Entry hydrodynamic

The fact that the appearance of a wall slip at sufficiently high shear rates is a property inwardly inherent in filled polymers or an external manifestation of these properties may be discussed, but obviously, the role of this effect during the flow of compositions with a disperse filler is great. The wall slip, beginning in the region of high shear rates, was marked many times as the effect that must be taken into account in the analysis of rheological properties of filled polymer melts [24, 25], and the appearance of a slip is initiated in the entry (transitional) zone of the channel [26]. It is quite possible that in reality not a true wall slip takes place, but the formation of a low-viscosity wall layer depleted of a filler. This is most characteristic for the systems with low-viscosity binders. From the point of view of hydrodynamics, an exact mechanism of motion of a material near the wall is immaterial, since in any case it appears as a wall slip. 

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). 

Cullen and Davidson ) studied the absorption of carbon dioxide into a laminar jet of water. When the water issued with a uniform velocity over the cross-section, the measured rate of absorption corresponded closely with the theoretical value. When the velocity profile in the water was parabolic, the measured rate was lower than the calculated value this was attributed to a hydrodynamic entry effect. 

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 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,... 

The Circular Tube Thermal-Entry-Length, with Hydrodynamically Fully Developed Laminar Flow... 

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. 

When the flow is constrained to a channel bounded on both faces, then at a sufficient distance from the entry point, xe, the two hydrodynamic boundary layers associated with each wall overlap and the fluid velocity varies parabolically with position across the channel. 

W.R. Bowen and A.O. Sharif, The hydrodynamic and electrostatic interactions on the approach and entry of a charged spherical particle to a charged cylindrical pore in a charged planar surface—with implications for membrane separation processes, Proc. R. Soc. Lond. A 452 (1996) 2121-2140. 

Of course, the rest of the star is plugged in at the entry, and offers a large resistance. But this resistance is finite and if the arm length N is long enough, the hydrodynamic force over N/gD blobs, ]DV N/gD, can be made very large at some point, it will be able to overcome the resistance from the (/-l) other arms. 

All our discussion has assumed that our branched structures are robust. But they are in fact subjected to significant hydrodynamic tensions. Consider a star polymer under a current J largerthan Jcl. Then we shall meet situations where one (or a few) arm(s) has entered the tube, while the central nodule is still stuck at the entry as in Fig. 3b. If is the length of tube which has been invaded by the arm(as in Sect. 2.3), the hydrodynamic force qv/, may be large, and can possibly reach the threshold force xm for rupture near the nodule. This leads to a critical value m ... 

Two injection techniques currently in use are hydrodynamic injection and electrokinetic injection. Hydrodynamic injection is pressure driven, and therefore all components in the sample are injected simultaneously. In contrast, with electrokinetic injection, the entry of the components of the sample into the column depends on ion mobility, charge, and concentration. 

After a distance (the entry length) from the entrance, the hydrodynamic layers from each wall merge, and the flow regime established is laminar in form (Compton and Coles, 1983 Albery and Bruckenstein, 1983) in which separate... 

We start this chapter with a general physical description of internal flow, and the average velocity and average temperature. We continue with the discussion of the hydrodynamic, and thermal entry lengths, developing flow, and fully developed flow. We then obtain the velocity and temperature profiles for fully developed laminar flow, and develop relations for the friction factor and Nusselt nmnber. Hinally we present empirical relations for developing and full developed flows, and demonstrate their use. 

B Have a visual understanding of different flow regions in internal flow, such as Ihe entry and the fully developed flow regions, and calculate hydrodynamic and thermal entry lengths,... 

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)]... 

The entry length is much shorter in turbulent flow, as expected, and its dependence on the Reynolds number is weaker. In many lube flotvs of practical interest, the entrance effects become insignificant beyond a tube length of 10 diameters, and the hydrodynamic and thermal entry lengths are approximately taken to be... 

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

S-18C Consider the flow of mercury (a liquid meial) in a tube. How will the hydrodynamic and thermal entry lengths compare if the flow is laminar How would they compare if (he flow uere turbulent ... 

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. 

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