THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. 02643-4-I AUTHORS' CLOSURE ON DISCUSSIONS ON "A STATISTICALLY VERIFIED MODEL FOR CORRELATING VOLUME LOSS DUE TO CAVITATION OR LIQUID IMPINGEMENT'' (To be published ASTM STP 474, 1970) by: F. G. Hammitt Y. C. Huang C.. L. Kling. T. M. Mitchell L. P. Solomon Financial Support Provided by: U. S. Naval Air'Development Center Contract No. N62269-69-C-0285 This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Commanding Officer, Naval Air Development Center. December 1969

CLOSURE ON DISCUSSIONS ON "A STATISTICALLY VERIFIED MODEL FOR CORRELATING VOLUME LOSS DUE TO CAVITATION OR LIQUID IMPINGEMENT " by y F G, Hammitt Y. C, Huang C. L. Kling T. M.. Mitchell L. P. Solomon The authors first of all would like to thank the various discussors for their very significant contributions to the subject matter of this paper. Much new data and many pertinent points have been added in these discussions for which the authors are grateful. Where additional elaboration on our part seems desirable, this is made in the following paragraphs, which consider the various discussions in alphabetical order. Both Prof. Elliot and Mr. Heymann, with respect to the first portion of the paper which involves very high velocity rocket sled "rain erosion" tests on rain erosion type materials which are generally not highly resistant to erosion (as compared with metals), make the point that for such materials at such velocities, fatigue is not an important erosion mechanism. Hence, the lack of success of the threshold velocity concept, proposed first by Pearson of C. E. G. B. for turbine blade erosion applications where fatigue failure is predominant, is not surprising. This point is further corroborated in the discussion of Messrs. Rao and Rao. We fully agree. We also agree with Mr. Heymann's comment in this regard that the threshold velocity must be a function of many variables other than material mechanical properties such as test duration, droplet size, surface roughness, and as Prof. Elliot points out, the extent of continued surface wetting, especially for very small drops.

Dr. Engel points out the probable necessity of dividing materials to be considered into various groupings if a good correlation with material mechanical properties is to be achieved. We agree that this is probably required if close correlations are to be achieved, since "there are as many mechanisms of multipledrop-impact erosion as there are broad groups of material properties". We have not been able as yet to pursue her suggestion that this might usefully be accomplished on the basis of brittleness and work-hardening capacity, but agree that this might be a useful approach. It is indeed encouraging to note the similarities in correlation of damage rates between our data set and that of Mr. Heymann with respect to mechanical property groupings. As he mentions, there is a maximum spread of a factor of about 4 around our best fit line (his Fig. 1) as applied to his data set (or to our own), giving an overall range of the data at a given ultimate resilience, e.g., of a factor of about 15. However, our "factorial standard error of estimate" is about 2. 5 for this case (Table 1 of Closure) indicating that approximately 2/3 of the data points will lie within this factor from the best fit line. This is of course still inconveniently large for predicting damage for engineering design purposes, though it should be useful in determining whether a given design is clearly in a feasible regime or clearly not so. Meaningful predictions for marginal cases are of course still not possible. However, this relatively large factor of uncertainty may not be surprising when it is realized that the damage rates of a resistant alloy such as Stellite 6-B and a non-resistant one such as soft aluminum differ by a factor of about 10, 000, and that the data set includes points from several different types of cavitation and

impingement facilities, all considered together. Partially as a result of Mr. Heymann's suggestion, we have tried a correlation of maximum damage rates with the mechanical property in question raised to an exponent, which is then adjusted to a best fit value (Table 1 of closure). Our best fit exponent for the term (URxE ) is then 0. 659 which agrees very closely with the value of 2/3 mentioned by Mr. Heymann in his discussion. We also found that the best exponent for UR is 0. 998, confirming the validity of the energy model approach when this term is used, i. e., a unity exponent is required for this model. As shown in Table 1 of closure the correlation coefficient for our data with Mr. Heymann's suggested term (URxE ) improves from 0.684 when this term is raised to unity exponent to 0. 744 when the term is taken to best fit exponent. However, each value is less than the correlation coefficient for our data with UR alone (raised to unity power), which is 0. 811. On the other hand, the factorial standard error of estimate improves from 2. 86 when the term is taken to unity power to 2. 35 when taken to best fit exponent. This compares with 2. 52 for UR alone. Hence the combined term provides a better fit in terms of standard error of estimate when raised to its best fit power than does UR alone, although its standard error is inferior when both are raised to the first power. Table 1 (closure) also indicates that Brinell hardness (BHN) provides a relatively good correlation when raised to unity power, and a better correlation when raised to its best fit power (0, 734). In this latter case the correlation coefficient is substantially inferior to that of UR and slightly inferior to that of (URxE ). This new information confirms the long-standing practice of using Brinell hardness as a correlating parameter. It is to be recommended still in the light of these results because of its simplicity

and ease of measurement, as well as the fact that its performance as a correlating term is only slightly inferior to results to be obtained with much more complex parameters which are also much more difficult to measure. A general conclusion from Table 1 is that in terms of a basic model the use of UR is justified by the fact that the best fit exponent is approximately unity as required by the energy model, and the best correlation coefficient, indicating that the best "explanation" of the data is obtained with this parameter. However, the data also indicates that the use of the strain energy (SE) rather than ultimate resilience in such an energy model, as suggested most recently in Dr. Eisenberg's paper in this symposium, is quite unjustified. The best fit exponent for this parameter (Table 1 of closure) is 0. 738 rather than unity as should be the case if its use in the energy model were valid, and the resulting correlation coefficient is only 0. 517 (vs. 0. 811 for UR). In addition the standard error with this parameter is substantially larger than that with any of the other parameters tried. Also, for the 0. 517 "sample correlation coefficient" with 33 points for SE, the "minimum population correlation coefficient"'() is only about 0. 2 (vs. 0. 64 for UR). Thus the statistical evidence for a good correlation with SE, even when raised to its best exponent, is weak. The minimum (and maximum) population correlation coefficients are shown in Table 1. The smallest factorial standard error (2. 25) for our data is provided with the term UR x BHN raised to its best fit exponent (0. 720). This term was suggested by Rao et al(2) as a result of their work with a venturi. Their data points are also incorporated into our own data set used for this paper. However, for this combined term the correlation coefficient is again slightly less than for JR al-one

Plots of our data against the various mechanical property groups discussed are not included here with the exception of the plot against UR which is Fig. 3 of the paper, since they have been published elsewhere(3) Messrs. Rao and Rao, in addition to providing some of the data points for the paper itself, have suggested empirical relations for a better fit of the rocket sled droplet impact data (discussed in the early part of our paper) as a function of velocity and angle of impact. They suggest dividing the overall velocity range for a typical material (Pyroceram) into a low velocity region where the damage rate is substantially independent of velocity and a higher velocity region where it is not. If this is done, and best fit values for K, V, and OC are chosen, the match between eq. (3) in the paper and O0 the actual data points is much improved over that obtained if eq. (3) is used for the entire region. We believe that this is a possible useful approach which should be applied to the remaining data if a better predicting relation for these rain erosion type materials is desired, However, it cannot be applied to a new untested material unless an understanding of the relation between the measurable material mechanical properties and the limiting velocity to divide the regions, can be found. The present state of the art unfortunately does not as yet allow such a prediction.

REFERENCES 1. "Biometrika Tables for Statisticians", Volo 1 (2nd edition), Cambridge University Press (1938); edited by E. S. Pearson and H. 0. Hartleyo 2. B. C. S. Rao, N. S. L. Rao, and K. Seetharamiah, "Cavitation Erosion Studies with Venturi and Rotating Disc in Water", ASME Paper No. 69-WA/FE-32, to be published Trans. ASME, J. Basic Engro, 1970. 3. F. G. Hammitt, Y. C. Huang, T. M. Mitchell, Discussion of "Cavitation Erosion Studies with Venturi and Rotating Disc in Water", B. C. S. Rao, N. S. L. Rao, and K. Seetharamiah, ASME Paper 69-WA/FE-32, to be published Trans. ASME, J. Basic Engr., 1970.

