Entry lengths

For the common problem of heat transfer between a fluid and a tube wall, the boundary layers are limited in thickness to the radius of the pipe and, furthermore, the effective area for heat flow decreases with distance from the surface. The problem can conveniently be divided into two parts. Firstly, heat transfer in the entry length in which the boundary layers are developing, and, secondly, heat transfer under conditions of fully developed flow. Boundary layer flow is discussed in Chapter 11. 

The number n must be an integer (i.e., n = 1,2,3,...) and channel flow is characterized by lengths that are at least several diameters in magnitude. The latter constraint does not preclude operation in the entry length mode. 

The thermal entry length should be considered by comparison between experimental and numerical results. 

Equation (26) gives values of Lp equal to 10-16 pipe diameters for Re between 2500 and 20 000. According to Davies [91] these are the lowest estimate of the hydraulic entry length. In practice the observed values are often in the range of 25 to 100 pipe diameters. 

H Slug to churn Entry length to develop stable slug... 

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). 

We consider steady-state, one-dimensional laminar flow (q ) through a cylindrical vessel of constant cross-section, with no axial or radial diffusion, and no entry-length effect, as illustrated in the central portion of Figure 2.5. The length of the vessel is L and its radius is R. The parabolic velocity profile u(r) is given by equation 2.5-1, and the mean velocity u by equation 2.5-2 ... 

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,... 

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 solution to the entry-length problem is illustrated in Fig. 4.17, where the nondimensional temperature profiles are shown at selected positions z along the channel. 

Figure 7.6 illustrates nondimensional velocity profiles that are computed for a Reynolds number of 1000. By solving the system for various Reynolds numbers we can determine that the entry length is... 

Since the velocity profile approaches the Hagen-Poiseuille profile asymptotically, the factor 0.05 depends on a criterion for deciding how close the profile needs to be to the Hagen-Poiseuille profile (e.g., the maximum velocity is within 1% of the steady state maximum velocity). In any case, the entry length scales linearly with the Reynolds number. 

The temperature and species profiles also have entry-region behavior. The fully coupled entry-region problem is easily formulated and can be solved using the method of lines. The details of the entry-region profiles depend on species and thermal boundary conditions as well as fluid properties. The entry length and the corresponding profile development also depend on the channel geometry. 

Consider the flow of an incompressible fluid in the entry region of a circular duct. Assuming the inlet velocity profile is flat, determine the length needed to achieve the parabolic Hagen-Poiseuille profile. Recast the momentum equation in nondimensional form, where the Reynolds number is based on channel diameter and inlet velocity emerges as a parameter. Based on solutions at different Reynolds numbers, develop a correlation for the entry length as a function of inlet Reynolds number. 

Assuming an initially flat velocity profile, calculate and plot the Nusselt number as a function of the inverse Graetz number. Compare the soultions for a range of Prandtl numbers. Explain why the Graetz number may not be an appropriate scaling for the combined entry-length problem. 

Laminar fluid flow in tubes has been described by Levich [ 3 ]. An entry length, le, is necessary to establish Poiseuille flow, given approximately by... 

The construction of tubular electrodes may be divided into two basic types integral and demountable. Channel electrodes are only of the latter type. Final dimensions must satisfy the entry length criterion for Poiseuille flow (pp. 370 and 372). 

Tailby and Portalski (T2), 1960 Reports on extension of Kapitsa theory (K7) to give increase in interfacial area due to waves experimental measurements of Am, for wave inception, entry length, and increase in interfacial area. 

Chien (C7), 1961 Investigation of liquid film structure and pressure drops in vertical downward cocurrent gas/film flow. Data on surface waves, entry length, energy dissipation in film, film thicknesses (local and mean), pressure drop. 

Shimizu, A., Echigo, R., Hasegawa, S. and Hishida, M. (1978). Experimental Study on the Pressure Drop and the Entry Length of the Gas-Solid Suspension Flow in a Circular Tube. Int. J. Multiphase Flow, 4, 53. 

In a tube or channel a certain entry length, /e, is necessary before the parabolic Poiseuille flow is obtained (Fig. 8.26). 

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

Figure 8.4 Sketch for the thermal-entry length problem.

Table 8.1 Infinite-series solution functions for the circular tube constant surface temperature thermal-entry length.

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. 

Azer, N.Z., Thermal Entry Length for Turbulent Row of Liquid Metals in Pipes with Constant Wall Heat Rux , Trans. ASME Serv. C, J. Heat Transfer, Vol. 90. pp. 483-485, 1968. 

---