AE 1350 Lecture 6-BSome definitions..First Law of ThermodynamicsFirst law continued..Adiabatic ProcessIntegration of First Law for Adiabatic Inviscid FlowsFirst Law, continued.Reversible FlowIsentropic FlowIsentropic Flow, continued..Slide 11SummaryFurther SimplificationsAE 1350Lecture 6-BCompressible FlowSome definitions..•Specific internal energy: Energy stored in random (linear, rotational) motion of molecules, per unit mass. For diatomic molecules, e = CvT = 5/2 RT, where Cv = 5/2 R•Specific enthalpy: e+p/ = e + RT = 7/2RT=CpT, where Cp = 7/2 R•Ratio of specific heats:  = Cp/Cv = 7/5 = 1.4 for air•System: A collection of particles of fixed identity.•Properties of a system: Quantifiable information such as p, , T, Velocity Vector V, etc.•Process: An event that causes changes to the properties of the system – e.g. flow of the particles over an airfoil will cause changes in velocity, pressure, density, and T.First Law of Thermodynamics•Change in the specific internal energy of a system is due to heat added to the system, and work done on the system.•This law can not be proved, but can be verified from observations.wqdeHeat added perUnit massWork done on theSystem per unit mass due to body forcesSuch as gravity, andPressure forcesFirst law continued..dpqdpp ddepeddhpdpd111)(1q de :becomes thuslawFirst volume.specific is where1d p- Mass / dV -pmassunit per done work Thusforces.magnetic and electrical gravity, assuch forcesbody neglect WeAdiabatic Process•An adiabatic process is one in which there is no heat addition or removal.•Examples of adiabatic flow are: Flow over a wing, outside the boundary layer, flow through a propeller or a turbine, etc.•In these cases, work is done by the pressure forces, but no heat is added.•Example of a non-adiabatic flow: Flow through a combustor or furnace, flow within the boundary layer where the wall is cooler or hotter than the fluid particles. •First law becomes: flows adiabaticin 1dpdhIntegration of First Law forAdiabatic Inviscid Flows•Recall Euler’s equation for conservation of momentum in a stream tube:enthalpy. stagnation (specific) called is h wherehConstant2V h :Integrate0dhVdV :get We1dh :slide previous thefrom flows, adiabaticfor lawfirst in the thisUse01002dpdpVdVFirst Law, continued.02vv002p022:get Wep/ e h Usere. temperatustagnation called is T where2COr,2TCVpTCpTCTCVThVhppInternal Energy “Pressure work”Kinetic EnergyGeneralization of Bernoulli’sEquation for compressible flowsReversible Flow•A reversible flow is one in which the system (i.e. collection of fluid particles moving over an airfoil or within a combustor, or through a turbine, or whatever) and the environment (i.e. surrounding particles), both may be restored to their original condition.•Example: Slow compression of air in a balloon does work on the air inside the balloon, and takes away energy from the surroundings. When the balloon is allowed to expand, the air inside and the surrounding air are both restored to original conditions.•Example of an irreversible Process: Heat flows from hot to cold, never in the opposite direction. Most conductive and viscous processes (i.e. flow inside the boundary layer) are irreversible.•A second example of an irreversible process: Flow across a shock wave where the Mach number abruptly decreases.Isentropic Flow•A reversible adiabatic flow is called an isentropic flow.•In such a flow,  1VVVlog1loglogC :Integrate1Cget toRT p , TC e :Use1dT1becomes 1pd-qde lawfirst The00CRRTRdTdTTpdeTqqIsentropic Flow, continued.. 111VVp1VT C :log-Anti Take)C1log(logCloglog(T)1-1:above (1)in equation thisUse(2) 1 Or,C :get WeCCp Also,2/52/7C :Use(1) CloglogCslide, previous theFromγ-VVVRCRCRCRRRRTIsentropic Flowequation.energy thefrom found is T once , and p find torelations theseuse willWe well.as Constant,p thatshowcan wep, of in terms T replace toRT p Law, Gas Ideal UsingConstant11TSummary•We will deal with inviscid, reversible, adiabatic flows.•For such flows, we get:ConstantConstant2211022TpTCVTCVhppFurther Simplificationslly.adiabatica and reversiblyrest brought to are they if have, willparticles theproerties theare Thesely.respective pressure, stagnationand density, stagnation pressure, stagnation called are T , ,p quantities The21-121-1:relations isentropic of aid With the21-1 ,21-1 :get to1)-RT/(by through Divide121get to1C