**2. Numerical Model**

A scheme of the steam pipeline connecting the boiler with the turbine in a 120 MW unit is shown in Figure 1. Superheated steam from the last superheater stage flows into the outlet chambers, connected from both sides of the boiler with steam pipelines (2 *rin =* 245 mm, *sw* = 30 mm). Next, the pipelines are connected through a T-pipe to the main steam pipeline (2 *rin =* 324 mm, *sw* = 40 mm).

**Figure 1.** The pipeline connecting the boiler and turbine.

First, the analyzed domain consisting of steam pipeline and steam was divided into control volumes (Figure 2). In the radial direction, the pipeline is divided into *n* finite volumes, and in the longitudinal direction into *m* finite volumes. Only half of the pipeline was analyzed due to symmetry.

**Figure 2.** Division of the computational domain into finite volumes.

The equation of the transient heat transfer for the pipeline wall has the form

$$\rho\_w \ c\_{p,w} \frac{\partial T\_w}{\partial t} = \nabla \cdot [k\_w(T\_w) \ \nabla T\_w] \tag{1}$$

The heat balance Equation (1) in a cylindrical coordinate system is as follows

$$\rho\_{w}\ c\_{pw}\frac{\partial T\_{w}}{\partial t} = \frac{1}{r}\frac{\partial}{\partial r}\left[r\,k\_{w}(T\_{w})\frac{\partial T\_{w}}{\partial r}\right] + \frac{\partial}{\partial z}\left[k\_{w}(T\_{w})\frac{\partial T\_{w}}{\partial z}\right] \tag{2}$$

Heat balance equations were formed for each node, including nodes in control volumes near the boundary with the steam. For example, the equation for a node *i* for a control volume located in the wall area is

$$\begin{split} \frac{dT\_{\overline{w}\_{i}}}{dt} = \frac{a\left(T\_{w\_{i}}\right)}{k\_{\text{w}}\left(T\_{w\_{i}}\right)} \left| \frac{r\_{j}}{\Delta r} \frac{k\_{\text{w}}\left(T\_{w\_{i}}\right) + k\_{\text{w}}\left(T\_{w\_{(i-1)}}\right)}{r\_{j+1}^{2} - r\_{j}^{2}} \left(T\_{w\_{(i-1)}} - T\_{w\_{i}}\right) + \frac{r\_{j+1}}{\Delta r} \frac{k\_{\text{w}}\left(T\_{w\_{(i+1)}}\right) + k\_{\text{w}}\left(T\_{w\_{i}}\right)}{r\_{j+1}^{2} - r\_{j}^{2}} \left(T\_{w\_{(i+1)}} - T\_{w\_{j}}\right) \right. \\ & \left. + \frac{k\_{\text{w}}\left(T\_{w\_{(i+1+n+1)}}\right) + k\_{\text{w}}\left(T\_{w\_{i}}\right)}{2\left(\Delta z\right)^{2}} \left(T\_{w\_{(i+1+n)}} - T\_{w\_{i}}\right)} \right. \\ & \left. + \frac{k\_{\text{w}}\left(T\_{w\_{(i-1-n)}}\right) + k\_{\text{w}}\left(T\_{w\_{i}}\right)}{2\left(\Delta z\right)^{2}} \left(T\_{w\_{(i-1-n)}} - T\_{w\_{i}}\right) \right] \end{split} (3)$$

where:

$$
\Delta r = \frac{r\_{\text{out}} - r\_{\text{in}}}{n}, \ \Delta z = \frac{L}{m} \tag{4}
$$

