CELE Hydraulics & Fluid Mechanics — Orifices, Weirs, Tubes and NozzlesCheat Sheet

Formula

v = C_v√(2gh)

Meaning

v = actual velocity (m/s); C_v = coefficient of velocity (~0.98); g = 9.81 m/s²; h = head to orifice center (m)

Watch Out

h is measured to the CENTER of the orifice, not the top or bottom edge. This is the single most common mistake.

When To Use

Finding jet velocity from an orifice under constant head

Formula

Q = C_d × A × √(2gh)

Meaning

Q = discharge (m³/s); C_d = coefficient of discharge (~0.60–0.62); A = orifice area (m²); h = head (m)

Watch Out

Do NOT use ideal velocity √(2gh) without C_d; also check whether the orifice is sharp-edged (C_d ≈ 0.62) or rounded (C_d higher, ~0.97)

When To Use

ANY orifice discharge calculation — this is the fundamental formula

Formula

C_d = C_v × C_c

Meaning

C_d = discharge coefficient; C_v = velocity coefficient (~0.98); C_c = contraction coefficient (~0.62)

Watch Out

The product is NOT linear; C_d is always less than either C_v or C_c alone

When To Use

Decomposing C_d into its physical components; rarely asked directly but helps conceptual understanding

Formula

Q_submerged = C_d × A × √(2g(h₁ − h₂))

Meaning

h₁ = upstream head; h₂ = downstream head; the effective head is their difference

Watch Out

The head is the DIFFERENCE, not absolute. If both surfaces are at the same level, h = 0 and Q = 0.

When To Use

Orifice discharging under water (both sides submerged) — e.g., drainage between two tanks

Value

0.60–0.62

Symbol

C_d

Quantity

Coefficient of discharge (sharp-edged orifice)

Value

0.95–0.98

Symbol

C_v

Quantity

Coefficient of velocity

Value

0.60–0.65

Symbol

C_c

Quantity

Coefficient of contraction

Value

9.81 m/s²

Symbol

g

Quantity

Standard gravitational acceleration

Term

Orifice

Example

A hole in the bottom of a water tank.

Definition

Small opening in a tank or pipe wall through which fluid discharges under pressure; typically sharp-edged or rounded.

Term

Coefficient of Velocity (C_v)

Example

C_v ≈ 0.98 for sharp-edged orifices.

Definition

Ratio of actual to ideal velocity; accounts for friction losses (typically 0.95–0.98).

Term

Coefficient of Contraction (C_c)

Example

C_c ≈ 0.62 for sharp-edged orifices.

Definition

Ratio of jet area at vena contracta to orifice area; typically 0.60–0.65 due to streamline convergence.

Term

Vena Contracta

Example

Water exiting a faucet narrows as it falls.

Definition

Narrowest section of the discharged jet, located about 0.5 orifice diameters downstream; jet area < orifice area.

Term

Submerged Orifice

Example

Drainage from one tank into another.

Definition

Orifice discharging into a fluid at rest (or a different head) on the downstream side.

Formula

Q = (2/3) × C_d × √(2g) × L × H^(3/2)

Meaning

Q = discharge (m³/s); C_d ≈ 0.62; L = weir length (m); H = head over crest (m); note EXPONENT IS 3/2

Watch Out

EXPONENT 3/2, NOT 2 or 1. Also, H is measured VERTICALLY from the weir crest to free surface, not to weir center.

When To Use

Rectangular weir under steady flow (most common weir type)

Formula

L' = L − 0.1nH (Francis formula)

Meaning

L' = effective length (m); L = actual length (m); n = number of end contractions (typically 0, 1, or 2); H = head (m)

Watch Out

If weir spans full channel width, n = 0 and L' = L. If side contractions present, n > 0 and L' < L.

When To Use

Accounting for end contractions (when weir does not extend full width of channel)

Formula

Q_corrected = (2/3) × C_d × √(2g) × L' × H^(3/2)

Meaning

Use L' instead of L to account for end contractions

Watch Out

Neglecting contractions overstates discharge; Francis formula reduces effective length by 0.1nH.

