Reaction nonlinear kinetic

The simplest manifestation of nonlinear kinetics is the clock reaction—a reaction exliibiting an identifiable mduction period , during which the overall reaction rate (the rate of removal of reactants or production of final products) may be practically indistinguishable from zero, followed by a comparatively sharp reaction event during which reactants are converted more or less directly to the final products. A schematic evolution of the reactant, product and intenuediate species concentrations and of the reaction rate is represented in figure A3.14.2. Two typical mechanisms may operate to produce clock behaviour. 

All catalytic reactions involve chemical combination of reacting species with the catalyst to form some type of inteniiediate complex, the nature of which is the subject of abundant research in catalysis. The overall reaction rate is often determined by the rate at which these complexes are formed and decomposed. The most widely-used nonlinear kinetic equation that describes... 

The autocatalytic reaction mechanism apparent at low temperatures is expected to apply to catalytic hydrogen oxidation at high pressures. In addition, the above study is the first to use STM to observe the formation of dynamic surface patterns at the mesoscopic level, which had previously been observed by other imaging techniques in surface reactions with nonlinear kinetics [57]. This study illustrates the ability of in situ STM to visualize reaction intermediates and to reveal the reaction pathway with atomic resolution. 

Solid-phase organic synthesis (SPOS) exhibits several shortcomings, due to the nature of the heterogeneous reaction conditions. Nonlinear kinetic behavior, slow reactions, solvation problems, and degradation of the polymer support due to the long reaction times are some of the problems typically experienced in SPOS [2], Any technique which is able to address these issues and to speed up the process of solid-... 

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

As in any other mass balance model of bioprocesses, a strongly nonlinear kinetic behavior is present due to the reaction rates. These rates are given by ... 

The important kinetic constants, V and Ku, can be graphically determined as shown in Figure E5.1. Equation E5.2 and Figure E5.1 have all of the disadvantages of nonlinear kinetic analysis. Kmax can be estimated only because of the asymptotic nature of the line. The value of Ku, the substrate concentration that results in a reaction velocity of Vj /2, depends on Kmax, so both are in error. By taking the reciprocal of both sides of the Michaelis-Menten equation, however, it is converted into the Lineweaver-Burk relationship (Equation E5.3). 

Astarita, G., Lumping Nonlinear Kinetics Apparent Overall Order of Reaction, AIChE J. 35, 529 (1989). 

For multisubstrate enzymatic reactions, the rate equation can be expressed with respect to each substrate as an m function, where n and m are the highest order of the substrate for the numerator and denominator terms respectively (Bardsley and Childs, 1975). Thus the forward rate equation for the random bi bi derived according to the quasi-equilibrium assumption is a 1 1 function in both A and B (i.e., first order in both A and B). However, the rate equation for the random bi bi based on the steady-state assumption yields a 2 2 function (i.e., second order in both A and B). The 2 2 function rate equation results in nonlinear kinetics that should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Chapter 6, Bardsley and Waight, 1978) and formation of the abortive substrate complex (Dalziel and Dickinson, 1966 Tsai, 1978). 

Although these arguments have been presented for reaction systems whose rates are forced by an external oscillator, they remain true for autonomous biochemical oscillations where ot and are nonlinear functions of metabolite concentrations. That is, the rate of removal of a labeled compound through a reaction step whose rate is oscillating due to nonlinear kinetics will be enhanced over an equivalent system that maintains the same mean chemical flux and mean concentrations of metabolites but does not oscillate. This has been demonstrated numerically ( 6) on the reaction system (1) from the previous section using the full kinetic equations... 

When the flow pattern is known, the conversion for a given reaction mechanism may be evaluated from the appropriate material and energy balances. When only the RTD is known (or can be calculated from tracer response data), however, different networks of reactor elements can match the observed RTD. In reality, reactor performance for a given reactor network will be unique. The conversion obtained by matching the RTD is, however, unique only for linear kinetics. For nonlinear kinetics, two additional factors have to be... 

Second-order reactions provide the simplest example of nonlinear kinetics, where micromixing limitations have significant effects on reactant conversion. We use the two-mode model to determine the same for a typical bimolecular second-order reaction of the type... 

As a next example, we consider the model of Termonia and Ross [193] for the nonlinear kinetics of glycolysis. This model simulates the kinetics of the following reactions... 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

In order to avoid the restrictions to complicated adsorptive reactions in the MOC3D, Selim et al. (1990) developed a simulation system based on the multireaction model (MRM) and multireaction transport model (MRTM). The MRM model includes concurrent and concurrent-consecutive retention processes of the nonlinear kinetic type. It accounts for equilibrium (Freundlich) sorption and irreversible reactions. The processes considered are based on linear (first order) and nonlinear kinetic reactions. The MRM model assumes that the solute in the soil environment is present in the soil solution and in several phases representing retention by various soil... 

The compound-soil interaction is reflected in the following parameters kd = 1.0 cm3/g (distribution coefficient) NEQ = 1.1 (Freundlich parameter) kt = 0.01 h (forward kinetic reaction rate) k2 = 0.02 h (backward kinetic reaction rate) U = 1.2 (nonlinear kinetic parameter) k3 = 0.01 hr1 (forward kinetic reaction rate) k4 = 0.02 Ir1 (backward kinetic reaction rate) W = 1.3 (non-linear kinetic parameter) k5 = 0.01 h (forward kinetic reaction rate) k6 = 0.02 Ir1 (backward kinetic reaction rate), ks = 0.005 (irreversible reaction rate). 

Be that as it may, there is a largely heuristic argument in support of the idea that the assumption of independent kinetics is reasonable only for first-order reactions. If the reactions are first order, the probability of a given molecule to undergo the reaction is a constant, so that the number of molecules of any given reactant that reacts per unit time is simply proportional to how many such molecules there are. For any nonlinear kinetics, the probability depends on the en-... 

Cicarelli et al. (1992) have developed the solution of Eq. (128) by a perturbation expansion, with e the perturbation parameter. They consider the special case of Langmuir isotherm kinetics, where F[ ] = 1/(1 -I- ). At the zero-order level, C = exp(r). This result simply reflects the fact that, in the distorted time scale r where the kinetics are linear, two species are formed from one at every reaction step, and hence the total concentration grows exponentially. This, however, does not include the fact that end products are being formed, and thus disappear from the spectrum of concentrations (at the zero-order level, e = 0 and no end products are formed). The critical warped time tq at which the zero-order approximation breaks down is estimated as —In e that is, it is well in excess of unity. Even for linear kinetics, there is an induction time significantly longer than the inverse of the kinetic constant during which very few end products are formed (this is even more true for nonlinear kinetics of the type considered). The solution can be obtained formally at all levels of perturbation the first-order level is of particular relevance because it yields (to within order e) the total amount of end products formed up to the critical time. 

Ocone, R., and Astarita, G., Lumping nonlinear kinetics in porous catalysts Diffusion-reaction lumping strategy. AIChEJ. 39,288 (1993). 

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