.635 T

³

A dh Q 56000 h2 1 dh 177 . B ³h1 h 3/ 2 2 u 56000 1/ 2 0.3 > h @ 0. 6 177 . B 1/ 2 5784>0.3 0.6 1/ 2 @ ³

h2

h1

3093 sec

5.5 Develop a formula for the discharge over a 90q V-notch weir in of head above the bottom of the V. A channel conveys 300 litres/sec of water. At the outlet end there is a 90q V-notch weir for which the coefficient of discharge is 0.58. At what distance above the bottom of the channel should the weir be placed in order to make the depth in the channel 1.30m? With the weir in this position what is the depth of water in the channel when the flow is 200 litres/sec? [0.755m, 1.218m]

Write the equation for the discharge in of the surface height change: Q Gt

A Gh A Gh Q


From the question: A = 10m2

Examples: Answers

b) Write the equation for the discharge in of the surface height change:

5.3 Show that the rate of flow across a triangular notch is given by Q=CdKH5/2 cumecs, where Cd is an experimental coefficient, K depends on the angle of the notch, and H is the height of the undisturbed water level above the bottom of the notch in metres. State the reasons for the introduction of the coefficient. Water from a tank having a surface area of 10m2 flows over a 90q notch. It is found that the time taken to lower the level from 8cm to 7cm above the bottom of the notch is 43.5seconds. Determine the coefficient Cd assuming that it remains constant during his period. [0.635] The proof for Q


h2

Derive this formula from the notes: Q

h1

From the question:

T = 90q

depth of water, Z = 0.3m

Q 137 . H 5/ 2

a) As H is the height above the bottom of the V, the depth of water = Z = D + H, where D is the height of the bottom of the V from the base of the channel. So

Q 137 . Z D

5/2

0.3 137 . 13 . D

5/2

D 0.755m b) Find Z when Q = 0.2 m3/s 0.2 137 . Z 0.755 . m Z 1218

From the question: Q = 1.77 B H 3/2

Q = 0.3 m3/s,

Cd 0.58

giving the weir equation:

5.4 A reservoir with vertical sides has a plan area of 56000m2. Discharge from the reservoir takes place over a rectangular weir, the flow characteristic of which is Q=1.77BH3/2 m3/s. At times of maximum rainfall, water flows into the reservoir at the rate of 9m3/s. Find a) the length of weir required to discharge this quantity if head must not exceed 0.6m; b) the time necessary for the head to drop from 60cm to 30cm if the inflow suddenly stops. [10.94m, 3093seconds] A = 56000 m2

8 T C tan 2 g H 5/ 2 15 d 2

5/ 2

Qmax = 9 m3/s

a) Find B for H = 0.6 9 = 1.77 B 0.63/2 B = 10.94m Examples: Answers


33

Examples: Answers


34


Examples: Answers

5.6 Show that the quantity of water flowing across a triangular V-notch of angle 2T is 8 Q Cd tan T 2 g H 5/ 2 . Find the flow if the measured head above the bottom of the V is 38cm, when 15 T=45q and Cd=0.6. If the flow is wanted within an accuracy of 2%, what are the limiting values of the head. [0.126m3/s, 0.377m, 0.383m]


Examples: Answers

Application of the Momentum Equation

6.1 The figure below shows a smooth curved vane attached to a rigid foundation. The jet of water, rectangular in section, 75mm wide and 25mm thick, strike the vane with a velocity of 25m/s. Calculate the vertical and horizontal components of the force exerted on the vane and indicate in which direction these components act. [Horizontal 233.4 N acting from right to left. Vertical 1324.6 N acting downwards]

Proof of the v-notch weir equation is in the notes. From the question: H = 0.38m T = 45q

Cd = 0.6

45q

The weir equation becomes:

25q

Q 1417 . H 5/ 2 1417 . 0.38

From the question:

5/ 2

0126 . m3 / s

a1

0.075 u 0.025 1875 . u 10 3 m 2

u1

25 m / s

Q 1875 . u 10 3 u 25 m 3 / s a1 a 2 , so u1 u 2

Q+2% = 0.129 m3/s . . H 5/ 2 0129 1417 H 0.383m

Calculate the total force using the momentum equation: FT x

Q-2% = 0.124 m /s . . H 0124 1417 H 0.377m

UQu 2 cos 25 u1 cos 45 1000 u 0.0469 25 cos 25 25 cos 45 233.44 N

3

5/ 2

FT y

UQu 2 sin 25 u1 sin 45 1000 u 0.0469 25 sin 25 25 sin 45 1324.6 N

Body force and pressure force are 0. So force on vane:

