GAS DYNAMICSM.S Process & Mechanical Engineering3rd Semester

Compressible Flow with Area Change

Steady 1-D Isentropic Flow With Area ChangeSO FAR WE HAVE DEVELOPEDGeneral equations governing steady 1-D flowThese equations have driving potential terms for,Area changeFrictionHeat TransferBody forcesWe Need To Quantify The Effects Of These Driving PotentialsGenerally all these potentials can be present In most of the practical cases one of them is more important and rest can be neglectedFirst we consider the effects of area change for adiabatic frictionless flowsRevise development of equations, given in Table 3.1 and 3.2

Steady, 1-D Isentropic Flow With Area Change, Governing Eqs.Governing equation for Steady 1-D Flow With Area Change;Assumptions:No body forces, gdz = 0No friction, Ff = 0No heat Transfer, adiabatic flow, Q = 0No drag force, D = 0No work done, W = 0Continuity EquationMomentum EquationEnergy EquationEntropy Equation

Steady, 1-D Isentropic Flow Process With Area ChangeFor An Isentropic Process all the thermodynamic states lie on a vertical line on h-s plotUpper limit of enthalpy is point a, showing zero KE and h=HLower limit of enthalpy is point b, showing h=0 but maximum isentropic speedAll the stagnation properties H, P, T, o and so are constant.All this is due to only one driving potential i.e area changeComparison Of Adiabatic, Frictionless And Adiabatic, Frictional Flow Differential equations are valid for both processesDifference lies in the integration of these equationsFor given set of initial conditions both processes have different final conditions

Comparison, Isentropic & Non-isentropic Processes, contd.Graphic representationExpansionCompression

Comparison contd.Integrate energy eq. for isentropic flow

Integrate energy equation for adiabatic flow with friction

Intuition tells that for frictional flow the maximum velocity thee fluid can attain will be less than that for isentropic flow Consequently one can say that the maximum enthalpy for frictional flow will always be greater than that for the isentropic flow

Note: point 6 and 7 is valid for positive or negative area changes, i.e.Expansion or CompressionComparison, Isentropic & Non-isentropic Processes

Comparison, Isentropic & Non-isentropic Processes, contd.Mathematical Representation of Specific Enthalpy ChangeIsentropic Flow, expansionIsentropic Flow, compressionAdiabatic Flow with friction, expansionAdiabatic Flow with friction, compression

Difference in specific enthalpy change for isentropic and adiabatic flow with friction = heat energy expanded to overcome frictionFor ExpansionFor CompressionComparison, Isentropic & Non-isentropic Processes, contd.

Effect of Area Change on Flow PropertiesContinuity Eq.Differentiate logarithmicallyMomentum Eq.Can be written asSpeed of SoundFor isentropic flowCombine the three equations and use the definition of Mach Number to getThese equations suggest the manner in which area should change to accomplish the required expansion or compression of a compressible fluid

Effect of Area Change on Flow Properties, contd. (a) Nozzle flow/action(b) Diffuser flow/actionRelationship between dA & dM for Steady 1-D Isentropic flowTable

dAM< 1.0> 1.0dA < 0dM > 0dM < 0dA > 0dM < 0dM > 0

Effect of Area Change on Flow Properties, ChokingWHAT WOULD HAPPEN ONCE THESONIC CONDITIONS ARE REACHED & AREA CHANGE CONTINUESFor subsonic/supersonic flowOnce M=1 at a particular section of converging section then What would be M2, whereas dA < 0There can be two possibilities/assumptionsM2 < M1M2 > M1For: M2 < M1Table suggests, that for dA < 0 dM > 0 Contradiction with basic assumption, M2 < M1

dAM< 1.0> 1.0dA < 0dM > 0dM < 0

For: M2 > M1Effect of Area Change, Choking, contd. -2The Table Is In Fact A Representation Of The Conservation Laws Under Isentropic ConditionsTHESE CONSERVATIONS ARE NEVER VOILATED BY NATURE

