numerical simulation of transport and agglomeration of diesel particles ...
ΡΟΗ 2008 6η
Επιστημονική Συνάντηση για τις
Ερευνητικές Δραστηριότητες στη Μηχανική Ρευστών στην Ελλάδα Κοζάνη, 28 Νοεμβρίου, 2008
NUMERICAL SIMULATION OF TRANSPORT AND AGGLOMERATION OF DIESEL PARTICLES UNDER THE INFLUENCE OF AN ULTRASONIC STANDING WAVE Charalambos D. Papadopoulos, Katerina Sardi, and George C. Bergeles National Technical University of Athens Department of Mechanical Engineering 5 Heroon Polytechniou, 15710 Zografos-Athens, Greece e-mail: [email protected]
Abstract A numerical investigation of the effect of ultrasound preconditioning on the motion and size distribution of particles, likely to exist at the exhaust flue gas of diesel engines, is presented. Two cases were investigated. In the former case, the effect of an ultrasonic field, in the form of a standing wave, on particle laden laminar and turbulent air mixtures travelling inside a straight rectangular channel was examined. In the latter case, the simple case of a particle laden homogeneous isotropic turbulence air field, without a convective flow component, inside a cube was considered. For this second case the modelling approach was extended to include inter particle collisions and agglomeration. In both cases a detailed parametric study related to the ultrasound system design (e.g. energy level and frequency) was carried out. Results showed that the application of an ultrasonic standing wave leads to increased particle concentration at the nodal planes, enhanced collision and agglomeration rates and thus to increased particle sizes. Higher frequencies and acoustic pressures are shown to promote agglomeration and larger particle formation.
Keywords: Ultrasound, agglomeration, diesel particles 1. INTRODUCTION Air pollution in the form of ultrafine particles with sizes ranging from 10 nm to 5000 nm is widely recognized as a major environmental challenge and a serious health hazard. The contribution of the transport and marine sectors to particulate matter (PM) accumulation in the atmosphere is significant and a large number of studies towards the reduction of PM emissions have been conducted (e.g. Colvile et al., 2001, Corbett and Koehler, 2003, Fridell et al. 2008). Although considerable progress has been made, particularly in the area of diesel on-road vehicles (improved fuels, improvement of the combustion processes), the installation of a Diesel Particulate Filter (DPF) is required in order to comply with the latest European Union standards (EURO5). The International Maritime Organisation in its 2005 review of the existing regulation for the prevention of air pollution from ships also considered controlling PM emissions, (MARPOL Annex IV, 2005). However currently available DPF’s are likely to become plugged with time and lead to high pressure drops. Moreover their efficiency in retaining ultrafine particles is questionable, Van Gulijk et al. (2001). Acoustic agglomeration via application of ultrasounds is a promising technique to be used for increasing the size of micron and submicron particles before they enter a conventional filter. During
1
Ch. Papadopoulos, K. Sardi & G. Bergeles
this process high-intensity standing waves are applied to the flue gas stream leading to increased particle concentration at the vicinity of the acoustic nodal planes and enhanced collision and agglomeration rates. This technique has already been tested in industrial sized chambers and it was shown that an up to 70% reduction of micron-sized fly-ash particles (in comparison to the initial number of particles) can be achieved, Hoffmann (2000). The purpose of the present work was to numerically investigate the effect of ultrasound preconditioning on the particle motion and size as a function of parameters related to the ultrasound system design (e.g. energy level and frequency). For this purpose, an in-house CFD code incorporating Lagrangian particle tracking has been extended to include a representation of the acoustic field and particle collision and agglomeration effects. Two cases were investigated. In the former, the effect of an ultrasonic field, in the form of a standing wave, on particle laden laminar and turbulent air mixtures traveling inside a straight rectangular channel was examined. In the latter case, the simple case of a homogeneous isotropic turbulence field without a convective flow component inside a cube was considered and inter particle collisions and agglomeration were computed as a function of acoustic frequency and amplitude. 2 MODELLING APPROACH The modeling approach is described in detail in Papadopoulos et al (2008). Only a brief outline is provided here. 2.1 Flow-field modeling In the first case (Case 1) the flow of a particle laden air mixture inside a straight rectangular channel with an inlet square section of h=0.0128m and length of L=0.1m was considered. The air stream was assumed to be moderately heated at 573 K so as to resemble conditions relevant to the flue gas stream travelling inside the exhaust system of a diesel engine. The three dimensional laminar and turbulent flow fields inside the channel were computed by solving the Reynolds averaged Navier-Stokes equations with the standard k-ε turbulence model. For the second case (Case 2) a cubic box of side of 0.01m was considered and the mean flow velocity was set equal to zero. Turbulent properties inside the box were prescribed to the values used in the study of Anh Ho and Sommerfeld (2002). The particle density in all computations was set to 1500 Kg/m3, a value that is typical of the density of particles emitted by diesel vehicles, Burtscher (2005), and cruise ships, Petzold et al. (2007). 