Calculating results

The accuracy of our calculations is strongly dependent on the accuracy of the experimental data used to obtain the necessary parameters. While we cannot make any general quantitative statement about the accuracy of our calculations for multicomponent vapor-liquid equilibria, our experience leads us to believe that the calculated results for ternary or quarternary mixtures have an accuracy only slightly less than that of the binary data upon which the calculations are based. For multicomponent liquid-liquid equilibria, the accuracy of prediction is dependent not only upon the accuracy of the binary data, but also on the method used to obtain binary parameters. While there are always exceptions, in typical cases the technique used for binary-data reduction is of some, but not major, importance for vapor-liquid equilibria. However, for liquid-liquid equilibria, the method of data reduction plays a crucial role, as discussed in Chapters 4 and 6. 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Figure 3 presents results for acetic acid(1)-water(2) at 1 atm. In this case deviations from ideality are important for the vapor phase as well as the liquid phase. For the vapor phase, calculations are based on the chemical theory of vapor-phase imperfections, as discussed in Chapter 3. Calculated results are in good agreement with similar calculations reported by Lemlich et al. (1957). ... 

There is a significant difference between the results shown in Figure 2 and calculated results given in Brit. Chem. Eng. Proc. Tech. 16 1036 (1971). We believe the latter to be in error. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Appendix C-6 gives parameters for all the condensable binary systems we have here investigated literature references are also given for experimental data. Parameters given are for each set of data analyzed they often reflect in temperature (or pressure) range, number of data points, and experimental accuracy. Best calculated results are usually obtained when the parameters are obtained from experimental data at conditions of temperature, pressure, and composition close to those where the calculations are performed. However, sometimes, if the experimental data at these conditions are of low quality, better calculated results may be obtained with parameters obtained from good experimental data measured at other conditions. 

Cutting petroleum into fractions is done by the method illustrated in Figure 4.2. The petroleum fractions should correspond to the characteristics described in article 4.1.2. If certain characteristics are estimated, it is mandatory to compare the calculational results with the properties of the cuts and to readjust the estimated characteristics. 

The constants k- enable the improved representation of binary equilibria and should be carefully determined starting from experimental results. The API Technical Data Book has published the values of constants k j for a number of binary systems. The use of these binary interaction coefficients is necessary for obtaining accurate calculation results for mixtures containing light components such as ... 

The development of Remote Field Eddy Current probes requires experience and expensive experiments. The numerical simulation of electromagnetic fields can be used not only for a better understanding of the Remote Field effect but also for the probe lay out. Geometrical parameters of the prohe can be derived from calculation results as well as inspection parameters. An important requirement for a realistic prediction of the probe performance is the consideration of material properties of the tube for which the probe is designed. The experimental determination of magnetization curves is necessary and can be satisfactory done with a simple experimental setup. 

Regardless of how many single-particle wavefiinctions i are available, this number is overwhelmed by the number of n-particle wavefiinctions ( ) (Slater detenninants) that can be constructed from them. For example, if a two-electron system is treated within the Flartree-Fock approximation using 100 basis fiinctions, both of the electrons can be assigned to any of the % obtained m the calculation, resulting in 10,000 two-electron wavefimctions. For water, which has ten electrons, the number of electronic wavefiinctions with equal numbers of a and p spin electrons that can be constructed from 100 single-particle wavefimctions is roughly... 

This set of calculations results in an output file containing all of the energies required to solve for the pH-dependent properties of interest. The form and content of the file are as follows ... 

Any set of one-eleelrori ftinctions can be a basis set in the IjCAO approximation. However, a well-defined basis set will predict electron ic properties using fewer leriii s th an a poorly-defiri ed basis set. So, choosin g a proper basis set in ah inilio calculation s is critical to the rcliabililv and accuracy of the calculated results. 

In making certain mathematical approximations to the Schrodinger equation, we can equate derived terms directly to experiment and replace dilTiciilL-to-calculate mathematical expressions with experimental values. In other situation s, we introduce a parameter for a mathematical expression and derive values for that parameter by fitting the results of globally calculated results to experiment. Quantum chemistry has developed two groups of researchers ... 

The calculated result as we have found it by this dissection of the G2 method agrees with the final result of the G2 script... 

Experience has shown that is better to obtain basis sets in electronic form than paper form since even slight errors in transposition will affect the calculation results. Some basis sets are included with most computer programs that require them. There is also a form page on the Web that allows a user to choose a basis and specify a format consistent with the input of several popular computational chemistry programs at http //www.emsl.pnl.gov 2080/forms/basisform.htm. The basis set is then sent to the user in the form of an e-mail message. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

You can also plot the electrostatic potential, the total charge density, or the total spin density determined during a semi-empirical or ab initio calculation. This information is useful in determining reactivity and correlating calculational results with experimental data. These examples illustrate uses of these plots ... 

