Minerals Engineering, Vol. 13, No. 2, pp. 183-191, 2000
Pergamon 0892-6875(99)00164--8
© 2000 Elsevier Science Ltd All rights reserved 0892-6875/00/$ - see front matter
A SIMPLE P R O C E D U R E F O R DESIGN A N D P E R F O R M A N C E
PREDICTION
OF BRADLEY AND RIETEMA HYDROCYCLONES
L.R. CASTILHO § and R.A. M E D R O N H O ¶* § Program of Chemical Engineering, COPPE, Federal University of Rio de Janeiro, Centre de Tecnologia, Bloco G, Ilha do Fundio, 21949-900 Rio de Janeiro-RJ, Brazil ¶ Department of Chemical Engineering, School of Chemistry, Federal University of Rio de Janeiro, Centre de Tecnologia, Bloco E, Ilha do Fund~o, 21949-900 Rio de Janeiro-RJ, Brazil * Corresponding author at present address: Gesellschaft fiir Bioteclmologische Forschung (GBF), Biochemical Engineering Division (BVT), Mascheroder Weg 1, 38124 Braunschweig, Germany Email:
[email protected] (Received 24 June 1999; accepted 25 October 1999) ABSTRACT Hydrocyclones were originally designed to promote solid-liquid separations but nowadays they are also used for solid-solid, liquid-liquid and gas-liquid separations. Although a hydrocyclone is a very simple equipment to build, the use of custom-made cyclones is not widely used. This is probably due to the lack era simple procedure for hydrocyclone design. In the present work a procedure is presented, which allows the design and performance prediction of hydrocyclones that follow Bradley and Rietema recommended geometries, that are the only two well-known families of geometrically similar hydrocyclones . The procedure proposed here resulted in little error, when the results were compared with experimental data. Additionally, a comparison of both families of hydrocyclones revealed that, for a given hydrocyclone diameter and at the same operational conditions, Bradley hydrocyclones provide higher efficiencies, while Rietema hydrocyclones give higher capacities. © 2000 Elsevier Science Ltd. All rights reserved
Keywords Hydrocyclones; thickening; t'me particle processing INTRODUCTION A hydrocyclone has no moving parts and consists of a conical section joined to a cylindrical portion, which is fitted with a tangential inlet and closed by an end plate with an axially mounted overflow pipe. The end of the cone terminates in a circular apex opening. Although the first patent of a hydrocyclone is more than 100 years old (Bretnei, 1891), its first industrial applications date from the late 40's. Hydrocyclones were originally designed to promote solid-liquid separations but nowadays they are also used for solid-solid (Klima and Kim, 1998), liquid-liquid (Capela et al., 1996; Smyth and Thew, 1996) and gas-liquid separations (Marti, 1996). Possible new applications, such as separation of microrganisms from fermented broth, Continue to be developed (Yuan et al., 1996; Cilliers and Harrison, 1997). The solution of simplified phenomenological models (Concha et al., 1996) and of the complete set of differential equations through computational fluid dynamics (CFD) are still in progress (He et al., 1999; Devulapalli and Rajamani, 1996), and will give in a near future a better understanding ofhydrocycloncperformance. When studying hydrocyclones, the dimensionless groups of interest are the Stokes (Stks0), Euler (Eu) and Reynolds (Re) numbers (Svarovsky, 1984), as defined below.
183
_84
L.R. Castilhoand R. A. Medronho
Stk5o = (19"-P~(d'5°)2
(1)
18,uD,~ AP
Eu -- PV z/--~2 Re =
(2)
Dcvp
(3)
The equilibrium orbit theory and the residence time theory are two well-known theories for particle separation in hydrocyclones. The former, first proposed by Driessen (1951), assumes that particles of a given size will reach an equilibrium radial orbit position inside the hydrocyclone where their outward terminal settling velocity is equal to the inward radial velocity of the liquid. The particle size whose equilibrium orbit is coincident with the locus of zero vertical velocity will have equal chance to escape the hydrocyclone either by the underflow or by the overflow. This particle is known as cut size (dso). The latter, proposed by Rietema (1961), assumes non-equilibrium conditions and considers whether a particle will be separated as a function of both the position it enters the cyclone and the available residence time. The cut size will be the size of the particle which entering the equipment exactly in the centre of the inlet pipe will just reach the wall in the residence time available. Several authors have used one of these two theories to derive different equations for the cut size. Svarovsky (1984) and Medronho (1984) have shown that most of these equations lead to the conclusion that the product between the Stokes number and the Euler number is constant for geometrically similar hydrocyclones. Medronho and Svarovsky (1984) have also shown that this product depends on the cyclone design, but is not affected by the relative size of the inlet orifice (D/De) for similarly shaped single inlets. Medronho (1984), in a massive experimental work, has shown that the product Stk50Eu is, however, a function of water flow ratio (Rw) and volumetric feed concentration (CO. The author has also proposed a model composed of Eqs. (4), (5), and (6) which describes the operation of geometrically similar hydrocyclones, when used to promote solid-liquid separations.
