Parameter estimation multiple reactions

Parameter Estimation and Statistical Testing of Models and Parameters in Multiple Reactions... 

Many of the models encountered in reaction modeling are not linear in the parameters, as was assumed previously through Eq. (20). Although the principles involved are very similar to those of the previous subsections, the parameter-estimation procedure must now be iteratively applied to a nonlinear surface. This brings up numerous complications, such as initial estimates of parameters, efficiency and effectiveness of convergence algorithms, multiple minima in the least-squares surface, and poor surface conditioning. 

Techniques for parameter estimation vary considerably. If consistent values for model parameters cannot be obtained, the investigators may decide that the model is itself unreliable and should be changed. Thus, model choice and parameter estimation are interactive. A number of workers have discussed generalised procedures [16—18]. Yeh [19] developed numerical algorithms and showed that multiple linear regression could be used successfully if the reaction scheme consisted of steps such as those shown in eqn. (41) or... 

Network of Reactions The statistical parameter estimation for multiple reactions is more complex than for a single reaction. As indicated before, a single reaction can be represented by a single con-... 

The previous sections described techniques employed for parameter estimation. These thermodynamic and kinetic parameters are input to a microkinetic model that is solved numerically to describe material balances in a chemical reactor (e.g., a PFR). This section describes tools for the subsequent model analysis, which can be used in multiple ways. Initially during mechanism development, they can be used to assess which reactions and reactive intermediates are important in the model, which helps the modeler to focus on important features of the surface reaction mechanism. During this process, simulated macroscopic observables, for example, global reaction orders and apparent activation energies can be compared directly to experimental data. Then, once the model describes experimental data reasonably well, analytical tools can be used to develop further insights into the reaction mechanism, with apphcations that include catalyst design [50]. 

However, in many cases the electrochemical systems present more complicated reaction mechanisms, including multiple electrochemical or chemical reactions. The reaction model is thus too complex to he translated into analytical equations. Besides, when gas bubbles are involved in the electrochemical process, a new transport model describing the two-phase mass transport is designed (Maciel et al., 2009 Nierhaus et al., 2009 Van Damme et al., 2010). Modeling these complex systems requires a numerical approach. In our group, focus is put on the development of numerical models for such complex systems. Nevertheless, in order to achieve a statistical and accurate parameter estimation, it is clear that also a numerical fitting procedure has to be introduced. This aspect is xmder development at present in our group. 

Many biochemists use the velocity equations for kinetic parameter estimates despite the fact that the rates are difficult to determine experimentally. In practice either the substrate depletion or the product formation is measured as a function of time and the rates are calculated by differentiating the data, leading to an inexact analysis (Schnell Mendoza, 1997,2000a). Alternatively, the differential equations governing the biochemical reactions may be solved or approximated to obtain reactant concentration as function of time. This approach decreases the number of experimental assays by at least a factor of live, as proved by Schnell and Mendoza (2001), because multiple experimental points may be collected for each single reaction. 

Except for very simple systems, initial rate experiments of enzyme-catalyzed reactions are typically run in which the initial velocity is measured at a number of substrate concentrations while keeping all of the other components of the reaction mixture constant. The set of experiments is run again a number of times (typically, at least five) in which the concentration of one of those other components of the reaction mixture has been changed. When the initial rate data is plotted in a linear format (for example, in a double-reciprocal plot, 1/v vx. 1/[S]), a series of lines are obtained, each associated with a different concentration of the other component (for example, another substrate in a multisubstrate reaction, one of the products, an inhibitor or other effector, etc.). The slopes of each of these lines are replotted as a function of the concentration of the other component (e.g., slope vx. [other substrate] in a multisubstrate reaction slope vx. 1/[inhibitor] in an inhibition study etc.). Similar replots may be made with the vertical intercepts of the primary plots. The new slopes, vertical intercepts, and horizontal intercepts of these replots can provide estimates of the kinetic parameters for the system under study. In addition, linearity (or lack of) is a good check on whether the experimental protocols have valid steady-state conditions. Nonlinearity in replot data can often indicate cooperative events, slow binding steps, multiple binding, etc. 

Using absorbance data collected as a function of time, the distribution function H(k,t) was calculated, and plotted versus In(time) to yield a curve providing a maximum for each first-order process in the reaction. The position of each maximum yields an estimate of the rate constant (t = 2/k), and the area under the maximum provides an estimate of the amount of metal dissociating by that process. Due to concerns of using higher-order derivatives to calculate rate constants for distributions of pathways in multiple first-order mechanisms as described in the literature (16-20), the kinetic spectrum method was used only to obtain initial estimates for the appropriate rate equations. The actual rate parameters reported herein were obtained from a simplex non-linear regression (21) of the original experimental data. When dissociation occurred in both fast (ti/2 > 30 s) and slow (t 1/2 < 30 s) time domains, each data set was treated independently. 

The above analysis is, of course, based on the assumption of simple order reactions under Tafel operation and on the availability of sufficiently accurate data ( 5-10%). With complex reaction kinetics, for example, those involving surface adsorption terms (Eq. 16), a nonlinear regression analysis would yield the best estimate of a, Uj, and for a possible kinetic model. In all cases, use of these parameters for predicting the performance of an electrochemical reactor or the selectivity of a reaction scheme should be restricted within the potential, concentration, and temperature range that they were determined. We should stress here that kinetic information is presently scanty for complex, multiple electrochemical reactions, yet it is essential for the design, optimization, and control of electrochemical processes. 

In this chapter, you have derived the equations governing chemical reaction equilibrium and seen how the key parameters can be estimated using thermodynamics. You have solved the resulting problems using Excel, MATLAB, and Aspen Plus. You also learned to solve multiple equations using MATLAB when there are several reactions in equilibrium. 

The linear model structures discussed in this section can handle mild nonlinearities. They can also result from linearization around an operating point. Simple alternatives can be considered for developing linear models with better predictive capabilities than a traditional ARMAX model for nonlinear processes. If the nature of nonlinearity is known, a transformation of the variable can be utilized to improve the linear model. A typical example is the knowledge of the exponential relationship of temperature in reaction rate expressions. Hence, the log of temperature with the rate constant can be utilized instead of the actual temperature as a regressor. The second method is to build a recursive linear model. By updating model parameters frequently, mild nonlinearities can be accounted for. The rate of change of the process and the severity of the nonlinearities are critical factors for the success of this approach. Another approach is based on the estimation of nonlinear systems by using multiple linear models [11, 82, 83]. 

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