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Erosion prediction of pure metals and carbon steels
1114
Wear, 162-164 (1993) 1114-l 122
Erosion prediction
of pure metals and carbon steels
R. R. R. Ellermaa
Abstract A survey of the existing erosion theories and their agreement with test results on erosion of pure metals and carbon steels is presented. It was found that the best correlation can be achieved with the use of Beckmann and Gotzmann’s erosion theory. The program for computing the erosion rate of metals is based on the modified Beckmann and Gotzmann model. The respective data bases for most common abrasives and steels have been prepared. This program allows the prediction of the erosion rate over the range of impact angles from 0” to 90” and . impact speeds from 10 to 200 m s-’ with an overall accuracy of +30% for annealed and heat treated carbon steels and f50% for pure metals.
1. Int~u~tion
The same ass~ption
A great number of installations and machines (disintegrators, separators, ventilators, etc.) are subject to the action of impacting abrasive particles. In order to select the best materials for wearing parts and to evaluate their working life, methods for erosion prediction of various metals working in different conditions are required. For more than 50 years attention has been concentrated on the process of erosion by solid particle impact, but owing to the complicated mechanism of erosion, we have until now no calculation procedures accurate enough for practical use. It is the aim of this paper to review the existing theories for predicting the erosion of metals and to choose the best of them for engineering calculations.
2. Selection of the best erosion theory 2.1. Existing erosion theories One of the first erosion theories was advanced by Lebedev [1] and was based on the assumption that the mechanism of erosion is micr~utting: 21’(sin QIcos LY - kf sin* o)
(1)
where f is the friction coefficient, k is the coefficient of restitution (by Newton), a,, is the yield point at pressure, E is the coefficient of chip thickening, m is a constant, a! is the angle of impact, and 2, is the particle velocity
I2
J,=cz12-2 2&%
K
sin 2a-
was also used by Finnie [2]: 2
11
- sin2 Q! P
a
PI2
(21 CVZ
(
cos= a
Ik== 2Cpum 1 + mr2fI
1
a> tan-’
PI2
P=Kf(1+mR2/1)
where c is the fraction of particles cutting in an idealized manner, 4 is the ratio of the vertical distance over which the particle contacts the surface to depth of cut, uT,, is the horizontal component of flow pressure, K is the ratio of the vertical force to the horizontal force on the particle, I is the moment of inertia of the particle about its center of gravity, m is the mass of an individual particle, and R is the particle radius Both theories have the same disadvantage - the erosion rate falls to zero when the impact angle is 90”. It is also incorrect to assume that most particles cut the target surface and to use a constant friction coefficient for all impact angles. In practice, both depend greatly on the impact angle [3, 41. Bitter [5] added the deformation wear mechanism to the microcutting mechanism for large angles of impact, hence the erosion process should be considered as the sum of these two mechan~ms:
Q 1993 - Elsevier Sequoia. All rights reserved
R. R R Ellermaa / Erosion prediction of pure metal!+ and carbon steels
Ik=Id+Ii2
when a>cu,
f=
Id = (v sin cr--K)‘/2~
(
J,2= [v” cos2 ar-K,(v
sin a-K)“](O.5/p)
El=
‘-c”M2 EM
v cos Ix-
; -l-l-b2 E‘4
C= 0.288 (P~~)‘/~~,
(8)
where vf is the particle velocity after impact and a, is the rebound angle. Their erosion model is very similar to the model of Nepomnyash~hy:
Cp(v sin CU-K)~ (v sin ff)ln 1
I = 2C(v sin a-K)’ 11 (v sin a)ln
v cos a-v, cos cyf v sin a-v, sin (Ye
1115
K= 0.0155 y5’2(glpA)‘nE‘= K; = 0.082 3’&~/p~)“‘~EI2
ff
(4)
where R is the maximum particle velocity at which the collision is still purely elastic, y is the elastic load limit 0, = 1.59 a,), g is acceleration due to gravity, pA is the abrasive density, jam and ,&Aare Poisson’s ratios of the material and abrasive, E, and EA are Young’s moduli of the material and abrasive, E is the deformation wear factor, C is a constant, K, is a constant, p is the cutting wear factor, and (Yeis the impact angle at which the horizontal velocity component has just become zero when the particle leaves the body. Practical use of this model is complicated by the necessity of making tests to obtain the unknowns (E and p), contained in the equations. Nepomnyashchy [6] assumed that erosive wear of metals is caused by low-cycle fatigue or microcutting. The erosion mechanism depends on the critical impact angle:
k’=2HVMoy
(9)
Both theories involve the same limitations: (1) 1-k’f>O (2) cot rX-f> 0 which means that the friction coefficient is expected to be f
exp( -N/k)
5R3Po
l=v cos a~+2 sin (~[(X-h)h]‘~ k’=2-IS&,
(5)
where KI/, is Vicker’s hardness of the material, e is the relative elongation at rupture, and co,=Hv,. It is low-cycle fatigue when the attack angle is below critical and microcutting in the opposite case. The erosion model was:
(6) 114 Ik= OS2 PA
In this theory the friction coefficient was also assumed to be constant for all impact angles, thus with an impact angle close to 90” the erosion rate becomes zero or negative. Styller and Ratner [7] improved the model of Nepomnyashchy by replacing the constant friction coefficient with a variable one:
h =v sin (Yexp( - (n&a q = (k? -rzy
7= r/q
tan(q/n))/k (11)
where N is the normal component of material reaction force, h, 1 are the crater depth and length, 7 is the contact interaction duration, n and k are coefficients which depend on the properties of the target material and abrasive, and PO and h are structure constants. The use of the given model is difficult because of the necessity of performing experiments to obtain the values of the unknown parameters (PO, A, n and k) and to determine the values of N for every impact angle and particle velocity. Beckmann and Gotzmann [9] derived an analytical expression for the erosion of metals from the hypothesis that in abrasive wear the volume removed is proportional to the work of shear forces in the surface layer. The basic model was formulated from the study of deformation caused by a single spherical particle:
R. R. R. Ellenma
1116
I, = 6.81fhlR)‘” 2PA;;;s=
/ Erosion prediction
of pure metais and carbon steels
In Beckmann and Gotzmann’s model (eqns. (12)) the specific shear energy density depends only on material melting temperature, ambient temperature and material constant (ez/T, = 3~/In(~~/~), while in Peter’s model
fy I, = 0.65 (h tRj2
M
e,*/70=2.5 [v”
sir? *-~]]“‘p=
E
(~)
70=p,c&T,
(12) where e: is the shear energy density, K is the material constant (for pure metals K=21), L is the latent heat of melting, phi is the material density, T” is the shearing stress in the target material, 71 is the mean shearing stress on indentation, TVis the mean shearing stress on rebound, TM is the melting temperature in degrees Kelvin, T is the ambient temperature in degrees Kelvin, HB, is Brinell’s hardness, & is the shape factor of the particle, and h is the depth of the impact crater. This theory also contains unknown parameters (I( and K,) which must be obtained experimentally or from nomograms, given in [9, lo]. Abramov [ll] supposed that deformation of the elastic material in the course of impact is governed by Hooke’s law, and its breaking occurs along the lines of maximum shear stresses: I,= 1.25 c,
P&M(TM - r) ‘I5 wh4
P,‘.~E “.2(R/r)o.6(v sin u)~.’ (13)
- T)o.33
(15)
where CM is the specific heat capacity of the material. Peter’s erosion model is I, =
+
2.17o;t/2y~cot* cyF(p)
tan p= 2
F(p)=5/3+/.&+
n(l + +/3)
(1+ t-v
/A= cpcot2
vu2 -
aIS
1-
~,)I’” 8,
P
(16)
where p, is the angle of the mean slope of the contact area between particle and material in impact and p2 is the angle of the mean slope of the contact area between particle and material in rebound. The only unknown coefficient is the shape factor Kf, which can be obtained from a nomogram [9].
