See

discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/284358391

Similitude in cyclone separators

Article in Powder Technology November 2015

DOI: 10.1016/j.powtec.2015.11.048

CITATIONS READS

0 27

1 author:

John S. Ontko
National Energy Technology Laboratory
11 PUBLICATIONS 87 CITATIONS

SEE PROFILE

All content following this page was uploaded by John S. Ontko on 09 December 2015.

The user has requested enhancement of the downloaded file.

 Powder Technology 289 (2016) 159162

Contents lists available at ScienceDirect

Powder Technology

journal homepage: www.elsevier.com/locate/powtec

Similitude in cyclone separators

J.S. Ontko
National Energy Technology Laboratory, U.S. Department of Energy, 3610 Collins Ferry Road, Morgantown, WV 26505, USA

a r t i c l e i n f o a b s t r a c t

Article history: Criteria for similitude in reverse ow cyclone separators are developed in this paper explicitly including the inlet
Received 2 June 2015 particulate probability distribution. The application of these criteria is demonstrated by example using data from
Received in revised form 18 November 2015 the literature. Some practical points to consider when using cyclone similarity relations are presented in the
Accepted 19 November 2015
Conclusion.
Available online 21 November 2015
Published by Elsevier B.V.
Keywords:
Cyclone separator
Similitude
Scaling
Stochastic

1. Introduction 2. Theory

The reverse ow cyclone separator, shown schematically in Fig. 1, We restrict our attention to a steady, incompressible, isothermal,
provides a simple and robust method of removing particles from a and chemically unreactive process in which gravitational and electro-
uid transporting them. It captures a fraction of the incoming particu- static effects as well as particle attrition may be neglected. Consider-
late mass, , and allows the remainder, v, to escape; v 1. It also ation of the characteristic variables (listed in the Nomenclature)
segregates the inlet mass cumulative probability distribution, , into inuencing the uid and particle mechanics in a cyclone under these
captured and escaped distributions, denoted G and H respectively. The restrictions leads to
fractional or grade efciency, , partitions into G and H, all of which  
may be functions of particle size and other variables. If (1) 0 1 and f f ; V; D; ; ; 1 ; ; j ; ; ; ; p 0:
(2) is continuous,
where f is an unknown function which is assumed to be dimensionally
Z 1 homogeneous, dimensionally invariant and in which the numerical
d values of the arguments are positive real numbers. It is understood
0
Z  that and stand here for the variables and parameters required to
 1
G d completely specify both functions. We may apply the theorem [2]
" 0Z  # to establish the conditions required for similitude, which yields

1 
H  d  
v 0 F NRe ; ; 1 ; ; j ; ; ; ; N p 0

where F is another unknown function. Now consider two cyclones, one

0 1. These equations can account for any number of random var- of which is denoted the model and the other the prototype. If the design
iates, which may be continuous or discrete, in the absence of attrition or ratios in the model, i , m, equal the corresponding ratios in the proto-
agglomeration [1]. To estimate fractional efciency from data, partition type, i/p (i.e.i ,m = i/p for all i = 1, , j) the model and prototype are
the domain of into disjoint subsets such that the union of the subsets said to be geometrically similar. If all corresponding arguments of
is equal to the domain. Let i and Gi denote the respective inlet and F are equal for model and prototype, the model and prototype are said
captured probabilities associated with subset i. The fractional efciency to be dynamically similar. If any of the restrictions specied at the be-
 Gi =i ; if i 0. A static pres-
in the ith subset is estimated by ^i ginning of this section are relaxed, additional variables accounting for
sure drop, p, is developed across the cyclone when in operation. the physical phenomena introduced must be included in the analysis.
Often one of the standard equivalent diameters [3] is chosen to
E-mail address: john.ontko@netl.doe.gov. transform a distribution of non-spherical particles (which in general

http://dx.doi.org/10.1016/j.powtec.2015.11.048
0032-5910/Published by Elsevier B.V.
160 J.S. Ontko / Powder Technology 289 (2016) 159162

one xed p. The overall height of the cyclones they considered was be-
tween 5.5 and 9.1 m, so D 2 m. Since does not depend on cyclone de-
sign or operating conditions while does, let = d. The functional
relationship among , , and is known:
Z Z 
 =
1e ex dx
x
x dx

