Limitation on reaction rate

Bench scale experiments. The reactors used in these experiments are usually designed to operate at constant temperature, under conditions that minimize heat and mass transfer limitations on reaction rates. This facilitates an accurate evaluation of the intrinsic chemical effects. 

Still another advantage of fluidized bed operation is that it leads to more efficient contacting of gas and solid than many competitive reactor designs. Because the catalyst particles employed in fluidized beds have very small dimensions, one is much less likely to encounter mass transfer limitations on reaction rates in these systems than in fixed bed systems. 

Thus, due to limitations on the available computer memory, DNS of homogeneous turbulent reacting flows has been limited to Sc 1 (i.e., gas-phase reactions). Moreover, because explicit ODE solvers (e.g., Runge-Kutta) are usually employed for time stepping, numerical stability puts an upper limit on reaction rate k. Although more complex... 

While the above criteria are useful for diagnosing the effects of transport limitations on reaction rates of heterogeneous catalytic reactions, they require knowledge of many physical characteristics of the reacting system. Experimental properties like effective diffusivity in catalyst pores, heat and mass transfer coefficients at the fluid-particle interface, and the thermal conductivity of the catalyst are needed to utilize Equations (6.5.1) through (6.5.5). However, it is difficult to obtain accurate values of those critical parameters. For example, the diffusional characteristics of a catalyst may vary throughout a pellet because of the compression procedures used to form the final catalyst pellets. The accuracy of the heat transfer coefficient obtained from known correlations is also questionable because of the low flow rates and small particle sizes typically used in laboratory packed bed reactors. 

On the other hand, use of whole cells as the vehicle for immobilization of enzymes is not without problems. These disadvantages include the susceptibility to mass transfer/diffusional limitations on reaction rates and possible losses in the yield of the desired product as a consequence of unwanted side reactions. In addition, there are potential problems associated with maintaining the integrity of the immobilized cells—supplying the nutrients, energy sources, or cofactors necessary to maintain the cells in a sufficiently viable condition to mediate the reaction(s) of interest. 

In Lhe case of slow reacLions for which k,. A di(Y, the overall rate constant Arnet is equal to the rate constant for transformation of the precursor complex, k. However, for very fast reactions, k is greater than A difr, so that the overall rate constant is close to Ardiff- Therefore, it is important to understand that there is an upper limit on reaction rates that can be measured experimentally for diffusion controlled processes. In section 7.5 it is shown that A difr for molecular reactants is given by... 

Fig. 7 Confirmation of mass transfer limitation on reaction rate using Mariotte setup O 825 rpm, X 300 rpm then 825 rpm.

Batch reactor studies with different agitator speeds have shown that there was no evidence for external mass transfer limitation on reaction rates in the examined cases when the agitator speed was beyond 200 rev/min [9,19,21,55]. 

Slurry reactors are commonly used in situations where it is necessary to contact a liquid reactant or a solution containing the reactant with a solid catalyst. To facilitate mass transfer and effective utilization of the catalyst, one usually suspends a powdered or granular form of the catalyst in the liquid phase. This type of reactor is useful when one of the reactants is normally a gas at the reaction conditions and the second reactant is a liquid (e.g., in the hydrogenation of various oils). The reactant gas is bubbled through the liquid, dissolves, and then diffuses to the catalyst surface. Mass transfer limitations on reaction rates can be quite significant in those instances where three phases (the solid catalyst and the liquid and gaseous reactants) are present and necessary to proceed rapidly from reactants to products. 

Onsager s reaction field model in its original fonn offers a description of major aspects of equilibrium solvation effects on reaction rates in solution that includes the basic physical ideas, but the inlierent simplifications seriously limit its practical use for quantitative predictions. It smce has been extended along several lines, some of which are briefly sunnnarized in the next section. 

Instead of concentrating on the diffiisioii limit of reaction rates in liquid solution, it can be histnictive to consider die dependence of bimolecular rate coefficients of elementary chemical reactions on pressure over a wide solvent density range covering gas and liquid phase alike. Particularly amenable to such studies are atom recombination reactions whose rate coefficients can be easily hivestigated over a wide range of physical conditions from the dilute-gas phase to compressed liquid solution [3, 4]. 

Reaction rate limited (zero-order kinetics). In this case, the biofilm concentration has no effect on reaction rate, and the biodegradation breakthrough curve is linear. 

If the rate of a reaction is governed by the encounter frequency, it is said to be diffusion-controlled. This frequency imposes an upper limit on the rate of reaction that can be evaluated by the use of Fick s laws of diffusion. The mathematical expression of this phenomenon was first presented by von Smoluchowski.2 We shall adopt a simple approach,3,4 although more rigorous derivations have been given.5... 

