Nonequilibrium stage

In Fig. 3.52, the column contactor is represented by a series of N nonequilibrium stages, each of which is of height H and volume V. The effective column height, Z, is thus given by Z = N H. 

If the same stages are equilibrium ones on all sites and the stoichiometric numbers of nonequilibrium stages are the same (namely, are equal to a certain number v), then (85) is applicable to all sites with the same v = v so that the ratio ril /r is identical for all sites. Then... 

Equation (83) may also be considered as applicable for a reaction on a nonuniform surface with different stoichiometric numbers of nonequilibrium stages in this case, however, the average stoichiometric number in (83) is determined not by (80), but in a more complicated manner the definition must involve averaging with respect not only to stages, but also to the kinds of the surface sites. 

If there are several nonequilibrium stages of a reaction, the transfer coefficient, a, may be taken identical for different stages. This assumption is justified by the fact already that often a = 1/2. 

For a further verification of this conclusion, we determined equilibrium constant, K, by combining the results of kinetic measurements in the region of r+ > r and in the region of r+ < r, having assumed that v = 1. The agreement of the obtained value with the value known from thermodynamic data (K = 0.0364 atm2 at 530°Q has confirmed that v = 1. This result means that stoichiometric numbers of all nonequilibrium stages of the reaction equal unity. 

When the stages can be considered to be nonequilibrium stages, then the model should... 

Let us test our two-phase model numerically. The model is captured in the two equations (6.99) and (6.102) that account for the gas and liquid molar balances. We shall compare these results for nonequilibrium stages with those of the one-phase equilibrium model given in equation (6.85) earlier. 

Powers MF, Vickery DJ, Arehole A, Taylor R. A nonequilibrium-stage model of multicomponent separation processes—V. Computational methods for solving the model equations. Computers Chem Eng 1988 12 1229-1241. 

Figure 12 is a schematic diagram of a nonequilibrium stage n in an absorber. The equations applying to this stage are described below. A more detailed description is given by Krishnamurthy and Taylor. 

Deg rees of Freedom Table 13-10 summarizes the equations for a single nonequilibrium stage. There are 6c -15 independent equations... 

Thus, quite generally, in calculating work performance, one must start with the initial equilibration state and end with the final equilibration state. The fact that the system passes through intermediate nonequilibrium stages, wherein not all of the alterations in potential energy of the piston are communicated to the gas, does not matter in determining functions of state of a system the details of the intervening processes are irrelevant in thermodynamic analyses based on equilibrium states. 

Two different approaches have evolved for the simulation and design of multicomponent distillation columns. The conventional approach is through the use of an equilibrium stage model together with methods for estimating the tray efficiency. This approach is discussed in Chapter 13. An alternative approach based on direct use of matrix models of multicomponent mass transfer is developed in Chapter 14. This nonequilibrium stage model is also applicable, with only minor modification, to gas absorption and liquid-liquid extraction and to operations in trayed or packed columns. 

Figure 14.1. Schematic diagram of a nonequilibrium stage. This stage represents a tray in a trayed column or a section of packing in a packed column.

The nonequilibrium stage in Figure 14.1 may represent either a single tray or a section of packing in a packed column. In the models described in this chapter the same equations are used to model both types of equipment and the only difference between these two simulation problems is that different expressions must be used for estimating the binary mass transfer coefficients and interfacial areas. 

In writing down the equations that model the behavior of this nonequilibrium stage, the flow rates of vapor and liquid phases leaving the jth stage are denoted by Vj and Lj, respectively. The mole fractions in these streams are y j and x j. The yT-y are the rates of mass transfer of species i on stage j. The temperature of the vapor and liquid phases are not assumed to be equal and we must allow for heat transfer as well as mass transfer across the interface. The symbol represents the rate of energy transfer across the phase boundary. 

In the nonequilibrium stage model of Krishnamurthy and Taylor (1985a) the total mass transfer rates are obtained by combining Eqs. 11.5.3a and lL5.5a and multiplying by the interfacial area available for mass transfer... 

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