The differential pressure calculator above does the arithmetic that sits behind almost every DP measurement in a process plant. It takes two pressure readings, gives you the difference in a choice of units, expresses it as a head of water and as a percentage of an instrument's calibrated span, and computes the flow rate that a differential-pressure flow element would infer from it.
Arb Digital publishes free engineering calculators that keep their boundaries clear. This page is about the differential pressure itself: what it is, how it reads on an instrument, and what flow it corresponds to. It is not a pressure-drop calculation along a length of pipe, which is friction and geometry and belongs with the pipe flow calculator, and it is not a unit conversion utility, which the pressure converter does properly. For full sizing of a restriction to a published standard, the orifice flow calculator is the right tool; this page is the instrumentation side of the same measurement.
What This Differential Pressure Calculator Does
Differential pressure is simply the difference between two pressures, but it is measured directly rather than by subtracting two separate readings, and that distinction is the reason it dominates process instrumentation. A DP cell has both tappings connected to opposite sides of a single sensing diaphragm, so it responds to the difference and largely ignores the common static pressure. That lets you resolve a few millibars of difference on a line running at forty bar, which no pair of independent gauges could do.
The hero figure is that difference. The grid gives the diameter ratio of the flow element, the volumetric flow the differential implies, the equivalent head of water — which is how a great deal of low-pressure work is still specified — and where the reading sits within a transmitter's calibrated span.
That last figure matters more than it looks, because the relationship between flow and DP is a square root. A reading at 25 per cent of span corresponds to 50 per cent of full-scale flow. Half the span is 71 per cent of flow. The number a technician reads on a percentage display is not the number they usually assume it is.
How to Use It
- Enter both pressures on the same basis. Two gauge readings or two absolute readings. Mixing one of each introduces a full atmosphere of error.
- Use the real internal diameter. Nominal pipe size is not bore. A 100 mm nominal pipe can have an internal diameter several millimetres either side of that depending on schedule.
- Take the discharge coefficient from your standard. It depends on the element type, the tapping arrangement, the diameter ratio and the Reynolds number, and the published standard for your geometry is the authority.
- Enter the density at operating conditions. For a liquid that means at temperature; for a gas it means at the actual line pressure and temperature, which can be many times the value at standard conditions.
- Set the span to what the transmitter is configured for, not to the maximum the sensor can withstand. The percentage figure is only meaningful against the configured range.
The Formula: How the Numbers Are Calculated
The differential pressure is ΔP = P1 − P2, converted into pascals internally so that every downstream figure is consistent. The equivalent head of water uses the conventional definition in which one millimetre of water is exactly 9.80665 Pa, which is the basis instrument scales use.
The flow relation is the standard differential-pressure form: Q = Cd A2 √(2ΔP ÷ (ρ(1 − β4))), where A2 is the throat area, ρ the fluid density and β = d/D the diameter ratio. The (1 − β4) term is the velocity-of-approach correction, which accounts for the fluid already moving before it reaches the throat. It is negligible for a small bore in a large pipe and becomes substantial as β rises.
The relation follows from Bernoulli's equation combined with conservation of mass. OpenStax University Physics Volume 1, section 14.6 on Bernoulli's equation, derives why pressure falls as speed rises in a constriction, and section 14.2 on measuring pressure covers manometers and the gauge-versus-absolute distinction that causes so many DP errors. The NIST Fluid Metrology Group maintains the primary flow standards that industrial flow meters are ultimately traceable to.
Work the defaults through by hand. With 3 bar upstream and 2.8 bar downstream, ΔP is 0.2 bar, which is 20,000 Pa. A 50 mm bore in a 100 mm pipe gives β = 0.5, so β4 = 0.0625 and 1 − β4 = 0.9375. The throat area is π × 0.025² = 1.9635 × 10−3 m². The bracket is 2 × 20,000 ÷ (998 × 0.9375) = 42.752, whose square root is 6.539 m/s. So Q = 0.61 × 1.9635 × 10−3 × 6.539 = 7.832 × 10−3 m³/s, that is 7.83 L/s or 28.2 m³/h. The head of water is 20,000 ÷ 9.80665 = 2,039 mm, and against a 0.5 bar span the reading is at 40 per cent.
Why the Square Root Ruins the Bottom of the Range
Because flow goes as the square root of DP, the differential falls off far faster than the flow does. At half of full flow the DP is a quarter of full scale. At a quarter of full flow it is a sixteenth. At a tenth of full flow it is one hundredth of full scale, and at that point the transmitter's own uncertainty — typically a fixed fraction of span — has become an enormous fraction of the reading.
This is the reason DP flow measurement is usually quoted with a turndown of only three or four to one, while a Coriolis or magnetic meter manages twenty or more. It is not that the DP element is inaccurate; it is that the physics compresses the whole lower half of the flow range into the bottom few per cent of the pressure signal.
It also explains the low-flow cutoff that DP flow computers apply. Below some threshold the square-root extraction amplifies noise so aggressively that the output becomes meaningless, and forcing it to zero is more honest than reporting a number. If your measurement spends most of its time near the bottom of the range, the answer is a smaller bore or a different measurement principle, not a better transmitter.
Where DP Readings Go Wrong in the Field
The classic failure is not the instrument but the impulse lines. A liquid-filled leg on one side and a gas-filled leg on the other produces a static offset that looks exactly like a real differential. Trapped gas in a liquid leg, or condensate in a gas leg, produces one that drifts. Plugged tappings produce a reading that is stable, plausible and completely wrong, which is worse than an obvious failure.
