Calculation sequence

The essential differences between sequential-modular and equation-oriented simulators are ia the stmcture of the computer programs (5) and ia the computer time that is required ia getting the solution to a problem. In sequential-modular simulators, at the top level, the executive program accepts iaput data, determines the dow-sheet topology, and derives and controls the calculation sequence for the unit operations ia the dow sheet. The executive then passes control to the unit operations level for the execution of each module. Here, specialized procedures for the unit operations Hbrary calculate mass and energy balances for a particular unit. FiaaHy, the executive and the unit operations level make frequent calls to the physical properties Hbrary level for the routine tasks, enthalpy calculations, and calculations of phase equiHbria and other stream properties. The bottom layer is usually transparent to the user, although it may take 60 to 80% of the calculation efforts. 

In the inner-loop calculation sequence, component flow rates are computed from the MESH equations by the tridiagonal matrix method. The resulting bottoms-product flow rate deviates somewhat from the specified value of 50 lb mol/h. However, by modifying the component stripping factors with a base stripping factor, S, in (13-109) of 1,1863, the error in the bottoms flow rate is reduced to 0,73 percent. 

For more precise values, computer programs can be used to calculate soluble recoveiy as weh as solution compositions for conditions that are typical of a CCD circuit, with varying underflow concentrations, stage efficiencies, and solution densities in each of the stages. The calculation sequence is easily performed by utihzing material-balance equations around each thickener. 

For an emission height, the calculational sequence begins by estiinabng tlie "effective" height of the emission, employing an applicable plume rise equation. The maximmn GLC may then be detennined using an appropriate abnospheric diffusion equtition (considered in tlie ne 1 Section). 

This includes recognizing the contribution from the feed (Rp), other feeds (Rqf). sidestreams (Rs). The Rp portion is determined a.ssuming no other feeds or side-streams are pre.sent. The Rop tmd R parts represent the summation of the contributions of other feeds and side-streams to the overall column minimum reflux ratio. The calculation sequence consists basically of three steps, here reproduced by permission of Chemical Engineering, Chou and Yaws, April 25, 1988, all rights reserved [96] ... 

A consistently anomalous (with respect to electrochemical evidence) position of Au has been found by two different groups. According to Kuznetsov etal.,437 the complete neglect of differential overlap (CNDO) method predicts for any given metal a weaker interaction on the more dense surface. Thus the predicted sequence is (111) < (100) < (110) for fee metals such as Cu, Ag, and Au and (0001) < (1100) for hep metals such as Zn and Cd. However, for the most compact surfaces, the calculated sequence is Hg < Ag(l 11) < Cu(l 11) =Zn(0001) < Au(l 11) < Cd(0001). 

To find the new gain margin, we need to, in theoiy, reverse the calculation sequence. We first use the phase equation to find the new crossover frequency (Dcg. Then we use the magnitude equation to find the new GOL, and the new GM is of course 1/ G0lI However, since we now know the values of td, xp, and KcKvKpKm, we might as well use MATLAB. These are the statements ... 

Successive substitution is the simplest although generally the least effective method for performing these calculations. Values are assumed for one or more of the unknown variables other variables are then determined from some of the equations new values of the assumed variables are determined from the remaining equations. The process is repeated until convergence is obtained. For flue gas desulfurization examples, many of the variables are highly constrained, and the calculation sequence can easily move into infeasible regions. The solution sequence frequently oscillates. 

The procedure will be described only for the case of bubblepoint temperature for which the calculation sequence is represented on Figure 13.5. Equations (13.8) and (13.32) are combined as... 

Normal coordinate analyses of a series of Cr(R2Dtc)3 complexes have been reported (81). In these analyses the complexes were treated as planar 1 1 metal-ligand systems of C2 symmetry and only the 11 in-plane vibrations (6At +5B2) were considered (81). The alkyl groups were treated as point masses. One interesting result of these studies was that the observed decrease in v(C-N) in the sequence R = Me > Et > Pr Bu, is paralleled by the calculated sequence obtained by increasing the point mass of the alkyl group... 

It is interesting that the observed sequence of reactivity (above, and Table 9) is strikingly similar to the calculated sequence for the ease of carbanion formation from various methyl derivatives, viz. Me2Hg < Me2Zn < MeLi (see Table 2, Chapter 2, p. 6). 

