**5. Result**

The calculations were carried out for the steam pipeline with the following data: *rout* = 0.162 m, *rin* = 0.122 m, *s*<sup>w</sup> = 0.04 m, and *L* = 45 m. The steam tube is made of steel 14MoV63. The pipeline wall was divided into 100 control volumes. The number of control volumes in the radial direction is *n* + 1 = 5 and *m +* 1 = 21 in the axial direction (Figure 2). The pipeline wall and steam temperatures were calculated for each node lying in the center of the control volumes.

Data from the direct heat conduction problem solution were treated as "exact measurement data". Both exact and measurement data were used to carry out many simulations. Figure 5 shows the temperature variation *Tmeas <sup>f</sup>* ,*m*+1(*t*), pressure *<sup>p</sup>*, and steam mass flow rate . *m* as functions of time *t* used in the first calculation test.

**Figure 5.** The temperature variation *Tmeas <sup>f</sup>* ,*m*+1(*t*), pressure *<sup>p</sup>* and steam mass flow rate . *m* as functions of time *t* at the pipeline outlet *z* = 45 m.

The temperature *Tcalc <sup>f</sup>* ,1 (*t*) at the pipeline inlet obtained from the IHP solution for "exact measurement data" is shown in Figure 6. Figure 6b compares the steam temperature determined by *Tcalc <sup>f</sup>* ,1 (*t*) with the expected steam temperature *<sup>T</sup>meas <sup>f</sup>* ,1 (*t*).

**Figure 6.** The steam temperature variation as a function of time (**a**) at the inlet of the pipeline *Tcalc <sup>f</sup>* ,1 (*t*) and at the turbine inlet *Tcalc <sup>f</sup>* ,*m*+1(*t*), (**b**) comparison of the temperature *<sup>T</sup>calc <sup>f</sup>* ,1 (*t*) obtained by solving IHP with measured temperature *Tmeas <sup>f</sup>* ,1 (*t*).

To assess the accuracy of solving IHCP, a relative difference between the inlet steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) obtained from the IHP solution and measured temperature *<sup>T</sup>meas <sup>f</sup>* ,1 (*t*) was calculated as follows.

$$\varepsilon\_T = \left| \frac{T\_{f,1}^{\text{meas}}(t) - T\_{f,1}^{\text{calc}}(t)}{T\_{f,1}^{\text{meas}}(t)} \right| \times 100\% \tag{25}$$

Root-Mean Square Error (*RMSE*) was calculated as follows.

$$\text{RMSE} = \sqrt{\sum\_{j=1}^{N\_{\text{meas}}} \frac{\left[T\_{f,1}^{\text{meas}} \left(t\_j\right) - T\_{f,1}^{\text{calc}} \left(t\_j\right)\right]^2}{N\_{\text{meas}}}} \tag{26}$$

where the symbol *Nmeas* stands for the number of measurement points.

The relative difference *ε<sup>T</sup>* as a function of time is depicted in Figure 7. The relative error does not exceed 0.016 %, while the Root-Mean Square Error (RMSE) is 0.087 K.

**Figure 7.** Relative difference between the inlet steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) obtained from the IHP solution and "exact" measured temperature *Tmeas <sup>f</sup>* ,1 (*t*).

The analysis of the results shown in Figures 5 and 6 shows excellent conformity of the calculated temperature *Tcalc <sup>f</sup>* ,1 (*t*) and measured temperature *<sup>T</sup>meas <sup>f</sup>* ,1 (*t*). The IHP was solved using data from a power plant to verify the model's effectiveness and accuracy. Based on the measured steam temperature at the end of the pipeline *Tmeas <sup>f</sup>* ,*m*+1(*t*), the steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) was estimated using the developed method.

Figure 8 illustrates the steam temperature variation *Tmeas <sup>f</sup>* ,*m*+1(*t*), pressure *p,* and mass flow rate . *m* as a function of time *t* obtained from the measurement.

**Figure 8.** The steam temperature variation *Tmeas <sup>f</sup>* ,*m*+1(*t*), pressure *<sup>p</sup>* and steam mass flow rate . *m* as a function of time *t* on the pipeline outlet.

