Exercise set

Water and Wastewater Pipe Headloss, Valve Capacity, and Surge Transient Exercises

Solved pipe hydraulics exercises for velocity, Reynolds number, headloss, valve Cv, orifice flow, water hammer, surge margin and release gates.

Branch: Environmental Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises practise water and wastewater pipe hydraulics: velocity, Reynolds number, friction headloss, minor losses, valve coefficient, orifice flow, pressure conversion, water hammer, surge margin and transient release evidence.

The goal is to prove that the pipe system can pass the required flow without excessive headloss, unsafe velocity, valve limitation or surge overpressure. A steady-state pass does not prove transient safety.

Assume screening formulas unless stated otherwise. Field release should use pipe material, diameter, roughness, valve positions, measured pressure logs, closure times, wave-speed basis, air-vacuum valves and surge-control settings.

Release Evidence Notes

Pipe evidence should state diameter, material, roughness, age, lining condition, valve state and whether the flow includes solids, air or high viscosity.

Valve evidence should use installed opening and pressure drop. Catalog coefficients can be misleading when valves are throttled, partially blocked or operating with wastewater solids.

Transient evidence should identify velocity change, closure time, wave speed, pipe pressure class, maximum operating pressure and surge protection.

Release evidence should include both steady-state hydraulic grade and transient maximum pressure.

Engineering Boundary Notes

This page covers pipe, valve and surge calculations. Pump duty, power, NPSH and VFD limits belong in the pump-station duty exercise set. Standby availability and monitoring evidence belong in the reliability exercise set.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Pipe capacity	1-5	area, velocity, Reynolds number and regime	Confirm hydraulic basis.
Headloss and valves	6-11	friction headloss, minor loss, Cv and orifice flow	Decide whether the pipe and valves pass flow.
Surge transient	12-16	Joukowsky pressure rise, total pressure and pressure margin	Add controls or restrict operation.
Pipe release	17-18	evidence closure and hard gates	Release, restrict or model in detail.

Exercise 1: Pipe Area

Pipe diameter is:

D=0.30\ \text{m}

Find flow area.

Solution

Area:

A=\dfrac{\pi D^2}{4}=\dfrac{\pi(0.30)^2}{4}=0.0707\ \text{m}^2

Engineering Comment

Area links measured flow to velocity and headloss.

Plausibility Check

A $300$ mm pipe area is a little above $0.07\ \text{m}^2$ .

Exercise 2: Pipe Velocity

Flow is:

Q=0.085\ \text{m}^3/\text{s}

Area is:

A=0.0707\ \text{m}^2

Find velocity.

Solution

Velocity:

v=\dfrac{Q}{A}=\dfrac{0.085}{0.0707}=1.20\ \text{m/s}

Engineering Comment

Velocity affects self-cleansing, abrasion, air entrainment, headloss and surge.

Plausibility Check

The value is near one meter per second, typical for municipal force mains.

Exercise 3: Reynolds Number

Velocity is:

v=1.2\ \text{m/s}

Diameter:

D=0.30\ \text{m}

Kinematic viscosity:

\nu=1.0\times10^{-6}\ \text{m}^2/\text{s}

Find Reynolds number.

Solution

Reynolds number:

Re=\dfrac{vD}{\nu}=\dfrac{1.2(0.30)}{1.0\times10^{-6}}=360000

Engineering Comment

The flow is turbulent. Friction-factor assumptions should match that regime.

Plausibility Check

Water pipes at meter-per-second velocity usually have Reynolds numbers far above turbulent threshold.

Exercise 4: Laminar Check for Thick Sludge

A sludge line has:

v=0.18\ \text{m/s},\quad D=0.08\ \text{m},\quad \nu=1.2\times10^{-4}\ \text{m}^2/\text{s}

Find Reynolds number.

Solution

Re=\dfrac{0.18(0.08)}{1.2\times10^{-4}}=120

Engineering Comment

This is laminar by a wide margin. Water-based turbulent assumptions would be wrong.

Plausibility Check

The viscosity is high and velocity is low, so a small Reynolds number is expected.

Exercise 5: Velocity Limit Check

Velocity is:

v=3.1\ \text{m/s}

The design limit is:

v_{max}=2.5\ \text{m/s}

Check gate.

Solution

Since:

3.1>2.5

the velocity gate fails.

Engineering Comment

High velocity may increase headloss, surge, abrasion and noise.

Plausibility Check

The measured velocity exceeds the limit by $0.6\ \text{m/s}$ .

Exercise 6: Friction Headloss Allowance

Friction headloss is:

18\ \text{m}

Static head is:

22\ \text{m}

Minor losses are:

4\ \text{m}

Find TDH.

Solution

Total dynamic head:

TDH=18+22+4=44\ \text{m}

Engineering Comment

The pump curve should be checked at this system head and design flow.

Plausibility Check

The three head components add directly.

Exercise 7: Minor Loss from K Value

A fitting has:

K=2.1

Velocity is:

v=1.8\ \text{m/s}

Compute minor headloss:

h_m=K\dfrac{v^2}{2g}

Solution

h_m=2.1\dfrac{1.8^2}{2(9.81)}=2.1(0.165)=0.347\ \text{m}

Engineering Comment

Many fittings and partially open valves can add meaningful headloss.

Plausibility Check

At moderate velocity, one fitting creates less than one meter of headloss.

Exercise 8: Pressure from Head

Pressure head is:

H=44\ \text{m}

Find pressure for water.

Solution

Pressure:

p=\rho gH=1000(9.81)(44)=431640\ \text{Pa}

So:

p=432\ \text{kPa}

Engineering Comment

Compare operating pressure with pipe class and surge allowance.

Plausibility Check

Each meter of water is about $9.8$ kPa, so $44$ m is about $430$ kPa.

