A couple of days ago, I used the speed of sound in air to determine whether
or not the top of a rotating tire at 300 mph was supersonic, i.e Mach 1 or
greater. I promised that if there were questions regarding the numbers and
methods I used, that I would post the derivations of those methods. Well, I
have received 2 requests, in the form of questions but not challenges, so I
figure maybe moe of you would like to know how I got to the speed of sound
in air. It is generally just a given, but here, 'tiz....
Oh, All of this is taken from one of my really olde aero text books, yeah, I
kept them all: "Airplane Aerodynamics, 3rd Edition", Daniel Dommasch, Sydney
Sherby, and Thomas Connolly, 1961, article 2:7, pages 46 - 48.
I need to establish some rules and symbology because this Outlook Express
does not give me the flexibility to use diffeent fonts for different
symbols.
So: Pressure = P
r = density (rho)
dVa = differential of Va
Temperature = T (deg Rankine)
Va = Velocity of sound disturbance
dP = differential of P
Gravity = g
dr = differential of r
A = cross sectional area
^ x means to the power of x
* means multiply
n = ratio of specific heats for air (1.4)
SQRT means square root of
R = gas constant for ait (53.3)
The analyses is set up considering that we "snap our fingers and we hear the
expanding sound wave. Because the sound wave is ever expanding outward in
all directions, it is hard to grasp the idea. So the use of a "pipe" is
which a disturbance is caused is the method of choice for a free body
diagram, so to speak...
"Imagine a long tube, open at one end through which we can fore air at any
velocity of our choosing. A presure disturbance is produced at one end, and
the flow velocity quickly adjusted to equal the rate of propagation at the
disturbance. This disturbance will be fixed in the tube and we may examine
it without regard to relative velocity effects." [since i cannot draw a
picture here, let me describe a picture: a pipe open at each end with flow
on one side of a disturbance equal to Va, with P and r as conditions. On the
other side of the disturbance are conditions Va + dVa, P+dP, and r +dr.
"From the continuity equation
r A Va = (r +dr) A (Va +dVa)
accross the wave (this is our disturbance) must equal the rate of change of
momentum across the wave, the rate of change of momentum being the
difference between the inflow and outflow of momentum per unit time.
The inflow momentum in unit time is
r A Va Va = r Va^2 A
and the outflow momentum in unit time is
(r + dr) A (Va + dVa) (Va + dVa)
but since the flow is continous, this is
(r A Va) (Va + dVa)
and the change in in momentum in unit time is
r A Va * dVa
the statically unbalanced pressure force is
P (A) - (P +dP) A = -dP A
and Newton's second law gives us
dP = - r Va dVa
If we expand the continuity equation we have
r A Va = r A Va + A Va dr + A r Va + A dr Va
or, because dr dV is insignificant (second order error, really small)
compareed to other terms, we have
dVa = - (Va / r) dr
hence our equation of motion is
dP = Va^2 dr
or
Va^2 = dP / dr
Thus we see that, were the fluid incompressible (dr = 0) that Va would be
infinite. Actually, however, we have that for frictionless adiabatic flow
P / r^n = constant
and
dP = n r^(n-1) (constant) dr = n r^(n-1) (P / r^n) dr
or
dP / dr = n P / r
Hence
Va = SQRT ( n P /r)
since
P / r = g R T
we may wrie also
Va = SQRT ( n g R T)"
[note that at this point the specific medium is not considered, could be a
mixture, a different gas, or with humidity. the ratio of specific heats for
the new medium woould need to be known as would the gas constsant for that
new medium]
Va is the sonic velocity and if we plug in the values for the variables n,
g, R then we get
Va = SQRT ( 1.4 * 32.2 * 53.3 * T)
Va = 49.018 SQRT (T)
Of course, you must keep the unit in sync. Temperaure must be in degrees
Rankine (459.6 + T fahrenheit), R must use the English system. Oh, and
notice no dependence on density, only temperature. This was one of the
fundamental notions of aero back when I started my education.
See, I told you it wouldn't be hard!
mayf, the tired, red necked ignorant desert rat in Pahrump, who now has door
hardwar to lock up the new house, cabinet doors, lots of gracel in the
driveways, ...
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