ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Part  A:  1  pts  each,  10  pts  total,  no  partial  credit.  

1) (Correct: 1 pt/ Wrong: -3 pts). The sum of static, dynamic, and hydrostatic pressures is ___constant _____________ when flow is steady, irrotational, incompressible, and when frictional effects are negligible.

2) (Correct: 1 pt/ Wrong: -3 pts). A streamline cannot terminate or start in the middle of the flow unless it is at a ___stagnation_____________ point.

3) (Correct: 1 pt/ Wrong: -3 pts). If the flow is inviscid then the Reynolds number is equal to ___infinity _____________

4) ___ Dynamic _____________ pressure ρV 2/2 is the pressure rise when the fluid in motion is brought to ____stop (or to zero velocity)____________ isentropically.

5) __ Hydrostatic ______________ pressure ρgz accounts for the effects of the fluid weight on pressure acting on a surface.

6) The pressure head P/ρg is the __elevation (or height)______________ of a fluid column that produces the static pressure P.

7) The velocity head V 2/2 is the __elevation (or height)______________ needed for a fluid to reach the velocity V during frictionless free fall.

8) The linear momentum equation is obtained by setting the intensive property n=_ velocity (U)____ and thus the extensive property N=_____ mass x velocity (mU)________ in the Reynolds transport theorem.

9) (Correct: 1 pt/ Wrong: -3 pts). If a body moves with a Mach number of M=0.25 then the speed

of sound of the fluid is equal to ___4 times______ the speed of the body.

10) (Correct: 1 pt/ Wrong: -3 pts). Pathlines, streaklines and streamlines coincide only if the flow is __steady_____________

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Part  B:  3  pts  each,  15  pts  total,  no  partial  credit.   1) Helicopter A hovers above sea level and a similar helicopter B, carrying the same weight, hovers

above a mountain top at 1 mile altitude. Select the correct answer: a. Power of helicopter A is higher than power for helicopter B b. Power of helicopter B is higher than power for helicopter A c. Power of helicopter A is equal to the power for helicopter B because the lift the same weight d. Cannot be determined because we do not know the rotation speed of the blades

2) A flat plate is held stationary while a jet is impinging normal to its surface. If the velocity of the jet

increases by a factor of 2 then the force required to hold the plate will: a. Increase by a factor of 2 b. Increase by a factor of 4 c. Stay the same d. Decrease by a factor of 2 e. Decrease by a factor of 4

3) A water jet of velocity V impinges on a plate moving toward the water jet with velocity ½V. If the force required to hold the plate stationary is F, then how much is the force required to move the plate towards the jet in terms of F.

a. 2.25 F b. 1.50 F c. 1.25 F d. 1.0 F (Stay the same) e. 0.5 F

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

4) Water enters a centrifugal pump axially at a specified volume flow rate of 0.09 m3/s and velocity 5

m/s, and leaves in the normal direction along the pump casing. The area of the intake is 0.018 m2. Determine the force acting on the shaft in the axial direction. Density of water is 1000 kg/m3 .

a) 45 N

b) 90 N c) 450 N

d) 900 N e) 4500 N

5) Water from a storm drain flows onto a porous bed that absorbs the water with uniform vertical velocity of 8mm/s as shown in the figure below. The system is 5m deep into the page. Find the length L of the bed needed in order to completely absorb the storm water.

a. 5 m b. 10 m c. 50 m d. 100 m e. 150 m

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Part  C:  10  pts  total   The drag coefficient as a function of the Reynolds number, for flow over a sphere and flow over a smooth cylinder is shown in the figure below. The Reynolds number for both cases is defined based on the free stream velocity and the diameter.

1) Using this figure and only the curve for the smooth cylinder

a. Identify the approximate Reynolds number for which the flow on the cylinder becomes turbulent.

Answer: (1pts) The drastic reduction in drag coefficient indicates the transition to turbulent flow at Re~300,000

b. Identify the range of Reynolds numbers for which the drag is not a function of the Reynolds number.

