This paper presents the optimization of vibrations of centrifugal pump considering fluidstructure interaction (FSI). A set of centrifugal pumps with various blade shapes were studied using FSI method, in order to investigate the transient vibration performance. The Kriging model, based on the results of the FSI simulations, was established to approximate the relationship between the geometrical parameters of pump impeller and the root mean square (RMS) values of the displacement response at the pump bearing block. Hence, multiisland genetic algorithm (MIGA) has been implemented to minimize the RMS value of the impeller displacement. A prototype of centrifugal pump has been manufactured and an experimental validation of the optimization results has been carried out. The comparison among results of Kriging surrogate model, FSI simulation, and experimental test showed a good consistency of the three approaches. Finally, the transient mechanical behavior of pump impeller has been investigated using FSI method based on the optimized geometry parameters of pump impeller.
Centrifugal pumps provide the energy to move fluids through piping systems, including equipment, piping, and fittings and through elevation changes in open systems. Centrifugal pumps have been widely used in various industrial applications, such as oil and gas, agriculture, chemistry, and marine industry as well as metallurgy. Because of the customers’ increasing demands of highquality pump, optimization design of centrifugal pump plays an important role in pump industry, and there have been many efforts to optimize the performance of centrifugal pump in recent years. Anagnostopoulos [
The vibration that occurs while centrifugal pump works can cause fatigue and damage of pump components and weaken the operation stability. Vibrations of centrifugal pump have attracted interest of researchers. For example Hodkiewicz and Norton [
Studies agree in considering the fluidstructure interaction (FSI) as the source of the highest vibration levels in large centrifugal pumps. Moreover, hydraulic excitation forces are due to the FSI and cause pressure fluctuations, mechanical vibrations, and alternating stresses in different components of centrifugal pump. In recent years, the application of FSI theory to centrifugal pumps became more popular and it is well documented in literature [
Vibration performance is one of the most important parameters in designing a centrifugal pump. Actually, experimental tests and CFD simulation are the two methods performed in order to obtain the centrifugal pump vibration response. However, both of the two methods cannot be considered in optimizing the vibration performance of the pump using an iterative method. Appropriate metamodels must be established between the decision variables and the concerned objective functions. Therefore, metamodel technique demonstrates its superiority in the optimization problem of engineering.
Kriging metamodels [
This paper presents an effective optimization method based on Kriging metamodel. The presented method optimizes the vibration performance of the centrifugal pump undergoing FSI phenomena, which reasonably take advantages of the FSI simulation, Kriging metamodel and experimental tests. Although considerable researches were devoted to investigating the vibration performance of centrifugal pump, it should be noted that there exists little literature evidence on the vibration optimization of centrifugal pump in particular that combines with FSI phenomenon. The second part of the paper deals with the study of the transient mechanical characteristics based on the optimized centrifugal pump using FSI method.
In this study, the fluidstructure interaction (FSI) problem’s domain
The fluid flowing through the centrifugal pump is treated as incompressible and isothermal. For
While working, the centrifugal pump undergoes large deformation and rotation. For
Closure for (
As mentioned above, the FSI occurs during the running process of centrifugal pump. Fluid pressure information transfers to the solid, while displacements information of the solid transfers to the flow. Furthermore, on the noslip fluidstructural interface, the information exchange between the fluid and solid should follow the equilibrium conditions
The working process of centrifugal pump involves vibrations, FSI, and energy conversion and loss. As the “heart” component for a centrifugal pump, impeller plays an important role in all these phenomena and transforms the mechanical energy into the kinetic energy of the fluid. Moreover, the geometry shape of impeller blade has strong effect on pump performance, including vibrations. This paper focuses on optimizing the impeller blade to minimize the vibration response of the centrifugal pump.
The recommended number of impeller blades for high head centrifugal pumps is usually between five and seven. In fact, too many blades lead to higher friction losses and may cause low blade loading; fewer blades may result in higher blade loading. Turbulent dissipation losses will rise because of the increased secondary flow and stronger deviation between blade and flow direction. Therefore, six blades are chosen.
Figure
Main dimensions of centrifugal pump’s impellers (unit: mm).
Figure
Decision variables and their boundaries.
Decision variable  Lower boundary  Upper boundary 


0  30 

0.02  0.98 

70  90 

0.02  0.98 

145  195 
Meridional section of the pump impeller.
Latin hypercube sampling (LHS) is a design of experiment (DOE) method originally developed by Mckay in 1979. LHS approach has the spacefilling character and can guarantee the sample points covering the entire design domains homogeneously. Hence, 119 simple points and 30 test points have been obtain by LHS method. The simple points are the input data of Kriging surrogate model, while the test points are used to validate the accuracy of the Kriging predictor.
Table
The sample points and corresponding results of FSI simulations.
Serial number  Decision variable  Objective  






RMS (mm)  
1  0.00  0.028  85.76  0.061  157.71  0.5411 
2  0.25  0.183  70.00  0.223  183.14  0.4068 
3  0.51  0.191  87.29  0.744  184.41  0.4228 
4  0.76  0.728  73.90  0.752  185.25  0.3867 
5  1.02  0.777  83.22  0.484  161.10  0.4108 
6  1.27  0.199  89.66  0.109  156.44  0.5271 
7  1.53  0.557  87.80  0.459  189.07  0.5251 
8  1.78  0.085  71.19  0.427  168.73  0.5812 
9  2.03  0.467  84.24  0.305  161.95  0.5672 
10  2.29  0.264  75.08  0.321  155.17  0.4128 
11  2.54  0.232  80.51  0.378  147.97  0.3603 
12  2.80  0.288  81.53  0.834  177.63  0.4529 







