AMEE 202 Instructor: Marios M. Fyrillas Introduction to Fluid Mechanics Email: eng.fm@fit.ac.cy

SOLUTION Homework Assignment #3

1. A horizontal water jet with diameter d= 5 cm impacts on a vertical plate

(Figure 4). After impact, the velocity normal to the plate is zero. Find the force F.

2

Momentum Equation

0.050 1000 10 10 196.25 N

4This is the force acting on the fluid. The force acting on the plate is

opposite 196.25 N

x x x

x

x out in out

R x

F M M m u

F F

2. To propel a light aircraft at an absolute velocity of against a head

wind of 48 km/hr a thrust of 10.3 kN is required. Assuming a theoretical efficiency of 90% and a constant air density of 31.2 kg/m determine the diameter of ideal propeller required and the power needed to drive it. Sketch the velocity and the pressure profile along the slipstream boundary.

(The efficiency is

240 km/hr

11

4 12( )th

u

u u

).

Figure: Velocity field due to a propeller (Question 2)

V=10 m/s F

Figure 4: Impinging jet

Consider the control volume enclosed by the slipstream boundary and planes 1 and 4. Momentum equation in the x-direction is:

out in 4 1( )x xx out inF m u m u m u u ,

where we have employed the mass conservation equation:

4 1 2out inm m m m m u A .

The pressure all around the control volume is atmospheric hence x RxF F , which is the force

from the propeller to the fluid. The thrust force from the fluid to the propeller is x RxR F .

We can obtain another equation for F if we take a control volume between sections 2 and 3,

and assume that 2 3u he momentum equation is simply

Rx ,

u . T

)A3 20 0 (x Rx px Rx pxF F F F F p p .

Hence,

4 1 3 2( ) ( )RxF m u u p p A .

Employing, Bernoulli’s equation between sections 1 and 2 gives:

2 21 1 2

1 1

2 2p u p 2u .

Similarly, between sections 2 and 3

2 23 3 4

1 1

2 2p u p 4u .

Note that, Bernoulli’s equation cannot be applied between sections 2 and 3 because the flow is unsteady.

Now, and also . Therefore adding the

last two equations gives: 2u u 3 1 4 pressure of undisturbed fluidp p

2 23 2 4 1

1( )

2p p u u .

Combining this with the momentum equation gives:

1 42 2

u uu

.

The velocity through the disc then is the arithmetic mean of the velocities upstream and downstream from it; in other words, half the change of the velocity occurs before the disc and half after it.

If the undisturbed fluid be considered stationary, the propeller advances through it with

velocity . The product of the thrust and the velocity gives the rate at which the propeller

does useful work: 1u

1 4 1Power Output = ( )thF u m u u u1 .

The efficiency is the ratio of the power output to the kinetic energy given to the stream

4 1 1 1 11 12 2

4 1 1 4 1 4 12 2

1 2 4 1 22 4 1

( )

( ) ( ) ( )

10300288, 320, 352, / 0.9 916kW,

( )

Fr

th

m u u u u u

m u u u u u u u

u u u W F u Au u u

.

3. An ideal windmill, 12 m in diameter, operates at a theoretical efficiency of 50% in a 14 m/s wind. If the air density is 31.235 kg/m determine the thrust on the windmill, the air velocity through the disc, the mean pressures immediately in front of and behind the disc, and the shaft power developed. Note: (the flow pattern of the windmill is the opposite of that of the propeller and takes energy from the fluid; the efficiency is

2 21 4

12

u

u

1 4

3). th

u u u

2 24 4

43

24 1 2 4 11 4 4 1 2 2

4 11 42

2 22

14 140.5 8.65 m/s

2 14

Momentum Equation between sections 1-4

121.235

2 411.33 m/s (see Exercise 1) 2

8.65 14121.235 8460 N

4 2

Be

x

x

u uu

F Q u u u A u uu u u u

F A uu uu

3214

u

2 2 2 21 1 2 2 2 2 1

2 23 4 1 2 3

rnoulli's equation between section 1-2

1 1 1 141.8 Pa

2 2 2 21

( ) (see Exercise 1) 33 Pa2

The shaft power is 50% of the kinetic energy of the stream. i.e.

atmp u p u p u p u

p u u p p

W

21 1

10.5 95.8 kW

2A u u

4. Air flows past an object in a pipe of 2 m diameter and exits as a free jet as shown in the figure. The velocity and pressure upstream are uniform at 10 m/s and 250 N/m respectively. At the pipe exit the velocity is non-uniform as indicated. The shear stress along the pipe wall is negligible. (a) Determine the head loss associated with a particle as it flows from the uniform velocity upstream of the object to a location in the wake at the exit plane of the pipe. (b) Determine the force that the air puts on the object.

