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Wear, 102 (1985) 309 - 330 309
THE ~FLUENCE OF TURBULENCE ON EROSION BY A
PARTICLE-LIEN FLUID JET
SUDIP DOSANJ H and J OSEPH A. C. HUMPHREY
~ateriu~s and Mo~ec~~r Research Division, Lawrence Berkeley Laboratory, University of
California, Berkeley, CA 94720 (U.S.A.)
(Received J uly 2,1984; accepted Febr ua ry ‘7, 1985)
Summary
Very few investigations on wear by particle impact have accounted forthe influence of turbulence on erosion. In this study, the effects of turbulentdiffusion on particle dispersion, and hence on erosion, are demonstratednumerically. Variations imposed on the level of turbulence intensity in aparticle-laden jet impinging normally on a flat wall show a strong depen-dence of wear parameters on this quantity. Impacting particle velocities,trajectories and surface densities are predicted from lagrangian equationsof motion. Eulerian equations are used to describe fluid motion, with theturbulence viscosity evaluated from a k-e model of turbulence. Nonessential
complexities are avoided by assuming one-way coupling between phases andusing an adjusted Stokes’ drag relation. Numerics calculations reveal thatthe relative rate of erosion decreases and that the location of maximum wearis displaced toward the stagnation point as the turbulence intensity increases,Efforts to remove present model limitations have been hindered by theunav~lability of suitable experiment data,
1.1. Theprablem of interestWhile it has been known for some time that mean flow conditions can
si~ifi~~tly affect erosive wear by particle imp~gement, the rule played byturbulent fluctuations, especially near walls, is neither appreciated norunderstood. As a result there is a significant risk when interpreting erosivewear results of incorrectly accounting for behavioral aspects of the wearprocess that are associated with the turbulent nature of the flow. In an earlystudy by Finnie 111, it was suggested that surface erosion by particle impactshould increase with increased turbulence levels in some flows. However, todate the phenomenon has not been investigated systematically and in depth.As a result there does not exist a satisfactory database for guiding fundamen-tal theoretical interpretation and the development of predictive modelswhich account for the influence of turbulence on erosive wear.
@ Elsevier Sequoial~inted in Switzerta nd
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In free shear flows it has been established that turbulent fluctuations
affect particle dispersion. Some of the experimental data available for the
particle-laden jet configuration have been reviewed and extended by Faeth
[2 J and Shuen et al. [3] and discussed by Melville and Bray [4] and Crowderet caE. [ 5 J. Following Pourahmadi and Humphrey [ 6 J, the latter workers
modeled particle dispersion in a turbulent mixing layer flow using eulerian
forms of the transport equations and allowing two-way coupling between
the continuous fluid and the solid phases.
As illustrated by, for example, Melville and Bray [73 and Po~~madi
and Humphrey [ 61, the in~ract~g continua approach facilitates the formu-
lation of two-phase flow turbulence modeling concepts, However, it does not
yield unambiguous specifications of impact velocity and impact angle on
collision of a particle with a surface. A knowledge of these two parameters is
essential for any useful model of erosive wear. For this reason, it was decided
to investigate the influence of turbulence on particle motion, and hence on
erosion, using a lagrangian description for the motion of the solid phase. To
simplify matters initially, particle motion was attributed entirely to the
mean flow (drag) differences between phases. As a result, turbulence-
enhanced diffusion of the particle phase can only occur as a consequence of
turbulence-enhanced diffusion in the fluid phase. It will be shown that in
spite of this weak one-way coupling, the approach demonstrates clearly the
importance of accounting for the influence of turbulence on erosive wear.
The flow chosen for analysis was a p~t~&le-lades jet impinging at rightangles to a flat solid surface also referred to as the “wall”. This configuration
was chosen because of its considerable practical importance in erosion test-
ing and its relative simplicity. The use of particle-laden jets for accelerated
erosion testing is discussed below,
Accelerated erosion testing of materials using particle-laden fluid jets
has been used extensively to characterize surface wear phenomena. A discus-
sion of the jet technique in relation to other types of testing has been given
by TiIIy [S]. Early examples of the use of jets for quantitative experimenta-
tion are given by, for example, Finnie [9] and Finnie et al. [lo]. In these
and later studies (see refs. 11 - 13) the importance was emphasized of
knowing the particle velocity and angle of attack with respect to the surface
at the instant of impact in order to correlate and model experimental obser-
vations. In contrast, the influence of turbulence on jet-induced erosion
remains unknown and, therefore, somewhat unpredic~ble.
