Reynolds pipe flow

The pressure drop accompanying pipe flow of such fluids can be described in terms of a generalized Reynolds number, which for pseudoplastic or dilatant fluids takes the form ... 

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

In laminar flow,/is independent of /D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydrauhcaUy smooth flow). For greater Reynolds numbers,/deviates from the smooth pipe cui ve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which / becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2-2.46ln( /D) ... 

Curved Pipes and Coils For flow through curved pipe or coil, a secondary circiilation perpendicular to the main flow called the Dean effect occurs. This circulation increases the friction relative to straight pipe flow and stabilizes laminar flow, delaying the transition Reynolds number to about... 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

Friction loss The pressure energy loss that takes place in duct or pipe flow. It is related to the Reynolds number, boundary layer growth, and the velocity distribution. 

Turbulence is known to occur in pipe flow at a Reynolds number (Re) above 2300. Beyond this stability limit, any disturbance will grow exponentially in time and the flow becomes fully chaotic at Re 4000, where Re is defined by ... 

Derive the Taylor-Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. 

The ratio of t/t, which is characteristic of the possibility of vortices, does not depend on the micro-channel diameter and is fully determined by the Reynolds number and L/d. The lower value of Re at which f/fh > 1 can be treated as a threshold. As was shown by Darbyshire and Mullin (1995), under conditions of an artificial disturbance of pipe flow, a transition from laminar to turbulent flow is not possible for Re < 1,700, even with a very large amplitude of disturbances. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

A polymer of the polyacrylamide type was injected as a 0.5% solution from an axially-placed nozzle at the bellmouth entrance. The experiments showed that the central thread provided drag reduction almost equivalent to premixed solutions of the same total polymer concentration flowing in the pipe. Overall concentrations of 1, 2, 4, and 20 ppm were used. Moreover, the effects were additive 2 ppm thread overall concentration plus 2 ppm premixed gave drag reductions equivalent to 4 ppm of either type. Reynolds numbers of up to 300,000 were investigated. In other experiments, a number of different polymer fluids were injected on the centerline of a water pipe-flow facility [857]. Two distinct flow regions were identified ... 

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

At high Reynolds numbers (high turbulence levels), the flow is dominated by inertial forces and wall roughness, as in pipe flow. The porous medium can be considered an extremely rough conduit, with s/d 1. Thus, the flow at a sufficiently high Reynolds number should be fully turbulent and the friction factor should be constant. This has been confirmed by observations, with the value of the constant equal to approximately 1.75 ... 

A general time-independent non-Newtonian liquid of density 961 kg/m3 flows steadily with an average velocity of 2.0 m/s through a tube 3.048 m long with an inside diameter of 0.0762 m. For these conditions, the pipe flow consistency coefficient K has a value of 1.48 Pa s0,3 and n a value of 0.3. Calculate the values of the apparent viscosity for pipe flow p.ap, the generalized Reynolds number Re and the pressure drop across the tube, neglecting end effects. 

Axial dispersion Low Z/D and low Reynolds number flow conditions Vessel with baffles or internals obstructing flows High ZID and high Reynolds flow in open pipes... 

In order to estimate to what extent degradation actually occurs in a turbulent pipe flow, we carried out measurements at different Reynolds numbers, as shown in Fig. 36. 

If Reynolds number is useful, it is because flow velocity and Reynolds number are related to the intensity and scale of turbulence in pipe flow. All the measurements... 

It is well known in fluid flow studies that below a certain critical value of the Reynolds number the flow will be mainly laminar in nature, while above this value, turbulence plays an increasingly important part. The same is true of film flow, though it must be remembered that in thin films a large part of the total film thickness continues to be occupied by the relatively nonturbulent laminar sublayer, even at large flow rates (N e ARecr J- Hence, the transition from laminar to turbulent flow cannot be expected to be so sharply marked as in the case of pipe flow (D12). Nevertheless, it is of value to subdivide film flow into laminar and turbulent regimes depending on whether (Ar6 5 Ar u). 

In plate-fin heat exchangers, the flow structure depends on the fin geometry. Continuous fins can be assimilated to rectangular channels and the flow is almost identical to pipe flows. For offset strip fins or louvered fins, there is a high degree of mixing (40), and the flow becomes turbulent even at low Reynolds number (Figure 25). 

Two common pipe flow problems are calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given the pressure drop. When flow rate is given, the Reynolds number may be calculated directly to determine the flow regime, so that the appropriate relations between/and Re (or pressure drop and flow rate or velocity) can be selected. When flow rate is specified and the flow is turbulent, Eq. (6-39) or (6-40), being explicit in/ may be preferable to Eq. (6-38), which is implicit in/and pressure drop. 

For the Reynolds number range typical of drag reduction (Re 105), / is about 0.02 from the Moody chart (see Fig. 11.7). The typical turbulent intensity of gas in a pipe flow is about 5 percent. Using the Hinze-Tchen model (see 5.3.4.1), the ratio of the velocity fluctuation of the particles to that of the gas may be given by Eq. (5.196) as... 

Equation (94) provides the means for rearranging all of the theoretical expressions for v) given above into expressions involving the friction factor. For example, when Eq. (75) for Newtonian pipe flow is so rearranged and one eliminates (v) in terms of the Reynolds number, Re = D v)p/fi, one obtains... 

The most important case of this transition for chemical engineers is the transition from laminar to turbulent flow, which occurs in straight bounded ducts. In the case of Newtonian fluid rheology, this occurs in straight pipes when Re = 2100. A similar phenomenon occurs in pipes of other cross sections, as well and also for non-Newtonian fluids. However, just as the friction factor relations for these other cases are more complex than for simple Newtonian pipe flow, so the criteria for transition to turbulence cannot be expressed as a simple critical value of a Reynolds number. 

There are two distinct modes of flow, laminar and turbulent. Fluid inertia tends to allow fluctuations to grow and give rise to turbulent eddies. Viscosity on the other hand, tends to damp out these fluctuations. A ratio of forces, inertial to viscous, is used to characterise the nature of the flow and is called the Reynolds Number, Re. For pipe flow this takes the form ... 

---