m ELSEVIER
PowdelTechnology87 ( 1996}93-104
Cyclone separator scaling revisited J.S. Ontko Morganlown EnL,rgy Tethnedogy Center. US S~partment of Energy, Morganlown. WV 26505. USA
Recei,.~.1 8 May 1995
Abstract It is customary to assume cyclone separator fractionalefficiencyis independent of the mass distributionof the incomingdust. To leStthis hypothesis and to clarify the role of other variablesknown to influencecyclone performance, an experimentalinvestigationof cyclone scaling using response surface techniques was undertaken. Both the overall collectionefficiencyand dimensionlesspressure drop were found to be smooth, well-estimatedfunctionsof the Reynoldsnumber, the dust loading,and their interactionover the experimentalregion,demonstrating Ihat the effects of these variablesmust be studiedconcurrently. A simplesimilaritycondition for fractionalefficiencyexpressed in terms of the inlet distribution is proposed and experimentally confirmed. Both local reduction in overall efficiency with increased loading and unclassifiedparticlecollection,that is, collectionindependent of particlesize, were observedexperimentallyand were described in a natural way using the present approach. Keyvtords: Cycloneseparator;Scaling;Similarity
1. I n t r o d u c t i o n
The reverse flow cyclone separator, shown in Fig. 1 with its characteristic design lengths and ratios, has long been used as a low cost device for removing dust from gas streams. Its simplicity and ruggedness make it especially useful in applications involvinghigh temperatures and aggressive chemical environments. Cyclones figure prominently in many advanced coal-fueled technologies currently being developed under the auspices of the US Department of Energy, Morgantown Energy Technology Center. These technologies demand accurate prediction of cyclone performance over a range of operating conditions (including excursions occasioned by process upsets) to provide satisfactory and economical performance, ensure regulatory compliance, and protect expensive downstream components. RepresentatiYe s~rai-empirical theories accounting for the overall particle collection efficiency ~ (the fraction of the particulate mass entering the eyeloee which is captured) and the pressure drop &p expended in aUaining this efficiency have been concisely summarized by Leith [ I ]. Fractional or grade efficiency, 't/,describes the variation of collection efficiency with particle size, or, more precisely, the relation between the inlet, captured, and exit cumulative mass distributions. The fractional efficiency is related to the overall efficiency by ElsevierScienceS.A. SSD10032-5910 (95) 03085-9
i
o where • is the cumulative distribution function of the incoming particulate mass. It is customary to express '9 instead as a function of the particle size dand calculate ~by integration over all particle sizes, assuming ~ to be independent of 4. If a force balance on an isolated particle immersed in a flow field such that the particle Reynolds number N~. p ~: I is nondimeasionalized, the particle trajectory is a function of the Stokes number, Nstk. The standard efficiency correlations are thus functions of the cyclone design lengths Ai and NStkPressure drop theories, on the contrary, typically relate a pressure coefficient to dl alone. Experiment has shown both collection efficiency and pressure drop to he influenced by the dr:st loading. Empirical expressions correcting for this factor acting alone are given in several sources [21. These convention,.', scaling and design methods, though widely used, predict cyclone performance with limited reliability, as has been shown by Abrahamson [3]. More recently, computational fluid dynamics (CFD) has been applied to the problem [4,5 ] and commercial codes are currently available which model flows and particle collection in cyclones. The procedure is first to calculate the gas flow field without the parliculate phase. Then, limited numbers of particles are injected and their traiecturies calculated, usually
J.S. Ontko / Powder Technotogy 87 ~1996) 93-104
94
C y c l o n e D e s i g n Ratios A l
A1 A2 •&3
A b
A4
aid bid Do/D s/D
0.45 015 050
075
A6
pl/U FIID
6 0o
A 7
BID
0.50
A 5
3.00
Exit
De
(rh=rt2) * (v~,=~:)
(2)
where subscript I might designate a model and subscript 2 a prototype. Since tile implication is one way, this rather restrictive condition requires experimental confirmation. Assuming the variables previously listed are sufficient to describe the problem, a set of dimensionless parameters m a y be found by use of the Buckingham 'n'-Theorem [7]. For collection efficiency this yields:
h -
density pp. the gas viscosity ~, the gas volumetric flow rate Q. the particulate mass loading C, the particle size d, the cyclone barrel diameter D, and the cyclone design lengths Ai. Considerations outlined in Appendix I show that it is preferable to express r/as a function of qbrather than d. A hypothetical similarity condition for collection efficiency then follows from Eq. ( 1):
~
D
H
Lt~D,C,
,
~'q[NR¢, C, O).q(pptv qb), A,I
(3)
where On is an unknown function introduced because both P~te. and ~,b were fixed in the present experiment once the particulate material was fixed. The dimensfimless pressure drop may be expressed as: ApD'* = Neu = NEu[ NRe ' C, 6)Eu(pp/g, q)), Ad pgQ:
Fig. I. Reverse flow cyclone separalor with characlenslic dimensions. includingnumericalvaluesused in the present study under the assumption that there is neither particle-particle interaction nor gas-particle momentum coupling. To be a useful tool for processdesign, it is essential that CFD methods predict global behavior accurately. Testing them definitively requires an adequate range of experimental data obtained systematically. Response surface methodology [61 permits eflicient experimental investigation of the response of a system to concurrent variations in any number of independent variables. This technique produces an empirical equation describing the response and a statistical assessment of the adequacy of the description. The present paper reports an experimental study of cyclone separator performance using this method combined with the principles of similitude. The aim '~as twofold: ( 1) to determine if vt must be expressed in terms of c0 for sealing; and (2) to clarify the response to variations in some independent variables known to influence cyclone performance.
