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Soluion%20to%20Control%20Volume%20Problem%20Set

Soluion%20to%20Control%20Volume%20Problem%20Set - Note The...

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Note: The soluitions to this problem set were provided by Dr. Shelplak. Problem 1 Given: A jet of water issuing into a moving cart, declined at an angle θ as shown. Find: 1. Draw the appropriate control volume, explicity stating unit normal vetors and flux areas. 2. List yourassumptions required to solve this problem. 3. What are the pressure boundary conditions on your control volume. 4. What is the time rate of change of the liquid level? 5. What is the thrust force acting on the car by the jet of water? 6. What is the frictional force acting on the car? Schematic: Basic Eqns: Integral form of the continuity equation: 0 d V t ρ = ( 29 CV CS V dA ρ + u uu v v Integral form of the momentum equation: ( 29 CV CS F VdV V V dA t ρ ρ Σ = + u u u u uu v v v v v Assumptions: a) Incompressible flow. b) Inviscid flow. c) Flow in cart has no velocity relative to CV. d) Uniform jet area and velocity. 1 HW 3
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e) Neglect body forces. Solution: 1. The control volume and normal area vectors are shown in the schematic. 2. The proper assumptions are listed above. 3. Pressure Boundary conditions: CV P Atm 4. The components of the velocity jet U in the control volume are ˆ ˆ cos sin jet jet jet U U U i U j θ θ = - + - Thus by mass continuity, ( 29 ( 29 2 2 0 sin 4 4sin jet dh t D d U dt π π ρ ρ θ θ = - - Which can be rearranged to ( 29 2 2 jet dh t d U dt D = 5. The momentum equation in the x direction becomes CV F udV t ρ - = ( 29 ( 29 c CS u V dA ρ + u uu v v Then the frictional force is ( 29 ( 29 2 cos sin 4sin jet jet d F U U U π ρ θ θ θ = - - - 2 HW 3
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The thrust force T acting on the car is the component of the force due to the jet in the x direction and will be balanced by the frictional force F since the car is moving at constant speed U . Thus, ( 29 2 cos 4 jet jet d T U U U ρπ θ = - 6. The frictional force is found from part (d) to be equal and opposite to the thrust force, ( 29 2 cos 4 jet jet d F U U U ρπ θ =- - 3 HW 3
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Problem 2 Given: A vane with a turning angle θ attached to a block which is sliding on a thin oil and water from a constant area nozzle flows over it as shown below Find: 1. List yourassumptions required to solve this problem. 2. Find an algebraic expression for A 2 (draw control volume). 3. Determined F x by assuming a linear velocity profile between the sliding block and the solid surface as a function of U T . 4. Determined the algebraic expression of U T . Schematic: Basic Eqns: Integral form of the continuity equation: 0 d V t ρ = ( 29 CV CS V dA ρ + u uu v v Integral form of the momentum equation: ( 29 CV CS F VdV V V dA t ρ ρ Σ = + u u u u uu v v v v v Assumptions: a) Incompressible flow. b) Inviscid flow. c) Steady d) Same velocity at the entrance and the exit of the vane e) Neglect body forces f) Constant pressure 4 HW 3
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Solution: 1. The proper assumptions are listed above.
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