Mean conversion

A number of methods have been described in earlier sections whereby the surface free energy or total energy could be estimated. Generally, it was necessary to assume that the surface area was known by some other means conversely, if some estimate of the specific thermodynamic quantity is available, the application may be reversed to give a surface area determination. This is true if the heat of solution of a powder (Section VII-5B), its heat of immersion (Section X-3A), or its solubility increase (Section X-2) are known. 

The mean conversion of all the groups is the sum of the products of the individual conversions and their volume fractions of the total flow. Since the groups are small, the sum is replaced by the integal. Thus,... 

Entry Position % Mean conversion Stdev % Mean conversion Stdev... 

The flow distribution in a cylindrical vessel has the shape of an isosceles triangle with apex on the axis, thus u = u0(j 3j, /3 = r/R. Find the mean velocity and the mean conversion of a reaction with a power law rate equation. Compare with laminar and uniform flows. 

Fig. 12 Visualization of mean conversion, X, within selected volumes located within the slice section identified in Fig. 11. The local volumes have in-plane dimensions of 1.5 mm x 1.5 mm and a depth (image slice thickness) of 500 pm. Data are shown for three feed flow rates (a) 0.025, (b) 0.05, and (c) 0.1 ml min-1. As flow rate increases, the residence time characterizing the bed decreases and, as expected, conversion also decreases. Reprinted from ref. 33, with kind permission of Springer Science and Business Media.

The regeneration problem when to stop a run and either discard or regenerate the catalyst. This problem is easy to treat once the first problem has been solved for a range of run times and final catalyst activities. Note each pair of values for time and final activity yields the corresponding mean conversion.)... 

When in plug flow all solids stay in the reactor for the same length of time. From this and the kinetics for whatever resistance controls, the conversion X R) for any size of particle R can be found. Then the mean conversion of the solids leaving the reactor can be obtained by properly summing to find the overall contribution to conversion of all sizes of particles. Thus,... 

Figure 26.4 Mean conversion versus mean residence time in mixed flow reactors, single size of solid.

Arbitrary Flow of Solids. Since particles flow as macrofluids, their mean conversion is given by Eq. 11.13. Thus with any RTD and with known gas composition we have... 

Find the mean conversion of the solids for a residence time of 15 min. 

A measure of the scatter or variability of data is the variance, as discussed earlier. We have seen that a large variance produces broad-interval estimates of the mean. Conversely, a small variability, as indicated by a small value of variance, produces narrow interval estimates of the mean. In the limiting case, when no random fluctuations occur in the data, we obtain exact identical measurements of the mean. In this case, there is no scatter of data and the variance is zero, so that the interval estimate reduces to an exact point estimate. 

Under steady-state conditions the conversion does not change to any appreciable degree, while under oscillatory conditions both the maximum and minimum conversions of the cycle as well as the mean conversion remain constant within 1-2%. This observation, of course,does not agree with prior observations by Schmitz and coworkers for the same reaction [l9]. We cannot explain at this point the apparent resistance of our catalyst system to deactivation. Possible reasons, we could think of, are the much higher ratio of surface area to geometric area of our catalyst, the use of a Pyrex-glass reactor and our lower experimental temperatures. 

A positive value of X arises because the trajectories in state space for chaotic behavior are diverging in the mean. Conversely, adjacent trajectories in a system which possesses a globally attracting limit cycle will converge. Values of X for periodic systems can be obtained by perturbing the reactor from a periodic state and observing the rate of convergence back to the periodic orbit. In Fig. 4 is shown the result of such an... 

Differences in the Responses of the Different Types of Models. The basic differences that exist in the heat and mass balances for the different types of models determine deviations of the responses of types I and II with respect to type III. In a previous work (1) a method was developed to predict these deviations but for conditions of no increase in the radial mean temperature of the reactor (T0 >> Tw). In this work,the method is generalized for any values of T0 and Tw and for any kinetic equation. The proposed method allows the estimation of the error in the radial mean conversions of models I and II with respect to models III. Its validity is verified by comparing the predicted deviations with those calculated from the numerical solution of the two-dimensional models. A similar comparison could have been made with the numerical solution of the one-dimensional models. 

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

The test requires the use of a standard batch of gas oil as a feedstock and a set of equilibrium fluid cracking catalysts with consensus mean conversion values assigned in a reactor of specified design. The gas oil and the set of equilibrium cracking catalysts are useful reference materials. Conversion for any equilibrium or laboratory-deactivated fluid cracking catalyst can be measured and compared to a conversion calibration curve. Conversion is measured by the difference between the amount of feed used and the amount of unconverted material. The unconverted material is defined as all liquid product with a boiling point above 216°C. 

Equation (5.41) assumes that the gas is of a uniform composition throughout the reactor at all times. If the gas composition changes with the time or position within the reactor, a different equation must be used. To account for the effect of particle size distribution in addition to the residence time distribution is difficult because different size particles can remain in the reactor for different periods of time. To account for these effects completely a population balance must be performed, where the conversion is an internal variable (see Chapter 3). This type of treatment is beyond the scope of this chapter. A simplified method of accounting for the effects of a particle size distribution, mQt), on the mean conversion, is by... 

Segregated Flow A real example is bead polymerization of styrene and some other materials. The reactant is in the form of individual small beads suspended in a fluid and retarded from agglomeration by colloids on their surfaces. Accordingly, they go through the reactor as independent bodies and attain conversions under batch conditions with their individual residence times. This is called segregated flow. With a particular RTD, conversion is a maximum with this flow pattern. The mean conversion of all the segregated elements then is given by... 

---