Lecture_37_notes

Lecture_37_notes - ChE 374—Lecture 37—Compressible...

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Unformatted text preview: ChE 374—Lecture 37—Compressible Flows o Compressible ﬂows occur at high speeds (B.E. AP = 1/2pAv2.) — Gas can convert signiﬁcant internal energy to kinetic energy. — Density decreases, temperature decreases. — Mach number: M = 11/6, where c is the local speed of sound. — Compressible flow for M > 0.3 —> density ratio R: 0.95 —> 5% diﬂerence. * Most scalings go with M2. 0 Sound Speed — Follow a pressure pulse (frame of the pulse). * Mass balance (pAv) and momentum balance (pressure force, ﬂow in, flow out) give (for ideal gases): _ g g ‘ (kRT U‘ 8p 5“:— M’ where k = Cp/CV, and M is the mean molecular weight. 7 Values for air: >1: k = 1.4 >0: Air at 25 C = 77 F: c: 346m/s 2 774mph * Air at 2000 K = 3140 F: c = 896 m/s 2 2000 mph - (heavier gases give slower speed, and higher temperatures give higher speed). — Water at 15 C c = 1490 m/s 2 3333 mph. — Steel at 15 C c = 5060 m/s 2 11318 mph. 0 Analysis: SS, no friction, 1D, Ideal Gas # Energy balance from a reservoir into some pipe /valve / etc. * No heat, no shaft work, no gravity effect, P/ p +11, = h, and dh = deT. Use deﬁnition of k, and 0 given above to obtain: T. _ Mm — 1) T1 2 where 7“ denotes the reservoir (no velocity), and subscript 1 is the point of interest. — For ideal gases: Pr/Pl = (Tr/T1)k/(k_1), pT/pl = (Tr/T1)1/(k’1). So: P _ k/(kAl) .1 = + 1) +1, P1 2 k _ 1 1/(Ic71) & = (M2— + 1) [’1 2 0 Flow in nozzles. — For incompressible ﬂow, the density is constant and have m = pAv. — For compressible ﬂow, the density drops. For M < 1, the density drops slower than velocity increases for a given change in area and we need a converging nozzle as usual to increase the velocity through the nozzle. For M > 1, the density drops faster than velocity increases for a given change in area and we need a diverging nozzel to increase the velocity through the nozzle. Hence the converging/ diverging nozzles on rockets. o Choked Flow: Pressure ratios below 0.53, result in choked ﬂow in valves, etc. For choked ﬂow, there is a maximum flow rate, independent of what is done downstream. — Most control valves, safety valves, bike tire valves are choked. — YOU MUST KNOW AND UNDERSTAND THIS CONCEPT 414:4 37~ (emprecsfblc Flows 'Pfam‘owb/ ; (0,04, ﬁlo“)! /D Val/2"“ ’- gffrr' ‘Nwd I [Mar 4)? ‘30, A? / / I“ (356 donor/2+4 DAMN”, may £0 kinn’r‘c ' T ‘Dfop; '/ / brorf V 7 Evy/mu”? 67’! “SL444 f I‘S Coma”! ‘— r/(a‘lﬁ “MM ﬂZ’ VA ; a Is 7m local «M __'____/ ‘3'“! C Omim “>171 {-twﬁptmﬁ/u I " Caqqrrgef‘vk (a, 2% 70.3 w—A» 20 N 0.75 a? 52 — M094" 'gta “4‘7 9 (9o Q Inf-L» 772 1' / 4am: grad ' f) a :+ F W f+3f v+cN c‘b's line.) A "Frees-am 'ka‘gf _ R'IJC m Pvdgc *—> Flux) Fﬁ/e-‘J 0'?- C'v‘ wank? «{ (Soc/‘46) £15653 7 C M Ma\$<7 Balcmq’ ; r91: -—--r /A\} : (/94-Jf)A(Vv‘Jv) dﬂ/1y#'fdv+VA/°+M ian VIOanJ'qv-dﬁalant( .' 6g, 0’) Pftsgw ﬁmff/o “L AP~A(PH?> = MUN”) * 9‘ \f 1% =‘/°A\) A (was I "AP -: fVCN fulfdv:f%—P—\ . Not: Ana I}; v'l’aLr Flo»; {Low-x 95‘” LIA/>76 Rwrww'lf, V“). ' Alec a) ' 7149 Ié A44 M ', twwy7 T5“ (9 6. 6+4? 3 ‘cmlr .‘ \}I‘&V\ Ca“.C)"/V"a*’\ CDIIC‘EPD’Ith'V {o 4 61‘ufm P/uk)_ (‘4: 6‘ (12’? um I‘ an} 726%?” 6~\(‘ ( (2+4Jfl I’D; 664-7 I no fifc7l/‘w’ "qnm 61045717 F/‘V 7!: A(:+u}%/¥€/Z> __ yk(:+u4i1%2) av 1 12, f 1 k. 1(w-LJ ZCP(Tr '17) y ‘1 ,ﬁ W a: 1;» : [AV " OWL A4 A f“ L([M <6 I / E'vMuc-lﬁ V = L go}, 6%) A, a): _ 1L4 kl : /0*\) /' ,‘MMJ f’ f5; T 4"] AJF *"' (.07 I *6 W910”)? {aura/MA / A Daran/Q F0? ‘2‘ ! A9 u 001/ harm/4’ A ;v\[{6't¢4_¢&l "chw /> " Der/M7 Fm+m TAM A at; DC {PM bf / M“ in. .1 £0 0 - w “- (0%{aw‘}, V 6 MM Le F m 74V “CkoLtJ Hod. W (mm a 1477!: \ / ‘P “>34 A" {q a Raceway A c1 .7)pr 'ﬂﬂ “Passw owhl'rjfl , Vela”? “gum in Acayk an Luv} gawk Valeria? I 1U; {Lani khl, {)M‘H‘J’l Rdww‘ (armof’ «56 («amnum‘gﬁJﬂJ wivng if I Calm-J 9%" “ﬂu mac] 5 "if 7L, pint! épad —” (Jam/Q éwcc’ I 77"" (eta-Ac)“, (QMMLAAI‘(¢_J{ " M/ V0.26 J WW WM a "inaximw w; #0 WM: um 7%) “'27 Chance) jflod‘ 9 (a7 [Adam axmo {a {'0’ MW Floyd, (cm ékcmat m ﬂwwuo‘m {0 5,154.79 We“). PF [7. L44 - ; P4? “L ; : 0‘93 “7}! ) Pa 13») I. : I}; 2 02333 156—71232 # Tf [0H ' L. :1: :7 .3: y“! \$0.437 fr; LH (av-«mom :« \LAvm, mfg/w, \FM‘UHM C/g-kwo, 40:1? 4% fr» MM :9 (Low) ft.» A #415,, W. > r109; 65/, (M40) Ma» M. Mr), W W9,‘,.& is. a} ﬂu wlim‘wxv-“ﬁ %\(€a 1‘4 TL: ‘4U‘L’ZV /\}tv{v{ ...
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