The Development of a Model to Predict the Erosion Of

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THE DEVEL OPMENT OF A MODEL TO PREDICT THE EROSION OF

MATERIALS BY NATURAL CONTAMINANTS

1%‘. J . HE AD

Civil En gineeri ng Depart ment, N orth Caroli na State U ni versit y, Ral eigh. N. C. (U.S.A.)

AN I ) M. E. HARR

Civ i l Engineering Department, Purdue University, Lafayette, Ind. (U.S.A.)

(Received October 28, 1969)

SUMMARY

Statistical models for predicting the erosion of both ductile and brittle target

materials were developed on the basis of laboratory erosion tests. The models

developed for ductile materials were considered adequate predictors of erosion

produced by natural contaminants. Insufficient data were available to assess the

predictive ability of the models for brittle materials.

Contaminants and target materials with varying physical properties were used

in the laboratory erosion tests. The apparent impingement angles of the contaminants

were controlled. In addition, the impact and rebound velocities of selected particle

size components of two contaminants were measured photographically with the aid

of a high-speed flash light source.

It was concluded that if the mechanism by which erosion was accomplished

were held constant, the primary cause of steady-state erosion was the energy trans-

mitted from the impinging particles to the target. I t was also concluded that a

certain quantity of energy must be transmitted to a target (threshold energy) before

erosion commences.

Factors which were found to influence the energy transmitted to a target

included the velocity, shape and hardness of the contaminant and the hardness and

resistance of the target to erosion.

NOMENCLATURE

A

A BS

A BW

A DS

ADW

AMS

Ao

AsAw

B

Erosion rate,

erosion rate for brittle materials, step-wise model,

erosion rate for brittle materials, WRAP model,

erosion rate for ductile materials, step-wise model,

erosion rate for ductile materials, WRAP model,

aeronautical materials specifications, Society of Automotive Engineers,

observed erosion rate,

erosion rate, step-wise model,erosion rate, WRAP model,

hardness of target,

radius of curvature of a corner of a particle,

constant,

Wear, rg (1970) 1-46



2 1%‘. .[. HE .\ I ), \ I . I :. H.kI<R

E erosion resistance per unit volume of target,

Eo Young’s modulus for plate glass,

E‘T energy transmitted per unit mass of particles impacted,

H effective hardness of dust,I Vickers indentation hardness,

Ih‘ radius of maximum inscribed circle,

k’ particle velocity at incipient erosion,

KE kinetic energy,

In natural logarithm,

log common logarithm,

Il/irP mass of particles,

Mmax maximum mass of particles,

MT modulus of toughness,

Mr total mass of particles per unit time,

:

number of particle corners measured,

indicated flow rate,

K degree of roundness of a particle in one plane,

r radial distance from centerline,

Kc Rockwell hardness, C scale,

YO radius of blast tube,

us ultimate strength,

V effective velocity of dust,

VA velocity of air,

V _4ave average air velocity,

Vhax maximum air velocity,

VEqui\requivalent particle velocity,

VHN Vickers hardness number,

VP particle velocity,

VP1 maximum particle impact velocity,

Vpmax maximum particle velocity,

VPR maximum particle rebound velocity,

X variable in regression analysis,

YPSyield point stress,

lx apparent particle impingement angle,

Bi Normalized airstream velocity; normalized particle velocity; normalized

particle mass flow,

6 constant,

& strain at rupture,

FX curve fitting parameter,

ium micron,

P2 square of multiple correlation coefficient,

eu curve fitting parameter.

INTRODUCTION

Erosion is a word meaning the act of eating away or destroying by disintegra-

tion. In the context of this work, erosion refers to the disintegration of materials

Wear, 1.5 (1970) 1-46



EROSION BY NATURAL CONTAMINANTS 3

due to impinging particulate matter in the form of dusts or other small-sized

particles.

Erosion may be a detrimental process. Recent experience with turbine engines

in helicopters is an example. The cost of maintaining such engines in dusty environ-ments is very greatl. Air filtration has alleviated the problem somewhat but filtration

reduces both payload and engine performance. If erosion could be incorporated as a

parameter in engine design, perhaps an erosion-resistant or at least an erosion-

tolerant engine could be produced.

Erosion can also be benificial. For example, sand-blasting techniques are

often employed to clean stone or masonry structures. Specialized erosion techniques

have been used successfully in rock cutting213.

The basic problem of the designer is to optimize the effects of erosion. However,

optimization cannot be accomplished efficiently, or perhaps at all, without an under-

standing of the factors which control erosion. It was the object of this study to

develop an understanding of these factors and, in addition, to develop a model for

predicting erosion with a minimum of laboratory tests. Such a model would provide,

hopefully, the missing link in the problem definition-optimization-design sequence.

REVIEW OF THE LITERATURE

Problems associated with the wear of surfaces due to the action of various

abrasives have been under investigation for some time*. Early investigators recog-

nized that properties of the abrasive medium as well as those of the abraded

surface were of importance in wear phenomena5p6.

In 1949, HAWORTH’ investigated the resistance of selected iron and steel

specimens to the cutting and rubbing action of quartz, feldspar and alumina abrasives

and concluded that both the shape and the hardness of the abrasive particles were

important in cutting and rubbing wear. Angular particles were found to be more

abrasive than rounded particles of the same material. Abrasiveness increased with

the hardness for particles with similar shapes.

STOKERS, concerned with erosion of fluid-type catalytic cracking plants,

employed a “sandblast” apparatus in his experiments in which dusts were entrained

in an air stream and impacted upon the target materials. He observed a pronounced

dependency of erosion rate* on air-stream velocity and on the apparent particle

impingement angle **. With brittle gypsum plaster targets, the erosion rate was

found to vary approximately as the cube of the air velocity; a similar relationship

was found for black iron when air velocities were less than 200 ft./set. When silica

sand eroded black iron, the greatest erosion rate occurred at an impingement angle

of 20’. All tests involving plaster targets were conducted at a go’ impingement angle,

i.e. normal to the surface of the targets.

FINNIE~, employing photographic techniques and a high-speed light source,

was the first to measure the speeds of erosive particles. He found that the weight

* Target weight loss/unit weight of dust impacted on the target.** The apparent particle impingement angle is the angle which the air stream makes with thetarget measured from the face of the target. The angle may vary from o0 (air stream parallel to

the face of the target) to go” (air stream normal to the target).

Wear, 1.5 (1970) 1-46



4 \ V. . HE.\l), \I. I<. H.Ilili

loss of an annealed steel target was proportional to the square of the speed of tllcl

eroding particles. Properties of the erosive agents considered important in erosion

of both ductile and brittle materials were particle size, shape, hardness and strengtll.

Some qualitative estimates of the effects of these properties c,n erosion were als0discussed.

As a consequence of his studies, FINNIE developed a mathematical model for

predicting the erosion of ductile materials. In this analysis, the target was assumed

to be a perfectly plastic material with a constant flow stress. The model indicated

that no erosion would occur at normal impingement; this was contrary to experimen-

tal evidence and empirical correction factors were proposed to improve the predictive

ability of the model. FINNIE indicated that it would be very difficult to predict the

erosion of brittle targets because of the complex nature of the origin and growth of

fracture in such materials.

BITTERNS also developed a model for predicting erosion; his model was moregeneral than the one noted above in that both brittle and ductile targets were

considered. The target-erosive agent system was characterized by the three para-

meters K, FR and QR. The parameter K was the particle speed at incipient erosion.

An expression was presented for computing h’ from a knowledge of certain elastic

plastic properties of the target and elastic properties of the erosive agent. Later,

HEAD et al.ll pointed out that mineral particles did not exhibit unique elastic proper-

ties, except possibly under very special conditions, because of crystal anisotropy;

thus, K, in general, was not single-valued. The parameters EH and P,V represented

the energy required to remove a unit volume of target material in cutting and

repeated deformation modes, respectively. These parameters were determined in-

directly by performing at least two erosion tests and back-calculating to determine

the necessary values. BITTER’S model successfully predicted erosion of both ductile

and brittle materials by dusts which were homogeneous with respect to particle

size, shape and hardness.

BITTER conducted his erosion tests in an evacuated free-fall tower, thereby

obviating the problem of non-uniform particle velocity. His results indicated that

erosion was proportional to the square of the particle velocity.

SHELDON~” and SHELDON AICD FINXIE 13,14 investigated the behavior of brittle

targets subjected to homogeneous erosive agents. They found good correlation

between erosion produced by normally impacting particles and a statistical descriptor

of the strength of brittle materials, the Weibull flaw parameter. Their results also

indicated the importance of particle size and speed. For sufficiently small particles,

nominally brittle targets were found to erode in a manner typical of ductile targets.

WOOD’” reported on the behavior of metals eroded by mixtures of a commer-

cial abrasive, silica flour. The mixtures varied in both maximum equivalent particle

size and in particle size distribution. He successfully used BITTER’S model to predict

erosion. An important finding was that erosion rates were essentially constant for

dust concentrations which ranged from about IO ~Jto 10-1 g dust/ft.3 of air. It appeared

that particle interference was not an important factor in erosion until the concen-

tration of particles became exceedingly large *. WOOD found no obvious correlation

between Knoop hardness and calculated strain energy of the targets and BITTER’s

* In engine testing, concentrations of 0.L g/ft.s are considcrcd sevcreL1

Wear, 15 (1970)r-46




cutting and deformation parameters. It was noted that this finding was at variance

with the findings of both FINNIE and BITTER.

THIRUVENGADAM~~ presented experimental evidence indicating that the time

rate of volume loss of materials due to cavitation or to erosion produced by normallyimpacting particles was inversely proportional to the strain energy to fracture of the

target materials. A nomograph was presented 17 based on his concept which aids in

the estimation of the depth of erosion which will occur with time.

HEAD et al.ll investigated the capabilities of five natural soils and three

commercial abrasives to erode stainless steel targets. They found that BITTER’S

model predicted the erosion rates produced by commercial (homogeneous) abrasives

very well. The model could not be used to predict accurately when natural soils

were employed as the erosive agents. It was concluded that BITTER’S model was

inadequate with natural soils because of the lack of adequate target and contaminant

descriptors. It was also concluded that the development of a model applicable to

natural soils required consideration of three main factors: the energy transferred

from the impinging particles to the target, the nature of the response of the target

and the nature of the erosive agent including pertinent descriptors of composition,

angularity, hardness and size distribution.

