Rate-based Stage Model

A more physically consistent way to describe a column stage is known as the rate-based approach [29, 30]. This approach implies that actual rates of multicomponent mass and heat transfer and chemical reactions are taken into account directly. 

Mass transfer at the gas/vapor-liquid interface can be described using different theoretical concepts [14, 16]. Most often, the two-film model [31] or the penetration/ surface renewal model [26, 32] are used, whereas the model parameters are estimated from empirical correlations. In this respect, the two-film model is advantageous since there is a broad spectrum of correlations available in the literature, for all types of internals and systems. 

In the two-film model (Fig. 9.4), it is assumed that all of the resistance to mass transfer is concentrated in thin films adjacent to the phase interface, and that transfer occurs within these films by steady-state molecular diffusion alone. Outside the films, in the fluid bulk phases, the level of mixing is so high that there is no composition gradient at all. This means that in the film region we have one-dimensional diffusion transport normal to the interface. 

Multicomponent diffusion in the films is described by the Maxwell-Stefan equations which can be derived from the kinetic theory of gases [33]. The Maxwell-Stefan equations connect diffusion fluxes of the components with the gradients of their chemical potential. In a generalized form, these equations can also be used for the description of real gases and liquids [16]  

Similar equations can also be written for the gas phase. Thus, the gas-liquid mass transfer is modeled as a combination of the two-film model presentation and Max-well-Stefan diffusion description. In this stage model, the thermodynamic equilibrium is assumed only at the phase interface. 

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In the dynamic rate-based stage model, molar holdup terms have to be considered in the mass balance equations, whereas the changes in both the specific molar component holdup and the total molar holdup are taken into account. For the liquid phase, these equations are as follows ... 

The so-called rate-based stage model presents a different way to the modeling of separation processes, by directly considering actual mass and heat transfer rates (Seader, 1989 Taylor and Krishna, 1993). A number of models fall into the general framework of the rate-based stage. In most cases, the film (Lewis and Whitman, 1924) or penetration and surface renewal (Higbie, 1935 Danckwerts, 1951) models find application, whereas the necessary model parameters are estimated by means of correlations. In this respect, the film model appears advantageous due to numerous correlation data available in the literature (see, e.g., Billet and Schultes, 1999). 

The rate-based stage model parameters describing the mass transfer and hydrodynamic behavior comprise mass transfer coefficients, specific contact area, liquid hold-up, residence time distribution characteristics and pressure drop. Usually they have to be determined by extensive and expensive experimental estimation procedures and correlated with process variables and specific internals properties. 

The methods described above are directly related to the application of the rate-based stage modeling and, more specifically, the two-film theory. As already mentioned, this theory is widely used however, some problems arise when it is applied to complex processes. A critical analysis shows that the difficulties are... 

The rate-based analysis introduced in Section 14.2 can be developed as outlined below to define a general, rigorous, multi-component rate-based stage model. It takes into account mass and energy transfer between the phases within a stage, and can be connected to other stages to form multistage columns. 

A rate-based stage can represent a section of a packed column or a tray. A stage model consists of a liquid phase and a vapor phase defined by the following relationships ... 

From the above list of rate-based model equations, it is seen that they total 5C -t- 6 for each tray, compared to 2C -t-1 or 2C -t- 3 (depending on whether mole fractious or component flow rates are used for composition variables) for each stage in the equihbrium-stage model. Therefore, more computer time is required to solve the rate-based model, which is generally converged by an SC approach of the Newton type. 

Based on an average tray efficiency of 90 percent for the hydrocarbons, the eqiiilibniim-based model calculations were made with 36 equilibrium stages. The results for the distillate and bottoms compositions, which were very close to those computed by the rate-based method, were a distillate with 0.018 mol % ethylbenzene and less than 0.0006 mol % styrene, and a bottoms product with only a trace of methanol and 0.006 mol % toluene. 

Seader JD. The rate-based approach for modeling staged separations. Chem Eng Prog 1989 85(10)4149. 

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