*2.2. The Developed Geometrical Variants*

Four selected geometric options are presented in this paper, with the first three reflecting the progress of the fourth and final concept. As mentioned earlier, this study analyzed the bypass duct transporting higher temperature flue gases introduced from the top. The hot flue inlet configuration is the most difficult from the point of view of mixing the flue gas streams before the SCR system. The difficulty in mixing the flue gas streams is mainly due to buoyancy forces and relatively slow flue gas velocities resulting from low boiler load. To obtain a complete representation of the flue gas flow through the analyzed duct section, a full 3D geometry was implemented for the numerical calculations. A general schematic of the examined flue gas duct section is shown in Figure 2. The section shows the inlets of the flue gas streams and the flow direction towards the SCR reactor.

**Figure 2.** A general schematic of the examined flue gas duct section.

The geometric variants of the turbulence flap are shown in Figure 3. The geometries presented show the same bypass inlet location in each variant and the evolving geometry of the exhaust gas mixing elements. The other elements in the flue gas duct are fixed vanes which regulate the flue gas flow. The geometric variant G1 involves placing a single flap directly behind the hot flue gas inlet. The flap is inclined at an angle of 45 degrees and its length corresponds to covering half of the duct cross-section. A longer flap could not be used in this solution due to limitations on the maximum flue gas velocity, which increases as the flow cross-section area decreases.

In the geometric variant G2, three U-profiles were placed, with a total width of approximately two-thirds of the channel width and a length corresponding to covering half of the flow cross-section resulting from the velocity condition mentioned earlier. Geometric variations G3 and G4 are a combination of G1 and G2. They use a flat flap in the upper part and three U-profiles in the lower part. Both variants are constructed to cover half of the main duct cross-section. Variant G3 uses shorter but wider U-profiles with a total width of about the main duct of two-thirds. In variant G4, the profiles have a total width corresponding to half the main channel width. The designed geometries of the turbulence flaps, as seen from above, are shown in Figure 4.

**Figure 3.** The four geometric variants developed, numbered from G1 to G4.

**Figure 4.** The four geometric variants seen from above, numbered from G1 to G4.

In order to adequately numerically reproduce the effects affecting the exhaust stream mixing, especially the buoyancy forces, the computational geometry was considered at a scale of 1:1. The basic geometrical dimensions of the analyzed channel section are shown in Table 1.

**Table 1.** The basic geometrical dimensions of the analyzed channel section.


*2.3. Flow-Governing Equations and Model Assumptions*

The main equations of fluid mechanics applied to numerical calculations of gas flow such as momentum, mass, energy and species conservation were used for the calculations. They are widely described in many works, e.g., in [24]. Momentum conservation is represented by Equation (1):

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho u\_i u\_j) + \frac{\partial P}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_i} \left\{ \mu \left[ \frac{\partial u\_j}{\partial \mathbf{x}\_i} + \frac{\partial u\_i}{\partial \mathbf{x}\_j} - \frac{2}{3} \delta\_{ij} \frac{\partial u\_i}{\partial \mathbf{x}\_i} \right] \right\} + \frac{\partial}{\partial \mathbf{x}\_i} (-\rho u\_j u\_i) - F\_{p\prime} \tag{1}$$

Mass conservation (Equation (2)) is presented below:

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho u\_i) = \sum\_{\mathbf{J}} \mathbf{S}\_{\mathbf{j}\prime} \tag{2}$$

Equation (3) represents conservation of energy:

$$\frac{\partial}{\partial \boldsymbol{x}\_{i}} (\boldsymbol{u}\_{i} [\rho \boldsymbol{E} + \boldsymbol{P}]) = \frac{\partial}{\partial \boldsymbol{x}\_{j}} \left( \lambda\_{eff} \frac{\partial T}{\partial \boldsymbol{x}\_{j}} \right) + S\_{j\prime} \tag{3}$$

The species conservation is given by Equation (4):

$$\frac{\partial}{\partial \boldsymbol{x}\_{i}} (\rho \boldsymbol{u}\_{j} \boldsymbol{Y}\_{k}) = -\frac{\partial}{\partial \boldsymbol{x}\_{j}} \boldsymbol{\overrightarrow{J}}\_{k} + \boldsymbol{\omega}\_{k} + \boldsymbol{S}\_{k\prime} \tag{4}$$

A realizable k-ε turbulence model was implemented for the calculations, with full buoyancy forces included. Detailed descriptions of the above equations and the k-epsilon turbulence model with experimental verification are presented in [24]. The k-ε realizable model was implemented since it is widely used to calculate free gas flows in relatively large domains, such as in power boilers [19,20]. In [25], this model was used to develop a flow in a large-scale coal-fired boiler. The analyzed channel geometry includes elements that can cause rotation, wall boundary layers and gas recirculation. According to [26], the applied turbulence model can provide improved numerical simulation results in the abovementioned phenomena. In the k-epsilon model, Reynolds stresses are supplemented by the Boussinesq relation according to Equation (5):

$$-\rho u\_j u\_i = \mu\_t \left(\frac{\partial u\_i}{\partial x\_j} + \frac{\partial u\_j}{\partial x\_i}\right) - \frac{2}{3} \left(\rho k + \mu\_t \frac{\partial u\_k}{\partial x\_k}\right) \delta\_{ij\prime} \tag{5}$$

With the turbulent viscosity calculated from Equation (6):

$$
\mu\_t = \rho \mathbb{C}\_{\mu} k^2 / \varepsilon,\tag{6}
$$

Unlike the standard and RNG models, the value of *Cμ* is not constant in the k-epsilon realizable model. *C<sup>μ</sup>* is a function of the mean strain and rotation velocities and the turbulence fields represented by k and epsilon. The effective thermal conductivity can be calculated from the following formula (Equation (7)):

$$
\lambda\_{eff} = \lambda + c\_P \mu\_t / Pr\_{t\prime} \tag{7}
$$

Based on the literature [27], the turbulent Prandtl number was assumed to be 0.85.

In keeping with the character of the flue gas flow passing through the duct in an industrial power boiler, the model adopts the simplifications and basic assumptions outlined below:


The discrete form of the equations above and all assumptions were implemented in ANSYS Fluent (18.2, ANSYS, Inc., Canonsburg, PA, USA).
