ESTIMATION OF THE WIDTH OF THE STATIONARY DETONATION WAVE IN THE MODEL OF ZEL'DOVICH- NEUMANN-DÖRING

The models of Chapman-Jouget and Zel'dovich-Neumann-Döring are considered to estimate the width of a detonation wave in gas mixtures. Improved software package was developed earlier. The calculations of stationary detonation wave width are in good agreement with the experimental data. Estimation of the stationary detonation wave width is important for the detonation control.

From the viewpoint of explosion protection the most important problems are the detonation initiation, the structure of detonation waves, the detonation limits (on concentration and initial pressure), and the limiting detonation regime (spin and a gallop). All of these problems are connected with the problem of stability of one-dimensional stationary detonation wave. A detailed analysis of the detonation wave instability allows to calculate the structure of multifront (cellular) detonation, estimating the cell size by the wave length of perturbations with maximum velocity for amplitude increase. The inversely proportional dependence between the detonation cell size and the initial combustible mixture pressure allows to control the detonation process, which is important for solving practical problems of explosion. For these cases a problem of estimation for a detonation wave width, including a chemical reaction zone and / or the induction zone, is important, especially if there are no experimental data. Solving of this problem is the purpose of this study. V.A. Michelson [1] was the first who gave physical explanation for observations of detonations in gas mixtures. Chapman and Jouget [2,3] developed idea about detonation wave as a shock wave with the small time of chemical transformation directly in the front. Reaction zone was considered simply as a surface of discontinuity. Their hypothesis that the velocity of detonation products is equal to the local velocity of sound closed the set of equations consisting of conservation laws and the equation of state. Coincidence of calculations and experimental data was the best confirmation of the validity of this classical hydrodynamic theory. However this theory doesn't explain existence of limits for detonation propagation, because the width of reaction zone simply equals zero.
According to the classical ZND-model, transfer of detonation is caused by the shock propagation on explosive, that could initiate an intense chemical reaction. Energy-release of chemical reaction maintains the constancy of wave's parameters and stationary character of detonation process as a whole. There are gradually expanding detonation products behind the chemical reaction zone. Thus, the propagation of the detonation wave consists of a shock wave, the chemical reaction zone and final detonation products. As it is shown in Fig. 1 the initial explosive is separated from the chemical reaction zone of the shock wave B-B, and the surface H-H separates the chemical reaction zone and the detonation products for stationary detonation. So after compression by shock wave the initial explosive gets into the point В (the Neumann peak), and then it goes into the point H (the Chapman-Jouget point) during the chemical reaction.
For a description of the stationary detonation process it is enough to compare the state of the initial explosive, the shockcompressed state and the reaction products: where 0 wdetonation velocity that equals to the velocity of the chemical reaction zone;  For the intermediate states in the chemical reaction zone there is dependence between the internal energy and chemical composition. Thus, the processes in the stationary detonation wave are considered with the kinetics of chemical reactions that produce heat. For simplicity, it is assumed that the chemical kinetics equations can be written with a single variable. Concentration by weight of a substance  is selected as such a variable. The dependence of all values on time is determined by the equation of chemical kinetics. Consequently, ZND detonation model can be written as a set of equations for conservation laws of mass, momentum and energy, applicable to any control surface inside the detonation wave, and equations of chemical kinetics: where the exponential quantity m is associated with the so-called order of reaction.
where     is a composite function, which is defined for each gas reaction. The total derivative of value  is: Автоматизація технологічних і бізнес-процесів Volume 9, Issue 1 /2017 www.journal-atbp.com It results from this that The equation of chemical kinetics in general can be represented as follows where l n m , , are some constants.
  x w can be obtained by numerical integration of the equations of gas dynamics and the chemical kinetics (2,8). First of all the profile of pressure distribution is interesting. Calculations for gas systems on the basis of Arrhenius kinetics in its different modifications were produced in [7].
To estimate the width of a detonation wave in ZND-model the following was proposed by Zel'dovich [8]. Thus, the width of a detonation wave can be estimated. Note that earlier this approach was proposed to use in [9], and also it was noted that most of the difference methods for calculating of detonation waves do not describe their structures, in particular, they do not estimate the width of detonation zone. The calculated pressure profile for stoichiometric hydrogen-oxygen mixture at p 0 =530 mm Hg is shown below in Fig.3. The resulting value of the width of the stationary detonation front 8,63 mm is in good agreement with the value of 8,32 mm from [10]. For a number of other gas mixtures the good agreement with experiment was also obtained. Correct estimation of the detonation zone width makes it possible to solve the problem of the detonation wave stability. This solution is used for control of detonation regimes.