Hydraulic network glossary

A working glossary of the terms and equations behind steady-state pipe-network analysis, written for practising engineers and anyone learning the field. Each entry is short on purpose: enough to define the term correctly and place it in context, with links to the worked examples, calculators and application guides where it shows up in practice.

These are the same methods Fluid Network Studio uses internally, stated openly. Where a term names a correlation or algorithm we actually solve with, we say which one. The tool is built to support a qualified engineer's judgement, not to replace it, so treat these definitions as a grounding rather than design advice. The terms are grouped into six sections below, and the primary sources for every named method are listed in the references at the foot of the page.

Friction and head loss

Darcy-Weisbach equation

The standard relation for friction head loss in a full pipe flowing steadily: h_f = f (L/D) (V^2 / 2g), where f is the Darcy friction factor, L the length, D the internal diameter, V the mean velocity and g gravity. It is physically based and valid across laminar and turbulent flow for any Newtonian fluid, which is why Fluid Network Studio defaults to Darcy-Weisbach over the empirical, water-only Hazen-Williams equation while supporting both. The whole method rests on getting the friction factor f right (see below), and you can work a single pipe on it with the pipe pressure drop calculator.

Head loss

The mechanical energy lost per unit weight of fluid as it moves through a pipe, expressed as a height of fluid column in metres. It splits into friction loss along the pipe wall (the Darcy-Weisbach term) and minor losses at fittings and changes of section. Head loss is what makes pressure fall along a real pipe run, and balancing it around every loop and path is the core of network analysis. See the supply line with a fitting example for friction and a fitting loss side by side, or compute it for one pipe with the pipe pressure drop calculator.

Hydraulic grade line (HGL)

The height to which fluid would rise in an open standpipe tapped into the pipe at each point: H = z + p / (rho g), where z is elevation, p is gauge pressure, rho is density and g is gravity. The HGL is total head minus the velocity head, and its slope along a pipe is the friction gradient. Fluid Network Studio solves the incompressible network in head and reports nodal pressure back from the HGL, so an HGL profile is the most natural way to read where pressure is being spent. Trace it along any run in the branched distribution example.

Friction factor (Churchill and Colebrook-White)

The dimensionless coefficient f in the Darcy-Weisbach equation, set by the Reynolds number and the relative roughness. The Colebrook-White equation is the classical implicit correlation for turbulent flow and is the reference most handbooks quote. Fluid Network Studio evaluates friction with the Churchill (1977) correlation, which is explicit, spans laminar, transitional and turbulent flow in one continuous expression, and agrees closely with Colebrook-White in the turbulent range - so there is no regime-switching discontinuity inside the solver. Read f directly for a Reynolds number and roughness with the Darcy friction factor calculator, and see how it works for where this sits in a solve.

Relative roughness

The internal wall roughness height divided by the pipe diameter, epsilon/D. It is dimensionless and is one of the two inputs (with Reynolds number) to the turbulent friction factor: a rougher or narrower pipe loses more head. Fluid Network Studio carries material roughness presets (commercial steel, PVC, drawn copper and so on) so you set a material and diameter rather than guessing epsilon. Note roughness only influences friction once flow is turbulent. The Darcy friction factor calculator lists typical values for common materials.

Nikuradse roughness (equivalent sand grain)

Johann Nikuradse's 1933 experiments on pipes roughened with uniform sand grains mapped how the friction factor moves from smooth-pipe behaviour to a fully rough limit where friction no longer depends on the Reynolds number. The roughness values quoted for commercial pipe are equivalent sand-grain roughnesses referenced back to those experiments, and they are the epsilon in the Colebrook-White equation and on the Moody chart. Fluid Network Studio's material presets (commercial steel, PVC and so on) are equivalent sand-grain values, so this is the same physics the solver evaluates on every pipe. See the Darcy friction factor calculator.

Hydraulically smooth pipe

A pipe behaves as hydraulically smooth when its roughness elements sit entirely inside the viscous sublayer at the wall, so the turbulent friction factor depends on the Reynolds number alone and not on roughness. Smoothness is a property of the flow as much as the material, because the sublayer thins as the Reynolds number rises, so the same pipe can behave smooth at low flow and rough at high flow. The Colebrook-White and Churchill correlations recover the smooth-pipe law as relative roughness tends to zero, which is why drawn tubing and plastics are commonly treated as smooth. See the Darcy friction factor calculator.

