Shape and operation optimisation of a supercritical steam turbine rotor

Shape and operation optimisation of a supercritical steam turbine rotor

Energy Conversion and Management 74 (2013) 417–425 Contents lists available at SciVerse ScienceDirect Energy Conversion and Management journal homep...

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Energy Conversion and Management 74 (2013) 417–425

Contents lists available at SciVerse ScienceDirect

Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Shape and operation optimisation of a supercritical steam turbine rotor G. Nowak ⇑, A. Rusin Silesian University of Technology, ul. Konarskiego 18, 44-100 Gliwice, Poland

a r t i c l e

i n f o

Article history: Received 24 October 2012 Accepted 13 June 2013 Available online 26 July 2013 Keywords: Ultra-supercritical turbine Optimisation Thermal stress

a b s t r a c t The presented study discusses the problem of shape optimisation of selected areas of the rotor of the high pressure part of an ultra-supercritical steam turbine together with the optimisation of the turbine startup method, using the maximum stress objective. The analysis relates to the rotor of a conceptual ultrasupercritical turbine which is characterised by high parameters of operation. The consequence is that the machinery components are subjected to significant stress, which further results in a substantial reduction in its life and reliability. These adverse effects can be contained in two ways, i.e. by optimising the shape of the rotor areas characterised by high stress values and by optimising the method of the turbine startup. In the case of the rotor under analysis, it is the thermal stress caused by large temperature gradients occurring in unsteady states of operation that has a predominant impact on the stress level. The performed research prove that the manner in which the power unit start-up is initiated and carried out depends largely on the limitations of the materials used to make the machinery components. This, in turn, has an impact on the assessment of the power unit in terms of energy and economy. The obtained optimisation results translate directly into the power unit energy effectiveness. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The need to meet environmental requirements concerning the reduction in greenhouse gas emissions necessitates the development of highly efficient technologies of power units for supercritical steam parameters. This in turn entails a further development in the design of both boilers and turbines. The components of these machines operate not only under a greater load but also in higher temperatures. To manufacture them, it is then necessary to use new materials resistant to high-temperature failure processes, such as creep and fatigue. Apart from the selection of the appropriate material, the choice of the design form is also essential [1]. The shape and the size of a component determine the level of thermal and mechanical stress that will arise at different phases of the operation, which has a decisive impact on the component life [2– 4]. From the point of view of operational safety, the turbine rotor is an element of particular importance. The temperature gradients appearing in it in unsteady states are the cause of significant stress [5]. One of the ways to reduce the stress and to improve the rotor life is the optimisation of its shape. In the further part of the paper, a mathematical model for the optimisation of the turbine components is defined. The basic objective function assumed here is the rotor life. The optimised values are selected rotor dimensions, especially in stress concentration zones. The fundamental problem ⇑ Corresponding author. Tel.: +48 322372822. E-mail addresses: [email protected] (G. Nowak), [email protected] (A. Rusin). 0196-8904/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2013.06.037

in the optimisation of the shape of components as complex as the rotor is the need to model the stress state in their subsequently changing forms. Consequently, the modelling has to be done with a variable numerical grid. In this study, the response surface method is used, which makes the problem easier to solve thanks to appropriate approximations.

2. Optimisation problem formulation Each structure treated as an object of design is characterised by a number of features which are given specific values in the design process. The selection of those features has a decision-making nature and is conditioned, to a large extent, by the system of objectives. To facilitate the decision-making processes and to make them more objective, the technique referred to as the optimum design method is often used. The optimum designing of a structure is aimed at the creation of the optimum design, i.e. one that not only makes it possible to meet all the requirements the structure is faced with, but also ensures that the structure is the best in respect of the selected optimisation objective. By assuming the optimisation objectives it is possible to build the mathematical model of the structure which includes: – design variables X = (x1, x2, . . ., xn), i.e. the values to be optimised; – constraint functions wi ðx1 ; x2 ; :::; xn Þ 6 0, defining the permissible area of the variability of the optimised values;

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– objective function V(x1, x2, . . ., xn), which depends on the variables under optimisation and which constitutes the mathematical notation of the optimisation objective. In the case of heat turbines operating in creep and fatigue conditions, one of the basic design objectives is the life objective, which can be written as:

V ¼ te

ð1Þ

where te is the component life, i.e. working time to failure. Therefore the aim of the optimisation is to maximise the component life.

