Entry region developing

Unfortunately, these equations cannot be modeled using the simple parallel-flow assumptions. In the entry region the radial velocity v and the pressure gradient will have an important influence on the axial-velocity profile development. Therefore we defer the detailed discussion and solution of this problem to Chapter 7 on boundary-layer approximations. 

The temperature and species profiles also have entry-region behavior. The fully coupled entry-region problem is easily formulated and can be solved using the method of lines. The details of the entry-region profiles depend on species and thermal boundary conditions as well as fluid properties. The entry length and the corresponding profile development also depend on the channel geometry. 

Consider the flow of an incompressible fluid in the entry region of a circular duct. Assuming the inlet velocity profile is flat, determine the length needed to achieve the parabolic Hagen-Poiseuille profile. Recast the momentum equation in nondimensional form, where the Reynolds number is based on channel diameter and inlet velocity emerges as a parameter. Based on solutions at different Reynolds numbers, develop a correlation for the entry length as a function of inlet Reynolds number. 

Figure 8.5 Development of the temperature profile in the thermal-entry region of a pipe.

Notter, R.H. and Sleicher, C.A., A Solution to the Turbulent Graetz Problem m. Fully Developed and Entry Region Heat Transfer Rates , Chem. ne Sci., Vol 27, pp. 2073-2093, 1972. 

When the pipe Reynolds number is greater than about 2100, the velocity boundary layer that forms in the entry region eventually turns turbulent as the gas passes down the pipe. The velocity profile becomes fully developed that is. the shape of the distribution ceases to change at about 25 to 50 pipe diameters from the entry. Small particles in such a flow are transported by turbulent and Brownian diffusion to the wall. In the sampling of atmospheric air through long pipes, wall losses result from turbulent diffusion. Accumulated layers of particles will affect heat transfer between the gas and pipe walls. 

For the entry region of the tube in the case of simultaneous development of the hydrodynamic and thermal boundary layers, the local Nusselt number Nu = Nu(2Q at constant temperature or heat flux at the wall can be calculated from the formulas [267]... 

As in the case of Newtonian flow, it is necessary to differentiate between the conditions in the entry region and in the region of fully (thermally) developed flow. 

In conclusion, a fundamental understanding of CFB hydrodynamics continues to lag behind commercial operating experience. Although CFB provides considerable advantages of operational flexibility, concerns about operational complexity and scale-up hinder commercialization of heterogeneous catalytic reactions. Concentrated research in the areas of gas-solid contact in the entry region, effect of internals, riser diameter, and solid fines content is required. The continued interaction between academia and industry, as seen in the four international conferences of CFB technology, will be necessary to further optimize commercial operations but also to further develop revolutionary industrial processes. 

Hawthorne first studied the development of flow in a straight tube to fully developed curved tube flow [63]. Development of velocity profile in the entry region... 

For smooth, cylindrical tubes (a model of straight blood vessels) with well-mixed entry flow, one can invoke the thin concentration boundary layer theory of L6v que [6] to estimate the Sherwood number in the entry region of the vessel where the concentration boundary is developing. This leads to... 

FIGURE 9.5 Schematic diagram showing the spatial distribution of the Sherwood number along the inner (I — toward the center of curvature) and outer (O — away from the center of curvature) walls of a curved vessel. In the entry region, before the secondary flow has developed, the Sherwood number follows a Leveque distribution. As the secondary flow evolves, the Sherwood number becomes elevated on the outer wall where the radial velocity of the secondary flow is toward the wall, and diminished on the inner wall where radial velocity of the secondary flow is away from the wall. 

Because of the thinness of the boundary layer, mass transfer in the entry region is very rapid, with Sherwood numbers in excess of 1000 attained near the tubular entrance (Figure 5.2). As we move away from the entrance in the downstream direction, the boundary layer gradually thickens and the Sherwood number diminishes with the one-third power of axial distance x. Eventually it levels off and attains a constant value as the fully developed region is reached (Figure 5.2). Table 5.2 lists some of the relevant Sherwood numbers obtained in ducts of various geometries and constant wall concentration. 

Duct Geometry Entry Region Fully Developed Region... 

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