Engineering reference // Fluid mechanics // Part 1

Drawing No. EH–FM–007 // Fluid Mechanics & Piping

Fluid Mechanics Formula Sheet

Part 1 of the EngineerHub fluid-mechanics reference series: internal flow and pump systems. It connects flow rate, velocity, pressure, friction, pump head and cavitation. Each section includes variable definitions, SI and US customary units, a worked example and a direct link to the corresponding EngineerHub calculator.

From continuity to NPSH

Use the sheet as a quick reference or follow it in sequence. The equations assume conventional engineering sign conventions; always use a consistent unit system inside a calculation.

Start with continuity Open formula summary

Reference conventions

g = 9.80665 m/s²Standard gravitational acceleration; ≈ 32.174 ft/s²

γ = ρgSpecific weight: N/m³ in SI, lbf/ft³ in US customary

SG = ρ/ρrefSpecific gravity: density ratio to the stated reference liquid, commonly water

ν = μ/ρKinematic viscosity: m²/s or ft²/s

Energy grade line along a pipe run

01 // Continuity

Conservation of mass links flow area and velocity. For an incompressible fluid of constant density, volumetric flow rate is conserved through a single streamtube.

Incompressible steady flow

Q = A v → A₁v₁ = A₂v₂

ṁ = ρQ = ρAv

Symbol	Meaning	SI units	US customary
Q	Volumetric flow rate	m³/s, L/s	ft³/s, gpm
A	Flow area normal to velocity	m²	ft², in²
v	Mean cross-sectional velocity	m/s	ft/s
ṁ	Mass flow rate	kg/s	lbm/s
ρ	Fluid density	kg/m³	lbm/ft³

For compressible flow, conserve mass rather than volume: ρ₁A₁v₁ = ρ₂A₂v₂. For a circular pipe, A = πD²/4. For non-circular internal flow, use hydraulic diameter D_h = 4A/P_w, where P_w is wetted perimeter.

Open Pipe Velocity Calculator Worked example ↓

Worked example — flow from pipe diameter and velocity

Given: water flows through a 100 mm internal-diameter pipe at an average velocity of 2.0 m/s.

A = π(0.100 m)² / 4 = 0.007854 m²

Q = Av = 0.007854 × 2.0 = 0.01571 m³/s

Answer: Q ≈ 0.0157 m³/s = 15.7 L/s.

US check: US equivalent: Q ≈ 249 US gpm.

Area–velocity relationship

Continuity does not create energy; it only enforces mass conservation. Pressure changes require Bernoulli / momentum relations.

02 // Bernoulli equation

Bernoulli expresses mechanical energy per unit weight along a streamline. The extended form includes pump head, turbine extraction and irreversible head losses.

Ideal mechanical-energy form

p/γ + αv²/(2g) + z = H

Extended engineering form

p₁/γ + α₁v₁²/(2g) + z₁ + Hₚ = p₂/γ + α₂v₂²/(2g) + z₂ + Hₜ + hL

Symbol	Meaning	SI units	US customary
p	Static pressure	Pa, kPa	psf or psi (convert psi × 144 before p/γ)
γ	Specific weight = ρg	N/m³	lbf/ft³
ρ	Mass density	kg/m³	lbm/ft³
α	Kinetic-energy correction coefficient	dimensionless	dimensionless
v	Mean fluid velocity	m/s	ft/s
z	Elevation datum	m	ft
Hₚ	Head added by pump	m fluid	ft fluid
Hₜ	Head removed by turbine	m fluid	ft fluid
hL	Total irreversible head loss	m fluid	ft fluid

The ideal form assumes steady, incompressible flow along a streamline. For ordinary turbulent pipe-flow screening, α ≈ 1 is commonly used; for fully developed laminar circular-pipe flow, α = 2. The extended form adds shaft work and irreversible head loss.

Open Bernoulli Calculator Worked example ↓

Worked example — reservoir to atmospheric discharge

Given: water is pumped from a large open reservoir to an atmospheric pipe discharge 10 m higher. The discharge velocity is 2.0 m/s, total head loss is 3.0 m, and the source-reservoir free-surface velocity is negligible.

Hₚ = Δz + v₂²/(2g) + hL

Hₚ = 10 + 2²/(2×9.80665) + 3 = 13.204 m

Answer: required pump head ≈ 13.2 m under the stated assumptions.

US check: US equivalent: required pump head ≈ 43.32 ft.

