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Lecture 7

Mechanics of Fluids 1: Lecture 6: Integral Relations for CV (Part 3) Department of Mechanical Engineering MEHB223

Integral RelationIntegral Relationfor a Controlfor a Control

Volume (Part 3)Volume (Part 3)


Integral Relation for CV :Integral Relation for CV : ForceForceMomentum EquationMomentum Equation


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Chapter Summary


Derivation of Momentum Equation Applications of Momentum Equation Moment of Momentum Equation

7.1. Introduction Motion of fluid is dependent on Newtons

MomentumPrinciple


Second Law Newtons 2nd Law :

Rate of Change of Momentum of the System in the direction of the force = Net External Force ON the system in the same direction

For fluid, since control volume approach is

?????

used, the General Control Volume Equation (Reynolds Transport Equations) need to be used.

Note that force and momentum are vector quantities


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7.2. Derivation

of

FME

Consider the general CV Equation :


oiSC XX

dtdX

dtdX

+=

Let X be the momentum in x-direction, M x :

o xi x

S xC x M M dt

dM

dt

dM ,,

,, +=

y ew on;s aw :dt

F S x x,=

So the FME becomes :

o xi x xC x M M F

dt

dM ,,

,

+=

7.2. Derivation of FME The FME for general non uniform inlets and exits :

Mechanics of Fluids 1: Lecture 67 Integral Relations for CV (Part 3) Department of Mechanical Engineering MEHB223

Steady Flow FME :

+=outflow All A

olow All A

i xCV oi

dAudAuF ud dt d

_

2

inf _

2

i xo x x M M F

,,

=

r or non-un orm ow : =

flow All Ai

outflow All Ao x

io

dAudAuF inf _

2

_

2

Note that these equation can be applied to y and zdirections


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7.2. Derivation

of

FME

Notes on Applicability :


The net force are net of all external forces acting ON the control volume

2 types forces need to be considered 1) Boundary forces such as pressure (act normal to the surface and towards the surface) and shear (act parallel to

across the CV boundary eg. Gravity or magnetic force

Application of FME to CV is analogous to drawing free body diagram in solid mechanics

7.3. Application of FME Example 1 : V1


Find the horizontal andvertical component of the

force exerted on the vane

Solution Steps :

A1 A2

- - Identify the inlets and exit positions- Identify all the external forces including pressure forces- Ensure that the MCE is satisfied - Apply the FME in x and y directions


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7.3. Application

of

FME

Example 2 :


Determine the external reactions in the x and ydirections needed to hold this fixed vane. Here V 1 is 28m/s, V 2 = 27 m/s and Q = 0.20 m 3 /s.

7.3. Application of FME Example 3 :


A water jet of diameter 30 mm and speed 20 m/s is filling atank. The tank has a mass of 20 kg and contains 20 liters ofwater at the instant shown. The water temperature is 15 oC.Find the force acting on the bottom of the tank and the forceacting on the stop block. Neglect frictions.


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7.3. Application

of

FME

Example 4 : (Nozzle)


Water flows through this nozzle at a rate of 2 m 3 /s. anddischarge into the atmosphere. D 1 = 0.4 m, D 2 = 0.3 m.

Determine the force required to hold the nozzle in place. Assume irrotational flow. Neglect the gravitational effect.

7.3. Application of FME Example 5 : (Pipe Bend Flange & Bolts )


When fluid in a pipenetwork pass through a

pipe bend, a force will beexerted on the bend. Forsmall and medium size P1

,by a flange bolts or bywelds. Determine the forcesthat will be experienced bythe bolts. (Assume the bendis in a horizontal plane)

P2


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7.3. Application

of

FME

Example 6 : (Pipe Bend Anchor Block )


For a large pipe thebolts joint may be notadequate and a speciallydesigned anchor blockmay be needed.

will be experienced by

the anchor block.

7.3. Application of FME Example 7 : (Non Uniform Velocity )


A missile is being tested in a cylindrical wind tunnel. The diameter of the windtunnel is 1 m. The measured upstream and downstream pressures are 2 kPagage and 0.5 kPa gage respectively. Assume velocity distribution as above. Airis assumed incompressible with density of 1.15 kg/m 3. Assume uniform

pressure and negligible viscous forces at the wall. Determine maximumvelocity at station C and the drag force experience by the missile ?

= 30 m/s


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7.3. Application

of

FME

Example 8: (Moving Control Volume )


Newtons 2 nd Law (& FME) is equally applicable to control volume

which is moving at a constant velocity. The same FME is applicablebut all taken relative to the moving control volume. Eg. A horizontal jetof water that is 6 cm in diameter and has a velocity of 20 m/s isdeflected by a vane. The vane is moving in the x direction at a constantvelocity of 7 m/s. Determine the force experience by the vane.

7.4. Moment of Momentum Equation The Newtons 2nd Law can also be applied to MOMENT rather than

force and state that :


( )dt

Mr d T S =

Substitution into the Reynolds Transport Equation give :

( ) ( ) ( )oi

C Mr Mr T dt

Mr d

+=

For steady flow :

( ) ( )io

Mr Mr T

=


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7.4. Moment of Momentum Equation Example 9:


Water enters a rotating lawn sprinkler through its base at a

steady rate of 1000 ml/s. The exit area of the 2 nozzles are 30mm2 and the sprinkler rotates at 500 rpm. Determine (a) Therelative speed of water leaving the nozzle (b) The resistive torqueof the sprinkler system.


End ofEnd of Lecture 7Lecture 7

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