Profile in the entrance region

FIGURE 1.5 The development of temperature profile in the entrance region of a tube. 

The rate of developments of velocity and temperature profiles in the entrance region depends on the Prandtl number defined as the ratio of fluid... 

Many additional studies have been conducted with the boundary layer model by taking into account the variation of physical properties with composition (or temperature) and by relaxing the assumption that Vy = 0 at y = 0 when mass transfer is occurring. Under conditions of high mass transfer rates one finds that mass transfer to the plate decreases the thickness of the mass transfer boundary layer while a mass flux away from the wall increases the boundary layer thickness The analogous problem of uniform flux at the plate has also been solved. Skelland describes a number of additional mass transfer boundary layer problems such as developing hydrodynamic and mass transfin- profiles in the entrance region of parallel flat plates and round tubes. 

Experimental and Theoretical Study of the Simultaneous Development of the Velocity and Concentration Profiles in the Entrance Region of a Monolithic Convertor... 

Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fuUy developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr, 71, 472 76 [1993])... 

Temperature Profile and the Nusselt Number 467 Constant Surface Heat Flux 467 Constant Surface femperaturc 468 I aminar Ftow in Noncircular Tubes 469 Developing laminar Flow in the Entrance Region 470... 

Consider steady flow of a fluid through a circular pipe attached to a large tank. The fluid velocity everywhere on the pipe surface is zero because of the no-slip condition, and the flow is two-dimensional in the entrance region of the pipe since the velocity changes in both the r- and z-directions. The velocity profile develops fully and remains unchanged after some distance from the... 

The developmeni of ihe velocity profile in a circular pipe. V V(r, z) and thus the flow is two-dimensional in the entrance region, and becomes one-dimensional downstream when the velocity profile fully develops and remains unchanged in the flow direction, = V r). 

Simultaneously developing flow is fluid flow in which both the velocity and the temperature profiles are developing. The hydrodynamic and thermal boundary layers are developing in the entrance region of the duct. Both the friction factor and Nusselt number vary in the flow direction. Detailed descriptions of fully developed, hydrodynamically developing, thermally developing, and simultaneously developing flows can be found in Shah and London [1] and Shah and Bhatti [2],... 

Parallel Plate Cross Section For laminar flows under the T boundary condition (see Convective Heat Transfer in Microchannels ) with a fully developed velocity profile (hydrodynamically fully developed and thermally developing flow Fig. 3), the following correlation, proposed originally by Shah for conventional channels, can be useful to calculate the mean Nusselt number in the entrance region of parallel plate microchannels as a function of the dimensionless axial coordinate z ... 

When the initial velocity profile is uniform and there are no abrupt contraction (Zc = 0) and expansion = 0), one can take into account the effects of momentum change in the entrance region by using K (x>) which includes the increased wall shear in the entrance region but separates out the downstream fully developed flow pressure drop due to wall shear to give... 

If the velocity profile at the entrance region of a tube is flat, a certain length of the tube is necessary for the velocity profile to be fully established. This length for the establishment of fully developed flow is called the transition length or entry length. This is shown in Fig. 2.10-6 for laminar flow. At the entrance the velocity profile is flat i.e., the velocity is the same at all positions. As the fluid progresses down the tube, the boundary-layer thickness increases until finally they meet at the center of the pipe and the parabolic velocity profile is fully established. 

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