(4)
where NE. is the Euler number and O~u is another unknown function, In both Eqs. (3) and (4), the parameters of the inlet distributiou are implicitly specilied by ~. A suitable function for fitting fractional efficiency data is developed in Appendix 2: k~-~u ' 0~< cO< I (5) rt=
[ I,
q~=l
where u= [ln[ l / ( 1 - qb) 1 }¢, k >0, ~'> 0. Eq. (5) has the useful property of approximating, for 0 < ~ < I, the limiting cases:
( 1) sharp cutoff, v/= 0 for ~b< I - ~, ~/= 1 for • >/1 (approached as ~'--<.~) and (2) unclassified collection. '0= ~ for 0 < qb< I ( approached as ~'~ 0), as well as those in between. Note that k is the value which the numerator of the upper member of Eq. (5) takes when '0= 0.5; k and sr may therefore be termed cut and sharpness parameters, respectively.
2. Cyclune similarity parameters
3. Experimental
Over six decades of investigation has shown that the fractional efficiency depends on the gas density pg, the particle
Fig. 2 is a schematic diagram of the cyclone test apparatus. Air was supplied at ambient temperature and a pressure suf-
J.S. Ontko I Powder Tecbnalagy 87 (1996) 93-104
ficient to obtain the desired flow through two parallel branches. One branch entrained the discharge of a twin screw particle feeder through an eductor and bypass arrangement. The other, separately metered, carried only air. The branches joined downstream to form a common header discharging into an acrylic model cyclone. The model (D = 83 mm (3.25 in) ) was constructed from a design evolved from the Stairmand high efficiency design [8]. The fractional efficiency models of Leith and Licht [9] and Enliang and Yingmin [ I 0] and the pressure drop models of Shepherd and Lapple [ I I ], Alexander [ 12], and Barth [ 13] were applied iteratively to obtain what was predicted to he a design of greater efficiency than the Stairmand at an acceptable pressure drop. The Ai of the resulting design are given in Fig. 1. The experiment was laid out in two blocks, each a 3 × 3 factorial design replicated twice. The blocked variable was the particulate type, represen=ed by ~ in Eq. (3) and ~ in Eq. (4). The materials tested were a pressurized fluidized bed fly ash and a styrene resin manufactured by Atuehem, Inc. The latter was sieved to reduce the mean particle size. Both distributions were adequately described by Rosin and Rammler's [ 14] cumulative mass distribution function:
• = l - e x p [ - ( d / d o ) ~] 7 > 0
(6)
Distribution parameters were estimated from Coulter counter data by linear regression, a suitable form for which may be obtained from Eq. (6). Each replication consisted of all combinations of three levels of Nn= (2500, 6250, 104) and C (0, 0.016, 0.032) executed in random order. At C = O , collection efficiency is not defined, so efficiency data was available for only two levcl.gof C. The apparatus was cleaned between (but not during) each replication. The particulate block to be run first, which happeaed to be fly ash, was chosen at random before the experintenl. Overall efficiency was measured by weighing the catch and comparing it to the particulate mass fed daring each
Table I Par0culalcmaterialproperties Propcuy
Fly~sh
Resin
pp (kg ra ') d~ ( #ml y (-) d (b~m) 4: (/.¢m) (-)
2740 19.08 1.76 16.99 9 97 0.960
1300 4550 2.49 40.38 17.14 0.976
individual run, i.e. each combination of Nm=and C. The pressure drop was obtained from the average of five manometer readings taken at equal intervals during each run. Response surfaces tbt N ~ and '~ were estimated by multiple linear regression with the independent variables suitably coded in order to equalize their scale and minimize variance inflation. The experimental design allowed the estimation of linear and quadratic effects.
4. Results An analysis of variance is prcsentod for each response surface estimr t.,'d. The F tests examine the significaacc of the linear and quadratic effects when incorporated sequentially into the model. To assert that a surface is adequately fitted by a polynomial regression, Box and Draper [ 15 ] suggest that the ratio oftbe model mean square/error mean square should exceed the F value required f'~r statist;ca] ¢~gnificonceby minimum factor of 10. The residuals of each surface wean subjected to a runs test for r~ndomacss [ 16,17] in the order that the points were obtained. All tests were consistent with the randomness hypothesis at a 95% coafidence levcl. Canonical analysis, in which the experimental origin is translated to an cxtrcmum or col and the experimental axes
Atmosphere
Manometer x ~
Rotameter
Pressure Regulator
Panicle Feeder
Air 845 kPa
Fig.2. Sch©raalicdiagramofeyclor.~mstapparams.
