# Optimized Analytical-Numerical Procedure for Ultrasonic Sludge Treatment for Agricultural Use
###### Abstract  
This paper presents an integrated approach based on physical-mathematical models and numerical simulations to optimize sludge treatment using ultrasound. The main objective is to improve the efficiency of the purification system by reducing the weight and moisture of the purification sludge, therefore ensuring regulatory compliance and environmental sustainability. A coupled temperature-humidity model, formulated by partial differential equations, describes materials' thermal and water evolution during treatment. The numerical resolution, implemented by the finite element method (FEM), allows the simulation of the system behavior and the optimization of the operating parameters. Experimental results confirm that ultrasonic treatment reduces the moisture content of sludge by up to 20% and improves its stability, making it suitable for agricultural applications or further treatment. Functional controls of sonication and the reduction of water content in the sludge correlate with the obtained results. Ultrasound treatment has been shown to decrease the specific weight of the sludge sample both in pretreatment and treatment, therefore improving stabilization. In various experimental conditions, the weight of the sludge is reduced by a maximum of about 50%. Processed sludge transforms waste into a resource for the agricultural sector. Treatment processes have been optimized with low-energy operating principles. Additionally, besides utilizing energy-harvesting technology, plant operating processes have been optimized, accounting for approximately 55% of the consumption due to the aeration of active sludge. In addition, an extended analysis of ultrasonic wave propagation is proposed. sludge treatment for agricultural uses; ultrasound; coupled temperature-humidity-pressure analytical model; FEM analysis 202417592024  
such as heavy metals, drugs, microplastics, or flame retardants, the potential use of sludge in agriculture applications is often limited [12; 13]. Thus, there is a growing interest in improving contaminant extraction to address resource recovery and pollution prevention in sewage sludge management [12; 13]. Untreated sludge has a complex structure and high moisture content, with particles retaining water. Reducing the water content results in a smaller sludge volume for transport and disposal; accordingly, drying becomes a critical step in sludge treatment, lowering its transport and disposal costs. Conditioning facilitates water removal, making dehydration and drying faster and less expensive. Without proper conditioning, these processes would require more resources, making them less sustainable and effective. Recent studies have explored different methods of sludge conditioning using various techniques to improve the effectiveness of treatment [14; 15]. A relatively simple technique that can be integrated with other treatment systems, now well established in the literature, is based on physical conditioning that applies mechanical forces to sludge to change its structure and facilitate dewatering and removal of contaminants [16]. While this technique reduces the viscosity of the material and improves drying, it requires large amounts of energy and expensive specialized equipment. However, physical conditioning can be ineffective for some types of sludge, especially in the presence of fine particles or chemical contaminants that require more specific treatments [17; 18]. Then, we are helped it their removal by sonication using high-frequency compression and rarefaction sound waves, which create bubbles in the liquid that, when they collapse, cause intense forces that disintegrate the sludge particles and facilitate the release of intracellular and extracellular materials [19]. This technique improves dewatering efficiency by increasing sludge biodegradability, resulting in rapid completion of anaerobic digestion [20; 21]. In addition, sonication is particularly effective in reducing viscosity and making the sludge more straightforward to process in subsequent stages.