# HW 2 - ASE 362K Assignment 2 Thursday January 24th 2008 1...

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Unformatted text preview: ASE 362K Assignment 2 Thursday, January 24th, 2008 1) The problem below requires use of the conservation equations, perhaps one of them, perhaps two, perhaps all three. Air (Cp = 1.0 kakg-K) at a pressure of 547 kPa and a temperature of 318K 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 Bemoulli’s equation can be used to calculate the stagnation pressure to acceptable accuracy. ie' P0 = Pm + 1/2 p Um; . . . . . . . . . . . . "(1) In compressible ﬂow we can use the more general equation given below to calculate stagnation pressure. ie. P0 = Pm [ 1 + I .21 Ma: ] W1 ............ ..(2) which applies in subsonic ﬂow, supersonic ﬂow and also incompressible ﬂow. Show that for small 1V1no (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 1 - 1 M001 is small, so that the expression given in 2) can be 2 reduced to P0 = P00 (1 + x )" where x is small, and 11 = constant. Such an expression can be “expanded”.] 3) On aircraft, Mach # is usually obtained from a pitot-statie tube. The pitot-static tube 4) (in subsonic ﬂow) measures the ﬂow static pressure and flow stagnation pressure. 3.5 From the relation Pa = P[1+ J’T_1M 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 dPo 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 15 the uncertainty in M for T)— = 0.01 and P = —0.01 . Suppose 0 gland? are both positive and equal to 0.01 — what is the uncertainty in M? 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; CC2) 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. CTa/aw‘ai B‘V‘ Li ﬁfdv“ i'PL/ofé + (20cm m a a __ __ v I J 1 TN ,1 (D S f 3. 1 F 5'! VZ‘I\B I \ i >— 6) Q T __ '1- 6: (e + \L} bf ( a. 5‘- V (9 S @ W 15‘ no FQC‘F cold 11:15, “ dz: =- O Q '— 0 ho Work, dcwe [0% (com rag}; ® :0 m: Was: ﬁnﬁ‘ (‘9’, we‘ S‘M’éwﬁet («.8 a“; PM? VOIWQ chow-mam Uoif’u fume -* {339\$ w We CM. 1:; 2-; ﬁreﬂng V Hence- : COW rimm‘ax Rina: @ GHAJ © M “cu-*2. V g e 1-? v at; 1 @S g e S O? > ._ d M(e) . ’x/ ~ swam in - e + P/P No.0 Hﬁﬁ L3, ‘1: P rem¥é ﬁxe e2". Era}er Wham; Fhé Ccmﬁai GE [Vin-e Cow'ﬁ-oi ft . “VO‘LLMC "— (Wﬁzﬁmﬁ>§n where mar-J 11 § 3 P 0 mass n: 0: a‘r ewol 0‘? (graces;- O m.‘ :1; C5! - a} of fraud; . Re “new Wmsge’rreok a“ anng w‘ii‘t... eano‘ipu\ 9x ( it: I 5...! mpg) m, wag; HM; Page (“45> H' (Jo-Q. w‘%FQXE FAQ, r k .5 We beaxk'ﬂwﬁxu agar-.51: ems»! aka Has; Qrgmég M 3% Q We dxﬁ/erewcg w; W anal e WU x3 0% qu 3cm ﬁn Hui EMF; CA” We L566\\A4nh+\$ 0M4 K.) 0| Q‘XLAI; ’_"L 9-; C"_H\ 4 ‘ H U NOE: We Cue Act-{r CQAWHQG’L 1.“.wa lewc‘i’ugda Marmara gate-5 ———- 3% m endy QEE. Hana; [ml—M4 F ‘-'— m-el "M‘Q‘ amo‘ So m7“ = m4 ivy—'84 / _ New: m4 =— P\\/ -: {oi "soc 4: ~ "— \ E M (2%?) (293/. :- 0‘304 Sugsmum¢ in (PT J __ 8 7'- Cu [ CP/CV—K 3° CV7— Cg/f'ﬁ“ m1 = <_..so4>(/ooox31<z — 414-: x 2°11) (/ooc)(3:8> '(?r%‘3>(71> we CAO meg?” ii'Dr-m; Cw) Bec'kuth PO "' P+"€1._(pu —"‘*',—' Q 2’73“: Cray—cud R = P({+ \(Zr [12) © ._ We (QM .fxpha‘ 1" lawmcqi heave... po- __' q . (WW: W m) N r 1 ‘6‘___ . g... 7, 7748' POTP '= XPHZ (+7512 + . . '3'. H— ..l. (Dal : .1 .. . mt!qu = _l‘ VIZ—l— 1 1- [21" 1 l2? —_- :6; P11" 7‘, Pﬁ’P =- i Pun-z +t11 ‘I" J 7' ‘1" I Po -= P L 1+‘6_" H‘- H 3.. \03(f’o) - lot) 0’) = 35: .103 (H—Yj Hz) . . x»! . 2__ w ago _. of = 3; alCl‘J—Yi'i‘ﬁ") 9° ’- , _ ._ ‘ P \( ‘ Hz 2. Y" l+—Y:J H’- I '2. of.» -433 —=. YHa‘I_H Pa P (ii-‘5“! ha a. > aw 0‘3 : IT‘S:‘I‘1‘¢ - M 2. . OLEM’L? __..._____......——-.~ (30 P 01:! =-_ §+HZ dF’o—dP -M 'J-H" Po l+ dPo MGR M GOPAQ‘ O‘Md‘ (Dark [wsl'W-N- ‘__.—Il' "Fe '1’: (9 O N d— n». D “:3 k u 'oL. (“k I"? amok l O o . :15— _. O 1 - O =- 0 . 4%: 1' 0°“:- >202.) \. 2i). , WILM- . . S;- Iim ( H’: ; M FOL— (4—K H. v J SW ’6 5%“ (P°"°°*3°~> 5' L‘s“? L1" a\- 0°C [(I-w)(28})(2?3)]"‘ 331...}; n .. 10°C [(I-u9(28})(2°|s)] ‘1 = 96.45 h " {00°C I (1.?)(2839C37-3>3V‘ -: 337345 ‘I 7 A [Z "-'- (2 = umweno.‘ 30D CWQP M 2' moieadoxr “What-W" Q (“Sung”) .. ﬁlm/'2. =- H-IS?-3'/h-©K ("\C‘h-Aw) ‘- 8’311/14— = 1099-: TS/fraﬁ. . WLRF okay-Jr \6 7' \ -__ni Lbdb'oapg 1:; d‘gRic \O’= l"'|‘ Chin- an} (14.) D- mmmccb 3-— pm U m. [chaonmwﬂl Bosh—Is 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 pomt 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: K4193 159 I—"p 6p: (') The speed of sound, expressed as a function of the isentropic compressibility, is therefore: _ 6p 1 = — = — 1.60 C J ( an J PK: ( ) In liquids and solids, changes in pressure generally produce only small changes in temperature. Consequently, in isentropic or isothermal processes: <> ~ e) 6p I 6 p T Therefore, the bulk modulus of elasticity E may be expressed in terms of the compressibility: . E m *- K: The speed of sound may now be expressed in terms of the bulk modulus: — E l 61 C — p { - ) The bulk modulus o-f-water at 15°C is 2 X 109 N/mz, and therefore: _ £_ 2X109_ 41 c— p— ———103 —15In/s 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 m/s. m «w is how. Saich (Cm/mslelc FHA """ SECFi—owi‘lo. ...
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HW 2 - ASE 362K Assignment 2 Thursday January 24th 2008 1...

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