Installation orientation matters for the same reason. A level measurement by DP relies on a known reference leg, and a wet leg that evaporates shifts the zero. Sloping runs, high points that collect vapour and low points that collect liquid all inject errors that no calibration can remove.
On the flow side, the assumption of a fully developed velocity profile is the one most often violated. A bend, valve or pump close upstream leaves swirl and asymmetry that changes the effective discharge coefficient. Published standards specify minimum straight lengths upstream and downstream for exactly this reason, and shortening them is the commonest cause of a flow meter that reads consistently high or low with no apparent fault.
Compressible Fluids Need One More Term
The relation used here assumes constant density through the element, which is fine for liquids and for gases where the pressure drop is a small fraction of the line pressure. When it is not, the gas expands as it accelerates through the throat, its density falls and the simple formula overstates the mass flow.
The standard treatment adds an expansibility factor, a number slightly below one that depends on the pressure ratio, the diameter ratio and the isentropic exponent of the gas. This tool does not apply one, so for gas service treat its output as an upper bound and use the expansibility term from the standard governing your element. As a rough guide, if the differential exceeds a few per cent of the absolute upstream pressure, the correction is worth carrying.
Density itself deserves care in gas service. A gas at 10 bar absolute is roughly ten times denser than at atmospheric pressure, so a density taken from a datasheet at standard conditions will be badly wrong. Use the density at line conditions, or a compensated flow computer that measures pressure and temperature alongside the differential.
How This Sits Next to the Other Fluid Tools
For pressure drop caused by friction along a run of pipe rather than by a deliberate restriction, use the pipe flow calculator. For converting a single pressure between units, the pressure converter and the pressure calculator are more direct, and for the pressure a column of liquid exerts, the hydrostatic pressure calculator is the right page.
On the flow side, the orifice flow calculator covers restriction sizing, the discharge coefficient calculator handles the coefficient itself, and the Bernoulli equation calculator gives the underlying energy balance. Check whether your flow is turbulent enough for a stable coefficient with the Reynolds number calculator, and convert the result with the flow rate calculator or the mass flow rate converter.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing gauge and absolute readings — the difference is only meaningful when both sides share a reference, and mixing them adds a full atmosphere.
- Using nominal pipe size as the internal diameter — schedule and wall thickness change the bore, and the diameter ratio is sensitive to it.
- Taking a density from standard conditions for a gas — line pressure and temperature change it by a large factor, and the flow scales with its square root.
- Ignoring impulse-line effects — unequal fill, trapped gas or a dried-out wet leg all produce offsets that look exactly like real differential pressure.
- Trusting the reading at the bottom of the range — the square-root relationship compresses low flows into a tiny pressure signal, which is why DP turndown is limited.
Related Free Tools From Arb Digital
Pair this with the orifice flow calculator for element sizing and the discharge coefficient calculator for the coefficient. Use the pipe flow calculator for frictional loss along a run, the hydrostatic pressure calculator for static head, and the pressure converter or pressure calculator for units. The Bernoulli equation calculator and Reynolds number calculator cover the underlying fluid mechanics, and the flow rate calculator and mass flow rate converter handle the output units. All of them are on the free online tools hub.
Frequently Asked Questions
Gauge pressure is measured against the local atmosphere, so it has a fixed reference. Differential pressure is measured between two arbitrary points and has no fixed reference at all. A DP cell senses both tappings across one diaphragm, which lets it resolve a small difference on a line whose static pressure is very much higher.
Because the pressure drop across a restriction comes from converting pressure into kinetic energy, and kinetic energy goes as the square of velocity. Inverting that relationship makes velocity, and therefore flow, proportional to the square root of the pressure difference. It is why doubling the flow quadruples the differential.
It sets the velocity-of-approach correction and, in practice, the permanent pressure loss. A small beta gives a large recoverable differential and good resolution but wastes more energy permanently. A large beta is gentler on the process but produces a weaker signal. Published standards also restrict the valid beta range for each element type.
Only as an upper bound. The relation here assumes constant density, and a gas expands as it accelerates through the throat. The standard treatment multiplies by an expansibility factor slightly below one, which this tool does not apply. If the differential is more than a few per cent of the absolute upstream pressure, that correction matters.
Because a tenth of full flow produces only a hundredth of full-scale differential pressure. Transmitter uncertainty is usually a fraction of span, so at the bottom of the range that fixed error becomes an enormous proportion of the reading. Three or four to one is a typical practical turndown for a single DP element.
Impulse lines, most often. Unequal liquid fill between the two legs, gas trapped in a liquid leg, condensate collecting in a gas leg or a wet reference leg slowly evaporating all shift the zero. Partially plugged tappings are worse still, because they give a stable and entirely plausible wrong answer.
A pipe pressure drop is caused by friction along a length of pipe and depends on roughness, length and Reynolds number. A differential across a flow element is caused by a deliberate restriction converting pressure into velocity. This page covers the second; frictional loss along a run belongs with a pipe flow calculation.
This tool is provided for educational and preliminary engineering use. It assumes incompressible flow with no expansibility correction, a fully developed velocity profile and a discharge coefficient you supply. Instrument selection, element sizing and installation must follow the governing standard and be verified by a qualified engineer.
