§ 2.4 Module 2 — Energy: The One Equation
A gauge measures force on an area. A pump gives energy to a weight. What joins them is a density, and it is not always yours.
By the end of this lesson
The energy equation of §2.1 is written in metres because it is energy per unit weight: divide joules by newtons and the newtons cancel, leaving length. That is what head is. A gauge reads force per unit area, because a gauge is a spring behind a diaphragm and force is all a spring can know. Same state of the same water; the exchange rate is the liquid's specific weight, γ = ρg, in newtons per cubic metre.
So p = γh and h = p/γ, and every difficulty in this lesson comes from γ being a property of a fluid at a temperature rather than a constant of arithmetic. For water at 15 °C the density correlation in src/core/hydraulics.js gives 999.13 kg/m³, so γ = 9798 N/m³ and one metre of water is 9.798 kPa. Ten metres is 97.98 kPa, 14.21 psi, 32.81 ft; one bar is 10.21 m of that water.
Why the industry rates pumps in head is the reason this lesson matters. An impeller adds energy per unit weight, set by its geometry and speed and not by what the liquid weighs: run the same pump at the same speed and flow on water, on diesel or on acid and you get the same number of metres. The static and pump terms are density-free exactly. The loss terms are dimensionless coefficients times v²/2g, which looks density-free as well — and is, for a turbulent water force main — but the friction factor is a function of Reynolds number, and Reynolds number carries density and viscosity together, so the same velocity in a thicker liquid costs more head. Written in head, then, the energy equation is free of density in its static and pump terms always, and in its loss terms only as far as f and the K values are Reynolds-independent: true of the force main, not true of a chemical feed line. Written in pressure, every term would need its own specific weight and one of them would be wrong. Head is the machine's property. Pressure is the liquid's answer to it — and the only one of the two a dial can show you.
One psi is one pound-force on a square inch — 144 lbf on a square foot. Water at 60 °F weighs 62.37 lb/ft³, the value tabulated in Crane Technical Paper 410 and every hydraulics text, so a column one foot tall presses on its base with 62.37 lbf/ft² and a column making 144 lbf/ft² stands 144 / 62.37 = 2.3088 ft high. That is the whole derivation of 1 psi = 2.31 ft, and of its reciprocal 0.4331 psi per foot.
Only one of those two numbers is a unit conversion. The 144 is exact and eternal. The 62.37 is a property of one substance at one temperature, and it moves: 2.3067 ft/psi at 4 °C, 2.3087 at 15 °C, 2.3461 at 60 °C — about 1.7% across the range a station sees. 2.31 is not a conversion factor. It is a division by the specific weight of cool water, with the specific weight left out of the notation.
A conversion table therefore has two halves. The first is definitions: exact, permanent, safe to memorise. The second is physics in a units costume, each row valid only for the liquid and temperature named in it.
| To convert | Multiply by | Status |
|---|---|---|
| psi → kPa | 6.894757 | exact by definition |
| bar → kPa | 100 | exact by definition |
| kgf/cm² → kPa | 98.0665 | exact by definition |
| in Hg → kPa | 3.386389 | exact — conventional mercury, 13 595.1 kg/m³ |
| mm Hg → kPa | 0.133322 | exact — same convention |
| m H₂O → kPa | 9.80665 | exact — conventional water, ρ = 1000 kg/m³ |
| ft H₂O → kPa | 2.989067 | exact — same convention |
| psi → ft of water at 60 °F | 2.3088 | a specific weight, 62.37 lb/ft³ |
| psi → m of water at 15 °C | 0.70368 | a specific weight, 9798 N/m³ |
| psi → ft of your liquid | 2.3088 ÷ SG | a specific weight you must look up |
The water row catches people reading datasheets. A "metre of water column" is a pressure unit, defined at exactly 1000 kg/m³ and 9.80665 m/s² — not a height of your water. Ten metres of real 40 °C water makes 97.31 kPa; 10 mH₂O means 98.07 kPa.
The conversion everyone memorises, 1 psi = 2.31 ft, embeds a
density: it is 144 / 62.37, water at about 60 °F. It is a
specific-weight conversion, not a unit conversion, so it drifts with
temperature and with specific gravity. For water the drift is small enough to
ignore in most municipal work (about 1% over 4–40 °C). For anything else it is not.
The trap bites on sludge and on chemical feed. Pumping 6% sludge at SG 1.03, or sodium hydroxide at SG 1.5, and converting pressure to head with 2.31 gives an error that goes the wrong way: the pump makes a fixed head, so a denser liquid means a higher discharge pressure and a higher power draw, both of which the 2.31 shortcut hides.
