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)

Return to Lori's EMT 725 Page