Orifices, Weirs, Tubes and Nozzles - Cheat Sheet
Your last-minute revision companion for discharge through openings, weirs, and orifices. Master the coefficients, exponents, and key distinctions that distinguish board-exam questions.
ORIFICES — Discharge Through Openings
Formulas
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
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
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
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
Common values
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
Important facts
- Sharp-edged orifice: C_d ≈ 0.62 (standard in exams unless otherwise stated)
- Rounded orifice: C_d ≈ 0.97 (much less contraction)
- Re-entrant (Borda) orifice: C_d ≈ 0.51 (smallest discharge)
- Ideal discharge formula (C_d = 1.0): Q = A√(2gh) — used for reference only
- Vena contracta forms ~0.5D downstream (D = orifice diameter)
- The discharge coefficient is EMPIRICAL and depends on orifice shape, Reynolds number, and surface roughness
Key definitions
Orifice
Example
A hole in the bottom of a water tank.
Small opening in a tank or pipe wall through which fluid discharges under pressure; typically sharp-edged or rounded.
Coefficient of Velocity (C_v)
Example
C_v ≈ 0.98 for sharp-edged orifices.
Ratio of actual to ideal velocity; accounts for friction losses (typically 0.95–0.98).
Coefficient of Contraction (C_c)
Example
C_c ≈ 0.62 for sharp-edged orifices.
Ratio of jet area at vena contracta to orifice area; typically 0.60–0.65 due to streamline convergence.
Vena Contracta
Example
Water exiting a faucet narrows as it falls.
Narrowest section of the discharged jet, located about 0.5 orifice diameters downstream; jet area < orifice area.
Submerged Orifice
Example
Drainage from one tank into another.
Orifice discharging into a fluid at rest (or a different head) on the downstream side.
Diagrams to know
- Orifice cross-section showing head h to center, vena contracta downstream
- Comparison of sharp-edged vs. rounded orifices (C_d difference)
- Free jet trajectory from orifice (parabolic path)
RECTANGULAR WEIRS — Overflow Notches
Formulas
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)
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)
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
Common values
Value
0.58–0.62
Symbol
C_d
Quantity
Coefficient of discharge (rectangular weir)
Value
4.429
Symbol
√(2g)
Quantity
√(2g)
Important facts
- Rectangular weir: Q ∝ H^(3/2) — discharge is proportional to head to the 3/2 power
- C_d ranges 0.58–0.62 depending on weir sharpness and upstream conditions
- √(2g) ≈ 4.429 (memorize for quick calculations)
- Effective length reduces by 0.1nH for each end contraction
- Nappe (falling water sheet) should break free; if it clings to the downstream face, velocity coefficients increase
- For precise discharge, consider velocity of approach: H_eff = H + v²/(2g)
Key definitions
Weir
Example
Dam spillway or channel gauge.
Overflow notch in a channel crest used to measure or control discharge; flow is in free fall over the crest.
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.
Vertical distance from weir crest to upstream water surface (NOT to the center of any opening).
End Contractions
Example
A weir 1 m wide in a 1.5 m channel has 2 end contractions.
Reduction in effective weir length due to side flows converging; each side loss reduces effective length.
Velocity of Approach
Example
May be significant in narrow channels; correction adds term √(v²/2g) to H.
Upstream channel velocity; neglected in basic weir formula if channel is wide or velocity is small (< 0.3 m/s).
Diagrams to know
- Rectangular weir elevation: L, H, nappe profile
- End contractions reducing effective length (top view)
- Flow over weir crest with nappe ventilation
TRIANGULAR (V-NOTCH) WEIRS — Precision Measurement
Formulas
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
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)
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)
Common values
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°)
Important facts
- V-notch: Q ∝ H^(5/2) — much steeper than rectangular (H^3/2), so small head changes yield large discharge changes → excellent for low-flow measurement
- 90° V-notch is standard in exams unless specified otherwise
- tan(45°) = 1, tan(30°) ≈ 0.577, tan(22.5°) ≈ 0.414
- C_d typically 0.55–0.60 for V-notches
- The H^(5/2) exponent provides better resolution at low flows compared to rectangular weirs
- V-notches are preferred for streams with low, variable discharge
Key definitions
V-Notch (Triangular) Weir
Example
Laboratory discharge measurement or stream gauging.
Weir with triangular opening (90°, 60°, 45°, etc.) used for measuring small flows with high precision.
Apex Angle (θ)
Example
90° V-notch: full right-angle opening.
Angle at the vertex of the V-notch; 90° is standard; smaller angles (45°, 60°) sharpen the notch.
