Calculation of the Global Rate

For use in reactor design the global rate should be calculable at all locations in the reactor. We suppose that the intrinsic rate equation is available. The problem is to evaluate the global rate corresponding to possible bulk concentrations Q, bulk temperatures 7, and flow conditions. If external and internal temperature differences can be neglected, the problem is straightforward and is essentially the reverse of the stepwise solution outlined in Sec. 12-1. The double-trial procedure is not necessary, because /(C) is known. The effective diffusivity of the catalyst pellet is required. The equations we need are Eq. (10-1) for external diffusion, 

Example 12-2 Using the intrinsic rate equation obtained in Example 12-1, calculate the global rate of the reaction o-Hj p- % at 400 psig and — 196°C, at a location where the mole fraction of ortho hydrogen in the bulk-gas stream is 0.65. The reactor is the same as described in Example 12-1 that is, it is a fixed-bed type with tube of 0.50 in. ID and with x -in. cylindrical catalyst pellets of Ni on AljOj. The superficial mass velocity of gas in the reactor is 15 lb/(hr)(ft ). The effective diffusivity can be estimated from the random-pore model if we assume that diffusion is predominately in the macropores where Knudsen diffusion is insignificant. The macroporosity of the pellets is 0.36. Other properties and conditions are those given in Example 12-1. 

Solution The intrinsic rate equation developed in Example 12-1 is 

It was shown that this expression could be written in terms of p-Hj concentrations [Eq. (F) of Example 12-1] as 

the particular form for Eq. (11-43) for the rate of this reaction is [Eq. (G) of Example 12-1] 

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It is seen that the calculation of the global rates involves only the solution of Eq. 4.125, from which Ac, Be, Pa(L), and Pb(L) can be directly calculated with the aid of Eqs. 4.131 and 4.132. The original split boundary value problem has been transformed into an initial value problem. 

It is obvious that the calculation of the reaction rate is very easy in this reactor. Now, let us sketch the reaction rate versus the rotation speed using the data obtained at 203C. As shown in Figure 5.9, the reaction rate is stabilized at rotation speeds above 400-450 rpm. This means that the external mass transfer does not affect the global rate, and thus values of the intrinsic rate can be safely considered to be obtained at those rotation speeds. 

For low conversions (less than 20%) of organic, the assumption of a differential reactor (dc/ct Ac/At=([C]in-[C]out)/ is a good one and rates are calculated on this basis. For higher conversions (greater than 20%), the lobal stoichiometric oxidation equation is numerically integrated and the inetic parameters are obtained by regression to conversion data. Rates are then calculated from the global rate expression. 

Example 10-3 For the oxidation of SOj calculate the ratio of the global rate to the rate if both temperature and concentration differences between gas and catalyst pellei are neglected. Use the data in Example 10-1. 

Also calculate the ratio y of the global rate and the rate evaluated at bulk conditions. 

SPSS version 12 was used for data analysis. Data are presented as the mean (standard deviation) or median (range). The mean change in LCQ score was calculated for each domain. In accordance with previous studies we expressed change of the global rating score as an absolute number, i.e. when the change was negative. 

Figure 7.8 Arrhenius plot of the global rate constant of glycerol degradation (first-order kinetics, 45MPa, water/ glycerol ratio 199) from experimental results and from model calculations. In addition, the ionic product for water at 45 MPa is given [31].

The above example is just for the calculations of net global rates at a point in the reactor. These global rates have to be used in conjunction with the reactor conservation equations for design and analysis. The complete design and analysis procedures for the reactor are outlined in Figure 10.27. 

Equations 12.6.2 to 12.6.4 and the relation between s, y, and are sufficient to calculate the global rate at specified values of TB and CB. Unfortunately, information on the last relation is rather limited. The curves presented in Figure 12.10 and reference 61 give the desired relation for first-order kinetics, but numerical solutions for other reaction orders are not available to this extent we will presume that numerical solutions may be generated if needed for design purposes. 

This procedure obviously requires machine computation capability if it is to employed in reactor design calculations. Fortunately, there are many reactions for which the global rate reduces to the intrinsic rate, which avoids the necessity for calculations of this type. On the other hand, several high tonnage processes (e.g., S02 oxidation) are influenced by heat and mass transfer effects and one must be fully cognizant of their implications for design purposes. 

Now, the global rate can be estimated at any conversion, since temperature can be calculated from eq. (5.232). Then, the conversion versus reactor depth or catalyst mass can be determined from the mass conservation equation (5.228). Only arithmetic solutions of the adiabatic model are possible. 

In order to calculate the theoretically predicted dependence of the global reaction rate on potential, the value of the symmetry factor p is missing. In many one-electron transfer reactions, p approximates a value of 0.5. A reminder should be given here that the intention is to find out whether a postulated reaction mechanism is possible or should be rejected. In this perspective, it is not necessary to have an exact value for p, and it is not indefensible to presuppose that p = 0.5. 

One more comment seems necessary. The Arrhenius expression [Eq. (32)] is commonly used to describe the rates of nonelementary reactions including several steps. In this case, the measured value of A is the apparent (global) activation energy, which is the resultant of sums and differences (with some coefficients) of activation energies of elementary steps whose rates contribute to the global rate (108). In our model approach, we calculate A for elementary steps only. Thus, there is no direct and simple way to compare our calculated barriers with the apparent barriers of nonelementary processes. This is particularly true for energy estimates made from the thermal-stability thresholds of chemisorbed species. 

Alternative expressions for the global rates of Reactions 14 and 16 were tried while developing the model. For the CO + CO2 conversion (Reaction 16) the overall correlation derived by Howard et al. (2) was initially used, but with this the calculated values of SL were considerably lower than the measured ones. For the methane disappearance rate (Reaction 14) the correlations proposed by Westbrook and Dryer (7) were tried, and these gave results negligibly different from those obtained by Equation 18. 

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