**5. Results of the superheater modeling using ANSYS//CFX**

The baseline *k*–*ω* model has the advantages of the Wilcox *k*–*ω*, and the *k*–*ε* but is not able to predict onset and amount of flow separation from smooth surfaces [53]. The main reason for this drawback is neglecting of the transport of the turbulent shear stress. This causes an overprediction of the eddy-viscosity. In order to eliminate this deficiency, the shear stress transport

> 1 max( , ) 1 2

a

a w

*k SF*

Because of the symmetry, only one-fourth of a tube cross section was analyzed (**Figure 19**). Heat is transferred by convection from the combustion chamber to the tube through the surface E–F. The surfaces A–F and B–C–D–E are thermally insulated. The tube inner surface A–B is cooled by the flowing live steam. The following boundary conditions were assumed for the

<sup>=</sup> (30)

is limited by

(SST) turbulence model was developed [53], in which the turbulent viscosity *ν<sup>t</sup>*

*t*

n

the following equation:

242 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 19.** One-fourth of a tube cross section.

tube wall (**Figure 19**):

The ANSYS CFX commercial software v13.0 [25] was used for the numerical calculations. Computer calculations were carried out for the second stage of the steam superheater which is made of omega tubes from 10CrMo 910 low alloy steel. At first, direct and inverse problems were solved for the single tube representing the first pass of the superheater. A finite element mesh contains 1 233 999 elements and 747 945 finite element nodes. The cross section of the tube is shown in **Figure 18**. The length of the modeled tube is 8 517 mm. Thermal properties of the superheated steam were calculated using the industrial standard IAPWS-IF97 [54]. Changes in thermal conductivity with temperature for the 10CrMo 910 steel are shown in **Figure 20**.

At first, test simulations were performed for the following data:


Two different steam inlet velocities *uin*: 16.7 m/s and 10 m/s were assumed at the inlet of the tube to illustrate the influence of the steam side conditions on the steam and tube wall

**Figure 20.** The thermal conductivity of 10CrMo 910 and T91 steels as a function of temperature.

**Figure 21.** Tube wall temperature; (a) changes of mean and maximum tube wall temperature over the tube length, (b) temperature of the tube wall at the nodes: 1, 2, 3, 4, 5, 7, 8, and 9 over the tube length; uin = 16.7 m/s, hg = 200 W/(m2 K).

temperature. The steam velocity of 16.7 m/s corresponds to 100% boiler load, while the steam velocity equal to 10 m/s corresponds to the partial boiler load. Then, to assess the impact of the gas side heat transfer coefficient *hg* on the temperature of the steam and tube, calculations for *uin* = 16.7 m/s and *hg* = 250 W/(m*<sup>2</sup>* K) were also carried out.

The results of the computer simulation for *uin* = 16.7 m/s and *hg* = 200 W/(m2 K) are shown in **Figures 21**–**23**.

Solving Inverse Heat Transfer Problems When Using CFD Modeling http://dx.doi.org/10.5772/63807 245

**Figure 22.** The temperature distribution in the tube wall and steam in two cross sections; (a) at the distance of 2839 mm from the inlet; (b) at the distance of 5678 mm from the inlet; uin = 16.7 m/s, hg = 200 W/(m2 K).

**Figure 23.** Mass-average steam temperature and the temperature of the steam at the tube axis.

temperature. The steam velocity of 16.7 m/s corresponds to 100% boiler load, while the steam velocity equal to 10 m/s corresponds to the partial boiler load. Then, to assess the impact of the gas side heat transfer coefficient *hg* on the temperature of the steam and tube, calculations

**Figure 21.** Tube wall temperature; (a) changes of mean and maximum tube wall temperature over the tube length, (b) temperature of the tube wall at the nodes: 1, 2, 3, 4, 5, 7, 8, and 9 over the tube length; uin = 16.7 m/s, hg = 200 W/(m2

K).

