Chapter 3 Fluid Statics
Outline
3.1 Pressure at a Point
3.2 Variation of Pressure with Depth
3.3 Pressure Expressed in Height of Fluid
3.4 Absolute and Gage Pressures
3.5 Measurement of Pressure
Objectives
1. Understand fluid pressure and how it relates to depth.
2. Understand absolute, atmospheric, and gage
pressures and how pressure is measured.
Definitions and Applications
• Statics: no relative motion between adjacent fluid layers.
–Shear stress is zero
–Only _pressure can be acting on fluid surfaces
• Gravity force acts on the fluid (_body force)
• Applications:
–Pressure variation within a reservoir
–Forces on submerged surfaces
–Tensile stress on pipe walls
–Buoyant forces
Motivation?
ÊWhat
are the pressure forces behind
the Hoover Dam?
Hoover Dam
Tall: 220 m
Crest thickness: 13.7 m
Base thickness: 201 m
WHY???
Hoover Dam in 1935
Hoover Dam
•
Example of elevation
head z converted to
velocity head V2/2g.
We'll discuss this in
more detail in
Bernoulli equation.
What do you think?
Lake Mead, the lake behind Hoover Dam, is the world's
largest artificial body of water by volume (35 km3). Is the
pressure at the base of Hoover Dam affected by the
volume of water in Lake Mead?
What do we need to know?
• Pressure variation with direction
• Pressure variation with location
• How can we calculate the total force on a
submerged surface?
3.1 Pressure at a point
•
•
•
•
Pressure is defined as a normal force exerted by a fluid
per unit area.
Units of pressure are N/m2, which is called a pascal (Pa).
Since the unit Pa is too small for pressures encountered
in practice, kilopascal (1 kPa = 103 Pa) and megapascal
(1 MPa = 106 Pa) are commonly used.
Other units include bar, atm
(1 bar = 105 Pa)
Pressure
T
ΔA
ΔA
ΔP
ΔP
p =
ΔA
P
Gate
ΔP
p = lim
ΔA→0 ΔA
Pressure at a Point: Pascal’s Law
Blaise Pascal (1623-1662)
Pressure is the normal force per unit area at a
given point acting on a given plane within a
fluid mass of interest.
How does the pressure at a point vary with orientation of the plane passing
through the point?
F.B.D.
Pressure Forces
Gravity Force
Wedged Shaped Fluid
Mass
p is average pressure in the x, y, and z direction.
Ps is the average pressure on the surface
θ is the plane inclination
V = (1/2δyδz)*δx
δ is the length is each coordinate direction, x, y, z
δs is the length of the plane
γ is the specific weight
Pressure at a Point: Pascal’s Law
For simplicity in our Free Body Diagram, the x-pressure forces
cancel and do not need to be shown. Thus to arrive at our solution
we balance only the the y and z forces:
Pressure Force
in the y-direction
on the y-face
Pressure Force
in the z-direction
on the z-face
Pressure Force
on the plane in
the y-direction
Pressure Force
in the plane in
the z-direction
Rigid body
motion in the ydirection
Weight of the
Wedge
Rigid body
motion in the zdirection
Now, we can simplify each equation in each direction, noting that δy and δz can
be rewritten in terms of δs:
Pressure at a Point: Pascal’s Law
Substituting and rewriting the equations of motion, we obtain:
Math
Now, noting that we are really interested at point only, we let
δy and δz go to zero:
Pascal’s Law: the pressure at a point in a fluid at rest, or in motion, is
independent of the direction as long as there are no shearing stresses
present.
Pressure at a Point: Pascal’s Law
p2δxδs
p1δxδs
psδxδs
ps = p1 = p2
Note: In dynamic system subject to shear, the normal stress representing
the pressure in the fluid is not necessarily the same in all directions. In
such a case the pressure is taken as the average of the three directions.
TWO important principles about pressure
• Pressure at any point is the same in all directions.
• In a fluid confined by solid boundaries, pressure acts
perpendicular to the boundary.
Fluid surfaces
Figure Pressure acting
uniformly in all directions
Figure Direction of fluid
pressures on boundaries
p1
p2
3.2 Variation of Pressure with Depth
•
•
In the presence of a gravitational field,
pressure increases with depth because more
fluid rests on deeper layers.
