Parabolic flow profile

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. 

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...

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. 

Time resolution can also be limited by the parabolic flow profile of a confined fluid in the low Reynolds number (laminar flow) regime. The fluid velocity at the walls approaches zero. If the probe beam sample molecules spread over the entire width of a channel, their differing velocities must be considered. Those in close proximity to the walls travel very slowly, whereas those at the center of the channel flow most rapidly. To compensate for this effect, we flow an extra layer of buffer against the walls... 

Fig. 26. Schematic design of field flow fractionation (FFF) analysis. A sample is transported along the flow channels by a carrier stream after injection and focusing into the injector zone. Depending on the type and strength of the perpendicular field, a separation of molecules or particles takes place the field drives the sample components towards the so-called accumulation wall. Diffusive forces counteract this field resulting in discrete layers of analyte components while the parabolic flow profile in the flow channels elutes the various analyte components according to their mean distance from the accumulation wall. This is called normal mode . Particles larger than approximately 1 pm elute in inverse order hydrodynamic lift forces induce steric effects the larger particles cannot get sufficiently close to the accumulation wall and, therefore, elute quicker than smaller ones this is called steric mode . In asymmetrical-flow FFF, the accumulation wall is a mechanically supported frit or filter which lets the solvent pass the carrier stream separates asymmetrically into the eluting flow and the permeate flow which creates the (asymmetrical) flow field...

Fig. 26. The velocity profile for flow of water through two different regions (highlighted in Fig. 25a) of the inter-particle space within a fixed bed of spherical glass beads. The velocity profiles are measured across the inter-particle space between two packing elements. Profiles are shown for local regions associated with fast and slow flow velocities, characterised by a local Reynolds number of 50 and 12, respectively. At low Re number, a parabolic flow profile typical of Poiseuille flow is seen (----). At the higher Re number, inertial effects in the flow are evident and the flow profile approaches that of plug flow (-------). For this particular bed, the Re number based on bed diameter is...

The "plug-like velocity flow profile for electrokinetically pumped capillary columns (see Figure 1) is important in minimizing resistance to mass transfer within the mobile phase (4). Hydrostatically-pumped capillaries, have parabolic flow profiles which tend to severely disperse solute bands unless extreme narrow-bore (i.d.s less than 10 pm) capillaries are employed (12). Fortunately, larger capillaries, with less stringent detector volume requirements, can be efficiently used in MECC. 

For laminar airflow in a tube, when 8 approaches the tube radius, Poiseuille flow or a parabolic flow profile is fully developed. This is accomplished by the acceleration of the central portion of the flow. However, when Re exceeds a value lying somewhere between 104 and 106, the laminar boundary layer becomes so thick that it is no longer stable, and a turbulent boundary layer develops. 

Fig-4. Mechanism of an FFF separation of two components X and Y across the parabolic flow profile resulting in different flow velocities of X and Y. Reproduced from [14] with kind permission of the American Association for the Advancement of Science... 

If the geometry of an FFF channel is known exactly and a parabolic flow profile in the channel can be assumed (see Sect. 1.2), it is possible to make exact predictions about the separation of the sample as well as the separation efficiency. In this section, only the general theoretical expressions universally applicable to all FFF techniques operating in the normal mode are provided. Specialities of the different FFF methods are given during their detailed discussion in Sect. 2. 

Complications in the theoretical description of retention in Th-FFF arise from deviation from isoviscous flow due to the temperature gradient resulting in a non-parabolic flow profile [194,217]. An exact analysis of the flow profile of a non-isothermal and thus non-isoviscous flow was published by Westerman-Clark [218]. The consequences of a temperature gradient on the form of the flow profile as well as on retention and peak broadening have been published by Gunderson et. al. [205]. 

The retention, which is determined via Eqs. (51) and (52), is larger than the value which would result from the corresponding simplified treatment in Sect. 1.4.1 using a parabolic flow profile which is explained by the reduced flow velocity near the cold wall. Exact velocity profiles in Th-FFF were numerically... 

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