When To Use

When end contractions are significant

Value

0.58–0.62

Symbol

C_d

Quantity

Coefficient of discharge (rectangular weir)

Value

4.429

Symbol

√(2g)

Quantity

√(2g)

Term

Weir

Example

Dam spillway or channel gauge.

Definition

Overflow notch in a channel crest used to measure or control discharge; flow is in free fall over the crest.

Term

Head (H)

Example

If water level is 0.5 m above the weir crest, H = 0.5 m.

Meaning

Vertical distance from weir crest to upstream water surface.

Definition

Vertical distance from weir crest to upstream water surface (NOT to the center of any opening).

Term

End Contractions

Example

A weir 1 m wide in a 1.5 m channel has 2 end contractions.

Definition

Reduction in effective weir length due to side flows converging; each side loss reduces effective length.

Term

Velocity of Approach

Example

May be significant in narrow channels; correction adds term √(v²/2g) to H.

Definition

Upstream channel velocity; neglected in basic weir formula if channel is wide or velocity is small (< 0.3 m/s).

Formula

Q = (8/15) × C_d × √(2g) × tan(θ/2) × H^(5/2)

Meaning

Q = discharge (m³/s); θ = apex angle (degrees or radians); H = head (m); note EXPONENT IS 5/2

Watch Out

EXPONENT 5/2, NOT 3/2. Also, tan(θ/2) is critical; for 90° notch, tan(45°) = 1.

When To Use

Triangular V-notch weir for small discharge measurement

Formula

Q_90° = (8/15) × C_d × √(2g) × H^(5/2) = 1.383 × C_d × H^(5/2)

Meaning

Special case for 90° V-notch (θ = 90°, tan(45°) = 1)

Watch Out

Memorize 8/15 × √(2g) ≈ 1.383 for quick 90° calculations.

When To Use

Standard 90° V-notch weir (most common in practice)

Formula

Q_60° = (8/15) × C_d × √(2g) × tan(30°) × H^(5/2) = 0.798 × C_d × H^(5/2)

Meaning

Special case for 60° V-notch (θ = 60°, tan(30°) ≈ 0.577)

Watch Out

tan(30°) ≈ 0.577, not 0.5.

When To Use

60° V-notch weir (less common but appears on exams)

Value

1.383 × C_d

Symbol

8/15 × √(2g) × 1

Quantity

90° V-notch coefficient

Value

0.55–0.60

Symbol

C_d

Quantity

Coefficient of discharge (V-notch)

Value

0.577

Symbol

tan(θ/2) for 60° notch

Quantity

tan(30°)

Term

V-Notch (Triangular) Weir

Example

Laboratory discharge measurement or stream gauging.

Definition

Weir with triangular opening (90°, 60°, 45°, etc.) used for measuring small flows with high precision.

Term

Apex Angle (θ)

Example

90° V-notch: full right-angle opening.

Definition

Angle at the vertex of the V-notch; 90° is standard; smaller angles (45°, 60°) sharpen the notch.

Formula

Q = C_d × A_exit × √(2gh)

Meaning

Same orifice formula; apply with tube/nozzle exit area and corrected C_d

Watch Out

Use EXIT area (not throat area for convergent nozzles). C_d varies: standard tube ≈ 0.82, re-entrant ≈ 0.51, convergent nozzle ≈ 0.98.

When To Use

Flow through tubes (standard, re-entrant, diverging) or nozzles

Formula

A_jet = C_c × A_orifice (for short tubes)

Meaning

Jet area at vena contracta; for tubes, jet may recover to full tube area or remain contracted

Watch Out

Inside a tube, the jet may re-expand; the contraction coefficient is modified by tube geometry.

When To Use

Analyzing flow inside tubes

Value

0.60–0.62

Symbol

C_d

Quantity

Sharp-edged orifice

Value

0.80–0.82

Symbol

C_d

Quantity

Short (standard) tube

Value

0.50–0.51

Symbol

C_d

Quantity

Re-entrant tube

Value

0.97–0.99

Symbol

C_d

Quantity

Convergent nozzle

Term

Short Tube (Pipe Orifice)

Example

Drainage pipe protruding slightly through a tank wall.

