Modular Representation

The number of degrees of freedom can also be determined by representing the col- 

These units are assumed to be at fixed pressure, determined independently based on material flow and equipment considerations. The feed rates, composition, and thermal conditions are either fixed or determined by upstream conditions, and therefore do not contribute to the degrees of freedom. 

The column section is a vertical multistage vapor-liquid countercurrent adiabatic column, with a liquid feed and a vapor product at the top, and a vapor feed and a liquid product at the bottom. At a constant pressure this unit has no variable that can be controlled independently, and therefore has zero degrees of freedom. 

The equilibrium stage can receive multiple feed streams and can have a heat duty (transferred in or out). Its products can be vapor or liquid, or vapor and liquid at equilibrium. The heat duty can be controlled independently therefore at constant pressure this unit has one degree of freedom. 

The stream splitter splits one feed stream into two products having the same composition and thermal conditions as the feed. The split ratio may be controlled independently, giving this unit one degree of freedom. 

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L. Domhoff, Group Representation Theory. Part A Ordinary Representation Theory. Part B Modular Representation Theory (1971,1972)... 

Papalexandri KP, Pistikopoulos EN. Generalized modular representation framework for process synthesis. AIChE J 1996 42(4) 1010. 

The proof of Glauberman s theorem requires Richard Brauer s Second Main Theorem on modular representations of finite groups cf. [5]. A proof of Glauberman s theorem is beyond the scope of this monograph. 

However, the proof of this latter fact is beyond the scope of this monograph. It is based on modular representation theory of finite schemes and follows from [13 Theorem 3], [6 Theorem 2], and [14 Theorem 1[. 

First, there are numerous results on characters of schemes of finite valency which have not found their way into this text, although they provide an impressive insight into the rich and prosperous theory of schemes. In particular, modular representation theory of schemes of finite valency, as it has been developed mainly by Akihide Hanaki, has been excluded completely.6... 

Johnson BA, Leon M. Modular representations of odorants in the glomerular layer of the rat olfactory bulb and the effects of stimulus concentration. J. Comp. Neurol. 2000 422 496-509. Laing DG, Legha PK, Jinks AL, Hutchinson 1. Relationship between molecular structure, concentration and odor qualities of oxygenated aliphatic molecules. Chem. Senses 2003 28 57-69. Rossiter K. Quantitative relationships between structure and the fruity odor of esters. Perfumer Flavorist 1996 21 33-36, 38-40, 42-44, 46. 

Two extremes are encountered in flowsheeting software. At one extreme, the entire set of equations (and inequalities) representing the process is employed. This representation is known as the equation-oriented method of flowsheeting. The equations can be solved in a sequential fashion analogous to the modular representation described below or simultaneously by Newton s method, Broyden s method, or by employing sparse matrix techniques to reduce the extent of matrix manipulations. Refer to the review by Evans and Chapter 5. ... 

Johnson BA, Leon M. 2000a. Modular representations of odorants in the glomerular layer of the rat olfactory bulb and the effects of stimulus concentration. J Comp Neurol... 

Figure 9.5 shows the distillation column and its modular representation. The feeds... 

Figure 17-5. Modular representation of crystal structures of superconductors with general formula TlBa2Ca iCUjj02n+3 for n = 1 (a) 2 (b) 3 (c) 4 (d) and 5 (e). Large and small black spheres represent Ba and Cu atoms, respectively large, intermediate and small blank spheres represent Tl, Ca and O atoms, respectively...

Maria, G. Reduced modular representations applied to simulate some genetic regulatory circuits. Rev. Chim. 59, 318-324 (2008)... 

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