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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/kgK) 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 pitotstatic 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 ﬁndé; 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
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"a". u— ..l. 2 :. l = —l (P [‘12 YR” u _. ,_ . w_ ._ l
1 (a 2 pr) 1 ‘2‘?) t; r11
3‘.
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[‘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; [oecmsacwbsl [(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 " “1244. " am w: a c 015:5 l 'L
9wch 0‘ Me‘s 3"”85 a_='(%‘;$= (a) useoi 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 (Canssilslc
Fluil'A SECFTowl'lO. ...
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 Spring '07
 DOLLING,D
 Fluid Dynamics, Aerodynamics, Speed of sound, ........., Stagnation pressure, Bulk Modulus

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