Straight laminar velocity profile

Enough space must be available to properly service the flow meter and to install any straight lengths of upstream and downstream pipe recommended by the manufacturer for use with the meter. Close-coupled fittings such as elbows or reducers tend to distort the velocity profile and can cause errors in a manner similar to those introduced by laminar flow. The amount of straight pipe required depends on the flow meter type. For the typical case of an orifice plate, piping requirements are normally Hsted in terms of the P or orifice/pipe bore ratio as shown in Table 1 (1) (see Piping systems). 

The velocity profile is replaced by a straight line in the laminar sublayer, b) In the completely turbulent region, a long distance away from the wall, y — 00 and therefore y+ — 00, the second term outweighs the first, and it holds that... 

This is the Stokes equation that has analytic solutions for certain types of simple laminar flows, for example the laminar flow in a straight channel known as Poiseuille flow (Fig. 1.2). Assuming that the flow is unidirectional along the x axis (vy = vz = 0) it follows from incompressibility that the velocity profile vx y) is uniform along the channel. When a pressure difference 5p is applied along a channel of length L and width d the Stokes equation reduces to... 

Figure 1.2 One of the simplest type of laminar flows shear flow with a parabolic velocity profile in a straight channel.

This is the classic parabolic or quadratic velocity profile for onedimensional laminar flow of an incompressible Newtonian fluid through a straight tube. The no-slip boundary condition (2) air = R yields... 

Figure 23-1 One-dimensional velocity profile for laminar viscous flow in a straight channel with square cross section.

With turbulent flow in a straight pipe, the velocity profile is blunter than with laminar flow but still quite different from the flat profile assumed for plug flow. The ratio of maximum velocity to average velocity is about 1.3 at Re = 10", and this ratio slowly decreases to 1.15 at Re = 10 . A pulse of tracer introduced at the inlet gradually expands, but the distribution of residence times at the exit is fairly narrow. The effect of the axial velocity profile is largely offset by rapid radial mixing due to the turbulent velocity fluctuations. 

FIGURE 213 Representations of laminar (a) and turbulent (b) flow through a straight tube. With laminar flow, the pressure drop across a length of tube is proportion to die flow. With turbulence, the pressure approaches proportionality with the square of flow. The velocity profile under laminar flow con-ditimis is parabolic, with higher flows near die center of the tube than near the boundary. 

FIGURE 7.2 Representative velocity profiles of laminar, oscillatory flow in a straight, rigid tube, at four phases of the flow cycle, (a) a = 3, (b) a = 13. 

Figure 4.3. Comparison of the velocity profile in a laminar regime with the average velocity profile in a turbulent regime for the flow in a straight pipe. U is the maximum velocity. For the turbulent case, the colored area indicates the boundary layer...

Figure 6.2.2. Axial dispersion of (a) a solute concentration pulse and (b) a solute concentration front introduced at the inlet of a straight circular tube having a laminar liquid flow and a parabolic velocity profile.

In a straight tube of uniform diameter (Fig. 2.SA), the parabolic profile (Fig. 3.4) formed by laminar flow remains undisturbed up to a flow velocity not normally reached in a typical FIA system, and since the radial diffusion occurring in the time frame oif an FIA experiment is not sufficient to offset the axial dispersion initially formed during sample injection, an asymmetrical peak is recorded (Figs. 2.4a and 2.10a). 

We wfll consider first the laminar flow of a Newtonian liquid in a straight circular pipe—radius a, length L—well away from its entrance and exit. The flow rate is Q and the pressure drop down the pipe is P. If we were able to visualise the velocity of the fluid, we would find that it has a parabolic profile, with a value at any radius u(r) given by... 

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