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Math problem set Grade 12 Science

Mid-Section Gauging on the Denbrook and a Withdrawal Limit

A grade 12 science problem set in which students compute stream discharge from nine measured verticals, then test a town's intake against two permit conditions.

Objectives

  1. Students will compute stream discharge by summing the flow through measured subsections.
  2. Students will derive cross-sectional area and mean velocity from a set of field measurements.
  3. Students will determine which of two permit conditions limits a withdrawal at a given streamflow.

Vocabulary

Key words: discharge, mid-section method, vertical, mean velocity, minimum instream flow, withdrawal permit

Teacher overview

The fictional town of Pentridge draws its drinking water from the Denbrook, and its permit ties how much it may take to how much the stream is carrying that day. Twice a month a technician wades the gauging section below the bridge, measures depth and velocity at a series of points across the channel, and adds up the flow those measurements represent. This problem set works through one of those measurement sets and then applies the two conditions in the town's permit.

The error to watch for is multiplying a single velocity by the whole cross-sectional area. Velocity is not uniform across a channel; it approaches zero at the banks and reaches its maximum near the deepest, most open part of the section. A student who takes the fastest reading of 2.6 ft/s and applies it to the entire area gets 65 cfs instead of the correct answer to problem 2, an overstatement of roughly 31 percent. The second habit worth naming is treating the recorded velocity as a surface reading. The technician sets the meter at six-tenths of the depth precisely because the velocity there approximates the average for that vertical.

Mid-Section Gauging on the Denbrook and a Withdrawal Limit — Grade 12 Science
Name: ____________________________ Date: ______________

Problems

The table gives one measurement set from the fictional Denbrook gauging section. The water's edge sits at 1.0 foot and at 19.0 feet from the left bank reference point, so each of the nine verticals represents a strip 2.0 feet wide and the wetted width is 18 feet. Treating all nine strips as the same width is a simplification of field practice, where the strips nearest each bank are narrower than the ones in midchannel.

Distance from left bank (ft) Depth (ft) Mean velocity (ft/s)
2 0.6 0.8
4 1.1 1.4
6 1.6 2.0
8 2.0 2.4
10 2.2 2.6
12 2.0 2.3
14 1.5 1.9
16 1.0 1.2
18 0.5 0.7

Use these relationships and permit conditions.

  • Subsection discharge = depth x mean velocity x strip width
  • Total discharge = the sum of all nine subsection discharges
  • Cross-sectional area = the sum of depth x strip width for all nine verticals
  • Mean velocity for the section = total discharge / cross-sectional area
  • Conversion: 1 MGD = 1.547 cfs
  • Permit condition one: the fictional Pentridge system may withdraw no more than 15 percent of the flow measured at the section
  • Permit condition two: at least 12.0 cfs must remain in the Denbrook below the intake at all times
  • The plant's intake capacity is 3.5 MGD

  1. Calculate the subsection discharge, in cubic feet per second, for the vertical at 10 feet from the left bank.

  2. Calculate the total discharge of the Denbrook at this section, in cubic feet per second, by summing all nine subsections. Round to two decimal places.

  3. Calculate the cross-sectional area of the flowing water, in square feet.

  4. Calculate the mean velocity for the whole section, in feet per second. Round to two decimal places.

  5. Convert the total discharge from problem 2 into million gallons per day. Round to one decimal place.

  6. Calculate 15 percent of the measured discharge, in cubic feet per second, which is the most the permit's first condition allows the town to withdraw. Round to two decimal places.

  7. Convert the plant's 3.5 MGD intake capacity into cubic feet per second. Round to two decimal places.

  8. Compare your answers to problems 6 and 7. On the day of this measurement, could the plant run at full intake capacity without breaking the first permit condition? Answer yes or no.

  9. In late summer the same section gauges only 14.0 cfs. Determine the maximum withdrawal the permit allows at that flow, in cubic feet per second, applying both conditions and reporting whichever is smaller.

  10. Convert your answer to problem 9 into million gallons per day. Round to two decimal places.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

  1. Answer: 11.44 cfs.
    2.2 feet of depth x 2.6 ft/s x 2.0 feet of strip width = 5.72 x 2.0 = 11.44 cfs. This is the largest of the nine subsections, since it is both the deepest and the fastest vertical.

