ATMO 551b
Spring 2010
Barometry
In our context, a barometer is an instrument designed to measure the hydrostatic (as opposed to
dynamic) pressure of the atmosphere.
Units of pressure:
Pressure is Force per unit area which is mass * acceleration / area.
In mks units this is kg m/s2 /m2 = Pa (Pascal)
In cgs units this is g cm/s2 /cm2 = dyne
Unfortunately, there are lots of different pressure units:
Some conversions
1 Pa = 10-2 mb
1 mb = 1 hPa (hectoPascals)
1 Atmosphere = 1 bar = 103 mb = 105 Pa.
Inches of mercury, inches of water
1 Torr = 1 mmHg (mm of mercury [0 °C]) at standard gravity
1 bar = 1000 mb = 760 mmHg (0oC) = 760 Torr
1 mmHg = 1.3158 mb
psi = pound per square inch
1 bar = 14.5 psi = 29.5 inches of mercury [0 °C]
Direct measurements of pressure:
http://en.wikipedia.org/wiki/Pressure_measurement#Liquid_column
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Kursinski 03/3/10
ATMO 551b
Spring 2010
General Theory
The basic concept for barometers uses the fact that pressure causes a displacement that
grows until a force balance is achieved. The displacement is then measured and then the known
relationship between displacement and force is applied to determine the pressure.
Mercury barometer
Manometer: a liquid column
hydrostatic instrument in which
the mercury sits in a horseshoe
shaped container.
Force balance: The basic force
balance is a combination of
gravitational and pressure forces
on the two top surfaces.
The gravitational force pulling
down on the fluid being balanced
by the pressure forces pushing on
each of the upper surface of the
liquid.
At the bottom of the liquid,
there is force balance. The force
on the left side is g ρm A hl + P1 A
where g is the acceleration of gravity, ρm is the density of the liquid, A is the crossectional area of
the tubes and h is the length of the left side of the tube and P1 is the (reference) pressure on the
top left surface of the liquid. This is equal and opposite to the force on right side: g ρm A hr + P2
A with analogous definitions. So we have
g" m Ahl + P1 A = g" m Ahr + P2 A
g" m hl + P1 = g" m hr + P2
g" m ( hl # hr ) = g" m H = P2 # P1
!
!
g" m ( hl # hr ) = g" m h = P2 # P1
P2 " P1
g# m
!
So the difference in the heights of the two columns of fluid is proportional to the difference in
the left and right hand surface pressures.
!
h=
!
Mercury barometer:
If we seal and evacuate one end of the manometer and use mercury as the liquid, we have
a standard mercury barometer or an absolute manometer. Note that while we would like P1 to be
zero, it actually becomes the saturation vapor pressure of the mercury which is a function of
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Kursinski 03/3/10
ATMO 551b
Spring 2010
temperature that should be corrected for depending on how accurately you want to measure
pressure.
Why is mercury used as a barometric fluid?
• Mercury remains a liquid over a wide range of temperatures: Mercury melts at -39°C, and
boils at 356.7°C, so it is useful over a wide range of temperatures.
•
•
•
Mercury has a high density ~ as dense than lead. This decreases H for a given pressure
difference which is good for both compact size and dynamic range.
Low vapor pressure: The pressure, P1, becomes the
saturation vapor pressure of mercury at the
temperature which is small
At 20°C, mercury’s saturation vapor pressure is
1.2 µmHg = 0.0016 mb.
At 100°C, mercury’s saturation vapor pressure
0.2729 mmHg = 0.36 mb.
Mercury is easily purified and chemically stable
Problems:
• Mercury is poisonous.
• Mercury barometers are difficult to automate
Sources of error
• Dynamic wind pressure (see below)
• Density is a function of temperature so we need to
correct for the thermal expansion coefficient of the
mercury
At 20°C, mercury’s density is 13.54562 g/cm3.
At 100°C, mercury’s density is 13.3522 g/cm3.
• The volume of the container also varies with
temperature which has to be considered as well. This
means again that the thermal expansion coefficient to
be used is the difference between that of mercury and
the material of the container like glass.
• Impurities affect density which is a relatively minor problem for mercury.
• Knowledge of local gravity
o Earth is not a sphere of constant density
o Corrections need to be made for latitude, altitude in particular (see gravity handout)
• Barometer must be kept vertical
• (Mercury) gas in the tube
At 20°C, mercury’s saturation vapor pressure is 1.2 µmHg = 0.0016 mb.
At 100°C, mercury’s saturation vapor pressure 0.2729 mmHg = 0.36 mb.
• Surface tension:
o Causes top surface of fluid to be curved.
o Curvature depends on the diameter of the tube
Vernier is used for precise measurement: see http://en.wikipedia.org/wiki/Vernier_scale
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Kursinski 03/3/10
ATMO 551b
Spring 2010
Aneroid (nonfluid) barometers
Coil Spring
Pressure compresses spring, coiling the spring, until the force from the compressed spring
matches that of the pressure. A pointer is attached to the spring so that it rotates as the spring
coil rotates and can be used to indicate the pressure
Flat diaphram
Based on deflection of a diaphragm covering an evacuated chamber. Force balance of
the force due to atmospheric pressure against the restoring force of an “elastic” material (e.g.
metal)
Following Brock and Richardson, the calibration equation for panel a is
$ y '316Et 4 * y
p= 4
, + 0.488& ) /
%t( .