Examples: Answers


35

Rx

Ft x

233.44 N

Ry

Ft y

1324.6 N

Examples: Answers


36


Examples: Answers


FRy

6.2 A 600mm diameter pipeline carries water under a head of 30m with a velocity of 3m/s. This water main is fitted with a horizontal bend which turns the axis of the pipeline through 75q (i.e. the internal angle at the bend is 105q). Calculate the resultant force on the bend and its angle to the horizontal. [104.044 kN, 52q 29’]

FTy FPy FBy 2.457 80.376 0

82.833 kN

These forces act on the fluid The resultant force on the fluid is

u2

y

Examples: Answers

FR

T x

FRx FRy § FRy · ¸ tan 1 ¨ © FRx ¹

104.44 kN 52 $ 29 '

6.3 A horizontal jet of water 2u103 mm2 cross-section and flowing at a velocity of 15 m/s hits a flat plate at 60q to the axis (of the jet) and to the horizontal. The jet is such that there is no side spread. If the plate is stationary, calculate a) the force exerted on the plate in the direction of the jet and b) the ratio between the quantity of fluid that is deflected upwards and that downwards. (Assume that there is no friction and therefore no shear force.) [338N, 3:1]

u1 θ

y u2

x

From the question: a u1

§ 0.6 · ¸ © 2¹

2

S¨

u2

0.283 m 2

3m / s

Q

d

0.6 m

h

30 m

0.848 m 3 / s

u1

Calculate total force.

FTx

UQ u 2 x u1x FRx FPx FBx

FTx

1000 u 0.848 3 cos 75 3 1.886 kN

FTy

UQ u2 y u1 y

FTy

1000 u 0.848 3 sin 75 0

θ

FRy FPy FBy

u3

2.457 kN From the question

Calculate the pressure force

u = 15 m/s

Apply Bernoulli,

p1 = p2 = p = hUg = 30u1000u9.81 = 294.3 kN/m2 FTx

a2 = a3 =2x10-3 m2

p1 u12 z Ug 2 g 1

p1 a 1 cos T1 p 2 a 2 cos T2 294300 u 0.2831 cos 75

p 2 u 22 z Ug 2 g 2

p 3 u 32 z Ug 2 g 3

. kN 6173

FTy

Change in height is negligible so z1 = z2 = z3 and pressure is always atmospheric p1= p2 = p3 =0. So

p1 a1 sin T1 p 2 a 2 sin T2

u1= u2 = u3 =15 m/s

294300 u 0.283 0 sin 75

By continuity Q1= Q2 + Q3

80.376 kN

u1a1 = u2a2 + u3a3

There is no body force in the x or y directions. FRx Examples: Answers

so

FTx FPx FBx 1886 . 61.73 0 63.616 kN

a1 = a2 + a3

Put the axes normal to the plate, as we know that the resultant force is normal to the plate. CIVE1400: Fluid Mechanics

37

Examples: Answers


38


Examples: Answers


-3

Examples: Answers

Q1 = a1u = 2u10 u15 = 0.03

FTx = UQ( 0 - u1x )

Q1 = (a2 + a3) u

FTx = 1000u0.11 ( 0 - 25 cos 30 ) = 2.39 kN

Q2 = a2u

6.5 The outlet pipe from a pump is a bend of 45q rising in the vertical plane (i.e. and internal angle of 135q). The bend is 150mm diameter at its inlet and 300mm diameter at its outlet. The pipe axis at the inlet is horizontal and at the outlet it is 1m higher. By neglecting friction, calculate the force and its direction if the inlet pressure is 100kN/m2 and the flow of water through the pipe is 0.3m3/s. The volume of the pipe is 0.075m3. [13.94kN at 67q 40’ to the horizontal]

Q3 = (a1 - a2)u Calculate total force.

FTx

UQ u 2 x u1x FRx FPx FBx

FTx

1000 u 0.03 0 15 sin 60

390 N

y

Component in direction of jet = 390 sin 60 = 338 N

p2 u 2

As there is no force parallel to the plate Fty = 0 FTy a1

x

Uu22 a2 Uu32 a3 Uu12 a1 cosT

a 2 a 3 a1 cosT

0

0

a2 a3

a 3 a1 cosT 4a 3

a1

a3

1 a 3 2

A2

1m

p1

a1 a 3

45°

u1

4 a 3 2

A1

Thus 3/4 of the jet goes up, 1/4 down 6.4 A 75mm diameter jet of water having a velocity of 25m/s strikes a flat plate, the normal of which is inclined at 30q to the jet. Find the force normal to the surface of the plate. [2.39kN]

