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The influence of labyrinth seal structure parameters and their interaction on the characteristics of leakage amount are numerically investigated by conducting a quadratic regression orthogonal experiment. To determine the optimal structure parameters of the steam seal for minimizing the leakage amount, a reliable regression equation that does not lack of fit is established. The flow characteristics of the fluid in the labyrinth seal are analyzed in detail. Results show that the leakage amount is greatly influenced by seal cavity depth, convex platform height, seal tooth thickness, and tooth tip clearance, with the tip clearance having the most significant effect. The interaction among the four items exerts a certain impact on the leakage amount. The proposed regression equation exhibits a good significance and does not lack of fit. After optimization, the labyrinth seal demonstrates increased entropy and energy dissipation at the tip of the seal tooth, as well as decreased speed and inertia effect in the cavity, suggesting that the resistance leakage performance of the optimized labyrinth seal is improved.

The diaphragm seal of steam turbine plays an important role in preventing steam leakage and reducing steam turbine loss. The diaphragm seal is widely used because of its simple structure, convenient installation, and high efficiency [

Many studies have investigated the influence of labyrinth seal structure parameters on the leakage amount, and reliable conclusions have been obtained. However, most of these studies were conducted by controlling a single variable factor. Choosing an appropriate research method is one of the important steps in analyzing sealing leakage characteristics. Orthogonal experiments have been widely used in biology, engineering, and other fields to study the effects of multiple factors on the test index [

Therefore, a regression equation is established in this paper by using quaternion quadratic regression orthogonal experiment to analyze the effects of multiple structure parameters on the leakage amount in consideration of the notion that basic geometrical parameters of the labyrinth seal have different influences on leakage characteristics. In addition, the solution of the regression equation is calculated by programming to confirm the optimal structure parameters that correspond to maximum sealing efficiency.

The cavity depth, convex platform height, tooth thickness, and tooth tip clearance parameters of a 300 MW condensing steam turbine are quoted according to the actual labyrinth seal structure parameters to identify the variation range of factors in regression orthogonal experiment that ensures the availability and reliability of the experiment in engineering. The concrete structure parameters are shown in

Calculation time is reduced by using an arc section with a periodic surface rotated 3.6 degrees around the center rather than the whole cycle section. Under the same grid number, the model has a higher grid quality.

In the calculation, water vapor is set in accordance with the real condition, which pressure boundary conditions are adopted in the inlet and outlet of the calculation domain. The pressure and temperature in the inlet is 10.7 MPa and 744.25 K, whereas that at the outlet is 10.263 MPa and 734.25 K. Standard k-ε equation is combined with SIMPLE algorithm in the 3D flow field of the labyrinth seal to solve the N-S equation. Gambit software is used to model the arc section, and the grid has a negligible effect on the calculation results of the steam leakage when the grid spacing reaches 0.05 - 0.1 and the grid number exceeds 2 million, as shown in

on the range of 200 - 300 million to improve the calculation speed.

Regression orthogonal experiment is chosen to arrange the simulated calculation statistics in a reasonable range of factors, and a regression equation with a certain degree of confidence is established by using finite experiment points. Consequently, the leakage amount can be estimated by using the regression equation when the structure parameters are determined, and the theoretical optimum value of test factors that corresponds to the minimum leakage amount of steam can be obtained in a certain range by solving the regression equation. Such an approach avoids a great deal of computation, reduces the testing workload, and improves the efficiency of the labyrinth seal design optimization.

Therefore, this paper uses the quaternion quadratic regression orthogonal 1/2 implementation method and determines the variation range of four control parameters on the basis of the general principles of labyrinth seal design: the cavity depth x_{1} is 5 - 13 mm; the convex platform height x_{2} is 1.5 - 4.5 mm; the tooth thickness x_{3} is 0.2 - 1.6 mm; the tooth tip clearance x_{4} is 0.2 - 1 mm. If x_{j} (j = 1, 2, 3, 4) is the level of each factor in the regression orthogonal experiment, and x_{1j} and x_{2j} are upper and lower limits of the factors, then the zero level of the factor can be represented by x_{0j} = (x_{1j} + x_{2j})/2. Based on the variation range of the factors, the level code is as shown in

The number of numerical calculations of the experiment program is determined by Equation (1), which, compared with a comprehensive experiment with five levels and four factors (i.e., 5^{4} = 625 times), only needs 17 times. The calculation period is reduced to 2.72% of the original one.

where m_{0}, the center tests’ number, is 1;

The change step of the test level can be determined by the upper and lower limits of the factors and asterisk arm length.

