3.4.2: Pressure Differential Methods
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- 44363
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Like flow in pipelines, the pressure differential concept can be used to measure with open channel flow. With open channel devices, velocity is usually not computed but is imbedded in the equations of flow. The equations of flow then account for both the shape of the metering section and the implied velocity. There are two general classes of pressure differential devices used in open channels: weirs and flumes. An example of a weir is shown in Figure 3.18. Figures 3.19 and 3.20 are pictures of flumes. With both classes of devices, a head or depth of water is measured upstream of the metering section. Since the metering section causes a contraction of flow, there is a lowering of the water surface elevation through the metering section, much like the decrease in pressure as water flows through a pipeline Venturi. Flow must pass through what is called critical depth for there to be a unique relationship between the upstream head and the flow rate.
Figure 3.18. A weir for measuring water flow in an open channel.
The contraction of flow is caused by either positioning the metering section above the channel floor (contraction from the bottom) or by having a narrower metering section than the channel (contraction from the side). Flow measurement flumes typically use a side contraction. Weirs always have a bottom contraction. Often, weirs use both a side and bottom contraction (Figure 3.18) while flumes sometimes have both side and bottom contractions.
Figure 3.19. A Parshall flume to measure open channel flow
Table 3.2 presents various shapes of weirs that are used to measure flow. These weirs have a relatively sharp edge (sharp crested). The edge where flow is measured is usually made out of metal or other rigid materials. The edge must retain its shape and maintain its sharp edge so that the correlation between flow and head will remain constant. Weirs come in various shapes and sizes: rectangular, trapezoidal, or triangular. The flow equations for these three types of weirs are shown in Table 3.2. While weirs are relatively simple devices, they have several disadvantages; a relatively large head loss is required to make them function properly and sediment accumulation upstream of the weir can lead to a change in the weir's head-flow relationship. The nappe of water leaving the crest of the weir must spring free of the weir for the unique head discharge relationship. If downstream water submerges a weir, the calculated flow may be incorrect. Flumes usually have a much higher tolerance to downstream submergence than weirs.
Figure 3.20. An RBC flume to measure open channel flow
| Measuring Device (all sharp crested) | Top View | Side View | Formula |
|---|---|---|---|
| Rectangular Weir (without side contraction) |  |
 |
Q = 3.33LH 3/2 |
| Trapezoidal Weir |  |
 |
Q = 3.37LH 3/2 |
| 90º Triangular Weir |  |
 |
Q = 2.49LH 3/2 |
Like weirs, there are many shapes and designs of flow measuring flumes available for flow measurement. The Parshall flume is common in irrigation; it is illustrated in Figure 3.19. Parshall flumes come in various sizes with throat widths from 1 inch to 50 feet. A big advantage is that sediment flows freely through Parshall flumes.
Another approach to flow measurement is the RBC flume. The RBC flume was designed to utilize a small ramp or bottom contraction within a prismatic channel or flume. This is illustrated in Figure 3.20. One advantage of this flume is that if an irrigator has a trapezoidal irrigation channel with stable sides, such as a concrete-lined ditch, the flow measuring device can be created by installing a ramp and a staff gauge upstream of the ramp section. An important feature of this type of flume is that the calibration is very predictable once the dimensions and materials of the metering section are known. Calibration equations and tables are available for Parshall and RBC flumes of numerous sizes. Example calibrations are given in Tables 3.3 and 3.4.
| Upstream Head (ft) | Flow Rate[a] (cfs) | Flow Rate[a] (gpm) |
|---|---|---|
| 0.25 | 0.46 | 207 |
| 0.50 | 1.35 | 605 |
| 0.75 | 2.53 | 1140 |
| 1.00 | 3.95 | 1770 |
| 1.25 | 5.58 | 2500 |
| 1.50 | 7.41 | 3320 |
| 1.75 | 9.40 | 4220 |
| 2.00 | 11.60 | 5190 |
|
Q = 3.95H 1.55 Q = flow rate in cfs and H = head in ft [a] Assumes that free flowing criteria is met. |
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| Upstream Head Flow Rate (ft) | Flow Rate[a] (cfs) | Flow Rate[a] (gpm) |
|---|---|---|
| 0.10 | 0.06 | 29 |
| 0.20 | 0.21 | 93 |
| 0.30 | 0.42 | 188 |
| 0.40 | 0.70 | 314 |
| 0.50 | 1.04 | 469 |
| 0.60 | 1.45 | 651 |
| 0.70 | 1.92 | 860 |
|
Q = 3.575 (H + 0.01259)1.8419 Q = flow rate in cfs and H = head in ft [a] Assumes that free flowing criteria is met. |
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