System 34 - sizes

Both (E) and Cy are extensive quantities and proportional to N or the system size. The root mean square fluctuation m energy is therefore proportional to A7 -, and the relative fluctuation in energy is... 

Since Cy and E are bodi extensive properties (ocN), the root-inean-square energy fluctuations are smaller, by a factor 1/y, than typical average energies E. As the system size increases, the relative magnitude of... 

Gray C G, Sainger Y S, Joslin C G, Cummings P T and Goldman S 1986 Computer simulation of dipolar fluids. Dependence of the dielectric constant on system size a comparative study of Ewald sum and reaction field approaches J. Chem. Phys. 85 1502-4... 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

This section presents some of the simulation results obtained by simulating systems of sizes 4000, 6912, 10976, 16384 and 32000 atoms on the IBM-SP/2. The simulations were performed on 4, 8 and 16 processors, respectively. Although, the simulated system size and the number of processors can be scaled easily, this section does not show all results. 

Table 1 describes the timing results (in seconds) for a system of 4000 atoms on 4, 8 and 16 nodes. The average CPU seconds for 10 time steps per processor is calculated. In the case of the force-stripped row and force-row interleaving algorithms the CPU time is reduced by half each time the number of processors is doubled. This indicates a perfect speedup and efficiency as described in Table 2. Tables 3, refibm table3 and 5 describe the timing results, speedups and efficiencies for larger systems. In particular. Table 4 shows the scaling in the CPU time with increase in the system size. These results are very close to predicted theoretical results.

Siepmann JI and IR McDonald 1993a. Domain Formation and System-size Dependence in Simulatio of Self-assembled Monolayers. Langmuir 9 2351-2355. 

Whether an adequate sampling of phase space is obtained Whether the system size is large enough to represent the bulk material Whether the errors in calculation have been estimated correctly... 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

The systems of interest in chemical technology are usually comprised of fluids not appreciably influenced by surface, gravitational, electrical, or magnetic effects. For such homogeneous fluids, molar or specific volume, V, is observed to be a function of temperature, T, pressure, P, and composition. This observation leads to the basic postulate that macroscopic properties of homogeneous PPIT systems at internal equiUbrium can be expressed as functions of temperature, pressure, and composition only. Thus the internal energy and the entropy are functions of temperature, pressure, and composition. These molar or unit mass properties, represented by the symbols U, and S, are independent of system size and are intensive. Total system properties, J and S do depend on system size and are extensive. Thus, if the system contains n moles of fluid, = nAf, where Af is a molar property. Temperature... 

The fluxes in hoUow-fiber membranes used in seawater desalination are 20—30-fold smaller, but the overall RO system size does not increase because the hoUow-fiber membranes have a much larger surface area per module unit volume. In use with seawater, their RR is about 12—17.5% and the salt rejection ratio is up to 99.5%. 

Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinefics) are frequently employed in optimization apphcations. These models are conceptually attractive because a gener model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input-output data without any physiochemical analysis of the process. For... 

II. MOLECULAR DYNAMICS SIMULATIONS OF MEMBRANES A. System Size and Construction... 

Calculation of the energies and forces due to the long-range Coulomb interactions between charged atoms is a major problem in simulations of biological molecules (see Chapter 5). In an isolated system the number of these interactions is proportional to N-, where N is the number of charged atoms, and the evaluation of the electrostatic interactions quickly becomes intractable as the system size is increased. Moreover, when periodic... 

We shall first consider the design of elevated flare systems. Sizing of flare systems is a function of maximum allowable back pressure on safety valves and other sources of release into the emergency systems. 

As has been previously mentioned, the minimum TAC can be identified by iteratively varying e. Since the inlet and outlet compositions of the rich stream as well as the inlet composition of the MSA are fixed, one can vary e at the rich end of the exchanger (and consequently the outlet composition of the lean stream) to minimize the TAC of the system. In order to demonstrate this opdmization procedure, let us first select a value of e at the rich end of the exchanger equal to 1.5 X 10 and evaluate the system size and cost for this value. 

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