POWER November 2020 - 26

HYDROPOWER
and b = 1.7328 from Figure 1 provides:
Vavg = (2g x 101.7328 / α)1/2 x S1/2 x (R0.7497)1/2
It is noted that the slope (m) is positive
as predicted earlier. Therefore:
Vavg = (108.1011g/α)1/2 x S1/2 x R0.3749
1. This chart shows the model flume at critical and uniform flow. Source: Lee H. Sheldon, PE
such a small flume would be impossible.
The best alternative was to only measure
slope, average velocity, and water depth
for critical and uniform flow.
At critical flow, where the Froude number
is equal to one, the least hydraulic energy
is contained for a given quantity of
moving fluid. Consequently, there should
not be any additional energy available to
form a non-constant velocity profile and
the velocity head correction factor should
be near one. In addition, because the
flume was short, the energy in the fluid
entering the flume needed to be matched
to the energy level desired for a given flowrate
in the flume, so that uniform or steady
state flow was immediately achieved.
It was not possible to adjust the
swimming pool pump that finely. Consequently,
the team of researchers opted
to bring in a second water tank, have the
pump discharge into that tank, and then
carefully siphon from that tank into the
flume. A sonic flowmeter connected to
the hose between the tank and flume
gave the volumetric flowrate. It took a
considerable amount of time and effort
to get everything balanced for a single
data point of steady state, uniform, and
critical flow in such a small flume. However,
ultimately three data points were
collected, which were sufficient to demonstrate
this method of data analysis
(Tables 1 and 2).
It is emphasized these data points
were closely spaced in terms of volumetric
flowrate. This is because a five-inchwide
flume-operated for both uniform
and critical flows-did not provide for a
wide range of flow variability. Also, this
experiment was done in a very smooth
Plexiglas flume where Manning's n was
measured as only 0.009, whereas, 0.012
is the smoothest value in the published
table of prototype water canals. Therefore,
any numerical results should be
viewed as applying only to this very narrow
hydraulic regime.
However, it is also emphasized that
the objective of this laboratory experiment
was only to demonstrate whether
26
this method could be used in future,
more-extensive research to provide further
insight and accuracy into the makeup
of the components of Chezy's and
particularly Manning's equations.
Data Reduction Technique
The plotting of these three data points
was done in the same manner as
the instrument calibration equation
described in an article I wrote titled
" A New Calibration Equation for the
Winter-Kennedy Piezometer System, "
which was published by Hydro Review
in October 2013. This method yields
a calibration equation directly in exponential
form for ready comparison
with the commonly used open-channel
equations, that is, log10(Hv/S) was plotted
as the ordinate or y-axis and log10R
was plotted as the abscissa or x-axis
(Figure 1).
These points closely approximated a
straight line and yielded an equation of
the form: y = mx + b.
log10(Hv/S) = mlog10R + b = log10(Rm) + b
Raising both sides of the equation as
powers of 10 yields:
10^(log10Hv/S) = 10^(log10Rm + b)
= 10b x 10^(log10Rm)
Then, by logarithmic identity:
Hv/S = 10b x Rm
or
Hv = 10b x S x Rm
Substituting for Hv results in:
αVavg
2/2g = 10b x S x Rm
Rearranging terms gives:
Vavg = (2g10b/α)1/2 x S1/2 x Rm/2
Substituting numerical values of m = 0.7497
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Resulting in the following equation,
which we'll call Equation 2 for future reference:
Vavg
= 10.3972(gS/α)1/2 x R3/8
Now, in this form, the open-channel
equation contains only parameters that
may be determined across an infinitely
thin cross-sectional area. Comparing
Equation 2 with Equation 1 provides insight
into the relationships of the parameters
in Manning's equation.
Vavg = 10.3972 x (gS/α)1/2 x R3/8 = (1.486/n)
x R2/3 x S1/2
Now, equating only the two expressions
and canceling the S1/2 terms gives:
10.3972 x (g/α)1/2 x R3/8 = (1.486/n) x R2/3
Combining the R terms, results in:
10.3972 x (g/α)1/2 = (1.486/n) x R7/24
Which results in the following, which
we'll call Equation 3 for future reference:
n = 0.1429 x (α/g)1/2 x R7/24
It is noted that Equation 2 does not
have exact dimensional homogeneity.
Neglecting the values of numerical coefficients,
if the exponent of R had been
4/8 instead of 3/8, and with the inclusion
of units for g (gravitational acceleration),
it would have had exact homogeneity.
Separately, it is noted that for Manning's
equation to have dimensional homogeneity,
the units of n in Equation 1 had
been historically assigned artificially as
seconds/feet1/3 or seconds/feet8/24. In
Equation 3, now, also including units for
g, n has units of seconds/feet5/24.
It is considered that these two differences
in Manning's equation and Manning's
n may be due to the uncertainty
or inaccuracy of the data measurement
in the limited test flume available to the
students. Therefore, again, it is emphasized
that the final numerical results of
this experiment probably have a degree
of uncertainty, but the method to more
POWER | November 2020

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POWER November 2020

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