m.where c, is the material coefficient, R is the particle radius, r is the particle tip radius, and E ’ are from eqns. (4). The model calls for microscopic measurements to determine the abrasive particle parameters (radius, tip diameter) necessary for imputing the erosion rate. Also the material coeffcient c, is obtainable only from experiments. Peter [12] uses in his model Beckmann and Gotzmann’s erosion theory, after the replacement of the equations for computing the indentation depth of the particle and the specific shear energy density e:/r,. The indentation depth of the particle is computed from the following equations:
2.2. ~ompa~on of the different theo~es Each erosion theory mentioned above contains one or more unknown parameters, i.e. parameters whose values are obtainable only from experiments (Table 1). In order to compare different theories and to select the best one for further use, a program was written for computing these parameters. In so far as some theories contain more unknown parameters than equations, it was necessary to use a simplex method. The Chenyakov and Ratner-Styller erosion theories contain unknown parameters (N and j’), whose values depend TABLE
1. Unknown
Theory
h = R(vA)“~
(14) where C is a constant (=0X9), (4).
and E’ is from eqns.
Lebedev Finnie Bitter Nepomnyashchy Styljer, Ratner Chervyakov Beckmann, Gotzmann Abramov Peter
parameters
of the different
Unknown
parameters
theories
R. R. R Ellermaa I Erosion prediction
on particle speed and impact angle. This makes their use impossible, without performing additional experiments for obtaining the values of these parameters. Experimental data, necessary for checking the erosion theories for correlation with experimental results, was obtained from [13] (heat treated carbon steels) and 1141(annealed pure metals). In both cases the abrasive material used was rounded quartz sand (d = 200.. .300 pm), with the following physical and mechanical properties (1) pA=2500 kg me3 (2) EA=60 GPa (3) /.&=0.3 Testing machines used were a centrifugal-type rig for pure metals and a sandblast rig for steels. Table 2 gives the materials’ physical and mechanical properties, necessary for calculations. The experimental data are given in Table 3, The values of the unknown parameters, found on the basis of the experimental data with the help of the simplex method are given in Table 4. In order to obtain better correlation, the elastic load limit y from Bitter’s theory was included in the unknown parameters. Given in Table 5 are the values of sums of the least squares, calculated from different erosion theories with the use of the following equation:
2
s=%y
( 1 I.-i.
i-l
(17)
e‘
where n is the number of tests, 1, are the calculated values of erosion rate, and Z, are the experimental values of the erosion rate. As shown in Table 5, the best results can be obtained from Beckmann and Gotzmann’s erosion theory, which was adopted for further analysis and modifi~tion. TABLE
2. Physical-mechanical
PM (kg ms3) EM (GW
?(J(kg K)-‘) TM (K) L (J kg-‘) @B tMpa) WY
e (%I WM
(kg
620/l b20D -
1117
metab and carbon steels
Figures 1 and 2 illustrate the comparison of the erosion rates computed from the best theories with the experimental values.
3. The modifications In its original form Bec~ann and Gotzmann’s model is given in eqn. (12). According to this model the particle indentation depth h depends, among other parameters, on the static hardness of the material. It would be more correct to use some dynamic parameter for describing the penetration process. In ref. 15 the indentation process is investigated and the equation below, which contains dynamic hardness, is proposed for computing the impact crater depth: l/z
(u” sin’ ff)
3
(18)
where HD is the dynamic hardness of the target material. According to ref. 15, comparison has shown good correlation between the computed and experimental values of h. The values of dynamic hardness can be obtained from special experiments or approximated from the equations given in Table 6. By Beckmann and Gotzmann’s theory: e,*/rO= 3K/ln( TM- 7)
(19)
Since this equation failed to give satisfactory correlation with experimental data, it was empirically modified by using eqn. (15): e*/r s
=
GdhdTtvl- l-3
0
H,,O.l
K = 0.0021 HEM + 0.3, for carbon steels K= 1, for pure metals
(20)
of the materials
Material
Properties
0~
properties
of pure
--*I
AISI AISI
1020, HV434. 1020, HV193.
Al
Ti
Fe
2011”
20Db
2730 71 0.31 896 933 3.7 x 16 80 30 35 23.7
4500 108.5 0.36 532 2073 4.2 x 105 610 470 20 160.2
7850 210 0.3 468 1812 2.7x 300 250 30 120.7
7850 210 0.3 468 1812 2.7 x 16 1300 1120 10 434
7850 210 0.3 468 1812 2.7x 600 450 26 193
l@
I@
1118 TABLE
R. R. R. Ellermau I Erosion prediction 3(a).
Values of experimental Erosion
V
Tdeg)
29 29 29 29 29 82 82 82 82 82 82 82
rate [mm3 kg-‘]”
6.7f9.0 ll.UlO.5 4.8/9.4 4.U7.1 124.8/120.2 143.01141.5 129.01148.4 93.z140.7 56.0,‘110.3 55.5l74.3 35.0160.5
‘Experimental/predicted TABLE
3(b).
