Gas Dynamics-Compressible Flow

ME 1303 GAS DYNAMICS ME 1303 GAS DYNAMICS

AND JET PROPULSIONAND JET PROPULSION

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11--D Flow: RevisitedD Flow: RevisitedStagnation QuantitiesStagnation Quantities

0

2

2TCUTC pP =+

Furthermore for Isentropic processFurthermore for Isentropic process

Relationship with Critical quantitiesRelationship with Critical quantities

20 )2

1(1 MTT −

+=γ

or

120 )2

1(1−

−

+=γγ

γ MPP 1

1

20 )2

1(1−

−

+=γγ

ρρ M

2121

2**2 URTURT+

−=+

− γγ

γγ 2*

2*2*22

)1(21

2121aaaUa

−+

=+−

=+− γ

γγγ

2*20

)1(21

1aa

−+

=− γ

γγ1

2

0

*2

0

*

+==

γT

Taa

1

0

*

12 −

+

=γγ

γPP

11

0

*

12 −

+

=γ

γρρ




Relationship between TRelationship between T00,T & M,T & MThe acoustic speed & Mach number

For a compressible flow, the speed of propagation of small disturbances, called the acoustic speed and the ratio of the flow velocity to the acoustic speed, called the Mach number .

The magnitude of K will depend on the process in which the compression is executed. For isentropic flow process,




Cont..Cont..

Mach numberMach number




Cont..Cont..




Basic characteristics of air Basic characteristics of air




Why is the speed of sound so important??

n Fluid particles send signals in form of acoustic (pressure) waves.

n If signals reach faster than the object itself, fluid particles will “hear” and “clear out” (Subsonic case)

n If the object is traveling faster than these acoustic waves (speed given by speed of sound), then there is “shock”. (Supersonic case)




Wave front

A sound wave, by definition,

ie: weak wave

( Implies that the irreversible,

dissipative conduction are negligible)

uContinuity equation

daddaadadaada ρρρρρρρ +++=++= ))((

ρρ

ddaa −=

1 2

T

pa

ρ

dTTddppdaa

++++

ρρ




Speed of Sound in different mediumSpeed of Sound in different mediumFor Liquids For SolidsFor Liquids For Solids

BB,,EE-- Bulk modulus andBulk modulus andYoung’s modulusYoung’s modulus




Effect of Mach number on CompressibilityEffect of Mach number on CompressibilityFrom Bernoulli eqn w.k.t, compressibility factor is unity From Bernoulli eqn w.k.t, compressibility factor is unity for incompressible fluid. for incompressible fluid.

For compressible flow this value deviates from unity; the For compressible flow this value deviates from unity; the magnitude of this deviation increases with the mach magnitude of this deviation increases with the mach number of the flow.number of the flow.




Compressibility Factor ComparisonCompressibility Factor Comparison




Wave PropagationWave PropagationQ. Q. An airplane is traveling while you are observing from the An airplane is traveling while you are observing from the

ground. How will you know whether it is subsonic or ground. How will you know whether it is subsonic or supersonic? supersonic?

Point disturbance is at restPoint disturbance is at rest Moving disturbanceMoving disturbanceM = 0 ( M = u/a = 0.5)M = 0 ( M = u/a = 0.5)

Always stays inside the family of circular sound waves




Wave fronts from Sonic disturbanceWave fronts from Sonic disturbance

ØØ All the wave fronts coalesce on the left side and move along with theAll the wave fronts coalesce on the left side and move along with thedisturbance.disturbance.

ØØ No region upstream is forewarned of the disturbance as the No region upstream is forewarned of the disturbance as the disturbance arrives at the same time as the wave front.disturbance arrives at the same time as the wave front.

Zone of Silence Zone of Silence Zone of action




Wave fronts from Supersonic disturbanceWave fronts from Supersonic disturbance

nn The wave fronts have coalesced to form a cone with the disturbance at The wave fronts have coalesced to form a cone with the disturbance at the apex.the apex.

nn The half angle at the apex is called Mach angle ( The half angle at the apex is called Mach angle ( µ )µ )

Always stays outside the family of circular sound waves




Disturbance PropagationDisturbance Propagationnn In Subsonic flow, both In Subsonic flow, both

Raj and Lisa can hear Raj and Lisa can hear Joy talking, since sound Joy talking, since sound waves travel from Joy’s waves travel from Joy’s mouth in all directions.mouth in all directions.

nn In Supersonic flow, sound In Supersonic flow, sound waves (and other waves (and other disturbances in the flow) disturbances in the flow) travel only in the travel only in the downstream direction; downstream direction; thus, while Lisa can hear thus, while Lisa can hear Joy talking, Raj can’t. Joy talking, Raj can’t. Disturbances can not Disturbances can not travel upstream in a travel upstream in a supersonic flowsupersonic flow







ProblemsProblems




Cont..Cont..




