Reynolds number definition

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

As stated in Eq. 4.156, it might appear that a the solution (i.e., u(r)) would exist for any value of the parameter Re/. However, the velocity profile must be constrained to require that the net mass flow rate is consistent with m = pU Ac, where U is the mean velocity used in the Reynolds number definition. Based on the integral-constraint relationship,... 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

The minus sign results from the definition of Ap, which is equal to p2 - Pi, a negative quantity. The term q is known as the seepage velocity and is equivalent to the velocity of approach v , which is also used in the definition of the Reynolds number. 

Starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (u, = /xRe/dpp). The following is an interpolation formula that can be applied over all three settling regimes ... 

Note that the absolute value of Pp - p has been assumed. The negative value of this difference indicates that the droplet displacement is centripetal. The value of the Reynolds number corresponding to Ar = 3650 from Figure 10 is 45. Hence we can determine the radial settling velocity from the definition of the Reynolds number ... 

The chart given in Figure 16 can be used in the following manner in order to size a relief valve for liquid service. First, determine the area required, A, without any viscosity correction (i.e., for K = 1). Then select the next larger standard orifice size from manufacturer s literature. Determine the Reynolds number, based on the following definition ... 

As indicated in Section 3.7.9, this definition of ReMR may be used to determine the limit of stable streamline flow. The transition value (R ur)c is approximately the same as for a Newtonian fluid, but there is some evidence that, for moderately shear-thinning fluids, streamline flow may persist to somewhat higher values. Putting n = 1 in equation 3,140 leads to the standard definition of the Reynolds number. 

Thus, the available data related to transition in circular micro-tubes testify to the fact that the critical Reynolds number, which corresponds to the onset of such transition, is about 2,000. The evaluation of critical Reynolds number in irregular micro-channels will entail great difficulty since this problem contains a number of characteristic length scales. This fact leads to some vagueness in definition of critical Reynolds number that is not a single criterion, which determines flow characteristics. 

The above analysis is restricted to high Reynolds numbers, although the definition of high is different in a stirred tank than in a circular pipe. The Reynolds number for a conventionally agitated vessel is defined as... 

Group N6 (or some multiple thereof) is also known as a friction factor (/), because the driving force (AP) is required to overcome friction (i.e., the energy dissipated) in the pipeline (assuming it to be horizontal), and N3 is known as the Reynolds number (N e). There are various definitions of the pipe friction factor, each of which is some multiple of N6 e.g., the Fanning friction factor is N6/2, and the Darcy friction factor is 2N6. The group N4 is also known as the Euler number. 

Get a first estimate for D from this value and the definition of the Reynolds number ... 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

In like fashion, the hydraulic diameter and the superficial velocity can be introduced into the definition of the Reynolds number to give... 

This result can also be applied directly to coarse particle swarms. For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, because the particle drag is a direct function of the local relative velocity between the fluid and the solid (the interstitial relative velocity, Fr), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Since Vr = Vs/(1 — Reynolds number and drag coefficient for the suspension (e.g., the particle swarm ) are (after Barnea and Mizrahi, 1973) ... 

In Fig. 2, the normalized model scalar energy spectrum is plotted for a fixed Reynolds number (ReL = 104) and a range of Schmidt numbers. In Fig. 3, it is shown for Sc = 1000 and a range of Reynolds numbers. The reader interested in the meaning of the different slopes observed in the scalar spectrum can consult Fox (2003). By definition, the ratio of the time scales is equal to the area under the normalized scalar energy spectrum as follows ... 

The value of the Reynolds number which approximately separates laminar from turbulent flow depends, as previously mentioned, on the particular configuration of the system. Thus the critical value is around 50 for a film of liquid or gas flowing down a flat plate, around 500 for flow around a sphere, and around 2500 for flow through a pipe. The characteristic length in the definition of the Reynolds number is, for example, the diameter of the sphere or of the pipe in two of these examples. 

For flow around a spherical particle of diameter dp, the appropriate definition of the Reynolds number is... 

The definition of high Reynolds number could thus be a Reynolds number for which Lujr > 380. Using (2.48), this condition yields Re > 8630 or R > 240. 

---