15 30 45 60 75 90 15 30 45 60 75 90
Ti
Fe
1.012.3 1.312.6 2.112.5 2.512.1 2.Ol1.4 12.6129.8 33.3138.8 49.0139s 55.6f34.0 47.1128.1 44.5125.7
1.111.6 2.111.9 2.611.8 1.911.5 1.Ul.O 11.1/20.1 28.3123.6 39.4126.5 39.5126.5 35X23.7 25.9119.7 22.9117.8
values (eqn. (21)).
Values of experimental V
Tdeg)
data
(m s-l) Al
20 30 45 60 90 15 20 30 45 60 75 90
and predicted
of pure metals and carbon steels
and predicted
Erosion
rate (mm’
data kg-‘)=
(m s-l)
40 40 40 40 40 40 80 80 80 80 80 80
20/P
20/2’
2.512.5 3.2f3.5 3.6f3.8 3.713.7 3.513.5 2.913.4 14N13.1 18.9118.9 21.3121.2 22.3121.4 20.8i20.7 18.1/20.0
4.813.4 4.914.6 4.814.5 4.314.0 3.613.3 2.6/3.0 19.6/18.5 22.9124.0 24.0124.9 23.3122.5 20.4119.1 15.7/1&O
‘~perimental/predicted values (eqn. (21)). b20/1 - AISI 1020, HV434. ‘20/2 - AISI 1020, HV193.
where K is the material constant and HB, is the Brine11 hardness of the material in GPa. Figure 3 illustrates the comparison of Beckmann and Go~mann’s equation (eqn. (19)) and eqns. (20) with values of ef/r, obtained from eqns. (12) modified with eqn. (18). Values of ezlrO were calculated from the experimental data from refs. 13 and 14 using a simplex method. In the course of calculations it became obvious that for carbon steels there exists linear correlation between material constant and Brine11 hardness. Therefore it was necessary to introduce into the eqns. (20) the material constant K. As shown in Fig. 3, values of e:/r,,, calculated from eqn. (20), correlate better with experimental results, obtained from eqns. (12), and accordingly eqn. (20) can replace the original eqn. (19) from the model. The modified erosion model for carbon steels and pure metals is as follows:
T=
1
+
W W’“H& (1 -h/R)r,, IZ= rc,,(ta IR)'
~=0.33
Lp,
ln(TJZJ
G~~ix.0’~
ez/q)=
- 7’)
O.lH,
(21) where K, and KeO are the material coefficients and K, is the abrasive coefficient. The comparison of the experimental values of erosion rate with values obtained from Beckmann and Gotzmann’s theory (Figs. 4 and 5) has shown that the best results can be obtained when some correction factors (KYand Km) are added into the final equation, to take into account the influence of particle velocity and greater (above 60”) impact angles. The equations were obtained empirically from experimental data [13, 141 by the simplex method K,,=O.O05 v+O.14
ifHB,<2
K, = 0.31(HB,)v”“057 K,, = 1_83v-O.”
GPa
if HBM gs2 GPa
(22)
for carbon steels
K,= 1 for pure metals
(23)
where 2, is in metres per second and HB, in gigapascals. The abrasive coefficient r(;, is a product of three factors: K,=K,+K,.K,, K,=d/120
(24)
if cl=0 . . . 120 pm
jK,=l
if 2>120 pm
K,=l
ifwM
(25)
KH = (HV, - HV,)I(OhHV,)
ifHVM
if HVM~HV,,
(26)
where Kd is a size factor, 2 is the mean particle diameter, KH is the hardness factor, and WA is the Vicker’s hardness of abrasive. The eqns. (25) and (26) are very approximate and meant to reduce the errors of calculations when the particle size is less than 120 pm and when the material is almost as hard as the abrasive.
1119
R. R. R. Ellemuaa / Erosion prediction of pure metals and carbon steels TABLE 4. Computed
values of the unknown
Lebedev
f k p c(&rh) -’ (MPa-*) K P (GPa) l @@a) Y (MPa) f k, k
Finnie
Bitter
Nepomnyashchy Beckmann, Gotzmann Abramov Peter
CY
kr A
Al
Ti
Fe
2011”
20l2b
0.021 0.4 12.3 1 x 10-6
0.046 0.5 4.8 39 x lo+
0.055 0.1 15.6 65 x lo+
0.29 0.001 20.8 34 x 10-6
0.29 8X 10-6 41.1 49x 10-6
2.14 4.3 10.7 2.25 0.20 39.0 10.6 1.2 0.36
4.65 28.2 19.1 14.4 0.14 37.6 20.1 1.2 0.10
6.82 21.4 29.2 12.3 0.09 50.4 58.5 1.2 0.46
3.55 29.1 20.9 19.5 0.005 11.4 60.8 1.2 0.30
3.42 17.4 20.2 22.3 0.077 25.1 55.7 1.2 0.58
AISI 1020, HV434. AISI 1020, Hv193.
TABLE 5. Sums of the least-squares different theories Theory
(eqn. (17)) computed
Sums of the least-squares,
Lebedev Finnie Bitter Nepomnyashchy Beckmann, Gotzmann Abramov Peter ‘20/l b20/2 -
Material
Parameters
Theory
“20/l b20/2 -
parameters
with
S (n=14)
Al
Ti
Fe
20/l”
20/2b
3.72 3.30 1.49 1.2x l@ 0.92 5.86 1.29
4.99 5.15 2.56 39.25 2.31 3.35 7.10
4.07 4.56 2.83 98.55 1.97 5.99 4.96
2.81 4.92 0.68 3.12 0.25 3.84 3.56
2.59 4.33 0.43 28.74 0.39 4.69 0.92
AISI 1020, HV 434. AISI 1020, HV193.
The model can be used to compute erosion rates of annealed and heat treated carbon steels and pure metals. Even better results will be obtained when the following limitations are observed: (1) impact angle 0”...90”, (2) particle velocity 10. ..200 m s-l, (3) abrasive hardness Hv, < 1.6Hv,, (4) particle size d > 120 pm, (5) abrasive is dry, (6) plastic deformation takes place. When using this model, the error will usually not exceed 30% for heat treated or annealed carbon steels with static hardness Hv,= 1...7 GPa, and 50% for pure metals. Figures 4 and 5 give the comparison of the original and modified erosion model. It is shown that the modified version yields a better correlation with experimental results. Table 3 gives the comparison of the modified erosion model and experimental results.