Hazen-Williams equation

An empirical head-loss formula for water in pressurised pipes, h_f = 10.674 L Q^1.852 / (C^1.852 D^4.871) in SI, with the roughness rolled into a single tabulated coefficient C (higher for smoother pipe). Textbooks often round the constant to 10.67 and the exponent to 4.87. The four-figure form above is the one given in the EPANET 2 Users Manual, and it is the constant the solver, the calculator and the published verification cases all use, so every page here quotes the same number. It is valid for ordinary-temperature water in turbulent flow and not for other fluids, gases or viscous liquids. Fluid Network Studio supports both Hazen-Williams and Darcy-Weisbach, defaulting to Darcy-Weisbach because it is physically based, and you can switch the project method to compare. See the Hazen-Williams calculator and the Hazen-Williams water main example.

Minor (fitting) losses and the K factor

The localised head losses at bends, tees, valves, entrances, exits and changes of section, lumped as h = K (V^2 / 2g) where K is a dimensionless loss coefficient for the fitting. They are called "minor" by convention, not because they are always small - in short, fitting-heavy runs they can dominate. Fluid Network Studio uses the fixed-K method with a 50-entry library of coefficients of the class published in Crane TP-410, and node types such as elbows, tees and crosses prefill their own K. Worth knowing where that ends: the fixed-K method is verified against independent references, but the individual K values are prefills you can edit, not verified numbers, and published handbook values for the same fitting differ by tens of per cent for the harder items. See the supply line with a fitting example, the orifice plate pressure drop calculator for the K of a sharp-edged restriction, and how it works for exactly what is and is not checked.

Fluids and flow regimes

Reynolds number

A dimensionless number, Re = rho V D / mu (or V D / nu with kinematic viscosity nu), that compares inertial to viscous forces and so predicts the flow regime. Below roughly 2300 the flow is laminar and friction is independent of roughness. Above about 4000 it is turbulent and roughness matters. The Reynolds number is the input that sets the friction factor at every iteration of a solve, and you can read it for a single pipe with the Reynolds number calculator. For non-Newtonian fluids the plain definition no longer applies and a generalised Reynolds number is used instead (see Metzner-Reed below).

Compressible vs incompressible flow

Incompressible flow treats density as constant, which is accurate for liquids and for gases at low velocity and small pressure drop, and the network can then be solved in head. Compressible flow lets density vary with pressure, which matters for gases over a meaningful pressure ratio - the velocity rises and density falls as pressure drops along the line. Fluid Network Studio solves liquids incompressibly in head, and gases compressibly in absolute pressure with mass-flow balances (isothermally, or with a temperature march when heat transfer is on). See the compressed-air line example and compressed air system design.

Pumps and machines

Pump curve

The relationship between the head a pump adds and the flow it passes, falling as flow rises. It is the pump's signature, usually given as a few catalogue points. Fluid Network Studio fits a least-squares head-flow curve through the points you enter and, for variable-speed pumps, scales it with the affinity laws. The pump curve is one half of finding the duty point, and the system curve is the other. The pump and system example shows the fit and the resulting chart.

System curve

The head the network demands as a function of flow: the static lift (elevation and pressure difference) plus the friction and minor losses, which grow roughly with the square of flow. It is a property of the piping, not the pump. Plotting it against the pump curve is the classic way to size a pump, and changing pipe diameter or length moves the whole curve. See pump system design.

Operating point

The single flow and head at which a pump actually runs in a given system: the intersection of the pump curve and the system curve, where supply equals demand. Move either curve - throttle a valve, change speed, alter the pipework - and the operating point shifts. Fluid Network Studio solves for it directly and marks it on the pump chart. The pump and system example is the canonical case.

Best efficiency point (BEP)

The flow at which a pump converts the most shaft power into useful hydraulic power, with efficiency falling away on either side. Running far from BEP wastes energy and, at low flow, can shorten pump life through recirculation and vibration. When you give a pump an efficiency curve, Fluid Network Studio reports its BEP and flags whether the duty point sits in a sensible operating region around it. This is covered in pump system design, and you can estimate the shaft power at a duty with the pump power calculator.

