Closure equations

Establishing the necessary constitutive and closure equations (the former relate fluid stresses with velocity gradients the latter relate unknown Navier-Stokes-equation coiTclations witli known quantities). 

If the reactor operates isothermally and if the pressure drop is sufficiently low, we have achieved closure. Equations (3.11) and (3.13) together allow a marching-ahead solution. The more common case requires additional equations to calculate pressure and temperature. An ODE is added to calculate pressure P z), and Chapter 5 adds an ODE to calculate temperature T z). 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

At least three species will be necessary to describe the behavior of the three end-members j=[Pg.6]

The disulfide sulfur of 263 acts as an electrophile with the enolate anion resulting in ring closure (Equation 54) <1980LA1917>. DMSO oxidizes the sulfur of 264 which then reacts with the enamine double bond (Equation 55) <2001JFC(108)51>. 

Kidwai et al. reported the preparation of a series of 3-substituted quinolino[3,2-< ]-l,2,4-triazines 240a-f by the nitrosation of the hydrazones 239 with sodium nitrite in acetic acid under microwave irradiation or reflux, leading to ring closure (Equation 23) <1998IJB1063>. 

The phosphorus macrocycles are made via template condensation of coordinated polyphosphine ligands and a dibromoalkane (equation 8).60 A more recently reported method involves a template-assisted single-stage ring closure (equation 9).61 The arsenic donor macrocycles are synthesized by reacting lithiated polyarsanes with a dichloroalkane (equation 10).62... 

Ni(CO)3py] catalyzes the carbonylation of 2-mercaptoethanol with ring closure (equation 100).471 The product is 0,S-ethylenethiocarbonate. A thiol complex, [Ni(SCH2CH2OH)2], was isolated from the reaction mixture and was also shown to catalyze the reaction, which occurred at a temperature and pressure only slightly above normal. 

C is-fused tetrahydrofurans are produced when 2-cycloalkenyl-substituted ethanol derivatives are cy-clized via 5-exo ring closure (equation 14 and Table 5). The related 1-cycloalkenyl alcohol systems also yield cri-fused tetrahydrofurans upon reaction with phenylselenyl reagents, similar to their carboxylic acid analogs (see equation 12).60 A sulfoetherification to a fused ring tetrahydrofuran from a system with an exocyclic methylene provided an 86 14 ratio of cis- and frans-fused isomers.61... 

In addition to products E and F, 16% (at 60% conversion) of 9-methyl-carbazole G has been isolated. Carbazole formation implies a loss of S02 from sulfone H and subsequent ring closure (Equation 30). 

The solution to this equation, the two functions h(r) and c(r), can be obtained if we have another equation to close, which is often referred to as the closure equation . 

The closure equations are approximations derived from statistical mechanics, meaning that the results are also given in an approximated manner. 

Both the 2-thiahex-5-enyl and the 2-sulfonylhex-5-enyl radicals exhibit a significant preference for 5-exo cyclisa-tion to the thiophene over formation of tetrahydrothiopyran by 6-endo ring closure (Equation 144) <2000TL7987> and only at higher I2 concentration does the formation of tetrahydrothiopyrans compete successfully with thietane and tetrahydrothiophene production when methylene-intermpted dienoates 455 react with dimethyl disulfide (Equation 145) <1994TL5575>. 

Thieno[2,3-Z>]pyrroles have been prepared from 2-thienylhydrazines and carbonyl compounds under the conditions of Fischer indolization (77S487). The Bu OCO group is eliminated, and the intermediate hydrazone can undergo ring closure (equation 15). Thieno[3,2-Z>]pyrroles were prepared similarly from acylated 3-hydrazinothiophenes (75CHE1133). 

Vinylsulphonium salts (375) are excellent precursors for the synthesis of electrophilic cyclopropanes via nucleophilic addition of active methylene functions followed by ring-closure (equation 123) . A similar reaction is observed using butadienylsulphonium... 

Altenbach used the Michael addition of sodium methyl malonate to allene (206) for a dia-stereoselective spiroannulation to a steroid (equation 72). Or, in imaginative work by Okamura, allenyl sulfoxides were transformed into enantiomerically pure hydrocarbons by pericyclic reactions like electrocyclic ring closure (equation 73) or intramolecular cycloaddition (equation 74). Note that the starting materials (propargylic alcohols) are readily accessible as single enantiomers. 

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenological, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section. 

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