Momentum Conservation

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# Momentum Conservation - PowerPoint PPT Presentation

Momentum Conservation. Newton’s Second Law Force = Mass * Acceleration Alternate method: From Reynold’s theorem Fluid Flow Force = Momentum flux + Momentum Accumulation rate. Flux. Accumulation Rate. 0. Example. Straight Pipe. Steady State. A 2 V 2. A 1 V 1. Assumption: No friction.

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Presentation Transcript
Momentum Conservation
• Newton’s Second Law
• Force = Mass * Acceleration
• Alternate method: From Reynold’s theorem
• Fluid Flow
• Force = Momentum flux + Momentum Accumulation rate

Flux

Accumulation Rate

0

Example
• Straight Pipe

A2

V2

A1

V1

• Assumption: No friction

1

2

Example
• Flow in a straight pipe. Realistic case

0

r1=1g/cm3

d1=8cm

V1=5m/s

d2=5cm

Example
• Problem: 5.32

d2,V2

d1,V1

1

2

h=58 cm

y

x

Weight of Fluid

Example
• Example 1 (bent pipe), page 47

r1,V1

By

Area=A1

Area=A2=A1

Q

Bx

r2,V2

out

in

in

out

Example
• d/dt =0
• From Eqn of Conservation of Mass
Example

V1

By

Area=A1

Area=A2=A1

Q

Bx

V2

Example
• Pipe with U turn

P1,V1

F

P2,V2

Use Gage Pressure!

In case of gas, use absolute pressure to calculate density

Example

N

• “L” bend

P1

E

F

Assume the force by the pipe on the fluid is in the positive direction

P2

What will the force be, if the flow is reversed (a) in a straight pipe? (b) in a L bend?

Conservation of Mass

X

Y

Example

Pbm. 5.13 & 5.19

V2

r1

P2

r2

Vw

P1

V1

Stationary CV

X

Y

Example

Pressure difference?

Pbm. 5.19

V1=-3 m/s

V2=0 m/s

Vw=Velocity of Sound in water

V2

r1

r2

Vw

P2

P1

V1

Stationary CV

~ 41 atm

y

x

Example

V2=?

Pbm. 5.1,5,3

V1=6 m/s

0.6 m

1.2 m

Momentum Conservation
• Angular Momentum
• In a moving system
• Torque = Angular Momentum flux + Angular Momentum Accumulation rate

Flux

Accumulation Rate

Q

Example
• Example 4 in book
• Find the torque on the shaft
• In a moving system
• Torque = Angular Momentum flux + Angular Momentum Accumulation rate

Q

Example
• Approach-1. Find effective Force in X direction
• Find the moment of Force
• Assume: No frictional loss, ignore gravity, steady state, atmospheric pressure everywhere
Example
• Approach-2. Using conservation of angular momentum
• Stationary CV
Example
• Consider a jet hitting a moving plate
• After 1 second
• Vnoz water has entered into the CV
• Plate has moved by Vplate
• In a control volume which moves with the plate, Vnoz-Vplate water has entered the CV (and exited at the bottom)
Example
• Pbm 5.24
• Thickness of slit =t, vol flow rate =Q, dia of pipe=d, density given
• Ignore gravity effects

3ft

6ft

Flux

Accumulation Rate

Example

1

3ft

• Pbm 5.31
• P1, P2, density, dia, vol flow rate given

2

• Calculate velocity at 1 (=2)

Friction Loss (Viscous)

Mechanical

Work

done by

the system

Heat

Work done

by pressure force

Energy Conservation
Energy Conservation
• No Frictional losses
• Incompressible
• No heat, work
• No internal energy change
Example
• Flow from a tank

Dia = d1

1

h1

0

2

Dia = d2

• Pressure = atm at the top and at the outlet

h3

3

• Velocity at 1 ~ 0
• Toricelli’s Law
• Sections 2 and 3
Example
• How long does it take to empty the tank?
• What if you had a pipe all the way upto level 3?

Dia = d1

1

h1

0

2

Dia = d2

h3

3

• Pressure @ section 2 != atm
• Pressure @ section 3 = atm
Example
• What if you had a pipe all the way upto level 3?

Dia = d1

1

h1

0

2

Dia = d2

h3

3

• More flow with the pipe
• Turbulence, friction
• Vortex formation
Example

Height is known

• Moving reference; Aircraft

60 km/h

• Find P and r (eg from tables)

150 km/h

• Flight as Reference

1

2

3

Example
• Pbm. 6.4
• Steady flow through pipe , with friction
• Friction loss head = 10 psi
• Area, vol flow rate given
• Find temp increase
• Assume no heat transfer
Example

D2

• Pbm. 6.10
• Fluid entering from bottom,

t

P2=atm

h2

D1

• Find Q, F on the top plate

P1=10 psig

Example

F

y

D1

P1=10 psig

• If the velocity distribution just below the top plate is known, then P can be found using Bernoulli’s eqn
Modifications to Eqn
• Unsteady state, for points 1 and 2 along a stream line

R

1

H

L

2

D

Draining of a tank
• We can obtain the time it takes to drain a tank
• (i) Assume no friction in the drain pipe
• (ii) Assume you know the relationship between friction and velocity
• Le us take that the bottom location is 2 and the top fluid surface is 1
• Incompressible fluid
Draining of a tank: Quasi steady state
• Velocity at fluid surface at 1 is very small
• i.e. R >> D
• No friction : L is negligible
• P1 = P2 = Patm
Draining of a tank: Quasi steady state
• Original level of liquid is at H = H0
• Integrating above equation from t=0, H=H0 to t=tfinal, H =0, we can find the efflux time
Draining of a tank: Unsteady state
• BSL eg.7.7.1
• At any point of time, the kinetic + potential energy of the fluid in tank is converted into kinetic energy of the outgoing fluid
• We still neglect friction
• Potential Energy of a disk at height z and thickness dz
Draining of a tank: Unsteady state
• Also, using continuity equation
• Substituting, you get a 2nd order non linear ODE with two initial conditions. Please refer to BSL for solution

R

1

H

L

2

D

Draining of a tank (accounting for friction)
• What if the flow in the tube is laminar and you want to account for friction?
• Bernoulli’s eqn is not used (friction present)
• Continuity
• Hagen-Poiseuille’s eqn
Draining of a tank (accounting for friction)
• Substituting and re arranging,
• Integrating with limits
• Note: The answer is given in terms of diameter of tube, so that it is easier to compare with the answer given in the book
Example
• A1,A2, initial height h1 known
• A1 >> A2

1

L

• Consider section 3 and 2

h1

3

2

• Pseudo Steady state ==> Toricelli’s law
Example
• Rearranging and solving, we get
• As t increases, the solution approaches the Toricelli’s equation
Appendix:Example
• Oscillating fluid in a U-tube

1

• Let h=h1-h2

h1

2

h2

L3

Appendix:Example
• Blood Flow in vessels
• Minimization of ‘work’
• Murray’s Law:
• Laminar Flow, negligible friction loss (other than that due to viscous loss in laminar flow) , steady
• Turbulent, pulsating flow
• Assume
Appendix:Example
• If the ratio of ‘smaller’ to larger capillary is constant
• And Metabolic requirement =m= power/volume
• Work for maintaining blood vessel
• Total work