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Unformatted text preview: Bibliotheek T» D /////1
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Artikel: 01’1 the resuspension of sma‘ ‘ partic1es by a turbulent flow Auteur; M W Reeks, J Reed and D Ha‘ ‘
Tijdschrift; JOURNAL OF PHYSICS D. APPLIED PHYSICS
Jaar: 1988 V01. 21 Aﬂevering:
Pagina(s): 574—589 Plaatsnr.: 803 D Met ingang van 1 april 2003 zullen de prijzen v00rf0t0k0pie levering baitenland stijgen met 6‘ 0,05 per pagina From April I 2003, prices for photocopy delivery abroad will increase by 6‘ 0.05 per page J. Phys. D: Appl. Phys. 21 (1988) 574—589. Printed In the UK M W Reeks, J Reed and D Hall Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley,
Gioucestershire, UK Received 15 July 1987, in final form 15 October 1987 Abstract. This work presents a new approach to the way small particles are
resuspended from a surface exposed to a turbulent flow. in contrast to current
force balance models, this approach recognises the influence of turbulent energy
transferred to a particle from the resuspending flow. This energy maintains the
particle in motion on the surface within a surface adhesive potential well. The
particle is detached from the surface when it has accumulated enough vibrational
energy to escape from the well. Such considerations lead to a formula for the rate constant p, for long—term resuspension of the form
p ~ too expl— O/(2(Pe))] where too is the typical frequency of vibration, Q the height of the surface adhesive
potential well, and (PE) the average potential energy of a particle in the well. (PE) is
shown to depend upon the fluid and mechanical damping and the energy spectrum
of the fluctuating aerodynamic lift force, particularly near the natural (resonant) frequency of the particle—surface vibration. Estimates of p, based on van derWaaIs adhesive forces for a particle on a surface (particle and surface being elastically
deformed), indicate that particles can be resuspended more easily from a surface
than anticipated on a balance of adhesive and aerodynamic lift forces. The
dependence of resuspension on flow and particle size is the same as that
observed in practice. Finally, resuspension rates from surfaces where there is a
spread in adhesive forces (due to surface roughness) are shown to decay almost
inversely as the time of exposure to the flow. This feature has been observed
experimentally and is particularly important in determining the longterm decay in
concentration of suspended particles in a recirculating flow (e.g. an Advanced Gas Cooled Reactor). 1. Introduction The suspension or resuspension of particles exposed
to a moving ﬂuid is a common occurrence in nature.
Familiar as the origin of dust storms, it is inﬂuential
in a diverse range of industrial and environmental
processes from the erosion of soil and river beds to
the ‘wave of darkening’ observed during the Martian
Springtime (Sagan and Pollack 1969). It has addition
ally become a key issue in the current debate over a
global Nuclear Winter (Turco et al 1984). The work we describe here is a new way of look—
ing at resuspension. Previous theoretical work has
considered only the balance of forces acting on a
resuspending particle. In contrast, our approach
recognises the importance of the turbulent energy
transferred to the particle from the resuspending
ﬂow. .It illustrates the similarity of particle resuspen
sion by a turbulent ﬂow to that of the desorption of
molecules from a surface. Eventually this approach
leads to some new and interesting results. 00223727/88/040574 + 16 $02.50 © 1988 IOP Publishing Ltd Resuspension is associated with the aerodynamic
detachment of small particles (generally less than
100 pm in size) for which the principal force holding
them onto a surface is the intersurface molecular
force (adhesive force). It is common experience that
particles of this size are difﬁcult to remove from a
surface aerodynamically, being immersed in the
viscous sublayer, adjacent to the surface, where the
aerodynamic forces are very small. In their classic studies on aerodynamic removal,
both Bagnold (1941) and Chcpil (1945) observed that
resuspension, in common with other types of aerody—
namic removal, was initiated by ﬂows whose velocity
exceeded some threshold value. More speciﬁcally this
threshold ﬂow velocity for resuspending particles
decreases with increasing particle size. It has become
traditional to explain this trend in behaviour in terms
of a balance of the aerodynamic and adhesive forces;
both forces increase with increasing particle size but
the aerodynamic force has a greater size dependence
(see, for example, Phillips 1980). . new _. (.2 However, this force balance approach takes no
account of the timescales over which particles are
resuspended. For instance many measurements of
resuspension in the environment indicate that particle
removal from a surface is not instantaneous but per
sists over a period of time (see, for example, Sehmel
1980). Several authors (Corn and Stein 1965, Aylor
1978) have suggested that this behaviour is evidence of
a statistical origin of resuspension intimately associated
with the random motion of ﬂuid close to the surface.
We shall call this resuspension longterm resuspension,
and it is the statistical process leading to this resus—
pension that we wish to investigate. There is, in fact, much evidence to suggest that
the aerodynamic force acting on a particle on a sur
face is random in time. For instance, the early ﬂow
visualisations of Kline et al (1967) and Corino and
Brodkey (1969) clearly show that ﬂuid motion in the
viscous sublayer of a turbulent boundary layer is
identiﬁable in terms of a deﬁnite sequence of coher
ent structures, i.e. ‘ejections’ followed by downward
‘sweeps’ with occasional ‘interactions’ (turbulent
bursting). Several models for resuspension, most
notably that of Cleaver and Yates (1973), have ident
iﬁed the frequency of such events with the resuspen
sion rate itself. They all contain an implicit
assumption that the resuspension rate is controlled by
the frequency of occurrence of an aerodynamic lift
force which exceeds the force of adhesion. Thus,
although such models are statistical in nature, they
still retain the essential character of a force balance
approach. Our approach, on the other hand, recognises the
inﬂuence of the transfer of turbulent energy to a par
ticle on a surface from the resuspending ﬂow. Such a
transfer takes place through the agency of the aero
dynamic lift force which ﬂuctuates randomly in time.
Thus, a particle in contact with a surface is in a con
stant state of vibration, building up energy until this
energy is sufﬁcient to detach the particle from the Potential energy, UM Particle resuspension surface (resuspension), This accumulation of energy
takes place most efficiently at driving frequencies
close to the natural frequency of the motion
(resonant energy transfer) where it is limited by the
energy dissipation in the local ﬂuid and substrate
(ﬂuid and mechanical damping). The analogy is a
familiar one of a vibrating spring, where repeated
applications of a relatively small force can, in time,
have an equal effect on the amplitude of vibration as
that of a much larger force applied at a much lower
frequency (quasistatic). It is not surprising, as shown
later, that the threshold ﬂow velocity for detachment
in this approach can be signiﬁcantly lower than that
based on a balance of aerodynamic and adhesive
forces. In this paper we present a statistical model for
longterm resuspension based on these simple ideas.
From this model we eventually obtain a general for
mula for the probability per unit time of a particle
being detached from a surface (the resuspension rate
constant). This formula is then used to evaluate the
resuspension rate constant of a particle on a surface
when both particle and surface are elastically defor
med under van der Waals adhesive forces. Such con
siderations enable us to estimate resuspension from
surfaces of practical interest where there is a spread
of surface adhesive forces (surface roughness). This
ﬁnal study reveals the interesting result (previously
observed in experiment) that after some initial time
(less than a second) the resuspension rate varies
almost inversely as the time. This feature we show is
particularly important in determining the longterm
decay in concentration of suspended particles in a
recirculating ﬂow. I 2. The statistical model We represent the interaction between particle and
substrate by a potential well formed from a surface Displacement,y Figure 1. Surface potential well diagrams in which the adhesive
energy is kept constant but the mean lift iorce increases in value from that in a to that in 0. D4 575 ____________—___———_—————I M W Fleeks et al adhesive force, an elastic restoring force and a mean
aerodynamic lift force (assumed independent of the
particle surface deformation). Figure 1 represents a
family of such potentials for different applied mean
lift forces. All the curves show the form of a typical
well that exhibits: (i) a position of stable equilibrium, A, where the
attractive adhesive force balances the repulsive elastic
force and mean lift force; (ii) a position of instability, B, where the attract
ive adhesive force balances the mean lift force. Increasing the mean lift force reduces both the width
of the potential well and Q, the height of the poten
tial barrier of B above A, as shown in ﬁgure 1. We suppose that the number of particles on the
surface is sufﬁcient to form an ensemble of all
realisable states of a single particle in a constant
potential well. A particle will leave the potential well
(be resuspended) when it receives enough energy
from the local turbulence to escape over the potential
barrier at B, beyond which the net force is directed
away from the surface. In practice, the motion of a particle in the well
(which causes the particle and surface to deform) can
be approximated by that of a very stiff, lightly
damped harmonic oscillator driven by random ﬂuctu
ations (of zero mean) in aerodynamic lift force. Thus
for a particle of mass m, the deformation (displace
ment) y(t) at time I, about the point of minimum
potential, A, is obtained from 5" + £3); + cozy = m'lfLO‘) (1) where fL(t) is the random ﬂuctuating component of
zero mean associated with the total aerodynamic lift
force FL(t), i.e. FL = (FL) +110) where (FL) is the mean lift force which in this
instance is assumed independent of time, a) is the
natural frequency based on the harmonic
approximation for small deformations about A
(ﬁgure 1), and [3 is the ﬂuid and mechanical damping
term ([3 is a function of the frequency of the driving
force but we shall see later that we may treat it as if it were a constant of motion). For a lightly damped,
stiff harmonic oscillator w/ﬁ >1 and (or >1 (2) where r is the typical timescale of the lift force
ﬂuctuations. In order to determine the resuspension rate con
stant we must ﬁrst establish the statistical motion of a
particle within the well, in particular the distribution
of y(t) and y(t) (particle velocity). We do this by first
examining the simple case of a harmonic oscillator
driven by a series of statistically independent random
impulses of ﬁxed duration. For zero damping the
distribution of y(t) is shown to be Gaussian in the
long term, i.e. after many impulses. Additional con
siderations, borrowed from the dispersion of ﬂuid 576 elements in stationary homogeneous turbulence,
suggest that this is a property common to resonant
energy transfer. In our eventual calculation of the resuspension
rate constant we use a Gaussian distribution for both
displacement and velocity with values for their mean
square displacement obtained Via the frequency
reSponSe of the system (this was found most useful in
identifying the resonant frequency contribution).
Only those values of the mean square displacement
and velocity at equilibrium are, in fact, required in
the formula for the rate constant. However, it was
found useful in illustrating the accumulation of
vibrational energy to determine the variation of these
quantities with time. The ﬁnal formula for the rate
constant shows how it depends explicitly on the
damping, [3, the energy spectrum of fL(t) at a), and
Q, the height of the potential well. 2.1. Distribution of displacements in a harmonic
potential We assume that the equilibrium distribution of
displacements for a harmonic oscillator driven by a
random stationary force is Gaussian. This assumption
is based upon the following argument. Consider the
behaviour of an undamped harmonic oscillator driven
by a sequence of random impulses, each lasting a
ﬁxed time 1'. Thus within any sequence of impulses
lasting a total time of T, the driving force f(t) at time
t, for t< T, is given by T/r
f0) = 21m:— (n we (3)
where 6(3) is a step function deﬁned as
6(s) = 1 0 < s < 1'
= 0 s S O, s a 1: (4) and [f,,; n = 1, 2 etc] a set of statistically independent
random forces of zero mean sharing the same prob—
ability of occurrence. We have thus for any f,I (fi) = (f2> (5) If to is the natural frequency we have for the dis
placement y(t) y(t) = Iii[7(5) sin (n(t — s) ds (6) 0 which, using equations (3) and (4), becomes y(t)=m_2fnfm Sinw(t—s)ds+~1— (n 1)r ma) I
X] stinw(t—s)ds forNr<!<(N+1)r
Nr 2 , cur N
._ ——mw2 Sin7'ng s1n w[t—(n—i)1:] + % X (1 — cos wAt) with At: t Nr. (7) l
i
{r
f
i For large N (t/ r > 1), we recognise that each moment
of y(t) is dominated by the contribution from the sum
of random variables 25:1 f". So for arbitrary values
of an we have, for example 2 2
W» = 2.04 0‘” (SW 923) (i + 0(1)) + £5.34.
(8) We note that for the special case when cor/7r is an
odd integer m 02(1)) = (9) The implication (by virtue of the central limit
theorem) is that y(t) in the limit of t/r>1 will be
normally distributed with a variance which increases
linearly with time. We recognise these features as
properties of a simple random walk diffusion process. In considering the random motion of a harmonic
oscillator driven by a stationary random force of a
more general nature (e.g. arbitrary form for the
energy spectrum), we exploit a well known argument
used by Taylor (1921) where he compared longterm
ﬂuid element diffusion in a homogeneous stationary
turbulent ﬂow to a random walk. If the diffusion time
of the harmonic oscillator is sufﬁciently long (t/r > 1)
it can be divided into a large number of subintervals;
the displacements induced by the driving force in
each of these subintervals are then statistically inde
pendent of one another and the diffusion process
equivalent to the impulsively driven random oscillator
previously described. This therefore implies that y(t)
is also normally distributed with a variance pro
portional to time. (See also the comments following
equation (25).) The longterm resuspension that concerns us here
is one for which damping plays a signiﬁcant role.
However, the fact that we consider the motion in the
well to be lightly damped means that a particle will
have been subjected to many ‘impulses’ (each lasting
a time I) before reaching equilibrium. So we would
expect the ﬁnal equilibrium distribution for y(t) to be
also Gaussian. 4<f2> (5) + (f2) mza)4 r m2w4' 2.2. Evaluation of mean square displacement and vel
ocity in a harmonic potential (the accumulation of
vibrational energy) To evaluate the mean square displacement and
velocity for a harmonic oscillator, we ﬁrst consider its
frequency response, C(v, t), to a forcing term eh”, i.e.
C(v,t) is the solution to equation (1) with a forcing
term ei'” on the righthand side, the initial conditions
being c<v,0) = two) = 0. (10)
Since the equation of motion is linear we can represent the displacement y(t) in response to an
arbitrary driving force fL(t) as a linear superposition Particle resuspension of the responses to the individual spectral
components of fL(t). Thus, representing the random
stationary force fL(t) by a Fourier—Stieltjes integral
(see, for example, Monin and Yaglom 1975), namely int) = (m eiv'dzw) (11)
(with dZ(v) = dZ*(—v) for the reality of fL(t)), we
have y(t) = m—1 (m ctv, r) dzm <12) V(t) = y(t) = m‘1 r 8(1), t) dZ(v). (13) To evaluate the mean square displacement (V2(t)), we
note that for a stationary random function fL(t),
dZ(v) in equation (11) has the property (dZ(v) dZ(U’)) = “(v + v’)EL(v) dv du’ (14) where EL(’U) is the energy spectrum of fL(t) (see
Monin and Yaglorn 1975). Thus using this property
we obtain from equations (12) and (13) W» = m'2 (m duEttumv, or (15)
0 (We) = m2 [m dumvnc‘w, m2. (16) To calculate C(v,t) and Zj(v,t) we note that the
conditions in (2) allow us to characterise the behav
iour of the system in terms of two separate responses: (i) A lowfrequency (force balance) response (v <
m) which is determined by a balance of the driving
force and harmonic restoring force. In general, C(v, t = £3 (eiu‘ — e'W2 cos wt). (17)
The second of the conditions in (2) indicates that this
response is typical of most of the energy spectrum of
fL(t). Thus in the long term (ﬁt > 1), the contribution
y,f(t) to y(t) from this region of the spectrum is deter
mined by an instantaneous force balance of the
driving force with the restoring force, i.e. meYIrU) 311(1) (18) (ii) A near—resonance response (2) ~ iw) of the
form eiiw __ e—ﬁr/z etiml E;(:tv,t) = 2 w  v2 : i/iv (19) For a)“ <t<< [3“, the amplitude of modulation is of
order sin (SUI/(5v, either side of the resonance (v =
:w), 62) being the beat frequency w  v (for v a 0).
Thus, whilst at the resonant frequency the response
grows linearly with time, the width of the resonance
decreases inversely with time. The integrated
response therefore remains constant, but as time pro— 577 M W Reeks er al gresses originates from an increasingly more localised
region of the energy spectrum of fL(t) around the
resonance. The ﬂuid damping inhibits the growth of resonant
energy through its suppression of the initial transient,
so that the ﬁnal state of equilibrium of constant
amplitude, gives extra: 6 iiévr
gi%0:2w®viMﬂ) conﬁning the nearresonant region to a narrow band of width ~13 about n).
Thus, separating out the lowfrequency and near— resonance contributions to both 012(0) and (Vim),
and using equations (15) and (16), we have: u .2 O (20) (i) low frequency (1f). (yz(t))ir = [um {1 ~ 6‘5/2’[cos(v ~ w)t
0 mzw“ + cos(v + (0):] + 2"” cos2 wt} EL(v) dv (21) "n1
(V2(t))lf = ;n2—1w—4 [0 {U2 — e'ﬂ‘ﬂvwkosw — w)t — cos(v + w)t] — e'ﬁ’ sin2 wt}EL(v) du. (22) (The upper limit of am is not critical, so long as it is
sufﬁcient to exclude the contribution from higher fre
quencies (v > to) damped out by the ﬂuid. The obvi—
ous choice for ’Um is the natural frequency, w.) (ii) near resonance (res). Writing the frequency
response [§(v, t)l2 around a) in terms of 62/ = v  a) zwimz I: [502 + (91—1 x (1 — C'ﬁl/Z cos 5m + e“ﬁ’)EL(w + 6v) d(5v).
(23) If ﬁr <1, then EL(w + 61)) may be assumed con
stant and equal to EL(a)) over the range of values of
612 for which the response §(v,t)2 is signiﬁcantly
greater than 1/ a)“. So after performing the necessary
integration, we have 02(t)>res z w»... = 25:,“ (1 — eﬁaEuw) (24) and similarly .7! (V2(t)>res = zﬁmz We note that in the limit of ﬁt< 1,012)”, is
proportional to time. This is the same as the domi—
nant response of the harmonic oscillator driven by
the series of random impulses (see the ﬁrst term on
the righthand side in equation (8)). Indeed, we can
formally identify the ﬁrst and second terms on the
righthand side of that equation with the nearreson (1 — crane). (25) 578 ance and low—frequency responses, respectively. It
suggests that Gaussian statistics are, in general,
exclusive to the nearresonance contribution. In con
trast, the low—frequency contribution is controlled by
the arbitrary probability of occurrence of fL, the ran
dom driving force. We note that the equilibrium values for (yz) and
(V2) based on the lowfrequency and near~resonance
contributions are given by ( 2>
oﬁ=ﬁgpﬂ+n) (m)
, <1” to
(V > 3 m2} (7’ + (dim) (27)
where
= EﬂEwZEJw). (28) E1401) refers to the normalised energy spectrum
EL(u)/(fi), so that f EJMdu=L
0
We have assumed [wﬁEJMdv=[wMEﬁMdv=db.(M)
0 0 2.3. Evaluation of the resuspension rate constant for a
harmonic potential To calculate the resuspension rate constant, p, we
ﬁrst calculate the net current of particles at the
detachment point B (ﬁgure 1) and then divide it by
the number of particles adhered to the surface (in the
well). We evaluate it ﬁrst for release from a harmonic
potential. Then, in the next section, we generalise the
result to include release from anharmonic potentials
(non—linear restoring forces). We assume that most of the particles on the sur—
face are to be found within the vicinity of A, the
point of minimum potential. This means that the
average potential energy is very much less than Q.
Such particles are almost in equilibrium at A, and
their release from the surface is long term, ie. on
timescales very much greater than the periodicity of
the well and the relaxation time of the particles. We further assume that the concentration of par
ticles at B, the point of detachment, is kept
sufﬁciently small to perturb only the particles in the
region of B from equilibrium. As a result there will
be a net current out of the well which we assume
takes place under conditions in which the concentra
tion at B, normalised with respect to the number of
particles in the well, is a constant. Most of the par
ticles, which are around A, are hardly affected and
behave only as a reservoir for those detached at B. m The point of detachment B is, in reality, a point
of discontinuity in slope of the potential or one for
which the relative curvature dZU 1/2
Elﬁn—1 dyzB) >1 where U3 is the potential at B (Reeks 1983). Under
such conditions the equilibrium distribution prevails
right up to the point of detachment, beyond which
the concentration is zero (Chandrasekhar 1943). In
this approximation (the transition approximatiOn) we
can use the equilibrium phase space distribution at B
(probability that a particle has a velocity V at
displacement ya) to calculate f”, the leakage current
out of the well.
Thus 1+ = f” VW<V,yB)dV (30)
0 where W(V, yB) is the phase distribution at B, and
integration is conﬁned to those particles with positive
velocities (i.e. in a direction out of the well). We have argued that the distribution of displace
ments is Gaussian. We assume that this is derivable
from a Gaussian distribution in velocities. At equilib
rium particle displacement and velocity are
statistically independent of one another; this is com—
patible with a Gaussian distribution for W(V, y) of the
form W(V, y) = ————1 ( V2)
22r(<iﬂ><y2>)1/2e"p W) x exp (— 232)) (31) where mean square values refer to equilibrium con
ditions, and [I W(V,y)dde=1. (32)
Thus
1 p=j+([°° yBW(V,y)dde)_ =j+ (33) .—oo —ua (since most of the particles are contained in the
region for which y S yB, i.e. yB/(y2)‘/2> 1). Substituting the form for W(V, yB) from equation
(31) into equation (30) we obtain (132)) <34) Finally, substituting in the equilibrium values for
(yz) and (V2) from equations (26) and (27) and
recognising that at the point of detachment mwzyB = fa — (FL) (35) where fa is the force of adhesion and (FL) the mean lift Particle resuspension force, we obtain for a harmonic potential _20_ __l(.fa_'(FL>)2
1’ ” 2n exp( 2(fi)(1+ 17)) (36)
with
‘2 2 2 1/2
600 = (001 + <fL>/<fl.>w ) (37) n+1 where 77 is given by equation (28).