FINNIE, WOLAK AND KABIL~~, when investigating the erosion of several

ductile targets by silicon carbide particles, found that the Vickers hardness of

annealed metals was proportional to the resistance of the metals to erosive wear at

small (approximately 20’) impingement angles.

FINNIE AND 0~19 investigated the problem of rock drilling by erosion. They

considered rock to be a brittle material, the strength of which was described by the

Weibull flaw parameter and demonstrated the importance of particle shape, size

and speed in rock drilling.

Using photographic methods, NEILSON AND GILCHRIST~%~~ measured particle

velocities in their study of erosion. They concluded that particle size, shape, density,

velocity and concentration, as well as the particle impact angle, were important

factors in erosion. They also observed that, for ductile targets, the maximum erosion

rate occurred at small impact angles: for brittle targets, the maximum rate occurred

at normal impact. These phenomena had been observed by other investigators

previously.

In summarizing the work reported in the literature, it appears that the

following variables have been identified as having major effects on erosion.

(I) Impact regime variables. Primary among these are the apparent particle

impingement angle, particle velocity and the distribution of velocities within an

aggregation of particles. Probably also of some importance are the temperature at

which erosion occurs as well as the characteristics of the medium within which

erosion takes place.

(2) Target variables. These include the nature of the target as reflected in

resistance to erosion. It is likely that, for a given target, erosion resistance is not a

constant. Intuitively, target resistance should be a continuous function of other

variables; for example, the apparent particle impingement angle.

(3) Contaminant variables. These include particle size, shape and hardness.

For heterogeneous erosive agents, some assessment must also be made of the distribu-

tion of these variables.

Wear, 15 (1970) I-46



HYPOTHESIS

The hypothesis that the primary cause of steady-state erosion* of target

materials subjected to impinging particulate matter is the energy transferred fromthe particles to the targets, forms the basis for the work reported here.

To verify this l~~p~~tl~esis,an extensive testing program was undertaken. In

the program, selected target materials were eroded by a series of artificial and

natural dusts (or “contaminants”). Measurements were made of the speeds of the

dust particles both just before and just after impact with the targets in order that

an assessment could be made of the energy transferred from the impinging particles

to the targets.

Other factors which were considered to influence erosion are

(a) the shape of the particles,

(b) the hardness of the particles,

(c) the hardness of the target and

(d) the resistance of the target to erosion.

\VAI)ELL'S concept of particle roundness~Z~~3 was adopted in determining the

shape of the dust particles; conventional indentation techniques were employed in

measuring the hardnesses of the dusts and the targets. Assessments of target resistance

to erosion were made on the basis of the calculated strain energy of the targets as

well as their indentation hardness.

Erosion device

The erosion device may be described generally as a miniature, precision sand-

blast apparatus. The device, depicted in Fig. I, consisted of nine major parts. Visible

in the figure are the glass-sided chamber in which the targets were eroded, the

5116 in. inside diameter steel blast tube which served to direct the shop air-dust

mixture onto the target, the target holder, with target in place, which permitted the

target to be rotated with respect to the air stream, a plastic hose which conducted

the air-dust mixture from the dust cup to the blast tube; the dust cup which held

the dust and the timing mechanism and air solenoid which controlled the time

duration of the erosion tests. Not shown in Fig. I is the air flow meter and air pressure

regulator which were used to monitor and control the volume flow rate of air into

the blast tube.

Photographic techniques were employed in the determination of particle

impact and rebound speeds. The equipment mciuded a camera, a light source and

a timing device. The camera was an 8 in. x I O in. format still camera with a Polaroid

3 in. x 5 in. film pack adapter. A 30 in. extension bellows and a 6.25 in. focal length

lens magnified the images of the dust particles two to three times. Polaroid black

and white film type 57 was used in photographing all but the smallest (16 ,um

equivalent diameter) particles. When photographing these small particles, it. was

* Steady-state erosion refers to erosion which OCCUM t a constant rate with respect to the weightof dust impacted on the target.



EROSION BY NATURAL CO~T~~~INA~TS 7

Fig. I. Erosion device

necessary to use a conventional sheet film which could be enlarged sufficiently so

that the particles were readily visible.

The light source was a dual flash device which employed a two-channel

control unit and two xenon flash lamps. The duration of each flash was approximately

0.5 psec at 60% peak intensity. The time interval between flashes could be set at

1, 3, IO, 30, I00 or 300 psec.

The timing device was an events/unit time meter. The meter, accurate to

less than I ,usec, was used to measure the time between flashes.

Three “artificial” dusts, blended from commercially available abrasives, and

a pre-blended abrasive were the erosive agents used in the majority of the erosion

tests reported here. The dusts were designated crystolon, glass beads, and alundum.

Crystolon, a hard, angular material, was silicon carbide. The glass beads were

spherical in shape and were about as hard as ordinary window glass. Alundum was

aluminum oxide and was also an angular material somewhat less hard than crystolon.

The pre-blended abrasive, silica flour, was silicon dioxide and was an angular material

of approximately the same hardness as alundum. Silica flour had been used pre-

viously by other investi~atorsllil5 in erosion testing, The maximum equivalent

Wear, rg (1970) r - 46



5

CM

C

E

P

S

T

SZ

IN

C

IWT

B

S

+Q\y - -




A Zeiss TGZ3 Particle-Size Analyzer was used to measure the shape factors

of the test dusts. Figure 3 shows a photograph of the instrument which consists of

a plexiglass plate on which photomicrographs of the particles were placed and anadjustable iris diaphragm which was connected through a footswitch to a series of

telephone counters. When the footswitch was depressed, a mark was entered on that

Fig. 3. Particle analyzer.

MAW IN

PROPER OSITION FORMEASURING RADII OF

CURVATUHE OF CORNERS

Fig. 4. Measurements necessary to determine particle roundness (after Boggs).

counter corresponding to the diameter of the diaphragm. Simultaneously, a pointed

marker descended and punched a small holein thephotomicrograph, thereby providing

the operator with a visible record of the particles measured. The shape parameter

employed in this study was particle roundness defined by WADELL~~,~~ as follows.

where R is the degree of roundness of a particle in one plane (dimensionless), c the

W&w, rg (1970) I-&



radius of curvature of the individual corners, I,; tire radius oi the maximum inscribed

circle and 11: the number of corners measured.

l;igure 4 presents a diagram showing the measurements necessary to determine

the roundness of a particle. Figure j is a typical I:)llotornicrogra~)t~ slrowing anassemblage of particles at approximately x 7~ magnification.

Fig. 5. ~ho~o~licr~~raph of ernsi vr parti cl es. J 7. 3

Uniaxial tension tests were performed on representative samples of the

metallic specimens used in this work. A Riehle testing machine of bo,ooo lb. capacity

was used in the tension tests. The strain rate was 0.050 in./min until the yield point

of the specimen was reached after which the strain rate was increased to 0.100 in./min

until rupture occurred. The approximate modulus of toughness was computed using

the following relationship suggested by M~-RI>HI-~~.

where PIT is the modulus of toughness (p.s.i.), F the strain at rupture (in.@.), YPS

the yield point stress (ps.i.) determined at z’:/ooffset and US the ultimate strength

(p.s.i.).

No tensile tests were performed on the glass target materials. Typical values

of properties of glass were assumed, based on published data”5 and a modulus of

toughness was computed and was assumed to be representative of the glass.

Total 9wm47e probe

The magnitude and distribution of the velocity of the shop air discharging



EROSION BY NATURAL CONTAMINANTS II

from the blast tube in the erosion chamber was used for assessing particle velocities.

Measurements of total pressure at various locations in the air stream were made

for indicated flow rates of 760, 660, 560 and 460 ft.z/h at horizontal distances of I,

z&, and 3& in. from the end of the blast tube. Total pressure was measured at0.030 in. increments along two mutually perpendicular diameters across the face of

the blast tube. The air velocities were calculated with the assumption that the air

flow was isentropic.

PROCEDURE

Erosion tests

An erosion test consisted of impinging a dust sample upon a target for a

preselected length of time and recording the resulting erosion rate*. The time

required for an erosion test varied from 15 to 60 min depending upon the type oftarget and the type of erosive agent.

The concentration of dust was assumed uniform throughout any one test.

The actual dust concentration was not of major concern because of the evident

insensitivity of erosion to reasonable variations in dust concentrations previously

reported by wOOD15 and also noted by HEAD et al. l l . However, efforts were made

to achieve dust concentrations of approximately 0.1 g dust per cubic foot of air to

simulate a “severe” dust conditionrr.

Particle velocity tests

A photographic technique was employed to determine the impact and rebound

velocities of the dust particles. Individually sized components of two erosion agents,

crystolon and glass beads, were photographed as the particles impacted and rebounded

from the targets. Crystolon and glass beads were chosen because it was believed

they represented extremes of available erosive agents as reflected by particle hardness

and shape.

The result of a successful trial was a doubly exposed photograph in which a

number of impacting and rebounding particles could be seen. Each flash produced

one image of an individual particle. The velocity of a particle could be computed by

multiplying the distance between the two successive images of the particles by the

appropriate scaling factor and dividing the result by the measured time between

Fig. 6. Double-exposed photograph of erosive agent.

* Erosion rate was defined as the volume loss of the target/g of dust impacted.

Wear, 15 (1970) 1-46



1, IV. J. HI'.\l), I. <. .ZRI;

Hardness tests

The Ilickers hardness numbers of the targets and of the artificial dusts,

crystolon and glass beads, were determined with the Durimet hardness tester. The

load applied to the diamond indenter was 50 g for 30 sec. The targets were tested

unmounted. The dusts were mounted in bakelite and polished before testing. Hardness

values were obtained for crystolon and glass beads but considerable scxtter existed

in the data. The scatter in the hardness data for crvstolon was attributed to crystal

00 1.0 20 3 0 4.0 5.0 6D 7.0 6.0 3.0 10.0

MOH'S SCRATCH HARDNESS

anisotropy and tllat for glass beads to the difficulties inherent in measuring the

hardness of convex particles. No rational method for predicting the “effective”

hardness of the particles in the erosion process is known. It was decided to use the

relationship between Vickers hardness and Moh’s scratch hardness proposed by

TABOR"S. This relationship is shown in Fig. 7. The Moh’s scratch hardness of both

crystolon and glass beads had been reported in the literature27,“8; thus, the Vickershardness number of these dusts could be determined.

