Reynolds number modified

Re Particle Reynolds number modified Reynolds number [Eq. (98)] s Distance, cm t Time, sec... 

The generahzed approach of Metzner and Reed (AIChF J., 1, 434 [1955]) for time-independent non-Newtouiau fluids defines a modified Reynolds number as... 

The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, S, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain ... 

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. 

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

In defining a 7-factor (jd) for mass transfer there is therefore good experimental evidence for modifying the exponent of the Schmidt number in Gilliland and Sherwood s correlation (equation 10.225). Furthermore, there is no very strong case for maintaining the small differences in the exponent of Reynolds number. On this basis, the /-factor for mass transfer may be defined as follows ... 

Mixtures of liquids with gas or vapour, flow 181 Modified Reynolds number, non-Newtonian flow 124 Molar units 8 Mole 8... 

For turbulent flow across tube banks, a modified Fanning equation and modified Reynolds number should be used. The following method is based on Reference 1. 

Recently, Celata et al. (1994c) modified Eq. (5-12) on the parameter C, together with a slight modification of the Reynolds number power, to give a more accurate prediction in the range of pressures below 5.0 MPa (725 psia). The modified equation is... 

The modified Reynolds number therefore is based on the velocity in the void fraction v/s, the kinematic viscosity v, and an equivalent diameter s/a, where s is total area per unit volume and a is the dimensional coefficient derived from a correlation of pressure drop data ... 

Using these parameters, the modified Reynolds number is... 

The modified Reynolds number based on the catalyst pellet diameter of 1/8 in. or 0.318 cm is then... 

The model for turbulent drag reduction developed by Darby and Chang (1984) and later modified by Darby and Pivsa-Art (1991) shows that for smooth tubes the friction factor versus Reynolds number relationship for Newtonian fluids (e.g., the Colebrook or Churchill equation) may also be used for drag-reducing flows, provided (1) the Reynolds number is defined with respect to the properties (e.g., viscosity) of the Newtonian solvent and (3) the Fanning friction factor is modified as follows ... 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

This result can also be applied directly to coarse particle swarms. For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, because the particle drag is a direct function of the local relative velocity between the fluid and the solid (the interstitial relative velocity, Fr), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Since Vr = Vs/(1 — Reynolds number and drag coefficient for the suspension (e.g., the particle swarm ) are (after Barnea and Mizrahi, 1973) ... 

The ratio between the bed and particle diameters and the Reynolds number based on bed diameter, superficial velocity, and solid density appear only in the modified drag expression, in which they are combined, see Eq. (40). These parameters form a single parameter, as discussed by Glicksman (1988) and other investigators. The set of independent parameters controlling viscous dominated flow are then... 

Reynolds number at minimum fluidization, dp UmfPjF = Particle Reynolds number, dp U pjjl = Modified Reynolds number ... 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

Govier (1959) developed a method for solving equation 3.55 for tw and the pressure gradient for a given value of the Reynolds number. He defined a modified Reynolds number ReB in terms of (3 ... 

FIG. 17-2 Schematic phase diagram in the region of upward gas flow. W = mass flow solids, lb/(h ft2) = fraction voids pP - particle density, lb/ft3 p, = fluid density, lb/ft CD = drag coefficient Re = modified Reynolds number. (Zenz and Othmer, Fluidization and Fluid Particle Systems, Reinhold, New York, 1960.)... 

Modified Froude group Modified Reynolds number... 

NFr Froude number Npr Substitution given in the text NFr Modified Froude group NRe Reynolds number Nrc Substitution given in the text Nr, Modified Reynolds number NwlCr Critical Weber number of maximum bubble size capable of survival... 

The modified Reynolds number Re is obtained by taking the same velocity and characteristic linear dimension d m as were used in deriving equation 4.9. Thus ... 

The friction factor, which is plotted against the modified Reynolds number, is Pi/pu, where R is the component of the drag force per unit area of particle surface in the direction of motion. R can be related to the properties of the bed and pressure gradient as follows. Considering the forces acting on the fluid in a bed of unit cross-sectional area and thickness /, the volume of particles in the bed is /(I — e) and therefore the total surface is 5/(1 — e). Thus the resistance force is R SH — e). This force on the fluid must be equal to that produced by a pressure difference of AP across the bed. Then, since the free cross-section of fluid is equal to e ... 

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