Water is flowing through a d = 10 cm diameter water pipe at a rate of Q = 0.1 m^3/s. Now a diffuser with an outlet diameter of D = 20 cm is bolted to the pipe in order to slow down water, as shown in the figure. Disregarding frictional effects, determine the force exerted on the bolts due to the water flow, F_R.
The Correct Answer and Explanation is :
To determine the force exerted on the bolts due to the water flow, we apply the principles of fluid dynamics, specifically the conservation of mass and momentum.
Step 1: Apply the Continuity Equation
The continuity equation states that for an incompressible fluid:
[
A_1 V_1 = A_2 V_2
]
where:
- ( A_1 = \frac{\pi d^2}{4} ) is the inlet area,
- ( A_2 = \frac{\pi D^2}{4} ) is the outlet area,
- ( V_1 ) and ( V_2 ) are the inlet and outlet velocities, respectively,
- ( Q = 0.1 ) m³/s is the volume flow rate.
Calculate the inlet and outlet areas:
[
A_1 = \frac{\pi (0.1)^2}{4} = 7.85 \times 10^{-3} \text{ m}^2
]
[
A_2 = \frac{\pi (0.2)^2}{4} = 3.14 \times 10^{-2} \text{ m}^2
]
Calculate velocities:
[
V_1 = \frac{Q}{A_1} = \frac{0.1}{7.85 \times 10^{-3}} = 12.74 \text{ m/s}
]
[
V_2 = \frac{Q}{A_2} = \frac{0.1}{3.14 \times 10^{-2}} = 3.18 \text{ m/s}
]
Step 2: Apply Momentum Equation
The force exerted by the water flow on the diffuser can be found using the momentum equation:
[
F_R = \dot{m} (V_1 – V_2)
]
where mass flow rate:
[
\dot{m} = \rho Q = (1000)(0.1) = 100 \text{ kg/s}
]
[
F_R = 100 (12.74 – 3.18) = 956 \text{ N}
]
Thus, the force exerted on the bolts is 956 N in the direction of the initial flow.
Now, I will generate an image illustrating the water flow through the diffuser.
The image illustrates the water flow through the diffuser, showing the velocity decrease and the reaction force on the bolts. Let me know if you need any modifications or further explanations!
