Intermediate Reynolds number

FIG. 11-14 Correction factor for adverse temperature gradient at intermediate Reynolds numbers. 

Although these wall correction factors appear to be independent of Reynolds number for small (Stokes) and large (> 1000) values of NRe, the value of Kxv is a function of both lVRe and d/D for intermediate Reynolds numbers (Chhabra, 1992). 

For intermediate Reynolds numbers, the wall factor depends upon the Reynolds number as well as d/D. Over a range of 10-2 < Re,pl <10,0 < d/D < 0.5, and 0.53 < n < 0.95, the following equation describes the Reynolds number dependence of the wall factor quite well ... 

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M. Drag force from lattice Boltzmann simulations of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres, Manuscript submitted to AIChE J. (2006). 

HAMIELEC, A. E. and Johnson, A. I. Can. J. Chem. Eng. 40 (1962) 41. Viscous flow around fluid spheres at intermediate Reynolds numbers. 

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... 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

Here we consider three theoretical approaches. As for rigid spheres, numerical solutions of the complete Navier-Stokes and transfer equations provide useful quantitative and qualitative information at intermediate Reynolds numbers (typically Re < 300). More limited success has been achieved with approximate techniques based on Galerkin s method. Boundary layer solutions have also been devised for Re > 50. Numerical solutions give the most complete and... 

As shown in Chapters 3 and 4, creeping flow analyses have little value for Re > 1. A number of workers (M4, M7, Mil, P5, R3) have obtained numerical solutions for intermediate Reynolds numbers with motion parallel to the axis of a spheroid. The most reliable results are those of Masliyah and Epstein (M4, M7) and Fitter et al (P5). Flow visualization has been reported for disks (K2, W5) and oblate spheroids (M5). 

Reestimate viscosity for intermediate Reynolds number. In the intermediate Reynolds number range, 60 < Re < 20,000 (for a helix impeller), the viscous power number is not constant, nor is the turbulent power number. Figure 12.8, a graph of the viscosity power factor as a function of the Reynolds number, can be used to correct the viscous power number in the transitional range. 

Intermediate Reynolds Number Flow, 1 < Re < 100. Hamielec and Johnson (75) obtained approximate expressions for the stream functions of both the inner and outer flows. Brounshtein et al. (69) used this result to analyze the heat and mass transfer for droplets and gas bubbles in liquid flows. 

Wu JC, Deluca RT, Wegener PP (1974) Rise speed of spherical cap bubbles at intermediate Reynolds number. Chem Eng Sci 29 1307-1309... 

Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number was studied numerically in [55],... 

Bozzi, L. A., Feng, J. Q., Scott, T. C., and Pearlstein A. J., Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number, J. Fluid Mech., Vol. 336, pp. 1-32, 1997. 

Nakano, Y. and Tien, C., Viscous incompressible non-Newtonian flow around fluid sphere at intermediate Reynolds number, AIChE J., Vol. 16, No. 4, pp. 554-569, 1970. 

Rimon, J. and Cheng, S. I., Numerical solution of a uniform flow over a sphere at intermediate Reynolds numbers, Phys. Fluid, Vol. 12, No. 5, pp. 949-959, 1969. 

Although considerable success in the development of the dynamic adsorption layer theory has been reached, there has been less progress experimentally. This is not marked for the transient state, where the theoretical advances are most impressive. It turns out that experimental works devoted to the stagnant cap theory are more or less of empirical interest as they are restricted to small Reynolds numbers. At small, and even intermediate, Reynolds numbers the bubble surface can initially behave immobile and the formation of a stagnant cap is almost impossible. 

This qualitative discussion cannot be reflected by Eq. (10.47) which is a very primitive approximation especially with respect to intermediate Reynolds number 1 < Re < 100. 

When deposition of emulsion drops is investigated, sedimentation can be neglected so that the first term in Eq. (10.27) dominates even at strong retardation. Retardation of the surface becomes less efficient with increasing bubble dimensions and, respectively, Reynolds number. Therefore, the experimental verification of the hypothesis of incomplete retardation of the surface at intermediate Reynolds numbers is of interest. A maximum removal of impurities fi om the water used is important in such experiments. 

T. Sundararajan and P. S. Ayyaswamy, Heat and Mass Transfer Associated with Condensation on a Moving Drop Solutions for Intermediate Reynolds Numbers by a Boundary Layer Formulation, J. Heat Transfer, 107, pp. 409-416,1985. 

Seeley, L.E., Hummel, R.L., and Smith, J.W., Experimental velocity profiles in laminar flow around spheres at intermediate Reynolds numbers, J. Fluid Mechs., 6S, 591-608 (1975). 

In practice, lower values than those calculated according to Eq. (30) are found for intermediate Reynolds numbers, only approaching them in the presence of oscillation. A compilation of experimental R factors is available (Cl). The experimental effective diffusivity in gas absorption with internal circulation has been plotted against the axial velocity representing the flow adjacent to the interface (G7), and shows a direct relation between circulation and internal resistance. Similar results are reported (C6) for other gas-liquid and liquid-liquid systems. 

Answer For boundary layer mass transfer across gas-liquid interfaces, X = and y =. In the laminar flow regime, 2 = 5. This problem is analogous to one where the bubble is stationary and a liquid flows past the submerged object at intermediate Reynolds numbers. 

Rivkind, V. Y. and Ryskin, G. M. Flow structure in motion of a spherical chop in a fluid medium at intermediate Reynolds numbers. Fluid Dynamics (English translation of Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza), 11, 5-12,1976. 

Clair, B. L. and Hamielec, A. Viscous flow through particle assemblages at intermediate Reynolds numbers. leEC Fundam. 7, 308-315, 1968. 

Tal, R. and Sirignano, W. Cylindrical cell model for hydrodynamics of particles assemblages at intermediate Reynolds numbers. AIChE J. 28(2), 233-237, 1982. 

Rivkind, V.Y., Ryskin, G.M., and Fishbein, G.A., Flow around a spherical drop in a fluid medium at intermediate Reynolds numbers, Appl. Math. Mech., 40, 687-691, 1976. 

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