Hugoniot adiabate

Einstein showed that when a reversible reaction is present sound dispersion occurs at low frequency the equilibrium is shifted within the time of oscillation, the effective specific heat is at a maximum, and the speed of sound c0 is at a minimum. At high frequency the oscillations occur so rapidly that the equilibrium has no time to shift (it is frozen ). The corresponding Hugoniot adiabate (FHA) is shown in the figure. Here the effective heat capacity is minimal, the speed of sound c is maximal cx > c0. From consideration of the final state and the theory of shock waves it follows that C>c0. 

System at rest in which the mixture is not 1 Hugoniot adiabate. moving, 2 explosive mixture, 3 reaction zone,... 

To each value of v thus corresponds a definite p. Taken together they form a curve in the p, u-plane—the so-called Hugoniot adiabate (Fig. 11). We find the value D corresponding to each v by the formula... 

We note that the use of the line drawn from the point A to one of the points on the Hugoniot adiabate CBG, e.g., the line AB, to geometrically determine the detonation velocity D is not related to any assumptions about the intermediate states of the substance in the wave or about the trajectory of the point which describes the state of the substance in the p, v-plane. On this see II.4. 

The state which corresponds to the minimum value of the detonation velocity possesses a number of remarkable properties in this state the extremum of the entropy is reached—a minimum on the Hugoniot adiabate and a maximum on the line joining the corresponding point with the initial one, pQ, v0, in the p, v plane. The detonation velocity in this state is equal to the sum of the velocity of the products and the velocity of sound in them. 

For a greater detonation velocity two regimes are possible, corresponding to the two points, G and C, of intersection with the line AGC and the Hugoniot adiabate (see Fig. 11). 

In particular, in the diagram of p, v the state of the mixture which has just been compressed by the shock wave and has not yet started to react is represented by the points Y or S (Fig. 15 here 1 denotes the original mixture and 2—the reaction products) (for two different values of the detonation velocity) of the Hugoniot adiabate, AYS, constructed without taking account of heat release. 

As a result of losses the state which occurs in fact differs from the point B on the Hugoniot adiabate calculated without losses we should have called it B. When drag is taken into account the line of state change in the reaction YB differs from a straight line. However, in the state B the basic property which distinguished the state B in the theory without losses is preserved, namely the condition of equality of the detonation velocity and perturbation velocity, D = w + c. 

In this formulation of the problem we may construct a regime with any flame speed with respect to the unperturbed—far from the flame front— mixture. In particular, all intermediate values of the velocity between slow combustion and detonation are possible. Why have they now become possible, while in the equations of the Hugoniot adiabate no real solutions corresponded to intermediate values of the velocity ... 

Completely unsatisfactory, however, are the arguments by which the lower branch BGE of the Hugoniot adiabate is excluded (see Fig. 1). Jouguet points out that the regimes described by the points on the segment BGE for which... 

Izmailov and Todes assumed that the point ascends directly from the initial state A, which obviously lies on the dynamic adiabate for the fresh chemically-unchanged gas, to the point B, located on the Hugoniot adiabate... 

In addition, in a shock wave (the Hugoniot adiabate) under strong compression the temperature increase significantly exceeds the heating in... 

It should be particularly emphasized that our considerations imply the possibility of propagation -of detonation with a velocity exceeding that calculated from Jouguet s condition, with the state G on the lower branch of the Hugoniot adiabate realized in the reaction products. This occurs if the igniting agent, e.g., radiation of the reaction products, travels faster than the shock wave and causes chemical reaction in the material which is in state... 

Fig. 68. Hugoniot adiabates of the initial mixture and of the reaction products (For explanations see text)...

A chemical combustion reaction proceeding at a constant rate occurs in a mixture ignited by shock compression. According to Eq. (47.3) this results in a linear change of the gas state. In Fig. 63 this corresponds to the transition of the gas along the straight line D K from the state represented by point D (and identical in composition to that of the initial mixture (D)) to the state of the reaction products represented by point pc, Vc(K), i.e. to the transition from the initial Hugoniot adiabate (initial mixture) to that of the reaction products. The latter is expressed by Eq. (47.2). 

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