Diagrams to know
- V-notch geometry showing apex angle θ and head H
- 90° V-notch vs. 60° V-notch profile comparison
- Discharge curve Q vs. H for V-notch (steeper slope than rectangular)
TUBES & NOZZLES — Modified Discharge
Formulas
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
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
Common values
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
Important facts
- Standard short tube: C_d ≈ 0.82 (higher than sharp-edged orifice, lower than convergent nozzle)
- Re-entrant tube: C_d ≈ 0.51 (smallest discharge due to internal vena contracta)
- Convergent nozzle: C_d ≈ 0.98–0.99 (nearly ideal; minimal losses)
- Nozzle exit area is always SMALLER than tank cross-section (nozzle throat)
- For nozzles, the formula applies with the OUTLET area, not throat area
- Jet velocity from nozzle is proportional to √h (same Torricelli law, but C_d modifies the constant)
Key definitions
Short Tube (Pipe Orifice)
Example
Drainage pipe protruding slightly through a tank wall.
Cylindrical tube of length ~2–3 diameters discharging from a tank; less contraction than sharp-edged orifice (C_d ≈ 0.82).
Re-Entrant Tube (Borda Tube)
Example
Internal suction pipe in a reservoir.
Tube that projects inward into the tank; vena contracta forms inside the tube, reducing discharge (C_d ≈ 0.51).
Convergent Nozzle
Example
Fire hose nozzle or jet engine intake.
Tapered nozzle that accelerates flow; exit velocity is higher than orifice velocity, minimal contraction (C_d ≈ 0.98).
Divergent (Diffuser)
Example
Draft tube in a turbine.
Nozzle that expands downstream; increases pressure and decreases velocity (low C_d, high pressure recovery).
Diagrams to know
- Sharp-edged orifice vs. short tube vena contracta location
- Re-entrant tube internal jet pattern
- Convergent nozzle acceleration profile
- Divergent nozzle (diffuser) expansion and pressure recovery
TIME TO EMPTY A TANK
Formulas
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
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
Important facts
- Time to empty depends on √h, not h; doubling the head does NOT double the drainage time
- Smaller orifice area (A_o) → longer drainage time (inverse relationship)
- Larger tank area (A_s) → longer drainage time (direct relationship)
- Higher C_d → faster drainage (e.g., C_d = 0.62 vs. 0.51 makes significant difference)
- Formula assumes constant tank area and orifice remains fully submerged throughout (h₂ ≥ orifice depth)
Key definitions
Plan Area (A_s)
Example
A 3 m × 4 m tank: A_s = 12 m².
Horizontal cross-sectional area of the tank (assumed constant); for rectangular tanks, A_s = length × width.
Head (h)
Example
If orifice is at depth 2 m and water surface is at height 5 m, h = 5 − 2 = 3 m.
Vertical distance from water surface to orifice center at any instant; h decreases as tank drains.
Diagrams to know
- Tank with orifice at bottom; head h decreasing over time
- Head vs. time graph (nonlinear curve, square-root relationship)
- Orifice location relative to tank bottom
Reactions or equations
dh/dt = −(C_d × A_o × √(2gh)) / A_s
Note
Integration of this differential equation yields the tank drainage formula with √h terms.
Conditions
Continuity equation: inflow (zero) = outflow; orifice discharge reduces tank level instantaneously
COEFFICIENT VALUES & RANGES — Quick Reference
Common values
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
Important facts
- C_d = 0.60–0.62 is the default for sharp-edged orifices (use 0.61 as midpoint if not specified)
- Rounded orifices: C_d ≈ 0.97–0.99 (approach ideal conditions)
- Rectangular weir: C_d ≈ 0.58–0.62 (typically 0.60 unless suppressed nappe)
- V-notch weir: C_d ≈ 0.55–0.60 (slightly lower than rectangular)
- Short tube (standard): C_d ≈ 0.80–0.82
- Re-entrant (Borda) tube: C_d ≈ 0.50–0.51 (smallest; internal vena contracta)
- Convergent nozzle: C_d ≈ 0.97–0.99 (nearly ideal)
- All C_d values are EMPIRICAL and depend on Reynolds number, surface roughness, and upstream conditions
Must remember
- Q = C_d A √(2gh) is the fundamental orifice discharge formula; C_d ≈ 0.61 for sharp-edged orifices unless otherwise stated.
- Head h is ALWAYS measured to the orifice CENTER (or weir CREST for weirs), not top or bottom edges.
- Rectangular weir has H^(3/2) exponent; V-notch weir has H^(5/2) exponent — confusing these guarantees exam failure.
