The Critical Value of

In a similar fashion it can be shown that the critical value of % is given by Equation 11-54  

This can be translated into a critical value of the solubility parameter difference by noting that for polymer solutions, to a first approximation we can use Equation 11-55  

Then at room temperature RT is of the order of 600 cal/mole, while many solvents have molar volumes Vr 100 cmVmole, giving Equation 11-56  

A rough rule of thumb would then be that if a polymer has a solubility parameter of say, 9, it should dissolve in solvents with solubility parameters between 8 and 10. Keep in mind that such considerations, approximate as they are, only apply to those systems that interact through dispersion and weak polar forces. 

FIGURE 11-23 Experimental and theoretical plots of critical value of % versus volume fraction of polymer [replotted from the data of Scholte, T. G.. 

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Based o the test data, the parameter a6 is correlating with the residual resistance (table 1). It is discovered that the less resistible samples have much higher value of a6. On the base of collected data it is possible to identify the critical value of the accumulation coefficient (which is a defective sign of the material (if aG> AiScR-the sample is defected if aG< a6cr - the sample is without defects). 

Show that the critical value of 0 in Eq. XVII-53 is 4, that is, the value of 0 above which a maximum and a minimum in bP appear. What is the critical value of 67... 

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

Since Fgxp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated. 

This value for fexp is smaller than the critical value of 2.26 for f(0.05, 9). Thus, there is no evidence for a systematic error in the method at the 95% confidence level. 

As an example of the quantitative testing of Eq. (5.47), consider the polymerization of diethylene glycol (BB) with adipic acid (AA) in the presence of 1,2,3-propane tricarboxylic acid (A3). The critical value of the branching coefficient is 0.50 for this system by Eq. (5.46). For an experiment in which r = 0.800 and p = 0.375, p = 0.953 by Eq. (5.47). The critical extent of reaction, determined by titration, in the polymerizing mixture at the point where bubbles fail to rise through it was found experimentally to be 0.9907. Calculating back from Eq. (5.45), the experimental value of p, is consistent with the value =0.578. 

The critical value of would be based on four degrees of freedom. This corresponds to (r — 1) — 1, since one statistical quantity X was computed from the sample and used to derive the expectation numbers. 

The transition to turbulent flow begins at Re R in the range of 2,000 to 2,500 (Metzuer and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K and n must be evaluated for the condition in question in order to determine Re R and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963] 14, 691 [1968] Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967] and Govier and Aziz, pp. 213-215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzuer and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of Re R increased to beyond 10,000 for some materials. 

Samples were tested on in a melt of salts (75% Na SO, 25% NaCl) at 950°C in an air atmosphere for 24 hours. Micro X-rays spectrum by the analysis found that the chemical composition of carbides of an alloy of the ZMI-3C and test alloys differs noticeably. In the monocarbide of phase composition of an alloy of the ZMI-3C there increased concentration of titanium and tungsten is observed in comparison with test alloys containing chemical composition tantalum. The concentration of more than 2% of tantalum in test alloys has allowed mostly to deduce tungsten from a mono carbide phase (MC) into solid solution. Thus resistance of test alloys LCD has been increased essentially, as carbide phase is mostly sensitive aggressive environments influence. The critical value of total molybdenum and tungsten concentration in MC should not exceed 15%. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Baturin " studied air jets supplied from rectangular nozzles at some angle to the plane with an edge of the nozzle coincident with the plane. The results of his studies indicate that the critical value of the angle of the jet supply direction to the plane is 45°. It was also shown that the jet supplied through a rectangular outlet with a nozzle kxated at some distance from the plane does not attach to the surface. 

Asphyxiant Simple asphyxiants are inert gases which deplete the oxygen supply in the breathing air to below the critical value of J8% by volume, such as gaseous fuels or nitrogen. Chemical asphyxiants, such as carbon monoxide and hydrogen cyanide, have a direct biological effect. 

