Explanation Given: The diameter of the water jet ( d ) is 10 mm. The range of flow rate is 0.5 × 10 − 3 m 3 /s ≤ Q ≤ 1 × 10 − 3 m 3 /s . The Increment in flow rate ( Δ Q ) is 0.1 × 10 − 3 m 3 /s . Calculation: The following assumptions are made in the problem: Assume the flow is steady, inviscid, and incompressible. The point A and B are open to atmosphere so the velocities at these points are zero. p A = p B = 0 The datum line pass through the point A so the elevation head at point A as zero ( z A = 0 ) and the elevation head at point B as h ( z B = h ) . Draw the control volume as shown in Figure (1). Calculate the area A A of the pipe at point A . A A = π 4 ( d ) 2 = π 4 ( 10 mm ) 2 = π 4 ( 10 mm × 1 m 1000 mm ) 2 = 0.0000785 m 2 Obtain the relationship between the velocity and the discharge of pipe. Q = A A V A Q = 0.0000785 V A V A = Q 0.0000785 m / s Calculate the velocity V B at point B by apply Bernoulli’s Equation between point A and B . p A γ w + V A 2 2 g + z A = p B γ w + V B 2 2 g + z B 0 9.81 kN / m 3 + ( Q 0.0000785 ) 2 2 ( 9.81 m / s 2 ) + 0 = 0 9.81 kN / m 3 + V B 2 2 ( 9.81 m / s 2 ) + h 162.278 × 10 6 Q 2 19.62 = 0.051 V B 2 + h 0.051 V B 2 = 162.278 × 10 6 Q 2 19.62 − h 19.62 ( 0.051 V B 2 ) = 162.278 × 10 6 Q 2 − 19.62 h V B 2 = 162.278 × 10 6 Q 2 − 19.62 h V B = 162.278 × 10 6 Q 2 − 19.62 h (I) Draw the free body diagram of the hemispherical bowl as in Figure (2). Write the expression for the Equation for linear momentum as follows, ∑ F = ∂ ∂ t ∫ c v V ρ w d V + ∫ c s V ρ w V ⋅ d A In diagram point B is the inlet of the control surfaces and C is outlet of the control surfaces. The velocity at point B and C is 162.278 × 10 6 Q 2 − 19.62 h . Apply linear momentum Equation along the y direction. ∑ F y = 0 + V B ρ w ( − Q B ) + ( − V C ) ρ w ( Q C ) − m g = V B ρ w ( Q ) + ( − V C ) ρ w ( Q ) (II) Substitute V C = V B in Equation (II)

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ISBN: 9780134649290

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 6, Problem 31P

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The height h of the bowl as a function of the volumetric flow Q of the water through the nozzle and Plot the height h versus flow rate Q.

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Chapter 6, Problem 30P

Chapter 6, Problem 32P

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The height h of the bowl as a function of the volumetric flow Q of the water through the nozzle and Plot the height h versus flow rate Q .

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The height h of the bowl as a function of the volumetric flow Q of the water through the nozzle and Plot the height h versus flow rate Q.

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Chapter 6 Solutions