1.2 Flow, Volume, and Detention

Build every problem around compatible units

Flow tells how much water passes a point per unit time; volume tells how much space or water is present; detention time relates the two. The current WPI formula table states detention time = volume ÷ flow, with the warning that units must be compatible. That warning is the heart of the calculation. Gallons divided by gallons per minute gives minutes. Cubic meters divided by cubic meters per day gives days. Mixing gallons with million gallons per day without converting produces a meaningless number.

Use the same relationship in three directions:

Here Q denotes flow, V volume, and t time. Average flow is appropriate only when the problem describes average conditions. A rapidly changing plant flow may require interval data or a totalizer rather than one instantaneous meter reading.

Calculate geometric volume first

WPI supplies these common forms:

• Rectangular tank: length × width × water depth.
• Cylindrical tank: 0.785 × diameter² × water depth.
• Cone: 1/3 × 0.785 × diameter² × height.

Dimensions must use one length unit before multiplication. The result is cubic feet when dimensions are feet and cubic meters when dimensions are meters. WPI lists 1 ft³ of water as 7.48 U.S. gallons and 1 m³ as 1,000 liters. Use water depth, not wall height, unless the tank is stated to be full. Subtract unusable or dead storage when the problem defines it.

Worked U.S. example

A rectangular basin is 40 ft long and 25 ft wide, with 12 ft of operating water depth. Flow is 0.50 MGD. Find theoretical detention time.

1. Cubic volume: 40 × 25 × 12 = 12,000 ft³.
2. Convert to gallons: 12,000 × 7.48 = 89,760 gal.
3. Convert flow to the same time basis: 0.50 MGD = 500,000 gal/day.
4. Time: 89,760 ÷ 500,000 = 0.1795 day.
5. Convert to hours: 0.1795 × 24 = 4.31 hours.

An alternate setup converts 0.50 MGD to about 347 gpm and then divides 89,760 gal by 347 gpm to obtain about 259 minutes. Both routes agree. WPI lists 1 MGD as approximately 694 gpm.

Worked metric example

A basin's operating dimensions are 18 m by 8 m by 4 m. Flow is 6,000 m³/day.

1. Volume: 18 × 8 × 4 = 576 m³.
2. Time: 576 ÷ 6,000 = 0.096 day.
3. Convert to hours: 0.096 × 24 = 2.304 hours.

If flow were stated in liters per second (L/s), convert either volume to liters or flow to m³/s. For example, WPI gives 1 L/s = 0.0864 MLD, where MLD is million liters per day. Thus 50 L/s equals 4.32 MLD, or 4,320 m³/day.

Flow from area and velocity

For water moving through a channel or pipe cross-section, WPI gives flow rate = area × velocity. Square feet multiplied by feet per second produces cubic feet per second (cfs); square meters multiplied by meters per second produces m³/s. A 3.0 ft² channel cross-section at 2.0 ft/s carries 6.0 cfs. WPI lists 1 cfs as about 448.8 gpm or 0.646 MGD, so 6.0 cfs is approximately 2,693 gpm. Use actual wetted area, not a full-pipe area when a channel or pipe is only partly full.

Storage-change check

Tank-level change can provide a reasonableness check on flow. In a vertical rectangular tank, a 0.5 ft drop over a 30 ft by 20 ft plan area represents 30 × 20 × 0.5 = 300 ft³, or 300 × 7.48 = 2,244 gal. If that occurs in 15 minutes with no inflow, average outflow is about 150 gpm. If inflow continues, the level change represents net flow, not total outflow.

Theoretical time is not guaranteed contact time

The formula calculates theoretical detention time from usable volume and flow. Real basins can short-circuit, contain dead zones, or mix non-ideally. Baffles and inlet/outlet configuration affect effective contact, and water level or flow changes affect available time. Never substitute geometric detention automatically for a regulator-approved contact-time method; use the plant's approved procedure and applicable authority's requirements.

Exam setup checklist

1. Sketch the tank or flow path and label the operating dimensions.
2. Calculate usable volume and state its units.
3. Convert flow to the matching volume and time basis.
4. Solve before converting the answer to requested minutes, hours, or days.
5. Check direction: higher flow shortens detention; greater usable volume lengthens it.
6. Distinguish theoretical tank time from effective process contact and from the hydraulic residence of an entire treatment train.

This sequence catches the most common Class I errors: using diameter as radius, omitting the 0.785 circle factor, using total shell depth instead of water depth, and dividing gallons by MGD without converting million gallons.

Flow that changes during the day

When flow varies, total volume is the sum of each interval's flow multiplied by its duration. If a plant runs at 0.60 MGD for 8 hours, 0.90 MGD for 8 hours, and 0.30 MGD for 8 hours, the volumes are 0.20, 0.30, and 0.10 MG, for 0.60 MG total. Dividing by one day gives a daily average of 0.60 MGD. A simple average works here only because the intervals are equal. For unequal intervals, weight each flow by its actual time. A totalizer is often the stronger operational record because it integrates changing flow directly.