Steady, 1-d, constant area, adiabatic flow with no external work but with friction Conserved quantities
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1 School of Aerosace Engineering Stead, -d, constant area, adiabatic flow with no external work but with friction Conserved quantities since adiabatic, no work: h o constant since Aconst: mass fluxρvconstant G combining: h o hg /ρ constant On h-s diagram, can draw Fanno ine line connecting oints with same h o and ρv ρ v τ x, f? As ou change h, ou change ρ (and v) since G and h o const. Fanno Flow - Coright b Jerr. Seitzman. All rights reserved.
2 School of Aerosace Engineering Velocit change (due to friction) associated with entro change Friction can onl increase entro can onl aroach friction alone can not allow flow to transition between sub/suersonic h v0 h < o (ρv) v > s (ρv) wo solutions given (ρv,h o,s): subsonic & suersonic change mass flux: new Fanno line s Fanno Flow - Coright b Jerr. Seitzman. All rights reserved.
3 School of Aerosace Engineering otal friction exerienced b flow increases with length of flow, e.g., h < duct length, h o For long enough duct, e ( ) (ρv) What haens if > flow alread choked > subsonic flow: must move to different Fanno line ( ), i.e., lower mass flux suersonic flow: get a shock ( ) Fanno Flow -3 Coright b Jerr. Seitzman. All rights reserved. s e
4 School of Aerosace Engineering Simlif (X.4-5) for δqda0 d d Fanno Flow -4 Coright b Jerr. Seitzman. All rights reserved. [ ( ) ] fdx ( ) can write each as onl f() o loss due to entro rise fdx (X.6) (X.7) d ds R dρ ρ dh h d o o dv v ( ) dx ( ) fdx 4 f fdx ( ) (X.8) (X.9) (X.0)
5 School of Aerosace Engineering ook at signs of revious equations to see how roerties changed b friction as we move along flow (- ) term makes < different than > s o h, ρ v < > Friction increases s, o dro Friction drives h o, o const: h, oosite to, ρ same as (like isen. flow) ρvconst: v oosite of ρ Fanno Flow -5 Coright b Jerr. Seitzman. All rights reserved.
6 School of Aerosace Engineering Need to integrate (X.6-0) to find how roerties change along length of flow ( fdx/ ) Fanno Flow -6 for examle, need to integrate Searate terms x x f ( ) x d f ( Re, surface ) 4 d ( Re, surface ) dx f ( x x ) Coright b Jerr. Seitzman. All rights reserved. x fdx f dx x x f function of Renolds number (e.g.,velocit) and surface roughness for simlicit, can aroximate f b average value
7 Fanno Flow -7 School of Aerosace Engineering Coright b Jerr. Seitzman. All rights reserved. o erform integral, redefine variables ( ) ( ) 4 d d x x d d ln ln ln
8 School of Aerosace Engineering Combine results into exression for change caused b friction f ln For examle, given f / and x x (X.) could solve X. for Can t invert X. analticall - can t write f(, f /) either use iterative (e.g., numerical or guessing) method or find f / as a function of and tabularize solution Fanno Flow -8 Coright b Jerr. Seitzman. All rights reserved.
9 School of Aerosace Engineering o get change in, use as reference condition (like Prandtl-eer and A/A* table solutions) f f (X.) f f ln f Find values in Aendix E in John,, is reference so if ou know f / and, ) look u f / at ) calculate f / at 3) look u corresonding Fanno Flow -9 Coright b Jerr. Seitzman. All rights reserved.
10 Fanno Flow -0 School of Aerosace Engineering Coright b Jerr. Seitzman. All rights reserved. o get changes in,, o,... can again use condition as reference condition (denoted as *) Integrate (X.7-0), e.g., ( ) * d d,, o,, *,, o
11 Fanno Flow - School of Aerosace Engineering Coright b Jerr. Seitzman. All rights reserved. Summarize results in terms of reference conditions OR in terms of initial and final roerties ( ) * (X.3) * * * v v ρ ρ (X.5) * * (X.4) ( ) * * o o (X.5) (X.7) ( o const) v v ρ ρ (X.0) ( ) o o (X.8) (X.9)
12 School of Aerosace Engineering Given: Exit of suersonic nozzle connected to straight walled test section. est section flows N at test 3.0, o 90. K, o 500. kpa, m, 0 cm, f0.005 test Find: -,, at end of test section - o,exit / o,inlet - for test section Assume: N is tg/cg,.4, stead, adiabatic, no work Fanno Flow - Coright b Jerr. Seitzman. All rights reserved.
13 School of Aerosace Engineering Analsis: e f (X.) e f f 3.0 f 0.005(00) e. (Aendix E) 70 ( o const) e 0.47 test another solution is 0.605, but since started >, can t be subsonic o 8K e Fanno Flow -3 Coright b Jerr. Seitzman. All rights reserved.
14 School of Aerosace Engineering (X.9) (X.7) 500 kpa 3.6kPa kPa.4 6.kPa.7 o,e / o,test o ( ) o o.7 (( ) ) (( ) ) (.4) test.4 5% loss in stagnation ressure due to friction e Fanno Flow -4 Coright b Jerr. Seitzman. All rights reserved.
15 School of Aerosace Engineering f m test 0.m f test 0 m long section would have at exit e Fanno Flow -5 Coright b Jerr. Seitzman. All rights reserved.
16 School of Aerosace Engineering ast roblem (suersonic duct), what would haen if calculated exit ressure ( e,f ) did not match actual back ressure ( b ) b < e,f : exansion outside duct (underexanded) e,f < b < e,sh : oblique shocks outside duct (overexanded) / o */ o shock inside shock at exit e,sh < b : shocks inside duct (until shock reaches ~throat) o test e e O U e,f x b e,sh Fanno Flow -6 Coright b Jerr. Seitzman. All rights reserved.
17 School of Aerosace Engineering Fanno Flow -7 Coright b Jerr. Seitzman. All rights reserved. Can t have flow transition to subsonic with ure Fanno flow shock in duct Shock location fixed b back ressure low enough b, e raise b, shock moves ustream until it reaches sonic location in nozzle / o b */ o o test e e e e b e x x
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