Critical point estimation

Above critical point, estimated or extrapolated. At saturation pressure, 288.72 K. 

The critical points of alkali halides such as NaCl are located at temperatures above 3000 K [48-50], while experimental data do not extend beyond 2000 K [48]. Critical point estimates are, however, often needed for comparative purposes. Matching results of molecular dynamics (MD) simulations to the available experimental data, Guissani and Guillot [51] developed an equation of state (EOS) for NaCl which predicts Tc = 3300 K and a critical mass density of dc = 0.18g cm-3. Pitzer [13] recommended a lower critical density, but, as discussed later, some MC data used in his assessment are questionable. 

The compilations of CRC (1-2), Daubert and Danner (3), Dechema (15), TRC (13-14), Vargaftik (18), and Yaws (19-36) were used extensively for critical properties. Estimates of critical temperature, pressure, and volume were primarily based on the Joback method (10-12) and proprietary techniques of the author. Critical density was determined from dividing molecular weight by critical volume. Critical compressibility factor was ascertained from application of the gas law at the critical point. Estimates for acentric factor were primarily made by using the Antoine equation for vapor pressure (11-12). 

Figure 1.19. Liquid-vapor coexistence line of SPC water and OPLS methanol (solid and dashed lines) following from the RISM/KH theory versus the simulation data (open circles and squares, respectively) and critical point estimates (closed symbols). Logarithmic and linear scales are used to resolve the density in gas and liquid state.

Our concepts of petroleum reserves and resources and their measurements are changing to reflect the uncertainty associated with these terms. Petroleum reseiwes have been largely calculated deterministically (i.e. single point estimates with the assumption of certainty). In the past decade, reseiwe and resource calculations have incorporated uncertainty into their estimates using probabilistic methodologies. One of the questions now being addressed are such as how certain arc you that the rcsciwcs you estimate arc the actual reseiwes and what is the range of uncertainty associated with that estimate New techniques arc required to address the critical question of how much petroleum we have and under what conditions it can be developed. 

Applying MD to systems of biochemical interest, such as proteins or DNA in solution, one has to deal with several thousands of atoms. Models for systems with long spatial correlations, such as liquid crystals, micelles, or any system near a phase transition or critical point, also must involve a large number of atoms. Some of these systems, including synthetic polymers, obey certain scaling laws that allow the estimation of the behaviour of a large system by extrapolation. Unfortunately, proteins are very precise structures that evade such simplifications. So let us take 10,000 atoms as a reasonable size for a realistic complex system. 

A question of practical interest is the amount of electrolyte adsorbed into nanostructures and how this depends on various surface and solution parameters. The equilibrium concentration of ions inside porous structures will affect the applications, such as ion exchange resins and membranes, containment of nuclear wastes [67], and battery materials [68]. Experimental studies of electrosorption studies on a single planar electrode were reported [69]. Studies on porous structures are difficult, since most structures are ill defined with a wide distribution of pore sizes and surface charges. Only rough estimates of the average number of fixed charges and pore sizes were reported [70-73]. Molecular simulations of nonelectrolyte adsorption into nanopores were widely reported [58]. The confinement effect can lead to abnormalities of lowered critical points and compressed two-phase envelope [74]. 

PARAMETER ESTIMATION USING BINARY CRITICAL POINT DATA... 

Prior work on the use of critical point data to estimate binary interaction parameters employed the minimization of a summation of squared differences between experimental and calculated critical temperature and/or pressure (Equation 14.39). During that minimization the EoS uses the current parameter estimates in order to compute the critical pressure and/or the critical temperature. However, the initial estimates are often away from the optimum and as a consequence, such iterative computations are difficult to converge and the overall computational requirements are significant. 

Five critical points for the methane-n-hexane system in the temperature range of 198 to 273 K measured by Lin et al. (1977) are available. By employing the Trebble-Bishnoi EoS in our critical point regression least squares estimation method, the parameter set (k , kb) was found to be the optimal one. Convergence from an initial guess of (ka,kb=0.001, -0.001) was achieved in six iterations. The estimated values are given in Table 14.8. 

Englezos, P., G. Bygrave, and N. Kalogerakis, "Interaction Parameter Estimation in Cubic Equations of State Using Binary Phase Equilibrium Critical Point Data", Ind. Eng Chem. Res.31(5), 1613-1618 (1998). 

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