**2.4. Flux tube for membrane water walls—fins attached to the adjacent water wall tubes**

The variation of the view factor on the surface of the flux tube, fin, and water wall tube is illustrated in **Figure 11**. The origin of the cylindrical coordinate system is at the center of the outer flux-tube surface (**Figure 2b**). Because of the symmetry, only the half of the flux tube was considered.

The measured temperatures: *f*1 = 418.28°C, *f*2 = 415.61°C, *f*3 = 374.05°C, *f*4 = 372.18°C, *f*5 = 318.00°C were generated artificially for the following input data *qm* = 250 000 W/m2 , *hf* = 30 000 W/(m2 K), *Tf* = 318.00°C. The inverse analysis yields the values of unknown parameters: *qm* = 249 999.43 W/m2 , *hf* = 29 999.60 W/(m2 K), *Tf* = 318.00°C, which are very close to the input values. The reconstructed temperature distribution is shown in **Figure 12**. The uncertainty analysis was omitted because the results are very similar to the results obtained in Section 4.2.

To show the influence of the measurement errors on the determined parameters, the 95% confidence intervals were estimated. The following uncertainties of the measured values were

*j* =1, …, 5, 2*σ<sup>k</sup>* = ± 1.0W/ (mK). For these test calculations, the 95% uncertainties in the param‐ eters measured directly were taken two times greater than the uncertainties in the previous

2 250 000.06 7102.46 W/m , *mq* = ±

2 30 000.05 4735.71 W/(m K), *<sup>f</sup> h* = ±

318.00 0.41 C. *Tf* = ±°

Despite the doubling of uncertainties, the results are quite good. Larger relative errors in determined the heat transfer coefficient are due to a small difference in temperature between the inner surface of the flux tube and the fluid temperature. For this reason, the impact of the uncertainties in direct measurements on the estimated heat transfer coefficient is greater.

The uncertainties (95% confidence interval) of the coefficients *xi* were determined using the error propagation rule (16). The calculated uncertainties are: ±4.1% for *qm*, ±27.3% for *hf* and

Then, the inverse analysis was carried out for perturbed data: *f*1 = 420.16°C, *f*<sup>2</sup> = 416.81°C, *f*<sup>3</sup> = 375.40°C, *f*4 = 372.69°C, *f*5 = 318.01°C. The reconstructed temperature distribution is illustrated

= 317.99°C. The influence of the error in the measured temperatures on the estimated param‐

The number of iterations in the Levenberg–Marquardt procedure is small in both cases

**2.4. Flux tube for membrane water walls—fins attached to the adjacent water wall tubes**

The variation of the view factor on the surface of the flux tube, fin, and water wall tube is illustrated in **Figure 11**. The origin of the cylindrical coordinate system is at the center of the outer flux-tube surface (**Figure 2b**). Because of the symmetry, only the half of the flux tube was

The measured temperatures: *f*1 = 418.28°C, *f*2 = 415.61°C, *f*3 = 374.05°C, *f*4 = 372.18°C, *f*5 = 318.00°C

= 318.00°C. The inverse analysis yields the values of unknown parameters: *qm* = 249 999.43

were generated artificially for the following input data *qm* = 250 000 W/m2

K), *Tf*

. The accuracy of the results obtained is acceptable.

in **Figure 10a**. The obtained results are *qm* = 250 118.613 W/m2

case analyzed in Section 4.1. The limits of the 95% uncertainty interval are as follows:

<sup>=</sup> <sup>±</sup> 0.4K, 2*σrj*

, *hf*

= 318.00°C, which are very close to the input values. The

= 30 050.041 W/(m2

K) and *Tf*

, *hf* = 30 000 W/(m2

= ± 0.10mm, 2*σφ<sup>j</sup>*

= ± 1.0° ,

assumed (at 95% confidence interval): 2*<sup>σ</sup> <sup>f</sup> <sup>j</sup>*

230 Numerical Simulation - From Brain Imaging to Turbulent Flows

±0.1% for *Tf*

eters is small.

considered.

K), *Tf*

W/m2 , *hf*

(**Figures 9b** and **10b**).

= 29 999.60 W/(m2

**Figure 11.** Comparison of view factor calculated analytically and by FEM for the flux tube shown in **Figure 2b**.

**Figure 12.** Temperature distribution in the flux tube obtained from the solution of the inverse problem for the unper‐ turbed data: f1 = 418.28°C, f2 = 415.61°C, f3 = 374.05°C, f4 = 372.18°C, f5 = 318.00°C; (a) temperature distribution; (b) itera‐ tion process.

#### **2.5. Boiler test**

The pulverized coal-fired boiler produces 58.3 kg/s superheated steam at 11 MPa and 540°C. Experimental studies were conducted in this boiler for a mass flow rate of live steam equal to 210 · 103 kg/h. Measuring heat flux tubes were installed in the middle of the water wall at different levels of the boiler combustion chamber. Despite the boiler is operated in a steady state load, slow time variations in the measured temperatures are observed. This phenomenon is characteristic for the combustion of coal in large boilers. However, time changes of the meter temperature are very slow and the temperature distribution in the meter can be considered as a steady state. Temperature measurement results for the heat flux tubes located at a level of

**Figure 13.** Measured temperature histories at five points for heat flux tube located at the level of 15.4 m (a) and esti‐ mated parameters: absorbed heat flux qm. Heat transfer coefficient hf and fluid temperature Tf (b).

