Arrhenius number

Inert gas pressure, temperature, and conversion were selected as these are the critical variables that disclose the nature of the basic rate controlling process. The effect of temperature gives an estimate for the energy of activation. For a catalytic process, this is expected to be about 90 to 100 kJ/mol or 20-25 kcal/mol. It is higher for higher temperature processes, so a better estimate is that of the Arrhenius number, y = E/RT which is about 20. If it is more, a homogeneous reaction can interfere. If it is significantly less, pore diffusion can interact. 

On this table, the Arrhenius number E/RT was designated by e, but this symbol is already used in this book for the empty fraction in a packed bed. The correct symbol for E/RT = y is used here. On the last line in this table the derivative of the heat removal rate is given ... 

Figure 10.3 shows results for an Arrhenius number of 20. With plausible estimates for and eff, the magnitude of r) can be calculated. For the special case of AHj( = 0 (i.e., P = 0), Equation (10.33) is an alternative to the pore diffusion model for isothermal effectiveness. It predicts rather different results. For example, suppose dpl2X) k[ f = = 1- Then Equation... 

The parameter W2 is called the Arrhenius number. It varies from values of about 15 for low-activation energy systems (e.g., thermosetting polyurethanes), to values of about 40 for high-activation energy systems (e.g., phenolic molding compounds). 

A->B- C has also been considered by Jorgensen (65). This last is a system of three equations for u and v, the concentrations of A and B, and w, the temperature, with seven parameters a, the Damkohler number for A->B 3, its dimensionless heat of reaction y, its Arrhenius number k, a dimensionless heat transfer coefficient v, the ratio of activation energies of the two reactions p, the ratio of the heats of reaction a, the ratio of the Damkohler numbers. They contain a characteristic non-linearity... 

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 Arrhenius number, which is a dimensionless representation of the intrinsic activation energy, related to the bulk temperature ... 

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).

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. 

Hence, as an alternative to Fig. 10, the overall effectiveness factor can also be plotted against the product rjDan, which again contains only measurable quantities, and which, as already stated, can never exceed unity. This leads to the representation shown in Figs. 11 and 12 for two different values of the Arrhenius number. 

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. 

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. 

At this point, it should be mentioned that there may be some doubt about how successful the nonisothermal criteria are at involving observable quantities only. This concerns the fact that in the nonisothermal case one has to specify the Arrhenius number which contains the true activation energy of the catalyzed reaction. The above statement would obviously define the true activation energy as a directly observable quantity in the nonisothermal criteria. However, this would presumably be an experimental value derived from studies in which intraparticle transport effects were absent, which is precisely what one is attempting to define [12]. 

EJRT, dimensionless energy of activation or Arrhenius number... 

The dimensionless group y is known as the Arrhenius number. Substitution of the dimensionless variables into the material and energy balances gives ... 

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... 

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