The objective of the study is to find the tolerance on vane pitch dimensions of a Vertical Turbine (VT) pump impeller. For this purpose, the study is divided into two parts viz. to find the critical hydraulic eccentricity of a VT pump impeller by way of numerical simulations and design of experiments to find the vane pitch tolerance using critical hydraulic eccentricity. The effect of impeller vane pitch deviations on hydraulic unbalance is examined for a vertical turbine pump using Design of Experiments (DOE). A suitable orthogonal matrix has been selected with vane pitch at different axial locations of an impeller as the control factors. Hydraulic eccentricity, which is the output of the DOE experiments is analyzed using S/N ratio, ANOM and regression analysis to find the significant control factor effecting the hydraulic unbalance and hence vibrations. The vane pitch deviation at outlet and inlet of impeller shroud geometry are found to be the most critical factor affecting the pump vibrations.

Optimization of certain parameters in industrial applications involves huge cost and time. For this purpose, design of experiments using orthogonal matrix method reduces the time and cost. The improvement of an impeller performance is always challenging due to involvement of various design, manufacturing factors and operating conditions. Douglas [

The special property of orthogonal matrices is the appearance of combination of variables between two pair of columns. It appears equal number of times. The effects of the other variables can be interrelated with a relative value compared to the levels of a variable [

The reasons for vibrations in a pump unit ranges over a broad range of causes which includes the pump type (radial, mixed flow or axial), operating points at site, system resistance of pumping system, type of fluid, design of impeller and flow intake conditions. The other important phenomenon which can cause vibrations is suction cavitation. All these factors are to be considered as the possible causes of the severe vibrations which may further lead to shaft failure and hence the operational loss to customer [

The impeller unbalance has been divided into two types viz. Mechanical or structural unbalance and Hydraulic unbalance. Mechanical unbalance is purely related to the manufacturing deviation due to mass distribution of rotating component. It is defined as the deviation between mass centerline of an impeller and the shaft axis. Hydraulic unbalance is defined as flow unevenness between impeller vanes passages which is due to variation in vane pitch dimensions. The non-uniform vane pitch in impeller which arises during manufacturing stage results into the hydraulic unbalance. This unbalance force rotates with rotor speed and it increases as the rotating speed increases. Apart from the speed of the impeller the flow rate also increases the unbalance forces [

The objective of a pump designer is to improve the performance of pump in operating region along with the reliability [

Impellers are designed and developed with combination of numerical simulations and design of experiments. It helps in developing new products with reliable quality. de Souza et al. [

In quality engineering, the generally used orthogonal arrays are L12, L18 and L36. Sung et al. [^{nd} factorial design has been used to identify the important design parameters. It is found that the incidence angles and the exit blade angle are the most significant parameters contributing to the performance of the pump.

The improvement of an impeller performance is always challenging due to involvement of multiple design factors and operating conditions. Ling et al. [

Abhijeet et al. [

The blade parameters which can impact the vibrations are blade exit angle, wrap angle. Han et al. [

Si et al. [

Dynamic balancing of a rotating system can be used to limit the mechanical unbalance which is a direct method [

Based on the literature study, the information is not available to control and define the tolerance on hydraulic dimensions like vane pitch of impellers. It is essential to define the appropriate tolerances on vane pitch considering the manufacturing process and its feasibility so that vibrations are controlled for reliable mechanical performance of vertical turbine pumps. Under this premise, current study attempts to investigate the effect of vane pitch deviations on hydraulic eccentricity and in turn on vibration.

The objective of the study is to find the tolerance on vane pitch dimensions of a Vertical Turbine (VT) pump impeller. The duty point of the VT pump considered for numerical simulations is given in

S. No. | Parameter | Value |
---|---|---|

1 | Flow rate (m^{3}/h) |
30,000 |

2 | Total head (m) | 24 |

3 | Speed (rpm) | 370 |

To find the vane pitch tolerance of present VT pump impeller, the present study is divided in two part:

eccentricity.

