## 1 Introduction
### Methods for Processing Kitchen Waste  
Biological methods for the processing of organic waste have a substantial role in waste management. Both composting and fermentation offer unique benefits and drawbacks as waste processing methods, and the decision to use one over the other is influenced by local conditions. Nonetheless, for biowaste treatment, methane fermentation appears to hold increasing potential due to its distinct advantages over composting. Over the past two decades, the technology and design of fermentation plants have seen steady optimization. Early operational challenges have been addressed, and technological adjustments have been made to suit the specific characteristics of solid waste. Consequently, methane fermentation has become a reliable, proven solution for biowaste processing, enabling the production of high-quality digestate. Anaerobic digestion (AD) is an effective and popular method for treating various types of organic waste, including kitchen waste. This biological process breaks down biodegradable material in the absence of oxygen, producing biogas--a renewable source of energy--and a nutrient-rich residue known as digestate. The process serves a dual purpose: waste management and resource recovery. Kitchen waste presents a unique challenge for anaerobic digestion due to its high moisture content, rapid biodegradability, and the presence of inhibitory substances such as fats, oils, and cleaning agents. To address these challenges, techniques such as pre-treatment, co-digestion, additive dosing, and process optimization have been developed. These strategies improve the microbial stability of the digestion process and reduce the inhibitory effects of harmful components. The efficiency of kitchen waste processing can also be enhanced by choosing suitable system configurations and adjusting parameters to match the local composition of kitchen waste. Reusing the treated residues, particularly the digestate, further adds value to the process and enhances the economic viability of biogas projects. One of the most promising innovations in this field is the use of anaerobic dynamic membrane bioreactors (DMR). Compared to traditional anaerobic digestion systems, DMR provide better solid-liquid separation, allowing for higher loading rates and improved retention of microbial biomass. These features help overcome many of the limitationstraditionally associated with kitchen waste treatment and expand the range of usable feedstocks [6]. The digestate produced in anaerobic digestion consists of undigested organic matter and microbial biomass. It is typically separated into liquid and solid fractions to facilitate storage, handling, and application. The liquid fraction generally contains a large share of nitrogen (N) and potassium (K), while the solid fraction is rich in residual fibers and phosphorus (P). The nutrient content and composition of digestate determine its agronomic value--that is, its effectiveness as a fertilizer or soil amendment [7; 8]. One of the key features of digestate is its high concentration of plant-available nutrients, especially nitrogen in the form of ammonium (NH\({}_{4}\)+). During anaerobic digestion, organic nitrogen compounds are decomposed by microbes into ammoniacal nitrogen, with only a small portion consumed for microbial growth. As a result, the digestate often contains high concentrations of ammonium and ammonia. The literature reports total nitrogen concentrations in digestates ranging from 11.2 to 25.7 g/kg of total solids. This readily available nitrogen enhances the short-term fertilization effect, promoting rapid plant growth following application [9; 10; 11]. In addition to nitrogen, digestates are also rich in phosphorus, derived primarily from adenylate acid, nucleic acids, and phospholipids present in the raw feedstock. The pH of the digestate, typically between 7.5 and 8.0, affects the solubility and form of phosphorus.