Reaction Prater number

Prandtl mixing length hypothesis, 11 779 Prandtl number, JJ 746, 809 13 246-247 Praseodymium (Pr), J4 631t, 634t electronic configuration, J 474t Praseodymium bromide, physical properties of, 4 329 Prater equation, 25 270, 299 Prater number, 25 299, 300-301, 303 effect on maximum dimensionless intrapellet temperature, 25 304, 309 effect on maximum intrapellet temperature, 25 306 Prato reaction, 12 244 Pratsinis aluminum nitride, 17 212 Pravachol, 5 143... 

Figure 10. Effectiveness factor ij as a function of the Weisz modulus ji. Combined influence of intraparticle and interphase mass transfer and interphase heat transfer on the effective reaction rate (first order, irreversible reaction in a sphere, Biot number Bim = 100, Arrhenius number y — 20, modified Prater number ( as a parameter).

This procedure yields the curves depicted in Fig. 10 for fixed values of Bim and y. and the modified Prater number fi" as a parameter. From this figure, it is obvious that for exothermal reactions (fi > 0) and large values of the Weisz modulus, effectiveness factors well above unity may be observed. The reason for this is that the decline of the reactant concentration over the... 

Whether or not such an effect occurs in a practical situation and if so, how pronounced it will be, depends basically on the modified Prater number fi (see eq 71), that is on the maximum amount of heat effectively produced inside the pellet, as compared to the maximum amount of heat transported across the external boundary layer. Additionally, the Arrhenius number plays an important role which, as a normalized form of the activation energy, is a measure for the increase of the reaction rate due to an increase of temperature. 

Any of the curves in Fig. 10, which refer to different values of the modified Prater number fi, tend to approach a certain limiting value of the Weisz modulus for which the overall effectiveness factor obviously becomes infinitely small. This limit can be easily determined, bearing in mind that the effective reaction rate can never exceed the maximum interphase mass transfer rate (the maximum rate of reactant supply) which is obtained when the surface concentration approaches zero. To show this, we formulate the following simple mass balance, analogous to eq 62 ... 

This effect will be particularly emphasized at small values of the Thiele modulus where the intrinsic rate of reaction and the effective rate of diffusion assume the same order of magnitude. At large values of , the effectiveness factor again becomes inversely proportional to the Thiele modulus, as observed under isothermal conditions (Section 6.2.3.1). Then the reaction takes place only within a thin shell close to the external pellet surface. Here, controlled by the Arrhenius and Prater numbers, the temperature may be distinctly higher than at the external pellet surface, but constant further towards the pellet center. 

Whether or not multiple steady states will appear, and how large the deviation of the effectiveness factors between both stable operating points will be, is determined by the values of the Prater and Arrhenius numbers. Effectiveness factors above unity generally occur when p > 0 (exothermal reactions). However, for the usual range of the Arrhenius number (y = 10-30), multiple steady states are possible only at larger Prater numbers (see Fig 13). For further details on multiple steady states, the interested reader may consult the monograph by Aris [6] or the works of Luss [69, 70]. 

Figure 16 shows an effectiveness factor diagram for a first order, irreversible reaction which has been calculated from eq 95 for various values of the modified Prater number / . From this figure, it can be seen that for exothermal reactions (/ > 0) effectiveness factors above unity may be observed when the catalyst operates at a temperature substantially above the bulk fluid phase temperature. This is caused by the limited heat transfer between the pellet and the surrounding fluid. The crucial parameters controlling occurrence and size of this effect are again the modified Prater number and the Arrhenius number. 

A result such as this is logical since it contains the three groups that determine the overall process. Ca determines the concentration drop over the film, the Prater number the maximum temperature rise or drop (exothermal or endothermal reaction), and the dimensionless activation energy expresses the sensitivity of the reaction to a temperature change. 

Figure 13. Internal effectiveness factor as a function of the Thiele modulus for nonisothermal reactions at different values for the Prater number and y, = 10 (numerical solutions for a first order reaction).

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the pellet... 

For exothermic processes the reactions cause a temperature rise inside the particle. This usually leads to increased values of the rate constants. This increase of the rate constants can sometimes overcompensate for the lower concentrations (compared to those in the bulk fluid) caused by the diffusion limitations in the particle. As a result, the reaction rate becomes higher than the reaction rate that is obtained with the concentrations in the bulk phase and temperature. Consequently, the effectiveness factor exceeds 1 This effect is particularly emphasized at small values of the Thiele modulus. The catalyst effectiveness factors as a function of Thiele modulus at different values of the Prater numbers are illustrated in Figure 9.15. 

The heat of reaction (—AkH ) can lead to a temperature increase/decrease inside the catalyst particles. The significance of this phenomenon is evaluated by means of the Prater number p, which is well known from the reaction engineering literature... 