2.2 Particle dynamics Particle motion inside the computational domains was computed by application of the Lagrangian approach. Considering spherical particles and neglecting heat and mass transfer phenomena and particle rotation, particle trajectories and local velocities were obtained from
du p.i dt
=
r r 3ρ Cd ρ Fac ui − u p,i ) u i − u p ,i + gi 1 − + + ni (t ) ( ρ m 4d p ρ p p p
dx p.i = ui dt
(1)
In equation (1) , xp, up,I, mp, dp and ρp are the particle position, particle velocity components, mass, diameter and density respectively, ui, and ρ are the velocity and density of the fluid flow, Cd is the aerodynamic drag coefficient (Allen & Raabe, 1985 Shimazaki et al., 2006) modified by the Cunningham correction factor for small particles, Cc so that Cdnew = Cd Cc , gi the gravitational acceleration, Fac the acoustic radiation force and ni(t) the Brownian force per unit mass. The fluid
2
Ch. Papadopoulos, K. Sardi & G. Bergeles
velocity components, also required for the computation of the drag force, were estimated by application of the stochastic separated flow (SSF) model of Gosman and Ioannides (1983). To include the effect of the Brownian motion in the simulations, the Brownian force per unit mass ni(t), was modelled as a Gaussian white noise random process with spectral intensity, Li and Ahmadi (1992). The acoustic radiation force Fac at a direction z, normal to the walls where the standing wave was applied was computed from Kawasima (1955) as
ur r F ac = −Vo Eac KG sin ( 2 Kz ) z
(2)
where Vo is the particle volume, Eac is a measure of the energy residing in a wave field and a function of the peak acoustic pressure amplitude Po and the speed of sound c, K = 2πc / f is the acoustic field wave number, f the acoustic wave frequency and G the acoustic contrast factor which determines the response of any solid particle to a resonant acoustic field. For the particular case of solid particles suspended in air G was set to 2.5, Anderson et al. (2006). Inter-particle collisions were modelled by application of the stochastic particle/particle collision model of Sommerfeld (2001). The model relies on the generation of fictitious collision partners and at each time step of the particle trajectory calculation, a collision probability, Pcoll, is estimated. The latter stems from gas kinetic theory formalism under the assumption that the fluctuating motion of the particles is analogous to the thermal motion of molecules in gas Pcoll =
r 2 r π d p1 + d p 2 ) u p1 − u p 2 N p ∆t ( 4
(3)
In equation (3) dp1 and dp2 are the diameters of the considered and the fictitious
r r urel = u p1 − u p 2
is their instantaneous relative velocity (Berlemont et al., 2001), and Np is the particle, number of particles per unit volume at the considered control volume. The time step Δt is selected so that the collision probability would be always less unity. The fictitious particle size is randomly chosen from the local particle size distribution and the fictitious particle velocity is computed from the sum of the local mean particle velocity and a fluctuating component. The latter is correlated to the fluctuating velocity of the real particle according to the formulation proposed by Sommerfeld (2001) . Following Anh Ho and Sommerfeld (2002), particle size distributions and velocity components at each computational cell were updated every 0.01sec. According to the model, a collision is assumed to take place when the collision probability computed from equation (3) is greater than a uniformly distributed random number in the range from 0 to 1. Agglomeration is assumed to occur when the normal component of the colliding particles relative velocity is smaller than a critical velocity ucrit computed from an energy balance, Anh Ho and Sommerfeld (2002) Kyoeng Lee (2003), Rong et al. (2006). The agglomerate, of diameter d 3p1,new = d 3p1 + d 3p 2
and mass
m p1,new = m p1 + m p 2
is assumed to retain the spherical shape of the original
particles. 3
RESULTS AND DISCUSSION
In the first case (Case 1) spherical particles of diameters in the range 1-10 μm were injected in the upstream end of the channel from two arbitrary selected locations and numerically tracked until they have reached the channel outlet. The acoustic field imposed was in the form of a planar standing wave created by opposite travelling sinusoidal sound waves along the channel vertical direction (z-axis) of length equal to the length of the channel. As the purpose of this case was to increase understanding on
3
Ch. Papadopoulos, K. Sardi & G. Bergeles
the effect of the acoustic radiation force on the particle motion, particle collision and agglomeration were not considered.
