Manometer: Working Principle, Types, Formula and Worked Examples

A manometer measures the pressure of a fluid by balancing it against a column of liquid of known density. The whole instrument rests on one hydrostatic relation, p = ρgh, where ρ is the density of the manometric liquid in kg/m³, g is 9.81 m/s² and h is the vertical height of the column in metres. Read a 300 mm mercury column and you have read 13,600 × 9.81 × 0.30 = 40,025 Pa, about 40 kPa, without any calibration, electronics or moving parts.

That is why manometers are still the reference against which pressure gauges and transducers are checked, and why they are the first pressure instrument taught in fluid mechanics. This article works through the principle, the four kinds of pressure a manometer can read, every standard type from the piezometer to the micromanometer, two fully worked numerical examples, the fluids used and the errors that spoil a reading.

The hydrostatic principle behind every manometer

Take a column of static liquid of density ρ and cross-sectional area A, of height h. The weight of that column is ρ × A × h × g. It is carried by the pressure acting over the area A at the bottom, so:

p = (ρ A h g) / A = ρ g h

Two consequences follow, and both are what make a manometer work:

• Pressure depends only on vertical depth, not on the shape or width of the tube. A 3 mm bore and a 300 mm cistern at the same depth read the same pressure.
• Pressure is the same at all points on the same horizontal level in one continuous body of the same liquid. This is the rule that lets you cut a manometer in half at the lower mercury surface and equate the two sides.

How to write a manometric equation

Do not memorise formulas for each type. Walk the fluid path from one free surface to the other and apply three rules:

1. Start at a point where you know the pressure, usually an open surface where the gauge pressure is zero.
2. Going down adds ρgh. Going up subtracts ρgh. Use the density of whichever liquid you are moving through in that leg.
3. Finish at the point whose pressure you want, and set the sum equal to that pressure.

Written this way, every type below is the same calculation with a different path.

Gauge, absolute, vacuum and differential pressure

Before reading any manometer, be clear about which of these you are measuring, because the same column height means different things.

Standard atmospheric pressure is 101.325 kPa, which is 760 mm of mercury or 10.33 m of water. Check the last one: 101,325 ÷ (1000 × 9.81) = 10.33 m. That number is also the reason a water manometer cannot read anything much above one atmosphere without a tube taller than a three-storey building.

Types of manometer

1. Simple piezometer tube

A plain vertical glass tube tapped into the side of the pipe, open at the top. The liquid rises until its own weight balances the pressure inside, and p = ρgh with ρ being the density of the flowing liquid itself.

Its limits are severe, and they are the reason the other types exist. It cannot measure gas pressure, because gas would simply escape. It cannot measure negative gauge pressure, because air would be drawn in. It cannot read high pressure, because the tube becomes impractically tall: 20 kPa of water pressure already needs 20,000 ÷ 9,810 = 2.04 m of tube. Its advantage is that nothing is simpler and there is no second fluid to account for.

2. Simple U-tube manometer

A glass U-tube, one limb connected to the point being measured and the other open to atmosphere, with a heavier manometric liquid such as mercury in the bend. Putting a dense liquid in the bend solves the piezometer’s height problem: the same 20 kPa that needed 2.04 m of water needs only 20,000 ÷ (13,600 × 9.81) = 0.150 m of mercury, a 150 mm tube.

For a liquid in the pipe, both legs count. With h₁ the height of pipe liquid in the connected limb below the pipe centre, and h₂ the mercury reading:

p_A + ρ_liquid g h₁ − ρ_man g h₂ = 0, so p_A = ρ_man g h₂ − ρ_liquid g h₁

For a gas in the pipe, the density of the gas is around 1.2 kg/m³ against 13,600 for mercury, more than four orders of magnitude smaller, so the ρ_gas g h₁ term is dropped and the equation collapses to p_A = ρ_man g h₂. That is the single most common simplification in manometry, and it is legitimate only because of that density ratio.

If the pressure in the pipe is below atmospheric, the manometric liquid is pushed up the limb towards the pipe instead, and both terms change sign: p_A = −(ρ_man g h₂ + ρ_liquid g h₁), a negative gauge pressure.

