7.1 Dimensionless variables - Water hammer pressure peak when closing a valve
A pressure peak
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
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
of the fluid and the fluid velocity
[m/s] before closing the valve.
Find two dimensionless combinations
. Can you from this guess the expression for the maximum
(which is obtained when the closing of the valve is infinitely fast) as a function of the other 3 variables ?
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
[kg/(m s)], Speed of pressure waves in fluid
[m/s], gravitational constant
[m/s^2] speed of object
[m/s], characteristic length of object
[m], fluid density
b) Show that you can construct the four dimensionless quantities mentioned in the book,
, Reynolds number =
, Mach number =
, Froude number=
(literature usually defines "Froude number
" 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
. (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 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
normal air pressure
10m water column
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
is proportional to
as a condition on the Froude number. What is the intuitive explanation to this limit speed ?
Some bonus material:
7.7 Hydralic jump phenomenon
Does the Froude number F_r=1 roughly predict the radius
for where the hydraulic jump occurs in the water in your water sink at home? Here
would be the (low) water depth [m] in the supercritical flow region near the center, and
its (radially outwards) water speed [m/s]. Use that flow rate Q at radius
] and guesstimate Q, d, r
Some related material: