Computers, digital

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

Because of its severe approximations, in using the Huckel method (1932) one ignores most of the real problems of molecular orbital theory. This is not because Huckel, a first-rate mathematician, did not see them clearly they were simply beyond the power of primitive mechanical calculators of his day. Huckel theory provided the foundation and stimulus for a generation s research, most notably in organic chemistry. Then, about 1960, digital computers became widely available to the scientific community. 

In this section, the conceptual framework of molecular orbital theory is developed. Applications are presented and problems are given and solved within qualitative and semi-empirical models of electronic structure. Ab Initio approaches to these same matters, whose solutions require the use of digital computers, are treated later in Section 6. Semi-empirical methods, most of which also require access to a computer, are treated in this section and in Appendix F. 

P. Baer and J. Frankie, Simulation of Interior Ballistic Peformance of Guns by Digital Computer Program, Rpt. 1183, BRL, Aberdeen, Md., 1962. 

D. J. WoUons, Introduction to Digital Computer Design, McGraw-HiU Book Co., Inc., London, 1972. 

C. L. Smith, Digital Computer Process Control, International Textbook Co., Scranton, Pa., 1964. 

D. Gnes, Compiler Construction for Digital Computers John Wiley Sons, Inc., New York, 1971. 

R. J. Kelly and R. E. Wheeling, A Digital Computer Program fior Optimifing Non-linear Functions, Mobil Oil Corp., Research Dept., Central Research... 

Vemuri, V, and W. Karplus. Digital Computer Treatment of Partial Differential Equations. Prentice Hall, Englewood Cliffs, NJ (1981). 

Method of Continuity (Homotopy) In the case of n equations in n unknowns, when n is large, determining the approximate solution may involve considerable eftoid. In such a case the method of continuity is admirably suited for use on digital computers. It consists basically of the introduction of an extra variable into the n equations... 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

The above FF controller can be implemented using analog elements or more commonly by a digital computer. Figure 8-33 compares typical responses for PID FB control, steady-state FF control (.s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, and Tl in the field to improve the performance oTthe FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting. s = 0 in Gj. s). This might be appropriate if there is uncertainty in the dynamic models for Gl and Gp. 

The subject of adaptive control is one of current interest. New algorithms are presently under development, but these need to be field-tested before industrial acceptance can be expected. It is clear, however, that digital computers will be required for implementation of self-adaptive controllers due to their complexity. An adaptive controller is inherently nonlinear and therefore more complicated than the conventional PID controller. 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

The widespread availabihty and utihzation of digital computers for distillation calculations have given impetus to the development of analytical expressions for iregression equation and accompanying regression coefficients that represent the DePriester charts of Fig. 13-14. Regression equations and coefficients for various versions of the GPA convergence-pressure charts are available from the GPA. 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

Although this procedure requires laborious calculations because of the number of samples normally needed, these computations and the determination of the best straight-hne fit to the data are readily programmed for digital computers. 

Mathematical modeling, using digital computers, aids in performing a systems-type analysis for either the entire system or parts of it. By means of integer or linear-programming techniques, optimum systems can be identified. The dynamic performance of these can then be determined by simulation techniques. 

There are few analytic solutions to the governing equations for interesting problems. The conservation equations are typically solved approximately on digital computers. It is assumed that the sound speeds are real and the system... 

Edmister, W. C., Absorption and Stripping-factor Functions for Distillation Calculation by Manual- and Digital-computer Methods, A.I.Ch.E. Journal, June 1957. 

The interpretation of the results and the quality of the design from the pulsation study, whether performed on the analog simulator or with digital computer simulation, depends quite heavily on the experience and skill of the analyst performing the study, A purchaser of a compressor system who may be a novice at this type of analysis should give serious consideration to using the services of a competent consultant. 

These results are the same as with the power series method, but difference equations are more suited to digital computation. 

The state-space approach is a generalized time-domain method for modelling, analysing and designing a wide range of control systems and is particularly well suited to digital computational techniques. The approach can deal with... 

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