95% Confidence Limits Correlating n Sample Correlation for Population Correlation Factorial Standard Relation (Where Applicable) Coefficient Coefficients Error of Estimate 1 n MDP1 = C(UR) 0o998 0.811 0.64 - 0.91 2.52 MDPR MDPR =C(UR) -- 0. 811 0. 64 - 0. 91 2. 52 MDPR MDPR = C(UR x BHN)n 0. 720 0. 798 0o 62 - 0. 89 2.25 MDPR 1 = C(UR x E Z) 0. 659 0. 744 0. 52 - 0.86 2o 35 MDPR = C (BHN) -- 0. 742 0.52 - 0.86 2o 75 MDPR 1 n 1.788 0o 734 0.52 - 0.85 2. 38 M = C(BHN) MDPR =MDPR C(UR x BHN) 0. 716 0o 49 - 0.84 2.57 MDPR DPR = C(UR x E) 0. 684 0. 44 - 0.82 2. 86 MDPR 1 = C(SE)n 0. 738 0. 517 0.21 - 0o 73 3.24 MDPR = C(SE) -- 0. 498 0o 17 - 0. 72 3.30 MDPR

A Statistically Verified Model for Correlating Volume Loss Due to Cavitation or Liquid Impingement by F. G. Hammitt, et al Discussion by F. J. Heymann Development Engineering, Westinghouse Electric Corporation Lester, Pennsylvania 19113

-1This paper gives me great satisfaction, because both its objectives and its findings are largely similar to those of my own contribution to this Symposium (1). Let me, therefore, underscore some of the agreements and point out some of the differences. The authors' statistical analysis of the Rocket Sled data gives further quantitative support to a conclusion which I had tentatively reached in a previous paper (2), and is now more thoroughly confirmed by the assemblage of data displayed in Fig. 8 of Ref. 1: namely, the velocity dependence of erosion can often be adequately expressed by a simple power law, without introducing a "threshold velocity". But there is an important proviso: these findings apply to conditions under which erosion proceeds rather rapidly, and may not be true at very low velocities or with very small drop sizes. Actually, two distinct approaches have been used at times to determine threshold velocities; the indirect method, by fitting an assume d velocity law to erosion rate data obtained at high velocities; and the direct method, involving low-speed tests to find the highest velocity at which no erosion sets in within a reasonable time. In my opinion there is no good reason for assuming that these two methods should yield the same results. Firstly, erosion mechanisms at high impact velocities are not identical to those at very low velocities, and may not be desscribed by quite the same simplified law. Secondly, the damage potential of impacts is affected by the surface roughness, and once erosion is started, it may be kept going by impact velocities which could

-2not initiate it on a smooth surface. (I am indebted to W. D. Pouchot for this observation). Thirdly, the "incubation period" has been shown to increase with a high power of the reciprocal impact velocity, making it difficult to run a test long enough to establish conclusively that erosion will not eventually begin at low velocities. Finally, if there is a physical threshold velocity, it may well be drop-size dependent (2). All of this tells us that it may be much more easy for us to predict the amount of erosion to be expected under severe conditions than under marginal conditions; unfortunately, in many practical instances -- particularly in long-life equipment -- the impingement conditions must be in the marginal zone, since even a very low erosion rate could lead to unacceptable erosion damage over a span of 10-20 years. Let us now turn to the second part of the authors' paper, the correlation between erosion rates and target material properties. Their major finding is that a proportionality between ultimate resilience (UR) and reciprocal erosion rate (MDPR ) provides the best correlation which is dimensionally consistent with the assumed energy transfer hypothesis (their Eq. 1). In other words, erosion resistance (~) is found proportional to UR. This is very similar to my qualitative findings in Ref. 1: the Normalized Erosion Resistance (N ), when plotted against UR on log-log coordinates, showed approximately a first power relationship (Fig. 6 of Ref. 1). I pointed out that this may be significant because it results in an erosion resistance which is dimensionally the same as other strength or energy properties.

