Multi-objective optimization of hot steam injection variables to control wetness parameters of steam flow within nozzles
Introduction
Steam turbines play a major role in power and energy industry; therefore, any development in their efficiency and performance saves costs of power generation. In order to meet this target, loss has to be reduced. One of the sources of loss in steam turbines is the formation of two-phase flow in the last stages of low-pressure steam turbines, which also causes mechanical damage to the blades of the last rows. Any improvement in the last stages of low pressure steam turbine causes a significant increment in the entire cycle efficiency [1].
During the expansion process in the last stages of low-pressure steam turbines, superheat steam crosses the saturation line and enters the two-phase flow region. After a short delay, a large number of fine droplets of liquids are formed. These droplets move with the steam and collide the moving blades, causing the erosion of rotor blades along the downstream of the turbine, which affects the safety and reliability of turbine operation. The extent of erosion damage depends on the number of droplets, size of droplets, impact velocity, and impact angle [2], [3]. Furthermore, the nucleation of liquid droplets and the release of latent heat create a thermal shock which is the source of a thermodynamic loss called wetness loss. This loss reduces the performance of the stages and decreases the adiabatic efficiency. Baumann [4] predicted that the increment of 1% mean wetness fraction leads to a decrease of about 1% stage efficiency.
The situation will be intensified when blade height is increased to allow higher steam mass flow rate, because a higher speed of the blades near the tip region makes the collision of liquid droplets more harmful.
Due to the above considerations, many research have focused on methods to prevent the increasing liquid mass fraction inside the systems by removing the liquid droplets, decreasing the droplet averaged radius, or reducing the number of liquid droplets. In this respect, various systems such as a suction slot [5], [6], [7], [8], hot air injection [9], the combination of blowing and suction slot [10], blades heating [11] and separator-reheater [12], [13] have been elaborated.
One of the methods to control the wetness of steam is the injection of dry steam into the passage of blades. Xu et al. [14] proposed the injection of hot steam at the trailing edge of blades to reduce the outlet liquid mass fraction. Furthermore, Gribin et al. [15], [16] investigated the injection effects on liquid droplet path and radius. However, the optimum location of steam injection and its thermodynamic properties have not been covered in detail in the aforementioned studies. In a previous attempt, authors of the present study developed a one-dimensional, in-house wet steam code to ensure the effects of steam injection in a supersonic flow within convergent-divergent nozzles. It was found that not only the size of liquid droplets can be reduced but also the wetness fraction is also decreased.
In the present paper, a two-dimensional hot steam injection within steam flow in convergent-divergent nozzles is modeled to control the level of steam wetness and droplet averaged radius. Steam is injected at different locations in the downstream and upstream of nozzle throats with various injection temperatures, injection angles, and even various mass flow rates. Results are studied in detail to find the optimal choice of different parameters of injection variables to exploit the maximum potential of steam injection method.
Due to the different behaviors of parameters, a multi-objective optimization is also employed to find the optimal solutions which are called Pareto-optimal solutions [17]. Following the research by Marler and Arora [17] on several methods and algorithms for multi-objective optimization, a two-dimensional analysis of wet steam flow is coupled with a multi-objective genetic algorithm to find the optimal variables of the injection in order to control the droplet averaged radius as well as the liquid mass fraction at nozzle outlet.
Section snippets
Governing equations
The Eulerian-Eulerian approach for modeling wet steam flow is selected because of its potential for developing three-dimensional models in future investigations. Compressible Navier-Stokes equations together with two additional transport scalar equations of droplet number per unit volume and liquid mass fraction are the governing equations of wet steam flow.
In the present mathematical modeling, the slip velocity between liquid droplets and dry steam is ignored and the interaction between
Numerical validation
In order to validate the basic numerical model of wet steam flow, the nozzle of Moses and Stein [28] is simulated and the results are compared with the experimental data. Moses and Stein conducted a series of tests by applying various total pressures and temperatures at the inlet of nozzle. Three sets of boundary conditions are given in Table 1.
It is worth saying that an extensive grid study has been conducted to find the appropriate number of grids. In all validation cases, 9056 quadrilateral
Hot steam injection
The main objective of the present study is to control the wetness fraction and size of liquid droplets by the injection of a required amount of hot steam into a suitable location in a nozzle. This is similar to extracting hot steam in the primary stages of a steam turbine and injecting it in the stages where steam nucleation is possibly formed. Therefore, choosing the appropriate injection pressure and temperature determines the appropriate stage of turbine when steam should be extracted.
Optimization of injection variables
There are different methods to optimize the function of a process such as gradient-based method, neural network, and genetic algorithm. In the present work, genetic algorithm (GA) is chosen due to its effectiveness. In order to provide the required data pool for optimization, 273 different combinations of design variables are modeled (more details on the optimization process can be found in Refs. [29], [30], [31]).
Optimization results
Reduction of the radius of droplets has a significant effect on the decrease of blade erosion, therefore, the optimal cases among Pareto-optimal solutions are selected by allocating a larger weighting factor to this parameter. The results of optimal design variables are given in Table 6. As it shows, injection of hot steam downstream of the throat at 1.327 (x = 0.109 m), which is near the Wilson point at an injection angle of 90° through a slot width of 0.3 mm, is the optimum case. The
Conclusions
Two dimensional analysis of hot steam injection in a convergent-divergent nozzle for the purpose of controlling the radius of liquid droplets and reducing the erosion damage are presented in this study. It was shown that the injection of steam with a higher temperature can reduce the size of droplets and decrease the liquid mass fraction at the exit plane of the nozzle. In order to find the optimum values of injection parameters, a multi-objective optimization of the injection is undertaken
Acknowledgements
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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