International journal of automotive technology, tập 11, số 1, 2010

International Journal of Automotive Technology, Vol 11, No 1, pp 1−10 (2010) DOI 10.1007/s12239−010−0001−9 Copyright © 2010 KSAE 1229−9138/2010/050−01 IMPROVED THEORETICAL MODELING OF A CYCLONE SEPARATOR AS A DIESEL SOOT PARTICULATE EMISSION ARRESTER P K BOSE , K ROY , N MUKHOPADHYA and R K CHAKRABORTY 1)* 2) 3) 4) 1) Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India Department of Mechanical Engineering, Central Calcutta Polytechnic, Kolkata 700014, India 3) Department of Mechanical Engineering, Jalpaiguri Government Engineering College, Jalpaiguri 735102, India 4) Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India 2) (Received July 2007; Revised 10 December 2008) ABSTRACT−Particulate matter is considered to be the most harmful pollutant emitted into air from diesel engine exhaust, and its reduction is one of the most challenging problems in modern society Several after-treatment retrofit programs have been proposed to control such emission, but to date, they suffer from high engineering complexity, high cost, thermal cracking, and increased back pressure, which in turn deteriorates diesel engine combustion performance This paper proposes a solution for controlling diesel soot particulate emissions by an improved theoretical model for calculating the overall collection efficiency of a cyclone The model considers the combined effect of collection efficiencies of both outer and inner vortices by introducing a particle distribution function to account for the non-uniform distribution of soot particles across the turbulent vortex section and by including the Cunningham correction factor for molecular slip of the particles The cut size diameter model has also been modified and proposed by introducing the Cunningham correction factor for molecular slip of the separated soot particles under investigation The results show good agreements with the existing theoretical and experimental studies of cyclones and diesel particulate filter flow characteristics of other applications KEY WORDS : Diesel soot particulate emission, Particulate filter, Cyclone separator, Cunningham correction factor NOMENCLATURE A H B D1 D2 Dd Dp50 Dp50m dp FC FD L1 L2 Li Lo Vθ Vθ Vr2 n : exhaust gas temperature in K : number of particles remain in the outer vortex at an angle of turn θ N0 : number of particles at the inlet of cyclone, at θ =0 Pref : reference pressure [pa] ∆P : pressure drop across cyclone [pa] Q : volume flow rate [m3/sec] r1 : vortex finder or Inner radius of cyclone flow [m] r2 : outer radius of cyclone flow [m] t : temperature of the exhaust gas [oC] ρc : density of the exhaust gas [kg/m3] ρp : density of the particle [kg/ m3] ηo : collection efficiency of outer vortex ηi : collection efficiency of inner vortex ηoverall : overall collection efficiency of the cyclone µ : dynamic viscosity of the gas [kg/m-sec] θ : angle of turn in traversing the cyclone [rad] θi : angle of turn of the inner vortex [rad] θo : angle of turn of the outer vortex [rad] Rgas : characteristic gas constant of the exhaust gas [Nm/kg/ok] Ru : universal gas constant, in N-m/kmolk Cp : concentration of the particles per unit area Cp(r1,θ ) : concentration of particles at inner radius r1 & at an angular position θ Cp(r2,θ ) : concentration of particles at outer radius r2 & at T Nθ : inlet cross sectional area of cyclone flow [m2] : inlet height of the cyclone [m] : inlet width of the cyclone [m] : outer diameter of the cyclone [m] : diameter of the vortex finder [m] : diameter of the dust exit [m] : cut size diameter of the particle [µ m] : modified cut size diameter of the particle [µ m] : diameter of soot particle [µ m] : centrifugal force [N] : drag force acting on the particle [N] : length of the cylindrical portion of the cyclone [m] : length of the conical portion of the cyclone [m] : inner vortex length [m] : outer vortex length [m] : tangential velocity of the exhaust gas and particle [m] : tangential velocity of the gas at outer vortex [m/ sec] : radial velocity of the particles at outer vortex [m/ sec] : vortex exponent *Corresponding author e-mail: pkb32@yahoo.com P K BOSE, K ROY, N MUKHOPADHYA and R K CHAKRABORTY an angular position θ Cp(θ )0 : mean value of particle concentration at outer vortex Cp(θ )i : mean value of particle concentration at inner vortex C* : cunningham correction factor λ u M m T in : mean free path of the gas molecules [µ m] : mean molecular velocity : molecular weight [kg/kmol] : mass of the soot particles : inlet temperature [K] INTRODUCTION The diesel engine is one of the most reliable, durable, and economical power plants extensively used to transport goods, services, and people The engine emits a significant level of particulate matter (PM), which is considered to be most harmful pollutant in the air The particulate matter is associated with carcinogenic compounds such as PAH (poly-nuclear aromatic hydrocarbons), nitro-PAH, and sulfates, and due to its extreme diameter range of 0.05 to 1.0 µm (Kittelson, 1998; Oh et al., 2002), such emissions can easily enter the human respiratory system Therefore, concern over the quality of air and, in particular, the implications for human health have led to continued tightening of particulate matter emission limits Hence, to achieve the existing particulate emission target, several after-treatment retrofit programs are being implemented Many solutions proposed to date suffer from high structural complexity, thermal cracking, cost, and increased backpressure, which, in turn, deteriorates diesel engine combustion performance On the other hand, a cyclone separator has tremendous potential to be applied to cheaper, easily fabricable diesel particulate filters (DPF) that are not subject to thermal failure in the exhaust gas operating temperature range Many studies of cyclone separators in other industries are already available in the literature, but very few theoretical and experimental studies have been reported (Mukhopadhya et al., 2006; Crane and Wisby, 2000) with cyclone separators as a diesel engine exhaust gas aftertreatment device Experimental studies shows that soot particles of 0.5 µm and higher in diameter can be effectively eliminated by a cyclone separator (Mayer et al., 1998) This paper presents a computer-aided improved analytical approach for controlling diesel soot particulate emissions by a cyclone separator with low back pressure, reasonably high particulate collection efficiencies and reduced regeneration problems The analysis of fluid flow and particle motion in a cyclone is very complicated The primary flow has been studied previously (Shepherd and Lapple, 1939; Stairmand, 1951) The aerodynamics inside the cyclone create a complex two-phase, three-dimensional, turbulent swirling flow with a confined outer free vortex (irrotational flow) and a low-pressure, highly turbulent inner forced vortex (solid body rotation) The transfer of fluid from the outer vortex to the inner vortex apparently begins below the bottom of the exit tube and continues Figure General flow pattern of a cyclone down into the cone along the natural length of the vortex of a cyclone (Alexander, 1949) Shepherd and Lapple concluded that the radius marking the outer limit of the inner vortex and the inner limit of the outer vortex was roughly equal to the exit duct radius The length of the inner vortex core is also referred to as the cyclone effective length, which does not necessarily reach the bottom of the cyclone, (Leith and Metha, 1973) Particle collection in the cyclone is due to the induced inertia force resulting in radial migration of particles suspended in the swirling gas to the walls and down the conical section to the dust outlet and the gas exits through the vortex finder Flow near the cyclone wall is assumed to be laminar, although it is usually somewhat turbulent In an earlier such work on the modeling of a cyclone, it was assumed that the soot particles