MathBench

Pipe Pressure Drop Calculator

Enter a flow rate, a pipe, and a fluid, and this calculator returns the pressure drop (wall friction and fitting losses computed separately) using the Darcy-Weisbach equation with the Colebrook-White friction factor. Fluid density and viscosity are evaluated at the temperature you specify.

The fluid

The pipe

The flow

Fittings and valves (resistance coefficients after Crane TP-410)

Result

—

—velocity
—Reynolds number
—flow regime
—friction factor f
—gradient (psi / 100 ft)
—head loss

Where the pressure goes

How this calculator works

A flowing fluid loses pressure for three distinct reasons, and this calculator accounts for each of them separately. The first is friction against the pipe wall along the entire run, known as the major loss. The second is the disturbance caused by fittings, valves, bends and the entry and exit of the pipe, known collectively as minor losses: a name that understates them, because in a short run with several valves they frequently exceed the wall friction. The third is elevation: raising a fluid costs pressure whether it is moving or not.

The calculation proceeds in a fixed order. From the volumetric flow rate and the true inside bore of the pipe, the mean velocity follows by continuity. Density and viscosity are then evaluated at your operating temperature (and for gases at your operating pressure as well), which fixes the Reynolds number and therefore the flow regime. The friction factor comes from that regime: exactly 64/Re in laminar flow, and the Colebrook-White equation solved by iteration in turbulent flow. Multiplying by the length-to-diameter ratio and the velocity head gives the major loss. The sum of the fitting resistance coefficients multiplied by the same velocity head gives the minor loss.

The second mode reverses the question. Given the pressure you can afford to spend, it searches for the flow rate whose total loss consumes that budget, which is the calculation you need when sizing a service line or checking whether an existing pipe can carry more. Because the friction factor itself depends on flow, this inverse problem has no closed-form solution and is solved numerically.

The formula

Major loss (wall friction), Darcy-Weisbach:
    Δp = f · (L / D) · ρV² / 2

Minor loss (fittings and valves):
    Δp = ΣK · ρV² / 2

Elevation:
    Δp = ρ · g · Δz                  g = 9.80665 m/s²

Reynolds number and friction factor:
    Re = ρ · V · D / μ
    Re < 2300     f = 64 / Re                     (laminar, exact)
    Re > 4000     1/√f = −2 log₁₀( ε/(3.7D) + 2.51/(Re·√f) )

Fluid properties:
    gases      ρ = p·M / (R·T)          R = 8.314462618 J/(mol·K)
               μ = μ₀ · (T₀+S)/(T+S) · (T/T₀)^1.5     (Sutherland)

The Darcy-Weisbach equation is dimensionally rigorous and applies to any fluid in any regime, so it is preferred here over empirical alternatives such as Hazen-Williams that are calibrated for water alone. Its friction factor comes from the implicit relation published by C. F. Colebrook in 1939 (Journal of the Institution of Civil Engineers 11, 133), the correlation that Lewis Moody plotted the following year as the chart still used today (Transactions of the ASME 66, 671). The laminar result f = 64/Re follows directly from the Hagen-Poiseuille solution for fully developed flow in a circular tube.

Resistance coefficients for the fittings are the conventional design values of the Crane Company's Technical Paper No. 410, Flow of Fluids Through Valves, Fittings and Pipe, the standard reference for this method. Pipe bores are the published dimensions of ASME B36.10M for steel, ASTM B88 for copper tube, ASTM D1785 for PVC and ASTM F876 for PEX. Wall roughness values are the tabulated design figures given by Moody and by Crane.

Water density uses the equation of G. S. Kell (Journal of Chemical and Engineering Data 20, 97, 1975), which reproduces measured values to about one part in ten thousand from 0 to 150 °C. The calculator returns 998.2 kg/m³ at 20 °C and 971.8 kg/m³ at 80 °C, matching the published table. Water viscosity uses the standard exponential correlation μ = 2.414×10⁻⁵ · 10^(247.8/(T−140)) pascal-seconds, accurate to roughly two percent. Gas viscosity uses the formula published by William Sutherland in 1893, with constants as tabulated by Frank White in Viscous Fluid Flow. The remaining liquids are interpolated between two published property points, cited in the assumptions below.

