The integration method

The basic problem of design was solved mathematically before any reliable kinetic model was available. As mentioned at start, the existence of solutions—that is, the integration method for reactor performance calculation—gave the first motivation to generate better experimental kinetic results and the models derived from them. 

Here the integration method will be shown that was used for the workshop program (Berty et al, 1989) to integrate the UCKRON rate equations. Since this is about methanol synthesis the reaction is shown here with the stochiometric coefficients ... 

Table 3-2 gives solutions for some ehemieal reaetions using the integration method. 

Level 1 in the ehart indieates that when the reaetion rate law is known, the integral method of analysis may be used after performing an experiment to evaluate the speeifie reaetion rate. This proeedure... 

ISIM uses four different integration methods. The user can select a particular method by assigning the variable ALGO a corresponding value. Other variables associated with the integration method are also explained below. 

Both the subtractive and the phase shift method are valid as long as the RF waveform remains sinusoidal [184]. For asymmetrical discharges the integral method is to be used. 

A much better approach for the estimation of specific rates in biological systems is through the use of the integral method. Let us first start with the analysis of data from batch experiments. 

The major disadvantage of the integral method is the difficulty in computing an estimate of the standard error in the estimation of the specific rates. Obviously, all linear least squares estimation routines provide automatically the standard error of estimate and other statistical information. However, the computed statistics are based on the assumption that there is no error present in the independent variable. 

As described in Section 7.3.2 when the integral method is employed, the average specific MAb production rate during any time interval of the batch start-up period... 

In order to implement the integral method, we must compute numerically the integrals [1 Xv (t)dt, tj=l,...,N where Xv is the smoothed value of the viable... 

The third period corresponds to the last five data points where it is obvious that the assumption of a nearly constant qM is not valid as the slope changes essentially from point to point. Such a segment can still be used and the integral method will provide an average qM, however, it would not be representative of the behavior of the culture during this time interval. It would simply be a mathematical average of a time varying quantity. 

Let us now turn our attention to the dialyzed continuous operation (212 to 570 h). By the integral method, the specific MAb production rate can be estimated... 

In this case besides the integral of Xv, the integral of MAb titer must also be computed. Obviously the same FORTRAN program used for the integration of Xv can also be used for the computations (see Appendix 2). The results by the integral method are shown in Figures 17.12 and 17.13. 

The integration method illustrated above becomes somewhat more complicated iftiHb detttifflihator botitiddStepeated line f Actors. Thus, if the denbmi-oator contains a factor such as (x — a), it identical factors would result which could of course be coipbined. To avoid this problem it is assumed (hat l/(x - a)" can bereplaced by... 

Test of reaction rate data by the integral method of analysis. 

Since the integral method described above is based on the premise that some rate function exists that will lead to a value of i/ Cj) that is linear in time, deviations from linearity (or curvature) indicate that further evaluation or interpretation of the rate data is necessary. Many mathematical functions are roughly linear over sufficiently small ranges of variables. In order to provide a challenging test of the linearity of the data, one should perform at least one experimental run in which data are... 

Much better results are obtained by using the parallel surface method (Section III.F.3), because the integral methods are used to determine (K). Nevertheless, PSM gives only approximate estimation of the Euler characteristic and is extremely time-consuming in comparison to the methods described below. 

Table II lists experimental halftimes obtained by extrapolation to 50% of initial concentration cf. Fig. 1 and The values within parentheses in Table II are calculated by integrating Equations (2) and (6) for initial, monodisperse conditions assuming c a = C = 0-20. It is clear that the adsorption halftime is considerably longer than the coagulation halftime, which results in a relatively long flocculation halftime. It is also seen that the integration method is a useful approximation to estimate halftimes.

The combination of (2) and (3) is referred to as a pseudo-first-order situation. H20 is present in great excess, but if it were not, its concentration change would likely affect the rate. We then use the integral method of Section 3.4.1.1.2 in conjunction with equation 3.4-11 to test assumption (3). 

Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods. 

There are two procedures for analyzing kinetic data, the integral and the differential methods. In the integral method of analysis we guess a particular form of rate equation and, after appropriate integration and mathematical manipulation, predict that the plot of a certain concentration function versus time... 

There are advantages and disadvantages to each method. The integral method is easy to use and is recommended when testing specific mechanisms, or relatively simple rate expressions, or when the data are so scattered that we cannot reliably find the derivatives needed in the differential method. The differential method is useful in more complicated situations but requires more accurate or larger amounts of data. The integral method can only test this or that particular mechanism or rate form the differential method can be used to develop or build up a rate equation to fit the data. 

To apply the integral method, separate variables and integrate Eq. 57. This gives... 

If the fit is reasonable, evaluate the constants and C. Solve by the integral method. 

There they will be examined individually using the integral methods and automatically moved on. Specimens found to be leaking will be shunted to the side. 

Differentiation of the experimental concentration-time curve would then need interpolation or smoothing, e.g.,by using splines. Parallelization in a typical robotic environment is easy when using the integral method with a few or even only one single well for characterization of one enzyme variant. 

This case was treated by Potter [ 18] by means of the integral method. Since for high values of the Schmidt number only the region in the vicinity of the planar fluid boundary contributes appreciably to the value of the mass... 

Fig. 2. Comparison for Pe = 103 between Eq. (137) (broken line) and the results of the integral method (solid line).

Assuming that the degradation reaction is represented by the Arrhenius relationship, the equation finally obtained by the integral method when the weight fraction is constant is as follows ... 

---