dAM< 1.0> 1.0dA < 0dM > 0dM < 0

Effect of Area Change, Choking, contd. -3WHAT ABOUT HAVING THE FOLLOWING CONFIGURATION AND;ONCE THE SONIC CONDITIONS ARE REACHEDNow consider the possibilitiesORM2 < M1M2 > M1Table suggests, that for dA > 0, dM < 0 : dA > 0, dM > 0 Table suggests, that for Both the possibilities the conservation laws are not violatedOnce M=1 is achieved at the throat, then how the downstream M will change in the diverging section. We need to answer this query

dAM< 1.0> 1.0dA > 0dM < 0dM > 0

Effect of Area Change, Choking, contd. -3Ans: The downstream flow can be either subsonic or supersonicWhich one of this is possibleThe subsonic or supersonic flow in the downstream section depends on the downstream physical boundary conditions at the exit section of the flow passageIMPORTANT INFERENCESFor supersonic flow starting fromrestOne needs a C-D geometryFor subsonic flow starting from supersonic flowThis CD geometry is named asDe Laval NozzleThroat will give the max mass flow rate corresponding to throat velocity.

Effect of Downstream Physical Boundary Conditions

Properties and Property RatiosProperty Ratios for isentropic flow witharea changeLimiting Values of the Property Ratios for the Steady One-D Isentropic Flow of a Perfect Gas

Properties and Property RatiosFor CHOKED flow where M=1The property ratios for isentropic flow leads to t*, p* and *Critical TemperatureCritical PressureCritical DensityCritical / Max Mass flow rateCritical Velocity

Effect of Area Change On Flow PropertiesThe Governing Equations For Isentropic Flow With Area ChangeEq. of ContinuityEq. of MotionEq. of EnergyEq. of StateEq. of Speed of SoundEq. of Mach NumberEq. of Stream Thrust

Effect of Area Change On Flow Properties, contd. 7 ODE relating, dp/p. dt/t. d/ , dV/V, da/a, dM/M, dF/F to dA/AAll the 7 equations are linear in derivative formHence if one of the 8 properties is specified all the other can be foundChoose dA/A as independent variable and write in matrix formThus you get all the dependent variables as function of dA/AeSolve to get the solution

Effect of Area Change On Flow Properties, contd. The coeff. of dA/A are called influence coefficientsEach coefficients of dA/A is the partial derivative of aParticular flow property w.r.t the driving potential dA/A++--dF/ F-++-da/a+--+dV/V-++-d/-++-dt/t-++-dp/p+--+dM/MM > 1M < 1M > 1M < 1dA > 0dA < 0Property Ratio

The Case of Converging NozzleFor given areas of nozzle and back pressure po, we are interested inThe Mach No at throat as function of pressure ratioThe pressure ratio that will cause the nozzle to chokeSince flow is assumed to be isentropic, write the energy equation for section 1 and 2If the gas is perfect

If po is decreased for a constant P thenthere will be increasing flow in nozzle and pt = po untill pt=po=p*Once pt = p* then any further reduction in po will have no effect on pt and pt will remain p*.At this condition the fluid will be flowing at its maximum speed, Vt =a*The critical pressure ratio p*/P does not get reduced to po/P but rather remains constant at p*/P, no matter how much po is reduced to.Even if P is increased it will raise the p* for the same constant p*/P ratioThus the throat Mach number will remain same equal to unityMass flow rate m* will also remain critical for the corresponding P & TThis applies to the converging nozzle and to throat section of CD nozzle

So for max isentropic throat speed and , complete nozzling & flow choked

The Case of Converging Nozzle, contd. -2IMPORTANT INFERENCES

The Case of Converging Nozzle, contd. -3Isentropic Mass Flow Rate:Apply continuity equationFor perfect gas density can be found asHence isentropic throat mass flow rate is

The Case of Converging Nozzle, contd. -4Effect of decreasing po with constant PThere can be 3 possibilitiespo = pb = p*po > p*po < p*Cases of po p*Jet issues from nozzle in parallel cylindrical streamCase of po < p*Gas expands outside the nozzleThis expansion is explosiveGas particles are accelerated radiallyCause low pressure in jet coreMaking particles move backThus giving a periodic thin & thick sections

The Case of Converging Nozzle, contd. -5Effect of increasing P with constant po Once P in slightly more than po flow starts as subsonicFurther increase in P increases flow rate till pt = p*Now further increase in P has no effect on the pressure ratio po/P

