Exercise 7

  • Due No Due Date
  • Points None
7.1 Dimensionless variables - Water hammer pressure peak when closing a valve
A pressure peak LaTeX: p can often be heard when closing a valve abruptly, stopping a fluid flow in a pipe.  We can  therefore assume it depends on propagation of pressure waves in the fluid, i.e. the speed  of sound LaTeX: v_{sound} of the fluid (340m/s for air, about 4-5 times faster in water). Our intuition might also say that it depends on the density LaTeX: \rho LaTeX: [kg/m^3]of the fluid and the fluid velocity LaTeX: v [m/s] before closing the valve.
Find two  dimensionless combinations LaTeX: \Pi_1 and LaTeX: \Pi_2 of LaTeX: p,LaTeX: v_{sound},LaTeX: \rho,LaTeX: v. Can you from this guess the expression for the maximum LaTeX: p (which is obtained when the closing of the valve is infinitely fast) as a function of the other 3 variables ?
Check your result against the Joukowski formula, described in https://www.youtube.com/watch?v=GfCCsdHKrF8
7.2 Dimensionless variables - Resistance when moving through air or water
a) Construct the dimension matrix A of size 3*7 when we consider the variables
Drag force LaTeX: F [N], Viscosity LaTeX: \mu [kg/(m s)], Speed of pressure waves in fluid LaTeX: v_{sound} [m/s],  gravitational constant LaTeX: g[m/s^2] speed of object LaTeX: v [m/s], characteristic length of object LaTeX: d [m], fluid density LaTeX: \rho[kg/m^3].
b) Show that you can construct the four dimensionless quantities mentioned in the book,
Pressure coefficient=LaTeX: P_c = \frac{F}{\rho v^2d^2}, Reynolds number = LaTeX: R_e = \frac{\rho vd}{\mu}, Mach number = LaTeX: M_m = \frac{v}{v_{sound}},  Froude number= LaTeX: \tilde F_r = \frac{v^2}{gd}
(literature usually defines "Froude number LaTeX: F_r" as the square root  of this expression...)
Note: The typical form of the function P_c=f(R_e) for the drag force of a sphere inside a fluid (for situations where M_m and F_r will not influence the result) can be found on wikipedia here: https://en.wikipedia.org/wiki/Drag_coefficient

7.3 Flow in a Tube and Reynolds numbers

 

a) At roughly what water flow rate v [m/s] would the water in your garden hose be laminar? Say diameter d=0.01 meter, and say R_e<1000.

b) What is approximately the Reynolds number when you use the water hose ?
c) Same question when you drink water through a straw ?
d) What about when you breathe air through a straw ?
Hint: You might find these links useful
Googling gives that max exhale pressure is around 50mbar (female) about double for men.
Water at room temperature has (dynamic) viscosity around LaTeX: \mu = 0.001 \textrm{ Pa s}. (Air has  about 20 times lower dynamic viscosity).
7.4 Energy intuition
Moving 100kg 1meter vertically requires 1kJoule = 1000Ws = 1000Nm. We also have 1kWh = 3.6MJoule.
Order the energy of the following according to size, and guess roughly how many kJ they correspond to
  • Heating 1kg water 100 degrees
  • Melting 1kg ice (0degree ice -> 0 degree water)
  • Standard 9V battery
  • Energy in 2mF capacitor charged to 230V
  • 1000kg car moving at 50km/h
  • 1 liter of car petrol
  • 1kg chocolate used as food LaTeX: \sim 5000kcal
  • Your daily electric energy consumption
  • A fully charged Tesla electric car
  • Daily consumption of food
  • 1 second usage of ESS

Also construct some rules of of thumb for heat energy:

  • For water, how many meters upwards movement does the energy in a 1 degree temperature increase correspond to ?
  • How many meters up of a 100kg body does daily consumption of food correspond to ?
7.5 About Pressure
SI Units: 1 Pa = 1N/m^2 and 1atm LaTeX: \approx1bar LaTeX: \approx normal air pressure LaTeX: \approx 10m water columnLaTeX: \approxLaTeX: 10^5Pa
Find out (Google) What is the typical water pressure at your home? Is it smaller, larger, or about the same as the air pressure in a bike tire? Roughly what pressure can you generate by your lungs? What is approximately the pressure difference inside a balloon compared to surrounding. What is the pressure under your foot if you stand on one leg ?
 
* 7.6 Froude number
A rule of thumb for sailing boats is that a longer boat can sail faster and that speed LaTeX: v_{hull} is proportional to
LaTeX: \sqrt{L}. Have a look at https://en.wikipedia.org/wiki/Hull_speed and formulate the
expression LaTeX: v_{hull}={\sqrt {L_{WL}\cdot g \over 2\pi }} as a condition on the Froude number. What is the intuitive explanation to this limit speed ?
Some bonus material:
7.7 Hydralic jump phenomenon
sink.png
Does the Froude number F_r=1 roughly predict the radius LaTeX: r for where the hydraulic jump occurs in the water in your water sink at home? Here LaTeX: d would be the (low)  water depth [m] in the supercritical flow region near the center, and LaTeX: v\left(r\right)  its (radially outwards) water speed [m/s]. Use that flow rate Q at radius LaTeX: rwould be LaTeX: Q = d 2\pi r v(r) [LaTeX: m^3/s] and guesstimate Q, d, r
Some related material:

 

Solutions

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