Integration interval

The conversion achieved in the vessel is obtained by the solution of the differential equation at the exit of the vessel where the hfe expectation is t = 0. The starting point for the integration is tj). When integrating numerically, however, the RTD becomes essentially 0 by the time becomes 3 or 4, and the value of the integral beyond that point becomes nil. Accordingly, the integration interval is from (f, t,<2> or 4) to (/effluent, = 0)-jl IS found from Eq. (23-46). 

Table 4.1. Corresponding integration intervals and levels of significance...

Table 43-1 Values of the integral between —0.01 and 0.01, for various values of the integration interval...

The accuracy of the integration was evaluated by comparing the values computed from the MATLAB program to the tables available ([14], p. 3) at several selected values of X (where X represents the number of standard deviations at which the truncated SD was evaluated from) as a function of the integration interval. The results are presented in Table 53-5. 

Note the parentheses in the derivatives. There is no choice of the number of integration intervals it is supposedly adjusted to obtain a good precision. The output can be a graph or a table with 20 divisions of the independent variable. The scale of the ordinate is selected automatically to fit the range of the abscissa. 

Divide the integration interval into four parts, evaluate the integrand I and apply the trapezoidal rule. The result is t =0.5(0.457)(22239) = 5081 sec... 

Assuming that ACP, A , and Ak are constant in the integration interval (which is fairly reasonable), the result is... 

If the step size (the integration interval) is small enough, this estimate of x will be very close to the correct value. 

Laeven and Smit presented a method for determining optimal peak integration intervals and optimal peak area determination on the basis of an extension of the mentioned theory. Rules of thumb were given, based on the rather complicated theory. Moreover, a simple peak-find procedure was developed, based on the derived rules. 

We want to use this for t > zc and therefore have to suppose atc <4 1. Then for tx> tc the upper limit in the integral may be replaced with oo as the integrand vanishes anyway. Although tt runs from 0 to tit is large compared to tc for the major part of the integration interval. Hence we have approximately for tc t ol 1... 

The integration intervals having been defined, this system of equations can be solved by calculating the nomogram of values that maintain as a constant the crystallinity, xc for a given function of disorder... 

After all corrections are completed, the diagrams of s2I vs. s are drawn. Convenient integration intervals are determined for calculating the nomogram of K values. For instance,... 

Figure 4 corresponds to a polypropylene sample and Figure 5 to a well-crystallized PET sample. Four integration intervals chosen for the diagrams of the main textile fibers. For PET fibers, they are ... 

Table II shows effects of the disorder parameter on the calculated crystallinity of cotton, nylon 66 (PA 66) and PET fiber samples. When k = 0, no correction for distortion is made during calculation of crystallinity. Accordingly, values of X become smaller as the integration intervals increase. On the nomogram of K values, it is possible to determine the disorder parameter value that maintains as approximately constant this crystallinity when using the different intervals the deviation from the constancy is used, in the computing program to determine the best value of k and to estimate the errors of these analyses. The disorder parameter k is higher in cotton and PA 66 than in all PET fibers. Besides, one can see that apparent crystallinity values (disorder parameter not considered) are lower than the true ones.

Figure 5.2 Integration intervals of the model of Van Der Voort. The deconvolution was based on following attributions ...

A noniterative numerical solution for ct X) is readily obtained by taking c[(X) to be constant over a small segment of the integration interval. The derivative is approximated by a finite difference. 

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