Head terms are all lengths

Bernoulli is especially convenient because pressure head, velocity head and elevation head are expressed in the same unit: metres or feet of the flowing fluid. Using specific weight γ keeps the head equation dimensionally clean in both SI and US customary systems.

Pressure head p/γ

Velocity head αv²/2g

Elevation z

hₚ = Δp/γ ; Δp = γh

US: h(ft) = 144·Δp(psi)/γ(lbf/ft³). For water-like liquids, h(ft) ≈ 2.31·Δp(psi)/SG.

Common mistake: do not insert psi directly into p/γ unless the 144 in²/ft² conversion is included. Gauge pressures may cancel in open-reservoir balances, but NPSH calculations require absolute pressure.

03 // Reynolds number

Reynolds number compares inertial and viscous effects and is the primary regime indicator for internal flow.

Internal pipe flow

Re = ρvD/μ = vD/ν

Symbol	Meaning	SI units	US customary
Re	Reynolds number	dimensionless	dimensionless
ρ	Density	kg/m³	lbm/ft³
v	Mean velocity	m/s	ft/s
D	Pipe internal diameter / hydraulic diameter	m	ft
μ	Dynamic viscosity	Pa·s	lbm/(ft·s)
ν	Kinematic viscosity = μ/ρ	m²/s	ft²/s

For circular pipe flow, a common engineering classification is laminar Re ≲ 2300, transition ≈ 2300–4000, and turbulent Re ≳ 4000. For non-circular ducts use D_h = 4A/P_w. Transition depends on inlet disturbances, roughness and geometry.

Open Reynolds Calculator Worked example ↓

Worked example — water in a 50 mm pipe

Given: 20°C water, ρ = 998 kg/m³, μ = 1.002×10⁻³ Pa·s, v = 2.0 m/s, D = 0.050 m.

Re = (998×2.0×0.050)/(1.002×10⁻³) ≈ 9.96×10⁴

Answer: Re ≈ 99,600, clearly in the turbulent regime for ordinary internal pipe flow.

US check: Using ρ ≈ 62.30 lbm/ft³, μ ≈ 6.733×10⁻⁴ lbm/(ft·s), v ≈ 6.562 ft/s and D ≈ 0.1640 ft gives the same Re ≈ 99,600.

Why Reynolds matters downstream

It determines whether laminar friction factor f = 64/Re applies.
It affects turbulent friction-factor correlations such as Colebrook or Swamee–Jain.
It influences heat-transfer and mass-transfer correlations.
It helps interpret whether a scale model is dynamically similar.

Open Moody Chart

04 // Darcy–Weisbach major loss

The Darcy–Weisbach equation gives frictional loss along a straight pipe. It is dimensionally consistent and widely applicable when the Darcy friction factor is known.

Head loss

hƒ = f (L/D) · v²/(2g)

Δpƒ = γhƒ

Symbol	Meaning	SI units	US customary
hƒ	Major head loss	m fluid	ft fluid
Δpƒ	Major pressure loss	Pa	psf; divide by 144 for psi
f	Darcy friction factor	dimensionless	dimensionless
L	Straight pipe length	m	ft
D	Internal / hydraulic diameter	m	ft
v	Mean velocity	m/s	ft/s
g	Gravitational acceleration	m/s²	ft/s²
γ	Specific weight	N/m³	lbf/ft³

Important: this page uses the Darcy friction factor. The Fanning factor is one quarter of the Darcy value: fF = fD/4.

Open Darcy–Weisbach Calculator Worked example ↓

Worked example — straight-pipe loss

Given: f = 0.020, L = 50 m, D = 0.100 m, v = 2.0 m/s, water density ≈ 998 kg/m³.

hƒ = 0.020 × (50/0.100) × 2²/(2×9.80665) = 2.039 m

Δpƒ = γhƒ = ρghƒ ≈ 998×9.80665×2.039 ≈ 19.96 kPa

Answer: major loss ≈ 2.04 m of water, or about 20.0 kPa.

US check: US equivalent: hƒ ≈ 6.691 ft and Δpƒ ≈ 2.895 psi.

Choosing the Darcy friction factor

Laminar circular pipe:

f = 64/Re

Turbulent pipe — Colebrook–White:

1/√f = −2 log₁₀[(ε/D)/3.7 + 2.51/(Re√f)]

Here ε is absolute roughness and ε/D is relative roughness. Colebrook is implicit in f; a Moody chart or validated explicit approximation is often more convenient.