(a)
(b)
Figure 1 (a) Trajectories of particles of diameter of 1,2 and 5 μm in laminar flow (Re=268); (b) Trajectories of 5 μm diameter particles in turbulent flow (Re = 5350). In both cases the pressure amplitude of the acoustic wave Po was 150 kPa and the acoustic wave frequency f to 18.5 kHz (λ=2h).
(α)
(β)
Figure 2 (a) Axial distance, xf, travelled by a particle until it reaches the nodal plane normalised by the channel length L as a function of the particle Stokes number St = Cc ρ P d PU o 18µh ; (b) Time required for a particle 2
to reach the nodal plane (time of flight) normalised by the Lagrangian timescale ( TL = L U o ), where Uo is the flow bulk velocity. In both figures the Reynolds number is 268 and the frequency of the acoustic field 18.5 kHz
Computations revealed that application of an acoustic field of wavelength equal to twice the channel height h results in the creation of a single nodal plane along the channel symmetry axis (Townsend et al. 2004) and to preferential particle concentration at the nodal plane, figure 1(a) and turbulent flow enhances particle dispersion figure 1(b). An increase in the acoustic frequency leads to a subsequent increase in the number of nodal planes inside the channel. Larger particles need to travel a shorter axial distance to reach the nodal plane (referred here as time of flight) than smaller particles, figure 2(a), and thus reach the nodal plane sooner in time, for the same Reynolds number and acoustic pressure amplitude, figure 2(b). Clearly, reduced time of flights can ensure prolonged particle residence times at the vicinity of the nodal planes and thus enhanced collision and agglomeration rates and larger particle formation. Thus the particle time response to the acoustic wave can be of significant practical importance to the design of ultrasound filter devices. Here, the computations reveal that the dependence of distance and time required for a particle to reach the nodal plane is quasi exponential to the particle diameter so that the collection of sub-micron particles would require extended, in length, channels particularly for the case of a laminar flow. For practical applications it is also of interest to note that the particle time of flight to the nodal plane is exponentially correlated to
4
Ch. Papadopoulos, K. Sardi & G. Bergeles
the acoustic pressure amplitude so that there is little benefit in further increasing the value of P0 above a certain value. (a)
(b)
Figure 3 (a) Normalised time of flight vs the Reynolds number. For the (moderately) turbulent flow calculations (Re of the order of 3000 and larger), the time of flight is the mean (averaged over 100.000 samples) time required for a particle to reach the nodal plane for the first time. The error bars denote standard deviations to the mean. (b) Normalised time of flight vs the acoustic pressure amplitude. The acoustic field frequency in both figures was set to 18.5 kHz and the particle diameter was equal to 5μm.
