effects cyclone separators

Aerosol Science 34(2003)937–955/locate/jaerosciE ects of ow and geometrical parameters on the collection e ciency in cyclone separatorsAtakan Avci,Irfan Karagoz ∗Department of Mechanical Engineering,Uludag University,Bursa 16059,TurkeyReceived 14November 2001;accepted 26February 2003AbstractA mathematical model has been developed for calculation of cut-o size and fractional e ciencies in cyclone separators,by taking into account the e ects of ow,particle and geometrical parameters,and acceleration assuming that the mixture of uid and particles is homogenous,and acceleration diminishes depending on the friction and geometry.Collection e ciency curves and cut-o size values predicted by the proposed model showed a good agreement with experiments over a wide range of inlet velocities for di erent types of parison of the obtained results with semi empirical models available in literature also indicated that the present model may be used successfully for determination of the performance of a tangential inlet cyclone.Analyses of the e ects of various parameters reveals that,in addition to ow and geometrical parameters,surface friction,vortex length and ow regimes play an important role on cyclone performance especially in small cyclones.?2003Elsevier Ltd.All rights reserved.Keywords:Cyclone separators;Vortex ow;Collection e ciency;Pressure losses1.IntroductionCyclones are widely used for various purposes,mainly for separating of the dense phase in a multi-phase ow.Entrance of ow into cyclone can be axial or tangential through inlet section,which can be in di erent shapes for each cyclone.Cyclone separators operate under the action of centrifugal forces.Fluid mixture enters the cyclone and makes a swirl motion and,due to the centrifugal forces,the dense phase of the mixture gains a relative motion in the radial direction and is separated from the main ow.It is di cult to analyze this problem since,in addition to its 3D character,there are many parameters in uenced this ow.∗Corresponding author.Tel.:+90-224-4428176;fax:+90-224-4428021.E-mail address:karagoz@.tr (I.Karagoz).0021-8502/03/$-see front matter ?2003Elsevier Ltd.All rights reserved.doi:10.1016/S0021-8502(03)00054-5938 A.Avci,I.Karagoz/Aerosol Science34(2003)937–955Nomenclaturea ow width,ma d modiÿed,dimensionless ow width,dimensionlessa0inlet width,ma0r average ow width(=a d D s a0),mA0cyclone inlet area(=a0b),m2A1cross-sectional area of cyclone body(= D21=4),m2A p cross-sectional area of particle,m2b inlet height,mB n mean acceleration,m=s2c cylinder body height,mC particle concentration,kg=m3C e particle concentration at the exit,kg=m3C cu Cunningham correction factor,dimensionlessC D drag coe cient,dimensionlessC0particle concentration at the inlet,kg=m3cs dimensionless parameter(=0:5f0L0=(a0−0:1D3))d particle diameter,md50cut-o size,mD d dimensionless cyclone diameterD L mean cyclone diameter,mD s dimensionless mean diameter of friction surfaces,dimensionlessD1diameter of cylindrical cyclone body,mD2outlet pipe diameter,mD3cone apex(or dust outlet)diameter,mD01,D02curvature diameters of elliptical cyclone body,mD10equivalent diameter of cyclone body,mf0average friction coe cient,dimensionlessh outlet height,mk particle shape factor,dimensionlessL streamwise distance from inlet,mL v vortex length,mL vs vortex length in streamwise direction,from inlet to end of vortex,m L0cyclone height,mL x vertical distance from inlet,mm p particle mass,kg˙m particle mass ow rate,kg/s˙m0particle mass ow rate at inlet,kg/s˙m e particle mass ow rate at exit,kg/sn vortex number,dimensionlessQ L volumetric ow rate at L,m3=sQ0inlet volumetric ow rate,m3=sA.Avci,I.Karagoz/Aerosol Science34(2003)937–955939 Q3volumetric ow rate at the cone apex,m3=sr radius of particle trajectory,mr0r average radius of particle trajectories(=D L=2),mRe Reynolds number(=Va= ),dimensionlessRe r particle Reynolds number in radial direction(=V r d= ),dimensionlesss streamwise coordinatet time,sV ow velocity in streamwise direction,m/sV r