Particle shape determninatiofzs

The Zeiss TGZ-3 Particle Analyzer was used in the particle shape deter-



EROSION BY NATURAL CONTAMINANTS I3

minations of crystolon and AC Coarse. Photomicrographs of the various size com-

ponents of the dusts were analyzed on the Zeiss machine. The results of a statistical

study of the shape parameter used in this study, particle roundness, indicated that

for a given size particle, approximately one hundred particles should be measuredin order to achieve a “good” measure of roundness2Q.

To insure that representative values of roundness would be obtained, more

than two hundred particles of each individual size component of crystolon and AC

Coarse* were analyzed. The glass beads were not analyzed; the roundness value

for the beads was assumed to be 1.0 because of the near-spherical shape of each

particle.

RESULTS

Erosion testsThe results of the erosion tests are shown in Table I. The erosion rate was

computed by dividing the volume loss ** of the target by the weight of dust impacted

on the target during the test. The average particle concentration was computed by

dividing the weight of dust impacted by the volume of air which passed through

the blast tube. In Figs. S-II are shown plots of target volume loss as a function of

apparent particle impingement angle. The Vickers hardness number (VHN) shown

on the plots are the nominal hardness values for target materials in the indicated

group. The dependency of erosion rate on apparent particle impingement angle,

target material and type of dust is evident.

Transferred energy tests

Air stream velocities

The velocity of the air emanating from the blast tube and the velocity distri-

bution as a function of position in the air stream was determined with the aid of

the total pressure probe. The results showed that the maximum air velocity occurred

near the blast tube centerline and that the velocity distribution was approximately

symmetrical about the maximum air velocity.

As an aid in assessing the effects of non-uniform particle velocity on erosion

rate, the air velocities were assumed symmetrical about the maximum velocity. The

data were then normalized with respect to the maximum air velocity, I’*max, and ~0,

the radius of the blast tube. It was found that a unique relationship existed between

normalized position in the air stream and normalized air velocity. This relationship

is shown in Fig. 12.

Relationship betueem maximum particle velocity alzd transmitted energy

A primary objective of this work was to develop a relationship between the

maximum particle velocity and the energy transmitted from the particles to a target.

The maximum particle velocity was considered because it was the velocity most

amenable to direct measurement. It was anticipated that particle velocities would

* AC Coarse w as fractionated in an AMINCO particle classifier into eight size categories asfollows: O--IO, IO--20, 20-30, 30-40, 40-50, 50-60, 60-70 ,um and all particles larger than 70 ,um.** Published values30 of densities of materials assumed representative of the densities of the

targets used in this study were as follows: aluminum alloy 7178 2.82 g/cm3; stainless steel 17.4PH

7.81 g/cm%; beryllium-copper alloy 8.23 g/cm3; and plate glass 2.47 g/cm3.

Wear, 15 (1970) 1-46



\\‘. J . HE>\ I), ;\ I. I:. H.\ I<I:

Target

wderial

illmninum

alloy 7178

Aluminum

alloy 7178

Aluminum

alloy 7 178

Aluminum

alloy 7178

Stainless

steel I 7-4PH

Stainless

steel I 7-4PH

Stainless

steel I 7-4PH

Stainless

steel I 7-4PH

Silica flour

Crystolon

Alundum

Glass beads

Silica flour

Crystolon

Alundum

Glass beads

Wear, ‘5 (1970) I-46

Target

kavdness

(kglnm2)

AL38 200AL38 200

AL39 194AL9 200

:\ Lg LOO

XL10 202

XL15 204AI, 16 209

AL16 209AL1 I 204Xl,12 204Al.29 214

AL22 202

AL22 202

A-L23 207AL23 207AL20 205AL20 205

AL7 208

AL7 208

AL5 208

hL6 205AL6 205

.\ I,1 2 202

PH4oIg 441PH4020 457PH4017 444PH4018 450PH4o18 450PH4023 453

PH4024

PH4015PH4015

PH4015

PH4016

PH4016

433

453453

453

440

440

PH4031 45’PH4031 451

PH4o3o 451PH40300 446

PH40300 446

PH4o31 I 453

PH4o32 445

PH4o32 445PH4o36 450

PH4o34 445

PH4o35 447

PH4o35 447

90

7500

45

3”

IS

90

75

0045

30

15

90

7560

45

30

‘5

9o

7560

45

30

15

90

7560

45

30

15

90

7560

45

30

15

Yo

7560

45

30

15

0.0614

0.0730

0.0760

0.0780

0.0770

0.0759

0.0807

0.0860

0.08720.0821

0.0787

0.0563

0.0959

o.ogsg

0.1013

o.og22

0.0890

0.0960

0.0901

0.0962

0.09360.0908

0.0967

0.0946

0.0798

0.0868

0.0930

0.09450.0998

o.ro4.T

0.0812

0.08600.0899

0.09470.1025

0.1079

0.1012

0.1020

0.0971

0.1062

0.09950.1026

0.0883

0.07930.0798

0.0789

0.0816

0.0796

“.0002’jZ

0.000287

0.000361

0.000472

0.000488

o.ooo471

0.0003430.000371

0.0004890.00057x

0.000693

0.000600

0.000226

0.000290

0.000348

0.000462

o.ooo504

0.000486

0.000035

0.000048

0.000069

0.000086

o.oooo71

o.oooo37

o.ooor29

o.ooo136

0.000158

o.000182

0.000189

o.oooI52

O.O”OI35

0.0001610.000197

0.000232

0.000255

0.000201

0.o00130

0.000148

0.000182

0.000213

0.000226

0.000193

0.000014

0.0000150.000015

0.000010

0.000006

0.000002



EROSION BY NATURAL CO~TA~I~A~TS

TABLE I (Continued)

I5

Target Erosive

material agent

Target Target

number hardness

(kglmmz)

Stainless

steei 17-4PH

Crystolon

Stainless

steel 17-4PH

Alundum

Beryllium-

copper alloy

Crystolon

Reryllium-

copper alloy

Beryllium-

copper alloy

Glass beads

Crystolon

Beryllium-

copper alloy

Plate glass

Glass beads

Silica flour

Plate glass Crystolon

Plate glass Alundum

Plate glass Glass beads

PH3o5PH306

PH306

PH3o7

PH3o7PH308

PH301

PH302

I’H303

PH.303I’H304

I’H3o4

B401

B402

B404

B4o3

B4o3

B4o7

B4o7B408

B201

B201

B202LB202

B2o3

B205

B206

B206

G-3

G-3

G-4

G-4

G-4

G-4

G-5G-j

G6

G-6

G-6

G-6

G-I

G-I

c-2

G-Z

G-2

G-2

G7

G-7G-8

G-8

G-9

G-9

337 90 0.0857

334 75 0.0895

334 60 0.0868

330 45 0.0879

330 30 0.0883

329 15 0.0892

0.0001680.000186

o.ooo227

0.000272

0.000301

0.000275

350

349

334

334

339

339

378

4’2

390

385

385

408

408

399

90 0.1039 0.000123

75 0.1057 0.000147

60 0.1007 0.000174

45 0.0958 0.000226

30 0.0929 0.000230

15 0.1029 o.oooIQg

::

45

30

15

0.0682

0.0818

0.0971

0.0902

0.0929

9060

45

0.0924

0.09450.0983

0.0608

0.0787

0.0834o.ogoo

0.0968

0.000254

0.000343

0.00040I

0.000452

0.000399

0.000028

0.000036

0.000029

220 90220 60

217 45217 30226 I5

0.000252

0.000364

0.0004250.000488

0.000457

221 90 0.0920 0.000034

225 45 o-0949 0.000042

225 30 0.0999 0.000028

660 90 0.0616

660 75 0.0663

626 60 0.0800

626 45 0.0801

626 30 0.0792

026 I5 0.0792

0.004589

0.004152

0.003321

0.002238

0.001222

0.000473

615615

634

634

634

634

618

618

667

667

667

667

623

623615

615

639

639

90 0.0637 0.00508875 0.0683 0.004719

60 0.0717 0.003890

45 0.0748 0.002711

30 0.0768 0.001614

15 0.0756 0.000719

9o 0.08x5 0.004326

75 0.0850 0.00399960 0.0862 0.003312

45 0.0875 0.002316

30 0.0825 0.001325

‘5 0.0720 0.000535

90 0.0813

75 0.082360 0.0820

45 0.0948

30 0.0883

=5 0.0892

0.001407

0.0011650.000556

0.000133

0.000006

0.000002

A#arent Average Average

particle part ic le erosion

im+ngement concentration rate

angle (gif t .3 air) (cmYg)(degrees)

-

Wear, 15 (1970) 1-46



’ 0.0 040 0.80 120 1.60 6.0 040 0.60 1.20 1.60

APPARENT PARTICLE IMPINGEMENT ANGLE, d (RADIANS) APPARENT RWTlCLE ,MPlNGEMENT ANGLE, d WMIANS)

be a function both of particle mass and of the position of the particle in the air stream.

To facilitate the development of the desired relationsllip, the following assumptions

were made.

(I) The normalized distributions of the impact and rebound velocities of the

particles and the mass flow of the particles were the same as the normalized distrihu-

tion of the air velocities shown in Fig. 12.

