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CIVE1400: An Introduction to Fluid Mechanics
Unit 2: Statics
Dr P A Sleigh: [email protected] Dr CJ Noakes: [email protected]
January 2008
Module web site: www.efm.leeds.ac.uk/CIVE/FluidsLevel
Unit 1: Fluid Mechanics Basics 3 lectures Flow Pressure Properties of Fluids Fluids vs. Solids Viscosity Unit 2: Statics 3 lectures Hydrostatic pressure Manometry / Pressure measurement Hydrostatic forces on submerged surfaces Unit 3: Dynamics 7 lectures The continuity equation. The Bernoulli Equation. Application of Bernoulli equation. The momentum equation. Application of momentum equation. Unit 4: Effect of the boundary on flow 4 lectures Laminar and turbulent flow Boundary layer theory An Intro to Dimensional analysis Similarity
Statics : Pressure
Hydrostatic Pressure:
linear change in pressure with depth
Measure depth, h, from free surface
Absolute pressure
p absolute = ρ g h + p atmospheric
Gauge pressure
p (^) gauge = ρ g h
p 2 − p 1 = −ρ g ( z 2 − z 1 )
p = ρ gh + p atmospheri c
Examples of pressure head calculations:
What is a pressure of 500 kNm-2^ in
head of water of density, ρ = 1000 kgm-^3
Use p = ρ gh ,
h =
In head of Mercury density ρ = 13.6× 103 kgm-^.
h =
In head of a fluid with relative density γ = 8.7.
(remember ρ = γ × ρwater)
h =
Manometers for Measuring Pressure
Manometers use the relationship between _____________and _______ to measure pressure
Piezometer Tube
A simple open tube attached to the top of a container with liquid at pressure.
The pressure measured is relative to
__________ so it measures _________ p ressure.
Problems with the Piezometer:
pA = ρgh
Liquid rises to
a height, h,
equal to the
pressure in
the container.
Equality of Pressure At
The Same Level In A Static Fluid
Fluid density ρ
pl , A
Area A
weight, mg
Face L (^) Face R
p (^) r, A
Horizontal cylindrical element
cross sectional area = A
mass density = ρ
left end pressure = p l
right end pressure = p r
For equilibrium the sum of the
forces in the x direction is zero.
=
=
Pressure in the horizontal direction is ________
This true for any fluid.
Consider these two tanks, one much larger than the other, and linked together by a thin tube:
P Q
L R
z z
We have shown
pl = pr
For a vertical pressure change we have
pl =
and
pr =
so
p (^) p = pq
Pressure at the two equal levels are _______.
“U”-Tube enables the pressure of both liquids and gases to be measured
Important points:
- “U”-Tube enables the pressure of both liquids and gases to be measured
- The manometric fluid density should be ________ ____ ____ _______ measured.
ρ man > ρ
- The two fluids should __________ _____ ___ they must be immiscible.
What if the fluid is a gas?
_________ changes.
The manometric fluid is liquid (usually mercury, oil or water)
And Liquid density is much greater than gas,
ρ man >> ρ
ρ gh 2 is negligible and pressure is given by
p A =
An example of the U-Tube manometer.
Using a u-tube manometer to measure gauge
pressure of fluid density ρ = 700 kg/m^3 , and the manometric fluid is mercury, with a relative density of 13.6. What is the gauge pressure if: a) h 1 = 0.4m and h 2 = 0.9m? b) h 1 = 0.4 and h 2 = -0.1m?
pressure at C =
pC =
pC =
pD =
=
Giving the pressure difference
pA - pB =
Again if the fluid is a gas ρ man >> ρ, then the terms
involving ρ can be neglected,
pA - pB =
An example using the u-tube for pressure
difference measuring
In the figure below two pipes containing the same fluid of density ρ = 990 kg/m^3 are connected using a u-tube manometer. a) What is the pressure between the two pipes if the manometer contains fluid of relative density 13.6?
ha = 1.5m
A
B
h = 0.5m hb = 0.75m
C D
E
Fluid density ρ
Manometric fluid density ρ man = 13.6 ρ
Fluid density ρ
volume of liquid moved from
the left side to the right
The fall in level of the left side is
2 2
2
2 2
1
/ 4
/ 4
⎟ ⎠
⎞ ⎜ ⎝
⎛
=
=
D
d z
D
z d
z
π
π
Putting this in the equation,
p p g z z
d D
gz
d D
1 2 2 2
2
2
2 1
− = +
⎛ ⎝
⎜
⎞ ⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
= +
⎛ ⎝
⎜
⎞ ⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
If D >> d then ( d/D)
2
is very small so
p 1 (^) − p 2 =
Inclined manometer Problem: Small changes difficult to see Incline the arm: same height change but bigger movement.
Datum line z 1
p 1 p^2
z 2
diameter D
diameter d
Scale Reader
x
θ
The pressure difference is still given by the height
change of the manometric fluid.
p 1 (^) − p 2 (^) = ρ gz 2
but,
z
p p
2
1 2
=
− =
The sensitivity to pressure change can be _________ further by a __________ inclination.
Choice Of Manometer
Take care when fixing the manometer to vessel Burrs cause local pressure variations.
Disadvantages:
- ______ response - only really useful for very slowly varying pressures - no use at all for fluctuating pressures;
- For the “U” tube manometer ____ measurements must be taken _______________ to get the h value.
- It is often difficult to measure ______ variations in pressure.
- It cannot be used for ____ _____ pressures unless several manometers are connected in series;
- For very accurate work the _____________ and relationship between temperature and ρ must be known;
Advantages of manometers:
- They are very __________.
- No ______________ is required - the pressure can be calculated from first principles.
Lecture 5: Forces in Static Fluids Unit 2: Statics
From earlier we know that:
1. A static fluid can have ______________ acting on it. 2. Any force between the fluid and the boundary must _________________________________________.
F
R
Fn
F 2
F 1
R 1
R 2
Rn Pressure force normal to the boundary. True also for
- curved surfaces
- any imaginary plane in the fluid
An element of fluid at rest is in equilibrium:
3. The sum of forces in any direction is _____
4. The sum of the moments of forces
about any point is ______