V ! V max

ð2Þ

Certain dimensions of the component may be the variables to be optimised. This means that the desired structure form of the rotor is the one with the longest operation time. This time is limited by the damage done to the component due to life degradation processes. Because of the fact that both creep and fatigue processes depend on the stress level, the optimisation objective related to working time may be replaced with the optimisation objective related to stress minimisation. Therefore the following may be assumed:

V ¼r

ð3Þ

V ! V min

ð4Þ

The dimensions to be optimised will only be those that have the most significant impact on the stress level of the component. Typically, these are the dimensions in stress concentration zones, e.g. curvature radii. The material properties, operation conditions, as well as the main dimensions of the component may be the optimisation parameters. Solving the optimisation problem presented above with mathematical methods, it is necessary to perform multiple calculations of the value of the objective function, which in this case is the maximum stress level in a given component. To determine the stress, the calculations of transient temperature distributions in the entire working cycle have to be made, and then, based on their results, stress distributions are calculated. Each time the calculations have to be performed for a different rotor form, i.e. for a different numerical grid. Therefore, obtaining the optimum solution is very time-consuming. One of the ways to avoid this particular inconvenience is the application of certain approximation techniques discussed in the further part of the paper.

3.1. The design of the experiments The process begins with setting the range of variability of individual design parameters of the simulation model. Based on that, the design of experiments is created that will be used to determine the response surface of the model [8]. Depending on the analysed problem and on the character of the correlation between design parameters and output values, various designs of experiments are assumed. The simplest is what is referred to as the screening design. It makes it possible to examine the impact of the assumed design variables on the system response, and to select for further analyses only those variables which affect the response in a significant way. In this design, permutations of the upper and lower constraints of all design variables are used. Because for each variable calculations are made for two levels of its value (the upper and the lower constraint), the design is referred to as two-level and marked as 2N, where N is the number of design variables. However, it allows a linear description of the input–output effect only. Also, the possible impact of the interaction between design variables on the result values is lost here. These inconveniences are removed by the application of three-level designs (3N) and multi-level, typically five-level ones (5N), which allow a non-linear description of the impact of parameters. The number of conducted experiments (in this case – numerical simulations) rises because it is equivalent to the number of levels raised to the power equal to the number of design variables. Thus, an increase in the number of design variables for a given design of experiments results in an exponential rise in the number of their possible combinations. The designs of experiments mentioned above are referred to as full factorial designs which include all possible combinations of the input parameter values. However, they become impractical for a bigger number of design variables and then designs referred to as fractional factorial designs are employed. These designs no longer include all possible combinations of the input values, but only a part of them. The reduction in the number of combinations of the values of variables may entail obtaining results of a poorer quality. Therefore, in order to minimise this effect, it becomes necessary to carefully select the combinations of the values of design variables, as well as their appropriate number. One of the fractional factorial designs which is often used is the Central Composite Design (CCD) [6], which is a two-level design expanded with an axial experiment design. This means that for each variable there are two extreme realisations available (at the edge of the area), plus a central realisation as well as axial ones in the middle of the intervals of all variables. In this case, the number of experiments is reduced compared to the full factorial design, e.g. for three design variables the fall is from 27 to 15 (Fig. 1).