4. Discussion The given erosion model contains many more or less empirical coefficients (K, K,, K,, &). The last two are not obligatory and were introduced only to give better correlation between experimental and theoretical erosion rate values for carbon steels. The best values for material constant K and abrasive coefficient K. are obtainable from experiments using a correspondingly standard abrasive or material whose parameters are known. A standard material must be sufficiently soft to erode with rounded and soft abrasive, for example annealed carbon steel AISI 1040, which is sufficiently soft (-200 HV) and whose material constant can be calculated from eqn. (20). Standard abrasive must be sufficiently hard to erode hard materials and non-shattering - for example corundum whose abrasive coefficient can be obtained from experiments with standard material.
5. The program for predicting the erosion rate The program package EROSION [16] is intended for specialists engaged in erosion problems. It enables the prediction of the erosion rate of metals, the computation of the exact values of K and K,, if the experimental values of the erosion rate are known, and the comparison of the predicted erosion rate values of any two different metals. The package also contains data bases for most commonly used abrasives and metals. The EROSION package has a used-friendly interface, with windows and pull-down menus. There are also
1120
R. R. R. Eliemaa
i Erosion prediction of pure metals and carbon steels
Fig. 1. Comparison of the erosion rates computed with different theories with the experimental values. Materiai - annealed Al, abrasive - rounded quartz sand, d=0.4...0.6 mm. Abrasive velocity v=29 m s-’ (a) and 82 m s-r (b).
case-sensitive help and error messages. The user can build up his own data bases and modify them by inserting, deleting or editing their entries. The computed results are given in the form of a table and graph of Z==f(a) or Z, =f(v). The EROSION package runs on an IBM PC and compatibles under MS-DOS and requires only one 360 Kb floppy disk.
6. Conc.Iusion
(1) The comparison of different erosion theories has led to the conclusion that the best correlation between experimental and computed erosion rate values can be obtained by means of Beckmann and Gotzmann’s erosion theory. (2) Beckmann and Gotzmann’s theory was modified by introducing the dynamic hardness of the material
(b) ’ Fig. 2. Comparison of the erosion rates computed with different theories with the experimental values. Material - heat treated steel AI.81 1020 (HV434), abrasive - rounded quartz sand, d = 0.4.. .0.6 mm. Abrasive velocity Y =40 m s-r (a) and 80 m s-l (b).
TABLE 6. Equations metals [15]
for calculating
Type of crystal lattice
Calculation
C8
Hn=1.3+1.8*HB, H,--5.06-Q.52~HBM HD= 1.7.H&, H,=O*58+1.8~HB,
Cl& H12 H6
and additional puting model. (3) In terms age was made erosion rate of
the dynamic hardness
of
of the Ho (GPa)
+0.24-HBZ,
if HBM < 2 GPa if HB, z=2 GPa
coefficients (K,, Ke5, K,) into the comof this modified theory a program packto computerize the prediction of the metals. The error of predicted erosion
R, R. R. Ellemaa
Fig. 3. Values of e:/ra,
obtained
from different
/ Erosion prediction of pure metals and carbon steels
equations.
“WV
.m Attack
45
60
75
M
angle. dqee
@)
90 Attack
an+
degree
Fig. 5. Comparison of the erosion rates computed with the original Beckmann and Gotzmann’s theory and the author’s modified theory. Material - heat treated steel AISI 1020 (HV434), abrasive - rounded quartz sand, d = 0.4.. .0.6 mm. Abrasive velocity v = 40 m s-’ (a) and 80 m s-’ (b).
rate values is usually less than f30% for heat treated carbon steels and f50% for pure metals.
References
0’)
-,:
?I2
At&k
mgle.
degree
Fig. 4. Comparison of the erosion rates computed with the original Beckmann and Gotzmann’s theory and the author’s modified theory. Material - heat treated steel AISI 1020 (HV193), abrasive - rounded quartz sand, d = 0.4.. .0.6 mm. Abrasive velocity u = 40 m s-’ (a) and 80 m s-’ (b).
I. K. Lebedev, Erosion in the boilers, Electric Power Stations, 11 (1958) 22 (in Russian). I. A. Finnie, Wear, 3 (1960) 87. I. R. Kleis and H. H. Uuemois, Wear Resistance of Impact Milling Equipment Parts, Machine building, Moscow, 1986, 160 pp. (in Russian). V. N. Vinogradov, V. I. Bijukov, S. I. Nazarov and I. B. Chervyakov, Experimental investigations of the friction coefficient with spherical particle, Friction and Wear, 2 (1981) 896 (in Russian). J. G. A. Bitter, Wear, 6 (1963) 169.
1122
R. R. R. Ellermaa 1 Erosion prediction of pure metals and carbon steels
6 E. F. Nepomnyashchy, Contact Interaction of Solid Bodies and Calculation of Friction Forces and Wear, Moscow, 1971, p. 190 (in Russian). 7 S. B. Ratner and E. E. Styller, Wear, 73 (1981) 213. 8 V. N. Vinogradov and I. B. Chervyakov, Semiempirical model of plastic materials wear in a stream of hard particles, fioc. 7th Int. Conf. on Erosion by Liquid and Solid Impact, Cambridge, 7-10 September, 1987, Paper 49. 9 G. Beckmann and J. Gotzmann, Wear, 73 (1981) 325. 10 G. Beckmann, Wear, 59 (1980) 421. 11 J. I. Abramov, Erosion of power engineering installations, Power Engineering, 7 (1985) 15 (in Russian). 12 R. Peter, Strahlverschleib an konventionellen Dampferzeugern Prognose und Verschleibschutz, Wus. Ber. THZ, 1004 (1989) Lecture No. V/3.
13 S. M. Levin, Investigation of the erosion of steels in different conditions, Dissertation, Moscow Institute of Petrochemical and Gas Industry, Moscow, 1978, p. 207 (in Russian). 14 J. Tadolder, Some correlations of the erosion of the technically pure metals, Proc. Tallinn Polytechnical Inst. 237A (1966) 3 (in Russian). 15 I. R. Kleis and H. F. Kangur, Resistance of metal surface to indentation by spherical projectile at impact, Proc. 7th Znt. Con& of Erosion by Liquid and Solid Impact, Cambridge, 7-10 September 1987, Paper 48. 16 EROSION, obtainable from Department of Mechanical Engineering, Tallinn Technical University, 200108 Tallinn, Estonia.
