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Reeks 1988 - Bibliotheek T D/1 W Uitsluitend VOOI eigen...

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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: [email protected] 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 identif...
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