Internal Flow in Pipes and Ducts

Introduction

Internal flow is the movement of a fluid through a confined passage such as a water-supply pipe, stormwater conduit, fire-service main, ventilation duct, or process pipeline. In construction technology, internal-flow analysis is used to size pipes, select pumps and fans, estimate operating costs, check available pressure, and prevent failures caused by excessive velocity or transient pressure.

The central design problem is to determine the relationship between discharge, pipe geometry, fluid properties, elevation change, and energy loss. For steady incompressible flow, the governing framework combines continuity, Bernoulli’s equation, and friction-loss relationships.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Calculate Reynolds number and classify pipe flow.
2. Determine major and minor head losses.
3. Use the Darcy–Weisbach equation and appropriate friction-factor methods.
4. Construct and interpret hydraulic grade lines and energy-grade lines.
5. Analyse equivalent pipes, series systems, parallel systems, and simple pipe networks.
6. Evaluate siphons and identify cavitation risks.
7. Explain water hammer and select practical mitigation measures.
8. Apply these methods to building and civil-construction systems.

1. Fundamental quantities and assumptions

The volumetric flow rate is

where Q is discharge (m³/s), A is cross-sectional area (m²), and V is mean velocity (m/s). For a circular pipe of internal diameter D,

Thus,

The hydraulic diameter is used for non-circular ducts:

where P is the wetted perimeter. For a full circular pipe, D_h = D. For a rectangular duct, D_h = 4ab/[2(a+b)] when width is a and height is b.

Most pipe calculations in this chapter assume steady flow, a single-phase Newtonian fluid, constant properties, a rigid conduit, and negligible compressibility. These assumptions must be reconsidered for gases at high pressure changes, slurry flows, rapidly varying flows, or flexible pipelines.

2. Reynolds number and flow regime

Reynolds number compares inertial and viscous effects:

Here ρ is density (kg/m³), μ is dynamic viscosity (Pa·s), and ν = μ/ρ is kinematic viscosity (m²/s). For a non-circular duct, D is replaced by D_h. Reynolds number is dimensionless.

For flow in a smooth, straight circular pipe, the conventional classification is:

The limits are indicative rather than universal. Disturbances, inlet geometry, vibration, and roughness can shift transition.

For fully developed laminar flow in a circular pipe, the Darcy friction factor is

and the velocity profile is parabolic. Turbulent flow has a fuller velocity profile and is affected by both Reynolds number and relative roughness, ε/D, where ε is absolute roughness.

3. Energy equation for internal flow

The extended Bernoulli equation between sections 1 and 2 is

where:

• z is elevation above a datum (m);
• p/(ρg) is pressure head (m);
• V²/(2g) is velocity head (m);
• α is the kinetic-energy correction coefficient;
• h_p is pump head added (m);
• h_t is turbine head removed (m);
• h_L is total head loss (m); and
• g is gravitational acceleration, approximately 9.81 m/s².

For turbulent flow in ordinary engineering systems, α is often taken as approximately 1. For fully developed laminar flow, α = 2 because of the non-uniform velocity profile.

The total loss is divided into major loss along straight pipe and minor losses at fittings and changes in geometry:

The term “minor” refers to localised losses, not necessarily small losses. In a short building pipe with many elbows and valves, minor losses may dominate.

4. Major loss: Darcy–Weisbach equation

The Darcy–Weisbach equation is

where L is pipe length (m), D is internal diameter (m), and f is the Darcy friction factor. It applies to fully developed flow in a conduit and is valid for both laminar and turbulent flow when the correct friction factor is used.

The pressure loss corresponding to friction is

4.1 Friction factor in laminar flow

For Re < 2300,

This relation does not depend on roughness because viscous effects control the flow.

4.2 Friction factor in turbulent flow

For turbulent flow, the Colebrook–White equation is widely used:

Because f appears on both sides, the equation is implicit. It can be solved iteratively, with a spreadsheet, or with a friction-factor chart. A useful explicit approximation is the Swamee–Jain equation:

This approximation is convenient for preliminary design but should not be confused with the exact Colebrook solution.