The symbol *n* denotes the number of the control volumes in radial directions in the pipeline wall. Over the length of the pipeline, we have *m* finite volumes of length Δ*z*. Similarly, the heat balance equations for the steam region were formed. For example, after transformation, the heat balance equation for a node *i* takes the form

$$\frac{dT\_{f\_{i+1}}}{dt} = -\frac{\dot{m}}{\rho\_f \left(T\_{f\_i}\right)} \frac{T\_{f\_{i+1}} - T\_{f\_{i\cdot}}}{\Delta z} - \frac{h\_{\text{in}} \left(T\_{\text{cz}\_{\text{ci}}}\right) \mathcal{U}\_{\text{in}}}{A \,\rho\_f \left(T\_{f\_{i\cdot}}\right) \, c\_{p,f} \left(T\_{f\_{\cdot}}\right)} \left[\frac{T\_{f\_{i\cdot+1}} + T\_{f\_{\cdot i}}}{2} - \frac{T\_{\text{w,i}(n+1)+1} + T\_{\text{w,i}(i-1)\cdot(n+1)+1}}{2}\right] \tag{5}$$

After formulating the energy conservation equation for all control volumes, a system of ordinary differential equations was obtained. The Runge-Kutta method of the fourth order was used to solve the formed system of ordinary differential equations.

The following boundary and initial conditions were assumed

$$\left.T\_f\right|\_{t=0} = T\_0\tag{6}$$

$$T\_w|\_{t=0} = T\_{w,0} \tag{7}$$

$$\left.T\_f\right|\_{z=0} = f(t) \tag{8}$$

$$\left.k\_w \frac{\partial T\_w}{\partial r}\right|\_{r=r\_{in}} = h\_{in} \left( |T\_w|\_{r=r\_{in}} - T\_f \right) \tag{9}$$

$$\left.k\_{w}\frac{\partial T\_{w}}{\partial r}\right|\_{r=r\_{out}}=0\tag{10}$$

$$\left.k\_w \frac{\partial T\_w}{\partial z}\right|\_{z=0} = 0\tag{11}$$

$$\left.k\_w \frac{\partial T\_w}{\partial z}\right|\_{z=L} = 0\tag{12}$$

The heat transfer coefficient *hin* (Figure 2) on the inside surface of the pipeline was determined from the following formula

$$h\_{in} = \frac{\mathcal{N}u \, k\_f}{d\_{in}} \tag{13}$$

The correlation proposed by Taler [28] was used to determine Nusselt numbers *Nu* in Equation (13)

$$Nu = Nu\_{m,q} (Re = 2300) + \frac{\left(\frac{\gamma}{8}\right) (Re - 2300) Pr^{1.008}}{1.08 + 12.4 \left(\frac{\gamma}{8}\right)^{\left(\frac{1}{2}\right)} \left(Pr^{\left(\frac{2}{3}\right)} - 1\right)} \left[1 + \left(\frac{d\_{in}}{L}\right)^{\left(\frac{2}{3}\right)}\right] \tag{14}$$
 
$$2300 < Re < 10^6, \ 0.1 < Pr < 1000$$

The friction factor *ξ* in Equation (14) is given by the correlation of Taler [29]

$$\xi = \left[ 1.2776 \log(Re) - 0.406 \right]^{-2.246} \tag{15}$$

The symbol *Num*,*q*(*Re* = 2300) designates the Nusselt number at *Re* = 2300 for laminar flow at the tube with constant wall heat flux [30]. At the beginning of the transitional flow, i.e., the end of the laminar flow for Reynolds number *Re* = 2300, the second term on the right-hand side of Equation (14) is equal to zero.

## **3. Inverse Problems in Heating up the Pipeline Connecting the Boiler with the Turbine during the Start-Up of the Unit**

The paper presents a numerical method to determine the steam temperature as a function of time at the pipeline inlet *Tf*(*t*)|z=0m, at which the steam temperature at the pipeline outlet *Tf*(*t*)|z=45m (turbine inlet) is known *f*(*t*) from the measurement. The problem formulated in this way is an inverse transient heat transfer problem. The inverse problem is much more difficult to solve than the direct one, as random errors influence the stability and accuracy of the determination of the inlet steam temperature in the measured transient steam temperature at the outlet of the pipeline.

