Weisz-Prater criterion

Checking the absence of internal mass transfer limitations is a more difficult task. A procedure that can be applied in the case of catalyst electrode films is the measurement of the open circuit potential of the catalyst relative to a reference electrode under fixed gas phase atmosphere (e.g. oxygen in helium) and for different thickness of the catalyst film. Changing of the catalyst potential above a certain thickness of the catalyst film implies the onset of the appearance of internal mass transfer limitations. Such checking procedures applied in previous electrochemical promotion studies allow one to safely assume that porous catalyst films (porosity above 20-30%) with thickness not exceeding 10pm are not expected to exhibit internal mass transfer limitations. The absence of internal mass transfer limitations can also be checked by application of the Weisz-Prater criterion (see, for example ref. 33), provided that one has reliable values for the diffusion coefficient within the catalyst film. 

Weisz-Prater criterion uses measured values of the rate of reaction to determine if internal diffusion is limiting the reaction. 

Weisz-Prater criterion. The relative significance of pore-diffusion resistance can be assessed by a criterion, known as the Weisz-Prater (1954) criterion, which requires only a single measurement of the rate, together with knowledge of De, Le, and the order of the surface reaction (but not of the rate constant). 

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 ... 

Transport Limitation For the estimation of the mass transport limitation, Equation (20) has an important drawback. In many cases neither the rate constant k nor the reaction order n is known. However, the Weisz-Prater criterion, cf. Equation (21), which is derived from the Thiele modulus [4, 8], can be calculated with experimentally easily accessible values, taking < < 1 for any reaction without mass transfer limitations. However, it is not necessary to know all variable exactly, even for the Weisz-Prater criterion n can be unknown. Reasonable assumptions can be made, for example, n - 1, 2, 3, or 4 and / is the particle diameter instead of the characteristic length. For the gas phase, De can be calculated with statistical thermodynamics or estimated common values are within the range of 10-5 to 10 7 m2/s. In the liquid phase, the estimation becomes more complicated. A common value of qc for solid catalysts is 1,300 kg/m3, but if the catalyst is diluted with an inert material, this... 

In Equation (35), an estimation of the mass transfer with the Weisz-Prater criterion is given. By taking always reasonable estimations or overestimated values, one obtains a good conclusion if mass transfer is present or not. For the characteristic length, 200 pm as particle diameter is used. The reaction order usually has the value of 1 to 4 a value of 4 would therefore be a worst case scenario. The catalyst density can be measured, or the common estimation of 1.3 kg/m3 can be used, which should not be too erroneous for Li-doped MgO. The observed reaction rate re is calculated from the concentration of CH4 at the inlet of the reaction cch4 0 multiplied with the highest observed conversion of 25% (the highest initial value for all tested catalysts), divided by the inverse flow rate, corrected by the reactor temperature. The calculation of re is shown in Equation (33) ... 

The obtained value is two orders of magnitude below 1 therefore, mass transport limitations can be considered to be absent. It is worth mentioning that the calculation for values such as gc is not absolutely correct in particular, the estimation of dr is not exact. But it is sufficient for an estimation, as done in with the Weisz Prater criterion. 

In agreement with other results (12,106), the steady-state diffusion of methanol in SAPO-34 was found to be a non-activated process, as shown in Fig. 14. This result means that the steady-state diffusivity measured at low temperatures can be used directly in the Weisz-Prater criterion at reaction conditions, namely, high temperatures. 

A lack of significant intraphase diffusion effects (i.e., 17 > 0.95) on an irreversible, isothermal, first-order reaction in a spherical catalyst pellet can be assessed by the Weisz-Prater criterion [P. B. Weisz and C. D. Prater, Adv. Catal., 6 (1954) 143] ... 

The results showed that the activity is improved by increasing the linear velocities and liquid-to-gas ratios. The absence of internal diffusion limitations was checked by means of the Weisz-Prater criterion. It appeared that internal gradients were negligible. 

The Weisz-Prater criterion uses measined values of the rate of reaction, —r tobs), to determine if internal diffusion is limiting the reaction. This criterion can be developed intuitively by first rearranging Equation (12-32) in the form... 

Are there aiy All the terms in Equation (12-61) are either measured or known. Consequently, internal diffusion can calculate Cwp. If limitations indicated fium the Weisz-Prater criterion ... 

P12C-1 Use the Weisz-Prater criterion to determine if the reaction discussed in AIChE J., 10, 568 (1964) is diffusion-rate-limited. 

However, since no intrinsic kinetics are measured, one cannot determine whether criterion (8.130) is satisfied. Introduction of the Wheeler-Weisz modulus, Eqn. (8.78), and series expansion, leads to the Weisz-Prater criterion to assess the importance of internal diffusion limitations. For an irreversible n order reaction and for a spherical particle, this criterion is ... 

Sec. 12.5 Eslimetion of Dftfusloo- and Reaction-Limited Regimes 12.5.1 Weisz-Prater Criterion for internai Diffusion... 

The Mears criterion, like the Weisz-Prater criterion, uses the measured rate of reaction. r, (kmol/kg cat-s) to learn if mass transfer from the bulk gas phase to the catalyst surface can be neglected. Hears proposed lhai when... 

By George, those regeneration eorrelations sure are good, but let s have another try. Figure 7.26 shows data on the rate of coke burning as a function of temperature for big beads and fine powder. How well would the Weisz-Prater criterion (Case 1, Table 7.3) predict the rate below which diffusion is not important if Cq were 3 x 10 mols/cm and 5 X 10 cm /s ... 

The other method, which can be used for a single particle size, is called the Weisz-Prater criterion [93] and is found for a first-order reaction by solving Eq. 3.6.a-16 for k ... 

Calculation shows that these are roughly equivalent, and if the inequality is not taken too literally, it is not really much different from the original Weisz-Prater criterion. However, for certain situations such as strong product inhibition, this is not the case—see Ex. 3.6.C-2. Finally, Brown [106] has considered macro-micro pore systems. For typical types of catalyst structure and diffusivities, the conclusion was that normally there will be no diffusional limitations in the micropores if there is none in the macroporcs. Thus, use of the standard criteria for the macropores should be sufficient to detect any pore diffusion problems however, the assumptions and calculations were probably not valid for zeolite molecular sieves, and so this case still needs special consideration. 

It is evident that the pellets must have had si ficant diffusional resistance at the higher (>450°C) temperatures. Using the Weisz-Prater criterion discussed in Chapter 3, with values of C, = 3 x 10 mol/cm and D, a 5 x 10" cm / ... 

We use the Weisz-Prater criterion, Equation 18.40, along with Equation 18.39 ... 

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