3. Differential U-tube manometer

Both limbs are connected to the process, one to point A and one to point B, with mercury in the bend. Neither limb is open, so the instrument reads a pressure difference, never an individual pressure. This is the arrangement used across a flow meter, and it is the main practical job manometers still do: the differential head read across an orifice meter or a venturi meter is what the flow rate is calculated from.

When both tapping points are at the same elevation and the same liquid of density ρ flows in both, the whole thing reduces to one line, with x the mercury reading:

p_A − p_B = x g (ρ_man − ρ)

Note the difference of densities, not the mercury density alone. The pipe liquid standing in the limbs partly cancels the mercury, so a mercury-water differential manometer is worth 12,600 kg/m³ of effective density, not 13,600.

4. Inverted U-tube differential manometer

The U is turned upside down above the two pipes, and the top is filled with a lighter fluid: air, or a light oil. It is used when the pressure difference is small, because a light fluid gives a long reading for a small Δp.

With both points at the same level, both pipes carrying a liquid of density ρ, a top fluid of density ρ_s and a reading x, the result mirrors the ordinary differential manometer exactly:

p_A − p_B = x g (ρ − ρ_s)

With air on top, ρ − ρ_s is about 999 kg/m³ against 12,600 for the mercury version. The same pressure difference therefore gives a reading roughly 12.6 times longer, which is the whole point of the design. The air at the top is bled in or out through a cock on the crown to set the levels at the start.

5. Single-column or cistern manometer

One limb of the U is replaced by a wide reservoir of area A, and the tube of area a stays narrow. Because A is perhaps 100 times a, the level in the reservoir barely moves when the level in the tube does, so only one scale has to be read instead of two. That removes one reading error and halves the work.

The small reservoir movement is δh = (a/A) h₂, and carrying it through the manometric equation gives:

p_A = ρ₂ g h₂ − ρ₁ g h₁ + (a/A) h₂ g (ρ₂ − ρ₁)

With A/a = 100 and mercury against water, the correction term is about 1 per cent of the main term, so many instruments simply compensate for it in the scale graduations and let you read the pressure directly.

6. Inclined single-column manometer

Tilt that narrow tube down towards the horizontal at an angle θ. The liquid now has to travel a length L along the tube to gain a vertical height h₂ = L sin θ. The pressure still depends only on the vertical height, so the reading is stretched by a factor of 1/sin θ:

• θ = 30°: magnification 1/0.5 = 2.0 times
• θ = 15°: magnification 1/0.2588 = 3.86 times
• θ = 10°: magnification 1/0.1736 = 5.76 times

So an inclined water manometer at 15° turns a 10 mm vertical rise, worth 98.1 Pa, into a 38.6 mm run along the scale. This is how draught across a furnace, a filter or an air duct is measured with usable resolution. Below about 5° the meniscus smears out along the tube and the gain is lost, which sets the practical limit.

7. Micromanometer

For pressure differences of a few pascals, the column is magnified further. Three approaches are used: a two-liquid micromanometer, where two immiscible liquids of nearly equal density face each other in an enlarged basin so the interface moves a long way for a tiny Δp; a hook or micrometer gauge that locates the meniscus optically or by a screw to a fraction of a division; and the Betz type, where a float carries a graduated glass scale projected onto a screen. Resolutions of the order of 0.1 mm of water, about 1 Pa, are normal. These are laboratory instruments, used for wind tunnel work, ventilation testing and calibrating low-range transducers.

Worked example 1: simple U-tube manometer with mercury

Problem. A pipe carries water. A mercury U-tube manometer is connected to it at point A, with the other limb open to atmosphere. The mercury surface in the connected limb stands 0.20 m below the centre of the pipe. The mercury in the open limb stands 0.30 m higher than that surface. Find the gauge pressure at A, and its head in metres of water. Take ρ_water = 1,000 kg/m³, ρ_mercury = 13,600 kg/m³, g = 9.81 m/s².

Step 1. Pick the datum. Take the horizontal level X-X through the lower mercury surface, the one in the connected limb. Below this level everything is mercury, so pressures on the two sides of X-X must be equal.