The influence of mean flow conditions on erosion by particle impact
has been reported by Tilly [S], and although the importance of turbulence
in relation to wear has been suspected, it remains virtually unexplored. In a
discussion on fluid flow conditions, Finnie [ l] notes that turbulent fluctua-
tions near walls may account for the increased rates of erosion observed in
some flows. While turbulent fluid-particle interactions have been extensively
studied in free flows (see ref. 14 for an enlightening discussion) correspond-
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ing detailed investigations in the presence of surfaces undergoing erosion are
practically nonexistent.
Other parameters affecting erosion such as particle rotation at impinge-
ment, particle size, shape, physical properties and concentration, surfaceshape and mechanical properties, nature of the carrier gas and its tempera-
ture, and surface temperature increase due to impact have been reviewed by,
among others, Finnie [15], Mills and Mason [16], Finnie et al. [17] andTilly [ 81. A comprehensive assessment of the state of knowledge pertaining
to solid particle erosion has been communicated by Adler [18]. Adler
performed a critical relative comparison of current erosion models forductile and brittle materials.
1.3. The present contributionThe main purpose of this study is to demonstrate, within the context of
a simple model, the extent to which variations in turbulent flow conditions
can influence erosion. Because the impinging jet configuration (Fig. 1) is so
popular among accelerated erosion testers it has been chosen for illustration.The study is numerical in nature and shows that mathematical methods andphysical models are presently available for predicting some of the main ef-fects of turbulence on erosion. Related studies include the work by Laitone[19, 201 and Benchaita et al. [13]. Both studies were concerned withpredicting the motion of particles near a stagnation point. However, as had
many others before them, they assumed potential flow for the fluid phaseand the effects of turbulence on erosion could not possibly be discerned.
The present research approach builds on and extends the earlier numeri-cal investigations by Pourahmadi and Humphrey [6] and Crowder et al. [5].However, the flows considered in this study are sufficiently dilute that the
consmt0 fll U,=U,=k=c=O
Fig. 1. Particle-laden impinging jet flow configuration with relative dimensions andboundary conditions indicated.
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equations of motion governing the fluid flow become uncoupled from those
equations governing the particle motion. This is because if the particle
volume fraction is small (for example, of order 10W3), the force exerted by
the particles on the fluid phase is negligible. Time-averaged, steady state,transport equations are solved numerically for the fluid phase. A two-
equation (k-e) model of turbulence is used to represent the turbulent
diffusion of fluid momentum.
After solving for the fluid flow field, lagrangian equations of motion
are solved for the particle motion and trajectories. Thus, the velocity at
which the particles strike the wall and the angle at which they strike can be
predicted. On collision with the wall a particle essentially disappears from
the flow field. Although this assumption is incorrect, for the dilute systems
studied here it is not indispensable to know the rebounding characteristics
of the particle in order to demonstrate how turbulence can affect erosion.
Prior to conducting the two-phase flow calculations referred to, extensive
testing of the calculation procedure was performed. The tests showed very
good agreement with the measurements of Araujo et al. [ 211 for a single-
phase turbulent jet flow impinging normal to a flat surface.
2. Transport equations and boundary conditions
The first step of the solution procedure is to solve for the fluid phase
flow field. A two-equation (K-e) turbulence model is used. A detailed
discussion of the transport equations and of the solution algorithm for this
model can be found in many papers (for example, ref. 22 and, more recently,
in ref. 6). A brief discussion follows.
2.1. Fluid phase
2.1 .l Transport equations
Decomposing the velocity and the pressure into mean and fluctuating
quantities and then time averaging the Navier-Stokes equations yields a setof equations for the mean quantities of interest. However, this procedure
introduces additional unknown quantities, the Reynolds stresses such as
U&X3 into the equations. The Reynolds stresses are modeled by an extension
of the Boussinesq assumption. This relates the turbulent stress to the rate of
strain through a turbulent viscosity pt. The turbulent viscosity can be written
in terms of the turbulent kinetic energy k and its rate E of dissipation. Speci-
fying transport equations for k and E closes the set of equations describing
the flow of fluid.For steady, incompressible, axisymmetric flow (assuming constant
properties) the conservation of mass is given by
av,.+au,+u,=o
& ax r(1)
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and the conservaticm of momentum by
au, av;
P G$- -I-U”ax
P
The Reynolds stresses are given by
au,-pz = 2/.iEr, x
The turbulent viscosity is found from the relation
h2Pt - C&lPT
The kinetic energy k of turbulence is
and the energy di~ipation E is
(6)
(7)
In eqns. (6) and (7), the generation G of turbulent kinetic energy is given by
For a round jet the constants are given by Pope 1231 as C, = 0.09, ok = 1,
u, = 1.3, C1 = 1.6 and Cz = 1.9.