3R (1" # 2 ) + t
where p is pressure in Pa, E is the modulus of elasticity in N/m2, y is the deflection at the
diaphram center in m, t is the diaphram thickness in m, R is the radius of the diaphram in m and v
is Poisson’s ratio which
!is related to the ratio of lateral and axial strain which is approximately
1/3 for metals. This response curve is curve a in the figure below.
a
b
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Kursinski 03/3/10
ATMO 551b
Spring 2010
dy r
and is clearly better at low pressures than high pressures.
dp
Using corrugation of the material surface in the aneroid barometer improves the sensitivity of the
barometer at higher pressures, and gives a simpler, more linear response overall.
The static sensitivity is
!
yr =
5
2
"1.52
y 4.5x10 R(1" # ) $
t'
=
1000
p
&
)
%
t
tE
D(
This is the straight line plotted in figure 2-6 above. The static sensitivity is 0.00106 hPa-1.
!
Capacitor
Simple flat surface: has nonlinear relation between pressure and displacement
Corrugated surface: larger displacement and more linear response
Displacement drives a mechanical arm or resistor or capacitor
Problems
• Dynamic pressure
• Temperature
• Hysteresis due to imperfections
• Nonlinearity
• Drift (not absolute)
Advantages of aneroid barometers:
• Very small size
• Readily automated
• Insensitive to orientation, motion, and shock (portable)
• No gravity correction required
• No toxic materials
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Kursinski 03/3/10
ATMO 551b
Spring 2010
Indirect measurements of Pressure
Hypsometer: Boiling point of a liquid
Boiling occurs when saturation vapor pressure equals atmospheric pressure. Therefore
measuring the temperature at which a liquid boils is an indirect measure of the air pressure.
dPs mL dT
= * 2
Ps
R T
Consider using water vapor to measure atmospheric pressure. How much would the boiling
temperature of water change when the surface pressure changes by 1 mb.
!
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Kursinski 03/3/10
ATMO 551b
Spring 2010
dT =
R*T 2 dPs
mL Ps
Plugging in representative values of T=300K, the resulting temperature sensitivity, dT, which
is required to measure a 1 mb change in surface pressure is 0.017K. This is a very tight
specification which means gross
!changes in pressure can be measured by measuring boiling
temperature but subtle pressure changes are best measured via other methods.
Pressure broadened Linewidth (Collisional or Lorentz lineshape)
Collisions between molecules cause the absorption line to broaden as well. Essentially the
line width is the inverse of the mean time between collisions. The line shape is given as
f (" ) =
1
"L
# (" $ " 0 ) 2 + " L2
where uL is the Lorentz linewidth and u0 is the line center. It is related to the mean time between
collisions, tc, by
!
1
"L =
2#t c
While uL depends on the particular molecules involved in the collisions, a good rule of thumb for
estimating uL is to use 3 MHz/mb times the atmospheric pressure in mb.
To understand how the collisional
! broadening works, we can estimate the average time
between collisions as the mean free path, Lc, between collisions of the molecules in the
atmosphere divided by the thermal velocity of the molecules. Think of a molecule as having a
collisional crossectional area, Ac, sweeping through the atmosphere as it moves. The average
distance it will move before colliding with another molecule defines a volume, Lc Ac. For Lc to
be the mean distance between collisions, this volume must be the average volume per molecule
in the gas, that is, the inverse of the number density of the gas, ng. So Lc Ac = 1/ng and
Lc = 1/(ng Ac) = kBT/(P Ac)
where kB is Boltzmann’s constant gas constant, T is the gas temperature and P is the gas pressure.
The thermal velocity, vT, capturing the motion between the two molecules is
kB T
m
vT =
where m is the mass per molecule (not per mole). So the average time between collisions is
tc !
=
Lc k B T
=
vT PAc
m
1
=
k B T PAc
mk B T
The collisional linewidth is then
1
=
2"t c
!
PAc
mk B T
So the collisional linewidth can be used to determine pressure.
!
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Kursinski 03/3/10
ATMO 551b
Spring 2010
Exposure error:
Barometers are supposed to be measuring the hydrostatic pressure. This means we have
to somehow eliminate the effects of the dynamic pressure which is
1
C" airVair2
2
To get some idea of how large the dynamic pressure can be, consider that for the case of C = 0.2,
and a wind speed of 20 m/s, the dynamic pressure is 40 Pa = 0.4 mb.
Reducing the dynamic pressure
error is accomplished by not allowing the wind into the
!
barometer somehow. A “static port” is designed to do just that. The flat-plate static port works
well but is sensitive depending on the angle of the wind to the flat plate.
tilt
angle
wind
The tilt of the wind direction with respect to the plate changes the constant, C, in the above
equation.
Tilt angle
+10º
0º
-10º
-15º
-30º
C
+0.10
-0.08
-0.20
-0.80
-1.70
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Kursinski 03/3/10
ATMO 551b
Spring 2010
A dual plate static port does better
Pitot tubes
Pitot tubes are used to measure wind speed for instance on aircraft. There are two tubes,
one is directed in the direction of the motion and the other is directed either orthogonal to or
away from the flow. The difference between the pressures measured by the two tubes is the
dynamic pressure. This is proportional to the wind or aircraft speed squared as noted previously.
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Kursinski 03/3/10