Open channels Reynolds number

An experimentally based rule-of-thumb is that laminar flow often occurs when the pipe Reynolds number, Vdjv, is less than 2,000, or when an open channel Reynolds number, Vhjv, is less than 500, where V is the cross-sectional mean velocity, d is the pipe diameter, v is the kinematic viscosity of the fluid, and h is the channel depth. The diameter or depth that would not be exceeded to have laminar flow by these experimental criteria is given in Table 5.1. 

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. 

For flow in an open channel, only turbulent flow is considered because streamline flow occurs in practice only when the liquid is flowing as a thin layer, as discussed in the previous section. The transition from streamline to turbulent flow occurs over the range of Reynolds numbers, updm/p = 4000 — 11,000, where dm is the hydraulic mean diameter discussed earlier under Flow in non-circular ducts. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

It is to be noted that, for laminar open channel flow, the Froude and Reynolds numbers are interrelated. Making use of (1), (3), and (19), it is easily shown that... 

In the flow of water in open channels, fluid friction is a factor as well as gravity and inertia, and apparently we face the same difficulty here. However, for flow in an open channel there is usually fully developed turbulence, so that the hydraulic friction loss is exactly proportional to V2, as will be shown later. The fluid friction is therefore independent of Reynolds number, with rare exceptions, and thus is a function of the Froude number alone. 

The Reynolds number does not ordinarily figure prominently in open-channel flow, as the viscous forces are generally far outweighed by roughness considerations. When the occasion calls for it, however, it is usually defined so as to be compatible with that for pipes, or Nr = ARV/I, where R is the hydraulic radius. 

As C and / are related, the same considerations that have been presented regarding the determination of a value for/apply also to C. For a small open channel with smooth sides, the problem of determining/or C is the same as in the case of a pipe. But most channels are relatively large compared with pipes, thus giving Reynolds numbers that are higher than those commonly encountered in pipes. Also, open channels are fre-... 

Behavior of the fluids in the microfabricated channels are different from those in the millimeter scale channels. Miniaturization of micro flow devices opens a new research field, microfluidics which represents the behavior of the fluid in the micro channel [8]. Since the Reynolds number in the micro channel is usually below 200, the flow is laminar and special design concepts are necessary for the fluidic elements of mixers, reaction coils etc. in the pTAS. Some components of flow switches and fluid filters were developed using laminar flow behavior. 

The letters R, F, and W stand for so-called Reynolds, Froude, and Weber numbers, respectively these are dimensionless numbers, as indicated. For example, if we make the Reynolds number the same in model and prototype, using the same fluid, the dimension of length is smaller in the model and hence the velocity v will have to be greater. In other words, the water would have to flow faster in the model. If we now consider the Froude number as the same in model and prototype, and that the same fluid is used in both, we see that the velocity would have to be less in the model than in the prototype. This may be regarded as two contradictory demands on the model. Theoretically, by using a different fluid in the model (thus changing p0 and p), it is possible to eliminate the difficulty. The root of the difficulty is the fact that the numbers are derived for two entirely different kinds of flow. In a fluid system without a free surface, dynamic similarity requires only that the Reynolds number be the same in model and prototype the Froude number does not enter into the problem. If we consider the flow in an open channel, then the Froude number must be the same in model and prototype. 

To give an impression of the validity of the relations presented in this chapter, two graphs of the friction factor vs. the Reynolds number for two lab-scale G/S BSR modules are taken from Ref. 7. The characteristic size in the calculation of Re, as well as the channel diameter in the calculation of 4/, is the hydraulic diameter of the reactor, i.e., four times the open area over the perimeter of the strings and the reactor wall. 

Figure 2 shows an example of a static mixer and a schematic representation of how such structures operate—see, for example, [1] and [2]. The open intersecting channels divide the main fluid stream into a number of substreams. In addition to the lateral displacement caused by the obliquity of the channels, a fraction of each substream shears off into the adjacent channel at every intersection. This continuous division and recombination of the substreams causes transition from laminar to turbulent flow at Reynolds numbers (based on channel hydraulic diameter) as low as 20Q-300 and results in... 

Open Channel Flow For flow in open channels, the data are largely based on experiments with water in turbulent flow, in channels of suflFicient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where hquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional losses and potential energy change. The mechanical energy balance reduces to = g(zi — z. In terms of the friction factor and hydraulic diameter or hydraulic radius,... 

Straub, L.G., Silberman, E., Nelson, H.C. (1958). Open-channel flow at small Reynolds numbers. Trans. ASCE 123 685-706 123 713-714. 

A semi-emperical turbulent one-equation model is developed for rectangular open channel flows of water and viscoelastic fluids. The model is used to predict friction factor vs. Reynolds number relations, velocity profiles, eddy viscosity distributions and turbulent energy budgets. Comparisons are made between the model and the measured results using a Laser Doppler Anemometer. 

Turbulent Reynolds Number Slope of the Open Channel Measuring time Absolute temperature Local axial velocity Local mean velocity Fluctuation velocity Shear velocity... 

Where H is the charmel height (the smaller dimension in a rectangular channel), tw,av the average wall shear stress, V the kinematic viscosity, and p the density of the fluid. In internal flows, the laminar to turbulent transition in abrupt entrance rectangular ducts was found to occur at a transition Reynolds number Ret = 2200 for an aspect ratio ac = I (square ducts), to Ret = 2500 for flow between parallel planes with = 0 [4]. For intermediate channel aspect ratios, a linear interpolation is recommended. For circular tubes. Ret = 2300 is suggested. These transition Reynolds number values are obtained from experimental observations in smooth channels in macroscale applications of 3 mm or larger hydraulic diameters. Their applicability to microchannel flows is still an open question. 

An electrowinning cell can be treated as an open channel (open top) with an equivalent diameter as the characteristic length, which depends on the geometry of the aoss-sectional area. This diameter is needed for determining the Reynolds Number. In general, this diameter is also known as hydraulic diameter defined as... 

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