Rate Equation Error Analysis

Rate of fructose disappearance experimental values (pmoiymin-mg enzyme) 

The uncertainty of the slope and intercept values can be calculated by standard statistical techniques. These calculations are straightforward in an EXCEL spreadsheet. An example is provided in Appendix 6-C. In fact, the calculations can be done automatically with the Data Analysis package in EXCEL. 

- be an experimentally measured value of the dependent variable y, and let y, be the value of y predicted by the equation obtained by linear least squares analysis. The error sum of squares (SSe) is given by 

In this summation, N is the number of data points that were used to establish the best fit straight line. The value of SSe is easily calculated in a spreadsheet, as shown in Appendix 6-C. The mean squared error (MSe) is closely related to the error sum of squares  

The values of the slope S and the intercept I that were estimated via the least-squares analysis contain some uncertainty because of errors in the data. Let x be the average of all of the values of x, and let Sxx be defined by 

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In a first step this control algorithem was tested by a simulation of the whole circuittaking into consideration reactor behavior,reaction rate equation. discontinuous analysis. analytical errors and control-strategy. 

A reaction rate constant can be calculated from the integrated form of a kinetic expression if one has data on the state of the system at two or more different times. This statement assumes that sufficient measurements have been made to establish the functional form of the reaction rate expression. Once the equation for the reaction rate constant has been determined, standard techniques for error analysis may be used to evaluate the expected error in the reaction rate constant. 

A major limitation of the linearized forms of the Michaelis-Menten equation is that none provides accurate estimates of both Km and Vmax. Furthermore, it is impossible to obtain meaningful error estimates for the parameters, since linear regression is not strictly appropriate. With the advent of more sophisticated computer tools, there is an increasing trend toward using the integrated rate equation and nonlinear regression analysis to estimate Km and While this type of analysis is more complex than the linear approaches, it has several benefits. First, accurate nonbiased estimates of Km and Vmax can be obtained. Second, nonlinear regression may allow the errors (or confidence intervals) of the parameter estimates to be determined. 

The determination of more comprehensive coking mechanisms and rate equations requires simultaneous treatment of all experimental data to enable all the relevant parameters related to coking to be considered. After analysing the experimental data, numerical values of the rate and adsorption equilibrium constants were determined by statistical tests, and models were rejected if a negative constant was estimated at more than one temperature. It was found that the hyperbolic type of decay, as described in Equation (1), gives the best fit from the 9 models tested because it gives the least error from the sum of squares analysis [8],... 

Vyazovkin and Lesnikovich [42] have emphasized that the majority of NIK methods involve linearization of the appropriate rate equation, usually through a logarithmic transformation which distorts the Gaussian distribution of errors. Thus non-linear methods are preferable [89]. Militky and Sest [90] and Madarasz et al. [91] have outlined routine procedures for non-linear regression analysis of equation (5.5) above by transforming the relationship ... 

Error Analysis — High Volatility Compounds. The test protocol requires measurements of both dissolved oxygen and chemical compound as a function of time. Equations 2 and 3, when integrated, show the linear relationship between ln[C] and time t for the chemical compound, and In ([O2] s [O2D an< time t for dissolved oxygen content. The respective slopes for each line are KyC and Ky0, the rate constants. Potential sources of error in this protocol are the individual dissolved oxygen and chemical compound concentration measurements. 

The error in the conversion of CO provided by this overall rate equation is virtually zero as compared with the exact microkinetic model, which points to the robustness of the reaction network analysis approach presented here. 

In theory, Equ.(2) can be rearranged into Equ.(5) as a linear function of Vm and Km- In Equ.(5), the instantaneous reaction time at the moment for Si is preset as zero so that there is no treatment of flag. When the signal for Si is not treated as a nonlinear parameter, kinetic analysis of reaction curve by fitting with Equ.(5) can be finished within 1 s with a pocket calculator. However, parameters estimated with Equ.(5) always have so large errors that Equ.(5) is scarcely practiced in biomedical analyses. Hence, the proper form of an integrated rate equation after validating should be selected carefully. 

However, the regression theory requires that the errors be normally distributed around (—7 a). and not around f as in the linearized version just described. Hence use the values determined as initial estimates to obtain more accurate values of the constants by minimizing the sum of squares of the residuals of the rates directly from the raw rate equation by nonlinear least squares analysis. 

Heardman et al. [11] described a radial flow method to measure permeability under transient flow, where neither the pressure nor the flow-rate is considered constant. They obtained good agreement with experiments using the normal constant pressure setup for assemblies of woven fabrics. They also showed that, using error analysis on the equation used to calculate permeability, the estimated error due to the parameters used to determine permeability (e g. pressure, viscosity and time) is in the range of 8-16%. However, they did not quantify the range of measured permeability values for their reinforcements and hence did not compare it to the estimated error. 

The rate of reaction at constant volume is thus proportional to the time derivative of the molar concentration. However, it should he emphasized that in general the rate of reaction is not equal to the time derivative of a concentration. Moreover, omission of the 1 / term frequently leads to errors in the analysis and use of kinetic data. When one substitutes the product of concentration and volume for nt in equation 3.0.3, the essential difference between equations 3.0.3 and 3.0.8 becomes obvious. 

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