Random measurement errors influence the solution of IHP significantly. Therefore, the measured temperature variation was approximated by a local polynomial of third-degree with respect to time. Similarly, the accuracy of approximation of the time derivatives from measured steam temperature *f*(*t*) is essential. In numerical methods, for example, moving digital filters or so-called future steps are used to reduce the impact of random measurement errors.

The following assumptions were adopted in the solution of the IHP:


Random measurement errors from the measured temperature and its first-order derivative after time are partly eliminated using moving digital filters. The Beck future time steps [24] in solving IHCP are also applied. Future steps are an effective tool for increasing the stability of solutions to inverse problems and making it possible to determine the time variation of the inlet temperature with a smaller time step. Beck's concept of future time steps is extended in this paper. The time step of the solution of the inverse problem is several to a dozen times larger than the integration step of the system of differential equations present in the solution of the direct problem. The single time step in solving the

inverse problem is divided into smaller time steps, the size of which is derived from the stability condition for the solution of the direct problem.

The temperature *Tf*(*t*)|*z*=0 = *Tf*<sup>1</sup> of the steam at the pipeline inlet was determined sequentially (Figure 2).

The steam inlet temperature in the time interval *t*M−<sup>1</sup> < *t* < *tM+F* (Figure 3) was determined using the least squares method with *F* future time steps. The following squared differences between the calculated steam temperature *Tcalc <sup>f</sup>* ,*m*+1(*t*), and measured fluid temperatures *Tmeas <sup>f</sup>* ,*m*+1(*t*) over the time interval [*t*M−1, *t*M+F] must be minimum (Figure 3).

$$S\left[T\_{f,1}^{\text{calc}}(t\_M)\right] = \int\_{t\_{M-1}}^{t\_{M+F}} \left[T\_{f,m+1}^{\text{calc}}(t) - T\_{f,m+1}^{\text{meas}}(t)\right]^2 dt + w\_r \left(\frac{dT\_{f,1}^{\text{calc}}(t)}{dt}\bigg|\_{t=t\_M}\right)^2 \to \min \tag{16}$$

**Figure 3.** Approximation the time changes of the steam at the pipe inlet by stepwise curve (**a**), and the concept of future time steps, *F* =3(**b**).

The second term on the right-hand side of Equation (16) represents the regularization term of order one in the Tikhonov regularization [31] that is used in this paper. The inlet steam temperature *Tf,*1(*tM*) was determined with a basic time step equal to Δ*tb* = *tM* − *tM*−<sup>1</sup> = *kb* Δ*t*. The time step Δ*tb* is a multiple of the step Δ*t* used to solve a direct heat transfer problem using the finite volume method. At the time *t* = *tM*−1, the steam inlet temperature *Tf,*1(*tM*−1) is known, while the steam temperature *Tf,1*(*tM*) at the time *t* = *tM* was sought. The time step Δ*tb* should be chosen so that the change of steam temperature at the pipeline inlet *Tf,*<sup>1</sup> at time *t* = *tM*−<sup>1</sup> caused the change of steam temperature at the pipeline outlet at time *t* = *tM*.

Step Δ*tb* is *kb* times larger than step Δ*t* used in determining the temperatures of the pipeline wall and steam from Equations (3) and (5), respectively. The step Δ*t* must not be too large for the solution of the system of Equations (3)–(5) to be stable.

To ensure the stability of determining the wall and steam temperature should be the Fourier stability condition for the wall, and the Courant-Friedrichs-Lewy condition for the steam should be satisfied. The allowable time step Δ*t* results from the Courant-Friedrichs-Lewy condition [32].

$$\frac{w\_{z,i}\Delta t}{\Delta z} \le 1, \qquad i = 1, \ldots, m+1 \tag{17}$$

The steam velocity *wi* at the *i*-th finite volume inlet is calculated from the following equation

$$w\_{z,i} = \frac{4\,\dot{m}}{\pi\,\rho\_i\,d\_{in}^2} \tag{18}$$

where the symbol *ρ<sup>i</sup>* denotes the steam density at the *i*-th finite volume inlet.