Wear, 162-164 (1993) 1114-l 122
Erosion prediction
of pure metals and carbon steels
R. R. R. Ellermaa
Abstract A survey of the existing erosion theories and their agreement with test results on erosion of pure metals and carbon steels is presented. It was found that the best correlation can be achieved with the use of Beckmann and Gotzmann’s erosion theory. The program for computing the erosion rate of metals is based on the modified Beckmann and Gotzmann model. The respective data bases for most common abrasives and steels have been prepared. This program allows the prediction of the erosion rate over the range of impact angles from 0” to 90” and . impact speeds from 10 to 200 m s-’ with an overall accuracy of +30% for annealed and heat treated carbon steels and f50% for pure metals.
1. Int~u~tion
The same ass~ption
A great number of installations and machines (disintegrators, separators, ventilators, etc.) are subject to the action of impacting abrasive particles. In order to select the best materials for wearing parts and to evaluate their working life, methods for erosion prediction of various metals working in different conditions are required. For more than 50 years attention has been concentrated on the process of erosion by solid particle impact, but owing to the complicated mechanism of erosion, we have until now no calculation procedures accurate enough for practical use. It is the aim of this paper to review the existing theories for predicting the erosion of metals and to choose the best of them for engineering calculations.
2. Selection of the best erosion theory 2.1. Existing erosion theories One of the first erosion theories was advanced by Lebedev [1] and was based on the assumption that the mechanism of erosion is micr~utting: 21’(sin QIcos LY - kf sin* o)
(1)
where f is the friction coefficient, k is the coefficient of restitution (by Newton), a,, is the yield point at pressure, E is the coefficient of chip thickening, m is a constant, a! is the angle of impact, and 2, is the particle velocity
I2
J,=cz12-2 2&%
K
sin 2a-
was also used by Finnie [2]: 2
11
- sin2 Q! P
a
PI2
(21 CVZ
(
cos= a
Ik== 2Cpum 1 + mr2fI
1
a> tan-’
PI2
P=Kf(1+mR2/1)
where c is the fraction of particles cutting in an idealized manner, 4 is the ratio of the vertical distance over which the particle contacts the surface to depth of cut, uT,, is the horizontal component of flow pressure, K is the ratio of the vertical force to the horizontal force on the particle, I is the moment of inertia of the particle about its center of gravity, m is the mass of an individual particle, and R is the particle radius Both theories have the same disadvantage - the erosion rate falls to zero when the impact angle is 90”. It is also incorrect to assume that most particles cut the target surface and to use a constant friction coefficient for all impact angles. In practice, both depend greatly on the impact angle [3, 41. Bitter [5] added the deformation wear mechanism to the microcutting mechanism for large angles of impact, hence the erosion process should be considered as the sum of these two mechan~ms:
Q 1993 - Elsevier Sequoia. All rights reserved
R. R R Ellermaa / Erosion prediction of pure metal!+ and carbon steels
Ik=Id+Ii2
when a>cu,
f=
Id = (v sin cr--K)‘/2~
(
J,2= [v” cos2 ar-K,(v
sin a-K)“](O.5/p)
El=
‘-c”M2 EM
v cos Ix-
; -l-l-b2 E‘4
C= 0.288 (P~~)‘/~~,
(8)
where vf is the particle velocity after impact and a, is the rebound angle. Their erosion model is very similar to the model of Nepomnyash~hy:
Cp(v sin CU-K)~ (v sin ff)ln 1
I = 2C(v sin a-K)’ 11 (v sin a)ln
v cos a-v, cos cyf v sin a-v, sin (Ye
1115
K= 0.0155 y5’2(glpA)‘nE‘= K; = 0.082 3’&~/p~)“‘~EI2
ff
(4)
where R is the maximum particle velocity at which the collision is still purely elastic, y is the elastic load limit 0, = 1.59 a,), g is acceleration due to gravity, pA is the abrasive density, jam and ,&Aare Poisson’s ratios of the material and abrasive, E, and EA are Young’s moduli of the material and abrasive, E is the deformation wear factor, C is a constant, K, is a constant, p is the cutting wear factor, and (Yeis the impact angle at which the horizontal velocity component has just become zero when the particle leaves the body. Practical use of this model is complicated by the necessity of making tests to obtain the unknowns (E and p), contained in the equations. Nepomnyashchy [6] assumed that erosive wear of metals is caused by low-cycle fatigue or microcutting. The erosion mechanism depends on the critical impact angle:
k’=2HVMoy
(9)
Both theories involve the same limitations: (1) 1-k’f>O (2) cot rX-f> 0 which means that the friction coefficient is expected to be f
exp( -N/k)
5R3Po
l=v cos a~+2 sin (~[(X-h)h]‘~ k’=2-IS&,
(5)
where KI/, is Vicker’s hardness of the material, e is the relative elongation at rupture, and co,=Hv,. It is low-cycle fatigue when the attack angle is below critical and microcutting in the opposite case. The erosion model was:
(6) 114 Ik= OS2 PA
In this theory the friction coefficient was also assumed to be constant for all impact angles, thus with an impact angle close to 90” the erosion rate becomes zero or negative. Styller and Ratner [7] improved the model of Nepomnyashchy by replacing the constant friction coefficient with a variable one:
h =v sin (Yexp( - (n&a q = (k? -rzy
7= r/q
tan(q/n))/k (11)
where N is the normal component of material reaction force, h, 1 are the crater depth and length, 7 is the contact interaction duration, n and k are coefficients which depend on the properties of the target material and abrasive, and PO and h are structure constants. The use of the given model is difficult because of the necessity of performing experiments to obtain the values of the unknown parameters (PO, A, n and k) and to determine the values of N for every impact angle and particle velocity. Beckmann and Gotzmann [9] derived an analytical expression for the erosion of metals from the hypothesis that in abrasive wear the volume removed is proportional to the work of shear forces in the surface layer. The basic model was formulated from the study of deformation caused by a single spherical particle:
R. R. R. Ellenma
1116
I, = 6.81fhlR)‘” 2PA;;;s=
/ Erosion prediction
of pure metais and carbon steels
In Beckmann and Gotzmann’s model (eqns. (12)) the specific shear energy density depends only on material melting temperature, ambient temperature and material constant (ez/T, = 3~/In(~~/~), while in Peter’s model
fy I, = 0.65 (h tRj2
M
e,*/70=2.5 [v”
sir? *-~]]“‘p=
E
(~)
70=p,c&T,
(12) where e: is the shear energy density, K is the material constant (for pure metals K=21), L is the latent heat of melting, phi is the material density, T” is the shearing stress in the target material, 71 is the mean shearing stress on indentation, TVis the mean shearing stress on rebound, TM is the melting temperature in degrees Kelvin, T is the ambient temperature in degrees Kelvin, HB, is Brinell’s hardness, & is the shape factor of the particle, and h is the depth of the impact crater. This theory also contains unknown parameters (I( and K,) which must be obtained experimentally or from nomograms, given in [9, lo]. Abramov [ll] supposed that deformation of the elastic material in the course of impact is governed by Hooke’s law, and its breaking occurs along the lines of maximum shear stresses: I,= 1.25 c,
P&M(TM - r) ‘I5 wh4
P,‘.~E “.2(R/r)o.6(v sin u)~.’ (13)
- T)o.33
(15)
where CM is the specific heat capacity of the material. Peter’s erosion model is I, =
+
2.17o;t/2y~cot* cyF(p)
tan p= 2
F(p)=5/3+/.&+
n(l + +/3)
(1+ t-v
/A= cpcot2
vu2 -
aIS
1-
~,)I’” 8,
P
(16)
where p, is the angle of the mean slope of the contact area between particle and material in impact and p2 is the angle of the mean slope of the contact area between particle and material in rebound. The only unknown coefficient is the shape factor Kf, which can be obtained from a nomogram [9].
