Ideal analytical model, deviations

In practice, there are hundreds of analytical equations of this form in the literature from which to choose It would be quite burdensome (and very impractical) to examine every one to decide when to use a particular form. Consequently, we will take a different tack. We will start with the friendliest equation of state, the ideal gas model. After examining its limitations, we will explore how we can describe deviations from ideal gas behavior. We will investigate the generalities of these different forms as well as tiy to develop an intuition about what equations to use and when to use them. To this end, we... 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

With a stack of low-rank bilinear data matrices such as the one above, a three-way array can be built. The underlying model of such an array is not necessarily low-rank trilinear if for example there are retention time shifts from sample to sample. However, the data can be still be fitted by a low-rank trilinear model. If the deviations from the ideal trilinearity (low-rank - one component per analyte) are small, directly interpretable components can be found. If not, components are still found in the same sense as in PCA, where the components together describe the data, although no single component can necessarily be related to a single analyte. 

To compute values for the deviation measures, we need volumetric data for the substance of interest such data are usually correlated in terms of a model PvTx equation of state. In 4.4 we develop expressions that enable us to use equations of state to compute difference and ratio measures for deviations from the ideal gas. Finally, in 4.5 we present a few simple models for the volumetric equation of state of real fluids. These few models are enough to introduce some of the problems that arise in attempting to analytically represent the PvTx behavior of real substances, and they allow us to compute values for conceptual, using the expressions from 4.5. However, more thorough expositions on equations of state must be found elsewhere [1-4]. 

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