The formula for p contains two extremes of
behaviour: (i) n —9 0 (zero resonance contribution) for which = i 1” ex <_ (f. — war)
2n (ff) p 2<ft> '
The formula reﬂects a force balance between a harmonic restoring force and a lift force normally distributed with
mean (FL) and variance (fﬁ). (ii) 17 > 1 (resonant energy transfer) for which (38) w f 3 ) p 2nexp< 2n<ft> (39)
for (FL)$(fﬁ)1/2. Resonant energy transfer increases
the resuspension rate constant in two ways: it ﬁrstly
increases the periodicity of the particle in the well and,
secondly, it increases the concentration of particles at
the detachment point. We note that for a given rate
constant p<w/2ar, we have (fﬁW2 <<f,. This is an
important feature which we will return to later. 2.4. Resuspension rate constant for a general potential In extending the analysis to resuspension from anhar
monic potentials, we recognise that the greatest inﬂu—
ence on the value of the rate constant arises from the
concentration of particles, p(yB), at the detachment
point relative to p(O), that at the point of minimum
potential. For the harmonic potential we can write this
as p(ya)/p(0) = exp(Q/2 (Nil) (40) where in referring to ﬁgure 1, Q is the height of the
potential barrier and (PE) is the average potential energy
of particles in the well; in the case of the harmonic
oscillator (PE) is given by (PE) = imaﬂyz). (41) The form in equation (40) suggests an identity with the
Arrhenius factor exp(Q/kT) for the relative con
centration of ideal gas molecules in equilibrium in a
conservative ﬁeld of force (where the thermal energy
kT, by virtue of equipartition, equals (PE)). We recall
that the Arrhenius form is appropriate to equilibrium
concentrations in a general conservative ﬁeld of force
(linear and nonlinear restoring forces alike). We shall
assume here that the form in equation (40) is likewise
suitable for the equilibrium of adhered particles in
anharmonic potentials. 579 M W Fleeks et a! We can, in fact, obtain this result by attributing a
pressure to the collective particle motion in the well
proportional to the local concentration. (See, for
example, Buyevich (1971) for the pressure associated
with a dispersed particle ﬂow arising from particle—ﬂuid
interactions.) Thus for a pressure P P = £90) (42) where a is a constant depending upon the intensity and
timescale of the aerodynamic lift force. (We note that
equation (42) contains an implicit assumption that the
mean square velocity is constant throughout the well.)
Thus at equilibrium we have 3P
8 3; + K(y)p(y) = 0 (43)
where K(y) is the conservative force acting on each
adhered particle at y. The solution at y = yB is Mr) 1 ’8 _ _ _Q_
pm”) = expgfo K(y)dy — exp( 8). (44) In evaluating a, we recognise that most of the particles
are in equilibrium around the point of minimum poten
tial and for these 1 wzyzm). (45) mmwmwﬂ~5£ Recognising this form as a Gaussian distribution, we
have a = mw2(y2) = 2(PE). (46) Substituting in equation (44) gives the form in
equation (40). Thus, in View of these arguments, we may
write more generally for the longterm resuspension of
particle in a conservative potential well _ ﬂ _. .9.
p _ 21: em ( 2(PE)) (47) where (PE) is deﬁned in equation (41) for small dis
placements about the point of minimum potential, Q is
the height of the potential well as in ﬁgure 1, and we is
as deﬁned in equation (37). (When 000 = w i.e. 17 > 1,
we note the similarity to the rate constant formula for
chemical activation processes and for the release of
Brownian particles from a potential well (Chand
rasekhar 1943).) Substituting in equation (47) the expression for (PE)
based on a harmonic oscillator, we obtain the more
general form of equation (36), namely _29 _km—unr
p‘mﬂﬁ emumd where k is a numerical constant dependent upon the
shape of the potential, and is given as _Qx
k“m—mw (m (48) 580 with x the stiffness deﬁned with reference to ﬁgure 1 as
dZU 2 _
dy2 x= mm = — (50) A where U(y) is the potential at y. We note from equation
(36) that k is 1/2 for a harmonic potential. 3. Evaluation of parameters determining the resus
pension rate constant In the previous section we derived a formula for the
resuspension rate constant which depended explicitly
on the height of the adhesive potential well, the natural
frequency of the particle in the adhesive potential well,
the mean, RMs, and energy spectrum of the aerodynamic
lift force, and the ﬂuid and mechanical damping of the
particle motion. In this section we evaluate all these
parameters. 3.1. Evaluation of the height of the potential barrier and
the natural frequency of the particle in the potential well Two ideally ﬂat clean surfaces in contact should adhere
through the action of intermolecular forces of adhesion.
The work which must be done to separate unit area of
two adhering surfaces is ZAy, where Ay is known as
the surface energy per unit area for the two surfaces
(typically, 0.04 < Ay < 0.2] m”2 for van der Waals
interaction, see, for example, Krupp (1967)). However,
the topographical roughness of real surfaces together
with the elastic properties ensure that surfaces are in
contact over an area which is generally small compared
with the apparent area of contact. This reduction in
area reduces the forces of adhesion by at least an order
of magnitude (see, for example, Gane et al 1974). In a detailed study on the effect of roughness on
adhesion, Tabor (1977) showed that a few high rough
ness peaks are the main points of contact. We shall thus
approximate the contact of a sphere, radius r, on a ﬂat
surface by assuming that this contact is made with a
single asperity point contact, radius r,, which has the
equivalent of the true adhesion, i.e. if fa is the measured
force of adhesion then zar
n where F, is the value of the force of adhesion for smooth
contact. Since we have taken a single point contact then we
can use the general analysis for the contact of a sphere
with a surface, if we replace the radius of the sphere
with our asperity radius r,, in the relevant equations. In the absence of surface forces, the Hertz equations
give the radius of the circle of contact, no, for an asperity
of radius r,, pressed onto a ﬂat surface with a positive
load, Po, as (Timoshenko and Goodier 1970) Fora
K r11 (51) a3: Where K_4(1—v%+1—V2)71 52)
‘3 E1 E2 ( v, and E, are Poisson’s ratio and Young’s modulus
respectively. (Note: for two spheres of radii R] and R2,
then ra = Rle/(Rl + R2). For a ﬂat surface Rz—a 00). Johnson, Kendall and Roberts (1971, hereafter
referred to as JKR) pointed out that under the influence
of surface forces the true contact radius, a1, will be
greater than a0 for the same applied load P0. This can
be considered as the application of an apparent Hertzian
force P1 without surface forces such that a? = Plra/K. Due to the effects of surface forces the stress ﬁeld
in the contact region will be modiﬁed from the Hertzian
stress ﬁeld. Johnson (1958) gave the pressure distri—
bution for the contact of two bodies where adhesive
forces are signiﬁcant’r. This total pressure distribution is equivalent to a
Hertz pressure distribution over radius a1 with a press
ure distribution for a rod of radius a1 with negative load,
P1 — P0, subtracted. To calculate the energy of the system JKR used a
similar load pattern to that described above in the
calculation for the pressure distribution. That is a
Hertzian load P1 is applied to give a contact radius (:1.
