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Article
Peer-Review Record

Characteristics of Air Resistance in Aerostatic Bearings

by Jianlong Yin 1,2, Jing Yu 2,*, Pengfei Cao 2, Dongsheng Li 2, Xiaoyan Shen 2 and Ming Li 1
Reviewer 1: Anonymous
Reviewer 2: Anonymous
Submission received: 8 October 2021 / Revised: 8 November 2021 / Accepted: 9 November 2021 / Published: 11 November 2021
(This article belongs to the Section Additive Manufacturing Technologies)

Round 1

Reviewer 1 Report

dear Authors, my comments in the attached file

Comments for author File: Comments.pdf

Author Response

The paper investigates the pneumatic capillary resistance for gas bearings. An empirical formula of the air resistance (8) is proposed and some experimental measures of flow rate in tubes of different diameter and length verify the formulation.

1.The authors should better explain the purpose of the activity. It is known that most gas bearings use feed holes with diameter D equal to 0.1-0.15 mm and length L generally not greater than 1 mm. When L / D <5-8 the isentropic expansion formula works well and is generally adopted to describe the behavior of the resistance. If the L / d ratio of the supply hole is further increased, the flow rate is better described by the well-known Poiseuille formula valid for the capillary tube and traced back to compressible fluids. In my opinion could be added in bibliography some papers about this theoretical formulation.

Answer: The purpose has been explained.

For aerostatic bearings, we want to get smaller orifices, which will undoubtedly increase the value of L/D. Some research results show that: reducing the diameter of the orifice will help improve the static and dynamic performance of the aerostatic bearing. Chen proposed an arrayed microhole restrictor (AMR) to suppress the vortex flow and reduce vibration of aerostatic bearings. By computational fluid dynamics analysis, the transient flow features are studied for aerostatic bearings with AMR and conventional restrictors, and static performances of the bearings are also compared. Vibration strength of the bearing is measured experimentally to validate the effectiveness of AMR. The results show that vortex shedding in the recess is suppressed and the vibration can be effectively reduced by AMR, while load capacity and stiffness of the bearing remain unchanged [1]. Belforte presents an experimental study on pneumostatic pads with micro holes realized with laser technology. The minimum orifice diameter is about 0.05mm. In order to optimize the performance of the pads, the influence of the diameter and the number of holes on load capacity, stiffness and consumption versus height air gap was analyzed. [2 ].

Due to the diverse structures of aerostatic bearings, and the length of the orifice has a small influence on the performance of the bearing, there is no strict requirement on the length of the orifice. In theory, as the diameter of the orifice decreases, the value of L/D may exceed the general design criteria (such as 5-8 you mentioned). Therefore, in this case, the formula we proposed may be more suitable.

In addition, the radial air film pressure distribution of the aerostatic bearing can be divided into the area near the orifice and the area of viscous resistance [3]. For the viscous resistance, it is laminar flow, which makes Hagen-poiseuille's formula still applicable.

And the reference has been added.

2.With Poiseuille formulation the speed is not constant along the tube, that is more realistic. You get:

 

From the same formula it also results that the pressure difference (? 1 − ? 2 ) is proportional to the volume flow Q= G/  . In formula 5 of the paper it appears that the pressure difference is proportional to ? 2 . Authors should better explain the assumptions made to obtain the proposed formula.

Answer:

We analysis the orifice restrictor in this article, it is mostly cylindrical. According to the flow required by the restrictor, the inner diameter is generally below 10 mm, which is a horizontal short straight pipe. The average density can be used to approximate the density of the inlet and outlet sections. Bernoulli equation of viscous compressible fluid flowing at high speed under adiabatic condition:

        (1-1)

And then:

        (1-2)

The first bracket on the right side of the equal sign is the change in kinetic energy, and the second bracket is the change in potential energy. When the flow of the viscous compressible fluid is adiabatic, the kinetic energy change and the potential energy change are both small and can be ignored, so the formula can be rewritten as

                    (1-3)

The energy loss of viscous fluid flowing  in the pipeline consists of pressure loss along the way  and local resistance loss . For the horizontal short straight pipe, there is no need to consider the local resistance loss , so the energy loss of the viscous fluid when moving in the horizontal short straight pipe  is equal to the pressure loss along the way . According to the dimensional analysis, the formula for the pressure loss of the viscous fluid in the horizontal short straight pipe is:

                   (1-4)

The pressure loss calculation formula in the horizontal short straight pipe can be obtained:

             (5)

Just like the determinant formula of resistance, , the resistance value is determined by the material. Therefore, we want to establish the air resistance similar to the resistance. The mass flow calculation formula you mentioned is indeed a good way to calculate mass flow. It can be seen that the mass flow  is proportional to .

However, if  is proposed on the right side of the equation, the right side of the equation is , and there are parameters other than the material property parameters, so this cannot be used as the determinant of air. resistance. From equation (6), it can be considered that are determined by the material, and  can be obtained by empirical formula. Therefore, consider  is a coefficient, and then is proportional to .

