CEEN598D Week 2 Statics - Inside Minesinside.mines.edu/~lbearup/CEEN598DF14/2.Notes.pdfPressure p is...
Transcript of CEEN598D Week 2 Statics - Inside Minesinside.mines.edu/~lbearup/CEEN598DF14/2.Notes.pdfPressure p is...
CEEN 598D– Fall 2015
Lindsay Bearup [email protected] Berthoud Hall 121
STATICS CEEN 598D: Fluid Mechanics for Hydro Systems
CEEN 598D– Fall 2015
From your homework: What is bigger:
kg or a slug? lb or a N?
CEEN 598D– Fall 2015
Fluid Statics • Fluids that are in a hydrostatic condition
– Hydrostatic Condition: means all fluid particles are in force equilibrium with one-another (i.e., at rest)
• Calculate forces due to distributions of pressure
Pressure: normal force per unit area
• Equilibrium of a fluid particle and surroundings – Fluid particle: body of finite mass and internal structure
but negligible dimensions
CEEN 598D– Fall 2015
Hydrostatic Condition
CEEN 598D– Fall 2015
Fluid Statics For fluids at rest, only normal forces (non-shear forces) exist which act over the entirety of each boundary surface. These are called pressure forces. Pressure p is a scalar – we can check by calculating the pressure in every direction for an odd-shaped volume.
dAdF
AFp
A=
Δ
Δ=
→Δ 0lim
CEEN 598D– Fall 2015
Pressure: Absolute, Gage and Vacuum
• Pressure in a perfect vacuum is considered absolute zero.
• All pressures relative to absolute zero are absolute pressures.
• Sea level standard pressure is 101 kN/m2 or 101 kPa as an absolute pressure.
CEEN 598D– Fall 2015
Measuring Pressure
• Most pressure monitoring devices measure the difference in the relative pressure, with atmospheric pressure being the reference pressure. – This is known as gage pressure. – 0 psi or 0 kPa gage pressure is atmospheric absolute
pressure ~ 101 kPa.
• Absolute pressure can be less than atmospheric pressure, so the gage pressure is negative. – This is called the vacuum pressure.
CEEN 598D– Fall 2015
CEEN 598D– Fall 2015
Pressure Transmission
• Fluids act as a continuum where forces/deformations are translated throughout the fluid system.
• Pascal’s Law: In a closed system, a pressure change at one point in the system will be transmitted throughout the entire system (Not instantaneously, however).
• In watersheds, which is faster – fluid flow or pressure propagation? What are the implications of this?
CEEN 598D– Fall 2015
CEEN 598D– Fall 2015
Pressure Variation with Elevation
As you move down the water column, the pressure increases due to the weight of more and more overlying fluid. For water this is called hydrostatic pressure.
pA = 0 gage
A
z
zp p zγ= +
CEEN 598D– Fall 2015
Pressure Variation with Elevation
Take an element of fluid:
A
Δz
zAW Δ−= γ
App )( Δ+−
pA
γγ
γ
γ
−=−=Δ
Δ
Δ−=Δ
+Δ+−Δ−==∑
dzdp
zp
pz
pAAppzAFz )(0
Fp
Fp Fw
CEEN 598D– Fall 2015
Solve ODE for General Solution
Czp
dzdpdzdp
+−=
−=−=
γ
γγ
IF THE DENSITY IS CONSTANT • Or, slightly rearranged, p/γ + z = constant. • The quantity is called the piezometric head
H =p/γ + z
separate and
integrate
*Later we will see that H is the total head and also includes a velocity component, but for now our fluid is at rest.
CEEN 598D– Fall 2015
Piezometric Head
1
2
z1
z2
22
11 zpzp
+=+γγ
Since this quantity is a constant in a static water body, we can apply this formula at any two points:
CEEN 598D– Fall 2015
Piezometric Head
• QUICK - What are the units of the piezometric head?
• What is it measuring, or keeping track of?
p zγ+
CEEN 598D– Fall 2015
Piezometric Head
• The last term is the potential energy of the packet of water with volume, V.
• Potential energy is a function of an objects position in space or arrangement of parts.
mgH =Vp+mgz
CEEN 598D– Fall 2015
• A decrease in elevation is accompanied by an increase in pressure. This is a simple conversion of energy.
• Take a packet of water deeper in a pool and the potential energy has been converted to pressure energy. Water under more pressure has the ability to do more work than water under less pressure – recall the car jack!
mgH =Vp+mgzPiezometric Head
CEEN 598D– Fall 2015
Let’s Summarize:
• Piezometric Head
• Hydrostatic Pressure
H =pγ+ z
zp p zγ= +pA = 0 gage
A
z
CEEN 598D– Fall 2015
Converting One to the Other
Recall, for constant H:
pz = Hγ
But what does this all mean?