The quantitative agreement between the authors' and my findings is actually quite remarkable, as can be seen on Fig. 1. This is the same as Fig. 6 of Ref. 1, except that superposed on it are the "best fit line" and "factorial standard error of estimate" boundaries taken from the author's Fig. 3. (In order to locate these lines uniquely, a conversion between my N and the authors' standardized MDPR was required. The value N = 1. 0 e e is defined as the erosion resistance of an austenitic stainless steel of hardness BHN 170. Such a material is found in the authors' Table IV (6th from bottom) and had an MDPR of 0. 653. Hence N = 0. 653/MDPR is the desired conversion.) e The most important thing to note, in Figure 1, is that the authors' data points and my data points show about the same scatter band; in both cases its vertical "height" encompasses a factor of about 15. Furthermore, in both cases some highly erosion-resistant materials, like stellites, have been left out, and would have increased the scatter if they had been included. By no stretch of the imagination, therefore, can this correlation be considered to give a useful tool for quantitative engineering predictions of erosion behavior. It is true that I found a somewhat (but not much) improved correlation with S E (or UR x E ), whereas the authors obtained a worse correlation u with that parameter. The reason may be that the author's correlation model permitted only a linear dependence on UR x E (see their Table VII), whereas Fig. 7 of Ref. 1 suggests a dependence of N on the 2/3 power of S E. It e u would be interesting to see what would result if the authors tried out the 2b 2 equation ~ = a (UR x E ), or log a +blog(UR x E ), compared to log~ = a +b log (UR).

-4Admittedly, the correlation with S E is dimensionally inconsistent U with the authors' Equation 1, as they point out. But this is not inevitably an impediment. While the energy transfer hypothesis of Eq. 1 is an attractive one, it is not the only one possible. In Ref. 1 I discussed this and suggested that new experiments must be carried out in order to discover the proper physical foundation for an erosion rate relationship, from which the dimensions of erosion resistance can then be deduced. Until that has been accomplished, we should not put any avoidable constraints on our correlation attempts. In fact, the authors' failure to improve their correlation by including the acoustic impedance ratio is an argument against the energy transfer theory, since the energy transmitted in an impact should be approximately proportional to the acoustic impedance ratio, if it is small. On the other hand, the impact stress is little affected by variations in the acoustic impedance ratio, again provided it is small as is true for the data considered. Thus, the authors' results provide no positive verification of their assumed "generalized erosion model." In summation, the authors' findings would lead me to precisely the same conclusions which I reached in Ref. 1: namely, that no correlations with conventional material properties have led to a useful prediction ability, and that at this stage of the game we ought to regard erosion resistance as an independent property, to be measured in erosion tests, and to be expressed quantitatively relative to one or more "standard materials" which should be incorporated in all test programs. This gives us the best opportunity of

-5gaining more knowledge and insight without being fettered by preconceived ideas and constraints. Even this approach cannot be expected to result in a unique repeatable erosion resistance value for each material, since in all probability more than one mechanism and more than one material parameter is involved in the physical erosion process. But the approach that I suggest promises to offer at least a first approximation of some practical usefulness. Refe rences 1. F, J. Heymann, "Toward Quantitative Prediction of Liquid Impact Erosion," presented at 72nd ASTM Annual Meeting, Atlantic City, N. J., 22-27 June, 19690 2. F. J. Heymann, "A Survey of Clues to the Relationship Between Erosion Rate and Impact Parameters, " in Proco Second Meersburg Conf. Rain Erosion and Allied Phenomena (16-18 Aug. 1967), Royal Aircraft Establishment, England, May 1, 1968, pp. 683-760.