- End contractions reduce effective weir length: L' = L − 0.1nH (n = number of contractions; typically 0, 1, or 2).
- Tank drainage uses √h terms: t = 2A_s(√h₁ − √h₂) / (C_d A_o √(2g)), NOT linear (h₁ − h₂) — this is the #1 algebra mistake.
- C_d varies by device: orifice 0.61, short tube 0.82, re-entrant 0.51, nozzle 0.98 — using wrong coefficient invalidates answer.
- V-notch weir (Q ∝ H^(5/2)) is far more sensitive to head changes than rectangular weir (Q ∝ H^(3/2)) — choose V-notch for small, variable flows.
- √(2g) ≈ 4.429 appears in every discharge formula; memorize this constant and 2√(2g)/3 ≈ 2.953 for weirs.
- Submerged orifice: h = (upstream level) − (downstream level); if both levels equal, Q = 0 (no flow).
- Vena contracta forms ~0.5 orifice diameters downstream for sharp-edged orifices; in re-entrant tubes, it forms inside the tube.
Last minute tips
- ALWAYS CHECK: Is the orifice submerged or free-discharging? Is the weir suppressed or free? Read the problem carefully; these details change coefficients.
- UNIT CHECK: Verify your final answer has correct units [m³/s for Q, s for time]. If units fail, the formula or substitution is wrong.
- MEMORY TRICK: Rectangular weir = 3/2 (sounds like 'weir has 3 sides' loosely); V-notch = 5/2 (steeper power for sharper notch). This helps avoid the most common swap.
- COEFFICIENT QUICK LOOKUP: 0.6 ≈ sharp orifice, 0.8 ≈ tube, 0.98 ≈ nozzle, 0.51 ≈ Borda (re-entrant, smallest). If not given, assume 0.60–0.62 for orifices and 0.58–0.60 for weirs.
- EXPONENT ALERT: Any formula with SQUARE ROOT (H^1/2) belongs to a point orifice; any with H^3/2 is rectangular weir; any with H^5/2 is V-notch. Identify the exponent first, then choose formula.
Comparison tables
ORIFICE vs. WEIR vs. V-NOTCH — Discharge Formula Comparison
| Device | Formula (without C_d) | Head Exponent | Area Term | C_d Range | Best Use | Key Formula |
|---|---|---|---|---|---|---|
| Orifice | A√(2gh) | H^(1/2) | Orifice area A | 0.60–0.62 | Tank discharge, point measurement | Q = C_d A √(2gh) |
| Rectangular Weir | (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) |
| 90° V-Notch Weir | (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) |
| 60° V-Notch Weir | (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) |
ORIFICE TYPES & COEFFICIENT OF DISCHARGE
| Orifice Type | Description | C_d | Key Characteristic | When It Appears on Exams |
|---|---|---|---|---|
| Sharp-Edged | 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 |
| Rounded (Bell-Mouth) | Smooth, curved inlet; minimal contraction | 0.97–0.99 | Nearly ideal discharge; used for precision | Rare on board exams; clarifies concept of contraction |
| Re-Entrant (Borda) | 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 |
| Standard Short Tube | 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 |
COMMON EXAM MISTAKES & HOW TO AVOID THEM
| Mistake | Why It Happens | Correct Approach | Example Impact |
|---|---|---|---|
| Using h to orifice TOP instead of CENTER | 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% |
| Forgetting C_d entirely; using Q = A√(2gh) | 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%! |
| Confusing H^(3/2) vs. H^(5/2) exponents | 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) |
| Not accounting for end contractions on weirs | 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 |
| Using (h₁ − h₂) instead of (√h₁ − √h₂) for tank drainage | 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) |
| Mixing up orifice area A and weir length L | 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 |
| Assuming constant head h in orifice discharge | 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 |
| Using wrong coefficient C_d for device type | 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% |
| Forgetting √(2g) ≈ 4.429 in weir formula | 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 |
| Not checking if weir is suppressed or contracted | 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% |
Diagrams
Flowchart: Which Formula to Use?
Decision flowchart to identify which discharge formula to apply based on device type (orifice, rectangular weir, V-notch weir) and flow conditions (submerged, free, head measurement).
Hydraulic Discharge: Core Concepts & Relationships
Mind map showing the hierarchy of hydraulic discharge concepts: orifices (with types and coefficients), weirs (rectangular vs. V-notch), tubes and nozzles, tank drainage, and key constants. Helps students see relationships and organize study material.
Coefficient of Discharge Distribution by Device Type
Pie chart showing typical coefficient of discharge values for various hydraulic devices. Heights represent C_d percentages; helps students quickly recall which device has which coefficient and see relative differences.