€
Δp = −γΔz2
21
1 zpzp+=+
γγ
CEEN 598D– Fall 2015
CEEN 598D– Fall 2015
Measuring Pressure: Bourdon-Tube Gage
• Measures pressure by sensing deflection of a coiled hollow tube
Pressure in tube balanced by spring rate of the tube
CEEN 598D– Fall 2015
Problem
For the closed tank with Bourdon-type gages tapped into it, what is the specific gravity of the oil and the pressure reading on gage C?
CEEN 598D– Fall 2015
Problem: Solution Part 1
Solve for γoil assuming γair is small (true value is 11.8 N/m3)
€
Δp = −γΔz
€
pa − pb = −γΔz
€
50000 N/m2 − 58530 N/m2 = −γ oil (1.0 m)
€
−8530 N/m3 = −γ oil
€
γ oil = 8530 N/m3
From Converting One to the Other Slide:
CEEN 598D– Fall 2015
€
pc = 58530 N/m2 + 8530 N/m3(0.5 m) + 9800 N/m3(1.0 m) = 72595 N/m2
Problem: Solution Part 2
Solve for pc by stepping from point C to point B: Pressure must be same at interface (bottom of oil) Substitute for pbottom_oil
€
pbottom _ oil − pc = −γwaterΔz
€
pb − pbottom _ oil = −γ oilΔz
€
pc = pb + γ oil (0.5m) + γwater(1.0m)
For water at 10◦C
CEEN 598D– Fall 2015
Measuring Pressure: Aneroid Barometer
http://usatoday30.usatoday.com/weather/waneroid.htm
CEEN 598D– Fall 2015
Measuring Pressure: Barometer
• Used to measure atmospheric pressure
• Most common types: mercury barometer and aneroid barometer
€
patm = γ Hgh + pv ≈ γ Hgh
CEEN 598D– Fall 2015
Measuring Pressure: Piezometer
• Vertical tube in which liquid (flowing or not) rises due to pressure in the pipe.
Where would you see something like this?
CEEN 598D– Fall 2015
Measuring Pressure: Pressure Transducers
• Converts pressure to electrical signal
CEEN 598D– Fall 2015
Measuring Pressure: Manometer
Pressure device that uses specific weight (or density) of a fluid to measure pressure.
CEEN 598D– Fall 2015
Manometry
• Pressure and elevation heads are transferrable in static liquids.
• Allows us to construct very simple pressure gauges called manometers.
• Attach a fluid-filled tube to the system in question and measure the height of the fluid.
CEEN 598D– Fall 2015
U-Tube Manometer
CEEN 598D– Fall 2015
hpp mγ+= 12
23 pp =
lhplpp
m γγ
γ
−+=
−=
04
34
Start at the open end and go around the horn:
U-Tube Manometer
CEEN 598D– Fall 2015
There’s almost no need for a general formula, but if you wish to, define a break point at every fluid change and find the subsequent elevation around a bend that is across from the break (if it exists). Then, • By the way, either pstart or pend must be known the solve
the other. • If you only care about the pressure difference in a pipe,
then you can use a differential manometer.
∑∑ −+=ups
jjdowns
iistartend hhpp γγ
U-Tube Manometer
CEEN 598D– Fall 2015
U-Tube Manometer with Two Fluids
1
2
3 4 h
γm
γ
5
6
Go round the horn in your head. • From 1 to 2, as much pressure
is added as is lost eventually going from 5 to 6 (those segments cancel out).
• Pressures at 3 and 4 are equal.
• From 2 to 3 add γmh and from 4 to 5 we lose γh.
So:
)(16
16
γγ
γγ
−=−
−+=
m
m
hpphhpp
CEEN 598D– Fall 2015
Uniform Pressure Distribution
• Uniform pressure distribution – For a panel or a plane or flat surface of arbitrary
shape the description of the pressure along all points along the surface is called a pressure distribution.
– If this pressure is the same at every point, it is called a uniform pressure distribution.
CEEN 598D– Fall 2015
CEEN 598D– Fall 2015
AF pdA pA= =∫
CEEN 598D– Fall 2015
Hydrostatic Pressure Distribution
When the pressure distribution is produced by a fluid in hydrostatic equilibrium the pressure distribution is called a hydrostatic pressure distribution.
