Geometry-Load Based Hybrid Correction Method for the Pre-Deformation Design of a Steam Turbine Blade

: To solve the problem of the slow convergence of the geometry-based correction (GC) method in the design of a steam turbine blade, this paper proposes a geometry-load-based hybrid correction (GLHC) method. In this method, the deformation of the blade caused by the centrifugal load is still corrected by the GC method, while the deformation caused by the aerodynamic load is corrected by the load-based correction (LC) method instead of the GC method. The LC method updates the cold shape of the blade by reversely applying the aerodynamic load to the ideal shape according to the balance between the internal force generated by the deformation of the blade and the aerodynamic load acting on surface of the hot blade shape, thereby reducing the number of iterations by reducing the shape deviation in each step of the iteration. The GLHC method, which combines the GC and LC methods, is used to improve the design process. The e ﬃ ciency of the GLHC and GC methods are compared with the maximum number of position deviations of the corresponding mesh nodes between the hot blade and ideal blade shapes, which acts as the criterion. The results show that the GLHC method reduces the number of iterations.


Introduction
Steam turbines are important power equipment for industrial production. During the operation of a steam turbine, high-temperature and high-pressure steam act on the blade surface and drive the spindle to rotate, thereby realizing energy conversion and output [1,2]. Therefore, the blades of the steam turbine are very significant for the efficient conversion of energy, especially those in the low-pressure stage [3][4][5].
The traditional design method mainly involves obtaining an ideal blade shape according to the aerodynamic performance of a steam turbine in operation [6,7], then the strength, modality and life of the blade are analyzed to ensure its reliability [2,[8][9][10][11][12][13].
However, the deformation of the blade under multiple complex loads causes the actual shape to deviate from the ideal shape obtained by theoretical design [14], which influences the aerodynamic performance of the blade and reduces the efficiency of the turbine [7,15,16]. Therefore, the ideal shape of the blade cannot be directly used for manufacturing [17].
The pre-deformation design method is proposed to solve this problem [5,[18][19][20][21][22][23][24][25], and the main process is as follows [26,27]: first, the ideal shape of the blade is designed by the theoretical calculation method; second, a hot shape is constructed based on the ideal shape according to the load acting on the running blade, and the geometric deviation between the hot blade shape and the ideal blade shape is calculated; finally, the deviation is reversely applied to the ideal shape to obtain a cold shape meeting the design expectations. (5) where ∆u cld is the node displacement increment, K u cld is the nonlinear internal force vector of u cld , and K cld T = (∂K/∂u) cld is the tangential stiffness matrix of the cold blade. To discuss the possible deformation of the blade in an iteration step without loss of generality, it is assumed that the aerodynamic load R idl E is slightly larger than the internal force K u idl caused by the deformation when the blade is deformed into the ideal shape u idl . The blade will then deviate from the equilibrium condition and continue to deform until it stabilizes as the hot shape u hot . The deformation caused by the deviation between u hot and u idl generates an internal force K(u) to balance the external force, that is, K(u) = K u hot − K u idl (6) It is difficult to directly obtain the internal force caused by the deformation of the blade due to its variable stiffness. However, the aerodynamic loads at some representative positions can be used as a reference for the internal forces at these positions according to the abovementioned analysis of the forces and the deformation of the blade.
If the hot shape constructed from the cold blade shape coincides with the ideal blade shape, the blade is in equilibrium under the action of the internal force caused by the deformation and the Energies 2020, 13, 2471 4 of 14 aerodynamic load; that is, the internal force of the hot shape and the aerodynamic load acting on the ideal shape have the same magnitude and opposite direction.
The ideal shape is taken as the cold shape in the first iteration step, and the data of the aerodynamic load are obtained by the numerical simulation and used for subsequent iteration steps.
Taking a rotor blade in the low-pressure stage of a steam turbine as an example, the material 2Cr13 with the characteristics shown in Table 1 is used to construct the blade model. Before the iteration of deviation analysis starts, the initial cold shape of the blade needs to be obtained. Considering that the initial condition of the iteration is an ideal shape, the aerodynamic load analysis starts from the ideal shape of the blade to obtain R idl E . Under the set boundary condition, the Computational Fluid Dynamics (CFD) solver is applied to perform numerical simulation analysis on the flow field around the blade and extract the aerodynamic load on the representative nodes on the blade grid surface. Then, R idl E is reversely applied to the ideal shape to obtain the initial cold shape u cld . The extracted aerodynamic load R idl E is reversely loaded to the corresponding representative nodes on the grid surface of ideal blade shape, so the ideal shape of the blade deforms under the load and the final shape of the blade is the initial cold shape u cld .The process is shown in Figure 1.
