Page 56 - Fluid mechanics, heat transfer, and mass transfer

COMPRESSIBLE FLUIDS 33

M ¼ 1 is reached and passed, the normal shock where q c is the impact pressure measured behind a
reaches the trailing edge and becomes a weak oblique normal shock. M is obtained by an iterative process.
shock, the flow decelerating over the shock but First determine if M > 1.0bycalculatingitfromthe
remaining supersonic. A normal shock is created subsonic equation. If M > 1.0 at that point, then use
ahead of the object and only subsonic zone in the the value of M from the subsonic equation as the
flow field is a small area around the leading edge of initial condition in the supersonic equation.
the object. . What is choked flow?
. How does a convergent–divergent nozzle work for & Choked flow of a fluid is a fluid dynamic condition
compressible flows? caused by the Venturi effect. When a flowing fluid at
& As flow in a channel crosses M ¼ 1, it becomes a certain pressure and temperature flows through a
supersonic, one significant change takes place. Com- restriction (such as the hole in an orifice plate or a
mon sense would lead one to expect that contracting valve in a pipe) into a lower pressure environment,
the flow channel would increase the velocity (i.e., under the conservation of mass, the fluid velocity
making the channel narrower) resulting in faster air must increase for initially subsonic upstream condi-
flow and at subsonic velocities this holds true. How- tions as it flows through the smaller cross-sectional
ever, once the flow becomes supersonic, the relation- area of the restriction. At the same time, the Venturi
ship between flow area and velocity is reversed; that effect causes the pressure to decrease.
is, expanding the channel actually increases the & Choked flow is a limiting condition that occurs when
velocity. the mass flux does not increase with a further de-
& The obvious result is that in order to accelerate a flow crease in the downstream pressure environment.
to supersonic, one needs a convergent–divergent & For homogenous fluids, the physical point at which
nozzle, where the converging section accelerates the choking occurs for adiabatic conditions is when the
flow to M ¼ 1, sonic speeds, and the diverging section exit plane velocity is under sonic conditions or at a
continues the acceleration. Such nozzles are called de Mach number of 1. For isothermal flow of an ideal
Laval nozzles, and in extreme cases, they are able to gas, choking occurs when the Mach number is equal
reach incredible hypersonic velocities (M of 13 at sea to the square root of C v /C p .
level).
& The choked flow of gases is useful in many engi-
. “In a converging nozzle, the velocity of the gas stream neering applications because the mass flow rate is
will never exceed the sonic velocity, though in a con- independent of the downstream pressure, depending
verging–diverging nozzle supersonic velocities may be only on the temperature and pressure on the upstream
obtained in the diverging section.” True/False? side of the restriction. Under choked conditions,
& True. valves and calibrated orifice plates can be used to
. Give equations for calculating Mach numbers at sub- produce a particular mass flow rate.
sonic and supersonic velocities in air. & If the fluid is a liquid, choked flow occurs when the
& Subsonic Velocities: Assuming air to be an ideal gas, Venturi effect acting on the liquid flow through the
the formula to compute Mach number in a subsonic restriction decreases the liquid pressure to below that
compressible flow is derived from the Bernoulli of the liquid vapor pressure at the prevailing liquid
equation for M < 1: temperature. At that point, the liquid will partially
boil into bubbles of vapor and the subsequent col-
lapse of the bubbles causes cavitation. Cavitation is
2=7
p ffiffi
M ¼ ½5ðq c =P þ 1Þ 1Š; ð2:23Þ quite noisy and can be sufficiently violent to phys-
ically damage valves, pipes, and associated equip-
where M is the Mach number, q c is the impact ment. In effect, the vapor bubble formation in the
pressure, and P is the static pressure. restriction limits the flow from increasing any
further.
& Supersonic Velocities: The formula to compute
Mach number in a supersonic compressible flow & Chocked flow (or critical flow) for a liquid occurs
is derived from the Rayleigh supersonic pitot when the mass flux through a restricted area is at its
equation: maximum (velocity is sonic). If the downstream
pressure is decreased, the mass flow will not increase.
Besides the possibility of physical equipment dam-
p ffiffi 2 2:5
M ¼ 0:88128485 ½ðq c =P þ 1Þf1 ð1=7M Þg Š; age due to flashing or cavitation, formation of vapor
bubbles in the liquid flow stream causes a crowding
ð2:24Þ