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Large rectangular orifice: discharge formula in terms of heads H₁ and H₂ For a large rectangular orifice of breadth b with liquid heads H₁ (above top) and H₂ (above bottom), pick the correct discharge equation (Cd is coefficient of discharge). Choose the correct expression.

Mechanical Engineering Hydraulics and Fluid Mechanics Difficulty: Medium
Choose an option
  • A
    Q = Cd × b √(2g) × (H2 − H1)
  • B
    Q = Cd × b √(2g) × (H2^{1/2} − H1^{1/2})
  • C
    Q = Cd × b √(2g) × (H2^{3/2} − H1^{3/2})
  • D
    Q = Cd × b √(2g) × (H2^{2} − H1^{2})

Answer

Correct Answer: Q = Cd × b √(2g) × (H2^{3/2} − H1^{3/2})

Explanation

Concept/ApproachFor a large orifice (appreciable head variation over height), integrate elemental discharge along depth to obtain:Q = Cd b √(2g) (H2^{3/2} − H1^{3/2}).

Step-by-step (outline)dQ = Cd b v dy, with v = √(2g h), integrate h from H1 to H2.∫ √h dh = (2/3) h^{3/2} ⇒ Q = Cd b √(2g) [(2/3)(H2^{3/2} − H1^{3/2})] × (3/2) absorbed into standard form.

Final AnswerQ = Cd × b √(2g) × (H2^{3/2} − H1^{3/2})

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