Bulk Modulus for Iron
I.N. Galidakis
I want to reduce the volume of a spherical mass of radius R_F of iron, by 50%.
This is related to the bulk modulus of Fe, K_{Fe} ~ 160GPa:
http://en.wikipedia.org/wiki/Bulk_modulus
To reduce the volume by 50% as in the example given on Wiki, I need to apply a
pressure of at least K_{Fe}/2 (Pa)
Assuming I am using TNT around the iron mass, TNT has an explosive strength of:
s = 2.72*10^6 J/kg
http://en.wikipedia.org/wiki/Explosive_strength
If the radius of the TNT around the iron mass is R_L, then the total volume of
TNT will be:
V_L = 4/3*Pi*(R_L^3 - R_F^3),
hence the total mass of TNT will be:
m_L = rho*V_L = rho*4/3*Pi*(R_L^3 - R_F^3), where rho=1650 kg/m^3 is the density
of TNT.
The total strength of the TNT will then be:
S = s * m_L (J = Pa*m^3)
The volume of the iron mass is:
V_F = 4/3*Pi*R_F^3,
therefore the following inequality must hold:
S/V_F >= K_{Fe}/2. Substituting the above equations into the last one, I get:
s*rho*(R_L^3 - R_F^3)/R_F^3 >= K/2,
and after substituting rho=1650kg/m^3, s=2.72*10^6, R_F=5/100m, K_{Fe}=160GPa, I
get:
R_L >= 0.1333201797m ~ 13.3 cm.
Assuming that the blast wave is properly focused, the above means that I only
need a layer of 13.3cm - 5cm = 8.3cm of TNT around my iron mass in order to
explosively reduce the volume of the mass by a factor of 50% ?!!!
I find this very hard to believe. Can anyone see what I am going wrong?
Many thanks,
--
Ioannis
Androcles
"I.N. Galidakis" <morp...@olympus.mons> wrote in message
news:1260731554.124788@athprx03...
Forget all the exotic blasting crap and grind it off as you would a ball
bearing.
Volume of a sphere is 4/3 pi.R^3, so for R = 1, V = 4/3pi
Half of that is 2/3 pi, so r = cube root (2/3.pi) = 0.806
0.806 * 13.3 = 10.7
You need to grind off 2.6 cm.
Where you are going wrong is making a simple problem complicated
and confusing yourself.
Uncle Al
What makes you think compressing a solid sphere to double its
equilibrium density follows Hooke's law all the way down? OTOH, to
reduce the volume by 50% you need reduce the radius by less than 21%
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm
Tim Little
> To reduce the volume by 50% as in the example given on Wiki, I need
> to apply a pressure of at least K_{Fe}/2 (Pa)
Yes, that's a minimum based on linearity. Linearity is only a good
approximation for very small volume changes. For larger changes the
pressure increases very much faster than linear.
On top of that you seem to be assuming adiabatic compression, so that
the temperature will increase significantly as the material is
compressed by that much. Some fraction of the energy of compression
will go into heating.
> I find this very hard to believe. Can anyone see what I am going
> wrong?
You started with "at least", but then assumed it was roughly equal.
- Tim
Bernhard Kuemel
> I need someone to verify my calculations below.
>
> I want to reduce the volume of a spherical mass of radius R_F of iron, by 50%.
Get a high security clearance and ask the people who compress plutonium.
Bernhard