introduction to compressible flow
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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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MOTIVATION TO LEARN COMPRESSIBLE FLOW
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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MOTIVATION TO LEARN COMPRESSIBLE FLOW
Applications in Engine Four Strokes
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MOTIVATION TO LEARN COMPRESSIBLE FLOW
Applications in Engine Four Strokes
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MOTIVATION TO LEARN COMPRESSIBLE FLOW
Applications in Intake Supersonic
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MOTIVATION TO LEARN COMPRESSIBLE FLOW
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?
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What is Compressible Flow?
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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What is Compressible Flow?
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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What is Compressible Flow?
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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What is Compressible Flow?
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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What is Compressible Flow?
• 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
Heat Addition and Specific Heat
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What is Compressible Flow?
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
Heat Addition and Specific Heat
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What is Compressible Flow?
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
Heat Addition and Specific Heat
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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
What is Compressible Flow?
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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
What is Compressible Flow?
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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
What is Compressible Flow?
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
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