radial velocity of particle,m/sV t tangential velocity,m/sV0average inlet velocity,m/sV3 ow velocity in streamwise direction at the cone apex,m/sÿdimensionless parameter,dimensionlessÁe ciency,dimensionlessdynamic viscosity,Pa skinematic viscosity,m2=sgas density,kg=m3p particle density,kg=m3The main performance characteristics of a cyclone separator are collection e ciency,fractional e ciencies and pressure losses.Many studies have been performed on this di cult problem for determination of these characteristics,but these studies are successful for only a certain range of Reynolds number and geometrical ratios.Therefore,developing more e cient cyclones have been essentially based on experiments rather than mathematical models(Moore&McFarland,1993;Dirgo &Leith,1985b;Kenny&Gussman,1995).A few examples of these cyclones are the Stairmand high e ciency cyclone(Stairmand,1951),the Lapple cyclone(1951),Southern Research Institute cyclones(Smith,Wilson,&Harris,1979),the German Z cyclone(K o nig,B u ttner,&Ebert,1991), the aerojet cyclone(Upton,Mark,Hall,&Gri ths,1994),and the Kim and Lee cyclone(1990). Any cyclone having small geometrical modiÿcations is supposed to belong to one of these cyclone families.The mathematical models for calculation of e ciency generally attempt to predict the cut-o diameter(or cut-o size),which corresponds to50%e ciency.Some models also include fractional e ciencies.Most of the models depend on empirical or semi-empirical correlation equations.The earlier mod-els,Shepherd and Lapple(1939),and Lapple(1951)are simple and include only a few parameters, but do not give satisfactory results.The improved models,such as the models of Barth(1956),Barth and Leineweber(1964),Chan and Lippmann(1977),Dietz(1981),Mothes and L o er(1988),Ioza and Leith(1990),LidÃe n and Kenny(1991),Gri ths and Boysan(1996),LidÃe n and Gudmundsson (1997)take into account several parameters and use some experimental constants.Although the re-sults of these models exhibit similar trends with experimental values,they presented poor prediction of the cut-o diameter(cut-o size),d50and fractional e ciencies.Some earlier experimental works(see,for example,Smith,1962;Muschelknautz&Krambrock, 1970)performed in similar test cyclones have given contradictory results,showing that some940 A.Avci,I.Karagoz/Aerosol Science34(2003)937–955parameters such as surface roughness,particle distribution,particle geometry,etc.might be more important than expected.On the other hand,many semi-empirical models(see,for example,Shepherd &Lapple,1939;Barth,1956)give either acceptable results only for a limited range of parameters or show combine e ects of many parameters.A literature survey showed that the vortex length can be important for the prediction of collection e ciency,especially in short cyclones.But in long cyclones,the cyclone vortex may not reach the cone apex.The collection e ciency increases with increasing cyclone length up to certain value and starts to decrease after that(see,for example,Alexander,1949;Zhu&Lee,1999).The natural vortex length wasÿrst deÿned and formulated by Alexander(1949).But the later experimental studies,Parker,Jain,Calvert,Drehmel,and Abbott(1981),B u rkholz(1984),Ji,Wu,and Shi(1991), showed that this formula cannot predict the vortex length correctly in many cases.Therefore,Ji et al.(1991)and Bryant,Silverman,and Zenz(1983)proposed di erent relations separately for vortex length,using more parameters,which depend on geometrical dimensions.But,the recent(Ozkoca, 2001)and some earlier experimental studies(Ho mann,Jonge,Arends,&Hanrats,1995;Smith, 1962;B u ttner,1988)showed that ow rate or inlet velocity also a ect the vortex length.An increase of the vortex length was observed by Ho mann et al.