For a hydraulically smooth turbulent pipe, roughness has little influence at moderate Reynolds number. At very large Reynolds number, the flow may become fully rough, in which case friction factor is governed mainly by ε/D. Manufacturer data should be used where pipe material, lining, ageing, or scale formation is important.

Worked Example 1: Friction loss in a water pipe

Water at approximately 20 °C flows at Q = 0.010 m³/s through a 100 mm internal-diameter pipe, 50 m long. Take ν = 1.00 × 10⁻⁶ m²/s, ε = 0.15 mm, and g = 9.81 m/s². Find velocity, Reynolds number, friction factor, and head loss.

Area:

Velocity:

Reynolds number:

The flow is turbulent. Relative roughness is ε/D = 0.00015/0.1 = 0.0015. Substitution in Colebrook–White gives approximately f = 0.0225. Therefore,

The friction pressure drop is

This result is reasonable: the pipe conveys water at a moderate velocity, and the 50 m friction loss is less than 1 m of water head.

5. Minor losses

A local loss is commonly expressed as

where K is the dimensionless loss coefficient and V is the appropriate reference velocity, usually the velocity in the pipe adjacent to the fitting. Values of K depend on fitting geometry, valve opening, bend radius, and flow direction. Reliable design uses manufacturer or code data rather than an unqualified generic value.

Typical sources include entrance, exit, sudden enlargement, sudden contraction, elbows, tees, reducers, strainers, check valves, and partially closed control valves.

For a sudden enlargement from area A₁ to A₂, the Borda–Carnot expression is

where V₁ and V₂ are upstream and downstream mean velocities. It is an idealised estimate and is most applicable to a reasonably abrupt enlargement.

The equivalent-length method represents a fitting by a length of straight pipe:

It is convenient when many fittings occur in a uniform pipe, but it becomes less reliable when fitting geometry, diameter, or flow regime changes substantially.

Worked Example 2: Total loss in a building branch

A 40 mm pipe carries water at Q = 0.002 m³/s. The branch is 30 m long and has Darcy friction factor f = 0.024. It contains four elbows with K = 0.30 each, one fully open gate valve with K = 0.15, and an entrance with K = 0.50. Find total head loss.

The area is A = π(0.04)²/4 = 0.001257 m², so

The straight-pipe loss is

The total local coefficient is

Thus,

The total loss is approximately h_L = 2.33 + 0.24 = 2.57 m. In this example, friction is the larger component, but that conclusion would change for a short branch with numerous valves and fittings.

6. Hydraulic grade line and energy grade line

The hydraulic grade line (HGL) is the locus of piezometric head:

The energy grade line (EGL) is

The vertical distance between EGL and HGL is the velocity head. In a constant-diameter pipe with constant velocity, the two lines are parallel. Along a real pipe in the direction of flow, both lines fall because energy is dissipated. A pump causes an upward jump in the EGL; a turbine causes a downward jump.

The HGL is especially important in water-supply design. If it falls below the pipe centreline, pressure is negative relative to atmospheric pressure. Negative gauge pressure may be acceptable in limited situations, but it can permit air ingress, contamination through leaks, or pipe collapse if the pipe is not designed for external pressure.

7. Equivalent pipes and pipe systems

7.1 Pipes in series

Pipes in series carry the same discharge through every segment, while total head loss is the sum of individual losses:

Each segment must be analysed using its own diameter, length, roughness, fittings, and velocity. A smaller-diameter segment can dominate the loss because velocity varies with 1/D², while Darcy loss also contains L/D.

7.2 Pipes in parallel

Parallel branches connect the same two junctions. Therefore, the head loss in every branch is equal, while the discharges add:

The discharge division is not generally proportional to pipe area. It depends on length, diameter, roughness, fittings, and the flow-dependent friction factor.

7.3 Equivalent pipe

An equivalent pipe is a single pipe that produces the same head loss for the same discharge as a compound system. For pipes in series, an equivalent diameter can be found from the governing loss equation, but because friction factor may vary with diameter and Reynolds number, there is no universally simple exact formula.

If friction factors are treated as known and equal, equality of Darcy losses for series pipes gives

with the same discharge and therefore velocity determined by each area. This relation is a modelling convenience, not a substitute for segment-by-segment analysis.

