Model series

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Equations 8-109, 8-110, 8-111, and 8-112 are redueed to an ordinary tanks-in-series model when N = i and h = 0. For the equivalent number of ideal CSTRs, N is obtained by minimizing the residual sum of squares of the deviation between the experimental F-eurve and that predieted by Equation 8-109. The objeetive funetion is minimized from the expression... 

A simple springs-in-series model represents the representative volume element loaded in the 2-direction as in Figure 3-11. There, the matrix is the soft link in the chain of stiffnesses. Thus, the spring stiffness for the matrix is quite low. We would expect, on this basis, that the matrix deformation dominates the deformation of the composite material. 

This function is intermediate between the parallel model and the series model and referred to as the logarithmic law of mixture shown in curve 3. The law of mixture is valid for a composite system when there is no interaction in the interface. However, it is natural to consider that interaction will occur in the interface due to contact between A and B. Then considering the creation of interfacial phase C, different from A and B, the following equation can be presented ... 

Fig. 9. Upper (parallel model) and lower (series model) bounds for the microhardness of a two-component composite as function of crystalline volume function. Hardness expressed as H/Hc for Hc/H = 120...

The Tanks-in-Series Model. A simple model having fuzzy first appearance times is the tanks-in-series model illustrated in Figure 15.2. The washout function is... 

FIGURE 15.2 The tanks-in-series model (a) physical representation (b) washout function. 

Solution Equations (15.27) and (15.28) give the residence time functions for the tanks-in-series model. For A =2,... 

The limits for part (b) are at the endpoints of a vertical line in Figure 15.14 that corresponds to the residence time distribution for two tanks in series. The maximum mixedness point on this line is 0.287 as calculated in Example 15.14. The complete segregation limit is 0.233 as calculated from Equation (15.48) using/(/) for the tanks-in-series model with N=2 ... 

Albanis TA, Pomonis PJ, Sdoukos AT. 1988a. Describing movement of three pesticides in soil using a CSTR in series model. Water Air Soil Pollut 39 293-302. 

Mathai and Singh have estimated the permeability coefficient P, using the formula P = kD where k is the partition coefficient and D is the diffusivity. They have used both parallel and series models to calculate P. The experimental values are always greater than measured values. The poor agreement between the experimental and calculated values is attributed to the polar-polar interaction between the epoxy group and nitrile group. 

The precision of time series predictions far into the future may be limited. Time series analysis requires a relatively large amount of data. Precautions are necessary if the time intervals are not approximately equal (9). However, when enough data can be collected, for example, by an automated process, then time series techniques offer several distinct advantages over more traditional statistical techniques. Time series techniques are flexible, predictive, and able to accommodate historical data. Time series models converge quickly and require few assumptions about the data. 

Building a Time Series Model Using Pilot Plant Data... 

Figure 2. Experimental trial used to Identify transfer function. In this experiment, the reactant flow rate was deliberately varied and the reactant temperature measured on-line in the pilot plant. This allowed us to identify the proper time series model.

At Rohm and Haas a committee of several experts contributed to the successes described In this paper. Discussions with Prof. John MacGregor (HcHaster University), Jeff Nathanson, Tom Shannon and Tom Throne were especially Important. Special thanks are due to Chris Altomare, who always had the proper equipment and Instrximentatlon ready for the pilot plant trials. We also would like to acknowledge Prof. Don Watts (Queens University) for assisting with the time series modeling and Prof. 

The Stirred Tanks in Series Model Another model that is frequently used to simulate the behavior of actual reactor networks is a cascade of ideal stirred tank reactors operating in series. The actual reactor is replaced by n identical stirred tank reactors whose total volume is the same as that of the actual reactor. 

The performance of a CSTR can be brought closer to that of a PFR, if the CSTR is staged. This is considered in Chapter 20 in connection with the tanks-in-series model. 

Apply the tanks-in-series model to the following kinetics scheme involving reactions in parallel ... 

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