Dynamic Characteristic Analysis of Centrifugal Pump Impeller Based on Fluid-Solid Coupling
Round 1
Reviewer 1 Report
Summary of the Work
The ultimate scope of this work is to study the vibration problems in a centrifugal pump during operation. More specifically, the authors investigated he centrifugal pump impeller during start-up and the unsteady flow induced impeller vibration. The dynamic characteristics of centrifugal pump impeller have been analysed by means of simulation experiment and numerical simulation. In particular, by fluid solid coupling method, the authors analysed the dynamic characteristics of impeller during the start-up of centrifugal pump whereas the structure and flow channel model of centrifugal pump are established by Creo software.
Main Results Obtained
Two kinds of analyses have been performed:
i) Analysis of centrifugal pump impeller during start-up;
ii) Analysis of unsteady flow induced impeller vibration.
The results are summarised well in Section 5. “Conclusion” (see pages 10 and 11 of the manuscript).
General Remarks
The work is interesting and challenging. However, there are some aspects that need to be clarified. The following tips aim to fill some gaps in the work.
Suggestions
1) Eq. (1) (on page 3 of the manuscript) provides the dynamic equation of centrifugal pump impeller. This equation pop-ups in the text without any explanation. Please explain the physical meaning of this equation (i.e., in terms of balance equation of conserved quantity).
2) For clarity, may the authors provide the characteristic curve related to the centrifugal pump model, Q25H52, studied in this paper (i.e., the curve of pump flow rate versus pump head)?
3) The authors obtained the dynamic equation of fluid by discretising the three-dimensional wave equation of the fluid (see page 4 of the manuscript). However, Eq. (3) is fully specified after having established the boundary conditions (BCs). For the sake of clarity, please specify (or describe) the BCs by showing that the dynamic equation (4) is compatible with such a choice.
4) Cavitation is an essential problem that occurs in any pump. Cavitation occurs in the actual operation process. In addition, it creates noise and causes strong vibrations in the pump. Have the authors studied the causes of the cavitation able to provoke impeller vibration during the start-up process of the centrifugal pump? If so, may the authors briefly describe, at least qualitatively, this phenomenon?
5) Figure 4 shows the rotating speed and the rate of flow of the centrifugal pump during start-up. What is the extent of the data errors? In fact, if the magnitude of the error were sufficiently high, the wave-behaviour would be absent (which would lead to the conclusion that the dynamic equations used by the authors have been oversimplified). Please, discuss.
6) In this work, the authors investigated the structural vibration characteristics of impeller under transient radial force. However, there are many other sources of vibration in pumps. Indeed, other common sources of vibration in pumps are bent shafts, unbalanced, misalignments, reaction forces, and contact between components. These factors have not been mentioned in the manuscript. For completeness (and clarity), the authors are asked at least to mention these factors, specifying that the main objective of this work is to investigate only some specific reasons that induce the vibration of the centrifugal pump.
7) Several studies - performed through numerical simulations - on pressure fluctuation characteristics of a centrifugal pump in its start-up process recently appeared in the literature. The results show that a large number of low-pressure areas and strong vortexes are formed within the impeller at the initial time of start-up. With the increase in rotating speed, the vortexes rapidly decrease and are concentrated on the blade non-working face. The vibration characteristics of the pumps are significantly influenced also by the presence of vortices. However, we may object that the dynamic equations studied by the authors are not sufficiently complete to take this problem into account as well. For the sake of completeness, the authors are invited to provide a (brief) comment on this matter.
8) Optional
As known, pressure head and efficiency of the centrifugal pump for different flow rates are data very useful to design the pumps in manufacturing industries (of course, together with the data from pump manufacturers). Other important factors are the instantaneous starting speed and power of the centrifugal pump. For a general view of the problem, it would be helpful to have a graphical representation of these factors i.e., the so-called "system curve".
Conclusions
The work is interesting and challenging. In my opinion, it deserves to be published. However, I encourage the authors to take into account the suggestions expressed above; this will increase the soundness of the work and will attract the interest of the reader more.
Author Response
Dear Editor,
We want to thank you and the reviewers for the valuable and constructive comments and suggestions for the improvement of this article. Please find herewith my responses, and revisions according to the review comments
Thank you very much.
With kind regards,
Yours sincerely,
Yufang Li
Reviewer 1：
- (1) (on page 3 of the manuscript) provides the dynamic equation of centrifugal pump impeller. This equation pop-ups in the text without any explanation. Please explain the physical meaning of this equation (i.e., in terms of balance equation of conserved quantity).