Figure 3(a) shows that the normalised time of flight tends to decrease in turbulent flow since the presence of the turbulent eddies promotes vertical particle movements, despite the fact that the mean flow velocity is considerably increased. This finding suggests that a careful choice of the turbulent characteristics of a flow field can, in principle, promote particle collection at the vicinity of the nodal planes, particularly because submicron particles are likely to rapidly respond even to the smallest turbulent scales. Further the increased particle dispersion due to turbulence, figure 1(b), is likely to further promote inter particle collisions and enhance the formation of larger particles. Figure 3(b) shows that the particle time of flight is loosely correlated to the pressure amplitude for the turbulent flow cases and that values of P0 in excess of 150 kPa are required to ensure a notable reduction in the time flight. This finding may corroborate the previous argument that a careful control of the turbulent flow parameters must be conducted in parallel to the acoustic pressure selection in a practical devices. The increase in the particle concentration, Ip, at the nodal planes due to the acoustic wave, computed from N − N p =0 Ip = p × 100% N total (4) is shown in figure 4 as a function of the acoustic pressure amplitude and frequency. In equation (4), Np is the number of particles that concentrate at the nodal planes at the outlet of the channel in the presence of an acoustic field, Np=0 the respective number in the absence of the acoustic field and Ntotal is the total number of particles injected at the inlet of the channel. For the sake of a quantitative analysis as a function of frequency and in order to allow for some particle dispersion around the nodal planes due to the turbulent fluctuations, particle concentration was computed at the outlet cross section, around the nodal points, and within a window of length equal to 5% of the wavelength of the acoustic field of frequency f=18.5 kHz. Figure 4 shows that an almost four fold increase in the acoustic frequency (from 18.5 to 74 kHz) results to an almost proportional increase in Ip, for pressure amplitudes of up to 100 kPa. For larger pressures and for the higher frequency case the value of Ip is almost constant revealing that all particles have already concentrated within the computational window.
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Ch. Papadopoulos, K. Sardi & G. Bergeles
f [kHz] 18.5 74
Ip [%]
80
40
0 0
100
200
300
Po [kPa]
Figure 4 Change in the particle concentration at the nodal planes Ip , versus the acoustic pressure amplitude for two different values of the acoustic field frequency (Re=5350).
From the cases presented so far it has been established that the application of acoustic forcing leads to the creation of areas with increased particle concentration and thus to locally increased collision probabilities as described in equation (3). In order to study the agglomeration process, a homogeneous isotropic turbulence field similar to the one studied by Anh Ho and Sommerfeld (2002) was considered (Case 2). An initial poly-disperse particle phase which consists of a normal size distribution in the range from 1 nm to 20 μm was considered. The distribution was selected to contain a wide range of particle sizes with differences of up to two orders of magnitudes so as to assess the effect of initial particle diameter to the agglomeration due to the ultrasound. Particles were homogenously distributed inside the cubic domain at the beginning of the calculation and were tracked for a time interval of 1-3 sec, Riera-Franco de Sarabia et al. (2003). Periodic boundary conditions were applied i.e. particles that left from one side of the domain were re-inserted from the opposite side with the same velocity. Following Anh Ho and Sommerfeld (2002) the volume fraction of the dispersed phase was set to 1.4x10-5.
Figure 5 Particle trajectories inside the computational domain of Case 2 (Po=50 kPa, f=23.6 kHz).
Figure 5 presents a typical snapshot of particle trajectories inside the cube for a time interval of 0.05 sec from the beginning of the calculation. Superimposed in the figure is a contour plot of the acoustic force normalized by it’s amplitude, along the central Y-Z plane of the computational domain. It can be seen that all particles eventually migrate to the vicinity of the nodal plane where they continue their motion under the effect of the turbulent eddies. The sound wave frequency was set to 23.6 kHz (and the intensity to 50 kPa) so that only one nodal plane appears to the centre of the domain. Figure 6(a) shows that the particle size distribution is shifted towards larger particle sizes suggesting that ultrasound enhances the agglomeration rate and promotes larger particle formation. As expected, increased residence times inside the computational domain also promote agglomeration. Clearly, the formation of larger particles leads to a reduction in the particle number concentration inside the box,
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Ch. Papadopoulos, K. Sardi & G. Bergeles