0 0 1 =

F becomes
 
F GS N Re ; ; i ; ; j ; =; D1 ; p = f ; ; Np 0:

The loci of constant / are loci of constant . The authors reported

that for the conditions they studied, Np depended only on the i; thus
for a given design Np was a constant.
Theodore and DePaola [5] proposed a two parameter model for
fractional efciency:

d=d50

1 d=d50

d N 0, N 0; at d = d50, = 0.5. As d , 1 asymptotically. If

1, is concave down everywhere. If N 1, there is an inection
point at d/d50 = [( + 1)/( 1)]1/ at or below which is concave
up and above which is concave down. As , [( + 1)/(
1)]-1/ 1 and the cut becomes sharper. The authors found the
model satisfactorily t smoothed laboratory and in-plant data provided
by Lapple [6] when = 2.
Iozia and Leith [7] used this model to t fractional efciency data
obtained from cyclones of various designs (D = 250 103 m) using
a mineral oil aerosol (p = 876 kg m3) at very low mass loading
(1 105 [8]) suspended in air. Tests at three volumetric ow
rates were made with a Stairmand high efciency design [9]. Eight addi-
tional tests were made at a xed volumetric ow rate with certain sys-
tematic variations of the i from the Stairmand design. d and were
not reported; because of this the similitude problem was inadequately
specied. Let dc denote a characteristic equivalent diameter of and
dene = d/dc. Then
Z
Fig. 1. A reverse ow cyclone separator shown with typical design lengths i. dc =d50 x

x dx
0 1 dc =d50 x

require more than one length to specify) into a distribution specied by and
an equivalent diameter d, or such a transformation is imposed by the  
particle sizing apparatus. Evidently and depend on whether such a F IL NRe ; ; 1 ; ; 6 ; dc =d50 ; ; dc =D; p = f ; ; ; Np 0:
transformation is applied and, if so, which equivalent diameter is chosen
or imposed. The following discussion is limited to joint probability dis-
The experiment was executed at ambient conditions which were not
tributions of the form = (d, p; 1,,k), where p is the particle den-
specied. For discussion purposes, suppose the discharge pressure and
sity and the i are distribution parameters, or = (, p/f; 1,,l),
temperature were 100 kPa and 288 K respectively. Using these values,
where is a dimensionless equivalent diameter, p/f is the particle-
NRe and Np were calculated from reported data and were used to con-
uid density ratio, the i are the corresponding dimensionless distribu-
struct Table 1. Averaged over all test numbers, p/f = 722 5; the
tion parameters and l k. If all particles to be separated have the same
density, we write d = d(d; 1,,k) or = (; 1,,l) if p/f
Table 1
is the same for all particles. It is useful to note Cyclone performance data [7].

Z Test NRe 103 1 2 3 4 5 6 d50 [m] Np

x dx 1 258 0.5 0.2 0.5 0.5 1.5 4 2.77 3.65 2.86
Z 0 2 127 0.5 0.2 0.5 0.5 1.5 4 2.15 4.91 2.93
d
x dx : 3 388 0.5 0.2 0.5 0.5 1.5 4 3.82 2.71 3.18
d 0 4 172 0.5 0.3 0.5 0.5 1.5 4 3.14 4.13 5.23
5 172 0.75 0.2 0.5 0.75 1.5 4 4.11 4.46 5.23
6 519 0.5 0.1 0.5 0.5 1.5 4 5.22 2.40 1.95
7 518 0.25 0.2 0.5 0.25 1.5 4 5.03 2.90 1.51
3. Discussion 8 260 0.5 0.2 0.3 0.5 1.5 4 4.82 2.23 7.80
9 257 0.5 0.2 0.7 0.5 1.5 4 2.83 4.34 1.61
Gallaer and Schindeler [4] treated the case where = 1 10 258 0.5 0.2 0.5 0.5 0.5 3 3.42 3.34 3.39
11 258 0.5 0.2 0.5 0.5 2.5 5 3.27 3.23 2.86
exp.(d), d = 1 exp.(d), 0 b d b , so = (d, p; ), with
 J.S. Ontko / Powder Technology 289 (2016) 159162 161

