## 1 Introduction  
The development of zero waste technology and municipal waste management is critical for sustainable development, as CO\({}_{2}\) emissions from fossil fuel combustion continue to be a global concern [1, 2]. In essence, conventional energy sources such as natural gas, coal, and petroleum are continuously exploited. Due to the current overdependence on these fuel sources, which are depleting but have significant ecological impacts, there is a rising need for alternative fuels [3]. Furthermore, anthropogenic CO\({}_{2}\) emissions, which contribute to global warming and climate change, have prompted the globe to seek out alternate actions to mitigate or control CO\({}_{2}\) emissions [4]. The wastewater treatment settings cannot be overlooked in this regard, particularly considering the prospect of boosting its circular economy [5, 6, 7, 8, 9]. Herein, energy-saving wastewater treatment systems have become essential, especially with the upgradation of biogas produced by anaerobic digesters (ADs).  
Anaerobic digestion (AD) is a technique that transforms organic matter in biomass resources into biogas with methane (CH\({}_{4}\)) as a source of sustainable energy [5, 10, 11]. Of essence, AD creates economic opportunities by producing bioenergy (60-70% methane) and stabilized digestate under suitable conditions to ease pollution [12]. The AD process utilises microorganisms' degradation potential in an ecologically responsible, odour-reducing, and pathogenic organism-degrading mode, particularly in bioreactors operating at temperatureswithin the mesophilic (25-45 \({}^{\circ}\)C) or thermophilic (>45 \({}^{\circ}\)C) ranges [5,10,13]. Additionally, produced biogas frequently contains pollutants such as H\({}_{2}\)S and CO\({}_{2}\), which reduce the calorific value of the biogas and corrode equipment such as pipelines and incineration engines [5,10]. Co-digestion, pre-treatment, reactor design, and additives to stimulate bacteria growth and prevent inhibitory effects are some of the strategies utilized to address this [5,13,14,15]. In this study, the concept of photoreduction of CO\({}_{2}\), which involves the conversion of the non-energy-rich component of the biogas into methane, is highlighted [11].  
In essence, biogas has been exploited as an alternative to fossil fuel for energy generation, and the concept of reducing the CO\({}_{2}\) component to methane is limited. Biogas produced via anaerobic digestion of organic waste is composed of 55-65% methane and 35-45% CO\({}_{2}\) and other gases such as H\({}_{2}\)S [16]. To solve the shortcomings, several abatement solutions have been proposed, whereby the AD process coupled with CO\({}_{2}\) photoreduction is gaining attention [12,17,18]. This technology involves the use of a photocatalyst in the presence of UV-light, visible light, or sunlight to convert the CO\({}_{2}\) to methane, which is rich in energy [4,19]. In photocatalytic CO\({}_{2}\) reduction, an incident light (UV, visible light, or sunshine) activates the photocatalyst surface. In doing so, it causes the electrons to shift from the valence band to the conduction band to form a hole and produce electrons [20,21]. The water molecules divide into hydroxyl and hydrogen ions, which then react with the released electron species. For example, CO\({}_{2}\) generated from fossil fuel combustion is absorbed and converted to energy-rich methane by catalytic electrolysis [3]. Subsequently, the addition of nanoparticles to the AD process to enhance biogas production has gained attention [22,23].  
Therefore, a biophotocatalytic reduction of CO\({}_{2}\) produced anaerobically into methane by the seeding of a magnetic photocatalyst is shown in this study (Fe-TiO\({}_{2}\)). Three distinct configuration systems were explored in this study for reducing sugar refinery wastewater contamination, biogas generation, and methane yield, as well as reducing CO\({}_{2}\) emissions. The modified Gompertz model was utilised to establish the model's operational variables kinematically.