,ler Cyclone Model
ZS. Ontko/Powder Technology 87 (1996) 93 104
rotated to eliminate the interaction terms, is a procedure which aids in visualizing response surfaces and extracting the maximum information from them. A detailed discussion of response surface interpretation is given in the previously cited text by Box and Draper [ 15]. The eanoaical equations for the surfaces estimated from this experiment are presented in Appendix 3. 4.1. Pressure drop
iiiI xa~ 4
~. j-'
The model m be fitted for the dimensionless pressure drop was: NE~ = ceo+ oqx, + ~x2x2 + a3x3
+ a, ixl' + a22x22 + ~2x,x2
(7)
°-

//
/i
where x~ =
N,~o- 6250 3750
(8) ×t
and x2=
Fig. 3. Contour plot, NE,. C-0.016 0.016
(9)
The variable x3 represents the value of the unknown function O~; x3 = 1 when the particulate material was fly ash, and x 3 = - 1 when it was resin. Thirty-six points were available to fit the surface. The data were regressed using weighted least squares, weighted as 1/NE.. The analysis of variance for the experiment is presented in Table 2. The block difference, measured by x3, was not significant. The fitted equation (with 95% confidence intervals on the coefficients) was. neglecting x3: NE. = 4 0 1 ( -I- 17) - 68( 5:11 )x, - 35( 5:10)x2 + 112( 5:17)x~2+ 17( + 17)x22
(10)
This is an elliptic paraboloid centered at xt = 0,45, x~.= 1.36, i.e. slightly outside the experimental region, with the major axis rotated 83 ° counterclockwise from the positive xraxis. The coefficient of multiple determination for the regression, r 2, was 0.935. Fig. 3 is a contour plot with the experimental points indicated by blackened circles. The Box and Draper criteflon required a value for Fo of 24.3 at the 95% confidence level, a value well exceeded here. The residuals are displayed in Table 3, with fly ash values denoted by F and resin values by R. Note that the variance was not constant with N ~ , but with Ap. i.e. related to the manometer graduation, hence the weighting chosen. 4.2. Overall collection efficiency
In the case of collection ofticiency, it was known a priori that pp/g and @ would be influential, so the fly ash and resin
- - 2 4 ( + 13)x,x2 Table 2 Analysis of variance, NL. Source of variation
Sum of squares
Degrees of freedom
Mean squaw
Fo
Model
501.52099
6
83.58683
69.03 h
257.19398
2
128.59699
106.20b
244.15326
3
81,38442
67.2] h
0.1737S
I
0.17375
0.14
35.11566 536.63665
29 35
1.21088
(a,. az. aj. all. '*a~.cqz[~)
Linear (cq, ~2[o~)
Quadratic ( a . . a22, a,z [o~, aL. c~2)
Block (of] I ~ , at. ¢¢2, <;flt, o22, aL2)
Enor Total a $ignificaat at Ihe P = 0,05 level. b Significant at the P = O . O I level,
J.S. Ontko/Powder Technology870996) 93-104 Table 3 Residuals about the estimated surface. N~ x, = - I
xx = 0
F
x2- I
R
xl ~ I
F
-43 -i6 18
R
-50 -13
R
-13 -2 -I 14
8
15
-3 4
where ~l = x t and
-18 -4
4 89
10
6~=
l
15
x2= - 1
-40 -
F
-14 -4
45 x2=0
data were regressed separately. T h e results o f the previous section provide assurance that the same loading was tested for both materials at each condition. Each surface was estimated from twelve points, obtained from two replications of what was in effect a 3 × 2 factorial. T h e model in this case was
14 II
-4
-
-I
38
-3
5
(12)
Because there were only t w o l e v e l s of C ( ~:2= + I ) available, any curvature o f the surface in the ~2 direction was measured through the interaction ~1~2. T h e analysis o f variance for the fly ash response is given in Table 4. The equation o f the response s u f f ~ e was
14
-8
3
C - 0.024 0,00--'---~
II
Table 4 Analysis of variance. Source of variation
Sum of squmes
Degrees of freedom
Mean square
Fo
Model (AI.A2.A..At2JAo)
0,0677423
4
0,0169356
42.12 =
Linear (A.A2IAo) Quttdratic (AH,AI2IAd, Az,A2)
0.0628351
2
0.0314176
78.14 •
0.0049072
2
0.0024536
6. I 0 b
Error Total
0.0028144
7
0.0004021
0.0705567
Il
• Significant at the P = 0.01 level. Significant at the P = 0.05 level.
i/i/1 ii !!1
....
