Critical value calculation

The excess free energy per solvent molecule of polymer solutions is characterized by a semi-empirical Flory-Huggins parameter, X) which is a function of temperature for a given polymer-solvent pair. To estimate the compatibility parameter experimentally, it is necessary to define the x value for each polymer-solvent pair and compare it to its critical value calculated by the equation... 

In Type 1, a bubble is considered to be unstable for the critical value of Fb, and assumed to detach from the plate. In fact, the critical values calculated from the models of Types 1 and 3 for a plate of 0a of 96° in the air-water system could satisfactorily predict the experimentally measured volume of detached bub-... 

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

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. 

The critical value for f(0.05, 5) is 2.57. Since the calculated value of fgxp is greater than f(0.05, 5) we reject the null hypothesis and accept the alternative hypothesis that the mean values for %w/w Na2C03 reported by the two analysts are significantly different at the chosen significance level. 

This value of fexp is compared with the critical value for f(a, v), where the significance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are interested in whether the larger of the two means is significantly greater than the other mean, the value of f(a, v) is that for a one-tail significance test. 

Using equation 14.25, we can calculate values of fexp for each possible comparison. These values can then be compared with the one-tailed critical value of 1.73 for f(0.05, 18), as found in Appendix IB. For example, fexp when comparing the results for analysts A and B is... 

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. 

Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from. 32-term MBWR equation of state transport properties are from extended corresponding states model, t = triple point c = critical point. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

Low-speed flames will only be quenched if die passageway diameter is below a certain critical value. This critical diameter can be calculated by the following equation (Mendoza et al. 1996) ... 

To evaluate the fibrillation behavior of dispersed TLCP domains according to the - 5 relation discussed previously, different - 5 graphs were calculated by eliminating the thickness variable x. The result is reported in Fig. 18. It is obvious that all the points obtained are found to be relatively close to the critical curve by Taylor. The Taylor-limit is also shown in the figure with a solid curve. One finds that all the values calculated on sample 1 are completely above the limit, while all those determined on sample 4 are completely below the limit. The other two samples, 2 and 3, have the We - 5 relation just over the limit. 

Turbulent flow occurs if the Reynolds number as calculated above exceeds a certain critical value. Instead of calculating the Reynolds number, a critical flow velocity may be calculated and compared to the actual average flow velocity [60]. 

If the calculated value of Q exceeds the critical value given in the Q table (Appendix 14), then the questionable value may be rejected. 

Based on a mechanical model in a time-independent flow, de Gennes derivation tries to extrapolate it to a time-dependent chain behavior. His implicit assumptions have been criticised by Bird et al. [55]. More recent calculations extending the de Gennes dumbbell to the bead-spring situation [56] tend, nevertheless, to confirm the existence of a well-characterized CS transition results with up to 100-bead chains show a critical value of the strain rate at scs = 0.5035/iz which is just 7% higher than the value predicted by de Gennes. 

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 ... 

Critical ( -values for p - 0.05 are available. " - In lieu of using these tables, the calculated -values can be divided by the appropriate Student s t(f, 0.05) and V2 and compared to the reduced critical -vdues (see Table 1.12), and data file QRED TBL.dat. A reduced -value that is smaller than the appropriate critical value signals that the tested means belong to the same population. A fully worked example is found in Chapter 4, Process Validation. Data file MOISTURE.dat used with program MULTI gives a good idea of how this concept is applied. MULTI uses Table 1.12 to interpolate the cutoff point for p = 0.05. With little risk of error, this table can also be used fo = 0.025 and 0.1 (divide q by t(/, 0.025) /2 respectively t f, 0.1) V 2, as appropriate. 

The phenomenon of critical flow is well known for the case of single-phase compressible flow through nozzles or orifices. When the differential pressure over the restriction is increased beyond a certain critical value, the mass flow rate ceases to increase. At that point it has reached its maximum possible value, called the critical flow rate, and the flow is characterized by the attainment of the critical state of the fluid at the throat of the restriction. This state is readily calculable for an isen-tropic expansion from gas dynamics. Since a two-phase gas-liquid mixture is a compressible fluid, a similar phenomenon may be expected to occur for such flows. In fact, two-phase critical flows have been observed, but they are more complicated than single-phase flows because of the liquid flashing as the pressure decreases along the flow path. The phase change may cause the flow pattern transition, and departure from phase equilibrium can be anticipated when the expansion is rapid. Interest in critical two-phase flow arises from the importance of predicting dis-... 

The reliability of multispecies analysis has to be validated according to the usual criteria selectivity, accuracy (trueness) and precision, confidence and prediction intervals and, calculated from these, multivariate critical values and limits of detection. In multivariate calibration collinearities of variables caused by correlated concentrations in calibration samples should be avoided. Therefore, the composition of the calibration mixtures should not be varied randomly but by principles of experimental design (Deming and Morgan [1993] Morgan [1991]). 

The formal transformation of the critical value into the sample domain is necessary to estimate correctly the limit of detection. If the sensitivity is known and without of any uncertainty (e.g. in case of error-free calibration constants fi), then the analytical value at CV is calculated by... 

Formally, an analytical result x,- can be calculated from y, by means of the corresponding calibration function. When this result (from repeated measurements) should be reported, it must be taken into account that the relative uncertainty amounts minimally 100% (see Sect. 7.5, item (1) p. 201) and, therefore, it holds that (x x)- That means, that the uncertainty interval of analytical results calculated from measured values nearby the critical value covers a range of about 0... 2x. As additional information, the limit of quantification, xLq, should be given. 

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