Untitled - Aerobib - Universidad Politécnica de Madrid

96 CHAPTER 4. COMBUSTION WAVES
Finally, at the Chapman-Jouguet point α H = α, that is
( ∂s2
= 0. (4.21)
∂τ 2
)H
From these three inequalities it results that the entropy of the burnt gases is
minimum at the Chapman-Jouguet point and increases from there on, both in strong
and weak detonation branches.
Since at point J condition (4.21) is satisfied, the isentropic passing through this
point is tangent to the Hugoniot curve. Therefore, at point J the following conditions
are satisfied
α S = α H = α = α min . (4.22)
These values, when taken into Eqs. (4.15) and (4.16), give at point J
a 2 = v 2 , (4.23)
thus resulting the following important property: in a Chapman-Jouguet detonation
the velocity of the burnt gases with respect to the wave is sonic. This property is
the one that best characterizes a Chapman-Jouguet detonation. As will presently be
seen, owing to this property, the detonations physically observed are of this same type.
Jouguet stated this property in 1905 [3].
For the purpose of determining the character of v 2 in strong and weak detonations,
it is necessary, as previously seen in Fig. 4.3, to compare α and α s . This
means that it is necessary to determine the relative position of the isentropic that passes
through each point of H, with respect to the straight line that joins P to that point. For
the determination of this relative position it is not sufficient to know the entropy variation
along H, but it is also necessary to know the entropy variation along another
direction, for instance, that of the radius vector with its origin at point P . This can
easily be attained by considering the function
F (p 1 , τ 1 ; p 2 , τ 2 ) ≡ h 2 − h 1 − 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) . (4.24)
On the Hugoniot curve H the value of this function is zero, as results from (4.10).
This curve divides the plane in two regions: the lower region, that contains point P ,
representative of the initial state, and the upper region. F takes opposite signs in both
regions. As can be easily be verified, in the lower region F < 0 and in the upper
region F > 0. For this purpose, it is enough to study, for instance, the sign of F in P .
In B we have (Fig. 4.2)
F (p 1 , τ 1 ; p 1 , τ 2 ) = h 2 (p 1 , τ 2 ) − h 1 (p 1 , τ 1 ) = 0. (4.25)