A two-layer diffusion-reaction model for quantifying internal carbon loop efficiency in algal-bacterial biofilms
Algal-bacterial biofilms function as stratified microecosystems in which photosynthetic oxygen production and heterotrophic respiration are coupled through counter-diffusive \(\:{\text{O}}_{2}-{\text{C}\text{O}}_{2}\:\) transport. Quantifying this internal carbon loop is critical for designing carbon-neutral and energy-efficient wastewater treatment systems. This study develops a mechanistic two-layer diffusion–reaction model representing an outer phototrophic layer, where CO 2 is assimilated and O 2 is produced, and an inner heterotrophic layer, where O 2 is consumed, and CO 2 is released. Steady-state mass balances with interfacial flux continuity are solved to obtain coupled O 2 and CO 2 profiles across the biofilm. A new dimensionless Carbon Loop Efficiency Index (CLEI) is introduced to quantify the degree of photosynthetic–respiratory coupling based on interfacial gas fluxes. By definition, CLEI = 0 represents a carbon-positive regime with no internal CO 2 recycling, CLEI = 1 denotes complete loop closure and carbon-neutral operation, and CLEI > 1 indicates a potentially carbon-negative regime in which CO 2 assimilation exceeds internally generated CO 2 under the modeled conditions. Model results suggest that, within the validated parameter ranges considered in this study, CLEI approaches unity only within a constrained design window characterized by intermediate total biofilm thickness (approximately 300–500 μm), where oxygen and carbon dioxide transport are balanced through coupled diffusion–reaction processes. Thinner biofilms are carbon-limited (CLEI > 1), whereas thicker biofilms develop oxygen diffusion limitation (CLEI < 1), preventing complete loop closure. Thinner biofilms are carbon-limited (CLEI > 1), whereas thicker biofilms develop oxygen diffusion limitation (CLEI < 1), preventing full loop closure. The framework demonstrates how stratified algal–bacterial biofilms can achieve self-oxygenation and intrinsic CO₂ mitigation through transport-controlled coupling, and provides a mechanistic basis for the rational design of carbon-neutral or potentially carbon-negative wastewater treatment and photobioreactor systems.