Sequential solutions examples

From a computational viewpoint, the presence of recycle streams is one of the impediments in the sequential solution of a flowsheeting problem. Without recycle streams, the flow of information would proceed in a forward direction, and the cal-culational sequence for the modules could easily be determined from the precedence order analysis outlined earlier. With recycle streams present, large groups of modules have to be solved simultaneously, defeating the concept of a sequential solution module by module. For example, in Figure 15.8, you cannot make a material balance on the reactor without knowing the information in stream S6, but you have to carry out the computations for the cooler module first to evaluate S6, which in turn depends on the separator module, which in turn depends on the reactor module. Partitioning identifies those collections of modules that have to be solved simultaneously (termed maximal cyclical subsystems, loops, or irreducible nets). 

Immersion Photografting As the name implies, the surface to be grafted is immersed in a solution containing sensitizer and monomer or alternatively solutions of each sequentially. The examples given thus far also fall into this category, regardless of the number of steps grafting has been conducted. 

While the voltammograms can be quantitatively analyzed to yield permeant transport parameters - for example, via numerical simulations - we have instead utilized rotating disk electrode (RDE) voltammetry for this purpose. Sequential solution and film transport problems are generally well described by a variant of the Koutecky-Levich analysis [11]. This analysis represents the series process Idnetically as a reciprocal sum of pure solution-phase convection/dilFusion (cd) and pure film permeation (p) rates or fluxes. Since detection is electrochemical, the fluxes can be written in terms of currents, where ilim is the observed overall, mass-transport limited, current [12] ... 

Although we can approach the problem of solving n equations in n unknowns directly, it may be easier to break the equations into a solution strategy where each equation can be used to solve for a single variable sequentially. For example, consider the following three equations in three unknowns ... 

Inclusions, occlusions, and surface adsorbates are called coprecipitates because they represent soluble species that are brought into solid form along with the desired precipitate. Another source of impurities occurs when other species in solution precipitate under the conditions of the analysis. Solution conditions necessary to minimize the solubility of a desired precipitate may lead to the formation of an additional precipitate that interferes in the analysis. For example, the precipitation of nickel dimethylgloxime requires a plT that is slightly basic. Under these conditions, however, any Fe + that might be present precipitates as Fe(01T)3. Finally, since most precipitants are not selective toward a single analyte, there is always a risk that the precipitant will react, sequentially, with more than one species. 

However, Pauli s Nobel Prize-winning work did not provide a solution to the question which I shall call the closing of the periods —that is why the periods end, in the sense of achieving a full-shell configuration, at atomic numbers 2,10, 18, 36, 54, and so forth. This is a separate question from the closing of the shells. For example, if the shells were to fill sequentially, Pauli s scheme would predict that the second period should end with element number 28 or nickel, which of course it does not. Now, this feature is important in chemical education since it implies that quantum mechanics cannot strictly predict where chemical properties recur in the periodic table. It would seem that quantum mechanics does not fully explain the single most important aspect of the periodic table as far as general chemistry is concerned. 

Transport systems can be described in a functional sense according to the number of molecules moved and the direction of movement (Figure 41-10) or according to whether movement is toward or away from equilibrium. A uniport system moves one type of molecule bidirectionally. In cotransport systems, the transfer of one solute depends upon the stoichiometric simultaneous or sequential transfer of another solute. A symport moves these solutes in the same direction. Examples are the proton-sugar transporter in bacteria and the Na+ -sugar transporters (for glucose and certain other sugars) and Na -amino acid transporters in mammalian cells. Antiport systems move two molecules in opposite directions (eg, Na in and Ca out). 

Several enamines also participate in these cycloaddition reactions. For example, the addition of methyl lithium to benzaldehyde 5 and the sequential introduction of the vinylogous amide and magnesium bromide results in the cycloaddition elimination product chromene 63 (method G, Fig. 4.33).27 The introduction of methyl magnesium bromide to a solution of the benzaldehyde 5 and two equivalents of the morpholine enamine produces the cycloadduct 64 in 70% yield with better than 50 1 diastereoselectivity (method F). Less reactive enamides, such as that used by Ohwada in Fig. 4.4, however, fail to participate in these conditions. 

EXAMPLE 14.3 SOLUTION OF AN ALKYLATION PROCESS BY SEQUENTIAL QUADRATIC PROGRAMMING... 

Example 14.3 Solution of an Alkylation Process by Sequential Quadratic... 

The solution of Example 6.1 is considered as the first stage of the sequential procedure. The following residuum in the balances and its corresponding covariance matrix are obtained ... 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

Because of the yield of only 16% in the synthesis of 239, the overall yields of cycloadducts with reference to indene according to Scheme 6.54 are rather low, however. A substantial improvement was achieved by the development of a one-pot procedure, which starts from indene and takes advantage of its dibromocarbene adduct (254) (Scheme 6.55). This was prepared at -60 °C with tetrabromomethane and MeLi as source for the carbene and remained unchanged in solution up to temperatures around 0 °C [92]. If an activated alkene and MeLi were added sequentially to such a solution at -30 C, cydoadducts of 221 were isolated in a number of cases in relatively good yields. In Scheme 6.55, this procedure is illustrated by the example of 1,3-cyclopentadiene, which furnished the [4 + 2]-cydoadducts 255 and 256, both as a mixture with endo exo= 2 1, in the ratio of 8 1 in 23% yield with reference to indene [67]. Analogously, the products from 221 and styrene, 1,3-butadiene [92] and 2,3-dimethylbutadiene [66], namely the compounds 240, 241, 246-249 and 250-253, were obtained in yields of 40, 24 and 25%, respectively, by means of the one-pot procedure from indene. 

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