1&2 Draw the control volume and the axis system

y

p1 = 100 kN/m2,

Q = 0.3 m3/s

d1 = 0.15 m

d2 = 0.3 m

A1 = 0.177 m2

A2 = 0.0707 m2

T = 45q

u2

x

3 Calculate the total force in the x direction

FT x

u1

UQu2 x u1 x UQ u2 cosT u1

θ

by continuity A1u1

A2u2

Q , so

u3

From the question,

djet = 0.075m

u1=25m/s

Q = 25S(0.075/2)2 = 0.11 m3/s

Force normal to plate is Examples: Answers


39

Examples: Answers


40


u1

0.3 . 2 / 4 S 015 0.3 0.0707

u2

Examples: Answers

CIVE1400: Fluid Mechanics FP x

16.98 m / s

Examples: Answers

100000 u 0.0177 2253614 . cos 45 u 0.0707 1770 11266.34

4.24 m / s

FP y

9496.37 kN

2253614 . sin 45 u 0.0707 11266.37

1000 u 0.3 4.24 cos 45 16.98

FT x

1493.68 N

5 Calculate the body force The only body force is the force due to gravity. That is the weight acting in the y direction.

FB y

and in the y-direction

UQ u2 y u1 y

FT y

1000 u 9.81 u 0.075 . N 1290156

UQ u2 sin T 0

There are no body forces in the x direction,

1000 u 0.34.24 sin 45 899.44 N

FB x

pressure force at 1 - pressure force at 2

FP x

p1 A1 cos 0 p2 A2 cosT

FP y

p1 A1 sin 0 p2 A2 sin T

0

6 Calculate the resultant force

4 Calculate the pressure force. FP

Ug u volume

FT x

FR x FP x FB x

FT y

FR y FP y FB y

p1 A1 p2 A2 cosT FR x

p2 A2 sin T

FT x FP x FB x 4193.6 9496.37 5302.7 N

We know pressure at the inlet but not at the outlet. FR y

we can use Bernoulli to calculate this unknown pressure.

p1

Ug

2 1

u z 2g 1

p2

Ug

FT y FP y FB y 899.44 11266.37 735.75 1290156 . N

2 2

u z hf 2g 2

And the resultant force on the fluid is given by

where hf is the friction loss

FRy

In the question it says this can be ignored, hf=0

FResultant

The height of the pipe at the outlet is 1m above the inlet. Taking the inlet level as the datum: z1 = 0

z2 = 1m

φ

So the Bernoulli equation becomes: 16.982 4.242 p2 100000 0 10 . 1000 u 9.81 2 u 9.81 1000 u 9.81 2 u 9.81 2 p2 2253614 . N /m Examples: Answers

FRx


41

Examples: Answers


42


FR

2 Rx

F

Examples: Answers


Examples: Answers

6.7 A curved plate deflects a 75mm diameter jet through an angle of 45q. For a velocity in the jet of 40m/s to the right, compute the components of the force developed against the curved plate. (Assume no friction). [Rx=2070N, Ry=5000N down]

2 R y

F

5302.7 2 1290156 . 2 13.95 kN

u2

y

And the direction of application is

I

§ FR y · ¸ tan 1 ¨ © FR x ¹

. · § 1290156 tan 1 ¨ ¸ © 5302.7 ¹

67.66$

x

The force on the bend is the same magnitude but in the opposite direction

R

FR

u1

6.6 The force exerted by a 25mm diameter jet against a flat plate normal to the axis of the jet is 650N. What is the flow in m3/s? [0.018 m3/s] y

θ

u2

From the question:

x u1

a1

S 0.0752 / 4 4.42 u 10 3 m2

u1

40 m / s

Q a1

4.42 u 10 3 u 40 01767 . m3 / s a2 ,

so

u1

u2

Calculate the total force using the momentum equation: FT x

UQ u2 cos 45 u1

40 cos 45 40 . 1000 u 01767 . N 207017 FT y u2

From the question,

djet = 0.025m

UQ u2 sin 45 0

40 sin 45 . 1000 u 01767 4998 N

FTx = 650 N

Body force and pressure force are 0.

Force normal to plate is

So force on vane:

FTx = UQ( 0 - u1x ) 650 = 1000uQ ( 0 - u ) Q = au = (Sd2/4)u

Rx

Ft x

2070 N

Ry

Ft y

4998 N

650 = -1000au2 = -1000Q2/a 650 = -1000Q2/(S0.0252/4) Q = 0.018m3/s

Examples: Answers


43

Examples: Answers


44


Examples: Answers

6.8 A 45q reducing bend, 0.6m diameter upstream, 0.3m diameter downstream, has water flowing through it at the rate of 0.45m3/s under a pressure of 1.45 bar. Neglecting any loss is head for friction, calculate the force exerted by the water on the bend, and its direction of application. [R=34400N to the right and down, T = 14q]

FT x

Examples: Answers

1000 u 0.45 6.365 cos 45 159 . 1310 N

and in the y-direction

ρ2 u2

y


UQ u2 y u1 y

FT y A2

UQu2 sin T 0 1000 u 0.45 6.365 sin 45 1800 N

x

4 Calculate the pressure force. ρ1

FP

u1 θ

A1

pressure force at 1 - pressure force at 2

FP x

p1 A1 cos 0 p2 A2 cosT

FP y

p1 A1 sin 0 p2 A2 sin T

p1 A1 p2 A2 cosT p2 A2 sin T

We know pressure at the inlet but not at the outlet.