o
~
o
~
l
' ~ = 0.878( + 0 . 0 2 4 ) + 0 . 0 8 9 ( + 0.017),~ -
0.000( :k 0,014) ~2
+ 0.013( =[:0,029) ~ 2 + O.024( =k0.017) bCzb¢ 2
(13)
T h i s is hyperbolic paraboloid centered at ~l = 0 , ~2 = - 3 . 7 5 with the canonical axes rotated 31 ° counterclockwise with respect to the original axes. An asymptote is coincident with the ~2 axis where ~ = 0 . 8 7 8 for all ~:2.Fig. 4 is a contour plo¢ of thi~ surface and Table 5 gives the residuals. Note that decreased with increased loading for ~ < 0 and increased with increased loading for ~l > 0. The Box and Draper criterion for this surface required F o > 4 1 . 2 , which was slightly exceeded here; r 2 = 0.960. The overallefficiency response with resin was ofadifferent character. The analysis o f variance is presented in Table 6. The equation o f the response surface was = 0 . 9 1 8 ( + 0 . 0 3 3 ) + 0 . 1 5 6 ( + 0 . 0 2 4 ) ~:=
1®
I
o~
eeo Fig, 4, Contour plot, ~ ,
I
e~
x~
- 0 . 0 1 7 ( 4- 0.019)~= - 0,127( + 0.041) ~=2 + 0.007( + 0.024) ~ 2
(14)
zs. Ontko/Powder Technolog'¢87 (1996193-104
98 Table 5 Residuals about tile estimated surface, fiF ~:~- - I
.~x= 0
~'~- 1
~-I
-0.0l o.oi
-0.02 0.04
~2=--1
-O.OI 0.a2
-0.01 n ~0
0.0o o.00 o.00 o.ol
In a recent experimental study of cyclone performance by Hoffmann c t a l . [ 19], in which the performance of four industrial scale cyclones ( D = 0.45 m) were compared, pressure drop results were given at several loading levels of air classified chalk dust with v,=Q/(AIA2D 2) fixed at 15 m s I. They found their pressure drop data to be in approximate agreement with an empirical equation attributed to Smolik see Ref. [ 191: Np = I - 1.26c°~' Np, o
This, too, is a saddle, but centered at ~ = 2.24, ~2 = 55.8. i.e. very remote fi'om the experimental region. A contour plot is given in Fig. 5 and the residuals in Table 7. For this surface, overall efficiency decreasec~ with increased loading for all ~. The Box and Draper criterion is the same as for the fly ash surface, here well exceeded; r2 = 0.978.
(15)
where No,. is the pressure coefficient obtained at zero loading, and c is the inlet dust concentration in kg m - 3. Here, xz = 0 corresponds to c = 0.019 kg m - ~, and x2 = 1 to c = 0.039 kg m -3. It may be verified by direct calculation that Eq. (15) over-predicts the reduclion in dimensionless pressure drop for xl = - 1, is in good agreement lbr xl = 0 , and underpredicts fro' x, ~ 1. Neither the Reynolds number effect nor the Reynolds number-loading inleraetion can, of course, be predicted by Eq. ( 1 5 ) . Nor can either be detected by the experimental arrangement of Hoffmann et al. 119], if'present in the region of their experiments.
5. D i s c u s s i o n
~.s Hunter [18] has pointed out, if the true response is a smooth function, then an eslimated response surface involving a polynomial in the independent variables may be identified with a Taylor series expansion of the true response in the neighborhood of the experimental origin. A well-estimated response surface is thus a reasonable approximation to the true response surface only near the experimental region.
5.2. Collection efficiency To check the hypothetical similarity condition, Eqs, (13) and (14) were set equal to estimate a locus of equal efficiency. Its projection on the ~a,~ plane is:
5. I. Pressure drop
0.140s¢~-'+ 0.016~'~ so2- 0.068~t The minimum estimated value in this experiment, Nsu = 383, correspoods to a pressure coefficient, N~=2N~(A,A2)2~Ap/(~pgv~:), of 3.49 and the maximum. N ~ u = 6 0 9 , corresponds to a pressure coefficient of 5.55. This compares to values predicted by the methods of Shepherd and Lapple [ 111, Alexander [ 121, and Barth [ 131 of 4.32, 3.17 and 3.38, respectively. These methods, then, provide an order-of-magnitede estimate in agreement with the present data, without resolving effects due to operating conditions.
+ 0.017,~2-- 0.041 = 0
(16)
which is a hyperbola. The form of the locus is obviously an artifact of the polynomials that estimate the response surfaces. Hence due caution is required to interpret its meaning; it is possible that ~/e = r/a nowhere on the locus or only at isolated points. Let us designate an experimental point by its coordinates (~], '~z). Two segments of the locus of equal efficiency pass through the experimental region, one of which passes very
Table 6 Analysisof vori~mce,~a Source of variation
Sum of squares
Degrees of freedom
Mean square
Fo
Model
0.2422533
4
0.0605633
76.42 b
0.1988921
2
0.0994461
125.48 b
0.0433612
2
0.0216806
27,36 h
0.0055474 0.2478007
7 II
0.0007925
(At,A2, A~,.A,~IAo) Linear (AI,A21Ao)
Quadratic
(AN,At2IAa, A~.A,.) Error Total Significant at tl~ P = 0.05 level. b Significant at the PfO.Ot level.
J.S. Ortt~ /Powder Teclmology 87 (1996) 93-104
IIID ~.......
