Experimental Design and Parameter Estimation

An isothermal, plug flow, fixed bed reforming pilot plant (shown in Fig. 14) was used to generate the kinetic data. The reactor was U shaped and contained roughly 70 ml of catalyst. Five sample taps were spaced along the reactor length to determine compositions over a wide range of catalyst contact times. The reactor assembly was immersed in a fluidized sand bath to maintain isothermal conditions. 

Data used to develop the start-of-cycle kinetics consisted of over 300 material balances on several commercial catalysts of three types Pt/Al203, Pt-Re/Al203, and Pt-Ir/Al203. 

Experimental conditions were 727, 756, and 794 K isothermal reactor temperature 827-, 1220-, and 2619-kPa hydrogen pressure 138- and 345-kPa hydrocarbon pressure and 1 to 26 liquid hourly space velocity. (See Section II for definition.) Charge stocks consisted of three C6 component blends (blends included 53/19/23/5, 25/75/0/0, and 0/0/50/50 wt. % hexane/methylcyclopentane/cyclohexane/benzene), C6 to C7 component naphthas (322-366 K TBP Kirkuk, Mid-Continent, and Nigerian), a C6 to C8 component naphtha (322-416 K TBP Mid-Continent), and C6 to C12 component naphthas (322-461 K TBP Arab Light, Mid-Continent, and 

Nigerian). In each experiment, single-pass hydrogen (100% H2 feed without recycle) was used. 

The fitting sequence is based on the following partitioning of the 13 x 13 selectivity rate constant matrix K  

---

This traditional approach starts with a number of preselected, measurable kinetic lumps and determines the best reaction network and kinetics through experimental design and parameter estimation. The number of lumps depends on the level of detail desired. The lumps, satisfying the conservation law and stoichiometric constraints, are usually selected based on known chemistry, measurability and physicochemical properties (boiling range, solubility, etc.). 

Procedures on how to make inferences on the parameters and the response variables are introduced in Chapter 11. The design of experiments has a direct impact on the quality of the estimated parameters and is presented in Chapter 12. The emphasis is on sequential experimental design for parameter estimation and for model discrimination. Recursive least squares estimation, used for on-line data analysis, is briefly covered in Chapter 13. 

Pinto, J. C., M. W. Lobao. and J. L. Monteiro, Sequential experimental design for parameter estimation analysis of relative deviations, Chem. Eng. Sci., 46, 3129-3138 (1991). 

Once a model has been fitted to the available data and parameter estimates have been obtained, two further possible questions that the experimenter may pose are How important is a single parameter in modifying the prediction of a model in a certain region of independent variable space, say at a certain point in time and, moreover. How important is the numerical value of a specific observation in determining the estimated value of a particular parameter Although both questions fall within the domain of sensitivity analysis, in the following we shall address the first. The second question is addressed in Section 3.6 on optimal design. 

The art of experimental design is made richer by a knowledge of how the placement of experiments in factor space affects the quality of information in the fitted model. The basic concepts underlying this interaction between experimental design and information quality were introduced in Chapters 7 and 8. Several examples showed the effect of the location of one experiment (in an otherwise fixed design) on the variance and co-variance of parameter estimates in simple single-factor models. 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

The true model parameters (ft, ftj(...) are partial derivatives of the response function / and cannot be measured directly. It is, however, possible to otain estimates, ft, bV], bVl, of these parameters by multiple regression methods in which the polynomial model is fitted to known experimental results obtained by varying the settings of xr. These variations will then define an experimental design and are conveniently displayed as a design matrix, D, in which the rows describe the settings in the individual experiments and the columns describe the variations of the experimental variables over the series of experiments. 

Ette, E.I. Howie, C.A. Kelman, A.W. Whiting, B. Experimental design and efficient parameter estimation in preclinical pharmacokinetic studies. Pharm. Res. 1995, 12, 729-737. 

M. K. Al-Banna, A. W. Kelman, and B. Whiting, Experimental design and efficient parameter estimation in population pharmacokinetics. J Pharmacokinet Biopharm 18 347-360 (1990). 

E. I. Ette, A. W. Kehnan, C. A. Howie, and B. Whiting, Efficient experimental design and estimation of population pharmacokinetic parameters in preclinical animal studies. Pharm Res 12 729-737 (1995). 

Figure 7.16 Histogram of posterior predictive check based on the observed data in Table 7.4. Concentration data were simulated for 26 subjects under the original experimental design and sampling times at each dose using population values and variance components randomly drawn from the bootstrap distribution of the final model parameter estimates (FOCE-I Table 7.5). The geometric mean concentration at 6-h postdose (top) and AUC to 12-h postdose (bottom) was calculated. This process was repeated 250 times.

At present these exercises are more interesting than definitive. At the same time, the appearance of the computer has created an opportunity for the application of complex rate expressions in reactor design and in data fitting and parameter estimation. In ways unthinkable before, it has provided us with the means of evaluating the kinetics of complex mechanisms. What computers per se cannot do is provide us with the massive amounts of experimental data required for the fitting of complex mechanistic rate expressions. For that a brand new approach to the measurement of reaction rates is required. 

Experimental data of CO conversion are presented in Figure 3.3. These data were normalized by dividing the experimental value by the reactor design conversion value. It is important to state that all experimental data were normalized to the design values in order to avoid numerical convergence issues along the modeling and parameter estimation tasks. 

The rate equations were determined by Dumez and Froment by means of sequentially designed experimental programs for model discrimination and parameter estimation, discussed and illustrated in Chapter 2. The following equations were obtained ... 

---