Linear process model variable scaling

Fixed cost effects are included in most production network design models but scale and scope effects related to variable costs and learning curve effects lead to concave cost functions (cf. Cohen and Moon 1990, p. 274). While these can be converted into piecewise linear cost functions, model complexity increases significantly both from a data preparation perspective (see Anderson (1995) for an approach to measure the impact on manufacturing overhead costs) and the mathematical solution process. Hence, most production network design models assume linear cost functions ignoring scale and scope effects related to variable costs. 

Chemometrics is a convenient mathematical tool for developing QSRR or scaling laws that can be embedded in process models. It is somewhat analogous to Wei and Kuo s treatment in that it contracts system dimensions through a projective transformation. Each lump is a linear combination of all original variables. The widely used principal component analysis (PCA) and partial least squares (PLS) are two examples of chemometric modeling. 

In PCR, a principal component analysis (PCA) is first made of the X matrix (properly transformed and scaled), giving as the result the score matrix T and the loading matrix P. Then in a second step a few of the first score vectors tg are used as predictor variables in a multiple linear regression with Y as the response matrix. In the case that the few first components of PCA indeed contain most of the information of X related to Y, PCR indeed works as well as PLS. This is often the case in spectroscopic data, and here PCR is an often used alternative. In more complicated applications, however, such as QSAR and process modeling, the first few principal components of X rarely contain a sufficient part of the relevant information, and PLS works much better than PCR. ... 

As with scale-up, two levels of implementation are possible. The first level only entails the ability to sense, and a directional characterization of the effect of variables. PAT methods can be extremely effective for this purpose by generating large datasets of process inputs and outputs that can then be correlated to generate statistical or polynomial control models. Provided that (i) deviations from desired set-points are small, (ii) interactions between inputs are weak, and (Hi) the response surface does not depart too much from linearity, such systems can provide the basis of an initial effort to control a system. 

In all of the above cases, a strong non-linear coupling exists between reaction and transport at micro- and mesoscales, and the reactor performance at the macroscale. As a result, the physics at small scales influences the reactor and hence the process performance significantly. As stated in the introduction, such small-scale effects could be quantified by numerically solving the full CDR equation from the macro down to the microscale. However, the solution of the CDR equation from the reactor (macro) scale down to the local diffusional (micro) scale using CFD is prohibitive in terms of numerical effort, and impractical for the purpose of reactor control and optimization. Our focus here is how to obtain accurate low-dimensional models of these multi-scale systems in terms of average (and measurable) variables. 

If a continuous process is to he used for commercial production, a similar small-scale reactor system should he utilized in this second stage of product development. There are a number of reasons for this recommendation. The earher discussion of the difference between batch reactors and CSTRs lists some of these reasons, if, for example, engineering data are to he obtained for design of a commercial unit, the variable relation ps might be quite different for the different reactors. The Smith-Ewart CSTR model predicts a linear relationship between or N and the surfactant concentration [5]. The same mechanistic model for a batch reactor predicts a 0.6 power relationship between Rp or N and... 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

Challenges in the future remain to model the high-order interactions among biogeochemical variables, to model complex linear and nonlinear processes, and to incorporate spatial and temporal autocorrelations into models. Scale is one of the most confounding factors when synthesizing and modeling wetland processes. Process-based models implicitly assume a specific scale at which... 

---