99
that ~ = ~/R= 0.96 at this point, which is within experimental error. In fact, this overall efficiency holds approximately all along the segment above die right half of the ~sez plane. To the same level of approximation, f/~= f~ = 0 . 8 5 along the segment in the left half-plane. Constant -~ along these segments, however, is not essential to what follows. The point ( 1, - 1 ) presents an opportunity to investigate the kypothetical similarity condition. Table $ runs the gamut of possibilities at this overall efficiency using Eq. ( 5 ) . Values as ~ ' ~ 0 and ~ are limiting cases, obtained fiom results described in Appendix 2. For a Rosin-Rammler distribution, Eq. (5) is equivalent to a function ofdproposed by T h ~ o r e and De Paola [20] (d/d~o) ~ ~l= 1 + (d/dsu) ~
I
T
~ > 0 . In this case k = (dso/do) ~ and sr=,6/y. A non-linear least squares fit to Eq. (5) was made using Eq. ( A 5 ) ; the best fit was obtained at r = 0.3. The result is shown in Fig. 7. Let us examine the consequences of assuming the fractional efficiency to be independent of the inlet distribution.
I
Fig 5 Contour plot, ~a-
Table 8 Parametersof Eq. (5) forO< ~<~ at ( I. - [)
Table 7 Residuals about the estimated surface, '~, 6,=-I
(]7)
~,=0
6,=1
62= I
- 0.02 0.01
0.00 0.02
- 0.o1 0.00
,~2~ - I
-0.04 0.05
-0.01 -0.0l
-0.01 0.0l
(-)
0 0~3 0.5 1.0 2.0 4.0
(~m)
(/,an)
(-)
0.03 0.29 1.37 2.47 2.93 3.10
0.46 2.35 7.0~ 10.72 12.1[ 12.59
0.065 0.123 0.194 0.230 0.242 0.246
' ' - ~ - ~ '
(-)
'~ . . . .
4 17X 10-2 3.25X 10-2 2.50X 10-z 9.75× 10-3 7.99X 10-~ 1.g7× 10-~
~
' i
'-~
o Rea4n • Fly ash
77 .4
Pig.6. Estimated ~ on the locusof equal efficiency,Eq. (16). .3 near the point ( l, - I ). Fig. 6 shows the overall efficiencies obtained on the locus plotted over a region of the response space containing the experimental region; the dotted segment beyond the plane ~ = 1 lies outside. The segment of the locus in the right half-plane runs fi'om l ~ z ~ - I between 0.64 ~g ~z ~g 1.011 the segment in the left half-plane runs from - 1 < ~:2~ 1 between - 0 . 4 1 ~ ~ ~ - 0 . 2 7 . Examination of the confidence intervals and residuals about ( 1, - 1) show
.~
'
'
'
'2
ra '
.'4
ts '
. . . . . .
'
Rg. 7. Calculated and expedmen~ ~ at ( I. - I ) using Eq. (5).
' i.e
chelknautz. as can be seen as t, proceeds from 0 to I. The fly ashefficiencyat (I, I) maybe takenasanothermanifestation, though the present experiment catnot provide any information on the passing of the crisis outside the boundaries of the experimental region.
6. Conclusions
ine Stokes number. NsLk,conventionally taken to be a similarity parameter for fractional efficiency, may be shown by these data to be inadequate for that purpose. Examination of Table 8 shows that the similarity condition required by Stokes number scaling, d,.,ld, R= (6. nl~O.r)“~.’ was never attaiied. This is not surprising. since for Stokes number SC& ing
where $ = drpldJN,k. so that @J~#@Joimplies &jFf V&for any tixed C and A,. This result is inconsistent with the data. Now Nsa is evidently a composite of several independent dimensionless variables. To generalize the relation, we again apply the ?r-Theorem to obtain ?= rl(p~/~. NH~.C. 8. 4)
(19)
We find [email protected]/d&=+(4 S, y), &=d,lD. This formulationcannotbedisposed ofconclusively with thepresent data, for pplL and @Jare confounded in S,. A decisive experiment may be performed by testing two distinct inlet disbibutions at ihe same P,~ and fixed NR, C, and A,. Eq. (19) requires V)to be similar under these conditions regardless of the inlet distribution. The rest of the available experimental points were litted to Eq. (5) as shown in Table 9: note that nowhere did pF= &. It is also pertinent that the Leith-Licht and Enliang-Yingmin equations predict fl> 0.98 at all conditions and ?jr,> qF everywhere I%?~ + nr P.!T zt (I. I) for fly ash, since there the estimated value for G = 1.JO. Two curiosities discussed in the literature were found within the exprimental region. Most studies have shown increasedefficiency with increased loading. Occasionally the opposite effect has been observed 1211. Both were found in the present experiment. Table 9 reveals another in!eresting feature alone the line &= (0. 1) Lh*O. . _,. __ 1. At the noint meaning that the resin was collected unclassified, i.e. without regard to particle size. The experiments of Hoffmann et al. [ 191 a!so revealed this phenomenon, which had been treated earlier by Muschelknautz [ 221 in his critical loading theory. This situation appeared to be a local crisis, contrary to Mus S=d/D.
