Prater numbers

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... 

The fraction surrounded by a box in eq. (14.25) is named Prater number p. It is formed with the gas-side values (cG, Tc). The fraction on the right-side of the box is the already known pi-number combination qextDan. When this correlation is introduced into the equation for T ext, then it follows that ... 

Prater number p represents a combination of pi-numbers which can be traced back to the already known ones ... 

For Da see the definition, eq. (14.7). From this it follows that the name Prater number is superfluous. 

Fig. 81 Outer catalyst effectiveness factor, r ext, as a function of the measurable quantity r)extDan for two values ofthe Arrhenius number, Arr, and for different values ofthe Prater number 3 A Dax Le-0+n)...

The Prater number (3 - in contrast to eq. (14.25) is related to Ts and not to TG - and the Arrhenius number have a major influence on the development of the T and c profiles. The pore utilization factor qp is therefore dependent upon Arr, (3 and Thiele modulus . The correlation between these four pi-numbers is represented in Fig. 83. For T]p and the following definitions apply ... 

The Prater number, defined as the maximum observable temperature difference between the pellet... 

By substituting hf from eq 69 into eq 64, the Biot number for heat transport can be replaced by the Biot number for mass transport, when additionally a modified Prater number /T is introduced ... 

According to eq 71 the temperature of the catalyst pellet can be calculated as a function of the Weisz modulus, for given values of the modified Prater number and the Biot number for mass transport. 

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. 

With the Thiele modulus, the Biot number for mass transport, the modified Prater number, and the Arrhenius number, four dimensionless numbers are necessary to fully characterize this problem. 

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 ... 

In the most general case, i.e. when intraparticlc and interphase transport processes have to be included in the analysis, the effectiveness factor depends on five dimensionless numbers, namely the Thiele modulus the Biot numbers for heat and mass transport Bih and Bim, the Prater number / , and the Arrhenius number y. Once external transport effects can be neglected, the number of parameters reduces to three, because the Biot numbers then approach infinity and can thus be discarded. 

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]. 

Although multiplicities of the effectiveness factor have also been detected experimentally, these are of minor importance practically, since for industrial processes and catalysts, Prater numbers above 0.1 are less common. On the contrary, effectiveness factors above unity in real systems are frequently encountered, although the dominating part of the overall heat transfer resistance normally lies in the external boundary layer rather than inside the catalyst pellet. For mass transfer the opposite holds the dominating diffusional resistance is normally located within the pellet, whereas the interphase mass transfer most frequently plays a minor role (high space velocity). 

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).

Ef molar flow of i at reactor inlet mol s 1 P Prater number -... 

The maximum relative temperature difference (external Prater number) is given by ... 

A new dimensionless grouping called the Prater number, jS, appears in the energy balance ... 

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... 

Notice that the dimensionless maximum temperature rise in the catalyst pellet is simply the Prater number j3 ... 

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