Shannon Method

Determined using AOAC method 26.026-26.031 (CB) for corn and peanuts, method 26.052-26.059 for cottonseed, and the Shannon method for mixed feed (J.Assoc. Off. Anal. Chem. 1983, 66, 582. 

A study of a 7 (2.13 m) high steel reheat furnace versus a 9 (2.74 m) high similar furnace (using the Shannon Method explained in chap. 8) showed that the 7 furnace required a higher average gas temperature than the 9 to heat the same load at the same rate—because of its shorter gas beam height. 

Suggested research project, described at the end of this chapter. No convection, conduction, or particulate radiation are included in Shannon Method calculations for steel reheat furnaces. 

Current practice requires engineers to have more than a feel for load heating patterns (time-temperature profiles). They must acquire an ability to determine the effects of many operating and design variables on various loads time-temperature curves. The Shannon Method, which enables one to calculate specific time-temperature curves, is discussed briefly several places in this book and then detailed in chapter 8. The reader is encouraged to adapt the Shannon Method for processes other than the steel reheat and forging cases illustrated here. 

By the Shannon Method explained in Chapter 8, a temperature-versus-time heating curve was calculated for the center piece, and the total heating time was found to be 23.5 hr. If the center piece were removed to give the two outer pieces better heat transfer exposure, the heating time for the two remaining pieces would be 20 hr. 

Many charts have been developed for predicting the time it takes to heat steel. (See figs. 3.12 and 4.21a.) The industry now has better methods for predicting required heating times (e.g., the Shannon Method, in chap. 8). It combines (a) the radiation heat transfer equation for the time it takes to transfer the required heat to the load, with (b) lag time theory. Together, (a) and (b) predict how fast and how uniformly a product can be heated, knowing the size and nature of the pieces to be heated and their location relative to the furnace gases and the refractory. 

Calculating furnace size and firing rate can be accomplished by the Shannon Method detailed in chapter 8. The required furnace length = required heating time multiplied by stock feed speed. Heating times and cooling times between barrels should be figured and plotted alternately. 

The objective of this exercise is to develop a set of time/temperature curves such as shown in figs. 6.26 to 6.33 and figure 8.1. In this book, the authors frequently urge the readers to use this Shannon Method to develop such curves for their own specific loads, processes, and heating equipment so that they can better analyze their furnace capabilities and requirements, and make good engineering judgments relative to their control. 

Sample Problem Shannon Method for Temperature-Versus-Time Curves... 

Fig. 8.4. Effect of carbon content in various steei grades on heat absorption is shown by these grade factors used in the iast steps of tabie 8.3 (worksheet) for the Shannon Method for piotting steei heating curves. The peaks in this graph show the effect of the dramatic increase in heat absorption for steeis containing various percentages of carbon, C, during the crystaiiine phase changes between 1200 F and 1900 F (650 C and 1038 C). SS = stainless steel.

In table 8.7, enter the difference between the black body radiation rate for furnace temperature [b] and load temperature [d], on line [i]. Multiply [i] by 3H, for the 3 unit group, and enter the resulting Btu/pound heat content rise of the load on line j]. The Shannon method s H factor reduces black body radiation by the effect of emis-sivity (absorptivity). In succeeding columns, use line [k] to totalize the cumulative Btu/pound. In figure 8.9, convert the new Btu/pound heat content to a new average temperature throughout the load (270 F for the first three time units), and record it on line [1],... 

Bottom temperature for top-only heating, but center temperature if using top and bottom heating. (Hus detailed explanation of the Shannon Method for plotting steel heating curves continues several pages later, after the worksheets and table 8.9.)... 

Shannon s method, expands a Boolean function of n variables in minterms consisting of all combinations of occurrences and non-occurrences of the events of interest. Consider a function of n Boolean variables XJ which may be expanded about X, as shown in Equation 2.2-3 where f(l, Xj,..., XJ where 1 replaces X,. This says that a function of Boolean variables equals the function with a variable set to I plus the product of NOT the variable limes the function with the variable set to 0. By extending Equation 2.2-3, a Boolean function may be expanded about all of its... 

Shannon, C. E., 190,195,219,220,242 Shapley, L. S316 Skirokovski, V. P., 768 Shortley, O. H., 404 Shot noise process, 169 Shubnikov, A. V., 726 Shubnikov groups, 726 Shubnikov notation for magnetic point groups, 739 Siebert, W. M., 170 Signum function, 313 Similar matrices, 68 Simon, A408 Simplex method, 292 Simulation, 317... 

According to Shannon, the uncertainty of the two methods can be expressed by means of ... 

Other methods explored internally to alter the form of nicotine delivered included the use of base- or acid-coated filters. For example, researchers at RJR applied sodium hydroxide-coated filters to a cigarette yielding only 0.06 mg of nicotine in order to heighten sensory impact (Shannon et al. 1992). Alternately, a filter coated with an acid (lactic, levulinic, citric) was used to reduce the impact of a high nicotine sheet, either by trapping the nicotine or by changing the pH of the smoke so there is not as much lucotine in the vapor phase (Shannon et al. 1992). The researchers noted that ... 

It is a method that samples the chemical space accessible to a library in a manner that, when combined with assay data, allows the elucidation of a set of pharmacophores consistent with biological activity. The library is selected in a manner that maximizes the Shannon entropy [75] for the possible outcomes being tested by the library [76]. 

Goetz, C.G., Tanner, C.M., Wilson, R.S., Carroll, V.S., Como, P.G., and Shannon, K.M. (1987) Clonidine and Gilles de la Tourette syndrome double-blind study using objective rating method. Neurol 31 307—310. 

C. E. Shannon (1916-2001) developed an information-theoretic definition of entropy that (although not equivalent to the physical quantity) carries similar associations with microstates and probability theory. Shannon recognized that Boolean bit patterns (sequences of l s and 0 s) can be considered the basis of all methods for encoding information. ... 

Figure 24.5 Measures of Shannon and Rao-Stirling diversity (in arbitrary units, a.u.) in relation to the publication disciplines of the references in the reviews. See Section 24.4, Data and Methods, for their definition. Both measures are normalized dividing by the largest value in the series.

Aquatic microcosms and mesocosms offer the ideal situation to investigate populations of species interacting in their natural environment (i.e., to study communities stressed in structured systems see Section 4.5.1). It is, however, only recently that these experiments have been analyzed at the community level (Van Wijngaarden et al. 1995 Sparks et al. 1999 van den Brink and Ter Braak 1999). Until 10 years ago, experiments were evaluated at the population level, largely ignoring species interactions and energy flows in the systems. The development of community-level endpoints offered the possibility to evaluate the experiments on a community level (i.e., they offered the opportunity to scale up the level of evaluation Kedwards et al. 1999). Summary community-level endpoints calculated from the results of these experiments are mostly structural ones measures of diversity (e.g. numbers of species, and the Shannon-Weaver diversity index) and similarity of the treated systems compared to the untreated controls (e.g., the principal response curves method, Bray-Curtis dissimilarity, or Stander s index see van den Brink and Ter Braak 1998 for a comparison). 

---