Over the region examined, the Euler numhcr and overall collection efficiency appeared to he smoo!h limctions of the dimensionless variables of Eqs. (3) and (4). well approximated by second order polynomials. There was no evidence that the Euler number depended on the nature of the particulate material, but a Reynolds number-loading interactton. indicated by the rotation of the canonical axes with respect to the experimental axes, showed that the effect of one of these two variables concot be determined without regard to the other. This result has implications f. r CFD methods, indicating that the assumption of no gas-particle momentum coupling is invalid cvcn for the moderate dust loadings invcsligated here. The present experiment also showed that the Stokes number cannot be a similarity variable for fractional ofliciency. It was inconclusive as to whether the fractional efficiency was independent of the inlet distribution. Circumstantial evidence, however, suggests that it was not. A locus of equal efhciency was estimated irom the data to iat the hypotbelical similarity condition, Eq. (2). which requires the inlet cumulative mass distribution to be considered explicitly. It was contirmed at the single poin! that it could be checkeddirectly. The fractional efficiency curves for fly ash and resin were not similar at any experimental point when expressed as a function of particle size. Since the locus was a proper subset of the response surfaces, Eq. (3) cannot satisfy eveywhere the similarity requirement of Eq. (2) and hence dora not constitute the scaling law for fractional efficiency. However, a transfommtlon mto a set ot natural smutanty coordinates may exist for distributions of the sane form. even if the hypothetical similarity condition is satisfied only at isolated points on the locus. 7% presw sr+ in inadequate to indicate the nat”re of the tmnsformation. To advance fzthertownrd a universal cyclone scaling law, additional experimental work will be necessary. An essential requirement is that A,~ and @ be specified independently. Apparatus such as that employed by Parker et al. [ 231, with which the pressure and temperature of the carrier gas may bc varied, would simplify the task. If used with a central cotnposite design, requiring five levels of each independent var. iable, or a special second order design due to Box and Behnken [%+I. requiting the minimum three levels, a larger region of the perfom~ance map could be explored and this would perhaps provide the basis for a more rigorous analysis than now possible. Such a program should also furnish a sufficiently large self-consistent data set to adequntely test CFD methods.
J.S. Omko/PowderTechnology87 (1996J93-104 7. List of symbols C c D d d~. d~ do d~o d F Fo G g H k N~,
Np NR~ NR~.p Nsm P Q r2 s n vg vi vp X, X2 x~ x2
inlet dust loading, particulate mass per unit mass of gas plus particulate material (-) inlet dust coneentratian (M L- 3) cyclone barrel diameter (L) particle size (L) minimum particle size (L) maximum particle size (L) Rosin-Rammlar location parameter (L) particle size at which '0=0.5 (L) mass mean particle size (L) Snedccor F-distribution function (-) realized value of Suedecor F-distribution (-) cumulative captured particulate mass distribution function (-) .:apmred particulate mass probability density function (-) cumulative exit particulate mass distribution function (-) cut parameter (-) Euler number, AplY*/ pgQ2 (-) pressure coefficient, Apl ( ~PsVt2) (-) cyclone Reynolds number, pwQIpD (-) particle Reynolds number, pg]vp- % Idlt2 (-) Stokes number, pplgNa#t21( 18A,A:D2) (-) probability o f F exceeding a given value (-) gas volumetric flow rate (L 3 T - ~) coefficient of (multiple) determination (-) standard deviation (L) function defined in Eq. (AI 1 ) (-) local gas velocity vector ( L T - i) nominal gas inlet velocity (L T t) local particle velocity vector (L T - t) canonical dimensionless pressure drop coordinates (-) coded cyclone Reynolds number in Eq. (7) (-) ceded dust loading in Eq. (7) (-)
o p+ --~J, --~'r2 '~'R~,~'R2 -'-s,. ~-s,_
101
fractional penetration (-) overall penetration (-) canonical fly ash efficiency coordinates (-) canonical resin efficiency coordinates (-) canonical locus of equal efficiency coordinates (-) coded cyclone Reynolds number in Eq. (11) (-) coded dust loading in Eq. ( I I ) (-) gas density (M L -a) particle density (M L - 3) density ratio, ppl pg (-) cumulative inlet particulate mass distribution function (-) value of • at which ~ = ~ (-) value of Oat which 7 = 0 . 5 (-) inlet particulate mass probability density function (-) l-O(-)
~ ~2 pg pp pp/g • O,~ Oso ~'
Api~endlx 1 Some general properties offractional efficiencyfunctions The segregation of an inlet distribution into captured and exit distributions is the fundamental physical process in particle separation. The fractioaal efficiency, ~], of a cyclone operating a steady state, without particle attrition or irreversible agglomeration, on a particulate material of eunstant density pp and polydisperse cumulative inlet mass distribution 4 ( d ) . may he developed from a diagram due to Lapplc [ 25 ], shown in Fig. 8. It is sufficient that candidate functions '0(O) have the following properties: (1) 7 7 = 0 a t O = 0 , ( 2 ) ~ = 1 at O = 1; and (3) '0 is continuous and increases monotoai-
Greek letters A 'g A Ap 8 8o
O~
e~ A~ P
regression coefficients in Eq. ( 1 ! ) (-) regression coefficients in Eq. (7) (-) Theodore-De Paola parameter (-) Rosin-Rammler dispersion parameter (-) increment (-) cyclone pressure drop (F L -2) d/D (-) do/O (-) sharpness parameter (-) fractional collection e?'iciency (-) overall collection efficiency (-) unknown function in Eq. (4) (-) unknown function in Eq. (3) (-) normalized cyclone design lengths (-) gas dynamic viscosity ( F T L -2)
I
1
~,T] ~ . . .I. . . ~. . . .-. . . . '. . ... . ... . .~. . . .I . . i l
+
,
~
~
•
%
+
•
.
