A multiscale mathematical model describing the genesis and ecology of algalbacterial photogranules and the metals biosorption on their solid matrix within a sequencing batch reactor (SBR) is presented. The granular biofilm is modelled as a spherical free boundary domain with radial symmetry and a vanishing initial value. The free boundary evolution is governed by an ordinary differential equation (ODE) accounting for microbial growth, attachment and detachment phenomena. The model is based on systems of partial differential equations (PDEs) derived from mass conservation principles. Specifically, two systems of nonlinear hyperbolic PDEs model the growth of attached species and the dynamics of free adsorption sites; and two systems of quasi-linear parabolic PDEs govern the diffusive transport and conversion of nutrients and metals. The model is completed with systems of impulsive ordinary differential equations (IDEs) describing the evolution of dissolved substrates, metals, and planktonic and detached biomasses within the granular-based SBR. All main phenomena involved in the process are considered in the mathematical model. Moreover, the dual effect of metal presence on the formation process of photogranules is accounted: metal stimulates the production of EPS by sessile species and negatively affects the metabolic activities of microbial species. To describe the effects related to metal presence, a stimulation term for EPS production and an inhibition term for metal are included in all microbial kinetics. The model is used to examine the role of the microbial species and EPS in the adsorption process, and the effect of metal concentration and adsorption proprieties of biofilm components on the metal removal. Numerical results show that the model accurately describes the photogranules evolution and ecology and confirm the applicability of algal-bacterial photogranules systems for metal-rich wastewater treatment.

A free boundary value problem related to the genesis of multispecies granular biofilms is presented. The granular biofilm is modelled as a spherical free boundary domain with radial symmetry. The proposed model is conceived in the framework of continuum mechanics and consists of: nonlinear hyperbolic PDEs which model the advective transport and growth of attached species that constitute the granular biofilm matrix; semilinear elliptic PDEs which govern the diffusive transport and conversion of nutrients; and semilinear elliptic PDEs describing the invasion phenomena and conversion of planktonic cells suspended in the surrounding environment. The evolution of the free boundary is governed by an ODE accounting for microbial growth, attachment, and detachment. By using the method of characteristics, the system of equations constituting the granular biofilm model is converted into an equivalent integral system. Existence and uniqueness of solutions are discussed and proved for the attachment regime using the fixed point theorem.