But isentropic (rev./adiab.), so = =

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Harold Riley
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1 School of erospace Engineering Isentropic Flow with rea Change Exaine ass and oentu equations for reersible and adiabatic conditions ass (VI.9) oentu (VI.0) Cobine dρ d + + ρ dρ + ρ ρ d d 0 0 d p ρ L p d d τx dx p p ρ 0 (no iscous stress/friction, reersible) ρ dρ τ x dx ρ p+ ρ+dρ +d +d Isentropic Flow with rea Change - Copyright by Jerry. Seitzan. ll rights resered. School of erospace Engineering ach Nuber Relation p d ρ ρ dρ p But isentropic (re./adiab.), so dρ ρ s d ( d ) ( ) (VI.6) ρ a p+ ρ+dρ +d +d Fro Isentropic Flow with rea Change - ρ Copyright by Jerry. Seitzan. ll rights resered. d d Deried using only ass/oentu conseration, and speed of sound Valid for all siple cop. substances

2 School of erospace Engineering Isen. Flow - ach Nuber Dependence d ρ d ( ) ( ) For <: d, sae sign p d, d opposite sign For >: d, opposite sign p d, d sae sign How does area change effect flow properties?, d always opposite signs ( T p ) < > Nozzle d>0 (d>0) Expansion <0 Diffuser d<0 (d<0) Copression >0 Isentropic Flow with rea Change -3 Copyright by Jerry. Seitzan. ll rights resered. d ρ Isentropic Flow with rea Change -4 School of erospace Engineering Copyright by Jerry. Seitzan. ll rights resered. Sonic Throat Requireent d ( ) ( ) Need to go through For :d or d 0 but d 0 (flow accel. or decel.) d0: axiu or iniu in area < gets saller > gets bigger then neer get So need a throat to transition, and at throat (sonic condition) How to transition fro subsonic to supersonic (or ica ersa)? < > > <

3 Isentropic Flow with rea Change -5 School of erospace Engineering rea Ratio For isentropic flow, look at effect of area change on by coparing at any point to area at sonic point () refers to properties of a flow if isen. accel./decel. to sonic conditions (e.g., ρ, T, ) alternatie to stagnation as ref. state Use ass conseration to find relation ρo γ ρ ρ + a ρ fro VI.8 Copyright by Jerry. Seitzan. ll rights resered. ρ ρ ρo a ρ o γ + ρ ρo ρ a ρ γ+ ( ) a T γ a T o + γ + (VI.7) for TPG, CPG T To with T T To γ + T To γ + T fro VI.6 Isentropic Flow with rea Change -6 School of erospace Engineering Two (isentropic) solutions for a gien / one subsonic one supersonic always ccel. to high requires large / ρ ( ) ρ Copyright by Jerry. Seitzan. ll rights resered. rea Ratio Results / Subsonic axiu ass flux at (throat) γ+ γ ( ) + γ + Supersonic γ

4 Isentropic Flow with rea Change -7 School of erospace Engineering ass Flux and Stagnation Properties Exaine ass flux in ters of stagnation conditions fro VI., TPG p p ρ a γrt RT RT fro VI.6,7 CPG To γ + T po γ γ+ po γ RT o ( ) + γ p + (VI.8) For gien isen. flow, all po stagnation (and sonic) or f ( γ,) properties constant, RTo including ass flow rate Copyright by Jerry. Seitzan. ll rights resered. γ Isentropic Flow with rea Change -8 School of erospace Engineering For fixed stagnation properties and flow area ax 0.4 at γ For nozzle with fixed 0. stagnation properties 0. if throat is sonic, Sub- Supersonic 0 can t get ore by changing downstrea conditions (e.g., back p pressure) o front t p back T o Choked Flow reseroir Copyright by Jerry. Seitzan. ll rights resered. Choked Flow f(γ,) po RT o f ( γ,)

5 Isentropic Flow with rea Change -9 School of erospace Engineering For nozzle with fixed stagnation properties and initially sonic throat Copyright by Jerry. Seitzan. ll rights resered. Choked Flow (con t) f(γ,) if reduce throat area 0.3 t, flow at throat stays 0. sonic ( front / t ) and 0. (since, & / sae) if increase t, and eentually throat not sonic ( front / ) not choked and ) ( ax Sub- / 3 0 Supersonic p o T o reseroir front t p back Isentropic Flow with rea Change -0 School of erospace Engineering Copyright by Jerry. Seitzan. ll rights resered. Choked ass Flowrate axiu flow rate when choked ( at throat) Choked ass flowrate fro f(γ,) ax po RT o γ γ+ o ax 0.7 throat RTo rule of thub for choked gas flows γ+ ( γ ) (VI.9) To increase ass flowrate increase (throat size) increase p o, decrease T o (increase stagnation density) f(γ,) typically near γ 5/3 p f ( γ,) γ γ.3 5

6 School of erospace Engineering Exaple For a supersonic wind tunnel with an incoing flow with << (fro a high pressure reseroir or copressor), need a throat to produce > Inlet Test Section > What area throat required to produce a test section ach nuber of 3 in test section with 0. cross-section? ssue isentropic flow, calor./therally perfect gas, γ.4. t Isentropic Flow with rea Change - Copyright by Jerry. Seitzan. ll rights resered. School of erospace Engineering Exaple Conerging-dierging supersonic nozzle with.0 and 3.0 What is /? ssue isentropic flow, calor./therally perfect gas, γ.4. Isentropic Flow with rea Change - Copyright by Jerry. Seitzan. ll rights resered. 6

7 School of erospace Engineering Exaple 3 For nozzle shown, inlet area 0.50, 0.0 and p o, T o fixed How far can be reduced fro without changing ass flowrate in nozzle? ssue isentropic flow, calor./therally perfect gas, γ.4. Isentropic Flow with rea Change -3 Copyright by Jerry. Seitzan. ll rights resered. School of erospace Engineering Exaple 4 For nozzle shown ( 0.060, 0.00 ) and p o, T o fixed For isentropic flow in the nozzle, what are the liits on the allowed inlet ach nubers ( )? ssue tpg/cpg, γ.4. p o T o Isentropic Flow with rea Change -4 Copyright by Jerry. Seitzan. ll rights resered. 7

8 School of erospace Engineering Exaple 5 Sae nozzle as Exaple 4, air and p o 500kPa, T o 300K What is ax. possible ass flowrate through the nozzle? ssue tpg/cpg, γ.4. p o T o Isentropic Flow with rea Change -5 Copyright by Jerry. Seitzan. ll rights resered. 8

Fanno Flow. Gas Dynamics

Fanno Flow. Gas Dynamics Fanno Flow Simple frictional flow ( Fanno Flow Adiabatic frictional flow in a constant-area duct * he Flow of a compressible fluid in a duct is Always accompanied by :- ariation in the cross sectional