Mech 280, Final Exam: April 24th 2020 Name: Student number: Instructions: 1. Check you have 16 pages 2. The exam is 2 hours long and has 4 questions. Answer all questions 3. Put your name and student number on the top of each sheet. 4. Please write neatly and structure your answers in a clear way. 5. Write your answer on the same sheet, do not use the back of the sheet. 6. At the end of the exam, upload your exam paper with solutions to Canvas within

2.5 hours, preferably in .pdf format Integrity pledge: I affirm that I will not give or receive any unauthorized help on this examination, that all work will be my own, and that I will abide by University of British Columbia policies and any special rules for conduct set out by the examiner.

I understand that any violation of University rules or those set out by the examiner will be investigated as a case of academic misconduct, and the usual consequences and penalties as set out in the UBC Academic Calendar will apply.

The rules that apply to this exam are as follows: • A non-programmable scientific calculator may be used • You may not use any electronic software • You may consult your notes and/or a course text • You may not communicate or collaborate with any other person

[ ] I agree to the above Integrity Pledge [ ] I understand that violation of the pledge will result in an academic misconduct investigation

Signature:

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number: Question 1: A pump is to be selected from the series in the pump chart on page 2 of this exam. You want to pump water through 50m of 4 inch diameter cast iron pipe (roughness 0.2 mm) from Tank 1 to Tank 2. Both tanks are at atmospheric pressure and have large surface area. Upstream of the pump the pipe enters Tank 1 at 1m below the surface (entry loss KL = 0.8) and runs horizontally for 15m through 2 x 90 degree long radius flanged bends before reaching the pump. After the pump the flow passes through a flanged globe valve and 1 x 90 and 2 x 45 degreee long radius flanged bends before exiting into Tank 2 (exit loss KL = 1.05). The surface of Tank 2 is 5m above that of Tank 1.

a) (3 marks) Above approximately what flow rate would the flow be in the fully rough regime? If the pump is to deliver a flow rate of Q =159 m3/hr, compute the friction factor from the Moody chart.

b) (3 marks) Using loss coefficients from the chart on page 2, write down expressions in terms of the flow rate Q for: (i) the minor losses; (ii) the major losses; (iii) the system curve hpump(Q).

c) (5 marks) Assume now a friction factor f=0.025. If Q =159 m3/hr, calculate the total pump head required. Discuss which impeller diameter would you choose? What is the shut off head required and how does this compare to that of the selected pump?

d) (4 marks) Calculate the actual NPSH and NPSHR for the above setup. Is there a risk of cavitation?

Assumptions/data: Assume ρ=1000kg/m3 and µ=10-3Pa.s. Atmospheric pressure 101.325 kPa and the vapour pressure of water = 1 kPa. 1 inch = 25.4mm

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number: Question 2:

(a) (5 marks) Consider the two-dimensional velocity u = (𝑢𝑢,𝑣𝑣) = (−𝐵𝐵𝐵𝐵,𝐵𝐵𝐵𝐵), where B is a constant.

i. If this flow possesses a stream function, find it. ii. If this flow has a velocity potential, find it

iii. Compute the vorticity and describe what the flow might represent.

(b) (10 marks) A 2D incompressible flow has stream function Ψ = (3𝐵𝐵𝐵𝐵2 − 𝐵𝐵3). i. Find the velocity: u = (𝑢𝑢,𝑣𝑣), and the

velocity potential. ii. This describes the flow within a “wedge”,

as illustrated. What is the angle 𝛼𝛼? Hint: a wall in a 2D potential flow is a streamline

iii. If the pressure 𝑝𝑝(𝐵𝐵,𝐵𝐵) = 𝑝𝑝0 at (0,0) and the fluid has constant density 𝜌𝜌 = 1, what is the pressure at (0.5,0.5)?

Explain any assumptions you make

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number: Question 3: Water flows from the reservoir through the penstock (length 𝐿𝐿, diameter 𝐷𝐷) and exits through the jet of diameter𝐷𝐷𝑗𝑗. The jet strikes the impulse wheel, of radius 𝑅𝑅, which rotates at angular velocity Ω. Assume that the full water flow rate 𝑄𝑄 hits the wheel and the power P imparted to the wheel is given by:

𝑃𝑃 = 2𝜌𝜌𝑄𝑄Ω𝑅𝑅�𝑉𝑉𝑗𝑗 − Ω𝑅𝑅�

(a) (3 marks) For fixed 𝑉𝑉𝑗𝑗, what rotation rate maximizes the power output? (b) (6 marks) Use the energy equation, accounting for the frictional losses in the penstock, but

neglecting all minor losses, to show that

𝑉𝑉𝑗𝑗2 = 2𝑔𝑔𝑔𝑔 − 𝑓𝑓𝐿𝐿𝐷𝐷 �

𝐷𝐷𝑗𝑗𝐷𝐷�

4

𝑉𝑉𝑗𝑗2 and hence that

𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 =𝜌𝜌𝜌𝜌𝐷𝐷𝑗𝑗2

8 𝑉𝑉𝑗𝑗 �2𝑔𝑔𝑔𝑔 − 𝑓𝑓𝐿𝐿𝐷𝐷 �

𝐷𝐷𝑗𝑗𝐷𝐷�

4

𝑉𝑉𝑗𝑗2�

(c) (6 marks) Assuming constant friction factor, show that the maximum power is generated when ℎ𝑓𝑓 = 𝑔𝑔/3 and the optimum jet velocity is then �4𝑔𝑔𝑔𝑔/3. What is the jet diameter that produces this velocity?

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number: Question 4: a) (7 marks) A simple flow-measurement device for streams and channels is a notch, of angle 𝛼𝛼,

cut into the side of a dam, as shown. The volume flow Q depends only on 𝛼𝛼, the acceleration of gravity g, and the height 𝛿𝛿 of the upstream water surface above the notch vertex. Tests of a model notch, of angle 𝛼𝛼 = 55°, yield the following flow rate data: 𝛿𝛿 (cm) 10 20 30 40 Q (m3/hr) 8 47 126 263

i. Find a dimensionless expression to describe the data

ii. Use your relation to predict the flow rate of a notch for which 𝛿𝛿 = 3.2m, also with angle 𝛼𝛼 = 55°.

Mech 280, Final Exam: April 24th 2020 Name: Student number:

Mech 280, Final Exam: April 24th 2020 Name: Student number: b) (8 marks) You drop a Timbit in a cup of Tim Hortons coffee. It floats with 50% of the Timbit

above the surface of the coffee. A short while later the sugar coating dissolves and the Timbit becomes saturated with coffee. Now only 10% of the Timbit is above the surface.

i. Estimate the density of the dry Timbit ii. What volume fraction of the dry Timbit is air and what fraction dough?

iii. Estimate the density of the dough

Assumptions: Assume TH coffee has the same density as water (and tastes like it - only joking), and that the sugar coating is so thin that its mass can be neglected. You should assume that the dry Timbit consists of solid dough (fraction φ, density ρd) and air (fraction (1-φ) and consider the air density negligible). Assume only the air fraction may be replaced by coffee.

Mech 280, Final Exam: April 24th 2020 Name: Student number:

The End