Transition region: friction is less predictable; do not treat Re = 2300 or 4000 as a perfectly sharp physical switch.

Moody Chart Calculator Full Head-Loss Tool

05 // Minor / local losses

Fittings, entrances, exits, valves and local geometry changes are commonly represented by dimensionless loss coefficients.

Local head loss

hₘ = ΣK · v²/(2g)

K = f · Leq/D → Leq = KD/f

Symbol	Meaning	SI units	US customary
hₘ	Minor / local head loss	m fluid	ft fluid
K	Loss coefficient for fitting / component	dimensionless	dimensionless
v	Reference velocity associated with K	m/s	ft/s
Leq	Equivalent straight-pipe length	m	ft
f	Darcy friction factor	dimensionless	dimensionless

Use the velocity specified for the coefficient source. For area changes, a coefficient may be based on upstream, downstream or throat velocity; mixing definitions can produce large errors.

Open Pipe Flow & Head Loss Calculator Worked example ↓

Worked example — fittings represented by ΣK

Given: combined fitting coefficient ΣK = 4.5 and pipe velocity v = 2.0 m/s.

hₘ = 4.5 × 2²/(2×9.80665) = 0.918 m

Answer: local losses add approximately 0.918 m of head loss.

US check: US equivalent: local head loss ≈ 3.011 ft.

“Minor” does not mean negligible

In short piping systems, suction lines, valve stations or heavily fitted process systems, local losses can equal or exceed straight-pipe friction.

Do not double count: if a vendor or design source already incorporates a component pressure drop explicitly, do not also add an independent K value for the same loss.

06 // Pump hydraulic and shaft power

Pump head becomes hydraulic power through flow rate and fluid specific weight. Pump and motor efficiencies determine the shaft and electrical input required.

Hydraulic power

Pₕ = γQH = ρgQH

Ppump,input = Pₕ/ηₚ ; Pelec = Pₕ/(ηₚηₘ)

US customary practical form

Pₕ(hp) = Q(gpm)·H(ft)·SG / 3960

Ppump,input(hp) = Q·H·SG / (3960·ηₚ)

Symbol	Meaning	SI units	US customary
Pₕ	Hydraulic power delivered to fluid	W, kW	hp
γ	Specific weight	N/m³	lbf/ft³
ρ	Mass density	kg/m³	lbm/ft³
Q	Volumetric flow rate	m³/s	ft³/s, US gpm
H	Total pump head	m fluid	ft fluid
ηₚ	Pump efficiency	fraction or %	fraction or %
ηₘ	Motor efficiency	fraction or %	fraction or %
SG	Specific gravity for practical US pump-power form	dimensionless	dimensionless

Use total dynamic head at the actual operating point, not only static elevation. Total head can include static head, pressure difference, velocity-head change and pipe/fitting losses. The 3960 constant is the conventional US gpm–ft–horsepower conversion used with specific gravity.

Open Pump System Calculator Worked example ↓

Worked example — shaft power

Given: water, ρ = 998 kg/m³, Q = 0.020 m³/s, pump head H = 25 m, pump efficiency ηₚ = 0.75.

Pₕ = 998×9.80665×0.020×25 = 4.894 kW

Pshaft = 4.894/0.75 = 6.525 kW

Answer: hydraulic power ≈ 4.89 kW; required shaft power ≈ 6.52 kW before any motor-efficiency allowance.

US check: US equivalent: hydraulic power ≈ 6.56 hp and pump shaft/input power ≈ 8.75 hp. The conventional 3960 form gives essentially the same result, subject to its rounded conversion constant.

Energy chain

Efficiency should be evaluated near the actual pump operating point rather than assumed constant across the full curve.

07 // Net positive suction head (NPSH)

NPSH measures how far the pump suction condition lies above the liquid vapor-pressure head. Use absolute pressures.