CEEN 598D– Fall 2015
CEEN 598D– Fall 2015
Another Representation
( sin )
sin
sin
A
A
A
c
c
F pdA
y dA
ydA
y A
p A
γ θ
γ θ
γ θ
=
=
=
=
=
∫
∫
∫
pressure at the centroid
To find the total force on the plane, integrate pdA
Integral of distance times area is the first moment of the area.
CEEN 598D– Fall 2015
Or, another way
where h=y sinθ for constant γ and θ. The integral is the first moment of the area with respect to the x-axis and can be defined as:
where yc is the y-coordinate of the centroid of the area A.
sin sinA A A A
F pdA hdA y dA ydAγ γ θ γ θ= = = =∫ ∫ ∫ ∫
€
ydAA∫ = ycdA
CEEN 598D– Fall 2015
where hc is the vertical distance from the fluid surface to the centroid of the area.
sin c cF y A h Aγ θ γ= =
Or, another way
CEEN 598D– Fall 2015
Example Concrete (23.6 kN/m3) poured into a vertical form 2.44m high by 1.22m by x = 0.5 m thick exerts what force?
2
23
1.22sin 90 1
2.44 1.22 3
23.6 1.22 3 85.7
( 19,000 )
cy m
A m m mkNF m m kNmlb
=
° =
= × =
= × × =
≈
If the thickness x doubles, how does the force change? x
2.44
m
CEEN 598D– Fall 2015
Line of Action
• In the last slides, we calculated the point at which the effective pressure is calculated (i.e., the centroid) to find the magnitude of the force on a submerged plane.
• Is this the point at which the force appears to be applied?
CEEN 598D– Fall 2015
Example
yc
x
2.44
m
yR
1.22
m
1.63m
0.81m
For the previous example, the resultant force for a triangular distribution (incompressible fluid) will act h/3 up from the bottom.
CEEN 598D– Fall 2015
• Coordinate of resultant force can be determined by a summation of moments around x-axis.
• Since F=γΑycsinθ we can cancel the RHS
2
xAR
c c
y dA Iyy A y A
= =∫
2sinR A AFy ydF y dAγ θ= =∫ ∫
Moments
Second moment of the area OR Moment of inertia
CEEN 598D– Fall 2015
Parallel Axis Theorem: Used to determine the moment of inertia about any axis given the body’s moment of inertia about a parallel axis through the centroid and the perpendicular distance between the axis. where Ixc is the second moment of the area with respect to an axis passing through the centroid and parallel to the x-axis. Inserting into previous equation:
So, resultant force does not pass through the centroid but always below it since Ixc/ycA > 0.
Parallel Axis Theorem
€
Ix = Ixc + Ayc2
xcR c
c
Iy yy A
= +
CEEN 598D– Fall 2015
Let’s go over one more time!
• The position of the “line of action” of a hydrostatic force lies below the centroid of a plane surface.
• This location is called the center of pressure. Call the distance along the plane surface, yR (R for resultant).
2( sin ) sinRA A A A
y F ydF ypdA y y dA y dAγ θ γ θ= = = =∫ ∫ ∫ ∫
CEEN 598D– Fall 2015
• Insert the parallel axis theorem:
• The above expression is finding the point at which the
total pressure force appears to be applied. • Insert the magnitude of the total force:
2sin ( )R xc cy F I y Aγ θ= +
2
2
sin sin ( )R c xc c
xc c xcR c
c c
xcR c
c
y y A I y AI y A Iy yy A y AIy yy A
γ θ γ θ= +
+= = +
− =
sin cF y Aγ θ=
CEEN 598D– Fall 2015
First, the magnitude of the hydrostatic force is , where pc is the pressure at the centroid, which is at a depth of 10m.
cF p A=
3
6
10 9810 2 2.51.541 10
F m N m m mN
π= × × × ×
= ×
Example What is the force F required to open the gate?
CEEN 598D– Fall 2015
The slant distance is calculated using similar triangles (no need to know α): The sum of torque moments around the hinge is zero:
The slant distance to depth ratio maintains the ratio 4:5.
mymy 5.12
45
10=∴=
mmm
ya
AyIyycp 125.0
)5.12(4)5.2(
4
22
====−
5 4 Fpush
F
€
Fpush × 5m − Fp × (2.5 + 0.125)m = 0
Fpush =1.541×106N 2.625m5m
= 8.09 ×105N
CEEN 598D– Fall 2015
Problem
Given: A 4-m diameter circular gate is located in the inclined wall of a large reservoir containing water (γ=9.8 kN/m3). The gate is mounted on a shaft along its horizontal diameter and the water depth is 10m above the shaft. Find: a) The magnitude and location of the resultant force
exerted on the gate by the water. b) The moment that would have to be applied to the shaft
to open the gate.
CEEN 598D– Fall 2015