Flow Rate vs Pressure in Pipes: How to Calculate (Bernoulli, Darcy-Weisbach, Hazen-Williams & Worked Examples)

Last Updated: 2026-11-17

Quick Answer: Pressure Does Not Set Flow Rate. Pressure Drop Does.

The flow rate through a pipe is set by two things: the pressure difference between the inlet and the outlet, and the resistance of everything in between (pipe length, diameter, roughness, fittings, valves). Static pressure on its own tells you nothing. A water tower at 6 bar with a closed valve delivers 0 L/s; a 1.5 bar garden line with an open end delivers plenty. To calculate flow from pressure you need the drop (ΔP = P₁ − P₂), the pipe geometry, and a friction model.

The formula you reach for depends on what you know and what fluid you have:

Three rules of thumb before any calculation, valid for turbulent water flow in a fixed pipe:

The rest of this guide gives every formula with units, a roughness table, a psi-per-100-ft reference table and four worked examples you can copy into a spreadsheet. If you would rather measure than calculate, the last section explains which instrument to use and why a single differential pressure transmitter beats two gauge transmitters.

Why “More Pressure = More Flow” Is Wrong (and When It Is Almost Right)

Pressure is stored energy per unit volume. Flow is the result of that energy being spent against resistance. Picture two systems:

• A closed valve on a 10 bar main. Pressure upstream is 10 bar, pressure downstream is 10 bar, ΔP = 0, flow = 0.
• A 2 bar supply feeding an open hose. Pressure at the hose end is atmospheric (0 bar gauge), ΔP = 2 bar, and the whole 2 bar is consumed by friction along the hose, so flow is whatever velocity produces 2 bar of friction loss.

So the question “what flow do I get at 3 bar?” has no answer until you say what the pressure is at the other end and what is between the two points. The statement becomes almost right in one common situation: a fixed system (same pump, same pipe, same open outlet) where raising the supply pressure raises ΔP across the system. Then flow rises with roughly the square root of ΔP. That is why a pump running 20% faster delivers more flow but nowhere near 20% more head efficiency: head goes as speed², flow as speed, and friction loss as flow².

The quantity that links them is the system curve, ΔP_system = K·Q², where K lumps together everything in the next section. Measure ΔP at one known Q, solve for K, and you can predict the whole curve.

The Formulas, With Units

All equations below are in SI. Use kg, m, s, Pa and m³/s and convert at the end. Our pressure units explained guide with converter covers bar, psi, kPa and feet of head if your data arrives in mixed units.

1. Continuity (flow = area × velocity)

Q = A · v
A = π · D² / 4

Q  volumetric flow rate      m³/s
A  internal cross-section    m²
v  average velocity          m/s
D  internal diameter (not nominal size)   m

Continuity is also why flow in a pipe is conserved: A₁·v₁ = A₂·v₂. Halve the area and velocity doubles. This is the entire principle behind a venturi.

2. Bernoulli (energy balance between two points, no friction)

P₁ + ½·ρ·v₁² + ρ·g·h₁  =  P₂ + ½·ρ·v₂² + ρ·g·h₂

P  static pressure        Pa (absolute or gauge, but the same on both sides)
ρ  fluid density          kg/m³   (water 20 °C ≈ 998)
v  velocity               m/s
g  9.81 m/s²
h  elevation              m

Bernoulli is exact only for an ideal fluid. In a real pipe you add a friction-loss term, ΔP_f, to the right-hand side, and that term comes from Darcy-Weisbach or Hazen-Williams. Bernoulli on its own is good for: elevation corrections, converting velocity to pressure (pitot tubes) and sizing restrictions where friction is negligible compared with the velocity change.