1&2 Draw the control volume and the axis system

we can use Bernoulli to calculate this unknown pressure. 5

2

3

p1 = 1.45u10 N/m ,

Q = 0.45 m /s

d1 = 0.6 m

d2 = 0.3 m

A1 = 0.283 m2

A2 = 0.0707 m2

T = 45q

p1 u12 z Ug 2 g 1

p2 u22 z hf Ug 2 g 2

where hf is the friction loss In the question it says this can be ignored, hf=0

3 Calculate the total force

Assume the pipe to be horizontal

in the x direction

z1 = z 2 So the Bernoulli equation becomes:

FT x

UQu2 x u1 x

. 2 159 6.3652 p2 145000 1000 u 9.81 2 u 9.81 1000 u 9.81 2 u 9.81 p2 126007 N / m 2

UQu2 cosT u1 by continuity A1u1

A2u2

Q , so FP x

u1

0.45 S 0.6 2 / 4

u2

0.45 0.0707

Examples: Answers

145000 u 0.283 126000 cos 45 u 0.0707 41035 6300

34735 N

. m/ s 159

FP y

6.365 m / s

126000 sin 45 u 0.0707 6300 N


45

5 Calculate the body force Examples: Answers


46


Examples: Answers


Examples: Answers

The only body force is the force due to gravity.

Laminar pipe flow.

There are no body forces in the x or y directions,

7.1 The distribution of velocity, u, in metres/sec with radius r in metres in a smooth bore tube of 0.025 m bore follows the law, u = 2.5 - kr2. Where k is a constant. The flow is laminar and the velocity at the pipe surface is zero. The fluid has a coefficient of viscosity of 0.00027 kg/m s. Determine (a) the rate of flow in m3/s (b) the shearing force between the fluid and the pipe wall per metre length of pipe. [6.14x10-4 m3/s, 8.49x10-3 N]

FB x

FB y

0

6 Calculate the resultant force FT x

FR x FP x FB x

FT y

FR y FP y FB y

FR x

FT x FP x FB x

The velocity at distance r from the centre is given in the question: u = 2.5 - kr2 P = 0.00027 kg/ms

Also we know:

We can find k from the boundary conditions: when r = 0.0125, u = 0.0 (boundary of the pipe) 0.0 = 2.5 - k0.01252 k = 16000

1310 34735 33425 N FR y

FT y FP y FB y

u = 2.5 - 1600 r2 a) Following along similar lines to the derivation seen in the lecture notes, we can calculate the flow GQ through a small annulus Gr: GQ ur Aannulus

1800 6300 8100 N

Aannulus

And the resultant force on the fluid is given by FRy

2r = 0.025m

GQ

FResultant

S (r Gr ) 2 Sr 2 | 2SrGr

2.5 16000r 2SrGr 2

0.0125

Q

2S

³ 2.5r 16000r dr 3

0

0.0125

ª 2.5r 2 16000 4 º 2S « r » 4 ¬ 2 ¼0

φ

. m3 / s 614

FRx

FR

b)

FR2 x FR2 y

The shear force is given by

334252 8100 2 34392 kN

From Newtons law of viscosity

W du dr F

And the direction of application is

I

§ FR y · ¸ tan 1 ¨ © FR x ¹

§ 8100 · tan 1 ¨ ¸ © 33425¹

F = W u (2Sr)

13.62 $

P

du dr

2 u 16000r

32000r

0.00027 u 32000 u 0.0125 u (2 u S u 0.0125) 8.48 u 10 3 N

The force on the bend is the same magnitude but in the opposite direction

R

FR

Examples: Answers


47

Examples: Answers


48


Examples: Answers

7.2 A liquid whose coefficient of viscosity is m flows below the critical velocity for laminar flow in a circular pipe of diameter d and with mean velocity u. Show that the pressure loss in a length of pipe is 32um/d2. Oil of viscosity 0.05 kg/ms flows through a pipe of diameter 0.1m with a velocity of 0.6m/s. Calculate the loss of pressure in a length of 120m. [11 520 N/m2]

