For an open channel

For the computation of NF, the length L must be some linear dimension that is significant in the flow pattern. For a ship it is commonly taken as the length at the waterline. For an open channel it is customary to take it as the depth. From Eq. (10.46), V varies as -(gL)V2, and if s be considered as a constant, as is usually the case, then from Eq. (10.42) ... 

If the limit is negative, then the channel is closed and the nuclei remain bounded to each other. The form of the solutions of the coupled equations, Eqs. (39 or 40), depends on the imposed boundary conditions. Consider first the case where there is at least one open channel. Scattering boundary conditions consist of imposing, for an open channel n, a combination of incoming (exp [ - ik R]) and outgoing (exp [ - - iknR]) waves, while all closed channel functions are constrained to vanish asymptotically. In the case... 

For an open channel flow, the dimensionless Froude number of the approaching flow can be defined as follows ... 

There are two basic (and interrelated) useful parameters for an open-channel slurry system design the minimum slope (to maintain slurry suspension) of straight lengths of launder, Smin (usually expressed as a percentage), and the velocity head corresponding to the minimum slurry velocity, K (expressed in metres), is often quoted as part of process technology, and may be arrived at directly by practical experience, whereas K is usually derived. The parameters have an approximate theoretical relationship, and the minimum slurry velocity Kmin is essentially the same as the minimum velocity to avoid settlement in full-flow pipes of comparable diameter, in terms of wetted perimeter. 

An important point is that with turbulence promoters, values are an order of magnitude greater than those for an open channel, although significantly more pumping energy is expended in achieving this increase. 

Typical dimensions for an open channel column are 20 pm x 60 pm x 10-15 cm. The disadvantage with open channel columns is that there is only a limited sized surface on which chromatographic interactions can take place, and which in its turn limits the capacity of the column. This can partly be circumvented by coating the walls with porous particles. 

Abulnaga, B. E. 1997. Channel 1.0 Computer Program For an Open Channel Slurry Flow. Fluor Daniel Wright Engineers. Vancouver, BC, Canada. Internal report. 

Since the flow does not frll the launder or pipe, the hydraulic diameter is the defined as the equivalent diameter of flow for an open channel. The hydraulic radius is defined as the ratio of the area of the flow to the wetted perimeter. It is also called the hydraulic mean depth in certain European books. 

Modifying it for an open channel in laminar flow yields... 

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. 

An empirical equation for the calculation of the velocity of flow in an open channel is the Cheiy equation, which may be expressed as ... 

Similarly it may be shown that, at the critical conditions, the flowrate is a maximum for a given value of the specific energy J. At the critical velocity, (ir/gD) is equal to unity. This dimensionless group is known as the Froude number Fr. For velocities greater than the critical velocity Fr is greater than unity, and vice versa. It may be shown that the velocity with which a small disturbance is transmitted through a liquid in an open channel is equal to the critical velocity, and hence the Froude number is the criterion by which the type of flow, tranquil or rapid, is determined. Tranquil flow occurs when Fr is less than unity and rapid flow when Fr is greater than unity. 

The energy dissipated in the hydraulic jump is now calculated. For a small change in the flow of a fluid in an open channel ... 

C. Here, B is assumed to combine mainly with the active forms of the receptor to form complexes (BR, ABR ) that are inactive. An example is the action of an open channel blocker. Note the convergence of the curves at low agonist concentrations (contrast with the pattern expected for noncompetitive antagonism, as in panel A and as shown in Figure 1.26). 

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. 

Slow-channel syndrome. Abnormally long-lived openings of mutant AChR channels result in prolonged endplate currents and potentials, which in turn elicit one or more repetitive muscle action potentials of lower amplitude that decrement. The morphologic consequences stem from prolonged activation of the AChR channel that causes cationic overload of the postsynaptic region - the endplate myopathy - with Ca2+ accumulation, destruction of the junctional folds, nuclear apoptosis, and vacuolar degeneration of the terminal. Some slow-channel mutations in the transmembrane domain of the AChR render the channel leaky by stabilization of the open state, which is populated even in the absence of ACh. Curiously, some slow-channel mutants can be opened by choline even at the concentrations that are normally present in serum. Quinidine, an open-channel blocker of the AchR, is used for therapy. 

Equation 2.73 is another way of writing equation 2.13 where, in this case, the pressure drop is expressed in height of fluid instead of in force per unit area. In equation 2.73, de is the equivalent diameter defined as four times the cross-sectional flow area divided by the appropriate flow perimeter,/is the Fanning friction factor for flow in an open channel and u is the mean velocity. Combining equations 2.72 and 2.73, and solving for u gives... 

Put in ordinary terms, the more successful we are in causing a separation, the more propensities there are for a re-mixing of the components. There are many ways this can occur but there are a fewer number of important routes to mixing. It seems reasonable that we examine these before we consider all the possible ways in which thermodynamics can be controlled in general terms. In almost all equilibrium separation systems, the separation can occur either in a packed bed of particles or fibers or in an open channel or tube. The stationary phase is either coated on the walls of the channel or on the particles/fibers of the packed bed. If there were no mixing mechanisms an infinitely narrow packet containing the components would become a series of infinitely narrow packets of pure components moving at different velocities toward the end of the packed bed or tube. 

There are thus three states for an ion channel, namely the resting state, activated state and inactivated state. Only the activated state is conducting in the resting state and in the inactivated state, the ion channel is closed. For renewed opening, the ion channel must move from the inactivated state to the resting state. 

Y. Nino, M.H. Garcia, Experiments on particle-turbulence interactions in the near-wall region of an open channel flow Implications for sediment transport, J. Fluid Mech. 326 (1996) 285-319. 

After sending and measuring a stream of photons Alice and Bob discuss the transmission using an open channel. Bob tells Alice which polarizers he used and Alice tells him if they match polarizers used by her. In the next step they discard all data for which there was no match. The remaining data can be converted to a bit string of 0s and Is and this string represents the transmitted encryption key. 

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. 

In a closed pipe conveying fluid under pressure, there is no relation between the slope of the pipe and the slope of the hydraulic gradient. In uniform flow in an open channel, the water surface must be parallel to the bed of the stream, and consequently S is then also the slope of the bed, S0 thus we arrive at the important relation, for uniform flow only, that... 

In the case of a real liquid in an open channel, it is necessary to differentiate between the behavior at subcritical and supercritical velocities. Subcritical flow in a rectangular channel has been investigated experimentally and has been found to conform fairly well to ideal conditions, especially within the first part of the bend [44], As the flow continues around the bend, the velocity distribution becomes complicated by the phenomenon of spiral flow, which for open channels is analogous to the secondary counterrotating currents found at bends in closed pipes. 

Figure 11 Schematic diagram of the experimental facility for simultaneous measurement of turbulent velocity field and free-surface wave amplitude in an open channel flow using PIV (Li et al., 2005c).

If T, were an exact solution of the Schrodinger equation, and total energy E is above the thresholds for n open channels, there would be n linearly independent solutions of the equations... 

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. 

---