General Instability Criterion

As shown before, K-plane(s) appear(s) to be stable if it (they) Correspond(s) to a negative slope of the velocity curve. A K-plane is considered to be unstable (virtual) if it is found at a positive slope of the velocity curve. Let us try to derive the general criterion for K-instabiHty taking into account the concentration dependence of dilFusion coefficients, which makes it somewhat difficult to find the expression for the velocity curve. 

A transformation of the expression obtained in Equation 6.79 leads to the inequality 

The interdifTusion coefficient relates to partial diffusion coefficients according to Darken (remembering ffiat partial volumes have different values)  

The above inequality is true when, for example, the value Da/Db — 1 turns from positive to negative as the B-content increases. 

We note that the instability criterion in the form of Equations 6.79-6.81 can be applied only for the K-plane of special composition Nr. This is the additional condition, similar to the sign of the second-order derivative determining the type of extremum (maximum, minimum, or inflection point) for some function (the first derivative being equal to zero). The condition Equation 6.76 can be used for finding the prospective unstable candidates.  

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Novozhilov (Ref 9) noted other instances where the instability criterion of Zel dovich is not satisfied. He also noted ZeTdovich s assertion that the form of the stability criterion may change if the variation in the surface temperature and the inertia of the reaction layer of the condensed phase are taken into account, and stability criteria obtained under the assumption that the chemical reaction zone in the condensed phase and all of the processes in the gas phase are without inertia. Novozhilov used a more general consideration of the problem to show that the stability region is determined by only two parameters Zel dovich s k and the partial derivative r of the surface temperature with respect to the initial temperature at constant pressure t=(dTi/(fro)p. Combustion is always stable if k 1, combustion is stable only when r >(k — l) /(k +1)... 

Developing a quantitative theory of the conditions for shear localization is the subject of ongoing numerical (finite element-based) research. Key earlier papers are (Recht 1964) in which the instability criterion dx/dy = 0 was first applied, (Semiatin and Rao 1983) in which it was argued that dx/dy needed to be substantially negative and (Hou and Komanduri 1997) in which the complexities of temperature distributions in shear localized chips were examined in more detail than in previous work. Adiabatic shearing has been the subject of a number of general reviews, for example, Walley (2007), and books, for example, Bai and Dodd (1992). These mention but do not have a main focus on machining. Walley (2007) mentions nine earlier reviews. 

Most substances which appear in the examples of this chapter are analysed In Part Two and their enthalpy of decomposition determined experimentally. This is because most of them are considered hardly stable. This is one of the reasons for assigning no Tow risk in the suggested classifications. But it is also indisputable that criterion Cf overestimates the instability risk. It is the case for all aromatic compounds that are generally very stable. In the examples above, N-methylaniline, dichlorobenzene... 

This criterion is the so-called T combustion instability. The stability criterion expressed by < 1 is not sufficient to obtain stable combustion when the flame temperature is dependent on pressure.lO In general, m is approximately zero in the high-pressure region for most propellants. However, l/of nitropolymer propellants such as single-base and double-base propellants decreases with decreasing pressure below about 5 MPa. Since direct determination of m is difficult, the heat of explosion, is evaluated as a function of... 

The degree of twist in a modem rifle is generally just about the minimum that will produce stability in a bullet passing thru an atmosphere of a density corresponding to a temperature of zero degrees centigrade. This is to ensure that entry into a denser medium — i.e. the target — will produce complete instability, which is a desired criterion for wound baLlistics... 

Equation 10.4.a-7 is a necessary but not sufficient condition for stability. In Other words, if the criterion is satisfied, the reactor may be stable if it is violated, the reactor will be unstable. (Aris [1] prefers to use the reverse inequality as a sufficient condition for instability.) The reason is that in deriving Eq. 10.4.a-7, it was implicitly assumed that only the special perturbations in conversion and temperature related by the steady-state heat generation curve were allowed. To be a general criterion giving both necessary and sufficient conditions, arbitrary perturbations in both conversion and temperature must be considered. Van Heerden s reasoning actually implied a sense of time ( tends to move... ), and so the proper criteria can only be clarified and deduced by considering the complete transient mass and energy balances. 

A few words on the stability of steady states of polymerization. This question arises immediately as soon as the multiplicity of steady-state conditions spears. It is well known that three solutions are possible in the flow of reactants. The general theory of thermal instability of reactors has been developed in detail in Refe. [16-20,30,31], and the theory of kinetic instability caused by peculiarities of the kinetic schenK (self-acceleration. gel-effect, etc. in Refs. [37-40]). The instability of steady states of poly-nKiization plug reactors of a hydrodynamic nature is more interesting for the present paper. It can be assumed that the state corresponding to the negative slopes of the P(Q) curve are unstable if P = const is maintained [30, 33, 34]. At Q = const, all states are stable and realizable. The analysis of this problem in zero-dimensional formulation [41], for a reactor determined by only one value of T, p, q and a complex variable hydrodynamic resistance has shown that the slope of the curve is not an exhaustive stability criterion. 

The limit equilibrium methods forming the framework of slope stability/instability analysis generally accept the Mohr-Coulomb failure criterion, which can be expressed in terms of effective or total stresses. The Mohr-Coulomb shear strength in terms of effective stresses is as follows ... 

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