CIVE1400: Fluid Mechanics ur

Examples: Answers

'p 1 R2 r 2 L 4P

The flow in an annulus of thickness Gr

GQ ur Aannulus Aannulus

See the proof in the lecture notes for Consider a cylinder of fluid, length L, radius r, flowing steadily in the centre of a pipe

S (r Gr ) 2 Sr 2 | 2SrGr

GQ

'p 1 R 2 r 2 2SrGr L 4P

Q

'p S R 2r r 3 dr L 2 P ³0

R

δr r

'p SR 4 L 8P

r R

So the discharge can be written Q

The fluid is in equilibrium, shearing forces equal the pressure forces.

u

'p r L 2

Newtons law of viscosity W

P

u du , dy

We are measuring from the pipe centre, so W

'p

P

'p Sd 4 L 128P

To get pressure loss in of the velocity of the flow, use the mean velocity:

W 2Sr L 'p A 'pSr 2 W

'p Sd 4 L128P

du dr

'p

Q/ A 'p d 2 32 PL 32 PLu d2 32 Pu d2

b) From the question

Giving: 'p r du P L 2 dr du 'p r dr L 2P

'p

per unit length

P= 0.05 kg/ms

d = 0.1m

u = 0.6 m/s

L = 120.0m

32 u 0.05 u 120 u 0.6 01 .2

11520 N / m2

In an integral form this gives an expression for velocity, u

'p 1 r dr L 2P ³

The value of velocity at a point distance r from the centre ur

'p r 2 C L 4P

At r = 0, (the centre of the pipe), u = umax, at r = R (the pipe wall) u = 0; C

'p R 2 L 4P

At a point r from the pipe centre when the flow is laminar: Examples: Answers


49

Examples: Answers


50


Examples: Answers

7.3 A plunger of 0.08m diameter and length 0.13m has four small holes of diameter 5/1600 m drilled through in the direction of its length. The plunger is a close fit inside a cylinder, containing oil, such that no oil is assumed to between the plunger and the cylinder. If the plunger is subjected to a vertical downward force of 45N (including its own weight) and it is assumed that the upward flow through the four small holes is laminar, determine the speed of the fall of the plunger. The coefficient of velocity of the oil is 0.2 kg/ms. [0.00064 m/s] F = 45N Q


Examples: Answers

vpiston = 0.00064 m/s 7.4 A vertical cylinder of 0.075 metres diameter is mounted concentrically in a drum of 0.076metres internal diameter. Oil fills the space between them to a depth of 0.2m. The rotque required to rotate the cylinder in the drum is 4Nm when the speed of rotation is 7.5 revs/sec. Assuming that the end effects are negligible, calculate the coefficient of viscosity of the oil. [0.638 kg/ms] r-1 = 0.076/2

From the question d = 5/1600 m

r2 = 0.075/2

Torque = 4Nm, L = 0.2m

The velocity of the edge of the cylinder is: ucyl = 7.5 u 2Sr = 7.5u2uSu0.0375 = 1.767 m/s udrum = 0.0 Torque needed to rotate cylinder

plunger

W u surface area

T

0.13 m

4 W 2Sr2 u L . N / m2 226354

W

Distance between cylinder and drum = r1 - r2 = 0.038 - 0.0375 = 0.005m Using Newtons law of viscosity:

cylinder

W

du dr 1767 . 0 0.0005 . P 3534 22635

P

0.64 kg / ms

W du dr

0.8m

P

( Ns / m 2 )

Flow through each tube given by Hagen-Poiseuille equation Q

'p Sd 4 L 128P

There are 4 of these so total flow is Q

4

'p Sd 4 L 128P

'p

4S (5 / 1600) 4 013 . u 128 u 0.2

u 10 10 'p3601 .

Force = pressure u area F

45

2 § § 0.08 · 2 § 5 / 1600 · · 'p¨ S ¨ ¸ 4S ¨ ¸ ¸ © 2 ¹ ¹ © © 2 ¹

'p

9007.206 N / m2

Q

3.24 u 10 6 m 3 / s

Flow up through piston = flow displaced by moving piston Q = Avpiston 3.24u10-6 = Su0.042uvpiston Examples: Answers


51

Examples: Answers


52


Examples: Answers


Examples: Answers

And the second group S2 :

Dimensional analysis

8.1 A stationary sphere in water moving at a velocity of 1.6m/s experiences a drag of 4N. Another sphere of twice the diameter is placed in a wind tunnel. Find the velocity of the air and the drag which will give dynamically similar conditions. The ratio of kinematic viscosities of air and water is 13, and the density of air 1.28 kg/m3. [10.4m/s 0.865N]

M]

0 = c2

L]

0 = a2 + b2 - 3c2 + 2

Draw up the table of values you have for each variable:

T]

variable

water

air

u

1.6m/s

uair

Drag

4N

Dair

Q

Q

13Q 3

1 a2

LT

M 0 L0 T 0

2

-2 = a2 + b2 0 = -a2 - 1 a2 = -1 b2 = -1

S2

u 1d 1 U 0Q

Q ud

3

U

1000 kg/m

1.28 kg/m

d

d

2d

So the physical situation is described by this function of nondimensional numbers, § D Q · , ¸ © Uu 2 d 2 ud ¹

I S1 , S 2 I ¨ Kinematic viscosity is dynamic viscosity over density = Q The Reynolds number = Re

Uud P

PU

Q

From Buckinghams S theorem we have m-n = 5 - 3 = 2 non-dimensional groups.