(2) The energy per unit mass of the particles transferred tcr a target was

~~~(~~(~rt~~)lla~o the difference between the squares of the impact and rehund

velocities. That portion of the energy of the particles expended in rotation, in

temperature changes and in deformation of the particles was negligible.

(3) The number of particles traversing an elemental area of the face of the

blast tube was both very large and uniform per unit of time. On this basis, the air-

dust system was assumed to bc a continuous rather tllan a discrete system.

These assumptions were utilized in the development of a relationship between

a fictitious particle velocity and the maximum particle velocity, either impact or

rebound. The fictitious velocity, 5’-l.:suiLX,was defined as the velocity wIiicli,wherr

squared and multiplied by half the mass of the particles, yielded the kinetic energ!

of the particles. This relationsllip, developed in Appendix A, accounted for the



EROSION BY NATURAL CONTAMINANTS I7

5000

TARGET: BERYLLIUM-COPPER ALLOY

0.0 040 0.60 120 I.60

APPARENT PARTICLE IMPINGEMENT ANGLE, D( (RADIANS)

WXJ

3000

2000

ssc, 000

500

300

200

”,

100

50

30 I

;20

IO -

0.0 0.40 0.80 120 1.50

APPARENT PARTICLE IMPINGEMENT ANGLE,o! lRAOIANS)

Fig. IO. Target crozion rate US. apparent particle impingement angle for beryllium-copper alloy.

Fig. II. Target erosion rate US. apparent particle impingement angle for plate glass.

assumed distributions of particle mass flow and velocity. Its form is as follows:

(VEquiv)‘=0.679(~ipm.,)~

where Vrmax is the maximum particle impact or rebound velocity.

(3)

Deternkation of particle velocities

Photographic methods were employed to determine the velocities of the dusts

used in this study. When photographing the dust particles, it was found necessary

to place the end of the blast tube approximately Z& in. horizontally from the face

of the target to allow the flash lamps to be positioned correctly. All erosion tests

had been conducted with the tube I in. from the target; examination of many

photographs showed that the increased spacing had no detectable effect on particle

velocity. It was also thought that a boundary layer of air might build up in front

of the targets and impede the particles. Comparisons of velocities of particles just

about to strike the target (thus presumably in the boundary layer) with velocities

of particles I-Z in. from the target showed no measurable differences in particle

velocity. The effect of particle shape on the impact velocities of the particles was

also studied. It was found that particles of crystolon (angular shape) and glass beads

(spherical shape) of approximately the same equivalent diameter exhibited similar

Wecw, 15 (1970) r-46



impact velocities, all other factors being the same. In addition, the data suggested

that the apparent impingement angle had no effect on the particle impact velocity.

Several photographs of impacting and rebounding particles were analyzed

so that the assumed normalized distribution of particle impact and rebound velocitiescould be compared with the actual normalized distribution. The results are shown

in Fig. 13 along with a curve which is a reproduction of the top half of Fig. 12. The

curve represents the relationship between the normalized position in the air stream

and the normalized air-stream velocity. The curve also represents the assumed

distribution of normalized particle impact and rebound velocities. Deviations from

0.80

1.00 _-- i00 020 040 060 0 SO 100

NORMALIZED AIRSTREW VELOCITY,‘AIR

/

“A, RAVE MAX

Fig. IL. Normalized position in airstream us. normalized airstream velocity.

w z,M ACTION

GLASS BEADS 100 IMPACT

NORMALIZE3 PARTICLE VELOCITY, VP, 1 VpMpx

Fig. 13. Normalized position in airstream US. normalized particle velocity.

Wear, 15 (Ig70):1-46



EROSION BY NATURAL CONTAMINANTS I 9

the curve are evident; however, it was felt that, in general, the results confirmed

the reasonableness of the assumed distribution.

The results of measurements of maximum velocities of individually sized

components of crystolon and glass beads are shown in Tables II and III. Data arepresented for particles impacting (Table II) and rebounding (Table III) from targets

for various apparent impingement angles and indicated air flow rates. In Fig. 14 is

shown a plot of maximum particle impact velocity, Vrr, as a function of particle

“size” (equivalent diameter) and indicated flow rate. The points representing

indicated air flow rates of 760, 660 and 560 ft.a/h were a fit with straight lines deter-

mined by a least squares procedure. Insufficient data were available to fit the data

for the indicated 460 ft.s/h flow rate; the straight line was estimated visually.

TABLE II

MAXIMUM PARTICLE IMPACT VELOCITIES

Type of@ar ti cle

Equivalent Indicatedpart ic le ai r f l ow

diameter rate, Q

(ccm) (ft.Yh)

Glass beads 100

85

2

760 ‘5

45

760 15

760 15

760 15

45660 I 5

45

560 15

45

460 ‘5

45

760 ‘5

760 ‘5

760 15

45

90

660 90

50

38

29

Crystolon 105

74

63

44

760 ‘5

45

90

760 I5

45

90

760 15

90

760 45

560 45

460 45

760 15

90

560 15

90

460 I516 760 15

90

31

Apparent Maximum

part ic le par t ic le

impingement impact

angle velocity VPI

(degrees) ( f t . + )

387

384

422

423

478

471

385

383

289

312

‘99

221

478

542

552

529

491

398

428

442

462

409

403

450

445

479

492

308

204

556

568

328

330

241

614

655

Wear, 15 (1970) 1-46



45

45

03

02

IO0

0,

or

29

$w

7f’O

700

* Restitution ratio is the maximum particle rebound velocity di\-itletl by the maximum p?rtiCk

impact velocitv taken from Fig. 11.

** Alum. = slum&unn alloy 7178.

t r7-xxx = Stainless steel 17.4PtI with indicated nominal \‘ickcrs hardness number.

tt Glass == plate ,@ass.



460

ro5 $I0

74 760

45 44 7Go

660

j60

‘$0

J 7-450 146 0.32

Glass 1% 0.41

Alum. 134 0.29

IF-450 93 0.20

C;lass 138 0.29

Alum. 143 0.26

I7---450 129 0.24

GlassAlum. 106 0.26

Glass123

0.30

Alum. 107 0.33Glass 129 O.‘pJ

Alum. 66 0.29

Glass 90 O.‘p

r y--220 200 0.32-~- -~-

* Restitution ratio is the maximum particle rebound veloci~divided~n~mnm particleimpact velocity taken from Fig. 12.

31 760

90 ro5 760

74 760

63 760

3’ 760

660

560

460

r6 760

(deg4ws)_ll- _-_..i- ._..--_.___C.rystolon I 5 ro5 760

660

5Go

21

--~

Target Ma&mum Restitut ion

material rebound rat io*

wEocity, vpn

f f t ' i =)

__ ~-

Ahim, 238 0.57

17-450 314 0.75X7--“LO 259 0.62

Glass 351 0.Q

r 7-450 344 0. j6

GlXS 326 0.72

Alum. 22.3 0.60

Glass 3’8 0.85

17-450 330 0.70

CXass 384 0.82AIUXI. 257 0.61

Glass 331 0.82

‘7-450 244 0.76

Glass 289 0.90

Al 1,111. r44 0.63Glass 186 0.82

Alum. arti 0.51

17-450 209 0.50

Glass 218 0.52

Alum. 208 0.46

x7-450 223 0.49

Glass ' 253 0.56Alum. 279 0.54

r 7-450 240 0.47Glass 280 0.55AlURl. 212 0.54Alum. 155 o-49Glass 18. 3 a.58

X7-450 119 o-53Glass 110 0.49Alum. 305 0.55

‘7-450 239 0.44Glass 310 0.56

r 7-450 ro8 0.2l5

I7--220 109 0.26

Glass I -17 0.35



SHELIION~", in his analysis of impact velocities, indicated that the velocities of 8 ,um

particles were essentially the same as the air velocity. hccordingljr, it was assumed

in the work reported here that the impact velocities of IO i&m particles were tlrc

same as the maximum air stream velocities. These assumed points appear in Fig. 14as black circles and in those cases where the impact velocity data were a fit with a

line determined by the least squares procedures, the assumed points were considered

data.

Figures Is-17 show plots of the ratio of maximum rebound velocity to

maximum impact velocity (restitution ratio) as a function of particle size and type

10 20 30 50 70 100 200

EO”,“AL ENi PfaRTlCLE DIBMETEF: ( MtCRONS)

$4, hkximum particle impact velocity z’s, equivalent partick diameter

GLASS BEADSI.00

5\ 0.60

>g

6 0.60

F

s040

g

; 020

8(L 0.0

0 20 40 60 80 100 12.0

EQUIVALENT PAR7lCl.E DIAMETER (MICRONS)

1.00CRYSTOLON

P-.=Oi?3

P

6 0.60F2p 040

g 020

2[L 00

0 20 so 60 80 100 120

EW”ALENT PPIRTICLE DIAMETER ,MICRONSl

I.00GLASS BEADS

_._0 20 40 60 GO $00 120

EOUiVACENT PARTICLE DIAMETER i MICRONS)

looCRYSTMDN

>‘DSO‘n :

Pd 060

Fs

040

$

z 020

F

iz 000 20 40 60 60 100 120

EQUIVALENT PARTICLE DIAMETER (MICRONS)

Fig. 15. Restitution ratio vs. equivalent particle diameter for aluminum alloy 7178.

Fig. 16. Restitution ratio vs. equivalent particle diameter for stainless steel 17.4PH (nominal.VHN =.~450).




of particle and target, The data for velocities of rebounding particles were considered

less reliable than the impact velocity data because the photographs of rebounding

particles were subject to more operator interpretation than the photographs of the

impacting particles. In consideration of the relative crudeness of the rebound data,it was felt appropriate to estimate visually the trends of the data. The data were

interpreted as indicating that the restitution ratio was independent of particle size.

Horizontal straight lines were drawn based on this interpretation and also on the

basis of work reported by TABOR31who studied the effects of velocity on the coefficient

of restitution of indenters striking various types of targets. TABOR showed that the

coefficient of restitution was very insensitive to impact velocity for targets exhibiting

a constant yield stress provided that the impact velocity was sufficiently high.

1.00GLASS BEADS

P_ 0.80

.a60

F

i 0.40

H

g O,*Oy 0.0

0 20 40 60 SO 100 120.EQUlVALENT PARTICLE DIAMETER (MICRONS)

CRYSTOLON

Q 20 40 So 80 loo 120

EOUIVALENT PAwrtc~E DiArdETm f MICRONS)

Fig. 17. Restitution ratio vs. equivalent particle diameter for plate glass.

Particles impacting targets at apparent impingement angles of 3o”, 60” and

75” were not pllotographed. The restitution ratios for particles impacting at these

angIes were estimated from the ratios established for impingement angles of r5’,

45” and 90~ on the assumption that the ratios were proportional to the apparent

impact angle. A complete list of the restitution ratios is presented in Table IV.

Relationsltip betweevz erosion rate and tra+vsmitted energy

It was asserted in the hypothesis of the work reported here that a causal

relationship existed between steady-state erosion of targets and the energy trans-mitted to the targets by impinging particles. To assess the validity of the hypothesis,

plots were made of the erosion rates experienced by aluminum alloy 717% and plate

