Predicting Cone-in-Cone Blender Efficiencies from Key Material Properties

Transcription

1 Pedicting Cone-in-Cone Blende Efficiencies fom Key Mateial Popeties By: D. Key Johanson Mateial Flow Solutions, Inc. NOTICE: This is the autho s vesion of a wok accepted fo publication by Elsevie. Changes esulting fom the publishing pocess, including pee eview, editing, coections, stuctual fomatting and othe quality contol mechanisms, may not be eflected in this document. Changes may have been made to this wok since it was submitted fo publication. A definitive vesion was subsequently published in Powde Technology, Volume 7, Issue 3, 4 Decembe 6, Pages 9-4, doi:.6/j.powtec Abstact: Blending of powdes and ganula mateials is a citical unit opeation in many industies, yet the ability to pedict blending effectiveness lags well behind ou ability to ceate new and novel blendes. As a esult of this, poduction plants must ely on vendo blending tests conducted on small scale model blendes to detemine if thei specific mateial will wok in the poposed blende design. Once these blending tests ae conducted, enginees must then use past expeience and consevative design pactices to scale-up to full scale units at pocess flow ates. The difficulty in pedicting blending efficiencies aises fom the fact that blending pefomance depends on basic mateial popeties, blende geomety, blende flow ates, and blende opeation paametes. These effects ae convoluted duing blending opeation. Successful scaleup would equie undestanding how to sepaate the influence of these fou effects. If this could be accomplished, blende pefomance could be detemined by measuing simple mateial popeties, pedicting blende velocity pofiles, and computing blende efficiencies fom pedicted velocity pattens. This method would allow sepaation of factos affecting blende pefomance and povide a means of eliable scale-up using simple mateial popeties and specified blendes geometies. This pape pesents a methodology of pedicting blende pefomance in simple in-bin blendes using easily measued mateial popeties. It discusses blende optimization and detemines the influence of gas pessue gadients on blende flow and opeation. The specific blende analyzed is the cone-in-cone blende and the analysis suggests that blende pefomance depends on wall fiction paametes fo conditions whee input concentation fluctuations occupy much of the blende volume. Howeve, blending action appeas to be independent of fiction angle fo conditions whee thee ae many concentation fluctuations within a blende volume. The analysis also shows that gas pessue gadients can lead to stagnant egion fomation. Key Wods: Mass flow, Mixing, Residence Time Distibutions, Powdes, Solid Mechanics

2 Intoduction Acceptable blending of powde and ganula mateials equies thee things. Fist, all mateial within the blende must be in motion duing blende opeation. Second, a distibution of mateial esidence times must exist within the blende. Thid, the blending shea and velocity pofiles must esult in mixing on a scale smalle than the size of the final poduct sample. It is obvious fom these thee citeia that the specific motion in a given blende configuation detemines the extent of blending caused by the pocess equipment. In fact, if flow pofiles in any given blende wee known, then they could be used to compute esidence time distibutions fo the given blende configuation. These esidence time distibution functions could then be used to evaluate blende pefomance. Blending of powde mateial can be accomplished by imposing a velocity pofile acoss a given piece of pocess equipment esulting in a distibution of esidence times within the blende. Fo example, in well designed mechanical blendes all of the mateial in the blende is in motion duing opeation. In these blendes the velocity flow field is complex, esulting in paticle flow paths that coss multiple times befoe exiting the blende. Since all paticle flow paths do not tavel the same distance befoe exiting the blende and individual paticle velocities ae diffeent, the complex flow paths esult in a esidence time distibution function. Ideally, adjacent paticles in a blende would have vey diffeent flow paths causing significant inte-paticle mixing and poduce wide esidence time distibution functions. Howeve, eal blendes always shea mateial, often poducing local zones which possess diffeent tajectoies and esult in mixing down to the scale of local shea zones poduced duing the mixing pocesses. These local shea zones ae caused by flow aound paddles o scew flights and, when combined with the dynamic mateial tajectoies, poduce the oveall blending in any given blende. In fact, the velocities in any given blende ae due to shea o dynamic effects that move goups of paticles aound. Thus, to undestand blending as a unit opeation, one must be able to estimate the velocity pofiles in both dynamic and shea flows. Evey blende will have a combination of these velocity types. The main pemise of this wok is that mateial popeties can be used along with specific blende geometies to pedict blending velocity pofiles. These velocity pofiles can then be used to compute the expected blende esidence time distibution functions and finally estimate the blende pefomance. This appoach de-convolutes the effects of mateial popeties, blende geomety, and blende opeation paametes, making scale-up possible. This appoach is pesented fo the simple case of in-bin blendes such as the cone-in-cone. The dynamic mateial tajectoies in this style blende occu only duing blende filling as mateial fee falls into the blende and distibutes on the esulting pile. Most of the blending occuing in this type of blende esults fom shea velocity pofiles caused by the specific blende geomety. The cone-in-cone blende will be used as an example to show how to de-convolute the effects of blende pefomance, mateial popeties, blende geomety, and blende opeation paametes and povide a methodology fo blende scale-up. This blende will also be analyzed elative to the thee citeia outlined above. Cone-in-cone blendes A cone-in-cone blende consists of a bin with a hoppe that compises two independent conical hoppe sections as shown in Figue. One conical hoppe section is inseted inside the othe to fom an inteio conical hoppe and an annula flow channel. Mateial flows though both the inne conical hoppe and the annula egion duing blending opeation. The vetical section!"#!$"%&! p

