## 3 Materials and Methods
### Mass Balance and Energy Balance  
In setting the mass balance for raw materials used in energy generation, the volume of waste streams resulting from the splitting of waste mass originating from MTP and GPP technology--\(m_{\beta}\), \(m_{\beta}\), \(m_{\beta}\), \(m_{i1}\), \(m_{i2}\), \(m_{i3}\)--must be calculated (Figure 1). In order to do this, it is necessary to determine volumes of waste supplied to the system--\(m_{SS}\), \(m_{BIO}\), \(m_{mix}\), \(m_{CS}\)--and know the results of the examination of waste morphological composition. The method of assessment of particular waste stream masses is shown in Table 3.  
Figure 1: Block diagram of the ERWP model. Two energy-generating processes have been provided in the system based on the ERWP model. These are methane fermentation in high- and low-hydration conditions and thermal waste transformation through combustion. The result of the oxidation of organic compounds in the municipal waste combustion process in the presence of oxygen is the liberation of thermal energy amounting to 10 MJ/kg (LHV) [63]. To calculate the parameters allowing for assessment of the energy potential, equations describing the relationship between the higher heating value (HHV) and lower heating value (LHV) as variables dependent on the contents of selected elements: HHV, LHV = \(f(C,H,N,O,S)\), can be used [64]. The elements are determined by application of elemental analysis and their contents are indicated as percentages of dry mass. Mutual relationships described in the literature are expressed by multiple regression equations having the general form of \(Y=a+b_{1}\cdot X_{1}+\ldots\ldots+b_{n}\cdot X_{n}\). The equations obtained are then approximated by application of the least squares method and in the majority of cases they have a high coefficient of determination (R\({}^{2}\)), which shows a good match of the estimated function (Table 4). Usage of the equations contained in Table 3 is associated with a necessity to analyse contents of the elements making up the independent variables. Arriving at a reliable result requires (i) application of the same methodology of preparation of the analytical sample and (ii) use of raw material procured in the same way as that used for the development of the given equation. Condition (ii) originates from the limitation of the impact of the so-called discreet variables that are not taken into account in the equations in Table 3. Finally, to estimate HHV, an equation was used which takes into account, as independent variables, contents of various materials in the waste mix making up the energetic material [68], in the following form:  
\[\text{HHV}=0.0535(\text{F}+32.6\cdots\text{CP})+0.3722\cdot\text{PLR},\text{ MJ}/\text{kg} \tag{7}\]  
Particular values in Formula (7) indicate contents of: \(F\)--bio fraction, \(CP\)--cardboard and paper, \(PLR\)--plastic, leather and rubber in the dry waste mixture expressed in \(\perm  
Approximation of the results based on determination of the _HHV_ value, being a dependent variable for sewage sludge samples after the methane fermentation process with variable organic substance contents, allowed for the development of a model for which coefficient R\({}^{2}\) was 0.87 [69]:  
\[\text{HHV}=0.2132\cdots Z_{\text{om}},\text{MJ}/\text{kg} \tag{8}\]  
\(Z_{om}\) in Formula (8) indicates contents of organic matter in % \(dm\). The higher heat value of wood waste was assessed from the following equation [70]:  
\[\text{HHV}=0.4373\text{C}-1.6701,\text{MJ}//\text{kg} \tag{9}\]  
Gas yield from methane fermentation is described, in practice, by two indicators. The first one is the rate of biogas production (GPR), counted as a quotient of the daily volume of generated gas and reactor volumetric capacity expressed in \(\text{m}^{3}\text{gas}/\text{m}^{3}\text{ reactor}/\text{d}\). This indicator is used directly to fix the calculated flow rate in the installation and to select proper biogas-processing devices.