(1995)when inlet velocity was high,but a possible upper limit should be investigated.On the other hand,surface resistance or friction losses seem to be important in cyclone performance.Recently,an experimental study performed by Ozkoca (2001)showed that the vortex length increases with the decrease of surface resistance.Therefore, di erent values for vortex length should be expected from these model equations as given by Zhu and Lee(1999).One of the problems in calculation of cyclone e ciency is the e ect of ow character in cyclones. In general, ow type is turbulent in big cyclones,and assumed friction factors and corresponding results are meaningful.But this is not the case for small cyclones in which ow and operational conditions,such as velocity,temperature,pressure,viscosity or cyclone diameter,may be more important and their e ects may di er from cyclone to cyclone,since the ow can be laminar, turbulent or transitional in these cyclones by Blachman and Lippmann(1974).The ow which is turbulent at the beginning may become laminar,even three type of ow may occur in a cyclone, Saltzman and Hochstrasser(1983).In laminar ow,operating parameters in uence cyclone e ciency more then turbulent case.Predictions of e ects of ow regimes on the e ciency and pressure losses are di cult especially in small cyclones,comparing with the e ects of geometrical parameters. The aim of present study is to develop new equations,for the prediction of cut-o size and collection e ciency,which are easy to use,applicable in big or small cyclones and simulate the e ect of all common geometrical and ow parameters.The e ects of ow regimes are also taken into account by means of friction coe cient in the present parison of the obtained results with previously published experimental values and semi-empirical models showed that the agreement of the present results with the experimental values is generally better than the other models.2.Theoretical basisA mathematical model has been developed for the ow in a tangential inlet cyclone(see Fig.1)to obtain cut-o size and collection e ciency.It is assumed that homogenous mixture of uid and particles having a mean particle concentration of C0and average velocity of V0,enters intoA.Avci,I.Karagoz /Aerosol Science 34(2003)937–955941D 2bFig.1.Critical dimensions of a cyclone.the cyclone,and all velocity components of particles,except radial velocity,equal to the velocity components of uid.Flow cross section may deform after entering the cyclone,depending on inlet and outlet pipe geometry.Since cyclone body may be cylindrical,elliptical or in a di erent shape,it should be better to deÿne an equivalent diameter,D 10,as2D 10=1D 01+1D 02;(1)which gives equivalent centrifugal forces in one turn.D 01and D 02are the ÿrst and second diameters of an elliptical cyclone body,respectively.Since,diameter of the cyclone changes along the cyclone due to conical part,a mean cyclone diameter,D L ,can be deÿned as a function of the equivalent diameter and the diameter of the cone apex,D 3,as D L =D d D 10D 3;(2)where the dimensionless diameter D d is a ratio of the diameter of a cylinder,whose height is L 0and surface is equal to the friction surface of the cyclone,to D 10,and can be calculated as D d =c L 0+0:5 1+D 3D 1 0:25 1−D 3D 1 2 D 1L 0 2+ 1−c L 02:(3)The ow cross section at the inlet has a height of b and width of a 0.This ow cross section may deform depending on the geometry of the vortex tube at the exit of the cyclone.A short length of the vortex tube causes some part of the mixture to leave from the main ow towards to the exit.The e ect of this phenomenon can be modelled as if the ow cross section were expanded.Therefore,modiÿed,dimensionless ow width deÿned as d = 1+0:5 1:25D d ÿ3 b h−1 ;(4)942 A.Avci,I.Karagoz/Aerosol Science34(2003)937–955whereÿ=2a0=(D10−D2)(Avci&Karagoz,2000).On the other hand,radial and tangential velocity components of the ow do not remain constant in cyclone.Gupta,Lilley,and Syred(1984)observed that tangential velocity is about2times the mean inlet velocity at the bottom of the cyclone,but actual value should be a function of ow and geometrical characteristics.During accelerating motion from the top to the bottom of a cyclone,some part of the mixture may separate from the outer vortex to the core which leads to decreasing in the ow