Process Evaluation by Dimensionless Numbers
For the reliable prediction of RD process performance it is crucial to identify the dominant local mass and energy transport resistances between the phases in a packed catalytic column section. Dimensionless parameter groups are an efficient tool for this purpose, which allow estimation of the relevance of certain transport resistances from experimental data. Here only the most important qualitative results are given .
1. As already used in the preceding subsections, the Damkohler number Da evaluates the effective rate of the chemical reaction in comparison to the feed flow rate kf Vcat effective rate of reaction
F feed flow rate
In (5.49), kf represents the effective reaction rate constant including internal transport resistances of the catalyst packing. V^ stands for the catalyst volume installed in the column. For Da ^ 1 the chemical reaction approaches chemical equilibrium, for Da ^ 1 the reaction is far from its equilibrium.
2. Since the separation in distillation columns is dominated by the vapor-liquid equilibrium, it is a reasonable assumption to neglect the mass-transfer resistances between the vapor phase and the liquid phase. However, in RD columns, the importance of vapor-liquid mass-transport resistances depends on the following dimensionless ratio
N^ov VL mass transfer rate
Da fcfgS effective rate of reaction which can be easily derived from a steady-state liquid-phase mass balance. NTUiJv is the overall number of mass-transfer units of a selected key component i referred to the vapor phase. For (5.50), the mass-transfer coefficient fc;ov and the specific mass-transfer area a have to be estimated from literature correlations, for example, from Onda's correlations, which are valid for packed towers . For NTUjv/Da » 1, the vapor-liquid mass-transfer resistances are negligible in comparison to the chemical reaction resistance. Therefore, only for very fast chemical reactions interfacial mass-transfer phenomena are of major importance.
3. Internal mass transport inside the catalyst particles should be evaluated with the help of the Thiele modulus rate of intraparticle diffusion rate of chemical reaction rate of intraparticle diffusion rate of chemical reaction
In (5.51), r stands for the intrinsic reaction rate at liquid bulk conditions. For worst-case-estimations, one should use a highest rate value possible in the considered RD column. In this respect it should be kept in mind that the reaction rates under RD conditions strongly depends on the operating pressure that influences the boiling temperatures, that is the reaction temperature. -Deff/ represents the effective diffusion coefficient of a selected reaction component inside the catalyst particles. One should use the component with the lowest mole fraction xi in the liquid bulk mixture as key component . Its effective diffusion coefficient can be estimated from the diffusion coefficient at infinite dilution: Desf = (s/r)D™ with the total porosity s and the tortuosity r of the applied catalyst. Based on (5.51) one can say that intraparticle diffusion resistances will be negligible, if 0 ^ 1.
4. The importance of external mass transport resistances at the outer surface of the catalyst particles are estimated with the following Biot number
D^ rate of intraparticle diffusion where fc;LS stands for the liquid-solid mass-transfer coefficient of the key component i, which can be estimated from published correlations [36, 37]. An evaluation of experimental data from different authors reveals that for most RD processes the Biot number is BimLS ^ 10-100. Therefore, external mass transport resistances are negligible. Only for extremely fast reactions at an egg-shell type catalyst, the transport of species towards the external catalyst surface may be important.
5. The heat of reaction (—ARH°) can lead to a temperature increase/decrease inside the catalyst particles. The significance of this phenomenon is evaluated by means of the Prater number /?, which is well known from the reaction engineering literature
_ (-ArH^XiD^ rate of intraparticle diffusion
Since the Prater numbers of RD processes were found to have values p < 10~3, the internal heat-transport effects can be neglected. The same is true for external heat transport effects .
6. The thermal sensitivity of the chemical reaction kinetics is an important effect that is characterized by the Arrhenius number oc rate of intraparticle heat conduction
Formulation of Reaction Rate Expressions
For process modeling proposes the effective chemical reaction rate reff has to be expressed as a function of the liquid bulk composition x, the local temperature T and the catalyst properties such as its number of active sites per catalyst volume cL, its porosity e, and its tortuosity r. As discussed in Section 5.4.2, the chemical reaction in the catalyst particles can be influenced by internal and external mass transport processes. To separate the influence of these transport resistances from the intrinsic reaction kinetics, a catalyst effectiveness factor ] is introduced by reff = ] ■ r
The catalyst effectiveness factor depends on the parameters defined in Section 5.4.2 n = v(<P,BiLJS,p,r) (5.56)
In correspondence with the guidelines given above, the catalyst effectiveness is mostly only a weak function of BimLS, p, and y. Therefore: n ^ n(^). The latter function is available in analytical form for first-order reactions . In the general case, it has to be determined from the numerical solution of the catalyst particle balances, as done for instance in [35, 39-42]. In the case of negative reaction orders, multiple effectiveness factors can be obtained . This is illustrated in Fig. 5.27 for the MTBE synthesis where the rate has a negative order with respect to the educt methanol.
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