2 21 1 2 2

2 2

Bernoulli's Equation

1 1 where are the losses

2 2

50 1 10 1 48.44 m

1.225 9.81 2 9.81 2 9.81

Momentum Equation

431 384.8 46.22 N

x x

x x x

x out in

in in in in

p u p uh h

g g g g

h

F M M

M m u u A

( ) ( )

2

2 2

41.225 10 384.8 N

4

1.225 4 1.225 4 431 N4 4

50

50

x

x x annulus x annulus disc x disc

in

out out out out out out out

x R b p R p R

R x

u

M m u m u m u

F F F F F F F

F F

46.22 157 110.78 N

110.78 NRR F

5. Steam flows smoothly onto a

stationary blade and turns through 600. The pressure across the blade is constant and the velocity at inlet is 150 m/s. The steam leaves with a velocity 145 m/s. Determine the force (magnitude and direction) exerted on the blade for a volumetric flow rate of 0.9 m3/s ( ). 310 kg/m

145 m/s

150 m/s

Mass Conservation

9 kg/s

Momentum Equation

9 145cos60 ( 150)

9 145sin 60 0

x x x x

y y y y

in out

x out in out in

y out in out in

m m m

m Q

F M M m u u

F M M m u u

6. A horizontal air jet having a velocity of 50 m/s and a diameter of 20

strikes the inside surface of a hollow hemisphere as indicated in the figure. How large is the horizontal anchoring force needed to hold the hemisphere in place? The magnitude of velocity of the air remains constant.

mm

Figure : Jet striking a hollow hemisphere

7. The figure shows an experimental flexible hose-nozzle designed to mix two

liquids entering at stations 1 and 2 prior to spraying the mixture to atmosphere at station 3. The mixture proportions depend upon the gap x that is controlled by axial expansion or contraction of the bellows under the action of the liquid and atmospheric forces on the nozzle. Both liquids have the same density 3 . 1000 kg/m a. Determine 1m if the stagnation pressure at station 1 is 5 bars and 2 0m .

b. f the mass flow rate at 1, 1m , remains unaltered determine that

given that the force applied by the bellows is 1722 N . 2 5.34 kg/sm

2 21 1 3 3

23

3

3 3 3 1

11

1

a. Bernoulli's equation

1 1= 500000 (ignore hydrostatic pressure)

2 21

500000 101300 3987002

2 398700797.4 28.2 m/s

100016

1000 28.2 45.12 kg/s10000

45.12

p u p u

u

u

m u A m

mu

A

1

23 3 1 1 1 1

(gage)

2231 1 1 1

3

23

2 3 1

100005.64 m/s

1000 80

b. Momentum equation

11722 (500000 ) 101300

2

11722 (500000 ) 101300

2

1594.84 1000 0.0016

p

m u m u u A

mu A m u

A

m

m m m

50.5 45.16 5.34 kg/s

x

1

21

01

0 cm

5 bars

m

A

p2m

3

23

3

16 cm

1.013 bar

m

A

p

8

Figure : Hose-nozzle

8. A set of guide vanes deflects a stream of warm air onto painted parts, as

illustrated in Figure 6. The guide vanes are rotated slowly to distribute the air evenly over the parts. Compute the force yR required to hold the guide

vanes in place when the angle 45o and the air is flowing at a velocity of 4 m/s . Assume that all the air that approaches a given guide vane is deflected to the angle of the vane. The air has density 3 . 1.225 kg/mair

R y

Guide vanes are 20in depth

Figure: Stream of air through guide vanes

Mass Conservation

1.225 4 (5 0.0254 20 0.0254) 0.316 kg/s

Momentum Equation

0.316 0 4sin 45 0.894 N

is the force acting on the fluid. On the guide van

y y y y

in out

y out in out in

y

m m m

m Q uA

F M M m u u

F

e it will be equal and

opposite

- 0.894 N

(Pressure is assumed uniform)

yR F

9. The propeller on a swamp boat produces a jet of air having a diameter of

0.91 m as illustrated in the figure. The ambient air density is 3/m and the axial velocity of the flow is 25.91 m/s relative to the

boat. What propulsive forces are produced by the propeller when the boat is stationary and when the boat moves forward with a constant velocity of 6.1 m/s ?

D 1.178 kg

Choose a control volume such that the left

side is far enough from the propeller such that

the relative velocity is zero. Then, the air mass flux

in would correspond to the mass flux from the sides.

