Momentum conservation
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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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Momentum conservation
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


Example

0

Example

  • Straight Pipe

  • Steady State

A2

V2

A1

V1

  • Assumption: No friction


Example1

1

2

Example

  • Flow in a straight pipe. Realistic case


Example2

0

r1=1g/cm3

d1=8cm

  • Steady State

V1=5m/s

d2=5cm

Example

  • Problem: 5.32

d2,V2

d1,V1

1

2

h=58 cm


Example3

y

x

Weight of Fluid

Example

  • Example 1 (bent pipe), page 47

r1,V1

By

Area=A1

Area=A2=A1

Q

Bx

r2,V2


Example4

out

in

in

out

Example

  • Steady State

    • d/dt =0

  • From Eqn of Conservation of Mass


Example5
Example

V1

By

Area=A1

Area=A2=A1

Q

Bx

V2


Example6
Example

  • Pipe with U turn

P1,V1

F

P2,V2

Use Gage Pressure!

In case of gas, use absolute pressure to calculate density


Example7
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?


Example8

Conservation of Mass

X

Y

Example

Pbm. 5.13 & 5.19

V2

r1

P2

r2

Vw

P1

V1

Stationary CV


Example9

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


Example10

y

x

Example

V2=?

Pbm. 5.1,5,3

V1=6 m/s

0.6 m

1.2 m


Momentum conservation1
Momentum Conservation

  • Angular Momentum

  • In a moving system

    • Torque = Angular Momentum flux + Angular Momentum Accumulation rate

Flux

Accumulation Rate


Example11

Q

Example

  • Example 4 in book

  • Find the torque on the shaft

  • In a moving system

    • Torque = Angular Momentum flux + Angular Momentum Accumulation rate


Example12

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


Example13
Example

  • Approach-2. Using conservation of angular momentum

  • Stationary CV


Example14
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)




Example17
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


Example18
Example

1

3ft

  • Pbm 5.31

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

2

  • Calculate velocity at 1 (=2)


Energy conservation

Friction Loss (Viscous)

Mechanical

Work

done by

the system

Heat

Work done

by pressure force

Energy Conservation


Energy conservation1
Energy Conservation

  • No Frictional losses

  • Incompressible

  • Steady

  • No heat, work

  • No internal energy change


Example19
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


Example20
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


Example21
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

  • Unsteady flow

  • Vortex formation


Example22
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



Example23
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


Example24
Example

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


Example25
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
Modifications to Eqn

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


Draining of a tank

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
Draining of a tank: Quasi steady state

  • 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


Draining of a tank quasi steady state1
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
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 state1
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


Draining of a tank accounting for friction

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 friction1
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


Example26
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


Example27
Example

  • Rearranging and solving, we get

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


Appendix example
Appendix:Example

  • Oscillating fluid in a U-tube

1

  • Let h=h1-h2

h1

2

h2

L3


Appendix example1
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 example2
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

  • Optimum radius


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