Radial convective flux

Solute flux within a pore can be modeled as the sum of hindered convection and hindered diffusion [Deen, AIChE33,1409 (1987)]. Diffusive transport is seen in dialysis and system start-up but is negligible for commercially practical operation. The steady-state solute convective flux in the pore is J, = KJc = where c is the radially... 

When radial convective fluxes due to transmembrane pressure occur, this model is no longer accurate. 

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 data of Fig. 20 also point out an interesting phenomenon—while the heat transfer coefficients at bed wall and bed centerline both correlate with suspension density, their correlations are quantitatively different. This strongly suggests that the cross-sectional solid concentration is an important, but not primary parameter. Dou et al. speculated that the difference may be attributed to variations in the local solid concentration across the diameter of the fast fluidized bed. They show that when the cross-sectional averaged density is modified by an empirical radial distribution to obtain local suspension densities, the heat transfer coefficient indeed than correlates as a single function with local suspension density. This is shown in Fig. 21 where the two sets of data for different radial positions now correlate as a single function with local mixture density. The conclusion is That the convective heat transfer coefficient for surfaces in a fast fluidized bed is determined primarily by the local two-phase mixture density (solid concentration) at the location of that surface, for any given type of particle. The early observed parametric effects of elevation, gas velocity, solid mass flux, and radial position are all secondary to this primary functional dependence. 

PROFILE is a biogeochemical model developed specially to calculate the influence of acid depositions on soil as a part of an ecosystem. The sets of chemical and biogeochemical reactions implemented in this model are (1) soil solution equilibrium, (2) mineral weathering, (3) nitrification and (4) nutrient uptake. Other biogeochemical processes affect soil chemistry via boundary conditions. However, there are many important physical soil processes and site conditions such as convective transport of solutes through the soil profile, the almost total absence of radial water flux (down through the soil profile) in mountain soils, the absence of radial runoff from the profile in soils with permafrost, etc., which are not implemented in the model and have to be taken into account in other ways. 

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 this case, SN is dependent on the axial coordinate the tubular electrode is not uniformly accessible. This complicates the mathematical description of partially kinetically controlled reactions at the TE. However, for total kinetic control (irreversible reaction at the foot of the wave), the flux is uniform as radial convection is uniformly zero along the tube. 

Let us consider the symmetrical burning of a spherical droplet with the radius rp in surroundings without convection. Assume that there is an infinitely thin flame zone from the surface of the droplet to the radial distance rn [137], which is much larger than the radius of the droplet, rp. The heat released from the burning is conducted back to the surface to evaporate liquid fuel for combustion. Because the reaction is extremely fast, there exists no oxidant in the range of rp< r < m while no fuel vapor is available at r > rn. At a quasi steady state the mass flux through the spherical surface with the radius r (>rp), Mfv, can be obtained with Fick s law as... 

Consider a thin region, in contact with the collector surface, within which convection and tangential diffusion are negligible. Then the radial flux, is inde-... 

For spherical symmetry, can vary only in the radial direction and not with any angle. The heat flux density at a sphere s surface for heat conduction across the air boundary layer followed by heat convection in the surrounding turbulent air then is... 

We see that the Ts depends on the rate of heat release q and that the overall effect of convection is to reduce the surface temperature and make it nonuniform, with the surface temperature being highest at q = 1 (i.e., the downstream stagnation point of the sphere) and lower at the front, q = — 1. The asymmetry is due to the fact that the radial temperature gradient is slightly increased at the front relative to the back and thus requires a slightly lower surface temperature to sustain the heat flux q, compared with the surface temperature that is required at the back. 

Different operating conditions may require some modification of the performed analysis. Ultrafiltration and/or osmosis can promote convective solute or water flux through the membrane wall. Should it happen, radial convection could compete with diffusion as the main substrate and product transport mechanism. The relative importance of the two transport mechanisms can be evaluated by comparing the radial convective velocity to the diffusive velocity, that is the ratio of the diffusion coefficient to the wall thickness. When the first one is negligible relative to the latter, the model applies without modification. The second possible effect of the radial flux is to remove enzymes from the fiber wall, resulting in the reduction of reactor efficiency. 

Local Interfacial Molar Flux. Results for P(0 and Sc(t) via (11-199) and (11-202), respectively, represent basic information from which interphase mass transfer correlations can be developed. Gas-liquid mass transfer of mobile component A occurs because it is soluble in the liquid phase, and there is a nonzero radial component of the total molar flux of A, evaluated at r = R(t). Even though motion of the interface induces convective mass transfer in the radial direction, there is no relative velocity of the fluid with respect to the interface at r = R t). It should be emphasized that a convective contribution to interphase mass transfer in the radial direction occurs only when motion of the interface differs from Vr of the liquid at r = R. Hence, Pick s first law of diffusion is sufficient to calculate the molar flux of species A normal to the interface at r = R t) when f > 0 ... 

Noble ( ) extended the one dimensional solution for facilitated transport to obtain an analytical solution for solute flux through a hollow fiber membrane. This result allows for convective transport through the lumen and radial transport through the membrane walls. The solution can also be used with planar geometry and... 

The model discussed here uses the effective transport concept, this time to formulate the fiux of heat or mass in the radial direction. This flux is superposed on the transport by overall convection, which is of the plug flow type. Since the effective diffusivity is mainly determined by the flow characteristics, packed beds are not isotropic for effective diffusion, so that the radial component is different from the axial mentioned in Sec. 11.6.b. Experimental results concerning D are shown in Fig. 11.7.a-l [61, 62,63]. For practical purposes Pe may be considered to lie between 8 and 10. When the effective conductivity, X , is determined from heat transfer experiments in packed beds, it is observed that X decreases strongly in the vicinity of the wall. It is as if a supplementary resistance is experienced near the wall, which is probably due to variations in the packing density and flow velocity. Two alternatives are possible either use a mean X or consider X to be constant in the central core and introduce a new coefficient accounting for the heat transfer near the wall, a , defined by ... 

The principal dimensions of the reactor are shown in Figure 6.1. The numerical values used here are D = 0.48 m //(= 1.02 m H, = 0.34 0 = 0.33 m s = 0.01 m r = 0.044125 m d = 0.0625 m d = 0.04 m and q = 0.07 m. Non-slip boundary conditions are assumed on the vessel wall. Both radial and axial velocities are set to zero on the shaft and impeller disk and the angular velocity is determined by the speed of rotation. On the free surface of the liquid, the axial component of velocity is zero with the other two components of velocity being stress free. Along the central line, below the impeller, the axial component of velocity is stress free and the other two components are zero. The temperature of the jacket at the vessel walls is fixed at 10 °C. Heat is lost by convection and at the free surface and there is an axis of symmetry along the centreline with no flux at the shaft and impeller boundaries. The flow is... 

The feed throat temperature is set at 15.6°C, the barrel temperatures are set at 288, 371, 371, and 371 °C, the screw shank is set at 93°C, the screw tip is at 371 C, and the heat flux at the screw centerline is zero in the radial direction. The program does not take into account viscous dissipation or convection the heat transfer is purely by conduction. The thermal conductivity of the FEP is taken as 0.246 J/ms°K. 

Radial diffusion gives very high rates of mass transport to the electrode snrface with a mass transport coefficient of the order of Dir. Therefore, even at rotation rates of Kf rpm, convective transport to a rotating macroelectrode is smaller than diffnsion to a 1-pm microdisk. The high flux at a microelectrode means that one does not observe a reverse wave under steady-state conditions (Figure 6.1.4.3A), because the electrolysis product leaves the diffusion layer at an enhanced rate. 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

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