Boundary layers fully developed region

Thin boundary layers provide the highest values of heat flow density. Because the boundary layer gradually develops upstream from the inlet point, the heat flow density is highest at the inlet point. Heat flow density-decreases and achieves its final value in the region of fully developed flow. The correction is noted in the equations by means of the quotients d/.L and d/x. 

Equation 17 gives the asymptotic limit of the boundary layer modulus in the far-downstream region of the fully developed region i.e., for a given value for the wall Pdcldt number, the modulus C /C eventually becomes independent of E and hence Z and... 

Figure 3. Asymptotic limit of boundary layer modulus in fully developed region fsee Equation 17)...

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],... 

Gas flow through the small channels of a honeycomb matrix is nearly always laminar, and analytical solutions are available for heat and mass transfer for fully developed laminar flow in smooth tubes. In the inlet region, where the boundary layers are developing, the coefficients are higher, and numerical solutions were combined with the analytical solution for fully developed flow and fitted to a semitheoretical equation [14] ... 

For pipe flow, velocity and temperature profiles in a developing and fully developed region are illustrated in Figure 22.9. The entrance length where the merging of the momentum boundary layer access is given by... 

The character of the fully developed region is determined by the character of the flow in the boundary layer at the cusp (Fig. 2.4). If it is laminar, the fully developed flow will be laminar (Fig. 2.4a). Or the flow may be transitional (Fig. 2.4b) or completely turbulent (Fig. 2.4c). The latter, in a pipe, channel, or annulus, has three zones of flow behavior which reflect those of the turbulent boundary layer. The velocity distribution depicted in Fig. 2.1 corresponds to the latter regime. 

Within the thermal entrance region, the local value of the heat transfer and the local Nusselt number N% steadily decreases as the thermal boundary layer increases until in the thermally fully developed region Nu reaches a constant value (Example 3.2.4). However, the value of the local Nusselt number is of little value in... 

Figure 3.2.23 Development of the velocity boundary layer of a fluid flowing in an empty tube. The velocity profile in the hydrodynamically fully developed region is parabolic in laminar flow (as shown) and somewhat blunt in turbulent flow (see Figure 3.2.22).

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. 

At some distance away from the entrance, the boundary layers meet and flow is assumed as viscous over the entire cross section of the channel. The internal flow is categorized into two distinct regions (i) hydrodynamic entrance region where velocity profile varies with the axial length of the channel and (ii) hydro-dynamic fully developed region where velocity profile remains invariable with the longitudinal distance along the channel, or becomes fully developed. 

For internal flows, thermal boundary layers develop from both top and bottom surfaces and develop into two regions thermal entry length and thermal fully developed regions similar to hydrodynamic internal flow as shown in Figure 6.8. 

Therefore, the average heat transfer coefficient over the boundary layer build-up region is equivalent to twice the local heat transfer coefficient in the fully developed region. The solution of the Polhausen equation clearly bounds the 13 percent increase in Nusselt number for compact heat exchanger modeling. In fact, increasing the Sellars solution (Nu=4.36) by a factor of two would also bound the proposed Nusselt number. 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

This expression is applicable only to the region of fully developed flow. The heat transfer coefficient for the inlet length can be calculated approximately, using the expressions given in Chapter 11 for the development of the boundary layers for the flow over a plane surface. It should be borne in mind that it has been assumed throughout that the physical properties of the fluid are not appreciably dependent on temperature and therefore the expressions will not be expected to hold accurately if the temperature differences are large and if the properties vary widely with temperature. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

For fully developed turbulent flow in a pipe, the whole of the flow may be regarded as lying within the boundary layer. The cross-section can then conveniently be divided into three regions ... 

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. 

When the Reynolds number based on tube diameter is greater than 2100, the boundary layer becomes turbulent at some distance from the inlet. The transition usually occurs at a Reynolds number, based on distance from the entrance, Rcj, of between 10 and 10, depending on the roughness of the wall and the level of turbulence in (he mainstream. As shown in Fig, 4,11, the deposition rate tends to follow the development of the turbulent boundary layer. No deposition occurs until Re is about 10- the rate of deposition then approaches a constant value at Re = 2 x 10 in the region of fully developed turbulence. On dimensional ground.s. the deposition velocity at a given pipe Reynolds number can be assumed to be a function of the friction velocity, if, kinematic viscosity, v, and the particle relaxation time, m/f ... 

Myoglobin, cytochrome-C, inulin, and vitamin B-12 were the solutes studied in saline, calf serum, and BSA systems at 37 C and pH 7.4. Observed solute rejections were corrected to intrinsic values by using uniform-wall-flux boundary layer theory for the developing and fully-developed asymptotic regions. The Splegler-Kedem equation ( ) for rejection versus volume flow was used to calculate reflection coefficients and diffusive permeabilities for each solute. There was no significant difference between rejection parameters measured in saline and protein solutions. 

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