The t

2

0

otal mass flux in must be equal to the total mass

flux out:

a. When the boat is stationary

0.91 1.178 25.91 19.851 kg/s

4

( ) 19.8511*25.91 514.34 x x x x

out in out out

x out in out in

m m m u A

F M M m u u

2

6.1

N

a. When the boat is moving

0.91Use relative velocities 1.178 25.91 19.851 kg/s

4

( ) 19.8511*(25.91 6.1) 393.25 Nx x x x

out in out out

x out in out in

m m m W A

F M M m W W

0.91D m

25.91 m/sW

Control Volume

inW

10. The thrust developed to propel the jet ski shown in the figure is a result of

water pumped through the vehicle and exiting as a high-speed water jet. For the conditions shown in the figure, what flowrate is needed to produce a 1334 N thrust? Assume the inlet and outlet jets of water are free jets.

0.089 m diameter

0.0161 m2

2 2out out

Mass Conservation

Momentum Conservation

( ) cos(30) cos(30)

Note: and / 4 0.0062 m

out outin out in in out out in

in

out outx out x in x out in out

in

x out

W Am m W A W A W

A

W AF m W W m W W m W

A

W W A D

( 1)

2 cos(30)cos(30) 1

cos(30) cos(30)1 1

cos(30)= 1 = 1000

/ /

/

out out x outx out out out out

in out in

x out x outout out

out in out in

outout x

in

W A F AF W A W W

A A A

F A F AW m A

A A A A

Am A F

A

0.0062 cos(30)* 0.0062 *1334 1 111 kg/s

0.0161/

inV

CV outV

yinV

in xV

11. Shown in the figure is a small decorative wheel fitted with flat paddles so the wheel turns about its axis when acted on by a blown stream of air. Assuming that all the air in a 15 mm diameter stream moving at 0.35 m/s strikes one paddle and is deflected by it at right angles, compute the force exerted on the wheel initially when it is stationary. The air has density

3 . Compute the force exerted on the paddle when the wheel rotates at 40 rpm . Estimate the power and torque and plot it in terms of the paddle speed.

1.20 kg/m

in in in

25

in in in

( ) ( )

*0.0151.2*0.35* 0 0.35 2.6*10 N

4

( ) ( )

In order to estima

y out y y y in out y y

y out y y y in out y y

F m V V V A V V

F m V V V A V V

If the wheel is stationar

y

If the wheel is moving

te we assume that there are

no mechanical losses hence .out

in out

V

W W

6in

in

in

From the velocity diagram we can deduce

40 20.075 0.314 m/s

60Hence

( ) 1.2 0.35 0.000177 (0.314 0.35) 2.67 10

Torque( ) ( )

Power( ) (

out y

y in in

in in

in

V U r

F V A U V

T F r r V A U V

W T r V A U

) ( )in inV mU U V

N

outW

inW

U

UoutV

inV

12. A Pelton wheel vane directs a horizontal, circular cross-sectional jet of

water symmetrically as indicated in Figure 1. The jet leaves the nozzle with a velocity of 30 m/s . Determine the x -direction component of anchoring force required to: a. Hold the vane stationary. b. Confine the speed of the vane to a value of 3 m/s to the right. The fluid

speed magnitude remains constant along the vane surface.

3 m/s

Figure: Jet impinging on a vane

2

cv

Mass Conservation

Momentum Equation

a. 30sin 45 30

0.0251000 30 30sin 45 30 805 N

4

Body forces 0

Pressure forces 0 (uniform pressure)

b. 30

x x x x

in out

x out in out in

b

p

in

m m m

F M M m u u uA

F

F

W V U

2

3 27

sin 45

0.0251000 27 27sin 45 27 652 N

4

x x x x xx out in out in in out inF M M m W W W A W W

2.5 cm 2.5 cm

m/s30

m/s30

13. In an experiment to determine the force due to the impact of a jet on a vane (see figure) the following measurements were taken: a. The volume of water gathered was 5 lts in 40 secs b. The diameter of the nozzle is 10 mmc. The vane used was a hemispherical cup

i. Determine the force on the vane by considering the following control volume.

ii. What mass is required to be hanged on the lever to restore the lever to its original position. The distance from the vane to the pivot is and from the hanging weight to the pivot is 210 .

150 mm mm

14. A static thrust stand as sketched in the Figure is to be designed for testing a jet engine. The following conditions are known for a typical test: Intake air velocity 200 m/s ; exhaust gas velocity 500 m/s ; intake cross sectional area 21 m ; intake static pressure 22.5 k Pa (abs);Pa=78.5 k

=101 kPa ( intake static

temperature 268 K ; exhaust static pressure abs)0 kPa . Estimate the thrust.

Note: Intake air density can be calculated using the ideal gas law 31

11

785001.02 kg/m

286.9 268

p

R T