Reynolds’ relation

Some experimental data on kinetics of film thinning confirm the conclusions from Eq. (3.60), i.e. a considerable acceleration can be expected to occur mainly in films from solutions of weak surfactants such as low molecular fatty alcohols [43] dodecyl alcohol in nitrobenzene/aniline [44]. Vice verse, films from typical surfactants such as NaDoS in water or silicone oil in nitrobenzene/hexadecane [44,52], thin with negligible deviations (less than 10%) from Reynolds relation. 

Reynolds relation requires liquid drainage from the film to follow strictly the axial symmetry between parallel walls. Rigid surfaces ensure such drainage through their non-deformability, while non-equilibrium foam films are in fact never plane-parallel. This is determined by the balance between hydrodynamic and capillary pressure. Experimental studies have shown that only microscopic films of radii less than 0.1 mm retain their quasiparallel surfaces during thinning, which makes them particularly suitable for model... 

These experimental results clearly indicate that Reynolds relation can be applied only to sufficiently small films (r < 100 pm), i.e. to films of a uniform thickness. This means that one of the main reasons for strong deviations from the theoretical relation should be attributed to the deformations of the film surfaces. 

Eq. (3.63) holds for large fdjns with strongly expressed non-homogeneity. When the radius decreases the film profile becomes parallel and its rate of thinning asymptotically satisfies Reynolds relation. The theory predicts also [66,67] that the transition between the... 

The question of the ( -potential value at the electrolyte solution/air interface in the absence of a surfactant in the solution is very important. It can be considered a priori that it is not possible to obtain a foam film without a surfactant. In the consideration of the kinetics of thinning of microscopic horizontal foam films (Section 3.2) a necessary condition, according to Reynolds relation, is the adsorption of a surfactant at both film surfaces. A unique experiment has been performed [186] in which an equilibrium microscopic horizontal foam film (r = 100 pm) was obtained under very special conditions. A quartz measuring cell was employed. The solutions were prepared in quartz vessels which were purified from surface impurities by a specially developed technique. The strong effect of the surfactant on the rate of thinning and the initial film thickness permitted to control the solution purity with respect to surfactant traces. Hence, an equilibrium thick film with initial thickness of about 120 nm was produced (in the ideal case such a film should be obtained right away). Due to the small film size it was possible to produce thick (100 - 80 nm) equilibrium films without a surfactant. In many cases it ruptured when both surfaces of the biconcave drop contacted. Only very precise procedure led to formation of an equilibrium film. 

Scheludko et al. [13,15,73,89,229,230] derived the Reynolds relation in a slightly generalised form and tested it experimentally. The agreement between experiment and theory was very reasonable. More recently, Chan and Horn [231] have used the surface force apparatus (SFA) and found that the Reynolds approach to hydrodynamic lubrication is very successful in describing the drainage of liquid films between smooth solid surfaces. 

Scheludko and Exerowa [14,155,228] assumed that the Reynolds relation holds even at thicknesses where disjoining pressure If acts. Knowing that A/ = p0 - n = Ug (see Sections 3.1 and 3.2) we can write... 

I think some light can be thrown on several of the puzzles mentioned earlier by looking at Reynolds relations with Stokes, who was In close contact with Lord Rayleigh. Beauchamp Tower, the discoverer of hydrodynamic lubrication, was acquainted with Rayleigh as Tower s cousin lived at Weald Hall, which adjoined Rayleigh s estate. 

The variation of viscosity ( ]) with temperature (7), in its simplest form, is given by Reynolds relation... 

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

The constant depends on the hydraulic diameter of the static mixer. The mass-transfer coefficient expressed as a Sherwood number Sh = df /D is related to the pipe Reynolds number Re = D vp/p and Schmidt number Sc = p/pD by Sh = 0.0062Re Sc R. ... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

The classical (and perhaps more famihar) form of dimensionless expressions relates, primarily, the Nusselt number hD/k, the Prandtl number c l//c, and the Reynolds number DG/ I. The L/D and viscosity-ratio modifications (for Reynolds number <10,000) also apply. 

Rothfus, Monrad, Sikchi, and Heideger [Ind. Eng. Chem., 47, 913 (1955)] report that the friction factor/g for the outer wall bears the same relation to the Reynolds number for the outer portion of the anniilar stream 2(r9 — A, )Vp/r9 l as the fricBon factor for circular tubes does to the Reynolds number for circular tubes, where / is the radius of the outer tube and is the position of maximum velocity in... 

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length AP/L to the pipe diameter D, density p, and average velocity V through two dimensionless groups, the Fanning friction factor/and the Reynolds number Re. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Power Consumption of Impellers Power consumption is related to fluid density, fluid viscosity, rotational speed, and impeller diameter by plots of power number (g P/pN Df) versus Reynolds number (DfNp/ l). Typical correlation lines for frequently used impellers operating in newtonian hquids contained in baffled cylindri-calvessels are presented in Fig. 18-17. These cui ves may be used also for operation of the respective impellers in unbaffled tanks when the Reynolds number is 300 or less. When Nr L greater than 300, however, the power consumption is lower in an unbaffled vessel than indicated in Fig. 18-17. For example, for a six-blade disk turbine with Df/D = 3 and D IWj = 5, = 1.2 when Nr = 10. This is only about... 

The comparison of the magnitude of the two resistances clearly indicates whether tire metal or the slag mass transfer is rate-determining. A value for the ratio of the boundary layer thicknesses can be obtained from the Sherwood number, which is related to the Reynolds number and the Schmidt number, defined by... 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

Figure 8-2. NsDs diagram for a turbine stage. Efficiency is on a total-to-total basis that is, it is related to inlet and exit stagnation conditions. Diagram values are suitable for machine Reynolds number Re > 10 . (Balje, O.E., A Study of Reynolds Number Effects in Turbomachinery, Journal of Engineering for Power, ASME Trans., Vol. 86, Series A, p. 227.)...

Impeller Reynolds Number a dimensionless number used to characterize the flow regime of a mixing system and which is given by the relation Re = pNDV/r where p = fluid density, N = impeller rotational speed, D = impeller diameter, and /r = fluid viscosity. The flow is normally laminar for Re <10, and turbulent for Re >3000. 

Rodi, W. A new algebraic relation for calculating the Reynolds stresse.s. ZAMM, vol. 56, pp. T219-T221, 1976. 

Archimedes number A dimensionless number that relates the ratio of buoyancy forces to momentum forces, expressed in many forms depending on the nature of the Reynolds number. 

Friction loss The pressure energy loss that takes place in duct or pipe flow. It is related to the Reynolds number, boundary layer growth, and the velocity distribution. 

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... 

Rec = Reynolds number related to inside diameter of pipe Qg = flow rate, gal/min... 

How does this relate to fluid turbulence The idea is that there exists a critical value of the Reynolds number, TZe, such that intermittent turbulent behavior can appear in the system for TZ > TZe- Moreover, if the behavior of the Lorenz system correctly identifies the underlying mechanism, it may be predicted that, as TZ changes, (1) the duration of the intermittently turbulent behavior will be random, and (2) the mean duration of the laminar phases in between will vary as... 

---