rate towards to the bottom.Ogawa and Seito(1986)proposed an empirical relation for the fraction of the ow rate which enters the dust bunker as follows:Q b Q0=2:621−0:36D3D1−0:56D3D11:16:(5)Losses in the ow rate also depend on time and surface resistance.Increase of surface resistance leads to decrease of acceleration and velocity,and eventually increase of ow losses.Thus,this is very important for determination of vortex length and mean velocity.During acceleration,the width of ow cross section a varies along the cyclone as a function of the cone apex diameter,and this variation has been modelled by Avci and Karagoz(2001)asa=a0I+0:1D3−a0a0L0L;(6)which is based on the assumption that the outer vortex reaches the cone apex.Many investigators have studied on the vortex length(see,for example,Parker et al.,1981;Cooper,1983;Dirgo& Leith,1985a;Ji et al.,1991;Patterson&Munz,1996),and they have proposed semi-empirical relations for the vortex length,such asL v=2:2D1A0A10:36D1D22:25(7)(Ji et al.,1991)which does not include the e ects of velocity and friction.The role of velocity was investigated by B u ttner(1988)who showed that the vortex length increases with the increase of the velocity.Similar results have been observed in experiments performed with a transparent telescopic cyclone(Ozkoca,2001).Velocity at the bottom of the cyclone can be calculated from Eqs.(5)and (6)but Eq.(5)may give negative ow rates for small cone apex diameter as in aerojet cyclones. Therefore,a di erent approach has been followed in the present study:Acceleration of the ow depends on the cyclone geometry and surface friction,thus the amount of mixture,which leaves the main ow and goes into the inner vortex before attaining to the cone apex should depend on the friction of the surface conÿned the ow.Supposing that the energy of friction losses is withdrawn from the ow and the same amount of kinetic energy leaves the main vortex,energy equilibrium for a di erential element inside uid ow can be written as−V22d˙m=˙mf0V22ad L;(8a)A.Avci,I.Karagoz /Aerosol Science 34(2003)937–955943where f 0is the average friction coe cient and depends on the ow type and surface roughness.f 0is calculated by using average surface roughness and the inlet Reynolds number Re 0=V 0a 0a d ;(8b)assuming that the wall friction in the cyclone may be estimated by the same equations as for laminar and turbulent ow in a circular ing Eq.(6)for the ow width,integration of Eq.(8a )from the inlet to any position L gives Q L Q 0= a a 0(f 0L=(a 0−a ));(9a)under the assumption of constant friction coe cient and density.Since,at the bottom of the cyclone,the width of the ow entering to the dust bunker is about 0:1D 3(Avci &Karagoz,2001)taking V =V 3,Q =Q 3,a =0:1D 3and L =L vs ,Eq.(9a )leads to Q 3Q 0= 0:1D 3a 0(f 0L vs =(a 0−0:1D 3)):(9b)Continuity equation gives the velocity of the ow entering to the dust bunker at the cone apex as V 3= Q 3Q 0 10V 0a 0D 3:(10)In order to see the e ects of velocity variations on the ow,Eqs.(9b )and (10)can be rearranged as ln Re 3Re 0 = f 0L vs a 0−0:1D 3 ln 0:1D 3a 0:(11)Frictional resistance is increased by the inner and outer surfaces of the outlet pipe.This e ect has been taken into account by the deÿnition of dimensionless mean diameter of the friction surfaces,D s D s =D d +D 2D 10h L 0;(12)which is a modiÿcation of D d .The vortex number,n ,which is the number of turns in outer vortex,can be calculated by the following semi-empirical relation (Avci,2000)n =0:38ln D 3D 10 ln L 0=D 10L 0=D 10−D 3=D 10+1(13)which gives slightly high values for cyclones of L 0¡(7–10)D 3.The spiral length of the vortex can be obtained as a function of n ,nothing that the pitch of the vortex varies in the conical partL vs = D 10D 2d n;(14)so that this length in a cyclone could be approximately equal to the length in equivalent cylin-der whose diameter is D 10D d .In the case of L 0¡(7–10)D 3,L vs =3D d L 0can be used for rough calculations.Eq.