Figure 6(b) and stronger, high frequency, ultrasonic fields result to a greater particle removal. It can be seen that the reduction rate of the particle number concentration of these smaller particles is greater than that of the respective whole particle ensemble rate indicating that the ultrasound promotes agglomeration of the smallest particles. (b)
1,2x10
9
1,0x10
9
8,0x10
8
6,0x10
8
4,0x10
8
2,0x10
8
0,99
t=0 sec No sound field t=1 sec t=3 sec With sound field t=1 sec t=3 sec
0,90 No Sound d:1nm-20μm d
Επιστημονική Συνάντηση για τις
Ερευνητικές Δραστηριότητες στη Μηχανική Ρευστών στην Ελλάδα Κοζάνη, 28 Νοεμβρίου, 2008
NUMERICAL SIMULATION OF TRANSPORT AND AGGLOMERATION OF DIESEL PARTICLES UNDER THE INFLUENCE OF AN ULTRASONIC STANDING WAVE Charalambos D. Papadopoulos, Katerina Sardi, and George C. Bergeles National Technical University of Athens Department of Mechanical Engineering 5 Heroon Polytechniou, 15710 Zografos-Athens, Greece e-mail: [email protected]
Abstract A numerical investigation of the effect of ultrasound preconditioning on the motion and size distribution of particles, likely to exist at the exhaust flue gas of diesel engines, is presented. Two cases were investigated. In the former case, the effect of an ultrasonic field, in the form of a standing wave, on particle laden laminar and turbulent air mixtures travelling inside a straight rectangular channel was examined. In the latter case, the simple case of a particle laden homogeneous isotropic turbulence air field, without a convective flow component, inside a cube was considered. For this second case the modelling approach was extended to include inter particle collisions and agglomeration. In both cases a detailed parametric study related to the ultrasound system design (e.g. energy level and frequency) was carried out. Results showed that the application of an ultrasonic standing wave leads to increased particle concentration at the nodal planes, enhanced collision and agglomeration rates and thus to increased particle sizes. Higher frequencies and acoustic pressures are shown to promote agglomeration and larger particle formation.
Keywords: Ultrasound, agglomeration, diesel particles 1. INTRODUCTION Air pollution in the form of ultrafine particles with sizes ranging from 10 nm to 5000 nm is widely recognized as a major environmental challenge and a serious health hazard. The contribution of the transport and marine sectors to particulate matter (PM) accumulation in the atmosphere is significant and a large number of studies towards the reduction of PM emissions have been conducted (e.g. Colvile et al., 2001, Corbett and Koehler, 2003, Fridell et al. 2008). Although considerable progress has been made, particularly in the area of diesel on-road vehicles (improved fuels, improvement of the combustion processes), the installation of a Diesel Particulate Filter (DPF) is required in order to comply with the latest European Union standards (EURO5). The International Maritime Organisation in its 2005 review of the existing regulation for the prevention of air pollution from ships also considered controlling PM emissions, (MARPOL Annex IV, 2005). However currently available DPF’s are likely to become plugged with time and lead to high pressure drops. Moreover their efficiency in retaining ultrafine particles is questionable, Van Gulijk et al. (2001). Acoustic agglomeration via application of ultrasounds is a promising technique to be used for increasing the size of micron and submicron particles before they enter a conventional filter. During
1
Ch. Papadopoulos, K. Sardi & G. Bergeles
this process high-intensity standing waves are applied to the flue gas stream leading to increased particle concentration at the vicinity of the acoustic nodal planes and enhanced collision and agglomeration rates. This technique has already been tested in industrial sized chambers and it was shown that an up to 70% reduction of micron-sized fly-ash particles (in comparison to the initial number of particles) can be achieved, Hoffmann (2000). The purpose of the present work was to numerically investigate the effect of ultrasound preconditioning on the particle motion and size as a function of parameters related to the ultrasound system design (e.g. energy level and frequency). For this purpose, an in-house CFD code incorporating Lagrangian particle tracking has been extended to include a representation of the acoustic field and particle collision and agglomeration effects. Two cases were investigated. In the former, the effect of an ultrasonic field, in the form of a standing wave, on particle laden laminar and turbulent air mixtures traveling inside a straight rectangular channel was examined. In the latter case, the simple case of a homogeneous isotropic turbulence field without a convective flow component inside a cube was considered and inter particle collisions and agglomeration were computed as a function of acoustic frequency and amplitude. 