Reacting numerical methods

Nylon resins are made by numerous methods (53) ranging from ester amidation (54) to the Schotten-Baumann synthesis (55). The most commonly used method for making nylon-6,6 and related resins is the heat-induced condensation of monomeric salt complexes (56). In this process, stoichiometric amounts of diacid and diamine react in water to form salts. Water is removed and further heating converts the carboxylate functions to amide linkages. Chain lengths are controlled by small amounts of monofunctional reagents. The molten finished nylon resin can be dkectly extmded to pellets. 

Although much as been done, much work remains. Improved material models for anisotropic materials, brittle materials, and chemically reacting materials challenge the numerical methods to provide greater accuracy and challenge the computer manufacturers to provide more memory and speed. Phenomena with different time and length scales need to be coupled so shock waves, structural motions, electromagnetic, and thermal effects can be analyzed in a consistent manner. Smarter codes must be developed to adapt the mesh and solution techniques to optimize the accuracy without human intervention. 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

Due to the difficulties of getting analytical solutions, many numerical methods were developed to simulate the solute transport and retention processes in the soil. Deane et al. (1999) analyzed the transport and fate of hydrophobic organic chemicals (HOCs) in consolidated sediments and saturated soils. Walter et al. (1994) developed a model for simulating transport of multiple thermodynamically reacting chemical substances in groundwater systems. Islam et al. (1999) presented a modeling... 

The rates of the overall reactions can be related to the rate law expressions of the individual steps by using the steady state approximation. However simple kinetic data alone may not distinguish a mechanism where, for example, a metal and an olefin form a small amount of complex at equilibrium that then goes on to react, from one in which the initial complex undergoes dissociation of a ligand and then reacts with the olefin. As a reaction scheme becomes more complex such steady state approximations become more complicated, but numerical methods are now available which can simulate these even for complex mixtures of reactants. 

Numerous methods can be used for the synthesis of trialkylsilyl ethers (Scheme 1.27). Alcohols react rapidly with trialkylsilyl chloride (R sSiCl) to give trialkylsilyl ethers (ROSiR 3) in the presence of an amine base like triethylamine, pyridine, imidazole or 2,6-lutidine (Table 1.2). 

Numerous methods have been described for the preparation of niobium(V) chloride, among them the reaction of niobium(V) oxide with thionyl chloride in a sealed system. In such a procedure some niobium(V) oxide trichloride, NbOCls, is almost always formed, and it is difficult to obtain the pentachloride completely free from this impurity, even by repeated sublimation. The simple, efficient method described here consists in allowing hydrous niobium(V) oxide to react with thionyl chloride at room temperature. Almost quantitative conversion is observed, the pentachloride dissolving in the thionyl chloride, from which it may be recovered, free of oxide trichloride, by vacuum evaporation... 

Solutions of tetracarbonylhydridoferrate(l — ) salts can be prepared by numerous methods. The most straightforward and widely used involves the reaction of Fe(CO)s and an excess of an alkaline base in protic media. The conventional method to prepare ethanolic solutions of K[FeH(CO)J is to react Fe(CO)5 and 3 equiv of KOH in EtOH at room temperature. The procedure described here uses 2KOH equivalents only and allows the rapid and quantitative preparation of ethanolic K[FeH(CO)4] solutions, which are stable provided that all traces of oxidizing agents are rigorously excluded. 

The main advantage of the separation is the reduction of the computational effort. Another aspect is the fact that the two sets of equations can be solved with different numerical methods and on different numerical grids. Due to the nature of the non-linearities in material and momentum balance equations, they usually require different grid refinements in different areas of the computational domain. Additionally, if the assumption of a stationary flow field is valid, the simulation of the coupled set of equations would be unnecessarily slowed down by solving the momentum equations. The material balance to be solved for each of the reacting components k reads... 

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