With = 1.4 , the critical pressure ratio remains constant at 0.52828Choked nozzle gas speed for at throatIncrease in P for constant T, increases the local density causing increase in mass flow rate. Increase in mass flow rate is linear with increase in PHowever for critical conditions at throat volumetric flow rate is independent of P, as Q* = m* / *. If T is also constant then Q also remains constant

MULTIDIMENSIONAL FLOW EFFECTSAssumption of 1-D flow is not valid for large angled nozzles, giving non uniform flowLarge angles cause the inward radial momentum to form vena contracta outside the nozzle, Vena contract is also called aerodynamic throatVena contracta shape is affected by the surrounding atmospheric conditions. Even for pressure ratios larger than the critical ratioThe Case of Converging Nozzle, contd. -6Line of constant M for 40o c-nozzle and r=4Expt. Sonic lines for 25o c-nozzle

The Case of Converging Nozzle, contd. -7Flow non-uniformities and vena contacta causes reduced flow ratesgiving rise to discharge coefficientsExperimental discharge coefficients for conical converging nozzles

Only CD nozzles can generate supersonic flowsThe converging part makes the flow criticalThe diverging part then makes the flow supersonicWe need to specify property ratios at exit of nozzle as function of pressure ratioThe Case of C-D Nozzle

The Case of C-D Nozzle, contd. -2We earlier calculated the isentropic discharge speed of gas asSimilarly the exit Mach numberSimilarly the exit Mass flow rateOR

Plot of various ratios of exit to stagnation conditions as function of pressure ratiosThe Case of C-D Nozzle, contd. -3

Area Ratio for Complete Expansion in terms of pressure ratioThe Case of C-D Nozzle, contd. -4Area Ratio for Complete Expansion in terms of exit Mach numberSimilarly Velocity ratio of exit to the throat or critical conditions

The Case of C-D Nozzle, contd. -4When mass flow rate is less than choked mass flow rate then throat area is greater than critical throat area and throat M

Effect of Back Pressure on C-D Nozzle

Effect of Back Pressure on C-D Nozzle

Under-expansion in CD NozzlesIf the nozzle expands the gas passing through it such that pe = po then the velocities in the diverging section are supersonic through out provided the nozzle is passing maximum mass flow rate through its throat.If for maximum flow rate condition and given fixed geometry, pe > po, then the gas expands outside the nozzle to po as in the case of only converging nozzle.Under such conditions the K.E of gas coming out of the nozzle < the K.E of gas expanding isentropically and completelyThrust produced by such nozzles is less than corresponding to complete expansionSuch CD nozzles are called under-expanding nozzles

Over-expansion in CD NozzlesSimilarly if the back pressure is greater than the exit pressure (po > pe), the gas is over-expanded and the CD nozzle is called over-expanding nozzlesThrust produced by such nozzles is also less than that corresponding to complete isentropic expansionIn order to meet with po ( which is not transmitted backward into the nozzle), the gas has to be compressed through series of shock waves, as discussed earlierThere are regions of subsonic flow in the Boundary LayersInformation of po gets transmitted through this regionIf po is slightly greater than pe then oblique shock waves are formed at the edge of nozzleAs po is raised further then these oblique shock waves get stronger and cause the jet to separate. Reason being the adverse pressure gradient that exits between the edge of nozzle and inside the nozzleSeparation of jet needs regulation for rocket motors, as they operate over an extremely large range of altitude

Over-expansion in CD Nozzles, contd. -2Plot of experimental data of over-expanding nozzlesOnly one isentropic complete expansion M and if p0 is increased from that corresponding to M then sharp increase in pressures / shock waves are observed.Shock waves become stronger & even move inside nozzle if po is raise sufficiently raisedThe separation pressure ps 0.4 po and depends on BL character & divergence angle For a given , ps depends on pressure ratio r