2 MODELLING APPROACH The modeling approach is described in detail in Papadopoulos et al (2008). Only a brief outline is provided here. 2.1 Flow-field modeling In the first case (Case 1) the flow of a particle laden air mixture inside a straight rectangular channel with an inlet square section of h=0.0128m and length of L=0.1m was considered. The air stream was assumed to be moderately heated at 573 K so as to resemble conditions relevant to the flue gas stream travelling inside the exhaust system of a diesel engine. The three dimensional laminar and turbulent flow fields inside the channel were computed by solving the Reynolds averaged Navier-Stokes equations with the standard k-ε turbulence model. For the second case (Case 2) a cubic box of side of 0.01m was considered and the mean flow velocity was set equal to zero. Turbulent properties inside the box were prescribed to the values used in the study of Anh Ho and Sommerfeld (2002). The particle density in all computations was set to 1500 Kg/m3, a value that is typical of the density of particles emitted by diesel vehicles, Burtscher (2005), and cruise ships, Petzold et al. (2007). 2.2 Particle dynamics Particle motion inside the computational domains was computed by application of the Lagrangian approach. Considering spherical particles and neglecting heat and mass transfer phenomena and particle rotation, particle trajectories and local velocities were obtained from
du p.i dt
=
r r 3ρ Cd ρ Fac ui − u p,i ) u i − u p ,i + gi 1 − + + ni (t ) ( ρ m 4d p ρ p p p
dx p.i = ui dt
(1)
In equation (1) , xp, up,I, mp, dp and ρp are the particle position, particle velocity components, mass, diameter and density respectively, ui, and ρ are the velocity and density of the fluid flow, Cd is the aerodynamic drag coefficient (Allen & Raabe, 1985 Shimazaki et al., 2006) modified by the Cunningham correction factor for small particles, Cc so that Cdnew = Cd Cc , gi the gravitational acceleration, Fac the acoustic radiation force and ni(t) the Brownian force per unit mass. The fluid
2
Ch. Papadopoulos, K. Sardi & G. Bergeles
velocity components, also required for the computation of the drag force, were estimated by application of the stochastic separated flow (SSF) model of Gosman and Ioannides (1983). To include the effect of the Brownian motion in the simulations, the Brownian force per unit mass ni(t), was modelled as a Gaussian white noise random process with spectral intensity, Li and Ahmadi (1992). The acoustic radiation force Fac at a direction z, normal to the walls where the standing wave was applied was computed from Kawasima (1955) as
ur r F ac = −Vo Eac KG sin ( 2 Kz ) z
(2)
where Vo is the particle volume, Eac is a measure of the energy residing in a wave field and a function of the peak acoustic pressure amplitude Po and the speed of sound c, K = 2πc / f is the acoustic field wave number, f the acoustic wave frequency and G the acoustic contrast factor which determines the response of any solid particle to a resonant acoustic field. For the particular case of solid particles suspended in air G was set to 2.5, Anderson et al. (2006). Inter-particle collisions were modelled by application of the stochastic particle/particle collision model of Sommerfeld (2001). The model relies on the generation of fictitious collision partners and at each time step of the particle trajectory calculation, a collision probability, Pcoll, is estimated. The latter stems from gas kinetic theory formalism under the assumption that the fluctuating motion of the particles is analogous to the thermal motion of molecules in gas Pcoll =
r 2 r π d p1 + d p 2 ) u p1 − u p 2 N p ∆t ( 4
(3)
In equation (3) dp1 and dp2 are the diameters of the considered and the fictitious
r r urel = u p1 − u p 2
is their instantaneous relative velocity (Berlemont et al., 2001), and Np is the particle, number of particles per unit volume at the considered control volume. The time step Δt is selected so that the collision probability would be always less unity. The fictitious particle size is randomly chosen from the local particle size distribution and the fictitious particle velocity is computed from the sum of the local mean particle velocity and a fluctuating component. The latter is correlated to the fluctuating velocity of the real particle according to the formulation proposed by Sommerfeld (2001) . Following Anh Ho and Sommerfeld (2002), particle size distributions and velocity components at each computational cell were updated every 0.01sec. According to the model, a collision is assumed to take place when the collision probability computed from equation (3) is greater than a uniformly distributed random number in the range from 0 to 1. Agglomeration is assumed to occur when the normal component of the colliding particles relative velocity is smaller than a critical velocity ucrit computed from an energy balance, Anh Ho and Sommerfeld (2002) Kyoeng Lee (2003), Rong et al. (2006). The agglomerate, of diameter d 3p1,new = d 3p1 + d 3p 2
and mass
m p1,new = m p1 + m p 2
is assumed to retain the spherical shape of the original
particles. 3
RESULTS AND DISCUSSION
In the first case (Case 1) spherical particles of diameters in the range 1-10 μm were injected in the upstream end of the channel from two arbitrary selected locations and numerically tracked until they have reached the channel outlet. The acoustic field imposed was in the form of a planar standing wave created by opposite travelling sinusoidal sound waves along the channel vertical direction (z-axis) of length equal to the length of the channel. As the purpose of this case was to increase understanding on
3
Ch. Papadopoulos, K. Sardi & G. Bergeles
the effect of the acoustic radiation force on the particle motion, particle collision and agglomeration were not considered.
