Turbulent flow circular pipe

In practice, it is found convenient to express the pressure loss for all types of fully developed internal flosvs (laminar or turbulent flows, circular or noncLrcu-lar pipes, smooth or rough surfaces, horizontal or inclined pipes) as (Fig. 8-19)... 

FIGURE 8-19 The relation for pressure loss (and head loss) is one of the most general relations in fluid mechanics, and it is valid for laminar or turbulent flows, circular or noncircular tubes, and pipes with smooth or rough surfaces. 

Nikitin, N. V., Direct numerical modeling of three-dimensional turbulent flows in pipes of circular cross-section. Fluid Dynamics, Vol. 29, No. 6, pp. 749-758, 1994. 

For turbulent flow in pipes with circular or rectangular cross section ... 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Miller Internal Flow Systems, 2d ed.. Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-seclion bends of constant area, as shown in Fig. 6-14 7, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is... 

Quite often the mixing units or elements are installed in a circular pipe however, they can he adapted to rectangular or other arrangements. Pressure drop through the units varies depending on design and whether flow is laminar or turbulent. Because of the special data required, pressure drops should he determined with the assistance of the manufacturer. Of course, pressure drops must he expected to he several multiples of conventional pipe pressure drop. 

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. 

For turbulent flow in a duct of non-circular cross-section, the hydraulic mean diameter may be used in place of the pipe diameter and the formulae for circular pipes may then be applied without introducing a large error. This approach is entirely empirical. 

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. 

HARTNETT and KOSTIC 26 have recently examined the published correlations for turbulent flow of shear-thinning power-law fluids in pipes and in non-circular ducts, and have concluded that, for smooth pipes, Dodge and Metzner S(27) modification of equation 3.11 (to which it reduces for Newtonian fluids) is the most satisfactory. 

For the heat transfer for fluids flowing in non-circular ducts, such as rectangular ventilating ducts, the equations developed for turbulent flow inside a circular pipe may be used if an equivalent diameter, such as the hydraulic mean diameter de discussed previously, is used in place of d. 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

Liquid core temperature and velocity distribution analysis. BankofT (1961) analyzed the convective heat transfer capability of a subcooled liquid core in local boiling by using the turbulent liquid flow equations. He found that boiling crisis occurs when the core is unable to remove the heat as fast as it can be transmitted by the wall. The temperature and velocity distributions were analyzed in the singlephase turbulent core of a boiling annular flow in a circular pipe of radius r. For fully developed steady flow, the momentum equation is given as... 

An approximate equation for the profile of the time-averaged velocity for steady turbulent flow of a Newtonian fluid through a pipe of circular... 

Consider a fully developed turbulent flow through a pipe of circular cross section. A turbulent boundary layer will exist with a thin viscous sublayer immediately adjacent to the wall, beyond which is the buffer or generation layer and finally the fully turbulent outer part of the boundary layer. 

Equation 2.69 fits the experimental data for turbulent flow in smooth pipes of circular cross section for y+ > 30 when 1 IK and C are given the values 2.5 and 5.5 ... 

As might be expected, the dispersion coefficient for flow in a circular pipe is determined mainly by the Reynolds number Re. Figure 2.20 shows the dispersion coefficient plotted in the dimensionless form (Dl/ucI) versus the Reynolds number Re — pud/p(2Ai). In the turbulent region, the dispersion coefficient is affected also by the wall roughness while, in the laminar region, where molecular diffusion plays a part, particularly in the radial direction, the dispersion coefficient is dependent on the Schmidt number Sc(fi/pD), where D is the molecular diffusion coefficient. For the laminar flow region where the Taylor-Aris theory18,9,, 0) (Section 2.3.1) applies ... 

Fig. 2.20. Dimensionless axial-dispersion coefficients for fluids flowing in circular pipes. In the turbulent region, graph shows upper and lower limits of a band of experimentally determined values. In the laminar region the lines are based on the theoretical equation 2.37...

Here, h is the enthalpy per unit mass, h=u + p/p. The shaft work per unit of mass flowing through the control volume is 8WS = Ws/m. Similarly, 8Q is the heat input per unit of mass. The factor a is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, a = 1. In turbulent flow, a is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circular pipe with a parabolic velocity profile, a = 2. 

Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D,DH = D-d, for a rectangular duct of sides a, b, DH=ab/[2(a+b)]. The hydraulic radius Rh is defined as one-fourth of the hydraulic diameter. 

It is now required to obtain the spatial distribution of the concentration of the tracer for two cases—center injection and wall ring injection. In general, it is difficult to theoretically obtain the spatial distribution of the concentration of tracer, and the number of experimental results with regard to this subject is insufficient because a very long pipe is required for performing experiments. However, it is possible to estimate the distribution of the tracer concentration based on the mean velocity profile, the intensity of velocity fluctuations, and so on, during the turbulent flow in a circular pipe. 

The pipe has not been treated as a chemical equipment but only as a means for transportation. However, recently, line mixing and inline reaction have been considered, and the scale-up rule for pipes comes into question. Therefore, the relationship between the pipe diameter and turbulent flow structure in a circular pipe must be clarified. 

Ogawa, K. and Kuroda, C. (1984). Local mixing capacity of turbulent flow in a circular pipe. Kagaku Kogaku Ronbunshu, 10, 268-271. 

The mathematical analysis of flow in ducts of noncircular cross section is vastly more complex in laminar flow than for circular pipes and is impossible for turbulent flow. As a result, relatively little theoretical base has been developed for the flow of fluids in noncircular ducts. In order to deal with such flows practically, empirical methods have been developed. 

Flow in circular tubes is of interest to many corrosion engineers. A large number of correlations exist for mass transport due to turbulent flow in a smooth straight pipe (4,9). The flow is transitionally turbulent at Re 2 X 103 and is fully turbulent at Re 105 (4). The most frequently used expression for turbulent conditions at a straight tube wall is that given by Chilton and Colburn using the analogy from heat transfer (13) ... 

An alternative version of the same expression describing flow-induced shear stresses for a circular pipe operating in the turbulent flow regime is described by the relation (6)... 

For turbulent flow in a conduit of noncircular cross section, an equivalent diameter can be substituted for the circular-section diameter, and the equations for circular pipes can then be applied without introducing a large error. This equivalent diameter is defined as four times the hydraulic radius RH, where the hydraulic radius is the ratio of the cross-sectional flow area to the wetted perimeter. When the flow is viscous, substitution of 4RH for D does not give accurate results, and exact expressions relating frictional pressure drop and velocity can be obtained only for certain conduit shapes. 

Despite many attempts made over more than a century, the exact route to turbulence is still far from clear. This prompted Morkovin (1991) to state One hundred eight years after O. Reynolds demonstrated turbulence in a circular pipe, we still do not understand the nature of the irregular fluctuations at the wall nor the formation of larger coherent eddies convected downstream further from the wall. Neither can we describe the mechanisms of the instabilities that lead to the onset of turbulence in any given pipe nor the Reynolds number (between about 2000 and 100 000) at which it will take place. It is sobering to recall that Reynolds demonstrated this peculiar non-larninar behaviour of fluids before other physicists started on the road to relativity theory, quantum theory, nuclear energy, quarks etc . While some additional researches have clarified some key concepts, the situation about flow transition in a pipe remains the same. 

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