Convection dimensional equations

A more rigorous treatment takes into account the hydrodynamic characteristics of the flowing solution. Expressions for the limiting currents (under steady-state conditions) have been derived for various electrodes geometries by solving the three-dimensional convective diffusion equation ... 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999)... 

We note that all the equations above are already written in dimensionless form in order to highlight the key parameters characterizing diffusion, reaction, and convection effects. Equations (8)-(14) form a coupled system of ordinary and partial differential equations in the unknowns X , Y , and Z (n = 2, N). Many additional details on the dimensional form and... 

The three-term convective-diffusion model provides the most accurate solution to the one-dimensional convective-diffusion equation for a rotating disk electrode. The one-dimensional convective-diffusion equation applies strictly, however, to the mass-transfer-limited plateau where the concentration of the mass-transfer-limiting species at the surface can be assumed to be both uniform and equal to zero. As described elsewhere, the concentration of reacting species is not uniform along the disk surface for currents below the mass-transfer-limited current, and the resulting nonuniform convective transport to the disk influences the impedance response. ... 

The result obtained with the Voigt measurement model shows that it is possible to obtain a regression using passive elements that describes the data to within the noise level of the measurement. The observation that the three-term model did not improve the regression shows that the regression cannot be improved by refining the solution to the one-dimensional convective-diffusion equation. Instead, the assumption of radial uniformity, implicit in the one-dimensional model, must be relaxed. 

Dispersion is most commonly modeled as a diffusive process. For flow in a packed column, dispersion is captured by the DV C term in the one-dimensional convection-diffusion equation. The longitudinal dispersion coefficient, D, is a function of the Peclet number, Pe = vR/Dj (where is the molecular... 

Based on Eq. [13-15], Selim et al. (1976b) assumed the reactions between exchangeable and nonexchangeable as well as those between nonexchangeable and primary minerals were first-order kinetic reactions. The authors coupled Eq. [13-15] with the one-dimensional convective-dispersive equation (one-site version of Eq. [6]) to describe the transformation kinetics of K during transport in soil. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Z. Zhang, An explicit fourth-order compact finite difference scheme for the three-dimensional convection-diffusion equation, Commun. Numer. Meth. Engng., vol. 14, pp. 219-218, 1998.doi 10.1002/(SICI)1099-0887(199803)14 3<219 AID-CNM140>3.0.CO 2-D... 

The convection-dispersion equation (CDE) is the most widely used of the velocity distribution models. For steady state, one-dimensional water flow, the CDE for a nonreactive solute can be written as (Fried Combarnous, 1971),... 

DIMENSIONAL EQUATIONS. Equations derived by empirical methods, in which experimental results are correlated by empirical equations without regard to dimensional consistency, usually are not dimensionally homogeneous and contain terms in several different units. Equations of this type are dimensional equations, or dimensionally nonhomogeneous equations. In these equations there is no advantage in using consistent units, and two or more length units, e.g., inches and feet, or two or more time units, e.g., seconds and minutes, may appear in the same equation. For example, a formula for the rate of heat loss from a horizontal pipe to the atmosphere by conduction and convection is... 

Since D plays the same role as the kinematic viscosity v, we may expect for large Schmidt numbers (v>D) that the viscous boundary layer thickness should be considerably larger than the diffusion boundary layer thickness. A consequence of this is that the velocity seen by the concentration layer at its edge is not the free stream velocity U but something much less, which is more characteristic of the velocity close to the wall (Fig. 4.2.1). We note also that since c is understood to be c, then in a multicomponent solution there may be as many distinct boundary layers as there are species, with the thickness of each defined by the appropriate diffusion coefficient. With this caveat in mind, we may write the convective diffusion equation for a two-dimensional diffusion boundary layer and estimate the diffusion layer thickness. 

Modeling particulate transport, or various process phases, has been attempted only relatively recently. Sayre (20) gave a very sound basis for further work by using a momentum solution of the two-dimensional convection-diffusion equation characterizing particle transport when additional terms for sedimentation, bed adsorbance, and re-entrainment (erosion) are included. He showed, with extensive hypothetical calculations, which hydrodynamic parameters are important and how they could be quantified. He was also able to show that his concept of bed adsorbance and re-entrainment requires further elucidation and indicated that there might be a turbulence effect on the sedimentation step. Hahn et al. 

This is one situation where the exact convective-diffusion equation can be solved and compared with the one-dimensional dispersion equation. The complete equation is... 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... 

The equation to be solved has been derived and given as Eq. (10), the three-dimensional (3-D) convective diffusion equation. The literature has many discussions about solution techniques for these equations, but they all fit three basic types integral transform methods, method of images, and numerical methods. Probably the single best reference on the first two techniques is the classic work... 

Kinematic similarity is concerned with the motion of phases within a system and the forces inducing that motion. For example, in the formation of boundary layers during flow past flat plates and during forced convection in regularly shaped channels, there are usually three dominant forces pressure, inertia, and viscous forces. If corresponding points in two different-sized cells show at corresponding times identical ratios of fluid velocity, the two units are said to be kinematically similar and heat and mass transfer coefficients will bear a simple relation in the two cells. It can be shown by means of dimensional analysis that for a closed system under forced convection the equation of motion for a fluid reduces to a function of Re, the Reynolds number, which we have met in Chapter 2. To preserve kinematic similarity under those circumstances, Reynolds numbers in the two cells must be identical. 

The "axial dispersion" or "axial dispersed plug flow" model [Levenspiel and Bischoff, 1963] takes the form of a one-dimensional convection-diffusion equation, allowing to utilize all of the clasical mathematical solutions that are available [e.g. Carslaw and Jaeger, 1959, 1986 Crank, 1956]. 

Convective Mass Transfer Convective Dispersion and the One-Dimensional Convective Diffusion Equation... 

Abstract. In this paper, the motion model of the two-component incompressible viscous fluid with variable viscosity and density is considered for modeling the process of the surface wave propagation. The model consists of the non-stationary Navier-Stokes equations with variable viscosity and density, the convection-diffusion equation and equations for determining the viscosity and density depending on the concentration of the components. Thus we model the two-component medium, one of the components being more dense and viscous liquid. The results of calculations for two-dimensional and three-dimensional problems are presented. 

Exact lineal flow solutions. For one-dimensional lineal flows, the convective-diffusion equation for a constant velocity U takes the form... 

The convective-diffusion Equation 7.8 in the case of two-dimensional system in a steady state [(dddt) = 0] in cylindrical coordinates is as follows ... 

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