Critical temperature calculation

Table 7.1. Ratio of the critical temperature calculated for an f.c.c. Ising lattice by different methods, assuming only constant nearest-nei bour interactions and using MC as the standard (after Kikuchi 1977)...

Fig. 15-2. Values of molar volume and chemical potential at a temperature below critical temperature calculated with a two-constant equation of state showing the "van der Waafs loop. ...

TABLE 16. The critical temperatures calculated from the Frank-Kamenetskii equation for a diverse set of explosives. 

To use Equation (10b), we require virial coefficients which depend on temperature. As discussed in Appendix A, these coefficients are calculated using the correlation of Hayden and O Connell (1975). The required input parameters are, for each component critical temperature T, critical pressure P, ... 

The critical temperature of methane is 191°K. At 25°C, therefore, the reduced temperature is 1.56. If the dividing line is taken at T/T = 1.8, methane should be considered condensable at temperatures below (about) 70°C and noncondensable at higher temperatures. However, in process design calculations, it is often inconvenient to switch from one method of normalization to the other. In this monograph, since we consider only equilibria at low or moderate pressures in the region 200-600°K, we elect to consider methane as a noncondensable component. 

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. 

There is no reason why the distortion parameter should not contain an entropy as well as an energy component, and one may therefore write 0 = 0q-sT. The entropy of adsorption, relative to bulk liquid, becomes A5fi = sexp(-ca). A critical temperature is now implied, Tc = 0o/s, at which the contact angle goes to zero [151]. For example, Tc was calculated to be 174°C by fitting adsorption and contact angle data for the -octane-PTFE system. 

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

Figure A2.5.6 shows a series of typical p, Fisothemis calculated using equation (A2.5.1). (The temperature, pressure and volume are in reduced units to be explained below.) At sufficiently high temperatures the pressure decreases monotonically with increasing volume, but below a critical temperature the isothemi shows a maximum and a minimum.

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

It is, however, possible to calculate the tensile strength of a liquid by extrapolation of an equation of state for the fluid into the metastable region of negative pressure. Burgess and Everett in their comprehensive test of the tensile strength hypothesis, plot the theoretical curves of T /T against zjp, calculated from the equations of state of van der Waals, Guggenheim, and Berthelot (Fig. 3.24) (7], and are the critical temperature and critical... 

Fig. 3.24 Test of the tensile strength hysteresis of hysteresis (Everett and Burgess ). TjT, is plotted against — Tq/Po where is the critical temperature and p.. the critical pressure, of the bulk adsorptive Tq is the tensile strength calculated from the lower closure point of the hysteresis loop. C), benzene O. xenon , 2-2 dimethyl benzene . nitrogen , 2,2,4-trimethylpentane , carbon dioxide 4 n-hexane. The lowest line was calculated from the van der Waals equation, the middle line from the van der Waals equation as modified by Guggenheim, and the upper line from the Berthelot equation. (Courtesy Everett.)...

By combining Eqs. (8.42), (8.49), and (8.60), show that Vi°(52 - 5i) = (l/2)RTj., where T. is the critical temperature for phase separation. For polystyrene with M = 3 X 10, Shultz and Floryf observed T. values of 68 and 84°C, respectively, for cyclohexanone and cyclohexanol. Values of Vi° for these solvents are abut 108 and 106 cm mol", respectively, and 5i values are listed in Table 8.2. Use each of these T. values to form separate estimates of 62 for polystyrene and compare the calculated values with each other and with the value for 62 from Table 8.2. Briefly comment on the agreement or lack thereof for the calculated and accepted 5 s in terms of the assumptions inherent in this method. Criticize or defend the following proposition for systems where use of the above relationship is justified Polymer will be miscible in all proportions in low molecular weight solvents from which they differ in 5 value by about 3 or less. 

Basic pure component constants required to characterize components or mixtures for calculation of other properties include the melting point, normal boiling point, critical temperature, critical pressure, critical volume, critical compressibihty factor, acentric factor, and several other characterization properties. This section details for each propeidy the method of calculation for an accurate technique of prediction for each category of compound, and it references other accurate techniques for which space is not available for inclusion. 

Critical Temperature The critical temperature of a compound is the temperature above which a hquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. 

For both hydrocarbons and nonhydrocarbon organic defined mixtures, the method of Li " is used with a relatively simple volumetric average mixing rule as shown in Eq. (2-5) to calculate the true critical temperature. 

Critical Pressure The critical pressure of a compound is the vapor pressure of the compound at the critical temperature. Below the critical temperature, any compound above its vapor pressure will be a liquid. The critical pressure is required for calculations discussed in the part of the section on critical temperature. 

Be is the critical pressure, MPa. Values of Ap from Table 2-383 are summed for each part of the molecule to yield X Ap. Calculation of the Platt number is discussed under Critical Temperature. Errors in average 0.07 MPa and are less reliable for compounds with 12 or more carbon atoms. 

Both equations (2-28) and (2-29) are also extrapolatable above the critical temperature where necessary for thermodynamic calculations. 

The method of Lee and KesleF is the preferred method if the critical temperature and the critical pressure of the hydrocarbon is known or can be reasonably predicted by the methods of the first section. The corresponding states method is shown in equation (2-31) with the simple fluid and correction terms to be calculated from equations (2-32) and (2-33), respectively, for any Tr-... 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

The gas pseudo critical pressures and temperatures can be approximated from Figure 2-16 or they can be calculated as weighted averages of the critical temperatures and pressures of the various components on a... 

The grand-thermodynamical potential is, like the temperature, calculated in units of b. Macroscopically, b is related to the critical temperature of the oil-water separation by kT = 3(1 — p )b. The coupling constants of O2 re... 

The application of information in Figure 6.19 requires some explanation. The decision as to which calculation method to choose should be based upon the phase of the vessel s contents, its boiling point at ambient pressure T its critical temperature Tf, and its actual temperature T. For the purpose of selecting a calculation method, three different phases can be distinguished liquid, vapor or nonideal gas, and ideal gas. Should more than be performed separately for each phase, and the... 

Step 3. Calculate the weight average critical temperature and critical pressure for the remaining heavier components to form a pseudo binary system. (A shortcut approach good for most hydrocarbon systems is to calculate the weight average T only.)... 

Figure 12-15 is a compressibility chart for natural gas based on pseudo-reduced pressure and temperature. The reduced pressure is the ratio of the absolute operating pressure to the critical pressure, P and the reduced temperature is the ratio of the absolute operating temperature to the critical temperature, T, for a pure gas or vapor. The pseudo value is the reduced value for a mixture calculated as the sum of the mol percentages of the reduced values of the pure constituents. 

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