Energies 2020, 13, x FOR PEER REVIEW 4 of 15 aerodynamic load; that is, the internal force of the hot shape and the aerodynamic load acting on the ideal shape have the same magnitude and opposite direction. The ideal shape is taken as the cold shape in the first iteration step, and the data of the aerodynamic load are obtained by the numerical simulation and used for subsequent iteration steps.
Taking a rotor blade in the low-pressure stage of a steam turbine as an example, the material 2Cr13 with the characteristics shown in Table 1 is used to construct the blade model. Before the iteration of deviation analysis starts, the initial cold shape of the blade needs to be obtained. Considering that the initial condition of the iteration is an ideal shape, the aerodynamic load analysis starts from the ideal shape of the blade to obtain . Under the set boundary condition, the Computational Fluid Dynamics (CFD) solver is applied to perform numerical simulation analysis on the flow field around the blade and extract the aerodynamic load on the representative nodes on the blade grid surface. Then, is reversely applied to the ideal shape to obtain the initial cold shape . The extracted aerodynamic load is reversely loaded to the corresponding representative nodes on the grid surface of ideal blade shape, so the ideal shape of the blade deforms under the load and the final shape of the blade is the initial cold shape .The process is shown in Figure 1. The iteration of the deviation analysis begins with the initial cold shape. In each step of the iteration, the deformation of the blade starts from the cold shape and ends with the hot shape . Thus, the internal force caused by the deformation is If the end of the iteration step is changed to the ideal shape , the next step will be close to the final shape of the blade in the pre-deformation design. The iteration of the deviation analysis begins with the initial cold shape. In each step of the iteration, the deformation of the blade starts from the cold shape u cld and ends with the hot shape u hot . Thus, the internal force K(u) caused by the deformation is If the end of the iteration step is changed to the ideal shape u idl , the next step will be close to the final shape of the blade in the pre-deformation design. The aerodynamic load R hot E acting on the representative nodes on the grid surface of hot blade u hot is extracted based on the CFD solver and then reversely loaded to the corresponding representative nodes on the grid surface of the ideal shape u idl so that the blade will be changed from the ideal shape u idl to the cold shape u cld .
When the blade is changed to u cld , the internal force caused by the deformation is still unbalanced with the external force R hot E , and the blade will continue to deform until the internal force is large enough to balance R hot E . The blade finally stabilizes as a new cold shape u new , as shown in Figure 2.
Energies 2020, 13, x FOR PEER REVIEW 5 of 15 The aerodynamic load acting on the representative nodes on the grid surface of hot blade is extracted based on the CFD solver and then reversely loaded to the corresponding representative nodes on the grid surface of the ideal shape so that the blade will be changed from the ideal shape to the cold shape . When the blade is changed to , the internal force caused by the deformation is still unbalanced with the external force , and the blade will continue to deform until the internal force is large enough to balance . The blade finally stabilizes as a new cold shape , as shown in Figure 2. In the abovementioned process, the internal force caused by the deformation from to is transferred to the deformation from to ; that is, the internal force is transferred from − to − . Therefore, the new cold shape can be derived from the ideal shape , which ends the current iteration step. Figure 3 shows the change in the blade shape during the iteration of the deviation analysis.  In the abovementioned process, the internal force caused by the deformation from u cld to u hot is transferred to the deformation from u idl to u new ; that is, the internal force is transferred from K u hot − K u cld to K(u new ) − K u idl . Therefore, the new cold shape u new can be derived from the ideal shape u idl , which ends the current iteration step. Figure 3 shows the change in the blade shape during the iteration of the deviation analysis.
