Convection convective-diffusion equation

The finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

Under conditions of limiting current, the system can be analyzed using the traditional convective-diffusion equations. For example, the correlation for flow between two flat plates is... 

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation... 

Another method often used for hyperbohc equations is the Mac-Cormack method. This method has two steps, and it is written here for the convective diffusion equation. 

Another approach to modeling the particle-collection process is based on the convective diffusion equation... 

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 ... 

Example 8.8 Explore conservation of mass, stability, and instability when the convective diffusion equation is solved using the method of lines combined with Euler s method. 

Radial motion of fluid can have a significant, cumulative effect on the convective diffusion equations even when Vr has a negligible effect on the equation of motion for V. Thus, Equation (8.68) can give an accurate approximation for even though Equations (8.12) and (8.52) need to be modified to account for radial convection. The extended versions of these equations are... 

The convective diffusion equations for mass and energy are given detailed treatments in most texts on transport phenomena. The classic reference is... 

The appropriateness of neglecting radial flow in the axial momentum equation yet of retaining it in the convective diffusion equation is discussed in... 

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. 

Include the radial velocity term in the convective diffusion equation and plot streamlines in the reactor. 

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

The unsteady version of the convective diffusion equation is obtained just by adding a time derivative to the steady version. Equation (8.32) for the convective diffusion of mass becomes... 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

A standard approach to modeling transport phenomena in the field of chemical engineering is based on convection-diffusion equations. Equations of that type describe the transport of a certain field quantity, for example momentum or enthalpy, as the sum of a convective and a diffusive term. A well-known example is the Navier-Stokes equation, which in the case of compressible media is given as... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The governing equation for mass transport in the case of an incompressible flow field is easily derived from the general convection-diffusion equation Eq. (32) with... 

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 ... 

A second approach is to assume that the drop surface approaching the electrode is a moving plane. This is appropriate, since the diffusion layer is almost always considerably smaller than the size of the drop, for most of its lifetime under practical conditions. To a good approximation, the convective effect close to the moving front is then calculated based on velocities which are twice those determined from Eq. (23), in order to account for the moving center of the drop. The convective-diffusion equation which describes this case is given by... 

At the RDE the velocity profile obtained by Karman and Cochran (see ref. 124) and depicted in Fig. 3.68a leads via solution of its differential convection-diffusion equation to the well known Levich equation ... 

At the RRE the derivation123 of the Levich equation requires reconsideration of the convection-diffusion equation, which results in... 

Our analytical interest now is to know iR as a direct measure of discgenerated red collected at the ring. Again the solution123 of the differential convection-diffusion equation needs a complex mathematical treatment, resulting in an involved equation for the so-called collection efficiency... 

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

Shah and Nelson [33] introduced a convective mass transport device in which fluid is introduced through one portal and creates shear over the dissolving surface as it travels in laminar flow to the exit portal. They demonstrated that this device produces expected fluid flow characteristics and yields mass transfer data for pharmaceutical solids which conform to convective diffusion equations. 

The convective diffusion equation is simply cast in terms of radius (r) instead of x. 

See Partial Differential Equations. ) If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. 

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