117  29.49  0.646  88.64  0.443  163.64  0.3627 
118  29.75  0.817  84.75  0.516  168.31  0.3667 
119  30.00  0.891  76.10  0.785  189.49  0.5792 
One case of FSI simulation models.
Full model
Cutaway view
Detail of the tongue region
The calculation of structural part of the pump has been carried out through computational structure dynamics (CSD) analysis, performed using Abaqus FEA software. The pump volute casing and impeller are both made of aluminumbronze alloy; the elastic modulus is 125000 MPa, the density is
The computational fluid dynamics (CFD) has been simulated using Fluent code. The fluid is water, with a temperature of 20°C, density of
Basic parameters for numerical simulations.
Parameter  Value 

Flow rate 
2000 (m^{3}/h) 
Rotational speed  1400 r/min 
Number of blades  6 
Inlet operating pressure  1 (atm) 
The information exchange between solid (Abaqus) and fluid flow (Fluent) at the coupling interface is performed in the platform of MpCCI. Figure
The process of FSI simulation.
As aforementioned, this paper mainly focuses on optimizing the vibration performance of the centrifugal pump using FSI. Hence, the root mean square (RMS) value of the displacement response at the pump bearing block is chosen as the objective function, which can be defined as follows:
Kriging predicts unknown values of a random function based on all of the observed points [
The predicted value and estimation error at point
Under the unbiased condition, the unknown parameters
As a matter of fact, once the types of regression model and correlation model have been chosen, the correlation matrix
Kriging metamodel is established according to Table
The parameters of the Kriging model.
Parameter  Value 




0.00453 


The Kriging metamodel can be applied to the vibration optimization only if the Kriging predictor’s estimated accuracy is higher enough. Otherwise, the metamodel should be rebuilt by adjusting the parameters. An additional set of 30 points obtained through LHS method is used as test points to verify the performance of Kriging’s predictor. The FSI analysis gives the RMS values of the displacement response corresponding with the test points. In addition, for the FSI simulations based on the test points, the basic parameters and boundary conditions are the same with the sample points.
Figure
The results of RMS values at test points.
The optimization problem of centrifugal pump in this paper can be given as follows:
The abovedefined problem can be resolved through multiisland genetic algorithm (MIGA), a modified version of genetic algorithm (GA). MIGA decomposes the population in one generation into several subpopulations. The subpopulations are also called “Islands,” and the genetic operations are executed on each “Island” independently. Furthermore, this independency can prevent the optimization solution from local optima. Table
The parameter settings of MIGA.
Parameters  Value 

Size of subpopulation  100 
Number of islands  10 
Number of generations  10 
Gene size  32 
Rate of crossover  1.0 
Rate of mutation  0.01 
Rate of migration  0.5 
Interval of migration  5 
Number of runs for the problem  30 
Table
The result of optimization.





RMS (mm)  Average time (s) 

26.37  0.938  83.31  0.934  156.89  0.3341  1836 
The prototype of centrifugal pump corresponding to the optimization result.
Table
Results of Kriging, FSI simulation, and experiment.
Kriging  FSI  Experiment  

RMS (mm)  0.3341  0.3296  0.3447 
This research shows that the predictive ability of the Kriging model has been well justified both by FSI simulations and by experimental test. Therefore, the well validated surrogate model can completely replace timeconsuming FSI simulations and substitute a great majority of expensive experiment tests. That is, the Kriging surrogate model provides great convenience in studying the vibration performance of centrifugal pump, especially for accumulating the practical experience of pump design. Moreover, the well validated surrogate model can benefit both the further development of centrifugal pump manufacturer and the improvement of the pump designer’s ability. Therefore, the surrogate model method makes the investigation of pump performance easy, which is of course on the promise that the model accuracy is high enough.
The mechanical characteristics of the pump impeller are significant for the working behaviors of centrifugal pumps. During the working process of a centrifugal pump, the periodic hydraulic loads imposed on the pump will lead to the dynamic deformation of the impeller and impeller shaft. Moreover, the dynamic deformation will further influence the flow field distribution. The analysis of the mechanical behavior of the impeller is a typical FSI problem. In general, there are mainly three types of loads acting on pump impeller: coupling pressure load from the fluid, gravity, and inertia force due to the circular motion. However, all these loads are finally balanced by the support reaction of the bearings and the input moment of the pump. FSI method allows investigating the dynamic force of the impeller and the input moment, and the calculation results are important to highlight the mechanical properties of the centrifugal pump. For example, the analysis results can help in choosing the appropriate sizes and types of impeller shaft and bearings.
Actually, either the radial force of the pump impeller or the input moment of the pump cannot be easily measured because of the expensive measuring equipment and complex multipoints installation. When the simulation model is accurate, FSI simulation method shows advantage in obtaining the radial force and input moment. Furthermore, as previously mentioned in Section
Figures
The radial force of the pump impeller.
The input moment of the pump.
This paper proposes a Krigingbased optimization method for the vibrations optimization of centrifugal pumps, which well integrates Kriging surrogate model, FSI simulations, and experimental tests. Moreover, the proposed method overcomes the faults of expensive computation and cost, and it has been proved to be effective on improving pump vibration performance in terms of minimum cost and reduction of development period.
The Kriging surrogate model of pump vibration performance has been established based on the sample points, and the results at the test points showed that the Kriging predictor well agreed with the FSI simulations. The final optimized decision variables have been obtained using MIGA; a prototype has been manufactured according to optimized values of geometrical parameters of the pump. Experimental tests carried out on prototype well agreed with the results of Kriging metamodel and FSI simulation.
Furthermore, based on the final optimized decision variables, the dynamic mechanical performance of pump impeller was further investigated using FSI method. The results showed that the radial force curve and moment curve exhibited cyclical fluctuation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (no. 11172108). This financial support is gratefully acknowledged.