Generalized Reynolds Numbers

Heat transfer at a wall is related to the Reynolds number, generally to the power 0.7. Thus, there is great interest in secnring a low viscosity when the heat of pyrolysis is to be supplied. Dissolving the feed in oil is a possible procedure. 

Unbafiled tanks. At low Reynolds numbers, below about 300, the power number curves for baffled and unbaffled tanks are identical. At higher Reynolds numbers the curves diverge, as shown by the dashed portions of curve D in Fig. 9.12 and curve B in Fig. 9.13. In this region of Reynolds numbers, generally avoided in practice in unbaffled tanks, a vortex forms and the Froude number has an effect. Equation (9.16) is then modified to... 

Figure 2.13 Packed bed pressure loss Euler number versus particle Reynolds number General Expression (1 < Rep <2000+)...

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

An outstanding advantage of common differential pressure meters is the existence of extensive tables of discharge coefficients ia terms of beta ratio and Reynolds numbers (1,4). These tables, based on historic data, are generally regarded as accurate to within 1—5% depending on the meter type, the beta ratio, the Reynolds number, and the care taken ia manufacture. Improved accuracy can be obtained by miming an actual flow caUbration on the device. 

Both wetted-sensor and clamp-on Doppler meters ate available for Hquid service. A straight mn of piping upstream of the meter and a Reynolds number of greater than 10,000 ate generally recommended to ensure a weU-developed flow profile. Doppler meters ate primarily used where stringent accuracy and repeatabiHty ate not requited. Slurry service is an important appHcation area. 

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 ... 

For laminar flow (Re < 2000), generally found only in circuits handling heavy oils or other viscous fluids, / = 16/Re. For turbulent flow, the friction factor is dependent on the relative roughness of the pipe and on the Reynolds number. An approximation of the Fanning friction factor for turbulent flow in smooth pipes, reasonably good up to Re = 150,000, is given by / = (0.079)/(4i e ). 

Dimensional analysis (qv) shows that is generally a function of the particle Reynolds number ... 

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically statistical methods are used to quantify turbulence. 

A turbine flowmeter consists of a straight flow tube containing a turbine which is free to rotate on a shaft supported by one or more bearings and located on the centerline of the tube. Means are provided for magnetic detection of the rotational speed, which is proportional to the volumetric flow rate. Its use is generally restric ted to clean, noncorrosive fluids. Additional information on construction, operation, range, and accuracy can be obtained from Holzbock (Instruments for Measurement and Control, 2d ed., Reinhold, New York, 1962, pp. 155-162). For performance characteristics of these meters with liquids, see Shafer,y. Basic Eng., 84,471-485 (December 1962) or May, Chem. Eng., 78(5), 105-108 (1971) and for the effect of density and Reynolds number when used in gas flowmetering, see Lee and Evans, y. Basic Eng., 82, 1043-1057 (December 1965). 

See Benedict, loc. cit., for a general equation for pressure loss for nozzles installed in pipes or with plenum inlets. Nozzles show higher loss than venturis. Permanent pressure loss for laminar flow depends on the Reynolds number in addition to p. For details, see Alvi, Sri-dharan, and Lakshamana Rao, J. Fluids Eng., 100, 299-307 (1978). 

Measurements by Harris and MagnaU [Trans. Jn.st. Chem. Eng. (London), 50, 61-68 (1972)] with a venturi (p = 0.62) and orifices wiSi radius taps (P = 0.60-0.75) indicate that the discharge coefficient for nonnewtonian fluids, in the range Nr (generalized Reynolds number) 3500 to 100,000, is approximately the same as for newtonian fluids at the same Reynolds number. 

Reentrainment is generally reduced by lower inlet gas velocities. Calvert (R-12) reviewed the hterature on predicting the onset of entrainment and found that of Chien and Ibele (ASME Pap. 62-WA170) to be the most reliable. Calvert applies their correlation to a liquid Reynolds number on the wall of the cyclone, Nrcl = 4QilhjVi, where is the volumetric liquid flow rate, cmVs hj is the cyclone inlet height, cm and Vi is the Idnematic liquid viscosity, cmVs. He finds that the onset of entrainmeut occurs at a cyclone inlet gas velocity V i, m/s, in accordance with the relationship in = 6.516 — 0.2865 lu A Re,L ... 

Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

Work done by Wiesner [6] is a much more accurate approach. The subject has also been reported on more recently by Simon and Bulskamper [71. They generally agree with Wiesner that the variance of performance with Reynolds number was more true at low value that at high values. The additional influence above a Reynolds number of 10 is not much. It would appear that if a very close guarantee depended on the Reynolds number to get the compressor within the acceptance range (if the Reynolds number was high to begin with), the vendor would be rather desperate. 

Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the ir-theorem is tc = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number Re = w Upl/i. Thus, we may write ... 

This expression represents the first form of the general dimensionless equation of sedimentation theory. As the desired value is the velocity of the particle, equation 64 is solved for the Reynolds number ... 

Experimental data on power eonsumption are generally plotted as a funetion of the Power number Np versus Reynolds number Np, that is by rearranging Equation 7-14. 

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