Page 77 - Aerodynamics for Engineering Students

60 Aerodynamics for Engineering Students

constituting the control surface of a ‘thermodynamic system’ or control volume. At
sections 1 and 2, let the fluid properties be as shown.
Then unit mass of fluid entering the system through section will possess internal
energy cVT1, kinetic energy $2 and potential energy gzl, i.e.

(2.9a)

Likewise on exit from the system across section 2 unit mass will possess energy

(2.9b)

Now to enter the system, unit mass possesses a volume llpl which must push against
the pressure p1 and utilize energy to the value of p1 x l/pl pressure x (specific)
volume. At exit p2/p2 is utilized in a similar manner.
In the meantime, the system accepts, or rejects, heat q per unit mass. As all the
quantities are flowing steadily, the energy entering plus the heat transfer must equal the
energy leaving.* Thus, with a positive heat transfer it follows from conservation of energy

However, enthalpy per unit mass of fluid is cvT +p/p = cpT. Thus

or in differential form
v2
d
-(cpT+l+gscosa) =$
ds (2.10)
For an adiabatic (no heat transfer) horizontal flow system, Eqn (2.10) becomes zero
and thus

V2
cp T + - = constant (2.11)
2

The equation of state
The equation of state for a perfect gas is

P/(m = R
Substituting forplp in Eqn (1.11) yields Eqn (1.13) and (1.14), namely

R
~p - cv = R, cP = - cy=- ‘R
Y-1 Y-1
* It should be noted that in a general system the fluid would also do work which should be taken into the
equation, but it is disregarded here for the particular case of flow in a stream tube.