2.1.2. Boundary conditions
Numerical solution of the system of elliptic equations describing theflow field requires the specification of the velocity components (U,, U, ), the
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turbulent kinetic energy k and its rate E of dissipation on a closed boundary
(see Fig. 1). The following conditions are used.
(a) Values for all the variables are prescribed at the jet inlet plane. For
r <d/2, U, is the measured value, U,. = 0, k = 0.01Ux2 and E = k3’2/0.2d. Forr > d/2, U, is the measured value (approximately 0), U, = 0, k = 0 and c = 0.
(b) On the axis of symmetry, a(any variable)/& = 0, except for U?
which is equal to zero.
(c) On the wall, V, = 0. The shear stress on the wall was prescribed
from the log~ithmic law of the wall velocity distribution. The values of k
and E were set at the grid points nearest the wall. If it is assumed that the
flow in the wall region is in local equilibrium, the transport equation for k
gives this quantity as a function of the wall shear stress and known quantities
in the flow (see ref. 22). The value of e was set by assuming that near the
wall the length scale for turbulent motion varies linearly with the distance
from the wall.
(d) Araujo et al. [Zl] indicate that the flow in the wall jet region is
approximately parallel to the wall. Hence, on the exit-flow plane, U, = 0,
~~~U~)/~r= 0, ~k ~a~ = 0 and ae,Qr = 0.
2.1.3. Finit e difference equations and num erical solut ion
The equations are discretized by control, or cell, volume integration
over a staggered interconnected grid. The general form of the resulting finite
difference equation for an arbitrary field variable @at a node p in the calcu-lation domain is
where EA& represents both the diffusion and the convection from the
nodes adjacent to the node p, S, and S, represent the sources and sinks of
the quantity 6. The system of difference equations generated by writing
eqn. (8) for the variable 4 at each node p in the calculation domain is readily
solved using the Thomas algorithm.
In the approximation leading to eqn. (8), a hybrid scheme is used todifference the convective terms. In this scheme, when diffusion dominates
convection, central differencing is used for the convective terms. However,
in regions of the flow where convection dominates diffusion, backwards
differencing is used for the convective terms. Because this procedure sacri-
fices accuracy for stability, it is necessary to ensure that a sufficiently
refined grid is used to reduce truncation error to an acceptable level. The
diffusion terms in the equations are always discretized using central dif-
ferencing.
The numerical solution procedure is based on the twodimensional
TEACH code (see, for example, ref. 24). In this procedure, the pressure field
requires special attention. Linearized expressions for the ve1ocit.y com-
ponents, in terms of pressure differences, are substituted into the continuity
equation. The resulting difference equation is solved for pressure using the
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SIMPLE scheme described by Patankar 125 1. The numerical solution se-quence is as follows. From an initial guess of the flow field (pressure andvelocity) the modified ~ont~uity equation is solved for a better estimate
of the pressure field. Using the updated pressure field, the momentumequations are solved for the velocity components. This is followed by thesolution of k and e from their respective equations in order to obtain abetter estimate of the turbulent viscosity pt. The solution sequence isrepeated until a pre-established convergence criterion is met, this being thatthe sum of the normalized residuals for each variable be less than 10V3.Space considerations have dictated the briefness of this section. However,much information is available on the TEACH code and the numerical prac-tices it embodies. The interested reader will find the reference by Patankar[ 253 especially informative,
2.1.4. Calculation grid
After the dependence of the numerical solution on the refinement anddistribution of the grid was explored, the 49 X 49 unevenly spaced gridshown in Fig. 2 was used for the fluid flow calculations. The spacing be-tween successive grid points differed by a multiplicative constant c. Givenna grid points along the 3c direction, the fo~owing algorithm fixed theirlocation:
Xl = -0.5 Ax
x2 = 0.5 Ax
x, = x,__~ + c”--~ AWL
Fig. 2. Calculation grid.
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X, = x,_~ + cm-4 Ax
where Ax is determined from the requirement that the computation domain
along x be of length X.x = i(x, +x,-I)
=L. “-4,,+~m_12c
1-cm-4 +
I_ p-3
= + 0.5 Ax2 l-c i
In the axial direction, the constant c was chosen to be 0.95 while in the
radial direction c = 1.058.
The final calculation grid was chosen by increasing the grid refinement
until the calculated fluid field was in good agreement with the experimental
results of Araujo et al. [211. Because of the thinness of both the developing
jet and the wall jet, the grid points were concentrated near the axis of
symmetry and near the wall. Estimates of false diffusion, relative to turbu-
lent diffusion, showed that pfaise/fit < 0.05 in the region of strongest. stream-
line curvature.
2.2. Particle phase
With the fluid flow field determined, it is possible to calculate the
motion of the pa_rticles. A detailed discussion of the relative magnitudes ofthe forces acting on an individual particle has been given by Laitone [20].