m.where c, is the material coefficient, R is the particle radius, r is the particle tip radius, and E ’ are from eqns. (4). The model calls for microscopic measurements to determine the abrasive particle parameters (radius, tip diameter) necessary for imputing the erosion rate. Also the material coeffcient c, is obtainable only from experiments. Peter [12] uses in his model Beckmann and Gotzmann’s erosion theory, after the replacement of the equations for computing the indentation depth of the particle and the specific shear energy density e:/r,. The indentation depth of the particle is computed from the following equations:
2.2. ~ompa~on of the different theo~es Each erosion theory mentioned above contains one or more unknown parameters, i.e. parameters whose values are obtainable only from experiments (Table 1). In order to compare different theories and to select the best one for further use, a program was written for computing these parameters. In so far as some theories contain more unknown parameters than equations, it was necessary to use a simplex method. The Chenyakov and Ratner-Styller erosion theories contain unknown parameters (N and j’), whose values depend TABLE
1. Unknown
Theory
h = R(vA)“~
(14) where C is a constant (=0X9), (4).
and E’ is from eqns.
Lebedev Finnie Bitter Nepomnyashchy Styljer, Ratner Chervyakov Beckmann, Gotzmann Abramov Peter
parameters
of the different
Unknown
parameters
theories
R. R. R Ellermaa I Erosion prediction
on particle speed and impact angle. This makes their use impossible, without performing additional experiments for obtaining the values of these parameters. Experimental data, necessary for checking the erosion theories for correlation with experimental results, was obtained from [13] (heat treated carbon steels) and 1141(annealed pure metals). In both cases the abrasive material used was rounded quartz sand (d = 200.. .300 pm), with the following physical and mechanical properties (1) pA=2500 kg me3 (2) EA=60 GPa (3) /.&=0.3 Testing machines used were a centrifugal-type rig for pure metals and a sandblast rig for steels. Table 2 gives the materials’ physical and mechanical properties, necessary for calculations. The experimental data are given in Table 3, The values of the unknown parameters, found on the basis of the experimental data with the help of the simplex method are given in Table 4. In order to obtain better correlation, the elastic load limit y from Bitter’s theory was included in the unknown parameters. Given in Table 5 are the values of sums of the least squares, calculated from different erosion theories with the use of the following equation:
2
s=%y
( 1 I.-i.
i-l
(17)
e‘
where n is the number of tests, 1, are the calculated values of erosion rate, and Z, are the experimental values of the erosion rate. As shown in Table 5, the best results can be obtained from Beckmann and Gotzmann’s erosion theory, which was adopted for further analysis and modifi~tion. TABLE
2. Physical-mechanical
PM (kg ms3) EM (GW
?(J(kg K)-‘) TM (K) L (J kg-‘) @B tMpa) WY
e (%I WM
(kg
620/l b20D -
1117
metab and carbon steels
Figures 1 and 2 illustrate the comparison of the erosion rates computed from the best theories with the experimental values.
3. The modifications In its original form Bec~ann and Gotzmann’s model is given in eqn. (12). According to this model the particle indentation depth h depends, among other parameters, on the static hardness of the material. It would be more correct to use some dynamic parameter for describing the penetration process. In ref. 15 the indentation process is investigated and the equation below, which contains dynamic hardness, is proposed for computing the impact crater depth: l/z
(u” sin’ ff)
3
(18)
where HD is the dynamic hardness of the target material. According to ref. 15, comparison has shown good correlation between the computed and experimental values of h. The values of dynamic hardness can be obtained from special experiments or approximated from the equations given in Table 6. By Beckmann and Gotzmann’s theory: e,*/rO= 3K/ln( TM- 7)
(19)
Since this equation failed to give satisfactory correlation with experimental data, it was empirically modified by using eqn. (15): e*/r s
=
GdhdTtvl- l-3
0
H,,O.l
K = 0.0021 HEM + 0.3, for carbon steels K= 1, for pure metals
(20)
of the materials
Material
Properties
0~
properties
of pure
--*I
AISI AISI
1020, HV434. 1020, HV193.
Al
Ti
Fe
2011”
20Db
2730 71 0.31 896 933 3.7 x 16 80 30 35 23.7
4500 108.5 0.36 532 2073 4.2 x 105 610 470 20 160.2
7850 210 0.3 468 1812 2.7x 300 250 30 120.7
7850 210 0.3 468 1812 2.7 x 16 1300 1120 10 434
7850 210 0.3 468 1812 2.7x 600 450 26 193
l@
I@
1118 TABLE
R. R. R. Ellermau I Erosion prediction 3(a).
Values of experimental Erosion
V
Tdeg)
29 29 29 29 29 82 82 82 82 82 82 82
rate [mm3 kg-‘]”
6.7f9.0 ll.UlO.5 4.8/9.4 4.U7.1 124.8/120.2 143.01141.5 129.01148.4 93.z140.7 56.0,‘110.3 55.5l74.3 35.0160.5
‘Experimental/predicted TABLE
3(b).