glass as functions of the energy transmitted by crystolon particIes and glass beads

W&W, g (1970) I-46



\I’. _1. ti1,.\11. \I. b._, ll.\lili

o.qo*0.82

0.;4*

O.(K)

0.04

0.01*

* Estimated from plots of data; remaining values proportional

impinging upon the targets at various angles. The results are shown in Figs. IS and

Ig and Table V. The plots were made on a rate basis (erosion rate ES. rate of energ)

transmitted) as a matter of convenience. The plots could have been presented equally

well on the basis of total amount of erosion 11s. total energy transmitted. Aluminum

alloy and glass were chosen as the target materials to represent ductile and brittle

targets, respectively. Crystolon was chosen to represent “angular’‘-shaped particles

and glass beads were chosen to represent “round” particles.

The bulk of the data were derived from tests in which single-sized particles



T

V

E

E

Y

T

A

S

F

M

I

P

A

E

E

SO

N

R

FOR

A

UR

NUM

A

OY

7

A

D

GL

T

E

T

g

T

o

p

ce

‘4

E

v

e

iv

n

me

p

ce

a

e

damee

(d

e

(

m)

In

c

e

a

o

w

rae

Q

IPV

P

ce

P

ce

imp

re

vPo

y

v

o

y

I*1

V

(W (+c

E

g

ta

mi

e

p

u

ma

o

p

ce

imp

e

E

(O

eg

g

E

m

rae A

(1

3

c

g

Ga

b

4

C

oo

9

C

oo

I5

Aumin

m

a

o

77

Mixu

7

IO

7

6

7

6

6

6

5

2

7

Mixu

7

7

7

3

6

3

5

3

4

Mixu

7

8

7

3

6

3

5

3

4

C

oo

C

oo

9

Mixu

3 3 3 3

I5

Mixu

8 7 3 3 3

Pae

ga

7 7 6 5 4 7 7 7 6 5

4

2

4

2

3

2

3o

I9

5

3

4

I2

4

IO

3

8

2

O

4

2

4

2

3

I9

2

I3

5

I8

4

I3

3

IO

2

7

4

3

4

3

4

3

3

2

2

1

65

08

43

02

42

05

35

02

2

0

00

6

3

07

1

1

34

6

4

I7

5

0

I5

3

3

08

1

3

04

7

59

00

43

24

39

28

2

77

16

I9

08

1

gg8

8

5

4

3

7

1

5

3

6

2

4

2

2

‘85 1

458

5

8

4

6

2

1

1

146

71

49

49

32

I5

05



*(II SO”(AORA)

ENERGY TRANSMITTED PER UNIT h%ASS OF PARTICLES

IMF’ACTED,ET(~O~ ERGS PER GRAM)

60

po

$ CRYSTOLON FARTICLESIy = W(BORA)

1 I

cc -==CRYSTOLON PARTICLESd= 15’ (0OR.l

ENERGY TRPiNSMlTTED PER UNIT MASS OF PARTICLESIMPACTEO, E,(lO* ERGS PER GRAM1

Fig. 19, Erosion rato us. transmitted energy for plate gll~ss argets

served as the erosive medium. Single-sized particle erosion tests were performed to

assure that the results would not be confounded by possible particIe-size-dependent

interactions. Additional data points were added on the basis of erosion tests per-

formed with crystolon and glass bead “dusts”. The grain-size distributions of the

Wear, I5 (1970) r-.$6




dusts were presented in Fig. 2. To compute the energy transmitted by the dusts, it

was assumed that each size fraction acted independently and that the energy trans-

mitted was the sum of the energies transmitted by each size group weighed according

to the respective grain-size distribution curves. The energies transmitted by thedusts are depicted on each figure by solid symbols; data from the single-size particle

tests are represented by open symbols.

In general, the results verify the previously unsubstantiated deduction that,

for an angular erosive agent and a given level of transmitted energy, the erosion

rate of a ductile target will be smaller at an apparent impingement angle of 90~

than at 15"; the reverse is true for brittle targets for angular as well as rounded

erosive media. The results also indicate that glass beads are a very inefficient erosive

medium.

While the trend of the data is clear, the indicated linear relationship between

erosion rate and transmitted energy is not immediately apparent. However, afterexamining log-log plots of the data, it was concluded that the trends were well

approximated by straight lines with non-zero intercepts on arithmetic plots. The

implication of this result is that, in general, some minimal threshold level of energy

exists, below which erosion will not occur. This phenomena has been noted by

otherslO~il,l6.

Intuitively, the order of the intercepts is satisfying; for the ductile aluminum

alloy, erosion at normal impact results in repeated deformation of the surface,

eventual embrittlement, and finally, a “flaking-off” of a portion of the target. This

process would seemingly require more energy to initiate than the micro-shearing of

the surface of the target which is the predominant mode of erosion at small im-pingement angleslo. In addition, the crystolon is very angular and probably quite

efficient in the shearing process. At an intermediate impingement angle, such as

45”, both micro-shearing and repeated deformation of the surface of the target occur,

When the erosive agent is glass beads, a relatively large amount of energy is required

to initiate erosion, probably because the beads are very inefficient in shear due to

their spherical shape. The beads, then, must accomplish erosion primarily through the

repeated deformation mechanism. At an impingement angle of 45”, only about half

of the initial energy of the beads can be expended in the repeated deformation mode;

the bulk of the remaining initial energy is expended in the inefficient shearing

process.

If the target is brittle, more energy is required to initiate erosion at a 90~

impingement angle than at 15’ as in the ductile case; however, once erosion at 90’

has started, it proceeds at a much greater rate than erosion at 15". The large erosion

rate is probably due to cracking of the surface of the target and the rapid spreading

of these cracks accompanied by spalling of the surface and the formation of new

cracks as the surface continues to be deformed by the impacting particles.

TO summarize, provided that the mechanism of erosion is held constant, the

resuhs presented in Figs. 18 and 19 affirm the basic hypothesis of this research; the

primary cause of steady-state erosion is the energy transmitted from the impinging

particles to the target.

The Vickers hardness numbers of the target materials used in this study are

Wear, 15 (1970) 1-46



2s L\.. _I. HI,:.\l), \I. I,__H.\lili

listed in Table I. The “effecti\,e” hardness values for the dusts are listed in Table \‘I

and were determined with the aid of published values of Mob’s scratcli I~ar-dness and

l:ig. 7 which shows the relationship between \‘ickers hardness and !Uol~‘s hardness

proposed by T_-\rror~~~. or all the dusts except I\(‘ Coarse, it was assumed thateffective hardness was the same as tile hardness of the individual particles irrespectiv~~

of particle size. A petrographic analysis” and the grain size distribution curve (I;ig, ZJ

were used in determining tlie effective hardness of AC Coarse. It was assumed that

the effective hardness was the sum of the hardnesses of each size component of the

dust weighed both by the size distribution curve and the estimated mineral content

of eacli size fraction.

T;\BI,li I-111

Aluminum alloy 7178 748.857Stainless steel 17.JI’H 996.08.&

Ueryllium-copper allo) 669.oooI’late glass 0.345*

* This value was computed by assuming that the yield point stress of glass was IO,OOO p.8.i.ad that the strain at rupture was O.OOI in./in.

Par t ic le sha$e tests

\‘alues of “effective” particle roundness, the particle shape parameter used

in this study, are listed in Table VII. Measurements of roundness were made for

the various size fractions of crystolon and AC Coarse. The “effective” values reported

wenv, 1.5 1970) I-JO




in Table VII were obtained by weighing the average values of roundness of each

size fraction with the percent of that fraction occurring in the mixture and summing

the results. The roundness value for glass beads is unity because of the spherical

shape of the beads.

Moduli of toughness

The modulus of toughness of the target materials used in this study are listed

in Table VIII. The moduli were computed using the relationship between modulus

of toughness, yield point stress and ultimate strength given in eqn. (2).

DEVELOPMENT OF A MODEL TO PREDICT EROSION

General

A primary objective of this study was the development of a model to predict

erosion produced by natural contaminants. As noted earlier in this report, BITTER’S

model for predicting erosion13 was adequate if the contaminant was homogeneous.

However, even when one worked with homogeneous contaminants, it was necessary

to perform at least two erosion tests so that BITTER’S curve-fitting parameters EB

and QB could be evaluated. As HEAD et al.ll pointed out, BITTER’S model was of

little value in predicting erosion produced by natural contaminants. Clearly, then,

a predictive model of erosion should be valid for natural soils if it is to represent a

significant improvement over existing models. Perhaps correlation of erosion rate

with transmitted energy for a wide variety of conditions is the most obvious approach

to the development of a predictive model in view of the relationship between the

variables demonstrated earlier in this study. Unfortunately, the determination of

transmitted energy is a laborious, tedious and time-consuming task; for these

reasons, it was decided to abandon the concept of transmitted energy per se and to

attempt to correlate erosion rate with the “effective” maximum initial velocity of

the dust. The effective maximum initial velocity of the dust is defined as the sum

of the maximum initial velocities of the individual size components of the dust

weighed on the basis of the grain-size distribution of the dust. The maximum initial

velocities of the particles were chosen because they were felt to be the velocities

most easily measured or estimated. Other variables known to have an effect on

erosion were target and particle hardness, particle impingement angle, the shape of

the particles, the erosion resistance of the target and the volume of the target affected.

Dimensional analysis of erosion

A dimensional analysis of the variables noted above was performed using the

Pi-theorem33333. The analysis, which may be found in Appendix B, indicated that

the functional form of the relationship between erosion rate and the variables was

as follows.

V2A =-fH B

E

R,oL,-

E’E

where A is the erosion rate (L3M-i), V the “effective” velocity of the mix (LT-i),

R the particle shape descriptor (“effective” roundness) (dimensionless), OL he apparent

particle impingement angle (dimensionless), H the “effective” hardness of the dust

Wear, 15 (1970) 1-46



(ML~rT~“), K the hardness of the target (ML 11-m2) and E the erosion resistance

per unit volume of the target (MI, -IT -‘). “Effective” roundness and hardness are

