Flow in open channels

Flow through an open channel of uniform cross section and slope (angle 0) can be calculated mathematically, when the depth of liquid is constant throughout the length of the channel. For a length of L, the accelerating force acting on the liquid 

Rearranging the preceding equations, and dividing both sides by pxF, 

The value of Rm/pv is almost independent of velocity but is a function of surface roughness (s) and is normally defined as 

Combining Equation 2.36 and Equation 2.37, the volumetric flow rate can be calculated as 

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Measurement by Liquid Level. The flow rate of Hquids flowing in open channels is often measured by the use of weirs (see Liquid-LEVEL measurement). The most common type is the rectangular weir shown in Figure 22e. The flow rate across such a weir varies approximately with the quantity. Other shapes of weirs are also employed. Standard civil engineering handbooks describe the precautions necessary for constmcting and interpreting data from weirs. 

Two cases are considered. The first, the laminar flow of a thin film down an inclined surface, is important in the heat transfer from a condensing vapour where the main resistance to transfer lies in the condensate film, as discussed in Chapter 9 (Section 9.6.1). The second is the flow in open channels which are frequently used for transporting liquids down a slope on an industrial site. 

Manning and others gave values of C for various types of surface roughness [Bama (1969)]. A typical value for C when water flows in a concrete channel is 100 m1/2/s. In general, liquids such as water which commonly flow in open channels have a low viscosity and the flow is almost always turbulent. 

The stability of flow in open channels has been investigated theoretically from a more macroscopic or hydraulic point of view by several workers (Cl7, D9, DIO, Dll, 14, J4, K16, V2). Most of these stability criteria are expressed in the form of a numerical value for the critical Froude number. Unfortunately, most of these treatments refer to flow in channels of very small slope, and, under these circumstances, surface instability usually commences in the turbulent regime. Hence, the results, which are based mainly on the Ch<5zy or Manning coefficient for turbulent flow, are not directly applicable in the case of thin film flow on steep surfaces, where the instability of laminar flow is usually in question. The values of the critical Froude numbers vary from 0.58 to 2.2, depending on the resistance coefficient used. Dressier and Pohle (Dll) have used a general resistance coefficient, and Benjamin (B5) showed that the results of such analyses are not basically incompatible with those of the more exact investigations based on the differential rather than the integral ( hydraulic ) equations of motion. The hydraulic treatment of the stability of laminar flow by Ishihara et al. (12) has been mentioned already. 

In addition to the theories reviewed above, there are many treatments in the literature which deal with the hydraulics of wavy flow in open channels. Most of these refer to very small channel slopes (less than 5°) and relatively large water depths. Under these conditions, surface tension plays a relatively minor part and is customarily neglected, so that only gravity waves are considered. For thin film flows, however, capillary forces play an important part (K7, H2). In addition, most of these treatments consider a turbulent main flow, while in thin films the wavy flow is often... 

Keulegan (K13) applied the semiempirical boundary-layer concepts of Prandtl and von K arm an to the case of turbulent flow in open channels, taking into account the effects of channel cross-sectional shape, roughness of the wetted walls, and the free surface. Most of the results are applicable mainly to deep rough channels and bear little relation to the flow of thin films. 

There are numerous reports of investigations of the effects of roughness on flow in open channels. For instance, Reinius (R4) has reported on the effects of surfaces covered with various types of roughnesses (spheres, sand, etc.) on the flow of water in open channels, while Hama (HI) has reported... 

It is well known that in turbulent pipe flow the parabolic profile present in laminar flow becomes blunter, so that the ratio uma,x/u decreases. A similar effect has been found for the relatively deep flows in open channels at small slopes by Jeffreys (J4), who obtained values of us/u down to 1.06, and by Horton et al. (H19), who measured values as low as 1.1. It can be expected that in the flow of thin films the ratio will decrease in turbulent flow from the value of 1.5, but by a very much smaller amount than observed in the deep flows noted above. 

Keulegan (Kl3), 1938 Extension of Prandtl-von KdrmSn turbulent flow theories to turbulent flow in open channels. Effects of wall roughness, channel shape, and free surface on velocity distribution are considered. 

Vedernikov (V2), 1946 Theoretical treatment of wavy flow in open channels. Wavy flow and turbulent flow clearly distinguished. 

Ishihara et al. (12), 1961 Gives summary of recent Japanese work on wavy flow in open channels, and semitheoretical analysis of problem (wave velocities, frequencies, heights, lengths). Mostly small channel slopes considered. 

Reinius (R4), 1961 Studies of water flows in open channels at small slopes, Nr, = 50-13,000. Data on film thicknesses, film friction factors, effects of wall roughness. 

R4. Reinius, E., Steady uniform flow in open channels, Trans. Roy. Inst. Technol. Stockholm 179 (1961). 

Ultrasonic meters are finding increasing application because of their ability to measure clear and dirty liquids in difficult situations. They are usually non-intrusive and present little or no obstruction to the flow. They are effective also in measuring flow in open channels (Section 6.2.5) and in partially filled pipes. They are, however, highly sensitive to flow conditions and should be calibrated with care. 

British Standard 3680 Methods of Measurement of Liquid Flow in Open Channels (1969-1983). 

Open Channel Flow For flow in open channels, the data are... 

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,... 

Liquid flow in open channels is not common in a process plant, because of health and safety considerations, apart from the potential risk of contamination of the liquid itself. Open channels, however, are sometimes to be found in cooling water systems where large volumes of water are involved. Nevertheless, for the sake of completeness a brief treatment of the subject is included in this entry. 

Hydraulic elements of unsteady flow in open channel change with time and space, and PIV... 

Keulegan G.H. 1939. Laws of turbulent flow in open channels. Journal of the Franklin Institute 227(1) 119-120. 

Liggett, J. A., and Cunge, J. A. (1975). Numerical methods of solution of the unsteady flow equations. In Unsteady Flow in Open Channels (K. Mahmood and V. Yevjovich, eds.). Chapter 4. Colorado State University, Fort Collins. 

Anderson, A.G. (1953). The characteristics of sediment waves formed by flow in open channels. Proc. Midwestern Conf. FluidMechanics aDsso a 379-395. 

Straub, L.G., Anderson. A.G. (1960). Self-aerated flow in open channels. Trans. ASCE 125 456-481 125 485-486. 

Boyer, M.C. (1935). The evaluation of values ofC in terms of the mean depth in the computation of flow in open channels. University of Colorado Denver. 

Houk, I.E. (1918). Calculation of flow in open channels. Technical Reports 4. The Miami Conservancy District Dayton OH. 

Matzke, A.E. (1937). Varied flow in open channels of adverse slope. Trans. ASCE 102 651-677. 

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