**Figure 20.** The thermal conductivity of 10CrMo 910 and T91 steels as a function of temperature.

244 Numerical Simulation - From Brain Imaging to Turbulent Flows

The results of the computer simulation for *uin* = 16.7 m/s and *hg* = 200 W/(m2 K) are shown in

K) were also carried out.

for *uin* = 16.7 m/s and *hg* = 250 W/(m*<sup>2</sup>*

**Figures 21**–**23**.

From the analysis of the results shown in **Figure 21a** and **b**, it can be seen that the steam temperature over the tube length grows almost linearly. Only, in the region close to the tube inlet the wall temperature is lower. This is due to much higher heat transfer coefficient in the developing flow which occurs at the inlet region of the tube. The steam in the boundary layer is cool at the inlet section, which contributes to a better cooling tube wall. It should be noted that the difference between the maximum and average temperature over the tube cross section is large but does not exceed 30 K (**Figure 21a**).

The rise of the tube wall temperature is almost linear over a length of the tube due to the constant value of the heat transfer coefficient in the region of the developed flow. At the entrance section of the tube, the heat transfer coefficient is higher because of developing a fluid flow. In this region, the temperature difference over the boundary layer is greater, so the heat flux and heat transfer coefficient are larger, and tube temperature is lower. **Figures 22a** and **6b** show the temperature distribution in the tube wall and steam at two cross sections away from the inlet, respectively at the distance of 2 839 mm and 5 678 mm. The maximum temperature of the wall is at the tube edge (**Figure 22a**). This temperature is 539.2°C at the distance of 2 839 mm and 551.3°C at the distance of 5678 mm from the inlet of the tube. The increase of the massaverage steam temperature over the entire length of the tube is 41 K (**Figure 23**).

**Figure 24.** Tube wall temperature; (a) changes of mean and maximum tube wall temperature over the tube length, (b) temperature of the tube wall at the nodes: 1, 2, 3, 4, 5, 7, 8, and 9 over the tube length; uin = 10 m/s, hg = 200 W/(m2 K).

The value obtained from the numerical simulation is in good agreement with the measured rise in the steam temperature. Steam temperature is highest near the inner surface of the tube.

Solving Inverse Heat Transfer Problems When Using CFD Modeling http://dx.doi.org/10.5772/63807 247

From the analysis of the results shown in **Figure 21a** and **b**, it can be seen that the steam temperature over the tube length grows almost linearly. Only, in the region close to the tube inlet the wall temperature is lower. This is due to much higher heat transfer coefficient in the developing flow which occurs at the inlet region of the tube. The steam in the boundary layer is cool at the inlet section, which contributes to a better cooling tube wall. It should be noted that the difference between the maximum and average temperature over the tube cross section

The rise of the tube wall temperature is almost linear over a length of the tube due to the constant value of the heat transfer coefficient in the region of the developed flow. At the entrance section of the tube, the heat transfer coefficient is higher because of developing a fluid flow. In this region, the temperature difference over the boundary layer is greater, so the heat flux and heat transfer coefficient are larger, and tube temperature is lower. **Figures 22a** and **6b** show the temperature distribution in the tube wall and steam at two cross sections away from the inlet, respectively at the distance of 2 839 mm and 5 678 mm. The maximum temperature of the wall is at the tube edge (**Figure 22a**). This temperature is 539.2°C at the distance of 2 839 mm and 551.3°C at the distance of 5678 mm from the inlet of the tube. The increase of the mass-

**Figure 24.** Tube wall temperature; (a) changes of mean and maximum tube wall temperature over the tube length, (b) temperature of the tube wall at the nodes: 1, 2, 3, 4, 5, 7, 8, and 9 over the tube length; uin = 10 m/s, hg = 200 W/(m2

The value obtained from the numerical simulation is in good agreement with the measured rise in the steam temperature. Steam temperature is highest near the inner surface of the tube.

K).

average steam temperature over the entire length of the tube is 41 K (**Figure 23**).

is large but does not exceed 30 K (**Figure 21a**).

246 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 25.** The temperature distribution in the tube wall and steam in two cross sections; (a) at the distance of 2839 mm from the inlet; (b) at the distance of 5678 mm from the inlet; uin = 10 m/s, hg = 200 W/(m2 ·K).