defined as the sum of the roundness or hardness of the individual size components

weighed on the basis of the grain-size distribution of the dust.Attempts were made to expand eqn. (4) in a multi-variable, finite form of

Maclaurin’s series in order to determine the functional relationship between the

variables. After numerous trials, this approach was abandoned. Statistical modeling

techniques, in which the ratios of the variables indicated in eqn. (4) were used, were

then employed successfully in the development of predictive models of erosion.

Details of the development may be found in a later section of this work.

Erosion resistance of the target

It seemed reasonable to assume that the resistance of a target to erosion at

small impingement angles was proportional to the hardness of the target. At normalimpingement, the ability of the target to absorb energy was thought to be an

appropriate measure of the resistance of the target to erosion. It was assumed that

ability of the target to absorb energy was proportional to the modulus of toughness

of the material; the modulus of toughness is represented by the area under the

stress-strain curve to the point of rupture. The stress-strain curves used in this

study were obtained from un-axial tension tests performed on representative

samples of the metallic targets. The areas under the curves were approximated by

a relationship proposed by MURPHY”~ and noted in eqn. (2). The modulus of toughness

of plate glass was approximated by assuming values for yield point stress and the

strain at rupture and assuming that the stress-strain relationship was a straight

line of slope EG to the point of rupture. The value of Ec: (Young’s modulus for plate

glass) was assumed to be 107 p.s.i.

\‘alues for the resistance of a target to erosion were thus established for low

and high particle impingement angles. In order to obtain values for resistance at

intermediate angles, it was assumed that target resistance was a continuous function

of the particle impingement angle. Two functional forms were assumed for the

variation of resistance with impingement angle; in the first of these, designated

“straight line” variation, resistance was assumed to vary linearly with impingement

angle. The form of the straight line variation was as follows.

where E is the erosion resistance per unit volume of the target, B the hardness of

the target, LX the apparent particle impingement angle, in radians, and MT the

modulus of toughness of the target material.

The second assumed functional form, designated “cosine variation” was

E = [(B - MT) (cos CX)] MT

where the symbols are the same as in eqn. (5).

Both eqns. (5) and (6) were developed on an intuitive basis. Equation (5)

was recommended by its simplicity and eqn. (6) came to mind after the shapes of

the erosion rates VS. apparent impingement angle were studied. For both equations,

the values of target resistance at zero impingement angle (predominately micro-

WPUV, 5 (1970) ‘P40




shear mode of erosion) are the same; namely, the hardness of the target. The values

of resistance at normal impingement (predominately repeated deformation mode of

erosion) are also the same, the modulus of toughness of the target. Both of the

assumed relationships were used in the development of the erosion model. Theresults indicated that both served equally well in describing the resistance of the

targets to erosion.

The erosion mode2

The data used to develop the erosion model included all the data generated

in those erosion tests in which crystolon and glass beads were the erosive agents.

To supplement these data, the results of erosion tests involving AC Coarse and

stainless steel 17-4PH taken from ref. II were also used.

The first step in the development of the model was to perform a correlation

study to determine if all the data could be considered as a single group of data orif variations existed which would require splitting the data into groups. It was

found necessary to consider the data in two distinct groups; the first group consisted

of erosion tests involving plate glass targets. The second group consisted of erosion

tests involving the metallic targets. The first group of data was used to develop a

model of brittle target erosion. The second group of data was used to develop a

model of ductile target erosion. It was thought that the necessity of developing two

erosion models was a consequence of using the modulus of toughness to characterize

the energy-absorbing ability of brittle materials. Perhaps descriptors of the density

and distribution of internal cracks would be a more appropriate measure of the

ability of brittle targets to absorb energy. Nevertheless, the data were analyzed on

the indicated bases and models of erosion were developed which appear useful.

The step-wise regression program

Two computerized statistical programs were used to develop the erosion

models. The first of these employed a multiple, linear, step-wise regression technique

and was used to develop an assumed power relationship between the erosion variables

grouped as indicated by the results of the dimensional analysis. The power relation-

ship assumed between the variables was

(7)

In logarithmic form

logA=logC1+61log ; +&logKfSsloga+8410g(~)+r)510g(~)( 1

(8)

where As is the erosion rate (step-wise model), V, E, R, e, H and B are as defined

in eqn. (4) and Ci, Si, 82, 83, 84, 65 are constants to be determined.

The step-wise regression program employed a model “build-up” procedure.

The program would first choose the most statistically significant independent variable

and enter that variable in the regression analysis. The second most significant

variable was then entered and so on until either all variables had been entered into

the regression analysis or the remaining variablesfailed to qualify as being statistically

significant.

Wear, r5 (1970) 1-46



11‘. J. HI:.\l,. \I. tr. H.\ ltfi

+---Q

-,-

_.-

-.-



EROSION BP WATURAL CONTAMINANTS 33

The program could be instructed to perform the regression analysis with the

constant Ci set equal to unity. The best erosion models were developed when this

option was used. However, the square of the multiple correlation coefficient, Pz,

was deceptively large when Ci was specified as unity because the confidence levelbands were centered about the origin rather than about the mean.

The step-wise model for brittle targets is as follows.

1.53~2.69 (J fpq2*08

ddBS = (v2’E’ cB,Ef3 6~

i .

Rearranging terms

(9)

AJ 73.06 &2.69 If2.08 Ea.03

Bg=-----fp.64 (10)

where Aus is the erosion rate for brittle materials (step-wise model). The square of

the multiple correlation coefficient, P 2, for this model was 0.985 which indicates

that the model fits the data very well; however, an inspection of Fig. 20, in which

a plot of fitted and observed erosion rates is shown, indicates clearly that the model

is a rather poor predictor of erosion rate for brittle materials. Neither is the model

complete in the physical sense. The effects of particle shape, the R factor, are not

incorporated in the model because R was not statistically significant at the 1%

significance level used arbitrarily for including variables in the regression analysis.

In addition, the model indicates that the erosion rate increases with increasing target

resistance. This is contrary to the anticipated dependency of erosion rate on target

resistance and may indicate that target resistance was ill-defined and/or that the

model was too simple to account for brittle target behavior. Both the straight-line

variation and the cosine variation of target resistance with impingement angle were

used in the regression analysis with very similar results. Equations (9) and (IO)

incorporated the straight-line variation.

The step-wise model for brittle targets is

A

(&‘2/~)2.17 &0.46 (~~E)O.lO

DR = ---.-gsqqpr---

Rearranging

(11)

Av4.34 &0.46 jfO.10 E0.21

DS =K2.34 p.43 (1%)

where ADS is the erosion rate for ductile materials (step-wise model). P2 for this

model was 0496. A plot of fitted and observed erosion rates is presented in Fig. zr.

The model fits the experimental data reasonably well. In addition, the model is

pleasing from a physical standpoint in that the particle velocity, impingement angle,

target and particle hardness, and particle shape terms are all properly located in

the model. For example, if all other variables are held constant, increasing the

effective velocity of the dust should increase the erosion rate; increasing the effective

hardness of the dust should have a similar effect. Increasing the effective roundness

of the dust should decrease the erosion rate as should an increase in target hardness.

The impingement angle term must be in the numerator of the proposed model

to prevent the predicted erosion rate from increasing without bound as the im-

Wear, rg (1970) 1-46



pingement angle approaches zero. The only undesirable aspect of the model is tliat

the predicted erosion rate increases with increasing target resistance. Again, this

result probably indicates a na’ive model and/or that the purported measure of target

resistance was not a good indicator of response of the target to erosion. The straight-line variation of target resistance with impingement angle was incorporated in

eqns. (II) and (12).

j I0.01

0.01 0.02 0.03 0.05 0.10 0.20 0.30 0.50 1.0 20 3.0 5.0 10.0

OBSERVED EROSlON RATE, A0 I lO%UBlC CENTIMETERS PER GRAM )

Fig. 2 I. Fitted us observed erosion rates step-wise model, ductile targets.

To summarize, the step-wise regression analysis of the data resulted in two

predictive models of erosion. The model for brittle materials neither fits the data

well nor was it totally sensible from a physical standpoint. The model for ductile

materials did fit the data reasonably well and, in addition, was physically sensible

excepting the anomaly involving the target resistance term. It should be noted that

the model for brittle targets was developed from only twelve sets of data. The

sample size was undoubtedly too small. The model for ductile targets incorporated

thirty-seven sets of data. It was noted that the method of determining the resistance

of the target erosion was perhaps fallacious. Other investigators16 have proposed

measures of target resistance to normal impingement similar to that incorporated

in this work; the proposed relationship between resistance to low-angle impingement

and target hardness seems reasonable. In addition, it seems reasonable to expect