Title: A Statistically Verified lodel for Correlating Volume Loss Due to Cavitation or Liquid Impingement Authors:?,O.GPamrnitt, Y.C.lluang, C.L.Kling, T.M.-itchell, L.P..olomon Discusser: Olive G. Fngel 1 The correlation found with ultimate resilience recalls the classification of materials into two groups on the basis of erosion resistance given by Von Schwsrz and Mantel 2. In the first group are materials for which the work of elastic deformation is lower than the energy delivered by a single drop 3 impact. If, in addition to being in this category, the material is brittle, the spots struck by the impinging dcrops are shattered.:ost metals are plastically d-formable and the surface metals at the spots where the drop impacts occurs is deformed and work-hardened until the limit of ability to deform is reached; when this limit is reached, the surface is broken, Von Schwars and Mantel found that the following proneties gave the greatest drop-irpact-erosion resistance to metals in the first group: hardness, ability to deform while cold, and extensive cold working properties. Von Schwarz and Iantel concluded that the high capacity for cold working of certain alloys gives tein good drop-iiiact-erosion resistance in spite of an inferior Drinell hardness and suggested that this explains -why B3rinell hardness is not a consistently good criterion of drop-impact-erosion resistance. In the second group Von Schwarz and Mantel placed all materials for which the elastic work ot deformation is so large that the enerpr delivered by a sinle drop impact is not sufficient to deform them. For these materials, dlmmage sets in first at imperfections. For materials in the second group, Von Schwarz and ianltel concluded that cdrop-impact-erosion resistance is determined by hardness and fatigue strength. The role that is played by the properties of a solid under erosive attack leads to the generalization that there are as many mechanisms of rnultiple-drop-impact erosion as there are broad groups of material properties * The fact that the

mechanismn by which erosion occurs affects the rate of erosion suggests that better correlations with erosion rate may result if tested materials are prouped on the basis of their properties. If highly resistant alloys, tool steel, and Stellite 63 are excluded froml the analysis, then, on the basis of the classification of Von Schwarz arnd Mantel, it night be informative to make an analysis of the renaiinng meta ls after they have been divided into groups on the basis of (1) brittleness, and (2) work-hardening capacity. The fact that the resistance of Stellite 63 is much greater than is expected on the basis of its mechanical properties stron-ly suggests that an understanding of the microscopic processes involved in drop-impact and cavitation erosion is required in order to be able to predict the resistance of materials to this form of attack and to be sable to formulate new alloys that will have a built-in resistance. Nuclear Systems Programs, General Electric Ccipany, Fvendale, Ohio 2 M. Von Schwarz and Wa. Mantel, Z. des Ver, Deut. Ing. 80, 863 (1936). 3 Von Schwarz and ttantel used a rotor and jet apparatus so that the drop is really a short section of a jet that is struck from the side. I am indebted to Dr. Albrecht ierzog for t.his insight given during a comr rsation at the WFright Air Develorment Center, xv!right-Patterson Air TForce Pase, Ohio in 1953.

22. A Statistically verified..odel for Correlating Volume Loss Due to Cavitation or Liquid Impingement. - by F. G. IIanmlitt et al. The use of the concept of a threshold velocity V~ was introduced by Pearson (C.E'G*B.) because of the similarity between erosion damage and fatigue where a relationship of the type shoyn in Equation I of the paper had proved very successful. Pearson correlated his data for low speed erosion experiments and found good agreement, thus giving support to the idea that, in this region, the process is similar to fatigue. However, much of the data that Professor Har.nitt has used is derived from high speed impact where the stress levels can be far above the fatigue strength of the materials. Thus we could not expect that the data would conform to the correlation proposed for low velocity imrpact.'fe may, therefore, have to think of erosion as a process which changes its character as the impact velocity increases. The initial process, when the velocity is near the threshold, being one of fatigue, changing to one where energy considerations are dominant when the velocity becomes high compared trith the threshold value. Furthermore when the size of the impacting droplet becomes snall (of the order of 100 microns) the impact stress levels can be significantly changed by the existence of water films on the surface as described in the paper by Y-r. Pouchot. It is therefore likely that the threshold velocity terh vill have to include a factor to account for thle attenuation of thIe stress level due to water filmts.