3. Response surface methodology The optimisation methodology used in the calculations is based on the Response Surface Method (RSM), which is a set of methods of the mathematical analysis and statistics [6,7]. For this purpose, experimental numerical studies are used, i.e. numerical simulations of the real process which is subjected to optimisation. The simulations, called the design of experiments (DoE) consist in finding the response (of the initial values) of a given process to independent input parameters (design variables) which affect the process [7]. In other words, the design of the experiments consists in a series of numerical simulations for changing input data with a view to identifying their impact on the output values. The procedure is to allow the approximation of the real (modelled) process with the use of an assumed approximation function, which in turn will be the object of the optimisation process. The aim of such an approach is to reduce the computational cost related to the optimisation processes which need time-consuming numerical analyses.

Fig. 1. Central composite design.

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This kind of design makes it possible to use a polynomial of the second order to build the system response model. 3.2. Response surface The performance of an appropriate type of the design of experiments allows the transition to another optimisation stage, namely – to the approximation of the obtained system responses. The standard response surface is made on the basis of polynomials of the second order with the use of the regression analysis [8]. This analysis is based on finding the mathematical dependence which describes the relationship between data. Generally, the mathematical model assumes the form:

Y ¼ f ðX; AÞ þ e

ð5Þ

where X is the vector of independent variables, A denotes the vector of regression coefficients, and e is the random error with expected value E(e) = 0. Notation f(X, A) in the equation is the set of basic functions of regression. A slightly different approach is realised in the Kriging algorithm [9], which assumes a relationship between variables in the following form of the polynomial and deviation combinations:

Y ¼ f ðXÞ þ ZðXÞ

ð6Þ

In the relation given above f(X) is the set polynomial function, while Z(X) is the realisation of a random process with a Gaussian distribution for an average equal to 0, variance r2 and non-zero covariance. The polynomial proposed in the above formula is a ‘‘global’’ model of the relationship between variables, whereas the other factor creates ‘‘local’’ deviations of approximated points. 3.3. Optimisation method The search for optimum solutions may be based on various optimisation methods, depending on the problem to be solved. One of the most commonly used optimisation algorithms, due to its universal nature, is the evolutionary approach in the form of the genetic algorithm [10]. It makes it possible to solve singleand multi-objective optimisation tasks with a continuous and discreet objective function. Genetic algorithm is given extensive attention in many works concerning engineering optimisation tasks [11,12]. This particular method is used in this paper to find the minimum of the objective function described in [12]. The approach generally requires that very many potential solutions be considered, especially at a large number of decision variables, which may involve a huge computational cost. This computational costs depends mainly on the computational model under consideration. In this case the genetic algorithm operates combined with the response surface method and therefore it is not the FEM model of the object under analysis that is directly applied for the optimisation purposes, but a meta-model which is a functional approximation of its response to a change in the decision variables of the task. Thus, even if it is necessary to analyse thousands of possible solutions, the computational cost related to this task is slight. 4. Shape optimisation of the rotor selected areas The object of the presented analyses is a conceptual rotor of the high pressure part of a supercritical steam turbine. The working steam parameters at the turbine inlet are assumed as: Tin = 650 °C and pin = 30 MPa. The rotor is a drum structure of a reaction type turbine with 25 stages (Fig. 2). The steam inflow is radial. The expansion line of steam in the flow system is determined through thermal-flow analyses [13]. Based on it, the conditions of the heat transfer on individual surfaces of the rotor are defined. These data