NPSH and NPSHa

Net Positive Suction Head is the margin between the absolute head at a pump suction and the fluid's vapour pressure, expressed as a height of fluid. NPSH available (NPSHa) is what the system provides. NPSH required (NPSHr) is what the pump needs to avoid cavitating, and safe operation needs NPSHa above NPSHr with margin. Fluid Network Studio computes NPSHa using the Antoine equation for vapour pressure and raises a cavitation-risk flag when the margin is thin. See cavitation and pump system design.

Cavitation

The formation and violent collapse of vapour bubbles when the local pressure in a liquid drops to its vapour pressure, typically at a pump suction or a sharp restriction. The collapsing bubbles erode metal, cut performance and make a characteristic gravelly noise. It is avoided by keeping NPSHa comfortably above NPSHr. Fluid Network Studio derives vapour pressure from the Antoine equation and warns when a pump is at risk. It does not predict the resulting damage, which remains an engineering judgement.

Affinity laws

The scaling relations that predict how a centrifugal pump or fan behaves at a new speed: flow scales with speed, head with speed squared, and absorbed power with speed cubed (at constant impeller diameter). They let one measured curve cover a whole range of variable-speed-drive operation. Fluid Network Studio applies the affinity laws to shift a pump's fitted curve when you set its speed, and to scale a fan's pressure-rise curve to the working density.

Non-Newtonian flow

Bingham plastic

A non-Newtonian fluid model for materials that behave as a solid until a yield stress is exceeded, then flow with a constant plastic viscosity - many sludges, drilling muds and concentrated suspensions approximate this. Below the yield stress nothing moves. Above it, the relation between shear stress and shear rate is linear but offset. Fluid Network Studio models homogeneous (non-settling) Bingham-plastic fluids using the Buckingham-Reiner relation in laminar flow and the Darby-Melson and Hanks methods for the transition and turbulent regimes. Settling slurries are out of scope. See the Bingham sludge line example.

Power-law (shear-thinning and shear-thickening) fluid

A non-Newtonian model where apparent viscosity changes with shear rate, set by a consistency index K and a behaviour index n. With n below 1 the fluid is shear-thinning, thinning as it is sheared, as many polymer solutions, pulps and pastes do. With n above 1 it is shear-thickening, and n equal to 1 recovers a Newtonian fluid. Fluid Network Studio handles homogeneous power-law fluids via the Metzner-Reed generalised Reynolds number and the Dodge-Metzner friction correlation. See the power-law fluid network example.

Metzner-Reed generalised Reynolds number

A redefinition of the Reynolds number for non-Newtonian fluids so that the familiar laminar friction relation (f = 16/Re, Fanning) still holds. It absorbs the consistency and behaviour indices into an effective viscosity at the wall shear rate, giving a single number that predicts the laminar-turbulent transition for power-law fluids. Fluid Network Studio uses it to choose the flow regime and friction model on every non-Newtonian pipe, and reports it per pipe so you can see the regime. See the power-law fluid network example.

Solution methods and scope limits

Hardy Cross method

The classical hand method for pipe-network calculation: assume flows that satisfy continuity at every junction, then correct each loop in turn until the head losses balance around it, repeating until the corrections die out. It made looped-network analysis practical by hand and is still how the problem is taught. Fluid Network Studio solves the same balance equations with the matrix-based Global Gradient Algorithm instead, which converges faster and handles pumps and valves in the same pass. See the looped network example and how it works.

Global Gradient Algorithm (GGA)

The incidence-matrix formulation of Todini and Pilati (1988) that solves a network's mass and energy balances simultaneously by Newton iteration, the algorithm behind EPANET and most modern network solvers. Each iteration linearises every element's head-flow relation and solves a sparse system for the nodal heads, then updates the flows. This is the algorithm Fluid Network Studio uses to assemble and solve every network. See how it works.

Hagedorn-Brown correlation

An empirical correlation from 1965 for the pressure drop and liquid holdup in vertical two-phase gas-liquid flow, built from oil-well test data and still a standard method in petroleum vertical-lift work. It is a multiphase method, and Fluid Network Studio models single-phase flow only (a liquid network or a gas network, never gas and liquid together in one pipe), so it does not implement Hagedorn-Brown and cannot analyse gas-liquid wells or flowlines.