For parallel branches under the same head loss, each branch’s discharge is obtained from its loss equation and the branch discharges are then summed. Iteration is usually required because f depends on Reynolds number.

8. Pipe networks

A network contains multiple junctions and loops. Its solution must satisfy two conservation principles:

1. At each junction, algebraic flow conservation applies:

1. Around every closed loop, the algebraic sum of head changes is zero:

The Hardy Cross method is a classical hand-calculation procedure. An initial flow distribution satisfying junction continuity is assumed. A correction is calculated for each loop, commonly using

where the loss in a pipe is represented as h = r|Q|ⁿ with sign assigned according to loop direction. For Darcy–Weisbach flow, n is approximately 2 if friction factors are held constant. Corrections are repeated until loop residuals are acceptably small.

Modern practice generally uses nonlinear equations solved by computer, with Hardy Cross retained as a useful teaching and checking method. Software output must still be checked for unit consistency, realistic velocities, correct valve states, and adequate residual convergence.

9. Siphons

A siphon carries liquid over a summit from a higher upstream reservoir to a lower downstream point. The liquid rises over the summit because atmospheric pressure and the downstream elevation difference establish the pressure distribution; it does not violate gravity or conservation of energy.

Between two reservoir free surfaces, where velocity is negligible and both surfaces are at atmospheric pressure,

if no pump is present. The summit pressure is found by applying the energy equation between the upstream free surface and the summit. If the absolute pressure falls to the liquid vapour pressure, cavitation or column separation may occur.

The condition for avoiding cavitation is approximately

where p_v is vapour pressure. Atmospheric pressure decreases with altitude, so siphon summits are more vulnerable at high elevations. Air leaks at joints can also interrupt siphon action.

10. Transient flow and water hammer

Water hammer is a pressure surge caused by rapid change in liquid velocity, commonly following sudden valve closure, pump stoppage, or check-valve slam. For a rapid closure relative to the pressure-wave travel time, the ideal Joukowsky pressure rise is

where a is pressure-wave speed (m/s) and ∆V is the change in velocity (m/s). The corresponding head rise is

For an elastic liquid and pipe, wave speed depends on fluid compressibility, pipe-wall elasticity, restraint, diameter, thickness, and support conditions. The simple formula is therefore a screening calculation, not a complete transient model.

The round-trip travel time for a pipeline of length L is approximately

A closure time shorter than approximately 2L/a is often treated as rapid closure; slower closure generally produces a smaller surge, although complex systems require transient analysis.

Mitigation measures include controlled valve closure, slow-closing check valves, surge tanks, air chambers, relief valves, variable-speed drives, pump-control logic, adequate pipe anchorage, and air-release valves located appropriately. Air-release devices must be selected carefully: uncontrolled air admission can create severe secondary transients.

11. Applied case: preliminary analysis of a building water-supply main

Background

Consider an illustrative, fictional design for a three-storey training facility. A storage tank supplies a remote washroom through a 60 m horizontal pipe. The design discharge is 0.006 m³/s. The pipe has internal diameter 75 mm, absolute roughness 0.15 mm, and four elbows, a strainer, and a fully open valve. The required residual pressure at the fixture is 150 kPa gauge. The tank water surface is 8 m above the fixture.

Problem and assumptions

Assume water at 20 °C, ρ = 998 kg/m³, ν = 1.00 × 10⁻⁶ m²/s, steady flow, and no pump. Use illustrative coefficients: four elbows at K = 0.25 each, strainer K = 1.0, valve K = 0.15, and entrance K = 0.5. The outlet discharges into a pressurised fixture connection, so the residual pressure must be included rather than treating the endpoint as atmospheric.

Analysis

Pipe area is A = π(0.075)²/4 = 0.004418 m², giving

The Reynolds number is

The flow is turbulent. Relative roughness is 0.00015/0.075 = 0.0020. Colebrook calculation gives approximately f = 0.024. Thus,

The local coefficient is

Therefore,

The required residual pressure head is

The available static head is 8 m, while friction and fitting losses add approximately 2.06 m. The total required head is therefore about 17.39 m, greater than the available 8 m. The system cannot meet the specified residual pressure without increasing the available elevation, reducing losses substantially, adding a pump, or revising the demand and design conditions.