[Reply] Thank you for your suggestion. Fs(t) is the external force on the impeller structure, including the gravity of the pump itself, the fluid force and the centrifugal force generated by rotation, indicating that the impeller achieves dynamic balance under the external force.
- For clarity, may the authors provide the characteristic curve related to the centrifugal pump model, Q25H52, studied in this paper (i.e., the curve of pump flow rate versus pump head)?
[Reply] Thank you for your suggestion. We measured the characteristics of Q25H52 centrifugal pump at 6400rpm, as shown in Table 1. The flow head curve of Q25H52 centrifugal pump is supplemented in the paper.( The characteristic curve of Q25H52 centrifugal pump has been added in the paper)
Table 1 Test results under different flow rates at 6400rpm
|
Speed（rpm） |
Voltage（V） |
Current（A） |
Power（kw） |
Rate of flow(m3/h) |
Lift（m） |
Efficiency |
|
6400 |
384 |
7.47 |
2.87 |
0.36 |
54.6 |
0.021 |
|
8.28 |
3.18 |
2.32 |
52.5 |
0.116 |
||
|
8.48 |
3.26 |
5.45 |
51.5 |
0.261 |
||
|
9.60 |
3.69 |
9.60 |
50.5 |
0.398 |
||
|
10.6 |
4.07 |
14.1 |
46.5 |
0.487 |
||
|
11.4 |
4.38 |
17.6 |
45.5 |
0.553 |
||
|
12.1 |
4.65 |
22.7 |
40.4 |
0.597 |
||
|
12.4 |
4.76 |
25.8 |
37.1 |
0.608 |
||
|
12.9 |
4.95 |
28.3 |
32.3 |
0.558 |
||
|
13.1 |
5.03 |
30.6 |
28.3 |
0.521 |
Fig.1 Flow head curve of Q25H52 centrifugal pump at 6400rpm
- The authors obtained the dynamic equation of fluid by discretising the three-dimensional wave equation of the fluid (see page 4 of the manuscript). However, Eq. (3) is fully specified after having established the boundary conditions (BCs). For the sake of clarity, please specify (or describe) the BCs by showing that the dynamic equation (4) is compatible with such a choice.
[Reply] Thank you for your suggestion. In formula (3), â–½ refers to divergence, using the divergence formula:
Equation (3) can be discretized to obtain the dynamic equation (4) of the fluid.
- Cavitation is an essential problem that occurs in any pump. Cavitation occurs in the actual operation process. In addition, it creates noise and causes strong vibrations in the pump. Have the authors studied the causes of the cavitation able to provoke impeller vibration during the start-up process of the centrifugal pump? If so, may the authors briefly describe, at least qualitatively, this phenomenon?
[Reply] Thank you for your suggestion. In this paper, the cavitation causes of impeller vibration caused by centrifugal pump startup are qualitatively added: cavitation mainly occurs at the front and back of the impeller and the inner surface of the front cover. During centrifugal pump startup, cavitation is due to the fact that when the fluid flows through the flow passage parts of the pump, with the decrease of fluid pressure, when the fluid pressure near the blade is lower than the saturated vapor pressure of the fluid itself, the fluid will vaporize and produce bubbles. As the bubble continues to increase, it collapses and disappears under external conditions such as gas dissolution and steam condensation, thus causing local water hammer and cavitation. When the centrifugal pump is cavitating, it will cause the change of the external characteristics of the pump and produce vibration. (Corrected text is marked in light green font in the paper)
- Figure 4 shows the rotating speed and the rate of flow of the centrifugal pump during start-up. What is the extent of the data errors? In fact, if the magnitude of the error were sufficiently high, the wave-behaviour would be absent (which would lead to the conclusion that the dynamic equations used by the authors have been oversimplified). Please, discuss.
[Reply] Thank you for your suggestion. Fig. 4 is a graph of the transient speed (left) and flow (right) of the centrifugal pump during the startup process obtained through simulation. There is a certain fluctuation within 2s of the startup of the centrifugal pump, and the speed (left) and flow (right) will gradually stabilize after 2s of startup.
