Power law kinetics

If we consider the irreversible reaction with two reactants forming a product 

The overall reaction rate has a temperature dependence governed by the specific reaction rate k(T) and a concentration dependence that is expressed in terms of several concentration-based properties depending on the suitability for the particular reaction type mole or mass concentration, component vapor partial pressure, component activity, and mole or mass fraction. For example, if the dependence is expressed in terms of molar concentrations for components A(Ca) and B(Cb), the overall reaction rate can be written as 

The temperature-dependent specific reaction rate k(T) is represented by the Arrhenius equation 

Effect of activation energy on temperature dependence of reaction rate. 

Specific reaction rates always increase as temperature increases and the higher the activation energy, the more sensitive the reaction rate is to temperature. 

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The model that best describes the mechanism is usually very complicated. For dynamic studies that require much more computation (and on a more limited domain) a simplified model may give enough information as long as the formalities of the Arrhenius expression and power law kinetics are incorporated. To study the dynamic behavior of the ethylene oxide reactor. 

Theorderofareactionmaynotbeassimpleas firstorsecondorder.Weoften findnonintegral orderinwhatiscalled "power-law" kinetics.Thistypicallyindicatesthatthe "reaction" rate... 

From the data shown in table 3, it is evident that the effect of Cs promotion on the power law kinetics is twofold First, the reaction order for NH3 is changed to essentially zero, and secondly, the apparent activation energy is higher by more than 20 kJ/mol in the presence of Cs. Contrary to the results obtained by Aika et al. [5], the reaction order for H2 was negative for all catalysts investigated. The positive reaction order for H2 reported by Aika et al. [5] for... 

Reaction order. One of the most widely used (particularly for homogeneous reactions) kinetic expressions is the power law kinetic equation. ... 

There might be various reasons that lead to finding an apparent instead of the true activation energy. The use of power-law kinetic expressions can be one of the reasons. An apparent fractional reaction order can vary with the concentration, i.e. with conversion, in one experimental run. Depending upon the range of concentrations or, equivalently, conversions, different reaction orders may be observed. As an example, consider the a simple reaction ... 

We used the experimental data of Miller et a/. (1981) to evaluate rate constants in the power law kinetic expression ... 

The values of x = 0.5 and = 1 for the kinetic orders in acetone [1] and aldehyde [2] are not trae kinetic orders for this reaction. Rather, these values represent the power-law compromise for a catalytic reaction with a more complex catalytic rate law that corresponds to the proposed steady-state catalytic cycle shown in Scheme 50.3. In the generally accepted mechanism for the intermolecular direct aldol reaction, proline reacts with the ketone substrate to form an enamine, which then attacks the aldehyde substrate." A reaction exhibiting saturation kinetics in [1] and rate-limiting addition of [2] can show apparent power law kinetics with both x and y exhibiting orders between zero and one. 

The analysis of simultaneous diffusion and chemical reaction in porous catalysts in terms of effective diffusivities is readily extended to geometries other than a sphere. Consider a flat plate of porous catalyst in contact with a reactant on one side, but sealed with an impermeable material along the edges and on the side opposite the reactant. If we assume simple power law kinetics, a reaction in which there is no change in the number of moles on reaction, and an isothermal flat plate, a simple material balance on a differential thickness of the plate leads to the following differential equation... 

The conclusions about asymptotic values of tj summarized in Tables 8.2 and 8.3, and the behavior of tj in relation to Figure 8.11, require a generalization of the definition of the Thiele modulus. The result for " in equation 8.520 is generalized with respect to particle geometry through Le, but is restricted to power-law kinetics. However, since... 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

To simplify the treatment for an LFR in this chapter, we consider only isothermal, steady-state operation for cylindrical geometry, and for a simple system (A - products) at constant density. After considering uses of an LFR, we develop the material-balance (or continuity) equation for any kinetics, and then apply it to particular cases of power-law kinetics. Finally, we examine the results in relation to the segregated-flow model (SFM) developed in Chapter 13. 

An important aspect of the micromixing models is that they define the maximum and minimum conversion possible for a given reaction and RTD. Zwietering (1959) showed that, for the reaction A - products, with power-law kinetics, ( -rA) = kAcA,... 

Comparison of the segregated-flow and maximum-mixedness models, with identical RTD functions, shows that the former gives better performance. This is consistent with the observations of Zwietering (1959), who showed that for power-law kinetics of order n > 1, the segregated-flow model produces the highest conversion. 