3 above the cone-in-cone is designed lage enough to povide a mixing zone but is usually limited to a height equivalent to twice the diamete. Distibution chutes at the top of the blende spead the input flow steam acoss the blende coss section thus helping educe possible segegation duing blende filling. Thee is a conical hoppe below the cone-in-cone section. The velocity pofile acoss the cone-in-cone section combines with the velocity pofile in the lowe cone to yield a combined velocity pofile esponsible fo axial blending in the in-bin blende. Mateial exiting this blende usually flows though some fom of feede to down steam pocess equipment. The cone-in-cone blende mixes well in the axial diection but povides only a small mixing capability in the adial diection. Hence, the feede below the cone-in-cone povides adial mixing that is lacking fom this style blende. DT dt i o DB db L Mass Flow in Cone-in-cone blendes Do Figue. Schematic of typical cone-in-cone geomety The cone-in-cone hoppe is a type of mass flow hoppe. The inteio cone and the lowe cone ae designed so the hoppe slope angle is compatible with standad mass flow citeia. Simply stated, mass flow is a condition that poduces significant mateial movement in the entie pocess equipment as mateial passes though o dischages fom it (BMHB []). Thee ae no stagnant egions in a mass flow bin. Howeve, depending on the hoppe shape and wall fiction angle, a significant velocity pofile can exist in a mass flow bin ceating a esidence time distibution. This popety has been used successfully to ceate mass flow blendes (Ebet, et al [], Johanson [3]). The adial stess theoy has been successfully used to compute the velocity in conical hoppes as well as to pedict the mass flow limit. This theoy will be extended to cone-in-cone hoppes as outlined below. These theoies pedict a elationship between the conical hoppe half angle and the fiction angle that is compatible with adial stess conditions. Figue shows the calculated elationship between conical hoppe angle and wall fiction angle based on the adial stess theoy (Jenike and Johanson [4], Neddeman [5]). It was shown that wall fiction and hoppe wall conditions that exceed the limiting line given in Figue esulted in no flow along the walls o funnel flow.!"#!$"%&! p 3

4 Wall Fiction Angle (deg) Conical Hoppe Angle (deg) Figue. Typical mass flow funnel flow limit fo effective intenal fiction angle of 5 degees The adial stess theoy assets that stess states within pocess equipment ae compatible with adial stess fields and will poduce mass flow in that pocess equipment. Non-adial stess states poduce funnel flow behavio (stagnant egion fomation). In addition, the moe fictional the wall suface, the steepe the velocity pofile acoss the bin. Figue 3 shows the computed velocity pofiles fom the adial stess theoy fo a conical hoppe with a hoppe slope angle of degees measued fom the vetical. This figue shows the incease in the steepness of the velocity pofile as the wall fiction angle inceases. This adial stess theoy will be discussed in moe detail below. Velocity atio (V(b) /Vcente) Dimensionless Radius (b/rb) Fiction Angle = 5 deg Fiction Angle =.5 deg Fiction Angle = deg Fiction Angle = 7.5 deg Fiction Angle = 5 deg b Rb Figue 3. Conical velocity pofiles in a degee hoppe with vaious fiction angle along the bin wall!"#!$"%&! p 4

5 !"#!$"%&! p 5 Computed adial velocities ae well established fo conical hoppes and could be extended to cone-in-cone geometies using the famewok of the adial stess theoy. Radial stess theoy can be applied to cone-in-cone geomety since the two hoppes have a common apex. The oveall objective of this wok is to undestand the elationship between blende velocities and the opeational paametes. Theefoe, the cone-in-cone adial theoy will include extenal foces such as gas pessue gadients so the elationship between pocess flow and velocity pofiles can be detemined. The stating point fo adial stess deivation is the equation of motion with the acceleation tems neglected. As a esult, the adial stess theoy applies to slow moving conditions in cone-in-cone geometies. g P = γ τ () Spheical coodinates will simplify the numeical epesentation of equation fo the case of conical hoppes. The adial diection oiginates fom the cone-in-cone apex esulting in equations and 3 descibing the elationship between the solid stess, bulk density, and gas pessue gadient. ( ) P g = γ φ τ τ φ φ ) sin( ) sin( ( ) sin( () γ τ φ τ τ φ φ = P g ) cot( ) ( ) sin( )) sin( ( ) sin( ) ( (3) The Haa-Von Kaman hypothesis can be used to eliminate some of the vaiables in equations and 3. This hypothesis states that the nomal stess ( φ ) is eithe a majo o mino pincipal stess. Nomal hoop stess ( φ ) equals majo pincipal stess if the mateial flows though conveging geometies, and equals the mino pincipal stess when flowing though diveging geometies. In eithe case, the shea stesses (τ φ ) and (τ φ ) acting on the φ plane equal zeo. This simplifies equations and 3 to yield equations 4 and 5. ( ) P g = γ τ φ ) sin( ( ) sin( (4) γ τ τ φ = P g ) cot( )) sin( ( ) sin( ) ( (5)

6 It is impotant to point out that thee ae fou unknown stesses in the two equations above. Additional elationships between stess components ae equied to solve these two equations. The equied elationship comes fom the effective yield locus of the bulk solid (see Figue 4), τ δ ω 3, τ Figue 4. Limiting stess state definitions Moh cicles ae a gaphical epesentation of the stess state acting on a bulk mateial. The effective yield locus is a line that is the envelope of the paticula states of stess that epesents a condition of continual defomation without volume change and is called the citical state of stess. This condition exists duing steady flow in conveging geometies. The elationship between stess tenso components can be deived using the definition of the effective yield locus and Moh cicle stess state geomety. This deived elationship povides the equied closue equation fo equations 4 and 5. The unknown stesses in these two equations can be epesented as functions of the mean stess () and the diection angle between majo pincipal stess and the spheical coodinate system (ω). These elationships ae given in equations 6 though. = ( sin( δ ) cos( ω)) (6) τ = ( sin( δ ) cos( ω)) (7) = sin( δ ) sin(ω) (8) = ( sin( )) (9) δ = ( sin( )) () 3 δ φ = ( sin( δ )) ()!"#!$"%&! p 6