For the two different steam velocities at the inlet to the pipe: 16.7 m/s and 10 m/s the difference between the mass-average temperature of the steam and temperature in the axis of the tube is small (**Figure 23**). Tube wall temperatures are higher when the steam velocity at the inlet is lower and equal to *uin* = 10 m/s. The results of the computer simulation for *uin* = 10 m/s and *hg* = 200 W/(m2 K) are shown in **Figures 24** and **25**. A similar effect on the tube wall temperature has the heat transfer coefficient on the flue-gas side. The CFD simulation was thus performed for the velocity of the steam at the tube inlet: *uin* = 16.7 m/s and the heat transfer coefficient on the gas side: *hg* = 250 W/(m2 K), that is, the heat transfer coefficient was increased by 50 W/(m2 K). The CFD simulation results are shown in **Figures 26** and **27**. If the heat transfer coefficient on the gas side is *hg* = 250 W/(m2 K) then the steam and tube wall temperature is higher, because the heat flow rate transferred from flue gas to the steam is larger (**Figures 26** and **27**).

**Figure 26.** Tube wall temperature; (a) changes of mean and maximum tube wall temperature over the tube length, (b) temperature of the tube wall at the nodes: 1, 2, 3, 4, 5, 7, 8, and 9 over the tube length; uin = 16.7 m/s, hg = 250 W/(m2 K).

**Figure 27.** The temperature distribution in the tube wall and steam in two cross sections; (a) at the distance of 2839 mm from the inlet; (b) at the distance of 5678 mm from the inlet; uin = 16.7 m/s, hg = 250 W/(m2 K).

The correctness of the results obtained using CFD simulation was examined using the energy balance equation. The heat flow rate *Q*˙ *out* transmitted through the outer surface of the tube with an area of *A*out, which is given by the integral

$$
\dot{Q}\_{\text{out}} = \int\_{A\_{\text{out}}} \dot{q}\_{\text{out}} dA \tag{35}
$$

should be equal to the heat flow rate *Q*˙ *in* transmitted through the inner surface

$$
\dot{Q}\_{\dot{u}} = \int\_{A\_{\ast}} \dot{q}\_{\dot{u}} dA \tag{36}
$$

and to the heat flow rate *Q*˙ *<sup>s</sup>* absorbed by the steam

**Figure 26.** Tube wall temperature; (a) changes of mean and maximum tube wall temperature over the tube length, (b) temperature of the tube wall at the nodes: 1, 2, 3, 4, 5, 7, 8, and 9 over the tube length; uin = 16.7 m/s, hg = 250 W/(m2

248 Numerical Simulation - From Brain Imaging to Turbulent Flows

K).

$$
\dot{Q}\_s = \dot{m}(h\_{\text{outlet}} - h\_{\text{lelet}}) \tag{37}
$$

where *hinlet* and *houlet* stand for the steam enthalpy at the inlet and the outlet of the superheater tube, respectively, and *m*˙ denotes the steam mass flow rate. The symbols *q*˙ *out* and *q*˙*in* designate the heat flux at the outer and inner tube surface. It has been proved that the solution is independent of the finite volume mesh. Twelve different meshes were examined. The finite element number ranged from 790 179 to 3 789 699. For the CFD simulations presented in the work, the mesh consisting of 1 233 999 elements with the total number of nodes equal to 747 945 was selected. The quality requirements for the finite element mesh, given in the CFX Reference Guide, were satisfied. The mesh statistics like the orthogonal angle, expansion factor, and aspect ratio were confirmed by the CFX solver that they are correct [25].

For the analyzed case: *uin* = 16.7 m/s, *hg* = 200 W/(m2 K) and for the mesh consisting of 1 233 999 elements, the following values of the heat flow rate were obtained:

$$
\dot{Q}\_{\alpha u} = \dot{Q}\_{\dot{\alpha}} = -\\$7.266\,\text{kW} \,\text{and} \,\text{S7.644\,\text{kW}}.
$$

The relative difference defined as:

$$\mathcal{E}\_{\mathcal{Q}} = \frac{0.5(\dot{\mathcal{Q}}\_{\text{in}} + \dot{\mathcal{Q}}\_{\text{out}}) - \dot{\mathcal{Q}}\_{s}}{0.5(\dot{\mathcal{Q}}\_{\text{in}} + \dot{\mathcal{Q}}\_{\text{out}})} 100,\% \text{s}$$

is:*εQ* = −0.656%

If the number of elements was increased to 3 789 699, then the relative difference is almost the same and equal to: *εQ* = −0.655%.

The consistency of the results is very good. Taking into account that in the inverse problem is solved iteratively, the direct problem is repeatedly solved so the element mesh cannot be too fine because of computing time.
