Liquid Calculations

Various modeling procedures have been proposed in the literature to predict the phase behavior of vapor-liquid systems at high pressures. (The designation vapor will be used synonomously with supercritical fluid in this chapter.) Regardless of the modeling procedure, the following thermodynamic relationships, or their equivalent relationships in terms of chemical potentials, must be satisfied for two phases to be in equilibrium. 

The most commonly used cubic EOSs are the Peng-Robinson (Peng and Robinson, 1976) and the Soave-Redlich-Kwong (Soave, 1972) equations. They produce essentially equivalent results since both equations are cubic in volume. We will use the Peng-Robinson (PR) EOS since we have the most experience with this equation. The PR equation is 

When dealing with gas and liquid mixtures, it is necessary to define combining rules for a iix and to use the equation of state to calculate mixture properties. In this development we will use the so-called van der Waals-1 mixing rules that assume random mixing of the components. These equations are used once for the gas phase mixture and once for the liquid phase mixture. 

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We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

The programs DRFLA for vapor-liquid and DRELI for liquid-liquid calculations are written in FORTRAN IV source language for the CDC 6400 of the Computer Center, University of California, Berkeley. Minor modifications, mostly with regard to input and output, will be required for implementation on most other computer systems. 

For vaporAiquid separators there is often a liquid residence (holdup) time required for process surge. Tables 1, 2, and 3 give various rules of thumb for approximate work. The vessel design method in this chapter under the Vapor/Liquid Calculation Method heading blends the required liquid surge with the required vapor space to obtain the total separator volume. Finally, a check is made to see if the provided liquid surge allow s time for any entrained water to settle. 

For rough sizing check of vapor/liquid separators and accumulators, see the Fluor method in Chapter 8, Separators/Accumulators—Vapor/Liquid calculation method. 

Determine bubble point temperature of bottoms and composition of vapor, ygj, up from liquid. Calculate relative volatility of light to heavy component at this temperature. 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

The fineness characteristic of a powder on a cumulative basis is represented by a straight line from the origin to 100 per cent undersize at a particle size of 50 un. If the powder is initially dispersed uniformly in a column of liquid, calculate the proportion by mass which remains in suspension in the time from commencement of settling to that at which a 40 i m particle falls the total height of the column. It may be assumed that Stokes law is applicable to the settling of the particles over the whole size range. 

Assuming that under these conditions, the rate of the above reaction is virtually independent of the o-xylene concentration and is thus pseudo first-order with respect to the concentration of dissolved O2 in the liquid, calculate the value of the pseudo first-rate constant. 

Thermodynamic stability. Let us consider a liquid binary mixture of two species, a and b, crystallizing to a solid solution. Let G be the free enthalpy (Gibbs) function of each phase, either solid or liquid. Calculate the change AG in molar free enthalpy when a liquid of molar composition XUqb crystallizes to a solid of molar composition... 

To assess the reproducibility of spray volume delivery, a series of preweighed weigh boats were placed on the instrument s carousel, and the instrument was programmed to spray a single time into each. After spray delivery, the boats were reweighed, and the volume of liquid calculated. The maximum amount of fluid dispensed was 105.1 pL, and the minimum was 65.7 pL The average of 20 separate determinations was 88.8 pL, yielding an error of 11.2%. 

At 200°C, a 50 50 Pb-Sn alloys exists as two phases, a lead-rich solid and a tin-rich liquid. Calculate the degrees of freedom for this alloy and comment on its practical significance. 

A clean dry 10-ml graduated cylinder weighs 37.6 g empty it weighs 53.2 g when filled to the 7.4 ml mark with an unknown liquid. Calculate the density of the liquid. 

The compositions of gas and liquid samples taken from a separator are given below. The separator was stabilized at 945 scf/STB. Laboratory measurement indicated a separator/stock-tank volume ratio of 1.052 SP bbl/STB. The density of the separator liquid, calculated with procedures given in Chapter 11, is 49.8 lb/cu ft at separator conditions of 115 psia and 100°F. 

A wet gas reservoir produces 170,516 scf/STB of 0.646 specific gravity separator gas, 492 scf/STB of 1.184 specific gravity stock- -tank gas, and 64.7°API stock-tank liquid. Calculate the specific gravity of the reservoir gas. 

Second, by trial and error, find a pressure which causes the fugacity of the liquid calculated with Equations 15-14, 15-15, 15-16, and 15-17 to equal the fugacity of the gas calculated with the same equations. Only the last calculation with a final trial value of p = 228.79 psia will be shown. 

Compositions and Quantities of the Equilibrium Gas and Liquid Phases of a Real Solution — Calculation of the Bubble-Point Pressure of a Real Liquid—Calculation of the Dew-Point Pressure of a Real Gas Flash Vaporization 362... 

The spectral function L(z) involved in Eq. (142) is determined by the profile of the model potential well (in this section it is the rectangular well). It follows from Eq. (148) that if we fix the dimensional quantities, such as frequency v and temperature, then the spectral function L(z) depends also on the lifetime x and the moment of inertia of a molecule I. We consider a gas-like reorientation of a polar molecule determined by a dipole moment p of a molecule in a liquid. Calculation of the moment of inertia I deserves special discussion. 

C, supercooled liquid, calculated-mole fraction of Aroclor mixtures, Murphy et al. 1987)... 

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