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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 kakgK) 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 pitotstatie tube. The pitotstatic 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
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POTP '= XPHZ (+7512 + . .
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7‘,
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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 'JH" Po l+ dPo MGR M GOPAQ‘ O‘Md‘ (Dark [wsl'WN ‘__.—Il' "Fe '1’: (9 O
N d— n». D “:3 k u 'oL. (“k I"? amok l
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H. v J SW ’6 5%“ (P°"°°*3°~> 5' L‘s“? L1" a\ 0°C [(Iw)(28})(2?3)]"‘ 331...}; n .. 10°C [(Iu9(28})(2°s)] ‘1 = 96.45 h " {00°C I (1.?)(2839C373>3V‘ : 337345 ‘I 7 A [Z "' (2 = umweno.‘ 30D CWQP M 2' moieadoxr “WhatW" Q (“Sung”) .. ﬁlm/'2. = HIS?3'/h©K ("\C‘hAw) ‘ 8’311/14— = 1099: TS/fraﬁ. . WLRF okayJr \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 ofwater 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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