In the following sections both the parts are discussed in detail.

The prediction of hydraulic eccentricity impact on VT pump vibrations is involved in multi-physics problem which is “Fluid Structure Interaction (FSI)”. The available two numerical methods of FSI are (1) One-way FSI (2) Two-way FSI. For the present research, one-way FSI methodology is selected due to the following major reasons,

One-way FSI is relatively less time consuming in terms of solver time compared to two way FSI. Though two-way FSI is more accurate in predicting the flow induced deformations, it involves more solver time due to information transfer between CFD and FE solver for each iteration. Since, the number of numerical experiments involved in this research are enormous, the time optimization is the prime factor.

The method is also consistent in predicting the vibration displacements with good accuracy. The solver is validated prior conducting the numerous experiments [

The process included the validation of the solver first and then conducting the numerical simulations with hydraulic eccentricity as a variable in the analysis.

Numerical analysis is successfully carried out and predicted the critical hydraulic eccentricity for the present VT impeller. But, in industries it is very difficult to predict the hydraulic eccentricity of a manufactured impeller directly. It needs a process of 3D scanning following up with 3D model preparation to find the hydraulic eccentricity. All these steps are involved with cost and time resources. Hence, a simple procedure is required, which relates the hydraulic eccentricity with vane pitch values of a manufactured impeller. For this purpose, a well-known statistical method, DOE is much applicable to decide the acceptable deviation of vane pitch dimensions at different locations on impeller periphery.

There are two aspects of any experimental problem, the design of experiments and the statistical analysis of the data. When many factors control the performance of a system, then it is essential to find out significant factors which need special attention either to control or to optimize the system performance. For this purpose, the use of statistical methods is applied, which has many applications to find industrial solutions. DOE is a statistical technique which has wide applications to study the effect of multiple parameters (factors or variables) simultaneously. It is highly preferred technique wherever and whenever it is suspected that the output performance is controlled by more than one parameter. Although DOE has the most applications in industries using experiments, it is equally preferred in applying for numerical simulations.

Taguchi Orthogonal method of design of experiments is planned to minimize the variation in impeller vane pitch dimensions. Generally, orthogonal method uses the original results of the prototype testing. However, considering the manufacturing process and number of experiments, it needs high accuracy and longer period, the error is also inevitable in the prototype test. Hence, for each run impeller, impeller geometry is created for experimentation with desired deviation and validated by regression method to predict the effect on response factor as eccentricity. By this approach error will be lower. Moreover, it saves the manufacturing time and cost.

The most fundamental step prior to commencement of the DOE study is the identification of the control factors to be varied. The primary objective of the research work is to determine the impact of vane pitch dimensions on hydraulic eccentricity and vibration. These control factors should be selected in such a way that the required objective is served with minimum number of experiments or attempts.

The impeller considered for DoE study consists of four vanes and the profile of the vane is mixed flow with semi-open type impeller. The four vanes of the impeller result in four hydraulic parts of the impeller. Hydraulic eccentricity is a function of the geometric variation of the hydraulic profile and hence the vane pitch. The control factors are vane pitch variation at the shroud side for the inlet portion, mid portion, and outlet portion along with the vane thickness. Considering the geometry of impeller, L27 matrix (thirteen-three level factors) is selected with thirteen control factors and three levels. The

For each experiment, three quality attributes are measured. The results of the experiments are analyzed to obtain the following objects:

To predict the optimum conditions by means of Signal to Noise (S/N) Ratio analysis.

To predict the contribution of individual parameters using ANOVA to prioritize the most important parameters.

Regression analysis.