(a)
(b)
Figure 1 (a) Trajectories of particles of diameter of 1,2 and 5 μm in laminar flow (Re=268); (b) Trajectories of 5 μm diameter particles in turbulent flow (Re = 5350). In both cases the pressure amplitude of the acoustic wave Po was 150 kPa and the acoustic wave frequency f to 18.5 kHz (λ=2h).
(α)
(β)
Figure 2 (a) Axial distance, xf, travelled by a particle until it reaches the nodal plane normalised by the channel length L as a function of the particle Stokes number St = Cc ρ P d PU o 18µh ; (b) Time required for a particle 2
to reach the nodal plane (time of flight) normalised by the Lagrangian timescale ( TL = L U o ), where Uo is the flow bulk velocity. In both figures the Reynolds number is 268 and the frequency of the acoustic field 18.5 kHz
Computations revealed that application of an acoustic field of wavelength equal to twice the channel height h results in the creation of a single nodal plane along the channel symmetry axis (Townsend et al. 2004) and to preferential particle concentration at the nodal plane, figure 1(a) and turbulent flow enhances particle dispersion figure 1(b). An increase in the acoustic frequency leads to a subsequent increase in the number of nodal planes inside the channel. Larger particles need to travel a shorter axial distance to reach the nodal plane (referred here as time of flight) than smaller particles, figure 2(a), and thus reach the nodal plane sooner in time, for the same Reynolds number and acoustic pressure amplitude, figure 2(b). Clearly, reduced time of flights can ensure prolonged particle residence times at the vicinity of the nodal planes and thus enhanced collision and agglomeration rates and larger particle formation. Thus the particle time response to the acoustic wave can be of significant practical importance to the design of ultrasound filter devices. Here, the computations reveal that the dependence of distance and time required for a particle to reach the nodal plane is quasi exponential to the particle diameter so that the collection of sub-micron particles would require extended, in length, channels particularly for the case of a laminar flow. For practical applications it is also of interest to note that the particle time of flight to the nodal plane is exponentially correlated to
4
Ch. Papadopoulos, K. Sardi & G. Bergeles
the acoustic pressure amplitude so that there is little benefit in further increasing the value of P0 above a certain value. (a)
(b)
Figure 3 (a) Normalised time of flight vs the Reynolds number. For the (moderately) turbulent flow calculations (Re of the order of 3000 and larger), the time of flight is the mean (averaged over 100.000 samples) time required for a particle to reach the nodal plane for the first time. The error bars denote standard deviations to the mean. (b) Normalised time of flight vs the acoustic pressure amplitude. The acoustic field frequency in both figures was set to 18.5 kHz and the particle diameter was equal to 5μm.
