fluid mechanics and energy transport bien 301 lecture 4 pressure distribution, hydrostatic forces,...

Fluid Mechanics and Energy TransportFluid Mechanics and Energy TransportBIEN 301BIEN 301

Lecture 4Lecture 4 Pressure Distribution, Hydrostatic Forces, and Pressure Distribution, Hydrostatic Forces, and

Pressure MeasurementPressure Measurement

Juan M. Lopez, E.I.T.Juan M. Lopez, E.I.T.

Research ConsultantResearch ConsultantLeTourneau UniversityLeTourneau University

Adjunct LecturerAdjunct LecturerLouisiana Tech UniversityLouisiana Tech University

12/12/200612/12/2006 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

PressurePressure

Definition Definition (White 2.1)(White 2.1)

A force applied over a A force applied over a surface area.surface area.

In fluid statics, the In fluid statics, the forces exerted by the forces exerted by the pressures on each of pressures on each of the faces of the fluid the faces of the fluid element must sum to element must sum to zero.zero.

zgpp

grearrangin

ppppF

nz

nzyx

2

1

,


PressurePressure

From this formula, we can see that a few From this formula, we can see that a few principal facts fall out:principal facts fall out:• Hydrostatic pressure is a function of height of the Hydrostatic pressure is a function of height of the

fluid columnfluid column• The pressure in any plane normal to the The pressure in any plane normal to the

gravitational field is identical.gravitational field is identical.• For hydrostatic conditions, the forces generated by For hydrostatic conditions, the forces generated by

the fluid must come from the gravitationally-induced the fluid must come from the gravitationally-induced weight component only.weight component only.


PressurePressure

What about if the fluid is moving?What about if the fluid is moving?• Strain rates will exist, and they will be out of Strain rates will exist, and they will be out of

balance.balance.• Viscous stresses will existViscous stresses will exist

element. theinto and

normal be toconsidered is Pressure

?3

1

signnegativeaWhy

p zzyyxx


PressurePressure

Forces from PressureForces from Pressure From our definition, pressure is From our definition, pressure is

a force over a surface area.a force over a surface area. What happens when we get What happens when we get

pressure variations?pressure variations?• What causes pressure What causes pressure

variations?variations? Because pressure variations Because pressure variations

can come from many different can come from many different sources, we need a more sources, we need a more generalized expression for generalized expression for expressing these forces.expressing these forces.

pf

dxdydzz

pk

y

pj

x

pifd

press

NOTE

press

:get weg,Integratin

:istion representa aldifferenti the

fluid, theof volumeOver the

p toequivalent is This:


PressurePressure

Surface and Body ForcesSurface and Body Forces Two types of forces that can act on our fluid elementTwo types of forces that can act on our fluid element Can you think of examples of surface vs. body forces?Can you think of examples of surface vs. body forces? Principal body force we’ll deal with here:Principal body force we’ll deal with here:

• GRAVITYGRAVITY• Integrating the force from gravitational effects over our element Integrating the force from gravitational effects over our element

volume:volume:

gf

dxdydzgfd

grav

grav


PressurePressure

We’re missing one principal forceWe’re missing one principal force Surface forces due to viscous effectsSurface forces due to viscous effects

Vf

z

V

y

V

x

Vf

VS

VS

2

2

2

2

2

2

2

:definition that extend now

we,(gradient) operator del for the

definition previousour recalling


PressurePressure

We can now combine all We can now combine all of our defined terms to of our defined terms to generate a more general generate a more general expression for the expression for the balance of forces on a balance of forces on a fluid element.fluid element.

This is a form of the This is a form of the differential momentum differential momentum equation from Chapter 4.equation from Chapter 4.

Vgp

fff

fa

VSgravpress

2


PressurePressure

For now, we assume that we know the velocity and For now, we assume that we know the velocity and acceleration acting on our fluid. If we have an acceleration acceleration acting on our fluid. If we have an acceleration vector, vector, aa, we can then re-express our equation as follows: , we can then re-express our equation as follows:

tzyxB

Vgpa

,,,

2


PressurePressure

With the knowns (V, a), we can solve for the With the knowns (V, a), we can solve for the pressure field via direct integration.pressure field via direct integration.

tzyxBz

ptzyxB

y

ptzyxB

x

pzyx ,,,,,,,,,

This form will be very useful. To illustrate this, This form will be very useful. To illustrate this, we’ll take a few exampleswe’ll take a few examples


PressurePressure

Flow under balanced forces (Flow under balanced forces (ΣΣF=0)F=0) HydrostaticHydrostatic Steady motionSteady motion

gpgp

Vgpa

0

0

2

0


PressurePressure

Rigid Body Motion, drops out the viscous termRigid Body Motion, drops out the viscous term RotationRotation TranslationTranslation Examples? Why does the viscous term disappear?Examples? Why does the viscous term disappear?

agp

Vgpa

0

2


PressurePressure

Irrotational MotionIrrotational Motion Different than rigid body flow…why?Different than rigid body flow…why?

Does not have to behave like a rigid bodyDoes not have to behave like a rigid body There simply are no rotational terms.There simply are no rotational terms.

