Rayleigh Flow - Thermodynamics

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Benjamin Carson
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1 Sl f Aerae Engineering Rayleig Flw - erdynai Steady, -d, ntant area, iniid flw wit n external wrk but wit reerible eat tranfer (eating r ling) ρ Cnered uantitie (a, entu e.) ine Ant: a fluxρntant G ine n fre but reure: ρ ntant bining: G /ρntant On - diagra, an draw ti Rayleig Line Q? Rayleig Flw - Cyrigt by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Rayleig Line ρ ntant, ig reult in lw dδq/ (reerible) eating inreae ling dereae at axiu Cange a flux new Rayleig line (ρ) < eating ling > lw, ig ig, lw ax (ρ) Rayleig Flw - Cyrigt by Jerry. Seitzan. All rigt reered.

Rayleig Flw - Cyrigt 00-00 by Jerry. Seitzan. All rigt reered.

2 Sl f Aerae Engineering Rayleig Line - Cking Heat additin an nly Q inreae entry < Fr enug eating, e (Q in Q ax ) eating (ρ) eat additin an lead t ked flw Wat aen if Q in >Q > ax ubni flw: ut e t ax different Rayleig line ( ), i.e., lwer a flux uerni flw: get a k ( ) Rayleig Flw -3 Cyrigt by Jerry. Seitzan. All rigt reered. tay n ae Rayleig line: ρ and ρ al nt. fr nral k e Rayleig Flw -4 Sl f Aerae Engineering Differential a Euatin Silify (X.4-5) fr fda0 ( ) δ( d ) δ (X.) d d d (X.) d d d Cyrigt by Jerry. Seitzan. All rigt reered. dρ ρ d d (X.3) (X.4) d fdx d da δ D A (X.5) d d ign ange (X.6) (X.7)

Seitzan. All rigt reered. tay n ae Rayleig line: ρ and ρ al nt. fr nral k e Rayleig Flw -4 Sl f Aerae Engineering Differential a Euatin Silify (X.

3 Sl f Aerae Engineering Prerty Variatin Wat an ange ign in reiu euatin (- ) and δ (- ) in d,d fr <, jut like ite f and Heating: ; ling ite, ρ ite f ( ρ nt), ae a,ρ unle </ (lw eed flw:.,ρ) < > δ <0 >0 <0 >0,,, ρ, < - > - δ<0ling δ>0eating Rayleig Flw -5 Cyrigt by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Rayleig Line Extreu Cbine X.3 and X.7 d d d d 0 fr ax < eat l -0.5 d d fr > ax Rayleig Flw -6 Cyrigt by Jerry. Seitzan. All rigt reered. 3

,ρ) < > δ <0 >0 <0 >0,,, ρ, < - > - δ<0ling δ>0eating Rayleig Flw -5 Cyrigt 00-00 by Jerry. Seitzan. All rigt reered.

4 4 Rayleig Flw -7 Sl f Aerae Engineering Cyrigt by Jerry. Seitzan. All rigt reered. Integrated a Euatin Integrating (X.-7) δ δ d d ρ ρ (X.3) (X.9) (X.3) (X.30) (X.8) ( ) (X.33) ( ) ln (X.34) (energy) Rayleig Flw -8 Sl f Aerae Engineering Cyrigt by Jerry. Seitzan. All rigt reered. Slutin Arae w etd t le Rayleig rble, i.e., fr tate ange fr In bt, need t knw ne tate (inluding ) and eting abut ter tate ( D rerty r r ) etd : diret lutin f integrated euatin (X.8-34) reuire iteratie lutin etd : tabular lutin uing referene nditin ue ni referene nditin (iilar t A/A and Fann etd) ni nditin baed n eat tranfer reuired fr flw t g ni;,

Slutin Arae w etd t le Rayleig rble, i.e., fr tate ange fr In bt, need t knw ne tate (inluding ) and eting abut ter tate ( D rerty r r ) etd : diret lutin f integrated euatin (X.

5 Sl f Aerae Engineering Diret Slutin Ue tw euatin t find ange e.g., if tate and knwn ue X.8 and X.33 a) get rati fr X.8 den t atter w ange alng lengt, nly ttal reuired Q b) get fr X.33 (ut knw ) reuire iteratie lutin fr ) get ret f rerty ange fr X.9-34, e.g., fr X.9 Rayleig Flw -9 Cyrigt by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Cled Dut Exale Gien: Nitrgen flwing in ntant area dut, enter at 3, 50 K, 50 kpa and exit wit teerature f 00 K Find:.. (inluding diretin) K 50 kpa Q 00K Aue: N i tg/g,.4, teady, reerible, n wrk Rayleig Flw -0 Cyrigt by Jerry. Seitzan. All rigt reered. 5

9 Rayleig Flw -9 Cyrigt 00-00 by Jerry. Seitzan. All rigt reered.

6 6 Rayleig Flw - Sl f Aerae Engineering Cyrigt by Jerry. Seitzan. All rigt reered. Cled Dut - Slutin Analyi: ine tarted >, an t be ubni (energy: in ut) ( ) 0.,3.4 00K Q K 50 kpa () K ( ) K( ) K ( )K kl kg klk J kg 36kJ K < > eat l (fr X.30) Rayleig Flw - Sl f Aerae Engineering Cyrigt by Jerry. Seitzan. All rigt reered. Referene Slutin etd Referene tate a,,,... (X.36) (X.35) ρ ρ (X.37) (X.38) (X.39) ln (X.40) Nte: r/r f() nly Value fr.4 in Aendix F, Jn ni, but nt ae tate a A r Fann State ni in integrated euatin (X.9-34)

4.4 kg 36kJ 9.4 50K < > eat l (fr X.30) Rayleig Flw - Sl f Aerae Engineering Cyrigt 00-00 by Jerry. Seitzan. All rigt reered.

