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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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• Newton’s Second Law

• Force = Mass * Acceleration

• Alternate method: From Reynold’s theorem

• Fluid Flow

• Force = Momentum flux + Momentum Accumulation rate

Flux

Accumulation Rate

Example

• Straight Pipe

A2

V2

A1

V1

• Assumption: No friction

2

Example

• Flow in a straight pipe. Realistic case

r1=1g/cm3

d1=8cm

V1=5m/s

d2=5cm

Example

• Problem: 5.32

d2,V2

d1,V1

1

2

h=58 cm

x

Weight of Fluid

Example

• Example 1 (bent pipe), page 47

r1,V1

By

Area=A1

Area=A2=A1

Q

Bx

r2,V2

in

in

out

Example

• d/dt =0

• From Eqn of Conservation of Mass

V1

By

Area=A1

Area=A2=A1

Q

Bx

V2

• Pipe with U turn

P1,V1

F

P2,V2

Use Gage Pressure!

In case of gas, use absolute pressure to calculate density

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?

X

Y

Example

Pbm. 5.13 & 5.19

V2

r1

P2

r2

Vw

P1

V1

Stationary CV

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

x

Example

V2=?

Pbm. 5.1,5,3

V1=6 m/s

0.6 m

1.2 m

• Angular Momentum

• In a moving system

• Torque = Angular Momentum flux + Angular Momentum Accumulation rate

Flux

Accumulation Rate

Example

• Example 4 in book

• Find the torque on the shaft

• In a moving system

• Torque = Angular Momentum flux + Angular Momentum Accumulation rate

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

• Approach-2. Using conservation of angular momentum

• Stationary CV

• 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)

• 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

1

3ft

• Pbm 5.31

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

2

• Calculate velocity at 1 (=2)

Mechanical

Work

done by

the system

Heat

Work done

by pressure force

Energy Conservation

• No Frictional losses

• Incompressible

• No heat, work

• No internal energy change

• 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

• 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

• 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

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

• 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

D2

• Pbm. 6.10

• Fluid entering from bottom,

• exiting at radial direction

• Steady, no friction

t

P2=atm

h2

D1

• Find Q, F on the top plate

P1=10 psig

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

• Unsteady state, for points 1 and 2 along a stream line

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

• Quasi steady state assumption

• Velocity at fluid surface at 1 is very small

• i.e. R >> D

• No friction : L is negligible

• P1 = P2 = Patm

• 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

• 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

• Also, using continuity equation

• Substituting, you get a 2nd order non linear ODE with two initial conditions. Please refer to BSL for solution

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

• 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

• A1,A2, initial height h1 known

• A1 >> A2

1

L

• Consider section 3 and 2

h1

3

2

• Pseudo Steady state ==> Toricelli’s law

• Rearranging and solving, we get

• As t increases, the solution approaches the Toricelli’s equation

• Oscillating fluid in a U-tube

1

• Let h=h1-h2

h1

2

h2

L3

• 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

• If the ratio of ‘smaller’ to larger capillary is constant

• And Metabolic requirement =m= power/volume

• Work for maintaining blood vessel

• Total work