Compressible Flow CHOKING

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Compressible FlowCHOKING
Jet Propulsion Systems
Vikas Kumar
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Compressibility

• What is Compressible Flow??
• How do we define flow as compressible or
  Incompressible??
• Mach Number, M (u/a)
• For an ideal gas,
• M lt 0.3 Incompressible (lt10 error)
• M gt 0.3 Compressible

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Compressibility

• Attributes of Compressible Flow
• Information propagates in the direction that
  depend on local Mach number
• Density can no longer be regarded as constant.
  Bernoullis principle doesnt hold for
  compressible flow.
• Coupling between Internal energy and Kinetic
  energy can no longer be ignored.
• Regions of mixed flow type including subsonic,
  sonic and supersonic can be present.

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Flow Regime Classification

• Subsonic Flow
• 0.8 lt M?
• Transonic Flow
• 0.8 gt M? gt 1.2
• Supersonic Flow
• M? gt 1.2
• Hypersonic Flow
• M? gt 5

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One-Dimensional Flow Governing Equations

• Assuming constant area duct
• Mass Conservation
• ?Uconstant
• Momentum Conservation
• P ?U2constant
• Energy Conservation
• hin (U2/2)in qin hout (U2/2)out

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Speed of Sound

• Flow through sound wave is one dimensional.

Mass conservation gives
Ignoring second order term
Similarly momentum conservation gives
Solving these two simultaneously
In fact
Why constant entropy?
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Entropy Quick Review

• For a perfect gas,
• e CVT h CPT Cp-CVR Cp/CV ??

Second Law of Thermodynamics
Equality holds for reversible process
For a reversible process and for a perfect gas
then, we have
Using 1st Law and PVRT
For Isentropic processes (ds0)
or
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Isentropic processes

• For an isentropic process then

The equation for speed of sound then becomes
Why is the speed of sound so important??

• Fluid particles send signals in form of acoustic
  (pressure) waves.
• If signals reach faster than the object itself,
  fluid particles will hear and clear out
  (Subsonic case)
• If the object is traveling faster than these
  acoustic waves (speed given by speed of sound),
  then there is shock. (Supersonic case)

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Mach Number Revisited

• Definition
• Ratio of the flow velocity (U) to the speed of
  sound (a)

Physical Significance Proportional to the ratio
of Kinetic Energy to the Internal Energy
Characteristic Mach number M U/a
a(?RT )0.5
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1-D Flow Revisited

• Stagnation Quantities

or
Furthermore for Isentropic process
Relationship with Critical quantities
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1-D Flow Revisited

• Characteristic Mach number

or
Furthermore for Isentropic process
Relationship with Critical quantities
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Shocks

• Shock Wave is a discontinuity in the flow
  quantities.

Normal Shock Relations
Using mass, momentum and energy equation of form
The quantities downstream then can be evaluated
using M1
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Shocks contd.

• Total temperature is constant across a normal
  stationary shock

Total Pressure decreases across a normal shock
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Quasi One-Dimensional Flow

• Area-Velocity relation

Mass conservation relation
Since dp-?udu
or
Area Mach Number Relation
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Quasi One-Dimensional Flow

• Flow Inside a Variable Area duct

• abe Subsonic
• acf Subsonic Isentropic
• acdg Supersonic with shock
• acdh Overexpanded
• acdi Underexpanded

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Choked Flow
Choked flow as the flow becomes sonic at the
throat. Maximum mass flux at choked flow.
Choked Flow
Variation of mass flow rate with exit pressure
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Choked Flow A Mathematical Explanation
It all has to do with weak shocks propagating
with the speed of sound. To simplify, we will
first create a single equation in the single
unknown ?
?UM P ?U2 I h (U2 /2) H
Eliminating U, H?I/(?-1)P (?1)M2/2(?-1) ?2
f(?)
Change the value of H an infinitesimal amount
.This changes the value of the flow density by a
corresponding amount . In terms of linear
algebra, we have a linearized problem'' fd
?dH, where f is the derivative It has a
unique solution d ?dH/f. At ? however, f0.
Now, consider shocks They have same enthalpy, H.
Result is a weak shock and it happens only at
Mach1.
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Example Problem

• A supersonic wind tunnel is designed to produce
  Mach 2.5 flow in test section at standard sea
  conditions. Calculate the exit area ratio and
  reservoir conditions necessary to achieve these
  design conditions.

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Example Solution

• Refer to slide 14 for area ratio formula.
• Answer Ae/A2.637.
• Known Mach no, Po/Pe(17.09), To/Te(2.25) known.