**Let:**

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• y**_{1} = headwater depth, i.e., flow depth upstream of sluice gate, under parallel flow.

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• y**_{2} = gate opening, or flow depth at sluice gate, under non-parallel flow.

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• y**_{3} = tailwater depth, i.e., flow depth downstream of sluice gate, under parallel flow.**
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****The conservation of energy between headwater and tailwater depths:**

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• y**_{1} + [v_{1}^{2}/(2g)] = y_{3} + [v_{3}^{2}/(2g)]

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****By conservation of mass:**

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• q = v**_{1}y_{1} = v_{3}y_{3}

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• y**_{1} + [q^{2}/(2gy_{1}^{2})] = y_{3} + [q^{2}/(2gy_{3}^{2})]

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• q = y**_{1} y_{3} [2g/(y_{1} + y_{3})]^{1/2}

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****To express this equation in terms of the
gate opening y**_{2},
a contraction coefficient is defined:

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• C**_{c} = y_{3}/y_{2}

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**

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• q = {C**_{c}/[1 + C_{c} (y_{2}/y_{1})]^{1/2}} (2gy_{1})^{1/2} y_{2}

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****Defining a discharge coefficient:**

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• C**_{d} = C_{c}/[1 + C_{c} (y_{2}/y_{1})]^{1/2}

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**

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• q = C**_{d} (2gy_{1})^{1/2} y_{2}

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****Defining a reference velocity:**

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• v**_{2} = C_{d} (2gy_{1})^{1/2}

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****The unit-width discharge under a rectangular sluice gate is:**

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**

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• q = v**_{2} y_{2}

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The contraction coefficient C**_{c} varies within a narrow range, between 0.598 and 0.611.
A value of C_{c} = 0.61 has been recommended for practical use
(Henderson: "Open channel flow," MacMillan, 1966).**
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