Class method bivariate

Class methods have been applied to bivariate PBE only for simple problems, and generally their extension to multivariate problems is quite complicated. In what follows selected examples are discussed to illustrate these difficulties. 

These findings provide for the first time a method to characterize the energetic topography (i.e., obtain the parameters from experimental measurements) of a class of heterogeneous surfaces that can be approximately represented as bivariate surfaces. 

As introduced in the previous section, class and sectional methods are based on a discretization of the internal coordinate so that the GPBE becomes a set of macroscopic balances in state space. Indeed, the fineness of the discretization will be dictated by the accuracy needed in the approximation of the integrals and derivative terms appearing in the GPBE. As has already been anticipated, the methods differ according to the number of internal coordinates used in the description and depend on the nature of the internal coordinates. Therefore, in what follows, we will discuss separately the univariate, bivariate, and multivariate PBE, and the use of these methods for the solution of the KE. 

Note that in the VPBD the parameter a permits the distribution width to be changed a > 1 leads to a narrower distribution. The fitting procedure is described in more detail by ledema et al. [20]. The VPBD method turns out to yield identical solutions to the classes (Section 9.5) and pgf (Section 9.7) methods as applied to the mixed-metallocene problem. The resulting full bivariate C LD/ D B D for the case of a CSTR is shown in Figure 9.9, together with the VPBD fit. 

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