Answer: (3pts) The almost constant drag between 1000< Re < 200,000

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

2) (2 pts) Draw the velocity profile at the point of separation for a turbulent flow over a smooth cylinder for a Reynolds number of Re = 1,000,000.

Re = 1,000,000 will the typical critical velocity profile with zero shear stress

3) (4 pts total) Draw the streamlines for a flow over a smooth cylinder for a Reynolds number of

Re=1,000,000. Show the streamlines for the vortices, the stagnation and the separation streamlines.

Part  IV  

1. Which  of  the  following  pump  curves  can  lead  to  an  unstable  operation?  Why?  (2  points)  

   Since  two  operating  points  are  possible      

2. Identify  (mark  ‘X’)  the  best  efficiency  operating  point  for  the  three  pumps  shown  in  the  figure  below.    (3  points)  

 3. A  gasoline  (specific  gravity  0.72)  pump  requires  P1  watts  of  power  at  a  certain  flow  rate.  

If  the  pump  is  now  used  to  pump  water  at  the  same  flow  rate  what  would  be  the  power  requirement?  Assume  that  the  pump  speed  is  the  same  in  both  cases.    (2  points)  

A. P1     B.      0.72  P1     C.  P1/0.72     D.  (0.72)2  P1      

Flow  rate  

Hea

d  

Flow rate

Hea

d

(A) (B)

   

4. Select  one  answer.  Consider  the  flow  through  a  pipe  as  shown  below.  The  minor  loss  for  case  1  would  be  greater  than  /  less  than  /  equal  to  the  minor  loss  for  case  2.    (2  points)  

   

 5. Water  flows  through  a  circular  pipe  of  length  L  and  diameter  D  at  a  certain  flow  rate.  If  

the  circular  pipe  is  replaced  with  a  square  duct  having  the  same  cross-­‐sectional  area,  the  major  loss  would  increase  /  decrease  /  stay  the  same.  Assume  flow  is  in  the  fully  rough  regime.  (2  points)    

6. Consider  viscous  flow  through  a  pipe  bend.  Draw  the  streamlines  and  identify  the  source  of  minor  loss.    (2  points)    

 

D 1.5D 2D D 2D Q Q

Case 1 Case 2

   

7. The  system  curve  for  a  pipe  system  is  as  shown  in  the  figure.  If  a  gate  valve  is  added  to  the  pipe  system,  draw  the  new  system  curve.    (2  points)    

   Problem  8  (15  points)  Pitch  (density  =  1100  kg/m3)  is  a  highly  viscous  fluid.  The  viscosity  of  pitch  can  be  estimated  using  data  from  the  famous  pitch  drop  experiment.  

 It  is  estimated  that  a  volume  V  =  4.7×10-­‐5  m3  of  pitch  drained  out  from  the  funnel  (dimensions  as  shown  in  the  figure)  over  a  period  of  582  months  (17708  days).  Estimate  the  viscosity  of  pitch.  Neglect  minor  losses.  Also  neglect  the  kinetic  energy  of  the  fluid.    Hint:  Assume  laminar  flow  and  check  the  validity  of  that  assumption  at  the  end.    Solution:    

Flow  rate  

Hea

d

 

System  curve  starts  at  the  same  point  but  gets  shifted  up  due  to  increase  in  minor  loss  

Assumptions:    1. Laminar  flow  2. Neglect  minor  losses  3. Neglect  kinetic  energy  of  fluid  

 

Apply  extended  Bernoulli’s  equation  (EBE)  from  1-­‐>2  2 2 2

1 1 2 21 1 2 22 2 2

p V p V l Vz z fg g g g d g

α αρ ρ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + − + + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠  

p1  =  p2  =  patm                            (2  points)    V1  =  V2  ≈  0    since  kinetic  energy  can  be  neglected            (2  points)  z1  =  h+l,  z2  =  0                      (2  points)    

For  laminar  flow    64 64Re

fVdµ

ρ= =                      (2  points)  

 

