Mech280final_2022

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University of British Columbia *

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280

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Mechanical Engineering

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Dec 6, 2023

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20

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Mech 280, Final Exam: April 20 th 2022 Name: Student number: Exam duration 2.5 hours, closed book Instructions: 1. Check that you have 20 pages and 5 questions. Answer all questions. 2. 3 single sided letter-sized sheets of notes are permitted. 3. You may use a calculator. 4. No wireless communication devices are permitted. 5. Write your answers on the same sheet as the question. 6. Put your name and student number clearly on the top of each sheet. 7. Write neatly and structure your answers in a clear way. State physical assumptions that you make. 8. Underline or box your final calculated answer for ease 9. Live long and prosper Useful data: 1 ft = 0.3048 m; 1 inch = 0.0254 m P atm = 101.325 kPa g = 9.81 m/s 2 . Centroid and centroidal moments of inertia for some common shapes
Mech 280, Final Exam: April 20 th 2022 Name: Student number:
Mech 280, Final Exam: April 20 th 2022 Name: Student number: Question 1. (20 marks) The pressure rise, P = P 2 -P 1 , across the sudden expansion shown can be expressed as P = f(A 1, A 2 ,V 1 , ρ ) where A 1 and A 2 are the upstream and downstream cross-sectional areas, respectively, ρ is the fluid density, and V 1 is the upstream velocity. a. (7 marks) Express the above relationship in dimensional terms, using the Buckingham Pi theorem
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Mech 280, Final Exam: April 20 th 2022 Name: Student number: Question 1. continued b. (13 marks) Water is circulated from a large tank, through a filter, and back to the tank as shown. The power added to the water by the pump is 300W. Determine the Reynolds number Re in the pipe, and the flowrate Q through the filter. Take ρ =1000kg/m 3 and µ =1.3x10 -3 Pa.s Use the Moody chart and iterate at most 3 times 100m of 5cm diameter pipe with ε /D = 0.001
Mech 280, Final Exam: Winter 2022 Name: Student number:
Mech 280, Final Exam: Winter 2022 Name: Student number: Q uestion 2. (20 marks) Water (density ρ =1000 kg/m 3 ) is to be pumped from a large reservoir, open to the atmosphere, into a pressurised tank that 10 m higher than the surface of the reservoir. The tank pressure is 98.1 kPa gage pressure . The reservoir and tank are connected by 50 m of 10 cm diameter pipe, and 4 elbows each with loss coefficient K=1.25. A centrifugal pump with impeller of size 199 mm and rotational speed n = 2900 rpm, with pump chart shown below, is to be used. You may assume a friction factor is f=0.018 for this question. P atm = 101.325 kPa, P vapor = 2.3 kPa a. (8 marks) Find the characteristic equation of the system (h vs Q) and plot it on the pump chart to find the operating point of the pump. b. (4 marks) How much electrical power 𝑊𝑊 ̇ , is required for the motor that drives the pump? c. (3 marks) We would like to change the impeller diameter so that the pump works more efficiently. What is your suggestion? d. (5 marks) The pump is placed a distance x pump = 10 m from the reservoir, after 2 elbows. Is there a risk of cavitation if the pump is located 1m below the surface of the reservoir? If so, what do you advise? Explain inn sufficient detail.
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Mech 280, Final Exam: Winter 2022 Name: Student number:
Mech 280, Final Exam: Winter 2022 Name: Student number:
Mech 280, Final Exam: Winter 2022 Name: Student number:
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Mech 280, Final Exam: Winter 2022 Name: Student number: Q uestion 3. (20 marks) A two-dimensional velocity field is given by 𝐮𝐮 = 𝐴𝐴 ( 𝛼𝛼 ( 𝑥𝑥 2 − 𝑦𝑦 2 ), 6 𝑥𝑥𝑦𝑦 ) , for constants 𝐴𝐴 and 𝛼𝛼 , where ( 𝑥𝑥 , 𝑦𝑦 ) are measured in m and 𝐮𝐮 is measured in m/s. a. (3 marks) What are the dimensions of the constants 𝐴𝐴 and 𝛼𝛼 ? b. (7 marks) If the flow is incompressible and 𝐮𝐮 = (0,2) m/s at ( 𝑥𝑥 , 𝑦𝑦 ) = (1,1) m, find 𝐴𝐴 and 𝛼𝛼 . Find the stream function 𝜓𝜓 . c. (5 marks) Show that the flow is irrotational and find the velocity potential, 𝜙𝜙 . d. (5 marks) If the fluid flowing is a gas with density ρ =2 kg/m 3 and if the gage pressure P= 500 Pa at ( 𝑥𝑥 , 𝑦𝑦 ) = (1,1) m, what is the pressure at (5,3) m?
Mech 280, Final Exam: Winter 2022 Name: Student number:
Mech 280, Final Exam: Winter 2022 Name: Student number:
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Mech 280, Final Exam: Winter 2022 Name: Student number: Q uestion 4. (20 marks) Up to an altitude of 11km, the mean atmospheric temperature decreases approximately linearly, represented by: 𝑇𝑇 = 𝑇𝑇 0 − 𝐵𝐵𝐵𝐵 , where 𝑇𝑇 0 = 288.16 𝐾𝐾 , 𝐵𝐵 = 6.5 𝐾𝐾 / 𝑘𝑘𝑘𝑘 and 𝐵𝐵 is the height above sea level (in km). You may assume that 𝑃𝑃 = 𝜌𝜌𝜌𝜌𝑇𝑇 with 𝜌𝜌 = 287 𝑘𝑘 2 /( 𝐾𝐾𝑠𝑠 2 ) , that P atm = 101.325 kPa at 𝐵𝐵 = 0 , and that gravitational acceleration is constant. a. (5 marks) find the atmospheric pressure and density 𝜌𝜌 , of air, both as a function of 𝐵𝐵 b. (5 marks) A large hot air balloon of diameter 𝐷𝐷 = 20 𝑘𝑘 is at sea level. Suspended below the balloon is a basket containing a professor, an air heater and a large white dog. The total mass of the balloon and its load is 300 kg. Assuming the balloon remains spherical, of diameter 𝐷𝐷 , and that air pressure inside the balloon is the same as that outside, what balloon air temperature is needed for the balloon to rise? c. (6 marks) Find the balloon air temperature for which the buoyancy force remains constant, as a function of 𝐵𝐵 , and therefore what temperature is needed to fly at 2000m? d. (4 marks) The hot air balloon is simply a submarine, insofar as how it controls depth. Discuss whether you agree or not.
Mech 280, Final Exam: Winter 2022 Name: Student number:
Mech 280, Final Exam: Winter 2022 Name: Student number:
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Mech 280, Final Exam: Winter 2022 Name: Student number: Q uestion 5. (20 marks) Tanks 1 & 2 are filled with water (density ρ =1000 kg/m 3 ) and open to the atmosphere. Tank 1 has a rectangular gate of size 4cm deep x 1 cm wide, hinged at the top: position A. Tank 2 has a 1cm diameter hole with a plug. The centre of the hole is 4cm above the base of the tank. Tank 2 has cross-sectional area 100cm 2 . a. (6 marks) Calculate the following. (i) The horizontal force 𝐹𝐹 𝐴𝐴𝐴𝐴 , exerted on the gate AB due to the static pressure of the water in the tank. (ii) The moment 𝑀𝑀 𝐴𝐴 that the force 𝐹𝐹 𝐴𝐴𝐴𝐴 exerts about the hinge at A. (iii) The force 𝐹𝐹 𝐴𝐴 that would need to be applied at B in order to prevent the gate from opening? b. (5 marks) If h=30 cm and the plug is removed from the hole, what is the initial flow rate 𝑄𝑄 , out of the hole in tank 2? Clearly state the assumptions you make. c. (6 marks) Suppose that tank 2 is moved close to tank 1, so that the jet from the hole in tank 2 travels horizontally and impacts directly on the wall of tank 1. Calculate the force of the jet 𝐹𝐹 𝐽𝐽 , exerted on the wall of tank 1, as a function of h . If this force is assumed to act at 0.5cm above B, what is the minimal height h that will prevent the gate from opening outwards? d. (3 marks) Calculate the Reynolds number Re , of the jet. Discuss if this should affect your answers above? Note: a jet is typically laminar for Re < 100, and is fully turbulent for Re > 1000-2000, assume µ =1.3x10 -3 Pa.s
Mech 280, Final Exam: Winter 2022 Name: Student number:
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Mech 280, Final Exam: Winter 2022 Name: Student number:
Mech 280, Final Exam: Winter 2022 Name: Student number: The End