A STATISTICALLY VERIFIED MODEL FOR CORRELATING VOLUME LOSS DUE TO CAVITATION OR LIQUID IMPINGEMENT, BY F. G. Hammitt, Y. C. Huang, C. L. Kling, T. M. Mitchell and L. P. Solomon Discussion by: B. C. Syamala Rao and N. S. Lakshmana Rao - The authors made a simple and elegant approach to understand the effect of velocity on rain erosion and to determine the material parameter ~_ and the energy transfer coefficient, q, by considering the erosion data from several devices. The predicted MDPR values from the best fit for Equation (3) in Table I, show a very wide deviation from the actual MDPR. In order to understand this further, we studied a plot of log (MDPR) + log (Sin 4) as a function of log (V Sin 0) shown in Figure 4. The trend of experimental points clearly indicate two different regions: (i) where the velocity has a negligible effect on the erosion, and (ii) where the velocity has a very significant influence on the erosion. The first region can be related empirically as MDPR - 4.5/Sin ~ (13) while the second region can be described by Equation (3). The value of V is chosen as the value of V Sin e at which a mean 0 line drawn through the data intersects the abscissa. The experimental results in the second region are then plotted with log (V Sin 9 - V ) as the abscissa. A curve which gives the least standard deviation on an arithmetic plot is fitted as a Department of Civil & Hydraulic Engineering, Indian Institute of Science, Bangalore, India.

: - straight line the log-log plot shown in FiuIre 4 sand thle values of K and cC are computed to be 0.178 and 1.50 respectively. it these values for K, Vo and o, ELquation (3) reduces to 17.8 x 10`2 M.iR = (V Sin G - 170So) 150/Sin G (14) Table IX shows that the values of NDPiR predicted from Equ stion(14) are much closer to the actual values compared.with th- e v,-lues froml. t,.e equation used by the -authors. The standr'd devi/. tons using Equation (3-i) in these cases are as follows: S t an d ar d Devi'C/ation With Eijquation (14) 451 PL/s With Equation (3) 1192 fr/s The existence of th e two regions where the effects of vciocity are very different arc also observed in our inve3tigt'ions on cavitation d(amie (2, 3). B.C. Syam la Rao, "Cavitataion Erosion Studies with Vn-tiuri anr. iot(-tijng Disc in Water", i~h.D. Thesis, Indi:an Insti ute of Science, Ban,'alore, Apri! 1969 3pK.Kandasmi, "WtUdics on the:ffect of velocity and Test Juratic on Cavitation Damage" I.I. Th1esis, Indian Insti'ute o:C Science, Binlgalore, Indias, Aurlust 1969.

Comparison of Actual and Predicted,iLDPR's for lIaterial -l1, lyroceram, using DlP!l = 5.34 x 10-5(V Sin V)6'27/,in G (3) Authors Standard deviation of Equation(3)= 1192 0/s 1TD-PR = 4.5/Sin @ For V Sin Q < 1650 fps (13) IvDPR = 0.178 (V Sin 3 - 1780)1 50/Sin 0 for V Sin 0>1780 fps (14: Standard devitation of Equations (13) and (14) = 451 A/s v fps......R F /s Predicted by Predicted by Equation 3 Actual Equations (13) of Authors and (14) of the discussers 1580 30.9 7.9 9.0 1580 30. 9 0 9 0 1580 45 5.5 10.5 6.4 1580 45 5.5 10.5 6.4 1580 60 16.1 0 5. 2 1580 60 16.1 5.3 50"2 2197 30 6.8 0 9.0 2197 30 6.8 0 9.0 2197 45 43.7 0 6.4 2197 45 43.7 3.6 6.4 2197 60 127.3 7.3 300 2197 60 127.3 80.6 300 2594 30 9.6 0 9.0 2594 30 9.6 0 9.0 2594 45 124.1 0 21.9 2594 45 124.1 4.3 21.9 2594 60 361.4 3,849 2,262 2594 60 361.4 2,240 2,262 2905 30 40.6 0 9.0 290r 30 40.6 14.5 9.0 2905 45 252.4 2,189 1,218 2905 45 252.4 179 2,218 2905 60 735.3 4,465 4,425

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