(4)
[ln(1/R w)]" exp(n 2 C v) Eu = k 2 Re"3 exp(n4 Cv )
(5)
R w = k 3 ( D " I " ' E u "6
(6)
Stk5oEu = k,
!,,D
Where k and n are parameters of the equations. Contrasting with gas cyclones, for which there are several families of geometrically similar cyclones, there are only two well-known families of geometrically similar hydrocyclones. These are due to Rietema (1961) and Bradley (1965). Table 1 gives the geometrical proportions of these two families, and Tables 2, 3, and 4 give the values for the constants in Eqs. (4), (5), and (6), respectively, according to Medronho (1984) and Antunes and Medronho (1992). TABLE 1 Geometric proportions of two well-known families of hydroeyclones
Hydrocyclone Bradley Rietema
D,/Oc
Do/De
1/7 0.28
1/5 0.34
L/Dc 5
LJD¢ 1/2 -
t/D~ 1/3 0.40
e 9° 20°
BradleyandRieternahydroclones
185
TABLE 2 Parameters of Equation (4) according to Medronho (1984) and Antunes and Medronho (1992) Hydrocyclone Bradley Rietema
kI 0.0550 0.0474
nI 0.66 0.74
ne 12.0 9.0
TABLE 3 Parameters of Equation (5) according to Medronho (1984) and Antunes and Medronho (1992) Hydrocyclone Bradley Rietema
ke 258.0 371.5
n~ 0.37 0.12
n4
0.0 -2.12
TABLE 4 Parameters of Equation (6) according to Medronho (1984) and Antunes and Medronho (1992) Hydrocyclone Bradley Rietema
k3 1.21 x 10° 1218
n5
2.63 4.75
n6 -1.12 -0.30
Manufacturers of hydrocyclones produce only a limited range of cyclone diameters and, in order to be able to cover a wide range of cut sizes and flow rates, each cyclone of a given size can be operated with different openings sizes (inlet, overflow and undertow) through the use of interchangeable parts. This approach requires an accurate knowledge of how geometric variables affect the equipment performance. An alternative approach to this is to use a custom-made hydrocyclone based on a geometrically similar family. Although a hydrocyclone is a very simple equipment to build, this approach is not widely used. This is due probably to the lack of a simple procedure for hydrocyclone design. The aim of the present work is therefore to propose a simple procedure for the design of Bradley and Rietema hydrocyclones. The procedure to be presented also allows the performance prediction of such equipments.
MATERIALS AND METHODS The method presented in this work was based in Eqs. (1) to (6), and for the calculations of the reduced total efficiency (E'T) eq. (7) was employed (Svarovsky, 1990).
E' r = ~ G' dy
(7)
For calculating E'T as given by eq. (7), it is necessary to know the relationship between the reduced grade efficiency (G') and the particle size distribution of the feed given as cumulative undersize fraction (y). This relationship can be obtained through the use of Eqs. (8) (Rosin and Rammer, 1933) and (9) (Plitt, 1976).
y:
(8)
186
L.R. Castilho and R. A. Medronho
G ' = 1 - exp
['
0.693
(9)
where k, m and n are parameters of the models and, for hydrocyclones, the value of n in eq. (9) is 2.9, according m Coellio and Medronho (1994). The total efficiency (ET) can then be calculated using eq. (10).
E' r
-
E r - Rw
(10)
1-R w In order to establish the method, the chosen fluid was water (p=l.0 g/cm3 and g=l.0 cp) and the solids had a density of 2.9 g/cm3. The volumetric concentration of solids in the feed was set to 1% and the flow ratio (R,) was fixed at 10%, since this Rw usually produces a good efficiency coupled with a relatively high underflow concentration.