15.4 m are depicted in **Figure 11a**, and the estimated parameters as functions of time are depicted in **Figure 11b**.

**2.5. Boiler test**

232 Numerical Simulation - From Brain Imaging to Turbulent Flows

210 · 103

The pulverized coal-fired boiler produces 58.3 kg/s superheated steam at 11 MPa and 540°C. Experimental studies were conducted in this boiler for a mass flow rate of live steam equal to

different levels of the boiler combustion chamber. Despite the boiler is operated in a steady state load, slow time variations in the measured temperatures are observed. This phenomenon is characteristic for the combustion of coal in large boilers. However, time changes of the meter temperature are very slow and the temperature distribution in the meter can be considered as a steady state. Temperature measurement results for the heat flux tubes located at a level of

**Figure 13.** Measured temperature histories at five points for heat flux tube located at the level of 15.4 m (a) and esti‐

and fluid temperature Tf

(b).

mated parameters: absorbed heat flux qm. Heat transfer coefficient hf

kg/h. Measuring heat flux tubes were installed in the middle of the water wall at

**Figure 14.** Temperature distribution in the cross section of the flux tube located at the level of 15.4 m. which was deter‐ mined on the basis of measured temperatures: f1 = 413.509°C; f2 = 412.227°C; f3 = 372.855°C; f4 = 372.227°C; f5 = 322.209°C. The estimated parameters are: qm = 230425.8 W/m2 ; hf = 24128.8 W/(m2 K); Tf = 319.19°C.

The differences between the measured and computed temperatures for the measurements at the elevation 15.4 m (**Figures 13a** and **14**) are reported in **Table 4**. The time points in the first column of **Table 4** are the same as indicated in **Figure 13a and b**.



**Table 4.** The differences between measured and calculated temperatures.

The residuals and the sum of temperature difference squares are small. The obtained results show that the proposed method can be successfully applied to identify the operating condi‐ tions of water walls in boilers.

Similar measurements and calculations were performed for the insert located at a height of 19.2 m. The results of measurements and calculations are shown in **Figures 15** and **16**.

Measuring insert at an elevation of 15.4 m is situated directly above the burners which makes the value of absorbed heat flux *q*m higher in comparison with the absorbed heat flux at a height of 19.2 m. At the level of 15.4 m, heat flux is approximately 230 000 W/m2 while at the height of 19.2 m is about 168 000 W/m2 . It is worth mentioning a very high lifetime of measuring heat flux tubes, which is more than 5 years.

**Time** *f***4, °C** *T***4, °C** *f***4-***T***4, K** *f***5, °C** *T***5, °C** *f***5-***T***5, K** *S***, K2** 00:05:00 373.88 373.76 0.12 319.49 319.49 0.00 0.04 00:11:00 372.18 372.38 −0.19 319.60 319.60 0.00 0.17 00:17:00 369.06 368.69 0.38 320.18 320.20 −0.02 1.58 00:23:00 371.17 371.14 0.04 319.96 319.96 −0.01 0.06 00:29:00 371.66 371.45 0.21 320.96 320.98 −0.01 0.46 00:35:00 372.23 371.67 0.56 322.21 322.23 −0.02 1.35 00:41:00 375.06 374.61 0.46 322.36 322.38 −0.01 0.62 00:47:00 373.82 373.42 0.40 321.41 321.43 −0.02 1.14 00:53:00 371.67 371.26 0.42 321.36 321.38 −0.02 1.69 00:59:00 371.88 371.66 0.22 322.39 322.39 0.00 0.14

234 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Table 4.** The differences between measured and calculated temperatures.

tions of water walls in boilers.

of 19.2 m is about 168 000 W/m2

flux tubes, which is more than 5 years.

The residuals and the sum of temperature difference squares are small. The obtained results show that the proposed method can be successfully applied to identify the operating condi‐

Similar measurements and calculations were performed for the insert located at a height of

Measuring insert at an elevation of 15.4 m is situated directly above the burners which makes the value of absorbed heat flux *q*m higher in comparison with the absorbed heat flux at a height of 19.2 m. At the level of 15.4 m, heat flux is approximately 230 000 W/m2 while at the height

. It is worth mentioning a very high lifetime of measuring heat

19.2 m. The results of measurements and calculations are shown in **Figures 15** and **16**.

**Figure 15.** Measured temperature histories at five points for heat flux tube located at the level of 19.2 m (a) and esti‐ mated parameters: absorbed heat flux *qm* heat transfer coefficient *hf* and fluid temperature *Tf* (b).

**Figure 16.** Temperature distribution in the cross section of the flux tube located at the level of 19.2 m. which was deter‐ mined on the basis of measured temperatures: T1 = 382.170°C; T2 = 379.040°C; T3 = 352.270°C; T4 = 351.460°C; T5 = 317.300°C. The estimated parameters are: qm = 166 577.5 W/m2 ; hf = 29 489.5 W/(m2 K); Tf = 315.21°C.