Worked example

Fifty gallons per minute of water at 20 °C through 100 feet of 2-inch Schedule 40 steel pipe, with four long-radius elbows, one fully open gate valve and a sharp-edged entrance, the calculator's default case:

  1. Bore and area: D = 2.067 in = 0.0525 m, so A = πD²/4 = 2.165 × 10⁻³ m²
  2. Velocity: Q = 50 gal/min = 3.1545 × 10⁻³ m³/s, so V = Q/A = 1.457 m/s (4.781 ft/s)
  3. Properties at 20 °C: ρ = 998.2 kg/m³, μ = 1.002 × 10⁻³ Pa·s
  4. Reynolds number: ρVD/μ = 76,230, firmly turbulent
  5. Relative roughness: ε/D = 0.045 mm ÷ 52.5 mm = 8.571 × 10⁻⁴, and Colebrook then gives f = 0.02234
  6. Velocity head: ρV²/2 = 1,060 Pa
  7. Major loss: f · (L/D) · ρV²/2 = 0.02234 × (30.48/0.0525) × 1,060 = 13,750 Pa = 1.994 psi
  8. Minor loss: ΣK = 4(0.30) + 0.15 + 0.50 = 1.85, so 1.85 × 1,060 = 1,960 Pa = 0.2843 psi
  9. Total: 15,710 Pa = 2.278 psi, equivalent to 5.264 feet of head

The fittings account for about an eighth of the loss in this run. Shorten the pipe to twenty feet and that share rises above forty percent, which is the practical reason the word "minor" deserves the scepticism it gets. Raising the water temperature to 90 °C drops the total to 2.022 psi, because viscosity falls by a factor of three and the friction factor with it.

Assumptions & tips

  • Enter straight pipe and fittings separately. The older practice of adding an "equivalent length" for each elbow was a convenience for slide rules. Counting fittings and applying their resistance coefficients is both more accurate and easier to audit, since you can see what each one costs.
  • Temperature deserves a real value. Chilled water at 5 °C is half again as viscous as water at 20 °C, and a glycol loop at winter start-up can be several times more viscous than the same loop running warm. A system sized at design temperature can fall short on the coldest morning of the year.
  • Check the regime before trusting the number. Viscous fluids in small pipes are often laminar, where friction depends on viscosity alone and roughness stops mattering entirely. The regime is reported with every result. The Reynolds number calculator explains what it means in more detail.
  • For gases, watch the pressure ratio. This calculation assumes constant density. That is sound while the loss stays under about a tenth of the absolute inlet pressure, and the calculator tells you when you have crossed that line. Long gas transmission lines require a compressible treatment such as the Weymouth or Panhandle equations.
  • Non-water liquid properties are interpolated. Seawater follows Sharqawy, Lienhard and Zubair (2010); the glycol mixtures follow the ASHRAE secondary-coolant tables; diesel, SAE 30 oil and gasoline use published typical values at two reference temperatures. Real oils vary between suppliers and batches, so for design work, enter the density and viscosity from your own datasheet using the custom fluid option.
  • Roughness changes with age. The tabulated value for commercial steel, 0.045 mm, describes clean new pipe. Scale, corrosion and biological growth can raise it by an order of magnitude over decades, and simultaneously reduce the bore. Where a system must still perform in thirty years, design it with degraded numbers.

Frequently asked questions

What is the difference between major and minor losses?

Major loss is the friction of the fluid against the pipe wall along the whole run, and it is calculated with the Darcy-Weisbach equation. Minor loss is the energy lost in fittings, valves, bends, entrances and exits, where the flow separates and re-forms. The name is misleading: in a short run with many valves, the so-called minor losses can easily exceed the wall friction. This calculator reports the two separately so you can see which one dominates your system.

Why does temperature change the pressure drop?

Temperature changes both properties that govern the flow. Viscosity falls steeply as a liquid warms (water is roughly three times less viscous at 90 °C than at 5 °C), which raises the Reynolds number and lowers the friction factor. Density falls more gently, which reduces the velocity head. For the same 50 gal/min of water in the worked example below, the total loss falls from 2.391 psi at 5 °C to 2.022 psi at 90 °C, a difference of about 15 percent. For gases the effect is larger still, because density depends directly on absolute temperature.

Can I use this calculator for compressed air and other gases?

