Fluid velocity profiles

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

At velocities greater than the critical, the fluid velocity profile in the conduit is uniform across the conduit diameter except for a thin layer of fluid at the conduit wall. This boundary layer continues to move in laminar flow. In connection with flow measurement, most flowmeters have constant coefficients under turbulent flow conditions. Some flowmeters have the advantage of constant coefficients over Reynolds Number ranges encompassing both turbulent and laminar flows. See also Fluid and Fluid Flow and Reynolds Number. 

The maximum fluid velocities for the different frequencies are shown in Fig. lb. For the case without cell activity Fig. 2 shows the effect of dispersion on solute content. For the case of a large limited solute Fig. 3b indicates that in correspondence with the fluid velocity profiles in Fig. lb, the solute penetration depth is largest for 0.001 Hz, while for 0.1 Hz solute concentrations are higher in the periphery. Concentration profiles for the small limiting solute are hardly affected by different dispersion parameters and loading conditions. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

The plug-flow model signifies that the fluid velocity profile is plug shaped, and is uniform at all radial positions, as explained earlier for the plug-flow adsorption reactor [38,49-53] (see Section 6.11.2). The fixed-bed ion-exchange reactor is packed randomly with particles from a solid ion... 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

There are two different philosophies about the optimum channel design for focusing-S-FFF reported in the literature. Whereas Janca et al. utilized channels with trapezoidal or parabolic cross sections [83,308-315], resulting in a variation of the fluid flow velocity across the channel width, Giddings [316] favored the classical rectangular cross section with a parabolic fluid velocity profile. 

Figure 2.10. Viscous spreading fluid velocity profile in the liquid wedge close to the triple line.

At relatively low fluid velocities, particularly for a viscous fluid (where turbulence is damped) in a small pipe one will normally obtain streamline flow (Fig. 1.3a). Under these conditions, the fluid is in a continuous state of shear with the fastest flow in the center of the pipe with low to zero flow right at the wall. The fluid velocity profile, along a longitudinal section of the pipe, is parabolic in shape. 

In practice, the fluid velocity profile is rarely flat, and spatial gradients of concentration and temperature do exist, especially in large-diameter reactors. Hence, the plug-flow reactor model (Fig. 7.1) does not describe exactly the conditions in industrial reactors. However, it provides a convenient mathematical means to estimate the performance of some reactors. As will be discussed below, it also provides a measure of the most efficient flow reactor—one where no mixing takes place in the reactor. The plug-flow model adequately describes the reactor operation when one of the following two conditions is satisfied ... 

If one does not control mixing volumes adequately, there will be an automatic increase in the volume of the product. If one utilizes a large volume taper at the end of the column to control fluid velocity, as has been done in the laboratory column technologies, one can get a smooth addition of sample onto the column at low linear velocities. However, in a production environment where one is going to try to optimally pump that bed structure, one can see from the Van Deemeter plot comparison to the fluid velocity profiles within this schematic of a column (Figure 3) that one will be operating at different linear velocities within that distribution oriface. This adds volume to the product, and consequently, there s a loss of resolving power within the system. 

Diffusive flow for neutrals The importance of convective vs. diffusive flow of neutrals is determined by the Peclet number Pe = uL/D, where L is a characteristic dimension of the system. Away from inlet and exit ports, the characteristic length will be on the order of the reactor dimension. The system will be primarily diffusive when Pe 1. For CI2 gas in a reactor with L 0.1 m and a neutral species diffusivity of D 5m s at 20mtorr, the Peclet number will be Pe 1 when M = 50ms. Convective gas velocities are not likely to be that high, except for a small region near the gas inlet ports. It follows that gas flow can be approximated as diffusive this obviates the need for solving the full Navier-Stokes equations which adds to the computational burden. It should be noted that both the diffusivity and the convective velocity scale inversely with gas pressure, so the Pe number is independent of pressure. However, as the pressure is lowered to the point of free molecular flow, the gas diffusion coefficient has no meaning any more. Direct Simulation Monte Carlo (DSMC) [41, 143] can then be applied to solve for the fluid velocity profiles. 

Obviously, integration constant A must be zero to satisfy the zero shear condition 3 at the gas-liquid interface. Now condition 2 is satisfied when 2B = —l Vbubbie-The final results for the stream function and the fluid velocity profile are... 

Answer. The following functional form of the low-Reynolds-number one-dimensional fluid velocity profile is based on solid-body rotation at r = R and conforms to the no-shp boundary condition at the fluid-solid interface ... 

FULLY DEVELOPED FLUID VELOCITY PROFILES IN REGULAR POLYGON DUCTS... 

When the liquid phase exhibits non-Newtonian behavior, the mass transfer coefficient will change due to alterations in the fluid velocity profile around the submerged particles. The trends for both mass transfer and drag coefficient are analogous. As before for Newtonian fluids, two types of interfacial behavior need to be considered. 

When the flow is fully developed (meaning that it is far enough from the entrance—and exit—such that entrance effects have disappeared), the fluid velocity profile between the two plates is linear and can be defined by the following relationship ... 

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