**7. Conclusions**

The chapter presents an effective method for solving nonlinear inverse heat transfer problems using CFD software. Application of the method is illustrated by the identification of the boundary conditions in water wall tubes and boundary conditions in a platen superheater of a CFB boiler.

A CFD-based method for determining heat flux absorbed by water-wall tubes, heat transfer coefficient at the inner flux-tube surface and temperature of the water-steam mixture has been presented. New heat flux tubes were proposed. The flux tubes are not welded to the adjacent water-wall tubes, so the temperature distribution in the measuring device is not affected by neighboring water-wall tubes. Based on the measured flux-tube temperatures the nonlinear inverse heat conduction problem was solved. The number of thermocouples placed inside the heat flux tube including the thermocouple on the rear outer tube surface is greater than the number of unknown parameters because additional measurement points reduce the uncer‐ tainty in determined parameters. To achieve a good accuracy of measurements, the uncer‐ tainties in measured heat flux-tube temperatures and the radial thermocouple locations should be small since they have the largest impact on the accuracy of the parameter estimation. The proposed flux tubes and the inverse procedure for determining absorbed heat flux can be used both when the inner surface of the heat flux tube is clean and when the scale or corrosion deposits are present on the inner surface what can occur after a long time service of the heat flux tube. The flux tubes can work for a long time in the destructive high-temperature atmosphere of a coal-fired boiler.

1 2 2

*<sup>x</sup> m i*

å (40)

*<sup>f</sup>* **<sup>1</sup> <sup>⋅</sup>100,** *% <sup>S</sup>***min, K2**

\* <sup>±</sup> *<sup>σ</sup>xi* ,

2 2 , 4, 1,2 *i j*

¶ <sup>=</sup> = = ¶ <sup>ø</sup> ë û

The 95% uncertainty in the estimated parameters can be expressed in the form *xi* = *xi*

1 366.54 414.89 463.02 508.45 0.240 0.0006

2 379.35 422.43 464.00 505.74 −0.087 0.3752

3 393.79 427.88 463.65 501.04 0.064 0.0384

**Table 7.** Comparison of measured and calculated steam temperatures at the inlet to the superheater and at measuring

The chapter presents an effective method for solving nonlinear inverse heat transfer problems using CFD software. Application of the method is illustrated by the identification of the boundary conditions in water wall tubes and boundary conditions in a platen superheater of

A CFD-based method for determining heat flux absorbed by water-wall tubes, heat transfer coefficient at the inner flux-tube surface and temperature of the water-steam mixture has been presented. New heat flux tubes were proposed. The flux tubes are not welded to the adjacent water-wall tubes, so the temperature distribution in the measuring device is not affected by neighboring water-wall tubes. Based on the measured flux-tube temperatures the nonlinear inverse heat conduction problem was solved. The number of thermocouples placed inside the heat flux tube including the thermocouple on the rear outer tube surface is greater than the number of unknown parameters because additional measurement points reduce the uncer‐ tainty in determined parameters. To achieve a good accuracy of measurements, the uncer‐

Assuming that the 95% uncertainty in measured steam temperatures is equal 2σfj = 0.5 K for *j* = 2, 3, 4 the uncertainties in calculated temperature T1 and heat transfer coefficient *hg* are:

, *i* =1, 2, 3 represent the value of the parameters obtained using the least squares

K). The results are quite satisfactory.

*<sup>e</sup>* **<sup>=</sup>** *<sup>f</sup>* **<sup>1</sup> <sup>−</sup>** *<sup>T</sup>***<sup>1</sup>**

2

=

*i x f j j*

é ù æ ö ê ú ç ÷ ê ú <sup>è</sup>

 s

*f*

*m*

s

254 Numerical Simulation - From Brain Imaging to Turbulent Flows

= 0.99 W/(m<sup>2</sup>

**No.** *f***1, °C** *f***2, °C** *f***3, °C** *f***4, °C**

**T1, °C T2, °C T3, °C T4, °C**

365.66 414.88 463.04 508.44

379.68 422.19 464.50 505.48

where *xi* <sup>=</sup> *<sup>x</sup>* \*

points 3, 4, and 5.

a CFB boiler.

**7. Conclusions**

method.

2*σT*<sup>1</sup>

*i*

<sup>=</sup> 0.74 K and 2*σ<sup>h</sup> <sup>g</sup>*

The CFD simulation of the thermal-hydraulic processes occurring in the platen steam super‐ heater, located in the combustion chamber of the fluidized bed boiler, was carried out.

The temperature distributions in the tube wall of complex shape and the flowing steam were computed. The steam velocity and pressure distributions can be determined with high accuracy. The CFD modeling is a useful tool to explore the real fluid and heat flow phenomena which occur in platen superheaters operating at high thermal loads. It is well known that even a small increase in tube operating temperature over the design temperature will reduce stress rapture life significantly. The CFD simulation allows calculation of the maximum tube wall temperature along the steam flow path for complex cross section shape of the tubes and complicated flow arrangement of the superheater. The detailed CFD prediction makes possible the proper selection of the steel grade for the analyzed superheater stage or superheater pass.

The temperature-dependent physical properties of the steam and tube material can be easily taken into account. The specific heat of water steam decreases significantly with increasing temperature. The calculations show that the temperature rise in each of the three passes is almost the same in spite of decreasing the temperature difference between the flue gas and steam. If the specific heat were constant, the steam temperature increase in the third pass would be much smaller compared to the first pass. It should be stressed, that it would be impossible to calculate accurately the steam and tube wall temperature if the classical methods for heat exchanger calculations, which assume constant physical steam properties, were used.

Accurate calculation of the heat transfer coefficient on the flue-gas side of the platen super‐ heater placed in the boiler furnace is critical for boiler design. The inaccurate calculation of the heat transfer coefficient is the reason for tubes damage due to an excessive steam temperature which frequently happens in power plants. A new method for determining of the flue-gas side heat transfer coefficient was developed.

The flue-gas side heat transfer coefficient or the inlet steam temperature and flue-gas side heat transfer coefficient were determined based on the measured steam temperatures at selected points along the steam flow path. To solve the inverse problems, the secant method was used when only one temperature measurement point was applied, and one unknown, that is, the heat transfer coefficient on the flue-gas side was searched. The Levenberg–Marquardt method was used to solve the over-determined heat transfer problem. For the solution of the direct conjugate heat transfer problem, which is encountered at every iteration step, while the inverse problem is being solved, ANSYS/CFX software was used.

Identifying the flue-gas side heat transfer coefficients for various boiler loads using the inverse method, a simple mathematical model of the platen superheater can be developed and used in the control system of the superheated steam temperature.

The proposed method of solution can be successfully applied to solve other inverse prob‐ lems occurring in industrial practice.

#### **Nomenclature**




