Information flow among equations

Decomposition leads to a rearrangement of the process equations from their flow chart sequence to a natural sequence based on the information flow among the equations. The ultimate goal is to set up an iterative scheme in which each equation is solved for a single variable (by some appropriate root identification method), and where values of unknown variables that must be assumed are checked cyclically. The greatest reduction in the number of iterates that must be assumed, and therefore the greatest reduction in computer storage and time requirements, takes place for those systems of process equations in which the number of variables per equation is small compared to the total number of variables in the system. Clearly, when each of the system equations contains every process variable, no effective decomposition can take place. Fortunately, most models used in the process industries are of such a character that extensive decomposition can be effected. 

Since the systems of equations to be considered are quite large, it is necessary to use some compact method to represent the information flow among them. A very convenient technique is to relate the system equations to a digraph and its associated Boolean matrix, which represent the structure of the information flow in the system. The Boolean matrix to be used is called the occurrence matrix (HI, S3), and is defind as follows (1) each row of the occurrence matrix corresponds to a system equation, and each column corresponds to a system variable (2) an element of the matrix, s -, is either a Boolean 1 or 0 according to the rule... 

In decomposing a system of equations, it is necessary to analyze the information flow among the equations concerning the values of the system variables (S3). In order to determine the direction of information flow in the system equations, one must first establish what information each equation is to supply, that is, the identity of the variable whose value is to be obtained from the equation. Further, the system equations together must supply all of the information about the system (the values of all of the variables). The variable for which an equation is to be solved is called its output variable and the set of all of the variables assigned to the equations as output variables is called an output set (S3, H2). Thus the information that one equation can... 

There are two distinct classes of analytic approximation that comprise the second and third approaches that were just mentioned. The first is based on the use of so-called macroscopic balances. In this approach, we do not attempt to obtain detailed information about the velocity and pressure fields everywhere in the domain, but only to obtain results that are consistent with the Navier Stokes equations in an overall (or macroscopic) sense. For example, we might seek results for the volumetric flow rates in and out of a flow system that are consistent with an overall mass or momentum conservation balance but not attempt to determine the detailed form of the velocity profiles. The macroscopic balance approach is described in detail in many undergraduate textbooks.2 It is often extremely useful for derivation of quantitative relationships among the average inflows, outflows, and forces (or rates of working) within a flow system but is something of a black-box approach that provides no detailed information on the velocity, pressure, and stress distributions within the flow domain. 

Gas phase properties As stated before, all the model equations involve parameters that are determined by the behavior of bubbles, either alone or in groupings, and the analysis becomes more of an exercise in bubble fluid mechanics than in reactor design. For plug-flow gas phase reactors there are a number of correlations that relate in-reactor bubble properties as a function of the inlet conditions. These are available for the bubble volume Vb, the bubble rise velocity Vb, the surface to volume ratio a, and the number of bubbles per unit volume N. In addition, if bubbles are spherical (or approximately so), information on db allows determination of a and Vb- However, these correlations are subdivided by the gross characteristics of bubble formation, namely whether there is a gas phase consisting of discrete bubbles, or whether there is interaction among bubbles with some coalescence, commonly termed a swarm bubble phase. 

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