CELE Hydraulics & Fluid Mechanics — Orifices, Weirs, Tubes and NozzlesConcept Map

Concept

Orifices

Sub Concepts

• Sharp-edged orifice
• Coefficient of velocity (Cv ≈ 0.98)
• Coefficient of contraction (Cc ≈ 0.62)
• Coefficient of discharge (Cd = Cv × Cc)
• Vena contracta
• Submerged orifice
• Discharge formula: Q = Cd·A·√(2gh)

Relationship To Central

Primary method for flow control and discharge measurement through small openings under pressure head

Concept

Tubes and Nozzles

Sub Concepts

• Standard tube
• Re-entrant tube
• External tube
• Convergent nozzle
• Divergent nozzle
• Pressure recovery
• Jet formation

Relationship To Central

Modified orifice configurations that alter discharge characteristics and velocity profiles

Concept

Weirs

Sub Concepts

• Rectangular weir: Q = (2/3)·Cd·√(2g)·L·H^(3/2)
• Triangular (V-notch) weir: Q = (8/15)·Cd·√(2g)·tan(θ/2)·H^(5/2)
• Trapezoidal weir
• End contractions (Francis formula)
• Head over crest (H)
• Effective length reduction
• Applications in small vs large flows

Relationship To Central

Open-channel flow measurement devices using overflow over a notch or crest

Concept

Discharge Coefficients

Sub Concepts

• Coefficient of velocity (Cv)
• Coefficient of contraction (Cc)
• Coefficient of discharge (Cd)
• Typical values: 0.60–0.62 for orifices
• Variation with Reynolds number
• Turbulence effects
• Empirical determination

Relationship To Central

Empirical factors accounting for real-world losses in all flow devices

Concept

Fundamental Energy Principles

Sub Concepts

• Torricelli's theorem: v = √(2gh)
• Bernoulli equation application
• Pressure head conversion to velocity head
• Energy losses (friction, separation)
• Ideal vs actual velocity
• Kinetic energy of jet

Relationship To Central

Theoretical foundation for all velocity and discharge calculations

Concept

Tank Drainage and Time-to-Empty

Sub Concepts

• Differential equation formulation
• Variable head over time
• Integration of discharge equation
• Time formula: t = [2·As·(√h₁ − √h₂)] / (Cd·Ao·√(2g))
• Constant plan area tanks
• Non-linear drainage rate

Relationship To Central

Practical application of orifice discharge to transient flow problems

Concept

Common Exam Pitfalls and Design Considerations

Sub Concepts

• Head reference (orifice center vs weir crest)
• Exponent confusion (H^3/2 vs H^5/2)
• Neglecting Cd (using ideal formula)
• End contraction effects
• Submergence conditions
• Scale and accuracy limits
• Selection criteria for device type

Relationship To Central

Critical factors for correct problem-solving and engineering practice

To

Ideal Velocity

From

Torricelli Theorem

Strength

strong

Relationship

Fundamental theoretical basis: v = √(2gh) provides reference for all real velocity calculations

To

Coefficient of Contraction

From

Coefficient of Velocity

Strength

strong

Relationship

Combined multiplicatively: Cd = Cv × Cc; both reduce ideal discharge independently

To

Coefficient of Contraction

From

Vena Contracta

Strength

strong

Relationship

Physical phenomenon: jet contracts beyond orifice edge to minimum area, quantified by Cc ≈ 0.62

To

Tank Drainage

From

Orifice Discharge Formula

Strength

strong

Relationship

Tank time-to-empty integrates discharge formula over variable head: dh/dt = -Cd·Ao·√(2gh) / As

To

Triangular Weir

From

Rectangular Weir

Strength

strong

Relationship

Both measure overflow discharge but with different head exponents: H^(3/2) vs H^(5/2); selection depends on flow range

To

Orifice Center

From

Head Measurement

Strength

strong

Relationship

Orifice head must be measured from surface to orifice center, not surface-to-surface; critical distinction from weirs

To

Effective Length

From

End Contractions

Strength

moderate

Relationship

Francis formula: L' = L - 0.1nH reduces weir length by contraction effects at sides

To

Standard Tube

From

Sharp-edged Orifice

Strength

moderate

Relationship

Tubes provide higher Cd (0.80-0.85) than sharp orifices (0.60-0.62) by reducing flow separation

To

Discharge Coefficient

From

Convergent Nozzle

Strength

moderate

Relationship

Nozzles achieve Cd ≈ 0.95-0.98 through streamlined design minimizing losses

To

Surface-to-Surface Head

From

Submerged Orifice

Strength

strong

Relationship

Submerged discharge uses upstream-minus-downstream water surface difference as head

To

Square Root Time Formula

From

Variable Head Integration

Strength

strong

Relationship

Mathematical integration of dQ = -As·dh yields t = [2As(√h₁ - √h₂)] / [Cd·Ao·√(2g)]

To

V-notch Weir

From

Small Flow Measurement

Strength

moderate

Relationship

H^(5/2) exponent in triangular weir provides greater sensitivity (resolution) at low heads compared to rectangular H^(3/2)

To

Divergent Nozzle

From

Pressure Recovery

Strength

moderate

Relationship

Divergent section allows static pressure to recover, reducing dynamic pressure losses after convergence

To

Torricelli Theorem

From

Bernoulli Equation

Strength

strong

Relationship

Torricelli is special case of Bernoulli with atmospheric pressure at surface and exit, uniform pressure at depth

To

Discharge Coefficient Variation

From

Reynolds Number Effects

Strength

weak

Relationship

Cd varies slightly with Reynolds number; higher Re (turbulent) gives stable Cd ≈ 0.60-0.62 for sharp orifices