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Unformatted text preview: Bibliotheek TÂ» D /////1
W Uitsluitend VOOI' eigen gebruik / for own use only Prometheusplein 1 Datum: 2 2 â€”OCt â€” 0 3
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Email: [email protected] Aan; T.N.O. 'I'ECHN. PHYSISCHï¬ DIï¬NSi POSTBUS 155
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Nï¬D *. RTAND Tav; A. van de Runstraat Aantal kopieÃ©n: 16 Uw referentie(s): 800 . 05105/01 . 01 Artikelomschrijving bij aanvraagnummer: 7 l 3 2 9 5
Artikel: 01â€™1 the resuspension of smaâ€˜ â€˜ partic1es by a turbulent flow Auteur; M W Reeks, J Reed and D Haâ€˜ â€˜
Tijdschrift; JOURNAL OF PHYSICS D. APPLIED PHYSICS
Jaar: 1988 V01. 21 Aï¬‚evering:
Pagina(s): 574â€”589 Plaatsnr.: 803 D Met ingang van 1 april 2003 zullen de prijzen v00rf0t0k0pie levering baitenland stijgen met 6â€˜ 0,05 per pagina From April I 2003, prices for photocopy delivery abroad will increase by 6â€˜ 0.05 per page J. Phys. D: Appl. Phys. 21 (1988) 574â€”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 longâ€”term resuspension of the form
p ~ too explâ€” 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 particleâ€”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 ï¬‚uid is a common occurrence in nature.
Familiar as the origin of dust storms, it is inï¬‚uential
in a diverse range of industrial and environmental
processes from the erosion of soil and river beds to
the â€˜wave of darkeningâ€™ 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 lookâ€”
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
ï¬‚ow. .It illustrates the similarity of particle resuspen
sion by a turbulent ï¬‚ow 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 Â© 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 difï¬cult 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 aerodyâ€”
namic removal, was initiated by ï¬‚ows whose velocity
exceeded some threshold value. More speciï¬cally this
threshold ï¬‚ow 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 ï¬‚uid close to the surface.
We shall call this resuspension longterm resuspension,
and it is the statistical process leading to this resusâ€”
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 ï¬‚ow
visualisations of Kline et al (1967) and Corino and
Brodkey (1969) clearly show that ï¬‚uid motion in the
viscous sublayer of a turbulent boundary layer is
identiï¬able in terms of a deï¬nite sequence of coher
ent structures, i.e. â€˜ejectionsâ€™ followed by downward
â€˜sweepsâ€™ with occasional â€˜interactionsâ€™ (turbulent
bursting). Several models for resuspension, most
notably that of Cleaver and Yates (1973), have ident
iï¬ed 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
inï¬‚uence of the transfer of turbulent energy to a par
ticle on a surface from the resuspending ï¬‚ow. Such a
transfer takes place through the agency of the aero
dynamic lift force which ï¬‚uctuates 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 sufï¬cient 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 ï¬‚uid and substrate
(ï¬‚uid 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 ï¬‚ow velocity for detachment
in this approach can be signiï¬cantly 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
ï¬nal 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 ï¬‚ow. 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 ____________â€”___â€”â€”â€”_â€”â€”â€”â€”â€”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 ï¬gure 1. We suppose that the number of particles on the
surface is sufï¬cient 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 ï¬‚uctu
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" + Â£3); + cozy = m'lfLOâ€˜) (1) where fL(t) is the random ï¬‚uctuating 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
(ï¬gure 1), and [3 is the ï¬‚uid 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/ï¬ >1 and (or >1 (2) where r is the typical timescale of the lift force
ï¬‚uctuations. In order to determine the resuspension rate con
stant we must ï¬rst 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 ï¬xed 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 ï¬‚uid 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 ï¬nal 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
ï¬xed 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:â€” (n we (3)
where 6(3) is a step function deï¬ned 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 probâ€”
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 â€” s) ds (6) 0 which, using equations (3) and (4), becomes y(t)=m_2fnfm Sinw(tâ€”s)ds+~1â€” (n 1)r ma) I
X] stinw(tâ€”s)ds forNr<!<(N+1)r
Nr 2 , cur N
._ â€”â€”mw2 Sin7'ng s1n w[tâ€”(nâ€”i)1:] + % X (1 â€” 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
WÂ» = 2.04 0â€˜â€ (SW 923) (i + 0(1)) + Â£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
ï¬‚uid element diffusion in a homogeneous stationary
turbulent ï¬‚ow to a random walk. If the diffusion time
of the harmonic oscillator is sufï¬ciently 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 signiï¬cant 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 â€˜impulsesâ€™ (each lasting
a time I) before reaching equilibrium. So we would
expect the ï¬nal 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 ï¬rst consider its
frequency response, C(v, t), to a forcing term ehâ€, i.e.
