Least squares batch

Differential equations Batch reactor with first-order kinetics. Analytical or numerical solution with analytical or numerical parameter optimisation (least squares or likelihood). Batch reactor with complex kinetics. Numerical integration and parameter optimisation (least squares or likelihood). 

One can apply a similar approach to samples drawn from a process over time to determine whether a process is in control (stable) or out of control (unstable). For both kinds of control chart, it may be desirable to obtain estimates of the mean and standard deviation over a range of concentrations. The precision of an HPLC method is frequently lower at concentrations much higher or lower than the midrange of measurement. The act of drawing the control chart often helps to identify variability in the method and, given that variability in the method is less than that of the process, the control chart can help to identify variability in the process. Trends can be observed as sequences of points above or below the mean, as a non-zero slope of the least squares fit of the mean vs. batch number, or by means of autocorrelation.106... 

Figure 1. Rate of Cs uptake from batch solutions at 25°C. Straight lines are least square regressions to Equation 1.

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

Ong, S. L. Biotechnoi. Bioeng. 25 (1983) 2347. Least-squares estimation of batch culture kinetic parameters. 

From Eqs. (160)—(162), the characteristic parameters of maximum capacity qm and equilibrium constant Kd can be determined173,326,415,416 by a least-squares fit of the experimental adsorption data generated using batch methods from the plot of 1 /q against 1/C0. Due to the nature of the data... 

An estimator (or more specifically an optimal state estimator ) in this usage is an algorithm for obtaining approximate values of process variables which cannot be directly measured. It does this by using knowledge of the system and measurement dynamics, assumed statistics of measurement noise, and initial condition information to deduce a minimum error state estimate. The basic algorithm is usually some version of the Kalman filter.14 In extremely simple terms, a stochastic process model is compared to known process measurements, the difference is minimized in a least-squares sense, and then the model values are used for unmeasurable quantities. Estimators have been tested on a variety of processes, including mycelial fermentation and fed-batch penicillin production,13 and baker s yeast fermentation.15 The... 

Batch reactors are used primarily to determine rate law parameters for homo, geneous reactions. This determination Ls usually achieved by measuring coa centration as a function of time and then using either the differential, integral, or least squares method of data analysis to determine the reaction order, a, and specific reaction rate, k. If some reaction parameter other than concentration i s monitored, such as pressure, the mole bMance must be rewritten in terms of the measured variable (e.g., pressure). 

In this approach, a least squares regression line is fit to stability data either from a single batch or from several batches. The expected result for any time point is given by the expression ... 

In this approach, a control chart for the slope at each time point is constructed. At each time point, a least squares regression is fit that includes all the earlier time points. The slope estimate for each batch is used to obtain the overall slope estimate for several lots. The OOT limit for the slopes at each time point are obtained from the tolerance interval, in which k is chosen to obtain the desired protection from false negative or false positive values. 

A new method for the determination of vulcanised rubber additives by FTIR spectrophotometry using partial least-squares regression (PLSR) for multivariate calibration was developed. The effect of various wavenumber ranges and the use of the absorbance and first-derivative spectral modes on performance were studied by applying the method to three different sample batches containing several additives in different proportions all of which were resolved with satisfactory results. 15 refs. 

Fig. 5.63 Strength-grain size relationship (a) and subcritical growth of radial indentation cracks (b) of sintered batches selected from Fig. 5.62. Straight lines are least squares fits. Strength ranking numbers associate specimens in (b) to batches in graph (a), numbers 1-3 are consistent with Table 5.2 and Fig. 5.62 [20]. With kind permission of Elsevier...

The values of the adsorption parameters used in this study were obtained as a result of the minimization of the sum of least squares deviation of calculated and experimental concentration profiles in a batch experiment [20]. The experimental concentration curve is given along with the calculated profiles (based on either the first-order or Langmuir kinetics) in Eigure 15.2. 

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