Aula Teórica 11

Integral Budgets: Momentum

General Principle & Mass• The rate of accumulation inside a Control Volume

balances the fluxes plus production minus consumption:

)(.. SiSodAncnudVt surfacevc

• Fluid Mass has no source/sink and no diffusion and consequently the accumulation balances the advective flux. If the fluid is incompressible there is no accumulation and thus mass flowing in balances the mass flowing out and so does the volume in incompressible flows:

SE

SE

SE

surface

AUAU

QQ

AUAU

dAnu

.0

Momentum

• In case of momentum

)(.. SiSodAncnudVt surfacevc

iu

dVolgx

pdAnunuudVu

t iisurface

ii

vc

i

..

• In lecture 4 we have seen that the Sources/Sinks of momentum are the pressure and gravity forces.

• Momentum diffusive flux is in fact the Shear stress that we can compute explicitly from the velocity derivative only if we know the velocity profile. For that reason we will call it shear stress in the integral budget approach.

Integral momentum budget

• Let us consider: incompressible and stationary flow.

dVolgx

pdAnunuudVu

t iisurface

ii

vc

i

..

dVolgx

pdAnuu i

isurface

ii

.0

• If we assume that the velocity is uniform at the inlet and the outlet of the volume:

• If p is uniform along inlet and outlet:

Intlets

iOutlets

i

surface

i QUQUdAnuu .

WeightdVolgi

dApndVolx

p

surface

ii

ioutletinlet

ji

surface

i FPApndApn

• And finally we get: VolgFVFPApnQUQU

iiiSurface

jiIntlets

iOutlets

i

Integral Momentum Budget

• This is an algebraic equation applicable if:– Stationary and incompressible flow,– Velocity is uniform at each inlet and outlet,– Pressure is uniform along surfaces (e.g. inlets and outlets).

• Can inlets be located in zones where streamlines have curvature?• Being a budget, this equation permits the calculation of a term

knowing all the others.

• Where is the summation of pressure forces other than those acting at inlet and outlet and is the summation of the friction forces.

VolgFVFPApnQUQUiii

Surfaceji

Intletsi

Outletsi

iFP

iFV

Example 1• Calculate the force exerted by the fluid over the

deflector neglecting friction V=2 m/s and jet radius is 2 cm and theta is 45º.

• We have a flow with an inlet and an outlet. • Velocity has a component at the inlet and

two at the outlet• Pressure is atmospheric at inlet and outlets

and thus the velocity modulus remains constant.

• We have to compute budgets along both directions x and y.

D= 2cm A= 0.000315m2

V= 2m/s Q= 0.000629m3/s

Theta 45 1.26N

0.88971N

0.88971N

Fx= -0.37N

Fy= 0.88971N

F= 0.963015N

VolgFVFPApnQUQUiii

Surfaceji

Intletsi

Outletsi

Example 2• The Jet is hitting the surface

perpendicularly (Vj=3m/s), but the surface is moving (Vc=1m/s). D=10 cm. Calculate the force and power supplied.

VolgFVFPApnQUQUiii

Surfaceji

Intletsi

Outletsi

Intlets

iOutlets

i

surface

i QUQUdAnuu .

• If the control volume is moving, fluxes depend on the flow velocity relative to the control volume:

Intletsjciij

Outletscii

Intletsi

Outletsi

surface

i

AUUUAUUU

QUQUdAnuu

.

Relative or absolute reference

• Discharge must be computed using relative velocity.• Transported velocity can be the relative or the absolute velocity. Usually the

relative velocity is more intuitive.

IntletsRiC

OutletsRiC

IntletsRiR

OutletsRiR

IntletsRi

OutletsRi

surface

i

RIntlets

jCjOutlets

C

Intletsi

Outletsi

surface

i

QUQUQUQU

QUQUdAnuu

So

QAUUAUU

but

QUQUdAnuu

.

.

Example 2 Vc= 1m/s

D= 10cm

Vj= 3m/s A= 0.007864m2

VRj= 2 Q= 0.015728m3/s

Theta 90 31.46N

0N

0Symmetrical

Fx= -31.46N

Fy= 0N

F= 31.456N

Symmetrical VolgFVFPApnQUQU

iiiSurface

jiIntlets

iOutlets

i

Summary

• The integral momentum equation states describes the momentum conservation principle (Newton law) assuming simplified solutions for momentum flux calculations.

• It is useful when flow is stationary and incompressible.

• Becomes more useful when associated to the Bernoulli Energy Conservation Equation.