Vapor-liquid equilibrium point pressure

Remember from Sec. 7.3 that while the the condition dP/dV)j = 0 on the van der Waals loop of an equation of state gave the conditions of mechanical stabiliQ, it did not give the vapor-liquid equilibrium points (that is, the vapor pressure). That had to be determined from the equality of species fugacities in each phase. The situation is much the same here in that the limit of stability from Eq. H.2-9 is not the equilibrium compositions found from the equality of species fugacities in the coexisting liquid phases. 

Figure 9.8 Pressure-temperature diagram for the alkane(l)-aromatic(2) mixture in Figures 9.4-9.7. Solid lines are pure vapor-pressure curves, ending at pure critical points (filled circles). Dashed line is the mixture critical line. Dash-dot lines are liquid constant-composition lines small dashed lines are vapor constant-composition lines. Filled square at A is a vapor-liquid equilibrium point it occurs at 14.5 bar, 386.7 K, Xj = 0.25, t/j = 0.75.

In a binary mixture, if two of the loop curves intersect, i.e., if the vapor curve of one crosses the liquid curve of the other, then the two compositions determine a vapor-liquid equilibrium point. This is due to the fact that, for a binary system of two phases, the phase rule allows two degrees of freedom. However, the value may not be unique, t.e., in the higher pressure region, particularly very near the critical, it is possible for a given vapor to have two possible equilibrium liquids of different compositions. These two conditions can be at the... 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

The computer subroutines for calculation of vapor-liquid equilibrium separations, including determination of bubble-point and dew-point temperatures and pressures, are described and listed in this Appendix. These are source routines written in American National Standard FORTRAN (FORTRAN IV), ANSI X3.9-1978, and, as such, should be compatible with most computer systems with FORTRAN IV compilers. Approximate storage requirements for these subroutines are given in Appendix J their execution times are strongly dependent on the separations being calculated but can be estimated (CDC 6400) from the times given for the thermodynamic subroutines they call (essentially all computation effort is in these thermodynamic subroutines). 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

The (vapor + liquid) equilibrium line for a substance ends abruptly at a point called the critical point. The critical point is a unique feature of (vapor + liquid) equilibrium where a number of interesting phenomena occur, and it deserves a detailed description. The temperature, pressure, and volume at this point are referred to as the critical temperature, Tc. critical pressure, pc, and critical volume, Vc, respectively. For COi, the critical point is point a in Figure 8.1. As we will see shortly, properties of the critical state make it difficult to study experimentally. 

The critical point is unique for (vapor + liquid) equilibrium. That is, no equivalent point has been found for (vapor + solid) or (liquid + solid) equilibria. There is no reason to suspect that any amount of pressure would eventually cause a solid and liquid (or a solid and gas) to have the same //m, Sm, and t/m. with an infinite o and at that point. mC02 was chosen for Figure 8.1 because of the very high vapor pressure at the (vapor + liquid + solid) triple point. In fact, it probably has the highest triple point pressure of any known substance. As a result, one can show on an undistorted graph both the triple point and the critical point. For most substances, the triple point is at so low a pressure that it becomes buried in the temperature axis on a graph with a pressure axis scaled to include the critical point. 

Pervaporation. Pervaporation differs from the other membrane processes described so far in that the phase-state on one side of the membrane is different from that on the other side. The term pervaporation is a combination of the words permselective and evaporation. The feed to the membrane module is a mixture (e.g. ethanol-water mixture) at a pressure high enough to maintain it in the liquid phase. The liquid mixture is contacted with a dense membrane. The other side of the membrane is maintained at a pressure at or below the dew point of the permeate, thus maintaining it in the vapor phase. The permeate side is often held under vacuum conditions. Pervaporation is potentially useful when separating mixtures that form azeotropes (e.g. ethanol-water mixture). One of the ways to change the vapor-liquid equilibrium to overcome azeotropic behavior is to place a membrane between the vapor and liquid phases. Temperatures are restricted to below 100°C, and as with other liquid membrane processes, feed pretreatment and membrane cleaning are necessary. 

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

An equilibrium-flash calculation (using the same equations as in case A above) is made at each point in time to find the vapor and liquid flow rates and properties immediately after the pressure letdown valve (the variables with the primes F , F l, y], x j,.. . shown in Fig. 3.8). These two streams are then fed into the vapor and liquid phases. The equations describing the two phases will be similar to Eqs. (3.40) to (3.42) and (3.44) to (3.46) with the addition of (1) a multi-component vapor-liquid equilibrium equation to calculate Pi and (2) NC — 1 component continuity equations for each phase. Controller equations relating 1 to Fi and P to F complete the model. 

A tabulation of the partial pressures of sulfuric acid, water, and sulfur trioxide for sulfuric acid solutions can be found in Reference 80 from data reported in Reference 81. Figure 13 is a plot of total vapor pressure for 0—100% H2S04 vs temperature. References 81 and 82 present thermodynamic modeling studies for vapor-phase chemical equilibrium and liquid-phase enthalpy concentration behavior for the sulfuric acid—water system. Vapor pressure, enthalpy, and dew point data are included. An excellent study of vapor—liquid equilibrium data are available (79). 

Vapor-liquid equilibrium experiments were performed with an improved Othmer recirculation still as modified by Johnson and Furter (2). Temperatures were measured with Fisher thermometers calibrated against boiling points of known solutions. Equilibrium compositions were determined with a vapor fractometer using a type W column and a thermal conductivity detector. The liquid samples were distilled to remove the salt before analysis with the gas chromatograph the amount of salt present was calculated from the molality and the amount of solvent 2 present. Temperature measurements were accurate to 0.2°C while compositions were found to be accurate to 1% over most of the composition range. The system pressure was maintained at 1 atm. 1 mm... 

