introduction to compressible flow

INTRODUCTION TO COMPRESSIBLE FLOW

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MOTIVATION TO LEARN COMPRESSIBLE FLOW

Applications of compressible flow in Aeronautics and Non Aeronautics 1. Jet engines 2. Intake Supersonic Fighter aircraft 3. Blended Wing Body 4. Engine Four strokes

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Applications in Jet Engines

T-s Diagram T

s T0

0

2 Tt0 =Tt2

Tt4

4

9’

3 Tt3

Tt9

Tt5

t5

t9

9

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Applications in Engine Four Strokes

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Applications in Engine Four Strokes

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Applications in Intake Supersonic

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Applications in Blended Wing Body

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What is Compressible Flow?

𝐾 = − 𝑑𝑃

𝑑𝑉/𝑉= −𝑉

𝑑𝑃

𝑑𝑉

∆𝒱

𝒱

𝑃 + ∆𝑃 𝑃

Compressible flow deals with fluids in which the fluid density varies significantly in response to a change in pressure

Modulus Bulk

𝐾 = 𝜌 𝑑𝑃

𝑑𝜌

waterK

At sea level (1 atm)

airK

5 x 10-10 m2/N

1 x 10-5 m2/N

Is it possible to change Δp with dynamic pressure?

http://en.wikipedia.org/wiki/Fluid

http://en.wikipedia.org/wiki/Density

http://en.wikipedia.org/wiki/Pressure

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Molecular Approach

𝑑𝑈 = 𝛿𝑄 + 𝛿𝑊

Change in Internal Energy : 1. Heat added to the system 2. Work done on the system

Molecule activities (e) increase or decrease by two things: Heat, 𝜹𝒒 Work, 𝜹𝒘

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State Condition

Perfect gas :

Intermolecular forces are neglected

10 x molecule diameter

Repulsive

force (+)

Attractive

force (-)

Distance from molecule

𝑝 = 𝜌𝑅𝑇

𝑝 = pressure [𝑁/𝑚2]

𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦[kg/𝑚3]

𝑇 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 [𝑜𝐾]

𝑅 = 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 constant = 287 [𝑚2/𝑠2𝑜𝐾]

𝑝𝑣 = 𝑅𝑇

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𝑑𝑈 = 𝛿𝑄 + 𝛿𝑊

Molecule activities (e) increase or decrease by two things: Heat, 𝜹𝒒 Work, 𝜹𝒘

• 1st Law: dU = dQ + dW o Find more useful expression for dw, in

terms of p and r (or v = 1/r)

• When volume varies → work is

done • Work done on balloon, volume ↓ • Work done by balloon, volume ↑

Change in

Volume (-) 𝛿𝑊 = − 𝑝𝑠 𝑑𝐴 = − 𝑝 𝑑𝑉

𝑑𝑈 = 𝛿𝑄 − 𝑝𝑑𝑉 𝑑𝑢 = 𝛿𝑞 − 𝑝𝑑𝑣 Per unit mass

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ℎ = 𝑢 + 𝑝𝑣 = 𝑢 + 𝑅𝑇

Enthalpy : Useful Quantity, h

Differentiate

𝑑ℎ = 𝑑𝑢 + 𝑝𝑑𝑣 + 𝑣𝑑𝑝

𝑑𝑢 = 𝛿𝑞 − 𝑝𝑑𝑣

𝑑ℎ = 𝛿𝑞 − 𝑝𝑑𝑣 + 𝑝𝑑𝑣 + 𝑣𝑑𝑝

𝑑ℎ = 𝛿𝑞 − 𝑝𝑑𝑣 + 𝑝𝑑𝑣 + 𝑣𝑑𝑝

𝛿𝑞 = 𝑑ℎ − 𝑣𝑑𝑝 𝛿𝑞 = 𝑑𝑢 + 𝑝𝑑𝑣

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Heat Addition and Specific Heat

• Addition of dq will cause a small change in temperature dT of system

dq

dT

Kkg

J

dT

qc

d

• Specific heat is heat added per unit change in temperature of system

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• Different materials have different specific heats

– Balloon filled with He, N2, Ar, water, lead, uranium, etc…

• For a fixed dq, resulting dT depends on type of process…

Kkg

J

dT

qc

d


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Process type I : constant volume

Kkg

J

dT

qc

d

dq

dT

dTcdu

dTcq

dT

qc

v

v

v

d

d

olumeconstant v

Tcu v


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Process type I I : constant pressure

Tch p

dq

dT

Kkg

J

dT

qc

d

dTcdh

dTcq

dT

qc

p

p

p

d

d

pressureconstant


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No Intermolecular

forces Real Gas

Intermolecular forces

P = 1000 atm

T = 30K

Thermally PG 800-2500 OK

Chemically reacting 2500-9000 OK

Calorically PG 0-800 Ok

For air


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ℎ = 𝑢 + 𝑝𝑣 For real gas and chemically reacting mixture of PG

𝑢 = 𝑢(𝑇, 𝑣) ℎ = ℎ(𝑇, 𝑝)

For thermally PG

𝑑𝑢 = 𝑐𝑣𝑑𝑇 𝑑ℎ = 𝑐𝑝𝑑𝑇

For calorically PG

𝑢 = 𝑐𝑣𝑇 ℎ = 𝑐𝑝𝑇


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For calorically PG

𝑐𝑝= 𝑐𝑣 + 𝑅

Ratio of specific heat

𝑐𝑝

𝑐𝑣= γ

𝑐𝑣 =𝑅

𝛾 − 1 𝑐𝑣 =

𝛾𝑅

𝛾 − 1


Relation of Spesific heats and Ratio of specific heat

Specific heat ratio

For air, g = 1.4

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Entropy

𝛿𝑞 + 𝛿𝑤 = 𝑑𝑢 𝑑𝑠 ≥ 𝑑𝑞/𝑇 𝑇𝑑𝑠 ≥ 𝑑𝑞

𝛿𝑤 = 𝑑𝑢 − 𝑇𝑑𝑠

𝑇𝑑𝑠 + 𝑣𝑑𝑝 = 𝑑ℎ

Helmholtz function : maximum work that can be obtained from a system

𝑇𝑑𝑠 − 𝑝𝑑𝑣 = 𝑑𝑢

Gibbs function : maximum useful work that can be obtained from a system

𝛿𝑤 = 𝑑ℎ − 𝑇𝑑𝑠

introduction to compressible flow

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