3. Darcy-Weisbach (friction loss, any fluid)

ΔP = f · (L / D) · (ρ · v² / 2)        [Pa]
h_f = f · (L / D) · (v² / (2·g))        [m of fluid]

f  Darcy friction factor (dimensionless)
L  pipe length           m
D  internal diameter     m

The friction factor depends on the Reynolds number and the relative roughness:

Re = ρ · v · D / μ        μ = dynamic viscosity, Pa·s (water 20 °C ≈ 1.002 × 10⁻³)

Laminar   (Re < 2,300):    f = 64 / Re
Turbulent (Re > 4,000):    Swamee-Jain explicit approximation of Colebrook
                           f = 0.25 / [ log₁₀( ε/(3.7·D) + 5.74/Re^0.9 ) ]²

Swamee-Jain is within about 1% of the Colebrook equation for 10⁻⁶ < ε/D < 10⁻² and 5,000 < Re < 10⁸, which covers every industrial water, oil and gas line you are likely to meet. Between Re 2,300 and 4,000 the flow is transitional; avoid designing there.

Absolute roughness ε values to use:

4. Hazen-Williams (water only, fast)

h_f = 10.67 · L · Q^1.852 / ( C^1.852 · D^4.87 )     [m of water]

L  length    m
Q  flow      m³/s
D  diameter  m
C  roughness coefficient (higher = smoother)

Hazen-Williams has no viscosity term. It is calibrated for water at ordinary temperatures in turbulent flow and nothing else. Fire protection, municipal distribution and irrigation designers use it because it is explicit and their fluid is always water. Do not use it for oil, glycol, steam or air.

5. Orifice plate and venturi (ISO 5167)

Q = C_d · A₂ · √( 2·ΔP / ( ρ · (1 − β⁴) ) )

β   = d / D   (bore diameter / pipe internal diameter), typically 0.2–0.75
A₂  = π · d² / 4   (bore area)
C_d  discharge coefficient: ≈ 0.60–0.62 for a sharp-edged orifice, ≈ 0.95–0.98 for a venturi
ΔP   differential pressure across the taps, Pa

This is the equation inside every differential-pressure flow meter. Flow goes with the square root of ΔP, which is why DP flow measurement has a limited turndown (typically 4:1 to 10:1): at 10% of full flow the ΔP is only 1% of full scale.

6. Torricelli (free outflow from a tank)

v = √( 2 · g · h )        h = liquid head above the outlet, m
Q = C_d · A_outlet · v     C_d ≈ 0.6 for a sharp hole, up to 0.97 for a rounded nozzle

Ten metres of head gives a theoretical jet velocity of 14.0 m/s. This is the same physics a submersible hydrostatic level transmitter uses in reverse: it reads ρ·g·h as a pressure and reports h.

Worked Example 1: Pressure Drop in a DN50 Steel Line at 3 L/s

Given: commercial steel pipe, internal diameter 52.5 mm (DN50 Schedule 40), length 50 m, water at 20 °C (ρ = 998 kg/m³, μ = 1.002 × 10⁻³ Pa·s), flow 3 L/s = 0.003 m³/s. Horizontal, no fittings.

Step 1 – Area and velocity

A = π · 0.0525² / 4 = 0.002165 m²
v = 0.003 / 0.002165 = 1.386 m/s

1.4 m/s is a comfortable velocity for a water line; 1–3 m/s is the usual design window.

Step 2 – Reynolds number

Re = 998 · 1.386 · 0.0525 / 1.002×10⁻³ = 72,500   → turbulent

Step 3 – Friction factor (Swamee-Jain)

ε/D = 0.045 / 52.5 = 0.000857
f = 0.25 / [ log₁₀( 0.000857/3.7 + 5.74/72,500^0.9 ) ]² = 0.0226

Step 4 – Pressure drop

ΔP = 0.0226 · (50 / 0.0525) · (998 · 1.386² / 2)
   = 0.0226 · 952.4 · 958.6
   = 20,650 Pa ≈ 0.21 bar ≈ 3.0 psi
h_f = 20,650 / (998 · 9.81) = 2.11 m of water ≈ 6.9 ft

Result: 3 L/s (47.5 US gpm) through 50 m of DN50 steel costs about 0.21 bar. Normalized, that is 0.126 bar (1.83 psi) per 100 ft, which sits between the 1 m/s and 2 m/s rows of the reference table further down, as it should at 1.39 m/s.