(14)is valid in cases provided that n could be predicted correctly.Any particle inside the uid makes an accelerating motion in outer vortex ow.It is assumed that any particle has a tangential velocity component (V t )equal to the one of the ow,and a radial944 A.Avci,I.Karagoz /Aerosol Science 34(2003)937–955Fig.2.Control volume for particle mass conservation.velocity component (V r )relative to the ow with a constant drag coe cient (C D )in this relative motion.By taking V t approximately equal to V for simplicity,equilibrium of forces applied to a particle gives (Avci &Karagoz,2000)m p V 2r =C D A p V 2r 2+m V 2r ;(15)where m p is the particle mass;m and are the mass and density of uid element which occupies the same volume with the particle,and A p is the particle cross section.Knowing the velocities at the beginning and at the end of the outer vortex,mean acceleration of the uid owing towards to the bottom of the cyclone,can be calculated asB n =V 23−V 202L vs:(16)The mass of second phase entering to the control volume (see Fig.2)can be written as ˙m =abVC and the distance travel during time d t is d s =V d t .The mass ow rate of the second phase (d ˙m =C d sbV r )reached the wall,is also equal to the change of the second phase mass in the control volume.Therefore,conservation of mass leads to−d ˙m ˙m =V r a d t:(17)Eqs.(15)and (16)can be combined with Eq.(17),to give −d ˙m ˙m = 4d 3rC D p −1 0:5V d V aB n:(18)Collection e ciency for particles can be deÿned asÁ=1−˙m 0˙m e =1−C 0C e:(19)Taking the mean values deÿned above for the variables of the ow width,the drag coe cient and the radius,Eq.(17)can be integrated to give Á=1−exp − 4d 3r 0r C D p −1 0:5L vs a 0r;(20)where r 0r =D L =2,and a 0r is the mean ow width deÿned as a 0r =a d D s a 0.A.Avci,I.Karagoz /Aerosol Science 34(2003)937–955945One of the most important parameters in cyclones is the cut-o diameter (or cut-o size)which corresponds to the diameter of particles collected with 50%e ciency.The cut-o diameter (d 50)can be obtained from Eq.(20)with Á=0:5,asd 50=0:36C D r 0r a 20r ( p − )L 2vs :(21)One important problem is the determination of the drag coe cient.For spherical particles,the value of C D =0:44can be used for 1000¡Re r ¡300000and C D =0:1for Re r ¿300000.However,the particle Reynolds number is generally smaller than 1000and the drag coe cient can be calculated byC D =24Re r (1+0:15Re 0:687r )(22)for Re r ¡1000(Parker,1993).On the other hand,the ow can be assumed as laminar for small particles especially in small cyclones.For laminar ows,it is proposed that the drag coe cient might be calculated by C D =24=Re k where k is a constant which depends on the shape of particles.Eq.(21)shows that the cut-o diameter is related to ow parameters via drag coe cient which depends on viscosity,velocity and temperature.In the case of high velocities and particle diame-ters,e ects of ow parameters are negligible since the drag coe cient is nearly constant at high Reynolds number.Therefore,excessive increase of velocity would not have big in uence on col-lection e ciency.On the contrary, ow parameters became dominant especially in small cyclones where Reynolds number is generally small.S shapes of fractional e ciency curves are similar but small cyclones exhibit steeper curves than the others.In order to demonstrate the e ects of size,velocity and viscosity on the cut-o size in Stokes range,Eqs.(15)and (21)together with C D =24=Re r =24 =(d V r )give d 50=2:5 V ( p − )C cu 0:5 D L a 0r VL vs0:5:(23)It would be more convenient to substitute Eqs.