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]. [Pg.482]

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. [Pg.266]

At this stage the fonnulated Galerkin-weighted residual Equation (2.52) contains second-order derivatives. Therefore elements cannot generate an acceptable solution for this equation (using C elements the first derivative of... [Pg.45]

Taylor, C. and Hood, P., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73-100. [Pg.69]

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... [Pg.85]

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... [Pg.131]

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... [Pg.133]

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... [Pg.143]

SOLVER Assembles elemental stiffness equations into a banded global matrix, imposes boundary conditions and solves the set of banded equations using the LU decomposition method (Gerald and Wheatley, 1984). SOLVER calls the following 4 subroutines. ... [Pg.212]

In these equations use has been made of the fact that gn and fk are functions of different sets of coordinates that S and R, respectively, act upon. [Pg.569]

Show by a senes of equations using any necessary organic or inorganic reagents how acetic acid can be converted to each of the following compounds... [Pg.826]

ZINDO/S is different from ZINDO/I because they use different algorithms in computing the Coulomb integrals. Hence the two equations used in the mixed model in ZINDO/1 are also employed... [Pg.296]

Ladder diagrams can also be used to evaluate equilibrium reactions in redox systems. Figure 6.9 shows a typical ladder diagram for two half-reactions in which the scale is the electrochemical potential, E. Areas of predominance are defined by the Nernst equation. Using the Fe +/Fe + half-reaction as an example, we write... [Pg.155]

You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an-... [Pg.176]

We conclude this section with an example illustrating the application of the WLF equation using the data in Fig. 4.17. [Pg.261]

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). [Pg.159]

Chemical Equations Chemical changes are discussed with the aid of the equations used to treat equiUbrium, ie, the reaction of reactants B, C, and so on, to produce products P, Q, and so forth. [Pg.507]

Indirect methods of estimating sorption have been used when actual measurement of sorption isotherm is impossible (44). For instance, sorption coefficients have been estimated from soil organic carbon and a specific surface of soil, and from semiempidcal equations using pesticide properties. [Pg.222]

These four equations, using the appropriate boundary conditions, can be solved to give current and potential distributions, and concentration profiles. Electrode kinetics would enter as part of the boundary conditions. The solution of these equations is not easy and often involves detailed numerical work. Electroneutrahty (eq. 28) is not strictly correct. More properly, equation 28 should be replaced with Poisson s equation... [Pg.65]

Curve fitting to data is most successhil when the form of the equation used is based on a known theoretical relationship between the variables associated with the data points, eg, use of the Clausius-Clapeyron equation for vapor pressure. In the absence of known theoretical relationships, polynomials are one of the most usehil forms to describe a curve. Polynomials are easy to evaluate the coefficients are linear and the degree, ie, the highest power appearing in the equation, is a convenient measure of smoothness. Lower orders yield smoother fits. [Pg.245]

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