- In this work, the authors investigated the structural vibration characteristics of impeller under transient radial force. However, there are many other sources of vibration in pumps. Indeed, other common sources of vibration in pumps are bent shafts, unbalanced, misalignments, reaction forces, and contact between components. These factors have not been mentioned in the manuscript. For completeness (and clarity), the authors are asked at least to mention these factors, specifying that the main objective of this work is to investigate only some specific reasons that induce the vibration of the centrifugal pump.
[Reply] Thank you for your suggestion. Indeed, there are many sources for centrifugal pump vibration, such as bent shafts, unbalanced, misalignments, reaction forces, and contact between components. The main objective of this paper is to study the structural vibration characteristics of impeller under transient radial force, which has been clearly defined in this paper. (Corrected text is marked in orange font in the paper)
- Several studies - performed through numerical simulations - on pressure fluctuation characteristics of a centrifugal pump in its start-up process recently appeared in the literature. The results show that a large number of low-pressure areas and strong vortexes are formed within the impeller at the initial time of start-up. With the increase in rotating speed, the vortexes rapidly decrease and are concentrated on the blade non-working face. The vibration characteristics of the pumps are significantly influenced also by the presence of vortices. However, we may object that the dynamic equations studied by the authors are not sufficiently complete to take this problem into account as well. For the sake of completeness, the authors are invited to provide a (brief) comment on this matter.
[Reply] Thank you for your suggestion. It is very meaningful to study the pressure fluctuation characteristics of centrifugal pump during start-up. As the pressure at the inlet of the impeller is lower than the saturation pressure of the working water temperature, it will cause a part of the liquid to evaporate (vaporize), and the gas dissolved in the liquid will also be released when the pressure and temperature change, forming steam pockets. When the internal pressure of the liquid drops below the saturated vapor pressure of the liquid at this temperature, bubbles or pockets are formed in local areas; At the place where the pressure rises, the steam bubble is suddenly crushed by the surrounding pressure, and the liquid flow is squeezed towards the center of the steam bubble at a very high speed due to inertia, causing hydraulic impact to the equipment. The instability of the cavitation process causes vibration and noise to the water pump. At the same time, a large number of bubbles block the impeller channel, which will reduce the flow and head to varying degrees, and the efficiency will decline accordingly. The main objective of this paper is to study the structural vibration characteristics of impeller under the action of transient radial force. Later, we will continue to study the pressure fluctuation characteristics during the start-up of centrifugal pump.
8) Optional
As known, pressure head and efficiency of the centrifugal pump for different flow rates are data very useful to design the pumps in manufacturing industries (of course, together with the data from pump manufacturers). Other important factors are the instantaneous starting speed and power of the centrifugal pump. For a general view of the problem, it would be helpful to have a graphical representation of these factors i.e., the so-called "system curve".
[Reply] Thank you for your suggestion. In fact, we have measured the characteristics of q25h52 centrifugal pump at different speeds, as shown in Table 2.
Table 2 test results of centrifugal pump operation at different speeds
|
Speed（rpm） |
Voltage（V） |
Current（A） |
Power（kw） |
Rate of flow(m3/h) |
Lift（m） |
Efficiency |
|
3200 |
200 |
4.0 |
0.80 |
13.0 |
13.3 |
0.654 |
|
3600 |
216 |
4.7 |
1.01 |
13.8 |
15.6 |
0.647 |
|
4000 |
242 |
5.5 |
1.34 |
14.9 |
18.1 |
0.607 |
|
4400 |
267 |
6.2 |
1.67 |
16.5 |
20.8 |
0.620 |
|
4800 |
291 |
7.3 |
2.12 |
17.8 |
23.8 |
0.606 |
|
5200 |
315 |
8.4 |
2.64 |
19.4 |
27.2 |
0.605 |
|
5600 |
340 |
9.5 |
3.23 |
21.0 |
31.0 |
0.610 |
|
6000 |
363 |
10.7 |
3.87 |
22.3 |
34.4 |
0.600 |
|
6400 |
384 |
11.8 |
4.53 |
23.8 |
37.9 |
0.601 |
|
6800 |
411 |
13.4 |
5.49 |
24.9 |
44.3 |
0.608 |
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Generally, the paper is poorly written, and I may not be the best person to fix it. On the other hand, the materials and results are not well explained. Massive revision is required for this paper. The introduction section, nevertheless, is sufficient but it can always be improved by adding information on outcomes and conclusions on each paper cited in this section.