From this a load P1 — P0 is subtracted, still keeping the
contact area the same. Thus the ﬁnal condition of a load
P0 over a radius a 1 is obtained. From this JKR give the
mechanical energy, Um, as Um = —K‘2/3r,1,/3(%P§/3P0 +§P5P;1/3) the elastic energy, UE, as UE = K’2/3r;1/3(315P§/3 + spam—V3) (53)
and the surface energy, Us, as
Us = ‘JTAY(Pira/K)2/3 (54) where Ay is the adhesive energy per unit area.
Using these three equations and the condition that
at equilibrium (1 JKR obtained P, = P0 + 3m yr, + [6m me, + (31mm )211/2.
(55) This equation connects the applied load P0 with the
apparent load P1. The force of adhesion is deﬁned as the opposite of
the force required to separate the two bodies, i.e. F, = —P0 = +%nAyr,. (56) “r Other adhesion models considered in Reeks el al (1985) give
similar results to JKR for the resuspension rate constant. I Particle resuspenslon 1.0 1.5 2.0 Narmulised approach diS'lunce,nl/l10 Figure 2. The variation of the total elastic and adhesive
surface potential energy, Us + Us, as a tunction of the
normalised approach distance, a/ao, for the JKR adhesion
model. U0 = (ismy)‘5’3r,,‘“3K"“’3 and wo is the approach distance for zero applied load. In JKR we = 2461:)“
r;/3(Ay/K)2/3. In this instance the contact area reduces discontinuously
to zero and the surfaces spontaneously separate. This
is because at this point the rate of release of mechanical
and elastic energy is greater than the surface energy
requirements (Tabor 1977). Using the distance of approach of the two bodies,
a, given by JKR as we show in ﬁgure 2 the energy of the potential well
(Us + UE) as a function of 0:. On this ﬁgure point A
represents the point of minimum potential, and point B
is the separation point where P0 = —%JrAyra. For a constant applied load (towards the surface)
the height of the potential well, Q, is the difference in energy (UE + Us) between points A and B plus the
mechanical energy in moving the load P0 from A to B.
Using JKR we obtain Q = (%JTAY)5/3r2/3K“2/3 1I)(P6) (57) where the function 111(P6) is given by
was) = —a + m — ans/3 + MariV3 +sP5P12/3 + §p 12/3 (58)
where the normalised loads P6 and P1 are given by
P; = P,/fa fori = 0,1. (59)
The stiffness of the system is given by
dPO
P = m
X( 0) d a 581 M W Fieeks et al H Figure 3. The motion of a particle when in Contact with a
flat surface changes the area of contact; the minimum
separation between surfaces is constant (~4 A). from which we ﬁnd P1+P0) (60) = 2/3 1/3 1/3
MP“) (M ’a P1 (5131+)?0 from which we can deduce the natural frequency to using
1/2
co = . (61)
m
Using equations (57), (58) and (60) we obtain
QX Pi + P6 P i” 3
= _.__———— = p' _..._.—_— _.____.
k (t—avr ii(°)ua+Psu+Pu2 (62) Evaluation of k as a function of normalised applied load
PO/fa shows k to vary from unity by the order of 10%
throughout the entire range —1 s Po/fa s cc. In view of
this relatively small variation in k, we set k = 1 in
subsequent calculationst. 3.2. Evaluation of mean and RMS lift force One of us (Hall 1988) has measured the mean lift force
on a captive sphere near to a surface in a fully developed
turbulent boundary layer. When the sphere is suf
ﬁciently close to the surface (less than 0.5% of its
diameter) the measurements show that for a sphere of
radius r, in a ﬂow of ﬂuid density pf, kinematic viscosity
11;, and friction velocity u,, a universal relationship exists
between the normalised sphere radius rig/vi and the
normalised average lift force (FL)/ 11% pf. A leastsquares
ﬁt performed on this data gives the empirical
relationship
2.31
(52 = 20. 9 . (63) 2
VrPr Vi This is valid over the range 1.8 > rut/up 70. T The JKR adhesion model was developed for the quasistatic
contact of two bodies. We have used it to describe the motion of a
deforming particle adhered to a surface. The application of the
model to this situation is justiﬁed if changes in deformation occur
quasistatically. Such conditions are appropriate if the timescale for
the displacement is much greater than the time taken for an elastic
stress wave to traverse the asperity many times. From parameters
we use later it can be shown that this condition is met for
particles < 100 um radius. 582 Unfortunately at present there are no suitable pub—
lished measurements of the RMS lift force. However,
what measurements that do exist (e.g. Chepil 1959)
indicate that the RMS lift force is comparable to the
mean lift force. In these calculations we shall assume
the RMS lift force is equal to the mean lift forcezi. 3.3. Evaluation of a normalised energy spectrum of
ﬂuctuating lift force There are, as yet, no reported measurements of the
energy spectrum associated with the ﬂuctuating lift force
suitable for our purpose. In the absence of such
measurements our estimation of the normalised energy
spectrum Elia») at the resonance is based on Schewe’s
(1983) measurements of the energy spectra of the wall
pressure ﬂuctuations beneath a fully developed tur
bulent boundary layer for a smooth wall with zero
pressure gradient. (It seems reasonable to assume that
the timescale of lift force ﬂuctuations is related to that of
the pressure ﬂuctuations.) We deﬁne, thus, a universal
energy spectrum Ef(v+) for normalised frequencies + = _
v v “E (64)
where
. v
am=ﬁawu (m We base E I (11*) upon the energy spectrum of pressure
ﬂuctuations exhibiting a decay ~(v+)‘7/3 in the high
frequency range of Schewe’s measurements (absence of
any spatial ﬁltering of the pressure ﬂuctuations due to
the size of the pressure sensor). The form of Eﬁ(v+)
we have chosen is a simpliﬁed form of Schewe’s
spectrum, in which E ff (W) is ﬂat up to some frequency
ug', beyond which it decays as (v+)‘7/3. Thus more
precisely 4
Ef(v+)=7v+ u” <U§’
s
= "12(Ui)“/3(v+)"7/3 vi 3 Us“. (66) In these calculations we take v; = 0.4. (We assume
that this value and —7/3 decay are unaffected by particle
size (absence of spatial ﬁltering).) This assumed energy
spectrum is considered appropriate for an aerody
namically smooth substrate in zero pressure gradient.
