# Bismuth Depletion Including Recycling

L. David Roper

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

2 July, 2016

Bismuth is used in cosmetics, pigments and pharmaceuticals.. The graph below shows the bismuth extraction data for the world and several Verhulst functions fit to the data in order to extrapolate into the future.

Bismuth extraction rate for the world and several Verhulst functions fit to the data.

The red curve is obtained by doing a best fit of two Verhulst functions to the the extraction data.

The crossover point at year ~2014 when the amount extracted is equal to the amount left to be extracted is shown here:

## Recycling

Assume that:

- Recycling of bismuth follows a hyperbolic tangent curve from 10% to 60% recycling with a break point of year 1990 and width 50 years, beginning in year 1970.
- The recycling is delayed by a Gaussian curve peaking at a delay of 25 years and a width of 10 years.

The effective bismuth available for making items after the first ten recycling cycles is shown in the following graph, along with the effective beryllium 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 beryllium available for making items peaked at about year 1975 and falls off rapidly after that. Humans will have taken concentrated beryllium 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