S1air

S 1water

S 2 air

S 2 water

For S1

I u, d , U , D,Q 0

§ D · ¨ 2 2¸ © Uu d ¹ air

I S 1 , S 2 0 u a1 d b1 U c1 D

S2

u a2 d b2 U c2 Q

Dair 128 . u 10.4 2 u (2d ) 2 Dair

As each S group is dimensionless then considering the dimensions, for the first group, S1: (note D is a force with dimensions MLT-2) 0

0

M LT

M]

0

1 a1

LT

3 c1

L ML b1

MLT

§ D · ¨ 2 2¸ © Uu d ¹ water

4 1000 u 16 . 2 ud2 0.865 N

For S2 §Q · ¨ ¸ © ud ¹ air

2

0 = c1 + 1

13Q uair u 2d

c1 = -1 0 = a1 + b1 - 3c1 + 1

L]

0

For dynamic similarity these non-dimensional numbers are the same for the both the sphere in water and in the wind tunnel i.e.

ud

Choose the three recurring (governing) variables; u, d, U

S1

c2

L b ML 3 L2 T 1

uair

§ Q · ¨ ¸ © ud ¹ water

Q 16 . ud 10.4 m / s

-4 = a1 + b1 T]

0 = -a1 - 2 a1 = - 2 b1 = -2

S1

u 2 d 2 U 1 D D

Uu 2 d 2 Examples: Answers


53

Examples: Answers


54


Examples: Answers

8.2 Explain briefly the use of the Reynolds number in the interpretation of tests on the flow of liquid in pipes. Water flows through a 2cm diameter pipe at 1.6m/s. Calculate the Reynolds number and find also the velocity required to give the same Reynolds number when the pipe is transporting air. Obtain the ratio of pressure drops in the same length of pipe for both cases. For the water the kinematic viscosity was 1.31u10-6 m2/s and the density was 1000 kg/m3. For air those quantities were 15.1u10-6 m2/s and 1.19kg/m3. [24427, 18.4m/s, 0.157]


M]

water

T]

0 = -a1 - 1 a1 = -1 b1 = -1 1

u d 1 U 0Q

air

Q

uair

ud

1.6m/s

p

pwater

pair

U

1000 kg/m3

1.19kg/m3

0 = a1 + b1 - 3c1 + 2 -2 = a1 + b1

S1

u

0 = c1

L]

Draw up the table of values you have for each variable: variable

Examples: Answers

And the second group S2 : (note p is a pressure (force/area) with dimensions ML-1T-2)

Q

u m s

u m s

U

1000 kg/m3

1.28 kg/m3

d

0.02m

0.02m

M 0 L0 T 0

M]

1 a1

LT

c1

L b ML3 MT 2 L1 1

0 = c2 + 1

c2 = -1

Kinematic viscosity is dynamic viscosity over density = Q

Uud P

The Reynolds number = Re

PU

L]

0 = a2 + b2 - 3c2 - 1 -2 = a2 + b2

ud

Q

T]

0 = -a2 - 2

Reynolds number when carrying water:

Re water

ud

Q

. u 0.02 16 . u 10 6 131

a2 = - 2 b2 = 0

24427

S2

To calculate Reair we know,

2

u U 1 p p

Uu 2

Re water

Re air uair 0.02 24427 15 u 10 6 uair 18.44m / s

So the physical situation is described by this function of nondimensional numbers, § Q p · , ¸ © ud Uu 2 ¹

I S 1 , S 2 I ¨

To obtain the ratio of pressure drops we must obtain an expression for the pressure drop in of governing variables.

0

For dynamic similarity these non-dimensional numbers are the same for the both water and air in the pipe.

Choose the three recurring (governing) variables; u, d, U From Buckinghams S theorem we have m-n = 5 - 3 = 2 non-dimensional groups.