Step 2. Walk from A down to X-X. Going down 0.20 m through water adds ρ_w g h₁:

• ρ_w g h₁ = 1,000 × 9.81 × 0.20 = 1,962 Pa
• Pressure at X-X on the left = p_A + 1,962

Step 3. Walk from the open surface down to X-X. Start at zero gauge pressure and go down 0.30 m of mercury:

• ρ_Hg g h₂ = 13,600 × 9.81 × 0.30 = 40,024.8 Pa
• Pressure at X-X on the right = 0 + 40,024.8

Step 4. Equate and solve.

• p_A + 1,962 = 40,024.8
• p_A = 38,062.8 Pa ≈ 38.06 kPa gauge, or 0.381 bar gauge
• Absolute: 38.06 + 101.325 = 139.39 kPa absolute

Step 5. Convert to head, carefully. The question asks for head in metres of water, so divide by the specific weight of water, not of mercury:

• h = p_A / (ρ_w g) = 38,062.8 ÷ 9,810 = 3.88 m of water
• Check by going the other way: 9,810 × 3.88 = 38,062.8 Pa. It closes.

Is the answer a pressure or a pressure difference? A pressure. One limb is open, so this is the gauge pressure at A and nothing else.

Worked example 2: differential manometer between two pipes

Problem. Two pipes, A and B, both carry water. A mercury U-tube differential manometer is connected between them. The mercury surface in the limb joined to A is the lower of the two, and the centre of pipe A stands 0.80 m above it. The mercury reading is x = 0.25 m, the higher surface being in the limb joined to B. Above that surface, 1.20 m of water stands in the right limb up to the centre of pipe B. Find p_A − p_B.

Step 1. Set the datum at X-X, the lower mercury surface. Take heights above it: centre of A is at 0.80 m; the right-hand mercury surface is at 0.25 m; the centre of B is at 0.25 + 1.20 = 1.45 m.

Step 2. Walk from A to B, term by term.

• Start at p_A.
• Down 0.80 m of water to X-X: + 1,000 × 9.81 × 0.80 = +7,848 Pa
• Up 0.25 m of mercury in the right limb: − 13,600 × 9.81 × 0.25 = −33,354 Pa
• Up 1.20 m of water to the centre of B: − 1,000 × 9.81 × 1.20 = −11,772 Pa
• Arrive at p_B.

Step 3. Collect the terms.

• p_A + 7,848 − 33,354 − 11,772 = p_B
• p_A − p_B = 33,354 + 11,772 − 7,848 = 37,278 Pa ≈ 37.28 kPa, or 0.373 bar
• As a differential head of water: 37,278 ÷ 9,810 = 3.80 m of water

Step 4. Check it against the general formula. Writing z for elevation above the datum, the same result should come from x g (ρ_Hg − ρ_w) + ρ_w g (z_B − z_A):

• x g (ρ_Hg − ρ_w) = 0.25 × 9.81 × 12,600 = 30,901.5 Pa
• ρ_w g (z_B − z_A) = 1,000 × 9.81 × (1.45 − 0.80) = 9.81 × 650 = 6,376.5 Pa
• Total = 30,901.5 + 6,376.5 = 37,278 Pa. It matches.

The check also shows what each part is doing. If A and B were at the same level, the answer would be only 30,901.5 Pa, that is 3.15 m of water, and you could get it straight from x × 12.6 = 0.25 × 12.6 = 3.15 m. The remaining 0.65 m of water head is pure elevation difference between the two tapping points and has nothing to do with the mercury.

Is the answer a pressure or a pressure difference? A difference. Both limbs are connected to the process, so neither p_A nor p_B can be found from this instrument alone.

Manometric head is not the pressure head of the flowing fluid

This is the mistake that costs marks and gives wrong flow rates. The manometer reading x is a height of manometric liquid. The head used in a flow equation is a height of the flowing fluid. They are different numbers and they must be converted.

For mercury against water, the conversion is:

h (metres of water) = x (metres of mercury) × (13.6 − 1) = 12.6 x

So a 0.25 m mercury reading is 3.15 m of water, not 0.25 m of water and not 3.40 m of water. The −1 is the buoyancy of the water standing in the limbs, and leaving it out overstates the head by about 8 per cent, which is a real and common error. In general, for a manometric liquid of relative density S_m reading across a fluid of relative density S_f:

h = x (S_m/S_f − 1)

Manometric fluids and the range they set

The liquid is the range selector. One metre of column is worth whatever ρg says it is worth.