Transients, surge and water hammer

Fluid Network Studio does not model water hammer or surge. It is the boundary we are asked about most, so rather than repeat the disclaimer we would rather explain the physics behind it, give you the estimate that tells you whether you have a problem, and name the class of tool that does the real work. Nothing in this section is something we sell.

Water hammer

The pressure wave that runs through a pipeline when the flow in it changes quickly: a valve slams, a pump trips, a check valve reverses onto its seat. Stopping a moving column of liquid has to convert its momentum into pressure, and the resulting wave travels at the acoustic speed of the fluid-and-pipe system - roughly 1000 to 1400 m/s for water in metal pipe, lower in plastic and much lower again with entrained air - reflecting off tanks, closed ends and junctions until friction damps it out. Peak pressures several times the working pressure are ordinary, which is why water hammer bursts pipework that was comfortably rated for its steady duty, and why the fittings and supports fail before the pipe does. The steady-state answer tells you nothing about it: two systems with identical steady results can behave completely differently on a valve closure.

Joukowsky equation

The first-order estimate of the pressure rise from a sudden flow change: dp = rho a dV, or as head, dh = a dV / g, where rho is density, a is the wave speed and dV is the velocity change. It is the one transient calculation worth doing by hand. For water at a = 1200 m/s, every 1 m/s of velocity you stop instantly is worth about 1.2 MPa, near enough 120 m of head, so a 2 m/s line that a fast valve shuts can see well over 200 m of surge head on top of its static pressure. Read it as an upper bound and a screening test, not an analysis: it holds only for a closure faster than the pipeline period 2L/a, it ignores friction and every reflection, and it says nothing about what the wave does after the first pass or about the negative pressures on the rebound. If Joukowsky puts you anywhere near the pipe rating, you need a surge study rather than a bigger safety factor.

Quasi-steady (extended-period) simulation

The time-simulation model Fluid Network Studio uses, and the reason it cannot predict surge. At each timestep it re-solves the verified steady network at the current tank levels, valve states and demands, then integrates tank storage forward to the next step. The equations it solves contain no fluid inertia and no wave speed, so there is nothing in them that could produce a pressure wave: this is a missing term, not a missing resolution, and shortening the timestep does not begin to approximate a transient. That makes it the right model for behaviour unfolding over minutes and hours - a tank filling, a pump cycling on level, demand moving through the day, which is what the tank and pump over time example shows - and the wrong model for anything unfolding in seconds. Extended-period simulation is not a weak surge analysis. It is a different question.

Surge analysis and surge protection

The study that does model water hammer solves the transient momentum and continuity equations together, usually by the method of characteristics on a fixed grid of pipe sections, so pressure waves travel at a finite speed and reflect correctly. What it gives you is the pressure envelope along every pipe over time: the maximum and minimum head each point sees, whether the minimum reaches vapour pressure and the column separates (the rejoining slam is frequently worse than the event that caused it), and how the waves interact around a network. That envelope is what sizes surge vessels, air chambers, one-way feed tanks, surge-relief valves, and slow-closing or damped check valves. Reach for one when you have long lines at appreciable velocity, fast-acting or automated valves, a pump that can trip on power loss, or unexplained noise and pipe movement in an existing system. It is a genuinely different solver and belongs in a dedicated transient package - see the comparison against desktop packages for which ones do it.

Hydraulic impedance

The ratio of an oscillating head (or pressure) to the oscillating flow that accompanies it at a point in a pipe system, the fluid analogue of electrical impedance and the working quantity of frequency-domain surge, water-hammer and pulsation analysis (reciprocating-pump pulsation and resonance in particular). Because Fluid Network Studio's time simulation is quasi-steady and carries no fluid inertia, hydraulic impedance never appears in its results, and any pulsation or surge study needs a dedicated transient tool.

