ATMO 551b Spring Barometry

Transcription

1 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/s 2 /m 2 = Pa (Pascal) In cgs units this is g cm/s 2 /cm 2 = 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 = 10 3 mb = 10 5 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 (0 o C) = 760 Torr 1 mmhg = mb psi = pound per square inch 1 bar = 14.5 psi = 29.5 inches of mercury [0 C] Direct measurements of pressure: 1 Kursinski 03/3/10

2 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 ρ m A h l + P 1 A where g is the acceleration of gravity, ρ m 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 P 1 is the (reference) pressure on the top left surface of the liquid. This is equal and opposite to the force on right side: g ρ m A h r + P 2 A with analogous definitions. So we have g" m Ah l + P 1 A = g" m Ah r + P 2 A g" m h l + P 1 = g" m h r + P 2 g" m ( h l # h r ) = g" m H = P 2 # P 1 g" m ( h l # h r ) = g" m h = P 2 # P 1 h = P 2 " P 1 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. 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 P 1 to be zero, it actually becomes the saturation vapor pressure of the mercury which is a function of 2 Kursinski 03/3/10

3 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 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, P 1, 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 = mb. At 100 C, mercury s saturation vapor pressure 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 g/cm 3. At 100 C, mercury s density is g/cm 3. 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 = mb. At 100 C, mercury s saturation vapor pressure 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 3 Kursinski 03/3/10

4 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 16Et 4 p = 3R 4 1"# 2 ( ) * y t $ y 3 ' -, & ) / + % t (. where p is pressure in Pa, E is the modulus of elasticity in N/m 2, 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 4 Kursinski 03/3/10

5 The static sensitivity is 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. ( ) y r = y t = 4.5x105 R 1"# 2 te $ 1000 t ' & ) % D( This is the straight line plotted in figure 2-6 above. The static sensitivity is hpa -1. "1.52 p 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 5 Kursinski 03/3/10

6 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. dp s P s = ml R * 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. dt T 2 6 Kursinski 03/3/10

7 dt = R* T 2 ml 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. dp s P s 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 ( ) 2 + " L 2 " $" 0 where u L is the Lorentz linewidth and u 0 is the line center. It is related to the mean time between collisions, t c, by " L = 1 2#t c While u L depends on the particular molecules involved in the collisions, a good rule of thumb for estimating u L 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, L c, 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, A c, sweeping through the atmosphere as it moves. The average distance it will move before colliding with another molecule defines a volume, L c A c. For L c 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, n g. So L c A c = 1/n g and L c = 1/(n g A c ) = k B T/(P A c ) where k B is Boltzmann s constant gas constant, T is the gas temperature and P is the gas pressure. The thermal velocity, v T, capturing the motion between the two molecules is v T = where m is the mass per molecule (not per mole). So the average time between collisions is The collisional linewidth is then t c = L c v T = k B T PA c k B T m m k B T = 1 PA c 1 2"t c = PA c mk B T So the collisional linewidth can be used to determine pressure. mk B T 7 Kursinski 03/3/10

8 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 2 C" V 2 air air 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 Kursinski 03/3/10

9 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. 9 Kursinski 03/3/10