Energies 2020, 13, x FOR PEER REVIEW 5 of 15 The aerodynamic load acting on the representative nodes on the grid surface of hot blade is extracted based on the CFD solver and then reversely loaded to the corresponding representative nodes on the grid surface of the ideal shape so that the blade will be changed from the ideal shape to the cold shape . When the blade is changed to , the internal force caused by the deformation is still unbalanced with the external force , and the blade will continue to deform until the internal force is large enough to balance . The blade finally stabilizes as a new cold shape , as shown in Figure 2. In the abovementioned process, the internal force caused by the deformation from to is transferred to the deformation from to ; that is, the internal force is transferred from − to − . Therefore, the new cold shape can be derived from the ideal shape , which ends the current iteration step. Figure 3 shows the change in the blade shape during the iteration of the deviation analysis.  There are two errors in the deviation analysis of the blade shape: one is caused by the change in the blade stiffness, and the other is caused by the change in the aerodynamic load acting on the blade.
On the one hand, the LC method constructs a new cold shape by transforming the deformation interval without the geometric deviation vector set, which is less affected by the change in the blade stiffness than the GC. On the other hand, the LC method uses the aerodynamic load acting on the hot shape to solve the internal force generated by the deformation and transfers it to the cold shape instead of the deformation, which is also less affected by the change in the aerodynamic load than the GC method. Therefore, the LC method has less errors than the GC method.

Implementation and Analysis of the GLHC Method
The GLHC method uses the GC and LC methods to correct the shape deviation caused by the centrifugal load and the aerodynamic load, respectively, and combines the results of the two methods to obtain the cold shape of the blade.
There are three main steps in the GLHC method, which are discussed as follows. First, the GC method is used to correct the blade shape deviation caused by the centrifugal load when the blade is in the ideal shape. Second, the LC method is used to extract the aerodynamic load acting on the hot shape and then apply it to the ideal shape, which has been corrected by the GC method. Third, the finite element method is used to solve the blade mesh, and the new cold shape obtained is the result of the current iteration step.
The material properties of a rotor blade taken from the low-pressure stage of a steam turbine are shown in Table 1. Unstructured tetrahedral meshes are used to mesh the flow field on the complex surface of the rotor blade. The mesh of the flow field model and the mesh of the blade model are shown in Figure 4. There are two errors in the deviation analysis of the blade shape: one is caused by the change in the blade stiffness, and the other is caused by the change in the aerodynamic load acting on the blade.
On the one hand, the LC method constructs a new cold shape by transforming the deformation interval without the geometric deviation vector set, which is less affected by the change in the blade stiffness than the GC. On the other hand, the LC method uses the aerodynamic load acting on the hot shape to solve the internal force generated by the deformation and transfers it to the cold shape instead of the deformation, which is also less affected by the change in the aerodynamic load than the GC method. Therefore, the LC method has less errors than the GC method.

Implementation and Analysis of the GLHC Method
The GLHC method uses the GC and LC methods to correct the shape deviation caused by the centrifugal load and the aerodynamic load, respectively, and combines the results of the two methods to obtain the cold shape of the blade.
There are three main steps in the GLHC method, which are discussed as follows. First, the GC method is used to correct the blade shape deviation caused by the centrifugal load when the blade is in the ideal shape. Second, the LC method is used to extract the aerodynamic load acting on the hot shape and then apply it to the ideal shape, which has been corrected by the GC method. Third, the finite element method is used to solve the blade mesh, and the new cold shape obtained is the result of the current iteration step.
The material properties of a rotor blade taken from the low-pressure stage of a steam turbine are shown in Table 1. Unstructured tetrahedral meshes are used to mesh the flow field on the complex surface of the rotor blade. The mesh of the flow field model and the mesh of the blade model are shown in Figure 4.  The CFD analysis conditions are as follows: the inlet pressure is 49.25 kPa, the temperature is 348.27 K and the outlet pressure is 0.0135 MPa. The settings for the CFD analysis is shown in Table 2. The CFD analysis conditions are as follows: the inlet pressure is 49.25 kPa, the temperature is 348.27 K and the outlet pressure is 0.0135 MPa. The settings for the CFD analysis is shown in Table 2.  The correction effects of the GLHC and GC methods are compared by constructing the hot shape of the blade and calculating the deviation of the hot blade shape from the ideal blade shape. The first iteration step is analyzed separately from the subsequent steps due to its initial input condition (the ideal shape is taken as the cold shape). The entire blade is analyzed based on an ANSYS CFD simulation, and the top trailing edge of the blade is used for comparison analysis because the deformation here is the largest under loading, which can best reflect the deviation of the hot blade shape from the ideal blade shape [51].
