Flow, Velocity & Continuity

Flow and velocity are basic hydraulic concepts used throughout water and wastewater operations. Operators use them to understand pipelines, distribution systems, collection systems, pumps, channels, treatment units, and many other processes where water moves from one location to another.

Flow describes the volume of water moving through a system over time. Velocity describes how fast that water is moving. The two are connected by the cross-sectional area available for flow.

What Is Flow?

Flow rate is the volume of liquid passing a point during a specified amount of time.

Common flow units include:

• gallons per minute, or gpm;
• gallons per day, or gpd;
• million gallons per day, or MGD;
• cubic feet per second, or cfs;
• cubic feet per minute, or cfm.

The correct unit depends on the application.

What Is Velocity?

Velocity is the distance water travels during a unit of time.

In operator hydraulics, velocity is commonly expressed in:

• feet per second, or fps;
• feet per minute.

Velocity depends on both flow rate and the area through which the water is moving.

The Basic Flow Equation

The fundamental relationship is:

Q = A × V

where:

• Q = flow rate;
• A = cross-sectional area;
• V = velocity.

This equation is one of the most important formulas in hydraulics.

Rearranging the Flow Equation

If two of the three variables are known, the third can be calculated.

To calculate velocity:

V = Q ÷ A

To calculate area:

A = Q ÷ V

Units Must Be Compatible

The flow equation works only when the units are compatible.

If area is in square feet and velocity is in feet per second, flow will be in cubic feet per second.

Therefore:

ft² × ft/sec = ft³/sec

If the problem gives flow in gpm, the operator may need to convert the flow before using the equation.

Common Flow Conversion

A useful conversion is:

1 cubic foot = approximately 7.48 gallons

Therefore:

1 cfs = approximately 448.8 gpm

For many operator calculations, 449 gpm per cfs is sufficiently accurate.

Example: Convert cfs to gpm

A pipeline carries 2.0 cfs.

Flow = 2.0 cfs × 448.8 gpm/cfs

Flow = 897.6 gpm

The flow is approximately 898 gpm.

Example: Convert gpm to cfs

A pipe carries 600 gpm.

Flow = 600 gpm ÷ 448.8 gpm/cfs

Flow = approximately 1.34 cfs

Area of a Circular Pipe

For a full circular pipe:

Area = π × D² ÷ 4

where:

• A = area;
• D = inside diameter.

When velocity will be calculated in feet per second, diameter should be converted to feet before calculating area in square feet.

Example: Area of a 12-Inch Pipe

A pipe has an inside diameter of 12 inches.

Convert diameter to feet:

12 in ÷ 12 in/ft = 1.0 ft

Calculate area:

A = π × 1.0² ÷ 4

A = 0.785 ft²

Example: Velocity in a Full Pipe

A 12-inch pipe carries 700 gpm.

First convert flow to cfs:

700 gpm ÷ 448.8 = 1.56 cfs

Area of a 12-inch pipe:

A = 0.785 ft²

Velocity:

V = Q ÷ A

V = 1.56 cfs ÷ 0.785 ft²

V = approximately 1.99 fps

The water velocity is approximately 2.0 feet per second.

Diameter Strongly Affects Area

Pipe area depends on the square of the diameter.

Because:

A ∝ D²

doubling the diameter does not double the area. It increases area by a factor of four.

For example:

• a 6-inch pipe has one-quarter the area of a 12-inch pipe;
• a 12-inch pipe has four times the area of a 6-inch pipe.

Effect of Diameter on Velocity

For the same flow:

• smaller pipe area means higher velocity;
• larger pipe area means lower velocity.

This relationship is important in distribution systems, collection systems, force mains, and treatment piping.

Example: Same Flow, Different Pipe Size

Suppose 700 gpm flows through two different pipes.

In a 12-inch pipe, velocity is approximately 2.0 fps.

Now consider a 6-inch pipe.

Diameter:

6 in ÷ 12 = 0.5 ft

Area:

A = π × 0.5² ÷ 4

A = approximately 0.196 ft²

Flow:

700 gpm = 1.56 cfs

Velocity:

V = 1.56 ÷ 0.196

V = approximately 8.0 fps

The same flow produces much higher velocity in the smaller pipe.

Continuity

The continuity principle states that for steady flow of an incompressible liquid, the flow entering a section must equal the flow leaving it, unless water is added or removed.

This leads to:

Q₁ = Q₂

and because:

Q = A × V

we also have:

A₁V₁ = A₂V₂

This is called the continuity equation.

