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Pipe Flow by the Numbers
Liquids in pipes: velocity, head, the energy equation, laminar and turbulent flow, friction, fittings, system curves, natural circulation and air
Size and check liquid piping with numbers: continuity, head and the energy equation, Reynolds number, Darcy friction, fittings, system curves, balancing and natural circulation.
Start readingAbout this book
Every factory has pipes full of moving liquid: cooling water to moulding machines and compressors, hot water to wash bays, glycol to chillers, oil to hydraulic power packs, process water to tanks. Most were sized once, by rule of thumb or by a supplier, and are fault-found many times: a machine that runs hot, a pump that cannot deliver, a branch that gets no flow, a pipe that bangs or gurgles. Each of those problems has a number behind it, and the numbers are not hard.
This book works through liquid pipe flow with one running case. It starts with flow, velocity and the energy equation, moves through laminar and turbulent flow, friction in straight pipe and the losses in fittings, builds a system curve, divides the flow between parallel branches and balances them, and then deals with natural circulation, air and two-phase flow, and measuring what the system actually does. Each chapter calculates with the running case, includes a plausible mistake and how it was found, and ends with an exercise and its worked answer.
It is for mechanical and maintenance engineers, technicians, plumbers and designers. It assumes school algebra and a calculator or spreadsheet. The Library’s Centrifugal Pumps by the Numbers takes the system curve from this book and matches pumps to it; Water Hammer by the Numbers deals with the pressure surges that sudden changes in flow cause; Ductwork by the Numbers applies the same ideas to air.
The running case
Every chapter follows one fictional system. This is an illustration. A plastics moulding plant cools six injection moulding machines with a water loop called CW-2. Water at 25 °C is pumped from an open 20 m³ tank through a steel supply header of 80.9 mm bore, overhead at 4 m, to the six machines, each taking 1.5 L/s through its mould and oil coolers. It returns through a matching header to the tank, where a separate cooler removes the heat. The design flow is 9.0 L/s, and the water returns at about 32 °C.
The plant, its dimensions, its equipment and its mistakes are invented. None describes KEVOS® work or a real business.
How to use the book
Read Chapters 1 and 2 first: they set out the units, the continuity equation and the energy equation, which every later chapter uses. Chapters 3 to 6 build the system curve from friction and fittings; Chapter 7 divides the flow between branches; Chapters 8 to 10 deal with natural circulation, air and measurement. Water properties are taken at the stated temperatures from public data, and every calculation was checked by running it.
Source limits and professional responsibility
Friction factors, fitting losses and equipment pressure drops vary with the actual pipe, fitting and product; the values marked as illustrations here must be replaced with the manufacturer’s or measured data for real work. Hot water, steam, pressurised systems and hazardous liquids bring risks of scalding, bursting and release that are governed by regulation and qualified design. This book teaches the hydraulics. It is not a design for any system, and qualified review governs real work.
Who this book is for
Mechanical and maintenance engineers, technicians, plumbers and designers who size, check or fault-find water and liquid piping in factories and plant rooms.
What you will learn
- Convert between flow, velocity, pressure and head, and apply the energy equation to a pipe run.
- Find a flow's Reynolds number and its friction factor, and calculate friction loss in straight pipe.
- Add fitting, valve and equipment losses, and build a system curve for a real loop.
- Predict how flow divides between parallel branches, and balance them.
- Estimate natural-circulation flow, find where air comes out of solution, and measure flow and pressure.
Inside this book
Chapters
- 01Flow, area and velocityVolume and mass flow, bore against nominal size, the continuity equation through CW-2's header and branches, the heat the water carries, and a branch velocity worked from its nominal size.
- 02Pressure, head and the energy equationGauge and absolute pressure, head as metres of liquid, the energy equation with pump head and losses, CW-2's pressures from pump to riser, and a 47 kPa 'loss' that was mostly height.
- 03Laminar and turbulent flowViscosity, the Reynolds number and the two regimes of flow; velocity profiles; CW-2's turbulent headers against a laminar oil line; and an oil line sized with a turbulent friction factor.
- 04Friction in straight pipeDarcy–Weisbach, roughness and the friction factor by Colebrook and Swamee–Jain, CW-2's header and branch losses, friction against flow and bore, and a Fanning factor in a Darcy formula.
- 05Fittings, valves and equipmentLoss coefficients and equivalent lengths, equipment drops scaled from data sheets, CW-2's losses added up, a clogging strainer, and a cooler's drop taken at the wrong flow.
- 06The system curveCW-2's system curve from static lift and losses, the square-law shortcut, aged and closed loops, a pump curve and its operating point, and a head scaled in proportion to flow.
- 07Branches, parallel paths and balancingHow CW-2's flow divides between six machines in parallel, why the nearest gets most, balancing valve settings that give each 1.5 L/s, what a shut-down machine does, and valves all set alike.
- 08Natural circulationHow density difference drives flow, the driving head of CW-2's roof cooler loop, its flow and heat, why heat grows with ΔT to the power 1.5, and a check valve that stopped it.
- 09Air, vapour and two-phase flowWhere air comes out of water, CW-2's return high point at −25.4 kPa, three remedies and their cost in flow, vapour, flashing and two-phase flow, and an air vent that let air in.
- 10Measuring flow and sizing the lineFlow meters from orifice plates to clamp-on ultrasonics, trustworthy gauges, sizing CW-2's header on whole-of-life cost in A$, and an orifice read with a venturi's coefficient.
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References and source notes
The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 27 October 2026.
- US Department of Energy. DOE Fundamentals Handbook: Thermodynamics, Heat Transfer, and Fluid Flow, Volume 3 of 3 (DOE-HDBK-1012/3-92) (1992). Supplied source, a US government training handbook approved for public release, read as a subject map of fluid flow: continuity, laminar and turbulent flow, Bernoulli's equation, head loss, natural circulation and two-phase flow. No text, figures, examples or exercises are reproduced; the running case, its numbers and its checks are this book's own.
- US National Institute of Standards and Technology. Thermophysical Properties of Fluid Systems (NIST Chemistry WebBook) (2026). Source for the properties of water used throughout: density, viscosity and vapour pressure at the running case's temperatures.
- L. F. Moody, Transactions of the ASME, vol. 66. Friction factors for pipe flow (1944). Source of the friction factor chart and of the absolute roughness of commercial steel pipe, 0.045 mm, used in Chapter 4.
- C. F. Colebrook, Journal of the Institution of Civil Engineers, vol. 11. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws (1939). Source of the Colebrook equation for the friction factor, solved by iteration in Chapter 4.
- P. K. Swamee and A. K. Jain, Journal of the Hydraulics Division, ASCE, vol. 102. Explicit equations for pipe-flow problems (1976). Source of the explicit friction factor approximation compared with Colebrook in Chapter 4.