Transcript Document

Liquids
o Static Liquids
 Some Basic Properties
 Pressure and pistons
 Viscosity, Density
 Evaporation
o Archimedes Principles
 Submerged Objects
 Floating Objects
Lecture 10
o Aim of the lecture
 Concepts in Basic Liquid Properties
 Incompressibility
 Pressure
 Viscosity
 Force Amplification
 Use of hydraulics
 Archimedes Principle
 Origin of upthrust
 Floating
o Main learning outcomes
 familiarity with
 buoyancy
 basic liquid properties
 Pressure gradient in liquids
Liquids
Liquids have many properties and varied behaviours
Here we consider only some of the
basic properties of liquids
Pressure
Temperature
Volume
Density
Pressure
Pressure, P is a force per unit area
P = F/A
The SI unit is Pascals, Pa,
1 Pa = 1N/m2 (a very small pressure)
P2
A cylindrical piston has a fluid on one side at pressure P2
The force F2 on that side is
F2 = P2pd22/4
Fnet
The piston also has a pressure on the other side of P1
(often will be air pressure) The force F1 on that side is
F1 = P1(pd22 - pd12)/4
The net force provided to the shaft is
Fnet = P2pd22/4 - P1(pd22 + pd12)/4
The Pascal is such a small unit of pressure that others are used:
bar
lbs/in2 = ‘pounds per square inch’
atm = ‘atmosphere’
mmHg = ‘millimetres of Mercury’
mmH2O = ‘millimetres of water’
1 bar = 100,000 Pa
1 lb/in2 = 6,894.7573 Pa
1 atm = 101,325 Pa
1mmHg = 133.3224 Pa
1mmH2O = 9.80638 Pa
o 1atm is the ‘standard’ pressure of the earth’s atmosphere at sea level.
o mm of Hg is a standard unit used in meteorology
A piston which uses a high pressure liquid
to produce a force is called a
Hydraulic piston
Hydrolytic pistons can use liquids at
Hundreds of bar – very high pressure
Compressibility – fixed volume – variable shape
o A liquid is an incompressible fluid
 [to a very good approximation]
o So its volume cannot change
 [which is one reason they are good for hydraulics]
A2
A1
NOT TRUE
FOR GASSES
•If the small pipe is area A1
•The large pipe area A2
•Then moving a distance d1 in the small pipe
•Will move the liquid a distance
d2 = A1d1/A2
This can be used like a lever – force amplification by a factor A2/A1
Evaporation
o All fluids will vaporise
Become the gaseous form of the fluid
How much depends on
Pressure
Temperature
oA dynamic equilibrium is formed, with
A fixed fraction of the total in gas
A fixed fraction of the total in liquid
o The amount of gas is described by the ‘partial pressure’
A fluid with a ‘low partial pressure’ is one which will stay as a liquid
•Oil has a very low partial pressure – it doesn’t evaporate at STP.
•Water has a medium vapour pressure – it will evaporate on a warm day
•Ether has a high vapour pressure – it will evaporate quickly
•Any liquid can only remain liquid provided that the
temperature and pressure are in the ‘allowed regime’
•Outside of that, no liquid can exist and the substance
will be all gas,
or all solid
For water, liquid can only
exist in the shown region
of pressure versus temperature
(and on the boundary)
At one place all three
states can exist at the
same time, called the
‘triple point’
(water, vapour, ice)
Viscosity
Liquids can be
thick and sticky
High Viscosity
thin and runny
Low Viscosity
Viscosity is measured in Pascal seconds and usually given the symbol m
Viscosity changes with temperature.
In this course we will not talk about viscosity in more detail
Density
o A liquid has a density, which is essentially fixed
o The symbol for density is r
 r = mass/volume in kg/m3
Mercury has a density of 13.5 tonnes per m3
Water has a density of 1 tonne per m3
Pressure and Depth
Consider a stack of lead plates
Place a finger between top two plates
no problem
With a liquid the pressure depends on depth
Because
of the
the bottom
WEIGHT of the liquid above it
Put a finger
between
thewill
weight
of lead sheets.
two platesJust
andlike
there
be pain
How does pressure change with depth?
Consider a cylindrical water tank
with radius r
Water, density r
r
Depth, d
This water has a mass of
m = pr2dr
With weight mg = pr2dgr
So this water must provide
an upwards force, F, to support the weight
F = pr2dgr
But pressure, P, is force/area ,
so P = F/pr2 = pr2dgr/pr2 = dgr
Pressure = d(gr)
Pressure = d(gr)
(note that r depends on temperature)
Pressure proportional to depth
gr = 9.81  9.96  103 N/m2/m
Pressure = 9770 Pa/m
The pressure increases by
~ 0.1atm per metre depth
Density
Kg/m3
Water Temperature
The pressure of the atmosphere needs to be added on, so that
at just 10m down a mermaid will experience twice the pressure of air.
Archimedes Principle
o The change in pressure with depth results in a buoyancy force
 Towards the surface
 This is the origin of Archimedes Principle
 the pressure on the bottom of an object is greater
 than the pressure on the top
 therefore there is a difference in force up and down
o Any submerged object experiences an upwards force which
is equal to the weight of the liquid it displaces.
o Any floating object displaces its own weight in the liquid it
is floating in.
Consider a cylinder submerged in a liquid
Because of symmetry, the pressure
Conversely,
thesurface
pressure
oncancel
the topout
on
the vertical
will
~
~
~
~
~
~
~
~
~
~
~
~
is lower than the pressure on the bottom
because
bottom
is
deeper
There the
is pressure
from
the water
The pressure
arrows
are in
pairs,
onaall
surfaces
each has
‘partner’
with the opposite direction
Liquid surface
Net upwards force, F is given by
P1 = rg d1
F = A(P2-P1)
where A is the area of the top/bottom
So F
= A(rgd2-rgd1)
= Arg(d2-d1)
= g rA(d2-d1)
= g rV
where V is the volume
=gm
and m the mass of the
liquid displaced
by the cylinder
So the upwards force is simply the WEIGHT
of the liquid being displaced
d1
d2
P2 = rg d2
o Calculation done for ‘easy’ shape
o Same result holds for any shape
 Mathematics to prove this involves
 surface integrals in 3-D
 beyond the scope of this course.
 (also its easier to prove using energy considerations for the general case)
Shape does
not affect the
result, only the
volume matters
Because that is
what determines
the weight of the
displaced liquid
o Upwards Force = weight of liquid displaced
o If weight of object < upwards force, then
 Net force upwards
 Object will float
Most wood has a density
 So objects less dense than less
water
willwater,
float r=996
than
 Also for balloons in air
 The amount of water displaced will be
 just enough to produce a force = to weight
 which is just the weight of the floating object
Wood floats in water
o The average density of a boat is also less than that of water
o The materials used to make the hull might be denser than water,
o But the average, including the space inside the hull is lower
Fills
sea with
bubbles,
Whatthe
happens
in the
Bermuda Triangle
which reduces the density of the ‘water’
Thetheory:
ship can become denser than the ‘water’ it is floating in
One
An underwater volcano