Isentropic flow with variableIsentropic flow with variablearea passagesarea passages

a) Nozzlea) Nozzleb) Diffuserb) Diffuser




OBJECTIVES OF THIS LECTUREOBJECTIVES OF THIS LECTURE

1. To examine different scenarios of nozzle& diffuser flows

2. Investigate the relation of flow velocity & pressure insubsonic & supersonic flow regimes

3. To understand how mass flow rate through a nozzle will change with the exit pressure

4. Determine the implications of choking




Distinction Between True 1-D Flow and Quasi 1-D Flow

• In “true” 1-D flow Cross sectional area is strictly constant• In quasi-1-D flow, cross section varies as a Function of the longitudinal coordinate, x• Flow Properties are assumedconstant across any cross-section• Analytical simplification very useful for evaluating Flow properties in Nozzles, tubes, ducts, and diffusersWhere the cross sectional area is large when compared to length




Nozzle Nozzle -- FunctionFunctionüü From an energy view point : Nozzle is a device From an energy view point : Nozzle is a device

that converts that converts static enthalpystatic enthalpy into into kinetic energykinetic energy

Expansion ProcessExpansion Processin nozzlein nozzle




Diffuser Diffuser -- FunctionFunctionüü From an energy view point : Diffuser is a device From an energy view point : Diffuser is a device

that converts that converts kinetic energy kinetic energy intointo static enthalpystatic enthalpy

Compression ProcessCompression Processin diffuserin diffuser




OneOne--Dimensional Isentropic Dimensional Isentropic FlowFlow

nn For flow through For flow through nozzles, diffusers, and nozzles, diffusers, and turbine blade passages, turbine blade passages, flow quantities vary flow quantities vary primarily in the flow primarily in the flow directiondirectionnn Can be approximated as Can be approximated as

1D isentropic flow1D isentropic flow




ApplicationsApplications

Ram-jet engine

Diffuser(compressor)

combustion chamber

nozzle




Applications Applications –– Cont..Cont..

Space Shuttle




OneOne--Dimensional Isentropic Flow Dimensional Isentropic Flow Variation of Fluid Velocity with Flow AreaVariation of Fluid Velocity with Flow Area

nn Relationship between Relationship between V , V , ρρ,, and and AA are complexare complexnn Derive relationship using continuity, energy, Derive relationship using continuity, energy,

speed of sound equationsspeed of sound equationsnn ContinuityContinuity

nn Differentiate and divide by mass flow rate (Differentiate and divide by mass flow rate (ρρAV)AV)





nn Derived relation (on Derived relation (on image at left) is the image at left) is the differential form of differential form of Bernoulli’s equation.Bernoulli’s equation.

nn Combining this with result Combining this with result from continuity givesfrom continuity gives

nn Using thermodynamic Using thermodynamic relations and rearranging relations and rearranging




OneOne--Dimensional Isentropic FlowDimensional Isentropic FlowVariation of Fluid Velocity with Flow AreaVariation of Fluid Velocity with Flow Area

nn This is an important relationshipThis is an important relationshipnn For For MaMa < 1, (1 < 1, (1 -- MaMa22) is positive ) is positive ⇒⇒ dAdA and and dPdP have have

the same sign. the same sign. nn Pressure of fluid must increase as the flow area of the duct Pressure of fluid must increase as the flow area of the duct

increases, and must decrease as the flow area decreasesincreases, and must decrease as the flow area decreases

nn For For MaMa > 1, (1 > 1, (1 -- MaMa22) is negative ) is negative ⇒⇒ dAdA and and dPdP have have opposite signs.opposite signs.nn Pressure must increase as the flow area decreases, and must Pressure must increase as the flow area decreases, and must

decrease as the area increasesdecrease as the area increases




Cont..Cont..








Comparison of flow properties in subsonic and supersonic nozzles and diffusers




Subsonic Vs Supersonic flowSubsonic Vs Supersonic flow

Nozzle DiffuserNozzle Diffuser DiffuserDiffuser NozzleNozzle




Property variation with area changeProperty variation with area change

At low Mach no. density variations isAt low Mach no. density variations isless and the velocity changes less and the velocity changes compensate for area changes.compensate for area changes.

At M = 1.0 , we reach a situation At M = 1.0 , we reach a situation where density changes and velocitywhere density changes and velocitychanges compensate for one changes compensate for one another and thus dA = 0another and thus dA = 0




Sonic PropertiesSonic PropertiesØØ Let [ * ] denote a property at the sonic state M = 1 Let [ * ] denote a property at the sonic state M = 1 ØØ Then giving M =1 in stagnation state set eqns,Then giving M =1 in stagnation state set eqns,

@ @ γγ = 1.4= 1.4




Sonic properties Sonic properties –– Cont..Cont..




Remarks on Isentropic Nozzle DesignRemarks on Isentropic Nozzle Design

nn Length of the nozzle is immaterial for an Length of the nozzle is immaterial for an isentropic nozzle.isentropic nozzle.

nn Strength requirements of nozzle material Strength requirements of nozzle material may decide the nozzle length.may decide the nozzle length.

nn Either Mach number variation or Area Either Mach number variation or Area variation or Pressure variation is specified variation or Pressure variation is specified as a function or arbitrary length unit.as a function or arbitrary length unit.

nn Nozzle design attains maximum capacity Nozzle design attains maximum capacity when the exit Mach number is unity.when the exit Mach number is unity.




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Gas Dynamics-Compressible Flow