The comparison of the node positions on the top trailing edge of the ideal shape, the uncorrected hot shape and the hot shapes corrected by different methods are shown in Figure 5.  The correction effects of the GLHC and GC methods are compared by constructing the hot shape of the blade and calculating the deviation of the hot blade shape from the ideal blade shape. The first iteration step is analyzed separately from the subsequent steps due to its initial input condition (the ideal shape is taken as the cold shape). The entire blade is analyzed based on an ANSYS CFD simulation, and the top trailing edge of the blade is used for comparison analysis because the deformation here is the largest under loading, which can best reflect the deviation of the hot blade shape from the ideal blade shape [51].
The comparison of the node positions on the top trailing edge of the ideal shape, the uncorrected hot shape and the hot shapes corrected by different methods are shown in Figure 5.
In the subsequent iteration steps, the input condition is not the ideal shape, but the cold shape obtained by the correction of the previous iteration step, and the other operations are similar. The analysis results of the second step are shown in Figure 6.  Figure 5 shows that the deviation in the hot shape constructed based on the cold shape corrected using the GLHC method from the ideal shape is smaller than that using the GC method in the axial, radial and tangential directions, especially in the axial direction. Therefore, the correction effect of the GLHC method is better than that of the GC method in the first iteration step. Figure 6 shows that the correction effect of the second iteration step is similar to that of the first iteration step, but the difference between them decreases; that is, the correction effect of the GLHC method in the subsequent iteration steps is not as good as that in the first iteration step, but is still better than that of the GC method. In the subsequent iteration steps, the input condition is not the ideal shape, but the cold shape obtained by the correction of the previous iteration step, and the other operations are similar. The analysis results of the second step are shown in Figure 6.  Figure 5 shows that the deviation in the hot shape constructed based on the cold shape corrected using the GLHC method from the ideal shape is smaller than that using the GC method in the axial, radial and tangential directions, especially in the axial direction. Therefore, the correction effect of the GLHC method is better than that of the GC method in the first iteration step. Figure 6 shows that the correction effect of the second iteration step is similar to that of the first iteration step, but the difference between them decreases; that is, the correction effect of the GLHC method in the subsequent iteration steps is not as good as that in the first iteration step, but is still better than that of the GC method.  Figure 5 shows that the deviation in the hot shape constructed based on the cold shape corrected using the GLHC method from the ideal shape is smaller than that using the GC method in the axial, radial and tangential directions, especially in the axial direction. Therefore, the correction effect of the GLHC method is better than that of the GC method in the first iteration step.
Energies 2020, 13, 2471 9 of 14 Figure 6 shows that the correction effect of the second iteration step is similar to that of the first iteration step, but the difference between them decreases; that is, the correction effect of the GLHC method in the subsequent iteration steps is not as good as that in the first iteration step, but is still better than that of the GC method.

Pre-Deformation Design Process Based on the GLHC
The GLHC method is used to improve the pre-deformation design process based on the GC method, as shown in Figure 7.

Pre-Deformation Design Process Based on the GLHC
The GLHC method is used to improve the pre-deformation design process based on the GC method, as shown in Figure 7. The convergence condition reflects the deviation between the hot shape of blades and ideal shape of blades. When the number of iterations is more, the deviation between the hot shape of blades and the ideal shape of blades is small, so the hot shape of blades is close to the ideal shape of blades. In addition, the blades made of alloy steel have sufficient strength and large stiffness through the surface hardening treatment, so the deformation is relatively small. Therefore, according to the computing power of the computer and the accuracy that can be achieved by CFD simulation, the convergence condition of the pre-deformation design is set to 0.00001 mm; that is, if the maximum of the position deviations of the corresponding nodes between the hot blade shape and the ideal blade shape is less than 0.00001 mm, the cold shape used to construct this hot shape is considered to meet the pre-deformation design requirements.
