A Model for the Optimal Recovery of Multiple Substances from Waste Water with a Focus on Phosphate

In recent times, the issue of a "Phosphate Peak" has entered the academic debate and it is widely present in social media. Arguments in favour and against are similar to those mentioned in the much earlier debate on the "Oil Peak". However, whereas the "Oil Peak" is supply driven, the "Phosphate Peak", if at all, is demand driven. In contrast with oil, most of which is chemically transformed in CO2, vapor and other constituting elements, phosphate is not "consumed" during its primary use as a fertilizer. Hence, whilst phosphate rock, from which phosphate is mined, is a depletable resource, phosphate itself is potentially recyclable and re-usable. Research on the technologies for such a recovery is manifold and, specifically, efforts are spent on waste water as a source of phosphate. This, if successful, could lead to a changing perception of waste water treatment plants as a set of backstop technologies to eliminate an environmental problem into a "secondary liquid mine" from which phosphate, but not only phosphate, could be extracted for re-use. Hence, for that purpose, an economic model of efficient extraction of phosphate and other elements from waste water in a waste water treatment plant could give guidance to operators. This paper presents such a model describing the optimal simultaneous extraction of several elements, including phosphate, from a "secondary liquid mine". The elements are assumed to be present in given proportions (ratios) in this "mine" and the model shows that these ratios have an impact on the optimal extraction path and on resulting "implicit" shadow pricing rules to be adopted by the waste water treatment plant operator.