The steam temperature at the pipeline inlet was determined with the time step Δ*tb* based on the preset steam temperature at the pipeline outlet. The steam temperature at the pipeline outlet was calculated with a time step Δ*t* using the mathematical model of the pipeline developed. The steam temperature *Tcalc <sup>f</sup>* ,1 (*tM*) was determined by minimizing the sum given by Equation (16).

$$\mathbb{E}\left[T\_{f,1}^{\text{calc}}(t\_M)\right] = \sum\_{i=1}^{k\_b(F+1)} \left[T\_{f,m+1}^{\text{calc}}(t\_i) - T\_{f,m+1}^{\text{mas}}(t\_i)\right]^2 + w\_r \left(\frac{T\_{f,1}^{\text{calc}}(t\_{M+F}) - T\_{f,1}^{\text{calc}}(t\_{M-1})}{t\_{M+F} - t\_{M-1}}\right)^2 \tag{19}$$

where *wr* = *w*, *<sup>r</sup>*/Δ*t*, *ti* = *tM*−<sup>1</sup> + *i*Δ*t*, *i* = 1, . . . , *k*2(*F* + 1).

Equation (19) is the discrete form of Equation (16), calculating the integral in Equation (16) using the rectangular method.

For too small values of basic step Δ*tb*, it is not possible to determine the steam temperature variation *Tf*,1 at the pipeline inlet in the time interval *tM*−<sup>1</sup> ≤ *t* ≤ *tM* based on the set or measured outlet steam temperature variation *Tmeas <sup>f</sup>* ,*m*+1(*t*). For a too small time step Δ*tb*, there are instabilities in the determined pipeline inlet temperature.

For the solution stabilization, the future time interval *tM* ≤ *t* ≤ *tM+F* is used (Figure 3). By increasing the analyzed time interval from *tM*−<sup>1</sup> ≤ *t* ≤ *tM* to *tM*−<sup>1</sup> ≤ *t* ≤ *tM+F,* there is a measurable change in steam temperature at the pipeline outlet *Tcalc <sup>f</sup>* ,*m*+1(*t*) as a result of the temperature change of the fluid at the pipeline inlet *Tcalc <sup>f</sup>* ,1 (*t*) during the time *tM*−1. After determination of the fluid temperature *Tf,*1(*tM+F*) at time point *tM+F*, it is assumed that this temperature value occurs only in the basic range *tM*−*<sup>1</sup>* ≤ *t* ≤ *tM.* The analysis at the next time step [*tM*, *tM*+1] is repeated with the time *tM* as the starting point and not the time point *tM+F*.

The temperature of the fluid *Tf*,1 (*tM*) was determined by the golden-section search method [32], for which the sum *S Tcalc <sup>f</sup>* ,1 (*tM*) given by Equation (19) attained minimum.

Figure 4 shows the block diagram of the program for the sequential steam temperature at the pipeline inlet *Tf,*<sup>1</sup> using the golden section method.

First, the limits of the interval [*Tf,MIN*, *Tf,MAX*] are set in which the desired inlet temperature *Tcalc <sup>f</sup>* ,1 (*t*) lies, for which the S sum is defined by Equation (19) reaches a minimum. The steam temperature calculated values at points *XL* and *XR* takes values in the range *Tf,MIN* < *XL* < *XR* < *Tf,MAX*. The value of the factor *k* is *k* ≈ 0.61803398. The golden division factor *k* is a constant factor that reduces the interval value at each iteration until the condition (*Tf,MAX* − *Tf,MIN*) ≤ is satisfied.

**Figure 4.** Block diagram of a program to determine the fluid temperature using the golden division method.
