**6. Inverse problem**

Ten platens of the superheater SH II are located at combustion chamber of the circulating fluidized bed boiler (**Figure 17**). Steam temperature is measured outside of the combustion chamber at the points 1–4. Because these points are located outside of a fire region, it is easy to measure steam temperature by attaching the thermocouple to the tube outer surface that is thermally insulated (**Figure 28**).

**Figure 28.** Three pass single superheater coil.

Solving Inverse Heat Transfer Problems When Using CFD Modeling http://dx.doi.org/10.5772/63807 251

**Figure 29.** Inverse problem for a three pass second stage platen superheater, (a) flue-gas side heat transfer coefficient α<sup>g</sup> is determined based on steam temperature measured at point number 4, (b) flue-gas side heat transfer coefficient hg and inlet steam temperature T1 are determined based on steam temperature measured at points 2, 3, and 4.


**Table 5.** Input data and results for the first inverse problem (**Figure 27a**).

Reference Guide, were satisfied. The mesh statistics like the orthogonal angle, expansion factor,

57.266 kW and 57.644 kW. *Q Q out in* = == & &

0.5( ) 100,% 0.5( ) *in out s*

*in out QQ Q Q Q*

If the number of elements was increased to 3 789 699, then the relative difference is almost the

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

Ten platens of the superheater SH II are located at combustion chamber of the circulating fluidized bed boiler (**Figure 17**). Steam temperature is measured outside of the combustion chamber at the points 1–4. Because these points are located outside of a fire region, it is easy to measure steam temperature by attaching the thermocouple to the tube outer surface that is

+ - <sup>=</sup> <sup>+</sup> && & & &

K) and for the mesh consisting of 1 233 999

and aspect ratio were confirmed by the CFX solver that they are correct [25].

elements, the following values of the heat flow rate were obtained:

*Q*

e

For the analyzed case: *uin* = 16.7 m/s, *hg* = 200 W/(m2

250 Numerical Simulation - From Brain Imaging to Turbulent Flows

The relative difference defined as:

same and equal to: *εQ* = −0.655%.

fine because of computing time.

thermally insulated (**Figure 28**).

**Figure 28.** Three pass single superheater coil.

**6. Inverse problem**

is:*εQ* = −0.656%

Two different inverse problems were solved. The first and second inverse problems are presented in **Figure 29a** and **b**, respectively. At first, a flue-gas side heat transfer coefficient *hg* was estimated based on the measured steam temperature at the outlet and at the end of the three pass platen superheater made from omega tubes with the length of 8517 mm (**Fig‐ ure 29b**). Also flue-gas temperature *Tg*, steam mass flow rate *m*˙ per one tube, and the outlet steam pressure *P*4 are known from measurements **Figure 27a**. The inverse problem was solved for three data sets (**Table 5**). The heat transfer coefficient *hg* was adjusted to obtain the outlet steam temperature *T*4 equal to the measured value *f*4. The following nonlinear algebraic equation

$$T\_4 - f\_4 = 0\tag{38}$$

was solved iteratively using the secant method [27, 28] to determine the flue-gas side heat transfer coefficient *hg*. At every iteration step, the temperature distribution in the tube wall and steam was determined using ANSYS/CFX. The three pass steam superheater was divided into finite element mesh which has 3 603 132 finite elements and 2 207 434 nodes (**Figure 30**). The solution of the inverse problem (**Table 6**) was obtained only after a few iterations. The CFD simulation also enabled the determination of pressure *P*1 at the inlet to the superheater (**Table 5**).

**Figure 30.** Finite volume mesh for a tube quarter.


**Table 6.** Input data and results for the second inverse problem (**Figure 29b**).

**Figure 31.** Steam and tube wall temperature distribution at the outlet of the three pass superheater for the data set number 1—three measurement points (**Table 6**).

In the second inverse problem, the flue-gas side heat transfer coefficient *x*1 = *hg* and steam temperature *x*2 = *T*1 at the inlet of the superheater are two unknowns which were determined based on measured steam temperatures *f*2, *f*3, and *f*4 at three points 2, 3, and 4 located at the outlets of each pass (**Figure 29b**). The number of measured temperatures is greater than the number of unknown parameters to increase the accuracy of the solution of the inverse problem. If the number of measuring points is greater, then the effect of random errors in temperature measurements on the results obtained from the solution of the inverse problem is smaller.

simulation also enabled the determination of pressure *P*1 at the inlet to the superheater

**·K)** *T***1, °C** *P***1, MPa**

(**Table 5**).