Definition

Cylindrical tube of length ~2–3 diameters discharging from a tank; less contraction than sharp-edged orifice (C_d ≈ 0.82).

Term

Re-Entrant Tube (Borda Tube)

Example

Internal suction pipe in a reservoir.

Definition

Tube that projects inward into the tank; vena contracta forms inside the tube, reducing discharge (C_d ≈ 0.51).

Term

Convergent Nozzle

Example

Fire hose nozzle or jet engine intake.

Definition

Tapered nozzle that accelerates flow; exit velocity is higher than orifice velocity, minimal contraction (C_d ≈ 0.98).

Term

Divergent (Diffuser)

Example

Draft tube in a turbine.

Definition

Nozzle that expands downstream; increases pressure and decreases velocity (low C_d, high pressure recovery).

Formula

t = (2 × A_s × (√h₁ − √h₂)) / (C_d × A_o × √(2g))

Meaning

t = time (s); A_s = tank plan area (m²); h₁ = initial head (m); h₂ = final head (m); A_o = orifice area (m²)

Watch Out

The formula uses (√h₁ − √h₂), NOT (h₁ − h₂). Square roots are inside; this is derived from integration of Q = C_d A_o √(2gh).

When To Use

Calculating drainage time from one water level to another

Formula

t_total = (2 × A_s) / (C_d × A_o × √(2g)) × √h₁

Meaning

Special case: draining from h₁ to 0 (h₂ = 0, √h₂ = 0)

Watch Out

h₂ = 0 when the tank is fully drained (water level reaches orifice).

When To Use

Emptying entire tank

Term

Plan Area (A_s)

Example

A 3 m × 4 m tank: A_s = 12 m².

Definition

Horizontal cross-sectional area of the tank (assumed constant); for rectangular tanks, A_s = length × width.

Term

Head (h)

Example

If orifice is at depth 2 m and water surface is at height 5 m, h = 5 − 2 = 3 m.

Definition

Vertical distance from water surface to orifice center at any instant; h decreases as tank drains.

Note

Integration of this differential equation yields the tank drainage formula with √h terms.

Equation

dh/dt = −(C_d × A_o × √(2gh)) / A_s

Conditions

Continuity equation: inflow (zero) = outflow; orifice discharge reduces tank level instantaneously

Value

9.81 m/s²

Symbol

g

Quantity

Gravitational acceleration (SI)

Value

4.429

Symbol

√(2g)

Quantity

√(2g)

Value

2.953

Symbol

Rectangular weir coefficient

Quantity

2√(2g)/3

Values

• A√(2gh)
• H^(1/2)
• Orifice area A
• 0.60–0.62
• Tank discharge, point measurement
• Q = C_d A √(2gh)

Property

Orifice

Values

• (2/3)L√(2g)H^(3/2)
• H^(3/2)
• Weir length L
• 0.58–0.62
• Channel flow measurement, spillways
• Q = (2/3) C_d √(2g) L H^(3/2)

Property

Rectangular Weir

Values

• (8/15)√(2g)H^(5/2)
• H^(5/2)
• tan(θ/2) = 1
• 0.55–0.60
• Small flow measurement, streams
• Q = (8/15) C_d √(2g) H^(5/2)

Property

90° V-Notch Weir

Values

• (8/15)√(2g)tan(30°)H^(5/2)
• H^(5/2)
• tan(30°) ≈ 0.577
• 0.55–0.60
• Alternative to 90°; narrower notch
• Q = (8/15) C_d √(2g) × 0.577 × H^(5/2)

Property

60° V-Notch Weir

Values

• Hole with sharp (unwelded) edges; vena contracta forms outside
• 0.60–0.62
• Vena contracta @ 0.5D downstream; standard in practice
• Default unless stated otherwise; use 0.61 if not given

