Estimating equilibria

Copeman Most provide the frame work for handling multi-component mixtures. Vodka et compared the (vapour-f liquid) equilibria estimated from cubic equations of state with measured values and preferred mixing rules of the form of eq 5.21. 

Consider the interface between a semiconductor and an aqueous electrolyte containing a redox system. Let the flat-band potential of the electrode be fb = 0.2 V and the equilibrium potential of the redox system o = 0.5 V, both versus SHE. Sketch the band bending when the interface is at equilibrium. Estimate the Fermi level of the semiconductor on the vacuum scale, ignoring the effect of dipole potentials at the interface. 

A bottle that you thought contained ethanol had oxidized to 10% ethylhydroperoxide. If the contents of this bottle suddenly went to ehemical equilibrium, estimate the temperature and pressure in the bottle (for a short time). The heat of formationof ethylhydroperoxide is -41 kcal/tnole and the heat of carbon oxidation to CO2 is -67 kcal/mole. 

In case 1 (upper panel of Fig. 15.9), the initial solution was determined by beginning with an equilibrium estimate at a reactor temperature of T = 1200 K. Then, using continuation, the solution at 1200 K was used as the starting estimate for a simulation at T = 925... 

A comparison of solid solubilities for TPP in supercritical pentane, supercritical toluene, and various liquid solvents is given in Table VI. The solid solubility of TPP in toluene corresponds to the liquid-phase concentration for SLG equilibrium, estimated from the results in Figures 2 and 4. Several important conclusions can be... 

Choose the thermodynamic methods for property and phase equilibrium estimations. 

Firstly the negative pressure appearance in fluid inclusions was found by Roedder (1967), who observed melting in pure water inclusion in quartz at + 6.5°C and from slope of L-S equilibrium estimate pressure in the inclusion as - 80 MPa. 

Describe heterogeneous equilibria and write their equilibrium constants Use the relationships between thermodynamics and equilibrium Estimate equilibrium constants at different temperatures... 

The rate of mass deposition and sublimation of CO2 is assumed to be proportional to the local deviation from the phase equilibrium, estimating the equilibration time constant ( ) at 1 X 10 s/m, which is assumed independent of temperature. The rate of sublimation of previously deposited CO2 is assumed to approach a first-order dependency on the mass deposition when this mass deposition approaches zero [13]. 

Wichner, R. P., Spence, R. D. A chemical equilibrium estimate of the aerosols produced in an overheated light water reactor core. Nucl. Technology 70, 376—393 (1985)... 

Using the following data, estimate the number of moles of ethyl acetate which may be obtained at 26 X" by mixing 1 mole each of acetic acid and ethyl alcohol and allowing the mixture to come to equilibrium. Estimate the corresponding yield which would be obtained at 200 °C under a pressxire sufficient for the system to remain liquid. What approximations are involved in these calculations ... 

The K+ concentration inside a nerve cell is much larger than the concentration outside it. Suppose the potential difference across the cell membrane is 90 mV. Assuming the system is in equilibrium, estimate the ratio of the K" " concentrations inside and outside the cell. 

Choose the thermodynamic methods for property and phase equilibrium estimations. Consider possible heat recovery and heat integration strategies for all the processes analyzed. Obtain a converged solution using a simulator for the mass and energy balances. 

A gas mixture containing 1 mol/s H2, 2 mol/s CO and 1 moEs CO2 is fed into a furnace at 1.5 bar. In addition to the species at the inlet, the oudet stream is measured to contain 0.40 moEs of H2O vapor that has formed through chemical reaction. Assuming the reaction to prcxluce water has reached equilibrium, estimate the temperature of the furnace. You may assume that Afiren = cxmst and that no other species is produced in the chemical reaction. 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

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. 

Substitution of Equations (2) and (3) into the equilibrium relations dictated by Equation (2-l)[Pg.99]

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

The subroutine is well suited to the typical problems of liquid-liquid separation calculations wehre good estimates of equilibrium phase compositions are not available. However, if very good initial estimates of conjugate-phase compositions are available h. priori, more effective procedures, with second-order convergence, can probably be developed for special applications such as tracing the entire boundary of a two-phase region. 

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. 

Subroutine VLDTA2. VLDTA2 loads the binary vapor-liquid equilibrium data to be correlated. If the data are in units other than those used internally, the correct conversions are made here. This subroutine also reads the estimated standard deviations for the measured variables and the initial parameter estimates. All input data are printed for verification. 

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). 

X(I) vector of estimated equilibrium liquid composition (mole fraction) if known (I = 1, N) otherwise can be any vector not summing to 1. 

Given the estimate of the reactor effluent in Example 4.2 for fraction of methane in the purge of 0.4, calculate the.actual separation in the phase split assuming a temperature in the phase separator of 40°C. Phase equilibrium for this mixture can be represented by the Soave-Redlich-Kwong equation of state. Many computer programs are available commercially to carry out such calculations. 

One can write acid-base equilibrium constants for the species in the inner compact layer and ion pair association constants for the outer compact layer. In these constants, the concentration or activity of an ion is related to that in the bulk by a term e p(-erp/kT), where yp is the potential appropriate to the layer [25]. The charge density in both layers is given by the algebraic sum of the ions present per unit area, which is related to the number of ions removed from solution by, for example, a pH titration. If the capacity of the layers can be estimated, one has a relationship between the charge density and potential and thence to the experimentally measurable zeta potential [26]. 

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