Suggested critical values

The exact numerical values of the criteria proposed here have no major significance because the interest is usually in knowing which extreme is close. The desire is mostly to be as close to zero as possible to avoid falsification of chemical rates by transport resistances. In spite of this, some numerical values are proposed in the final table to give some orientation about the magnitudes. These estimates are based on the 

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The critical value for F(0.05, 6, 4) is 9.197. Since Fexp is less than F(0.05, 6, 4), the null hypothesis is retained. There is no evidence at the chosen significance level to suggest that the difference in precisions is significant. 

Introduction of the local Archimedes criterion helped to clarify nonisothermal jet design procedure. Grimitlyn suggested critical local Archimedes number values, Ar , below which a jet can be considered unaffected by buoyancy forces (moderate nonisothermal jet) Ar, 0.1 for a compact jet, Ar, < 0.15 for a linear jet. 

Estimate a unit based upon suggested values of U from Tables 10-15 and 10-18Aand the known LMTD. Check to be certain that does not exceed critical value between shell side and tube wall or the tube side temperatures (however expressed). 

Based on associations between abnormal serum J32m, -globulin and urinary albumin values and the number of times the PbB was above critical values, Chia et al. (1995a) suggested that the threshold of 70 pg/dL for PbB may not prevent the occurrence of nephropathy. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of The results of this analysis are shown in Figure 3.7. This relationship has been tested for shear thinning and for Bingham... 

The exact method by which fracture occurs is not known, although it is suggested by I>r r i i 51 that the compressive force produces small flaws in the material. If the energy concentration exceeds a certain critical value, these flaws will grow rapidly and will generally branch, and the particles will break up. The probability of fracture of a particle... 

In contrast, the single fiber composite model predicts that the IFSS concentration becomes higher at the embedded end than at the loaded end if fiber Kf is greater than a critical value, suggesting the possibility of debond initiation at the embedded fiber... 

This crude analysis is based on the behavior postulated by the Born equation. However, ion-pair formation equilibrium constants have been observed to deviate ma edly from that behavior (22/ -222)1 Oakenful, and Fenwick (222) found a maximum in the ion-pair formation constants of several alkylamines with carboxylic acids when determined at various methanol-water solvent compositions as shown by their data in Fig. 54. The results demonstrate that in this system the stability constant decreases with increasing organic solvent concentration above a.critical value which yields maximum stability. The authors suggested that this was due to a weakening of hydrophobic interactions between the ion-pair forming species by increased alcohol concentrations. In practice the effect of added organic solvent has been either to decrease the retention factor or to have virtually no effect. 

An interpretation of the fact that, for some explosives at least, the detonation velocity does not continue to rise with rise in density, but goes thru a maximum and detonation finally fails when the density exceeds a critical value is reptd by Dunkle (Ref 5) and Price Refs 9 10). Roth (Ref 4), on the basis of results reported in Refs 1, 2 St 3, suggests the existence of a property he calls Widerstand ("resistance or "impedance ) of value equal to the product of loading density and detonation velocity, analogous to acoustic impedance and shock impedance (See abstract of Roth s paper at the end of this item)... 

An approach suggested by Williams et al. (117) uses an energy balance equation for initiation of flaw growth in a linearly viscoelastic material. A spherical flaw geometry was selected for simplicity since the expressions for the critical values of applied stress to cause fracture were similar for several flaw geometries. The critical conditions are based on a power (energy rate) balance... 

The concentrations of elastically active network chains related to the dry state vd series A-F were [26] vd = 3.6, 5.7, 6.3, 7.1, 10.9 and 15.2 x 10 5 molcm 3 (structure formation at high dilution in the system. Using vd values together with other molecular parameters, the dependences of y vs ip 2 were calculated and both the extent of the collapse, A, and the critical value,... 

Levedahl (106) noted that aliphatics from acetylene to octane all gave hot flames at 600° C. He suggested that some reaction, such as thermal decomposition, which was common to all aliphatics, became important. Benzene showed a very high, inconsistent hot flame limit, while cyclohexane was low and variable. Levedahl believed the cool flame and subsequent reaction, along with compression, served to raise the mixture temperature to the critical value. Acetylene was believed to play a major role in the ignition reaction. 

From the point of view of the ideas discussed above concerning the variability on the free-volume fraction at Tg, even for the same modes of molecular motion in different polymers, there is great interest in some new concepts about the free-volume distribution, in the system, first proposed in 24. The starting point is the suggestion that all molecular motions, like transfer phenomena, can take place only when the size of the voids or holes in the system exceeds a critical value v. This critical volume appears as a result of redistribution of the free-volume within the system. 

A similar procedure can be applied with the negative weight of harmonic added. In this case a critical value emin corresponds to the minimum of the propagation velocity when the jet thickness at x = oo tends to zero, and the stagnation zone occupies the whole channel cross section. One may suggest... 

Figure 8 also shows the mass balance model predictions for two runs. The mass balance model does a reasonable job of predicting the product concentration for the low cycle time. Here the predicted values are lower than the experimental values, differing by a factor of approximately two. However, for a cycle time of 12 minutes, the prediction falls short of the experimentally determined value. The critical value of y 3 1.0 was used in the model calculations. If 1.3 were used for the purge to feed ratio as suggested by Skarstrom (4), or a value closer to 3 as found in these experiments, the agreement would be slightly better for both runs. 

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