Simultaneous equations, partitioning

Once the complete system of equations has been partitioned into the irreducible subsystems of simultaneous equations, it is desirable to decompose further these irreducible blocks of equations so that their solution can be simplified. The decomposition of the irreducible subsystems is called tearing. In the remainder of this section the subsystems of irreducible equations found by partitioning will be referred to as blocks to distinguish them from the smaller subsystems of simultaneous equations obtained within a block after the tearing is accomplished. 

The advantage of using the time lag method is that the partition coefficient K can be determined simultaneously. However, the accuracy of this approach may be limited if the membrane swells. With D determined by Eq. (12) and the steady-state permeation rate measured experimentally, K can be calculated by Eq. (10). In the case of a variable D(c ), equations have been derived for the time lag [6,7], However, this requires that the functional dependence of D on Ci be known. Details of this approach have been discussed by Meares [7], The characteristics of systems in which permeation occurs only by diffusion can be summarized as follows ... 

To improve the formatting of the equations that represent a plant, many commercial codes partition the equations into groups of irreducible sets of equations, that is, those that have to be solved simultaneously. If a plant is represented by thousands of equations, the overall time consumed in their solution via either a GRG or SQP algorithm is reduced by partitioning and rearranging the order of the equations with the result indicated in Figure 15.6. Organization of the set of equations into irreducible sets can be carried out by the use of permutation matrices or by one of... 

This section treats the partitioning of the system equations into the smallest irreducible subsystems, that is, the smallest groups of equations that must be solved simultaneously. Partitioning represents the first and easiest of the two phases of decomposition tearing (which is discussed in Section VI) is more difficult. 

In tearing, the objective is to wind up with less computation time required to solve the torn system compared with the time required to solve the entire block of equations simultaneously. However, the criteria for evaluating the effectiveness of the tearing are by no means so well defined as those for partitioning, where the objective is clearly to obtain the smallest possible subsystems of irreducible equations. There is no general method for determining the time needed to effect a solution of a set of equations it is necessary to consider the particular equations involved. Any feasible method of tearing, then, must be based on criteria that are related to the solution time. Some of the more obvious criteria are ... 

A process can also be represented by its adjacency matrix, which can be formed without first drawing the graph, by just assigning each unit a number and placing a nonzero entry in column j and row i if there is a stream directed from unit j to unit /. Once the adjacency matrix is formed, it can be partitioned into blocks of units that must be solved simultaneously exactly as described in Section IV. It is the tearing of these blocks that is of interest here. In both of the methods of tearing discussed here the objective is to tear a block so that the minimum number of variables will have to be assumed in solving the process equations involved in the block. 

SVOCs may be simultaneously present in meaningful amounts in the gas phase and on the surface of airborne particles. Therefore it is important to sample both the gas phase and the partide phase. This distribution between the gas phase and particles is known as partitioning and may roughly be illustrated by Junge s equation (Junge, 1977) derived on the basis of adsorption theory ... 

If heat is removed (e.g., via an intercondenser) or added (e.g., via an interreboiler). the foregoing equations still apply when each source of heat removal or addition is treated as two simultaneous partitions, one from which a stream leaves the column and one to which a stream of the same composition reenters. The two partitions are denoted k- and k, respectively. For heat removal, the stream leaving the column is saturated vapor, and the stream reentering is condensed liquid the reverse applies for heat addition. The following equations are used ... 

Here a and b are occupied MO s of systems A and B. Equation (6,32) is easily expressible in terms of integrals over atomic basis functions and elements of the density matrix. In eqn. (5.31) two terms may be distinguished. The first one is due to single electron excitations of the type a r") and (b —->s"), where a and r", respectively, are occupied and virtual MO s in the system A, and b and s" are occupied and virtual MO s in the system B, Contribution of these terms corresponds to the classical polarization interaction energy, Ep, Two-electron excitations (a r", b — s"), i.e. simultaneous single excitations of either subsystem, may be taken as contributions to the second term - the classical London dispersion energy, Ep, If the Mjiller-Ples-set partitioning of the Hamiltonian is used, Ep may be expressed in... 

The partitioning of H2O among all simultaneous and competitive reactants is not known at all. A system of kinetic equations to describe well this network is not known again and empiricism is the only solution known by the authors. 

A method for precedence ordering so as to partition a model into a sequence of smaller models containing sets of irreducible equations (equations that have to be solved simultaneously as illustrated in Fig. 5.6). 

Figure 5.6 Partitioning and tearing. The equations are partitioned into blocks containing common variables, as in (c). Equations hz and /14 (set I) are solved simultaneously for xz and xs first, then hs (set II) is solved for JC4 ( and lastly hi and hz are solved simultaneously. For example, assume a value for Xs solve hi for jci then check to see if Equation hs is satisfied. If not, adjust X3, solve hi for Xt, recheck hz, and so on until both h and hz are satisfied.

Suppose that instead of a simultaneous solution of the equations, a sequential solution is wanted. In what order should the equations be solved Partition the equations so that a sequential solution can be executed. Lump together blocks of equations that still have to be solved simultaneously. 

The constants were then used to predict partition coefficients for the four component benzene-hexane-water-ethanol system. This prediction was accomplished in the following way. The hydrocarbon phase composition was assumed to be known. The activity coeifficient yes was calculated by the Wilson equation. Equation (10) was then solved simultaneously with the partition coefficient equation ... 

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