Vapor-liquid equilibrium data at atmospheric pressure (690-700 mmHg) for the systems consisting of ethyl alcohol-water saturated with copper(II) chloride, strontium chloride, and nickel(II) chloride are presented. Also provided are the solubilities of each of these salts in the liquid binary mixture at the boiling point. Copper(II) chloride and nickel(II) chloride completely break the azeotrope, while strontium chloride moves the azeotrope up to richer compositions in ethyl alcohol. The equilibrium data are correlated by two separate methods, one based on modified mole fractions, and the other on deviations from Raoult s Law. 

In this system, C = 2. If we choose a point which does not fall on the vapor-liquid equilibrium line, then all three variables must be known to describe the system. However, by choosing a point on the vapor-liquid line phases, P=2 and thus, degrees of freedom F = 2-2+2 =2. In other words, only two of the three degrees of freedom (variables) must be known. Referring to Figure 2.3b, if we have a 50/50 mole fraction solution of A and B, the mixture boils at 92°C and the vapor contains 78 mole % of B. In Figure 2.3a the dotted lines indicate the partial pressure of each of the components, that is, the equation of each line defines Raoult s law ... 

Minimum boiling point azeotrope with no data given Vapor-liquid equilibrium data are given in the original reference Azeotropic concentration is given in volume per cent. Unless so indicated, all concentrations are weight per cent Pressure in mm. of mercury absolute Approximate Greater than Less than... 

Pressure has a marked effect on the azeotropic composition and vapor-liquid equilibrium diagrams of alcohol-ketone systems (J). This is due to the fact that the slopes of the vapor pressure curves of alcohols are appreciably greater than for ketones it results in an unusually larger change in the relative boiling points of the components of an alcohol-ketone system with change in pressure. 

Table 4.4a presents the parameters of Equation 4.2, with an indication of the correlation coefficient. The Kvsi-value charts or equations are used to determine the temperature or pressure of three-phase (Lw-H-V) hydrate formation. The condition for initial hydrate formation from free water and gas is calculated from an equation analogous to the dew point in vapor-liquid equilibrium, at the following condition ... 

Figure 14.9 is a three-dimensional graph that shows the extension of (vapor + liquid) equilibrium isotherms or isobars to the critical region. Line ab at X2 = 0 is the vapor pressure line for pure component 1, with point b as the critical point. In a like manner, line cd at x2 = 1 is the vapor pressure line for pure component 2, with point d as the critical point. Note that at temperatures and pressures below points b and d, the isotherms and isobars (shown as the shaded areas) intersect the vapor pressure curves.k However,... 

From the point of view of the potential for a fire, the closed cup flash point determination is usually the most important. In a perfect closed cup test, the vapor pressure is in equilibrium with the liquid at the temperature of the test. At the flash point, the vapor composition is at the lower flammable limit. In fact, the lower flammable limit can be estimated from vapor pressure data (for a pure compound). Open cup flash points are generally higher and, thus less conservative, than closed cup determinations. The value determined in an open cup test is subject to air movement at the open face of the cup and true vapor-liquid equilibrium probably does not occur. 

The vapor-liquid equilibrium for the nitric acid / water system at atmospheric pressure is shown in Figure 9.4. This figure shows that a concentration of 68.4 weight % nitric acid is the maximum (i.e., the azeotropic point) that can be obtained by simple distillation of the weak acid220. 

In most industrial processes coexisting phases are vapor and liquid, although liquid/liquid, vapor/solid, and liquid/solid systems are also encountered. In this chapter we present a general qualitative discussion of vapor/liquid phase behavior (Sec. 12.3) and describe the calculation of temperatures, pressures, and phase compositions for systems in vapor/liquid equilibrium (VLE) at low to moderate pressures (Sec. 12.4).t Comprehensive expositions are given of dew-point, bubble-point, and P, T-flash calculations. 

Calculate dew-point equilibrium for the feed. A vapor is at its dew-point temperature when the first drop of liquid forms upon cooling the vapor at constant pressure and the composition of the vapor remaining is the same as that of the initial vapor mixture. At dew-point conditions, K, = A = Ki Xt, or Xj = Nj /Kj, and Nj /Kj = 1.0, where Yj is the mole fraction of component i in the vapor phase, Xi is the mole fraction of component i in the liquid phase, A is the mole fraction of component i in the original mixture, and Kj is the vapor-liquid equilibrium K value. 

Related Calculations. The convergence-pressure K -value charts provide a useful andrapid graphical approach for phase-equilibrium calculations. The Natural Gas Processors Suppliers Association has published a very extensive set of charts showing the vapor-liquid equilibrium K values of each of the components methane to n-decane as functions of pressure, temperature, and convergence pressure. These charts are widely used in the petroleum industry. The procedure shown in this illustration can be used to perform similar calculations. See Examples 3.10 and 3.11 for straightforward calculation of dew points and bubble points, respectively. 

Two early studies of the phase equilibrium in the system hydrogen sulfide + carbon dioxide were Bierlein and Kay (1953) and Sobocinski and Kurata (1959). Bierlein and Kay (1953) measured vapor-liquid equilibrium (VLE) in the range of temperature from 0° to 100°C and pressures to 9 MPa, and they established the critical locus for the binary mixture. For this binary system, the critical locus is continuous between the two pure component critical points. Sobocinski and Kurata (1959) confirmed much of the work of Bierlein and Kay (1953) and extended it to temperatures as low as -95°C, the temperature at which solids are formed. Furthermore, liquid phase immiscibility was not observed in this system. Liquid H2S and C02 are completely miscible. 

---