Keep head in metres or feet of the pumped liquid throughout, and convert to pressure only at the end using the actual specific weight. When you hand a pressure rating to a mechanical engineer, state the SG you used.
Specific gravity — relative density — is the ratio of a liquid's density to water's. Catalogues quote it at 15.6 °C (60 °F) or 20 °C, against water at 4 °C or at the same temperature; those conventions differ by a couple of tenths of a percent, negligible here, but say which you mean. What is not negligible is the range a pump station contains.
1/SG = fs/SGs + (1 − fs)/SGw, 5% solids of specific gravity 1.4 gives 1.0145 and 8% of gritty solids at 2.5 gives 1.050. Thick sludge is neither water nor Newtonian.Now put a number on the betrayal. Divide by the specific gravity and 2.3088 ft/psi becomes 2.716 for diesel, 1.924 for hypochlorite, 1.626 for ferric chloride, 1.509 for 50% caustic, 1.262 for 93% sulphuric acid. Use water's 2.31 on a gauge reading from the acid line and you overstate head by 83%. The error is exactly the specific gravity — and no gauge face, unit symbol or column heading will tell you it happened, because psi is psi and feet are feet.
A pump delivers 40 L/s at 30 m of head, 75% efficient.
On water at 15 °C
gamma = 999.13 x 9.80665 = 9 798 N/m3
p = 9 798 x 30 = 293.9 kPa ( 42.63 psi )
shaft = rho.g.Q.H / eta = 15.7 kW ( 21.0 hp )
On 40% ferric chloride, SG 1.42, same speed, same flow
head = 30 m UNCHANGED (the impeller does not care)
gamma = 1.42 x 9 798 = 13 913 N/m3
p = 13 913 x 30 = 417.4 kPa ( 60.54 psi )
shaft = 15.7 x 1.42 = 22.3 kW ( 29.9 hp )
Read 60.54 psi back with water's 2.31 ft/psi
h = 60.54 x 2.31 = 139.9 ft = 42.6 m <-- 1.42x the truth
and 42.6 m is a head this pump cannot produce at any flow.The same pump, the same duty, two liquids — worked in SI and checked in US customary.
Three things fall out of that, and each has broken equipment. The head is unchanged, so the pump curve, the system curve and the operating point of §7.2 are as they were. The pressure rises by the specific gravity, so casing, flanges, gaskets and relief setting see 42% more than the water test suggested. And the shaft power rises by the specific gravity, because power is ρgQH/η — a motor chosen on water at 21 hp is asked for 30 hp and trips. That last is the classic chemical-feed failure: head carried across from a water calculation without its density.
The sign of the error tells you what kind of trouble you are in
Water's conversion on a denser liquid overstates head and understates pressure: the pump looks better than it is and the piping looks safer than it is — the dangerous direction. On a lighter liquid the errors reverse and you merely buy too much pump. Work out which way a mistake points before you work out how big it is.
Interactive 3D instrument
One pressure, six units, and the density hiding in the conversion
A 3D instrument you drive yourself, one variable at a time. It needs JavaScript and WebGL, so it is not shown in this static copy of the page.
A gauge cannot read pressure. It reads a difference: the process on one side of its element, the atmosphere of the room on the other. Zero on the dial means "the same as outside", which at sea level is 101.325 kPa — 14.696 psi, 10.34 m or 33.93 ft of water. That is more static lift than many stations have, and gauge and absolute pressure differ by exactly it.
For most of this course that offset does not matter, and it is worth knowing why: every term in the energy equation is a difference of pressures, and a common offset cancels out of a difference. Head loss, static lift and pump head are happily computed in gauge. It stops cancelling the moment you compare a pressure against something inherently absolute — and then gauge is not inconvenient, it is wrong by 101 kPa.
Altitude is the term people underestimate. By the ISA troposphere formula in hydraulics.js, atmospheric pressure is 101.33 kPa at sea level, 84.56 kPa at 1500 m and 70.11 kPa at 3000 m — as water, 10.34 m, 8.63 m, 7.16 m. Take the vapour pressure out and the ceiling on static suction lift for 15 °C water is 10.17 m at sea level, 8.46 m at 1500 m; for 60 °C water at sea level, 8.44 m. Those ceilings hold no friction and no NPSH margin, which is why a real suction lift is kept to a few metres — and why a mountain station and a coastal one are not the same design.
