Problem:
This should be a very simple problem, I just want to make sure I am doing it correctly. I have a box of sand. I am adding aluminum shavings to it. How do I determine the effective thermal conductivity of the mixture.

Known:
Density of sand and aluminum
Mass of sand and aluminum in box
K values of sand and aluminum

I have seen several methods online how to do this so I want someones expert advice on the correct way to figure out the theoretical effective thermal conductivity of the mixture.

It's not an area I deal with much, but you've been hanging now for a few days so I'll give it a shot if only to get shot down myself. I'd use a simple rule-of-mixtures approach. The bits of aluminium will conduct well and dry sand particles poorly, so I'd expect the 1/K for the combination 1/K_comp ~= Vf_alu/K_alu + Vf_sand/K_sand and K_comp to be closer to K_sand (after all, if the sand does not conduct heat at all K_comp will also be zero). So for 25% aluminium by volume (not mass) maybe K_comp = 1/(0.25/250 + 0.75/0.15) =0.2 W/(m.K). K values from from  http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html .

In 'Introduction to Composite materials Design' Barbero gives an alternative: K_comp = K_sand * (1+eta*Vf_alu)/(1-eta*Vf_alu) and eta = (K_alu/K_sand-1)/(K_alu/K_sand+1). For 250 and 0.15 and 25% Vf aluminium this gives K_comp = 0.25 W/(m.K). (At least, I think it does.)

I tend to trust Barbero, but I must admit this is my simplification of his formula for 2-direction or though-thickness conductivity of a long fibre composite.

My own googling turned up
ftp://ftp.deu.edu.tr/muh/turhan/public_html/papers_Turhan_Coban/Thermal_conductivity_2002.pdf
and
http://wvuscholar.wvu.edu:8881//exlibris/dtl/d3_1/apache_media/L2V4bGlicmlzL2R0bC9kM18xL2FwYWNoZV9tZWRpYS8yMDk0Nw==.pdf
so it looks like it might be quite a bit more complicated and slightly higher K_comp is possible, but for a reasonable real-world value 'close to the sand plus a bit' is probably good enough as an initial stab, and as usual trial and error with your actual application (whatever it is) is the most practical way forward. If it's important then measuring the K_comp would seem quite advisable (particularly if the sand can be wet), though it would quite awkward to do well. Let's hope someone with a more practical or aerothermal-type background is prompted to contribute.
 

viderde (Mechanical)
You also have this posted in heat transfer and thermodynamics engineering.
B.E.

The good engineer does not need to memorize every formula; he just needs to know where he can find them when he needs them.  Old professor