1History of the Standard Atmosphere• With a little digging, you can discover that the Standard Atmosphere can betraced back to 1920. The constant lapse rate of 6.5° per km in the tropospherewas suggested by Prof. Toussaint, on the grounds that– … what is needed is … merely a law that can be conveniently applied and which issufficiently in concordance with the means adhered to. By this method, corrections dueto temperature will be as small as possible in calculations of airplane performance, andwill be easy to calculate. …– The deviation is of some slight importance only at altitudes below 1,000 meters, whichaltitudes are of little interest in aerial navigation. The simplicity of the formula largelycompensates this inconvenience.• The above quotation is from the paper by Gregg (1920). The early motivations forthis simplified model were evidently the calibration of aneroid altimeters foraircraft, and the construction of firing tables for long-range artillery, where airresistance is important.• Unfortunately, it is precisely the inaccurate region below 1000 m that is mostimportant for refraction near the horizon. However, the Toussaint lapse rate,which Gregg calls “arbitrary”, is now embodied in so many altimeters that itcannot be altered: all revisions of the Standard Atmosphere have preserved it.• Therefore, the Standard Atmosphere is really inappropriate for astronomicalrefraction calculations. A more realistic model would include the diurnal changesin the boundary layer; but these are still so poorly understood that no satisfactorybasis seems to exist for realistic refraction tables near the horizon.http://mintaka.sdsu.edu/GF/explain/thermal/std_atm.htmlInternational Standard Atmosphere• The ISA model divides the atmosphere into layers with linear temperature distributions.[2] The othervalues are computed from basic physical constants and relationships. Thus the standard consists of a tableof values at various altitudes, plus some formulas by which those values were derived. For example, at sealevel the standard gives a pressure of 1.013 bar and a temperature of 15°C, and an initial lapse rate of -6.5 °C/km. Above 12km the tabulated temperature is essentially constant. The tabulation continues to18km where the pressure has fallen to 0.075 bar and the temperature to -56.5 °C.[3][4] * U.S. Extension to the ICAO Standard Atmosphere, U.S. Government Printing Office, Washington, D.C., 1958. * U.S. Standard Atmosphere, 1962, U.S. Government Printing Office, Washington, D.C., 1962. * U.S. Standard Atmosphere Supplements, 1966, U.S. Government Printing Office, Washington, D.C., 1966. * U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.Geopotential HeightICAO Standard AtmosphereLecture Ch. 2a• Energy and its properties– State functions or exact differentials– Internal energy vs. enthalpy• First law of thermodynamics• Heat/work cycles– Energy vs. heat/work?– Adiabatic processes– Reversible “P-V” work• Homework problem Ch. 2, Prob. 2Curry and Webster, Ch. 2 pp. 35-47Van Ness, Ch. 2Internal Energy vs. Enthalpy• Difference b/w U and H– U depends on v– H depends on p• Specific heats [a.k.a. heat capacity]– cv is constant v– cp is constant p2Heat CapacityFor an ideal gas• Simplify to• [Types of processes]– Constant pressure– Constant volumeLord Kelvin(a.k.a William Thomson)James P. Joule• The First Law of Thermodynamics• ConsequencesΔE = mc2Q = 0,ΔE = 0 ⇒ W = 0(Relativity)Impossibility of perpetual motion machineConservation of energyDefinition of energyUniqueness of work valuesQ = 0,W = 0 ⇒ Δ E = 0 ⇒ E2= E1Q = 0 ⇒ ΔE = WWrev= − pdv∫ReversibleAdiabaticState functionSee also 2nd law!Proof follows..Other Kinds of Energy• In addition to changes in internal energy, asystem may change– Potential energy for height change Δz– Kinetic energy for velocity change Δv– Nuclear energy for mass change ΔmVan Ness, p. 13€ ΔE = ΔU p,V,T( )+ mgΔz +12mΔv2− c2Δm = Q + Wif ΔE ≈ ΔU p,V,T( ), then ΔU p,V ,T( )= Q + WWork• Expansion work W=-pdV or w=-pdv– Lifting/rising– Mixing– Convergence• Other kinds of work?– Electrochemical (e.g. batteries)3• Work and heat are path-dependenttransfers– W work– Q heat• State functions are unique “states”– U internal energy– H enthalpy– η (also S) entropy– A Helmholtz free energyCyclesExact Differentials• State functions are exact differentialsHeat/Work Cycles• The efficiency with which work is accomplished in a reversible cyclic processdepends only on the temperature of the reservoirs to which heat istransferredQ1Q2WFLUIDT1T2STEP 1: Expand isothermally and reversibly at T1STEP 2: Expand adiabatically and reversiblySTEP 3: Compress isothermally and reversibly at T2STEP 4: Compress adiabatically and reversiblyW1= Q1= RT1lnPAPBW2= Q2= RT2lnPCPDW = CvT2− T1( )W = CvT1− T2( )THE CARNOT CYCLECarnot was an engineer inNapoleon’s defeated armywith an interest in engines.Efficiency:HotColdTT−= 1η4P-V diagrams of work• Work is determined by pathwayOther Work CyclesNikolausOttodevelopedthe Ottocycle in1876.RudolfDieseldevelopedthe Dieselcycle in1892.The Diesel Cycle worksby compressing air andthen adding fuel directlyto the piston. Thecompressed air thencombusts the mixture.The Otto Cycle worksby compressing amixture of air and fuelin a piston and thenigniting the mixturewith a spark.The compression ratio of the Diesel Cycle rangesfrom 14:1 to 25:1, while the Otto Cycle range issignificantly lower, from 8:1 to 12:1.Efficiency: Efficiency:DACBTTTT−−−= 1η−−−=DACBTTTTγη1135=γfor monatomic ideal gas4 Steps of Carnot “Engine”1:Add Heat(isothermally)2:Adiabatic3:Lose Heat(isothermally)4:AdiabaticHurricane as Carnot Cycle1:Add Heat(isothermally)2:Adiabatic3:Lose Heat(isothermally)4:AdiabaticIdeal Gases€ cp≡∂h∂T p=dhdT=∂h∂T vReversible-Adiabatic-WorkReversible-Adiabatic-WorkAdiabaticFirst LawReversibleInternal EnergyIdeal Gasp1v1T1= R =p2v2T2Δu = cvdTW = − pdvΔu = Q + WQ = 0thick wallsLow P, Low TFrictionlessReversible, Adiabaticmass is conserved€ T2T1=P2P1 Rcp5Reversible ProcessesReversible ProcessesReversible€ Wrev= − pdvFrictionlessmass is conserved• Always at or infinitesimally close to equilibrium• Infinitesimally small steps• Infinite number of steps•
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