To obtain a relation for the variation of
pressure with depth, consider rectangular
element
• Force balance in z-direction gives
∑F
z
= maz = 0
P2 Δx − P1Δx − ρ g ΔxΔz = 0
• Dividing by Δx and rearranging gives
ΔP = P2 − P1 = ρ g Δz = γ s Δz
or
p2 = p1 + ρgh
Variation of Pressure with Depth
Variation of Pressure with Depth
•
•
Pressure in a fluid at rest is independent of the shape of the
container.
Pressure is the same at all points on a horizontal plane in a given
fluid.
3.3 Pressure Expressed in Height of Fluid
ƒ If a fluid is incompressible...
p = γh
h=
p
γ
When pressure is expressed as a height, it is referred to as pressure
head (units of ft or m).
Figure : Relationship of pressure and height.
3.3 Pressure Expressed in Height of Fluid
kPa
h( m of H 2O ) =
= 0 ⋅1020 × kPa
9.81
For an incompressible static fluid, the sum of the pressure
head and the elevation head are equal a constant...
p
γ
+z=
p1
γ
Figure : Pressure and elevation head relationship.
+ z1 = Constant
Equipressure Surface
(Equality Of Pressure At The Same Level In A Static Fluid)
Fluid density, ρ
A
A
PL
PR
Figure Horizontal element
cylinder of fluid
W = mg
ƒ Consider the horizontal cylindrical element of fluid with cross sectional
ƒ
ƒ
area, A, in a fluid of density ρ, pressure PL at the left end and PR at the
right end.
Fluid is at equilibrium, so the sum of forces acting on the x-direction is zero.
(→) ΣF =0.
pLA – pRA = 0
∴
pL = pR
This proof that pressure in the horizontal direction is constant.
In the flow field under gravity, equipressure surface must
satisfy the following conditions：
1.At rest；
2.Connection；
3.Medium in connection is the same homogenous fluid；
4.The only mass force is gravity；
5. At the same level
which surface in fig is
equipressure surface ?
surface C-C ;
surface B-B
3.4 Absolute, gage, and vacuum pressures
•
•
•
Actual pressure at a give point is called the absolute pressure.
Most pressure-measuring devices are calibrated to read zero in the
atmosphere, and therefore indicate gage pressure, Pgage=Pabs - Patm.
Pressure below atmospheric pressure are called vacuum pressure,
Pvac=Patm - Pabs.
3.4 Absolute, gage, and vacuum pressures
Absolute pressure equals the atmospheric pressure plus
the gage pressure
pabs = patm + pgage
Barometric pressure is another word for atmospheric pressure
Unit of Pressure
a. Stress unit : Pa（N/m2）， kPa（KN/ m2）
b. Barometric pressure
Barometric pressure，1atm=1.013×105Pa=101.3 kPa
1at=98000Pa
c. Height of liquid column
Height of water column （mH20）：
For 1at:
Height of mercury column mmHg
for 1at:
Example 1
ƒ What will be the gauge pressure and absolute pressure of
water at a depth 12m below the surface? Take ρwater = 1000
kg/m3 and Patm = 101 kN/m2
•Solution:
Pgauge = ρgh
= 1000 x 9.81 x 12
= 117.7 kN/m2 (kPa)
Pabs
= Pgauge + Patm
= (117.7 + 101) kN/m2
= 218.7 kN/m2
Example 2
ƒ A cylinder contains a fluid at a gauge pressure of 200 kN/m2.
Express this pressure in terms of
ƒ head of water (ρ =1000 kg/m3)
ƒ head of mercury (SG=13.6)
ƒ What would be the absolute pressure if the atmospheric
pressure is, Patm = 101.3 kN/m2.
•Solution:
h= P/ρg
a) for water:
h = 200x103/(1000x9.81)
= 20.39 m of water.
b) for mercury
h = 200x103/(13.6x1000x9.81)
= 1.5 m of mercury
Absolute pressure = Patm + Pgauge
= 101.3 + 200 = 301.3 kN/m2.
