Balance analogy designs

Adsorption The design of gas-adsorption equipment is in many ways analogous to the design of gas-absorption equipment, with a solid adsorbent replacing the liqiiid solvent (see Secs. 16 and 19). Similarity is evident in the material- and energy-balance equations as well as in the methods employed to determine the column height. The final choice, as one would expect, rests with the overall process economics. 

If we return to our notched beam analogy, as shown in Figure 15-5, we find that we numbered the notches, 1, 2, 3, . These numbers serve as natural identifying designations. They are the quantum numbers of the balance beam. 

The balance equation for (26), which is an analog of (30), is aimed at designing the difference scheme, making it possible to write on the interval < f < D+1/2 = D the difference scheme... 

Before developing specific relationships to describe cooling tower operations, it is worthwhile to review some elementary principles in developing material and energy balances. In addition, we need to review heat and mass transfer analogies before tackling design problems. The more experienced reader may wish to proceed to Chapter 4 or try the example problems at the end of the chapter as a refresher. 

The mass-balance constraint has the form [FjJ+ [0] + [Se] = const. [O] + [Se] = const. In these relations we use designations analogous to those in Eqs. (1-5) Jq is the flux of oxygen atoms (radicals) incident on a two-dimensional crystal, is the surface concentration of oxygen adatoms. No is the concentration of oxygen atoms in the surface layer, and [O] is the net oxygen concentration in the adlayer and surface layer. 

In the next sections, the design equations for the three ideal reactor t)q>es will be derived for isothermal conditions. In practice, the heat effects associated with chemical reactions usually result in non-isothermal conditions. The application of the law of conservation of energy leads to the so-called energy balance equation. This derivation is analogous to the derivation of the mass balance equations, and will not be treated here (see for instance, [4,5]). However, it should be noted that under non-isothermal conditions, the energy balance equation should always be solved simultaneously with the corresponding mass balance equation, since the reaction rate depends not only on composition but also on temperature. 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

In ihe three idealized types of reactors just discussed (the perfectly mixed batch reactor, the plug-fiow tubular reactor (PFR). and the perfectly mixed con-tinuous-siirred tank reactor (CSTR), the design equations (i.e.. mole balances) were dei doped based on reactor volume. The deris ation of the design equation for a packed-bed catalytic reactor (PBR) will be carried out in a manner analogous to the development of the tubular design equation. To accomplish this derivation. we simply replace the volume coordinate in Equation (MO) with (he catalyst weight coordinate H (Figure - 4). 

Liquid-solid adsorption, and ion-exchange equilibrium data and material balances, are handled in a manner completely analogous to gas-solid systems. An example of a liquid-solid ion-exchange design calculation is included in Chapter 8. 

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