Ch11_4_CD - MAE130B Ch 11 (4) Compressible Flow Changzheng...

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MAE130B – Ch 11 (4) Compressible Flow Changzheng Huang
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Ch 11 (4) Changzheng Huang 2 Compressible Flow Non-isentropic adiabatic flow (Fanno flow) Non-isentropic frictionless flow (Rayleigh flow)
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Ch 11 (4) Changzheng Huang 3 2. Non-isentropic adiabatic flow (Fanno flow) V 1 p 1 A V pA τ w Mass 11 VA VA m ρ = = ± Momentum ( ) w p Ap A D l m VV τπ −− = ± D l Energy 22 1 mh g z h g z Q W ⎡⎤ ⎛⎞ + +− + + = + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ ± ± ± Since flow is adiabatic (Q=0), steady (W=0) and horizontal (gz=0), Mass Constant V = Energy 2 0 constant 2 V hh += = Note the conditions of validity of this.
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Ch 11 (4) Changzheng Huang 4 2. Non-isentropic adiabatic flow (Fanno flow) V pA V+dV (p+dp)A τ w D dx T s Temperature-entropy diagram T a p a M<1 Second law of thermodynamics states that entropy must be constant or increasing for adiabatic flow. Three typical Fanno flow cases, M>1 M=1 Fanno line 1 2 a 1 2 a 1 2 a M<1 M>1 Shock (iii) Normal shock Fanno flow T T T ss s (ii) Supersonic Fanno flow (i) Subsonic Fanno flow
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Ch 11 (4) Changzheng Huang 5 2. Non-isentropic adiabatic flow (Fanno flow) V pA V+dV (p+dp)A τ w Mass () 0 dV ρ = D dx Energy 2 0 2 V dh ⎛⎞ += ⎜⎟ ⎝⎠ T s Temperature-entropy diagram T a p a Momentum w Adp Ddx VAdV τ πρ −− = Entropy Tds dh vdp =− pp dp dp p Tds c dT c dT p 0 p cdT VdV + = 0 dd V V 1 Question, for a given flow condition (say M ), how far can the flow extend before it reaches the chocked condition ( l *- l )? 0 l l *
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Ch 11 (4) Changzheng Huang 6 2. Non-isentropic adiabatic flow (Fanno flow) V pA V+dV (p+dp)A τ w D dx T s Temperature-entropy diagram T a p a Momentum w Adp Ddx VAdV τ πρ −= 1 0 l l * From pipe flow, 2 2 2 2 w p lrR lV pf D ρ == ∆= 2 8 w V f τ= 22 2 48 4 DV D dp f Ddx V dV π −− = 2 dx dp dV f V +
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Ch 11 (4) Changzheng Huang 7 2. Non-isentropic adiabatic flow (Fanno flow)
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This note was uploaded on 03/10/2009 for the course MAE 130B taught by Professor Huang during the Spring '09 term at UC Irvine.

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Ch11_4_CD - MAE130B Ch 11 (4) Compressible Flow Changzheng...

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