L. David Roper

http://www.roperld.com/personal/RoperLDavid.htm

2 July, 2016

Bauxite is the most important ore that contains aluminum. The major extractors of bauxite are Australia, China and Brazil. The United States imports almost all bauxite it uses.

The graph below shows the bauxite extraction data for the world and double Verhulst function fits to the data in order to extrapolate into the future for two different amounts of eventual extraction.

Bauxite extraction rate for the world and two Verhulst-functions fits to the data.

The blue curve is obtained by restricting the total extraction to the amount already extracted plus the estimated reserve of 28x10^{9} tonnes; the redll curve is obtained by restricting the total extraction to the amount already extracted plus the estimated reserve base of 38x10^{9} tonnes.

It appears that the recent rapid rise in extraction rate is unsustainable for more that a few decades or so from now.

Taking an average extraction curve of the two fits, the crossover point at year ~2050 when the amount extracted is equal to the amount left to be extracted is shown here:

Aluminum is the second most recycled of all minerals after iron. Assume that:

- The bauxite extraction curve is the average of the two curves given above.
- Recycling of aluminum follows a hyperbolic tangent curve from 30% to 90% recycling with a break point of year 2000 and width 25 years.
- The recycling is delayed by a Gaussian curve peaking at a delay of 15 years and a width of 10 years.

The effective bauxite available for making items after the first ten recycling cycles is shown in the following graph, along with the effective bauxite available for each cycle:

The equation for a recyling cycle is

,

where E_{i} is the amount available from the previous cycle. Here is an example of the Excel coding:

{=(($J$2+$I$2)/2+(($J$2-$I$2)/2)*TANH((A27-$K$2)/$L$2))*SUM($I$27:I27*(EXP(-1*((A27-$A$27:A27-$N$2)/$O$2)^2/2))/$O$2/SQRT(2*PI()))} (The curly bracket surrounding the term makes it into an array; it must be entered by holding down the SHIFT & CTRL keys while pressing the ENTER key.)

Of course, the recycling could be extended to more cycles, skewing the curve further into the future. However, the peak and fall off after it will not change because further cycles are essentally zero in that time region.

Thus, under the assumptions given above, the effective amount of bauxite available for making items peaks at about 2035 and falls off rapidly after that. Humans will have taken concentrated bauxite deposits and scattered them across the surface of the earth.

The Excel spreadsheet is set up to make it easy to calculate with different recycling assumptions.

Dividing the bauxite recycling curve above by the projected world population (asymptote ~10^{10}), one gets:

The recycling peak and per-capita peak are both at year ~2045.