Laitone shows that for high speed incompressible air flows in which the
particulate phase is dilute the dominant force acting on a particle is the
drag force. Particle-particle interactions, virtual mass effects, lift, viscous,
pressure, gravity, Basset and Magnus forces are all relatively small and are
assumed to be negligible,
22.1. Equ ation of m otion
The particles are assumed to be spherical so that Stokes’ drag formula
can be used. Since this formula is only valid when the particle Reynoldsnumber (based on the relative velocity between the two phases) is much less
than unity, an empirically determined correction factor f is used when the
particle Reynolds number is of order unity or larger. For the conditions
stipulated, the particle equation of motion is
d2r f-=-dt2 T
(9)
where r = r(xp , p) refers to the position of the particle, the particle velocity
is U, = dr/dt and U is the velocity of the fluid at r. The particle response
time is given by
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The cor rect ion factor f given by Booth royd [261 is
1
1 + 0.15Re,0*687 O<Re,<200
f =, 0.914Re, o*282 O.O135Re, 200 < Re, G 2500 01)
O.O167Re, Re, > 2500
wher e Re, is th e pa rt icle Reynolds nu mber . Equa tion (9) can be rewr itt en
as a set of first -ord er differen t~ equa tions:
dr-=dt
u*
dUP=
-Qu- p)dt r
(12)
(13)
The solution of t his init ial-value problem requ ires the specificat ion of the
initia l position an d velocity of th e par ticle.
2.2.2. Solution algorithm
One of th e second -ord er Run ge-Kut ta schemes, th e midpoint met hod,
was us ed to solve eqns. (12) an d (13). The midpoint met hod is a mu ltist ep
techn ique in which th e slopes ar e evalua ted at the midpoint of th e time
interval (t,, tn+l). A det ailed discussion of th e midpoint met hod is con-
ta ined in most element ar y nu merical a na lysis textbooks (see, for example,ref. 27). The solut ion algorit hm is as follows (th e super scripts refer to th e
time step).
The first st ep is to determ ine th e position an d the velocity of th e
part icle at t ime t, + h/2. Thus,
hrn+l’2=rn + -p*”
u P n+1/2 xc r/,n + _1 f (U” - Up”)
(14)
When deter min ing th e position an d velocity of th e pa rt icle at tim e t, + h, the
slopes a re evaluat ed a t time t, + h/Z:
r n+l. = r” + hupn+lf2
hfn+1/2 (15)
UPn+Lqpn+
-(Un+1/2 _u n +l/Z
P )7
The t ru ncat ion err or of th is met hod is of ord er h2. The ~plementationof higher order discretizat ion schem es did n ot a lter th e calculat ed p a r t i c l e
trajectories.
The time step h was decreased until further decreases in the time step
did not alter th e solution. A typical time st ep used was lo-' .
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22.3. Particle impact velocity
To find the impact velocity, Euler’s method was used to discretize
eqns. (12) and (13). The primary reason for choosing this discretization
scheme was that it facilitates computing the time at which a partiele col-lides with the wall. If a particle collides with the wall during the time step
iz + 1, the location s and the velocity 4 of the particle at the moment of
impact can be determined from
s=r”+KU P (16)
q=U/+Kf”(U”-Up”)7
(17)
The time K at which the collision takes place can be determined using the
x component of eqn. (16) by noting that x, = L at the moment of impact.
2.2.4. Interpolating the fluid velocity
Since the fluid velocity is calculated at fixed grid points using an
eulerian formulation of the equations of motion, the fluid velocity along
the trajectory of a particle must be found by interpolation. Figure 3 shows
the four fluid velocity calculation grid points ((i,j), (i,j + l), (i + 1,j) and
(i + 1, j + 1)) nearest a particle located at (xP, r,). The variables A i (with
i = 1,2,3,4) represent the areas shown in the figure.
A linear interpolation for the fluid velocity around the point (x,, rp) is
given by
wxv d =AlUi+l.j + AzUi,j + AsUi+l,j+l +A~ui j+l
I
EAi(18)
In order to use eqn. (18), the grid point (i, j) must be found (given the
location of the particle).
i , i + 1 i + l , j + l
Fig. 3. The four fluid flow calculation grid points ((i,]), (i,j + I), (i + 1, j) and (i + 1,
.i + 1)) nearest the particle located at (x,, rP). The variables A; represent the areas used in
the weighting scheme given by eqn. (18).
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Solving for the grid point number n from the calculation grid relations,
i - 2 = intlog{1 - (1 c)(3tP/A~ - 0.5))
log c 1j- = intlog{1 - (1 c)(rPfAr - 0.5))
log c 1 cw
where the int(x) function takes the integer portion of x.