Factors Levels | cavity depth x_{1}/mm | convex platform height x_{2}/mm | tooth thickness x_{3}/mm | tooth tip clearance x_{4}/mm |
---|---|---|---|---|

(+r) | 13 | 4.5 | 1.6 | 1 |

(+1) | 12 | 4 | 1.4 | 0.9 |

(0) | 9 | 3 | 0.9 | 0.6 |

(−1) | 6 | 2 | 0.4 | 0.3 |

(−r) | 5 | 1.5 | 0.2 | 0.2 |

Δ_{j} | 3 | 1 | 0.5 | 0.3 |

The factors should be kept in a consistent dimension and should not substantially differ in the orthogonal experiment. Thus, the natural variables of each factor should be treated in the centralization. Suppose z_{j} is code value obtained after the centralization of each factor level value, then

A unified form of regression equation can be obtained by the data processing of the orthogonal table:

Here, y denotes the steam leakage amount, which is the inspection index of the test object. Given that the leakage amount of the partial arc section of the labyrinth seal model is small, the steam leakage of that arc section is converted to a leakage of the whole cycle section; i.e., the leakage is magnified one hundred times. b is a constant term. b_{j} is the first regression coefficient. b_{ij} is the partial regression coefficient of the interaction term. b_{jj} is the quadratic regression coefficient.

No. | z_{0} | z_{1} | z_{2} | z_{3} | z_{4} | z_{1}z_{2} | z_{1}z_{3} | z_{1}z_{4} | z_{2}z_{3} | z_{2}z_{4} | z_{3}z_{4} | Y × 100 | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | −1 | −1 | −1 | 1 | 1 | 1 | −1 | 1 | −1 | −1 | 0.314 | 0.314 | 0.314 | 0.314 | 2.5074 |

2 | 1 | −1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | 0.314 | 0.314 | 0.314 | 0.314 | 0.8839 |

3 | 1 | −1 | 1 | −1 | −1 | −1 | 1 | 1 | −1 | −1 | 1 | 0.314 | 0.314 | 0.314 | 0.314 | 0.9102 |

4 | 1 | −1 | 1 | 1 | 1 | −1 | −1 | −1 | 1 | 1 | 1 | 0.314 | 0.314 | 0.314 | 0.314 | 2.9027 |

5 | 1 | 1 | −1 | −1 | −1 | −1 | −1 | −1 | 1 | 1 | 1 | 0.314 | 0.314 | 0.314 | 0.314 | 0.8372 |

6 | 1 | 1 | −1 | 1 | 1 | −1 | 1 | 1 | −1 | −1 | 1 | 0.314 | 0.314 | 0.314 | 0.314 | 2.5457 |

7 | 1 | 1 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 | −1 | 0.314 | 0.314 | 0.314 | 0.314 | 2.4717 |

8 | 1 | 1 | 1 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 0.314 | 0.314 | 0.314 | 0.314 | 0.9971 |

9 | 1 | −r | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.145 | −0.686 | −0.686 | −0.686 | 1.8152 |

10 | 1 | r | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.145 | −0.686 | −0.686 | −0.686 | 1.8457 |

11 | 1 | 0 | −r | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | 1.145 | −0.686 | −0.686 | 1.8105 |

12 | 1 | 0 | r | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | 1.145 | −0.686 | −0.686 | 1.8407 |

13 | 1 | 0 | 0 | −r | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | −0.686 | 1.145 | −0.686 | 1.6193 |

14 | 1 | 0 | 0 | r | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | −0.686 | 1.145 | −0.686 | 1.7169 |

15 | 1 | 0 | 0 | 0 | −r | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | −0.686 | −0.686 | 1.145 | 0.6195 |

16 | 1 | 0 | 0 | 0 | r | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | −0.686 | −0.686 | 1.145 | 3.1487 |

17 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.686 | −0.686 | −0.686 | −0.686 | 1.8603 |

D_{j } | 17.000 | 11.661 | 11.661 | 11.661 | 11.661 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 6.705 | 6.705 | 6.705 | 6.705 | |

B_{j } | 30.333 | −0.3112 | −0.4487 | 0.7349 | 10.221 | −0.3357 | −0.1351 | −0.4329 | 0.4329 | 0.1351 | 0.3357 | −0.0492 | −0.067 | −0.644 | 0.147 | |

b_{j } | 1.7843 | −0.0267 | −0.0385 | 0.0630 | 0.8765 | −0.0420 | −0.0169 | −0.0541 | 0.0541 | 0.0169 | 0.0420 | −0.0073 | −0.0100 | −0.0960 | 0.0220 | |

S_{j} | 0.0083 | 0.0173 | 0.0463 | 8.9590 | 0.0141 | 0.0023 | 0.0234 | 0.0234 | 0.0023 | 0.0141 | 0.0004 | 0.0007 | 0.0618 | 0.0032 | ||

