Non-Dimensional Groups, Film Thickness Equations and Correction Factors for Elastohydrodynamic Lubrication: A Review

Round 1

Reviewer 1 Report

1. The latest papers are not referenced.
2.  V. Simon has published many papers on the full and mixed thermo-elastohydrodynamic lubrication analysis in different types of gears, not referenced.
3.  For a full thermo-elastohydrodynamic lubrication analysis the use of non-dimensional groups is not convenient.

Author Response

Dear editors,

we greatly appreciate the positive feedback provided on our invited review paper and the opportunity to resubmit a revised version of our manuscript. Also, we would like to take this opportunity to express our thanks to the reviewers for their excellent feedback.

Based upon their comments, we have improved our manuscript. All changes made have been  highlighted in yellow. Furthermore, we have prepared a point-by-point response.

We believe that the manuscript has substantially improved after making suggested edits. We hope our revised version will be received favorably and are looking forward to receive a positive response from your side.

On behalf of all co-authors,

Andreas Rosenkranz

Point-by-point responses:

Reviewer #1:

The latest papers are not referenced.

1. Simon has published many papers on the full and mixed thermo-elastohydrodynamic lubrication analysis in different types of gears, not referenced.

For a full thermo-elastohydrodynamic lubrication analysis the use of non-dimensional groups is not convenient.

A: We thank the reviewer for his feedback and absolutely agree that there have been undisputed and significant advances in the numerical modeling of the TEHL problem by various researchers, also for application-oriented conditions including transient effects or mixed friction. We have extended the works cited in the manuscript in this respect. Nevertheless, we would like to point out that this review is mainly focused on non-dimensionless groups, analytically solvable film thickness equations and their correction factors. We therefore believe that we covered most of the relevant work. The aspect that not all effects are sufficiently covered by these analytically solvable equations is also part of our discussion about applicability, limitations and potential future directions.

Reviewer #2:

This review paper goes over non-dimensional groups relevant to EHL and film thickness analytical predictions using standard formulae and correction factors. The latter are used to account for shear-thinning, thermal, starvation and surface roughness effects. Overall, the manuscript is well-written, and the proposed review is of interest and value to the EHL community. It may benefit though from some minor additions, modifications and corrections, as listed below.

A: The authors are particularly grateful for the positive feedback as well as the careful and specific comments. We have addressed all raised points below and believe that the manuscript substantially improved after suggested edits.

1. The nomenclature should be entirely revisited for omissions. Many parameters throughout the manuscript are missing from the nomenclature e.g. T_0 in equation (58), u and h_n in equation (64) (besides, “u” should be the mean entrainment speed “u_m” here, I believe), h_liq in equation (77), etc.

A: Many thanks for the detailed review. We have thoroughly rechecked the nomenclature and added the missing parameters.

2. Page 1, lines 30-32: the authors state that in EHL, elastic deformations are of the same order of magnitude as film thickness. This is not very accurate. Typically, in EHL, elastic deformations can be orders of magnitude higher than the film thickness.

A: The original statement particularly aimed at a differentiation to conformal contacts with macro-elastic deformations, for example journal sliding bearings, which are also frequently referenced as EHL. However, we are grateful for the comment and have modified the statement (line 31).

3. (5): in the definition of M_2D, W should be W_2D and in the definition of M_3D, it should be W_3D.

A: We have adapted equation (5) as suggested. For better clarity, we have added this in all the appropriate formulas.

Section 2: In reference [55], there was a set of dimensionless groups that were introduced to identify / delineate regimes of friction in EHL. These groups should be listed in this section, out of completeness.

A: Many thanks for this valuable advice. We have added the groups from the mentioned reference (lines 92–102)

4. Section 3: The authors are advised to split this section in two parts: 2D and 3D. Going back-and-forth between 2D and 3D film thickness equations is rather confusing.

A: We have rearranged and subdivided section 3 into 2D (3.1; lines 106–148) and 3D contacts (3.2; lines 149–175).

