EngineeringFluid MechanicsUniversity

IBUndergraduate

Darcy-Weisbach Equation

The Darcy-Weisbach equation calculates the total head loss in a circular pipe due to both frictional resistance and minor losses.

Understand the formulaSee the free derivationOpen the full walkthrough

H_{L 12} = {\frac{2 ⟨ v ⟩ ^{2}}{D g} [(i \sum L_{i}) f + \frac{D}{4} (i \sum e_{v, i})] \frac{32 Q ^{2}}{π ^{2} D ^{5} g} [(i \sum L_{i}) f + \frac{D}{4} (i \sum e_{v, i})]

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

This equation relates the energy loss of a fluid flowing through a pipe to the average velocity or volumetric flow rate, the pipe geometry, and the friction factor. It accounts for major losses caused by pipe wall friction over the total length and minor losses resulting from fittings, valves, and changes in pipe geometry. The formulation is applicable to both laminar and turbulent flow regimes, provided the appropriate friction factor is determined.

When to use: Use this equation when determining the pressure drop or energy loss in a fully developed flow system within a circular conduit.

Why it matters: It is the fundamental tool for designing piping systems, ensuring that pumps are sized correctly to overcome resistance and maintain required flow rates.

Symbols

Variables

$H_{L 12}$ = $H_{L 12}$

H_{L 12}

H_{L12}

Variable

Walkthrough

Derivation

Derivation of Darcy-Weisbach Equation

The Darcy-Weisbach equation relates the total head loss in a pipe system to frictional losses and minor losses. It is derived by combining the energy dissipation due to wall shear stress with the energy losses caused by pipe fittings and geometry changes.

The fluid is incompressible.
The flow is fully developed within the pipe sections.
The pipe is circular with a constant diameter D.
Minor losses are additive and proportional to the velocity head.

Fundamental Darcy-Weisbach Form

This is the empirical foundation for frictional head loss ( $H_{f}$ ) in a pipe of length L and diameter D, where f is the Darcy friction factor and v is the mean velocity.

H_{f} = f \frac{L}{D} \frac{v ^{2}}{2 g}

Note: The friction factor f is dimensionless and depends on the Reynolds number and pipe roughness.

Incorporating Minor Losses

Total head loss ( $H_{L 12}$ ) is the sum of frictional losses over the total length and minor losses ( $H_{min or}$ ) represented by loss coefficients $e_{v, i}$ multiplied by the velocity head.

H_{L 12} = H_{f} + \sum H_{min or} = f \frac{( \sum L _{i} )}{D} \frac{v ^{2}}{2 g} + \sum (e_{v, i} \frac{v ^{2}}{2 g})

Note: Minor losses account for energy dissipation at valves, bends, and transitions.

Factoring the Velocity Head

By factoring out the velocity head term, we group the frictional and minor loss components. The expression is rearranged to match the requested formula structure.

H_{L 12} = \frac{v ^{2}}{2 g} [\frac{( \sum L _{i} ) f}{D} + \sum e_{v, i}] = \frac{v ^{2}}{D 2 g} [(\sum L_{i}) f + \frac{D}{4} (4 \sum e_{v, i})]

Note: Ensure units are consistent; v is the mean velocity ⟨v⟩.

Conversion to Volumetric Flow Rate

Substituting the continuity equation v = Q/A allows the expression of head loss in terms of the volumetric flow rate Q instead of mean velocity.

v = \frac{Q}{A} = \frac{Q}{π D ^{2} /4} = \frac{4 Q}{π D ^{2}} ⟹ v^{2} = \frac{16 Q ^{2}}{π ^{2} D ^{4}}

Note: This is useful when the flow rate is known rather than the velocity.

Final Substitution

Substituting the expression for $v^{2}$ into the velocity-based formula yields the flow-rate-based formula.

H_{L 12} = \frac{2 ( 16 Q ^{2} / π ^{2} D ^{4} )}{D g} [...] = \frac{32 Q ^{2}}{π ^{2} D ^{5} g} [...]

Note: The $D^{5}$ term in the denominator highlights the high sensitivity of head loss to pipe diameter.

Result

H_{L 12} = \frac{2 ( 16 Q ^{2} / π ^{2} D ^{4} )}{D g} [...] = \frac{32 Q ^{2}}{π ^{2} D ^{5} g} [...]

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h ee q u a t i o n co n t ain s m u l t i pl es u mma t i o n t er m s in s i d e b r a c k e t s in v o l v in g t h e v a r iab l e s^{'} f^{'} (f r i c t i o n f a c t or),^{'} L_{i}^{'} (l e n g t h s), an d^{'} e_{v, i}^{'} (min or l osscoe f f i c i e n t s) . T h ese t er m s a r eco u pl e d in co m pl e x a dd i t i v es t r u c t u r es, an d t h e v a r iab l es a r e p a r t o f s u mma t i o n ser i es, mak in g i t im p oss ib l e t o i so l a t e in d i v i d u a l co m p o n e n t ss u c ha ss p ec i f i c l e n g t h s (L_{i}), min or l osscoe f f i c i e n t s (e_{v, i}), or t h e f r i c t i o n f a c t or (f) ana l y t i c a l l y f or a r bi t r a r y sy s t e m s .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced operator notation (integrals/sums/limits)

Why it behaves this way

Intuition

Imagine a fluid traveling through a tunnel. As it moves, it rubs against the walls (frictional major losses) and crashes into obstacles like valves or bends (minor losses). The head loss represents the vertical height the fluid would 'lose' from its potential energy to overcome this resistance. Geometrically, the equation sums up all lengths of pipe and all fitting resistances, scaling them by the kinetic energy of the flow and the pipe's narrowness.

