Q1. (a) If the flow rate, Q, through an orifice depends on the diameter of the orifice D, the kinematic viscosity v (1.14x10-6 m/s2) and the product (gH), where H is the head above an orifice and g is the acceleration due to gravity, show by dimensional analysis that: Q = D2√(gH)Φ(D√(gH)/v) where Φ means a function.
(b) Use the experimental results (see table Q1) for the flow of water, Q, through a 12mm orifice, to determine the function Φ(D√(gH)/v) to 3 significant figures.
Hence derive the actual flow rate equation for this orifice and determine the head of water above the orifice needed to produce a flow rate of 8.2 l/s
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Head H(m)
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0.15
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0.3
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0.45
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0.6
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Flow Rate Q(l/s)
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4.98
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7.04
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8.63
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9.95
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Table Q1
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Q2. (a) The product (SFg), where St is the energy gradient (= hf/L) due to the flow of a fluid through a pipe and g is the acceleration due to gravity, is related to the velocity of flow (V), the diameter of the pipe (D), the kinematic viscosity (v) and the effective roughness (k). Show by dimensional analysis that:
SFgD/V2 = Φ[Re, k/D] where Φ means a function.
(b) The head loss over a 6.2m length of pipe, 125mm in diameter, was investigated at varying flow rates. Use the experimental results (see table Q2) to show that the function Φ[Re, k/D] is approximately constant to 2 significant figures. Hence find the flow rate that causes a head loss of 250mm.
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Discharge (l/s)
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4.4
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6.1
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7.9
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9.4
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11.3
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13.0
|
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Head Loss (mm)
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44
|
83
|
136
|
203
|
284
|
378
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Table Q2
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