## 2 Methodology
### Numerical Model  
#### 2.2.1 Governing Equations  
A comprehensive three-dimensional thermal model conjugate with an incompressible turbulent flow model is developed to determine the thermo-fluid characteristics of the proposed heated water considering deposit layer. The governing equations include applying conservation of energy on solid parts and conservation of mass, momentum, and energy with the turbulence model on fluid parts. Solid Regions: Tube Wall and Deposit Layer  
The conservation of energy governs the thermal characteristics and energy transfer through the solid domain, including the tube wall and deposit layer, and can be written as follows:  
* Conservation of Energy: \[\
abla\cdot(k_{s}\
abla T)=0\] (1) where \(k_{s}\) is the thermal conductivity of the solid part (i.e., tube material or deposits material), and the term on the left-hand side refers to the heat flux by conduction. Fluid Region  
The flow inside the tube is considered to be single-phase, steady, Newtonian, incompressible, and turbulent. Therefore, the flow characteristics can be represented using the conservation of mass, momentum-based on Reynolds-averaged Navier-Stokes (RANS) equations coupled with turbulence model, and conservation of energy as indicated by Equations (2)-(7):  
* Conservation of mass and momentum: \[\frac{\partial\overline{u_{i}}}{\partial x_{i}}=0\] (2) \[\overline{u_{i}}\frac{\partial\overline{u_{i}}}{\partial x_{i}}=-\frac{1}{ \rho_{f}}\frac{\partial\overline{P}}{\partial x_{i}}+\frac{\mu_{f}}{\rho_{f}} \frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}^{2}}-\frac{\partial}{ \partial x_{j}}\left(\overline{u_{i}^{\prime}u_{i}^{\prime}}\right)\] (3) where \(\overline{u_{i}}\) and \(x_{i}\) refer to the mean velocity of the flow and coordinate system, respectively, with \(i=1,\ 2,\ \text{and}\ 3\). \(\overline{P}\) is the mean pressure of the flow whereas \(\rho_{f}\) and \(\mu_{f}\) represent water density and viscosity, respectively. \(\overline{u_{i}^{\prime}u_{i}^{\prime}}\) is the turbulent Reynolds stress associated with the imposed turbulence over the mean flow, modelled and solved via turbulence models. * Turbulence Model: Realizable \(k\)-\(\epsilon\) turbulent model the realizable \(k\)-\(\epsilon\) turbulent model with enhanced wall function is used following [26; 27] to account for the existing fully turbulent flow while obtaining an accurate solution at a reasonable computational time. The model is based on model transport equations for the turbulence \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Domain** & **Deposit Material** & **Density** & **Specific Heat** & **Thermal** & **Viscosity** & **Dimensions** \\  & **[kg/m\({}^{3}\)]** & **[j/kg\(\cdot\)K]** & **[mm]** & **Vism** & **[kg/m\(\cdot\)s]** & **[mm]** \\ \hline Deposit & Magnetic & 5175 & 586 & 0.625-1.39 & NA \({}^{*}\) & \(t_{d}\) = from 0 to 0.2 \\ \hline \multirow{3}{*}{Tube} & Carbon Steel & & & & & \\  & Seamless Pipe & 7830 & 465 & 53 & NA \({}^{*}\) & \(t_{p}\) = from 1 to 6 \\  & (SA210 C) & & & & & \(L_{p}\) = 500 \\ \hline \multirow{3}{*}{Fluid} & Water at & & & & & \\  & \(P\) = 210 bar & & & & & \\  & \(T_{mergo}\) = 300 \({}^{\circ}\)C & & & & & \\ \hline \hline \end{tabular} \({}^{*}\)NA refers to not applicable.  
\end{table} 
Table 1: Material properties and related dimensions. kinetic energy (\(k\)) and its dissipation rate (\(\epsilon\)).