Lori Pearman
EMT 725
For the full problem, see Flowing
Stream Problem on Jim Wilson's EMT 725 web page.
(Use the back key to return here.)
Problem: Assume that a stream is about 30 ft. wide and no more than
5 ft. deep. How would you estimate for the cross-sectional area?
Solution: To get the cross-sectional area, you could stretch a rope
across the stream with, say, 1 foot markings on it. Go out in a boat, or
if you're adventurous (and tall) you could walk out into the stream. Take
a rod (maybe a surveyers rod) which also has ft. markings on it. For every
foot you go out across the stream (the rope will tell you how far across
you are) stick the rod down to the bottom of the stream to measure the depth
of the stream. Use these measurements to break the cross-sectional area
up into vertical trapezoidal strips (which will be used to estimate the
actual area).
For an even better approximation, you could find the depth at every 1/2
ft. across the stream. (The more trapezoids that are used, the more accurate
the approximation will be.)
Example:
The above picture shows a cross-sectional view of a stream. Here, the
width of the stream is 9 ft. The numbers correspond to ft. markings on the
rope. To find the cross-sectional area between the 4 ft. mark and the 5
ft. mark, you could find the heights H1 and H2, and use the formula for
the area of a trapezoid. Area = W[(H1)+(H2)/2] where W = width = 1 ft. (for
this example). Notice that the two end areas are more triangular than trapezoidal.
In those cases, you can still use the same formula, you'll just have a 0
for one of the heights. ( When one of the heights = 0, the formula reduces
to the formula for the area of a triangle.)
Flow Rate Extensions:
Q = V*A, where Q = flow rate (ft^3/sec), V = average velocity (ft./sec),
and A = cross-sectional area (ft.^2)
One way to estimate the flow rate is by placing some floats on the surface
and timing how long it takes them to travel a given distance. However, the
velocity decreases closer to the bottom of the stream. If this method is
used to get the velocity of the stream, the flow rate will be over estimated.
A better method would be to find the liquid velocity for each strip at a
60% depth. This will give a closer approximation to the actual average velocity.
To get this depth of a strip, average the heights of each vertical side
of the strip and multiply this by 0.6. This will give you the depth below
the water surface where the velocity should be found. (Measure at this depth
between each two rope markings.) A pitot or Fecheimer probe are instruments
that can be used to obtain a velocity at this level. The method for estimating
the flow rate is as follows:
For each stip, obtain the velocity at a 60% depth (using an appropriate
instrument). Multiply the velocity obtained in each strip by the area of
the strip. Do this for each strip, and add these to get the total flow rate.
An even better method than this is to obtain two velocity measurements for
each strip, and average the two. Find the velocity at a 20% depth and a
80% depth and take the average of the two. Take the average velocity of
each strip, multiply it by the area of the strip, and sum up these products
for the flow rate.
References: Fundamentals of Fluid Mechanics (2nd Edition)
P. Gerhart, R. Gross, J. Hochstein
Addison- Wesley Publishing Co. (1992)
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