(9)and (10)into this equation,as explained below,in order to demonstrate clearly the e ects of ow and geometrical parameters as well as ow type.The Reynolds number,deÿned asRe = Va =Va (24)is not constant because velocity and characteristic length a vary along the cyclone.The Reynolds number decreases generally from the inlet (Re 0=V 0a 0a d = )to a minimum value at the bottom (Re 3=0:1V 3D 3= ).Laminar,turbulent or transitional ow may exist especially in small cyclones.Although several studies concerning the ow regimes have been reported in literature (Lippmann &Chan,1979;Saltzman &Hochstrasser,1983;Moore &McFarland,1990;DeOtte,1990),there is no certain criterion to identify transition to turbulence.It was assumed that,if Re 0¿3000and Re 3¿3000the ow is turbulent;Re 0¡2300and Re 3¡2300the ow is laminar,otherwise transitional ow may exist in the cyclone.Di culties in calculating the friction coe cient arise when transitional ow is dominant.For simplicity,the friction coe cient can be calculated by using the Reynolds number based on the inlet conditions,but this may cause signiÿcant errors especially in transitional ow.946 A.Avci,I.Karagoz /Aerosol Science 34(2003)937–955As an approximation,the following deÿnition of mean ow velocity V 0r ∼= Q 3Q 0 10V 0a 0a 0r (25)can be written and Eq.(14)can be used in Eq.(23),for V and L v ,respectively.Substituting Eqs.(2),(14)and (25)into Eq.(23)and using Eq.(9),the cut-o size can be obtained as d 50=0:446 v ( p − )C cu 0:5 √D 3D 10a 0V 0nD 10 0:5 10a 0D 3cs a d D s (26)for the Stokes region,wherecs =0:5f 0L vs a 0−0:1D 3:(27)It should be note that when the natural vortex does not reach the bottom of cyclone,cs becomes larger than 1and the actual vortex length is required so that cs =1.In this case,cs can be set to 1for rough calculations.Similar equations can be derived for outside the Stokes region.In order to obtain the e ciency curves from the deÿnition of the e ciency,the drag coe cient (Eq.(15))may be substituted into Eq.(20).For certain geometry,the cut-o size of particles and,consequently,e ciency depending on the Cunningham correction can be obtained in the Stokes region.As the diameter increases,the Cunningham correction becomes ine ective,and the drag coe cient mainly depends on the geometry and ow parameters.3.Results and discussionA mathematical model has been developed for the tangential inlet cyclones and applied to cyclones in di erent geometries,in order to predict cyclone performances and investigate the e ects of various parameters such as inlet velocity,cyclone height and viscosity.The results predicted by the present model were compared with experiments and semi-empirical models for seven cyclones given by Stairmand (1952),Breuer (1961),Dirgo and Leith (1985a),Kim and Lee (1990),Upton et al.(1994),Bohnet (1995)and Xiang,Park,and Lee (2001),which have been all tested at di erent sizes and ow rates.The e ects of cyclone size and length,inlet velocity,viscosity,and the diameter of the cone have been investigated and summarized as follows.3.1.E ects of cyclone sizeIt is important to know the e ects of cyclone size on the cut-o diameter under certain operational conditions and geometry.It is generally accepted that the cut-o diameter varies with the square root of size,which is the similar result when friction coe cient is zero in the present model.For comparison purposes,the Stairmand cyclone with the diameter of 305,205and 30:5mm at the mean inlet velocity of 15m =s,was considered.The test conditions have been used in the model with the assumption of smooth surface.Table 1shows the computed results and experimental values given in literature.Table1The e ect of cyclone size on d50for the Stairmand high-e ciency cycloned50( m)Experimental Present modelD1=305mm 3.0(Dirgo&Leith,1985a) 2.9D1=205mm 2.0(Stairmand,1952) 2.5D1=30:5mm 1.3(Zhu&Lee,1999) 1.3Table2The e ect of inlet velocity on the cut-o size for the Stairmand cyclone of305mm diameterVel.(m/s)(d50; m)Experimental(Dirgo&Leith,1985a)Present model5 6.0 6.010 3.8 3.815 2.8 2.920 2.1 2.425 1.8 2.1As can be seen from the table,the model results are in good agreement with the experimental values.The discrepancies,which are less than10%,may be caused by the surface roughness,local losses and also experimental errors.Although this discrepancy may be high for bigger cyclones,the results are acceptable for the practical point of view.3.2.E ects of average inlet velocityThe average inlet velocity re ects di erent e ects in di erent ow regimes of the uid-cyclone wall interaction.Similar to the size e ects,friction factor depends on the inlet