i ~ .u.~l
*
,.w
)--->
Fig. 8. Cyclonefractionalefficiencyand petmtr0fiondlagranL
J.S. Oatko/ PowderTechnology87(1996) 93-104
102
cally on 0~< ~ < I. The overall efficiency, ~, is given by Eq. (AI), which is the same as Eq. ( 1):
and dqb dd*
I
~=!r/d~
(A1)
The li'action of captured particles v.'dh ~ < ~*, i.e. d ~
Sic:~ilarly, if the fractional penetration, v = l - */, is expressed as a function of ~ = 1 - ~, 1
~ = J v dtP'= l - ~
(A3)
and the fraction of escaping particles with ~ < ~'*, ~c. d>~d*, is ,i,° H(qt*) =
1 fvd*
(A4)
o Properties ( 1) through (3) and the convergence of Eq. ( 1) guarantee that G and H are cumulative distribution functions. This requirement is the origin of the empirical observatien that 7/increases monotonically with d, which follows from the way q~ is defined, -0(~(d*)) may be interpreted as the probability that partieles of size d ~<=' 1=' divided=' into=' n=' subintervals,=' allows=' estimation=' fractional=' efficiency=' at=' some=' point=' qb*=' jth=' subinterval=' between=' g=' _=' ,=' +=' qb=' -=' g(qb~='>
(A5)
j = l, ..., n. This expression and the analogous one which may be written in terms of H form the basis of the experimental determination of fractional efficiency and penetration. From Fx]. (A5) *l(qb) may be expressed in terms of G: rl(~) = ~ lim
G ( ~ + Aqb) - G(q~)
if this limit exists. Noting that dO dG d f dd* dd ~ g(d) dd=g(d*)
dG
(A6)
(AT)
a~ d f dd* q~(d) dd=d~(d *) dm,
(Ag)
if g and ~/, are continuous on d,m~d<~d~ and dram< d* < dm~, parametric differentiation yields , _ dG/dd* g(d*) ~/td ~) ~ , F / ~ = ~ dq~/dd 6(d )
(A9)
So long as d.~m>10 and d ~ > din,., they may take on otherwise arbitrary values. Evidently d may he regarded as a parameter, with q~and ~ parametrized by the form of the inlet cumulative mass distribution. Since we have assumed both r/and • to be continuous, we may cast Eq. (1) as a function of d by a change of variables: dm~ r~= / r/(d)~bdd
(AI0)
where 4' = ddP/dd is the probability density of q~, Taking Eq. (AI0) as a postulate with d ~ , = 0 and d . ~ =o0, we might hypothesize that ~ is independent of 4~. This is implicitly assumed in the conventional analysis. The experiment presented in this paper was undertaken in part to test this hypothesis.
Appendix 2 Derivation and properties of Eq. {5)
The empirical equation proposed by Theodore and De Paola, Eq. (17), may be expressed in terms of the RosinRammler cumulative distribution function, Eq. (6). as follows. Eq. (6) is first solved for d. Substitution oftbe resulting expression into Eq. ( 17 ) yields the upper member ofEq. (5), where k = (dso/do) a and sr=3/y, Noting that ~.~ I as • ~ 1 -, we include the point r/= 1 at @= 1 to satisfy property (2) and the hypotheses of the theorem of the mean. It can be shown that this does not alter the value of Eq. (1) when integrated. Eq. (5) has certain valuable mathematical properties which suggest it would be useful with any continuous inlet distribution, despite its empirical and slightly ad hoe parentage. These properties are developed briefly below. In general, k and ~'would not be related so neatly to the distribution parameters; they nevertheless have an interpretation of a universal character. For convenience let
,MI,
II can be shown thar
Eq. (1) convergs for {> 0. When
~=l-e-‘,u=Iandg=Il(k+l).Thevalueof1)there depend? on the value of b, for example ‘I=
I Ik for k:*
I,
and
~=lfork~l.Whsn~=0.5,u=u(~~50)=k,~ndth~~kis recognized as a generalized cut parameter. The sbapc of thh function may be better appreciated by the examination of its first derivative, du
kz d?) [email protected]
(Al2)
(k+~)~
As @+O+,
dq/[email protected]+O
and [email protected]+m
for I>
for << />
1. dv/dG’-+
For 9+0
Ilk for l=
or I, as c+O,
1.
d,t
d~jOandrl~ll(k+l)~t),soEq.(5)approximatesthe case 1)= ii as 5-O. To sketch the properties of Eq. (5) as [email protected]+ Since
consider the behavior of drj/[email protected]
evaluated at @S,b
k= [I+$--1;
(.413)
we obtain, after some manipuMon, (A14)
The denominator is finite and positive for 0 i &, < 1. fore knowledge of the exact value of $,
There-
is unnecessary to
obtain lim d?l (_-dO *a=.=
(A15)
To complete the sketch. Eq. (5)
is solved for k al Q= @+
,_? (--I
and the resulting expression set equal tu Eq. (AU).
