Question 12.6: The sluice gate in Fig. 12–15 is used to control the flow of......

Fluid Mechanics [1012629]

The sluice gate in Fig. 12–15 is used to control the flow of water from a large reservoir that has a depth of 6 m. If it opens into a channel that has a width of 4 m, determine the maximum flow that can occur through the channel and the associated depth of flow.

fig 12-15

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Fluid Description. The water surface in the reservoir is assumed to be at a constant elevation so we have steady flow. Also, the water is assumed to be an ideal fluid.
Analysis. Maximum flow occurs at the critical depth, which is determined from Eq. 12–14.

y_2  =  \frac{2}{3} y_1  =  \frac{2}{3}(6  m)  =  4  m

The flow at this depth is determined from Eq. 12–15.

Q_{max}  =  q_{max} b;             Q_{max} =  (\sqrt{\frac{8}{27} gy_1 ^3}) b

=  (\sqrt{\frac{8}{27} (9.81  m/s^2)(6  m)^3}) (4  m)

$= 100.23 m^3/s = 100 m^3/s$
As expected, the Froude number for this flow is

Fr  =  \frac{V_c}{\sqrt{gy_c}}  =  \frac{(100.23  m^3/s)/[4  m(4  m)]}{\sqrt{(9.81  m/s^2)(4  m)}}  =  1

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