Flow, laminar Gradient, concentration

Laminar flow reactors have concentration and temperature gradients in both the radial and axial directions. The radial gradient normally has a much greater effect on reactor performance. The diffusive flux is a vector that depends on concentration gradients. The flux in the axial direction is... 

Gas-phase reacdotis are carried out primarily in tubular reactors where the flow is generally turbulent. By assuming that there is no dispersion and ttiere are no radial gradients in either temperature, velocity, or concentration, we can model the flow in the reactor as plug-flow. Laminar reactors are discussed in Chapter 13 and dispersion effects in Chapter 14. The differential form of the design equation... 

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. 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

Such effects principally cannot be observed in multi band detectors such as a UV diode array detector or a Fourier transform infrared (FTIR) detector because all wavelengths are measured under the same geometry. For all other types of detectors, in principle, it is not possible to totally remove these effects of the laminar flow. Experiments and theoretical calculations show (8) that these disturbances can only be diminished by lowering the concentration gradient per volume unit in the effluent, which means that larger column diameters are essential for multiple detection or that narrow-bore columns are unsuitable for detector combinations. Disregarding these limitations can lead to serious misinterpretations of GPC results of multiple detector measurements. Such effects are a justification for thick columns of 8-10 mm diameter. 

With respect to general corrosion, once a surface film is formed the rate of corrosion is essentially determined by the ionic concentration gradient across the film. Consequently the corrosion rate tends to be independent of water flow rate across the corroding surface. However, under impingement conditions where the surface film is unable to form or is removed due to the shear stress created by the flow, the corrosion rate is theoretically velocity (10 dependent and is proportional to the power for laminar flow and... 

The above equations are applicable only when the Schmidt number Sc is very close to unity or where the velocity of flow is so high that the resistance of the laminar sub-layer is small. The resistance of the laminar sub-layer can be taken into account, however, for equimolecular counterdiffusion or for low concentration gradients by using equation 12.118. 

A highly concentrated suspension of flocculated kaolin in water behaves as a pseudo-homogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald-de Waele power law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced even though the flowrate of suspension is kept constant, Explain how this is possible in slug flow and estimate the possible reduction in pressure gradient for equal volumetric flowrates of suspension and air. 

Gas-phase reactors are often in laminar flow but have such high diffusivities that radial concentration gradients tend to be unimportant. Combustion reactions are fast enough to be exceptions. See... 

Figure 3 Schematic diagram of transportation of sample via the stream of carrier by (a) turbulent flow, (b) laminar flow, and (c) radial diffusion of the sample in response to radial concentration gradient, within the flowing stream.

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

It is of interest to consider first what is happening in pipe flow. Random molecular movement gives rise to a mixing process which can be described by Fick s law (given in Volume 1, Chapter 10). If concentration differences exist, the rate of transfer of a component is proportional to the product of the molecular diffusivity and the concentration gradient. If the fluid is in laminar flow, a parabolic velocity profile is set up over the cross-section and the fluid at the centre moves with twice the mean velocity in the pipe. This... 

In order to ensure that the solution flow is indeed laminar past a tubular electrode, we choose a solution of known and fixed concentration, and measure /iim as a function of Vf. A plot of log(/um) as y against log(Vf) as x should be linear with a gradient of 1/3. Deviation from this line at fast flow rates indicates those flow rates that should be avoided. The analyst now knows the range of flow rates to keep within. If the solution flow is not completely laminar, then the flow rate is amended until it is - amend in this context almost always means decrease o). ... 

In Sect. 3.2, the development of the design equation for the tubular reactor with plug flow was based on the assumption that velocity and concentration gradients do not exist in the direction perpendiculeir to fluid flow. In industrial tubular reactors, turbulent flow is usually desirable since it is accompanied by effective heat and mass transfer and when turbulent flow takes place, the deviation from true plug flow is not great. However, especially in dealing with liquids of high viscosity, it may not be possible to achieve turbulent flow with a reasonable pressure drop and laminar flow must then be tolerated. 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

Mori and Ototake (M17) have presented a mathematical analysis of the laminar flow of Bingham-plastic materials in the annulus between two concentric pipes. The complex results have been shown in convenient graphical form which enables one to solve for the flow rate corresponding to a given pressure gradient. 

When transport is not able to do its job adequately and there is a change in the interfacial concentrations of electron acceptors and donors from the bulk values, there is a variation of concentration with distance from the interface toward the bulk of the solution. What matters, however, as far as the charge-transfer reaction is concerned, is the gradient of concentration at the interface because it is this gradient that drives the diffusion flux Jjy Even when there is convection with a laminar flow of electrolyte, the transport in the (assumed) stagnant layer adjacent to the electrode is by diffusion... 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

For thermal conductivity, the SI units are W/(m K). In laminar flow, the thermal conductivity, A, and the diffiisivity, D, are constant with respect to their respective gradients. Eqn. (3.4-3) indicates that the diffusion flux of solute [mol A/(m2 s)] is proportional to the transverse concentration gradient, with D as the proportionality constant. The dimensions of D are length2/ time, and its units are m2/s in the SI. Eqn. (3.4-2) states that the heat flux [in J/ (m2.s) = W/m2] is proportional to the temperature gradient, with a constant a = A/(p cp) that is called the thermal diffusivity. Its dimensions are length2/time and its SI units are m2/s. Thus, it is not unexpected that the coefficient v = p/p has the same dimensions and units, m2/s. The coefficient v is called the kinematic viscosity, and it clearly has a more fundamental significance than the dynamic viscosity. The usual unit for kinematic viscosity is the Stokes (St) and submultiples such as the centistokes (cSt). In many viscometers, readings... 

Thus the electrolysis process is controlled by a combination of (1) mass transport of O to the edge of the stagnant layer by the laminar flow, and (2) subsequent diffusion of O across the stagnant layer under the influence of the concentration gradient (caused by electrolysis of O to R at the electrode surface) to satisfy the Nernst equation. 

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