Then the inverse problem was solved based on actual measurement data. Figure 8 depicts the measured steam temperature *Tmeas <sup>f</sup>* ,*m*+1(*t*), pressure *<sup>p</sup>*, and mass flow rate . *m* of steam measured at the end of the pipeline. The IHP was solved to determine *Tcalc <sup>f</sup>* ,1 (*t*). The basic time step Δ*tb* was equal to Δ*tb* = *kb*·Δ*t* = 30 × 0.04 = 1.2 s with the time step Δ*t* = 0.04 s used for the solution of the direct problem. The number of future time intervals and the regularization factor were *F* = 2, *wr* = 0.01, respectively.

Figure 9a shows the calculated fluid temperature at the inlet *Tcalc <sup>f</sup>* ,1 (*t*) and outlet *Tcalc <sup>f</sup>* ,*m*+1(*t*) of the pipeline. A comparison of the calculated steam temperature *<sup>T</sup>calc <sup>f</sup>* ,1 (*t*) and the measured steam temperature *Tmeas <sup>f</sup>* ,1 (*t*) is shown in Figure 9b.

**Figure 9.** The steam temperature variation as a function of time (**a**) on the inlet pipeline *Tcalc <sup>f</sup>* ,1 (*t*) and on the turbine inlet *Tcalc <sup>f</sup>* ,*m*+1(*t*), (**b**) comparison of the temperatures determined *<sup>T</sup>calc <sup>f</sup>* ,1 (*t*) and the expected *Tf*<sup>1</sup>*,m*.

For the determined steam temperature *Tf* at the pipeline inlet, the relative error was determined using Equation (25); the change in time of this error is depicted in Figure 10.

**Figure 10.** The relative difference between the inlet steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) obtained from the IHP solution and the actual measured temperature *Tmeas <sup>f</sup>* ,1 (*t*).

The analysis of the results shown in Figure 9b reveals that the steam temperature at the pipeline inlet *Tcalc <sup>f</sup>* ,1 (*t*) determined from the IHCP solution, differs slightly from the steam temperature obtained from the *Tcalc <sup>f</sup>* ,1 (*t*) measurements. The RMSE = 0.038 K.

Figure 11 shows the time variations in temperature, pressure, and steam mass flow rate obtained from the measurements used in the next calculation test. The calculations were performed for a basic time step equal to Δ*tb* = *kb*·Δ*t =* 1.2 s (Δ*t* = 0.04 s), where the number of basic time steps was *kb* = 30. The number of future time intervals *F* = 2 and the regularization factor *wr* = 0.01 were assumed.

**Figure 11.** The temperature variation *Tf*, pressure *<sup>p</sup>* and steam mass flow rate . *m* as a function of time *t* on the pipeline outlet.

Figure 12a presents the time variations in the outlet steam temperature *Tcalc <sup>f</sup>* ,*m*+1(*t*) of the pipeline and the variations in the inlet steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) determined by solving the IHP. A comparison between the IHP solution *Tcalc <sup>f</sup>* ,1 (*t*) and the measured steam temperature *Tmeas <sup>f</sup>* ,1 (*t*) is shown in Figure 12b.

**Figure 12.** The steam temperature variation as a function of time (**a**) at the pipeline inlet *Tcalc <sup>f</sup>* ,1 (*t*) and at the turbine inlet *Tcalc <sup>f</sup>* ,*m*+1(*t*), (**b**) comparison of the temperatures *<sup>T</sup>calc <sup>f</sup>* ,1 (*t*) by solving the IHP and the measured temperature *Tmeas <sup>f</sup>* ,1 (*t*).

Analyzing the results depicted in Figure 12, it can be seen that there is very good agreement between the calculated steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) and the measured temperature *Tmeas <sup>f</sup>* ,1 (*t*). The average value of the RMSE is 0.322 K for this case.

The relative difference *ε<sup>T</sup>* between the fluid temperature obtained from the inverse solution and the measured temperature at the pipeline inlet is shown in Figure 13. The maximum value of the relative difference is *ε<sup>T</sup>* = 0.296%.

**Figure 13.** The relative difference between the inlet steam temperature *Tcalc <sup>f</sup>* ,1 (*t*) obtained from the IHP solution and the actual measured temperature *Tmeas <sup>f</sup>* ,1 (*t*).

The analysis of the results (Figures 11 and 12) shows that the differences between the steam temperature obtained from the inverse solution and the fluid temperature assumed in the direct solution are small. The small value of RMSE indicates a very good agreement between the inverse solution and measured steam temperature.