The thirteen control factors and levels are mentioned in

A | Vane1_Shroud inlet | F | Vane2_Shroud mid | K | Vane3_Shroud at outlet |

B | Vane2_Shroud inlet | G | Vane3_Shroud mid | L | Vane4_Shroud at outlet |

C | Vane3_Shroud inlet | H | Vane4_Shroud mid | M | Vane thickness |

D | Vane4_Shroud inlet | I | Vane1_ Shroud outlet | Levels 1. X% 2. Y% and 3. Z% variation from design dimensions | |

E | Vane1_Shroud mid | J | Vane2_Shroud outlet | Response factor_{h} μm |

The results of are discussed in two parts in line with the objective of the work as below.

The numerical simulation approach adopted in this work is a multi-physics continuum solver approach. The reliability of numerical simulation output depends on the variation of solver output with respect to mesh quantity. Hence, mesh independence study is also carried out on both solvers (CFD and FE-Structure) independently. Ansys-CFX has been used for CFD purpose and Ansys-Work bench is used for FE analysis purpose. Following sub sections describes the mesh independence study of each solver in detail.

The unstructured mesh is created using triangular elements for 2D surfaces and tetrahedron elements for 3D flow domain using Ansys ICEM CFD. The hydraulic geometry of pump is divided into four domains: bell mouth, impeller, bowl and column pipe. 3 layers of prism layers are generated near impeller blade wall surfaces to resolve the boundary layer near rotating wall. The mesh is independently created in fluid domains like impeller, bowl geometry, discharge column pipes and assembled for analysis purpose. Denser mesh is created near impeller vanes and bowl vanes regions. In order to restrict the influence of grid number on the numerical results, a mesh independence study has been conducted with four different sets of mesh. The value of the relative deviation of total head for each mesh with its subsequent mesh for 4 points are shown in

Mesh | Bell mouth | Impeller | Bowl | Column pipe | Total nodes (million) | Head (m) | Absolute deviation with test data (%) |
---|---|---|---|---|---|---|---|

Mesh 1 | 0.017 | 0.23 | 0.31 | 0.12 | 0.68 | 22.6 | – |

Mesh 2 | 0.02 | 0.265 | 0.365 | 0.17 | 0.82 | 23.1 | 2.2 |

Mesh 3 | 0.0287 | 0.32 | 0.415 | 0.206 | 0.97 | 23.5 | 1.7 |

Mesh 4 | 0.033 | 0.375 | 0.47 | 0.235 | 1.12 | 23.6 | 0.5 |

The FE mesh is used to subdivide the CAD model into smaller areas called elements, based on which a set of equations are solved. These equations approximately present the governing equation of interest with a set of functions which are polynomial defined over each element. As the mesh is refined by making them smaller and smaller, the computed solution will converge to the true solution. However, computation or simulation time increases as the mesh elements decrease in size. Therefore, an optimum mesh is required to validate any FE simulation through mesh independence.

In this analysis, the mesh independence study is carried out on FE solver using three different mesh sizes to estimate the variation in results such as displacements, stresses, etc. Firstly, a coarse mesh was taken for analysis and subsequent finer meshes are taken with same loading and boundary conditions.

The results from mesh independence study are plotted between ‘Total number of elements’ in model (abscissa)

One-way Fluid Structure Interaction (FSI) numerical simulations are conducted using Mesh 3 of CFD and Mesh 2 of FE solver. The simulation results are discussed quantitatively in terms of vibration displacement magnitudes and qualitatively in terms of displacement contours on motor component. The selected components for monitoring the vibration displacements are shaft, impeller, suction bearing (rotating parts) and bell mouth, discharge bend and motor (stationary parts). The numerical simulation has been conducted and results are noted for a set of hydraulic eccentricity magnitude varying between 0 and 0.688 mm. The value zero corresponds to the design geometry with no sign of hydraulic unbalance whereas 0.688 mm is corresponding to maximum hydraulic unbalance at which heavy vibrations in pump observed at site [

The maximum magnitude of vibration displacements is given in

Sr. No. | Hydraulic eccentricity (mm) | Maximum vibration displacement (Micron) X (μm) | |||||
---|---|---|---|---|---|---|---|