tolerance is twice the sample standard deviation. Thus, p/f was essen- particle size for the RosinRammler distribution, 1= 1, in-
tially constant. The remaining arguments of F(I-L) were not specied cludes both distribution parameters. Hence, the relative mean particle
quantitatively and may have inuenced the contents of Table 1 in 3
mass, N m p = f =6, may be dened for use as a variable charac-
ways which cannot be evaluated. Tests 13 assessed the performance
terizing the block difference. The regressed model for Np was a well-
of the original Stairmand design. Tests 45 investigated an inlet area in-
estimated quadric surface (coefcient of multiple determination r2 =
crease, Tests 67 an inlet area decrease, Tests 89 an exit area variation,
0.935) which did not depend signicantly on Nm. To be precise, the pres-
and Tests 1011 a cylinder height variation.
ent experiment showed that Np did not vary measurably between two
Test 1 and Tests 411 were all made at the same volumetric ow
particular combinations of d0/D, , and p/f over the tested domain.
rate. When rendered dimensionless, Tests 47, which involved inlet
Estimated Np are shown in Table 3.
area changes and in which 1 and 4 were confounded, exhibited differ-
The data for were regressed with data for = 1 omitted, since is
ent Reynolds numbers from Test 1, indicating different uid and particle
not dened there. The effect of Nm was highly signicant but r2 = 0.850
dynamics. Tests 6 and 7, with a higher NRe, showed higher and smaller
for the complete regression. When the particulate material blocks were
d50, indicating greater fractional efciency and, if were unchanged,
regressed individually the response surfaces were much better estimat-
greater total efciency. This was accompanied by a lower Np, indicating
ed: r2 = 0.978 for the resin and r2 = 0.960 for the y ash. Estimates of
relatively less energy per unit volume was expended in attaining great-
and d50 at each condition may be obtained from data using linear regres-
er collection efciency. If the Reynolds number of the standard
sion [5], but the estimate of d50 is dependent on . It was found did not
Stairmand design (Tests 13) were raised to the value of Tests 6 and 7
depend signicantly on NRe, , and Nm . Consequently, was best esti-
and the trends going from Test 1 to 3 continued to hold, one would ex-
mated by the mean value = 0.7. The corresponding d50 required for
pect higher fractional and overall collection efciency but at higher Np
closure of at each condition was obtained by iterated numerical inte-
compared to Tests 6 and 7. Tests 4 and 5 involved an increase in inlet
gration and these results are summarized in Table 4. Note that R
area and consequently a lower NRe than Test 1. In Test 4, 2 was in-
decreased with decreasing for all tested NRe, but F decreased, stayed
creased and consequently d50 was slightly reduced while was slightly
the same, or increased with depending on the Reynolds number.
increased compared to Test 2. The net effect would be slightly less col-
Thus far has been modeled as monotonically increasing. This is not
lection of nes for xed . In Test 5, 1 and 4 were increased, resulting
mathematically necessary, as can be inferred from Conditions (1) and
in increased d50 and , further reducing the collection of nes. Both
(2) postulated in the Introduction. Parker et al. [13] studied the effect
Tests 4 and 5 produced a greater Np. Test 8, which involved an exhaust
of inlet gas temperature (20 C T 693 C) and pressure
diameter reduction without a change in NRe, increased and reduced
(141 kPa p 2520 kPa) on and p with V approximately constant
d50, but at the expense of an appreciably increased Np compared to
(V = 2.46 0.60 m s1) for twenty ve temperature and pressure com-
the unmodied Stairmand design. Test 9, in which the exhaust diameter
binations in a cyclone (D = 50 103 m) operating on particles with
was increased, showed degraded fractional efciency performance with
p = 2300 kg m3 conveyed by nitrogen and with unspecied. Six
reduced Np. Tests 10 and 11, in which the cylinder height 5 and the
additional conditions were reported with V = 5.22 0.02 m s1 at
overall cyclone height 6 were shortened or lengthened, showed
room temperature and high pressure (p 2030 kPa).
minor performance differences. In short, changes in the i produced var-
Only four of nineteen measured fractional efciency curves present-
ious measurable effects on cyclone performance, but assessing their in-
ed by Parker et al. were monotonically increasing. The authors sug-
dividual effects was complicated by confounding among them or with
gested that the minimum fractional efciencies observed, generally
NRe and the unknown variation, if any, in and .
found between 2 m and 4 m, may have been an artifact caused by
Ontko [10] reported the results of a study in which two particulate
breaking up agglomerates during sample preparation. To estimate ,
materials, sized by Coulter counter and distributed according to the
RosinRammler distribution [11], they reconstructed an effective inlet distribution obtained from the rela-
tion eff = G vH. eff for one run was compared with a sample of the
  distribution discharged from their uidized bed dust generator, with
d 1 exp d=d0
satisfactory agreement reported. This supported their contention of
reversible agglomeration. In general, if eff, f and F must be modied
N 0, were tested in a cyclone (D = 83 103 m) discharging through a
to account for the transformation of to eff.
lter to the atmosphere. The particle properties are summarized in
This experiment is noteworthy from the standpoint of similitude
Table 2. Let = d/d0. Using the Theodore-DePaola model for ,
since it encompassed an extraordinary range of Reynolds numbers
Z (2,700 NRe 433,000). However, p/f was highly positively correlated
d0 =d50 x