7 The last equation equied fo the adial stess model comes fom wok done by Sokolovski [6]. He states that the mean solid stess nea the apex of a wedge shaped section of a bulk mateial is popotional to the adial distance fom the wedge apex to the point of inteest. This yields equation fo the mean stess. = γ g s( ) () Substitution of equations 6 though into equations 4 and 5 esults in two equations in tems of the mean solids stess () and the pincipal stess diection angle (ω). These deived equations assume that the effective angle of intenal fiction is constant and that the pessue gadients ae only functions of adial position fom the hoppe apex. sin( ω) cot( ) dω sin( δ ) s sin( δ ) s cos( ω) cos( ω) d = (3sin( δ ) cos( ω) ) s ds sin( δ ) sin( ω) ( A cos( )) d (3) dω sin( ω) d (sin( δ ) cos( ω) ) cos( ω) cot( ) sin( δ ) s sin( ω) = (3 sin( δ ) sin( ω)) s ds ( A sin( )) d (4) P A = (5) γ g P A = (6) γ g Equations 3 and 4 can be solved simultaneously to yield equations 7 and 8. sin( ω) sin( δ ) cot( ) cos( ω) sin( δ ) cot( ) sin( δ ) sin( ω) A cos( ω) cos( ω) sin( ) sin( ω) sin( ω) cos( ) ds = A d ( cos( ω) sin( δ )) s (7)!"#!$"%&! p 7

8 dω = d sin( ω) cot( ) cos( δ ) sin( δ ) cos( ω) A (sin( δ ) cos( ω) ) sin( ) sin( ) A ω δ cos( ω) cos( ) sin( δ ) cos( ) sin( ω) sin( ) s sin( δ ) ( cos( ω) sin( δ )) (sin( δ ) ) s (8) With the exception of the tems containing A and A, these two equation ae identical to the adial stess used by Jenike [7] [8], Johanson [4], and Neddeman [5]. The tems A and A include the gas pessue effects and can be used to estimate the influence of fluid pessues of the behavio of the bulk. It should be pointed out that the methodology used in this pape assumes the use of the taditional Moh-Coulomb yield citeia. One could also apply a Ducke-Page yield citeia as outlined in Jenike s wok [9]. This will be a topic of futhe study. Up to this point we have not made any distinction between the conical geomety and the annula geomety in the cone-in-cone. Consequently, equations 7 and 8 will apply to both geometies. Howeve, the bounday condition fo each geomety will be diffeent. In the case of the conical geomety, the value of ω is known at the centeline of the hoppe and at the hoppe wall. In the case of the annula egion, the value of ω is known at both the inne and oute hoppe walls. The stess at the centeline of an axial symmetic conical hoppe must be aligned with the pincipal stess diection. This implies that the angle between the coodinate axis and the diection of majo pincipal stess (ω) equals zeo. ω = at = (9) The othe equied bounday condition comes fom the state of stess at the hoppe wall. The stess condition at the wall must satisfy both the citical state of stess and the taditional columbic fiction condition. Figue 5 shows a Moh cicle compatible with the citical state of stess. The line in Figue 5 indicates the columbic fiction condition. Two intesection points satisfy these conditions. If the hoppe conveges, then the intesection at point A, epesenting the passive state of stess, applies. If the hoppe diveges, then the active stess at intesection point B applies.!"#!$"%&! p 8

9 δ Point B, τ φw ωb ωa Point A, τ 3 Figue 5. Two possible wall fiction states fo conveging and diveging conical hoppe geometies The cone-in-cone is a conveging geomety so the passive stess state applies. The concepts of passive and active stess states ae boowed fom the discipline of soil mechanics. They aise fom etaining wall teminology. In ode to activate the stess state behind a etaining wall, the wall is pulled away fom the mateial causing the stesses acting on the wall suface to be lowe than the stesses acting vetically. Convesely, a passive state of stess esults in wall stesses that ae lage than the vetical stesses. The situation descibing flow in a hoppe is simila to a etaining wall, except the mateial moves and the wall is stationay. If the mateial flow diection causes mateial to pull away fom the wall, then the active stess state applies, causing the wall stesses to be lowe than the vetical stesses. This phenomenon occus in diveging hoppes. Conveging hoppes ae then subject to passive stess states whee the wall stesses ae geate than the vetical stesses. The stess state epesented by point A implies that the diection between the coodinate axis and the majo pincipal stess is given by equation. sin( φ δ sin( ) we ω = φ we a sin at = iw () In non-aeated conditions, the wall fiction angle is exclusively known and equations 7 and 8 can be integated diectly to poduce a elationship between the pincipal stess diection angle (ω) and the hoppe slope angle. Howeve, the additional body foces caused by the gas pessue gadients nea the hoppe wall change the effective wall fiction angle (i.e. the angle of slide against the wall). Gas pessue gadient is a body foce tem and is a vecto quantity that can be added vectoally with the gavitational vecto tems to poduce a new vecto component that acts at a diection (β) fom the coodinate system (see Figue 6.). This implies that the total effective body foce tem acts in a diection othe than the diection of gavitational pull. In effect the addition of gas pessue gadients causes the effective fiction angle to act pependicula to the β- diection in ode to maintain equilibium of foces. Consequently, the net effect of the gas!"#!$"%&! p 9

