Turbulent Flows in Tubes

Turbulent flow reactors are modeled quite differently from laminar flow reactors. In a turbulent flow field, nonzero velocity components exist in all three coordinate directions, and they fluctuate with time. Statistical methods must be used to obtain time average values for the various components and to characterize the instantaneous fluctuations about these averages. We divide the velocity into time average and fluctuating parts  

There is nothing as unstable as water, Gen. 49. 4, and as soon as we have any reasonable flow rate in a pipe for water-like liquids, the flow becomes turbulent and elements of fluid undergo local chaotic movement because inertial-driven instabihties can no longer be damped out by viscosity. In this situation the following equations are useful. For Reynolds numbers. Re, from around 2300 up to 60,000, the pressure drop dovm the tube is given by [8] 

Even though the bulk of the flow is turbulent, there is laminar sublayer at the wall, the thickness of which is given by 

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Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

Geometrically Similar Scaleups for Turbulent Flows in Tubes. Integrating Equation (3.15) for the case of constant density and viscosity gives... 

Const ant-Pres sure Scaleups for Turbulent Flows in Tubes. Equation (3.34) gives the pressure drop ratio for large and small reactors when density is constant. Set AP2 = APi to obtain 1 = Equation (3.31) gives the inventory... 

Deissler (D3) recently extended the analysis of thermal and material transfer associated with turbulent flow in tubes to include the behavior of fluids with high molecular Prandtl and Schmidt numbers. If the variation in molecular properties of the fluid with position are neglected, the following expression for the temperature distribution was suggested (D3) ... 

Equations (6-4) and (6-5) apply to fully developed turbulent flow in tubes. In the entrance region the flow is not developed, and Nusselt [3] recommended the following equation ... 

Entrance effects for turbulent flow in tubes are more complicated than for laminar flow and cannot be expressed in terms of a simple function of the Graetz number. Kays [36] has computed the influence for several values of Re and Pr with the results summarized in Fig. 6-6. The ordinate is the ratio of the... 

Petukhov, B.J. Popov, N.V. Theoretical calculation of heat exchange and frictional resistance in turbulent flow in tubes of an incompressible fluid with variable physical properties. High Temperature 1 (1963) 69-83... 

Detailed information about heat transfer in turbulent flows in tubes and channels, as well as various relations for determining the mean flow rate temperature and Nusselt number, can be found in the references [185, 196, 406], which contain extensive literature surveys. 

For fully developed turbulent flow in tubes subject to a uniform heat flux, the experimental data are correlated with... 

B. S. Petukhov, and V. V. Kirillov, The Problem of Heat Exchange in the Turbulent Flow of Liquids in Tubes, (in Russian) Teploenergetika, (4/4) 63-68,1958 see also B. S. Petukhov and V. N. Popov, Theoretical Calculation of Heat Exchange in Turbulent Flow in Tubes of an Incompressible Fluid with Variable Physical Properties, High Temp., (1/1) 69-83,1963. 

Studies have shown that the transition from laminar to turbulent flow in tubes is not only a function of velocity but also of density and viscosity of the fluid and the tube diameter. These variables are combined into the Reynolds number, which is dimensionless. 

For fully developed turbulent flow in tubes with uniform heat flux the following equation can be used (LI) ... 

Turbulent flow in tubes. For turbulent flow of power-law fluids through tubes, Clapp (C4) presents the following empirical equation for heat transfer ... 

Apparently, in well developed turbulent flow in tubes the Bodenstein number is ateut 3-5. Note that the coefficient of axial mixing decreases slightly with increasing velocity of flow. 

Geometrically Similar Scaleups for Turbulent Flows in Tubes. Integrating... 

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