Calculation Of Equilibrium Distribution Research Articles

Lattice Boltzmann method for rarefied channel flows with heat transfer

Abstract A thermal lattice Boltzmann method (TLBM) is presented for the analysis of fluid flow and heat transfer in two-dimensional channels with non-continuum effects. The relaxation times ( τ f , τ g ) are linked to the Knudsen number which accounts for the rarefaction that can be present at micro geometries or at low density conditions. The TLBM used here employs inlet/outlet boundary conditions to generate a forced convection problem where the calculation of equilibrium distributions at the wall surfaces are modified to incorporate the velocity slip and temperature jump conditions. Numerical simulations are obtained for thermal micro-Couette and thermal micro-Poiseuille channel flows and the effect of the Knudsen number on the velocity and temperature profile is investigated.

Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes

Quasi-Birth-and-Death (QBD) processes have been analysed in detail by many authors. This paper considers the calculation of equilibrium distributions in level dependent QBD processes which are an extension of the classical QBD process. In addition to the general case, we consider a number of special cases of level dependent QBD processes, presenting algorithms for computing the equilibrium distribution

Use of analytic expressions for the calculation of equilibrium distributions of polycyclic aromatic hydrocarbons in benzene flames

The thermodynamic properties of twenty isomer groups in six series of benzenoid polycyclic aromatic hydrocarbons and first members of seven higher series have been used to explore their thermodynamic stabilities in a near-sooting benzene flame studied by Bittner and Howard. Since the thermodynamic properties of isomer groups are linear in carbon number within a homologous series, analytic expressions for the equilibrium distribution can be used. On the basis of the assumption that the successive isomer groups in an homologous series are formed from benzene by the addition of acetylene and the production of molecular hydrogen, equilibrium distributions of isomer groups within six series have been calculated at various heights in this flame. The competition between these six isomer groups and the first members of higher series up to circumcircumcoronene (C 96 H 24 ) has been studied by calculating equilibrium partial pressures. These calculations show that the first members of series above coronene remain very stable at levels above 8 mm, where C 10 H 8 , C 14 H 10 , and C 16 H 10 decline for thermodynamic reasons. These calculations show the usefulness of isomer group properties and emphasize the simplifications resulting from the linear dependence of the standard Gibbs energies of formation on carbon number. They also show the usefulness of calculations based on observed partial pressures of acetylene, molecular hydrogen, and benzene.

Calculation of equilibrium distributions of chemical species in aqueous solutions by means of monotone sequences

A numerical method for the calculation of equilibrium distributions of chemical species in aqueous solutions of electrolytes is presented. This method is constructed by transforming the problem of determining the set of unknown concentrations satisfying the mass balance and mass action equations into the equivalent problem of finding the limits of certain well-behaved mathematical sequences in a multivariable direct iteration scheme. The total (analytical) concentrations are taken as the starting estimates, and the recursive equations are constructed from the starting equations. It is shown that the sequences so constructed are both monotonic and bounded, hence convergent in an orderly fashion by a mathematical axiom. Each unknown free ion concentration (that is, each limit) is approached simultaneously from above and from below, being effectively “sandwiched” in an ever tightening manner. Strict error bounds therefore are easily constructed. The method has been found to be exceedingly efficient in practice, with the error bound (maximum fractional error in taking the upper estimate as the final answer) decreasing in an approximately exponential fashion with respect to iteration number, commonly 0.5–1.5 orders of magnitude per iteration. Correction for nonideality is presented, and the possibility of this giving rise to more than one solution is discussed in connection with three examples of natural waters of low-, medium-, and high-ionic strengths.