10 !"#!$"%&! p pessue gadient is to otate the effective coodinate system elative to the gavitational diection. The new effective wall fiction angle (φ we ) is given by equation. β φ φ = w we () P P g z γ Body Foce β - Figue. 6. Foce balance at wall suface including gas pessue gadient tems Angle (β) is a function of the pessue gadient in the adial and -diections. This leads to equation, descibing the new effective wall fiction angle in tems of the gas pessue gadient tems. = tan cos sin wall wall wall wall w w we P g P g P P a a γ γ φ φ φ () Equations 9 and povide the equied bounday conditions fo standad conical hoppes. Equation also applies to the oute wall in the annula egion of the cone-in-cone geomety yielding the following equation fo the oute wall bounday condition. ow we we at a δ φ φ ω = = ) sin( sin( sin (3)

11 Howeve, the inne wall in the annula egion of the cone-in-cone equies diffeent bounday conditions. Figue 7 shows the expected contous of majo pincipal stess diection within the annula egion. This figue implies that the diection of majo pincipal stess is in the negative ω-diection along the inne cone wall. This suggests that the bounday condition along the oute suface of the inne cone wall is given by Equation 4: sin( φ δ sin( ) we ω = φ we a sin at = iw (4) ψ ψ is the angle between the majo pincipal stess diection and the -coodinate axis ψ Contos of majo pincipal stess diection Figue 7. Diection of pincipal stess in annula egion Equations 7 and 8 can be integated using bounday conditions given in equations 3 and 4 to poduce adial stess solutions fo annula geometies. Figue 8 shows some of the solutions fo the case of a degee inne cone. Caeful examination of the solution space of these diffeential equations eveals no adial stess solutions to the ight of the annula solution dotted line shown in Figue 8. This is simila to the esult fo the solution of adial stess field in standad conical hoppes. These standad conical hoppe solutions also showed no adial solution to the ight of the solid line shown in Figue 8. Expeimental evidence suggests that this line is the limiting line between mass flow and funnel flow behavio fo typical conical hoppes. It stands to eason that the limiting line fo the annula geomety would also be the limit between mass flow and funnel flow behavio in the annula geomety. This figue eveals how a cone-incone hoppe can be used to extend the mass flow limit to flatte hoppes fo a given wall fiction condition. Conside, fo example, a mateial that has a fiction angle of 33 degees. The conical mass flow limit line shown in Figue 8 indicates that conical hoppes must be steepe than degees measued fom the vetical to poduce flow along hoppes walls. Howeve, the solution fo the same wall fiction angle in an annula flow channel flow whee the inne cone is degees is given by the dotted line in Figue 8. This figue implies that mass flow is possible if!"#!$"%&! p

12 the extenal hoppe is about degees measued fom the vetical. The net esult is that this combination of hoppes shifts the limiting adial stess line towads flatte hoppes. Thus, the cone-in-cone geomety extends the mass flow limit to flatte oute cone angles. If mateial will flow in mass flow in a degee conical hoppe then the cone-in-cone hoppe can be almost degees and still poduce mass flow. Wall Fiction Angle (degees) Hoppe Half Angle (degees) Conical Mass Flow Limit Inne Cone deg Figue 8. Mass flow limit fo cone-in-cone hoppe with inne cone of degees This same analysis can be pefomed fo vaious combinations of inne and oute conical hoppes. Figue 9 shows the pedicted mass flow limit lines fo seveal inne conical hoppe configuations. Geneally, if fictional chaacteistics of the inne and oute cone ae the same and the inne hoppe is designed fo mass flow with a hoppe angle of c, then the oute hoppe angle can be designed with a slope angle given by the balanced fiction angle limit line found in Figue 9. Convesely, if an existing conical hoppe has a fictional condition that puts it in the funnel flow egion, then Figue 9 could be used to detemine the size and shape of the inne cone that would be equied to cause flow along the walls. Fo example, conside the case of a 35 o conical hoppe with a wall fiction angle of o. This cone would be in the funnel flow egime as indicated by the plus sign in Figue 9. Howeve, if an inne cone with a o slope was placed within the existing funnel flow hoppe, then mass flow could be achieved.!"#!$"%&! p

13 Wall Fiction Angle (degees) Hoppe Half Angle (degees) Figue 9. Conical Mass Flow Limit Inne Cone 5 deg Inne Cone deg Inne Cone 5 deg Inne Cone deg Inne Cone 5 deg Inne Cone 3 deg Balanced fiction mass flow limiting line fo cone-in-cone geomety (intenal fiction angle δ=5 deg) The analysis above suggests that cone-in-cone geometies could be designed to poduce mass flow duing opeation povided hoppe walls ae steep enough. Thus, designing the hoppe angle in accodance with these limits will satisfy the fist blende citeia that states that all of the mass within the blende must be in motion duing blende opeation. The above analysis detemines the elationship between wall fiction angle and the steepness of both the inne and oute hoppe walls that will poduce mass flow. Walls that ae too fictional o too flat will esult in flow pattens whee at least some of the mateial in the blende is stagnant. Howeve, imposing just the concept of mass flow is not sufficient to guaantee significant blending. Taditional adial stess theoy applied to standad conical hopes indicates that nealy unifom flow pofiles can exist in hoppes whee mateials have low fiction angles. It stands to eason that applying adial velocity pattens to the cone-in-cone geometies will also poduce unifom flow pattens fo low fiction angle o steep hoppes. These nealy unifom velocity pofiles will esult in poo mixing. A method of computing velocity pofiles in cone-in-cone geometies will be pesented below. It is impotant to note this mass flow citeion was detemined solely fom knowledge of basic mateial popeties. Consequently, this analysis allows us to elate the geneal flow patten to mateial popeties. This is the fist step in de-convoluting mateial popeties and blende opeation. The next step involves computing the expected velocity pofile in the blende fom just knowledge of mateial popeties and blende geomety.!"#!$"%&! p 3

