## 1 Introduction  
Iron and manganese are natural components in the crust. High levels of iron and manganese in groundwater are common. In China, 20% of groundwater resources have excess iron and manganese [1]. In water supply networks, iron and manganese in tap water are oxidized to high valence during disinfection. The oxide precipitation formed in the pipeline is easily adsorbed into the water supply network, affecting the quality of the drinking water supply [2]. Although iron and manganese are necessary trace elements for the human body, drinking high-iron and -manganese surface water or groundwater for a long time will lead to chronic poisoning and damage to human health [3, 4, 5]. Physiologically, a large amount of iron ingested by the human body cannot be removed through metabolism. Excessive iron accumulation will induce diabetes, skin diseases, and other diseases, while excessive manganese can cause pathological changes in human organs and even cause neurotoxicity [6, 7, 8].  
The coexistence of iron and manganese is common in groundwater. In recent years, how to efficiently and stably remove iron and manganese has become the focus of research. Iron usually exists in a soluble ferrous (Fe(II)) state. The most stable oxidation state of manganese is +2 valence [9, 10]. The traditional removal methods of Fe(II) and Mn(II) in groundwater include natural oxidation, biological, and adsorption [11, 12, 13, 14]. The natural oxidation method is to oxidize Fe\({}^{2+}\) to Fe\({}^{3+}\) via aeration and then generate Fe(OH)\({}_{3}\) precipitation. The removal of manganese requires adding alkali based on aeration to improve pH. The procedure flow of this method is complex, and the high pH in effluent needs acidification treatment, which increases the treatment cost and management difficulty.  
The biological method mainly depends on microorganisms to reduce iron and manganese concentration in groundwater. However, the metabolism of microorganisms is influenced by the oxygen content, and insufficient oxygen slows down the metabolism of microorganisms [15]. Although oxidation and biological methods are widely used, their disadvantages are unstable effluent quality and difficulty to control reaction conditions. In contrast, the adsorption method has the advantages of large capacity, less energy consumption, and less pollution [16; 17]. It is widely used in removing iron and manganese from groundwater and is considered one of the most effective methods to remove iron and manganese [18; 19]. Manganese sand is the most widely used adsorbent for water adsorption of iron and manganese ions [20].  
This study aimed to determine the best conditions for the adsorption effect of modified manganese sand. Response surface methodology can effectively optimize the optimal process parameters and evaluate the interactions between various influencing factors [21; 22]. We established a response surface model (RSM) based on single-factor experiments to determine the optimal adsorption factors. The model helped analyze various factors affecting the experiment in a limited number of experiments. Meanwhile, we analyzed the interaction between the factors and proved the rationality of the response surface model through experiments.