Estudo sobre a pressão manométrica, Exercícios de Mecânica dos fluidos

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CHAPTER 2

Pressure and Head

FLUID MECHANICS

Dr. Khalil Mahmoud ALASTAL

Gaza, Sep. 2012

• Introduce the concept of pressure.

• Prove it has a unique value at any particular

elevation.

• Show how pressure varies with depth according to

the hydrostatic equation.

• Show how pressure can be expressed in terms of

head of fluid.

• Demonstrate methods of pressure measurement

using manometer.

Objectives of this Chapter:

This is also true for:

• Curved surfaces (the force acting at any point is normal to the surface at that point).
• Any imaginary plane.

2.1 Statics of Fluid Systems:

• We use this fact in our analysis by considering elements of fluid bounded by solid boundaries or imaginary planes.
• And since the fluid at rest: (the element will be in equilibrium)  The sum of the components of forces in any direction will be zero.  The sum of the moments of forces on the element about any point must also be zero.

• As mentioned above a fluid will exert a normal force

on any boundary it is in contact with.

• Since these boundaries may be large and the force

may differ from place to place

it is convenient to work in terms of pressure “ p ”

which is the force per unit area.

2.2 Pressure:

How does the pressure at a point vary with orientation of the plane passing through the point?

p is average pressure in the x, y, and z direction. Ps is the average pressure on the surface q is the plane inclination  is the length is each coordinate direction, x, y, z s is the length of the plane g is the specific weight

Wedged Shaped Fluid F.B.D. Mass

Pressure Forces Gravity Force

V = (1/2  y  z)*  x

Blaise Pascal (1623-1662)

2.3 Pascal’s Law for Pressure

at a Point:

Remember:

• No shearing forces
• All forces at right angles to the surfaces For simplicity, the x-pressure forces cancel and do not need to be shown. Thus to arrive at our solution we balance only the y and z forces: Pressure Forcein the y-direction on the y-face

Pressure Forceon the plane in the y-direction

py = ps

0

2.3 Pascal’s Law for Pressure at a Point:

px = py = ps

Thus:

Pressure at any point is the same in all directions.

This is known as Pascal’s Law

and applies to fluids at rest.

2.3 Pascal’s Law for Pressure at a Point:

Vertical elemental cylinder of fluid

The pressure at:  the bottom of the cylinder is p 1 at level z 1  the top of the cylinder is p 2 at level z 2

The fluid is at rest and in equilibrium so all the forces in the vertical direction sum to zero.

2.4 Variation of Pressure Vertically in a

Fluid under Gravity

Horizontal cylinder elemental of fluid

For equilibrium the sum of the forces in the x direction is zero. pl A = pr A

pl = pr

Pressure in the horizontal direction is constant

This result is the same for any continuous fluid

2.5 Equality of Pressure at the Same Level

in a Static Fluid:

• It is still true for two connected tanks which appear not to have any direct connection.

Equality of pressures in a continuous body of fluid

We have shown: pR = pS For a vertical pressure change we have:

pS = pQ +  g z

pR = pP +  g z and

so (^) pP = pQ

Pressure at the two equal levels are the same.

2.5 Equality of Pressure at the Same Level

in a Static Fluid:

If q = 90o, then s is in the x or y directions, so:

Confirming that pressure on any horizontal plane is zero.

If q = 0o^ then s is in the z direction so:

Confirming the result

Horizontal

Vertical

• In a static fluid of constant density we for vertical pressure the relationship:
• This can be integrated to give
• measuring z from the free surface so that z = -h

g dz

dp  

p    gz constant

p   gh constant

2.13 Pressure And Head

pabsolute  pgauge  patmospheric

pabsolute   gh  patmospheric

+^ -

Summary: Absolute and Gauge Pressure

Pressure measurements are generally indicated as being either absolute or gauge pressure.

Gauge pressure

• is the pressure measured above or below the atmospheric pressure (i.e. taking the atmospheric as datum).
• can be positive or negative.
• a negative gauge pressure is also known as vacuum pressure.

Absolute pressure

• uses absolute zero, which is the lowest possible pressure.
• therefore, an absolute pressure will always be positive.
• a simple equation relating the two pressure measuring system can be written as: Pabs = Pgauge + Patm