target resistance to erosion to be a continuous function of particle impingement

Wear, rg (1970) I-lfl




angle. It was strongly suspected that the difficulty in successfuly modeling with

the step-wise procedure was the consequence of an inadequate initial model. To

verify this suspicion it was decided to develop erosion models which would initially

incorporate, not only the erosion variables, but also a large number of terms re-presenting interactions among the variables. The procedure chosen to accomplish

this task was a computerized statistical program called WRAP.

The ~e~g~~~~ egress~~~ nal ysis progral n (WRAP)

WRAP, like the step-wise program, is a multiple, linear, regression analysis

program; however, WRAP incorporates a model “tear down” procedure in which

terms of an initial model are deleted from the regression analysis on the basis of a

specified probability level. The probability level was arbitrarily chosen at 95%.

The initial model proposed for both ductile and brittle targets was

Aw = So + & XI + . & X0 + 67X12 + . + d,zXs2 + &3X1X2 + . . . + 817X1X6

+&&2X3+...+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+(S26~4~6+~2?~5~0+~28~12~2+...+~32~12~6+833~1~22$....

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~...+654X3X'62+~55X4X52+~56~4~62f8.57~5~63+~56~213

+s~9x23+860~~14+861x24 (13)

where AW is the erosion rate (WRAP model), XI = W/E, X2= R, X3 =sinn, X4=

In (H/E), X&=ln (B/E) and Xg=cos K

The resulting model for the brittle targets was

VZA~~=O.OOj~~j+O.OOOOO~ - +0.0OyjO7 R+oOo9335sinx

( I

- 0.000630 InH

( J- - 0.004706 cos a-o.oogr14 sin%E

112-0.0x0888 R sin n-0.0038ro K coscx+0.000495 In 7;

f ( 1);cosa (14)

where ABW is the erosion rate for brittle materials (WRAP model).

A plot of fitted and predicted erosion rates is presented in Fig. 22. The P2

value for eqn. (14) was 0.998. In the WRAP procedure, the confidence bands are

centered about the mean; hence, the indicated value of P2 is not misleading as in

the case of the step-wise program. A P2 value of 0.998 indicates an excellent fit of

the data.

The model for ductile materials was

Anw= O.C00233-o~oooI~o~--0.000238 1n

i-o.001577 G: ’ +0.00829 sin 01+0.000034 In $( > f i

)II? fo.Ooorrg R In

2

cos Ly (15)

Wear, 5 (1970) -4d



E

C

R

P

C

B

WR

M

F

D

L

T

A

lo

c

C

C

M

P

G

E

O

R

P

C

WR

M

F

rC

C

M

P

F

M

f3

e



EROSION BP NATUIML CO~T.~~I~~~TS 37

To summarize, the WRAP procedure provides models of erosion which fit

the data very well; this tends to confirm the suspicion concerning the inadequacy

of the simple power models. However, the power models are simpler to assess physic-

ally than the WRAP models. This is especially evident if one considers the step-wisemodel of ductile target erosion; the model provides a good insight into the relative

importance of most of the variables which have been shown to effect erosion. The

proper method of characterizing target resistance to erosion has not been firmly

established; perhaps additional data would clarify this question and establish the

validity of the method to measure target resistance used in this work.

Assessment of the jmdictive ability of the erosion models

The final step in the development of predictive models of erosion was to

assess the ability of the models to predict erosion. Of special interest was the ability

of the models to predict erosion produced by natural soils. Previously publisheddata11 was used, in which three natural soils and silica flour eroded stainless steel

targets. The natural soils were designated Longstreet, Allen and Eglin. The Longstreet

and Allen soils were obtained from Fort Rucker, Alabama; the Eglin sample was

from Eglin Air Force Base in Florida. The grain-size distribution curves for these

soils, a petrographic analysis of each sample, and a description of the target materials

may be found in an earlier paper”.

Unfortunately, measurements of the effective roundness parameter were

available only for silica flour. Some objective method of estimating roundness of

each of the natural soils was needed and it was decided to use the model for ductile

TABLE IX

ESTIMATED VALUES OF ROUNDNESS PARAMETER FOR SELECTED NATURAL SOXLS

Allen

Eglin

15 O-53930 0.51x45 0.524

60 0.536

90 0.487

&WXage 0.519

I5 0.4’330 0.403

:: 0.4040.413

90 0.397

Average 0.406



targets developed with the aid of the step-wise regression technique for tliis. Accortl-

ingly, eqn. (II) was solved for h’ giving the following expression.

Equation (16) was solved for each reported impingement angle; the results are

listed in Table IX. The average value of ZZ was considered representative of the

indicated soil.

Values of effective velocity and effective particle hardness were computed

for each soil and for silica flour. These values, along with appropriate values of

impingement angle, effective roundness, and target hardness were then substituted

into the WRAP model for ductile targets, eqn. (15). Predicted and observed values

of erosion are listed in Table X. A plot of predicted and observed erosion rates as a

function of apparent particle impingement angle is presented in Fig. 24 for theLongstreet and Allen samples and Fig. 25 for the Eglin and silica-flour samples.

A comparison of predicted and observed erosion rates reveals that, for the

natural soils, the model for ductile materials predicts erosion reasonably well for

impingement angles ranging from 15” to 75”. The model predicts erosion rates which

Egl in

Silica flour ‘5 ‘0.3 119

30 IO8 I IO

45 ‘07 9900 98 X8

75 XIYO 5’ 76

Wear, rg (1970) I--$’




are too high by factors ranging from about 1.3 to 2.0. The predicted values are

therefore on the safe side for the natural soils studied but not grossly so. The model

ranks the soils properly; that is, the predictions indicate that sample

least erosive soils and, in little difference inabilities and Eglin to erode target material. Another

feature is that, the natural soils, maximum erosion

an apparent impingement angle of 45”. This prediction is verified by

I20

pso

Eiz N (OBSERVED1

8

Y40

9

y;c

zE!

I::: IS’ 30’ 4!? 66 75

0OD 0.40 0.60 120 1.60

APPARENT PAfiTICLE IMPINGEMENT ANGLE (RADIANS1 APPARENT PARTlClE IMPINGEMENT ANGLE I RADIANS)

0.0 0.40 0.60 120 1.60

Fig. 24. Predicted and observed erosion rates for Allen and Longstreet samples.

Fig. 25. Predicted and observed erosion rates for Eglin and silica flour samples.

the observations. The model does not accurately predict erosion by natural soilstested for the normal (90”) impingement case. No explanation is evident for this

deficiency. For ductile target materials, the inadequacy of the model is not of great

consequence because the model can be used to predict erosion rates for impingement

angles to approximately 75”. The difference between the erosion rate experienced by

a ductile target at 75” and the erosion rate experienced at normal impingement is

small (see Figs. 8-10). The model adequately predicted the silica-flour erosion rates.

The maximum discrepancy between predicted and observed erosion rates for silica

flour occurred for normal impingement where the observed value was larger than

the predicted value by a factor of approximately 1.5. The model predicted that

the maximum erosion rate would occur at an impingement angle of approximately30”; the maximum observed erosion rate occurred at a 15” mpingement angle.

Despite the shortcomings and deficiencies noted above, it is felt that the

WRAP model for ductile targets represents a major advancement in the dust-tech-

r,Vear, g (rgyo) I-46



40 \ \ ‘. J . HiS.\ I), \ I. Ii. H.\ I<li

nologv field. Machine designers and filtration technologists can now assess tllc

effects of changes in air-dust systems on the erosion experienced byturbine macliinery,

reciprocating engines, and the like. The assessments can be made with some c‘cjn-

fidenre on the basis of a minimum number of laboratory tests. The WRAP modelrepresents a significant improvement (approximately one order of magnitude) over

the only other model which has been used to predict erosion by natural soils”. In

addition, the WRAP model obviates the need for repetitive erosion testing.

Data were not available to permit an evaluation to he made of the predictive

models for brittle targets nor were independent data available for assessing the

ability of the step-wise model to predict the erosion of targets by natural soils.

However, if the estimated values of the roundness parameter listed in Table IS

were substituted into eqn. (x3), the results listed in Table XI were obtained. It

should be emphasized that the good agreement between predicted and observed

erosion rates is a manifestation of necessity; the predictive model was used pre-viouslv in the determination of the roundness parameter.

Eglin

To summarize, four predictive erosion models have been developed, two

models for ductile target materials and two for brittle materials. For the erosive

agents tested, the WRAP model for ductile targets overestimates erosion rates by a

maximum factor of about 2.0; the WRAP model did not accurately predict erosion

at normal (go”) impingement for the natural soils tested. This inadequacy does not

seriously impair the usefulness of the model because the erosion rate at 75” impinge-

Weav, 15 (1970) I 40



EROSION BYN;ilTURAL CONTAMINANTS 4r

ment (for which the model predicts reasonably well) is quite similar in magnitude

to the erosion rate experienced at 90~ impingement. The step-wise model for ductile

targets has not been thoroughly tested. It appears, however, that the model may

prove to be an excellent predictive tool despite the fact that the model predicts anincrease in erosion rate if the erosion resistance of the target is increased. The predictive

ability of the models for brittle materials remains undetermined at this time.

CONCLUSIONS

For the ranges of variables incorporated in this work, the following conclusions

appear warranted.

(I) Provided the mechanism by which erosion is accomplished is held constant,

steady-state erosion is directly proportional to the energy transmitted from impinging

particles to the target.(2) A certain quantity of energy must be transmitted to the target material

before erosion commences; i.e. a threshold energy level must be exceeded before the

target will experience erosion.

(3) The major factors which influence the energy transmitted to a target are

