ASE_362K_Assignment_2_Solutions

ASE_362K_Assignment_2_Solutions - ASE 362K Assignment 2...

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Unformatted text preview: ASE 362K Assignment 2 Thursday February 15‘, 2007 1) The problem below requires use of the conservation equations, perhaps one of them, perhaps two, perhaps all three. Air (Cp = 1.0 kJ/kg-K) at a pressure of 547 kPa and a temperature of 31 SK flows through a pipeline. A tank, which is connected to the pipeline through a valve (see sketch below), initially contains air at a pressure of 101.3 kPa and a temperature of 293K. The tank has a volume of 0.25m3. Assuming an adiabatic process, determine the mass of air which flows into the tank when the valve is opened if the final pressure in the'tank is the pipeline pressure. 2) In incompressible flow Bernoulli’s equation can be used to calculate the stagnation pressure to acceptable accuracy. ie. P. = R. + a p U...“ ............ ..(1) ressible flow we can use the more general equation given below to calculate In comp stagnation pressure. 16- P0 = Poo [ 1 + ILL Ma; 1 W ............ ..(2) 2 which applies in subsonic flow, supersonic flow and also incompressible flow. Show that for small Mm (ie. < about 0.3 say) that the second, more general expression, can be reduced to essentially Bernoulli’s equation. [ Hint: when M is small, then y - 1 M001 is small, so that the expression given in 2) can be 2 reduced to P0 = PCL. (1 + x )“ where x is small, and n = constant. Such an expression can be “expanded”.] 3) 4) On aircraft, Mach # is usually obtained from a pitot—static tube. The pitot-static tube (in subsonic flow) measures the flow static pressure and flow stagnation pressure. 3.5 From the relation Pa = P[l + LEM 2] the Mach # can be obtained. (To get velocity a separate measurement is made of To. From M and To, one can calculate T, then a, and finally from a and M, get V). Any measurement always involves uncertainties. Suppose the uncertainties in static pressure P and stagnation pressure P0 are dP and (1?0 respectively. First, determine an expression for the resulting dP uncertainty in M (namely, dM) in the form %- = f[M,d—; , P” ]. At a nominal dP dP Mach # of 0.8, what is the uncertainty in M for 73— = 0.01 and = —0.01 . Suppose 0 find-é;- are both positive and equal to 0.01 - what is the uncertainty in M? P 0 Calculate the speed of sound in air at 0°C, 20°C and 100°C. Calculate the speed of sound in gaseous helium, hydrogen and Freon (CF; CCz) at 20°C. Do a little research and find out how the speed of sound is calculated in liquids, then calculate the speed of sound in water at 20°C. M IS no ‘nec‘k— mold 11:15, 7. Ci =- 0 O HO work dam-Ia (00% Eovcgfi, © :.0 “are [S (1‘59 no {mad—7c Famsgzfi r F V - Moss. VH6 {Ca/a SxMfi-wflei in. ca.» Hag («62' erg? Hne VOIWQ ckomdaz woth hue -' Hue $54.5 CV, . I“: 2., fins/«wdx‘ l5'-’\Cf-’— COW“ OMgm‘k Rim-5 @ aa/«ol g) m have. V g C86 1"?)1 d3 z@§<€€) s ' V 5‘: [Va—30' H‘fi [L3 mpfiwwh Hue enhqifx CCMWQCI-GT‘J‘ nhu’u. “we Cantata“: gwfcm ME “(\Q Cojfic‘ Vo‘wfi‘e "—'- szwm4>k Luke-e mlg' mass \vé’ CV a‘r and 0-? Carmina C) «~14 =2 madb 1»; C5! - 0" of frames: . We mas“: WMS-gea’tfid a“ ear-e}; Lu.fo 6%anpr in (:14 Hang We Md»; Hne; P-F‘ae. Inge, air EfiaxUAMggL r e u . 