1. Methods and Results
It is not clear to me on the approach used in this paper. In the PDE modeling or lump modeling, does the model use 1 DOF (degree of freedom) or multi DOFs? If it is multi-DOF, what would be matrices for the mass, stiffness, and damping? And what are the DOFs? If the model is single DOF, what is the DOF here? It must be a displacement or rotation, but of what part? If the impeller is assumed as one (1) mass, then I think a one-DOF model is less than sufficient.
It is also not clear to me where the implementation of the PDE. Is it solved using Flomaster? What is being solved using Flomaster? It looks like that the model used in Flomaster is lump model or zero-degree model, but you need to explain that. Please elaborate results from Flomaster. Anyway, the diagram is too small and it needs detail explanation.
What is being solved by the CFD software? Where are the fluid simulation and fluid flow results? What is the stability criteria for the transient and the fluid-structure interaction simulation? Did you carry out any verification of the model? Did you carry out validation of the model? What is the software DM ?
BTW – all software (Creo, DM, Flomaster) need to have clear and detail citations.
The results refer to the radial displacement. Which node or location? And what is the radial direction that we talk about here?
In Figure 4 – the steady flowrate shows 8 m3/h but the rated flowrate of the pump is 25 m3/h. Which one is correct here? The fonts in this figure are too small.
Figure 2 – please add dimension.
Figure 5 – the figure and fonts here are too small. The notation for displacement is too small.
Figure 6 – What is the node or location represented by the “Radial Displacement”? What is the angle of this displacement with respect to the flow direction at the inlet.
Figure 7 shows the “impeller vibration amplitude” – again, which node here? Is it just the maximum? So, its location can be different from one case to another? The fonts are too small here. The lines are too thin. Use black lines on all figures and graphs.
Figure 8 – same question? Which part of the impeller ? The entire impeller? Are modeling it as one-DOF then ? Not sure.
Figure 9 – the results show “stead” outcome – but the section title is “Analysis of Unsteady Flow …” This is confusing .
Many sentences in this paper are not well-written. They are either too long, illogical, or simply unclear. Some sentences contain repeated words. The following are just example. The authors have to consult experts for improvement.
In the “Purpose”
1. “The centrifugal pump is prone to vibration problems during operation, as the only running part of the centrifugal pump, the dynamic characteristics of the impeller during operation are the main reasons for the vibration of the centrifugal pump.”
Change to
“Centrifugal pumps are prone to vibration problems during operation due to poor dynamic characteristics of its impellers that serve as the only running parts of such devices”
2. “Therefore, it is necessary to study the influence of the internal fluid flow of the centrifugal pump on the dynamic characteristics of the impeller and explore the causes of the impeller vibration during the start- up process and unsteady flow of the centrifugal pump, so as to guide the design of the impeller of the centrifugal pump.”
Change to
“Therefore, it is important to study the internal fluid flow and its influence on the dynamic characteristics of the pump impeller and to explore the causes of vibration during the transient start-up process. The understanding of such phenomena may lead to better design of such impellers”
In the “Methods”
3. “The structure and flow channel model of centrifugal pump are established by Creo software”
Change to
“The geometry of the flow channel inside the centrifugal pump is established using Creo software (add citation here)”
4. “The centrifugal pump simulation experiment platform is built by flomaster software to obtain the variation law of speed and flow during the start-up of centrifugal pump, which is loaded into the simulation calculation of centrifugal pump.”
The last sentence was very unclear. Please change to
“The internal fluid flow computer simulation is carried out using Flomaster software, developed by …. (add citation here). The variation of speed and flow during the start-up process was further processed using …. “
5. “The vibration characteristics of impeller during the start-up of centrifugal pump are analyzed by fluid solid coupling method, and the structural vibration characteristics of impeller under transient radial force are obtained by harmonic response analysis”
Please explain the “fluid solid coupling method” and split the above sentence to two shorter sentences.
Author Response
Dear Editor,
We want to thank you and the reviewers for the valuable and constructive comments and suggestions for the improvement of this article. Please find herewith my responses, and revisions according to the review comments
Thank you very much.
With kind regards,
Yours sincerely,
Yufang Li
Reviewer 2：
Comments and Suggestions for Authors
- Methods and Results
It is not clear to me on the approach used in this paper. In the PDE modeling or lump modeling, does the model use 1 DOF (degree of freedom) or multi DOFs? If it is multi-DOF, what would be matrices for the mass, stiffness, and damping? And what are the DOFs? If the model is single DOF, what is the DOF here? It must be a displacement or rotation, but of what part? If the impeller is assumed as one (1) mass, then I think a one-DOF model is less than sufficient.
