Reeks 1988 - Bibliotheek T» D /////1 W Uitsluitend VOOI'...

Info iconThis preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 16
Background image of page 17
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Bibliotheek T» D /////1 W Uitsluitend VOOI' eigen gebruik / for own use only Prometheusplein 1 Datum: 2 2 —OCt — 0 3 Postbus 98 2600 MG DELFT Bonnummer: 7 1 3 2 9 5 Telefoon: 015 - 2784636 Fax: 015 - 2785673 Email: Klanten-Service@Library.TUDelft.NL Aan; T.N.O. 'I'ECHN. PHYSISCHfi DIfiNSi POSTBUS 155 2600 AD DELFT NfiD *. RTAND Tav; A. van de Runstraat Aantal kopieén: 16 Uw referentie(s): 800 . 05105/01 . 01 Artikelomschrijving bij aanvraagnummer: 7 l 3 2 9 5 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 long-term 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. 0022-3727/88/040574 + 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 inter-surface 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 long-term 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 (quasi-static). 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 long-term 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 long-term 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 Sin-7'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 long-term 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 sub-intervals; the displacements induced by the driving force in each of these sub-intervals 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 long-term 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 right-hand 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) = m-2 [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 low-frequency (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) ~ i-w) 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 near-resonant region to a narrow band of width ~13 about n). Thus, separating out the low-frequency 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 — e-fiaEuw) (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 right-hand side in equation (8)). Indeed, we can formally identify the first and second terms on the right-hand side of that equation with the near-reson- (1 — crane). (25) 578 ance and low—frequency responses, respectively. It suggests that Gaussian statistics are, in general, exclusive to the near-resonance 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 low-frequency 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 non-linear 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 long-term 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 + Mari-V3 +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 least-squares 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 quasi-static 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 quasi-statically. 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 semi-infinite 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 m-3 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 = 6-1,, 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 near-resonance 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 expl-p(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 ‘longer-term’ 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 longer-term 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 larger-size particles. There is, there- fore, less availability of the 50 pm size particles than the 25 um particles for longer-term 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, op-l]. 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).) Longer-term 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 longer-term 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 gas-borne con- centration of small particles in a recirculating turbulent flow. A striking example of the influence of longer-term 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 longer-term 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 longer-term resus- pension reduces the decay in a manner which depends on the history of the gas-borne 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 right-hand side of equation (81) represents the contribution to the decay rate from longer-term 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 gas-borne 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 gas-borne 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 gas-borne concentration,clf)/r(0l 10 i 101 103 Decoy timer (5) Figure 11. Injection experiment results for gas-borne 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 longer-term 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 best-fit 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 longer-term 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 longer-term 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 longer-term resuspension but, more precisely, evidence of a longer-term 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- mission of the Central Electricity Generating Board. References Aylor D E 1978 Plant Disease (New York: Academic) Bagnold R A 1941 The Physics of Wind Blown Sand and Desert Dunes (London: Methucn) Batchelor G K 1979 An Introduction to Fluid Mechanics (Cambridge: CUP) pp 353—8 Blackwelder R F and Haritonidis J H 1983 J. Fluid Mech. 132 87—103 Bdhme G, Krupp H, Rabenhorst H and Sandstede G 1962 Trans. Inst. Chem. Eng. 40 252—9 Buyevich Yu A 1971 J. Fluid Mech. 49 489~506 Chandrasekhar S K 1943 Rev. Mod. Phys. 15 1—89 Chepil W S 1945 Soil Sci. 60 305—20 w 1959 Soil Sci. Soc. Am. Proc. 23 422—8 Cleaver J W and Yates B 1973 J. Colloid Interface Sci 4 464—74 Corino E R and Brodkey R S 1969 J. Fluid Mech. 37 1-30 Corn M and Stein F 1965 Am. Ind. Hyg. Assoc. J. 26 325—36 Gane N, Pfaelzer P F and Tabor D 1974 Proc. R. Soc. A 340 495—17 Garland J A 1979 UKAEA Report AERE R9452 Hall D 1988 J. Fluid Mech. 187 451—66 Hunter S C 1957 J. Mech. Phys. Solids 5 162*71 Johnson K L 1958 J. Appl. Phys. 9 199—200 Johnson K L, Kendall K and Roberts A D 1971 Proc. R. Sac. A 324 301—13 Kline S J, Reynolds W C, Schraub F A and Runstadler P W 1967 J. Fluid Mech. 30 741—73 Krupp H 1967 Adv. Colloid Interface Sci. 1 111—239 Miller G F and Pursey H 1954 Proc. R. Soc. A 223 521—41 Monin A S and Yaglom A M 1975 Statistical Fluid Mechanics vol 2 (Massachusetts: MIT) Phillips M 1980 J. Phys. D: Appl. Phys. 13 221—33 Please C and Wilmott P 1987 Math. Eng. Ind. 1 21—32 Reed J 1985 J. Phys. D: Appl. Phys. 18 2329—37 ~— 1986 unpublished Particle resuspension Reeks M W 1983 Atmospheric Dispersion of Heavy Gases and Small Particles ed. G Ooms and H Tenneke (Berlin: Springer) pp 221—40 Reeks M W and Hall D 1986 Symp. Gas—Solid Flows, FED vol 35 (ASME) pp 1—8 Reeks M W, Reed J and Hall D 1985 CEGB Report TPRD/B/0639/N85 Sagan C and Pollack J B 1969 Nature 223 791-4 Schewe G 1983 J. Fluid Mech. 134 311—28 Sehmel G A 1980 Environ. Int. 4 107—27 Tabor D 1977 J. Colloid Interface Sci. 58 2—13 Taylor G I 1921 Proc. London Math. Soc. 2 196—212 Timoshenko S P and Goodier J N 1970 Theory of Elasticity (New York: McGraw-Hill) pp 409—13 Turco R P, Toon O B, Ackerman T P, Pollack J B and Sagan C 1984 Sci. Am. 251 23—33 Wells A C, Garland J A and Hedgecock J 1984 Proc. CSNI Specialist Meeting Nuclear Aerosols in Reactor Safety, Karlsruhe (CSNI) pp 366~75 589 ...
View Full Document

Page1 / 17

Reeks 1988 - Bibliotheek T» D /////1 W Uitsluitend VOOI'...

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online