3. The model for erosion
Given the velocity r~ at which the particles strike the wall surface, theangle 8 at which they strike (measured with respect to the perpendicular tothe surface) and the number N of particles (per area per time) hitting thesurface, the cutting model proposed by Finnie [9,28] can be used to predictthe erosion of a ductile metal plate. In the model, Q is the volume of materialremoved per unit time per unit area by N particles each of mass m. Therelations derived by Finnie are
Nmq*Q=_...m_2p J/ {sin(20) - 3 cos*0}
sfor 71.5” < 8 < 90”
and (21)Nmq*
Qc- sin20 for 0” < B < 71.5”@‘&
Since relative rates of erosion are of interest here, precise values for thewear model constants P and $J are not required. In deriving these equationsFinnie assumed that the fluid phase was of low density and low viscosity(e.g. air) and also that the abrasives strike the metallic surface with a sharpleading edge.
Throughout the remainder of this paper, q, 8 and N are referred to as
the erosion parameters.
4. Results and discussion
4.1. Single-phase impinging jet
For testing purposes, predictions of a single-phase fluid flow field werefirst made for conditions of the experiment of Araujo et al. [21], corre-sponding to an air jet impinging on a smooth wall. Araujo et aE, used laserDoppler ~emomet~ to measure the mean and ~uctuating velocity com-ponents in the free jet and wall jet regions. They made measurements in jetsimpinging both normally and obliquely, with impingement angles rangingfrom 0 = 0” to 8 = 20”. The predictions discussed below were made for anormal impingement angle (0 = 0”).
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tc1 Y/r
Fig, 4. (a) Single-phase impinging jet flow axial velocity component as a function of radial
position at three axial locations showing the best fit through the data of Araujo et al.
[ 2 1 1 ( ) and predictions from the present work (a, x/d = 6: 9, x/d = 8; 3, s/d = IO).
(b) Single-phase impinging jet flow radial velocity component as a function of the distame from the wall at various radial positions showing the best fit through data of Araujo
et al. [Zl] ( -) and the predictions from the present work (0, r/d = 3; “., r/d = 6; :I,
r/d = 9). (c) Single-phase impinging jet flow turbulence intensity as a function of distance
from the wall at various radial positions. U is the local x component velocity: -,
predictions from the present work; 7’. from the measurements of Araujo at of. f 21 J.
Figure 4(a) shows the predicted axial velocity distribution as a function
of radial position at x/d = 6, 8 and 10. As expected, the velocity profiles are
self-similar. The velocity has been non-dimensionalized by the center-line
velocity at the particular axial location, while the radial position has been
Nan-dimension~ized by the jet half-width (the distance at which the velocity
falls to one-half of its center-line value). Table 1 provides a comparison of
the jet spreading rate and the decay of the center-line velocity (both as a
function of axial position) with the results of Araujo et al.
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TABLE 1
xld
0
4
6
a
10
The width of the jet ~2rl~~ld~ The center-line uelocity (Ue/Uinf
Araujo et al. Predictions Arcmjo et al. Predictions
1.00 1.00 1.00 1.00
1.17 1.23 0.98 0.99
1.20 1.26 0.93 0.93
1.40 1.42 0.81 0.85
1.69 1.61 0.70 0.72
Figure 4(b) shows the radial velocity as a function of the distance from
the wall at various positions in the wall jet. Both the me~~ernen~ and thepredictions show that the velocity profiles are self-similar. In the figure, thevelocity has been non-dimensionalized by the maximum velocity in the waI1jet (at the radial position in question) and the distance from the wall by thewidth of the wall jet (defined by Araujo eE al. as the distance at which thevelocity falls to one-half the maximum velocity).
Figure 4(c) shows the turbulence intensity as a function of the distancefrom the wall at various radial positions along the wall,
For the purposes of this study, the predictions are in acceptable agree-ment with the experimental results.
4.2. Tkuo-phase impinging jet
The two-phase calculations presented and discussed in this section wereobtained with fluid flow conditions set to the normal impingement jetconfiguration of Araujo et at. discussed above,
4.2.1. Cold gas flowThe effects of various parameters, such as the inlet turbulence intensity
and the particle size, on the motion of sand-like particles (&Jpf = 1709) inair at 300 K are discussed in this section.
Figures 5(a) and 5(b) show the effect of turbulence on the speed Q atwhich 5 pm particles strike the wall surface, the angle 0 at which they strike(measured from the normal to the wall) and the number N of particles (perarea per time) that hit the surface.
The greater the inlet turbulence intensity, the more mixing occurs inthe jet and consequently the faster the jet spreading rate. The conservationof mass requires that the center-line velocities decrease faster. Thus, thegreater the turbulence, the smaller the speeds of impact (as is shown inFig. 5(a)).
