## 2 Materials and Methods
### Mathematical Model
The equation of heat balance of the U-shaped flu gas trajectory furnace is:  
\[Q_{in\ i}+Q_{rec\ i}+Q_{bur\ i}=Q_{Me\ i}+Q_{lost\ i}+Q_{egg\ i} \tag{1}\]  
where  
\(Q_{in\ i}\)--heat entered into the 1 st zone with fuel, air and fuel combustion products, (W);  
\(Q_{rec\ i}\)--heat of combustion products passing through the furnace gas flow interface on the straight and reverse path, (W);  
Figure 4: Calculation scheme of the furnace and the movement of flue gases. \(Q_{bur}\) --heat released by the fuel combustion, (W);  
\(Q_{Me}\) --heat consumed to heat up the material in the zone, (W);  
\(Q_{lost}\) --heat loss through furnace walls, (W);  
\(Q_{exg}\) --heat loss of furnace gases leaving the \(i\)th zone, (W). Heating components of the heat balance calculation are calculated according to Equations (2)-(4). \[Q_{in}\quad i=\left\{ \begin{array}{l}B_{f}\cdot t_{f}\cdot c_{f}+L_{n}\cdot B _{f}\cdot t_{air}\cdot c_{air}\quad\text{for}\quad i=1\\ \left(1-\sum\limits_{k=1}^{i-1}m_{k}\right)\cdot\left(B_{f}\cdot t_{i-1}\cdot c _{f}+L_{n}\cdot B_{f}\cdot t_{i-1}\cdot c_{air}\right)+\\ +\sum\limits_{k=1}^{i-1}m_{k}\cdot v_{exg}\cdot B_{f}\cdot t_{i-1}\cdot c_{ exg}+\sum\limits_{k=1}^{i-1}(\Delta G_{2n-k+1}-\Delta G_{k})\cdot t_{i-1} \cdot c_{exg}\quad\text{for}\quad i=2,\quad\ldots\quad,\quad 2n\\ \end{array} \right. \tag{2}\]  
\[Q_{rec}\quad i=\left\{ \begin{array}{l}\Delta G_{2n-i+1}\cdot t_{2n-i+1}\cdot c _{exg}\quad\text{for}\quad i=1,\quad\ldots\quad,\quad n/2\quad\text{and}\quad i =n+1,\quad\ldots\quad,\quad 3n/2\\ -\Delta G_{2n-i+1}\cdot t_{2n-i+1}\cdot c_{exg}\quad\text{for}\quad i=n/2+1, \quad\ldots\quad,\quad n\quad\text{and}\quad i=3n/2+1,\quad\ldots\quad,\quad 2n\\ \end{array} \right. \tag{3}\]  
\[Q_{bur\ i}=m_{i}\cdot B_{f}\cdot Q_{low} \tag{4}\]  
where  
\(B_{f}\) --consumption, (kg/s);  
\(t_{f},\ t_{air}\) --temperature of heated fuel and air, (\({}^{\circ}\)C);  
\(c_{f},\ c_{air},\ c_{exg}\) --heat capacity of fuel, air and flue gases, (J/(m\({}^{3}\) K));  
\(L_{n}\) --actual combustion air consumption, (kg air/kg fuel);  
\(t\) --temperature of the gases in combustion chamber, (\({}^{\circ}\)C);  
\(m_{i}\) --fraction of fuel burned in the \(i\)th zone;  
\(v_{exg}\) --specific exhaust gases output during combustion, (kg/kg). \(\Delta G_{i}\) --consumption of gases that move between local recirculation zones, (kg/s);  
\(Q_{low}\) --lower operating heat of combustion of fuel, (J/kg);  
\(n\) --number of calculated zones along the length of the furnace, (pcs). The value of \(\Delta G_{ij}\) is determined by the law of the distribution of mass flow density of combustion products through the interface between the forward and reverse trajectories of their movement. For this purpose, the following formula can be used:  
\[\Delta G_{i}=\left\{ \begin{array}{l}0\quad\text{for}\quad 1\leq i\leq n/2 \quad\text{and}\quad n<i\leq 3n/2\\ A\cdot\left(\frac{2\cdot y_{i}}{L_{tr}}\right)^{b}\quad\text{for}\quad n/2<i\leq n \\ A\cdot\left(\frac{2\cdot y_{i-n}}{L_{tr}}\right)^{b}\quad\text{for}\quad 3n/2<i \leq 2n\end{array} \right. \tag{5}\]  
where  
\(y_{i}=\Delta y\cdot(i-1/2)\) --coordinate of zone \(i\) centre, (m);  
\(\Delta y=L_{tr}/(2\cdot n)\) --length of the \(i\)th zone, (m);  
\(L_{tr}\) --total length gas movement trajectory, (m);  
\(b\) --distribution function exponent;  
\(A\) --the regularity coefficient of the distribution of the specific mass flow of combustion products across the interface of the forward and reverse trajectories of their movement, which is determined by the Equation (6):  
\[B_{f}\cdot v_{exg}\cdot\left(k_{p}-1\right)-A\cdot\left[\sum\limits_{i=\frac{n }{2}}^{n}\left(\frac{2\cdot y_{i+1}}{L_{tr}}\right)^{b}\right]=0 \tag{6}\]  
where  
\(k_{p}\) --coefficient of the large scale (recirculation through all zones of the combustion chamber) recirculation. The share of fuel that burns in the \(i\)th calculation zone of the furnace is determined as:  
\[m_{i}=\left\{ \begin{array}{ccc}1-C(L_{i})&\text{for}&i=1\\ C(L_{i-1})-C(L_{i})&\text{for}&i=2,&\ldots\end{array}