Worked Example 2: Flow Rate From a Measured 0.5 bar Pressure Drop

Given: the same 50 m DN50 steel line. A YL1800 differential pressure transmitter connected across the two ends reads ΔP = 0.50 bar = 50,000 Pa. What is the flow?

Because f depends on v, which is what we are solving for, we guess f, solve, and refine.

Pass 1 – guess f = 0.020

v = √( 2 · ΔP · D / ( f · L · ρ ) )
  = √( 2 · 50,000 · 0.0525 / ( 0.020 · 50 · 998 ) )
  = 2.29 m/s
Re = 998 · 2.29 · 0.0525 / 1.002×10⁻³ = 119,900
f (Swamee-Jain) = 0.0215

Pass 2 – use f = 0.0215

v = √( 2 · 50,000 · 0.0525 / ( 0.0215 · 50 · 998 ) ) = 2.21 m/s
Re = 115,600 → f = 0.0215  (converged)
Q = A · v = 0.002165 · 2.21 = 0.00479 m³/s ≈ 4.8 L/s ≈ 76 US gpm

Result: 0.5 bar across that line corresponds to about 4.8 L/s. Note the square-root law: ΔP went from 0.21 bar to 0.50 bar (×2.4) and flow went from 3.0 to 4.8 L/s (×1.6 ≈ √2.4). One iteration was enough because turbulent f is a weak function of Re.

Worked Example 3: Orifice Plate Flow From ΔP

Given: DN100 pipe (take D = 100 mm internal), sharp-edged orifice with β = 0.6 (bore d = 60 mm), corner or flange taps, water (ρ = 998 kg/m³), measured ΔP = 25 kPa, C_d = 0.61.

A₂ = π · 0.060² / 4 = 0.002827 m²
1 − β⁴ = 1 − 0.6⁴ = 1 − 0.1296 = 0.8704

Q = 0.61 · 0.002827 · √( 2 · 25,000 / ( 998 · 0.8704 ) )
  = 0.61 · 0.002827 · √( 57.56 )
  = 0.61 · 0.002827 · 7.587
  = 0.01308 m³/s ≈ 13.1 L/s ≈ 47 m³/h ≈ 207 US gpm

Result: 25 kPa across a β = 0.6 orifice in a 100 mm line is about 47 m³/h of water. A DP transmitter spanned 0–25 kPa with square-root extraction switched on would output 20 mA at 47 m³/h and 12 mA (50% flow) at 6.25 kPa. The real C_d from ISO 5167 (Reader-Harris/Gallagher equation) varies slightly with Re and tap location; 0.61 is the usual first estimate.

Worked Example 4: Hazen-Williams for a PVC Irrigation Main

Given: PVC pipe, 100 mm internal diameter, 100 m long, 10 L/s (0.010 m³/s) of water, C = 150.

h_f = 10.67 · 100 · 0.010^1.852 / ( 150^1.852 · 0.100^4.87 )
    = 1.46 m of water per 100 m

Convert: 1.46 m × 998 × 9.81 = 14,300 Pa = 0.143 bar = 2.07 psi per 100 m. Per 100 ft (30.48 m): 0.63 psi/100 ft.

Cross-check with Darcy-Weisbach using ε = 0.0015 mm: v = 1.27 m/s, Re = 126,800, f = 0.0171, h_f = 1.41 m. The two methods agree within 3%, which is typical for clean water in smooth pipe. For old cast iron (C = 100) the Hazen-Williams answer would be (150/100)^1.852 = 2.1 times higher: about 3.1 m per 100 m.

Reference Table: Water Pressure Drop per 100 ft of Steel Pipe

Assumptions: new commercial steel (ε = 0.045 mm), water at 20 °C, straight pipe, no fittings, Darcy-Weisbach with Swamee-Jain. Internal diameters are Schedule 40. Add 10–30% for a normal count of elbows and valves, or calculate fittings separately as equivalent lengths.

