Mean reaction rate

J.P. Dumont, D. Durox, and R. Borghi 1993, Experimental study of the mean reaction rates in a turbulent premixed flame. Combust. Sci. Technol. 89 219-251 (more informations through www.informaworld.com). 

As a measure of how much the effective rate is lowered by the resistance to pore diffusion, the effectiveness factor qpore is used. This factor is defined as the ratio of the actual mean reaction rate within the pore to the maximum rate if not... 

This result could be improved by assuming a more appropriate distribution function of T instead of a simple sinusoidal fluctuation however, this example—even with its assumptions—usefully illustrates the problem. Normally, probability distribution functions are chosen. If the concentrations and temperatures are correlated, the rate expression becomes very complicated. Bilger [47] has presented a form of a two-component mean-reaction rate when it is expanded about the mean states, as follows ... 

For each of the following terms, write a sentence that shows your understanding of its meaning, reaction rate average rate... 

Such a method has been additionally developed. The method uses the mean values of temperature and species concentrations and applies a presumed PDF, to obtain the mean reaction rate. The PDF takes into account the effect of fluctuating temperature on the mean reaction rate. 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

Figure 2 shows the temperature probability distributions along a radial traverse at the axial position L/D = 65. When one contemplates these probability distributions convolved with the highly nonlinear temperature dependence of the reaction rates, the futility of attempting to model the mean reaction rates with any single temperature is obvious. 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

Method of Prediction of Deviations of Type I and II Models. The radial mean conversion is proportional to the volumetric mean reaction rate ... 

Comparison between One and Two-Dimensional Models. In a recent article ( 2), the differences between the responses of the one and two-dimensional versions of the same type of model were compared. It was found that they depend to a greater extent on kinetic parameters (j3 and y) than on physical parameters (Re and dp/dt). In order to explain this behaviour an analysis of the error in the evaluation of the reaction rate for a one-dimensional model is carried out. Since the reaction rate at radial mean concentration and temperature generally differs from the radial mean reaction rate, we can define ... 

The nature and arrangement of the pores determine transport within the interior porous structure of the catalyst pellet. To evaluate pore size and pore size distributions providing the maximum activity per unit volume, simple reactions are considered for which the concept of the effectiveness factor is applicable. This means that reaction rates can be presented as a function of the key component. A only, hence RA(CA). Various systems belonging to this category have been discussed in Chapters 6 and 7. The focus is on gaseous systems, assuming the resistance for mass transfer from fluid to outer catalyst surface can be neglected and the effectiveness factor does not exceed unity. The mean reaction rate per unit particle volume can be rewritten as... 

To examine the dependence of the mean reaction rate on the porous structure we should describe the relationships between all parameters of Equation 8.1 and the structural parameters mean pore radius, void fraction and surface area per unit particle volume 5. 

First to be considered are the limiting forms of this dependence. In the kinetic regime, that is without any diffusion limitations (2 1, and hence usually An0 1 and Anx 1), the effectiveness factor approaches unity and the mean reaction rate according to Equation 8.1 is proportional to the specific surface area ... 

Table XI. 1. Variation op the Mean Reaction Rate k Em) with Critical Eneroy /i and Number of DEcmEEs of Freedom s...

Assuming that the covariance is null would thus lead to an overprediction of the mean reaction rate (S). 

Separate deposit samples were oxidised in six different water vapour pressures between 38 and 362 mm Hg (Figure 3). The oxidation kinetics were all of a similar form, in that initially, the rate decreased with burn off up to 5-10% removal and then was essentially linear up to nearly 90% burn off before decreasing with greater attack. The last change was undoubtedly associated with the diminution of the sample size so that the proportionality relationship with the initial sample weight or surface area no longer applied. The mean reaction rates increased in direct proportion with pj o (Figure 4), as observed at 815°C. 

To better judge this situation, a degree of pore use is introduced (also known as degree of catalyst use Figure 2.1-17, Equation 2.1-58), which expresses the ratio of the mean reaction rate to the maximum possible reaction rate, that is, without the influence of diffusion. Inserting the course of the concentration in the cylindrical pore (Equation 2.1-54) into this equation gives, after integration. Equation (2.1-59) ... 

Note that the long-time rate and the mean rate differ by a factor of in 2 0.7. This discrepancy indicates that in general the mean reaction rate can be a poor characteristic of nonexponential kinetics. For activated reactions, the kinetics is practically exponential and the mean reaction rate coincides with the long-time rate. Note also that for a reversible reaction, the decay is usually normalized with respect to the stationary population, P co) = [1 + exp(AG)] ... 

Where is mean reaction rate over the time At if the number of substances moles i during this time changed by AN. .. Rate of the reactions is not constant in time and is equal to derivative of the change in content... 

For reactions with positive reaction order, the flattening of concentration profile diminishes the mean reaction rate in the tubular reactor and the conversion will decrease at constant space time, constant Dal, respectively. 

Various operations in the field of chemical engineering and in combustion can be characterized by the simultaneous interaction of two processes, namely the transfer of heat and mass and chemical reaction. However, the determination of mean reaction rates in turbulent flows requires detailed knowledge about fluctuations of scalar quantities such as species concentrations and enthalpy. Due to the non-linear character of chemical reaction the calculation of mean reaction rates based on mean values of temperature and species concentrations is only possible in special circumstances, such as practically infinitely fast micro-mixing-rates or very small fluctuations of the scalar variable around its mean value. 

According to this model, the influence of diffusion is restricted to a shell free of solid reactant, whereas in reality the solid reactant is still present to a certain extent in this outer zone. This leads to an underestimation of the (mean) reaction rate. Conversely, the assumption of a constant solid concentration in the core (equivalent to the initial value) overestimates the (mean) rate as in reality the concentration of the solid and of the gaseous reactant decrease in the core. As shown in Section 6.9.4 by the example of the regeneration of a coked catalyst, these effects compensate each other quite well. 

The concept used to derive such criteria is the following Since gradients of concentration and temperature always exist in and around a particle (although sometimes they are negligibly small), an assumption has to be made about the deviation up to which the reaction can considered to be uninfluenced by mass and heat transport. Commonly the criterion is that the deviation of the mean reaction rate from the zero-gradient rate should be within 5%. 

If we compare the values of Xa calculated by Eqs. (4.10.19), (4.10.20), (4.10.25) and (4.10.26) for a given value of Da, we see that for a positive reaction order the conversion in a PFR is always higher than in a CSTR (see also Section 4.10.2.7). This effect can also be explained without any mathematics The mean concentration in a PFR is somewhere between the in- and outlet value, whereas in a CSTR we have a constant but always lower reactor concentration that equals the outlet concentration. Thus for a positive value of the reaction order, the mean reaction rate in a PFR is higher. For a zero-order the difference in the reaction rate vanishes, and only for the rare case of a negative reaction order is the CSTR superior to a PFR. 

A model for the mean reaction rates is needed, similarly as the turbulent diffusion fluxes. This model can be quite complicated, because the molecular reaction rate itself can include a large number of species, depending on the mechanism it has to include the effects of fluctuations of species mass fractions, and of correlations of these fluctuations, but without direct influence of velocity fluctuations. 

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