Second-order cone optimization of the gradostat

Josh A. Taylor, Alain Rapaport

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


The gradostat consists of multiple chemostats interconnected by mass flows and diffusion. It has been used to model biochemical systems such as wastewater treatment networks and microbial activity in soil. In this paper we maximize the production of biogas in a gradostat at steady state. The physical decision variables are the water, substrate, and biomass entering each tank and the flows through the interconnecting pipes. Our main technical focus is the nonconvex constraint describing microbial growth. We formulate a relaxation and prove that it is exact when the gradostat is outflow connected, its system matrix is irreducible, and the growth rate satisfies a simple condition. The relaxation has second-order cone representations for the Monod and Contois growth rates. We extend the steady state models to the case of multiple time periods by replacing the derivatives with numerical approximations instead of setting them to zero. The resulting optimizations are second-order cone programs, which can be solved at large scales using standard industrial software.

Original languageEnglish (US)
Article number107347
JournalComputers and Chemical Engineering
StatePublished - Aug 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Chemical Engineering
  • Computer Science Applications


  • Biogas
  • Convex relaxation
  • Gradostat
  • Second-order cone programming
  • Wastewater treatment


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