[Reply] Thank you for your suggestion. The model is a single degree of freedom. No slip boundary conditions are used on the contact boundary between the impeller and the pump casing and the fluid. Under the relative coordinates of rotation, all the walls on the impeller are relatively static. Set the inner surface of the front and rear cover plates of the impeller and the blade surface as the fluid structure coupling surface, and restrict the impeller to the cylinder, and the speed is the rated speed.
- It is also not clear to me where the implementation of the PDE. Is it solved using Flomaster? What is being solved using Flomaster? It looks like that the model used in Flomaster is lump model or zero-degree model, but you need to explain that. Please elaborate results from Flomaster. Anyway, the diagram is too small and it needs detail explanation.
[Reply] Thank you for your suggestion. Fig. 3 has marked the name and composition of each part in detail. Through the flomaster simulation experiment, the data of the speed changing with time during the startup process is obtained. The MATLAB curve fitting is performed to obtain the speed changing formula. The function with time as a variable is applied to the simulation of the centrifugal pump startup process, so as to obtain the dynamic characteristics of the impeller during the startup process.（Fig. in the text have been modified）
Fig. 3 Flomaster simulation experiment platform of centrifugal pump
- What is being solved by the CFD software? Where are the fluid simulation and fluid flow results? What is the stability criteria for the transient and the fluid-structure interaction simulation? Did you carry out any verification of the model? Did you carry out validation of the model? What is the software DM ?
[Reply] Thank you for your suggestion. The CFD software is used to simulate the fluid flow of the centrifugal pump. In the CFX flow field calculation, the time step is set to 0.005s, the total calculation time is 2s, and each time step is saved once. At the same time, it is coupled with the solid domain of the transient structural analysis to obtain the changes of the radial force on the impeller and the changes of the vibration amplitude, velocity and acceleration of the impeller structure. The convergence accuracy of transient and fluid structure coupling simulation is 10-4, and the grid independence is verified. DM refers to Design Modeler, a modeling module of ANSYS Workbench. The following is the distribution of pressure field and velocity field of centrifugal pump.
- BTW – all software (Creo, DM, Flomaster) need to have clear and detail citations.
[Reply] Thank you for your suggestion. Creo 4.0，ANSYS Workbench Design Modeler 19.0，Flomaster V9 are applied. (Corrected text is marked in oxblood red in the paper)
- The results refer to the radial displacement. Which node or location? And what is the radial direction that we talk about here?
[Reply] Thank you for your suggestion. Radial refers to the diameter direction of centrifugal pump impeller. F1, F2, F3 and F4 in the figure below are the radial force distribution of centrifugal pump impeller under 0.8q, Q, 1.2Q and 1.5Q working conditions respectively. The magnitude and direction of radial force change with the change of flow.
- In Figure 4 – the steady flowrate shows 8 m3/h but the rated flowrate of the pump is 25 m3/h. Which one is correct here? The fonts in this figure are too small.
[Reply] Thank you for your suggestion. The rated flow of the centrifugal pump is 25m3/h. The unit in Figure 4 is wrong and has been corrected. (The corrected figure is shown in Figure 5 in the paper)
Fig. 5 Speed and flow curve of centrifugal pump during startup
- Figure 2 – please add dimension.
[Reply] Thank you for your suggestion. Figure 2 has added a three-dimensional coordinate system.
- Figure 5 – the figure and fonts here are too small. The notation for displacement is too small.
[Reply] Thank you for your suggestion. The figure and font in Figure 5 have been adjusted to fit.
- Figure 6 – What is the node or location represented by the “Radial Displacement”? What is the angle of this displacement with respect to the flow direction at the inlet.
[Reply] Thank you for your suggestion. Radial refers to the diameter direction of centrifugal pump impeller. The magnitude and direction of radial force change with the change of flow. The displacement is 90 ° to the inlet flow direction.
- Figure 7 shows the “impeller vibration amplitude” – again, which node here? Is it just the maximum? So, its location can be different from one case to another? The fonts are too small here. The lines are too thin. Use black lines on all figures and graphs. Figure 8 – same question? Which part of the impeller ? The entire impeller? Are modeling it as one-DOF then ? Not sure.
[Reply] Thank you for your suggestion. The impeller amplitude shown in Figure 7 is the maximum value of the overall amplitude of the impeller. We mainly analyze the overall impeller. Figure 8 is also the overall impeller. All the lines in the figure have been changed to black.