Solution approach and outcome

A preliminary redesign might use a larger main, but increasing diameter alone cannot supply 15.33 m of pressure head from only 8 m of static elevation. A booster pump or a higher tank elevation is necessary. If a pump is installed, its duty point must be selected from the complete system curve, including static head, friction, local losses, and required residual pressure. Pump power is estimated by

where H_p is pump head and η is pump efficiency. A final design must also check fire-flow requirements, local plumbing codes, backflow protection, water quality, noise, maintenance access, and emergency operation.

Learning points

The case demonstrates that velocity and friction calculations are only part of design. The HGL and required residual pressure often control feasibility. A mathematically correct friction calculation cannot compensate for an inadequate elevation or pump head.

12. Design practice, limitations, and professional responsibility

Good internal-flow design follows a traceable sequence: establish design flow; select realistic internal dimensions; determine fluid properties at operating temperature; calculate Reynolds number; select friction and local-loss data; apply the energy equation; check pressure, velocity, noise, and transient conditions; and verify compliance with codes and manufacturer limits.

Common errors include using nominal rather than actual internal diameter, confusing Darcy and Fanning friction factors, omitting fittings, applying a local coefficient with the wrong reference velocity, treating transitional flow as reliably turbulent, mixing gauge and absolute pressure, and ignoring future roughness or fouling.

Construction professionals must coordinate hydraulic calculations with structural supports, pipe expansion, fire safety, water hygiene, accessibility, and environmental requirements. Water hammer, leakage, contamination through negative pressure, and discharge of relief devices are safety and governance issues, not merely numerical details.

Computational fluid dynamics and digital twins can provide detailed predictions for complex manifolds, ventilation systems, and transient events. However, they require validated geometry, boundary conditions, turbulence models, and reliable material data. Software does not remove the need for conservation checks and engineering judgement.

Chapter Summary

Internal pipe-flow analysis begins with continuity and Reynolds number. Reynolds number identifies whether viscous laminar relations or turbulent friction methods are appropriate. The extended Bernoulli equation accounts for elevation, pressure, velocity, pumps, turbines, and head loss. Major loss is evaluated with Darcy–Weisbach, while local losses are represented using loss coefficients or equivalent lengths. Turbulent friction depends on Reynolds number and relative roughness and is commonly obtained from Colebrook–White or an explicit approximation.

The hydraulic grade line represents piezometric head, while the energy grade line includes velocity head. Pipes in series carry equal discharge and have additive losses; parallel branches have equal end-to-end loss and additive discharge. Networks require continuity at junctions and zero algebraic head change around loops. Siphons must be checked for low summit pressure and cavitation. Rapid changes in velocity can cause water hammer, for which the Joukowsky relation provides an initial screening estimate. Practical design integrates these calculations with codes, equipment data, constructability, pressure requirements, and transient protection.

Key Terms

Reynolds number: Dimensionless ratio comparing inertial and viscous effects.

Hydraulic diameter: 4A/P, used to represent non-circular conduits.

Laminar flow: Flow with predominantly orderly motion and limited mixing.

Turbulent flow: Flow with fluctuating motion and strong mixing.

Darcy friction factor: Dimensionless factor used in the Darcy–Weisbach equation.

Relative roughness: Absolute roughness divided by internal diameter.

Major loss: Head loss caused by friction along straight pipe.

Minor loss: Local head loss caused by fittings, entrances, exits, and geometry changes.

Loss coefficient: Dimensionless K relating local loss to velocity head.

Hydraulic grade line: Elevation plus pressure head.

Energy grade line: Hydraulic grade line plus velocity head.

Equivalent pipe: Single pipe representing the loss behaviour of a compound system.

Series system: Pipes connected so that the same discharge passes through each segment.

Parallel system: Branches connected between common junctions with equal branch head loss.

Pipe network: Interconnected pipes requiring junction and loop conservation.

Siphon: Conduit carrying liquid over a summit between differing elevations.

Cavitation: Formation and collapse of vapour cavities when pressure falls sufficiently.