velocity and the cut-o diameter varies inversely with the square root of the velocity.In high ow rates and big particle diameters,the friction coe cient approaches a constant value,therefore,e ects of the inlet velocity diminish.Table2shows the comparison of the computed results with experimental values (Dirgo&Leith,1985a)for the Stairmand cyclone with a diameter of305mm,atÿve di erent inlet velocities which cover turbulent region.It is possible to say that the results are very promising.The cut-o size varies with the velocity to the power of about0.72.Another comparison has been made with the experimental values of Kim and Lee(1990)at the inlet velocities of1.35,2.26,and3:35m=s(Table3)which are in the range of laminar regime. Keeping in mind the possible errors in predicting the friction factor,the agreement is very good especially at high velocities,and discrepancies are less than10%.On the other hand,cs becomes greater than1and collection e ciency diminishes rapidly in low velocities,most probably causedTable3The e ect of inlet velocity on the cut-o size for the Kim andLee cyclone of31:1mm diameterVel.(m/s)(d50; m)Experimental(Kim&Lee,1990)Present model2.03 6.2 6.42.86 5.0 5.04.25 2.9 2.6by insu cient vortex length.Therefore,cs was set to1in this case.The di erences between exper-imental and computed values become smaller as friction coe cient increases.Therefore,calculation of the vortex length is more important especially in long cyclones and in low velocities which is the case of cs¿1.One interesting example in this matter was given by Blachman and Lippmann (1974)for a10mm nylon cyclone.Variation of the cut-o diameter was observed in laminar and transitional ow regions by changing the inlet velocity up to15times,and variation between0.26 and6:5 m in diameter was obtained.A linear relation was observed between the cut-o diameter and inlet velocity,in logarithmic scale.It was found that the cut-o diameter varies with the velocity to the power of about0.55in laminar region and is proportional to the mean velocity in transitional and turbulent ow regions.Two regions can be deduced from these results.Theÿrst one covers transitional and laminar regions(cs¡1).In the transitional region,determination of friction coef-ÿcient for the model is di cult due to unstable behavior of the ow.However,observation shows that the cut-o diameter varies with the inlet velocity to the power of0.9–0.95for laminar ow, whereas,the exponent ranges from approximately1.3–1.5for transitional ow.In the second region cs is higher than unity.When cs=1,experimental values of the cut-o diameter agree with the present model where the cut-o diameter is proportional with the square root of the inlet velocity. The present mathematical model gives very satisfactory results except in transitional region where di erent behaviors are observed,as can be seen in the experimental results given by Zhu and Lee (1999).These experiments were carried out with the Stairmand cyclone of30:5mm,at the mean inlet velocities varied between13.4and24:6m=s.A comparison the experimental results to the computed ones shows that agreement is violated towards to high velocities which cause to a ow dominated by transitional region(Table4).For example,at V0=24:6m=s,the present model gives f0=0:0318which leads to d50=0:89 m.If f0=0:01is used in the model,the cut-o size obtained as0:45 m which is closer to the experimental value given in Table4.The e ects of transitional ow create most probably an oscillating ow whichÿnally causes an increase in collection e ciency. Therefore,this type of ow needs to be analyzed in detail as a future work.3.3.E ects of cyclone heightHigh collection e ciency in a cyclone requires high residence time and mean velocity of the ow inside the cyclone.Therefore,it is possible to deÿne the optimum cyclone height depending on these two parameters.In the present model,the e ect of cyclone height is a function of the ratio Q L=Q0 (Eq.(9a))which depends on the geometry and ow conditions.Fig.3shows the relation between。