## 2 Materials and Methods
### Numerical Modelling
Setting the boiler construction details (lengths, diameters, number of pipes, etc.). 2. Setting the combustible characteristics (composition, properties from the burning process point of view, thermal properties, physical properties). 3. Defining the useful thermal power and the secondary agent temperatures. 4. Burning process calculus. 5. Furnace heat transfer calculus considering the combined radiation heat transfer from flame and flue gases and convective heat transfer. The furnace output temperature is calculated with Equation (1), [13]. \[\mathrm{T_{F}}=\frac{\mathrm{T_{ad.}}}{\left(\mathrm{M}\cdot\frac{\mathrm{C_{0 }}\cdot\mathrm{z}\cdot\mathrm{T_{ad.}}^{3}\mathrm{g}}{\left(1-\mathrm{q_{ext}} \right)\cdot\mathrm{B}\cdot\mathrm{V_{g}}\cdot\mathrm{c_{pg}}}\right)^{0.6}+1} \mathrm{[K]}\] (1)  
Furnace thermal load is determined using Equation (2), [13]. \[\mathrm{\dot{Q}_{F}}=\mathrm{B}\cdot\mathrm{V_{g}}\cdot\left(1-\mathrm{q_{ext} }\right)\cdot\left(\mathrm{c_{pg}}\cdot\mathrm{t_{g}}-\mathrm{c_{pg}}\cdot \mathrm{t_{f}}\right)\mathrm{[W]} \tag{2}\]
6. Verification computation for heat transfer in each convective section determining the flue gases output temperature and the useful heat load corresponding for each discretization section. In this purpose, Mendeleev convective heat transfer coefficient--Equation (3)--was used. \[\mathrm{h_{c}}=\varepsilon\cdot 0.0263\cdot\mathrm{C}\cdot\frac{\mathrm{k}\cdot \mathrm{Pr}^{0.35}}{\mathrm{d_{i}}}\cdot\mathrm{Re}^{0.8}\left[\frac{\mathrm{W }}{\mathrm{m^{2}K}}\right]2300<\mathrm{Re}<10,000 \tag{3}\]
7. Radiation heat transfer coefficient is estimated according to Equation (4). \[\mathrm{h_{r}}=\mathrm{C_{0}}\cdot\frac{\varepsilon_{\mathrm{wall}}+1}{2} \cdot\varepsilon_{\mathrm{g}}\cdot\mathrm{T_{gm}}^{3}\cdot\left[\frac{1-\left( \mathrm{T_{wall}}\right)}{1-\left(\mathrm{T_{gm}}\right)^{3.6}}\right]\mathrm{ \left[\frac{\mathrm{W}}{\mathrm{m^{2}K}}\right]} \tag{4}\]  
Total heat transfer coefficient to surface is the sum of the convection and radiation coefficients. The flue gases exit temperature is determined through Equation (5) implementation into the computational program. \[\mathrm{T_{fg,out}}=\mathrm{T_{wall}}+\left(\mathrm{T_{fg,in}}-\mathrm{T_{wall }}\right)\cdot\mathrm{e}^{-\frac{\mathrm{h_{TOT}}\cdot\mathrm{S_{R}}}{\mathrm{ EV_{g}}\cdot\mathrm{c_{pg}}\cdot\left(1-\mathrm{q_{ext}}\right)}}\mathrm{[K]} \tag{5}\]
8. Numerical determination of the condensing heat transfer:  
- Initial (input) flue gases temperature and composition. - Condensing heat transfer computation steps: condensing condition for the wall temperature with regard to flue gases water vapor saturation temperature; dimensionless criterions (invariants) specific for the mass transfer process (Schmidt,Reynolds, and Sherwood) calculation; determination of the condensing coefficient using Equation (6). \[\mathrm{J_{MASS}}=2.157\cdot 10^{-3}.\frac{\mathrm{Sh\cdot C_{diff}}}{\mathrm{FL}} \left(\mathrm{P_{H20-flue\text{ gases}}}-\mathrm{P_{H20-pipewall}}\right)\cdot\frac{1}{\mathrm{T_{gm}}} \left[\frac{\mathrm{kg}}{\mathrm{m^{2}s}}\right] \tag{6}\]  
- Convection coefficient determination for the secondary agent. - Overall heat exchange coefficient determination considering sensible and latent heat transfer from the flue gases to the \"wall\"; surface extension geometry (for finned pipes only); surface extension efficiency (for finned pipes only); secondary agent heat transfer resistance. - Computation loop closing errors with iteration decision. - Final determination for the thermal and physical parameters. For the finned pipes body boiler, the computational model validation was realized by direct comparison of computational results with experimental results obtained after testing on an experimental stand accredited by the Romanian National Accreditation Body.