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are used further to carry out the presented optimisation calculations which are performed using the Ansys software. The solving of the optimisation task starts with preliminary thermal-strength calculations of the turbine rotor. As a result, two stress concentration zones are selected: the blade grooves and the shaft undercut in the balance piston area. Fig. 2 presents the contours of the maximum stress in stress concentration zones. Preliminary calculations were performed for the reference start-up marked as ‘‘initial curve’’ in Fig. 9. The results of these calculations indicate that the values of the maximum unsteady stress in the groove exceed 550 MPa, whereas in the undercut they are close to 500 MPa. Considering the materials which are used at present, such big values of stress are unacceptable in terms of safety and reliability of the turbine operation. 4.1. Shape optimisation of the blade groove It is preliminarily assumed that the blades are mounted on the rotor by means of typical hammer grooves. However, the thermalstrength numerical simulations of the rotor heating process show that the obtained maximum stress level is very high (Fig. 2), which might impair the machine operation reliability and limit its life. The maximum stress values are obtained in the bottom corners of the groove. The appearing stress results from the radial gradient of temperature in the rotor. Due to it, the deformations close to the rotor external surface, where the temperature is higher, are much bigger than those inside. In this way, the bending of the grooves occurs and the maximum bending moment appears on the groove bottom edge. The corner causes an additional stress concentration in its area. To minimise this effect, measures are taken to reduce this stress by redesigning (optimising the shape of) the groove. The shape optimisation task is solved using the response surface methodology. The determination of the response surface is implemented by means of the Kriging model based on the CCD type of the design of experiments. Depending on the number of design variables, the plan requires calculations for a different number of design points: 15 in the case of the smallest number of 3 variables, and 151 for the biggest case under analysis with 11 variables. Consequently, for each design point, i.e. for the set values of design variables, thermal-strength simulations of the analysed rotor are carried out with a view to defining the dependence (or its approximation) of the stress resulting during the turbine start-up process on the decision variables. Example curves illustrating the dependence of stress in the groove bottom on selected design variables are shown in Fig. 3. Each of the curves is defined for its own domain, and its shape is determined at fixed values of the remaining decision variables of the task. As all decision variables have an impact on the groove bottom shape, a change in any of them results in a change in the shape of the curves presented in Fig. 3. The optimisation of the obtained response surface is carried out using the genetic algorithm implemented in Ansys software. The groove bottom redesigning process calls for assuming a description of the shape of this area by means of design variables, whose values become the object of the optimisation. The idea of the optimisation is to obtain the biggest possible reduction in maximum stress in the groove bottoms which occurs at the turbine start-up only by means of changes in the design form of these areas. To achieve this, numerous different variants of the optimisation calculations are performed, using a different mathematical description of the groove shape. Fig. 4 shows the analysed groove modelling variants with marked dimensions which constitute decision variables of the optimisation task. The constraints are in this case defined as constants in the form of the upper and lower bounds of the domain of each variable to eliminate shapes of the groove surface which are physically impossible. This is essential

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Fig. 2. The rotor geometrical form with marked areas to be optimised.

Fig. 3. Dependence of maximum stress on selected design variables (markings as in Fig. 4f).

from the point of view of the automation of optimisation computations, where one incorrect (e.g. loop) configuration causes the computational process to stop. The objective function in the presented optimisation task is to minimise the maximum values of von Mises stress during the unsteady phase of the turbine operation – the start-up.

reqm ðx1 ; x2 ; . . . ; xn ; tÞ ! min

ð7Þ

Fig. 5 is an overall presentation of the results of this optimisation for the configuration shown in Fig. 4. The analysis starts with the optimisation of the fillet radii of the corners of the hammer groove and of its height. It is quickly found that it is the radius of the bottom corner that has the biggest impact on the stress value, and, which seems fairly obvious, the bigger the radius, the smaller the stress values (b). However, increasing this radius is limited due to the need to fix the blade root mounted in the groove. Therefore, in the next step it is decided to perform deeper undercuts of the groove corners, thus lengthening the curvature radii (c). In this case, the undercut centre is located at the same place as the centre of the initial groove curvature and it is the radius which is optimised. The use of an undercut with a constant radius does not bring the desired results and even though it is possible to use curvatures with longer radii than before, the stress level is a bit higher. This is probably the effect of the rise in the groove bending moment caused by an increase in the arm of the acting force. The observations made in