No measurements are available for the effect of aero~ dynamic roughness and pressure gradient 0n pressure
ﬂuctuations. 3.4. Evaluation of the damping constant There are two ways in which the vibrational energy of
the particle in contact with a surface can be dissipated; 1 Initial measurements being carried out at present indicate that
the RMS 11ft Will be slightly less than the mean lift force, but at present this will not seriously affect qualitative results presented
later. by ﬂuid damping opposing the particle motion and by
the propagation of elastic waves in the solid substrate
(mechanical damping). The total damping associated
with motion in the direction normal to the area of
contact is assumed here to be the sum of the two. In order to estimate the ﬂuid damping it is necessary
to know how the actual contact geometry varies under
the inﬂuence of an applied external load. We refer thus
to the contact geometry shown in ﬁgure 3, based on the
JKR adhesion model. Throughout the motion of particle
and substrate the distance between particle surface and
substrate in the contact zone remains constant and it is
the contact area that changes. There is thus no signiﬁ
cant squashing of the ﬂuid between particle and sub
strate (ﬂuid gap effect). For the system considered here the main source of
ﬂuid damping is associated with the thin layers of posi
tive and negative vorticity close to the particle surface,
generated by the oscillatory motion of the particle. The effective thickness of this region, 6, (boundary
layer thickness) is given by Batchelor (1979) as where a) in this case is the natural frequency (since the
damping is only signiﬁcant around the resonance) and
Vi is the kinematic viscosity of the ﬂuid. The calculation of the damping (energy dissipation)
is a standard example in the use of boundary layer
theory (see, for example, Batchelor 1979). For a sphere
of mass m, and radius r, oscillating in an unbounded
ﬂow (dynamic viscosity m) with small amplitude of
vibration (68) _ 6m2uf __ 6nr2uf < a) )1/2
'Bf _ m6 ~ m 212‘
and is valid for 6/r< 1. For a 10 um radius particle
vibrating in air with a frequency of 107 rad 5‘1 (see ﬁgure
4), 6 ~ 2 pm. Therefore, the use of this expression for
such particles in unbounded ﬂow is justiﬁed. When considering a particle near a surface there will
be additional dissipation of energy due to the oscillation
of ﬂuid in and out of the narrow annular gap between
the particle and surface. In most cases of interest, this
gap is increased in height by the presence of the asperit—
ies between the particle and surface. This reduces the Particle resuspension velocity of the ﬂuid in and out of the gap relative to the
velocity of the particle normal to the surface. In such
cases the dissipation in this annular region does not
signiﬁcantly alter the total energy lost. We note that the displacements of the particle are
very small compared to its diameter. In such situations
the changes in the ﬂuid lift force will be negligible. The damping constant in equation (68) is a function
of the frequency of the motion, whereas in our equation
of motion (equation (1)) we assumed it was constant.
However, the damping is only signiﬁcant over a very
narrow range of frequencies around the resonances for
which the damping is effectively constant. In evaluating the mechanical damping, it has been
shown that the elastic impact of a particle onto a surface
involves an energy loss due to the propagation of elastic
waves into the massive, plane body (Hunter 1957, Reed
1985). In a similar manner, we would expect the propa
gation of elastic waves to contribute to the damping of
a vibrating particle on a surface. We shall use directly
results obtained by Miller and Pursey (1954) for the
absorption of vibrational energy in an isotropic elastic
semiinﬁnite material. The source is assumed to be a
circular disc of ﬁnite radius vibrating normally to the
surface of the medium under the inﬂuence of a periodic
force. From Miller and Pursey (1954, equation (31)), the
net energy lost per second is wz a
W: . —
4 8.71: (2”) p262, (69) where the periodic force, f= f0 cos wt, and C2 is the
elastic wave velocity =(E2/p2)1/2. For a lightly damped forced harmonic oscillator of
the form considered, it is readily shown that the energy
dissipated per oscillation, W0,c is W... = J—L (70)
2mm?“ + col) where m is the mass of the oscillator and ﬁn, the damping
constant. Equating W0SC with W, and recognising that,
in practice, a) > ﬁm, we have 2.4 mm4 Table 1. Material and flow properties used to calculate the resuspension rates in air of glass spheres on steel surfaces. Diameter of channel
Particle density, pp Gas density. p, Substrate density, p2 Gas kinematic viscosity, :2, Friction velocity, ut for 60 m s~1 bulk ﬂow Adhesive surface energy, Ay
Young’s modulus of particle (glass), E1 Young’s modulus of substrate (stainless steel), E2 Particle Poisson’s ratio, a,
Substrate Poisson's ratio, 02 0.2 m 2470 kg m3
1.18 kg m“3 7.8 X 103 kg m~3
1.54 X10“5 rn2 s“
2.19 m s"1 0.15 J m‘2 8.01 X 1010 Pa
2.15 X 1011 Pa
0.27 0.28 583 M W Reeks el al 4. The dependence of resuspension on flow, 4.1. Resuspension from identical adhesive sites particle Size and surface mughness As a preliminary to evaluating resuspension for rough surfaces, we consider resuspension from identical
adhesive sites. In this situation the value of the rate
constant, p, is the same for each particle in contact with
the surface. For these particles the fraction fR remaining
on the surface will decay exponentially with the time
they have been exposed to the ﬂow, t, i.e. For illustration we consider the resuspension of spheri
cal glass particles from a stainless steel substrate exposed
to a fully developed turbulent air ﬂow in an aero
dynamically smooth channel. The relevant ﬂuid and
material properties are given in table 1. Throughout these illustrations we have taken the adhesive radius, ra, to be 1/10 of the particle radius and f = 61,,
have neglected any gravitational effects. R ' (72)
From these properties we have evaluated the ﬂuid Figure 5 shows the variation of fR (after one second
and mechanical damping constants as a function of exposure) as a function of friction velocity for a range
particle radius for zero applied load. These are shown Of partieie Sizes. The iranSitiOH 0f fa from unity to
in ﬁgure 4, together with the corresponding values of effeCtively zero with increasing ﬂow is extremely sharp
the natural frequency, m. It is clear that, for this com in all cases; this demonstrates the eXiStenCC 0f a
bination of materials, the major contribution to the total threshold friction velocity for resuspenSion which
damping (/3 = ﬁr +. .3111) arises from the ﬂuid damping, decreases with increasing particle size. We recall this
ﬁr dependence is a feature traditionally explained in terms of a force balance of aerodynamic and adhesive forces. Figure 6 shows, in more detail, the variation of threshold friction velocity with particle radius for 50% r 10% removal after one second exposure. Also shown are the '3 corresponding values of n (the factor indicating the 42 war contribution of the near—resonance region) and the ratio I
g of the force of adhesion to the RMS lift force. The near
ff 2 resonance region dominates the resuspension through
E10 out the range of particle size such that an RMS lift force
as of order 5% of the force of adhesion will remove a
W \ particle from the surface in times of order a second.
10% w . , “16’s A . L , . “L104. Finally, ﬁg'ure7 illustrates the inﬂuence of the near
resonance region on the energy spectrum of the particle P f‘l d'  
“l'” "l '“5 ‘m’ mean square displacement/potential energy at equi librium. The case illustrated is that for a 10 ,um radius Figurg 4_ The variation of ﬂuid damping ﬁt. mechanical particle using the ﬂow conditions for 50% removal after
damping ﬁn}, and natural frequency of vibration on, with one second (see ﬁgure 6). Even though the relative
pﬁggﬁtgadlus. for glass Spheres "1 all, on a Steel contribution of the nearresonance region to the mean square lift force is so small (see curve A), the high selectivity of the frequency response results in most of
We note that in the cases considered, the ratio w/p the contribution coming from this region. > 1 conﬁrms our initial assumption that the system is lightly damped. Value Fraction remaining after 1 s 1‘ \‘x. ‘ ............ "a ...... ms ..... WNW». .... '0 105 to"
Friction velocity (m 5") PM“ mm (m)
Figure 5. The fractional resuspension of glass particles in Figure 5_ Conditions for 50% resuspension of glass air from a steel surface. Curves: A, 20 um. spread a; =
4; B. 20 am, no spread; C, 50 ,um, spread 0; =4; D.
