Flow in Tubes

For practical estimates, we present the distribution of the axial velocity component in the plane wake behind the moving body [427]  

Here Cf is the drag coefficient of the body, d is the center section diameter, and b(X) is the local half-width of the wake. The coordinate X is measured from the rear point of the body and Y is the transverse coordinate. The definition of the local half-width b(X) of the wake is a matter of convention it can be estimated as 

Formulas (1.4.20), (1.4.21) hold for X 50 Cfd, that is, describe only the so-called remote wake. 

The similar relation for the wake behind a body of revolution has the form 

Laminar steady-state fluid flows in tubes of various cross-sections were studied by many authors (e.g., see [179,276, 427]). Such flows are often encountered in practice (water-, gas- and oil pipelines, heat exchangers, etc.). It is worth noting that in these cases the corresponding hydrodynamic equations admit an exact closed-form solution. In what follows we describe the most important results in that direction. 

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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. 

Narezhnyy, E. G., and Sudarev, A. V. (1971). Local heat transfer in air flowing in tubes with a turbulence promoter at the inlet. Heat Transfer 3, 62-66. 

L10. Lowdermilk, W. H., Lanzo, C. D., and Siegel, B. L., Investigation of boiling burnout and flow stability for water flowing in tubes, NACA Tech. Note 4382 (1958). 

The /-factor of Chilton and Colburn for flow in tubes Heat transfer... 

For the most part of the experiments one can conclude that transition from laminar to turbulent flow in smooth and rough circular micro-tubes occurs at Reynolds numbers about RCcr = 2,000, corresponding to those in macro-channels. Note that other results were also reported. According to Yang et al. (2003) RCcr derived from the dependence of pressure drop on Reynolds number varied from RCcr = 1,200 to RCcr = 3,800. The lower value was obtained for the flow in a tube 4.01 mm in diameter, whereas the higher one was obtained for flow in a tube of 0.502mm diameter. These results look highly questionable since they contradict the data related to the flow in tubes of diameter d> mm. Actually, the 4.01 mm tube may be considered... 

Lockhart RW, Martinelli RC (1949) Proposed correlation of data for isothermal two-phase two-component flow in pipes. Chem Eng Prog 45 39-48 Lowdermilk WH, Lanzo CD, Siegel BL (1958) Investigation of boihng burnout and flow stability for water flowing in tubes, NACA TN 4382. National Advisory Committee for Aeronautics, Washington, DC... 

Geometrically Similar Scaleups for Laminar Flows in Tubes. The pressure drop for this method of scaleup is found using the integrated form of the Poiseuille equation ... 

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

Constant-Pressure Scaleups for Laminar Flows in Tubes. As shown in the previous section, scaling with geometric similarity, Sr = Sr = 5 /, 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... 

With 5 per cent vapour, liquid velocity (for liquid flow in tube alone)... 

The combustion gases flow across the tube banks in the convection section and the correlations for cross-flow in tube banks can be used to estimate the heat transfer coefficient. The gas side coefficient will be low, and where extended surfaces are used an allowance must be made for the fin efficiency. Procedures are given in the tube vendors literature, and in handbooks, see Section 12.14, and Bergman (1978b). 

In the following sections, the flow patterns, void fraction and slip ratio, and local phase, velocity, and shear distributions in various flow patterns, along with measuring instruments and available flow models, will be discussed. They will be followed by the pressure drop of two-phase flow in tubes, in rod bundles, and in flow restrictions. The final section deals with the critical flow and unsteady two-phase flow that are essential in reactor loss-of-coolant accident analyses. 

Steady two-phase flow. In rod (or tube) bundles, such as one usually encounters in reactor cores or heat exchangers, the pressure drop calculations use the correlations for flow in tubes by applying the equivalent diameter concept. Thus, in a square-pitched four-rod cell (Fig. 3.51), the equivalent diameter is given by... 

Bergles, A. E., R. F. Lopina, and M. P. Fiori, 1967b, Critical Heat Flux and Flow Pattern Observations for Low Pressure Water Flowing in Tubes, Trans. ASME, J. Heal Transfer 59 69-74. (6)... 

Wurtz, J., 1978, An Experimental and Theoretical Investigation of Annular Steam-Water Flow in Tubes and Annular Channels, Riso Natl. Lab., Oslo, Norway. (5)... 

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

In summary, the plot suggested by Baker (Fig. 4) appears to predict approximately the probable flow pattern, for horizontal flow in tubes. Additional investigation is still needed, to give more universally applicable flow-pattern charts. 

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) ... 

Laminar fluid flow in tubes has been described by Levich [ 3 ]. An entry length, le, is necessary to establish Poiseuille flow, given approximately by... 

Grimley (G10), 1945 Film flow in tubes and channels (water, water + surfactant), co- and counter-flow of air. Wave observations, onset of rippling, surface velocity, velocity distribution, film thicknesses, effects of surface tension and surfactants. 

Calvert (Cl), 1952 Theory for case of upward cocurrent gas/film flow in tubes. Numerous experimental results on film thicknesses, pressure drops, etc. 

Theoretical and experimental studies of pressure drops in gas streams flowing in tubes with wavy walls (longwave and short-wave roughnesses). Importance to case of flow of gas adjacent to wavy film pointed out. 

Tube flow is encountered in several polymer processes, such in extrusion dies and sprue and runner systems inside injection molds. When deriving the equations for pressure driven flow in tubes, also known as Hagen-Poiseuille flow, we assume that the flow is steady, fully developed, with no entrance effects and axis-symmetric (see Fig.5.13). 

Ujhidy, A., Nemeth, J., Szepvolgyi, J., Fluid flow in tubes with helical elements, Chem. Eng. Proc. 2003, 42,1-7. 

A number of kinds of emulsions, foams, and suspensions may be made to flow in tubes or pipes, at scales ranging from the laboratory (e.g., capillary viscometers, Section 6.2.1) to full-scale industry (e.g., transportation pipelines, Sections 10.2 and... 

Example 3.5 The CEF Equation in Steady, Fully Developed Flow in Tubes The viscosity functions in both the Power Law model GNF fluid and the CEF fluid are expected to be... 

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