Water is flowing vertically upward through a brass pipe. It then flows through a 90 degree elbow, changing direction from flowing vertically, to flowing horizontally at which point it eventually ejects to atmosphere in the horizontal direction (at the outlet). The entire brass pipe assembly is firmly fixed at its base to a wall. You may take the "inlet" to be at a point where the pipe attaches to the base plate and the water is still flowing vertically.Which of the following correctly expresses the vertical reaction force (Ry) at the base plate?Useful Info: The inlet velocity and pressure are Ui and Pi respectively. The outlet velocity and pressures are Uo and Po respectively. The inlet and outlet areas are Ai and Ao respectively.Group of answer choicesRy=-(density*(Ui^2)*Ai)-(Pi*Ai)None of the available optionsRy=(density)*(Uo^2)*Ao+(Pi*Ai)Ry=(density)*(Uo^2)*AiRy=(density)*(Uo)^2*Ao
Question
Water is flowing vertically upward through a brass pipe. It then flows through a 90 degree elbow, changing direction from flowing vertically, to flowing horizontally at which point it eventually ejects to atmosphere in the horizontal direction (at the outlet). The entire brass pipe assembly is firmly fixed at its base to a wall. You may take the "inlet" to be at a point where the pipe attaches to the base plate and the water is still flowing vertically.Which of the following correctly expresses the vertical reaction force (Ry) at the base plate?Useful Info: The inlet velocity and pressure are Ui and Pi respectively. The outlet velocity and pressures are Uo and Po respectively. The inlet and outlet areas are Ai and Ao respectively.Group of answer choicesRy=-(density*(Ui^2)Ai)-(PiAi)None of the available optionsRy=(density)(Uo^2)Ao+(PiAi)Ry=(density)(Uo^2)AiRy=(density)(Uo)^2*Ao
Solution
The vertical reaction force (Ry) at the base plate can be calculated using the principle of conservation of momentum. The momentum change in the vertical direction is due to the change in velocity of the water from the inlet to the outlet.
At the inlet, the momentum is the product of the mass flow rate and the inlet velocity, which is in the upward direction. The mass flow rate can be calculated as the product of the density, inlet velocity, and inlet area (densityUiAi). Therefore, the momentum at the inlet is (densityUi^2Ai).
At the outlet, the water is flowing horizontally, so there is no momentum in the vertical direction.
The pressure forces at the inlet and outlet also contribute to the vertical reaction force. The pressure force at the inlet is the product of the inlet pressure and the inlet area (Pi*Ai), and it acts in the upward direction. The pressure at the outlet does not contribute to the vertical reaction force because it acts in the horizontal direction.
Therefore, the vertical reaction force at the base plate is the sum of the momentum change and the pressure force at the inlet, both of which act in the upward direction. However, since the reaction force is in the opposite direction, we take the negative of this sum.
So, the correct expression for the vertical reaction force (Ry) at the base plate is:
Ry=-(density*(Ui^2)Ai)-(PiAi)
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