14 Velocity pofiles in Cone-in-cone blendes Thee ae two ways velocity pofiles can be affected in cone-in-cone hoppes. Since the cone-incone actually consists of two independent hoppe sections, it is possible that these two hoppes sections have diffeent aveage flow ates though them based on the aeas of the top and bottom outlets. In fact, the cone-in-cone hoppe dimensions can be specified to ceate a desied flow atio between the inne cone and the annula egion. Equation 5 shows how the inne cone outlet diamete can be chosen to specify the faction (R) of global flow ate diected though the inne cone. This faction is based on the atio of flow aeas at the top and bottom diametes of the cones. Fom a pactical point of view, a typical cone-in-cone can achieve a maximum velocity atio (R) between inne and oute cones of about five to one. Equation 5 can be used to estimate the inne cone diamete (d B ) equied fo a given velocity atio (R vel =V inne /V oute ). d B = D sin( o ) sin( ) i R vel B (5) In addition to this global velocity patten, the velocity acoss the hoppe section will vay based on the wall fiction conditions. This is a diect esult of applying the adial stess and velocity theoy to a cone-in-cone geomety. The cone-in-cone geomety with a common apex suggests that adial velocity pofiles could also exist in the annula hoppe egion as well as the inne cone egion. The adial velocity patten is a function of the diection of pincipal stess in both the inne cone and annula hoppe section. None of the assumptions egading adial velocities ae violated in annula conical flow so the standad adial velocity equations (Jenike and Johanson [4], Neddeman [5]) can be used to pedict the velocities in annula conical geometies. The adial velocity in the inne cone can be appoximated by equation 6 while the adial velocity in the annula egion can be appoximated by equation 7. iw V ( ) = V () exp 3 tan( ω) d (6) ow V ( ) = V () exp 3 tan( ω) d (7) iw These equations yield an appoximation to the velocity pofile acoss both the inne and annula hoppe sections that depend on wall fiction angles, effective intenal fiction angles, and adial gas pessue gadients. These velocities can be combined with the inne cone velocities to give the complete adial velocity patten in a cone-in-cone hoppe. Scaling factos can be applied to the inne and oute velocity pofiles to adjust the flow atio in the inne and oute geometies. Figue shows two expected velocity pattens in a cone-in-cone hoppe with a two-to-one velocity flow atio (R vel )!"#!$"%&! p 4

15 fom the inne to oute cones. These velocities ae a function of the wall fiction angle. Figue a pedicts vey flat velocity pofiles with low wall fiction angle mateials. Figue b shows steepe velocity pofiles with moe fictional wall mateials. The flow pofiles in Figue assume that mateial exits the blende at the bottom of the conein-cone section. Howeve, in good blendes thee exists a conical hoppe below the cone-incone section. The velocity pofile in this conical hoppe extension influences the oveall blende velocity pofile. The adial stess theoy can be used to compute the expected velocity pofile in the lowe cone. This pofile can be nomalized to poduce an aveage velocity equal to one. This nomalized velocity pofile can then be used as a multiplie on the cone-in-cone velocity to poduce a new oveall velocity pofile that includes the effect of the lowe conical hoppe. Figue shows how the velocity pofiles above would be influenced by the conical section below the cone-in-cone. Figue. a Wall fiction angle of deg b Wall fiction angle of 3 deg Velocity pofiles in typical cone-in-cone geometies at two diffeent fiction conditions a Wall fiction angle of 3 degees; with a 3: flow ate atio between inne and oute cones and no lowe conical hoppe placed below the blende. b Wall fiction angle of 3 degees; with 3: flow ate atio between inne and oute cones and lowe conical hoppe. Figue. Velocity pofiles showing the effect of the lowe conical hoppe on velocity pofiles in cone-in-cone blendes!"#!$"%&! p 5

16 Residence time distibution functions in cone-in-cone blendes Undestanding blending in a cone-in-cone will equie using these computed velocity pofiles to detemine the esidence time distibution function (E(t)) fo the blende and then using this esidence time distibution function to compute the expected vaiance eduction facto in the blende. The standad way of evaluating continuous powde blende pefomance is to place a sudden impulse of makes in the blende, then opeate the blende while maintaining the level in the blende and obseve the dischage pofile of makes leaving the blende. This exiting make pofile, when nomalized to poduce a total integated concentation of one, is the esidence time distibution function. The poblem with this expeimental analysis is that changing the mateial, adjusting the ate, o slight modifications to blende geomety equies a whole new blende test to develop the new esidence time distibution function. Thus scale-up of blende pefomance is nealy impossible fo most blendes. Howeve, if the velocity pofile is known, then it is a modeately simple task to numeically place a unit impulse in the cone-incone blende and integate the flow field to compute the distibution of makes exiting the blende as a function of time. In the cone-in-cone geomety, calculated velocity pofiles do not coss each othe and, consequently, can be integated along flow steam lines to detemine the time that a paticle placed at the top of the blende will exit the blende outlet. Computing the exit times fo a laye of paticles placed at the top of the blende gives ise to a method of computing the esidence time distibution function of the blende. The pocedue fo computing esidence time in the blende fom mateial popeties is as follows. The time equied fo one complete blende volume to pass though the blende is divided into small equal incements. This incemental time unit is then used to compute the thickness of a mathematical make laye placed at the top of the blende. The thickness of this laye epesents the aveage distance taveled fo mateial at the top of the bin in one time incement. The coss sectional aea at the top of the blende is divided into ings of equal aea, ceating egions of constant volume. These egions will be subject to adial flow steamline velocities. The position of any of the egions ove time can be computed by multiplying the adial velocity by the incemental time and vectoally adding this value to the last make volume position. The position of these individual make volumes can be continually monitoed to detemine the time when they exit the blende. The numbe of volume elements exiting the blende divided by the total numbe of initial volume elements in the blende is an appoximation of the blende esidence time distibution. Pefect plug flow will esult in all the make volumes exiting the blende at the same time (afte one blende volume dischage). Any deviation fom this will poduce a distibution in esidence times and esult in blending. Figue shows a calculated esidence time distibution function fo a cone-in-cone hoppe with a degee inne cone and a degee oute cone and a unifom velocity pofile imposed below the cone-in-cone hoppe. The elative flow ate vaiation between the inne and oute cone was two-to-one and the wall fiction angle was 3 degees measued fom the hoizontal. This figue also contains the velocity pofile computed fom this blende geomety. The initial make laye comes fom the cente of the blende and exits the blende afte only.5 blende volumes have passed though the system. The lage peak is fom the sudden change in the slope of the blende pofile and exits the blende afte about.5 bende volumes have passed though the system. It is!"#!$"%&! p 6