**Figure 30.** Finite volume mesh for a tube quarter.

252 Numerical Simulation - From Brain Imaging to Turbulent Flows

number 1—three measurement points (**Table 6**).

**No. Input data Results** *m***˙ , kg/s** *T***g, °C** *P***4, MPa** *f***2, °C** *f***3, °C** *f***4, °C** *hg***, W/(m2**

**Table 6.** Input data and results for the second inverse problem (**Figure 29b**).

1 0.2644 812.70 8.73 414.89 463.02 508.45 114.95 365.66 8.82 2 0.3148 826.78 10.12 422.43 464.00 505,74 121.65 379.68 10.22 3 0.4616 866.11 13.71 427.88 463.65 501.04 158.81 393.54 13.85

**Figure 31.** Steam and tube wall temperature distribution at the outlet of the three pass superheater for the data set

The least-squares method was used to determine parameters *hg* and *T*1. The sum of squares

$$S = \sum\_{i=2}^{4} (f\_i - T\_i)^2 \tag{39}$$

was minimized using the Levenberg–Marquardt method [26]. At every *k*-th iteration step, the steam temperature distribution **T**(**x**(*k*) , **r***i* ) is calculated, where **r**<sup>i</sup> designates the position vector of the steam measurements points. The steam and tube wall temperature were calculated at each iteration step using ANSYS/CFX software. The CFX program was called and controlled by an external program written in Python language [28]. The Levenberg–Marquardt algorithm was also included in the Python program. Three data sets were used in the inverse analysis (**Table 6**). From the analysis of the results shown in **Table 7**, it can be seen that the agreement between the calculated and measured steam temperature at the superheater inlet is very satisfactory. Also, the minimum value of the sum of squared temperature differences (39) is very small (**Table 7**). The similar values of the heat transfer coefficients *hg* were obtained from the solution of the first and second inverse problems (**Tables 5** and **7**).

The actual measured data were adopted for the inverse analysis: tube material—T91 steel, pass length—8517 mm. The thermal conductivity of T91 steel as a function of temperature is shown in **Figure 20**. The experimental data sets adopted for the inverse analysis and the estimated heat transfer coefficients *αg* and steam inlet temperatures *T>*1 are shown in **Table 6**. The computed temperatures at the measurement locations are very close to the measured values (**Table 7**). The minimum sum of the temperature difference squares is small.

The analysis of the results listed in **Table 7** shows that steam temperature increase is almost the same in each pass. Despite the decreasing temperature difference between the flue gas and steam the increases in steam temperature are: (*T*2 − *T*1) = 42.51 K, (*T*3 − *T*2) = 42.31 K, and (*T*4 − *T*3) = 40.98 K for the first, second, and third pass, respectively (**Table 7**, data set no. 2). The reason for this is a large reduction in the specific heat of the steam with the temperature. To achieve the same steam temperature rise in the third pass, much less heat flow rate is needed in comparison with the first pass.

The uncertainties of the determined parameters **x**\* were estimated using the error propagation rule of Gauss [29–33]. The propagation of uncertainty in the independent variables, that is, in measured steam temperatures *f*2, *f*3, and *f*4, is estimated from the following equation:

$$2\sigma\_{\mathbf{x}\_i} = \left[\sum\_{j=2}^{m} \left(\frac{\partial \mathbf{x}\_i}{\partial f\_j} 2\sigma\_{f\_j}\right)^2\right]^{\frac{1}{2}}, \quad m = 4, \quad i = 1, 2\tag{40}$$

The 95% uncertainty in the estimated parameters can be expressed in the form *xi* = *xi* \* <sup>±</sup> *<sup>σ</sup>xi* , where *xi* <sup>=</sup> *<sup>x</sup>* \* *i* , *i* =1, 2, 3 represent the value of the parameters obtained using the least squares method.

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: 2*σT*<sup>1</sup> <sup>=</sup> 0.74 K and 2*σ<sup>h</sup> <sup>g</sup>* = 0.99 W/(m<sup>2</sup> K). The results are quite satisfactory.


**Table 7.** Comparison of measured and calculated steam temperatures at the inlet to the superheater and at measuring points 3, 4, and 5.