Bell mouth | Shaft | Impeller | Suction bearing | Discharge bend | Motor (top) | ||

1 | 0 (Design) | 166 | 78 | 74 | 60 | 20 | 44 |

2 | 0.2 | 204 | 125 | 117 | 93 | 30 | 71 |

3 | 0.3 | 223 | 149 | 139 | 109 | 36 | 84 |

4 | 0.4 | 242 | 172 | 160 | 126 | 41 | 98 |

5 | 0.5 | 261 | 196 | 182 | 142 | 47 | 112 |

6 | 0.688 | 296 | 239 | 222 | 173 | 57 | 137 |

Sr. No. | Hydraulic eccentricity (mm) | Maximum vibration displacement (Micron) Y (μm) | |||||
---|---|---|---|---|---|---|---|

Bell mouth | Shaft | Impeller | Suction bearing | Discharge bend | Motor (top) | ||

1 | 0 (Design) | 236 | 138 | 132 | 111 | 27 | 47 |

2 | 0.2 | 273 | 185 | 174 | 143 | 38 | 74 |

3 | 0.3 | 292 | 208 | 196 | 159 | 43 | 88 |

4 | 0.4 | 310 | 232 | 217 | 176 | 49 | 101 |

5 | 0.5 | 329 | 255 | 239 | 192 | 54 | 115 |

6 | 0.688 | 364 | 298 | 279 | 222 | 64 | 141 |

To verify the numerical analysis results, the vibration displacement values are compared with actual test data of five pumps.

To find the critical hydraulic eccentricity to comply with HIS standard, the vibration displacement near the motor is limited to 100 μm in both X and Y directions as per guidelines of Hydraulic Institute standard. For the limiting value of 100 μm vibrational displacement in X direction, the critical hydraulic eccentricity is found 405 μm. Similarly, for the limiting value of 100 μm vibrational displacement in Y direction, the critical hydraulic eccentricity is found 400 μm. From the _{2} and P_{3} are having higher vibration than limit of 100 microns given in HIS.

The next part of research work is to correlate the critical eccentricity to vane pitch of the pump. Measuring a critical eccentricity for a manufactured impeller is not a direct step as practically it is not feasible to measure eccentricity. The alternative method is to measure the vane pitch dimensions of an impeller and predict the hydraulic eccentricity of respective impeller. But, the estimation of eccentricity from vane pitch measurements needs a set of experiments based on which the factors affecting critical hydraulic eccentricity is to be estimated.

In general, the vane pitch dimensions are different in a mixed flow type impeller at different sections. Hence, there are many parameters in terms of vane pitch which can impact the hydraulic eccentricity. To study the effect of these different parameters on the hydraulic eccentricity, a set of experiments can be examined by using the orthogonal experimental design. All parameters or control factors envisaged to affect hydraulic eccentricity (e_{h}) are ranked in perceived order. The top thirteen factors selected are listed in

Generally, the orthogonal method uses the original results of the experimental testing. However, considering the manufacturing process of impeller and number of experiments, it needs a longer period with high accuracy. The error is also unavoidable during testing. Hence, for each run of DOE, the 3D impeller geometry (

