Rough pipes velocity profile

Example 6 Losses with Fittings and Valves It is desired to calculate the liquid level in the vessel shown in Fig. 6-15 required to produce a discharge velocity of 2 m/s. The fluid is water at 20°C with p = 1,000 kg/m and i = 0.001 Pa - s, and the butterfly valve is at 6 = 10°. The pipe is 2-in Schedule 40, with an inner diameter of 0.0525 m. The pipe roughness is 0.046 mm. Assuming the flow is tiirhiilent and taking the velocity profile factor (X = 1, the engineering Bernoulli equation Eq. (6-16), written between surfaces 1 and 2, where the... 

For rough pipes, the velocity profile in the turbulent core is given by... 

Each term has the dimensions of energy per unit of mass - in this case, ft-lbp/lbM. The factor, a, in the kinetic energy term, Av /2agc, corrects for the velocity profile across a duct. For laminar flow in a circular duct, the velocity profile is parabolic, and a = 1/2. If the velocity profile is flat, a = 1. For very rough pipes and turbulent flow, a may reach a value of 0.77 [10]. In many engineering applications, it suffices to let a = 1 for turbulent flow. 

The velocity head is v /2g(,. For any velocity profile the true velocity represents the integral of the local velocity head across the pipe diameter. It is found by dividing the volumetric flow rate by the cross-sectional area of the pipe and multiplying by a correction factor. For laminar flow this factor is 2 for turbulent flow the factor depends on both the Reynolds number and the pipe roughness and (1 -i- 0.8fp) [4]. Therefore, when the pipe size changes, the velocity head also changes. Because the velocity is inversely proportional to the flow area and thus to the diameter squared, K is inversely proportional to the velocity squared. 

Flow is laminar for Re < 2300, and in the ideal case (no disturbances due to pipe roughness, internals, etc.) can be described by the Hagen-Poiseuille law. In the derivation of this law, equating the surface force with the shear force gives a parabolic velocity profile u(r) (Equation 2.4-9), as shown in Figure 2.5-2 ... 

Figure 1 shows the particulate loading of a pipe containing gas and particulates where the nonuniformity induced by a disturbance, ie, a 90° bend, is obvious (2). A profile of concentration gradients in a long, straight, horizontal pipe containing suspended soHds is shown in Figure 2. Segregation occurs as a result of particle mass. Certain impurities, eg, metal-rich particulates, however, occur near the bottom of the pipe others, eg, oily flocculates, occur near the top (3). Moreover, the distribution may be affected by Hquid-velocity disturbances and pipe roughness.

---