Untitled - Aerobib - Universidad Politécnica de Madrid

4.3. VELOCITY OF THE BURNT GASES 97
Similarly, in P
F (p 1 , τ 1 ; p 1 , τ 1 ) = h 2 (p 1 , τ 1 ) − h 1 (p 1 , τ 1 ) . (4.26)
Now subtracting (4.25) from (4.26), results
F (p 1 , τ 1 ; p 1 , τ 1 ) = h 2 (p 1 , τ 1 ) − h 2 (p 1 , τ 2 ) . (4.27)
But, τ 1 is smaller than τ 2 , and, since to reduce the volume of the burnt gases from τ 2
to τ 1 without varying their pressure p 1 it is necessary to cool the burnt gases, that is to
reduce their enthalpy, then
h 2 (p 1 , τ 1 ) < h 1 (p 1 , τ 2 ) , (4.28)
that is
F (p 1 , τ 1 ; p 1 , τ 1 ) < 0. (4.29)
Therefore F is negative in the lower region and positive in the upper region.
By means of the function F , the entropy variation can be expressed in a simple
manner by differentiating Eq. (4.24), keeping p 1 and τ 1 constant and then combining
the result with Eq. (4.17), thus, obtaining
T 2 ds 2 = dF + 1 2 (p 2 − p 1 ) dτ 2 + 1 2 (τ 1 − τ 2 ) dp 2 . (4.30)
But the variations dp 2 and dτ 2 corresponding to the straight line that joins P with
(p 2 , τ 2 ) must satisfy condition
dp 2
dτ 2
= p 2 − p 1
τ 2 − τ 1
, (4.31)
this expression, when taken into Eq. (4.30), gives for entropy variation along a radius
vector
T 2 ds 2 = dF. (4.32)
Therefore, along a radius vector, F and s 2 increase and decrease in the same direction.
Now, at point E (Fig. 4.5), F increases from E to E ′ . Therefore, s 2 also
increases from E to E ′ . In the same manner, at point E ′ , F and s 2 increase in the
direction E ′ E.
Furthermore, s 2 decrease along H in the directions E ′ J and EJ. Thereby it is
deduced that the isentropic curves that pass by E and E ′ must, necessarily, have the
positions shown in Fig. 4.5. There results that:
a) In E, α S < α and the velocity of the burnt gases is supersonic.
b) In E ′ , α S > α and the velocity of the burnt gases is subsonic.