Machine calculation

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

Cooley J W and ] W Tukey 1965. An Algorithm for the Machine Calculation of Complex Fourier Series Aiathemalics of Computation 19 297-301. 

J. W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput., 19 (1965) 297-301. 

McDonald, I. R. Singer, K., Machine calculation of thermodynamic properties of a simple fluid at supercritical temperatures, J. Chem. Phys. 1967, 47, 4766-4772... 

Ibid, UCRL-7797 (1964) (Description of sphere test) 6) M.L. Wilkins, "The Use of One- and Two-Dimensional Hydrodynamic Machine Calculations in High-Explosive Research", 4thONRSympDeton (1965), pp 519-26 7) J.M. Kury et al, "Metal Acceleration by... 

See Note). The relationships among certain parameters were machine-calculated and plotted (Ref 32)... 

Another evident trend is the application of mechanical computers for the rapid calculation of distillation problems. At least two papers have already appeared, one by Weil (57) on the use of an electronic computer, and the other by Rose and Williams (47) on the application of the IBM card punch technique. If a short rigorous method is not forthcoming, machine calculations may be the answer to accurate design of fractionating equipment. 

The flame temperature, Tf, can be evaluated by thermodynamic methods. A simplified treatment of a method is given in detail by Lichty (40). Excellent extensions of this method are available in a series of machine calculated reports (50). 

The identification of probable combustion product species for all but the most exotic propellant combinations has been well established. In conjunction with machine calculations which consider large numbers of possible product species, the likelihood of Introducing error into equilibrium performance calculations through omission of significant product species is small. With the completion of the JANAF Thermochemical Tables (18) which currently contain some 800 possible species and continues to be expanded, the task of taking all possible significant product species into consideration is well defined, although not necessarily simple. The problem of identification of probable species has been reduced to the identification of possible species which for most propellant combination is a routine task. 

Choose progressively smaller values of Ax and observe the behavior of the solution. If the problem has been correctly formulated and solved, the nodal temperatures should converge as Ax becomes smaller. It should be noted that computational round-off errors increase with an increase in the number of nodes because of the increased number of machine calculations. This is why one needs to observe the convergence of the solution. 

For more detailed calculations (e.g. if the presence of free carbon must be assumed), the reader is referred to M. A. Cook The Science of High Explosives, Chapman Hall, London 1958 and, by the same author The Science of Industrial Explosives, copyright 1974 by IR-ECO CHEMICALS, Salt Lake City, USA. They contain basic data on heat capacities and equilibria constants concerned, as well as computing programs for hand and machine calculations. 

Chum HL, Ratcliff, Schroeder HA, Sopher DW (1984) Electrochemistry of biomass-derived materials Characterization, fractionation, and reductive electrolysis of ethanol extracted explosively depressurized aspen lignin J Wood Chem Technol 4 505-532 Compton DAC, Young JR, Kollar RG, Mooney JR, Grasselh JG (1987) In McClure GL (ed) Computerized quantitative infrared analysis ASTM, Philadelphia, 36-57 Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series Math Comput 19 297-301... 

The problem of cross-section calculation for various inelastic collisions is mathematically equivalent to the solution of a set (in principle, infinite) of coupled wave equations for nuclear motion [1]. Machine calculations have been done recently to obtain information about nonadiabatic coupling in some representative processes. Although very successful, these calculations do not make it easy to interpret particular transitions in terms of a particular interaction. It is here that the relatively simple models of nonadiabatic coupling still play an important part in the detailed interpretation of a mechanism, thus contributing to our understanding of the dynamic interaction between electrons and nuclei in a collision complex. 

If we wished to make full, accurate machine calculations we would never need to make this distinction we could simply look at the results of the full calculation to check for the presence of an energy gap. Instead, our methods are designed to result in intuitive understanding and approximate calculations of properties, which will allow us to guess trends without calculations in some cases, and which will allow us to treat complicated compounds that would otherwise be intractable by full, accurate calculation in other cases. 

Table I, in the column headed HSE-VW, shows the results of using Equations 2 through 6 to define the diameters with the HSE method to calculate the properties of an equimolar mixture of LJ fluids. The reference is a pure LJ fluid. Other columns show comparison with the machine-calculated results of Singer and Singer (8) in column MC. The van der Waals (VDW) one-fluid theory (9) and the VDW two-fluid theory (10) are in columns VDW-1 and VDW-2. The GHBL column gives the Grundke, Henderson, Barker, Leonard (GHBL) (11) pertu-bation theory results with each diameter determined by Equation 3.

The GHBL perturbation procedure is remarkably accurate and the HSE-VW method is only slightly better in its overall agreement with the machine-calculated results. This comparison is not completely valid in that the conformal solution theory uses pure component data while in the perturbation theory each term is calculated from molecular parameters. 

Hard sphere theories can be compared in a special sense with "experiment . This is not because any real substance possesses a fluid phase which over its whole existence region could be adequately described by this simple model. The "experimental data, rather, are the result of extensive machine calculations of the equation of state of large collections of hard spheres undertaken by Alder and Wainwright and Wood, Parker, and Jacobson, among others. 

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