17 impotant to point out that this figue epesents the esidence time distibution in tems of the total amount of mateial passing though the blende instead of the elapsed time in the blende itself. In othe wods, the numbe of blende volumes (BV) passing though the system would be equal to the esidence time (t esidence ) multiplied by the aveage mass flow ate (Qs avg ) of the mateial in the blende and divided by the blende mass capacity (Cap blende ). BV Qsavg = tesidence (8) Cap blende 7 Residence Time Distibution Blende Volume Thoughput Figue. Residence time distibution function fo cone-in-cone geomety shown in Figue b with wall fiction angle of 3 degees Aveage esidence time T avg can be used to chaacteize blendes o othe pocess equipment. Howeve, two blendes may have the same aveage esidence time but cause vey diffeent blending behavios. In fact, it is possible to constuct a bin whee the mateial flows in nealy pefect unifom velocity mass flow. Residence time fo this case would coespond to the time equied fo one blende volume to pass though the system. The distibution function in pefect mass flow would be a shap concentation spike centeed aound one blende volume. This type of blende pofile would not lead to blending and a unifom velocity mass flow bin would be a poo choice fo a blende. A blende could also be constucted with an aveage esidence time of one but distibute mateial ove seveal bin volumes and esult in good blending. Theefoe, the width of the esidence time distibution is moe impotant in quantifying blending opeation. It is the ange of esidence times in any blende that causes blending. The esidence time distibution function could be consideed a blende finge pint. With this distibution function, enginees can compute the expected concentation pofiles fo any input condition. Nomally this esidence distibution function is a measued quantity. The stength of this pape is the ability to compute the distibution function solely fom measued mateial!"#!$"%&! p 7

18 popeties fo a given blende geomety. Once the esidence time distibution function is known, any output concentation can be pedicted by combining the input concentation function C in (t) and the esidence time distibution function (E(t)) using the convolution integal given in Equation 9. C out = ( τ ) C ( τ t) E( t) dt (9) in Enginees can now compute the expected output concentation fom a given cone-in-cone blende fo any pescibed input concentation based only on a knowledge of basic mateial popeties. Fo example, compae the input and output concentation pofiles shown in Figue 3. The continuous blende opeation smoothes the input concentation pofile and poduces a concentation pofile with the same fequency but smalle vaiation exiting the blende. This esult suggests that cone-in-cone blendes satisfy the second blending citeia which states that thee must be significant esidence time distibutions within the blende fo mixing to occu. Make Concentation (faction) Blende Volume Thoughput (a) Figue 3. Make Concentation (faction) Blende Volume Thoughput (b) Input (a) and output (b) concentations fo blende with computed esidence time distibution given in Figue and input concentation peiod of.!"#!$"%&! p 8

19 Figue 3 shows the expected eduction in output concentation fluctuation fo the case whee the peiod of concentation fluctuation extends ove one complete bin volume. The numbe of fluctuations in a blende volume affects the degee of smoothing obseved in the output concentation. This effect is shown in Figue 4 whee the peiod of the input concentation function is less than one blende volume. This shote peiod esults in smalle output concentation fluctuations. One could compute the vaiance of the output concentation and divide that by the vaiance of the input concentation to povide a measue of blending effectiveness. This vaiance eduction facto can povide a means of anking blende pefomance. Lowe values of the vaiance eduction facto will esult in bette blending. Sdevout VR = (3) Sdevin Make Concentation (faction) Blende Volume Thoughput (a) Make Concentation (faction) Blende Volume Thoughput (b) Figue 4. Input (a) and output (b) concentations fo blende with computed esidence time distibution given in Figue and input concentation peiod of.33!"#!$"%&! p 9

