Mathematical Models for Gas-Liquid-Solid Reactors

In Chap. 2, the gas liquid-solid reaction process based on the film theory was analyzed. In this chapter, some of the reported models for three-phase reactors are presented. Some models consider only the effectiveness of contact between the liquid and solid, while others consider the roles of the gas-liquid and liquid-solid mass-transfer resistances in three-phase gas-liquid-solid reactors. A large number of models consider the role of the RTD on the reactor performance. Both isothermal and non-isothcrmal models are considered here. 

4-1 MODELS BASED ON EFFECTIVENESS OF CONTACT, WITH NO EXTERNAL MASS-TRANSFER RESISTANCES (MODELS FOR TRICKLE-BED REACTORS) 

When (a) there are no external mass-transfer resistances (such as gas-liquid, liquid solid, etc.), (b) catalysts are all effectively wetted, ( ) there is no radial or axial dispersion in the liquid phase, (d) a gaseous reactant takes part in the reaction and its concentration in the liquid film is uniform and in excess, (e) reaction occurs only at the liquid-solid interface, (/) no condensation or vaporization of the reactant occurs, and (g) the heat effects are negligible, i.e., there is an isothermal operation, then a differential balance on such an ideal plug-flow trickle-bed reactor would give 

In the above equation, C s are assumed to be in grams per cubic centimeter. Any other convenient units for C can also be chosen. The above equations can be used to correlate the data obtained in large-scale isothermal reactors such as hydrodesulfurization, hydrodenitrogenation reactors, etc. 

In pilot-scale hydroprocessing trickle-bed reactors, low liquid flow rates make the catalyst effectiveness dependent upon the liquid flow rate. Henry and Gilbert15 proposed that this may be due to insufficient liquid holdup in the reactor. For the first-order reaction, they modified Eq. (4-1) as 

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Development of a mathematical model for the gas-liquid-solid reactor where significant evaporation of the liquid occurs. The reaction could occur either only in liquid phase or in both liquid and gas phases. Both isothermal as well as nonisothermal operations should be considered. A practical example of such a reactor is the reactor used in high-severity hydrocracking operations. 

The momentum balance will not be discussed in more detail here, because the first simulation tests for periodic process control of trickle-bed reactors do not consider the momentum balance. A complete mathematical model for a three-phase reactor would thus be made up of the respective material, heat and momentum balances for the gas phase , for the liquid phase , and for the solid phase (catalyst) , but their complete solution currently encounters major difficulties. 

We now look at the mathematical equations for a general isothermal steady-state model for the trickle-bed reactor, which takes into account external mass-transfer resistances, i.e., gas-liquid and liquid-solid, axial dispersion, and the intraparticle mass-transfer resistances, along with the intrinsic kinetics occurring at the catalyst surface. Since many practical reactions can be characterized as... 

In the monograph by Ramachandran and Chaudhari [27] regarding catalytic three-phase reactors, it is generally pointed out that the balance calculations for the creation of mathematical models in many international papers regard the liquid outside the catalyst grain as the liquid phase and the grain filled with liquid and/or gas pseudohomogeneously as the catalyst (solid phase) . 

Also, Valerius et al. [28, 29] show by exact formulation that the balance of the solid-phase (catalyst) actually is a balance of the liquid phase and/or gas phase in the catalyst pellet. Nevertheless, this text will follow the international trend and designate the trickle-bed reactor balance equations for the gas/liquid phase in the catalyst grain as balance for the solid phase or catalyst . The complex hydrodynamic relations in trickle-bed reactors together with the various assumptions or neglects result in widely different structured mathematical models. Two basic types of models, the differential models and the cascade or cell models, are most frequently used in the literature. 

In general, it can be concluded that substantial progresses have been made in the experimental and theoretical analysis of trickle-bed reactors under unsteady-state conditions. But until now these results are not sufficient for a priori design and scale-up of a periodically operated trickle-bed reactor. The mathematical reactor models, which are now available are not detailed enough to simulate all of the main transient behavior observed. For solving this problem specific correlations for specific model parameters (e.g. Hquid holdup, mass transfer gas-solid and liquid-solid, intrinsic chemical kinetic, etc.) determined under dynamic conditions are required. The available correlations for important hydrodynamic, mass-and heat-transfer parameters for periodically operated trickle-bed reactors leave a lot to be desired. Indeed, work for unsteady-state conditions on a larger scale may also be necessary. 

The mathematical model was constracted on the basis of a three-phase plug-flow reactor model developed by Korsten and Hoffmaim [63]. The model incorporates mass transport at the gas-liquid and liquid-solid interfaces and uses correlations to estimate mass-transfer coefficients and fluid properties at process conditions. The feedstock and products are represented by six chemical lumps (S, N, Ni, V, asphaltenes (Asph), and 538°C-r VR), defined by the overall elemental and physical analyses. Thus, the model accounts for the corresponding reactions HDS, HDN, HDM (nickel (HDNi) and vanadium (HDV) removals), HD As, and HCR of VR. The gas phase is considered to be constituted of hydrogen, hydrogen sulfide, and the cracking product (CH4). The reaction term in the mass balance equations is described by apparent kinetic expressions. The reactor model equations were built under the following assumptions ... 

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