Entry lengths laminar flow

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. 

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 ... 

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

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... 

Entry length Fully developed laminar (L) flow over electrode surface... 

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)]... 

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 ... 

The unusual behavior of Nu decreasing with increasing Re in the laminar regime in microchannels may alter the status of thermal development and hence the conventional thermal entry length, since the variation of the heat transfer coefficient along the flow is a variation of the boundary condition. 

Step 2 Set up the entry problem as was done for laminar flow, except put in a kinematic viscosity of 10 , a radius of 0.025, an inlet velocity of 2 m/s, and a length of 3 m. Note that the momentum equations has been divided by density ... 

Engineering correlations have been presented for pressure drop and entry length in common geometrical flow channels with laminar flow. The correlations differ from their counterparts with turbulent flow (available in handbooks, depending upon density and velocity) since in slow flow the pressure drop is proportional to the velocity and the fluid viscosity. The entry length in laminar flow is also longer than is the case for turbulent flow, which means that some section of flow in microfluidics is in the entry region. 

Finally, the true mean heat transfer coefficient that also takes into account the laminar flow entry length effect is given by ... 

Laminar Flow. Thermal entry length solutions with developed velocity profiles are summarized in Refs. 19 and 44 for a large number of practically important flow passage geometries with extensive comparisons. 

Turbulent Flow. The thermal entry length solutions for smooth ducts for several cross-sectional geometries have been summarized [46]. As for laminar flow, the Nusselt numbers in the thermal region are higher than those in the fully developed region. However, unlike laminar flow, Nu,x and NuxHi are very nearly the same for turbulent flow. The local and mean Nusselt numbers for a circular tube with and boundary conditions are [46] ... 

Entry Lengths and Boundary Layers It is the case that most flow investigations at the microscale happen in the laminar flow regime with developing boundary layer (combined entry lengths). As for macroscale cases, the velocity entry length Le is often simply calculated by... 

In the trajectory analysis, fiJly developed laminar flow is assumed to exist which may be justified if Reynolds numbers are kept low. The entry length required for the development of laminar flow is well known to be given by ... 

When a fluid flows through a pipe, boundary layers form at the entry point and grow in thickness along the length of the pipe until they meet in the centre. The boundary layer thus fills the entire pipe and the flow is termed fully developed. If the boundary layer is still laminar then laminar flow persists. 

The thermal entry length 1th for laminar flow can be calculated by [40]... 

If the velocity profile at the entrance region of a tube is flat, a certain length of the tube is necessary for the velocity profile to be fully established. This length for the establishment of fully developed flow is called the transition length or entry length. This is shown in Fig. 2.10-6 for laminar flow. At the entrance the velocity profile is flat i.e., the velocity is the same at all positions. As the fluid progresses down the tube, the boundary-layer thickness increases until finally they meet at the center of the pipe and the parabolic velocity profile is fully established. 

The pressure drop or friction factor in the entry length is greater than in fully developed flow. For laminar flow the friction factor is highest at the entrance (L2) and then decreases smoothly to the fully developed flow value. For turbulent flow there will be some portion of the entrance over which the boundary layer is laminar and the friction factor profile is difficult to express. As an approximation the friction factor for the entry length can be taken as two to three times the value of the friction factor in fully developed flow. 

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 correlations provide good predictions and are suitable for electrode lengths of up to about 20d. Equation 2.21 can also be used for developing laminar flow. For design purposes, entry effect can be ignored. 

For laminar flow, the thermal entry length is approximately given by (Cengel, 2002) ... 

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

The entry length Leto achieve a near steady-state velocity profile in laminar flow where inertia is important is given approximately by... 

With such a flat plate, the boundaiy layer will increase in thickness in-dcfinitely if slowly (Fig. 1.10(c))l On the other hand if the flow is in a restricted channel (e.g. a circular-tube or a paraltel-plate cell) the boundary layers at the two walls must merge at some point and beyond, a steady-state situation or fully developed laminar flow will result (Fig. Ltl). Fundamental mass transport studies in electrolytic cells are usually carried out in cells with an entry length without electrodes so that the boundary-layer thickness is uniform over the current-carrying surface. 

For a suitably small Reynolds number. Re, to establish laminar flow, the transition from plug flow (on entry) to parabolic flow is established over an entry length of... 

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