Sr. No. | Notation | Control factors |
---|---|---|

1 | A | Vane1_Shroud inlet |

2 | B | Vane2_Shroud inlet |

3 | C | Vane3_Shroud inlet |

4 | D | Vane4_Shroud inlet |

5 | E | Vane1_Shroud mid |

6 | F | Vane2_Shroud mid |

7 | G | Vane3_Shroud mid |

8 | H | Vane4_Shroud mid |

9 | I | Vane1_Shroud outlet |

10 | J | Vane2_Shroud outlet |

11 | K | Vane3_Shroud at outlet |

12 | L | Vane4_Shroud at outlet |

13 | M | Vane thickness |

Response factor_{h} μm |
||

Levels 1. X%, 2. Y% and 3. Z% variation from design dimensions |

To fix the exact levels, initially it is planned to conduct 3 different experiments. Experiment 1 with vane deviation of 5% in all vanes. Experiment 2 is with vane deviation of 3% in all vanes. Experiment 3 with vane deviation of 1% in all vanes. From the experiments it is noted that with these variations, the eccentricity is noted as 741, 533, and 378 μm, in Experiments 1, 2 and 3, respectively. These hydraulic eccentricities are quite higher than critical eccentricity as deduced earlier. Hence, based on the outcome of these experiments, three levels are selected viz. 3%, 2% and 1%.

Name | Notation | Definition |
---|---|---|

Levels | 1 | 3% of design dimensions |

2 | 2% of design dimensions | |

3 | 1% of design dimensions | |

Response factor | Hydraulic eccentricity, e_{h} (μm) |

_{ch}) due to change in vane pitch. The data will be used to find the control factors which affect the hydraulic eccentricity.

Run | A | B | C | D | E | F | G | H | I | J | K | L | M | e_{h} (μm) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 533 |

2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 401 |

3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 234 |

4 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 303 |

5 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | 216 |

6 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 315 |

7 | 1 | 3 | 3 | 3 | 1 | 1 | 1 | 3 | 3 | 3 | 2 | 2 | 2 | 258 |

8 | 1 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 3 | 3 | 341 |

9 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 280 |

10 | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 214 |

11 | 2 | 1 | 2 | 3 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 235 |

12 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 310 |

13 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 2 | 3 | 1 | 3 | 1 | 2 | 250 |

14 | 2 | 2 | 3 | 1 | 2 | 3 | 1 | 3 | 1 | 2 | 1 | 2 | 3 | 383 |

15 | 2 | 2 | 3 | 1 | 3 | 1 | 2 | 1 | 2 | 3 | 2 | 3 | 1 | 400 |

16 | 2 | 3 | 1 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 2 | 3 | 1 | 305 |

17 | 2 | 3 | 1 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 3 | 1 | 2 | 305 |

18 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 2 | 3 | 1 | 1 | 2 | 3 | 193 |

19 | 3 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 192 |

20 | 3 | 1 | 3 | 2 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 267 |

21 | 3 | 1 | 3 | 2 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 288 |

22 | 3 | 2 | 1 | 3 | 1 | 3 | 2 | 2 | 1 | 3 | 3 | 2 | 1 | 275 |

23 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 163 |

24 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 2 | 1 | 3 | 163 |

25 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 3 | 2 | 1 | 2 | 1 | 3 | 179 |

26 | 3 | 3 | 2 | 1 | 2 | 1 | 3 | 1 | 3 | 2 | 3 | 2 | 1 | 282 |

27 | 3 | 3 | 2 | 1 | 3 | 2 | 1 | 2 | 1 | 3 | 1 | 3 | 2 | 329 |

The hydraulic eccentricity value is measured for total 27 3D models and tabulated (

In the present case, since the lower the response factor (which is critical hydraulic eccentricity) lower the vibrations. Smaller the better option is used when the objective is to minimize the output value and there are no negative data. Hence as per Taguchi’s method smaller is better option is selected to minimize the vibrations.

For each experiment, the hydraulic eccentricity is calculated and signal to noise ratio is obtained considering the hydraulic eccentricity as response factor.