5. Equation (57): This correction factor was originally proposed by Venner and Bos (Wear, 1994, vol. 173, pp. 151-165) under steady-state isothermal Newtonian pure-rolling conditions. It was later shown to also hold under steady-state thermal non-Newtonian rolling-sliding conditions by Habchi and Bair (ASME Journal of Tribology, 2013, vol. 135 (1), 011502).

A: We are thankful for making us aware of that. We have corrected this accordingly (formerly equation (57), now (61)) and lines 190–191.

6. Figure 5 caption: “using eq. (60)” should be “using eq. (61)”.

A: The caption of figure 5 has been updated (now eq. (65)).

7. Equations (69) and (70): These correction factors were derived for double-Newtonian modified Carreau fluids. This should be specified in the text to differentiate with earlier formulae provided in this section for Carreau fluids.

Many thanks for this comment. We have added this information ((72) and (73)).

Equation (74): “Phi_s,min” should be “Phi_s,c”.

A: Equation (74) (now (77)) has been corrected.

Section 4.5, lines 250-251: The definition of the Peklenik factor should be provided, out of completeness. Also, “the ratio of the quadratic mean surface roughness to the smooth lubricant gap” should be “the ratio of the smooth lubricant gap to the quadratic mean surface roughness”. The lambda ratio is h/sigma, not sigma/h.

A: We are grateful for the indication that the definition of the Peklenik factor would be beneficial for reading comprehension. Therefore, we have added this accordingly. In addition, we have adapted the description of the lubricating film ratio lambda, which is now correct and also fits the x-axis in figure 6 (lines 260–262).

Equation (81): This equation is out of context in this section and even in the entire manuscript. I suggest removing it.

A: We followed the reviewer's comment and removed the former equation (81) as well as the subsequent related correlations. Section 4.5 now ends after equation (83), formerly eq. (80).

Figure 7: Line contact domains are not clearly identifiable. I could only see a vertical line for each of DH and Moes, but no clearly identifiable rectangular domain. Also, in the legend, Masjedi and Khonsari should be [81] instead of [68].

A: We have improved the visibility of the domains in figure 7 (they had overlapped with the edge of the diagram) and updated the references to the current version.

In equation (84), M should be M_2D and in equation (85), it should be M_3D.

A: We have adapted the equations accordingly and fully agree that reading comprehension is now improved.

Reference [88]: the original reference should be listed here instead of (or in addition to) the corrigendum reference.

A: Thank you very much for this advice. We are sorry about this error and have corrected it.  

Thanks again for the valuable and detailed comments. This is highly appreciated. We are convinced that the manuscript has gained in quality and hope our revised version will be received favorably.

Author Response File: Author Response.docx

Reviewer 2 Report

This review paper goes over non-dimensional groups relevant to EHL and film thickness analytical predictions using standard formulae and correction factors. The latter are used to account for shear-thinning, thermal, starvation and surface roughness effects. Overall, the manuscript is well-written, and the proposed review is of interest and value to the EHL community. It may benefit though from some minor additions, modifications and corrections, as listed below:

• The nomenclature should be entirely revisited for omissions. Many parameters throughout the manuscript are missing from the nomenclature e.g. T_0 in equation (58), u and h_n in equation (64) (besides, “u” should be the mean entrainment speed “u_m” here, I believe), h_liq in equation (77), etc.
• Page 1, lines 30-32: the authors state that in EHL, elastic deformations are of the same order of magnitude as film thickness. This is not very accurate. Typically, in EHL, elastic deformations can be orders of magnitude higher than the film thickness.
• Equation (5): in the definition of M_2D, W should be W_2D and in the definition of M_3D, it should be W_3D.
• Section 2: In reference [55], there was a set of dimensionless groups that were introduced to identify / delineate regimes of friction in EHL. These groups should be listed in this section, out of completeness.
• Section 3: The authors are advised to split this section in two parts: 2D and 3D. Going back-and-forth between 2D and 3D film thickness equations is rather confusing.
• Equation (57): This correction factor was originally proposed by Venner and Bos (Wear, 1994, vol. 173, pp. 151-165) under steady-state isothermal Newtonian pure-rolling conditions. It was later shown to also hold under steady-state thermal non-Newtonian rolling-sliding conditions by Habchi and Bair (ASME Journal of Tribology, 2013, vol. 135 (1), 011502).
• Figure 5 caption: “using eq. (60)” should be “using eq. (61)”.
• Equations (69) and (70): These correction factors were derived for double-Newtonian modified Carreau fluids. This should be specified in the text to differentiate with earlier formulae provided in this section for Carreau fluids.
• Equation (74): “Phi_s,min” should be “Phi_s,c”.
• Section 4.5, lines 250-251: The definition of the Peklenik factor should be provided, out of completeness. Also, “the ratio of the quadratic mean surface roughness to the smooth lubricant gap” should be “the ratio of the smooth lubricant gap to the quadratic mean surface roughness”. The lambda ratio is h/sigma, not sigma/h.
• Equation (81): This equation is out of context in this section and even in the entire manuscript. I suggest removing it.
• Figure 7: Line contact domains are not clearly identifiable. I could only see a vertical line for each of DH and Moes, but no clearly identifiable rectangular domain. Also, in the legend, Masjedi and Khonsari should be [81] instead of [68].
• In equation (84), M should be M_2D and in equation (85), it should be M_3D.
• Reference [88]: the original reference should be listed here instead of (or in addition to) the corrigendum reference.

Author Response

Dear editors,

we greatly appreciate the positive feedback provided on our invited review paper and the opportunity to resubmit a revised version of our manuscript. Also, we would like to take this opportunity to express our thanks to the reviewers for their excellent feedback.

Based upon their comments, we have improved our manuscript. All changes made have been  highlighted in yellow. Furthermore, we have prepared a point-by-point response.

We believe that the manuscript has substantially improved after making suggested edits. We hope our revised version will be received favorably and are looking forward to receive a positive response from your side.

On behalf of all co-authors,

Andreas Rosenkranz

Point-by-point responses:

Reviewer #1:

The latest papers are not referenced.

1. Simon has published many papers on the full and mixed thermo-elastohydrodynamic lubrication analysis in different types of gears, not referenced.

For a full thermo-elastohydrodynamic lubrication analysis the use of non-dimensional groups is not convenient.

A: We thank the reviewer for his feedback and absolutely agree that there have been undisputed and significant advances in the numerical modeling of the TEHL problem by various researchers, also for application-oriented conditions including transient effects or mixed friction. We have extended the works cited in the manuscript in this respect. Nevertheless, we would like to point out that this review is mainly focused on non-dimensionless groups, analytically solvable film thickness equations and their correction factors. We therefore believe that we covered most of the relevant work. The aspect that not all effects are sufficiently covered by these analytically solvable equations is also part of our discussion about applicability, limitations and potential future directions.

Reviewer #2:

This review paper goes over non-dimensional groups relevant to EHL and film thickness analytical predictions using standard formulae and correction factors. The latter are used to account for shear-thinning, thermal, starvation and surface roughness effects. Overall, the manuscript is well-written, and the proposed review is of interest and value to the EHL community. It may benefit though from some minor additions, modifications and corrections, as listed below.

A: The authors are particularly grateful for the positive feedback as well as the careful and specific comments. We have addressed all raised points below and believe that the manuscript substantially improved after suggested edits.

1. The nomenclature should be entirely revisited for omissions. Many parameters throughout the manuscript are missing from the nomenclature e.g. T_0 in equation (58), u and h_n in equation (64) (besides, “u” should be the mean entrainment speed “u_m” here, I believe), h_liq in equation (77), etc.

