## 1 Introduction  
The recovery of phosphorus (P) from wastewater is increasingly important in the water process engineering sector. More than 90% of the P demand corresponds to the fertilizer industry, while the remainder is used in detergents, food supplements, pesticides, and medicine industries [1, 2, 3]. However, due to the excess of P in the diet of people and animals, the surplus of this element is eliminated through human exceta, manure, and solid waste [4, 5]. Detergents also contribute to the concentration of P in domestic wastewater. In addition to this relevant environmental aspect, P and phosphate rock were reconfirmed in the 2023 Critical Raw Materials (CRM) List [6]. Thus, it is important to use less P (e.g., by minimizing the input in goods) and promote recycling (e.g., by maximizing the recovery and reuse of P in waste streams). The CRM Act (in force on 23 May 2024) set goals to improve the sustainability and circularity of CRM in the European Union (EU) market by taking measures to increase the collection of CRM-rich waste and ensure its recycling into secondary CRM [7]. In addition, the new Urban Wastewater Treatment Directive(Directive (EU) 2024/3019) introduced updated guidelines, including stricter monitoring of chemical pollutants, pathogens, and antimicrobial resistance, as well as promoting the reuse of treated urban wastewater to address water scarcity. Specifically, concerning P in wastewater treatment plants (WWTTP), the directive mandates tertiary treatment by 2039. New discharge limits and minimum P removal percentages relative to influent load are proposed: 0.7 mg P/L or 87.5% reduction (10,000-150,000 population equivalent (pe)) and 0.5 or 90% reduction (>150,000 pe). In the context of regulations for agricultural reuse of treated wastewater, some countries have set specific P concentration limits. For example, Portugal has set a limit of 5 mg P/L, while Cyprus allows concentrations up to 10 mg P/L [8]. To meet these goals, various strategies and technologies are available for P recovery from wastewater and for preparing water for reuse, with chemical precipitation and adsorption being among the most commonly employed methods. Previous studies have investigated an innovative chemical precipitation approach to recover P in the form of high-purity struvite while minimizing interference from calcium compounds [9]. Despite achieving promising results and producing a high-quality product suitable for soil application, the P concentration in the treated wastewater decreased from around 100 mg P/L (concentration in the inlet stream of anaerobic digestion) to 10 mg P/L. However, this concentration still exceeds the previously mentioned regulatory limits. Therefore, integrating both technologies presents an effective strategy to enhance the P removal efficiency. In this context, it is worth noting that our previous studies on batch adsorption using eggshells calcined at 700 \({}^{\circ}\)C for 60 min demonstrated promising results for P removal from wastewater, achieving an adsorption capacity of approximately 5 mg P/g at 25 \({}^{\circ}\)C. These experiments revealed that the main mechanism of P removal in this adsorbent/adsorbate system is physisorption based on electrostatic attraction. In addition, ligand exchange and microprecipitation may contribute to this phenomenon. In addition, the buffering capacity of wastewater helps stabilize the pH when the adsorbent meets the liquid. Although batch experiments offer valuable insights into the potential of calcined eggshells as adsorbents for P removal, the results may not translate directly to large-scale applications. When treating larger volumes of wastewater, the adsorption capacities observed in batch tests are likely overestimated, as scale-up introduces changes in the operating conditions and system dynamics that affect performance. Thus, studying adsorbent behavior under dynamic conditions is essential for accurately predicting breakthrough curves and effectively scaling up the process to an industrial level [10; 11].