Entry region

From Fig. 5.3 it may be seen that this exceeds the shear rate for non-laminar flow (approximately 30 s ) so that the entry to this region would need to be streamlined. Fig. 5.3 also shows that the extensional strain rate, k, in the tapered entry region should not exceed about 15 s if turbulence is to be avoided. 

Velocity Profile Effects Many variables can influence the accuracy of specific flow measurement methods. For example, the velocity profile in a closed conduit affects many types of flow-measuring devices. The velocity of a fluid varies from zero at the wall and at other stationary solid objects in the flow channel to a maximum at a distance from the wall. In the entry region of a conduit, the velocity field may approach plug flow and a constant velocity across the conduit, dropping to zero only at the wall. As a newtonian fluid progresses down a... 

Fig. 4.8 Illustration of the axial-velocity profile in the entry region of a circular duct.

We have just discussed several variations of the flow in ducts, assuming that there are no axial variations. In fact there well may be axial variations, especially in the entry regions of a duct. Consider the situation illustrated in Fig. 4.8, where a square velocity profile enters a circular duct. After a certain hydrodynamic entry length, the flow must eventually come to the parabolic velocity profile specified by the Hagen-Poiseuille solution. 

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. 

As illustrated in Fig. 5.2, the classic Jeffery-Hamel flow concerns two-dimensional radial flow in a wedge-shaped region between flat inclined walls. The flow may be directed radially outward (as illustrated) or radially inward. The flow is assumed to originate in a line source or terminate in a line sink. Velocity at the solid walls obeys a no-slip condition. In practice, there must be an entry region where the flow adjusts from the line source to the channel-confined flow with no-slip walls. The Jeffery-Hamel analysis applies to the channel after this initial adjustment is accomplished. 

Fig. 7.1 Illustration of the velocity and temperature profiles in the entry region of a cylindrical channel. Gases enter the channel with uniform velocity and temperature profiles. The no-slip condition causes a zero velocity at the wall, and the heat transfer from the hot wall increases the gas temperature.

Consider the flow of an incompressible fluid in the entry region of a circular duct, assuming that the inlet velocity profile is flat. As is often the case, the problem can be generalized by casting into a nondimensional form. A set of nondimensional variables may be chosen as... 

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. 

Dynamics of a Vaporizing a (10-50 (mu) Water) Droplet in Laminar Entry Region of a Straight Channel (with Isothermal Walls, as in a Dry Cooling Tower)... 

J. L. White and A. Kondo, Rheological Properties of Polymer Melts and Flow Patterns During Extrusion through a Die Entry Region, J. Appl. Polym. Sci., in press. 

Melt fracture occurs when the rate of shear exceeds a critical value for the melt concerned at a particular temperature (that is, the critical shear rate ). There is a corresponding critical shear stress and the relevant point on the flow curve (or the shear rate-shear stress diagram) is known as the critical point. It is believed that it is reached in the die entry region (that is, where material is being funnelled from the die reservoir into the capillary of a capillary rheometer)—which, in an extruder, corresponds with the point at which melt moves into the die parallel portion of the die. Some further complicating effects may occur at the wall of the die. 

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. 

Our discussion on the aspiration of linear chain has been basedon the behavior inside the tube. It is also helpful to consider what happens outside (in the entry region). Here, in the most naive picture, we have a convergent flow of velocity v(r)=J/r2, where r is the distance to the center of the entry disc. The velocity gradient is y = Vv J / r3. At a certain radius rc, this is such that yTz = 1, where Tz is the relaxation time of a coil Tz = T)R3 /kT. At distances rdeforms affinely the final lateral dimension of the distorted coil is Tj D/rcRg. When Tj becomes as small as D, the chain can get in. This corresponds to rc Rg and J Jc, where Jc is again given by Eq.(3). Thus this different picture leads to the same conclusion. 

The spinneret is a type of die principally used in fiber manufacture. It is usually a metal plate with many small holes (or oval, etc.) through which a melt is pulled and/or forced. They enable extrusion of filaments of one denier or less. Conventional spinneret orifices are circular and produce a fiber that is round in cross section. They can contain from about 50 to 110 very small holes. A special characteristic of their design is that the melt in a discharge section of a relatively small area is distributed to a large circle of spinnerets. Because of the smaller distance in the entry region of the distributor, dead spaces are avoided, and the greater distance between the exit orifices makes for easier threading.143... 

Chum flow-segmentedflow transition. This transition is an entry region phenomenon associated with the existence of slug flow further along the pipe, and is described by Taitel et al. [3] as follows ... 

A cross-flow and a parallel-channel structure are prepared in such a way that colorization of the surface takes place upon instantaneous chemical reaction with ammonia, which is fed as a pulse to an air flow passing over the investigated structures. It can be observed with the parallel-channel structure that there is strong colorization at the inlet, due to the flow phenomena associated with the entry region. The colorization decreases rapidly very soon thereafter, due to the establishment of a laminar boundary layer. Mass transfer is by molecular diffusion only, and the reactor dimensions necessary to transfer all of the ammonia from the bulk gas to the surface are considerably greater than those of the body examined. 

Proof that the low intensity of colorization downstream of the entry region in the parallel-channel structure is due to formation of a laminar boundary layer, rather than to the lack of further ammonia for transfer, is offered by a second experiment conducted by Gaiser and illustrated in Fig. 8. Here a cross-flow structure is placed immediately downstream of the parallel-channel structure, and the mass transfer rate rises dramatically, indicating that much ammonia remained in the gas at the parallel-channel structure outlet. 

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