Entrance length, hydrodynamic

As a rough guide for plate heat exchangers, the rate of the hydrodynamic entrance length to the corresponding thermal entrance length is given by... 

This expression has been used to correlate results obtained in a rectangular channel, Eq. (14) in Table VII, the hydrodynamic entrance length of the channel (Le = 0.0575 d Red) being too short to assure a fully developed flow. The results were still 24/ high, compared with Eq. (31) modified by a relaxation assumption ... 

Equation (31) has also been used to correlate results obtained in annular channels with insufficient hydrodynamic entrance length (B2, B3). Some of the data fit Eq. (31) with a slightly smaller coefficient. A similar expression, including terms for the hydrodynamic entrance length, was used by Pickett and Ong (P3a). 

Hydrodynamically developing flow is isothermal fluid flow in which the velocity profile varies in the flow direction. Fluid flow from the entrance of the duct to the location at which the fully developed velocity profile forms is referred to as hydrodynamically developing flow. The distance over which the velocity distribution changes and the hydrodynamic boundary layer develops is referred to as the hydrodynamic entrance length. The friction factor in the hydrodynamic entrance is a function of the axial location. 

The hydrodynamic entrance length Lhy is defined as the axial distance required to attain 99 percent of the ultimate fully developed maximum velocity when the entering flow is uniform. The dimensionless hydrodynamic entrance length is expressed as Lty — LhyIDh Re. 

Corresponding to the hydrodynamic entrance length, the thermal entrance length L,h is defined as the axial distance needed to achieve a value of the local Nusselt number Nu which is 1.05 times the fully developed Nusselt number value. The dimensionless thermal entrance length is expressed as L+,h = L,hl(Dh Pe). 

Based on these results, the hydrodynamic entrance length was found to be ... 

Solutions for the Flow with Re < 400. It has been found that the effects of axial momentum diffusion and radial pressure variation are significant only in the duct inlet of x < 0.005. Chen [11] obtained the dimensionless hydrodynamic entrance length L y and the fully developed incremental pressure drop number K(< >), which are given by... 

Solutions for Small Reynolds Number (Re — 0) Flows. For small Reynolds number flows, such as creeping flow, in which viscous forces completely overwhelm the inertia forces, the hydrodynamic entrance length L y has been found to approach the value of 0.60 as Re 0 with the uniform flow at the inlet of the circular duct [1]. 

The hydrodynamic entrance length LUDh can be calculated by the following equation [87] ... 

In addition, the hydrodynamic entrance length Vhy, recommended by Shah and London [1], is given in Table 5.15. 

TABLE 5.29 Turbulent Flow Hydrodynamic Entrance Lengths for Smooth, Eccentric Annular Ducts [120]... 

The expression for hydrodynamic entrance length Vhy developed by Bhatti [181] is as follows ... 

Prakash and Liu [266] have numerically analyzed laminar flow and heat transfer in the entrance region of an internally finned circular duct. In this study, the fully developed / Re is compared with those reported by Hu and Chang [265] and Masliyah and Nandakumar [267]. The incremental pressure drop K(°°) and hydrodynamic entrance length L+hy together with /Re are given in Table 5.48, in which the term n refers to the number of fins, while / denotes the relative length of the fins. 

Simultaneously developing flow in annular sector ducts for air (Pr = 0.7) has been analyzed by Renzoni and Prakash [287]. In their analysis, the outer curved wall is treated as adiabatic, and the boundary condition is imposed on the inner curved wall as well as on the two straight walls of the sector. The fully developed friction factors, incremental pressure drop numbers, hydrodynamic entrance lengths, and thermal entrance lengths are presented in Table 5.62. The term L y used in Table 5.62 is defined as the dimensionless axial distance at which /app Re = 1.05/ Re. The fully developed Nusselt numbers are represented by Nu/< in order not to confuse the reader since the thermal boundary condition applied in Renzoni and Prakash [287] is different from those defined in the section. 

L dimensionless hydrodynamic entrance length=LkyIDh Re Ln thermal entrance length, m... 

TABLE 10.4 Hydrodynamic Entrance Length in Laminar Pipe Flow [39]... 

The hydrodynamic entrance length for purely viscous fluids in turbulent pipe flow is approximately the same as for newtonian fluids, being of the order of 10 to 15 pipe diameters... 

This behavior can be seen in Fig. 10.22, which shows the fully established turbulent friction factor as a function of Reynolds number Re for concentrations ranging from 10 to 1000 wppm of polyacrylamide in Chicago tap water. This series of measurements, which were taken in a tube 1.30 cm in diameter, revealed that the hydrodynamic entrance length varied with concentration, reaching a maximum of 100 pipe diameters at the higher concentrations. Therefore, the friction factors shown in Fig. 22 were measured at values of xld greater than 100. The asymptotic friction factor is reached at concentrations of approximately 50 wppm of polyacrylamide in tap water for the tube diameter used in the test program [50, 93]. The... 

Analytical values of L hy and K(°°) are also listed in Table 17.14. The hydrodynamic entrance length Lhy [dimensionless form is L+hy = Lhyl(Dh Re)] is the duct length required to... 

Turbulent Flow. /app Re for turbulent flow depends on Re in addition to x+. A closed-form formula for/app Re is given in Refs. 46 and 48 for developing turbulent flow. The hydrodynamic entrance lengths for developing laminar and turbulent flows are given by Refs. 44 and 46 as... 

In microstructured channels, laminar flow can be considered when the hydrodynamic entrance length remains short compared to the channel length. Therefore, the axial dispersion coefficient can be estimated with a relation developed by Aris [20] and Taylor [21] ... 

The hydrodynamic entrance length (Lhy) is defined as the channel length required to achieve... 

In Table 1 are quoted the maximum Reynolds number, the corresponding hydrodynamic entrance length, and the length to entrance length ratio (L/Lhy) as a function of the internal diameter of the microtube for water as working fluid. The results quoted in Table 1 allow one to highlight the following main aspects ... 

It has been numerically demonstrated that Eq. 5 no longer holds for creeping flows where the dimensionless hydrodynamic entrance length seems to be a function of the Reynolds number. It has been demonstrated that for very low Reynolds numbers, the hydrodynamic entrance length has to be calculated by means of the following equation ... 

Equation 6 is useful in order to calculate the hydrodynamic entrance length for liquid flows characterized by very low Reynolds numbers (Re < 10) through small microchannels. 

The hydrodynamic entrance length is also influenced by the cross-sectional geometry of the microchannel. In order to take into account the role played by the geometry of the microchannel cross section on the hydrodynamic entrance length, Eq. 5 can be replaced by the following approximate equation ... 

Consider a fluid moving through a pipe in the laminar flow regime. The wall of the pipe contains an electrode, located at a certain distance from the entry (Figure 4.26). The flow rate at the wall is zero. In the vicinity of the walls, viscous forces slow down the fluid as soon as it enters the pipe. Thus a gradient in flow rate is established across a layer referred to as the hydrodynamic boundary layer. Its thickness increases with the distance from the inlet. The boundary layers of opposing walls eventually meet after a distance L, called the hydrodynamic entrance length. From this point onward, the flow profile is observed to be parabolic. For a tube, Lh has a value of about 70 times its diameter. 

The hydrodynamic entrance length (Lhy) is defined as the channel length required to achieve a maximum value of the velocity of 99% of the corresponding fully developed magnitude when the entering flow is uniform. 

For conventional channels the hydrodynamic entrance length can he estimated for laminar flows hy means of the following approximated equation ... 

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