Turbulent Flow in Circular Pipes

Gilliland and Sherwood (1934) studied the vaporization of nine different liquids into air. Their correlation is 

In a subsequent study, Linton and Sherwood (1950) extended the range of Schmidt number when they investigated the rate of dissolution of benzoic acid, cinnamic acid, and -naphthol. The combined results were correlated by 

Notice that equation (2-75) is the best correlation for all the data, including both gases and liquids. However, the data for gases only are best correlated by equation (2-74). 

Example 2.11 Simultaneous Heat and Mass Transfer in Pipe 

Mass transfer may ocur simultaneously with the transfer of heat, either as a result of an externally imposed temperature difference or because of the absorption or evolution of heat, which generally ocurs when a substance is transferred from one phase to another. In such cases, within one phase, the heat transferred is the result not only of the conduction or convection by virtue of the temperature difference which would happen in the absence of mass transfer, but also includes the sensible heat carried by the diffusing matter. 

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Estimate convective mass-transfer coefficients for the following situations (a) flow paralell to a flat surface, (b) flow past a single sphere, (c) flow normal to a single cylinder, (d) turbulent flow in circular pipes, (e) flow through packed and fluidized beds, and (f) flow through the shell side of a hollow-fiber membrane module. 

Hsu, S. T., A. V. Beken, E. Landweber, and J. F. Kennedy. 1971. The distribution of suspended sediment in turbulent flows in circular pipes. Paper presented at the American Institute of Chemical Engeering Conference on Solids Transport in Slurries, Atlantic City, NJ. 

Momentum eddy diffusivity For turbulent flow in circular pipes. Re = 50 000 to... 

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... 

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 ... 

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...

The head loss /i(, represents the additional height that the fluid needs to be raised by a pump in order to overcome the frictional losses in the pipe. The head loss is caused by viscosity, and it is directly related to the wall shear stress. Equation 8—45 is valid for both laminar and turbulent flows in both circular and noncircular tubes, but Eq. 8-46 is valid only for fully developed laminar flow in circular pipes. 

The Moody chart for the friction factor for fully developed flow in circular pipes for use in the head loss relation -----. Friction factors in tlie turbulent flow... 

Priimak, V. G., Results and potentialities of direct numerical simulation of turbulent viscous fluid flows in circular pipe, Physics Doklady, Vol. 36, No. 1,1991. 

Torrance, B. McK. 1963. Friction factors for turbulent non-Newtonian fluid flow in circular pipes. South African Mechanical Engineer, 13, 4, 89-91. 

Dittus-Boelter correlation A dimensionless equation used in heat transfer for forced convection. For a fluid with turbulent flow in a pipe of circular 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. 

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. 

Laminar flow ceases to be stable when a small perturbation or disturbance in the flow tends to increase in magnitude rather than decay. For flow in a pipe of circular cross-section, the critical condition occurs at a Reynolds number of about 2100. Thus although laminar flow can take place at much higher values of Reynolds number, that flow is no longer stable and a small disturbance to the flow will lead to the growth of the disturbance and the onset of turbulence. Similarly, if turbulence is artificially promoted at a Reynolds number of less than 2100 the flow will ultimately revert to a laminar condition in the absence of any further disturbance. 

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. 

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. 

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

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) ... 

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. 

Clayton, C. G., Ball, A. M. and Spackman, R. (1968) Dispersion and Mixing during Turbulent Flow in a Circular Pipe. UK Atomic Energy Authority Res. Group Report AERE-R 5569. 

The foregoing arguments may be applied to turbulent flow in noncircular ducts by introducing a dimension equivalent to the diameter of a circular pipe. This is known as the mean hydraulic diameter, which is defined as four times the cross-sectional area divided by the wetted perimeter. The following examples are given ... 

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. 

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