I u, d , U , Q , p 0 I S 1 , S 2 0

S1air

S1water

S 2 air

S 2 water

We are interested in the relationship involving the pressure i.e. S2 a1

b1

c1

S1

u d U Q

S2

u a2 d b2 U c2 p

As each S group is dimensionless then considering the dimensions, for the first group, S1: M 0 L0 T 0

Examples: Answers

1 a1

LT

c1

L b ML3 L2 T 1 1


55

Examples: Answers


56


Examples: Answers

§ p · ¨ 2¸ © Uu ¹ air

§ p · ¨ 2¸ © Uu ¹ water

pwater pair

2 water 2 air

Uwater u Uair u

S1

1 0158 .

ML-3

u

LT-1

d

L

P

ML-1T-1

M]

1

-2 = 3a2 + b2 0 = -a2 - 2 a2 = - 2 b2 = 4

S2

Q 2 d 4 U 1 p d4 p UQ 2

So the physical situation is described by this function of non-dimensional numbers, § dP d 4 p · , ¸ © QU UQ 2 ¹

I S 1 , S 2 I ¨


I Q, d , U , P , p

0

or

I S 1 , S 2

0

dP QU

S1

Q a1 d b1 U c1 P

S2

Q a2 d b2 U c2 p

1 a1

L T

§ d4 p· ¸ © UQ 2 ¹ § dU 1/2 p 1/2 · § d 2 p 1/2 · f¨ ¸¨ ¸ P ¹© U ¹ ©

c1

L b ML3 ML1T 1 1

0

I1 ¨

The question wants us to show : Q

As each S group is dimensionless then considering the dimensions, for the first group, S1:

Take the reciprocal of square root of S2:

0 = c1 + 1

1

U 1/2 Q

S2

d 2 p1/2

S 2a ,

Convert S1 by multiplying by this number

c1 = -1 0 = 3a1 + b1 - 3c1 - 1

S1a

-2 = 3a1 + b1 T]

c1

L b ML3 MT 2 L1

0 = 3a2 + b2 - 3c2 - 1

T]

We are told from the question that there are 5 variables involved in the problem: d, p, U, P and Q.

L]

3

0 = c2 + 1

L]

Choose the three recurring (governing) variables; Q, d, U

M]

1 a1

L T

c2 = -1

i.e. Re is dimensionless.

3

Q d U P

M 0 L0 T 0

Re = ML-3 LT-1L(ML-1T-1)-1 = ML-3 LT-1 L M-1LT = 1

M 0 L0 T 0

Examples: Answers

1

(note p is a pressure (force/area) with dimensions ML-1T-2)

The dimensions of these following variables are U

1


6.327

Show that Reynold number, Uud/P, is non-dimensional. If the discharge Q through an orifice is a function of the diameter d, the pressure difference p, the density U, and the viscosity P, show that Q = 1/2d2/U1/2 where C is some function of the non-dimensional group (dU1/2d1/2/P). Draw up the table of values you have for each variable:

1

dP UQ

1000 u 16 . 2 119 . u 18.44 2


S1S 2 a

dP U 1/ 2 Q QU d 2 p 1/ 2

P dU 1/ 2 p1/ 2

then we can say

0 = -a1 - 1 a1 = -1 b1 = 1

Examples: Answers


57

Examples: Answers


58


Examples: Answers 1/ 2

1/ 2

2

§p U d d p · , 1/2 ¸ P U ¹ ©

I 1 / S 1a , S 2 a I ¨

M 0 L0 T 0

0

M]

or Q


1/ 2

§ p 1/2 U 1/2 d · d 2 p 1/ 2 ¸ P ¹ U 1/ 2 ©

3 c1

L ML

ML1T 1

0 = c1 + 1

L]

Chimney:

U = 1.12kg/m3

P = 16u10-6 kg/ms

Model:

U = 1000kg/m3

P = 8u10-4 kg/ms

0 = a1 + b1 - 3c1 - 1 -2 = a1 + b1

8.4 A cylinder 0.16m in diameter is to be mounted in a stream of water in order to estimate the force on a tall chimney of 1m diameter which is subject to wind of 33m/s. Calculate (A) the speed of the stream necessary to give dynamic similarity between the model and chimney, (b) the ratio of forces.

