Reynolds number, formula

The mass flow is found using the continuity equation riv= punrd /4 and the Reynolds number formula Re = dm/firp dv)-. 

Starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (u, = /xRe/dpp). The following is an interpolation formula that can be applied over all three settling regimes ... 

But in order to prediet ly, knowledge of cdoo is required as a funetion of the fluid properties and partiele eharaeteristie dimension, viz. the partiele Reynolds number. This is straightforward for partieles experieneing laminar flow for whieh there exists an analytieal solution. In turbulent eonditions, however, the flow is mueh more eomplex and analytieal solutions are not available. Fortunately, in these eases resort ean be made to semi-empirieal (or semi-theoretieal) formulae or eharts that eorrelate reported experimental data over a wide range of eonditions. 

The average Nusselt number is not very sensitive to changes in gas velocity and Reynolds number, certainly no more than (Re)I/3. The Sherwood number can be calculated with the same formula as the Nusselt number, with the substitution of the Schmidt number for the Prandtl number. While the Prandtl number of nearly all gases at all temperatures is 0.7 the Schmidt number for various molecules in air does depend on temperature and molecular type, having the value of 0.23 for H2, 0.81 for CO, and 1.60 for benzene. 

To save computational effort, high-Reynolds number models, such as k s and its variants, are coupled with an approach in which the viscosity-affected inner region (viscous sublayer and buffer layer) are not resolved. Instead, semiempiri-cal formulas called wall functions are used to bridge the viscosity-affected region between the wall and the fully turbulent region. The two approaches to the sublayer problem are depicted schematically in Fig. 2 (Fluent, 2003). 

Chapter 2 reviews the statistical theory of turbulent flows. The emphasis, however, is on collecting in one place all of the necessary concepts and formulae needed in subsequent chapters. The discussion of these concepts is necessarily brief, and the reader is referred to Pope (2000) for further details. It is, nonetheless, essential that the reader become familiar with the basic scaling arguments and length/time scales needed to describe high-Reynolds-number turbulent flows. Likewise, the transport equations for important one-point statistics in inhomogeneous turbulent flows are derived in Chapter 2 for future reference. 

Finlayson and Olson (1987) used the Galerkin finite element numerical method to explore heat transfer to spheres at low to intermediate Reynolds numbers (1 < Re < 100) and for Prandtl numbers in the range 0.001-1,000. They found that the best correlation of their data was an interpolation formula of the form proposed by Zhang and Davis their correlation is... 

Reynolds numbers are an order of magnitude below turbulence. The D/1 ratios are less than 1/10 except for the 10 hi" pore membrane in which Foiseuille flow formulas need to be corrected for end effects to determine true viscosities or shear rates. 

Reynolds number, p 46), etc 61-72 (Shock relationships and formulas) 73-98 (Shock wave interactions formulas) 99-102 (The Rayleigh and Fanno lines) Ibid (1958) 159-6l(Thermal theory of initiation) 168-69 (One-dimensional steady-state process) 169-72 (The Chapman-Jouguet condition) 172-76 (The von Neumann spike) 181-84 (Equations of state and covolume) 184-87 (Polytropic law) 188, 210 212 (Curved front theory of Eyring) 191-94 (The Rayleigh transformation in deton) 210-12 (Nozzle thepry of H. Jones) 285-88 (The deton head model) ... 

Comparison of McMillen s results with those for Newtonian fluids is instructive in two important respects. First, the non-Newtonian entrance loss was felt 40 diameters downstream from the entrance. Since the Reynolds number of the flow was only 50, the entrance loss for a Newtonian fluid (P3) would have only been felt downstream for a distance of 3 diameters. Second, comparison of the foregoing formula with the usually recommended procedure for Newtonian fluids (P3) indicates that the non-Newtonian pressure drop was approximately six times as great as that for Newtonian fluids under the same conditions. This figure was checked reasonably closely by the later work of Mooney and Black (M16), who found entrance losses of up to seven times those for comparable Newtonian fluids. Since this entrance loss (P3) is due to the energy required to set up the velocity profile, it might appear logical that... 