2 flag 1 @ NoE: WE We no?!" (bummed wlw lewok‘ud“ \ "H—J ata‘Fé. Hence. _[ml-—M4 k '-'- mjel—‘M‘Q\ GAME?x 9° N; = FY14 krve" NW MA '3' va -:— [of 3:00 '1)— 12 TA (2??) (2% 3,) 7'- O‘ZOA Sugstums in: (PT I E 7"- CuT CP/Cu =lf 39 C, = Cp/t-q— m2 5 _.<-_3-o4>(looo><3\% - #lH-‘s x 1‘11) (/oooDC’égg/ --(?u+-3>(71> U39, do We? (00%") g; k I _- 7 J 1 ha“) m1 "-1 .. .1 Fig“ 12 ‘2. E 2 lvg l —— . 3' I guiss‘waA‘funfdx I; w ear--~ ouefleggfw R _ W . U ml 3 ('10: g/glglooo— 2001 2010 935000 fl (1r%-1>(5"1flxto§\:§ 0.52 2%? m; _______._...__—.-——---~—--—-— 3183000 -' a Q awe/N.Z ml «— 2172!? a .‘\ M; T. l HO O . Am = I-f}'3"'§ol a “$72. m Bernaull; Po " Pry-'11,} (DH? _,____.__.— Q) ' b’45“! G-cmud R = P(l+ kg} [12) _______® when: ac- <<. 1 and ,_ hr— c9»)?- ., We (QM .!.K(=c.~.al 1" low-{qui heave“. F; =. + 9'2 ' +.. , Ea; 3(3) .5 6 .4. .. r 4'3. PofP... = *5 RH" l 4- ff + "a". u— ..l. 2- :. l = —l (P [‘12 YR” u _. ,_ . w_ ._ l 1 (a 2 pr) 1 ‘2‘?) t; r11 3‘. PO’P = l I. gig: amok M QoFAq‘ and ‘baHn-Pou'fibv. 69 01f -;. .o-o-\ GM die 1° 1’0 01-31 __-_.=_ "3"" M CDC-8s) .__.__ é; “DC-o 02..) in [‘1 =0 Umwéréa‘ 0.) (Ma‘- g'll‘f/z H—.5}-5'/h_©|< 333'? /H- 1039‘s TS/kSK. 7 LbAJ'OW, I). Amman \6 1"? C‘ka w (He) \5 m~«vs»'.—.-c> {-— Ma} 5%! la; [o-ecmsacwbsl [(l.sp,)(zm9(2a93"‘ .m- - |ooKMJ5. '\ g “Hf/non = 6E'973/koh'. 0‘ (CAM/(22‘ Wal. he” 0 Iowa: IA _5° .. a: L(I>((?-}9D(2‘)9)hz "- “1-244. " am w: a c 015:5 l 'L 9wch 0‘ Me‘s 3"”85 a_='(%‘;$= (a) use-oi a lame> On the other hand, in an incompressible medium there is no change in density as the pressure changes. Therefore, the velocity of sound is infinite, and a pressure pulse generated at any point is sensed instantaneously at all other points of the medium. . ' The speed of sound through a medium is related to the isentropic compressibility of the medium. The isentropic compressibility K, is defined as: K -‘—I 53—” 1 59 I“ ,0 6p S ( - ) The speed of sound, expressed as a function of the isentropic compressibility, is therefore: 6p 1 = — = -— .60 c J(ap), Jsz (1 ) In liquids and solids, changes in pressure generally produce only small changes in temperature. Consequently, in isentmpic or isothermal processes: (2% (Q) 6p 5 6p T Therefore, the bulk modulus of elasticity E may be expressed in terms of the compressibility: 1 . E as -— Ks The Speed of sound may now be expressed in terms of the bulk modulus: — E 1 61 C - p ( - ) The bulk modulus of water at 15°C is 2 X 109 N/mz, and therefore: _ .13- m_1415m, c_ P‘ 103 fl 3 This is about four times the speed of sound in air at the same temperature. At this same temperature, sound travels through quartz at 5500 m/s and through steel at 6000 mfs. me about \5 Paw- Samd (Can-ssilslc Fluil'A SECFTowl'lO. ...
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This note was uploaded on 02/07/2011 for the course ASE 362K taught by Professor Dolling,d during the Spring '07 term at University of Texas at Austin.

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ASE_362K_Assignment_2_Solutions - ASE 362K Assignment 2...

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