are uniformly distributed within the cyclone turbulent flow field both in the outer and the inner vortex However, that was a strong assumption, leading to conservative results Therefore, to make the analysis more physically realistic, this paper proposes an improved analytical approach to calculate the overall collection efficiency of a cyclone by considering a particle distribution function due to non-uniform distribution of soot particles across the cyclone turbulent flow field Because of the extreme size of the soot particles, a molecular slip correction factor (Crawford, 1976; Strauss, 1975) has been introduced The cut section diameter model (Mukhopadhya et al., 2006; Lapple, 1951) has been modified by introducing the molecular slip correction factor for the calculation of actual viscous drag of the soot particles under investigation The back-pressure of this system is found to be within the recommended limit and less than that of other methods of particulate filtration Studies of earlier and present work through computer-aided graphical analysis have been presented, compared, and discussed FORMULATION OF THE MODEL C- , C=Constant Vθ = rn n=0.5, (Shepherd and Lapple, 1939) or n=0.4-0.8 for the outer vortex (Cortes and Gil, 2007) IMPROVED THEORETICAL MODELING OF A CYCLONE SEPARATOR AS A DIESEL SOOT PARTICULATE Figure Turbulent cyclone flow Figure Cyclone geometry n=(−) 1, for the inner vortex (First, 1950) n =1 − ( – 0.67D2 0.14 )⎛ ⎝ t + 273⎞ -⎠ 283 0.3 (1) (Alexander, 1949) For a cyclone with the outer free vortex [Vθ =C/r] having turbulent swirling flow and with a particle distribution function and effective turn angle made by the gas in traversing the cyclone, as proposed by Crawford, the collection efficiency is: ηcollection=1 – exp (−) ρ Qd θ θ -⎞ ⎛ + -r ⎝ θ 1⎠ 18 µ Hr2 ( r2 – r1 ) ln -p p r1 for < θ < θ1, where the effective turn angle made by the gas in traversing the cyclone is: π θ = ( 2L1 + L2 ) H θ =2 π (Crawford, 1976) or L1+ {( r2 – r1 ) L2 / ( r2 – D /2 ) } -d H (Mukhopadhya , 2006) The angle of turn under laminar flow, at which the efficiency is unity, is given by: The tangential velocity in the annular section of the cyclone can be determined by the following equation (Crawford, 1976; Ter Linden, 1949; Leith and Licht, 1972) et al 2 r2 µ H ( r2 – r1 ) ln /r1 θ1 = -ρp Qd2 (Crawford, 1976) 2.1 Collection Efficiency Model Over the Outer Vortex For the proposed mathematical model of collection efficiency of diesel soot particles with cyclone flow in the outer vortex of both the cylindrical and the conical parts, the effect of the boundary layer and secondary circulation in the flow due to the presence of side walls are neglected Furthermore, the effects of particle-gas interaction, interaction between particles, particle-wall interaction, and gravitational force on exhaust two-phase flows are also ignored The following assumptions were made for formulating the model: (1) Laminar particle motion in the radial direction (2) Exhaust gas flow rate in the cyclone is constant, i.e., steady flow (3) The flow of the exhaust gas is turbulent in nature (4) Soot particle distribution across the cyclone vortex crosssection is non-uniform (5) Distribution of the particles across the cyclone vortex section is linear (6) Soot particles begin sequestering at the outer wall immediately as the exhaust gas enters the cyclone (7) Cyclone separator flow field is 2-D axi-symmetric (8) Stokes’ law can be applied to the movement of the particles relative to the gas stream (9) Buoyancy effect is neglected (10) The tangential velocity of particles is constant and independent of position The effect of strong turbulent swirling flow at any given angle θ will lead to a transfer of gas between the outer and inner vortex, which is important for particle separation A laminar sub-layer forms adjacent to the outer edge of the cyclone, such that all particles entering it are captured From Figure 3, the distance a particle travels in the direction ‘θ ’ or ‘ θ ’ of angular distance, within the thin laminar layer ‘ ’ over a time interval ‘ ’ becomes: = 2θ θ2 By substituting dt, the thickness of the captured zone where particle removal occurs is: d dr V dt dt r d V dr=Vr2dt = r-2 r2 d θ Vθ2 (2) After entering the cyclone, the soot particles are subjected to a strong centrifugal force, leading to non-uniform distribution of the particles across the cross-section of outer vortex Therefore, a general particle distribution function was used in the analysis of collection efficiency, set in P K BOSE, K ROY, N MUKHOPADHYA and R K CHAKRABORTY ln Nθ ( ( r2 θ ) r2 - -V 2- C ∫ r2 r1 V θ C ( θ ) d θ θo )=(− ) , p r – p +C o Evaluating this constant of integration at the inlet, θ =0, where ‘N ’ is the total numbers of particles and at θ =θ (actual angle of turn of the exhaust gas at the outer vortex): o θ C ( r2 θ ) r V - r2 r1 Vθ ∫0 C (θ ) dθ Nθ N0 )= ( o (−) ( ) exp , p r – p (5) o Substituting equation (5) into the collection efficiency (η ) of the outer vortex at a angle of turn θ =θ o is expressed as: o Figure Variable particle density terms of particle concentration, and was defined as the number of particles per unit area (Crawford, 1976) Let Cp(r,θ ) be the number of particles per unit area (particle concentration) at a radius ‘r’ and at a angle of turn ‘θ ’ of the exhaust gas, and let ‘dr’ be the radial thickness (captured zone) of unit depth within the cross-section of the outer vortex Then, the total number of particles ‘Nθ’ at the same co-ordinate from the inner radius r to the outer radius r of cyclone can be written as: r , p 1) If Cp (r , θ ) is the particle concentration at the outer radius ‘r ’ of the cyclone, then the fractional diminution of soot particles over the angle ‘θ ’ in the outer vortex is: 2 C ( r2 θ )( dr)dN -Nθ ∫ C ( r θ ) dr r ) , p r (3) , p = ( Substituting ‘dr ’ into equation (3) we obtain: (−) C (r2 θ ) × -V 2dN - -N(θ ) V θ r2 d θ ∫ C ( r θ ) dr = , p r r (4) The mean value of particle concentration between radius r and r over an angle ‘θ ’ is: r , p – p (6) o C ∫ V θ dA H ∫ V θ dr H ∫ r dr r = r = = A r n r 1 HC- [ r1 – r1 – ] Q = n n 1– n – Q( n) C H[ r12 – r11 – ] A generalized expression for tangential velocity is: Q( n) Vθ (7) Hr2[ r12 – r11 – ] Next, the expression for radial velocity considering the Cunningham correction factor ‘C ’ (Strauss, 1975) due to molecular slip of very small soot particles under investigation, acting to decrease resistance to particle motion, can be written as: C* ρ ( n )2 Q d2 (8) V -2 µ H2r22 + 1[ r12 – r11 – ] where: C (r θ )dr C ( θ ) -r2 r1 Substituting into equation (4): o= n n – 1– = n n – p , – C ( r2 θ ) - -V 2dN - -N ( θ ) ( r2 r1 ) C ( θ ) × V θ r2 d θ = p – , r p Integrating the above equation gives: n * C* p = 18 =1+2 1– n -λd( p n n – 1.257 + 0.400 p e –0.55 dp / λ ), R T πM r2 V C*( n)ρ Qd2 - -( r2 r1 ) V θ µ H ( r12 – r11 – ) r2 ( r2 r1 ) µ λ = 0.499 ρ u p ∫ r (−) Q r , p r p θo r 1– ( ) (−) ( − ) r2 V C ( r2 θ ) - dθ ( r r1 ) V θ ∫ C ( θ ) The rate of flow in the cyclone is: = N θ = ∫ C ( r θ )(dr × r N N0 ηo =1 − -θ- =1 − exp r – and u 1– = = n 18 – u p n (9) p n – Subsequently, substituting equations (7), (8), and (9) into equation (6), the collection efficiency of the outer vortex becomes: ηo=1−exp C (1 – n)ρ Qd ( r ,θ ) -dθ ∫ C -* (−) 18µH( r12 – n – r p p 1–n n ) ( 2 θo p r r – r ) C p ( θ )o (10) IMPROVED THEORETICAL MODELING OF A CYCLONE SEPARATOR AS A DIESEL SOOT PARTICULATE Figure Linear particle distribution function Assuming the particle distribution function is linear, as shown in Figure 5, therefore, the mean particle concentration is: Cp ( r1 θ ) C p ( r2 θ ) Cp( θ)o= (11) , + , =1 Using equations (11) and (12), we obtain: Cp(r2 θ -) -θ -(13) θ1 Cp ( θ )o Therefore, the modified collection efficiency of the outer vortex, considering the Cunningham correction factor by substituting equation (13) into equation (10), becomes: , =1+ ηo =1 − exp C*( n)ρpQd2pθo - ⎛ ( − ) -µ H (r12 – n r11 – n ) rn2 ( r2 r1 ) ⎝