One more trap lives here: suction gauges are often calibrated in inches of mercury of vacuum, counting downwards from atmospheric. One inch of mercury is 3.386 kPa — 0.346 m or 1.134 ft of water — so "10 in Hg vacuum" is −33.86 kPa gauge, 67.46 kPa absolute at sea level, 3.46 m of water lift. Three unit families and two datums in one reading, on a scale that runs backwards.
Stevin, Torricelli, and a barometer carried up a volcano
That the pressure at the bottom of a column depends on its height and the liquid's density — and not on the shape or the total weight of the vessel — was published by Simon Stevin of Bruges in De Beghinselen des Waterwichts (Leiden, 1586). It is the hydrostatic paradox, and it is why a pressure gauge measures a height.
In 1643 Evangelista Torricelli (1608–1647), Galileo's successor as mathematician to the Grand Duke of Tuscany in Florence, had a glass tube of mercury inverted in a dish — the experiment is usually credited to his associate Vincenzo Viviani — and found the column standing about 760 mm with an apparent emptiness above it. He described it in letters to Michelangelo Ricci in June 1644, reasoning that the weight of the air held it up. Mercury stands at a fourteenth of water's height for the same pressure because it is 13.6 times as dense — the first specific-gravity conversion.
If air has weight, there is less of it on a mountain. On 19 September 1648 Florin Périer, Blaise Pascal's brother-in-law, carried a Torricellian barometer from the Minim convent garden at Clermont-Ferrand to the summit of the Puy de Dôme, about 1465 m, and found the mercury standing three pouces and one and a half lignes lower — 84.6 mm in modern units. Pascal published the account as Récit de la grande expérience de l'équilibre des liqueurs. The ISA atmosphere behind this lesson's instrument predicts 87.7 mm for that ascent — 3.7% more than Périer measured, which is about what one September afternoon's weather is worth; the instrument's verifier checks the size of that gap rather than waving at it. The 14th Conférence Générale des Poids et Mesures named the SI unit of pressure the pascal in 1971.
Ranked by cost, and by how easily they survive review: 2.31 applied to a liquid that is not water is worth up to 83% and looks like arithmetic. Gauge read as absolute is worth 101 kPa, 10.3 m of water — on a suction calculation, the difference between a pump that primes and one that cavitates. kgf/cm² read as bar is worth 1.97%: 6 kg/cm² is 588.4 kPa, so 5.884 bar and 85.34 psi, not 6 bar and 87.02 psi. It survives review because it is small, and because 1 kgf/cm² is exactly 10 m of conventional water, which makes the substitution feel like a rounding.
Below those sit the quiet ones: "feet of head" with no liquid named on a datasheet for a sludge pump, and a drawing in metres beside a pump curve in feet. And waiting in Module 3, the same disease in another organ: the Hazen-Williams coefficient is 10.67 in SI and 4.73 in US customary, and Manning's carries 1.486 between the two — factor-class errors that look plausible on the page, which is why hydraulics.js will not offer either formula without its unit system in the function name.
The habits that prevent all of this are dull and they work. Write the datum on every pressure. Name the liquid and its specific gravity beside every head. Convert once, at the boundary, and do the physics in one system. And when a number surprises you, take the ratio of what you got to what you expected before hunting for the mistake: 1.42 names the liquid, 2.31 names the conversion, 1.97% names the gauge, 10.3 names the datum, 1.486 names the unit system. The magnitude of an error is usually its confession.
What this lesson has not accounted for is everything that happens once the water moves: no friction, no velocity head, no minor losses. §2.5 assembles those into the total dynamic head a pump is bought against, and Module 3 sizes the friction. None of these conversions change when it does — which is the point of getting them right first.
Check your understanding
Check your understanding
3 auto-graded questions with an explanation for every wrong answer. Requires JavaScript. (m2-l4-q1)
Lab 2.4
A pressure-and-head converter you can trust
Three functions. You will use the first two at work every week, and the third the first time somebody asks you about a pump at altitude. psiFromHeadFt(headFt, sg) — pressure in psi at the bottom of headFt feet of a liquid of specific gravity sg . The specific weight of water is provided as GAMMA_WATER_LBF_FT3 (62.37 lb/ft³ at 60 °F); remember that a psi is a pound-force per square inch and the specific weight is per cubic foot . headFtFromPsi(psi, sg) — the inverse: feet of that liquid , not feet of water. absoluteKpa(gaugeKpa, elevationM) — absolute pressure in kPa, given a gauge reading and a site elevation. atmosphericKpa(elevationM) is provided. Graded in the browser against 5 assertions; the editor and harness require JavaScript.
Head, Loss and Lift · Module 2, Lesson 4 — still no friction. Only the exchange rates, and the density inside them.