The positions of the contour nodes at the top and the top trailing edge of the cold shape of the blade designed by the abovementioned method are shown in Figures 8 and 9, respectively. The convergence condition reflects the deviation between the hot shape of blades and ideal shape of blades. When the number of iterations is more, the deviation between the hot shape of blades and the ideal shape of blades is small, so the hot shape of blades is close to the ideal shape of blades. In addition, the blades made of alloy steel have sufficient strength and large stiffness through the surface hardening treatment, so the deformation is relatively small. Therefore, according to the computing power of the computer and the accuracy that can be achieved by CFD simulation, the convergence condition of the pre-deformation design is set to 0.00001 mm; that is, if the maximum of the position deviations of the corresponding nodes between the hot blade shape and the ideal blade shape is less than 0.00001 mm, the cold shape used to construct this hot shape is considered to meet the pre-deformation design requirements.
The positions of the contour nodes at the top and the top trailing edge of the cold shape of the blade designed by the abovementioned method are shown in Figures 8 and 9, respectively. Energies 2020, 13, x FOR PEER REVIEW 10 of 15  The data obtained by the GLHC and GC methods are compared with the same ideal shape, numerical calculation method and convergence condition as the design input. The maximum deviations in each iteration step are shown in Table 3.   The data obtained by the GLHC and GC methods are compared with the same ideal shape, numerical calculation method and convergence condition as the design input. The maximum deviations in each iteration step are shown in Table 3.  The data obtained by the GLHC and GC methods are compared with the same ideal shape, numerical calculation method and convergence condition as the design input. The maximum deviations in each iteration step are shown in Table 3. Table 3 shows that the GLHC and GC methods satisfy the convergence condition in the 18th and the 20th steps of the iteration, respectively. To verify the accuracy of convergence, five additional steps are performed, and the maximum deviations continue to decrease without exceeding the convergence condition. Therefore, the GLHC method reduces the iteration steps by nearly 10% with 0.00001 mm as the convergence condition compared with the GC method for the pre-deformation design of a blade.
The maximum deviations varying with the iteration steps are shown in Figure 10. Figure 10 shows that the GLHC method reduces the iteration steps and improves the efficiency of the pre-deformation design.  Table 3 shows that the GLHC and GC methods satisfy the convergence condition in the 18th and the 20th steps of the iteration, respectively. To verify the accuracy of convergence, five additional steps are performed, and the maximum deviations continue to decrease without exceeding the convergence condition. Therefore, the GLHC method reduces the iteration steps by nearly 10% with 0.00001 mm as the convergence condition compared with the GC method for the pre-deformation design of a blade.
The maximum deviations varying with the iteration steps are shown in Figure 10.  Figure 10 shows that the GLHC method reduces the iteration steps and improves the efficiency of the pre-deformation design.

Conclusions
This paper proposes a GLHC method to solve the slow convergence of the iteration based on the GC method in the current pre-deformation design.
First, the coupling relationship between the blade deformation and change in the aerodynamic load was analyzed to establish the iteration process based on the LC method. The internal force caused by the deformation from the cold blade shape to the hot blade shape was transferred to the deformation from the ideal shape to the new cold shape; thus, the LC method had less errors than the GC method.
Second, the GLHC method was established by combining the LC and GC methods. The GC and LC methods were used to correct the shape deviation caused by the centrifugal load and the aerodynamic load, respectively, and the results of the two methods were combined to obtain the cold shape of the blade. The shape deviation of the blade obtained by the GLHC method was smaller than that of the GC method.
Third, the GLHC method was used to improve the pre-deformation design process. The data obtained by the pre-deformation design using two methods were compared with the same design input, calculation method and convergence condition. The results show that the GLHC method reduced the iteration steps by nearly 10%, with 0.00001 mm as the convergence condition compared

Conclusions
This paper proposes a GLHC method to solve the slow convergence of the iteration based on the GC method in the current pre-deformation design.
First, the coupling relationship between the blade deformation and change in the aerodynamic load was analyzed to establish the iteration process based on the LC method. The internal force caused by the deformation from the cold blade shape to the hot blade shape was transferred to the deformation from the ideal shape to the new cold shape; thus, the LC method had less errors than the GC method.
Second, the GLHC method was established by combining the LC and GC methods. The GC and LC methods were used to correct the shape deviation caused by the centrifugal load and the aerodynamic load, respectively, and the results of the two methods were combined to obtain the cold shape of the blade. The shape deviation of the blade obtained by the GLHC method was smaller than that of the GC method.
Third, the GLHC method was used to improve the pre-deformation design process. The data obtained by the pre-deformation design using two methods were compared with the same design input, calculation method and convergence condition. The results show that the GLHC method reduced