Reynolds analogy number

This closely resembles the inertial range of the energy cascade process in homogeneous turbulence at high Reynolds numbers. Analogously, assuming... 

Kramers(581 carried out experiments on heat transfer to particles in a fixed bed and has expressed his results in the form of a relation between the Nussell, Prandtl and Reynolds numbers. This equation may be rewritten to apply to mass transfer, by using the analogy between the two processes, giving ... 

For laminar flow of a power-law fluid through a packed bed, Kemblowski et alS25) have developed an analogous Reynolds number (Rei) , which they have used as the basis for the calculation of the pressure drop for the flow of power-law fluids ... 

The (Ta - Ts)/8 term is the temperature gradient, which correlates (dTIdy through the boundary layer thickness. The fact that 8 can be correlated with the Reynolds number and that the Colburn analogy can be applied leads to the correlation of the form... 

Obviously, correlations of one of these friction factors with an analogous Reynolds number, or with two-phase pressure-drop, throws little light on the other variables concerned, and these quantities will appear as parameters in any proposed relationship. However, Govier and Omer point out that plots of such a form do give a systematic spread of data above the single phase lines and allow easy comparison of trends. 

Yabannavar et al. [81] proposed a proportionality relationship valid for spin-filters based on an analogy to Eq. (15). They defined the Reynolds number based on the tangential velocity at the screen surface. Since in spin-filters the permeation velocity, or perfusion flux, is given by Eq. (16), and it can be assumed that the screen porosity e will be maintained constant throughout the scale-up process, it is possible to write a proportionaHty relationship for the ratio from drag to lift force in spin-filters as given by Eq. (17). 

Here is a function of Re, analogous to used for steady motion in Chapter 5, and may be evaluated using the correlations in Table 5.2. Since = 24Rexs for a spherical particle at its terminal velocity, Re s fixes the terminal Reynolds number Re via the correlations in Table 5.3. The relationship between Rex and Re s is shown by the uppermost curve in Fig. 11.11. In view of the complex dependence of A and Ah on Re and Re, Eq. (11-33) must be... 

In the central core for Reynolds numbers above 35,000 an expression analogous to Eq. (11) but based on different experimental information may be written (S7) as... 

The Reynolds analogy is equivalent to setting the turbulent Prandtl number as defined in Eq. (24) equal to unity. Figure 13 shows the effect of Reynolds number upon a space average value of the turbulent Prandtl number (C3, P3, S7). Information presented in Fig. 13 is open to uncertainty since it is based upon measurements for air and represents only the space average value of this ratio throughout the turbulent portion of the stream. The turbulent Prandtl number is undoubtedly a function of position as well as of the Reynolds number for a given stream (PI). 

Data acquired by many investigators have shown a close analogy between the rates of heat and mass transfer, not only in the case of packed beds but also in other cases, such as flow through and outside tubes, and flow along flat plates. In such cases, plots of the /-factors for heat and mass transfer against the Reynolds number produce almost identical curves. Consider, for example, the case of turbulent flow through tubes. Since... 

J5 Consider the wall-injection problem in an axisymmetric setting, where a uniform injection velocity flows through the wall of a cylindrical tube. There is a mean velocity U that enters through one end of the tube. Following a procedure analogous to the flow-between-plates problem (Section 5.6), develop a solution for the velocity profiles and the wall shear stress as characterized by the product of a Reynolds number and a friction factor. 

The equation is nearly analogous to the thermal energy equation with the Schmidt number replacing the Prandtl number, although the density dependence is different. The Schmidt number and Reynolds numbers are defined as... 

By analogy with the Newtonian relation, / = 16/Re, the denominator of Eq. (6.52) is designated as a modified Reynolds number,... 

The higher the initial concentration, the greater is the effectiveness of the polymer and, by analogy, the maximum of effectiveness is displaced in the direction of increasing Reynolds numbers. At equal concentrations, the long-chain macromolecule is more effective, as in fact one would expect for reasons of the chain-length influence. 

Horizontal Reactors. Horizontal reactor flow may involve both transverse and longitudinal rolls, as well as time-periodic flows. Insights into these phenomena may be gained from previous analysis of idealized, analogous systems, as well as from recent experiments and computations. Analytical studies of flow between two plates of infinite size differentially heated from below (180) and horizontal channel flow (181) indicate that the development of transverse and longitudinal rolls depends on the relative and absolute magnitudes of the dimensionless Rayleigh and Reynolds numbers, Ra and Re, as well as the aspect ratio. 

The rise velocity of a single spherical cap bubble in an infinite liquid medium can be described by the Davies and Taylor equation [Davies and Taylor, 1950] (Problem 9.6). Experimental results indicate that the Davies and Taylor equation is valid for large bubbles (4oo > 0.02 m, in general) with bubble Reynolds numbers greater than 40, while for bubbles in fluidized beds, the bubble Reynolds numbers are typically on the order of 10 or less [Clift, 1986]. By analogy, the rise velocity of an isolated single spherical cap bubble in an infinite gas-solid medium can be expressed in terms of the volume bubble diameter by [Davidson and Harrison, 1963]... 

Air flows over a wide t-m long flat plate which has a uniform surface temperature of 80°C, the temperature of the air ahead of the plate being 20°C. The air velocity is such that the Reynolds number bas J on the length of the plate is 5 x 106. Derive an expression for the local wall heat flux variation along the plate. Use the Reynolds analogy and assume the boundary layer transition occurs at a Reynolds number of 10 ... 

Numerically determine the local Nusselt number variation with two-dimensional turbulent boundary layer air flow over an isothermal flat plate for a maximum Reynolds number of 107. Assume that transition occurs at a Reynolds number of 5 X 105. Compare the numerical results with those given by the Reynolds analogy. 

Pe is a Peclet number, analogous to the Reynolds number, which is defined for this flow as... 

Qfis may ask the reason for the functional form of Eq. (6-4). Physical reasoning, based on the experience gained with the analyses of Chap. 5, would certainly indicate a dependence of the heat-transfer process on the flow field, and hence on the Reynolds number. The relative rates of diffusion of heat and momentum are related by the Prandtl number, so that the Prandtl nunfber is expected to be a significant parameter in the final solution. We can be rather confident of the dependence of the heat transfer on the Reynolds and Prandtl numbers. But the question arises as to the correct functional form of the relation i.e., would one necessarily expect a product of two exponential functions of the Reynolds and Prandtl numbers The answer is that one might expect this functional form since it appears in the flat-plate analytical solutions of Chap. 5, as well as the Reynolds analogy for turbulent flow. In addition, this type of functional relation is convenient to use in correlating experimental data, as described below. 

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