Convective flow profile

Convective Flow Profile in a Hollow-Fiber Membrane 14.4.4.1 Without Cake and Polarization Layers... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

Kelsey LJ, Pillarella MR, Zydney AL (1990) Theoretical analysis of convective flow profiles in a hollow-fiber membrane bioreactor. Chem Eng Sci 45 3211-3220... 

Fig. 21 Convective flow profile associated with a rotating-disc electrode.

Fig. 25 (a) Schematic representation of a wall jet electrode and (b) convective flow profile associated with a wall jet electrode. 

Chen, T., and Goodson, R. E. "Computation of Three-Dimensional Temperature and Convective Flow Profiles for an Electric Glass Furnace." Glass Technology 13, no. 6 (1972) 161-67. 

Figure 14 (a) Schematic representation of a channel electrode, and (b) convective flow profile associated with a channel electrode having a maximum flow velocity, Vq, at the center of the channel (adapted from Adv. Phys. Org. Chem. 1999, 32, 1 this reference should be consulted for further details). 

Effect of dispersion on a sample s flow profile at different times during a flow injection analysis (a) at injection and when the dispersion is due to (b) convection ... 

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. 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

To overcome thermal entry effects, the segments may be virtually stacked with the outlet conditions from one segment that becomes the inlet conditions for the next downstream section. In this approach, axial conduction cannot be included, as there is no mechanism for energy to transport from a downstream section back to an upstream section. Thus, this method is limited to reasonably high flow rates for which axial conduction is negligible compared to the convective flow of enthalpy. At the industrial flow rates simulated, it is a common practice to neglect axial conduction entirely. The objective, however, is not to simulate a longer section of bed, but to provide a developed inlet temperature profile to the test section. 

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. 

Now imagine that a protein solution enters the flow chamber at time t = 0. The protein solution (of uniform concentration and flow velocity) begins displacing the buffer solution (Fig. 5 a). Given a parabolic velocity profile, Fig. 5 b shows the development of the concentration profile in the cell at various times after entrance. A bulletshaped concentration profile develops. This is easily observed using a dye solution or blood. No protein reaches the surface by convective flow alone. Protein is transported to the interface by diffusion. 

The study of rotating disk electrode behavior provides a unique opportunity to develop a model that predicts the effect of diffusion and convection on the current. This is one of the few convective systems that have simple hydrodynamic equations that may be combined with the diffusion model developed herein to produce meaningful results. The effect of diffusion is modeled exactly as it has been done previously. The effect of convection is treated by integrating an approximate velocity equation to determine the extent of convective flow during a given At interval. Matter, then, is simply transferred from volume element to volume element in accord with this result to simulate convection. The whole process repeated results in a steady-state concentration profile and a steady-state representation of the current (the Levich equation). 

The flow pattern itself can vary depending on the source of the flow. Ordinary pump-driven flow will create a parabolic (bullet-shaped) profile in narrow tubes, with a high velocity in the center and low velocities at the tube edges (see Section 4.4). Convective flow, on the other hand, is often countercurrent, with dense fluid sinking and less dense fluid rising (Section 4.8). 

As was done in dealing with forced convective flow over a uniform temperature plate, it is assumed that the velocity and temperature profiles are similar at all values of x, i.e., that ... 

As previously discussed, there are two limiting cases for natural convective flow through a vertical channel. One of these occurs when /W is large and the Rayleigh number is low. Under these circumstances all the fluid will be heated to very near the wall temperature within a relatively short distance up the channel and a type of fully developed flow will exist in which the velocity profile is not changing with Z and in which the dimensionless cross-stream velocity component, V, is essentially zero, i.e., in this limiting solution ... 

Velocity profiles in fully developed mixed convective flow in a vertical plane channel. Results are for assisting flow. 

Returning to the Nissan and Hansen model, they use a finite difference numerical analysis model to determine both the temperature profile of a sheet material and the subsequent water removal as it passes over the cylinder. Their experimental results match well with the predicted values. However, their experiments were limited to a cylinder surface temperature of 93.3°C. Accordingly, the maximum vapor pressure of the evaporated water is less than one atmosphere. The diffusion model advanced by Hartley and Richards is in close agreement with experimental work of Dreshfield(12l. However, the boundary conditions are still relatively uncertain since the convective flow region outside the sheet is relatively unknown. Later work by this author studies this convective flow. 

The foregoing analysis of free-convection heat transfer on a vertical flat plate is the simplest case that may be treated mathematically, and it has served to introduce the new dimensionless variable, the Grashof number, which is important in all free-convection problems. But as in some forced-convection problems, experimental measurements must be relied upon to obtain relations for heat transfer in other circumstances. These circumstances are usually those in which it is difficult to predict temperature and velocity profiles analytically. Turbulent free convection is an important example, just as is turbulent forced convection, of a problem area in which experimental data are necessary however, the problem is more acute with free-convection flow systems than with forced-convection systems because the velocities are usually so small that they are very difficult to measure. Despite the experimental difficulties, velocity measurements have been performed using hydrogen-bubble techniques [26], hot-wire anemometry [28], and quartz-fiber anemometers. Temperature field measurements have been obtained through the use of the Zehnder-Mach interferometer. The laser anemometer [29] is particularly useful for free-convection measurements because it does not disturb the flow field. 

Flow profile is perturbed by electrodes which are not flush or smooth Generally steady state. Apart from RDE, models are two-dimensional due to convection EPR, spectroscopic and photochemical methods are easily incorporated into the small, transparent channel flow cell. RDE is less versatile... 

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