This assumes that the concentration at any value of x is not a function of radius. Ca is the concentration of reacting species A, u the mean convective velocity, which is assumed to be neither a function of axial or radial position, and Ta is the reaction rate of A based on unit volume. If nth-order power law kinetics pertain, i.e. 

Assuming that we have an irreversible reaction with a single reactant and power-law kinetics, r = kC, the concentration in a constant-volume isothermal batch reactor is given by integrating the expression... 

Thus from this procedure we have the simplest method to analyze batch-reaction data to obtain a rate expression r (CA, T) if the reaction is irreversible with a single reactant and obeys power-law kinetics with the Arrhenius temperature dependence. 

The experiments were carried out at a pressure of 1.5 bar and a flow rate of 80-270 Ncm3 min-1. At 200 °C no deactivation of the catalyst was observed. As the rate of reaction was found to show a linear dependence on the residence time, differential conditions were assumed for the measurements. Because of the determined high activation energy of 5 6 kj mol-1, mass transport limitations were excluded. A power law kinetic expression of the following form was determined for methanol steam reforming ... 

Power-law kinetic rate expressions can frequently be used to quantify homogeneous reactions. However, many reactions occur among species in different phases (gas, liquid, and solid). Reaction rate equations in such heterogeneous systems often become more complicated to account for the movement of material from one phase to another. An additional complication arises from the different ways in which the phases can be contacted with each other. Many important industrial reactors involve heterogeneous systems. One of the more common heterogeneous systems involves gas-phase reactions promoted with porous solid catalyst particles. 

Figure 1.3 shows a plot of 0 versus partial pressure for various values of the adsorption equilibrium constant. These show that as the equilibrium constant increases for a given pressure, we increase the surface fraction covered, up to a value of 1. As the pressure increases, we increase the fraction of the surface covered with A. But we have only a finite amount of catalyst surface area, which means that we will eventually reach a point where increasing the partial pressure of A will have little effect on the amount that can be adsorbed and hence on the rate of any reaction taking place. This is a kind of behavior fundamentally different from that of simple power-law kinetics, where increasing the reactant concentration always leads to an increase in reaction rate proportional to the order in the kinetic expression. 

The assumption of which step is slowest governs the form of the final kinetic expression. For the purposes of this simple example, we assume that the second step is the slowest and is first-order with respect to the adsorbed A species. Therefore the rate r is determined by a rate constant and the concentration of A absorbed on the surface (Cas) according to standard power-law kinetics ... 

Primarily, the reforming reaction rate depends on the anode gas composition according to the following power law kinetics ... 

The pyrolysis reactor can be simulated in Aspen Plus as PFR with power-law kinetics and temperature profile or heat duty. To validate the kinetic data, we consider an initial flow rate of 73000kg/h EDC at a reaction temperature of 530°C and 18 bar. The reactor consists of 16 tubes in parallel with an internal diameter of... 

Modern catalysts for vinyl-acetate synthesis contain Au in the chemical formulation, which manifests in much higher activity and selectivity. This is reflected by fundamental changes in the kinetics, such as for example switching the reaction order of ethylene from negative to positive [8]. As a consequence, in more recent studies the formation of vinyl acetate can be described conveniently by a power-law kinetics involving only ethylene and oxygen ... 

In the same vein and under dimensionally restricted conditions, the description of the Michaelis-Menten mechanism can be governed by power-law kinetics with kinetic orders with respect to substrate and enzyme given by noninteger powers. Under quasi-steady-state conditions, Savageau [25] defined a fractal Michaelis constant and introduced the fractal rate law. The behavior of this fractal rate law is decidedly different from the traditional Michaelis-Menten rate law ... 

Overall, a large number of drugs that exhibit apparently multiexponential kinetics obey power-law kinetics. The cogent question is why many of the observed time-concentration profiles exhibit power function properties. Although the origin of the power function remains unclear, some empirical explanations could elucidate its origin ... 

Beard, D. and Bassingthwaighte, J., Power-law kinetics of tracer washout from physiological systems, Annals of Biomedical Engineering, Vol. 26, 1998, pp. 775-779. 

Table 2. Experimental diagnostic criteria for the absence of intraparticle transport effects in simple, irreversible reactions (power law kinetics only).

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