- Figure 9 – the results show “stead” outcome – but the section title is “Analysis of Unsteady Flow …” This is confusing .
[Reply] Thank you for your suggestion. Figure 9 belongs to Section 4.1 and shows the steady-state radial force distribution under different working conditions. This paper first simulates the steady-state conditions, and then simulates the change of transient radial force with time under different flow conditions.
- Many sentences in this paper are not well-written. They are either too long, illogical, or simply unclear. Some sentences contain repeated words. The following are just example. The authors have to consult experts for improvement.
[Reply] Thank you for your suggestion.The sentences and English language in this paper were carefully reviewed and edited. (Corrected text is marked in red font in the paper)
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors answered the questions raised in my previous report. Question 3) was not answered. The reviewer's question was: "The authors obtained the dynamic equation of fluid by discretising the three-dimensional wave equation of the fluid (see page 4 of the manuscript). However, Eq. (3) is fully specified after having established the boundary conditions (BCs). For the sake of clarity, please specify (or describe) the BCs by showing that the dynamic equation (4) is compatible with such a choice." The authors replied: "In formula (3), â–½ refers to divergence, using the divergence formula:
div (\rho u) = \partialx (\rho ux) + \partialy (\rho uy) + \partialz (\rho uz).
Equation (3) can be discretised to obtain the dynamic equation (4) of the fluid". Of course, this reply does not provide the requested answered. This is a pity, as this may be source of objections from the reader. Apart question 3), the authors replied satisfactorily to the other questions. Anyhow, apart from the aforementioned flaw, in my opinion, the work deserves to be published.
Author Response
[Reply] Thank you for your suggestion. The specific formula is derived as follows:
The dynamic equation of centrifugal pump impeller is:
(1)
In formula (1), Ms is the mass matrix of centrifugal pump, Cs is the damping matrix of centrifugal pump, Ks is the stiffness matrix of centrifugal pump; is the acceleration vector, is the velocity vector and X is displacement vector; Fs(t) is the prestress of centrifugal pump, namely: fluid force, self gravity and self rotating centrifugal force.
The mode of centrifugal pump impeller in air is dry mode and undamped mode. Therefore, Cs =0, Fs(t)=0, and the dynamic equation of equation (1) can be simplified as:
(2)
For the case of linear small disturbance, the ideal fluid is inviscid, uniform and irrotational. According to Euler equation, the dynamic balance equation or motion equation of fluid can be derived as follows:
(3)
Where, ρ Is the density of the fluid, u, v and w are the displacement components of the fluid particle respectively, vx, vy and vz are the velocity components of the fluid particle respectively, and P is the pressure of the fluid particle.
Set k as the compression modulus of the fluid, and the continuity equation of the compressible fluid is:
(4)
From equation (3) to equation (4), the derivative of time t can be obtained, and the fluid motion control equation is:
(5)
In formula (5), c is the velocity of sound in the fluid, and its magnitude is determined by the formula c=(k/ρf)1/2 , K is the fluid compressibility modulus, ρf is the density of the fluid. is Laplace operator.
The effect of fluid medium on impeller structure can be expressed as:
(6)
In equation (5), â–½P is the pressure gradient along the normal vector n, is the velocity vector of a liquid particle.
The centrifugal pump impeller structure is discretized by finite element method to obtain the fluid dynamic equation:
(7)
In formula (7), Mf and Kf are mass matrix and stiffness matrix of fluid respectively, R is the coupling matrix between the fluid and the structure, and Ff is the external force acting on the fluid.
For the action of acoustic fluid, the dynamic equation of impeller structure is:
(8)
In formula (8), Fp(t) is the surface force vector on the fluid solid contact surface, and F0 is the fluid force applied to the impeller structure.
The coupling three-dimensional equation of centrifugal pump impeller structure and flow field can be obtained by combining equations (7) and (8):
(9)
In formula (9), Mfs and Kfs are equivalent coupling mass matrix and stiffness matrix respectively.
Author Response File: Author Response.pdf
Reviewer 2 Report
This is my approval for the revised one. Thank you for the revision. I hope you realized the different. Graphics and tables are already improved. It looks better but the English still can always be improved.
Author Response
We want to thank you and the reviewers for the valuable and constructive comments and suggestions for the improvement of this article. The sentences and English language in this paper were carefully reviewed and edited.