At pump suction reference

NPSHₐ = pₛ,abs/γ + vₛ²/(2g) − pᵥ/γ

Open suction reservoir — useful system form

NPSHₐ = pₐtm/γ + zstatic − hL,suction − pᵥ/γ

Symbol	Meaning	SI units	US customary
NPSHₐ	Net positive suction head available	m liquid	ft liquid
γ	Liquid specific weight	N/m³	lbf/ft³
pₛ,abs	Absolute static suction pressure	Pa absolute	psia (×144 for psf in p/γ)
vₛ	Suction-pipe mean velocity	m/s	ft/s
pᵥ	Liquid vapor pressure at temperature	Pa absolute	psia
pₐtm	Atmospheric pressure	Pa absolute	psia
zstatic	Liquid-surface elevation above pump datum; negative for lift	m	ft
hL,suction	Suction-line losses	m liquid	ft liquid

NPSH required is a pump characteristic supplied by the manufacturer for a defined test criterion. Published rotodynamic-pump curves commonly use NPSH3: the NPSH available at which total head has fallen by 3% at the specified condition. NPSHA equal to NPSH3 therefore does not mean zero cavitation or full non-cavitating performance. Establish margin from applicable pump/vendor/project guidance rather than assuming one universal margin.

US customary: convert pressure to head as h(ft) = 144·p(psia)/γ(lbf/ft³). Use absolute pressure for atmospheric, suction and vapor-pressure terms.

Open Pump NPSH Calculator Worked example ↓

Worked example — open tank suction

Given: water at about 20°C, ρ = 998 kg/m³, atmospheric pressure 101.325 kPa absolute, vapor pressure 2.34 kPa absolute, liquid surface 2.0 m above the pump, and suction-line loss 1.0 m.

NPSHₐ = (pₐtm−pᵥ)/γ + zstatic − hL,suction

NPSHₐ = (101.325−2.34)×10³/(998×9.80665) + 2.0 − 1.0

NPSHₐ ≈ 11.11 m

Answer: NPSH available ≈ 11.1 m of water.

US check: US equivalent: NPSHA ≈ 36.46 ft of water.

This is not a complete cavitation guarantee. Compare the result with the pump manufacturer's NPSHr at the actual flow and apply the project/vendor margin criteria.

What lowers NPSHa?

Higher liquid temperature → higher vapor pressure.
Higher elevation above sea level → lower atmospheric pressure.
Longer / smaller suction piping → greater suction friction loss.
Higher flow → greater velocity and usually greater suction losses.
Liquid level below the pump centerline → negative static contribution.

08 // How the equations fit together

A pipe-and-pump calculation usually moves through these relationships in sequence rather than treating them as isolated equations.

GeometryArea and diameter

ContinuityQ ↔ velocity

ReynoldsIdentify regime

FrictionMajor + local losses

BernoulliTotal system head

Pump powerγQH / efficiency

NPSHCheck suction margin

System head

Hsystem = Hstatic + Hpressure + hL,major + hL,minor

For many turbulent systems: Hsystem ≈ Hstatic + KQ²

The pump operating point is the intersection of the pump curve and the system curve. Friction-factor variation means the exact system curve is not always a perfect Q² parabola.

Iteration is normal. Pipe diameter changes velocity, Reynolds number, friction factor, head loss, pump duty and NPSH. A realistic design often requires several passes or a coupled system-curve calculation.

09 // Quick formula summary

Use this table as the compact print reference. The detailed sections above contain definitions and assumptions.

Topic	Equation	Main purpose	EngineerHub tool
Continuity	Q = Av	Relate area, velocity and volumetric flow	Pipe Velocity
Hydraulic diameter	D_h = 4A/P_w	Characteristic diameter for non-circular internal flow	Reynolds Number
Bernoulli	p/γ + αv²/2g + z = H	Mechanical energy / head balance	Bernoulli
Reynolds	Re = ρvD/μ	Classify internal-flow regime	Reynolds Number
Friction factor	f = 64/Re (laminar); Colebrook for turbulent	Determine Darcy friction factor	Moody Chart
Major loss	hƒ = f(L/D)v²/(2g)	Straight-pipe friction loss	Darcy–Weisbach
Minor loss	hₘ = ΣK v²/(2g)	Fittings / valves / local geometry	Pipe Flow & Head Loss
Pressure ↔ head	Δp = γh	Convert pressure loss and head loss	Pipe Flow & Head Loss
Pump power	Pₕ = γQH = ρgQH	Convert hydraulic duty to power	Pump System
NPSH available	pₛ,abs/γ + vₛ²/2g − pᵥ/γ	Assess suction condition above vapor pressure	Pump NPSH

10 // Assumptions, units & reference notes

The formulas are simple; choosing the correct inputs and conventions is the real engineering work.

11 // Related fluid mechanics tools

Use the formula sheet as the entry point, then move to the full calculators for input handling, unit conversion and expanded assumptions.