A: Many thanks for the detailed review. We have thoroughly rechecked the nomenclature and added the missing parameters.

2. Page 1, lines 30-32: the authors state that in EHL, elastic deformations are of the same order of magnitude as film thickness. This is not very accurate. Typically, in EHL, elastic deformations can be orders of magnitude higher than the film thickness.

A: The original statement particularly aimed at a differentiation to conformal contacts with macro-elastic deformations, for example journal sliding bearings, which are also frequently referenced as EHL. However, we are grateful for the comment and have modified the statement (line 31).

3. (5): in the definition of M_2D, W should be W_2D and in the definition of M_3D, it should be W_3D.

A: We have adapted equation (5) as suggested. For better clarity, we have added this in all the appropriate formulas.

Section 2: In reference [55], there was a set of dimensionless groups that were introduced to identify / delineate regimes of friction in EHL. These groups should be listed in this section, out of completeness.

A: Many thanks for this valuable advice. We have added the groups from the mentioned reference (lines 92–102)

4. Section 3: The authors are advised to split this section in two parts: 2D and 3D. Going back-and-forth between 2D and 3D film thickness equations is rather confusing.

A: We have rearranged and subdivided section 3 into 2D (3.1; lines 106–148) and 3D contacts (3.2; lines 149–175).

5. Equation (57): This correction factor was originally proposed by Venner and Bos (Wear, 1994, vol. 173, pp. 151-165) under steady-state isothermal Newtonian pure-rolling conditions. It was later shown to also hold under steady-state thermal non-Newtonian rolling-sliding conditions by Habchi and Bair (ASME Journal of Tribology, 2013, vol. 135 (1), 011502).

A: We are thankful for making us aware of that. We have corrected this accordingly (formerly equation (57), now (61)) and lines 190–191.

6. Figure 5 caption: “using eq. (60)” should be “using eq. (61)”.

A: The caption of figure 5 has been updated (now eq. (65)).

7. Equations (69) and (70): These correction factors were derived for double-Newtonian modified Carreau fluids. This should be specified in the text to differentiate with earlier formulae provided in this section for Carreau fluids.

Many thanks for this comment. We have added this information ((72) and (73)).

Equation (74): “Phi_s,min” should be “Phi_s,c”.

A: Equation (74) (now (77)) has been corrected.

Section 4.5, lines 250-251: The definition of the Peklenik factor should be provided, out of completeness. Also, “the ratio of the quadratic mean surface roughness to the smooth lubricant gap” should be “the ratio of the smooth lubricant gap to the quadratic mean surface roughness”. The lambda ratio is h/sigma, not sigma/h.

A: We are grateful for the indication that the definition of the Peklenik factor would be beneficial for reading comprehension. Therefore, we have added this accordingly. In addition, we have adapted the description of the lubricating film ratio lambda, which is now correct and also fits the x-axis in figure 6 (lines 260–262).

Equation (81): This equation is out of context in this section and even in the entire manuscript. I suggest removing it.

A: We followed the reviewer's comment and removed the former equation (81) as well as the subsequent related correlations. Section 4.5 now ends after equation (83), formerly eq. (80).

Figure 7: Line contact domains are not clearly identifiable. I could only see a vertical line for each of DH and Moes, but no clearly identifiable rectangular domain. Also, in the legend, Masjedi and Khonsari should be [81] instead of [68].

A: We have improved the visibility of the domains in figure 7 (they had overlapped with the edge of the diagram) and updated the references to the current version.

In equation (84), M should be M_2D and in equation (85), it should be M_3D.

A: We have adapted the equations accordingly and fully agree that reading comprehension is now improved.

Reference [88]: the original reference should be listed here instead of (or in addition to) the corrigendum reference.

A: Thank you very much for this advice. We are sorry about this error and have corrected it.  

Thanks again for the valuable and detailed comments. This is highly appreciated. We are convinced that the manuscript has gained in quality and hope our revised version will be received favorably.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The corrections are accepted.