Relationships Among Distribution Functions

The two distribution functions n and are not equal but are related as follows For a spherical particle, the volume and particle diameter are related by the expression 

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The characteristic separation curve can be deterrnined for any size separation device by sampling the feed, and coarse and fine streams during steady-state operation. A protocol for determining such selectivity functions has been pubHshed (4). This type of testing, when properly conducted, provides the relationships among d K, and a at operating conditions. These three parameters completely describe a size separation device and can be used to predict the size distribution of the fine and coarse streams. 

A summary of the relationships among the age-distribution functions is given in Table 13.1. This includes the results relating E(6) and F(6) from Section 13.3.2, together with those for W(6), 1(6), and H (these last provide answers to problems 13-3, -4 and -5(a), (b)). Each row in Table 13.1 relates the function shown in the first column to the others. The means of converting to results in terms of Eft), Fft), etc. is shown in the first footnote to the table. 

To analyze and measure the reliability and maintainability characteristics of a system, there must be a mathematical model of the system that shows the functional relationships among all the components, the subsystems, and the overil system. The reliability of the system is a function of the reliabilities of its components. A system reliability model consists of some combination of a reliability block diagram or cause-consequence chart, a definition of all equipment failure and repair distributions, and a statement of spare and repair strategies (Kapur 1996a). All reliability analyses and optimizations are made on these conceptual mathematical models of the system. 

Table 6.4 lists the 31 optional objective functions, and Table 6.5 shows the 48 reaction activity factors for selection. Aspen H YSYS Petroleum Refining combines the input plant product distribution to construct the reactor effluent and partition the reactor effluent into Cl, C2, C3, C4, C5, and four square cuts , namely, naphtha (C6 to 430 °F cut), diesel (430-700 °F cut), bottom (700-1000 °F) cut and residue (1000 °F-i- cut) which are shown in Table 6.4. AU of the objective functions listed in Table 6.4 are either the prediction errors of crucial operations or important product yields for the HCR process. Aspen HYSYS Petroleum Refining allows us to select the desired objective functions during calibration. After selecting the objective functions, we choose appropriate activity factors to calibrate the reactor model. Figure 6.12 illustrates the relationships among the activity factor, catalyst bed, and reactor type, and Table 6.5 shows the major effect of each activity factor on the model performance, such as global activity (Rgiobai) on the bed temperature profile to help the selection of activity factors.

Statistical models. A number of statistical dose-response extrapolation models have been discussed in the literature (Krewski et al., 1989 Moolgavkar et al., 1999). Most of these models are based on the notion that each individual has his or her own tolerance (absorbed dose that produces no response in an individual), while any dose that exceeds the tolerance will result in a positive response. These tolerances are presumed to vary among individuals in the population, and the assumed absence of a threshold in the dose-response relationship is represented by allowing the minimum tolerance to be zero. Specification of a functional form of the distribution of tolerances in a population determines the shape of the dose-response relationship and, thus, defines a particular statistical model. Several mathematical models have been developed to estimate low-dose responses from data observed at high doses (e.g., Weibull, multi-stage, one-hit). The accuracy of the response estimated by extrapolation at the dose of interest is a function of how accurately the mathematical model describes the true, but unmeasurable, relationship between dose and response at low doses. 

The fact that or 2 can be used as the independent variable for the log normal distribution creates confusion among students because the log normal density function can take on different forms depending on which variable is used. Thus, if we let jc be the actual particle size and z be the In transformation then we have the following relationships ... 

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