50 am, no spread. particles from a steel surface after exposure In air flow.
Lines: A, friction velocity (m s"); B, resonant factor, n; C,
ratio of the force adhesion to the RMS lift force. 584 ‘4___—__4 mSELlulw]  y21<rf> 10' 10" Particle resuspension 10 10" uELlululeE) 10'a ‘i 10 Relative frequency, u/w Figure 7. The energy spectrum of the particle mean
square displacement at equilibrium (C) as the product of
the energy spectrum of the driving force (B) and the
frequency response for a 10 )um radius glass particle in air
(A) on a steel surface (1‘;I = 0.1, a; = 4). Flow conditions
for 50% removal are given in figure 6. 4.2. Resuspension for rough surfaces Most surfaces involved in resuspension are rough; they
possess a topography which can be characterised by a
distribution in height and radius of curvature of surface
asperities. This will produce both a reduction and spread
in the force of adhesion compared with that for the
contact of smooth surfaces. There will also be a spread
and reduction of the natural frequency but it is clear
from the formula for p, in equation (36), that it is
the distribution of adhesive forces which will have the
greatest inﬂuence on the resuspension rate. Therefore, let us consider the resu5pension, at an
exposure time t, of an ensemble of rough spherical
particles of radius r, all initially distributed uniformly
on a rough surface, Suppose that any one single particle
from the ensemble experiences a force of adhesion fa,
corresponding to an adhesive radius ra. Assuming the
same material constants, let us normalise this force of
adhesion on the force of adhesion for a smooth sphere
of radius r on a perfectly smooth surface. This is equiv
alent to normalising ra on the radius of the sphere. We
deﬁne thus r; = ra/r <73) and let cp(r;) be the probability density for the occur
rence of r;. The fraction of particles, fR(t), remaining on the
surface at time t, is thus given by rat) = [a explvptrntirprrn dri (74) 0 and the fractional resuspension rate A(z) by A0) = vita) = [warn explp(rl)tl<p(rl) dri
0
(75) Using these expressions for fR(t) and Mr), we have
investigated the behaviour of both these quantities for
a log—normal distribution of normalised adhesive radii,
r5. Such distributions are typical of the distributions
normally encountered with surface adhesive forces
(Bijth et al 1962, Reed 1986). Thus ¢(r,’,) is of the form (r'>— 1 i——~——1
(p a (2701/2 r; (in 0,5,)2 1
x exp(— [met/aw). (76) Here, r‘; is the geometric mean of r; and is a measure
of the reduction in adhesion due to surface roughness
(adhesion reduction factor). a; is a measure of the
spread in adhesive forces (adhesion spread factor).
Typical values are r"; = 0.1, and a; = 4 (Reed 1986). If
the force of adhesion scales on the particle radius then
cp(r,’,) will remain invariant to changes in particle size. 4.2.1. The fraction remaining, fR(t). Figure 5 shows how surface roughness, characterised by a reduction and
spread in the force of adhesion, inﬂuences the effect of 585 M W Reeks et a! particle size and ﬂow on the fraction remaining after
one second exposure. The friction velocity for the
chosen rough surface (0;, = 4) is no longer a well deﬁned
threshold velocity. In practice a threshold velocity
occurs only for a narrow spread of adhesive forces
(or; s 2). The friction velocity for a remainder of 50%
is almost identical to that given in ﬁgure 6 for a distri
bution of identical adhesive sites (a; = 1, r,’, = 0.1). 4.2.2. The fractional resuspension rate, A(t). The
expression for Mt) in equation (75) has been evaluated
numerically for the standard case of resuspension of
glass spheres from a rough stainless steel surface. As an
example, ﬁgure 8 shows A(t) for the resuspension of
25 pm and 50 pm radius spheres exposed to an air ﬂow
of 60 m s”. The calculations were performed for a log—
normal distribution of normalised adhesive radii with
roughness parameters a; = 4, r"; = 0.1. We note that in the deVelopment of the resuspension
model the particles are assumed to have had many
interactions with the ﬂuid lift force. This assumption
will no longer be valid when particles are weakly bound
to the surface (i.e. are predicted by the formula to
resuspend in times less than the ﬂuid timescale).
Accordingly, we have limited the maximum value of p
to the bursting frequency VB of turbulent motion in
a turbulent boundary layer. Measurements by
Blackwelder and Haritonidis (1983) show 113 to be of
order iii/(300 vf). Essentially we have assumed that
weakly bound particles have to encounter a burst before
they resuspend. The variation in A(t) is far from the exponential
decay associated with identical adhesive sites (smooth
surfaces) and naturally divides into two regions. (i) A short ‘initial’ resuspension (over times s 10”2 s
in this example) in which the resuspension rate is very
high. Such resuspension is usually responsible for a
significant fraction of the total resuspension from a
rough surface. In the examples chosen the fraction
resuspended at 10’23 is 59% and 87% for the 25 ,um
and 50 pm radius spheres. (ii) A region of ‘longerterm’ resuspension in which
the resuspension rate is seen for both particle sizes to vary almost inversely with the time of exposure to the
ﬂow, i.e. A(t) = 5t“ (77) where E is a constant, and a = 1. In these particular
examples 8 ~ 1.07. Figure 8 indicates that this inverse
relationship is maintained over a wide range of exposure
time (10‘25 < t < 105 s). In fact, our numerical cal—
culations show that this inverse relationship is extremely
robust to wide variations in flow, particle diameter and
surface roughness. It is of no practical signiﬁcance that
we have chosen a logenormal distribution for our distri
bution of adhesive forces. (Note, for longerterm resus—
pension in the example chosen, A(t) for 50 pm radius
particles is less than that for 25 pm radius particles. This
is a consequence of the corresponding higher initial 586 resuspension of the largersize particles. There is, there
fore, less availability of the 50 pm size particles than
the 25 um particles for longerterm resuspension. The
difference in behaviour shown in ﬁgure 8, is not univer
sal; it depends critically upon the relative proportion
of initial resuspension; little initial resuspension would
reverse this trend in behaviour.) The non—exponential decay is related to the fact that,
though the fraction of particles on a rough surface
decreases with time, the relative proportion of strongly
adhered particles increases. That it decays so close to
1/t relates to the form of p and its sensitivity to the
force of adhesion. For example, let us write equation
(48) for p as a) r; — r" 2
Jexp ( " (78) where r6 and F6 are constants depending on ﬂow, par
ticle size and surface energy. p is most sensitive to
variations of r; in its exponent rather than in too. For
simplicity therefore, we neglect the slight dependence
of (no on r,’,. Consider thus the integral form for Mt) in
equation (75). For root» 1, the values of r; for which
the function p 6“” in the integrand is signiﬁcantly dif
ferent from zero, is very sharply deﬁned. The maximum
values of p e"” is e“l/t and occurs when _ mo, 1/2
r,’, = r5 + r6 [in . (79) Its width is of order The value of A is thus typically 3% (wow {, , war In
A all“ 2,. ‘P “Mimi‘s?” } (80) where We have assumed in practice that the variation in
(p is small over the width of p e“”‘. Clearly, the width
of pe‘l’r changes on a much longer timescale, O[ln
(wot/2n)]'1/2, than does its height, opl]. It suggests
that the dependence of A on t is extremely close to t’x
in the limit of wot» 1. These features are illustrated in
ﬁgure 9, where the function p e‘P’ is shown as a function
ofrg/n’, att= ls, and t= 10s. Evaluation of p was based on the form for p given
in equation (78) with coo/2:: = 107 Hz, and F5 = 0 for
convenience. The argument, though not rigorous, gives a result
which is consistent with our numerical calculations of
A(t). (For a more rigorous derivation of the form of A(t) in this region we refer to Please and Wilmott
(1987).) Longerterm resuspension —————— l 1. l ___ l 4... I—J_.. .L
10 10'3 10‘1 103
f (s) Figure 8. The variation of the fractional resuspension rate,
A(t), with exposure time, t, for glass spheres on a steel
surface exposed to an air flow of 60 m s“. Roughness,
a; = 4,?; = 0.1. Curves: A, 25 pm radius; B, 50 um radius. 4.3. Experimental conﬁrmation of Ur decay law 4.3.1. Measurements of resuspension from grass (Gar
land 1979). Garland has measured the longerterm
resuspension rate of small particles from grass exposed
to various air ﬂows in a wind tunnel. The particles
(radioactively labelled) ranged from submicrometre
tungsten oxide powder to silt from sea beaches.