Then
“C
ii
=- I”( I - &‘u) In( 1 - @,J
As 5-m. &,off, where vj =
(Aib)
@+ This case approaches l$at of sharp cub
0 for @< 1 - ?j and v= I for O> 1 - ij. The chief features of IQ. (5) are shown in Figs. 9 and 10. Fig. 9 is an example of 1)calculated with several values of C for +j=O.8. Fig. 10 is a parametric surface which shows log, k fol 0.1 G6~0.95 and 0.25 d (~4.50. The required values of k were obtained by iterated numerical integration of Eq. (5) for fixed values of sj and c.
Fig 10. Pa¶ame,Iic surfaL
Overall
~-0.87S=2.0XiO-‘~~,2-7~10~‘~~*
(Al8) and
&=
Overall efficiency with resin &R: lo-“&
’ - 0.127&’
where & =O.OZW(~, -2.237) +0.9996(&-55.77) &= -0.9996(& -2.237) +O%K!;& 55.77).
(Al9) and
Locus of ;qual efficiency:
Dimensionless pressure drop NE.: NE. - 362= 114X,‘+ 15X2’
(5,.
where & =0.8598& +0.5106(&+3.751) -0.5106&+0.8598(5,+3.751).
ljR-0.633= Appendix 3
forEij.
efficiency with fly ash qF:
(A17)
where XI = -0.9923(x, -0.450) +0.1252(~,- 1.363) and X,=0.1252(x, -0.450) +0.9923Cr- 1.363).
3x 10“~s~-0.813~s,2=
0.9983(& + 1.015) +0.0578(&21.63) -&05i8(~,+I.015)+0.9983(&-21.63).
where &=
&-
I
(AZ’3 and
104
Z S Ontko / Powder Technolagy 8711996) 93-104
References [ I ] D. Leith, in M.F. Payed and L. Otren (eds 1. Handbook aJ F.wder Science and Techn~doA~v.Van Noslrand Rheinhold. New York. 1984, p. 730 [21 AC. Slern, K.J. Caplan and P,D, Bush, C~vclone Du.¢! Collectors, American Petroleum lnslilute. New York, 1955. p. I. [3] J. Abmhamson, in R J Wakeman led.l, Pt~gress bl Filtration and Separation 2, Elsevier, Amsterdam, 1981. p, 1. [4] F. Boysan, W . H Ayers and J. Swithenbank, Trans. hist. C6em. Eng.. 6O (1982) 222. [5] S.J. Kim and C.T. Crowe, in LT. Jurewies led.), Gas-Solid Flows FED, Vol 10, American Society of Mechanical Engineers. New York, t984, p. 5 7 16] O.L. Davies (ed.I, 7heDesignandAnalysis vflndustrialExperiments, Hornet, New York, 1960, p, 495. [ 7 ] H.L. Langhaar, Dimensional Analysis und Thevry .]'M~dels, Krieger. Malabar. FL. 1983, p. 47. [8] CJ. Slairmand, Trmhv. hist. Chem. Eng., 2911951) 356. [9] I1. Leith and W, Lieht. AIC6ESymp, Set;, 68 (1972) 196. [ 10] L. Enliang and W. Yingmin, AlChEJ. 35 11989) 666. [ 11 ] C.B. Shepherd and C.E. Lapplu, hid Eng C6em., 32 (1940) 1246.
,b
[ 12] R.MeK. Alex ander, Pra¢. Australasian Inst. Mining Met., 152 (1949) 203. [ 131 ',¢. Barth, Brenn.~t.-Wiirme-Krafi, 8 (1956) I. [ 141 P. Rosin and E. Rammler, Z Inst. Ftlel, 7 (1933) 29. ] 151 G.E.P Box avd N.R. Draper. gmpiricalMadeI-BuildingandResponse Surfaces, Wiley, New York, 1987, p. 279. [ 16] E.L. Crow, F.A, Davis and M.W. Maxfield, StutLvtics Manual. US Naval Ordinance Test Slalian, China Lake, CA, 1955, p. 244, [ 17J C. Eisenhart and P. Swed, Ann. Math. Stal., 14 ( 1943 ) 83. 1181 J.S. Hunter, C6em. Eng, Prog. Syrup. Set, 56 (1960) I0. [19] A.C. Hoffmann, A, van Santen, R.W.K. Allen and R, CUff, Powder TechnoL 70 (1992) 83. [20] L. Theodore and V. De Paola, Z Air PoL Contr Assoc.. 30 (1980) 1132. [21 ] T.M. Knowbon and D.M. Bachovchin, Coal Prr~'essing Technology. Vol. IV, American Inslilute of Chemical Engineers, New York, 1978, p. 122. [221 E, Muschelknautz, VDI-Ber., 363 11980) 49. [ 23 ] R. Parker. R. Jain, S. Calvert, D, Dtchmel and J. Abbott. Environ. Sci. TechnoL. 15 ( 1981 ) 451. [241 G.E,P, BOX and D.W. Behnken, Technometrics. 2 0 9 6 0 ) 455. Corrections, 3 ( 1961 ) 576, 125] C.E. Lapple, Am. Ind. Hyg. Ass. Quarl.. II (1950) 40.