Jet Propulsion EnginesBased on the Newton third Law.Examples, all types of sea ships or aircrafts etc.How to get required reaction force. By giving sufficient momentum to a mass of fluidReaction to time rate of increase in momentum of fluid produces a force called thrustWhat are the available means for propelling a body in a fluid medium differ only in the manner increase in momentum is imparted to the working fluidThe working fluid to receive the increase in momentum , water, steam, air, gases produced by chemical reactions, charged particles or their combinationsSelection of the working fluid depends on the nature of propulsion problemAll the propulsion systems are classified by the way the working fluid is handled Propeller propulsion: where the momentum is imparted to the working fluid when it passes through the propeller. The working fluid moves around the body to be propelledJet Propulsion: where the momentum is imparted to the working fluid by ejecting it as a high speed jet from with in the body to be propelledThe jet propulsion devices are further classified into two classesThe Air Breathing EnginesThe Rocket Engines

Jet Propulsion Engines, contd. -2THE AIR BREATHING ENGINESThey involve the normal combustion process. Fuel is mixed with air drawn into the engine from the atmosphereThe air fuel mixture is burnt and the temperature of the gases produced after combustion is increased to the desired value.These hot gases are then ejected out of the engine through nozzles to get the required propulsion and hence thrustTurbojet, turbofan, ramjet and SCRAM jet are typical examples of this classChoose control volume given by dashed lineDA = Additive DragF =Net force acting on Engine surfacePressure forces will be acting on the inlet and outlet areas of nozzlesMomentum will be coming in and going out from inlet and outlet of nozzle

Jet Propulsion Engines, contd. -3Net force acting on propulsion devicePressure force acting on propulsion device inlet/outlet areasAdditive drag force if stream lines are curvedMomentum leaving / enteringthe propulsion deviceNet force acting on propulsion device =Mass of fluid at outlet =Mass of fuel is generally 2-5% of airThrust per unit mass flow rate air =Another performance parameter, thrust specific fuel consumption =TSFC should be as low as possible for economical reasons

Jet Propulsion Engines, contd. -3The following assumptions are very much valid

Mass of fuel is negligible compared to the mass of air usedDA is generally only few percent of the total thrustFor a well designed nozzle pe = po hence the pressure thrust should be zero

Hence the net thrust produced can be approximated as

Rocket EnginesThey differ from the Air Breathing Engines in the following waysThey do not draw air from the atmosphereTotal mass ejected out of the engine to produce thrust is carried with in the engineBasic Features of A rocker EngineA similar analysis on a control volume givesPressure ThrustJet ThrustNet surface force acting on the rocketSimilarly Specific thrust = Similarly for Max Thrust Nozzle Force =

Performance of Propulsive NozzlesNet thrust developed by a propulsive nozzle depends onMass flow rate through the nozzleThe exit areaThe exit static pressureThe ambient pressureThe velocity of fluid at exitEssential features of propulsive nozzlesThe converging propulsive nozzles

Performance of Propulsive Nozzles, contd. -2The converging propulsive nozzlesTwo Commonly Used Performance Criteria for Propulsive NozzlesSpecific ImpulseThrust CoefficientOne can use the isentropic flow relations and represent CF in terms of m* and Ve

Performance of Propulsive Nozzles, contd. -3Doing the necessary algebra leads to

Performance of Propulsive Nozzles, contd. -4Plot of CF as function of e (area ratio) for = 1.2 IMPORTANT INFERENCES For a given po/P there is a unique value of area ratio that gives max thrustMaximum thrust occurs when po = 0, for given eClearly max thrust occurs when such an arearatio is chooses which gives pe = poAround maximum thrust CF curve is quite flatThus slight adjustment in area ratios are possible without losing CF. Thus one can design for reduced area and thus less weight and sizeOne can use the correction coefficients to account for the multidimensional effects

Thrust Reduction by Flow Divergence in NozzlesAxial thrust is first calculated on a differential exit area and then integrating over the total areaIntegration and algebraic manipulation gives

Design Criteria for Propulsive Nozzles for Max ThrustThrust developed by a propulsive nozzle is given asWe need to find the nozzle exit area which should yield max thrust. Differentiating we getZero: Bernoullis equationZero: For maximum thrusti.e pe = p0Hence Area must be chosen so that

AssignmentChapter 1: 3, 4, 5, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 23Chapter 3: 1, 5, 6, 8, 9, 13, 14, 15, 16, 17, 18, 19Chapter 4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19,20

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