As an indication of the changes in deformation modes that can be produced in ionomers by increase of ion content, consider poly(styrene-co-sodium methacrylate). In ionomers of low ion content, the only observed deformation mode in strained thin films cast from tetra hydrofuran (THF), a nonpolar solvent, is localized crazing. But for ion contents near to or above the critical value of about 6 mol%, both crazing and shear deformation bands have been observed. This is demonstrated in the transmission electron microscope (TEM) scan of Fig. 3 for an ionomer of 8.2 mol% ion content. Somewhat similar deformation patterns have also been observed in a Na-SPS ionomer having an ion content of 7.5 mol%. Clearly, in both of these ionomers, the presence of a... 

A. Lubinski also found [171] that to drill a vertical hole in homogeneous formations, it is best to carry less weight on the bit than the critical value of the first order at which the drill string buckles. However, if such weight is not sufficient, it is advisable to avoid the weight that falls between the first and second buckling order and to carry a weight close to the critical value of the third order. 

The critical value of the Reynolds number (Remit) for the transition from laminar to turbulent flow may be calculated from the Ryan and Johnson001 stability parameter, defined earlier by equation 3.56. For a power-law fluid, this becomes ... 

As an example, the flow of air at 293 K in a pipe of 25 mm diameter and length 14 m is considered, using the value of 0.0015 for R/pu2 employed in the calculation of the figures in Table 4.1 R/pu2 will, of course, show some variation with Reynolds number, but this effect will be neglected in the following calculation. The variation in flowrate G is examined, for a given upstream pressure of 10 MN/m2, as a function of downstream pressure P2. As the critical value of P /P2 for this case is 3.16 (see Table 4.1), the maximum flowrate will occur at all values of P2 less than 10/3.16 = 3.16 MN/m2. For values of P2 greater than 3.16 MN/m2, equation 4.57 applies ... 

Similar results have been obtained by Bonilla and Perry 79>, Insinger and Bliss 801, and others for a number of organic liquids such as benzene, alcohols, acetone, and carbon tetrachloride. The data in Table 9.9 for liquids boiling at atmospheric pressure show that tile maximum heat flux is much smaller with organic liquids than with water and the temperature difference at this condition is rather higher. In practice the critical value of AT may be exceeded. Sauer et al.m] found that the overall transfer coefficient U for boiling ethyl acetate with steam at 377 kN/m2 was only 14 per cent of that when the steam pressure was reduced to 115 kN/m2. 

From equation 9.267, the critical value of hr/k, below which the heat loss may be increased by a thin layer of lagging, is 1. For hr/k > 1, as in this problem, the. situation will not arise. 

Li et al. (2003) studied the flow in a stainless steel micro-tube with the diameter of 128.76-179.8 jm and relative roughness of about 3-4%. The Poiseuille number for tubes with diameter 128.76 and 171.8 jm exceeded the value of Po corresponding to conventional theory by 37 and 15%, respectively. The critical value of the Reynolds number was close to 2,000 for 136.5 and 179.8 pm micro-tubes and about 1,700 for micro-tube with diameter 128.76 pm. 

Thus, the critical value of R-A parameter a Is not the same nor Is It as clearly defined. Moreover, It Is possible to experience Insensitive (potentially stable) R-A. Sample experimental results showing sensitive and Insensitive R-A have been plotted In Figures 1 and 2, respectively. 

Criteria for sensitivity, B and b, are also criteria for validity of the early R-A approximation (ERA), which says that R-A occurs virtually when m = 1 = I. While B for most free-radical polymerizations lies within a narrow range, which exceeds the critical value, b varies widely from subcritical to critical values, depending strongly uponcholceof Initiator and feed parameters [lio and Tq. Decreasing values of b generally depress the critical value of a slightly. Computed R-A... 

Based on the distributions studied so far, the simulation results show that at low TEA/Tl ratios the site distribution is probably unlmodal or bimodal with predominantly HAFD sites (l.e., > 90 %) undergoing first order decay. Beyond the critical value of TEA/Ti (10,8) a different distribution must exist for 6 to be > 1-0... 

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