Let us consider a fixed radial position on the wall. For the high turbu-lence case, the particles striking the wall at that position originated at apoint closer to the center of the nozzle. Thus, the angle 8 at which theparticles strike the wall increases with increasing turbulence intensity (as isshown in Fig. 5(b)). This point is illustrated schematically in Fig. 5(c).
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(b)
(cl
Fig. 5. (a) Effect of inlet tu rbu fenee int ensit y on th e impa ct velocity of 5 ,um par ticles
(x = 0.41) in an air jet with Rej = 20 000, H /d = 12 and pPfpf = 1709 for var ious values of
(k”2/U)i~. (b) Effect of inlet tu rbulence intens ity on part icle impact a ngle an d sur face
“flux” for 5 p pa rt icles fh = 0.41) in an air jet with Rej = 20 000, N/d = 12 an d ~~~~~ =
1709. (c) Qualita tive sketch of th e effect of inlet turbu lence intens ity on par ticle tr ajec-
tories.
If it is assumed that the inlet particle flux Ni, is uniform across the
nozzle, the number of particles (per area per time) striking the surface at a
given location decreases with increasing turbulence intensity (see Fig. 5(b)).
The number of particles striking the area nrc2 per unit time is nra2Ni, for the
high turbulence case and nri,‘Ni,, for the low turbulence case (r, is the
impact radial location).
Figure 6(a) shows the effect of particle size on particle trajectory.
Figures 6(b) and 6(c) show the various erosion parameters for 5, 10 and
20 E.tm particles. The results confirm that the motion of larger particles is
relatively unimpeded; thus they strike the surface at greater speeds and at
smaller angles.
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s 0.5Fluid etnemline
(a) 2 r/d
60
00 1 .o 2.0
2rld
0
fbf ’
1,O
2 rid
Fig, 6. (a) E ffect of part icle size (or moment um equilibrat ion par am eter h) on part icle
tr ajectories in an air jet. All par ticles wer e released from th e sam e initial ra dial position at
th e inlet pla ne (x/d = 0). Flow condit ions ar e Rej = 20 000, H/d = 12, pp/pf = 1709 and
(k”*/U)in = 0.30. (b) Effect of pa rt icle size (or momen tu m equilibra tion par am ete r h)
on impa ct velocity in a n a ir jet with Rej =: 20000, H/d = 12, ppJpf = 1709 and
(Jz~/~/U)~, = 0.30. All pa rt icle sizes had t he sam e inIet m ass flow ra te. (c) Effect of
par ticle size (or momentu m equilibrat ion par am eter k) on impa ct an gle an d sur face flux
in an air jet wit h Rej = 20 000, H /d = 12, &,/pf = 1709 and (k”“/U)in = 0.30. All pa rt icle
sizes had th e same inlet m ass flow ra te, an d Nin for t he 5 pm par ticles ha s been used for
nondimension~ization.
A non-dimensional particle response time or momentum equilibrationconstant h is useful when quantifying the sensitivity of a particle to changesin the mean fluid flow. It is indicated in Fig. 6 and is defined as
ruinX=-..--d
w%?-4,=--181.1 d
The quantity h is the ratio between a time scale characteristic of the meanparticle motion and a time scale characteristic of the mean fluid flow. Theresults of this study are in good agreement with Laitone’s [ZO] prediction
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that for X G 0.1 the particles follow the streamlines and very few collide with
the wall, whereas for h > 10 the particles are virtually unaffected by the mean
fluid flow. Since this study is concerned with the effects of fluid mechanics
on erosion, particles with response times in the range 0.1 < h < 10 wereinvestigated. For the conditions of this study, 5 pm, 10 pm and 20 pm
particles correspond to response times of 0.41,1.64 and 6.55 respectively.
4.22. Hot gas flow
Figure 7 shows the effect of temperature on the erosion parameters for
10 I.tm particles (p, = 2000 kg mW3) in an air jet with U, = 30 m ss’~ In one
test case the air was at 1200 K (Rej = 2000) and in the other test case it
was at 300 K (Rej = 20 000). In each case, the system was assumed to be
isothermal so that heat transfer between phases could be ignored. The
primary effects of increasing the temperature are to increase the viscosityand to decrease the density of the air. The viscosity increases by a factor of
2.5 and the kinematic viscosity by a factor of 10.
The increase in temperature decreases h by a factor of 2.5. Thus, the
particles follow the fluid more closely.