A working limit follows from the table. A 1 m mercury column reads about 1.33 bar, a 1 m water column about 9.8 kPa. Above a few bar the tube gets impractical, which is where Bourdon gauges and transducers start.

Sources of error in a manometer reading

Where manometers are used, and where they have been replaced

They are still preferred where accuracy at low pressure matters and nothing needs to be logged:

• Differential head across an orifice plate, venturi or nozzle in a laboratory flow rig.
• Furnace and boiler draught, chimney draught and fan static pressure.
• Pressure drop across air filters, coils, cyclones and packed beds in HVAC and process plant.
• Wind tunnel and pitot-static work, where an inclined or micromanometer gives fine resolution.
• Calibrating and checking Bourdon gauges and pressure transducers, because a manometer is a primary standard: its reading depends only on density, gravity and a measured length. The NIST primary pressure standards are mercury and oil manometers for exactly this reason.
• Medical sphygmomanometers, where the mercury column is the reference type even though aneroid and digital units have taken over routine use.

Where they have lost ground, and why:

• Bourdon gauges for anything above a few bar. A 10 bar reading would need a 7.5 m mercury column; a Bourdon tube fits in a 100 mm dial.
• Electronic transducers and digital manometers wherever the signal has to be transmitted, logged, alarmed or fed to a controller. A liquid column gives no output but a number on a scale.
• Anywhere mercury is banned. Occupational health rules have pushed mercury manometers out of many plants and hospitals, replaced by oil-filled or digital units.
• Vibrating, pulsating or fast-changing service, where the liquid cannot settle.

References

• NIST – Pressure measurement and primary pressure standards.
• Bansal, R.K., A Textbook of Fluid Mechanics and Hydraulic Machines, chapter on pressure and its measurement.
• Modi and Seth, Hydraulics and Fluid Mechanics, pressure measurement by liquid columns.
• NPTEL – Fluid Mechanics lecture series, IIT.

FAQs

What is a manometer and how does it work?

A manometer is an instrument that measures fluid pressure by balancing it against a column of liquid of known density. It works on the hydrostatic relation p = ρgh: the pressure at the bottom of a static liquid column equals density times gravity times the vertical height of the column. Measure the height, know the density, and you have the pressure. A 300 mm mercury column, for example, represents 13,600 × 9.81 × 0.30 = 40,025 Pa, about 40 kPa.

What is the formula of a U-tube manometer?

Walk the liquid path and add ρgh going down, subtract it going up. For a liquid-filled pipe at A with an open mercury U-tube, p_A = ρ_man g h₂ − ρ_liquid g h₁, where h₂ is the mercury reading and h₁ is the depth of pipe liquid in the connected limb. For a gas-filled pipe the second term is negligible and it reduces to p_A = ρ_man g h₂.

What is the difference between a simple and a differential manometer?

A simple manometer has one limb open to atmosphere, so it reads the gauge pressure at a single point. A differential manometer has both limbs connected to the process, so it reads only the pressure difference between two points and can never give either pressure by itself. With both tapping points at the same level and the same liquid in both pipes, p_A − p_B = x g (ρ_man − ρ), where x is the manometer reading.

Why is an inclined manometer more sensitive?

Pressure depends only on the vertical height of the column, but an inclined tube makes the liquid travel a longer distance along the scale to gain that height, since h = L sin θ. The reading is magnified by 1/sin θ: 2.0 times at 30°, 3.86 times at 15° and 5.76 times at 10°. That is how small draught pressures of a few tens of pascals are read to a usable resolution.

Why is mercury used in manometers instead of water?

Because its density of 13,600 kg/m³ makes the column short. One metre of mercury is worth 133.4 kPa against 9.81 kPa for water, so a pressure needing a 2.04 m water column needs only 0.15 m of mercury. Mercury also does not wet glass, evaporates very little and gives a sharp meniscus. Its drawback is toxicity, which is why water, light oil and digital instruments have replaced it in many places.

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