Stage equilibrium, defined

Fig. 7. Distillation column with stacked multiple equilibrium stages. Terms are defined in text.

When it is desired to compute, with rigorous methods, actual rather than equilibrium stages, Eqs. (13-69) and (13-94) can be modified to include the Murphree vapor-phase efficiency T ij, defined by Eq. (13-29). This is particularly desirable for multistage operations involving feeds containing components of a wide range ol volatility and/or concentration, in which only a rectification (absorption) or stripping action is provided and all components are not sharply separated. In those cases, the use of a different Murphree efficiency for each component and each tray may be necessary to compute recovery accurately. 

Thermodynamic control (Figure 1, right) is based on adsorption of substances until quasi-equilibrium stage. In this case, the surface ratio of the adsorbed species is defined by the ratio of products of their concentration and binding constant. This deposition is much less influenced by poorly controllable fluctuations of external conditions and provides much better reproducibility. The total coverage can be almost 100%. Because of these reasons, the thermodynamic control is advantageous for preparation of mixed nanostructured monolayers for electrochemical applications including a formation of spreader-bar structures for their application as molecular templates for synthesis of nanoparticles. 

A model was first developed for the Petlyuk system. Consider a generic equilibrium stage as shown in Figure 2. The model for the distillation system is based on the following set of equation (with all symbols defined in the Nomenclature section). 

The reciprocal 1/Ka of Ka measures how far the system is from equilibrium. The equilibrium stages are naturally the most efficient ones and they occur for high Ka values. For lower Ka values the system efficiency decreases. Here the tray efficiency is defined as the ratio between the separation efficiency for specific low Ka values and that for high Ka values. Note that high Ka values indicate that the system is very close to equilibrium. 

The simple stage model. The ideal or equilibrium stage was defined and discussed earlier (Sec. 2,1). The material and energy flows in and out of a simple stage, with no feeds or side products, is depicted in Fig. 

Equilibrium composition of the vapor for the Murphree vaporization efficiency, component i, stage / defined by Eq. (4.20). 

The equations describing a steady-state equilibrium stage system without chemical reaction are summarized here. The matrix notation has been previously discussed (1,2,3), and all symbols are defined below. 

A stage of a separation process is defined as an operation in which components of one or more feed streams divide themselves between two phases, and the phases are taken off separately. In an ideal stage or equilibrium stage, the effluent (exiting) streams are in equilibrium with each other. 

Emin is an engineering design quantity and is the flow rate of solvent needed for the extraction given an infinite number of equilibrium stages. E -, therefore, defines the lower limit on the required flow rate of the solvent to achieve the extraction. As the solvent flow rate is increased from E. , the number of required stages decreases however, the operation cost increase ... 

The above equation defines an operating line for this system. Since this operation is assumed to occur at equilibrium, it is defined as an equilibrium stage. The equation therefore relates the liquid and vapor compositions leaving an equilibrium stage. 

The values of and (L/D) - have been previously defined as the minimum number of equilibrium stages (Fenske equation) and minimum reflux ratio (Underwood equation). 

The characteristic value for a single-stage separation system is the isotopic separation factor a which is defined by Eq. (12). Because a deviates not very much from one, a number of separation stages must be connected in series for a sufficient total isotopic enrichment. The total isotopic separation factor S, using N equilibrium stages, is given by ... 

If an infinite or nearly infinite number of equilibrium stages is involved, a zone of constant composition must exist in the fractionating column. In this instance, there is no measurable change in the composition of liquid or vapor from stage to stage. Under these conditions, the reflux ratio can be defined as the minimum reflux ratio, R j, with... 

The calculation of multi-stage separation processes involves the solution of phase equilibrium relationships, mass balances, and energy balances. Energy balances require the computation of enthalpies of streams entering and leaving an equilibrium stage. The enthalpy is a function of state, defined in Section 1.1.2 as H = U -i- PV. It is a function of the stream composition, its temperature, and its pressure. 

The number of independent variables required to define the state of a system is determined by the degrees of freedom. The degrees of freedom in an equilibrium stage are consistent with the phase rule, which may be expressed as follows ... 

In order to define the quantitative relationships among the various parameters in a nnit operation, a mathematical model is employed, in which the physical relationships are expressed as mathematical equations. Thus, the equilibrium stage may be simulated by a model for which the mathematical solution represents physical performance. When physical relations are translated into analytical expressions, certain assumptions mnst be made and the accuracy of the simulation model depends on the validity of these assumptions. For an equilibrium stage model, it is assumed that the stage is essentially at equilibrium. Additionally, it is assumed that the models used for predicting the thermodynamic properties, namely the distribution coefficients and enthalpy, are accurate. To the extent that these assumptions are met, the performance of the equilibrium stage can be accurately predicted. 

The last set of equations defining the equilibrium stage consists of the phase equilibrium relations, which may be expressed in terms of the distribution coefficients as... 

In this method, a mixed A -value is defined as the ratio of the mole fraction of a component in the vapor to its mole fraction in the mixed liquid phase (Schuil and Bool, 1985). Applied to an equilibrium stage or a flash drum, the phase equilibrium is solved using the mixed /f-valucs instead of the usual vapor-liquid X-values to determine the flow rates and compositions of the vapor and the total liquid. The liquid phase split is then calculated on the basis of A -values for each liquid phase to determine the flow rates and compositions of the two liquid phases. An energy balance may also be included to determine the temperature or the heat transfer for the unit. 

If only one point on the operating line or only its slope is known, the number of equilibrium stages must be known in order for the column section to be defined. The operating line would then be determined by trial and error to match the specified number of stages. 

Continuing with the procedure of defining equilibrium stages down the column, the pivot point is switched from D to B when the feed tray is crossed. Alternatively, the procedure could be started at the reboiler, defining stages up the column and switching from B to D as the feed tray is crossed. 

The extraction model is shown in Figure 11.3. The equation variables Lq and are the solvent and extract component flows, and are the feed and raffinate component flows, and N is the number of equilibrium stages. Flow rates are on a mass or mole basis. The extraction factors E are defined as... 

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