Vapor bubble point, calculation

For the special case of a bubble-point calculation (incipient vaporization), a is 0 (also Q = 0) and Equation (7-13) becomes... 

If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble-point calculation is to be performed on a liquid of known composition using an equation of state for the vapor-liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. 

Solution To determine the location of the azeotrope for a specified pressure, the liquid composition has to be varied and a bubble-point calculation performed at each liquid composition until a composition is identified, whereby X = y,-. Alternatively, the vapor composition could be varied and a dew-point calculation performed at each vapor composition. Either way, this requires iteration. Figure 4.5 shows the x—y diagram for the 2-propanol-water system. This was obtained by carrying out a bubble-point calculation at different values of the liquid composition. The point where the x—y plot crosses the diagonal line gives the azeotropic composition. A more direct search for the azeotropic composition can be carried out for such a binary system in a spreadsheet by varying T and x simultaneously and by solving the objective function (see Section 3.9) ... 

The vapor-liquid x-y diagram in Figures 4.6c and d can be calculated by setting a liquid composition and calculating the corresponding vapor composition in a bubble point calculation. Alternatively, vapor composition can be set and the liquid composition determined by a dew point calculation. If the mixture forms two-liquid phases, the vapor-liquid equilibrium calculation predicts a maximum in the x-y diagram, as shown in Figures 4.6c and d. Note that such a maximum cannot appear with the Wilson equation. 

This equation is familiar to us from bubble point calculations. In this formulation of the MESH equations, the vapor-phase mole fractions no longer are independent variables but are denned by Eq. (13-52). This formulation of the MESH equations has been used in quite a number of algorithms. It is less useful if vapor-phase nonideality is important (and, therefore, the K values depend on the vapor-phase composition). 

For a total condenser, the vapor composition used in the equilibrium relations is that determined during a bubble point calculation based on the actual pressure and liquid compositions found in the condenser. These vapor mole fractions are not used in the component mass balances since there is no vapor stream from a total condenser. It often happens that the temperature of the reflux stream is below the bubble point temperature of the condensed liquid (subcooled condenser). In such cases it is necessary to specify either the actual temperature of the reflux stream or the difference in temperature between the reflux stream and the bubble point of the condensate. 

After calculating the temperature of the top and bottom products, obtain a new estimate of the colmnn relative volatility for each component. Find the relative volatihty of each conponent in the bottom and top product. Assuming that we have a total condenser, the conposition of the vapor rising above the top tray is equal to the conposition of the top product. The calculation for the dew-point tenperature will give the composition of the hquid on the top tray as well as the temperature. The temperature and hquid composition at the bottom tray is obtained from a bubble point calculation. Next, calculate the relative volatility of each conponent at the top and bottom tray. Using these values of the relative volatihty and the values for the feed, calculate the geometric average volatihty, (oCj)avg, of each component from Equation 6.26.19. This calculation is summarized in Table 6.7.2... 

The dew point and bubble point calculations do not present peculiar problems, but the flash calculation does. Let X fx) be the mole fraction distribution in the feed to a flash, and let a be the vapor phase fraction in the flashed system. The mass balance is ... 

This would correspond to the bubble-point calculation as performed for vapor-liquid equilibrium, the object being to determine the temperature at a given pressure, or vice versa, whereby the first drop of vapor ensues from the vaporization of the liquid phase. That is, it would correspond to a point or locus of points on the saturated liquid curve. 

Note When V/F = 0, essentially only a drop of the feed has permeated through the membrane. The determination corresponds to a bubble-point calculation as practiced in vapor-liquid phase separations (where V —> 0). When V/F = 1, essentially all the feed material has permeated through the membrane, leaving a drop of reject as calculated. The calculation corresponds to that of a... 

The equations above are not the only formulations in which vapor-liquid calculations are conducted. For example, for the bubble point calculation, an equation equivalent to Eq. (3.40) would be... 

Example 3 Detv and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x, and t/i, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoults law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. 

The equilibrium vapor composition y has been estimated from a bubble point calculation on to be... 

A conventional bubble point calculation involves the specification of the liquid mole fractions and pressure the subsequent computation of the vapor-phase mole fractions and the system temperature. For a binary system (and only for a binary system) we may specify the temperature and pressure and compute the mole fractions of both phases. Thus, our first step is to estimate the interface temperature T. The second step is to solve the equilibrium equations for the mole fractions on either side of the interface. This step is, in fact, equivalent to reading the composition of both phases from a T-x-y equilibrium diagram. 

The first step is to assume a set of temperatures Tj and vapor flows Vj. The temperatures can be obtained by linear interpolation between the condenser and reboiler temperatures, determined by dew point and bubble point calculations of products estimated by shortcut methods or on the basis of past experience with similar columns. The vapor rates are estimated from the specified distillate and reflux rates. Constant vapor rates are assumed above and below the feed. The... 

The calculated mole fractions on a given stage generally may not sum up to unity due to inaccuracies, truncation errors, and so on, and must therefore be normalized. With the normalized compositions, a bubble point calculation is performed on each stage to determine the temperatures Tj, and the vapor mole fractions Ty . This is equivalent to solving Equations 17.27 and 17.28. (The solution of similar equations is discussed in Chapter 2.)... 

A more accurate estimate of N i can be obtained using a mean relative volatility based on values at the top, middle, and bottom of the column. The top temperature is about 75"C, the boiling point of n-hexane at 1.2 atm, and the relative volatility is 2.53 from the vapor pressures in Fig. 19.1. The bottom temperature is about 115 C, by a bubble-point calculation for the bottoms product, giving a relative volatility of 2.15. From Eq. (19.14),... 

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