x1 ex dx with 1/NRe (correlation coefcient r = 0.98). Throughout the experi-
0 1 d0 =d50 x ment, p/f was coupled, though not perfectly, with NRe. If, for example,
values of p/f and were selected so all values of each were tested at
and F becomes every NRe, the available range of NRe would be considerably smaller for
  a cyclone of xed size.
F O NRe ; ; 1 ; :::; j ; d0 =d50 ; ; d0 =D; p = f ; ; ; Np 0:
4. Conclusion
The experiment was 3 3 factorial design in NRe and , replicated
twice, and run in blocks for each material [12]. Because of this structure, The experimental data reviewed were consistent with the hypothe-
d0/D, , and p/f were confounded. Note the dimensionless mean sis that the arguments of F determine the operating state of a cyclone,

Table 2 Table 3
Particulate material properties [10]. Estimated Np [10].

Property Fly ash Resin NRe

p [kgm3] 2740 1300 37,000 92,500 148,000

d0 [m] 19.08 45.50 0.968 5.38 3.43 3.68
[] 1.76 2.49 0.984 5.42 3.68 4.14

d [m] 16.99 40.37 1 5.46 3.94 4.61
162 J.S. Ontko / Powder Technology 289 (2016) 159162

Table 4 i an index, dimensionless


Estimated and d0/d50 ( = 0.7) [10]. j, k, l positive integers, dimensionless
NRe 37,000 92,500 148,000 m _f uid mass ow rate, kgs1

R 0.61 0.91 0.94
m _p inlet particulate mass ow rate, kgs1

(d0/d50)R 2.42 37.2 70.2 Np pressure coefcient, p=1 2 f V 2 , dimensionless
3
0.968

F 0.78 0.88 1.00 Nm relative mean particle mass, p = f =6,dimensionless
(d0/d50)F 9.30 27.6 NRe Reynolds number, fVD/, dimensionless

R 0.63 0.93 0.96 n argument in (n), dimensionless
(d0/d50)R 2.75 55.5 130 p inlet pressure, Pa

0.984 F 0.83 0.88 0.95 r correlation coefcient, dimensionless
(d0/d50)F 15.1 27.6 113
r2 coefcient of multiple determination, dimensionless
T uid temperature, K or C
subject to the restrictions enumerated in Section 2, and they can be V inlet uid velocity, ms1
measured repeatably using statistical experimental design methods. If x a dummy variable, dimensionless
any of the restrictions are relaxed, additional variables must be taken efciency parameter, m1
into consideration. Because of the large number of variables involved distribution parameter, m1

