Characteristic values, approximate

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

The second RPT criterion relates to the temperature of the hot liquid. That is, this temperature must exceed a threshold value before an RPT is possible. From one theory of RPTs, the superheated-liquid model (described later), this criterion arises naturally, and the threshold hot-liquid temperature is then equal to the homogeneous nucleation temperature of the colder liquid T. This temperature is a characteristic value for any pure liquid or liquid mixture and can be measured in independent experiments or estimated from theory. From alternate RPT theories, the threshold temperature may be equated, approximately, to the hot fluid temperature at the onset of stable film boiling. 

Thus, differences among samples due to differences in Jc° and rj0 are removed in this reduced form. Since tj0 and Je° are sensitive characteristics of the long relaxation time processes, and since the initial departure of t] 0 — rjs from t]0 — s is governed by these same processes, one would espect significant departures from rj0 to occur near some roughly constant characteristic value of c% with a value of approximately unity. Finally, if the onset of shear rate dependence depends on the same processes, then Eq.(8.8) also gives an appropriate reduced form for f ... 

To estimate the plastic deformation, assume that the deformation occurs mainly in the shear bands. Introduce into the plastic deformation rate. Equation (5) nominal characteristic values for RDX, ° so that the plastic deformation rate reduces to dy/dt = 2T(t,U)Pc(t)x10 s. Approximating dy/dt = Ay/At and letting At = 5 x lO s, which is the measured time of the impact, gives Ay = T(t,U)Pc(t)x10 s. It will be shown in the following paragraph that for mild impacts slightly less than that required for initiation of chemical reaction in RDX, T(t,U)Pc(t) = 10 in crystals at the outer perimeter of the impacted sample. The plastic strain is Ay 10 which is close to the measured values of Ay = 10 to 20, (1000% to 2(KX)%). 

The basis for the assessment is identical in both cases, namely the knowledge of all characteristic values of the chemical reaction itself. These are mainly the heat of reaction and the formal kinetics. In the introduction to Section 4.1 it was shown that a variety of different formal kinetic rate laws may be approximated by a power rate law with sufficient accuracy. In these cases the reaction order n has to be interpreted as an... 

To each value of q, a countable infinite set of characteristic values of a is associated for which x t) is an odd or even function that is njr-periodic in time, n being an integer. Series approximations for the characteristic values are obtained by expressing the integral-order Mathieu function as a series of harmonic oscillations, plugging the resultant expression into Eq. (20.8), and equating coefficients of each (orthogonal) frequency component to zero. These laborious calculations yield infinite series in q where each coefficient of q can be expressed as a continued Iraction [4]. 

Note that in the small double layer thickness approximation, the character of motion of liquid in the capillary is that of plug flow with the velocity U. If the thickness of the double layer is small, but finite, the velocity profile looks like the one shown in Fig. 7.9. For the characteristic values C = 0.1 V, = 10 m, we have for water U = m/s. Thus, electroosmotic motion has a very low velocity. 

The formulated boundary value problem has an analytical solution [46], but it is unwieldy and is not presented here. It should be noted that the necessary condition for the considered approximation is Pej> 1, otherwise it would be impossible to formulate the condition at the infinity. At centrifugation, the characteristic values of the Pedet number are 10 -10, therefore this assumption is justified. Another assumption refers to the short time intervals. Time enters the expression for the Strouhal number... 

For the characteristic values of parameters entering we have 1 ranging between 1 and 10. This condition allows us to find an approximate solution of the... 

Among all the non-numerical approximation methods, the effectiveness of the variational methods is perhaps most surprising [11]. This method serves to determine characteristic values of linear operators. Since the time variable can be eliminated from almost every reactor equation by transforming it into a characteristic value problem, the variational method should have wide applications in reactor theory. Its use has been limited, so far, because Boltzmann s operator is not self adjoint or normal. Whether this limitation is a necessary one, remains to be seen. The reason for the great accuracy of the variational principle in simple problems of quantum mechanics is that any function which is positive everywhere and has a single maximum can be so well approximated by any other similar function. Thus... 

That is to say, RATE considers acceptable for SMEs an approximate approach where risks are related to characteristic values based on statistical data and combined with simultaneousness coefficients to create reasonable but not precise sceneries. 

The procedure described allows analyst to obtain characteristic risk values that are approximate but representative as regards to real occurrences data one risk could have a high characteristic value because a lot of accidents have been recorded for the related contact, or because there is a high incidence of death cases even if total number of accidents is low. 

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