  2. Answer: 49.48 cfs.
    Working across the section, the nine subsection discharges are 0.96, 3.08, 6.40, 9.60, 11.44, 9.20, 5.70, 2.40, and 0.70 cfs. Adding them gives 49.48 cfs. Each is the product of depth, velocity, and the 2.0-foot strip width.

  3. Answer: 25.0 square feet.
    The nine depths sum to 0.6 + 1.1 + 1.6 + 2.0 + 2.2 + 2.0 + 1.5 + 1.0 + 0.5 = 12.5 feet. Multiplying by the 2.0-foot strip width gives 12.5 x 2.0 = 25.0 square feet.

  4. Answer: 1.98 ft/s.
    49.48 cfs / 25.0 square feet = 1.9792 ft/s, which rounds to 1.98 ft/s. Note that this is well below the 2.6 ft/s maximum reading, which is why applying the fastest velocity to the whole area overstates discharge.

  5. Answer: 32.0 MGD.
    49.48 cfs / 1.547 cfs per MGD = 31.98 MGD, which rounds to 32.0 MGD.

  6. Answer: 7.42 cfs.
    0.15 x 49.48 cfs = 7.422 cfs, which rounds to 7.42 cfs.

  7. Answer: 5.41 cfs.
    3.5 MGD x 1.547 cfs per MGD = 5.4145 cfs, which rounds to 5.41 cfs.

  8. Answer: Yes.
    The plant's full intake of 5.41 cfs is below the 7.42 cfs the first condition allows, so the plant may run at capacity. Checking the second condition as well, withdrawing 5.41 cfs leaves 49.48 - 5.41 = 44.07 cfs in the stream, far above the 12.0 cfs floor.

  9. Answer: 2.0 cfs.
    The first condition allows 0.15 x 14.0 = 2.10 cfs. The second condition requires 12.0 cfs to remain, which allows only 14.0 - 12.0 = 2.0 cfs to be withdrawn. The smaller of the two governs, so the maximum withdrawal is 2.0 cfs. The minimum instream flow becomes the binding condition once the stream falls this low, which is the point of writing both conditions into the permit.

  10. Answer: 1.29 MGD.
    2.0 cfs / 1.547 cfs per MGD = 1.2928 MGD, which rounds to 1.29 MGD. That is roughly a third of the 3.5 MGD the plant can treat, so the fictional town would have to draw the rest from storage or reduce demand.

Differentiation

Support

Give students the fictional Denbrook table with a fourth column already added and headed "subsection discharge," and complete the first two rows together so the pattern of depth times velocity times width is visible. Provide a channel cross-section sketch with the nine verticals drawn and the strip widths shaded, which makes problem 3 read as a set of rectangles rather than an abstract sum. Check problem 2 before students continue, since every later answer depends on it.

Extension

Ask students to recompute problems 2 through 4 using only five verticals, dropping the readings at 4, 8, 12, and 16 feet and widening the three interior strips (6, 10, and 14 feet) to 4.0 feet and the two edge strips (2 and 18 feet) to 3.0 feet, so the strips still span the 18-foot wetted width, then compare that result with the nine-vertical answer and state which is more trustworthy and why. Have them also determine the streamflow at which the two permit conditions would allow exactly the same withdrawal.

ELL supports

Provide a labeled cross-section diagram of the fictional gauging site with the terms vertical, depth, strip width, and wetted width placed directly on it. Give a sentence frame for problems 8 and 9, such as "The binding condition is ___ because it allows ___ cfs, which is less than ___ cfs." Allow students to lay out problem 2 as a labeled table of nine products rather than as a written equation.

Extensions

Students can compare the fictional Pentridge permit with a permit that sets a single fixed withdrawal all year, listing which approach protects the stream better in a dry summer and which is easier for a utility to plan around. A second option is to find a real gauging station near the school in the United States Geological Survey's current-conditions data, read today's discharge, and calculate what a 15 percent withdrawal would amount to in million gallons per day at that flow.