Yes, within limits. Gas density is computed from the ideal gas law at the operating pressure and temperature you enter, and the same incompressible Darcy-Weisbach method is applied. That treatment is accurate while the pressure drop stays small relative to the absolute inlet pressure. The usual engineering guideline is below about 10 percent. Beyond that the gas expands appreciably along the pipe, its velocity rises, and a compressible method such as the Darcy or Weymouth equation is required. The calculator computes the ratio for you and warns when the assumption no longer holds.

Which friction factor correlation does this use?

For laminar flow, below a Reynolds number of about 2,300, the friction factor is exactly 64/Re, a result derived from theory rather than measured. For turbulent flow above 4,000 it solves the Colebrook-White equation by Newton iteration, the implicit correlation that underlies the Moody chart. Between those limits the flow is unpredictable, so the calculator interpolates and labels the result transitional.

How accurate are the fluid properties?

Water density comes from the Kell equation, which reproduces published values to within about 0.01 percent between 0 and 100 °C, and water viscosity from a standard exponential correlation good to roughly 2 percent. Gas viscosity uses Sutherland's formula, which agrees with tabulated values to about 1 percent over ordinary temperature ranges. The other liquids are interpolated between two published data points and are labelled with the temperature range over which that interpolation is meaningful. Every source is listed at the foot of this page, and you can always enter measured density and viscosity directly.

Sources

  1. Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws. C. F. Colebrook, Journal of the Institution of Civil Engineers 11(4), 133–156, 1939. DOI 10.1680/ijoti.1939.13150. The implicit friction-factor relation solved by Newton iteration for every turbulent result on this page.
  2. Friction Factors for Pipe Flow. Lewis F. Moody, Transactions of the ASME 66(8), 671–678, 1944. DOI 10.1115/1.4018140. The chart form of the Colebrook relation, and the source of the conventional wall-roughness design values and the 2,300 / 4,000 regime boundaries used here.
  3. Technical Paper No. 410: Flow of Fluids Through Valves, Fittings and Pipe. Crane Co. tp410.comThe resistance coefficients K for every fitting and valve in the list (0.30 for a long-radius elbow, 10.0 for an open globe valve, 0.50 for a sharp-edged entrance) and the ΣK·ρV²/2 method used to sum them.
  4. ASME B36.10M, Welded and Seamless Wrought Steel Pipe. American Society of Mechanical Engineers. asme.orgThe published inside diameters for the Schedule 40 and Schedule 80 steel sizes in the pipe menu, including the 2.067 in bore of nominal 2-inch Schedule 40 used in the worked example.
  5. Copper Tube Handbook. Copper Development Association. copper.orgThe Type L copper tube dimensions of ASTM B88 that supply the copper bores in the size menu.
  6. Density, Thermal Expansivity, and Compressibility of Liquid Water from 0° to 150 °C: Correlations and Tables for Atmospheric Pressure and Saturation Reviewed and Expressed on 1968 Temperature Scale. George S. Kell, Journal of Chemical and Engineering Data 20(1), 97–105, 1975. DOI 10.1021/je60064a005. The water-density equation coded here, which returns 998.2 kg/m³ at 20 °C.
  7. Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary Water Substance (R12-08). International Association for the Properties of Water and Steam, 2008. iapws.orgThe international reference formulation against which this page's simpler exponential water-viscosity correlation is checked, and the basis of the "roughly two percent" accuracy claim.
  8. LII. The Viscosity of Gases and Molecular Force. William Sutherland, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 36(223), 507–531, 1893. DOI 10.1080/14786449308620508. The gas-viscosity formula μ = μ₀·(T₀+S)/(T+S)·(T/T₀)^1.5 applied to air, methane, nitrogen, oxygen, carbon dioxide, hydrogen and helium.
  9. Thermophysical Properties of Seawater: A Review of Existing Correlations and Data. M. H. Sharqawy, J. H. Lienhard and S. M. Zubair, Desalination and Water Treatment 16(1–3), 354–380, 2010. doi.orgThe density and viscosity of 35 g/kg seawater at the two reference temperatures the seawater preset interpolates between.
  10. 2021 ASHRAE Handbook — Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers. ashrae.orgThe secondary-coolant property tables behind the 50 percent ethylene glycol and 50 percent propylene glycol presets.