The time step, At, is used to switch the method from being a relaxation method to a global Newton method. When the time step is small, e.g., if = 0.1, then the changes in the independent variables are small. The method performs like a damped Newton-Raphson method, where the steps are small but in the direction of the solution and without any oscillation. When the value of At is large, i.e., At = 1000, the method performs like a Newton-Raphson method. The value of At at each column trial determines the speed and stability of the method, The units of the time step are the same as the flows to and from the column. The calculation sequence of the Ketchum method is as follows ... 

This calculational sequence is repeated until there is no significant change i from one iteration to the next. Final values of Pf are found from the Anto equations, and final yk values come from Raoult s law. 

The calculations are done in Excel worksheet Table D.l. The calculation sequence is ... 

Here the number 1 reappears, so another recycle loop has been found. This loop includes all the operations in this list. Note that the first loop, unit 10, is nested within this new loop. All the listed operations can now be combined into one new unit, label it 11. At operation 8, stream S14 was followed, it returned to operation 1 if stream S15 had been followed instead, operation 9 would have been transferred to the calculation list-the same result as obtained later. The choice of which output to follow first has no fundamental effect on the result except to perhaps change the calculation sequence. Continue following the outputs (there is only one from unit 11, 515) and listing the operations encountered ... 

The styrene feed was used as the starting point in this example the stream feed could have been chosen just as well. If it is, the same recycle loops will be identified, but a somewhat different calculation sequence will be identified. This is recommended as an exercise for the reader. 

Many flow-sheeting programs perform the partitioning, solution ordering, and tearing functions discussed above and present the user with one or more choices of solution sequence and tear variables. FLOWTRAN, however, does not do this. The user must identify the recycle loops, the calculation sequence, and the tear streams. The preceding example illustrated their identification and selection. 

For the methanol synthesis process illustrated in Fig. 4-1, Example 1, assume that there are algorithms for calculating the outputs of each process unit from the inputs. Determine how many stream variables must be specified and decide what these should be so that a unique solution exists for the mass and energy balances. Identity all recycle loops, tear streams for these loops, and a calculation sequence. 

Repeat Example 4 for the styrene synthesis process with a different selection for the tear streams to determine a new calculation sequence. 

For a process flow sheet obtained in Problem 3, assume that algorithms are available to calculate the outputs from each process unit from known inputs. Determine the number of stream variables that must be specified, decide what they should be, identify all recycle loops, select tear streams for these loops, and establish a calculation sequence. 

Related Calculations. Sequences of distillation columns are often used for separating multicomponent mixtures. The question arises In what order should the individual components be separated For example, should a given component be separated in the first column, the second column, the... 

The set of independent variables which satisfies these requirements is the vapor flow vector V, the liquid phase compositions, X2 to Xm, and the temperature vector T. The calculation sequence is,... 

The only undefined step in the calculation sequence proposed above is the correction of the independent variables in step f. A linear correction process is given by... 

Experimental and theoretical studies on the solvent influence on molecular geometries and cis/trans isomerization processes of other so-called push-pull ethenes, R2N CH=CH A R2N+=CH CH=A- (with A = NO2, CHO, CN, etc.), have been collected in references [287, 288], The barriers to isomerization about the C=C bonds of these acceptor-substituted enamines are considerably smaller than those for simple ethenes such as 2-butene Ea = 259 kJ/mol [82]), owing to a significant contribution of the mesomeric zwitterionic structure to the electronic ground state. Increasing solvent polarity increases the contribution of this dipolar mesomeric structure, and hence leads to a decrease in the barrier to C=C isomerization and a simultaneous increase in the barrier to rotation about the C—N bond [288], The calculated sequence of solvent stabilization for such acceptor-substituted enamines is (activated complex for C=C rotation) E form > Z form > (activated complex for C—N rotation). Obviously, the activated complex for isomerization about the C=C bond corresponds to a full zwitterion with maximal solvent stabilization [288]. 

Obviously, what we would really like to do is not just have a feel for tendencies, useful as this is, but also calculate copolymer composition and sequence distributions, things that can also be measured by spectroscopic methods. We will start by using kinetics to obtain an equation for the instantaneous copolymer composition (it changes as the copolymerization proceeds). Later we will use statistical methods to describe and calculate sequence distributions. In deriving the copolymer equation, we only have to consider the propagation step and apply our old friend, the steady-state assumption, to the radical species present in the polymerization, and... 

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