Parabolic flow

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Patankar, S. V., and D. B. Spalding. 1972. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat and Mass Transfer. 15 1787-1806. 

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... 

In addition, the effects of pulsatile flow cannot be ignored. One measure of the impact of oscillary flow is the Wcmersley parameter (a) a= h/2tt f/v where r is the tube radius, f the frequency of oscillation and v is the kinematic viscosity of the fluid (Wcmersley, 1955). The degree of departure from parabolic flow increases with and frequency effects may become important in straight tubes when a > 1 (Ultman, 1985). For conditions of these experiments, a exceeds one to beyond the third generation. 

FIGURE I The flow profiles showing the plug or flat flow in CE (A) and the laminar or parabolic flow in HPLC (B). 

These results suggest that simple parabolic flow is ensured only in the conductive airways where the Reynolds number is less than 1.0. There, the fluid inertia is negligible, and the convective fluid transport is less than the molecular transport. ... 

Since the driving force of the flow is uniformly distributed across the diameter of the capillary, the flow profile is essentially flat. This flat profile contributes to the very high separation efficiency of CZE. Electroosmotic pumping therefore is beneficial, in contrast to laminar flow generated by a HPLC pump, where a parabolic flow profile is established. The electroosmotic flow rate and its flat profile are generally independent of the capillary diameter. However, if the internal diameter of the capillary exceeds 250 pun, the flat profile is increasingly disrupted. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

Different species, belonging to the same sample, form exponential distributions or layers of different thickness I (see Figure 12.5c) the greater the thickness I, the higher the mean elevation above the accumulation wall and the further the penetration into the fast streamlines of the parabolic flow profile. The thickness is inversely proportional to the force exerted on the particle by the field (see Equation 12.8). Usually, this force increases with particle size and this defines the so-called normal mode of elution smaller particles migrate faster and elute earlier than larger particles (see Figure 12.4a). This sequence is referred to as the normal elution order. The above-described equilibrium-Brownian mode will behave as normal mode. However, Brownian, equilibrium, and normal concepts are strictly interrelated. 

Steric elution mode occurs when the particles are greater than 1 jm. Such large particles have negligible diffusion and they accumulate near the accumulation wall. The mean layer thickness is indeed directly proportional to D and inversely proportional to the field force F (see Equation 12.3). The condition is depicted in Figure 12.4b. The particles will reach the surface of the accumulation wall and stop. The particles of a given size will form a layer with the particle centers elevated by one radius above the wall the greater the particle dimension, the deeper the penetration into the center of the parabolic flow profile, and hence, larger particles will be displaced more rapidly by the channel flow than smaller ones. This behavior is exactly the inverse of the normal elution mode and it is referred to as inverted elution order. The above-described mechanism is, however, an oversimplified model since the particles most likely do not come into contact with the surface of the accumulation wall since, in proximity of the wall, other forces appear—of hydrodynamic nature, that is, related to the flow—which lift the particles and exert opposition to the particle s close approach to the wall. 

The classical FEE retention equation (see Equation 12.11) does not apply to ThEEE since relevant physicochemical parameters—affecting both flow profile and analyte concentration distribution in the channel cross section—are temperature dependent and thus not constant in the channel cross-sectional area. Inside the channel, the flow of solvent carrier follows a distorted, parabolic flow profile because of the changing values of the carrier properties along the channel thickness (density, viscosity, and thermal conductivity). Under these conditions, the concentration profile differs from the exponential profile since the velocity profile is strongly distorted with respect to the parabolic profile. By taking into account these effects, the ThEEE retention equation (see Equation 12.11) becomes ... 

FIG. 4.5 Flow in a cylindrical capillary (a) a volume element in the flowing liquid and (b) the parabolic flow profile. 

Reynolds number, a parabolic flow profile typical of Poiseuille flow is evident (--------). At the higher... 

In reality, additional sources of zone broadening include the finite width of the injected band (Equation 23-32), a parabolic flow profile from heating inside the capillary, adsorption of solute on the capillary wall (which acts as a stationary phase), the finite length of the detection zone, and mobility mismatch of solute and buffer ions that leads to nonideal elec-... 

Laminar How is characterized by a parabolic flow profile where the maximum velucily at or near the center of the conduit is approximately twice the average velocity in the protile. Laminar flow often is referred to as vi.wrm.i flow, streamline flows, and low-Reynolds number flow. Special attention must be paid to the constancy of coefficient of most flowmeters in the region uf laminar flow. Sec also Reynolds Number. 

Fig. 3. The parabolic flow profile in a thin wall channel. In addition to flow, mass transport can occur by molecular diffusion and by thermal convention...

Shape factors of a different sort are involved in the Taylor dispersion problem. With parabolic flow at mean speed U through a cylindrical tube of radius R, Taylor found that the longitudinal dispersion of a solute from the interaction of the flow distribution and transverse diffusion was R2U2/48D. The number 48 depends on both the geometry of the cross-section and the flow profile. If, however, we insist that the flow should be laminar, then the geometry of the cross-section determines the flow and hence the numerical constant in the Taylor dispersion coefficient. 

The three-dimensional, fully parabolic flow approximation for momentum and heat- and mass-transfer equations has been used to demonstrate the occurrence of these longitudinal roll cells and their effect on growth rate uniformity in Si CVD from SiH4 (87) and GaAs MOCVD from Ga(CH3)3 and AsH3 (189). However, gas expansion in the entrance zone combined with flow obstructions, such as a steeply sloped susceptor, can also produce... 

Tubular and radial Parabolic flow Dissociation rate coefficient treated as a parameter 73... 

In addition, dynamic studies were performed including moving fluids. Generally, similar results were obtained [140], Different from the stationary case, the width of the inner lamellae is decreased relative to the outer layers owing to the parabolic flow profile. 

Tracey, M. C., Cox, T. I., Davis, J. B., Microfluidic mixer employing temporally interleaved liqiuid slugs and parabolic flow, in Ramsey, J. M., van den Berg, A. (Eds.), Micro Total Analysis Systems, Kluwer, Dordrecht, 2001,141-141. 

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