What Continuity Means in a Changing Pipe Diameter

If the same flow moves from a large pipe into a smaller pipe:

• area decreases;
• velocity increases.

If the flow then enters a larger pipe:

• area increases;
• velocity decreases.

The flow itself remains the same as long as no water enters or leaves between the two sections.

Example: Continuity Through Two Pipe Sizes

Water moves from a 12-inch pipe into a 6-inch pipe.

The 12-inch pipe velocity is 2 fps.

The area ratio is:

A12 ÷ A6 = 4

Because the smaller pipe has one-quarter the area, velocity must become four times greater to carry the same flow.

V6 = 2 fps × 4 = 8 fps

Branching Pipes

At a junction, total incoming flow must equal total outgoing flow if storage at the junction is not changing.

For example:

Qin = Q1 + Q2 + Q3

If 1,000 gpm enters a junction and two branches carry 300 gpm and 250 gpm:

Third Branch Flow = 1,000 - 300 - 250

Third Branch Flow = 450 gpm

Combining Flows

If several lines combine into one:

Qtotal = Q1 + Q2 + Q3 + ...

Example:

Three wastewater lines carry:

• 0.6 MGD;
• 0.9 MGD;
• 0.4 MGD.

Total flow:

0.6 + 0.9 + 0.4 = 1.9 MGD

Flow Through Tanks and Treatment Units

Continuity also applies to treatment units.

If inflow equals outflow and the tank level is stable:

Qin = Qout

If inflow exceeds outflow:

• tank volume increases;
• water level rises.

If outflow exceeds inflow:

• tank volume decreases;
• water level falls.

Storage Changes the Simple Continuity Balance

When storage is changing:

Change in Storage = Inflow - Outflow

For example, if 800 gpm enters a tank and 650 gpm leaves:

Net Storage Increase = 800 - 650 = 150 gpm

The tank volume increases at 150 gallons per minute.

Example: Tank Filling Volume

A tank receives 900 gpm and discharges 600 gpm.

Net increase:

900 - 600 = 300 gpm

Over 2 hours:

300 gal/min × 120 min = 36,000 gal

The tank gains 36,000 gallons.

Velocity and Friction

Higher velocity generally increases friction loss.

This is one reason small pipes carrying high flows can require much more pumping energy.

As velocity increases:

• friction losses generally increase;
• pressure loss increases;
• energy requirements may increase.

The exact relationship depends on the pipe and hydraulic equation used.

Velocity and Water Hammer

Higher velocity can also increase the severity of hydraulic transients when flow changes rapidly.

Sudden valve closure or pump shutdown can create a rapid pressure change commonly called water hammer.

Hydraulic transients are covered in a later article.

Velocity in Distribution Systems

Distribution-system velocity affects:

• friction loss;
• pressure;
• water age;
• sediment movement;
• flushing effectiveness.

Very low velocity may contribute to long residence time, while very high velocity can increase head loss and transient risk.

Velocity in Wastewater Collection Systems

Wastewater collection velocity affects the ability of flow to carry suspended solids.

If velocity is too low, solids may settle and contribute to:

• deposits;
• blockages;
• odor;
• reduced hydraulic capacity.

Actual design and cleaning criteria depend on the collection system and applicable engineering requirements.

Velocity in Force Mains

Force-main velocity depends on pump flow and pipe diameter.

Operators may use velocity to evaluate:

• solids transport;
• friction loss;
• pump operating conditions;
• surge potential.

Velocity in Treatment Channels

Open channels and treatment basins also have flow and velocity relationships.

For a channel:

Q = A × V

where area is the wetted cross-sectional area of the flowing water.

If a rectangular channel is 4 feet wide and water depth is 2 feet:

A = 4 ft × 2 ft = 8 ft²

If velocity is 1.5 fps:

Q = 8 ft² × 1.5 ft/sec = 12 cfs

Convert Channel Flow to MGD

A useful conversion is:

1 cfs = approximately 0.646 MGD

For 12 cfs:

12 × 0.646 = approximately 7.75 MGD

Flow Measurement

Operators may measure flow using devices such as:

• magnetic flow meters;
• ultrasonic meters;
• venturi meters;
• orifice meters;
• weirs;
• flumes;
• positive-displacement meters;
• turbine meters;
• other facility-specific instruments.

Each device has installation, calibration, and operating requirements.

Instantaneous Flow Versus Totalized Flow

An instantaneous flow rate shows the current rate of flow, such as 1.5 MGD.

A totalizer records accumulated volume over time.

For example:

• instantaneous flow = 1.5 MGD;
• daily total = 1.42 million gallons.

