Download Fluid Dynamics Quiz 2 Practice Problems and more Exams Fluid Dynamics in PDF only on Docsity!
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
2.06 Fluid Dynamics
Practice Problems for Quiz 2, Spring Term 2013
Problem 1
A volcano erupts. A constant-density lava flows down the volcano’s slope at an angle θ = 45 o^. The steady volumetric flow rate is known to be 300 m3/s at point A (elevation: 3.5 km). The lava spreads as it flows down the slope, covering an angle α=10o, as shown on the figure. In steady- state, what is the average velocity vB of the lava at point B if the height of the lava flow at B is 10 cm?
v B =
α = 10o
B
3.5 km
2 km
θ = 45o
1 km
A
A
B
Side View Top View
v out =
v out
v in
θ = 20
o
(Not to scale)
t =
Problem 2 : Surfboard In a motorized surfboard design, water (ρ= 1000 kg/m3) enters the horizontal board at average velocity v =5 m/s at a nominal anglein θ=20 oand through an area (perpendicular to vin) of A (^) in =0.01 m2. The water is pumped through the board and ultimately exists at high speed v (^) out. What is the required mass flow rate and constant average velocity v (^) out of the exit jet if the total drag forces along the board is to be F=1500 N?
Page 4 of 6
ng an angle α=10o, as shown on the figure. In lava at point B if the height of the lava flow = 1000 kg/m^3 ) 5 m/s at a icular to v in ) of rd and ultimately flow rate and tal forward sum
http://www.memagazine.org
th thermal-fluid systems. She asks you to
ch that it is rare in the middle (T=60 oC). You =0.20 m. The roast is initially at uniform maintained at air in the oven is = 0.5 W/m-K,
SHORT QUESTIONS
v B =
α = 10o A
o (Not to scale)
t =
Pa
B
at B is 10 cm?
- ( 3 points ) In a motorized surfboard design, water (ρ= 1000 kg/m^3 ) enters the horizontal board at average velocity v in =5 m/s at a nominal angle θ=20o^ and through an area (perpendicular to v in ) of Ain =0.01 m 2. The water is pumped through the board and ultimately exists at high speed v out. What is the required mass flow rate and constant average velocity v out of the exit jet if the total forward sum of forces along the board is to be F=1500 N?
http://www.mem
11. ( 3 points ) A good friend knows you are familiar with thermal-fluid systems. She asks y
compute the time t required to cook a long roast such that it is rare in the middle (T= approximate the roast as a slab of total thickness R =0.20 m. The roast is initially at unif temperature of Ti = 10 oC and is inserted in an oven maintained at 165 oC. The convection heat transfer coefficient of air in the oven is h = 200 W/m 2 -K. The properties of the roast are: k = 0.5 W/m-K, c= 3000 J/kg-K, ρ= 1200 kg/m 3.
THIS IS THE END OF THE SHORT QUESTIONS
v B =
α = 10o B 3.5 km 2 km θ = 45o 1 km A A B Side View Top View v out = v out v in θ = 20
o (Not to scale)
t =
Problem 4 : Oil Spill from an Oil Tanker
tankers can result in oil spills that can have catastrophic effects on sea life and require expensive cleanup operations. Some recent examples include spills near New Zealand and South Korea from tanker accidents as shown here. The goal of the problem is to determine the flow rate of the leaking oil in terms of measurable parameters of the oil films that form.
Consider oil leaking out of a crack in an oil tanker at sea as shown in the cross section schematic below. The crack has a width w into the page and is located at a depth H from the surface. The density of the seawater ρ w is greater than that of the oil ρ o, causing the oil to rise along the wall of the tanker, up to the ocean’s surface. The leaked oil forms a thin film along the wall of the tanker, which is inclined at an angle θ from the horizontal. The thickness of the oil film is h ( << H ) and the viscosity of the oil is μ o. Assume that the viscosity of the sea water μ (^) w is much smaller than that of the oil, i.e. μ (^) w << μ (^) o. You can also assume that the flow in the oil film is fully-developed. Note that the velocity at the oil-water interface is not zero.
Accidents involving breach of the structure of oil
Considering the steady state oil flow in the inclined film of thickness h ,
a). Provide the velocity and pressure boundary conditions for the oil flow in the inclined film, from the crack to the free surface ( Hint: four boundary conditions in total ).
∂ p
b). Determine the pressure gradient for this oil flow in the inclined film, in terms of the
∂ x
parameters of the problem.
c). Derive an expression for the velocity of this oil flow and sketch the velocity profile, in terms of the parameters of the problem.
d). Derive an expression for the volumetric oil flow rate Q per unit length of the ship and in terms of the parameters of the problem.
e). What is the velocity of the water next to the oil film, i.e. at y = h , in terms of the parameters of the problem?
