Pipes streamlining

Extrusion Resins. Extmsion of VDC—VC copolymers is the main fabrication technique for filaments, films, rods, and tubing or pipe, and involves the same concerns for thermal degradation, streamlined flow, and noncatalytic materials of constmction as described for injection-molding resins (84,122). The plastic leaves the extmsion die in a completely amorphous condition and is maintained in this state by quenching in a water bath to about 10°C, thereby inhibiting recrystallization. In this state, the plastic is soft, weak, and pHable. If it is allowed to remain at room temperature, it hardens gradually and recrystallizes partially at a slow rate with a random crystal arrangement. Heat treatment can be used to recrystallize at controlled rates. 

The copolymers have been used in the manufacture of extruded pipe, moulded fittings and for other items of chemical plant. They are, however, rarely used in Europe for this purpose because of cost and the low maximum service temperature. Processing conditions are adjusted to give a high amount of crystallinity, for example by the use of moulds at about 90°C. Heated parts of injection cylinders and extruder barrels which come into contact with the molten polymer should be made of special materials which do not cause decomposition of the polymer. Iron, steel and copper must be avoided. The danger of thermal decomposition may be reduced by streamlining the interior of the cylinder or barrel to avoid dead-spots and by careful temperature control. Steam heating is frequently employed. 

The interior of piping systems should be streamlined for easy drainage. Stubs and dead ends should be avoided, and pipelines should be sloped continuously downstream to their outlets. Elbows should be sloped for drainage purposes. 

Fcr piping with air in streamline flow at absolute pressures in the range between 50 microns and 1 millimeter of mercury, the following is a recommended method. Calculation procedures in pressure regions below atmospheric are very limited and often not generally applicable to broad interpretations. 

Particles of fluid flowing in pipes act in the same manner. The flow is streamlined if the fluid flows slowly enough, and remains streamlined at greater velocities if the diameter of the pipe is small. If the velocity of flow or size of pipe is increased sufficiently, the flow becomes turbulent. 

The geometry of the system is a further factor that enables control to be effected, and in general, reducing the velocity of the solution and ensuring that flow is laminar and not turbulent will reduce the tendency for attack by erosion-corrosion. Thus the pipe diameter should be as large as possible, consistent with other considerations bends should have a large radius and inlet and outlets should be streamlined so that there is not a sudden change in section. 

It is seen that it is important to be able to determine the velocity profile so that the flowrate can be calculated, and this is done in Chapter 3. For streamline flow in a pipe the mean velocity is 0.5 times the maximum stream velocity which occurs at the axis. For turbulent flow, the profile is flatter and the ratio of the mean velocity to the maximum... 

For flow in a pipe of circular cross-section a will be shown to be exactly 0.5 for streamline flow and to approximate to unity for turbulent flow. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

If a turbulent fluid passes into a pipe so that the Reynolds number there is less than 2000, the flow pattern will change and the fluid will become streamline at some distance from the point of entry. On the other hand, if tire fluid is initially streamline (Re < 2000), the diameter of the pipe can be gradually increased so that the Reynolds number exceeds 2000 and yet streamline flow will persist in the absence of any disturbance. Unstable streamline flow has been obtained in this manner at Reynolds numbers as high as 40,000. The initiation of turbulence requires a small force at right angles to the flow to promote the formation of eddies. 

Region l (Re < 2000) corresponds to streamline motion and a single curve represents all the data, irrespective of the roughness of the pipe surface. The equation of the curve is R/pu2 = 8/Re. 

The velocity over the cross-section of a fluid flowing in a pipe is not uniform. Whilst this distribution in velocity over a diameter can be calculated for streamline flow this is not possible in the same basic manner for turbulent flow. 

Since in the energy balance equation, the kinetic energy per unit mass is expressed as a2/2a, hence a = 0.5 for the streamline flow of a fluid in a round pipe. 

WHiTEm) found that stable streamline flow persists at higher values of the Reynolds number in coiled pipes. Thus, for example, when the ratio of the diameter of the pipe to the diameter of the coil is 1 to 15, the transition occurs at a Reynolds number of... 

Streamline Flow in pipes and channels of regular geometry... 

In order to predict Lhe transition point from stable streamline to stable turbulent flow, it is necessary to define a modified Reynolds number, though it is not clear that the same sharp transition in flow regime always occurs. Particular attention will be paid to flow in pipes of circular cross-section, but the methods are applicable to other geometries (annuli, between flat plates, and so on) as in the case of Newtonian fluids, and the methods described earlier for flow between plates, through an annulus or down a surface can be adapted to take account of non-Newtonian characteristics of the fluid. 

Thus, the pipe friction chart for a Newtonian fluid (Figure 3.3) may be used for shearthinning power-law fluids if Remit is used in place of Re. In the turbulent region, the ordinate is equal to (R/pu2)n 0 fn5. For the streamline region the ordinate remains simply R/pu2, because Reme has been defined so that it shall be so (see equation 3.140). More recently, Irvine(25j has proposed an improved form of the modified Blasius equation which predicts the friction factor for inelastic shear-thinning polymer-solutions to within 7 per cent. 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

Under streamline conditions, the velocity at the axis will increase from a value u at the inlet to a value 2u where fully-developed flow exists, as shown in Figure 11.9, because the mean velocity of flow u in the pipe is half of the axial velocity, from equation 336. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

For the inlet length of a pipe in which the boundary layers are forming, the equations in the previous section will give an approximate value for the heat transfer coefficient. It should be remembered, however, that the flow in the boundary layer at the entrance to the pipe may be streamline and the point of transition to turbulent flow is not easily defined. The results therefore are, at best, approximate. 

A Bingham plastic material is flowing under streamline conditions in a pipe of circular cross-section. What are the conditions for one half of the total flow to be within the central core across which the velocity profile is fiat The shear stress acting within die fluid Ry varies with velocity gradient du,/dy according to the relation ... 

A liquid is pumped in streamline flow through a pipe of diameter d. At what distance from the centre of the pipe will the fluid be flowing at the average velocity ... 

A liquid w hose rheology can be represented by the power law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow in I nt/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n ... 

If a pitot tube is inserted in a circular cross-section pipe in which a fluid is in streamline flow, calculate at what point in the cross-section it should be situated so us to give a direct reading representative of the mean... 

The flowrate of a fluid in a pipe is measured using a pilot tube which gives a pressure differential equivalent to 40 mm of water when situated at the centre line of the pipe and 22.5 mm of water when midway between the axis and the wall. Show that these readings are consistent with streamline flow in the pipe. 

White, C.M., "Streamline Flow through Curved Pipes", Proc. Royal Soc., A 123, 645 (1929)... 

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