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Lab 3 2004 Coriolis – II, Equation of State, Phase Change, Latent Heat, Thermal Energy Equation, Pδv Work

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Lab 3 2004 Coriolis – II, Equation of State, Phase Change, Latent Heat, Thermal Energy Equation, Pδv Work 1 Geophysical Fluid Dynamics I 22 Jan 2004 Lab 3 2004 Coriolis – II, equation of state, phase change, latent heat, thermal energy equation, pδv work. 1. Coriolis: We want to see geostrophic balance. Use the free-surface ‘pressure gauge’ to see the pressure variation in eddies on the big rotating table. Using reflected light, viewed with the video camera overhead, quite small deflections of the free surface can be seen. The appearance is not unlike the TOPEX/Poseidon satellite altimetry images of the world ocean. 2. Equation of state: water The curved function ρ(T,S,p), density as a function of temperature, salinity and pressure can be explored. In Lab I one of the experiments measured the buoyancy force on a glass bulb immersed in water: ice was added to the water and the bulb then weighed less! Today we take an Erlenmayer flask (sloping sides) with a ‘chimney’, that is, a glass tube sticking out the top. The system is filled with water, initially in two layers of differing temperature. The height of the water in the chimney is marked, and then the two layers are mixed together with a magnetic stir-bar. The height of the fluid in the chimney would stay the same if the equation of state were linear, but (see plots in previous handout) it is not. The resulting change is a very sensitive measure of the curvature of the equation of state. The effect of the nonlinear equation of state in Nature can be quite important. Now the experiment is repeated for salt stratification, a fresh layer floats on a salty layer, and the change in volume of the entire fluid is observed when the two are mixed…same idea. So the equation of state is curved (nonlinear) in both directions on the ρ(T,S) plane. There are also pressure nonlinearities. A fascinating form of convection occurs in the ocean: if two water parcels with the same density, yet different T and S, are mixed, the mixture will be more dense than either parcel was initially, so this can produce vertical convection. 3. Equation of state: gas The perfect gas equation of state is p = ρRT where R = 287.04 Joule kg-1 0C-1 for dry air. For water vapor, R = 461.5 Joule kg-1 0C-1. The state of an ideal gas is determined by two state variables, and if we introduce entropy, η, the equation of state becomes γ p =ηρ exp( /Cv ) (1) where γ is the ratio of specific heats, γ = Cp/Cv = 1.4 for a diatomic gas (also, Cp-Cv=R). For a gentle ‘reversible’ process, entropy is defined by δ 'qT=ηδ So equation (1) is very useful, because often we have no heat-source or heat-sink in a moving fluid, so that η is constant following a fluid parcel. In this case, p=× const.ρ γ Compare this with an isothermal (T=const) process where p = const x ρ. Adiabatic heating/cooling occurs when we squeeze/expand a gas in an insulated container, and is a dominant atmospheric process. We explore this with a glass vessel with a rubber bulb that can be squeezed, a pressure gauge and a crude thermometer. 2 4. Phase change, latent heat: In the same apparatus we can witness phase change due to adiabatic cooling: formation of a cloud. Do this by inserting a drop of water in the glass vessel, sealing it up, and changing the pressure. Initially you will not see any result. This is because condensation requires a nucleus on which to form a water droplet. So, if you add some nuclei, the result should be visible. Note how rapid the phase change occurs. Sometimes you will see a cloud form above the wing of an airplane as you are landing in moist air: this is because of the rapid decrease in pressure in the air flowing over the wing (which lifts the plane). The phase change is nearly instantaneous. The heat capacity of a fluid is the heating in Joules required to raise 1 kg. of fluid 10 C. For water at 280 C this is 4000 J kg-1 K-1 (see Gill, Appendix 1). For dry air it is -1 -1 cp = 7/2 R = 1004.6 J kg K ; 3 the number 7/2 is for diatomic molecules. For a polyatomic molecule, with more than 3 atoms, the number is 4. This leads to a specific heat coefficient for moist air of cp = 7/2 R(1 - q +8q/7ε) (Gill, 1982, p43.). These are measured at constant pressure; at constant volume, the specific heat cv = cp - R. The specific heat capacity (at constant pressure) of fluids and solids normally exceed that for gases; for air cp is about 1/4 that of water, With