The Reynolds Number

The Reynolds number, which distinguishes between inertial and viscous regimes and 

the Proude number, which is associated with the appearance of ripples. 

The Reynolds number compares inertia and viscosity. It is defined by 

Whenever the Reynolds number is large, the equation of motion of a ridge is fundamentally altered. Consider the case of a one-dimensional ridge. The 

When i e 1, dewetting is viscous and we then have Fm = Fy. This limit corresponds to the analysis in section 7.2 where we have neglected the inertial term d MV)/dt. When 1, dewetting is inertial and Fy is negligible. The second term in equation (7.48) no longer depends on V. The ridge of mass M(t) = peR advances at a constant velocity V that is the solution of equation (7.48). That velocity is 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have  

Another common geometry used for laboratory measurement of viscos- 

In a tube we use the volumetric flow rate, Q, to calculate a mean velocity along the tube and we have 

It is important that we know at what Reynolds number our instrumental configurations give turbulent flow and work below this figure or we will think that shear thickening is occurring A figure of Re 3000 to 10,000 is usually satisfactory for cone and plates or capillary viscometers, but values as low as 300 may be the maximum for some cup and bob units. 

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It is essential for the rotating-disc that the flow remain laminar and, hence, the upper rotational speed of the disc will depend on the Reynolds number and experimental design, which typically is 1000 s or 10,000 rpm. On the lower lunit, 10 s or 100 rpm must be applied in order for the thickness of tlie boundary layer to be comparable to that of the radius of the disc. 

The Reynolds number for flow in a tube is defined by dvpirj, where d is the diameter of the tube, V is the average velocity of the fluid along the tube, p is the density of the fluid, and rj is its dynamic viscosity. At flow velocities corresponding with values of the Reynolds number of greater than 2000, turbulence is encountered. 

An outstanding advantage of common differential pressure meters is the existence of extensive tables of discharge coefficients ia terms of beta ratio and Reynolds numbers (1,4). These tables, based on historic data, are generally regarded as accurate to within 1—5% depending on the meter type, the beta ratio, the Reynolds number, and the care taken ia manufacture. Improved accuracy can be obtained by miming an actual flow caUbration on the device. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

As the Reynolds number rises above about 40, the wake begins to display periodic instabiUties, and the standing eddies themselves begin to oscillate laterally and to shed some rotating fluid every half cycle. These still laminar vortices are convected downstream as a vortex street. The frequency at which they are shed is normally expressed as a dimensionless Strouhal number which, for Reynolds numbers in excess of 300, is roughly constant ... 

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 phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

The dimensionless quantities in brackets are, respectively, the reciprocal of the Froude number, the Euler number, and the reciprocal of the Reynolds number for the system. 

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

Noncircular tubes are often used in various compact heat exchangers and the Reynolds number in these tubes is of interest. For noncircular tubes such as square, rectangular, eUiptic, and triangular tubes, the so-called hydrauhc diameter, defined as... 

The flow distribution in a manifold is highly dependent on the Reynolds number. Figure 14b shows the flow distribution curves for different Reynolds number cases in a manifold. When the Reynolds number is increased, the flow rates in the channels near the entrance, ie, channel no. 1—4, decrease. Those near the end of the dividing header, ie, channel no. 6—8, increase. This is because high inlet velocity tends to drive fluid toward the end of the dividing header, ie, inertia effect. 

Fig. 17. Heat-transfer coefficient comparisons for the same volumetric flow rates for (A) water, 6.29 kW, and a phase-change-material slurry (O), 10% mixture, 12.30 kW and ( ), 10% mixture, 6.21 kW. The Reynolds number was 13,225 to 17,493 for the case of water.

To the extent that radiation contributes to droplet heatup, equation 28 gives a conservative estimate of the time requirements. The parameter ( ) reflects the dependence of the convective heat-transfer coefficient on the Reynolds number ... 

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

For laminar flow (Re < 2000), generally found only in circuits handling heavy oils or other viscous fluids, / = 16/Re. For turbulent flow, the friction factor is dependent on the relative roughness of the pipe and on the Reynolds number. An approximation of the Fanning friction factor for turbulent flow in smooth pipes, reasonably good up to Re = 150,000, is given by / = (0.079)/(4i e ). 

Viscous Drag. The velocity, v, with which a particle can move through a Hquid in response to an external force is limited by the viscosity, Tj, of the Hquid. At low velocity or creeping flow (77 < 1), the viscous drag force is /drag — SirTf- Dv. The Reynolds number (R ) is deterrnined from... 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

A = 4.05 X lO " cm/(s-kPa)(4.1 X 10 cm/(s-atm)) and = 1.3 x 10 cm/s (4)//= 1 mPa-s(=cP), NaCl diffusivity in water = 1.6 x 10 cm /s, and solution density = 1 g/cm . Figure 4 shows typical results of this type of simulation of salt water permeation through an RO membrane. Increasing the Reynolds number in Figure 4a decreases the effect of concentration polarization. The effect of feed flow rate on NaCl rejection is shown in Figure 4b. Because the intrinsic rejection, R = 1 — Cp / defined in terms of the wall concentration, theoretically R should be independent of the Reynolds... 

Based on such analyses, the Reynolds and Weber numbers are considered the most important dimensionless groups describing the spray characteristics. The Reynolds number. Re, represents the ratio of inertial forces to viscous drag forces. 

The Reynolds number is sufficient as a parameter for describing the internal flow characteristics, such as discharge coefficient, air core ratio, and spray angle at the atomizer exit. 

Suppose that an experiment were set up to determine the values of drag for various combinations of O, p, and ]1. If each variable is to be tested at ten values, then it would require lO" = 10, 000 tests for all combinations of these values. On the other hand, as a result of dimensional analysis the drag can be calculated by means of the drag coefficient, which, being a function of the Reynolds number Ke, can be uniquely determined by the values of Ke. Thus, for data of equal accuracy, it now requires only 10 tests at ten different values of Ke instead of 10,000, a remarkable saving in experiments. 

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

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. 

Values of and m for various configurations are hsted in Table 5-5. The characteristic length is used in both the Nusselt and the Reynolds numbers, and the properties are evaluated at the film temperature = (tio + G)/2. The velocity in the Reynolds number is the undisturbed free-stream velocity. 

For falling films applied to the outside of horizontal tubes, the Reynolds number rarely exceeds 2100. Equations may be used for falling films on the outside of the tubes by substituting 7TD/2 for L. 

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... 

The Reynolds number of the condensate film (falling film) is 4r/ I, where F is the weight rate of flow (loading rate) of condensate per unit perimeter kg/(s m) [lb/(h ft)]. The thickness of the condensate film for Reynolds number less than 2100 is (SflF/p g). ... 

The film thickness 6g depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables ... 

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