For the same configuration and the same inlet velocity, the Reynolds
number for the hot gas is smaller by a factor of 10. Thus, the hotter jet will
tend to spread slightly faster and consequently the conservation of mass
requires the axial velocity to decrease faster.All these consequences, due to increasing the temperature of the flow
system, tend to decrease the speed of impact, to increase the angle at which
the particles strike the surface and to decrease the number of particles
striking the surface per area per time at a given position on the plate.
m
fb) 2 r /d
Fig. 7. (a) Effect of tem pera tu re on th e impa ct velocity of 10 pm par ticles. At 300 K,Rei = 20 000 (Vi, = 30 m s-l), A= 1.64 and pp/pf = 1709. At 1200 K, Rej = 2000 (Vi, =
30 m s-l), X = 0.66 an d pP /pi = 6840. In addit ion, H/d = 12 and (k”2/U)i, = 0.30. (b)
Effect of tempera tu re on t he impact an gle and surface flux of 10 pm pa rt icles. Flow
condit ions ar e as given in (a).
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4.2.3. Liquid flow
Figure 8 shows the erosion parameters for 200 pm (h = 0.587) and
400 pm (h = 2.35) sand-like particles (pp/& = 2) in a water jet with an inlet
velocity of 1.25 m s-i co~esponding to Rel = 20 000. As expected, the
larger particles strike the surface at greater speeds and smaller impact angles.
In liquid flows, as in gas flows, particles corresponding to h = 0.1 follow thestreamlines while particles with h > 10 are virtually unaffected by theturning flow. Because of the assumptions in the particle motion equation,these findings are of qualitative value only.
5B
(a)
(A = 0.587)
0 1I I
02%
2.0
m
(b) 2 r/d
Fig. 8. (a) Effect of particle size (OF momentum equil ibration parameter A) on impactvelocity in a water jet with Rej = 20 000, H/d = 12, pp /& = 2 and (klf2/U)i~ = 0.30. (b)Effect of particle size (or momentum equil ibration parameter h) on impact angle andsurface flux in a water jet with Rej = 20 000, H/d = 12, &,/pf = 2 and (k”2/U)in = 0.30.
Nin far the 5 /tm particles has been used for non-dimensionalization.
4.3. Applications t o surface erosion by particle impact
Figure 9 shows the effect of turbulence on erosion by 5 lum particles(pp/pt = 1709) in an air jet at a temperature of 300 K. Finnie’s f9, 231
ductile metal erosion model was used to illustrate the influence of turbulent
diffusion on erosion.The figure shows a decrease in the amount of erosion with increasing
turbulence intensity. Although perhaps counterintuitive, the phenomenon
can be explained and is due to two effects. One effect is that the impactvelocity decreases with increasing turbulence intensity (see Fig. 5(a)), the
other effect is that the number of particles (per area per time) striking thesurface also decreases with increasing turbulence intensity (see Fig. 5(b)).
Figure 9 also shows that the position of maximum wear is significantlyaffected and is moved nearer the symmetry axis with increasing turbulenceintensity. This is due to the fact that, according to the erosion model,maximum wear occurs at approximately 8 = 74”. Since the line 8 = 74” inFig. 5(b) intersects the high turbulence intensity impact angle curves at small
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0 1.0 20
2 r/d
Fig. 9. Effect of turbulence intensity on the wear of a ductile metal by 5 pm particles
(h = 0.41)inanair jet with Rej = 20 000, H/d = 12 and pp/pf = 1709. Finnie’s [ZS] modelwas used to calculate the erosion.
values of r, the location of maximum wear is moved progressively closer to
the symmetry axis with increasing turbulence intensity.
In concluding this section attention is drawn to two points. First, the
radial wear patterns implied by the shapes of the profiles in Fig. 9 are in
good qualitative agreement with the patterns observed experimentally by Li
et al. [12] and Benchaita et al. [131. In the case of ref. 12, the experimental
conditions were Rej = 3700 - 12 500, H/d = 12, ,op/pf = 2.4 and X = 10 - 50,while in ref. 13, Rej = 140 000, H/d = 2.5, pp/pf = 2.4 and X = 7 - 30. For a
jet oriented normal to the test specimen surface these workers observed
erosion at the stagnation point, a feature not predicted by the wear model
chosen here for illustration. However, there is very good agreement with the
radial location for maximum wear which was 2r/d = 0.9 in Li et al. [ 121,
2r/d = 0.75 - 0.9 in Benchaita et al. [13] and 2r/d = 0.8 - 1.2 in the present
work. Interestingly, the potential flow model of Benchaita et al. overpredicts
the experimental location of maximum erosion by about 50% for large
values of X. We attribute this discrepancy partly to the inability of their
model to account for turbulent diffusion of momentum along the shearlayer evolving from the nozzle edge (see also Fig. 9). As explained earlier,
turbulent diffusion works to move the location of maximum wear nearer the
wall surface stagnation point.
