Convection velocity profile

For a vertical isothermal flat plate at 93°C exposed to air at 20°C and 1 atm, plot the free-convection velocity profiles as a function of distance from the plate surface at x positions of 15, 30, and 45 cm. 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

The wall boundary condition applies to a solid tube without transpiration. The centerline boundary condition assumes S5anmetry in the radial direction. It is consistent with the assumption of an axis5Tnmetric velocity profile without concentration or temperature gradients in the 0-direction. This boundary condition is by no means inevitable since gradients in the 0-direction can arise from natural convection. However, it is desirable to avoid 0-dependency since appropriate design methods are generally lacking. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

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. 

Delete the radial convection term but otherwise run the full simulation. This gives avgC = 0.5197. Now add the radial term to get 0.5347. The change is in the correct direction since velocity profile elongation hurts conversion. 

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 ... 

Note that when the fluid velocity (v) is constant, the description of convection given by the second term on the right-hand side of this equation is identical to that of the plug flow model [Eq. (8)]. In more complex systems, a spatially varying fluid velocity may by incorporated by using the Navier-Stokes equations [Eqs. (10)—(12)] to describe velocity profiles. 

Gibaldi et al. [45] postulated that convective forces may be present in the GI tract during in vivo dissolution. This study took advantage of the well-defined hydrodynamics of the rotating disk, incorporating the solutions for the velocity profile and transport equations of Cochran [50] and Levich [51] to obtain... 

For a number of flow situations, the mass-transfer rate can be derived directly from the equation of convective diffusion (see Table VII, Part A). The velocity profile near the electrode is known, and the equation is reduced to a simpler form by appropriate similarity transformations (N6). These well-defined flows, therefore, are being exploited increasingly by electrochemists as tools for the kinetic characterization of electrode reactions. Current distributions at, or below, the limiting current, transient mass transfer, and other aspects of these flows are amenable to analysis. Especially noteworthy are the systematic investigations conducted by Newman (review until 1973 in N7 also N9b, N9c, H6b and references in Table VII), by Daguenet and other French workers (references in Table VII), and by Matsuda (M4a-d). Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and mass-transfer experiment. 

With the carrier stream unsegmented by air bubbles, dispersion results from two processes, convective transport and diffusional transport. The former leads to the formation of a parabolic velocity profile in the direction of the flow. In the latter, radial diffusion is most significant which provides for mixing in directions perpendicular to the flow. The extent of dispersion is characterized by the dispersion coefficient/). 

The first step in solving convective diffusion problems is the derivation of the velocity profile. In this case, the flow arriving with velocity v is modified by... 

It should be highlighted that equation (47) holds for solid particles. In the case of liquid particles, e.g. with emulsions, the convective diffusion process is very different due to interfacial momentum transfer which gives rise to a different velocity profile. Consequently, convective diffusion to/from a liquid particle is more effective than that for a solid particle. Starting again from equation (43),... 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

In well-developed fires, the convective heat fraction is typically measured at more than about 65% of the total heat release rate (Heskestad, 2002). This heat is carried away by the plume above the flames. Prediction of plume velocity and temperatures above the flames serve as the basis for convective heat transfer calculations where overhead equipment exists. Widely used fire plume theory assumes a point source origin, and uniformity throughout the plume relative to air density, air entrainment, velocity profile, and buoyancy. 

For laminar flow in short tubes or laminar flow of viscous materials these models may not apply, and it may be that the parabolic velocity profile is the main cause of deviation from plug flow. We treat this situation, called the pure convection model, in Chapter 15. 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

Diffusive transport with convection occurring simultaneously can be solved more easily if we orient our coordinate system properly. First, we must orient one axis in the direction of the flow. In this case, we will choose the v-coordinate so that u is nonzero and v and w are zero. Second, we must assume a uniform velocity profile, u = U = constant with y and z. Then, equation (2.33) becomes... 

Fig. 5. Gas flow by forced convection through a wetted-wall tower, (a) with flat velocity profile, and (b) with parabolic velocity profile.

As stated, this equation finds limited practical application. It requires knowledge of both velocity profiles and its solution requires vorticity boundary conditions that also depend on the velocity profiles. The principal reason to write the equation is to make the point that vorticity is transported within the boundary layer by convection and diffusion in a manner analogous to momentum transport. 

The convection term is obtained by analysis of the velocity profile of the system, which is derived from two equations. The first of these is the equation of continuity... 

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