These values describe different things and should not be confused.

Average Flow

Average flow can be calculated as:

Average Flow = Total Volume ÷ Time

Example:

A plant treats 2.4 million gallons in 24 hours.

Average Flow = 2.4 MG ÷ 1 day

Average Flow = 2.4 MGD

Peak Flow

Peak flow is the highest flow occurring during a specified period.

Peak flows may occur because of:

• high customer demand;
• fire flow;
• stormwater infiltration and inflow;
• industrial discharge;
• pump cycling;
• operational changes.

Peak flow can affect pressure, detention time, treatment loading, and equipment capacity.

Peak Factor

A simple peak factor is:

Peak Factor = Peak Flow ÷ Average Flow

If average wastewater flow is 2.0 MGD and peak wet-weather flow is 5.0 MGD:

Peak Factor = 5.0 ÷ 2.0 = 2.5

The peak flow is 2.5 times the average flow.

Flow Balance

A flow balance compares measured inflows and outflows.

For a treatment plant:

Influent Flow ≈ Effluent Flow + Removed Water + Storage Change

Large unexplained differences may indicate:

• meter error;
• unmeasured recycle flow;
• storage change;
• leakage;
• incorrect data.

Recycle Flows

Wastewater plants often contain internal recycle flows.

Examples include:

• return activated sludge;
• filtrate;
• centrate;
• supernatant;
• backwash return.

Internal recycle flow can be much larger than plant influent in some parts of the facility, so operators should understand which flow is being measured.

Pump Flow and System Conditions

A pump does not always produce the same flow.

Actual pump flow depends on:

• pump characteristics;
• system head;
• valve position;
• pipe resistance;
• pump speed;
• equipment condition.

Changing system resistance can change pump flow and therefore pipe velocity.

Flow Through Parallel Pipes

When flow divides into parallel paths, total flow equals the sum of the branch flows.

However, the division is not necessarily equal.

The branch with lower hydraulic resistance generally carries more flow.

Operators should not assume that two parallel pipes carry equal flow unless system conditions support that assumption.

Flow Through Series Components

When water passes through components in series without branches:

the same flow passes through each component

For example, water moving through:

• a pump;
• then a meter;
• then a filter;
• then a pipe;

has approximately the same flow through each component under steady conditions.

Common Flow and Velocity Mistakes

• Using pipe diameter in inches without converting to feet when area must be in square feet.
• Forgetting to square the diameter in the pipe-area formula.
• Using radius where the formula requires diameter.
• Using gpm directly with square feet and expecting velocity in fps.
• Confusing flow rate with total volume.
• Assuming larger pipe diameter produces higher velocity at the same flow.
• Forgetting that flow is conserved through a changing pipe diameter.
• Assuming branch flows are equal without hydraulic justification.
• Ignoring changing tank storage when comparing inflow and outflow.
• Confusing instantaneous flow with totalized flow.
• Using nominal pipe size instead of actual inside diameter when high accuracy is required.

A Practical Flow and Velocity Problem Method

1. Identify the flow, area, and velocity information provided.
2. Convert all units into a compatible system.
3. Convert pipe diameter from inches to feet when needed.
4. Calculate cross-sectional area.
5. Use Q = A × V.
6. Apply continuity if pipe size changes.
7. Add or subtract branch flows when junctions are involved.
8. Account for changing storage when inflow and outflow differ.
9. Convert the final answer into the requested units.
10. Check whether the result is physically reasonable.

What to Remember for the Exam

• Flow is volume per unit time.
• Velocity is distance traveled per unit time.
• The basic hydraulic flow equation is Q = A × V.
• Velocity equals flow divided by area.
• For a full circular pipe, area equals πD²/4.
• Convert pipe diameter to feet before calculating square-foot area when using cfs and fps.
• 1 cubic foot is approximately 7.48 gallons.
• 1 cfs is approximately 448.8 gpm.
• 1 cfs is approximately 0.646 MGD.
• For steady incompressible flow, continuity requires the same flow through sections in series.
• The continuity equation is A₁V₁ = A₂V₂.
• For the same flow, decreasing pipe area increases velocity.
• At a junction, incoming flow equals the sum of outgoing flows when storage is not changing.
• If tank inflow exceeds outflow, storage increases.
• Higher velocity generally increases friction loss.
• Instantaneous flow and totalized volume are different measurements.
• Average flow equals total volume divided by time.
• Peak factor equals peak flow divided by average flow.
• Always verify that the units used in Q = A × V are compatible.