Hydrodynamic entrance effect

Suppose A + B — C with kajj = 2 in a laminar flow tubular reactor with unmixed feed. Component A with initial concentration Oju = 1 is injected at the center of the tube and component B with initial concentration fcin = 1 in injected in an annnlar ring between component A and the wall. The molar flow rates of A and B are eqnal so that the overall stoichiometry is perfect. Ignore hydrodynamic entrance effects and assume that the velocity profile is parabolic immediately after injection at location z = 0. 

The hydrodynamic entrance effect is important on the heat transfer coefficient in the coolant pass of each crossflow. Under the experimental conditions, the entrance effect may penetrate more than halfway into the crossflow. Approximations to the Graetz problem for flow entering a sinusoidal duct are available ( 8) although many of these are in the form of infinite series. Hawthorn ( ) has proposed an cina-lytical approximation to the Graetz problem of the form. 

The characteristic curve of extrudate flow including adherence to the walls, and hence representative of shghtly to moderately entangled polymer flow in sudden two-dimensional or axisymmetrical contractions [7, 32], is represented in Fig 2. It shows a slope discontinuity above a certain pressm-e level, which depends on the pol3uner-die pair considered. With low flow rates, the flow is stable. Indeed, for these regimes, allowing for entrance effects, the flow curve is in fact representative of the shear rheometry of the polymer imder consideration, at low shear rates [34]. The slope discontinuity of the head loss curve indicates a modification in the structure of flow. It will be seen that this corresponds to the triggering of a hydrodynamic instability upstream of the contraction. 

The entry length is much shorter in turbulent flow, as expected, and its dependence on the Reynolds number is weaker. In many lube flotvs of practical interest, the entrance effects become insignificant beyond a tube length of 10 diameters, and the hydrodynamic and thermal entry lengths are approximately taken to be... 

In this hydrodynamic entrance region, the apparent friction factor /app is employed to incorporate the combined effects of wall shear and the change in momentum flow rate due to the developing velocity profile. Based on the total axial pressure drop from the duct inlet (x = 0) to the point of interest, the apparent friction factor is defined as follows ... 

Solutions for the Flow with Re < 400. It has been found that the effects of axial momentum diffusion and radial pressure variation are significant only in the duct inlet of x < 0.005. Chen [11] obtained the dimensionless hydrodynamic entrance length L y and the fully developed incremental pressure drop number K(< >), which are given by... 

For values of n less than 1, this gives a velocity that is flatter than the parabolic profile of new-tonian fluids. As n approaches zero, the velocity profile predicted by this equation approaches a plug flow profile. Figure 10.5 shows the velocity profile generated by Eq. 10.43 for selected values of the power-law index n. It should be noted that the velocity profiles given in Fig. 5 are valid in the hydrodynamically fully developed region where the entrance effect can be neglected. 

Basic Consideration - To predict accurate catalyst and gas temperature distributions, some basic calculations and considerations of effective area, flow condition, and the hydrodynamic entrance region are made. 

Hydrodynamic Entrance Region. - Flow conditions for almost all analyses described above are assumed to be fully developed laminar, although there would be a hydrodynamic entrance region for each channel. In this section, the effect of hydrodynamic entrance region is reviewed. The ratio of the local mass transfer coefficient for the entrance region to that for a fully... 

Cell Size Effect. - The mass transfer coefficient is inversely proportional to width of a cell for fully developed laminar region, and surface area is proportional to cell width. This means conversion through a cell is constant. However, as flow is proportional to the square of cell width as velocity is held constant, the conversion efficiency decreases as cell width increases. This tendency is shown in Figure 10. A large cell size may offer a moderate conversion rate and a longer hydrodynamic entrance region. These are favorable characteristics for a catalyst. On the other hand, for fully developed turbulent flow, the mass transfer coefficient increases with cell width to the 0.2 power, and the surface area is proportional to the width. This means conversion increases as the cell width increases to the 0.8 power. Therefore, the cell size effect for turbulent flow is rather small compared with laminar flow. However, the conversion efficiency decreases as cell width increases as is for laminar flow. 

Two-phase models could not be directly applied to account for the entrance effects of the reactor. For shallow beds or reactors with strong jetting, the distributor effects dominate the hydrodynamic behavior of the bed and thus have to be considered differently from that of the in-bed region. The two-phase model that applies to the in-bed region could be modified to describe the entrance effects, as conceptually depicted in Fig. 25b. Detailed descriptions of distributor effects on hydrodynamics are given in Chapter 6. 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

Length Effect. The heat transfer coefficient can vary significantly in the entrance region of the laminar flow. For hydrodynamically developed and thermally developing flow, the local and mean heat transfer coefficients h, and h, for a circular tube or parallel plates are related as [19]... 

In the same vein, the variation of the diffusion coeflBcient of species across the lipid membrane cannot be explained by employing hydrodynamic expressions, such as the Stokes-Einstein relation. Here one would need to consider the free-energy barrier for entrance into the layer for each species, charged (positive or negative) and neutral the free-energy barrier is expected to be different even for same-sized species. The lipid bilayer diffusion series (LBDS) given by Eq. (12.2) is a manifestation of such microscopic effects. 

As the initially injected sample plug is normally a distance away from the capillary inlet in capDlaiy electrophoresis, the entrance region should have negligible influence on the species transport. In the region of fully developed (denoted by the subscript fd) flow field, the thermally induced pressure-driven flow causes additional hydrodynamic dispersion to the species diffusion. Analogous to Eq. 17, the effective dispersion coefficient is given by... 

Commercialization of new CFB processes for the production of chemicals— specialty or commodity—has been hindered due to scale-up concerns and operational complexity. In particular, the effect of diameter, high solids mass flux, and high pressure on gas hydrodynamics are undocumented in the open literature. Innovative research aimed at the design of new solids feeding devices and gas-solids separation techniques may reduce operational complexities. However, studies on small scale equipment must be performed prudently and documented concisely to be useful for scale-up. Scale effects at the entrance region are considerable, and sufficient attention has not been devoted to this region. 

The dispersing effect of HP systems is related to the intense turhulence in and behind the accelerating zone, as well as to cavitation (mainly for radial difiusers and nozzles) and laminar elongation at the entrance to the dispersion unit (especially for nozzles and counter-jet dispersers). The computation of the turhulent flow field in the dispersion unit commonly requires numerical tools, in particular when the effect of cavitation is to be adequately considered. Nonetheless, it is possible to provide simple estimates for the hydrodynamic stress on the particles because the power density Py can be approximated by that of a pipe flow ... 

The steady-state net flux of the fluid, due to EOF, is given by Equation 8.90. Since the fluid flow must be continuous in the system, all the fluid elements in the cis compartment must flow into the pore, assuming that all nonlinear hydrodynamic effects such as the intermittency and turbulence are absent. This means that at a distance R away from the pore entrance, the surface area of the... 

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