Two things jump out. Going from 1 m/s to 3 m/s (×3 flow) costs ×8–9 in pressure drop, close to the Q² law with the small decrease in f taken into account. And going from DN25 to DN100 at the same velocity cuts the loss per foot by a factor of 5, because ΔP ∝ 1/D at fixed velocity and f also falls with larger pipe.

Measuring Instead of Calculating: Which Instrument, and Where the Errors Hide

Every formula above assumes you know the pipe condition. After ten years of service you do not: scale, corrosion and partially closed valves all change K. In a running plant, the honest answer to “what is the flow rate?” is a measurement. Three approaches, in order of cost:

Two pressure transmitters, subtract. Tempting because gauge transmitters are cheap and you may already have them. The problem is span. Suppose the line runs at 8 bar and you want the 0.2 bar drop from Example 1. Two 0–10 bar transmitters at ±0.5% FS, such as the YL805 industrial pressure transmitter at the economical end of the range, can each be off by ±50 mbar. Worst case the difference is off by ±100 mbar on a 200 mbar reading: a 50% error in ΔP and a 25% error in flow. Upgrading to a YL0500 pressure transmitter at ±0.1% FS reduces that to ±10 mbar each, which is workable for trending but still not a flow measurement.

One differential pressure transmitter. A DP transmitter such as the YL808B differential pressure transmitter has both ports exposed to the same static pressure, so the 8 bar cancels and the instrument is spanned to, say, 0–500 mbar. Accuracy of ±0.1–0.5% now applies to 500 mbar, not 10 bar: ±0.5–2.5 mbar. That is a 20× improvement for roughly the price of one good gauge transmitter. Put it across an orifice plate and you have a DP flow meter, the most widely installed flow technology in the process industries. Most smart DP transmitters do the square-root extraction internally and output 4–20 mA proportional to flow.

A direct flow meter. When you need better than ±1–2%, wide turndown, or no pressure loss, skip the inference. An electromagnetic flowmeter reads conductive liquids at ±0.5% with no obstruction and no pressure drop. A vortex flowmeter for liquid, gas and steam handles the fluids a mag meter cannot. A clamp-on transit-time ultrasonic flowmeter (DN15–6000, ±1%) straps onto the outside of an existing pipe, which makes it the usual choice for verifying a calculation on a line you cannot shut down. Our comparison of electromagnetic vs turbine vs ultrasonic flow meters goes through the selection in detail, and the full flow meter range lists every model SIBXSUN supplies.

The pump curve shortcut. If a centrifugal pump is the driver, you can read flow without any flow meter: measure the pressure rise across the pump with the DP transmitter, find that head on the manufacturer’s pump curve, and read across to flow. The operating point is where the pump curve crosses the system curve K·Q². This is how many plants estimate flow on older lines, and it is accurate to about ±5% provided the impeller has not been trimmed or worn.

Whichever instrument you choose, the pressure type matters for the calculation. Formulas that subtract two pressures work equally in gauge or absolute, but gas density calculations need absolute pressure. Our guide to gauge vs absolute vs differential pressure explains which transmitter reads which, and the pressure transmitter supplier page lists the gauge, absolute and DP models available with lead times.