Control factors | Levels | Delta | Rank | ||
---|---|---|---|---|---|

1 | 2 | 3 | |||

A | 10.212 | 11.292 | 12.779 | 2.566 | |

B | 10.983 | 11.886 | 11.414 | 0.903 | 9 |

C | 11.521 | 11.719 | 11.043 | 0.676 | 10 |

D | 10.245 | 11.701 | 12.337 | 2.092 | |

E | 11.537 | 11.364 | 11.382 | 0.173 | 13 |

F | 10.918 | 11.401 | 11.964 | 1.047 | 8 |

G | 10.812 | 11.506 | 11.965 | 1.152 | 6 |

H | 10.851 | 11.183 | 12.249 | 1.398 | 5 |

I | 9.801 | 11.395 | 13.087 | 3.285 | |

J | 11.698 | 11.281 | 11.305 | 0.417 | 11 |

K | 11.981 | 11.423 | 10.879 | 1.102 | 7 |

L | 11.621 | 11.225 | 11.437 | 0.396 | 12 |

M | 10.412 | 11.338 | 12.533 | 2.122 |

Analysis of means (ANOM) is a graphical substitute to ANOVA which tests the equality of data means. Data mean is all elements average that meet the selection criteria for a group. The graph (

Control factors | Levels | Delta | Rank | ||
---|---|---|---|---|---|

1 | 2 | 3 | |||

A | 0.3201 | 0.2787 | 0.2373 | 0.0828 | |

B | 0.2969 | 0.2647 | 0.2745 | 0.0322 | 8 |

C | 0.2857 | 0.2647 | 0.2857 | 0.021 | 9 |

D | 0.3226 | 0.2647 | 0.2488 | 0.0738 | |

E | 0.2789 | 0.2784 | 0.2789 | 0.0004 | 13 |

F | 0.301 | 0.2784 | 0.2568 | 0.0442 | 7 |

G | 0.3011 | 0.2784 | 0.2566 | 0.0445 | 6 |

H | 0.3048 | 0.2813 | 0.25 | 0.0549 | 5 |

I | 0.3303 | 0.2813 | 0.2245 | 0.1057 | |

J | 0.2774 | 0.2813 | 0.2774 | 0.0039 | 11 |

K | 0.2685 | 0.2802 | 0.2875 | 0.019 | 10 |

L | 0.278 | 0.2802 | 0.278 | 0.0022 | 12 |

M | 0.3127 | 0.2802 | 0.2433 | 0.0694 |

To describe the statistical relationship between the control factors and response factor in terms of an equation, a linear regression analysis has been conducted.

Source | DF | Seq SS | Contribution (%) | Adj SS | Adj MS | ||
---|---|---|---|---|---|---|---|

Regression | 13 | 0.162391 | 95.13 | 0.162391 | 0.012492 | 19.55 | 0 |

A | 1 | 0.030821 | 18.06 | 0.030821 | 0.030821 | 48.24 | 0 |

B | 1 | 0.002252 | 1.32 | 0.002252 | 0.002252 | 3.53 | 0.083 |

C | 1 | 0 | 0.00 | 0 | 0 | 0 | 1 |

D | 1 | 0.024508 | 14.36 | 0.024508 | 0.024508 | 38.36 | 0 |

E | 1 | 0 | 0.00 | 0 | 0 | 0 | 1 |

F | 1 | 0.008786 | 5.15 | 0.008786 | 0.008786 | 13.75 | 0.003 |

G | 1 | 0.008903 | 5.22 | 0.008903 | 0.008903 | 13.94 | 0.003 |

H | 1 | 0.01354 | 7.93 | 0.01354 | 0.01354 | 21.19 | 0 |

I | 1 | 0.050282 | 29.46 | 0.050282 | 0.050282 | 78.7 | 0 |

J | 1 | 0 | 0.00 | 0 | 0 | 0 | 1 |

K | 1 | 0.001618 | 0.95 | 0.001618 | 0.001618 | 2.53 | 0.136 |

L | 1 | 0 | 0.00 | 0 | 0 | 0 | 1 |

M | 1 | 0.021681 | 12.70 | 0.021681 | 0.021681 | 33.94 | 0 |

Error | 13 | 0.008306 | 4.87 | 0.008306 | 0.000639 | ||

Total | 26 | 0.170697 | 100.00 |

S | R-sq (%) | R-sq (adj) | R-sq (pred) |
---|---|---|---|

0.025276 | 95.13% | 90.27% | 79.63% |

Based on regression analysis, Regression Equation is formulated as below:

The equation clearly indicates that the coefficients of I, A, D and M are higher and showing the major contribution to the hydraulic unbalance (Bold highlighted in equation). These results are in alignment with the output obtained from S/N ratio analysis and ANOM. Hence the vane pitch at inlet and outlet of the vane near shroud is having more impact on the vibrations. To limit vibration displacement to 100 μm, the hydraulic eccentricity should be limited below 400 μm. From the statistical technique it is found that the locations at impeller vane inlet (Control factor I) and outlet (Control factor A) at shroud side are sensitive to the vane pitch deviation. From the

The above statistical study reflects that, the prime location of vane pitch deviation to be noted to limit of deviation. With these inputs, two impellers P2 and P3 are manufactured adhering to the tolerance limits of 2% on impeller shroud. With these tolerances, the hydraulic eccentricity is less than 400 microns. The eccentricities obtained are 60 microns and 72 microns respectively with vane pitch tolerance of 1.2% and 1.3% on shroud surface, respectively. The impellers are placed in P2 and P3 locations at site and retested under similar conditions. The vibration displacements on motor top surface are measured for both the pumps. _{2} and P_{3} pumps is shown

Pump No. | Hydraulic eccentricity (μm) | Vibration displacement X direction (μm) | Vibration displacement Y direction (μm) | ||
---|---|---|---|---|---|

Predicted | Tested | Predicted | Tested | ||

2 | 60 | 42 | 38 | 50 | 48 |

3 | 72 | 55 | 64 | 55 | 49 |

The results clearly indicate the importance of vane pitch tolerances in a centrifugal pump impeller which further defines the hydraulic eccentricity and hence the vibration magnitudes.

The hydraulic unbalance of an impeller in terms of hydraulic eccentricity defines the quality with respect to pump vibrations. The hydraulic eccentricity of any impeller should be below a specific limit to control the vibrations known as critical hydraulic eccentricity (e_{ch}). The critical hydraulic eccentricity of an impeller and the vane pitch tolerances are found using numerical simulations. A validated numerical methodology approach which is “One-way FSI” has been used to find the critical hydraulic unbalance of a vertical turbine impeller. The critical hydraulic eccentricity e_{ch} achieved is 400 microns for a selected VT pump impeller rotating at 370 rpm.

The vane pitch tolerance of an impeller is function of e_{ch}. Hence, Design of Experiments (DOE) approach is used to find the vane pitch tolerance to limit the hydraulic eccentricity and vibrations. Taguchi’s method of design of experiments method is followed to identify the control factors affecting hydraulic eccentricity. The control factor ‘I’ (shroud outlet) has highest contribution (29.46%) followed by ‘A’ (shroud inlet) (18.06%) and D (14.36%). The effect of these control factors is significant in increase of hydraulic eccentricity, which needs to be controlled to minimize the vibration. Hence, the shroud portion of an impeller is the primary control factor to limit the tolerance. From the orthogonal matrices of the 27 models with hydraulic eccentricity, 3 models are found to be crossing the e_{ch}. In all these models the limiting tolerance is Level 2 at shroud outlet which is 2%. Hence, the tolerance limit is 2% on shroud outlet for e_{ch} of 400 μm. Moreover, based on regression equation given, if tolerance of 2% is considered, then eccentricity is less than critical eccentricity of 400 μm.

By controlling the geometrical vane pitch dimensions, two impellers are casted for P_{2} and P_{3} pumps with reduced hydraulic eccentricities of 60 and 72 μm, respectively, which are less than critical eccentricity as estimated. The predicted and tested vibration displacements are within 10% deviation.

It is found that, if the dimensional variation of the vane pitch is restricted within 2%, then it is possible to control the hydraulic eccentricity and vibrations within specified limit.

The authors would like to acknowledge Kirloskar Brothers Limited for facilitating the research work.