For exothermic reactions the temperature inside the pellet will be higher than the surface temperature. Because of the exponential increase of the reaction rate, the temperature effect can overcompensate the lower concentration in the pellet. An example is shown in Figure 2.29 where the effectiveness factor is plotted versus the Thiele modulus for an Arrhenius number of y = 20 and different Prater numbers. 

For majority of industrial catalysts the effective heat conductivity is in the order of 0.2 < Ag < 0.5 and efficient diffusion coefficients for gas phase reactions are in the order of 10 to 10 m s . Therefore, Prater numbers seldom exceed values of = 0.1 and the maximum temperature in the pellet center is seldom higher than... 

Belfiore LA. Effects of the collision integral, thermal diffusion, and the Prater number on maximum temperature in macroporous catalysts with exothermic chemical reaction in the diffusion-controlled regime. Chemical Engineering Science 2007 62 655-665. 

Figure 4.5.15 External effectiveness factor as a function ofthe ratio ofthe (measurable) effective reaction rate to the maximum rate (complete control by external mass transfer) fora constant Arrhenius number of 20. For a Prater number jSex < 0, the reaction is endothermic, for /Sex > 0 exothermic, and for /Sex = 0 we have isothermal conditions. Arrows and dashed line indicate ignition, as explained in the text.

The left-hand side of this equation leads to a sigmoidal curve in a plot of this term versus the temperature of the gas phase, while the right-hand side leads to lines as shown in Figure 4.5.16 for two different cases. In the first case (Figure 4.5.16a), the Prater number ex is small, (< about 0.2) for example, the reaction is only weakly exothermic or cooling is efficient or the feed gas is diluted with inert gas. For a low gas temperature, r/ x is >1, that is, krn.eff (T p) [> fA(T Am,ex (state 1 in Figure 4.5.16a]. If the temperature is increased, rjex becomes < 1 (state 2 in Figure 4.5.16a). 

If a second-order reaction is chosen as a reasonable upper limit, then with n = 2, p = 0.95, and the dimensionless Weisz criterion, Nw-p, frequently referred to as the Weisz-Prater number [21], is obtained for negligible diffu-sional limitations ... 

The final result is again a dimensionless number that compares the rate of reaction to the rate of mass transfer to the active sites in the pores. This is a conservative estimate because the concentration gradient depicted by curve 1 is likely to be steeper than the linear plot shown if the rate is very high. Furthermore, for a value of t s 0.95, the inequalities required in equation 4.88 for l -order and zero-order reactions are 0.6 and 6, respectively therefore, if a Weisz-Prater criterion (equation 4.93) of 0.3 or less is used, rates for all reactions with an order of 2 or less should have negligible mass transfer limitations. In addition, if the Weisz-Prater number is greater than 6, pore diffusion limitations definitely exist [20]. This transition region from kinetic to diffusion control occurs frequently around a TOF 1 s for many supported metal catalysts [22]. 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

This is the same case with which in Eqs. (2)-(4) we demonstrated the elimination of the time variable, and it may occur in practice when all the reactions of the system are taking place on the same number of identical active centers. Wei and Prater and their co-workers applied this method with success to the treatment of experimental data on the reversible isomerization reactions of n-butenes and xylenes on alumina or on silica-alumina, proceeding according to a triangular network (28, 31). The problems of more complicated catalytic kinetics were treated by Smith and Prater (32) who demonstrated the difficulties arising in an attempt at a complete solution of the kinetics of the cyclohexane-cyclohexene-benzene interconversion on Pt/Al203 catalyst, including adsorption-desorption steps. 

To ensure the system is probing reactions in a kinetically controlled regime, the reaction conditions must be calculated to determine the value of the Wiesz-Prater criterion. This criterion uses measured values of the rate of reaction to determine if internal dififusion has an influence. Internal mass transfer effects can be neglected for values of the dimensionless number lower than 0.1. For example, taking a measured CPOX rate of 5.9 x 10 molcH4 s g results... 

The Weisz-Prater criterion can be used for detecting drop in effectiveness due to internal diffusion. This is a number representing the ratio of actual reaction rate to a diffusion rate, and is given by ... 

An elegant, general solution for first-order networks has been provided in a classic publication by Wei and Prater [22]. In essence, the mathematics are developed for a reaction system with any number of participants that are all connected with one another by direct first-order pathways. For example, in a system with five participants, each of these can undergo four reactions, for a total of twenty first-order steps. Matrix methods are used to obtain concentration histories in constant-volume batch reactions, and a procedure is described for determination of all rate coefficients from such batch... 

The analytical problems associated with differential reactors can be overcome by the use of the recirculation reactor. A simplified form, called a Schwab reactor, is described by Weisz and Prater . Boreskov.and other Russian workers have described a number of other modifications " . The recirculation reactor is equivalent kinetically to the well-stirred continuous reactor or backmix reactor , which is widely used for homogeneous liquid phase reactions. Fig. 28 illustrates the principle of this system. The reactor consists of a loop containing a volume of catalyst V and a circulating pump which can recycle gas at a much higher rate, G, than the constant feed and, withdrawal rates F. 

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