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Scale Models by the Numbers

Dimensional analysis, similitude and model testing for engineers, worked through one pump-station sump model

Use dimensions, dimensionless groups and similarity laws to plan a scale model, scale its results to full size, fit power laws to test data and know when the model stops telling the truth.

10 chapters · About 101 minutes · Free to read · Intermediate engineering fluid mechanics and test practice

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KEVOS® LibraryENGINEERINGScale Models bythe NumbersKEVOS® ORIGINAL BOOK10 CHAPTERS

About this book

Engineers test small things to learn about big ones. A pump sump is modelled at a sixth of full size, a ship hull in a towing tank, a building in a wind tunnel, a spillway in a flume. The model is cheaper, faster and safer to change than the real thing, but it only helps if its results can be carried to full size, and only if the engineer knows which results cannot. The tools for that are old and simple: the dimensions of physical quantities, the dimensionless groups built from them, and the similarity laws that say how each quantity scales.

This book teaches those tools for practical use. It starts with the dimensional check that catches unit errors in any formula, builds dimensionless groups by the Buckingham pi method, and then sets up a real model: choosing its scale, scaling flows and times by the Froude number, and facing the Reynolds and Weber numbers that a small model cannot match. It shows how to fit a power law to test data, how to spot a change of regime in the data, how the same similarity ideas give the pump and fan affinity laws, and how to plan and report a model test so that its results can be trusted.

It is for mechanical, civil and hydraulic engineers, test-lab staff and technical managers who plan model tests, commission them or read their reports, and for students who want to see dimensional analysis used on a real problem. It assumes algebra, logarithms and basic fluid mechanics: pressure, flow, velocity and head. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional project. This is an illustration. A designer is finalising a stormwater pump station, called PS-3 here, for a low-lying industrial estate. A station of similar layout elsewhere is said to have suffered from surface vortices, so the designer commissions a physical model of the wet well before the concrete is poured.

  • Pumps: three identical vertical mixed-flow pumps, two duty and one standby, each 0.45 m³/s at 18 m head, running at 1,450 rpm with an efficiency of 82%, so about 96.7 kW at the shaft.
  • Intakes: a bellmouth of 600 mm diameter on each pump, 300 mm above the floor, with 1.20 m of water above the bellmouth lip at the lowest operating level.
  • Approach: a channel 4.0 m wide carrying 0.90 m³/s to the duty pumps at a depth of 1.8 m at the lowest level, an average approach velocity of 0.125 m/s.
  • Storage: a 6.0 m diameter circular storage tank that empties back into the wet well through a 300 mm outlet.
  • Model: a 1:6 geometric scale model in a hydraulics laboratory, with the pumps replaced by suction pipes of 100 mm bore, each with a flow meter, and the lab’s water at about 20 °C.

The station, its dimensions, the lab and every measurement are invented. None describes KEVOS® work, a real station or a real laboratory.

How to use the book

Read Chapters 1 to 4 in order: they set out dimensions, dimensionless groups, similarity and Froude scaling, which every later chapter uses. Chapters 5 and 6 deal with the scale effects that limit a small model, and Chapters 7 and 8 with turning test data into scaling laws without being misled by a change of regime. Chapter 9 applies the same ideas to pumps and fans, and Chapter 10 brings everything together in a test plan and report.

Water properties come from the NIST Chemistry WebBook and are carried to more figures than a model test could justify, so that a reader can follow each step. KEVOS® tools are linked where they perform a step of the method, with a note of exactly what each one calculates.

Source limits and professional responsibility

The similarity laws in this book are exact consequences of the governing physics when their assumptions hold, and approximations when they do not; each chapter says which. The model study criteria that apply to a real pump intake, such as limits on vortices, swirl and velocity variation and minimum model Reynolds and Weber numbers, are set by standards such as ANSI/HI 9.8 and by the project’s specification. They are named here and not quoted, and none of the illustrative thresholds in the running case should be taken as theirs.

Pump stations are hazardous places: wet wells are confined spaces, and pumps, electrical equipment and fast water can kill. The design, model study, commissioning and operation of a real station are matters for competent engineers and accredited laboratories under the project’s specifications and Australian work health and safety law. This book teaches how scaling and model testing work. It is educational and is not a design for any station or a substitute for a model study.

Who this book is for

Mechanical, civil and hydraulic engineers, test-lab staff, technical managers and students who plan scale-model tests, read model study reports or need to scale measured results to full size.

What you will learn

  • Check any formula or spreadsheet for dimensional consistency and catch unit errors before they reach a decision.
  • Build dimensionless groups with the Buckingham pi method and decide which ones a model must match.
  • Choose a model scale and scale flows, velocities, times, forces and powers by Froude and Reynolds similarity.
  • Recognise viscous and surface-tension scale effects, and know whether a model result is conservative or not.
  • Fit a power law to test data on logarithmic axes, judge its uncertainty and avoid extrapolating across a change of regime.
  • Plan, run and report a model test so that its results can be scaled and trusted.