20 The example in Figue 3 above would esult in a eduction facto of.86 while the example in Figue 4 would esult in a eduction facto of.36, indicating bette blending. It is impotant to point out that, in geneal, the geate the fequency o the shote the peiod of the input concentation, the smalle the output concentation fluctuation. Since the above method allows calculation of the blende esidence time distibution as a function of mateial popeties and specific blende design constaints, it can be used fo a paametic study of blend efficiencies based on flow popeties and design constaints without having to measue blende distibution functions expeimentally using expensive scale blende models. Fo example, conside the case of a cone-in-cone blende with a degee oute cone and a degee inne cone. Conside also the effect of changing the input fluctuation fequency in two nealy identical blendes whee the diffeence in these two blendes is the wall fiction angle. Figue 5 shows the effect of blende vaiance eduction factos (VR) fo two blendes whee the numbe of input concentation fluctuation in one blende volume was vaied. The numbe on the x-axis of this gaph is the numbe of fluctuation peiods of a sinusoidal vaying concentation that entes the blende duing the time it would take fo one blende volume to pass though the system. The esulting vaiance eduction facto depends stongly on the numbe of fluctuations pe blende volume. This figue clealy shows that, the lage the numbe of fluctuations pe blende volume, the lowe the vaiance eduction facto will be. The esults of this paametic study also suggest that the wall fiction angle does not make a significant diffeence in blende effectiveness if the numbe of concentation input cycles is geate than about seven. Howeve, thee is a significant diffeence in blende efficiency in this blende when the numbe of input concentation cycles pe blende volume is small. Thus, conein-cone blende opeation becomes sensitive to mateial popeties (i.e. wall fiction) when attempting to blend fluctuations that occu ove lage blende volumes..3 Output Vaiance Reduction Facto # Input Concentation Cycles pe Bin Volume Wall Fiction Angle = 5 degees Wall Fiction Angle = 3 degees Figue 5. Vaiance eduction factos fo identical blendes with a diffeence in wall fiction angle!"#!$"%&! p

21 The influence of small changes in blende design can also be studied without conducting a blende test. Conside compaing two blendes whee the wall fiction angle is 3 degees. Assume that the blende is designed with a degee inne cone and a degee oute cone. Howeve, assume one blende dischages diectly fom the cone-in-cone section to the downsteam pocess. Also assume that the inne cone diamete, fo this blende, is designed with a two-to-one flow ate atio. The othe blende has a conical hoppe below it that is imposing an additional blende velocity on the mateial and the inne cone diamete is inceased to geneate a thee to one flow ate atio between the inne cone and annula egion. This second hoppe configuation will esult in a steepe velocity pofile acoss the blende and should esult in bette blending. Figue 6 shows a compaison between the computed blende velocity pofiles fo these two blendes. The blende with the highe flow ate atio and lowe conical hoppe has a centeline velocity that is 5 times geate than the velocity at the side wall. The blende with the two-to-one flow ate atio between the inne and oute cones has a velocity that is only 4.6 times geate than the side wall velocity. 6a Wall fiction angle of 3 degees; with a : flow ate atio between inne and oute cones and no lowe conical hoppe placed below the blende. 6b Wall fiction angle of 3 degees; with 3: flow ate atio between inne and oute cones and lowe conical hoppe. Figue 6. Velocity pofiles showing the effect of the lowe conical hoppe on velocity pofiles and flow ate atio in cone-in-cone blendes The vaiance eduction atio fo these two blendes can be computed and compaed fo a vaiety of peiod input concentation fluctuations. Figue 7 shows these esults. The steepe velocity pofile shows a significant decease in the vaiance eduction facto fo blending. Howeve, fo the case of a peiodic input concentation thee also exists a complex elationship between the input concentation fequency and the blende vaiance eduction facto. Thee appea to be zones of input concentation fluctuation fequencies that will esult in low vaiance eduction factos. This effect may be due to the peiodic natue of the input fluctuation and may not exist, o at least be educed, fo andom concentation fluctuations. It is clea fom this Figue that one of the pimay vaiables influencing blending is the flow ate atio between inne and oute cones. In geneal, the geate this atio is, the bette the blending will be. Intuitively, this makes sense. The geate the flow ate atio between the inne and oute cones, the moe distibuted the layes become in the axial diection. Hence, blending is bette.!"#!$"%&! p

22 Output Vaiance Reduction Facto # Input Concentation Cycles pe Bin Volume 3: Flow Rate Ratio : Flow Rate Ratio Figue 7. Vaiance eduction factos fo blendes with diffeences in the flow ate atio showing a decease in blende vaiance eduction with lage flow ate atios This analysis also suggests that the optimal blende configuation will be one whee the blending velocity pofile is continuous and whee the velocity in the cente of the bin is an ode of magnitude faste than the velocity at the blende wall. This type of velocity pofile can only be accomplished by using a conical hoppe in conjunction with a cone-in-cone hoppe section. The stength of the analysis pesented in this pape is the ability to make small modification to the blende design based on knowledge of the basic mateials popeties. Blende optimization can then be caied out without extensive and expensive pilot scale blende tests. Of couse, pudence suggests that pilot scale testing should not be completely eliminated. Howeve, the analyses suggest a method of at least educing the pilot scale tials to achieve optimal blende design. Much of this design pocess can be accomplished numeically and then a small scale pilot unit constucted based of the optimal blende design. This design methodology should help enginees intelligently select o design a cone-in-cone blende to wok with thei paticula mateials. Compaison to measued esidence time distibution functions A small scale cone-in-cone blende was constucted. The top diamete was 3.5 cm. The hoppe section consisted of a degee inne cone and a degee oute cone that connected on to a. cm diamete conical hoppe with a hoppe slope of degees. The inne cone in the blende section was designed to poduce a flow ate atio of thee to one. The wall fiction angle against the hoppe wall suface was 3 degees. This blende configuation is simila to the configuation poducing the velocity pofile shown in Figue 6b. The analysis outlined above was caied out to compute the esidence time distibution function fo this blende configuation. The faction of total makes passing though the blende outlet as a function of!"#!$"%&! p

