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

In recent times, the issue of a &ldquo;Phosphate Peak&rdquo; 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 &ldquo;Oil Peak&rdquo;. However, whereas the &ldquo;Oil Peak&rdquo; is supply driven, the &ldquo;Phosphate Peak&rdquo;, if at all, is demand driven. In contrast with oil, most of which is chemically transformed in CO<inf>2</inf>, vapor and other constituting elements, phosphate is not &ldquo;consumed&rdquo; 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 &ldquo;secondary liquid mine&rdquo; 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 &ldquo;secondary liquid mine&rdquo;. The elements are assumed to be present in given proportions (ratios) in this &ldquo;mine&rdquo; and the model shows that these ratios have an impact on the optimal extraction path and on resulting &ldquo;implicit&rdquo; shadow pricing rules to be adopted by the waste water treatment plant operator.