Entry lengths hydrodynamic

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. 

The entry-length region is characterized by a diffusive process wherein the flow must adjust to the zero-velocity no-slip condition on the wall. A momentum boundary layer grows out from the wall, with velocities near the wall being retarded relative to the uniform inlet velocity and velocities near the centerline being accelerated to maintain mass continuity. In steady state, this behavior is described by the coupled effects of the mass continuity and axial momentum equations. For a constant-viscosity fluid, 

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. 

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It should be emphasized that these results are applicable only to fully developed flow. However, if the fluid enters a pipe with a uniform ( plug ) velocity distribution, a minimum hydrodynamic entry length (Lc) is required for the parabolic velocity flow profile to develop and the pressure gradient to become uniform. It can be shown that this (dimensionless) hydrodynamic entry length is approximately Le/D = 7VRe/20. 

Just as for laminar flow, a minimum hydrodynamic entry length (Le) is required for the flow profile to become fully developed in turbulent flow. This length depends on the exact nature of the flow conditions at the tube entrance but has been shown to be on the order of Le/D = 0.623/VRe5. For example, if /VRe = 50,000 then Le/D = 10 (approximately). 

The Graetz problem considers the thermal entry of an incompressible fluid in a circular pipe with a fixed velocity profile. The situation is illustrated in Fig. 4.16. The Graetz problem is a classic problem in fluid mechanics, and one that permits an analytic solution. After some hydrodynamic entry length, the velocity profile approaches a steady profile that is,... 

Shah, R.K., A Correlation for Laminar Hydrodynamic Entry Length Solutions for Circular and Non-Circular Ducts ,Fluids Eng. Trans. ASME, Vol. 100, p. 177,1978. 

During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr = I, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr > I, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, the hydrodynamic entry length is smaller than the thermal entry length. The opposite is tnie for fluids with Pr < 1 such as liquid metals. 

The hydrodynamic entry length is usually taken to be the distance from the lube entrance where the wall shear. stress (and thus the fficliou factor) reaches within about 2 percent of the fully developed value. In laminar flow, the hydrodynamic and thermal entry lengths are given approximately as (see Kays and Crawford (1993) and Shah and Bhatli (1987)]... 

For Re = 20, the hydrodynamic entry length is about the size of the diameter, but increases linearly with velocity. In the limiting case of Re = 2300, the hydrodynamic entry length is 115D. 

In turbulent flow, the intense iqjxing during random fluctuations usually overshadows the effects of molecular diffusion, and therefore the hydrodynamic and thermal entry lengths ate of about the same size and independent of the PflLndil number. The hydrodynamic entry length for turbulent flow can be detennined from [see Bbatti and Shah (1987) and 7.hi-qing (1982)]... 

How is the hydrodynamic entry length defined for flow in a tube Is the entry length longer in laminar or turbulent flow ... 

Figure 20 shows the test section and its instrumentation. Both ends are equipped with 90° manifolds for the fluid distribution. The tube diameter used for these manifolds is ten times that of the minichannels in order to suppress fluid distribution problems. The test section is made of two functional parts an adiabatic section for the hydrodynamic entry length and a heating zone placed between two pairs of electrodes brazed on the tube to produce a Joule effect heating. 

In turbulent flow, the boundary conditions constant wall temperature and constant heat flux lead to approximately the same mean Nusselt numbers. Correlations in the far turbulent regime (Re> 10 ) are noted here. The hydrodynamic entry length is approximately independent of Re, so that an approximation for fully turbulent flow after length x can be made for... 

Doughty, J.R., Perkins, H.C., 1970. Hydrodynamic entry length for laminar flow between parallel porous plates. ASME J. Appl. Mech. 37, 548-550. 

Hydrodynamic Entry Length The length required for the velocity profile to become fully developed. 

Criterion for entry length Criterion for hydrodynamic entry length for laminar flow ... 

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