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Fig. 5.27 Simulated effectiveness factor of a ring-shaped catalyst body plotted versus the Thieie modulus for MTBE synthesis ([32], reprinted from Chem. Eng. Technol., Vol 21, Sundmacher, Kunne and Kunz, Pages 494-498, Copyright 1998, with permission from Wiley-VCH)

Fig. 5.27 Simulated effectiveness factor of a ring-shaped catalyst body plotted versus the Thieie modulus for MTBE synthesis ([32], reprinted from Chem. Eng. Technol., Vol 21, Sundmacher, Kunne and Kunz, Pages 494-498, Copyright 1998, with permission from Wiley-VCH)

In (5.55), r is the intrinsic reaction rate per volume of catalyst at liquid bulk conditions r = cL ■ k(T) ■ f (x) (5.57)

where k(T) is the intrinsic reaction rate constant at reaction temperature, and J(x) contains the dependency of r on the liquid-phase composition. The intrinsic rate r has to be determined in the absence of any transport limitations. For this purpose, the use of separate kinetic measurements in a continuous stirred tank reactor [31, 43] or a differential tube reactor [44] are strongly recommended. In order to excluded transport limitations, measurements with increasing stirrer speed (CSTR) or increasing recycling rate (differential reactor), and decreasing catalyst particle size are indispensable [45, 46].

When formulating a microkinetic rate expression r(T,x) one has to account for the sorption equilibria of the reaction components between the liquid phase and the active catalyst phase. For this purpose, since the liquid-phase reaction mixtures have a strongly non-ideal behavior, one should use generalized Langmuir sorption isotherms in terms of liquid-phase activities as proposed first in the pioneer work of Rehfinger and Hoffmann [45]. According to these authors, the sorption equilibria of the N species A¡ on one active site S is given by

where is the fraction of active catalyst sites occupied by component A;, Ksi is the sorption equilibrium constant of A;, and a; is the liquid-phase activity at T (a; = x; y;) with the activity coefficient which can be estimated from the UNIQUAC method, for example.

In order to formulate an expression J[x) in (5.57), the rate determining step of the reaction mechanism has to be identified. For many heterogeneously catalyzed liquid-phase reactions the rate limiting step is found to be the reaction of sorbed molecules. For example, in the synthesis of the fuel ethers MTBE, TAME, and ETBE at acid ion-exchange catalyst the rate limiting step can be expressed as follows

Alcohol ■ S + Olefin ■ S ^ Ether ■ S + S (5.60)

where 8o is the fraction of unoccupied sites, and k the rate constant of forward (+) and reverse reaction (—). Inserting (5.59) into (5.60) leads to am expression for the reaction rate in terms of the liquid-phase activities, which in turn are dependent on the liquid-phase composition k+-Ks,Akohol-Ks,01efinaAkohol«01efin k--Ks,Ether«Ether /r

Due to the highly selective sorption of the polar alcohol molecules at the acid ionexchange catalyst, the rate expression in (5.62) can be simplified considerably into

\aAlcohol —a aAlcohol/

The temperature dependence of the reaction rate constant k is given by the Arrhe-nius equation k(T) = k(Tref)exp(- |(I - ¿)) (5.64)

where k(Tref) is the rate constant at Tref and EA is the activation energy.

In Fig. 5.28a experimental and simulated rates for the synthesis of MTBE from methanol and isobutene are depicted, which show that the rate expression (5.63) is valid for the MTBE synthesis [45]. Fig. 5.28b illustrates its validity for the ETBE synthesis from ethanol and isobutene [41] compared with experimental data reported by Francoisse and Thyrion [47]. In analogous manner this rate approach can be applied to the synthesis of the fuel ether TAME from methanol and isoam-lyenes [43, 46]. Activity-based rate expressions were also applied for other reactions carried out in strongly non-ideal liquid mixtures, for example for butyl acetate synthesis [48] and for dimethyl ether synthesis [49].

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