3.2: Basic Principles of Flow Measurement
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Velocity-Flow Area Relationship
The flow rate in an irrigation water conduit can be expressed as: Q = Vm Af
where: Vm = mean velocity of flow in the channel or pipeline and
Af = cross-sectional area of flow.
The concept of this equation is shown in Figure 3.2. This equation is called the continuity equation and is fundamental to water measurement. Velocity (Vm) is the average or mean velocity within the pipeline or channel. The use of this equation is illustrated in Example 3.3.
Figure 3.2. The continuity principle for flow.
Measurement of Mean Velocity
With most water measuring devices, the fundamental measurement is the velocity of the flowing water. Using the continuity principle (Equation 3.6), flow velocity is converted to flow rate. There are many methods used to estimate flow velocity. These include mechanical devices such as impellers, paddle wheels, bucket wheels, vanes, floats, and the measurement of pressure differences within hydraulic structures to infer the flow velocity. Newer devices, e.g., ultrasonic meters, use either the Doppler principle or the time of travel of an ultrasonic wave to estimate the velocity. These devices will be discussed in more detail in Sections 3.3 and 3.4.
Determine the flow rate (gpm) in a circular pipeline that has an inside diameter (ID) of 8 in and a mean velocity of flow of 5 ft/s.
Given: ID = 8 in
Vm = 5 ft/s
Find: The flow rate (Q) in gallons per minute (gpm)
Solution
\(A_f=\dfrac{[\pi(ID^2)]}{4}\)
\(\text{ID}=8\text{ in}\)
\(A_f=\dfrac{\pi(8)^2 \text{ in}^2}{4}=\dfrac{64\pi \text{ in}^2}{4}=50.27 \text{ in}^2\)
\(A_f=50.27\text{ in}^2\dfrac{1\text{ ft}^2}{144\text{ in}^2}=0.349\text{ ft}^2\)
\(Q=(5 \text{ ft/s})(0.349 \text{ ft}^2)=1.75 \text{ cfs}\)
\(=(1.75 \text{ cfs})\dfrac{450 \text{ gpm}}{1 \text{ cfs}}=785 \text{ gpm}\)
Distribution of Velocity
The water velocity in a pipeline or in an open channel is not constant throughout its cross section. Typically, the velocity in a closed circular pipeline is highest in the middle of the pipeline and then gradually goes to zero at the wall of the pipeline. This is illustrated in Figure 3.3. Likewise, open channels also have nonuniform velocities within the flow area. Again, the velocity is zero at the wall of the channel and then gradually increases towards the center. In Figure 3.4 you see the illustration of the nonuniform distribution of velocity in an open channel. The variations of velocity within the flow conduit affect where velocity should be sensed or how to correct the measured velocity to obtain the mean velocity as required to use the continuity equation.
Figure 3.3. Velocity distribution in a closed circular pipeline (pipe is full).
Figure 3.4. Velocity distribution in a circular pipe with open channel flow (pipe is not full).