Laminar flow in a tube

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

Careful study of various fluids in tubes of different sizes has indicated that laminar flow in a tube persists up to a point where the value of the Reynolds number (NRt = DVp/n) is about 2000, and turbulent flow occurs when NRe is greater than about 4000, with a transition region in between. Actually, unstable flow (turbulence) occurs when disturbances to the flow are amplified, whereas laminar flow occurs when these disturbances are damped out. Because turbulent flow cannot occur unless there are disturbances, studies have been conducted on systems in which extreme care has been taken to eliminate any disturbances due to irregularities in the boundary surfaces, sudden changes in direction, vibrations, etc. Under these conditions, it has been possible to sustain laminar flow in a tube to a Reynolds number of the order of 100,000 or more. However, under all but the most unusual conditions there are sufficient natural disturbances in all practical systems that turbulence begins in a pipe at a Reynolds number of about 2000. 

By analogy with laminar flow in a tube, the friction factor in laminar flow would be... 

The solution to the problem of determining the wall shear rate for a non-Newtonian fluid in laminar flow in a tube relies on equation 2.6. 

Recall that the wall shear rate for a Newtonian fluid in laminar flow in a tube is equal to —8w/d,. In the case of a non-Newtonian fluid in laminar flow, the flow characteristic is no longer equal to the magnitude of the wall shear rate. However, the flow characteristic is still related uniquely to tw because the value of the integral, and hence the right hand side of equation 3.17, is determined by the value of tw. 

Fig. 3. Establishment of Poiseuille flow and diffusion layer under laminar flow in a tube.

Solution Refer to Figure 4.2 and set Q = Q = Qout = 0.25m3/s, q = 0.75 m3/s, and V = I m3. Then t = 4 s for the overall system and 1 s for the once-through distribution. The differential distribution corresponding to laminar flow in a tube was found in Section 8.1.3. The corresponding washout function can be found using Equation (15.7). See also Section 15.2.2. The once-through washout function is... 

Fig. 5-3 Velocity profile for (a) laminar flow in a tube and (to) turbulent tube flow.

Using the velocity distribution for developed laminar flow in a tube, derive an expression for the friction factor as defined by Eq. 5-112. 

Figure 8-1 Schematic Diagram for Analysis of Laminar Flow in a Tube. Vj is the velocity in the axial direction, r is the radial coordinate, and r is the radius of the tube.

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. 

Therefore, the velocity profile in fiiUy developed laminar flow in a tube is parabolic with a maximum at the centerline and minimum (zero) at the tube wall. Also, the axial velocity h is positive for any r, and thus the axial pressure gradient dPIdx must be negative (i.e., pressure must decrease in the flow direction because of viscous effects). 

In laminar flow in a tube with constant surface temperature, both the friction factor and the heat tranter coefficient remain constant in the fully developed region. 

Geometrically Similar Scaieups for Laminar Flow in a Tube... 

Why does the preceding analysis not work for turbulent flow Equation 6.3 is correct for steady laminar or turbulent flow of any kind of fluid, but the substitution of //, d ldy for the shear stress is correct only for laminar flow of newtonian fluids. In laminar flow in a tube, there is no motion perpendicular to the tube axis. In turbulent flow, there is no net motion perpendicular to the tube axis, but there does exist an intense, local, oscillating motion perpendicular to the tube axisi The transfer of fluid perpendicular to the net axial motion causes an increase in shear stress over the value given above for laminar flow of newtonian fluids. This is seen most easily in an analogy. Consider two students playing catch with baseballs. One is standing on the ground. The other is on a railroad car (see Fig. 6.8). 

In one-dimensional flow, the direction of flow at every point in the flow is the same, although the velocity may not be the same at every point (e.g., laminar flow in a tube). In two- and three-dimensional flows, the velocity and direction both change from place to place. For unsteady (i.e., time-varying) flows, they also change from one instant to the next. For steady flow we can map out the velocity and direction at any point see Fig. 10.2, in which the velocity at any point is represented by an arrow showing the relative velocity and direction of the flow. 

As discussdd in Sec. 6.3, the velocity profile for laminar flow in a tube is parabolic. For turbulent flow it is much closer to plug flow, i.e., to a uniform velocity over the entire pipe cross section. Furthermore, as seen from Fig. 11.7, as the Reynolds number is increased, the velocity profile approaches closer and closer to plug flow. At the wall the turbulent eddies disappear so the shear stress at the wall for both laminar and turbulent flow of newtonian fluids is given byj dVJdy. Although it is ve difficult experimentally to... 

The way we have presented the one-dimensional dispersion model so far has been as a modification of the plug-flow model. Hence, u is treated as uniform across the tubular cross section. In fact, the general form of the model can be applied in numerous instances where this is not so. In such situations the dispersion coefficient D becomes a more complicated parameter describing the net effect of a number of different phenomena. This is nicely illustrated by the early work of Taylor [G.I. Taylor, Proc. Roy. Soc. (London), A219, 186 (1953) A223, 446 (1954) A224, 473 (1954)], a classical essay in fluid mechanics, on the combined contributions of the velocity profile and molecular diffusion to the residence-time distribution for laminar flow in a tube. 

Determine Np according to the Taylor-Aris model for a component of a solution in laminar flow in a tube of 0.5 cm radius at an average velocity of lOm/h. The molecular diffusivity is 2.3 x 10 cm /s. What would the F l) curve look like for this situation ... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

Here Dg is the diffusivity of the salt, and the Reynolds number is defined by Eq. tl7-35bl with d = height of feed channel. This equation is similar to Eq. tl7-35aT but predicts a higher mass transfer coefficient. For laminar flow in a tube of length L and radius R with a bulk velocity U], the average mass transfer... 

Determine p for laminar flow in a tube. Solution Using Eq. (2.8-16),... 

Equations for flow in a tube. In order to predict the frictional pressure drop Ap in laminar flow in a tube, Eq. (3.5-4) is solved for Ap. 

Mass transfer in laminar flow in a tube. We consider the case of mass transfer from a tube wall to a fluid inside in laminar flow, where, for example, the wall is made of solid benzoic acid which is dissolving in water. This is similar to heat transfer from a wall to the flowing fluid where natural convection is negligible. For fully developed flow, the parabolic velocity derived as Eqs. (2.6-18) and (2.6-20) is... 

The frictional pressure drop in both phases was calculated from the Hagen-Poiseuille equation for laminar flow in a tube. The interfacial pressure drop is described by the Bretherton s solution for the pressure drop over a single bubble in a capillary (Eq. 9.40) [63] ... 

Figure 3.2.8 Local and average Nusselt number (Nu Nu) for laminar flow in a tube (fully developed laminar flow, that is, without hydrodynamic entrance region, see Section 3.2.1.2.1).

Figure 2-18. Laminar flow in a tube. (Adapted with permission from reference 6. Copyright I960, John Wiley and Sons.)...

---