Steady-state Numerical Solutions Research Articles

A two-dimensional numerical model of estuarine circulation: The effects of altering depth and river discharge

Steady-state numerical solutions are obtained for a two-dimensional, vertically stratified model of a partially mixed estuary. The boundary at the seaward end of the estuary is considered to be open, with the profiles of salinity, vorticity and streamfunction obtained by extrapolating interior dynamics out to the boundary. A salinity source is maintained at the bottom at the mouth. Zero salt flux is required at a free-slip top and no-slip bottom boundary. Zero salinity and a parabolic velocity profile are maintained at the head of the estuary. A number of cases are run for various estuarine parameters; the river transport and Rayleigh number being the two parameters that have the most pronounced effect. The river transport is varied by adjusting the mean freshwater velocity, U f. Decreasing U f allows salt as well as the stagnation or null point to penetrate upstream. The estuarine circulation weakens, but expands over a larger portion of the estuary. The position of the stagnation point, with respect to the seaward boundary, varies as U f − 5 8 for U f > 1 cm/s and as U f − 5 6 for U f < 1 cm/s. Increasing the Rayleigh number, by deepening the estuarine channel, H, results in an increased circulation as well as strong intrusion of salinity and inward migration of the stagnation point. The horizontal location of the stagnation point is found to be proportional to Ra and therefore, varies as H 3.

Numerical treatment of time dependent advection and diffusion of air pollutants

A mixed Lagrangian-Eulerian finite-difference scheme has been developed to study the time dependent advection and diffusion of air pollutants for variable emission rates, wind velocity and diffusion coefficients. The Lagrangian and Eulerian procedures are applied, respectively, to the advective and diffusive transport. The Lagrangian technique is mass conserving and completely avoids the artificial diffusion errors associated with conventional Eulerian finite-difference approximations. Furthermore, it does not require any condition of numerical stability. Compatibility with the Eulerian grid geometry is achieved by using a convergent approximation of the wind profile providing a translation of the concentration field to positions always coincident with grid points. The method is applied to the horizontal advection-vertical diffusion transport of line source pollutants. The diffusion is computed with an implicit finite-difference scheme allowing for a variable grid spacing. A Gaussian vertical distribution of the grid point “density” is chosen with its maximum at the source height. Typical numerical steady-state solutions obtained with different approximations of given velocity profiles are compared with corresponding analytical solutions.

An analysis of countercurrent exchange with emphasis on renal function.

An analysis of countercurrent exchange in a U-tube is presented for a single-solute, constant-volume flow rate system with spatially varying source fluxes and permeabilities. Analytical solutions are given for the steady-state equations and numerical solutions for the unsteady-state equations. The solutions indicate that an external source of solute delivered to the stream flowing away from the U-tube bend can be distributed by the exchanger so that the concentration in both limbs increases toward the bend. In particular, there exist source fluxes whose magnitude decreases monotonically toward the bend for which the maximum solute concentration occurs at the bend. The point at which a concentration maximum occurs is governed principally by the solute permeability of the barrier separating the two limbs and by the volume flow rate through the exchanger. The system dynamics depend strongly on the relative cross-sectional areas of the two limbs or, equivalently, on the flow velocities within them. The model is used as a basis for discussion of various functional aspects of the renal vasa recta system.

Role of Singlet and Triplet Excitons in Extrinsic Photocurrent Production in the Anthracene–Gold System

AbstractExtrinsic hole photocurrents in a gold coated anthracene crystal are measured as a function of the absorption depth of the exciting light. Charge pairs are formed at the gold‐anthracene interface as a result of exciton–gold interactions. It is found that both singlet and triplet excitons are involved as precursors to charge formation. The contributions of singlets and triplets to the observed photocurrent quantum yields are determined for 394 nm excitation light (a and b polarized) and 340 nm excitation light (a and b polarized). Since reabsorption is known to have a significant effect on the distribution of excitons and on the effective singlet lifetime, a steady state numerical solution for the reabsorption problem is offered for this system. The inclusion of the reabsorption effect allows the calculation of a “reabsorption free” singlet diffusion length (316 Å) and an evaluation of the electron transfer efficiency ratio (ηT/ηS ≈ 3.5) where ηS is the probability of electron injection per singlet exciton quenched by the surface and ηT is the corresponding probability for a triplet exciton. The magnitude of the ratio ηT/ηS is interpreted in terms of singlet energy transfer to the gold layer.

Numerical analysis of a forward-biased step-junction P-L-N diode

Exact numerical steady-state solutions are presented for a forward-biased step-junction one-dimensional p+-n-n+rectifier over a current range of 10-2to 104A/cm2The versatility of the numerical approach is demonstrated by the inclusion in the physical model of injection-dependent carrier lifetimes due to Auger recombination or injection-dependent mobilities due to electron-hole scattering. The solutions obtained are compared to those given by conventional analytical treatments at low and high injection levels. Among the major results established are the validity of the Boltzmann's relations and of the quasi-neutrality assumption at all injection levels of interest. It is shown that at the upper end of the current range, a complete analytical solution for the whole device is intractable. However, partial analytical solutions are available for the quasi-neutral regions, and these have enabled us to obtain a better understanding of some of the numerical solutions. Examples are the early onset of non-linearity effects in the n+region because the minority carrier current there is small compared to the total current, and the shift in the minimum of the base carrier distribution towards the n-p+junction as the injection efficiency from the p+region into the base decreases with increasing forward bias.

Accurate numerical steady-state solutions for a diffused one-dimensional junction diode

A set of accurate steady-state solutions for a diffused junction diode is presented. The internal consequences of biasing of the diode are demonstrated by the spatial distributions of the mobile charge carriers, the electrostatic potential and the space charge. The terminal characteristics presented include the V- I characteristic and the low frequency capacitance.

An accurate numerical steady-state one-dimensional solution of the P- N junction

A numerical method of solution of the fundamental semiconductor steady-state one-dimensional transport equations, already available in the literature, is improved and extended, and is applied to a single-junction device. A reduced set of ‘exact’ relations is derived directly from the fundamental set with none of the conventional assumptions or approximations, and is solved numerically by a simple iterative procedure. Freedom is available in the choice of the doping profile, recombination law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. In spite of the generality of the original method, its analytical formulation is shown to be unsuitable for generating a sound numerical algorithm sufficiently acccurate and valid for high reverse-bias conditions. Difficulties and limitations are exposed, and overcomeby an improved formulation extended to any bias condition. Emphasis is on the selection of a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties present in the numerical analysis, and on achieving a high degree of accuracy in the final results (the most delicate problem). As a simple application of the improved formulation, ‘exact’ and first-order theory results for an idealized structure are presented and compared. The poorness of some of the basic assumptions of the conventional first-order theory is exposed, in spite of a satisfactory agreement between the exact and first-order results of the terminal properties for particular bias conditions. The computation time for the achievement of one set of very accurate solutions for a specified applied voltage amounts to approx 1 min on an IBM 7094/7040 shared-file system.