Stage and Column Models

Figure 4.1 Stage and column models, (a) Overall column model (6) simple stage model...

Figure 4.1 (Continued) Stage and column models, (c) feed sage model (d) sidestream product withdrawal stage model.

The performance of a chromatographic system is generally evaluated by the time required for the elution of each solute retention time) and the width of the elution curve peak width), which represents the concentration profile of each solute in the effluent from a column. Although several models are used for evaluation, the equilibrium, stage, and rate models are discussed here. 

In reactive distillation, chemical reactions are assumed to occur mainly in the liquid phase. Hence the liquid holdup on the trays, or the residence time, is an important design factor for these processes. Other column design considerations, such as number of trays, feed and product tray locations, can be of particular importance in reactive distillation columns. Moreover, since chemical reactions can be exothermic or endothermic, intercoolers or heaters may be required to maintain optimum stage temperatures. Column models of reactive distillation must include chemical reaction... 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

In extraction column design, the model equations are normally expressed in terms of superficial phase velocities, L and G, based on unit cross-sectional area. The volume of any stage in the column is then A H, where A is the cross-sectional area of the column. Thus the volume occupied by the total dispersed phase is h A H, where h is the fractional holdup of dispersed phase, i.e., the droplet volume in the stage, divided by the total volume of the stage and the volume occupied by the continuous phase, in the stage, is (1-h) A H. 

The model balance equations are developed by considering both the individual tank stages and the absorber. Component balances are required for all components in each section of the reactor column and in the absorber, where the feed and effluent streams are located. Although the reaction actually proceeds in the biofilm phase, a homogeneous model apparent kinetics model is employed, which is justified by its simplicity. 

For the data of streams and equipment models, ASPEN utilizes a plex data structure of the type proposed by Evans, et al. (3) Information is stored in blocks of contiguous locations known as beads. Beads of any length are created dynamically from a pool of free storage which may be thought of as a lengthy FORTRAN array. The combination of the preprocessor approach and the plex data structure has resulted in the absence of dimensional constraints on the system. There are no maximum numbers of streams, components, models, stages in a column, etc. except as limited by the total memory available. 

A first model is used to compute the flowrates allowing to perform the separation with the greatest productivity. Then, the "mixed cell in series" model takes into account thermodynamic, hydrodynamic and kinetic properties of the system and compute the concentration profile inside the columns [14], In this model, we make the assumptions that the pressure drop inside the column is negligible compared to the pressure drop realized and controlled with the analogical valves, and we model the true moving bed assuming that the performance of SMB and TMB are equivalent. A mass balance equation is written for each stage and a classical Newton Raphson numerical method is used to solve the permanent state of the process [14],... 

The classic papers by Lewis and Matheson [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes [Ind. Eng. Chem., 25, 290 (1933)] represent the first attempts at solving the MESH equations for multicomponent systems numerically (the graphical methods for binary systems discussed earlier had already been developed by Pon-chon, by Savarit, and by McCabe and Thiele). At that time the computer had yet to be invented, and since modeling a column could require hundreds, possibly thousands, of equations, it was necessary to divide the MESH equations into smaller subsets if hand calculations were to be feasible. Despite their essential simplicity and appeal, stage-to-stage calculation procedures are not used now as often as they used to be. 

Computer software for equilibrium-stage and nonequilibrium column models is available from a number of suppliers. Many other models have been implemented primarily for research purposes and are not available commercially. 

The overall concentration distribution of solids in a multistage column depends upon the axial dispersion of the solids in each stage and the back flow of liquid (and solids) at the perforated plate. Kubota and Sekizawa69 proposed a model to describe the exchange of solid particles through the perforated partition plate. The model used a parameter K° which was defined as... 

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