the velocity, shape and hardness of the particles and the hardness and the intrinsic

ability of the target to absorb energy.

(4) Developed statistical models for ductile targets employing the factors

listed above in conclusion (3) can be used as predictors of erosion produced by natural

soils or contaminants.

SUGGESTIONS FOR FUTURE RESEARCH

The following are recommendations for further research.

(I) Establish a more positive method of measuring the particle restitution

ratio. An optical device, perhaps employing a system of mirrors, might be used to

insure that actual particle velocities are measured rather than the components of

the velocities.

(2) Determine a method for accurately measuring the erosion resistance of

materials. Relative to the erosive process, the modulus of toughness parameter (see

eqn. (2)) is probably too gross a measure of the ability of a target to absorb energy.Some measure of the ability of the surface of the target to absorb energy, along with

an assessment of the fatigue properties of the surface, is indicated. For brittle targets

eroded at normal impingement, descriptors of the density and distribution of surface

flaws might serve as measures of the energy-absorbing capacity of the target.

(3) A sufficient number of erosion tests should be performed to assess the

predictive ability of the step-wise model for ductile targets and the models for brittle

targets. In addition, the maximum particle velocity for which the models are validshould be determined.

ACKNOWLEDGEMENTS

The authors wish to express their sincere appreciation to W. H. PERLOFF,

Associate Professor of soil mechanics at Purdue University, for his valuable advice.

Weav, 5 (1970) -46



42 iv. _). HE.\I), \I. E. blAlui

The authors arc verv grateful to Allison I)ivision of (kneml Motors ~VIIOS~~

financial support made this‘project possible.

Anno., The military significance of the dust problem, PE.w. I)usf Tt~huo/ .~~wimcr. 1qh6,General Motors, Paper 4.

G. B. CLARK, J. CV. HROWN, c‘.J. HAAS AKD 1). .\. SUMMERS, Rock PW~PY~GS f~&&df~j h’apfii

Escflvu ti mz, Rock M ~clmzi rs und E splos!ves fi ~~w a~ch Cenfm , l’niversity of Missouri l’rcss,1z011u, Rio., Ic)O<).1 cfensc Documentation Center, Scicrzt~fic nprif T~,cizra!cc~l .4~pliciltIori.~ i:0vl~cus~- -I<)&$“.

lixcnvafiora,Dept. of the Army, r9o+

1;. ROBIN. The wear of steel with abIasivcs, Garwgir Sciaolurshi~ :lf?f?z. ZJW~ZSlt~:l I~sf,, 2

(191o) 0.

J. .i. BUNNELL, :\n investigation of the resistance of iron and steel and of some other materials

to war, J er nk onfcrets Am ., 76 (rgzr ) 347.

S. J. ROSENBERG, ICesistance of steels to abrasion by sand, Tvaw. .4 .S.S.T , 18 (1930) Iog3.

I(. 1). HAWORTH, The abrasion resistance of metals, Tvar zs. Am . SK. M et& , $1 (1gJ g) 81g-85,+.I< . iL . STOKER, lcrosion due to dust particles in a gas stream, Ind. Enfi. Ckrnz., 41 (1g,+9)

1 9(1~ .1 99.

I . F‘INNIE, Erosion of engineering materials: Mechanism of material removal, TechIa. &:pt.

-\b. 202.56, Shell Development Company, Emeryvillc, Calif., November, I956

f. G. .\. BITTER, .\ study of erosion phenomena, ?Z’ectr, 6 (I903) 5- 2I and IO9rqo.

\V. J. HEAD, T. PACALK APZD J, POOLE, Final report on Phase 1 of .-WisoII~Purdue dust tcch-

nology program, Paper presented at the ~~~,,r- :~lobiEi ty-SPrai ceubi l l fy Fovrtun . _4$vi l I I -12.

XC& ?, General Motors.(;. 1,.SHELDON, Erosion of brittle materials, 11. E~ag. Thesis, l’niversity of California, Berkeley,

19Oj.

G. L. SHELDON AND 1. FICYIE, On the ductile behavior of nominally brittle materials during

erosive cutting, Paper presented to Wider Meeting of d4.‘5ME, ovembev, 1965.

G. I,. SHELDOK AND I. FINNIE, The mechanism of material removal in the erosive cutting of

brittle materials, Paper presented at the Winlev Meeting of .4SME, Aravrnzbrv, 1~6.5.C’. 11. \Voon, EIosion of metals by the high-speed impact of dust particles, Prnr . Inst . I : ‘m~i m~%.

.%I., (I9M)) 55 03.

A. ?‘HIKUVEAIGADA~I, The concept of erosion strength. In EvosioIz hv Cavitat ion OI wzpi~zgrwz~~rl ,

XSTN Spzc. Tech. Pub. No. 408, 19f6, pp. 22 jr,.

.I. THIRUVEXGADAM, Now there’s a way to work out erosion strength of materials, Prod. E:ltg.,

31 (17) (I96o) 5.j -5’,.I, ~TINNIE, J. WOLAK ,AKD \i. I\;ARIL, l<rosion of metals by solid particles, ,I. .?fofev. 2 (T907)

08r-700.

I. PIXNIE AND 1.1. OH, .in analysis of rock drilling by erosion, I-‘rot. 1st Iztrw c‘orrg’r. ock

Muck. , 1967, 999104.

J. H. NEILSON AND .\. GILCHRIST, Erosion by a stream of solid particles, ll~~vzr, I I (1908)

III-121.

J, 1-I NEILSON ANU 4. (;ILCWRIST, An experimental investigation into aspects of erosion inrocket motor tail nozzles, Wear, II (1968) rz3--14.3.

Il. WADELL, Volume, shape and roundness of rock particles, ,I, Geol., jo (1032) 443-.l,jI,

1-I WADELL, Sphericity and roundness of rock particles, J. Geol., 4r (1933) 310 ~3.31.

G. %lURPHY, Pr ope7+i es of Er tgi ~zesvin g M ateazls, 1st. edn., International Textbook Co.,

Scranton, 1939.

c’. I ,. MANTEL. Ed., E~~~gim evin g Water i a ls H andbook, 1st. edn., McGrawHill, New York, I9.G.

II. TAWOR, The physical meaning of indentation and scratch hardness, &‘,I. .f. .‘I@l. P&Y,?.,

7 (19jO) 159105.

H igh Temperatur e M ater i a ls, The Norton Co., Worcester, 1904.

Physical and chemical propertics of glass beads, :2finiz. : I l& ir zg and Mar auf . Co., Tech. Ih l a

Shwf , I9OI>,

Ii. ITINcH, personal communication, September, 1908.

I:. li. DARKER, ;2lateviaZs ata Shet, 1st. edn., AlcGran-H i l l , New York, 1907.

1). TAuOR, T!M H ard ness of etals, Clarendon Press, Oxford, lgjl.

II. BUCKIXGHAM, On physically similar systems: Illustrations of the we of dimensional

equations, Pkys. Rev., 4 (1941) 345 -376.li. BUCKINGHAM, Model experiments and the forms of empirical equations, Trans. A.s.wL’,



EROSION BY NATUHAL CONTAMINANTS 43

APPENDIX A

DEVELOPMENT OF THE RELATIONSHIP BETWEEN THE MAXIMUM PARTICLE VELOCITY,

V~max, AND THE “EQUIVALENT” PARTICLE VELOCITY, T/Equiv

It is assumed that the distribution of both normalized particle mass flow

and normalized particle velocity were the same as the normalized distribution of

air-stream velocities shown in Fig. 13. Two views of the assumed distribution are

shown in Fig. 26.

“.“...~ll.*I..

SIDE VIEW END VIEW

Fig. 2G. Assumed distributions of normalized particle flow and velocities.

From Fig. 26, the elemental area, A, is given by the following expression.

Area A =$(<b@

and the area of the ith annuius is then

Now the mass 3f particles flowing in the ith annulus per unit of time i is

(MPd)(Area of ith annulus) =m$?rM~

m=(:Pi:i

The total mass of particles flowing per unit time is

The velocity of the particles in the ith annulus is

v&V=@iVRnax

From eqn. (A-I)

(A-2)

(A-3)

W&W* 5 (‘970) I-46



Putting

‘j

KE =

Assuming that a fictitious velocity, 17nc(uiv, exists such that KE=+MTI/i+iuIv’ and

substituting into eqn. (A-4),

I-&Juiv.l.= ?jk-pmax”

Having the /&‘s and the values of (r/r*) from Fig. 13, the value of q was determined

numerically. For the work reported here, rl~o.679. Therefore,

~Equiv2N0.679 T’rmax’ (A-5)

Equation (A-5) is of considerable importance. It implies that the kinetic

energy of an aggregation of particles, whose velocities and mass flow vary in a

prescribed fashion, can be computed by multiplying the square of the maximum

particle velocity by an appropriate constant and then simply multiplying the result

by half the total mass of particles flowing per unit time.

APPENnIx n

The Pi-theorem elucidated by Bnc~r~cn~r a%:33 formed the basis of the

following analysis. The procedure was that suggested by STKEETEK~*.

The variables which effect the erosion process are listed in Table XII.

By inspection, zi=R and x2=01. Six variables remain with three units M,

L, and T; three additional n terms are required for a complete, dimensionless system.

Taking I/, E and C as convenient repeating variables,

7Ca=l/Xl $1 (‘5 4




TABLE XII

VARIABLES AFFECTING THE EROSION PROCESS

Var iab le

Erosion rateEffective velocity of the dust

Particle shape

Apparent particle impingement angle

Effective hardness of the dust

Hardness of the targetErosion resistance per unit volume

of the target

Volume of target effected

Fuvzdamental

U &S *

L%-1

LT-’

ML-IT-2

MI,-IT-2

ML-IT-2

L3

* >f = mass, L = length and T = time

Substituting the appropriate fundamental units

n3=(LT-+ (ML-1T-2)Yr (L3)“” L3M-1 (B-1)

zq = J LT-‘)x2 (~~L-lT-2)y2 (L3)‘” ML-‘T-2 (B-2)

iz 4LT-rT3 (ML-1T-2jY3 (L3)‘3 ML-IT-2 (B-3)

Expanding the equations and imposing dimensional homogeneity. From eqn. (B-I)

L-terms: XI-Y1+321+3=0

M-terms : Yt - I = o

T-terms : -X1 - ZYI = o

Solving these three equations simultaneously, i”r3 = $

From eqn. (B-2)

L-terms : X2 - Y2 + 322 - I = 0

M-terms : YZ + I = o

T-terms : -x2-2Y2-2=0

Solving these three equations simultaneously, ~‘4 = $

From eqn. (B-3)

L-terms: X3-Y3+3.&-I==0

M-terms : li8 + I = o

T-terms : -x3-2Y3-2=0

Solving these three equations simu~tan~usly, ~5 = g

Summarizing,F(nl, zz, na, x4, 7~5) =o

P-4 1

Wear, 15 (1970) r-46



Kc-writing eqn (13-q) in a more con\-enient form

/I I<c;; = G (I<, x, p;m) ;;I

therefore l/2A =pG

H BK,(\ ,- -

E E ’ I : ‘

APPENDIX C

SAMPLE CALCULATIONS

Sample calculations will be carried out to determine the “effective” impact

velocity and “effective” roundness of the crystolon erosive agent.

From the grain-size distribution curve for crystolon, measurements of the

roundness of each component and from the relationship between particle size and

impact velocity for an indicated flow rate of 760 ft.s/h shown in Fig. 14, we have

the values appearing in Table XIII.

To find the weighted roundness and weighted velocity, we multiply the

percent of the size category in the mix by the roundness and velocity of that size.

To compute the effective roundness and effective velocity, the individual weighted

values are summed as shown in Table XIV.

TXBLE XIII

ROUNDNESS AND IMPACT VELOCITYDATAFORCRYSTOLON

Part ic le

size

(p )

Amount

i n m i x t u r e

(%i

Roundness

ofindicated

size

Impact

velocity of

par t ic le

(ft . lsecl

105 I 0.423 420

88 3 0.47’ 436

74 4 0.483 455

63 4 0.352 470

44 12 0.438 5’0

31 15 0.500 550

16 61 0.508 630

TABLE XIV

U'EIGHTED Roui-mNEss AND VELOCITY DATA FOR CRYSTOL~N

Pa&k size

(PI

Weighted Weighted velocity

voun dn ess (ft. I sec)

105 0.00423 4

88 0.014’3 13

74 0.01932 18

63 0.01408 19

44 0.05256 61

3’ 0.07500 83

The Development of a Model to Predict the Erosion Of

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