0 V



Absolute, Vacuum, and Absolute, Vacuum, and Gauge pressureGauge pressure

Can anyone tell me what the Can anyone tell me what the most commonly used most commonly used pressure reporting method pressure reporting method is?is?

Is there a better scientific Is there a better scientific way of reporting the way of reporting the pressure?pressure?

Why?Why?



Hydrostatic conditionHydrostatic condition As we mentioned before, the hydrostatic (no motion, As we mentioned before, the hydrostatic (no motion,

or balanced forces) condition reduces to:or balanced forces) condition reduces to:

gp

For a fluid at rest, the horizontal components drop out.For a fluid at rest, the horizontal components drop out.

z

pkgk

z

pji

kgkz

pj

y

pi

x

pp

00



We can also express this in terms of an We can also express this in terms of an integralintegral

2

112

2

1

2

1

variablesof Sep.

dzpp

dzdp

z

p



For liquids, incompressibility is a good For liquids, incompressibility is a good assumptionassumption

121212

12

1212

2

112

2

112

1ppz

ppz

or

zpp

dzppdzpp

ZZ

Z


Pressure MeasurementPressure Measurement It is this simple It is this simple

relationship which is relationship which is exploited in many exploited in many pressure measurement pressure measurement devices and/or devices and/or calculations.calculations.

For example, calculating For example, calculating the rise of mercury in a the rise of mercury in a tube (a mercury tube (a mercury barometer)barometer)

mercury

catmospheri

mercury

catmosphericatmospheri

catmospherimercurycatmospheri

catmospherireferencemercuryvacuumcatmospheri

ph

pz

zp

zzpp

00


Pressure MeasurementPressure Measurement What if the medium is a gas (compressible)?What if the medium is a gas (compressible)?

How about introducing the ideal gas law?How about introducing the ideal gas law?

A sufficiently accurate assumption for most cases.A sufficiently accurate assumption for most cases.

gRT

pg

dz

dp


Pressure MeasurementPressure Measurement Pressure is not something we can measure directly, but Pressure is not something we can measure directly, but

rather derive it from some other measurement.rather derive it from some other measurement. What forms of measurement of pressure are used, and What forms of measurement of pressure are used, and

how do we have to derive the signal from these?how do we have to derive the signal from these?


Pressure MeasurementPressure Measurement Common pressure measurement instrument designations:Common pressure measurement instrument designations:

Gravity-based: Gravity-based: • barometer, manometer, dead weight pistonbarometer, manometer, dead weight piston

Elastic deformation: Elastic deformation: • bourdon tube, diaphragm, bellows, strain gauge, displaced optical beamsbourdon tube, diaphragm, bellows, strain gauge, displaced optical beams

Gas behavior: Gas behavior: • Gas compression, thermal conductance, molecular impact, ionization, Gas compression, thermal conductance, molecular impact, ionization,

thermal conductivity.thermal conductivity. Electric output: Electric output:

• Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency.inductance, linear variable differential transformer, resonant frequency.

Luminescent coatingsLuminescent coatings• Surface PressuresSurface Pressures



Common pressure measurement instrument Common pressure measurement instrument designations:designations:

Electric output: Electric output: • Resistance, diffused strain gauge, piezoelectric, potentiometric, Resistance, diffused strain gauge, piezoelectric, potentiometric,

magnetic inductance, linear variable differential transformer, magnetic inductance, linear variable differential transformer, resonant frequency.resonant frequency.

Luminescent coatingsLuminescent coatings• Surface PressuresSurface Pressures

Examples of where we might choose different pressure Examples of where we might choose different pressure measurement devices in biomedical applications?measurement devices in biomedical applications?



Let’s apply what we’ve learned to some Let’s apply what we’ve learned to some flow examples:flow examples: Ex. 2.4Ex. 2.4 Ex. 2.7Ex. 2.7 P2.23P2.23


AssignmentAssignment HW 4 has been posted on blackboardHW 4 has been posted on blackboard

Exam 1Exam 1 Reviews have been postedReviews have been posted In-class exam materials allowed:In-class exam materials allowed:

• 1 Calculator1 Calculator• Writing ImplementsWriting Implements• Chapter Reviews from course documents, NO NOTES.Chapter Reviews from course documents, NO NOTES.• Lecture slides, 6 to a page, NO NOTES.Lecture slides, 6 to a page, NO NOTES.• Fluid Mechanics, fifth edition, by White (class textbook), NO NOTES.Fluid Mechanics, fifth edition, by White (class textbook), NO NOTES.• If I find handwritten notes in these sections, your materials will be removed. If I find handwritten notes in these sections, your materials will be removed.

Any additional cheating will result in failing the test, and maybe the course.Any additional cheating will result in failing the test, and maybe the course. Take-Home OptionTake-Home Option

• Sign-Up E-mail due by Wednesday, 11:59 pm.Sign-Up E-mail due by Wednesday, 11:59 pm.


Questions?Questions?

fluid mechanics and energy transport bien 301 lecture 4 pressure distribution, hydrostatic forces,...

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