the process lead to an attempt to model this area with a curve with a variable curvature, without deepening the groove unnecessarily, i.e. without increasing stress. Therefore, an analysis is performed of the possibility of modelling the groove bottom by means of spline curves whose shape is controlled using a specific number of control points which belong to these curves. A variant with two curves (one curve in each corner (d)) and a variant with one curve describing the entire groove bottom (e and f) are analysed. Both approaches prove to be equally good. Nevertheless, the lowest maximum stress values are obtained for variant (f) with one curve defined by 7 control points. This gives 11 design variables (coordinates of the points) because 3 out of 14 coordinates are pre-determined (the axial coordinates of the extreme points and of the central point of the curve). In variant (e) 6 coordinates are predetermined, which gives 8 design variables, and the obtained stress values are higher than in the case with two curves. The variant with the lowest stress level (f) is selected as the final solution to the set optimisation task. In it, relatively gentle curvatures in the stress concentration zones are obtained owing to the assumed way of the groove bottom modelling. This shape of the groove bottom causes that the obtained level of the maximum stress of 336 MPa during the analysed start-up is by almost 200 MPa lower than in the initial phase (an almost 36% improvement). It should be emphasised that the presented stress values do not take account of the loads caused by the blade inertial force and they include only the thermal stress resulting from heat exchange between steam and the shaft. Taking account of the loads caused by the blade impact causes a reduction in von Mises stress in the first stage of the rotor at nominal rotation speed by approximately 45 MPa. This results from the fact that the dominating thermal and inertial components feature opposite senses. The groove shape assumed as the final one is presented in Fig. 6 together with maximum von Mises stress distribution. The groove shape obtained in the optimisation process is asymmetrical, which results from the temperature gradient asymmetry on both sides of the groove. This shape ensures almost identical stress values on both sides of the groove. These values are the highest at the flattest parts of the bottom. Therefore the relatively big curvatures at the ends and in the centre of the curve which models the bottom do not cause the effect of stress concentration. It should also be noted here that the thermal stress occurring at the groove bottom is a compressive stress which is partly lessened by tensile components resulting from the blade inertial forces. 4.2. The shaft undercut optimisation The next step after the design structure of the blade groove is determined is to redesign the shaft undercut in the balance piston area. In this case, the change in the rotor diameter is more than double. Moreover, this is an area that separates the hot inlet zone

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421

Fig. 4. Groove bottom modelling variants with marked decision variables.

550

Mises stress MPa

a

c

500

e

450 400

b

350

f

d

300 1

2

3

5

6

Fig. 5. Optimisation results for different variants of the groove shape description. Fig. 6. The final groove shape with maximum stress contour.

of the rotor from the much cooler zone of the end sealing and of the bearing, which results in considerable gradients of temperature. Due to that, stress concentration arises in the analysed corner. The stress values may be close to the material yield point (at small radii of the fillet curvature). The optimisation process starts with the selection of the fillet curvature radius of the corner under consideration. This, however, does not give satisfying results, i.e. a sensible increase in the curvature radius does not result in a significant reduction in stress. Therefore it is decided to model the undercut by means of a spline curve based on 4 control points (Fig. 7). A modification of the location of these points causes a change in the corner undercut shape, which results in – contrary to a simple fillet – a fillet with a variable curvature. The presented method of the corner modelling leads to the formulation of the shape optimisation task using 6 design variables which are the coordinates of the control points belonging to the

curve; the extreme points can move in one plane only. The location of the inside control points is determined in relation to the location of the extreme points so that correct curves (without loops) may be obtained at any configuration of the coordinates. Fig. 7 presents all the design variables on the optimised geometry of the rotor undercut. Owing to this shape, the maximum von Mises stress may be reduced by about 30%.

4.3. Optimisation of the rotor inner chambers Despite the conducted optimisation processes of the groove shape and of the rotor undercut, which result in a significant reduction in stress during the unsteady phases of operation, the maximum stress level in the rotor during the turbine start-up is

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Fig. 7. The final shape of the rotor undercut with marked design variables.

