## 1 Introduction  
Around 50,000 metric tons of synthetic dyes are discharged with wastewater worldwide every year [1, 2]. Generally, dyeing wastewater is characterized by high pH, deep color, strong toxicity, various components, and wide range of water quality and amount variation [3, 4]. Dyeing wastewater is commonly treated by economic and effective biochemical methods. However, the recalcitrant nature of azo dyes, together with their toxicity to microorganisms, makes aerobic treatment difficult to meet the discharge criterion. In anaerobic treatment, microbial hydrolyzation and acidification by microorganisms may convert recalcitrant organic matter in dyeing wastewater to biologically degradable ones, improve biodegradability of the wastewater, and remove a portion of organisms to create an advantageous nutritional condition for the subsequent aerobic treatment in dyeing wastewater [5, 6, 7]. A few types of anaerobic reactors, such as up-flow anaerobic sludge blanket (UASB) [8, 9, 10, 11], expanded granular sludge bed (EGSB) [12], anaerobic baffled reactor (ABR) [13, 14], strengthened circulation anaerobic reactor (SCAR) [15], anaerobic membrane bioreactors (An MBRs) [16], anaerobic sequential batch reactor (ASBR) [17], and anaerobic side-stream reactor (ASSR) [18], have been employed to process dyeing wastewater. Recent studies on dyeing wastewater treatment by pilot-scaleanaerobic reactors are summarized in Table 1, which have achieved chemical oxygen demand (COD) removal rate between 10 and 45.8% with organic loading rate (OLR) at 0.53-7.4 kg COD/(m\({}^{3}\).h). Among these technologies, ABR takes advantage of baffle-board structure to form several separate compartments inside the reactor, the flow regime in which is mainly plug flow. ABR also forms multilevel and multiphase anaerobic environment, and has fine buffer capability resistant to shock loads and adaptability to influent toxicants [19]. It uses up-and-down water stream to stir anaerobic sludge to mix with wastewater; its structure and flow regime make a certain microorganism community cultivated in each of its compartments, which acclimates its inflowing wastewater quality and environmental condition. Thus, nicer biological population distribution will be formed to improve treatment efficiency and operation stability due to the unique structure and flow regime of the reactor [20; 21]. However, solids loss is a disadvantage of ABR, especially at high hydraulic loads [22]. Up to now, most reported studies are laboratory scale, and limited information is available on full or pilot-scale ABR implementations [23]. In this study, a pilot-scale HABR was built to treat dyeing wastewater after FeSO\({}_{4}\) coagulation pretreatment. In order to create cross-flow conditions inside the reactor to maintain high biomass concentration in this study, cross-flow packing was developed and utilized in each compartment of the ABR to form a hybrid anaerobic baffled reactor (HABR). The improvement of the structure led to the adherence of the microorganisms on the packing to form biofilms, and further increased the microbial richness and diversity [18; 24]. Recently, a future research direction from a review conducted on the state-of-the-art anaerobic biological technologies for processing dyeing wastewater has been clearly pointed out, which is to determine the boundary conditions for high-performance operations of anaerobic reactors [16]. Researchers have employed a few methods to evaluate the performance of the anaerobic reactor in processing wastewater, including tests of organic and iron contents in the wastewater based on gas chromatography mass spectrometry and spectrophotometer [12; 25], analysis of microbial communities by fluorescence in situ hybridization, polymerase chain reaction and denaturing gradient gel electrophoresis [26; 27], and biological toxicity tests [13]. However, limited literature has been found on the reactor performance evaluation based on mathematical diagnostic tools [28].