Exercise 9: Valve Cv Flow

A valve has:

C_v=120

Pressure drop:

\Delta p=9\ \text{psi}

Specific gravity is $1.0$ . Estimate flow:

Q=C_v\sqrt{\dfrac{\Delta p}{SG}}

Solution

Q=120\sqrt{9}=360\ \text{gpm}

Engineering Comment

Installed valve position and solids risk should be checked before accepting the catalog coefficient.

Plausibility Check

Square root of nine is three.

Exercise 10: Required Cv

Required flow is:

Q=300\ \text{gpm}

Allowed pressure drop is:

\Delta p=4\ \text{psi}

Find required $C_v$ for water.

Solution

Rearrange:

C_v=\dfrac{Q}{\sqrt{\Delta p/SG}}

So:

C_v=\dfrac{300}{\sqrt{4}}=150

Engineering Comment

If installed $C_v$ is lower, the valve will throttle the line or require more pump head.

Plausibility Check

At a square-root factor of two, $C_v$ must be half the gpm value.

Exercise 11: Orifice Outlet Flow

An orifice has:

C_d=0.62,\quad A=0.018\ \text{m}^2,\quad h=1.6\ \text{m}

Estimate flow.

Solution

Use:

Q=C_dA\sqrt{2gh}

Then:

Q=0.62(0.018)\sqrt{2(9.81)(1.6)}=0.0625\ \text{m}^3/\text{s}

Engineering Comment

Orifice flow is sensitive to blockage, submergence and head measurement.

Plausibility Check

The result is tens of liters per second, reasonable for this opening and head.

Exercise 12: Water-Hammer Pressure Rise

A valve closure changes velocity by:

\Delta v=0.9\ \text{m/s}

Wave speed:

a=950\ \text{m/s}

Find pressure rise.

Solution

Joukowsky relation:

\Delta p=\rho a\Delta v

Thus:

\Delta p=1000(950)(0.9)=855000\ \text{Pa}=855\ \text{kPa}

Engineering Comment

This is a severe transient and needs closure-time and surge-control review.

Plausibility Check

High wave speed times nearly one meter per second gives near one megapascal.

Exercise 13: Surge Head Equivalent

Pressure rise is:

\Delta p=855\ \text{kPa}

Convert to meters of water head.

Solution

Head:

\Delta H=\dfrac{\Delta p}{\rho g}=\dfrac{855000}{1000(9.81)}=87.2\ \text{m}

Engineering Comment

Surge head can exceed steady head and dominate pipe-pressure rating.

Plausibility Check

About $10$ kPa per meter means $855$ kPa is about $87$ m.

Exercise 14: Maximum Transient Pressure

Operating pressure is:

432\ \text{kPa}

Surge rise is:

855\ \text{kPa}

Find maximum pressure.

Solution

Maximum:

p_{max}=432+855=1287\ \text{kPa}

Engineering Comment

Transient pressure should be compared with pipe pressure class and surge allowance.

Plausibility Check

The surge contribution is about twice operating pressure, so total exceeds one megapascal.

Exercise 15: Pressure-Class Margin

Maximum pressure is:

1287\ \text{kPa}

Allowable pressure is:

1200\ \text{kPa}

Find margin.

Solution

Margin:

M=1200-1287=-87\ \text{kPa}

The pipe fails the pressure-class screen.

Engineering Comment

The response may be slower closure, surge tank, air valve correction, pressure class review or operating restriction.

Plausibility Check

Maximum pressure is above allowable, so margin is negative.

Exercise 16: Reduced Closure Velocity Change

Surge controls reduce velocity change to:

\Delta v=0.55\ \text{m/s}

Wave speed is:

950\ \text{m/s}

Find revised pressure rise.

Solution

\Delta p=1000(950)(0.55)=522500\ \text{Pa}=523\ \text{kPa}

Engineering Comment

Reducing velocity change directly reduces Joukowsky pressure rise.

Plausibility Check

The velocity change is about sixty percent of the original, so pressure rise also falls to about sixty percent.

Exercise 17: Revised Pressure-Class Margin

Operating pressure is:

432\ \text{kPa}

Revised surge rise is:

523\ \text{kPa}

Allowable pressure is:

1200\ \text{kPa}

Find margin.

Solution

Maximum pressure:

p_{max}=432+523=955\ \text{kPa}

Margin:

M=1200-955=245\ \text{kPa}

Engineering Comment

The revised transient screen passes, but field settings and valve timing must be validated.

Plausibility Check

The total is now below allowable by several hundred kilopascals.

Exercise 18: Pipe and Surge Release Gate

A pipe release package has:

Gate	Requirement	Current result
velocity	below limit	fail
valve capacity	pass required flow	pass
pressure class	positive transient margin	pass
surge-control setting	validated	open

Decide whether to release.

Solution

Velocity fails and surge-control validation is open. The pipe package is not releasable.

Engineering Comment

Steady flow and pressure margin do not compensate for velocity and evidence failures.

Plausibility Check

Two hard gates fail or remain open, so release is blocked.

Validation Package Checklist

A strong pipe, valve and surge solution should check:

whether pipe diameter, material, roughness and fluid properties are explicit;
whether velocity and Reynolds number match the assumed flow model;
whether friction and minor losses are included in TDH;
whether valve coefficients reflect installed position and service;
whether orifice and outlet flow assumptions are valid;
whether transient pressure includes wave speed and velocity change;
whether pressure class, surge-control settings and validation records are closed.

Common Release Mistakes

Common mistakes include checking pipe flow without velocity limits, using turbulent assumptions for viscous sludge, omitting minor losses, using catalog valve coefficients at the wrong opening, treating steady pressure as surge evidence, ignoring air valves and closure time, and releasing a pipe system while surge-control validation remains open.

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