3.5 Measurement of Pressure
●
●
●
●
●
Barometer
Bourdon gage
Pressure transducer
Piezometer column
Simple manometer
Measurement of Pressure: Schematic
+
-
+
+
Barometer
Evangelista Torricelli
(1608-1647)
The first mercury barometer was constructed in 1643-1644 by Torricelli. He
showed that the height of mercury in a column was 1/14 that of a water barometer,
due to the fact that mercury is 14 times more dense that water. He also noticed
that level of mercury varied from day to day due to weather changes, and that at
the top of the column there is a vacuum.
Torricelli’s Sketch
Schematic:
Animation of Experiment:
Note, often pvapor is very small,
0.0000231 psia at 68° F, and
patm is 14.7 psi, thus:
The Barometer (气压表 )
•
•
•
PC + ρ gh = Patm
Patm = ρ gh
Atmospheric pressure is measured
by a device called a barometer;
thus, atmospheric pressure is
often referred to as the barometric
pressure.
PC can be taken to be zero since
there is only Hg vapor above
point C, and it is very low relative
to Patm.
Change in atmospheric pressure
due to elevation has many effects:
Cooking, nose bleeds, engine
performance, aircraft performance.
Manometry
Manometry is a standard technique for measuring pressure using
liquid columns in vertical or include tubes. The devices used in this
manner are known as manometers(测压表）.
3 common types of manometers including:
1) The Piezometer Tube
2) The U-Tube Manometer
3) The Inclined Tube Manometer
Piezometer Tube(测压管)
po
•Pressure can be estimated
by measuring fluid elevation
Closed End “Container”
Move Up the
Tube
pA (abs)
Moving from left to right:
Rearranging:
Disadvantages:
1)The pressure in the container has to
be greater than atmospheric pressure.
2) Pressure must be relatively small to
maintain a small column of fluid.
3) The measurement of pressure must
be of a liquid.
pA(abs) - γ1h1 = po
p A − po = γ 1h1
Gage Pressure
Then in terms of gage pressure, the equation for a Piezometer Tube:
Note: pA = p1 because they are at the same level
U-Tube Manometer
Note: in the same fluid we can
“jump” across from 2 to 3 as
they are at the sam level, and
thus must have the same
pressure.
Closed End
“Container”
pA
The fluid in the U-tube is known
as the gage fluid. The gage fluid
type depends on the application,
i.e. pressures attained, and
whether the fluid measured is a
gas or liquid.
Since, one end is open we can work entirely in gage pressure:
Moving from left to right:
pA + γ h - γ2h2 = 0
1 1
Then the equation for the pressure in the container is the following:
If the fluid in the container is a gas, then the fluid 1 terms can be ignored:
U-Tube Manometer
Measuring a Pressure Differential
Closed End
pB “Container”
Closed End
“Container”
pA
Final notes:
1)Common gage fluids are Hg and
Water, some oils, and must be
immiscible.
2)Temp. must be considered in very
accurate measurements, as the gage
fluid properties can change.
3) Capillarity can play a role, but in
many cases each meniscus will cancel.
Moving from left to right: pA + γ1h1 - γ2h2 - γ3h3 = pB
Then the equation for the pressure difference in the container is the following:
Inclined-Tube Manometer
This type of manometer is used to measure small pressure changes.
pB
pA
h2
l2
θ
h2
θ
sin θ =
h2
l2
h2 = l2 sin θ
Moving from left to right: pA + γ1h1 - γ2h2 - γ3h3 = pB
Substituting for h2:
Rearranging to Obtain the Difference:
If the pressure difference is between gases:
Thus, for the length of the tube we can measure a greater pressure differential.
Other Pressure Measurement Devices
•
Mechanical and electronic pressure measuring devices
(a) Liquid-filled Bourdon pressure gages for various pressure ranges.
(b) (b) Internal elements of Bourdon gages. The “C-shaped” Bourdon tube
is shown on the left, and the “coiled spring” Bourdon tube for high
pressures of 1000 psi and above is shown on the right.
Bourdon Pressure Gage
•
•
Bourdon tube pressure gage uses a hollow, elastic, and
curved tube to measure pressure.
As the pressure within the tube increases the tube tends
to straighten, and although the deformation is small, it
can be translated into the motion of a pointer on dial.