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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.
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 Fall '12
 DavidLignell

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