7 Rayleig Flw -3 Sl f Aerae Engineering Ue f able get ange in, ue ange in rerty rati (like uing A/A), e.g., Cyrigt by Jerry. Seitzan. All rigt reered. ) again get rati fr X.8 Fr exale, if tate and knwn Q ) lk u / at 3) alulate / at 4) lk u tat rrend t alulated / Sl f Aerae Engineering Heated Dut Exale Gien: Air flwing in ntant area dut, enter at 0. and gien inlet nditin and a eating rate f 765 kj/kg Find:. e, e, e. ax fr e i 0. i 300K i 600 kpa Q e Aue: i tg/g,.4, teady, reerible, n wrk Rayleig Flw -4 Cyrigt by Jerry. Seitzan. All rigt reered. 7

8 Fr exale, if tate and knwn Q ) lk u / at 3) alulate / at 4) lk u tat rrend t alulated / Sl f Aerae Engineering Heated Dut Exale Gien: Air

8 Analyi: Rayleig Flw -5 Cyrigt by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Heated Dut - Slutin i 0. i 300K e i 600 kpa e i i (X.8, energy) J / kg J 004 ( 300K ) kgk e e i i 0. e ( ) i 765kJ e 0.45 kg e anter lutin fr 3.53, but ine tarted wit <, an t be uerni e e i kpa kpa / / ο (Aendix F) e ax Sl f Aerae Engineering Heated Dut Slutin (n t) e e e e i i e ax i 0. i 300K i 600 kpa ( 300K ) K 765kJ reuire ked exit, e K 300 ( ) 78K K J J K. 43 kgk kg kg e Rayleig Flw -6 Cyrigt by Jerry. Seitzan. All rigt reered. 8

0 0.7355.3460 0. 0.8943.34 0.44 0.59748.3936 0.45 0.6393.3508 0.46 0.63007.308 3.53 0.6393 5.47054 (Aendix F) e ax Sl f Aerae Engineering Heated Dut Slutin (n t) e e e e i i e 3. 54 0 0.

9 Sl f Aerae Engineering Cbutr Exale Gien: Air entering ntant butr, enter at 0. and gien inlet nditin. fuel eating alue f fuel (45.5 J/kg fuel ) Find: i 0. i 500K. e and are t alue yu wuld alulate if negleted kineti energy ange. l fuel Aue: i tg/g,.4, teady, reerible, n wrk, fuel << ( an neglet a additin, jut ue ) e Rayleig Flw -7 Cyrigt by Jerry. Seitzan. All rigt reered. Analyi: e Rayleig Flw -8 Cyrigt by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Cbutr Slutin (energy) i 0.? butin jut e i 500K anter kind f i eat additin i fuel kgfuel J J kg kg fuel kg i i 500K(.008) 504K e i 504K( 4.75) 376K 004(504) (Aendix F) e e i 4.75(0.736) 0.88 e 0.60 i e 376K e 6K ( ) ( ) e een fr ti ubni flw, kineti energy redue final by 60K 9

Seitzan. All rigt reered. Analyi: e Rayleig Flw -8 Cyrigt 00-00 by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Cbutr Slutin (energy) i 0.

10 Sl f Aerae Engineering Cbutr Slutin (n t) i 0. e / i e i 500K e e.0753 i i fuel (Aendix F) S eat additin (burning fuel) in an engine gie l a wa ae wit fritin A we le (~3% l ere) le trut le wrk utut Rayleig Flw -9 Cyrigt by Jerry. Seitzan. All rigt reered. Sl f Aerae Engineering Cking Heating f a flw an lead t king jut like fritin Fr uerni flw, deending n bak reure, an get underexanded flw erexanded flw k inide dut / / > Q k inide e e k at exit O U x b e, e,r Rayleig Flw -0 Cyrigt by Jerry. Seitzan. All rigt reered. 0

Flw -9 Cyrigt 00-00 by Jerry. Seitzan. All rigt reered.

Recall Gibbs eqn. ds. Using h version. for ideal gas. integrate AE3450. for ideal gas. integrate s. s(t,p) behavior? AE3450. T p.

Recall Gibbs eqn. ds. Using h version. for ideal gas. integrate AE3450. for ideal gas. integrate s. s(t,p) behavior? AE3450. T p. Enti Shl f Aeae State Engineeing Eqn. Ideal Gae du eall Gibb eqn. d f ideal ga d integate d d d Ideal Ga Enty State elatin - Cyight 03 by Jey M. Seitzman. All ight eeed. d d d d d ln d ln Enti Shl f Aeae