Solution for Reaction Kinetics

Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The cui ves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The cui ve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about cf= nf = 0.5, N — Ti) = 0]. The hnear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. 

In general, fiiU time-dependent analytical solutions to differential equation-based models of the above mechanisms have not been found for nonhnear isotherms. Only for reaction kinetics with the constant separation faclor isotherm has a full solution been found [Thomas, y. Amei Chem. Soc., 66, 1664 (1944)]. Referred to as the Thomas solution, it has been extensively studied [Amundson, J. Phy.s. Colloid Chem., 54, 812 (1950) Hiester and Vermeiilen, Chem. Eng. Progre.s.s, 48, 505 (1952) Gilliland and Baddonr, Jnd. Eng. Chem., 45, 330 (1953) Vermenlen, Adv. in Chem. Eng., 2, 147 (1958)]. The solution to Eqs. (16-130) and (16-130) for the same boimdaiy condifions as Eq. (16-146) is... 

In the following sections, the solutions of the models as well as examples will be presented for the case of trickle-bed reactors and packed bubble bed reactors. Plug flow and fust-order reaction will be assumed in order to present analytical solutions. Furthermore, the expansion factor is considered to be zero unless otherwise stated. Some solutions for other kinetics will be also given. The reactant A is gas and the B is liquid unless otherwise stated. 

Interest in the kinetics of alkaline hydrolysis of esters in DMSO + water mixtures was stimulated by the observation that the rate constant often increased gradually as x2 increased. This is observed, for example, in the alkaline hydrolysis of ethyl acetate. For higher esters, e.g. ethyl p-nitrobenzoate, the rate constant drops slightly at low x2 but then rises again until k/k x2 = 0) > 1 (Tommila, 1964). The rate of alkaline hydrolysis of esters of benzoic acid is accelerated when DMSO is added (Tommila and Palenius, 1963), as also is the rate of alkaline hydrolysis of 2,4-dinitrofluorobenzene. In the latter case the effect is less dramatic because the rate constant for spontaneous hydrolysis also increases (Murto and Hiiro, 1964). The rate constants also increase when DMSO is added to aqueous solution for reactions between hydroxide ions and benzyl chloride (Tommila... 

Whereas the theoretical treatment of gas-phase reactions is comparatively simple, the calculation of rate constants for reactions in the liquid phase is very complicated. This is essentially due to the complexity of the many possible intermolecular solute-solvent interactions [cf. Section 2.2). When investigating solution-phase reaction kinetics, the problems to be faced include deciding which property of the solvent to use when setting up mathematical correlations with the reaction rates. Another problem is deciding which characteristics of the reacting molecules are to be considered when the effects of the solvent on their reactivity is determined. A quantitative allowance for the solvent effects on the rate constants k for elementary reactions involves establishing the following functions ... 

For first-order reactions and moderate values of less than the value of i] for a slab with the same volume/surface ratio. As shown in Table 4.2, the maximum difference is about 14%. This small difference means that solutions for complex kinetic models that were obtained for the flat-slab case can be used to get approximate effectiveness factors for spherical catalysts. 

The time-dependent currents which arise in sweep experiments in anodic oxidations which proceed in the presence of, or through reaction with, surface oxides (e.g., methanol oxidation, oxidation of hydrocarbons at Pt) were considered by Conway, Kozlowska, Klinger, MacDougall, and Tilak, who obtained numerical solutions for the kinetics in selected cases. 

For reaction kinetics other than first, the analytical work is even more scarce and numerical solution is the more common... 

Figure 5.24a shows an interesting effect for reaction kinetics of the order p, with respect to the reactant, even multiple solutions can be obtained for the catalyst particle balance equation, if the reaction order is negative. This phenomenon is called multiple steady states. The same effect appears in Langmuir-Hinshelwood kinetics, which is illustrated in Figure 5.24b. 

An extension pack DAEP (Data Analysis Extension Pack) includes some functions that might be useful for solving inverse problems requiring nonlinear fitting. This pack is automatically installed with the latest Mathcad versions. In Chap. 1, we obtained a solution for direct kinetic problem of the following consecutive second-order reaction ... 

Experience shows that in many cases pseudo-steady state modes are acceptable solutions for the kinetic results of reactions such as the expression of speed as a function of different variables. 

FMS is an elegant application of fluorous chemistry. Its solution-phase reaction kinetics, the split-mix technique, and separative fluorous tags expedite synthesis by increasing the speed, efficacy, and purity (Scheme 7.3). It is especially suitable for the combinatorial synthesis of a large number of compounds in larger quantities than existing mixture synthesis techniques. The synthesis usually begins with... 

---