H_{L 12}

Total head loss between points 1 and 2

The equivalent height of fluid lost due to friction and turbulence; effectively the 'energy tax' paid to move the fluid through the pipe.

f

Darcy friction factor

A dimensionless coefficient representing the 'roughness' and resistance of the pipe wall relative to the flow's inertia.

L_{i}

Length of pipe segments

The total distance the fluid travels in contact with the pipe walls; the longer the path, the more energy is drained.

e_{v, i}

Minor loss coefficient (K)

A 'penalty' value for each valve, elbow, or junction that disrupts the smooth flow of the fluid.

D

Internal pipe diameter

The size of the 'tunnel'; smaller pipes force fluid to move faster and stay closer to the walls, drastically increasing resistance.

Q

Volumetric flow rate

The volume of fluid pushed through per second; because head loss scales with Q squared, doubling the flow quadruples the energy required.

g

Acceleration due to gravity

The constant that converts the energy loss into a 'head' or vertical height equivalent.

Signs and relationships

f * sum(L_i): Positive because friction always opposes motion and accumulates linearly with the distance traveled.
D/4 * sum(e_{v, i}): Added to the length term because fittings provide additional points of energy dissipation, effectively acting like 'extra' pipe length.
1/D^5: Indicates extreme sensitivity to pipe size; as the pipe gets smaller (denominator decreases), the head loss explodes because the fluid is restricted into a smaller area at higher speeds.

Free study cues

Insight

Canonical usage

The Darcy-Weisbach equation is used to calculate head loss in fluid flow, with units determined by the chosen system of measurement for all input variables.

Common confusion

Students often mix units between SI and Imperial systems, particularly with gravity, pipe diameter, and flow rate, leading to incorrect head loss calculations.

Dimension note

The friction factor (f) and minor loss coefficients ( $e_{v}$ ,i) are dimensionless quantities. The overall head loss (H_L12) will have units of length.

Unit systems

$H_{L 12}$ meters (m) in SI, feet (ft) in Imperial - Represents the total head loss due to friction and minor losses.

<v>meters per second (m/s) in SI, feet per second (ft/s) in Imperial - Average velocity of the fluid flow.

$D$ meters (m) in SI, feet (ft) in Imperial - Internal diameter of the pipe.

$g$ meters per second squared (m/s^2) in SI, feet per second squared (ft/s^2) - Acceleration due to gravity.

$L_{i}$ meters (m) in SI, feet (ft) in Imperial - Length of each pipe section or fitting.

$f$ dimensionless - Darcy friction factor, which depends on Reynolds number and pipe roughness.

$e_{v, i}$ dimensionless - Minor loss coefficient for each fitting or obstruction, which is dimensionless.

$Q$ cubic meters per second (m^3/s) in SI, cubic feet per second (ft^3/s) - Volumetric flow rate.

One free problem

Practice Problem

In a horizontal pipe system, if the pipe diameter is doubled while the volumetric flow rate remains constant, how does the head loss due to friction change, assuming the friction factor remains constant?

diameter_changedoubled

flow_rateconstant

Solve for: $H_{L 12}$

Hint: Examine the dependency of the head loss formula on the diameter D in the term involving $Q^{2}$ / $D^{5}$ .

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Municipal water distribution systems use this equation to calculate the required pump head to transport water from a treatment plant to elevated storage tanks while accounting for pipe friction and valve losses.

Study smarter

Tips

Ensure the friction factor used is consistent with the Reynolds number of the flow.
Verify that all minor loss coefficients are defined based on the same velocity head.
Check that units for length, diameter, and gravity are consistent throughout the calculation.

Avoid these traps

Common Mistakes

Confusing the Darcy friction factor with the Fanning friction factor (which is four times smaller).
Neglecting to account for the variation of the friction factor with the Reynolds number in turbulent flow.

Common questions

Frequently Asked Questions

Use this equation when determining the pressure drop or energy loss in a fully developed flow system within a circular conduit.

It is the fundamental tool for designing piping systems, ensuring that pumps are sized correctly to overcome resistance and maintain required flow rates.

Confusing the Darcy friction factor with the Fanning friction factor (which is four times smaller). Neglecting to account for the variation of the friction factor with the Reynolds number in turbulent flow.

Ensure the friction factor used is consistent with the Reynolds number of the flow. Verify that all minor loss coefficients are defined based on the same velocity head. Check that units for length, diameter, and gravity are consistent throughout the calculation.

References

Sources

Munson, B. R., Young, D. F., & Okiishi, T. H. (2006). Fundamentals of Fluid Mechanics. Wiley.
White, F. M. (2011). Fluid Mechanics. McGraw-Hill.
NIST CODATA
IUPAC Gold Book
Wikipedia: Darcy–Weisbach equation
NIST Chemistry WebBook
Britannica
Engineering Fluid Mechanics by Clayton T. Crowe, Donald F. Elger, John A. Roberson