Reynolds-number

The Reynolds number is dependent on the fluid velocity, density, viscosity, and some characteristic length of the system or conduit for pipes, this characteristic length 

ILLUSTRATIVE EXAMPLE 3.9 Calculate the Reynolds number for a fluid with a density of 50 Ib/ft and a viscosity of 0.65 cP flowing through a 5-inch diameter reactor at 10 fjps (feet per second). Is the flow turbulent or laminar  

An important chemical property of an aqueous solution is its pH. The pH measures the acidity or basicity of the solution. In a neutral solution, such as pure water, the hydrogen (H ) and hydroxyl (OH ) ion concentrations are equal. At ordinary temperatures, this concentration is 

The unit g ion stands for gram ion, which represents an Avogadro number of ions. In all aqueous solutions, whether neutral, basic, or acidic, a chemical equilibrium or balance is established between these two concentrations, so that 

The numerical value for Teq given in Equation (3.16) holds for room temperature and only when the concentrations are expressed in gram ion per liter (g ion/L). In acid solutions, Ch+ is Cqh- in basic solutions, Cqh- predominates. 

For high Reynolds numbers (Re 2000) the inertial forces dominate the viscous forces, and the flow is turbulent. An example of turbulent flow would be a whale swimming that would have a Reynolds number of around 200000000 [2]. In this turbulent regime there would be a wake with vortices in the water behind the whale. For low Reynolds numbers (Re 1) the viscous forces dominate, and the flow is steady, or laminar. An example of laminar flow would be a paramedum swimming. In this laminar regime there would be no wake behind the paramecium, and the flow lines would be smooth and steady, with no formation of vortices. In microfluidics, the flow is usually laminar. In laminar flow conditions the mixing of fluids occurs by diffusion, which is slow compared to turbulent mixing. 

The degree of turbulence In pipes and channels is characterised by the value of the Reynolds number. For a pipe  

In a channel, the pipe diameter. Up. is substituted by a hydraulic mean 

However the annulus of the pond in a decanter does not have its inner surface wetted , and thus the hydraulic mean diameter becomes  

If the values for dm from equation (4.121) and velocity from equation (4.11 7) are introduced into the Reynolds number in equation (4.119). this would 

Equations (4.124) and (4.125) can be substituted into equation (4.119) to find the Reynolds number lor helical flow. 

What this transcription into dimensionless variables means physically is very interesting. It means that, if expressed in terms of the dimensionless variables v, x and t, any two fluid problems will have essentially the same flow solutions whenever their Reynolds numbers are equal. This is of considerable practical importance. of course, since it implies that the air flow past an airplane wing, for example, 

In order to give the Reynolds number a physical interpretation, we can look at the typical magnitudes of individual terms of the Navier-Stokes equation. Since I V Vv V /L and vVjl , we see that 

In other words, TZ can be thought of as a gauge , measuring he relative importance of the purely inertial forces (as embodied by the v Vv term) and the viscous forces (as embodied by the term vV v, which provides the mechanism by which energy may be dissipated). 

The simplified circuit model discussed in the previous section depends on the validity of fully developed flow assumption. The present section discusses the fully developed flow criteria from nondimensionalization of the governing equation. 

The proper way to verify whether the nonlinear term (v V)v in the N-S equation can be neglected is to make the equation dimensionless. This means that we express all physical quantities, such as length and velocity, in units of the characteristic scales, for example, Lq for length and Vq for velocity. Reynolds number appears in the nondimensionalized N-S equation and the relative importance of different terms in the can be verified for small Reynolds number assumption. We explain this in the following sections using two examples. 

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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 convection term in the equation of motion is kept for completeness of the derivations. In the majority of low Reynolds number polymer flow models this term can be neglected. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

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. 

Reynolds number Re Reynolds numbers Reynolds stresses Rezipas... 

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

Pressure Drop. The prediction of pressure drop in fixed beds of adsorbent particles is important. When the pressure loss is too high, cosdy compression may be increased, adsorbent may be fluidized and subject to attrition, or the excessive force may cmsh the particles. As discussed previously, RPSA rehes on pressure drop for separation. Because of the cychc nature of adsorption processes, pressure drop must be calculated for each of the steps of the cycle. The most commonly used pressure drop equations for fixed beds of adsorbent are those of Ergun (143), Leva (144), and Brownell and co-workers (145). Each of these correlations uses a particle Reynolds number (Re = G///) and friction factor (f) to calculate the pressure drop (AP) per... 

Reynolds dumber. One important fluid consideration in meter selection is whether the flow is laminar or turbulent in nature. This can be deterrnined by calculating the pipe Reynolds number, Ke, a dimensionless number which represents the ratio of inertial to viscous forces within the flow. Because... 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Fig. 1. Flow profiles, where N is velocity (a) laminar, and (b) turbulent for fluids having Reynolds numbers of A, 2 x 10, and B, 2 x 10 .

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. 

The wedge design maintains a square root relationship between flow rate and differential pressure for pipe Reynolds numbers as low as approximately 500. The meter can be flow caUbrated to accuracies of approximately 1% of actual flow rate. Accuracy without flow caUbration is about 5%. 

This equation is appHcable for gases at velocities under 50 m/s. Above this velocity, gas compressibiUty must be considered. The pitot flow coefficient, C, for some designs in gas service, is close to 1.0 for Hquids the flow coefficient is dependent on the velocity profile and Reynolds number at the probe tip. The coefficient drops appreciably below 1.0 at Reynolds numbers (based on the tube diameter) below 500. 

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 ... 

Both wetted-sensor and clamp-on Doppler meters ate available for Hquid service. A straight mn of piping upstream of the meter and a Reynolds number of greater than 10,000 ate generally recommended to ensure a weU-developed flow profile. Doppler meters ate primarily used where stringent accuracy and repeatabiHty ate not requited. Slurry service is an important appHcation area. 

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 pressure drop accompanying pipe flow of such fluids can be described in terms of a generalized Reynolds number, which for pseudoplastic or dilatant fluids takes the form ... 

The transition from laminar to turbulent flow occurs at Reynolds numbers varying from ca 2000 for n > 1 to ca 5000 for n = 0.2. In the laminar region the Fanning friction factor (Fig. 2) is identical to that for Newtonian fluids. In the turbulent region the friction factor drops significantly with decreasing values of producing a family of curves. 

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. 

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