15 30 45 60 75 90 15 30 45 60 75 90
Ti
Fe
1.012.3 1.312.6 2.112.5 2.512.1 2.Ol1.4 12.6129.8 33.3138.8 49.0139s 55.6f34.0 47.1128.1 44.5125.7
1.111.6 2.111.9 2.611.8 1.911.5 1.Ul.O 11.1/20.1 28.3123.6 39.4126.5 39.5126.5 35X23.7 25.9119.7 22.9117.8
values (eqn. (21)).
Values of experimental V
Tdeg)
data
(m s-l) Al
20 30 45 60 90 15 20 30 45 60 75 90
and predicted
of pure metals and carbon steels
and predicted
Erosion
rate (mm’
data kg-‘)=
(m s-l)
40 40 40 40 40 40 80 80 80 80 80 80
20/P
20/2’
2.512.5 3.2f3.5 3.6f3.8 3.713.7 3.513.5 2.913.4 14N13.1 18.9118.9 21.3121.2 22.3121.4 20.8i20.7 18.1/20.0
4.813.4 4.914.6 4.814.5 4.314.0 3.613.3 2.6/3.0 19.6/18.5 22.9124.0 24.0124.9 23.3122.5 20.4119.1 15.7/1&O
‘~perimental/predicted values (eqn. (21)). b20/1 - AISI 1020, HV434. ‘20/2 - AISI 1020, HV193.
where K is the material constant and HB, is the Brine11 hardness of the material in GPa. Figure 3 illustrates the comparison of Beckmann and Go~mann’s equation (eqn. (19)) and eqns. (20) with values of ef/r, obtained from eqns. (12) modified with eqn. (18). Values of ezlrO were calculated from the experimental data from refs. 13 and 14 using a simplex method. In the course of calculations it became obvious that for carbon steels there exists linear correlation between material constant and Brine11 hardness. Therefore it was necessary to introduce into the eqns. (20) the material constant K. As shown in Fig. 3, values of e:/r,,, calculated from eqn. (20), correlate better with experimental results, obtained from eqns. (12), and accordingly eqn. (20) can replace the original eqn. (19) from the model. The modified erosion model for carbon steels and pure metals is as follows:
T=
1
+
W W’“H& (1 -h/R)r,, IZ= rc,,(ta IR)'
~=0.33
Lp,
ln(TJZJ
G~~ix.0’~
ez/q)=
- 7’)
O.lH,
(21) where K, and KeO are the material coefficients and K, is the abrasive coefficient. The comparison of the experimental values of erosion rate with values obtained from Beckmann and Gotzmann’s theory (Figs. 4 and 5) has shown that the best results can be obtained when some correction factors (KYand Km) are added into the final equation, to take into account the influence of particle velocity and greater (above 60”) impact angles. The equations were obtained empirically from experimental data [13, 141 by the simplex method K,,=O.O05 v+O.14
ifHB,<2
K, = 0.31(HB,)v”“057 K,, = 1_83v-O.”
GPa
if HBM gs2 GPa
(22)
for carbon steels
K,= 1 for pure metals
(23)
where 2, is in metres per second and HB, in gigapascals. The abrasive coefficient r(;, is a product of three factors: K,=K,+K,.K,, K,=d/120
(24)
if cl=0 . . . 120 pm
jK,=l
if 2>120 pm
K,=l
ifwM
(25)
KH = (HV, - HV,)I(OhHV,)
ifHVM
if HVM~HV,,
(26)
where Kd is a size factor, 2 is the mean particle diameter, KH is the hardness factor, and WA is the Vicker’s hardness of abrasive. The eqns. (25) and (26) are very approximate and meant to reduce the errors of calculations when the particle size is less than 120 pm and when the material is almost as hard as the abrasive.
1119
R. R. R. Ellemuaa / Erosion prediction of pure metals and carbon steels TABLE 4. Computed
values of the unknown
Lebedev
f k p c(&rh) -’ (MPa-*) K P (GPa) l @@a) Y (MPa) f k, k
Finnie
Bitter
Nepomnyashchy Beckmann, Gotzmann Abramov Peter
CY
kr A
Al
Ti
Fe
2011”
20l2b
0.021 0.4 12.3 1 x 10-6
0.046 0.5 4.8 39 x lo+
0.055 0.1 15.6 65 x lo+
0.29 0.001 20.8 34 x 10-6
0.29 8X 10-6 41.1 49x 10-6
2.14 4.3 10.7 2.25 0.20 39.0 10.6 1.2 0.36
4.65 28.2 19.1 14.4 0.14 37.6 20.1 1.2 0.10
6.82 21.4 29.2 12.3 0.09 50.4 58.5 1.2 0.46
3.55 29.1 20.9 19.5 0.005 11.4 60.8 1.2 0.30
3.42 17.4 20.2 22.3 0.077 25.1 55.7 1.2 0.58
AISI 1020, HV434. AISI 1020, Hv193.
TABLE 5. Sums of the least-squares different theories Theory
(eqn. (17)) computed
Sums of the least-squares,
Lebedev Finnie Bitter Nepomnyashchy Beckmann, Gotzmann Abramov Peter ‘20/l b20/2 -
Material
Parameters
Theory
“20/l b20/2 -
parameters
with
S (n=14)
Al
Ti
Fe
20/l”
20/2b
3.72 3.30 1.49 1.2x l@ 0.92 5.86 1.29
4.99 5.15 2.56 39.25 2.31 3.35 7.10
4.07 4.56 2.83 98.55 1.97 5.99 4.96
2.81 4.92 0.68 3.12 0.25 3.84 3.56
2.59 4.33 0.43 28.74 0.39 4.69 0.92
AISI 1020, HV 434. AISI 1020, HV193.
The model can be used to compute erosion rates of annealed and heat treated carbon steels and pure metals. Even better results will be obtained when the following limitations are observed: (1) impact angle 0”...90”, (2) particle velocity 10. ..200 m s-l, (3) abrasive hardness Hv, < 1.6Hv,, (4) particle size d > 120 pm, (5) abrasive is dry, (6) plastic deformation takes place. When using this model, the error will usually not exceed 30% for heat treated or annealed carbon steels with static hardness Hv,= 1...7 GPa, and 50% for pure metals. Figures 4 and 5 give the comparison of the original and modified erosion model. It is shown that the modified version yields a better correlation with experimental results. Table 3 gives the comparison of the modified erosion model and experimental results.