its density some 800 times less, the heat capacity of the entire column of atmosphere is equivalent to that of just 5m of ocean beneath. For this reason the ocean is an active heat source and sink for the atmosphere, and generally more exchange occurs as latent heat of evaporation than as ‘sensible’ conducted heat It is difficult to have an intuitive quantitative sense of heating. Perhaps the best visualization is the heating of a tea-kettle, where a 500W stove-top heating unit (500 J sec-1) takes about 300,000 J. to boil 1 kg. of water initially at 250C, and 10 minutes to do it. As a liquid is heated, it develops an increasing vapor pressure. At at a given temperature and pressure, a vessel of liquid has an equilibrium vapor pressure which describes the escape of water molecules from the surface. At the boiling point, the vapor pressure has risen to equal the atmospheric pressure. Raising the water to the higher energy level requires a great deal of heat: the latent heat of evaporation is 6 3 -1 Lv = 2.50 x 10 -2.3 x 10 T J kg , where T is the Celsius temperature. For this reason, it takes longer to boil a tea kettle dry, than the time initially to bring it to a boil. Energy units are all round us; caloric value of foods, horsepower of automobiles, heating of a liquid, radiation of an electric heater. It takes about 1.5 horsepower to take a hot shower, and a small (250 food ‘calories’) candy bar eaten in 1 minute represents about a million J. of energy/60 seconds, or 22.3 horsepower! (our conversion of food into mechanical work is far from perfectly efficient). (1 hp. = 745.7watts = 745.7J sec -1). Note that food ‘calories’ are actually kilocalories; 1 thermodynamic calorie = 4.184 J, yet 1 food ‘calorie’ is 4186.8 J, which is strange. How much power output can the human body produce for a short time? Running upstairs, you can readily calculate your change in gravitational potential energy, and it’s difficult to do much better than 1 horsepower. The ratio of heat flux carried by water vapor and that carried directly as ‘sensible’ heat (ρcpT) is a crucial part of atmospheric dynamics. Generally in the tropics the latent heat flux upward from the heated sea surface greatly exceeds the sensible: if it were not so, clouds would contain be the dominant vertical motion. The delayed dynamical impact of the latent heating (which appears when the water vapor condenses) makes a peculiar forced fluid dynamics problem (see, e.g., the book Atmospheric Convection by Kerry Emmanuel). At higher latitude, for example in the Labrador Sea, cold, dry winds from the continents flow over the ocean and are warmed; the sensible and latent heat fluxes there are more comparable. Visit the Clausius-Clapyron equation in Gill to learn more. The saturation vapor pressure is the contribution of gaseous water to the pressure, and it is a function of temperature of the water. This is famous Clausius-Clapyron equation. We can measure the s.v.p. using the two-glass-bulb apparatus in the lab, which has almost no air in it. Swirl the water around so that the 4 inside of one bulb is wet. Warm that bulb in your hand with the glass tube downward. Watch the water column rise, and measure the hydrostatic pressure difference. This should give a point on the C.C. curve below! The cooling by evaporation is used in the sling-psychrometer to measure humidity of the air: two thermometers, one with a moistened fabric surrounding the bulb, are whirled in the air. The ordinary thermometer comes to equilibrium with the atmosphere (for small Mach number!), this is the ‘dry bulb’ temperature. The other thermometer cools due to evaporation. Due to a complex process of evaporative cooling yet conductive warming, this thermometer reaches a stable ‘wet bulb’ temperature. The difference indicates how much water the air can hold…the humidity. 5. Thermal energy equation: The first law of thermodynamics for a parcel of fluid is δ E =−δδ'qpv where δE is a small change in thermal internal energy, δ’q is a heating or cooling, p is pressure and δv is a small volume change. This internal energy equation combines with the ‘mechanical energy’ equation derived from the momentum equation, to describe the total energy balance of a fluid. pδv is the work done by the fluid on its surroundings. For a perfect gas the internal thermal energy is simply E = Cv T where T is temperature and Cv is specific heat capacity at constant volume. This makes sense because if we heat a gas while keeping its volume constant (δv = 0) then δE = δ’q = CvδT which is really a definition of Cv .
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