Figure 3(a) shows that the normalised time of flight tends to decrease in turbulent flow since the presence of the turbulent eddies promotes vertical particle movements, despite the fact that the mean flow velocity is considerably increased. This finding suggests that a careful choice of the turbulent characteristics of a flow field can, in principle, promote particle collection at the vicinity of the nodal planes, particularly because submicron particles are likely to rapidly respond even to the smallest turbulent scales. Further the increased particle dispersion due to turbulence, figure 1(b), is likely to further promote inter particle collisions and enhance the formation of larger particles. Figure 3(b) shows that the particle time of flight is loosely correlated to the pressure amplitude for the turbulent flow cases and that values of P0 in excess of 150 kPa are required to ensure a notable reduction in the time flight. This finding may corroborate the previous argument that a careful control of the turbulent flow parameters must be conducted in parallel to the acoustic pressure selection in a practical devices. The increase in the particle concentration, Ip, at the nodal planes due to the acoustic wave, computed from N − N p =0 Ip = p × 100% N total (4) is shown in figure 4 as a function of the acoustic pressure amplitude and frequency. In equation (4), Np is the number of particles that concentrate at the nodal planes at the outlet of the channel in the presence of an acoustic field, Np=0 the respective number in the absence of the acoustic field and Ntotal is the total number of particles injected at the inlet of the channel. For the sake of a quantitative analysis as a function of frequency and in order to allow for some particle dispersion around the nodal planes due to the turbulent fluctuations, particle concentration was computed at the outlet cross section, around the nodal points, and within a window of length equal to 5% of the wavelength of the acoustic field of frequency f=18.5 kHz. Figure 4 shows that an almost four fold increase in the acoustic frequency (from 18.5 to 74 kHz) results to an almost proportional increase in Ip, for pressure amplitudes of up to 100 kPa. For larger pressures and for the higher frequency case the value of Ip is almost constant revealing that all particles have already concentrated within the computational window.
5
Ch. Papadopoulos, K. Sardi & G. Bergeles
f [kHz] 18.5 74
Ip [%]
80
40
0 0
100
200
300
Po [kPa]
Figure 4 Change in the particle concentration at the nodal planes Ip , versus the acoustic pressure amplitude for two different values of the acoustic field frequency (Re=5350).
From the cases presented so far it has been established that the application of acoustic forcing leads to the creation of areas with increased particle concentration and thus to locally increased collision probabilities as described in equation (3). In order to study the agglomeration process, a homogeneous isotropic turbulence field similar to the one studied by Anh Ho and Sommerfeld (2002) was considered (Case 2). An initial poly-disperse particle phase which consists of a normal size distribution in the range from 1 nm to 20 μm was considered. The distribution was selected to contain a wide range of particle sizes with differences of up to two orders of magnitudes so as to assess the effect of initial particle diameter to the agglomeration due to the ultrasound. Particles were homogenously distributed inside the cubic domain at the beginning of the calculation and were tracked for a time interval of 1-3 sec, Riera-Franco de Sarabia et al. (2003). Periodic boundary conditions were applied i.e. particles that left from one side of the domain were re-inserted from the opposite side with the same velocity. Following Anh Ho and Sommerfeld (2002) the volume fraction of the dispersed phase was set to 1.4x10-5.
Figure 5 Particle trajectories inside the computational domain of Case 2 (Po=50 kPa, f=23.6 kHz).
Figure 5 presents a typical snapshot of particle trajectories inside the cube for a time interval of 0.05 sec from the beginning of the calculation. Superimposed in the figure is a contour plot of the acoustic force normalized by it’s amplitude, along the central Y-Z plane of the computational domain. It can be seen that all particles eventually migrate to the vicinity of the nodal plane where they continue their motion under the effect of the turbulent eddies. The sound wave frequency was set to 23.6 kHz (and the intensity to 50 kPa) so that only one nodal plane appears to the centre of the domain. Figure 6(a) shows that the particle size distribution is shifted towards larger particle sizes suggesting that ultrasound enhances the agglomeration rate and promotes larger particle formation. As expected, increased residence times inside the computational domain also promote agglomeration. Clearly, the formation of larger particles leads to a reduction in the particle number concentration inside the box,
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Ch. Papadopoulos, K. Sardi & G. Bergeles
Figure 6(b) and stronger, high frequency, ultrasonic fields result to a greater particle removal. It can be seen that the reduction rate of the particle number concentration of these smaller particles is greater than that of the respective whole particle ensemble rate indicating that the ultrasound promotes agglomeration of the smallest particles. (b)
1,2x10
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t=0 sec No sound field t=1 sec t=3 sec With sound field t=1 sec t=3 sec
0,90 No Sound d:1nm-20μm d