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Where these methods come from

Fluid Network Studio assembles the network with the Global Gradient Algorithm (Todini and Pilati, 1988) - the same incidence-matrix formulation behind EPANET - and solves it by Newton iterations. Incompressible results are cross-checked against an independent EPANET implementation, within the published verification cases spanning liquid, gas, heat and non-Newtonian flow. The named correlations above (Churchill, the Crane TP-410 class of fixed-K coefficients, Antoine, the affinity laws, Metzner-Reed, Dodge-Metzner, Buckingham-Reiner, Darby-Melson and Hanks) are stated so the method is auditable. Naming a source is not the same as verifying against it, so it is worth being explicit about which is which: the solver's methods are checked case by case on the verification page, while the fitting-library K values and the tabulated property and roughness figures are sourced prefills that no published case checks. The tool does not claim compliance with any particular standard, and every result should be checked by a qualified engineer before it is relied upon.

For a step-by-step of how a network goes from sketch to solved result, see how it works. To see how the same physics meets a real design task, read the application guides for compressed air, pipe heat loss and pump systems.

References

The primary sources for the methods named above. Where we could not point at a single authoritative source, we say so rather than dressing up a guess.

• Colebrook, C. F. (1939), "Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws", Journal of the Institution of Civil Engineers. doi.org/10.1680/ijoti.1939.13150
• Moody, L. F. (1944), "Friction factors for pipe flow", Transactions of the ASME. ASME Digital Collection
• Churchill, S. W. (1977), "Friction-factor equation spans all fluid-flow regimes", Chemical Engineering, Vol. 84, No. 24, pp. 91-92.
• Nikuradse, J., Laws of Flow in Rough Pipes, NACA Technical Memorandum 1292, the English translation of the 1933 rough-pipe experiments. NASA NTRS
• Reynolds, O. (1883), "An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels", Philosophical Transactions of the Royal Society. doi.org/10.1098/rstl.1883.0029
• Todini, E. and Pilati, S. (1988), "A gradient algorithm for the analysis of pipe networks", in Computer Applications in Water Supply, Volume 1: Systems Analysis and Simulation, John Wiley and Sons. Its use in EPANET is set out in the EPANET analysis algorithms documentation.
• Cross, H. (1936), Analysis of Flow in Networks of Conduits or Conductors, Bulletin No. 286, University of Illinois Engineering Experiment Station, for the Hardy Cross method. IDEALS
• Rossman, L. A. (2000), EPANET 2 Users Manual, US Environmental Protection Agency, the source of the SI Hazen-Williams constant quoted above. epa.gov/water-research/epanet
• OWA-EPANET 2.2, the Open Water Analytics build used as the incompressible-water cross-check. github.com/OpenWaterAnalytics/EPANET
• NASA/TP-2016-218218, Generalized Fluid System Simulation Program (GFSSP). NASA NTRS
• Crane Co., Technical Paper No. 410: Flow of Fluids Through Valves, Fittings and Pipe, the published source for the class of fixed-K coefficients the fitting library carries. We do not reproduce its tables, and no verification case checks a library K against it. tp410.com
• Metzner, A. B. and Reed, J. C. (1955), "Flow of non-Newtonian fluids: correlation of the laminar, transition and turbulent-flow regions", AIChE Journal. doi.org/10.1002/aic.690010409
• Dodge, D. W. and Metzner, A. B. (1959), "Turbulent flow of non-Newtonian systems", AIChE Journal. doi.org/10.1002/aic.690050214
• Gnielinski, V. (1976), "New equations for heat and mass transfer in turbulent pipe and channel flow", International Chemical Engineering, and Dittus, F. W. and Boelter, L. M. K. (1930), University of California Publications in Engineering, for the inner-film coefficients.
• Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, Wiley.
• Wylie, E. B. and Streeter, V. L., Fluid Transients in Systems, Prentice Hall, for the water-hammer and surge material above, including the Joukowsky relation and the method of characteristics. Joukowsky's own account (1898, on the hydraulic shock in water mains) is the origin of the pressure-rise estimate. Fluid Network Studio implements none of this - the section exists to explain a boundary, not a feature.
• The Buckingham-Reiner, Darby-Melson and Hanks methods for Bingham-plastic pipe flow, and the Antoine vapour-pressure constants, are drawn from the standard non-Newtonian and physical-property literature. We name the methods rather than attach a single citation to each, because the forms in common use differ slightly between sources.

Put a term to work

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