and their wide domain of denition, only a restricted operating sub- (n) Gamma function, 0 xn1 ex dx, dimensionless
space is ordinarily explored experimentally. Over these limited regions, RosinRammler shape parameter, dimensionless
F was generally smooth and well estimated. The condition required p cyclone static pressure drop, Nm2
for similitude between a model and prototype is that each argument equivalent particle diameter, dimensionless
of F for the model must equal the corresponding argument for the d=d0 , dimensionless
inlet uid mass fraction, m _ f m
_ f =m _ p , dimensionless
prototype.
Scale effects can present practical difculties. For example, since the fractional efciency, dimensionless
pressure drop scales with the square of the inlet uid velocity, in practi- estimated fractional efciency, dimensionless
cal cyclone operation with gasses around atmospheric pressure, total efciency, dimensionless
V 30 m s1 is typically maintained, regardless of the size of the cy- distribution parameter, dimensionless
clone, to obtain a reasonable p. Suppose we wish to perform an exper- distribution parameter, various units
iment using the cyclones which have been discussed in this paper and design ratio, /D, dimensionless
are willing, if need be, to sacrice the p limitation. Taking one of design length, m
Gallaer and Schindeler's cyclones as the prototype, and assuming uid dynamic viscosity, kg m1 s1
i,p = i,m for all i, we nd Dp/Dm = 8, 24, or 40 for the other cyclones v 1, dimensionless
discussed. To maintain a constant NRe we must increase the inlet density f uid density, kg m3
f (and therefor the inlet pressure p), or the inlet velocity V, or their p particle density, kg m3
product, by the scale factor Dp/Dm. This will reduce p/f or increase V fractional efciency exponent, dimensionless
in proportion. The latter may introduce compressibility, given the mag- inlet particulate cumulative probability distribution function,
nitude of the scale factors. This effect may be moderated somewhat by dimensionless
judiciously varying the gas temperature and choosing the test points inlet particulate probability density function, dimensionless
carefully. Also, for a given inlet distribution, the particulate material
appears relatively ner in the prototype than in the model, to an extent
depending on Dp/Dm. Moreover, for industrial scale cyclones, Np has been References
reported to depend only on the design whereas at smaller scales it clearly [1] J.S. Ontko, Multivariate fractional efciency, Aerosol Sci. Technol. 31 (1999)
depended on operating conditions as well. All these issues should be care- 194197.
[2] S. Drobot, On the foundations of dimensional analysis, Stud. Math. 14 (1953) 8499.
fully considered when extrapolating small scale model data.
[3] F.A.L. Dullien, Introduction to Industrial Gas Cleaning, Academic Press, Inc., San
F can play a useful role in assessing the tradeoffs required to effec- Diego, CA, 1989 28.
tively design and execute an experimental test campaign. The tradeoffs [4] C.A. Gallaer, J.W. Schindeler, Mechanical dust collectors, J. Air Pollut. Control Assess.
to be made depend on the objectives of the campaign. F can also provide 13 (1963) 574580.
[5] L. Theodore, V. DePaola, Predicting cyclone efciency, J. Air Pollut. Control Assoc. 30
a framework for laying out the domain for computational simulations (1980) 11321133.
and their experimental validation. [6] C.E. Lapple, Fluid and Particle Mechanics, University of Delaware, Newark, DE, 1956
305309.
[7] D.L. Iozia, D. Leith, The logistic function and cyclone fractional efciency, Aerosol Sci.
Nomenclature Technol. 12 (1990) 598606.
D cyclone cylinder diameter, m [8] J. Dirgo, D. Leith, Cyclone collection efciency: comparison of experimental results
d equivalent particle diameter, m with theoretical predictions, Aerosol Sci. Technol. 4 (1985) 401415.
[9] C.J. Stairmand, The design and performance of cyclone separators, Trans. Inst. Chem.
d0 RosinRammler scale parameter, m Eng. 29 (1951) 356.
d50 50% cut diameter, m [10] J.S. Ontko, Cyclone separator scaling revisited, Powder Technol. 87 (1996) 93104.
d mean particle diameter, m [11] P. Rosin, E. Rammler, The laws governing the neness of powdered coal, J. Inst. Fuel
7 (1933) 2935.
F, f unknown functions
[12] G.E.P. Box, N.R. Draper, Empirical Model-Building and Response Surfaces, John Wiley
G captured particulate cumulative probability distribution func- & Sons, New York, 1987 7 (Ch.).
tion, dimensionless [13] R. Parker, R. Jain, S. Calvert, D. Drehmel, J. Abbott, Particle collection in cyclones at
high temperature and high pressure, Environ. Sci. Technol. 15 (1981) 451458.
H escaped particulate cumulative probability distribution func-
tion, dimensionless

View publication stats