1. Introduction

Biogas plants are environmental friendly due to their capability of methane emission reduction. The fermentation process in such biogas plants is a very complex process which can be compared to the digestion process in mammals (e.g. cows and pigs) stomach because during chemical digestion, which occurs in the gastrointestinal tract, food is digest into small molecules by digestive enzymes. Consequently, the biogas plants often use piggery manure1 or cattle-manure2 as an inoculum. Such material, containing ready culture of bacteria and partially digested, allows the fermentation process in the tank to begin3. In addition to the inoculum, temperature (psychrophilic, mesophilic, thermophilic)4,total solid5, carbon in glucose6 and pH7, which influence archaea 8 survival capability also influence the methane production9.
It is known that magnesium (Mg) takes part in some important processes in our life such as carbohydrate and glucose metabolism10. The glucose metabolism for the formation of thiamine diphosphate (TDP) from thiamine requires Mg 11. Therefore, the microorganisms that are involved in methane production may also need the Mg. For instance, Li et al. reported that magnesium oxide in the co-fermentation process was an effective additive12. In another similar study by Wang et al., it was discovered that hydrogen production was linearly increased by using Mg element13. Not only Mg could have influence on methane production, but other elements like sodium (Na) and calcium (Ca) may also influence this process. This assumption was backed from the previous reports in literature indicating that methane production could be increased by using potassium hydroxide, calcium hydroxide and sodium hydroxide 14-17.
Besides, the information about influence these elements on methane production can be used for improving efficiency of mathematical model which could be a good tool for biogas plants. Such way could be used for prediction methane production before real production would be started. Unfortunately, many presented models took into account not sufficient amount of factors or not crucial factors for methane production. For instance model presented by Masse and Droste contained a large amount of factors like for example: requirement of ammonia-N, total concentration of acetate or rate of hydrolysis. Besides for this model was conducted only few tests and average error for calculation methane production was 28% 9. However, most of the studies focused on all four stages18-21 (hydrolysis, acidogenesis, acetogenesis, methanogenesis22). This generate a large amount of variables but it does not guaranty a high model efficiency. For instance model created by Minot included a large amount properties and based on bacterial growth curve23. Moreover in this work was found only one sample for testing. Therefore this model cannot be reliable. Similar situation can be found in model presented by P. Axaopoulos and P. Panagakis24. Situation looks better for next work, where relative error was smaller than 20%25.
Obviously, the information presented above does not fully describe the factors and models of methane production, but it can be claimed that most of them need very precise information about the process. Besides, because information about the influence of crucial elements during fermentation process is limited, it was decided to extend this knowledge and improve mathematical model. Therefore this study aimed to present mathematical model for continuous technology in biogas plant, which take into account influence of the crucial elements: initial carbon content in the dry mass of the substrate, influence Mg on methane production, volumetric load, temperature, pH, etc.