Although the solvent effect on reaction rate could, in principle, be large, the limited availability of... 

The treatment of chemical reaction equilibria outlined above can be generalized to cover the situation where multiple reactions occur simultaneously. In theory one can take all conceivable reactions into account in computing the composition of a gas mixture at equilibrium. However, because of kinetic limitations on the rate of approach to equilibrium of certain reactions, one can treat many systems as if equilibrium is achieved in some reactions, but not in others. In many cases reactions that are thermodynamically possible do not, in fact, occur at appreciable rates. 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

The flow terms represent the convective and diffusive transport of reactant into and out of the volume element. The third term is the product of the size of the volume element and the reaction rate per unit volume evaluated using the properties appropriate for this element. Note that the reaction rate per unit volume is equal to the intrinsic rate of the chemical reaction only if the volume element is uniform in temperature and concentration (i.e., there are no heat or mass transfer limitations on the rate of conversion of reactants to products). The final term represents the rate of change in inventory resulting from the effects of the other three terms. 

In this case the reaction rate will depend not only on the system temperature and pressure but also on the properties of the catalyst. It should be noted that the reaction rate term must include the effects of external and intraparticle heat and mass transfer limitations on the rate. Chapter 12 treats these subjects and indicates how equation 8.2.12 can be used in the analysis of packed bed reactors. 

When a solid acts as a catalyst for a reaction, reactant molecules are converted into product molecules at the fluid-solid interface. To use the catalyst efficiently, we must ensure that fresh reactant molecules are supplied and product molecules removed continuously. Otherwise, chemical equilibrium would be established in the fluid adjacent to the surface, and the desired reaction would proceed no further. Ordinarily, supply and removal of the species in question depend on two physical rate processes in series. These processes involve mass transfer between the bulk fluid and the external surface of the catalyst and transport from the external surface to the internal surfaces of the solid. The concept of effectiveness factors developed in Section 12.3 permits one to average the reaction rate over the pore structure to obtain an expression for the rate in terms of the reactant concentrations and temperatures prevailing at the exterior surface of the catalyst. In some instances, the external surface concentrations do not differ appreciably from those prevailing in the bulk fluid. In other cases, a significant concentration difference arises as a consequence of physical limitations on the rate at which reactant molecules can be transported from the bulk fluid to the exterior surface of the catalyst particle. Here, we discuss... 

Before terminating the discussion of external mass transfer limitations on catalytic reaction rates, we should note that in the regime where external mass transfer processes limit the reaction rate, the apparent activation energy of the reaction will be quite different from the intrinsic activation energy of the catalytic reaction. In the limit of complete external mass transfer control, the apparent activation energy of the reaction becomes equal to that of the mass transfer coefficient, typically a kilocalorie or so per gram mole. This decrease in activation energy is obviously... 

At 250 °C, the reaction rate was shown to be affected by pellet sizes as small as 0.2mm [31]. At 210 °C, the reaction rate was found to be by-product diffusion limited at a particle size >16-18mesh (ca. 1.3mm) [27], while at 160°C no effect of pellet size was seen for particle sizes <2.1 mm [29], Therefore, it can be seen that the influence of the pellet size on reaction rate becomes more pronounced as the temperature increases. Under normal industrial SSP conditions, where the pellet size is between 2 and 3 mm and temperatures are >200 °C, decreasing the pellet size will lead to an increasing of the reaction rate. 

It will be demonstrated in this section that a narrow pore structure limits the reaction rate to an extent which casues the reaction rate to be either proportional to the square root of the specific surface area (per unit mass) or independent of it, depending on the mode of diffusion within the pore structure. Lest this departure of the reaction rate from direct proportionality with specific surface area might be thought to be accounted for in terms of a non-uniform distribution of surface energy over the catalyst surface, it should be pointed out that such in situ heterogeneity is usually only a small fraction of the total chemically active surface and cannot therefore explain the observed effects. 

It was assumed that there were no limitations on the rates of oxidation due to mass transport as discussed in detail by Schwartz and Freiberg (1981), this assumption is justified except for very large droplets (> 10 yarn) and high pollutant concentrations (e.g., 03 at 0.5 ppm) where the aqueous-phase reactions are very fast. It was also assumed that the aqueous phase present in the atmosphere was a cloud with a liquid water content (V) of 1 g m-3 of air. As seen earlier, the latter factor is important in the aqueous-phase rates of conversion of S(IV) thus the actual concentrations of iron, manganese, and so on in the liquid phase and hence the kinetics of the reactions depend on the liquid water content. 

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