Mechanics of Fluids

492 Compressible flow of gases
For gases the gz term in eqns 11.8–11.11 can be omitted as negligible
compared with the remaining terms because ρ is small and changes of z are
usually small also. (The concept of piezometric pressure p + ρgz cannot of
course be used where the density is variable.) Equation 11.9 may then be
reduced to
h + 1
2 u2 = constant along a streamline SF, A (11.12)
It is clear from eqn 11.12 that in steady adiabatic flow an increase of
velocity must be accompanied by a decrease of enthalpy and a decrease
of velocity by an increase of enthalpy. For a given streamline the specific
enthalpy is a maximum when the velocity is zero (at a stagnation point), and
this maximum value is termed the stagnation enthalpy, h0. From eqn 11.7
the corresponding stagnation temperature T0 is h0/cp and so the energy
equation may be written
cpT + 1
2 u2 = cpT0 SF, A, PG (11.13)
If an attempt is made to measure the temperature of a flowing gas by placing
a thermometer or similar device in the stream, the temperature recorded will
be greater than T. Equation 11.13 shows that the stagnation temperature
exceeds T by u 2 /2cp. For air cp = 1005 J · kg −1 · K −1 and so the stagnation
temperature for an air stream at, say, 200 m · s −1 exceeds the ordinary
static temperature by about 20 K. And the nose cone of a rocket travelling
through air at, say, 2 km · s −1 must withstand a temperature rise approaching
2000 K! For these extreme conditions the use of eqn 11.13 involves
some inaccuracy because of the variation of cp during the large increases
of temperature and pressure. However, although the stagnation temperature
would be reached at the stagnation point on a thermometer bulb, the
temperature would rise less at other points on it, so the mean temperature
recorded by an ordinary thermometer would be somewhat less then the stagnation
temperature. The static temperature cannot be directly measured by
any stationary instrument. (It could be measured only by a thermometer or
other instrument moving at the same velocity as the gas.)
Example 11.1 Air flows adiabatically through a pipe. At a plane,
denoted by suffix 1, the temperature is −2 ◦ C, its pressure is 1.50 ×
10 5 N · m −2 , and the air moves at a speed of 270 m · s −1 . At plane 2 the
pressure is 1.20 × 10 5 N · m −2 and the speed of the air is 320 m · s −1 .
Calculate the following properties of the air:
(a) the density at plane 1
(b) the stagnation temperature
(c) the temperature and density at plane 2.