Substitute  into  EBE  2642l Vh l

Vd d gµ

ρ+ =  

Solve  for  µ  2

132gd hV l

ρµ ⎛ ⎞= +⎜ ⎟⎝ ⎠        (2  points)  

The  average  velocity  can  be  obtained  from  the  flow  rate  

2

4Q QVA dπ

= =  

4

1128gd hQ l

πρµ ⎛ ⎞⇒ = +⎜ ⎟⎝ ⎠  

The  flow  rate  can  be  obtained  using  the  data  provided  3

14 34.7e-5 m 3.07 10 /17708*3600*24

VQ m sT s

−= = = ×      (2  points)  

Plug  in  all  numbers    4(1100)(9.81)(0.0094) 2.91

(128)(3.07e-14) 7.59.3e7 Pa s

πµ ⎛ ⎞= +⎜ ⎟⎝ ⎠=

       (1  point)  

Check  for  Reynolds  number    4Re 4.9e-17Vd Qd

ρ ρµ µπ

= = =      Laminar  flow  assumption  OK!                          (2  points)  

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Problem 1 (3 pts)

1) Consider the two nozzle in figure. They both work between two environments at given pressure with air. Assume 1D flow and neglect both friction and heat transfer effects. How is the mass flow rate of nozzle a compared with the nozzle b? Nozzle A Nozzle B a) 𝑚! = 0.75  𝑚! b) 𝑚! = 𝑚! c) 𝑚! = 2  𝑚! d) 𝑚! =

!!!!!!!

𝑚!

p1 = 200kPa T1 = 100 °C

p1 = 200kPa T1 = 100 °C

p2 = 150kPa p2 = 150kPa

LB = 2 LA LA

1 m2 0.5 m2 1 m2 0.5 m2

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Problem 2 (11 pts) 2A (6 pts) For which exit pressure sonic conditions are reached for the flow inside this Convergent-Divergent nozzle operating with air? Assume 1D isentropic flow conditions.

a) Sonic condition cannot be established for any value of exit pressure b) Sonic condition are always established for any value of exit pressure c) For any outlet pressure below 97.3 kPa d) Only for a pressure between 4.7 kPa and 97.3 kPa e) Only below 4.7 kPa

P1 = 100 kPa A1 = 6 mm2

A2 = 6 mm2

Amin = 2 mm2

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

2B (5 pts) Represent realistic trends for the pressure and the Mach number inside the nozze for

A. p2 = 99 kPa; B. p2=97 kPa; C. p2=50 kPa; D. p2=1 kPa. Clearly indicate the curves A,B,C,D in the diagrams below

x xA1 xAmin xA2

p

100 kPa

50 kPa

0 kPa

x xA1 xAmin xA2

M

1

0

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Problem 3 (3 pts) The coupling in figure can be considered as a minor loss in a pipe flow problem. Since the minor loss coefficient k is not given, an experiment on a smaller model (1:5 geometry scale) is performed. Determine which input velocity you would use for the model case.

A. 0.6 m/s B. 6.71 m/s C. 15 m/s D. 75 m/s

3 m/s

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

Problem 4 (18 pts) A rigid tank of volume 10-5 m3 contains fluid with k = 1.2 initially at 700 kPa and 100 °C. At t = 0 s, a valve connected to the tank is opened. The valve can be assumed as nozzle with minimum area 10-7 m2. The pressure outside the tank is 100 kPa. Assuming isentropic flow conditions:

a) Determine the mach number at nozzle exit, at t = 0 s (3 pts) b) Determine the temperature of the fluid at the exit section of the nozzle,

at t = 0 s (2 pts) c) Determine the flow rate at nozzle exit, at t = 0 s (2 pts) d) If the temperature in the tank remains constant, what is the pressure in the tank after

10 s? (10 pts) e) What is the M number after 10 s? (1 pt)

0.1 m3 RIGID TANK k=1.2;

R=290 Nm/kgK 700kPa 100 °C

10-5 m2

Ambient 100 kPa 25 °C

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________

ME 309 – Fluid Mechanics LAST NAME: ___________________________

Spring 2014 FIRST NAME: ___________________________