Inside this book

Chapters

  1. 01Dimensions, units and the homogeneity checkExpress engineering quantities in mass, length and time, check any formula or spreadsheet for dimensional consistency, and find the hidden constants in unit-specific formulas.
  2. 02Building dimensionless groups with the pi theoremList the variables that govern a problem, reduce them to dimensionless groups by the Buckingham pi method, and read the Froude, Reynolds and Weber numbers as ratios of forces.
  3. 03Similarity and choosing the model scaleDistinguish geometric, kinematic and dynamic similarity, see why a small model cannot match every group at once, and choose a model scale from flow, space, measurement and scale-effect limits.
  4. 04Froude scaling: flows, velocities, times and forcesDerive the Froude scaling ratios for velocity, flow, time, pressure, force and power, apply them to the PS-3 model, and check them on a draining tank and an orifice.
  5. 05The Reynolds conflict and viscous scale effectsSee where viscosity acts in a Froude model, judge Reynolds independence, raise the model Reynolds number with temperature and flow, and tell conservative scale effects from unsafe ones.
  6. 06Surface tension, vortices and the smallest useful modelUnderstand how surface tension and viscosity weaken vortices in small models, grade and record vortex observations, test for size effects and decide how small a model can be and still be trusted.
  7. 07Power laws from test dataFit a power law to measured data on logarithmic axes, read the exponent and its uncertainty, test it against theory, and use the fit to set a model's flows without fooling yourself.
  8. 08Regimes, crossovers and the danger of extrapolatingRecognise when a flow changes regime, read a crossover from the local slope of test data, carry pipe losses to full size through the friction factor, and avoid extrapolating across a crossover.
  9. 09Pumps, fans and similarity: affinity laws and specific speedDerive the affinity laws from dimensionless groups, apply them to a variable-speed pump and a ventilation fan, classify machines by specific speed, and see why model pump efficiency is stepped up.
  10. 10Planning, measuring and reporting a model testTurn the questions a model must answer into a test matrix, size the number of readings, carry uncertainties and scale effects into the results, and write a report a designer can act on.

Apply This in Practice

Use these KEVOS® tools to try the method from this reading with your own numbers.

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 31 October 2026.

  • J. Cardy, Cambridge University Press, Cambridge (Cambridge Lecture Notes in Physics 5, ISBN 0 521 49959 3). Scaling and Renormalization in Statistical Physics (1996). Supplied source, in copyright. Read only as a subject map of scaling ideas: dimensions and scaling laws, crossover between regimes and finite-size effects. It is a physics text about critical phenomena; no text, equations, figures, examples or exercises are reproduced or adapted, and the engineering treatment, running case, numbers and checks in this book are original.
  • E. Buckingham, Physical Review, vol. 4, pp. 345–376. On physically similar systems; illustrations of the use of dimensional equations (1914). The original statement of the pi theorem used in Chapter 2. Its argument is described, not quoted.
  • Bureau International des Poids et Mesures. The International System of Units (SI Brochure), 9th edition (2019). Reference for base quantities, derived units and the dimensions of derived quantities used in Chapter 1.
  • National Institute of Standards and Technology. Thermophysical Properties of Fluid Systems, NIST Chemistry WebBook, SRD 69 (2026). Source of the density, viscosity and surface tension of water at 10, 20 and 30 °C used for the Reynolds and Weber numbers throughout (for example 998.21 kg/m³, 1.0016 mPa·s and 72.74 mN/m at 20 °C).
  • C. F. Colebrook, Journal of the Institution of Civil Engineers, vol. 11, pp. 133–156. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws (1939). Named as the origin of the friction-factor equation used by the pipe friction tool in Chapter 8.
  • Hydraulic Institute. ANSI/HI 9.8, Rotodynamic Pumps for Pump Intake Design (Current edition). Named by number and scope only, as a standard that sets out intake design and physical model study requirements for pump sumps. No values or criteria from it are quoted; a real model study must use the edition specified for the project.
  • International Electrotechnical Commission. IEC 60193, Hydraulic turbines, storage pumps and pump-turbines: Model acceptance tests (Current edition). Named by number and scope only, as the standard that governs model tests of hydraulic machines and the scaling of their efficiency to full size, discussed in Chapter 9. No values are quoted.
  • International Organization for Standardization. ISO 5167-2, Measurement of fluid flow by means of pressure differential devices: Orifice plates (Current edition). Named by number and scope only, as the standard for orifice-plate flow measurement whose discharge coefficient depends on Reynolds number, the effect examined in Chapter 7. No values are quoted.
  • Safe Work Australia. Model Code of Practice: Confined spaces (2020). Cited because pump-station wet wells are typically confined spaces. The book does not cover pump-station safety; this code, the owner's procedures and competent people do.

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