You can obtain detailed results of calculations by using Start Log on the File menu. The amount of calculational results is controlled by the QuantumPrintLevel. The QuantumPrintLevel can be changed in the CHEM.INI file before you run HyperChem or by a script command after you are running HyperChem. For more details of the log file and information saved in the log file, see the HyperChem Reference marwiaX. 

Foster s neglect of the role of the adsorbed film was unavoidable in the then absence of any reliable information as to the thickness of the film. It is now known that in fact the effect of the film on the calculated result is far from negligible, as will be demonstrated shortly. Since, however, all the methods of calculating pore size distributions involve a decision as to the upper limit of the range to be studied, this question needs to be discussed first. In effect one has to choose a point corresponding to point G in Fig. 3.1, where the mesopores are deemed to be full up. If the isotherm takes the course GH there are no further cores to be considered in any case but if it swings upwards as at GH, the isotherm is usually so steep that the Kelvin-type approach becomes too inaccurate (cf. p. 114) to be useful. 

Propagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are sr, sa, sb) and sq. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 

For both 1 = 4 and 1 = 8, the polymer is favored over the ring to a far greater degree than for 1 = 6. In the latter case the difference between AH° and T AS° is so small that there is probably no justification for retaining as many figures as shown in the calculated results. 

Under 0 conditions occurring near room temperature, [r ] = 0.83 dl g for a polystyrene sample of molecular weight 10. f Use this information to evaluate tg and for polystyrene under these conditions. For polystyrene in ethylcyclohexane, 0 = 70°C and the corresponding calculation shows that (tQ /M) = 0.071 nm. Based on these two calculated results, criticize or defend the following proposition The discrepancy in calculated (rQ /M) values must arise from the uncertainty in T>, since this ratio should be a constant for polystyrene, independent of the nature of the solvent. 

Nonisothermal Gas Absorption. The computation of nonisothermal gas absorption processes is difficult because of all the interactions involved as described for packed columns. A computer is normally required for the enormous number of plate calculations necessary to estabUsh the correct concentration and temperature profiles through the tower. Suitable algorithms have been developed (46,105) and nonisothermal gas absorption in plate columns has been studied experimentally and the measured profiles compared to the calculated results (47,106). Figure 27 shows a typical Hquid temperature profile observed in an adiabatic bubble plate absorber (107). The close agreement between the calculated and observed profiles was obtained without adjusting parameters. The plate efficiencies required for the calculations were measured independendy on a single exact copy of the bubble cap plates installed in the five-tray absorber. 

Relevant hydrological fundamentals are utilized (21) to take account of the complex interaction of physical and chemical processes involving sod or rock, water, and contaminant. Attention is paid to uncertainties in calculated results. 

Analysis of this type of classifier (15) suggests that the sharpness iadex is between 0.5 and 0.6, consistent with calculated results, because the degree of turbulence can be high ia these devices (16). A DSF Dorr classifier (1.8 m x 7 m), operating at 19 strokes per minute and having a weir depth of 100 cm and a slope of 19.4 cm/m, produced a cut size equal to 240 p.m, a sharpness iadex of 0.5, and an apparent bypass of approximately 26% when the water spht was 26% (15). 

Belt Presses Belt presses were fiiUy described in the section on filtration. The description here is intended to cover only the parts and designs that apply expression pressure by a mechanism in adchtion to the normal compression obtained from tensioning the belts and pulling them over rollers of smaller and smaller diameters. The tension on the belt produces a squeezing pressure on the filter cake proportional to the diameter of the rollers. Normally, that static pressure is calculated as P = 2T/D, where P is the pressure (psi), T is the tension on the belts (Ib/hnear in), and D is the roller diameter. This calculation results in values about one-half as great as the measured values because it ignores pressure created by drive torque and some other forces [Laros, Advances in Filtration and Separation Technology, 7 (System Approach to Separation and Filtration Process Equipment), pp. 505-510 (1993)]. 

L.N. Kmetyk and P. Yarrington, Cavity Dimensions for High Velocity Penetration Events—A Comparison of Calculational Results with Data, SAND88-2693, Sandia National Laboratories, Albuquerque, NM, 1989. 

Figure 3.6.1 (Berty 1979) is a Sankey (1898) diagram, used in power engineering, where the bandwidth is proportional (here qualitatively only) to the flowing masses. This illustrates the calculation results for a rather extreme case of an NOx reduction problem. The case is extreme because the catalyst particle has a dp=0.2mm, i.e., 200 microns. Flow resistance is very high, therefore an L=1 mm deep bend is used only. Per pass concentration drop is still high, Ci-C=1.2ppm, or Dai=0.11. This was tolerated in this case, since it is between 11.2 and 10.00 ppm concentration, and nothing better could have been achieved.

---