23 time is defined as the accumulated esidence time distibution function (F(t)) fo this scale blende and is shown in Figue 8 along with a measued cumulative esidence time distibution pofile fo this blende configuation. The initial make appeaance is the same fo these two distibutions. In addition, the esidence time distibutions end afte the same volume input though the blendes. Ageement of these stating and ending points fo the esidence time distibution functions imply that the theoy outlined above accuately captues the elationship between the centeline velocity and the velocity at the bin wall. Howeve, the discepancy between the measued and computed cumulative distibution functions indicate that the actual velocity pofile does deviate somewhat fom the computed pofile. Faction Passing Bin Volume Computed F Function Measued F Function Figue 8. Cumulative esidence time distibution function showing the cumulated faction of makes exiting a model blende due to a unit impulse of makes The diffeences between these distibution functions can be bette seen when plotted as instantaneous esidence time distibution functions (E(t)) as shown in Figue 9. The measued initial peak is lage than the calculated peak and thee is a peak nea one bin volume that does not match the calculated pofile. Caeful obsevation of the measue data may help explain these discepancies. Duing the testing it was obseved that the flow pofile acoss the hoppe was not completely symmetic and flow was slightly faste on one side of the hoppe than the othe. This non-symmetic flow pofile could explain the spuious peak aound. bin volume and the noticeable lack of makes aound.8 bin volumes. The likely cause fo this non-symmetic flow is an off-cente conical inset. The elative flow between the inne and oute hoppes can be detemined by the atio of bin diametes. Howeve, if the inset is positioned slightly off-cente the local velocity on one side of the bin would speed up based on the inceased gap between the inne and oute cones while the flow on the opposite inside would decease due to the smalle gap between the inne and oute cones. The lack of dimensional toleance could then account fo the discepancies in these two esidence time distibution functions.!"#!$"%&! p 3

24 3 Residence Time Distibution Function Bin Volume Computed E Function Measued E Function Figue 9. Computed and measued esidence time distibution function fo model bin Blende effectiveness in the cone-in-cone blende depends on the velocity distibution acoss the blende. Theefoe, anything that changes the velocity distibution will also change the blende effectiveness. Diffeences in wall fiction angle change the velocity distibution and will esult in changes to blende pefomance depending on the blende opeation. Howeve, Equations 3 and 4 also contain the effect of the local pessue gadient. These pessue gadient values can esult in diffeences in velocity pofiles in the cone-in-cone blende. Significant gas pessue gadients can affect mixing in cone-in-cone blendes when dischaging fine powde mateials. Gas pessue gadients aise fom the fact that bulk mateials ae compessible and change poosity as solids contact stesses within the equipment change and ae impemeable. Fo example, conside a bulk powde mateial flowing at a slow ate though a cone-in-cone blende. The solid s stesses incease as mateial flows though the bin. This causes some consolidation of the bulk mateial and squeezes some ai out of the solid s poes. Gas leaves the system pedominately though the top mateial fee suface. Howeve, the consolidation pessue in the bin o hoppe deceases as the mateial appoaches the outlet. Deceasing consolidation pessue esults in expansion of the bulk mateial. Howeve, some of the gas within the solid s poes has aleady left the system though the top of the bin. This esults in a net gas deficit nea the hoppe outlet, ceating negative gas pessues within the mateial and poducing significant negative gas pessue gadients nea the outlet. These gas pessue gadients could be esponsible fo changing the local velocity pofiles in the bin. The cone-in-cone hoppe situation is complicated by the fact that the inne cone wall is impevious to gas flow. Thus, thee exists a diffeence in adial gas pessue gadients between the inne cone and the annula egion in the cone-in-cone blende. This could significantly change the blende effectiveness of cone-in-cone blendes. Figues and show the expected gas pessue and gas pessue gadient pofiles in both the inne cone and annula egion fo a typical cone-in-cone blende.!"#!$"%&! p 4

25 Height (m) Qs = 36.8 kg h Gas Pessue (Pa) Cylinde Inne Cone Annula Region Figue. Gas pessue pofile in cone-in-cone blende when opeating with fine plastic powde at a flow ate of 36.8 kg/h (about 3% of the expected limiting flow ate fo this powde). Velocity atio between inne and oute cone is : 5 Height (ft) Dimensionless Gas pessue gadient Cylinde Inne Cone Annula Region Qs = 36.8 kg h Figue. Radial gas pessue gadient (dp/d / γg) in cone-in-cone blende when opeating with fine plastic powde at a flow ate of 36.8 kg/h (3% of expected limiting flow ate fo this powde). Velocity atio between inne and oute cone is :!"#!$"%&! p 5

26 Once the local gas pessue gadients ae known, the adial velocity analysis outlined above can be implemented to compute the expected velocity pofiles fo the case of the local adial pessue gadients given in Figue. This analysis can be applied to both the inne cone and the annula egion to detemine the velocity pofiles at vaious local gas pessue gadients. Figue shows the expected velocity pofiles in the degee inne cone at diffeent adial gas pessue gadients. Note that the negative gas pessue gadient at the bottom of the inne cone is lage enough to esult in a funnel-flow velocity patten. If this high gas pessue gadient pesisted thoughout the inne cone section, it would induce a funnel flow velocity patten in the blende and esult in a pefeed flow channel. Luckily the gadients in othe hoppe elevations ae smalle than this value. Likewise, Figue 3 indicates that if gas pessue gadients in the annula section have lage enough negative values, then pefeed flow channels would develop along the oute wall of the inne cone. Each of the velocity pofiles in these figues has been nomalized so the velocity at the centeline equals one.. Ro Velocity Ratio (V/V()) Radial Distance /Ro A =-.45 Figue. A =-. A = -. A =. A =. Nomalized velocity pofiles fo inne cone at vaious local gas pessue gadients A = (dp/d / γg)! " #!$ " % &! p 6