4. Discussion The given erosion model contains many more or less empirical coefficients (K, K,, K,, &). The last two are not obligatory and were introduced only to give better correlation between experimental and theoretical erosion rate values for carbon steels. The best values for material constant K and abrasive coefficient K. are obtainable from experiments using a correspondingly standard abrasive or material whose parameters are known. A standard material must be sufficiently soft to erode with rounded and soft abrasive, for example annealed carbon steel AISI 1040, which is sufficiently soft (-200 HV) and whose material constant can be calculated from eqn. (20). Standard abrasive must be sufficiently hard to erode hard materials and non-shattering - for example corundum whose abrasive coefficient can be obtained from experiments with standard material.
5. The program for predicting the erosion rate The program package EROSION [16] is intended for specialists engaged in erosion problems. It enables the prediction of the erosion rate of metals, the computation of the exact values of K and K,, if the experimental values of the erosion rate are known, and the comparison of the predicted erosion rate values of any two different metals. The package also contains data bases for most commonly used abrasives and metals. The EROSION package has a used-friendly interface, with windows and pull-down menus. There are also
1120
R. R. R. Eliemaa
i Erosion prediction of pure metals and carbon steels
Fig. 1. Comparison of the erosion rates computed with different theories with the experimental values. Materiai - annealed Al, abrasive - rounded quartz sand, d=0.4...0.6 mm. Abrasive velocity v=29 m s-’ (a) and 82 m s-r (b).
case-sensitive help and error messages. The user can build up his own data bases and modify them by inserting, deleting or editing their entries. The computed results are given in the form of a table and graph of Z==f(a) or Z, =f(v). The EROSION package runs on an IBM PC and compatibles under MS-DOS and requires only one 360 Kb floppy disk.
6. Conc.Iusion
(1) The comparison of different erosion theories has led to the conclusion that the best correlation between experimental and computed erosion rate values can be obtained by means of Beckmann and Gotzmann’s erosion theory. (2) Beckmann and Gotzmann’s theory was modified by introducing the dynamic hardness of the material
(b) ’ Fig. 2. Comparison of the erosion rates computed with different theories with the experimental values. Material - heat treated steel AI.81 1020 (HV434), abrasive - rounded quartz sand, d = 0.4.. .0.6 mm. Abrasive velocity Y =40 m s-r (a) and 80 m s-l (b).
TABLE 6. Equations metals [15]
for calculating
Type of crystal lattice
Calculation
C8
Hn=1.3+1.8*HB, H,--5.06-Q.52~HBM HD= 1.7.H&, H,=O*58+1.8~HB,
Cl& H12 H6
and additional puting model. (3) In terms age was made erosion rate of
the dynamic hardness
of
of the Ho (GPa)
+0.24-HBZ,
if HBM < 2 GPa if HB, z=2 GPa
coefficients (K,, Ke5, K,) into the comof this modified theory a program packto computerize the prediction of the metals. The error of predicted erosion
R, R. R. Ellemaa
Fig. 3. Values of e:/ra,
obtained
from different
/ Erosion prediction of pure metals and carbon steels
equations.
“WV
.m Attack
45
60
75
M
angle. dqee
@)
90 Attack
an+
degree
Fig. 5. Comparison of the erosion rates computed with the original Beckmann and Gotzmann’s theory and the author’s modified theory. Material - heat treated steel AISI 1020 (HV434), abrasive - rounded quartz sand, d = 0.4.. .0.6 mm. Abrasive velocity v = 40 m s-’ (a) and 80 m s-’ (b).
rate values is usually less than f30% for heat treated carbon steels and f50% for pure metals.
References
0’)
-,:
?I2
At&k
mgle.
degree
Fig. 4. Comparison of the erosion rates computed with the original Beckmann and Gotzmann’s theory and the author’s modified theory. Material - heat treated steel AISI 1020 (HV193), abrasive - rounded quartz sand, d = 0.4.. .0.6 mm. Abrasive velocity u = 40 m s-’ (a) and 80 m s-’ (b).
I. K. Lebedev, Erosion in the boilers, Electric Power Stations, 11 (1958) 22 (in Russian). I. A. Finnie, Wear, 3 (1960) 87. I. R. Kleis and H. H. Uuemois, Wear Resistance of Impact Milling Equipment Parts, Machine building, Moscow, 1986, 160 pp. (in Russian). V. N. Vinogradov, V. I. Bijukov, S. I. Nazarov and I. B. Chervyakov, Experimental investigations of the friction coefficient with spherical particle, Friction and Wear, 2 (1981) 896 (in Russian). J. G. A. Bitter, Wear, 6 (1963) 169.
1122
R. R. R. Ellermaa 1 Erosion prediction of pure metals and carbon steels
6 E. F. Nepomnyashchy, Contact Interaction of Solid Bodies and Calculation of Friction Forces and Wear, Moscow, 1971, p. 190 (in Russian). 7 S. B. Ratner and E. E. Styller, Wear, 73 (1981) 213. 8 V. N. Vinogradov and I. B. Chervyakov, Semiempirical model of plastic materials wear in a stream of hard particles, fioc. 7th Int. Conf. on Erosion by Liquid and Solid Impact, Cambridge, 7-10 September, 1987, Paper 49. 9 G. Beckmann and J. Gotzmann, Wear, 73 (1981) 325. 10 G. Beckmann, Wear, 59 (1980) 421. 11 J. I. Abramov, Erosion of power engineering installations, Power Engineering, 7 (1985) 15 (in Russian). 12 R. Peter, Strahlverschleib an konventionellen Dampferzeugern Prognose und Verschleibschutz, Wus. Ber. THZ, 1004 (1989) Lecture No. V/3.
13 S. M. Levin, Investigation of the erosion of steels in different conditions, Dissertation, Moscow Institute of Petrochemical and Gas Industry, Moscow, 1978, p. 207 (in Russian). 14 J. Tadolder, Some correlations of the erosion of the technically pure metals, Proc. Tallinn Polytechnical Inst. 237A (1966) 3 (in Russian). 15 I. R. Kleis and H. F. Kangur, Resistance of metal surface to indentation by spherical projectile at impact, Proc. 7th Znt. Con& of Erosion by Liquid and Solid Impact, Cambridge, 7-10 September 1987, Paper 48. 16 EROSION, obtainable from Department of Mechanical Engineering, Tallinn Technical University, 200108 Tallinn, Estonia.