Water hammer: Pressure transient caused by rapid velocity change.

Joukowsky equation: Screening relation ∆p = ρa∆V for rapid transients.

System curve: Relationship between required head and discharge for a piping system.

Review Questions

Short Answer Questions

1. Define Reynolds number and explain its significance in pipe-flow analysis.
2. Distinguish between major and minor head losses.
3. What physical quantities are represented by the HGL and EGL?
4. State the continuity and head-loss conditions for pipes in parallel.
5. Why can a siphon summit be vulnerable to cavitation?

Long Answer Questions

1. Explain the Darcy–Weisbach method for calculating friction loss, including the determination of friction factor in laminar and turbulent flow.
2. Describe how the extended Bernoulli equation is applied to a pumped building-water system.
3. Explain the Hardy Cross concept and the conservation requirements governing pipe networks.
4. Discuss the causes, consequences, and mitigation measures for water hammer.
5. Compare the analysis of series, parallel, and equivalent-pipe systems, noting the limitations of simplified methods.

Analytical / Application Questions

1. Water flows at 0.008 m³/s through a 100 mm pipe. Given ν = 1.0 × 10⁻⁶ m²/s, determine velocity and Reynolds number, classify the flow, and identify the appropriate friction-factor relation.
2. A 40 m pipe has D = 0.08 m, V = 1.5 m/s, and f = 0.025. Calculate Darcy friction head loss and pressure loss for water of density 998 kg/m³.
3. A valve closes in a pipeline with wave speed a = 1000 m/s while velocity decreases by 1.2 m/s. Estimate the pressure rise and explain why the result should be treated as a transient screening value rather than a final design prediction.

Multiple Choice Questions with Answers

Question 1

For a circular pipe, which expression gives the mean velocity from discharge?

A. V = QπD²/4

B. V = 4Q/(πD²)

C. V = πD²/(4Q)

D. V = QD/π

Correct Answer: B. V = 4Q/(πD²)

Question 2

A pipe has Re = 1500. Under the conventional classification used in this chapter, the flow is:

A. Laminar

B. Transitional

C. Turbulent

D. Fully rough

Correct Answer: A. Laminar

Question 3

Which equation is used for major head loss in a straight pipe?

A. h_f = f(L/D)V²/(2g)

B. h_f = K V²/(2g)

C. h_f = ρgQ

D. h_f = Q/A

Correct Answer: A. h_f = f(L/D)V²/(2g)

Question 4

In pipes connected in series, which quantity is the same in every segment?

A. Pressure

B. Velocity head

C. Discharge

D. Diameter

Correct Answer: C. Discharge

Question 5

For parallel branches connected between the same two junctions, which condition applies?

A. Each branch has the same discharge

B. Each branch has the same head loss

C. Branch discharges subtract to zero

D. All branches must have equal diameters

Correct Answer: B. Each branch has the same head loss

Question 6

The hydraulic grade line is defined by:

A. z + V²/(2g)

B. p/(ρg) + V²/(2g)

C. z + p/(ρg)

D. p/(ρV)

Correct Answer: C. z + p/(ρg)

Question 7

A local fitting loss is commonly calculated using:

A. h_m = K V²/(2g)

B. h_m = 64/Re

C. h_m = ρgQ

D. h_m = πD²/4

Correct Answer: A. h_m = K V²/(2g)

Question 8

If Q = 0.01 m³/s and D = 0.1 m, the mean velocity is approximately:

A. 0.127 m/s

B. 0.785 m/s

C. 1.27 m/s

D. 12.7 m/s

Correct Answer: C. 1.27 m/s

Question 9

The ideal Joukowsky pressure rise for a rapid velocity change is:

A. ∆p = ρg∆V

B. ∆p = ρa∆V

C. ∆p = a∆V/g

D. ∆p = fL/D

Correct Answer: B. ∆p = ρa∆V

Question 10

Which condition is required at a pipe-network junction?

A. The algebraic sum of flows is zero

B. All pipes have identical friction factors

C. The pressure must be atmospheric

D. The velocity must be zero

Correct Answer: A. The algebraic sum of flows is zero