For the empirical formula (Equation 4), Davies divides the Reynolds number range into two parts. The data for Re are fitted with a fourth-order polynomial in 4Re2 for the higher values. To integrate Equation 3 analytically, one must fit the same data with second-order polynomials. The results are as follows ... 

The results of our tentative calculations [formulas (41)—(43)] of the distance at which a stabilized regime is established and the braking and heat transfer cover the entire cross-section show the opposite whereas in the stabilized flow the dependence on the Reynolds number disappears, the distance at which this stabilization occurs is very strongly dependent on the Reynolds number. At our large Reynolds numbers, long before stabilization, at a distance of 5 105d/Re turbulization of the boundary layer takes place. 

Alternatively, the friction loss can be estimated using the formula and friction factor (Fp) chart presented in Ref. P3 (p. 143). In this case the friction factor is a function of Reynolds number (NRe). Friction loss per metre = 4 FDp u2/D... 

When an SRV is sized for viscous liquid service, it is suggested that it would be sized first as for a non-viscous type application in order to obtain a preliminary required effective discharge area (A). From the manufacturer s standard effective orifice sizes, select the next larger orifice size and calculate the Reynolds number, Re, per the following formula ... 

Churchill also provided a single equation that may be used for Reynolds numbers in laminar, transitional, and turbulent flow, closely fitting/= 16/Re in the laminar regime, and the Colebrook formula, Eq. (6-38), in the turbulent regime. It also gives unique, reasonable values in the transition regime, where the friction factor is uncertain. 

An empirical formula for the turbulent-friction factor up to Reynolds numbers of about 2 x 10s for the flow in smooth tubes is... 

Calculate the impeller Reynolds number. The formula for impeller Reynolds number is... 

AVe modified Peclet number, Gdp/E Niu, Reynolds number, Gdp/p P pressure (ML 1r>) p , gw coefficients in recursion formulas Q heat of reaction (—AH) mole-1)... 

This formula is valid for Reynolds numbers up to 50 and for sphericities from 1 to 0.8. As the gelatinous hydroxide floes have an avarage sphericity of 0.8 (3—5), the drag coefficient can be approximated in laminar conditions (Re<0.1)by(7--9) ... 

There are two main approaches to modeling the near-wall region. In one approach, the so-called wall function approach, the viscosity-affected inner regions (viscous and buffer layers) are not modeled. Instead, semi-empirical formulae (wall functions) are used to bridge the viscosity-affected region between the wall and the fully turbulent region. In another approach, special, low Reynolds number turbulence models are developed to simulate the near-wall region flow. These two approaches are shown schematically in Fig. 3.5(b) and 3.5(c). 

A number of numerical calculations was carried out for various Reynolds numbers up to 2000, and the duct s initial region length was found as a graphical function Lx = f(Re h, A) represented in Fig. 3.11. The dependence Lx vs A significantly differs for small and big Reynolds numbers Re. The value A = 0 corresponds to the case where the EPR is absent. It can be seen that, for A -> 0, all the curves arrive at the constant value c 0.03, but the curves associated with small Reynolds numbers 1, 5, and 10. Hence, it can be concluded that the principal Schlichting s formula (3.45) has been justified for large Re despite it was derived from the boundary-layer approach, rather than from the complete Navier - Stokes equations. 

It is possible to solve a flow problem in either dimensional or dimensionless form. The variables can be assigned values using a consistent set of dimensions, which must be the SI system for turbulent flow. The dimensional formula is convenient since the problem is usually specified in that way, but in some cases the iterations may not converge. Alternatively, the equations can be made dimensionless. The dimensionless formulations are good when you are having trouble getting the iterations to converge, since you have a better sense of the problem when you specify the Reynolds number. This section takes the dimensional Navier-Stokes equation, Eq. (10.40), and derives two different dimensionless versions ... 

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