Reynolds exponent

In the literature, values ranging from 0.67 to 0.7 are mostly reported and, in order to improve comparability, in all of the follo ving discussions the Reynolds exponent is approximated by 0.67. Nusselt functions are typically determined for air, vhich is the typical medium used during application. 

For all the detectors, the limiting current increases with Reynolds number. Therefore it is desirable to construct detectors with small cross-sectional areas (or high electrolyte speeds) in order to increase the mean linear fluid velocity, and to choose an electrode geometry with a high Reynolds exponent... 

Because of the wide range of appHcations and complexity of the physical phenomena, the values of the exponents reported in the Hterature vary significantly. Depending on the range of Reynolds and Weber numbers, constant a ranges between 0.25 and 0.6, constant b between 0.16 and 0.25, constant (/between 0.2 and 0.35, and constant dfiom 0.35 to 1.36. 

Both effects can produce coarser atomization. However, the influence of Hquid viscosity on atomization appears to diminish for high Reynolds or Weber numbers. Liquid surface tension appears to be the only parameter independent of the mode of atomization. Mean droplet size increases with increasing surface tension in twin-fluid atomizers (34). is proportional to CJ, where the exponent n varies between 0.25 and 0.5. At high values of Weber number, however, drop size is nearly proportional to surface tension. 

Leva s correlation (Leva 1949) is the easiest to use in manual calculation, especially when the particle diameter-based Reynolds number is high, i.e., above Rep>l,000. A changing exponent n in the Leva expression, shown below, accounts for the transient region as turbulence of flow increases. The dependence of n on Rep was specified by Leva graphically (1949) as n growing with the Rep between 1.0 and 2.0. The value reaches n = 1.95 at Rep =1,000 thus approximating 2.0 closely. 

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 ... 

Coefficient A and exponent a must be evaluated experimentally. Experiments have shown that A and a are themselves functions of the Reynolds number. Equation 47 shows that the resistance force increases with increasing velocity. If the force field (e.g., gravity) has the same potential at all points, a dynamic equilibrium between forces P and R develops shortly after the particle motion begins. As described earlier, at some distance from its start the particle falls at a constant velocity. If the acting force depends on the particle location in space, in a... 

Coefficient A and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function = f (Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope P = 135 with respect to the x-axis. This section corresponds to the critical Reynolds number, Re < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, p, in equation 46 vanishes. 

The turbulent regime for Cq is characterized by the section of line almost parallel to the x-axis (at the Re" > 500). In this case, the exponent a is equal to zero. Consequently, viscosity vanishes from equation 46. This indicates that the friction forces are negligible in comparison to inertia forces. Recall that the resistance coefficient is nearly constant at a value of 0.44. Substituting for the critical Reynolds number, Re > 500, into equations 65 and 68, the second critical values of the sedimentation numbers are obtained ... 

Exponent given by equations Mass of particle, lb mass Constant given in table Reynolds number, dimensionless (use consistent units)... 

Figure 5-41 indicates the mixing correlation exponent, X, as related to power per unit volume ratio for heat transfer scale-up. The exponent x is given in Table 5-6 for the systems shown, and is the exponent of the Reynolds number term, or the slope of the... 

Determine from experimental data the Reynolds number exponent for the s) stem, or use information in Figure 5-40 if the systems of Table 5-8 can be considered similar, use proper coefficients and solve for outside film coefficient, hg. 

For rotating cylinders the exponent x for Reynolds number is very often unity for turbulent flow, and therefore L may be included in the constant term for a particular geometry of cynlinder. 

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 ... 

At Reynolds numbers above 400, the exponent n on the Reynolds number is. 61, very close to the. 67 in the case of a turbine. At Reynolds numbers below 400, where the flow is presumably laminar, the exponent drops to. 43. As a result, heat transfer coefficient varies inversely with the viscosity to the. 28 and. 1 power at Reynolds numbers above and below 400 re spec tively. 

Hubbard and Lightfoot (HI la) earlier reported a Sc,/3 dependence on the basis of measurements in which the Schmidt number was varied over a very large range. The data did not exclude a lower Reynolds number exponent than 0.88, and reaffirmed the value of the classical Chilton-Colburn equation for practical purposes. Recent measurements on smooth transfer surfaces in turbulent channel flow by Dawson and Trass (D8) also firmly suggest a Sc13 dependence and no explicit dependence of k+ on the friction coefficient, with Sh thus depending on Re0,875. The extensive data of Landau... 

Primary device Discharge coefficient at infinite Reynolds number Coefficient b Exponent n... 

From dimensional considerations, the drag coefficient is a function of the Reynolds number for the flow relative to the particle, the exponent, nm, and the so-called Bingham number Bi which is proportional to the ratio of the yield stress to the viscous stress attributable to the settling of the sphere. Thus ... 

Conventional dimensional analysis uses single length and time scales to obtain dimensionless groups. In the first section, a new kind of dimensional analysis is developed which employs two kinds of such scales, the microscopic (molecular) scale and the macroscopic scale. This provides some physical significance to the exponent of the Reynolds number in the expression of the Sherwood number, as well as some bounds of this exponent for both laminar and turbulent motion. 

Equations (15) and (17) show that the exponent of the Reynolds number is... 

In addition, Eq. (37b) can be modified to correlate the data of James and Acosta using the observation that in the lower range of Reynolds numbers employed by them the exponent of the Reynolds number is nearer to 1/3 than... 

The exponent m is a function of particle Reynolds number based on the minimum fluidization velocity. It can be estimated by the following correlation ... 

This type of equation comes from dimensional analysis. The coefficient and exponents are found by experiment. If forced convection is used, the Reynolds number has the conventional meaning of DVp/p. If free convection is used, the Reynolds number is replaced by the Grashof number, which can be shown to have a meaning of a Reynolds number, owing to free convection (B8). 

In order to obtain the coefficient and the two exponents —0.7), Rohsenow used experimental data for five systems water boiling on a 0.024-in.-diam. platinum wire (Al), water on a 1.5-in.-diam. horizontal tube (C6), and benzene, ethyl alcohol, and n-pentane on a chromium-plated horizontal surface (C2). The exponents for the Reynolds and Prandtl numbers were constant at % and —0.7 as shown. Unfortunately the coefficient varied from 0.006 to 0.015 from system to system. 

Although the Reynolds number occurs to about the same exponent in Eqs. (9) and (20), the meaning and numerical values of these Reynolds numbers are quite different. In addition the two equations disagree completely on the effect of the Prandtl number. Comparing Eqs. (11) and (22) shows that the two semitheoretical equations have little in common in respect to prediction of effects of variables on h. However, any change in AT in Eq. (21) must be accompanied by a change in pY — so a comparison of the effects of AT is not so simple as first appears. 

In one respect Eq. (20) is satisfying. The exponents on the Reynolds and Prandtl numbers are roughly the same as those used for ordinary forced-convection heat transfer. The negative sign on the Prandtl-number exponent in Rohsenow s equation has seemed illogical to many scientists. The logical exponents found by Forster and Zuber... 

The expressions given by Eqns. (3.4-9) correspond to the theory of the boundary layer. Similar expressions are obtained with different theories. In practical work, expressions of the type given below are used for different arrangements. Generally, the exponent of the Reynolds number is less than unity, while the exponent of the Schmidt and Prandtl numbers has been kept as 1/3. The usual expressions for the Nusselt and Sherwood numbers are ... 

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