C(v,t) is the solution to equation (1) with a forcing
term ei'â€ 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 Fourierâ€”Stieltjes integral
(see, for example, Monin and Yaglom 1975), namely int) = (m eiv'dzw) (11)
(with dZ(v) = dZ*(â€”v) for the reality of fL(t)), we
have y(t) = mâ€”1 (m ctv, r) dzm <12) V(t) = y(t) = mâ€˜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(Uâ€™)) = â€œ(v + vâ€™)EL(v) dv duâ€™ (14) where EL(â€™U) is the energy spectrum of fL(t) (see
Monin and Yaglorn 1975). Thus using this property
we obtain from equations (12) and (13) WÂ» = m'2 (m duEttumv, or (15)
0 (We) = m2 [m dumvncâ€˜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 = Â£3 (eiuâ€˜ â€” 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 (ï¬t > 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 nearâ€”resonance response (2) ~ iw) of the
form eiiw __ eâ€”ï¬r/z etiml E;(:tv,t) = 2 w  v2 : i/iv (19) For a)â€œ <t<< [3â€œ, 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 proâ€” 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 ï¬‚uid damping inhibits the growth of resonant
energy through its suppression of the initial transient,
so that the ï¬nal state of equilibrium of constant
amplitude, gives extra: 6 iiÃ©vr
gi%0:2wÂ®viMï¬‚) conï¬ning the nearresonant region to a narrow band of width ~13 about n).
Thus, separating out the lowfrequency and nearâ€” 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 ~ 6â€˜5/2â€™[cos(v ~ w)t
0 mzwâ€œ + cos(v + (0):] + 2"â€ cos2 wt} EL(v) dv (21) "n1
(V2(t))lf = ;n2â€”1wâ€”4 [0 {U2 â€” e'ï¬‚â€˜ï¬‚vwkosw â€” w)t â€” cos(v + w)t] â€” e'ï¬â€™ sin2 wt}EL(v) du. (22) (The upper limit of am is not critical, so long as it is
sufï¬cient to exclude the contribution from higher fre
quencies (v > to) damped out by the ï¬‚uid. The obviâ€”
ous choice for â€™Um is the natural frequency, w.) (ii) near resonance (res). Writing the frequency
response [Â§(v, t)l2 around a) in terms of 62/ = v  a) zwimz I: [502 + (91â€”1 x (1 â€” C'ï¬l/Z cos 5m + eâ€œï¬â€™)EL(w + 6v) d(5v).
(23) If ï¬r <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 Â§(v,t)2 is signiï¬cantly
greater than 1/ a)â€œ. So after performing the necessary
integration, we have 02(t)>res z wÂ»... = 25:,â€œ (1 â€” eï¬aEuw) (24) and similarly .7! (V2(t)>res = zï¬mz We note that in the limit of ï¬t< 1,012)â€, is
proportional to time. This is the same as the domiâ€”
nant response of the harmonic oscillator driven by
the series of random impulses (see the ï¬rst term on
the righthand side in equation (8)). Indeed, we can
formally identif...
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