Fig. 8. The rotor with hollowed out chambers.

still high, which may reduce its life. In order to prevent this, steps have to be taken with a view to a further reduction in the level of unsteady stress. Because the large gradients of temperatures during the initial heating phase are directly responsible for the thermal stress values, methods have to be found to level the temperature fields. One of the solutions may be a reduction in the amount of material in the areas of the highest gradients of temperature. Therefore it is decided that the rotor should be hollowed out at the steam inlet area. Two cases are analysed, i.e. with one bigger chamber and with two smaller ones. The introduction of these chambers causes that in their areas the temperatures are levelled relatively fast in the radial direction, which substantially reduces the thermal stress observed in the rotor blade grooves. Because the location and the size of the inner chambers have an impact on the thermal and strength state of the rotor, an optimisation task is formulated in this case as well. The object of the task is to find the optimum location and size of the chambers. The optimisation objective, like previously, is to minimise the stress during the turbine start-up. The introduction of the chambers results in a new stress concentration zone – the corners of these chambers. Therefore the radii of the curvatures of the rotor inner chambers are included in the set of the design variables of the optimisation task which is being solved. Fig. 8 presents a model of the rotor fragment with two inner chambers with additionally marked design variables of the optimisation task. The presented configuration of the chambers is

already an optimisation result. An analysis of the cases with one and two chambers gives very similar results in terms of the maximum stress values. However, in order to ensure the rotor appropriate rigidity, the length of a single chamber is limited and, for this reason, the use of two chambers seems to be more favourable in this respect. The reach of the chambers (in the axial direction) determines the areas where stress occurs in the blade grooves because the maximum thermal stress appears in the first groove right after the chambers, i.e. in the presented case – in the groove of the fifth stage. Moving this maximum further into the flow system brings measurable benefits because lower and lower gradients of the temperature in the rotor are observed in subsequent stages (a smaller heat flux is received by the rotor due to a smaller temperature difference between the flowing steam and the rotor). Besides, the maximum stress occurs at a lower temperature of the rotor material. Because the material features better mechanical properties in a lower temperature, the safety margin in this case is bigger.

5. Optimisation of the turbine start-up curve After the optimisation of the critical areas of the turbine rotor, the next task is to optimise the turbine start-up curve. The analysis in this case is focused on the curve of the change in temperature at the turbine inlet. The aim of the task is to shorten the time of the start-up process, maintaining safe operation of the turbine.

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Fig. 9. The curves of the changes in the live steam temperature during start-up.

Fig. 10. Dependence of the safety factor on the value of multipliers.

Shortening the start-up process is of significant importance because it limits the non-productive phase of the power unit operation. On the other hand, it intensifies the unsteady processes of the heat transfer between the working medium and the material of the machinery components, which entails higher thermal loads of these components. Therefore, from the point of view of the operational safety and reliability, the start-up process should be run more slowly. This means that two contradictory objectives have to be dealt with here. The solution to this problem has to be found by means of optimisation in such a way that the start-up is carried out fast enough without exposing the power unit components to excessive (hazardous) loads. The task is solved through a modification of the previously assumed curve of the steam temperature change at the turbine inlet. A characteristic with 76 time steps (3 min each) is divided into three ranges: 0–15, 15–30, 30–76, in which the rate of the increase in the steam temperature is optimised (Fig. 9). In each of the defined ranges, the curve local slope coefficient is multiplied by an appropriate multiplier (ARG1, ARG2 and ARG3) which is different for each range, and the resulting stress is observed. The value of each multiplier may vary in the range of 0.8–1.5. The task is thus defined by three design variables – the start-up curve multipliers in individual ranges, and the optimisation objective is the safety factor defined as:

(UB = 1.5). The impact of decision variables on each other and – in consequence – on the safety factor can be seen in the figure. It is therefore impossible to present the system response in the form of a single function. Instead, a set of approximation functions has to be employed to do it. The search for the optimum configuration is conducted using a screening method which results in sets of multipliers that satisfy the task objective of the safety factor level. The obtained results compared to the initial start-up curve are presented in Fig. 9. The chart also shows the area where the optimised curve could appear. The boundaries of this area present the start-up curves for extreme values of the multipliers (0.8 and 1.5). The optimised curve features a higher rate of the increase in the steam temperature (ARG1 = 1.4) in the initial phase of the startup and a lower rate – in the subsequent phases (ARG2 = 0.85, ARG3 = 0.89). With the start-up process run in this way, the safety factor reaches the assumed limit value of s = 1.2. As it can be seen in the presented chart, the shortening of the entire start-up process is slight. The live steam nominal temperature at the turbine inlet is reached only a few minutes faster. However, the stress maximum is shifted in time and occurs at a lower temperature of the rotor material. In respect of the deterioration in the strength parameters resulting from temperature, the time shift mentioned above is beneficial because the safety margin between the operational and permissible stress values is bigger. Globally, the highest von Mises stress level is observed in the corner of Chamber 1, but the other selected areas also feature high stress values (Fig. 12). The optimisation does not result in stress reduction. In some cases the stress level is even higher, but in each case there is at least an hour’s shift of the stress maximum towards the start-up beginning. Analysing the heating process, it can be seen that this shift results in the occurrence of the biggest loads at the material temperature which is by at least 150 °C lower than in the case before optimisation (Fig. 11). Such a temperature difference raises for example the material yield point by approximately 20% (Fig. 13).



ry reqm

ð8Þ

where ry is the material yield point (Fig. 10) for the nominal working temperature (650 °C), and reqv denotes the operational von Mises stress. The minimum safety factor assumed during the calculations is 1.2. Besides the safety factor, all three design variables are the task additional objectives. In this way, the optimisation is carried out for objectives related to the safety factor and the curve multipliers maximisation. Like before, the set task is solved using the RSM with the Kriging meta-model which precisely maps the relationships between the input data and the safety factor (correlation coefficient R = 1). The relations between the start-up curve multipliers and the stress coefficient are presented in Fig. 10. The course of the curve for each multiplier is shown at a fixed value of the other two, for the lower bound of the domain (LB = 0.8), mean value from the interval (MEAN = 1.15) and for the upper bound of the set of solutions

6. Summary The paper presents an optimisation of the rotor shape and of the method of running the start-up process of the steam turbine for ultra-supercritical parameters. The analysed turbine is a conceptual

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Fig. 11. The temperature and von Mises curves in the rotor selected areas for the initial and optimised start-ups.

Fig. 12. Maximum von Mises stress in the rotor after shape and start-up optimisation.

Fig. 13. The rotor material yield point depending on temperature.

machine and the presented calculations relate to the rotor of its high pressure part. The very high working parameters of the machine result in the occurrence of hazardous thermal stress in the transient phases of the operation. Within the study, the stress concentration zones (the blade grooves and the shaft undercut in the balance piston area) are selected, and an optimisation of the rotor

shape in these areas is performed. These measures allow a substantial reduction in the maximum stress that arises during the turbine start-up. Additionally, chambers are hollowed in the front part of the rotor, which results in a further decrease in stress. In the next stage, the method of running the turbine start-up process is optimised so that, keeping a safe stress level in the rotor, this non-productive phase of operation can be shortened as much as possible. Although the conducted optimisation does not result in a significant shortening of the time of the start-up process, another beneficial effect of the start-up characteristic modification can be observed. The start-up process may be run faster in its initial phase and then it may slow down. This results in a shift of the stress maximum towards the beginning phase of the start-up with a lower temperature of the rotor material. At a lower temperature the rotor features higher strength parameters, which makes its operation safer. The issues discussed here are really essential because, as experience shows, the first supercritical units made recently are beset by continual failures and their operation continually creates significant problems. Therefore, further research and testing in this area are by all means desired and each new experience gained in the field is very important. Acknowledgements The results presented in this paper were obtained from research work co-financed by the Polish National Centre of Research and Development in the framework of Contract SP/E/1/67484/10 – Strategic Research Programme – Advanced technologies for

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obtaining energy: Development of a technology for highly efficient zero-emission coal-fired power units integrated with CO2 capture.

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