F_{j} | 8.3069 | 17.268 | 46.322 | 8959.0 | 14.087 | 2.2815 | 23.425 | 23.425 | 2.2815 | 14.087 | 0.3614 | 0.6692 | 61.807 | 3.2334 | ||

significance | * | * | ** | *** | * | ** | ** | * | ** | * |

According to the above coefficients and the data in the table, the specific calculation equation can be expressed as follows:

where

To conduct the significant test and the lack of fit test of the regression equation, three center experiments are conducted in this paper. The error sum of squares _{13}, b_{24}, b_{11}, and b_{22}, of which the level of satisfies

The influence of factors on the inspection index can be arranged in the values of b_{j}, b_{ij}, and b_{jj} by testing the regression coefficient of the equation. The array of linear terms is_{4} is the largest, suggesting that z_{4} has a very large effect on the leakage of steam, and the main factor that influences leakage in the steam turbine separator is the tooth tip clearance. Moreover, b_{4} is positive, indicating that the steam leakage is smaller when the tooth tip clearance is smaller. This relationship is consistent with the theory that the flow area and the leakage is proportional to that derived from the empirical equation of steam leakage. In addition, cavity depth, convex platform height, tooth thickness, and their interaction exert certain effects on the leakage of steam. Positive regression coefficients indicate the positive correlation between the inspection index and the factors, whereas negative coefficients indicate the converse. The effects of the test terms on the inspection index are shown in

According to the significance test equation of the regression orthogonal experiment:

The equation accords with the significance test of quaternion quadratic re-

gression orthogonal experiment indicating that the confidence level of the regression equation is

Similarly, according to the equation of the lack of fit:

The regression equation established in this paper can be determined to have good not lack of fit, indicating that the leakage, the linear term, the quadratic term, and the interaction of factors have a very good fit, and it has no significant relationship with higher-order terms.

The regression equation can be obtained by the abovementioned method of calculation and data.

In view of the verification test and the optimization design, the code of regression equations should be transformed into natural variables, and the test code is transformed inversely into the function of x_{j} by the central processing equation:

To verify the accuracy of the regression equation, the calculation of theoretical equation is implemented through a group of parameters in the reasonable range of each factor. When x_{1} = 9.5 mm, x_{2} = 3.5 mm, x_{3} = 0.7 mm, and x_{4} = 0.5 mm, the calculation result of the regression equation is 1.4246 kg/s, and the error is 6.14% than the result calculated by Fluent. These findings suggest that the equation has high accuracy and can meet the needs of engineering application to a certain extent.

The regression Equation (7) is calculated by C# under the constraints of a given range of factors and is locally optimized to determine the steam leakage. In this way, the best value that corresponds to the minimum value can be obtained for each factor to realize design optimization.

The calculated step size is set to 0.1, and each factor x_{j} is calculated to determine the minimum value of

meridional plane before and after optimization. As shown in the diagram, the change in the pressure gradient mainly occurs at the top of the seal, showing that the kinetic energy of working fluid at the tooth tip clearance is greater than that in the cavity. Comparison of (a) and (b) in

At the same time, the enlarged picture of the tip clearance at the inlet and the outlet shows that the pressure drops evenly when the working fluid flows through the seal tooth after optimization. However, the pressure of tip clearance at the inlet and the outlet “initially decreased and subsequently increased” before optimization. This shows that part of the kinetic energy of the working fluid in the accelerating process is not dissipated but reconverted into pressure energy. Consequently, the throttling effect of the sealing teeth is not obvious, and the leakage increased before optimization.

model declines faster. Entropy changes smoothly before arriving at the next seal tooth. Then, the energy loss of the steam is reduced. The main reason of this is that the steam is stagnant, the velocity of the steam is smaller, and most of the kinetic energy is converted into pressure energy, leading to the reduction of energy dissipation. However, the reduced energy dissipation does not cause increased leakage, because the inertia effect almost disappeared due to the stagnant steam, which greatly reduces the leakage amount.

The regression equation of the structure parameters and leakage amount of labyrinth seal is established by performing a quaternion quadratic regression orthogonal experiment. The equation satisfies the test of significance and not lack of fit. In this paper, the theoretical calculation results are in good accordance with the simulation results, which can meet the needs of the actual project.

The cavity depth, convex platform height, tooth thickness, and tooth tip clearance of the labyrinth seal exerts a certain influence on the sealing performance. The tip clearance is the main factor that influences the leakage amount, and leakage amount was positively correlated with the change of tip clearance. Under the influence of multiple factors, the leakage amount of the labyrinth seal has the minimum value.

After the optimization of the structure parameters of the labyrinth seal, the leakage amount of optimized labyrinth seal is reduced by approximately 78% when the optimal structure ratio is

This work was supported by National Natural Science Foundation of China (51576036).

Cao, L.H., Si, H.Y., Li, P. and Li, Y. (2017) Labyrinth Seal Design Optimization Based on Quadratic Regression Orthogonal Experiment. Energy and Power Engineering, 9, 204-215. https://doi.org/10.4236/epe.2017.94B025