Common Mistakes That Wreck the Calculation

1. Using gauge pressure where absolute is needed. For liquids the error cancels in any ΔP. For gases, density ρ = P·M/(R·T) needs absolute pressure: 2 bar gauge is 3.01 bar absolute, a 50% density difference. Also, a vacuum line at −0.5 bar gauge is 0.51 bar absolute, not negative.
2. Ignoring elevation. 10 m of rise is 0.98 bar for water. A pump pushing uphill may show a large ΔP with modest flow; subtract ρ·g·Δh before attributing the rest to friction.
3. Mixing units. 1 psi = 6,895 Pa = 0.0689 bar = 2.31 ft of water. A spreadsheet that takes bar in one cell and psi in another will be off by a factor of 14.5. Convert everything to Pa, m, kg/m³ and m³/s first.
4. Nominal size instead of internal diameter. DN50 Schedule 40 steel has a 52.5 mm bore; Schedule 80 is 49.3 mm; DN50 PVC can be 55 mm or more. D appears to the fifth power in Darcy-Weisbach (through area² and L/D), so a 5% diameter error is a 25% ΔP error.
5. Hazen-Williams for non-water fluids. Hot glycol, diesel, lube oil and compressed air all have viscosities far from water. Use Darcy-Weisbach with the real μ and ρ.
6. Forgetting fittings. Each 90° elbow adds roughly 30 pipe diameters of equivalent length; a fully open globe valve adds about 340. On a short line with many fittings, fittings dominate.
7. Applying the orifice equation outside its range. ISO 5167 requires β between 0.1 and 0.75, Re above about 5,000 (higher for large β), and straight runs of 10–40 diameters upstream. Outside that, C_d drifts and the equation is a guess.
8. Reading mass flow as volumetric. The formulas here give volume. If your meter or your material balance is in kg/h, you need density; see mass flow rate vs volumetric flow rate for the conversion and the standard-vs-actual gas trap.

Which Technology Fits Which Calculation Problem

The nine technologies are compared head to head in our types of flow meters guide. For sourcing, SIBXSUN supplies the pressure, differential pressure and flow instruments above from partner factories in China, with datasheets and calibration certificates shipped with each unit; the electromagnetic flowmeter supplier page explains the sizing information we need to quote correctly.

Frequently Asked Questions

Does higher pressure mean higher flow rate?

Not by itself. Flow rate is driven by the pressure difference between two points and limited by the resistance of the pipe between them. A closed valve on a 10 bar main carries zero flow, while an open 2 bar line can carry a lot. If you raise the supply pressure and the pipe, fittings and outlet stay the same, the pressure drop across the system rises and flow increases roughly with the square root of that drop; doubling the drop gives about 1.4 times the flow, not double.

What is the formula for calculating flow rate in a pipe?

The base formula is Q = A × v, flow rate equals cross-sectional area times average velocity. To get velocity from pressure you need a friction model: Darcy-Weisbach (ΔP = f × L/D × ρ × v² / 2) for any fluid, or Hazen-Williams (h_f = 10.67 × L × Q^1.852 / (C^1.852 × D^4.87)) for water only. For a restriction such as an orifice plate, use Q = C_d × A₂ × √(2ΔP / (ρ(1 − β⁴))) per ISO 5167.

How do I calculate flow rate from a pressure drop reading?

Measure the differential pressure across a known length of pipe or across an orifice plate with a differential pressure transmitter, then solve the Darcy-Weisbach or orifice equation for Q. For a pipe you have to iterate once or twice because the friction factor depends on velocity; for an orifice the equation is explicit. In practice most plants skip the calculation and install a flow meter that outputs flow directly, or an orifice plate with a DP transmitter whose square-root extraction is done in the transmitter.

Why is a differential pressure transmitter better than two pressure transmitters for measuring pressure drop?

Because the pressure drop is usually a small number sitting on top of a large static pressure. Two 0–10 bar transmitters at ±0.5% of full scale can each be off by 50 mbar, so a 200 mbar drop could be read with a 50% error. A differential pressure transmitter spanned to 0–250 mbar measures the difference directly at ±0.1–0.5% of that much smaller span, and both ports see the same static pressure so it cancels.

Can I use the Hazen-Williams formula for oil, glycol or air?

No. Hazen-Williams is an empirical fit for water at roughly 5–25 °C in turbulent flow, with the fluid viscosity baked into the C coefficient. For oil, glycol mixtures, chemicals, steam or gas, use Darcy-Weisbach with the actual density and viscosity; it is dimensionally correct and works for any Newtonian fluid in laminar or turbulent flow.

Need a DP transmitter spanned for your orifice, or a flow meter sized for a specific line? Send the pipe size, fluid, flow range and operating pressure and we will recommend the instrument and span, with the calibration certificate included.