**1. Introduction**

In inverse problems, the boundary conditions are identified, or the fluid physical properties are determined based on the measured responses of the system.

Typically, inverse heat conduction problems (IHCP) are solved, and much less attention is paid to inverse convective heat transfer issues in the literature.

A method for solving non-linear inverse heat conduction using the space marching method is presented in [1]. The internal surface temperature of the cylindrical element and the heat flux was determined from the measured wall temperature using a temperature sensor near the inner surface of the element. The resulting temperature distribution across the component wall was used to calculate the thermal stresses at the inner surface. The excellent accuracy of the method presented in [1] was achieved by eliminating random disturbances of measured wall temperature and its first-order derivative using moving digital filters.

Inverse problems are often solved for unsteady heat conduction [2,3]. The paper [2] presents the solution to the IHCP using the Trefftz method. The authors presented two methods for solving IHCPs. The conjugate gradient method with the Tikhonov regularization method was used to stabilize the inverse solution when measured temperatures were perturbed with random errors. They showed that the regularization results in a shorter computation time, while methods using iterations do not always lead to convergent solutions. The paper [2] shows that Trefftz numerical functions can be used to solve non-linear IHCP.

**Citation:** Kaczmarski, K. Identification of Transient Steam Temperature at the Inlet of the Pipeline Based on the Measured Steam Temperature at the Pipeline Outlet. *Energies* **2022**, *15*, 5804. https://doi.org/10.3390/en15165804

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 21 July 2022 Accepted: 8 August 2022 Published: 10 August 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

An algorithm is presented in [3] for the solution of an IHCP: for determining the steady-state distribution of the heat transfer coefficient (HTC) on one surface of a slab wall from the known temperature distribution in a plane inside the slab. It was assumed that the thermal boundary conditions on the other wall surfaces were known.

The algorithm is based on the finite volume discretization of the slab and on the formulation and the subsequent inversion of square matrices linking the wall surface temperature and heat flux to that of measured temperature at the inner plane [3].

The paper [4] presents a sequential gradient-based method for non-linear one-dimensional heat conduction. A quasi-Newton update strategy was used to determine the transient HTC on the solid surface. Compared to the traditional sequential conjugate gradient method, the proposed method gave more accurate, reliable, and stable results.

There are considerable stability problems with the solutions of the IHCPs, which are very sensitive to random temperature measurement errors. The paper [5] developed a hybrid algorithm for selecting regularization parameters which give low error variances for estimated parameters. Consequently, the algorithm can reduce the total error and provide better stability for the IHCP solution.

Determining the unsteady temperature of flowing fluid at high pressure from thermometer indications is also an inverse problem [6,7]. Accurate fluid temperature measurement is critical since correct identification of thermal stresses depends heavily on it.

A new method for determining the fluid temperature based on measuring the temperature of the pipeline wall near its inner surface and the readings of a thermometer is presented in [6]. The thermometer for measuring the temperature of a flowing fluid has the form of a solid cylinder in the axis of which the temperature is measured using a thin thermocouple. The fluid temperature determined from the pipeline wall temperature measurement and the temperature determined from the thermometer readings should be equal. From this temperature equality condition, the correlation to Nusselt's number was determined on the outer surface of a thermometer transversely swept by steam or another fluid.

Jaremkiewicz et al. [7] present a method for measuring the unsteady steam temperature based on a new fast response thermometer. The proposed thermometer can be adapted to a wide range of temperatures and steam pressures by optimum design and suitable materials for the thermometer housing. This paper [7] demonstrates the effect of temperature measurement accuracy on the values of stresses calculated in pressure components.

The permissible time variations of fluid temperature and pressure in pipelines and other pressure components can be determined using the procedure outlined in [8], which is based on the European Standard [9] for calculating allowable heating and cooling rates for thick-walled components. The fluid temperature was determined from the solution of the first-order ordinary differential equation for time, considering that the allowable rate of fluid temperature change is a linear function of fluid pressure.

An essential issue of the flexibility of thermal power blocks is to optimize pipeline heating and cooling so that the sum of circumferential thermal and pressure stress at the edge of the pipeline opening is less than the allowable stress. The optimum time variations of the fluid temperature are determined using the solution to the IHP [10]. The heating time of the cylindrical element from the initial temperature to the given final temperature is about 40% shorter than the heating time determined using the European Standard [9].

