Pipe wall, curvature

The discrepancy between the coefficients in equations 11.45 and 11,46 is attributable to the fact that the effect of the curvature of the pipe wall has not been taken into account in applying the equation for flow over a plane surface to flow through a pipe. In addition, it takes no account of the existence of the laminar sub-layer at the walls. 

Effect of curvature of pipe wall on shear stress... 

A dent may be defined as a depression that produces a gross disturbance in the curvature of the pipe wall (as opposed to a scratch or gouge, which reduces the pipe wall thickness). The depth of a dent shall be measured as the gap between the lowest point of the dent and a prolongation of the original contour of the pipe in any direction. 

The Reynolds number characterizing laminar-turbulent transition for bulk flow in a pipe is about Re 2300 provided that the fluid moves unidirectionally, the pipe walls are even and behave in a hydraulically smooth manner, and the internal diameter remains constant. However, intestinal walls do not fulfill these hydraulic criteria due to the presence of curvatures, villi, and folds of mucous membrane, which are up to 8 mm in the duodenum, for instance (Fig. 18). Furthermore, the internal diameter of the small intestine is estimated to... 

Overall Heat Transfer Coefficients with Curvature, for Example, Transfer through a Pipe Wall... 

Figure 18 shows the solids concentration profile 22 pipe diameters downstream of a short-radius elbow. The concentration profile is symmetrical, and a minimum solids concentration appears at the center of the pipe. Also, the solids concentration gradually increases toward the pipe wall. This variation in concentration across the pipe is evidently a consequence of the centrifuging action of the secondary flow that is generated by the bend upstream. Figure 18 also shows that the concentration profiles are concentration dependent, and as the solids concentration is increased, the profiles become flatter. Other results (55) showed that these profiles are also functions of the particle size and the radius of curvature of the elbow. 

U being the total heat transfer coefficient and dAx = Pdx the differential heat transfer area. Recalling Chapter 2, when the effect of pipe curvature is negligible, say r0jri <2, the total heat transfer coefficient based on the mean of the inner and outer surface areas of the pipe wall is given by... 

From Table 1.2, the heat transfer coefficient for condensation hf, S 120,000 W/m2 K isfound to be an order of magnitude greater than ht. Thus, neglecting the effect of curvature, the conductive resistance of the pipe walls, and the convective resistance of the steam, the overall heat transfer coefficient is U = hc = 12,000 W/m2-K. Then from Eq. (7.39)... 

An optimum layout is arrived at, based on the requirement of flexibility to accommodate thermal expansion and the availability of space in the cold pool and compactness. The optimized layout consists of a short straight portion connected to a single curvature bend having a radius equal to 945 mm (1.5 times diameter). Stress analysis has been carried out for an internal pressure of 0.8 MPa and seismic loads imposed at the nozzle in the horizontal direction (peak value of 20 t under SSE). Analysis shows that the maximum Pm and (Pm+Pb) values are 76 and 80 Mpa, respectively, for the pipe wall thickness of 8 mm. These values are less than the primary stress limits of RCC-MR, 104 and 156 Mpa respectively. Considering the possible wall thinning during fabrication of the pipe bend, a plate thickness of 10 mm is used for the manufacture of the pipes, ensuring the minimum requirement of the wall thickness of 8 mm after fabrication. 

Flow in bends and elbow fittings is more turbulent than in straight pipe, thus increasing corrosion and erosion. This can be countered oy selecting a component with greater radius of curvature, thicker wall, or smoother interior contour, but this is seldom economical in miter-elbows. 

Close to the wall of a pipe, the effect of the curvature of the wall has been neglected and the shear stress in the fluid has been taken to be independent of the distance from the wall. However, this assumption is not justified near the centre of the pipe. 

Let / b be the radius of curvature of the pipe axis and R4 be the radius of the circular cross section of the pipe. Define U as the axial velocity component and (=/ d — r) as the distance normal to the wall. Denote 0 as the angle in the transverse plane with respect to the outward direction of the symmetry line and 4> as the angle measured in the plane of the curved pipe axis, as shown in Figs. 11.9(a) and (c). Assume that the changes of the flow pattern along the axis of the bend can be neglected. Thus, the momentum integral equations... 

The situation is quite different for particle diffusion. In this case, v/D )s> I and even weak fluctuations in the viscous sublayer contribute significantly to transport. Consider a turbulent pipe flow. In the regions near the wall, the curvature can be neglected and the instantaneous particle flux can be written as follows ... 

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