Measurements of the resuspension factor (as deﬁned
by, for example, Sehmel 1980), proportional to the
fractional resuspension rate, were made over a period
of many hours exposure to a constant air ﬂow. For all
particles and ﬂows considered the resuspension rate
varied almost inversely with the exposure time. As
an example, ﬁgure 10 shows Garland’s results for the
resuspension of silt from grass. 4.3.2. Measurement of the decay in gasborne con
centration of small particles in a recirculating turbulent
flow. A striking example of the inﬂuence of longerterm
resuspension is seen in the decay of gas—borne particle Arbitrary tplrg/ro’l [‘0' [1'5 Figure 9. An illustration of the factors leading to a 1/1‘
decay law for longerterm resuspension values for p are
based on equation (78) with too/21:: 107 Hz. and F6 =0. Particle resuspension Resuspension factor (tn—‘ldlth‘l Accumulated running timeJ lhl Figure 10. The variation of the resuspension factor with
running time for silt on grass at wind speeds of 5 m s‘1 (O)
and 10 m s‘1 (+). Reproduced from Garland (1979), with
permission. concentration in the coolant ﬂow of a Civil Advanced
Gas Cooled Reactor (CAGR). In such a recirculating
turbulent ﬂow, deposition and resuspension at surfaces
exposed to the ﬂow occur simultaneously. The initial
resuspension increases the decay time by reducing the
net deposition to a surface; the longerterm resus
pension reduces the decay in a manner which depends
on the history of the gasborne concentration. The appropriate equation for the concentration C(t) at time t is an integro—differential equation of the form
(Reeks and Hall 1986) ac —a—t = — ac(t) + b for” A(t — s)c(s) ds (81) where a is a decay constant modelling all particle
removal mechanisms including deposition and initial
resuspension from surfaces plus natural leakage; b is a
decay constant modelling only the deposition to surfaces
exposed to the ﬂow; and ti is the duration time for initial
resuspension. (The value of ti is not critical so long as
it is sufﬁcient to include all initial resuspension.) The integral term on the righthand side of equation
(81) represents the contribution to the decay rate from
longerterm resuspension. Our previous considerations
imply that the fractional resuspension rate, A(t  s),
will have the form 5
(t—S)E A(t — s) = (82)
where s = 1. Numerical solutions of equation (81) with A given
by equation (82) behave in a manner consistent with
the observed decay of gasborne particles injected into
the coolant of a CAGR (Wells et al 1984). The injected
particles were labelled with radioactivity for identi
ﬁcation, and measurements made of sampled particle
gasborne activity over a period of 1—200 min from the
start of the injection. 587 M W Fieeks et al .. 8 O
I I
n _. .s
o
J: Relative gasborne concentration,clf)/r(0l 10 i 101 103
Decoy timer (5) Figure 11. Injection experiment results for gasborne
concentration of 2 pm diameter iron oxide particles. (The
initial rise in concentration is due to mixing.) Symbols: 0.
experiment; —X, deposition plus initial resuspension
plus longerterm resuspension; ———><, deposition plus
initial resuspension. As an example the results for the decay in sampled
activity (corrected for background) for 2 pm size iron
oxide injected particles are shown in ﬁgure 11. Shown
also are the corresponding values obtained from the
bestﬁt solution of equation (81) using equation (82) for
A(r  s) with a = 1.1 (numerical solutions for c(t) with
6 outside the range 1 < a s 1.1 gave unacceptably poor
ﬁts). For comparison we haVe also shown the decay aris
ing from deposition and initial resuspension alone (i.e.
no longerterm resuspension). Of the two decays, that
based on the integro—differential equation (with longer
term resuspension) is clearly a much better ﬁt to the
data. It shows the marked inﬂuence of longerterm
resuspension on the ﬁnal stages of the decay. (Please
and Wilmott (1987) have shown that in the limit of
t——> 00, C(t) from equation (81), arising from an instan
taneous point source, behaves ~t‘1.) Further investigations suggest that forms of A(t  s)
in equation (81), other than the form proposed in
equation (82), give unacceptable ﬁts to the experimental
data. Therefore, not only are these data evidence of
longerterm resuspension but, more precisely, evidence
of a longerterm resuspension rate which decays almost
inversely with the time of exposure on the surface. 5. Summary and conclusions Let us now summarise the basic features of this new approachrto particle resuspension and discuss its impli
cation for the behaviour of similar systems. We have considered the ﬂuctuating lift force as an
agency for the transfer of turbulent energy to a particle 588 on a surface from the surrounding flow. In this way a
particle is able to accumulate vibrational energy until
this energy is sufﬁcient to detach it from the surface.
This transfer of energy takes place most efﬁciently at
forcing frequencies close to the natural frequency of the
vibration (resonant energy transfer). The formula for the resuspension rate constant is
similar in form to the Arrhenius formula for the
desorption rate constant of molecules from a surface
and for chemical activation rate constants in general.
As a consequence of resonant energy transfer, estimates
of the resuspension rate constant in practical cases indi
cate that particles are resuspended more easily than on
a balance of aerodynamic lift and adhesive forces. The
commonly observed feature of decreasing threshold
ﬂow velocity for resuspension with increasing particle
size is explained here in terms of the balance of the
adhesive energy (height of surface potential well) and
the mean vibrational potential energy. We recall that our estimates for resuspension were
based upon the adhesion of small elastic particles on an
elastic substrate under the inﬂuence of van der Waals
adhesive forces. However, other situations and removal
mechanisms should, in principle, be capable of analysis
using the same statistical approach. Possible con
siderations are the resuspension of charged or anelastic
particles from surfaces and turbulent agglomeration. Finally, application of this approach to resuspension
for rough surfaces revealed a feature previously
observed experimentally, namely that for ﬂow exposure
times typically greater than a second, the resuspension
rate varies almost inversely with time. This result, we
ﬁnd, is extremely robust to wide variations in particle
size, How and surface roughness. Acknowledgment The work described was carried out by Berkeley
Nuclear Laboratories and is published with the per
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