Reactors statistical model

Turbulent flow reactors are modeled quite differently from laminar flow reactors. In a turbulent flow field, nonzero velocity components exist in all three coordinate directions, and they fluctuate with time. Statistical methods must be used to obtain time average values for the various components and to characterize the instantaneous fluctuations about these averages. We divide the velocity into time average and fluctuating parts ... 

The statistical data shown in Table 5.2 were obtained for an isothermal continuously stirred reactor (CSR) with a spatial time of 1.5 h. With these experimental data, we can formulate a relationship between the reactant conversion (y) and the input concentration (x). For the establishment of a statistical model based on a... 

What follows is an attempt to describe the performance of nonideal reactors using some simple statistical models that can be represented by analytical solutions. This... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

Before collecting data, at least two lean/rich cycles of 15-min lean and 5-min rich were completed for the given reaction condition. These cycle times were chosen so as the effluent from all reactors reached steady state. After the initial lean/rich cycles were completed, IR spectra were collected continuously during the switch from fuel rich to fuel lean and then back again to fuel rich. The collection time in the fuel lean and fuel rich phases was maintained at 15 and 5 min, respectively. The catalyst was tested for SNS at all the different reaction conditions and the qualitative discussion of the results can be found in [75], Quantitative analysis of the data required the application of statistical methods to separate the effects of the six factors and their interactions from the inherent noise in the data. Table 11.5 presents the coefficient for all the normalized parameters which were statistically significant. It includes the estimated coefficients for the linear model, similar to Eqn (2), of how SNS is affected by the reaction conditions. 

This study shows that the optimization of process conditions could be achieved rapidly by a judicious use of statistics and parallel reactors. A two-level factorial method with two center points was used to limit the total number of experiments to ten. Using two identical high-pressure reactors in parallel further shortened the time required to conduct these experiments. For the model reaction of phenol hydrogenation over a commercially available Pd/C, it was experimentally determined that the optimal yield was 73% at 135 °C, 22.5 bar, and 615 ppm w/w NaOH... 

The development of on-line sensors is a very costly and time-consuming process. Therefore, if one has available a dynamic model of the reactor which predicts the various polymer (or latex) properties of interest, then this can be used to guide one in the selection and development of sensors. Ideas from the optimal statistical design of experiments together with the present model expressed in the form of a Kalman filter have been successfully used (58) to select those combinations of existing or hypothetical sensors which would maximize the information that could be obtained on the states of the polymerization system. Both the type of sensors and the precisions necessary for them are easily investigated in this way. By changing the choice of the measurement matrix and... 

The first combination - well macromixed and well micromixed - is just the CSTR model for a stirred reactor wherein the scalar is assumed to be constant at every point in the reactor. The second combination - well macromixed and poorly micromixed - corresponds to a statistically homogeneous flow and is often assumed when deriving CRE micromixing models. The third combination - poorly macromixed and well micromixed - is often... 

The 100(1 —a) % point of the F statistic with (vltv2) degrees of freedom Mass feed rate to reactor Partial derivative of nonlinear rate model with respect to the ith parameter and evaluated at the nth experimental point Matrix of unity diagonal terms and zero off-diagonal terms... 

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