# Vanadium Depletion Including Recycling

L. David Roper

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

2 July, 2016

Vanadium is used in making steel. The graph below shows the vanadium extraction data for the world and two Verhulst function fits to the data in order to extrapolate into the future.

Vanadium extraction rate for the world and two Verhulst function fits to the data.

The red curve is obtained by restricting the total extraction to the amount already extracted plus the estimated reserve of ~14x10^{6} tonnes; the blue curve is obtained by restricting the total extraction to the amount already extracted plus the estimated reserve base of ~38x10^{6} 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 ~2009 when the amount extracted is equal to the amount left to be extracted is shown here:

## Recycling

Assume that:

- The vanadium extraction curve is the average of the two curves given above.
- Recycling of vanadium follows a hyperbolic tangent curve from 10% to 60% recycling with a break point of year 2020 and width 25 years, beginning in year 2000.
- The recycling is delayed by a Gaussian curve peaking at a delay of 25 years and a width of 10 years.

The effective vanadium available for making items after the first ten recycling cycles is shown in the following graph, along with the effective vanadium 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.)

Thus, under the assumptions given above, the effective amount of vanadium available for making items peaks at about year 2040 and falls off rapidly after that. Humans will have taken concentrated vanadium 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.

Minerals Depletion