The interesting IHP was solved in [11]. The rotational speed of the fan forcing air through the heat exchanger was sequentially determined so that the calculated water temperature at the heat exchanger outlet was equal to the set temperature. The golden section method was used to solve the non-linear IHP. Three methods were used to stabilize the determined fan speed: the Tikhonov regularization method, Beck's future time steps, and smoothing of the measured outlet water temperature using a moving averaging filter.

IHPs also occur in identifying leakages in various types of aboveground and underground pipelines. Leakages are usually identified by measuring the flow and thermal parameters of the flowing fluid or medium around the pipeline.

A review and comparative study of computer-based methods for pipeline leak detection were carried out in [12].

A method for leak detection in buried pipelines based on measurements of the temperature and moisture of the soil was developed in [13]. A CFD (Computational Fluid Dynamics) modelling was used to validate the proposed method.

During start-up, shut-down, and load changes, thick-walled steam boiler elements operating under high pressure and high-temperature conditions cause high thermal stresses. In thick walls, significant temperature differences occur during transient operation, accompanied by the formation of high thermal stresses. The largest, in value, stresses and deformations usually occur at the edges of openings. Circumferential stresses at the hole's edge of varying signs, compressive during heating and tensile during cooling, causing fatigue cracking. The stresses lead to low-cycle cracking, failure, and accelerated degradation of the block components [14]. For this reason, thick-walled boiler elements limit the maximum heating and cooling rates during the start-up or shut-down of the boiler.

The heating of the steam pipeline connecting the boiler to the turbine is essential for the start-up of the boiler and turbine [15]. The pipeline design, internal and external diameters, length, and material for a steam boiler type are different and depend on the operating parameters of the steam. The working fluid, i.e., superheated steam fed to the turbine, must have appropriate parameters (temperature and pressure). Moreover, the large wall pipeline thickness and the length pipeline influence the fluid temperature drop. Significant changes in the working fluid parameters substantially affect the lifetime of turbine components. It is not only the turbine rotor at risk but also the turbine casing as thick-walled components.

The literature on modelling thermal-flow phenomena in pipelines is scarce despite its very high practical relevance.

It is difficult to find information in the literature on modelling steam pipelines' transient operation. The steam and pipeline wall temperatures at the given boundary and initial conditions were determined numerically in [16]. The direct heat transfer problem was solved using the finite volume method.

Flow and thermal phenomena in superheater tubes are much more frequently analyzed [17–21]. Due to the small wall thickness of superheater tubes, little attention is paid to the temperature distribution over the wall thickness. The wall thickness of the pipeline is much greater. There are often holes in the walls of the pipelines with a high concentration of stresses at their edges. A high concentration of thermal stresses also occurs in Y- or T-shaped tees in the pipelines. For this reason, pipelines' flow-thermal and strength analysis is highly important for their safe long-term operation.

Two types of models analyze heat transfer in pipelines and tube heat exchangers. The first is a model with distributed parameters, in which the system of partial differential equations is solved to determine the fluid and wall parameters [21,22]. The second model is a model with concentrated parameters, described by the system of ordinary differential equations [23]. The solution to the IHCP can be used to determine the temperature distribution in the wall of pressure thick-walled elements [24–26]. Solving inverse problems is very sensitive to random measurement errors [24,25]. Therefore, the measured time changes in the temperature are approximated by an appropriate function, or digital filters are used to eliminate random errors from the input data [18,27].

This paper developed a new numerical model of the steam pipeline. The pipeline connects the boiler to the turbine. The steam turbine works at specific input parameters of superheated steam. To ensure its safe and trouble-free start-up operation, the changes in time of the input steam parameters cannot be rapid, and the steam temperature cannot differ more from the rotor and turbine casing temperature compared to allowed values. Too rapid a steam temperature change may cause high stresses in the pipeline and the turbine's structural components. The steam temperature at the outlet of the pipeline connecting the boiler and the turbine, i.e., before the turbine, depends on the steam temperature at the steam pipeline inlet.

In this work, the new IHP was solved to determine the time variation of temperature at the pipeline inlet with a known temperature at the turbine inlet. The time variation of the temperature at the turbine inlet is due to the conditions of safe turbine start-up or the steam temperature before the turbine is known from measurements. To the author's knowledge, the IHP solved in this paper has not yet been analyzed in the available literature.