T]

0 = -a1 - 1 a1 = -1 b1 = -1

S1

Draw up the table of values you have for each variable:

i.e. the (inverse of) Reynolds number

variable

water

air

u

uwater

33m/s

F

Fwater

Fair

U

1000 kg/m3

1.12kg/m3

P

u kgms

ukg/ms

d

0.16m

1m

And the second group S2 : M 0 L0 T 0

M]

c2

L b ML 3 ML1T 2 2

c2 = -1 0 = a2 + b2 - 3c2 - 1 -3 = a2 + b2 T]

Uud P

1 a2

LT

0 = c2 + 1

L]

Kinematic viscosity is dynamic viscosity over density = Q

u 1d 1 U 1 P

P Uud

[11.55m/s, 0.057]

0 = -a2 - 2

PU

a2 = - 2

ud

b2 = -1

Q

S2

For dynamic similarity: Re water 1000uwater 016 . 8 u 10 4 uwater

LT

Examples: Answers b1

c1 = -1

I¨

The Reynolds number = Re

1 a1

2

u d 1 U 1 F F u 2 dU

Re air 112 . u 33 u 1 16 u 10 6 . m/ s 1155

So the physical situation is described by this function of nondimensional numbers, § P F · , ¸ © Uud Udu 2 ¹

I S 1 , S 2 I ¨

To obtain the ratio of forces we must obtain an expression for the force in of governing variables.

0

Choose the three recurring (governing) variables; u, d, U F, P For dynamic similarity these non-dimensional numbers are the same for the both water and air in the pipe.


I u, d , U , P , F 0 I S 1 , S 2 0 S1 S2

b2

S1water S 2 water

To find the ratio of forces for the different fluids use S2

u a1 d b1 U c1 P a2

S1air S 2 air c2

u d U F

As each S group is dimensionless then considering the dimensions, for the first group, S1: Examples: Answers


59

Examples: Answers


60


S 2 air

Examples: Answers

S 2 water § F · ¨ 2 ¸ © Uu d ¹ water

§ F · ¨ 2 ¸ © Uu d ¹ air

§ F · ¨ 2 ¸ © Uu d ¹ water

M] L] T]

a2 = - 2

P 2 § Uur · f¨ ¸ U © P ¹

S2

u 2 r 1 U 1 R R u 2 rU

So the physical situation is described by this function of nondimensional numbers, § P R · , ¸ © Uur Uru 2 ¹

I S1 , S 2 I ¨

§ P · ¸ © Uur ¹

R

I1 ¨

Uru 2

0

I S 1 , S 2 0

he question asks us to show R

u a1 r b1 U c1 P

S2

u a2 r b2 U c2 R

LT

3 c1

L ML b1

P 2 § Uur · RU f¨ ¸ or 2 U © P ¹ P

§ Uur · f¨ ¸ © P ¹

Multiply the LHS by the square of the RHS: (i.e. S2u(1/S12) ) R

As each S group is dimensionless then considering the dimensions, for the first group, S1: 1 a1

0

or


Uru 2

ML1T 1

u

U 2u2r 2 P2

RU

P2

So

0 = c1 + 1

RU

c1 = -1

P2

0 = a1 + b1 - 3c1 - 1

T]

0 = -a2 - 2 b2 = -1

Choose the three recurring (governing) variables; u, r, U R, P

L]

0 = a2 + b2 - 3c2 - 1 -3 = a2 + b2

Hence show that if at very low velocities the resistance R is proportional to the velocity u, then R = kPru where k is a dimensionless constant. A fine granular material of specific gravity 2.5 is in uniform suspension in still water of depth 3.3m. Regarding the particles as spheres of diameter 0.002cm find how long it will take for the water to clear. Take k=6S and P=0.0013 kg/ms. [218mins 39.3sec]

M]

2

0 = c2 + 1

0.057

R

S1

c2

L b ML3 ML1T 2

c2 = -1

8.5 If the resistance to motion, R, of a sphere through a fluid is a function of the density U and viscosity P of the fluid, and the radius r and velocity u of the sphere, show that R is given by

M 0 L0 T 0

1 a2

LT

M 0 L0 T 0

. u 332 u 1 112 . 2 u 016 . 1000 u 1155

I u , r , U , P , R

Examples: Answers


§ F · ¨ 2 ¸ © Uu d ¹ air

Fair Fwater


§ Uur · f¨ ¸ © P ¹

The question tells us that R is proportional to u so the function f must be a constant, k

-2 = a1 + b1

RU

0 = -a1 - 1

P2 R

a1 = -1

k

Uur P

Pkru

b1 = -1

S1

u 1r 1 U 1 P

The water will clear when the particle moving from the water surface reaches the bottom.

P Uur

At terminal velocity there is no acceleration - the force R = mg - upthrust. From the question: V = 2.5

i.e. the (inverse of) Reynolds number Examples: Answers


61

Examples: Answers

so

U = 2500kg/m3

P = 0.0013 kg/ms

k = 6S CIVE1400: Fluid Mechanics

62


r = 0.00001m mg

Examples: Answers

depth = 3.3m

4 S 0.000013 u 9.81 u 2500 1000 3 616 . u 10 11

Pkru 0.0013 u 6S u 0.00001u 616 . u 10 11 u

t

Examples: Answers

2.52 u 10 4 m / s

3.3 2.52 u 10 4

218 min 39.3 sec


63

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