Pipes center, fluid flows

This unit consists of two pipes or tubes, the smaller centered inside the larger as shown in Figure 10-92. One fluid flows in the annulus between the tubes the other flows inside the smaller tube. The heat transfer surface is considered as the outside surface of the inner pipe. The fluid film coefficient for the fluid inside the inner tube is determined the same as for any straight tube using Figures 10-46-10-52 or by the applicable relations correcting to the O.D. of the inner tube. For the fluid in the annulus, the same relations apply (Equation 10-47), except that the diameter, D, must be the equivalent diameter, D,.. The value of h obtained is applicable directly to the point desired — that is, the outer surface of the inner tube. ... 

The amount of fluid flowing between R and R + dR is 2n R dR, and p t) is foimd by integrating from R = 0 at the center of the tube to R = at the pipe wall. We will leave the derivation of this equation as a homework problem This gives... 

Since the molecules that make up the fluid can interact with each other and the pipe, the resulting flow pattern may be surprising. The speeds of the molecules next to the pipe walls are essentially zero, while those molecules near the center of the pipe are moving the fastest. The closer a molecule is to a wall, the slower it moves. We thus see that adjacent regions of the fluid will have different speeds and the faster regions will flow past the slower ones hence the shape of the flow pattern. 

The kinetic-energy terms of the various energy balances developed h include the velocity u, which is the bulk-mean velocity as defined by the equati u = m/pA Fluids flowing in pipes exhibit a velocity profile, as shown in Fi 7.1, which rises from zero at the wall (the no-slip condition) to a maximum the center of the pipe. The kinetic energy of a fluid in a pipe depends on actual velocity profile. For the case of laminar flow, the velocity profile parabolic, and integration across the pipe shows that the kinetic-ertergy should properly be u2. In fully developed turbulent flow, the more common in practice, the velocity across the major portion of the pipe is not far fro... 

Thin L-shaped probes are commonly used to measure solids concentration profile in slurry pipelines (28-33), However, serious sampling errors arise as a result of particle inertia. To illustrate the effect of particle inertia on the performance of L-shaped probes, consider the fiuid streamlines ahead (upstream) of a sampling probe located at the center of a pipe, as shown in Figure 2. The probe has zero thickness, and its axis coincides with that of the pipe. The fluid ahead of the sampler contains particles of different sizes and densities. Figure 2A shows the fluid streamlines for sampling with a velocity equal to the upstream local velocity (isokinetic sampling). Of course, the probe does not disturb the flow field ahead of the sampler, and consequently, sample solids concentration and composition equal those upstream of the probe. 

This distinguishes it from the velocity of the fluid at a certain point (since fluids flow faster in the center of a pipe). The bulk velocity is about the same as the instantaneous velocity for relatively fast flow, or especially for flow of gasses. 

Figure 48.5(a) Parabolic-shaped velocity gradient laminar flow fluid flows fastest at the center of the pipe. 

Figure 2.12 Fluid flow in twin counter flowing 1.765 cm diameter pipes. The velocity is averaged over a 1 mm thick slice. The in-plane resolution is 195 p.m, and the velocity resolution is 40 m s . The maximum velocity is 520 pm s . (a) Measured velocity profile in both pipes, (b) Measured velocity (circles) along a line going through the centers of both pipes. The theoretical velocity profile based on the flow rate and the diameter of the tube is shown as a dotted line. There are no fitted parameters, (c) Measured distribution of velocity through both pipes (solid) and theoretical velocity distribution (dotted) with no fitting parameters.

The shear rate at the wall of the pipe is related to the volumetric flow rate Q, which can be measured experimentally. The volumetric flow rate also can be obtained by integrating the velocity profile over the cross-sectional area of the pipe. Figure 8.13 shows the velocity profile of a polymer fiuid in a pipe with radius R. The velocity of the fluid at distance r from the pipe center is u. The velocity at the wall i.e., r = R)is zero since there is no slip at the wall. Shear rate is the same as the velocity gradient, and hence the shear rate at any point in the pipe is defined as -du Jdr. 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Fig. 1.13 Left schematic plot of the distribu- water flowing under laminar conditions in a tion of flow velocities, vz, for laminar flow of a circular pipe the probability density of dis-Newtonian fluid in a circular pipe the max- placements is constant between 0 and imum value of the velocity, occurring in the Zmax = i>ZimaxA, where A is the encoding time center of the pipe, is shown for comparison. of the experiment.

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