Pressure equilibrium calculations

To determine the need for recombination or dissociation processes in a flame, one must first consider the mole number of the final equilibrium composition. A constrained enthalpy and pressure equilibrium calculation will determine the adiabatic flame temperature and the species distribution at that temperature. If the mean molecular weight (IT = Ylk WkXk) is larger than that of the reactants, then recombination must occur. If the W is smaller for the products, then dissociation must take place. Note that the mole number (moles per mass of gas) is the reciprocal of the mean molecular weight. At the adiabatic flame conditions there will be the expected stable products as well as a distribution of other species, including free radicals. 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

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. 

Large errors in the low-pressure points often have little effect on phase-equilibrium calculations e.g., when the pressure is a few millitorr, it usually does not matter if we are off by 100 or even 1000%. By contrast, the high-pressure end should be reliable large errors should be avoided when the data are extrapolated beyond the critical temperature. 

For mixtures, the calculation is more complex because it is necessary to determine the bubble point pressure by calculating the partial fugacities of the components in the two phases at equilibrium. 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

Later, we will make equilibrium calculations that involve activities, and we will see why it is convenient to choose the ideal gas as a part of the standard state condition, even though it is a hypothetical state/ With this choice of standard state, equations (6.94) and (6.95) allow us to use pressures, corrected for non-ideality, for activities as we make equilibrium calculations for real gases.s... 

The vapor pressure of Pu(g) in equilibrium with various phases was also determined from ion intensity measurements with the mass spectrometer. The pressure was calculated from the equation... 

This expression provides the basis for vapor-liquid equilibrium calculations on the basis of liquid-phase activity coefficient models. In Equation 4.27, thermodynamic models are required for cf>y (from an equation of state) and y, from a liquid-phase activity coefficient model. Some examples will be given later. At moderate pressures, the vapor phase becomes ideal, as discussed previously, and fj = 1. For... 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

Example 6.4 Following Example 6.2, the reactor temperature will be set to 700 K. Examine the effect of increasing the reactor pressure by calculating the equilibrium conversion of hydrogen at 1 bar, 10 bar, 100 bar and 300 bar. Assume initially ideal gas behavior. 

When the VOC-laden gas stream contains a mixture of VOCs, then the calculations must be performed using the methods described for single-stage equilibrium calculations in Chapter 4. The temperature at the exit of the condenser must be assumed, together with a condenser pressure. The vapor fraction is then solved by trial and error using the methods described in Chapter 4, and the complete mass balance can be determined on the basis of the assumption of equilibrium. 

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. 

The solution requires the concentration of the heptane and toluene in the vapor phase. Assuming that the composition of the liquid does not change as it evaporates (the quantity is large), the vapor composition is computed using standard vapor-liquid equilibrium calculations. Assuming that Raoult s and Dalton s laws apply to this system under these conditions, the vapor composition is determined directly from the saturation vapor pressures of the pure components. Himmelblau6 provided the following data at the specified temperature ... 

Advances continue in the treatment of detonation mixtures that include explicit polar and ionic contributions. The new formalism places on a solid footing the modeling of polar species, opens the possibility of realistic multiple fluid phase chemical equilibrium calculations (polar—nonpolar phase segregation), extends the validity domain of the EXP6 library,40 and opens the possibility of applications in a wider regime of pressures and temperatures. 

Mixture property Define the model to be used for liquid activity coefficient calculation, specify the binary mixture (composition, temperature, pressure), select the solute to be extracted, the type of phase equilibrium calculation (VLE or LLE) and finally, specify desired solvent performance related properties (solvent power, selectivity, etc.)... 

Although one can probably find exceptions, most equilibrium calculations involving flue gas slurries are performed with temperature as a known variable. With temperature known, the numerical values of the appropriate equilibrium constants can be immediately calculated. The remaining unknown variables to be determined are the activities, activity coefficients, molalities, and the gas phase partial pressures. The equations used to determine these variables are formulated from among the equilibrium expressions presented in Table 1, the expressions for the activity coefficients, ionic strength, material balance expressions, and the electroneutrality balance. Although there are occasionally exceptions, the solution sequence generally is an iterative or cyclic sequence. 

Most of the equations of state are pressure explicit, and Eq. (10.6) can be used for equilibrium calculations. As the integration is made from F to oo the EOS has to be valid in the density range from zero to the actual density. 

PAH PAN PBN PCT PES PHREEQC PIC PM PMATCHC PM-10 PM-2.5 PRB PUREX PW PWR PZC Polycyclic aromatic hydrocarbon Peroxyacetylnitrate Peroxybenzoylnitrate Product consistency test Plasma emission spectroscopy pH redox equilibrium calculations (computer program) Product of incomplete combustion Particulate matter Program to manage thermochemical data, written in C++ Particulate matter with an aerodynamic diameter <10 p,m Particulate matter with an aerodynamic diameter <2.5 p,m Powder River Basin Pu-U-recovery-extraction Purex waste Pressurized water reactor Point of zero charge... 

In the preceding discussion we considered equilibrium void stability however, actual processing conditions involve changing temperature and pressure with time. Whereas equilibrium calculations provide bounds on void growth, it is the time-dependent growth process that is most important from a product quality viewpoint. 

A quite different approach to the detonation product state has been to treat it as solidlike. Jones and Miller6 performed equilibrium calculations on TNT with this idea in mind. They used an equation in which the volume was a virial expansion in the pressure. Other solidlike equations are cited in Ref. 2, but these have mostly been used for computing the state parameters with an assumed product state. The modified Kistia-kowsky-Wilson equation of interest to us liere appears to be one of several possible compromises between the hard-sphere molecule approach and the solid state approach. 

These equations are applied for the determination of the equilibrium solid loading qt for the specified inlet concentration. To do this, the partial pressure of toluene at inlet conditions is needed. This pressure is calculated by using the ideal gas law ... 

In the time scale covered by the calculation, the 137I in the particles results from in situ decay of 137Te. The low surface concentration is associated with the high volatility of iodine in an oxidizing atmosphere. In this same figure the 137Xe curves reflect their parent 137I curves to some extent. The surface concentration of this element probably does not represent a true vapor pressure equilibrium but just the thermody-... 

It is not known in advance what the real physical state of a system of given composition at given pressure P and temperature T is. For this reason a bubble pressure (Pbubi) calculation of a hypothetical liquid of composition z, at the temperature T, and a dew pressure (Pdew) calculation of a hypothetical vapour of composition z, at the temperature T, is performed. Only for pressures between Pdew and Pbubi is the system an equilibrium mixture of vapour and liquid, and flash calulations make sense. 

Chapter Two deals with the basic concepts of high-pressure thermodynamic and phase equilibrium calculations. Experimental methods and theoretical modelling are described briefly in order to give both a comprehensive view of the problems, and suggestions and references to more detailed treatments. 

A convergence pressure of 10,000 psia should have been used for this mixture. See Figure 14-4. Obtain K-factors at pk = 10,000 and repeat the gas-liquid equilibrium calculation. 

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