Untitled - Aerobib - Universidad Politécnica de Madrid

4.3. VELOCITY OF THE BURNT GASES 95
α S
α > : supersonic α = : sonic α < : subsonic
α S
α S
τ ,
α
α S
( τ , )
( τ , )
( τ , )
2 p 2
S
α
2 p 2
α
α S
S
2 p 2
αS
P( τ 1,
p 1 )
P( τ 1,
p 1 )
P( 1 p 1 )
S
Figure 4.4: Schematic diagram showing the line connecting the initial and final states and
the corresponding isentropic curve.
relative to the wave will be supersonic, sonic or subsonic. Hence, the problem reduces
to a comparison between the slope of the isentropic that passes through each point of
H, and that of the radius vector that joins this point with point P of the initial state.
The three possible cases appear schematically in Fig. 4.4.
To perform this comparison let us start by studying the variation of the entropy
s, along the Hugoniot curve. The elemental entropy variation ds corresponding to the
variations dh and dp of enthalpy h and pressure p, is
T ds = dh − τ dp. (4.17)
Now, by differentiating (4.10), keeping p 1 and τ 1 constant and taking the result into
(4.17), the following expression for the entropy variation along H, is obtained
( ∂s2
T 2 =
∂τ 2
)H
τ [ ( ]
1 − τ 2 p2 − p 1 ∂p2
+
2 τ 1 − τ 2 ∂τ 2
)H
(4.18)
= (τ 1 − τ 2 ) (tan α − tan α H )
,
2
where α H is the angle between the tangent to H at point p 2 , τ 2 , and the negative
direction of τ axis.
Fig. 4.3 shows that in the strong detonation branch α H
virtue of (4.18), in this branch
( ∂s2
∂τ 2
)H
> α. Therefore, in
< 0. (4.19)
On the contrary, in the weak detonation branch α H < α, that is
( ∂s2
> 0. (4.20)
∂τ 2
)H