The second point relates to the first and is offered here as an explana-
tion for the “halo” effect discussed by, among others, Lapides and Levy
[29]. These workers find that the test specimens from jet impingement tests
show two regions of substantially different erosion characteristics. In the
center of the erosion area large amounts of material are removed by the
impacting particles. Surrounding this heavily eroded zone is an annulus of
considerably less-damaged surface, but which still shows clear signs of
particle impact. With reference to Fig. 1 we see that for small values of H/d
the test specimen surface will sense two types of flow. One type is highly
monodirectional and is of large speed in the potential core of the jet. The
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other type, of lower speed, is embedded in the shear layer generated at thenozzle edge. Relative to the core flow, the shear layer is highly turbulent,
with diffusion working to deviate particle trajectories from their original
paths in a random manner. We suggest that the halo arises as a consequence
of the shear layer surrounding the potential core of the jet and that the sharpdistinction between the halo and the heavily eroded central region should
diminish do~stre~ of the location where the potential core ceases to
exist. Experiments are currently underway in our laboratory which willhelp to confirm or disprove this hypothesis.
5. Conclusions and r~ ommendations
Although limited by the simplifications made in the analysis, thepresent numerical study demonstrates the following.
(1) Fluid turbulence can have a significant influence on erosion by
particle impact in jet flow con~~ations.(2) Numerical methods and physical models are available for predicting
the approximate mean motion and trajectories of particles suspended inturbulent flow. Some of the restrictions imposed in this study can readilybe removed; for example, by including two-way coupling between the phasesand allowing for various particle sizes. Other restrictions, due to particle-particle and particle-wall interactions, are not so easily removed.
(3) The erosion parameters Q, 8 and N vary significantly with fluidtemperature.
Because of the simplicity of the model, the present study demonstra~sthe relative effects of turbulence on erosion on a qualitative basis. The lackof accurate experimental data to guide theoretical developments and to test
predictive numerical models in general has imposed the modest objectives ofthis study. However, from the results provided it is clear that in futureexperiments, aimed at qu~tifying p~icle-fluid motions near surfaces anderosive wear, it will be indispensable to measure and control the turbulent
characteristics of the flows investigated. This will ensure that the charac-teristics of erosion associated with turbulent flow are separated from (as
opposed to confounded with) the ch~ac~ristics more directly associatedwith material properties and particle-surface mechanical interactions.
Acknowledgments
This work was funded by the Technical Coordination Staff of theOffice of Fossil Energy of the U.S. Department of Energy, under Contract
DE-AC03-76SF00098 and through the Fossil Energy Materials Program,Oak Ridge National Laboratory, Oak Ridge, TN. The authors are grateful tothe agency for its support. Thanks are due to Mrs. Judy Reed and Ms. Loris
Donahue for the preparation of this manuscript. The authors’ names appearin alphabetical order.
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Appendix A: Nomenclature
b
c
c1, c2
c,
d
d,fh
H
i
jk
I(f
4,
P
PS
4
QQref
r
r
1^1/2
Rej
Re,
U&Xu,2
.1X
width of the wall jetgrid spacing expansion co~st~tt~bulen~e model constantsturbulence model constantnozzle diameter
particle diameterdrag force correction factorintention time stepdistance from nozzle tip to wall surfacegrid index in the axial directiongrid index in the radial directionturbulent kinetic energyparticle massnumber of particles per area per time striking the wallparticle flux through the nozzle
mean pressurewear model constantparticle impact velocityvolume of material removed per unit time per unit areaNi,mUi,2/6PsU/, normalization constant
radial coordinate directionvector position of the particlejet half-widthUi,dfVp jet Reynolds’ number
IU-U,ld,/v,
particle Reynolds’ number based on the relativevelocity of the phasesshear stress componentradial stress componentaxial stress component
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fluid mean (vector) velocityjet center-line velocity (varies with x)
jet velocity at nozzle tip
particle mean (vector) velocityradial component of fluid mean velocityaxial component of fluid mean velocity
axial coordinate direction {pointing toward wall surface)axial coordinate direction (pointing away from wall surface)
Greek symbols
;dissipation of turbulent kinetic energyangle at which the particle strikes the wall (with respect to thenormal to the wall)
h non-Dimensions particle response time
lu fluid phase dynamic viscosity
fJt fluid phase turbulent viscosityV fluid phase kinematic viscosity
P fluid density (pure phase)
P P particle density (pure phase)
ITk, a, “Prandtl” numbers for k and E7 particle response time
J, wear model constant
5%bscrip t s
f fluid phase
max maximum value
P particle phaser radial coordinate directionX axial coordinate direction