3.The authors calculate the resistance Rg using equation 7. It would be appropriate to calculate Rg also using the well-known Poiseuille formula to compare with the previous results.

Answer: In formula (5), the head loss  is mentioned. Its calculation formula is closely related to Hagen Poiseuille’s law. It can be obtained by substituting the calculation formula of Reynolds number into Hagen Poiseuille’s law (refer to Lin Jianzhong's "Fluid Mechanics" P226). Therefore, in the process of deriving the air resistance value, the Hagen Poiseuille's law is actually used. So, our main job is give the definition of air resistance.

4.Insert nomenclature at the beginning of the paper.

Answer: It has been inserted.

5.The graph of figure 5 shows the trend of k (eq. 8) as ? 1 varies. Results should also be shown for pressures below 0.12 MPa, as in theoretical model was made.

Answer: In the theoretical model the input pressure is (0.01-0.15) MPa. Figure 5 shows the measurement results. But because of the experimental device limitations, the pressures below 0.12MPa can not be measured. We have designed another measurement device; this would be improved.

6.Explain formula (9) and formula of Re (page 8, line 255).

Answer:

The pressure and temperature of the inlet and outlet of the measuring pipeline are measured by the sensor. At the same time, according to the density form of the gas state equation , the calculation formula for the average density of the stagnant gas in the measuring pipeline is:

                       (1)

——The average density of stagnant gas in the measured pipeline;

——The inlet air pressure of the measured pipeline;

——The air pressure at the inlet of the measured pipeline.

Since the air in the measurement pipeline is in a flowing state, compared with stagnant air, the density of flowing air is slightly lower than that of stagnant air, which can be calculated according to the following formula:

                    (2)

——The average density of the gas flowing in the measured pipeline;

——Mach number.

According to the continuity equation of air:

                    (3)

——Average density in the measurement pipeline;

——The average flow velocity in the measurement pipeline.

Among that, the average flow rate of the air in the measurement pipeline can be obtained from the average flow density and average flow rate of the measured pipeline.

The mathematical model is as follows:

,       (9)

where  is the hydraulic diameter of the measured pipeline and  is the local speed of sound.

Theoretically, the parameter  changes with the Reynolds number , showing a negative exponential relationship, setting the parameter expression as . Using the least-squares method, the parameter  and the Reynolds number  are curve-fitted to obtain the best estimated value of the coefficients  and  in the expression of the parameter :

 

where —— the Reynolds number.

7.In paragraph 3 it is highlighted that Rg has a minimum in graphs 6-9 but the reason is not explained.

Answer: The Rg has a minimum in graphs 6-9. The Mach number at the turning point is approximately 0.3, which is the boundary point between the low-speed flow and high-speed flow. The minimum value indicates the low-speed and high-speed interchanges.

References:

[1].     Chen, X., et al., Vortex suppression and nano-vibration reduction of aerostatic bearings by arrayed microhole restrictors. Journal of Vibration and Control, 2017. 23(5): p. 842-852.

[2].     Belforte, G., et al., Experimental Analysis of Air Pads with Micro Holes. Tribology Transactions, 2013. 56(2): p. 169-177.

[3].     Belforte, G., et al., Discharge coefficients of orifice-type restrictor for aerostatic bearings. Tribology International, 2007. 40(3): p. 512-521.

 

 

Author Response File: Author Response.docx

Reviewer 2 Report

Dear Authors

The paper is interesting and it is dealing with the air resistance calculation in aerostatic bearings. Although there are several papers in the literature dealt with this topic, this can be also a good aspect in the field since a set of mathematical developments was done by the authors would be helpful for the others.

It would have the potential to be considered in a publication if the authors address all reviewer’s concerns. The language must be reviewed, there are several typos and grammar issues were found.

DETIALS ON NUEMRICAL SIMULATIONS ARE MISSED. PLEASE INCLUDE IT IN THE REVISED VERSION.

Major Comments:

  • There are several sentences beginning with “WE”. It is recommended to use the passive tense in the academic manuscript. Therefore, it would be great if such sentences rephrased with the passive tense.
  • English must be reviewed;
  • What is the novelty of the work? It was not described in the introduction nor abstract. This is the main Reviewer’s concern. It is highly recommended that the Authors state clearly the main contribution of this work and pointing out the innovative aspects of this work.
  • When presenting equations, it is better to cite a reference for them. Therefore, the readers could easily reach them.
  • The manuscript lacks a table of nomenclature including all symbols and variables.
  • The organisation of the paper must be improved; too many subsections. Please simplify and merge the subsections.
  • Section 3, it was stated that the results presented in Figure 4 are obtained by simulation. What kind of simulation? Please clarify this matter in detail.

Very Best

The Reviewer

Author Response

  1. There are several sentences beginning with “WE”. It is recommended to use the passive tense in the academic manuscript. Therefore, it would be great if such sentences rephrased with the passive tense.