Property

Sharp-Edged

Values

• Smooth, curved inlet; minimal contraction
• 0.97–0.99
• Nearly ideal discharge; used for precision
• Rare on board exams; clarifies concept of contraction

Property

Rounded (Bell-Mouth)

Values

• Tube projects INTO the tank; vena contracta forms INSIDE
• 0.50–0.51
• Smallest C_d; internal jet forms before exit
• Often used as contrast to sharp-edged; tests understanding

Property

Re-Entrant (Borda)

Values

• Cylindrical pipe ~2–3D long; moderate contraction recovery
• 0.80–0.82
• Discharge higher than sharp-edged; jet may re-expand inside
• Common in real systems; bridges orifice & nozzle concepts

Property

Standard Short Tube

Values

• Orifices are point openings; head should be measured to CENTER by Torricelli's law
• Always measure h from water surface to orifice CENTROID (for circular orifice, its center)
• Measuring to top → h too large → Q overestimated by ~5–10%

Property

Using h to orifice TOP instead of CENTER

Values

• Ideal formula is theoretical; students sometimes skip empirical coefficient
• ALWAYS include C_d: Q = C_d A√(2gh). Ideal is reference only.
• Omitting C_d (≈0.61) → Q overestimated by 62%!

Property

Forgetting C_d entirely; using Q = A√(2gh)

Values

• Rectangular weir has H^(3/2); V-notch has H^(5/2). Easy to mix up.
• MEMORIZE: Rectangular (straight-sided) = 3/2; Triangular (V-notch) = 5/2
• Using wrong exponent → completely wrong answer (off by 50–100% depending on H)

Property

Confusing H^(3/2) vs. H^(5/2) exponents

Values

• End contractions reduce effective weir length; students sometimes ignore them
• Use L' = L − 0.1nH (n = number of contractions, typically 0, 1, or 2)
• Ignoring contractions → Q overestimated; L' ≈ 0.9L to 0.8L for full contraction

Property

Not accounting for end contractions on weirs

Values

• Drainage formula has square roots; students sometimes linearize incorrectly
• t = 2A_s(√h₁ − √h₂) / (C_d A_o √(2g)). Square roots INSIDE numerator.
• Using h₁ − h₂ → t massively underestimated (example: h₁ = 4, h₂ = 1: √4 − √1 = 1, but 4 − 1 = 3)

Property

Using (h₁ − h₂) instead of (√h₁ − √h₂) for tank drainage

Values

• Orifice formula uses area A; weir formula uses length L. Easy confusion in multi-part problems.
• Orifice: Q = C_d A√(2gh); Weir: Q = (2/3) C_d √(2g) L H^(3/2). Check units (area vs. length).
• Mixing units → dimensional analysis fails; answer nonsensical

Property

Mixing up orifice area A and weir length L

Values

• If tank level drops, h decreases → Q decreases. Not constant.
• For steady-state orifice: assume constant pump input. For draining tank: use integration (time formula).
• Assuming h constant → underestimates time to empty tank

Property

Assuming constant head h in orifice discharge

Values

• Each device has a different C_d range; students may use generic value
• Check orifice sharpness (0.62), weir type (0.60), nozzle (0.98), tube (0.82)
• Using 0.62 for nozzle (should be 0.98) → underestimates discharge by 36%

Property

Using wrong coefficient C_d for device type

Values

• The constant √(2g) appears in nearly all formulas; students sometimes omit or miscalculate
• MEMORIZE: √(2g) ≈ 4.429. Also 2√(2g)/3 ≈ 2.953 for rectangular weirs.
• Omitting √(2g) → Q off by factor of 4.4

Property

Forgetting √(2g) ≈ 4.429 in weir formula

Values

• Nappe may cling to downstream face (suppressed) or fall free (free); changes C_d
• Free nappe (standard): C_d ≈ 0.60–0.62. Suppressed: C_d ≈ 0.70–0.75 (higher). Problem must state.
• Using C_d = 0.60 for suppressed weir (C_d actually ≈0.72) → underestimates Q by 20%

Property

Not checking if weir is suppressed or contracted