Page 30 - Aerodynamics for Engineering Students

Basic concepts and definitions 13

It is often convenient to link the enthalpy or total heat above to the other energy of
motion, the kinetic energy w; that for unit mass of gas moving with mean velocity Vis
- v2
K=- (1.16)
2
Thus the total energy flux in the absence of external, tangential surface forces and
heat conduction becomes
VZ +
- cp T = cp TO = constant (1.17)
2
where, with cp invariant, TO is the absolute temperature when the gas is at rest.
The quantity cpTo is referred to as the total or stagnation enthalpy. This quantity is an
important parameter of the equation of the conservation of energy.
Applying the first law of thermodynamics to the flow of non-heat-conducting
inviscid fluids gives

(1.18)

Further, if the flow is unidirectional and cvT = E, Eqn (1.18) becomes, on cancelling
dt,
dE +pd(i) = 0 (1.19)

but differentiating Eqn (1.10) gives

(1.20)

Combining Eqns (1.19) and (1.20)
1
dh = -dp (1.21)
P
but

(1.22)
R
which, together with Eqn (1.21), gives the identity

P
*+ypd(j) =O (1.23)
Integrating gives
(3
In p + yln - = constant

or
p = kpy (1.24)
which is the isentropic relationship between pressure and density.