Answer: We modified it, and also find Editage (www.editage.cn) for English language editing.

  1. English must be reviewed;

Answer: We have found Editage (www.editage.cn) for English language editing.

  1. What is the novelty of the work? It was not described in the introduction nor abstract. This is the main Reviewer’s concern. It is highly recommended that the Authors state clearly the main contribution of this work and pointing out the innovative aspects of this work.

Answer: Air resistance is proposed in this study to illustrate the air flow characteristics in aerostatic systems. In this study, the capillary tubes to analyze and measure the air drop and determine the mechanism of the throttle to define air resistance. The introduction and abstract have been modified.

  1. When presenting equations, it is better to cite a reference for them. Therefore, the readers could easily reach them.

Answer: The reference is cite in the describe of the equations. otherwise, the equations are calculated.

  1. The manuscript lacks a table of nomenclature including all symbols and variables.

Answer: It has been inserted.

  1. The organisation of the paper must be improved; too many subsections. Please simplify and merge the subsections.

Answer: Thank you for your advice, the subsections have been simplified and merged.

 

  1. Section 3, it was stated that the results presented in Figure 4 are obtained by simulation. What kind of simulation? Please clarify this matter in detail.

Answer: 

First, the simulation model is established. Pipes with the following dimensions (inner diameter [mm] × length [m]): 2 × 0.5, 2 × 1.0, 3 × 0.5, 3 × 1.0, 4 × 0.5, and 4 × 1.  We use COMSLO to simulation, and the air pressure at the end of the pipeline under test and the average flow velocity in the pipeline under test can be obtained.

The details for 2 × 0.5 pipes:

The medium is air, and the flow state is turbulent, so the model is selected. We first import the tested pipeline and the measured pipeline drawn by SolidWorks, and then obtains the internal flow channel model through the Boolean operation in Comsol Multiphysics 5.4. Figure 1 shows the measured pipeline-measured pipeline model.

 

Figure 1 The measured pipeline-measured pipeline model.

Click on the grid in the toolbar to create a solid geometry containing the measured pipeline model, click Boolean operation and segmentation-subtraction to obtain the internal flow channel model of the measured pipeline-measured pipeline, as shown in Figure 2.

 

Figure 2 Internal flow channel model of the measured pipeline-measured pipeline

And then, the simulation model is established. The main branch of the channel is the pressure chamber of the pressure sensor, and the pressure drop is basically negligible. Therefore, the air in the pipeline is basically flows in one direction.

The inlet and outlet are added to the  model. Since the flow velocity under actual working conditions is uncontrollable, the boundary conditions of the inlet and outlet are set as pressure conditions, and the remaining boundaries are set as non-slip walls.

Since the simulation model is not particularly complicated, the free tetrahedral mesh is selected for meshing, and the maximum element size is set to 0.02 mm, and the minimum element size is mm. After setting, click to build the selected object to mesh the entire geometry, such as shown in Figure 3.

 

Figure 3 mesh the entire geometry,

The quality of the meshing quality greatly affects the quality of the simulation results. The average element quality is 0.691, the mesh quality is good and can be simulated in the next step.

Then, the air resistance of the pipeline under test is calculated using Eqs. (7).

Finally, the air pressure and average flow at the inlet and outlet of the measurement pipeline are simulated in the air resistance model of the pipeline under test, and the air resistance of the pipeline under test is calculated. The results are compared, as shown in Figure 4.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

dear authors,

it is clear you use a different formulation of flow rate than Poiseuille formula. In fact, with Poiseuille formula, assuming an average density of the flow through the tube, the pressure difference p1-p2 is proportional to Q. With the proposed formula (7) the pressure difference is proportional to Q^2. Of course the behavior of the resistance can be described with an empirical function and then experimentally identifying the coefficient k to obtain the correct result.

The upstream and downstream pressures and the flow rate through the pipe are known from the experimental measurements. Although a study has already been done by Lin Jianzhong's and the current objective is the calculation of resistance, in my opinion you can easily calculate the theoretical flow rate with Poiseuille's formula and compare it with the experimental flow rate. I think that a comparison of the proposed formulation and Poiseuille formula improves the quality of the work, probably strengthening the use of the proposed formulation. In paragraph 3.2 it is possible to add, at least in some cases, a comparison graph between the experimental flow values, those of the Poiseuille formula and those obtainable from the empirical formula. The field of applicability of the proposed formula can be discussed in conclusion.

Author Response

   Thank you very much for your advice, we are trying to calculate the theoretical flow rate with Poiseuille's formula and compare it with the experimental flow rate in our next work. And then, we will compare the Poiseuille formula and those obtainable from the empirical formula. Maybe the results of the Poiseuille formula and the empirical formula are the same. This will help to establish the air resistance theory.

Author Response File: Author Response.docx

Reviewer 2 Report

Dear Authors

Good improvement! Accepted!

Author Response

Thank you very much for your work and help

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