RESULTS AND DISCUSSION
Comparison of performance of Bradley and Rietema hydrocyclones Figures 1 and 2 were plotted based on Eqs. (4) and (5). These Figures show, for Bradley and Rietema hydrocyclones respectively, the reduced cut size as a function of flow rate, for some values of pressure drop and cyclone diameter. It can be observed that for the same pressure drop Rietema hydrocyclones produce flow rates 2 to 3 times higher than Bradley hydrocyclones of the same size. On the other hand, at these same conditions, Bradley hydrocyclones produce lower reduced cut sizes and, consequently, greater total efficiencies than Rietema hydrocyclones. 11
- -
Hydrocyclone diameter ]
o5
- ressu'Oo0'ar' / , E::L
9
/
I
\
Al.o ~
\ 1.5
.N 7 O9 --I
O -o ~D
5
O "O 0~
¢:
3
2 cm 1 cm
1 I0
I O0
1000
10000
Flowrate (cma/s) Fig. 1 Reduced cut size for Bradley hydrocyclones (Cv=1%, p,=2.9 g/cm3 and R,=I 0%). These features of both families of hydrocyclones may be better visualised if, beyond the conditions mentioned in the Materials and Methods section, the desired flow rate and pressure drop are set to 3000 cm3/s and 2.5 bar, respectively. Figure 3 shows, for this situation, the cyclone size (De) and the reduced cut
Bradley and Rietema hydroclones
187
size (d'50) as a function of the number of hydrocyclones used in parallel. It can be seen that for processing the same flow rate at a given pressure drop Rietema hydrocyclones are always smaller in diameter than Bradley hydrocyclones. Therefore, it is possible to state that Rietema hydrocyclones are high capacity separators.
16
Hydrocyclone diameter Pressure Drop (bar)
•14
A/~\0\ .5 \
E
=L 12 O N 10 •"o
,1.0
\\ ~ x \
1.5
/ \
8
&
/
0
4 2
,
. . . .
,
10
,'r
100
1000
10000
Flowrate (cm3/s) Fig.2 Reduced cut size for Rietema hydrocyclones (C~=1%, ps=2.9 g/cms and Rw=10%). 20
10
\
-\.-> \ \
Eo 16
Bradley Rietema
--
8
\
A
E v
6
121
03 ¢3
t'-
_o
8
4
~,
4
2 cc
0 o
02 O 03
I
t
0
2
c
q
t
I
4 6 8 10 12 Number of Hydrocyclones
I
14
0
16
Fig.3 Hydrocyclone diameter and reduced cut size as a function of number of cyclones in parallel (Q=3000 cm3/s, AP=2.5 bar, Cv=l%, p~=2.9 g/cm3 and Rw=10%). According to Svarovsky (1984), the reduced cut size (d'50) will be lower and, consequently, the total separation efficiency will increase as the hydrocyclone diameter of a given geometry is reduced. In Figure 3 it may be observed that, for a given flow rate, pressure drop, and number of cyclones, although Rietema hydrocyclones are smaller, the reduced cut size obtained with Bradley hydrocyclones is always lower,
188
L.R. Castilho and R. A. Medronho
showing that the latter is more efficient than the former. Therefore, it is possible to state that Bradley hydrocyclones are high efficiency separators.
Design and performance prediction Figure 4 presents the reduced total efficiency (E'T) as a function of the reduced cut size (d'50) and the parameters of eq. (8). This figure is valid for any hydrocyclone design and not only for Bradley and Rietema hydrocyclones, provided that the feed size distribution can be represented by eq. (8) and that eq. (9) with n=2.9 fits well the reduced grade efficiency curve. 100 90
o~ ¢,to
80 70
LU 0
60
I"0 0
50
0
rr
40
--
Values of m !
30 0
2
4
6
8
10
~d'~
Fig.4 Reduced total efficiency of hydrocyclones as a function of the reduced cut size and of the parameters of Equation (8). The use of Figures 1, 2 and 4 allows the design and performance prediction of Bradley and Rietema hydrocyclones to be performed. In a typical design problem, the total feed flow rate is known and a given reduced total efficiency is desired. Once the particle size distribution of the feed is known, the reduced cut size can be determined using Figure 4. If the pressure drop is known, Figure 1 or 2 can then be used to fred the diameter of Bradley or Rietema hydrocyclone, respectively, and the flow rate capable of being processed by this cyclone. If the pressure drop is not previously known, it can be selected according to Figure 1 or 2. The ratio between the total suspension feed flow rate and the flow rate given by one hydrocyclone gives the number of cyclones to be used in parallel. Equation (6) can then be used to calculate the undertow orifice size. The performance prediction is done in the same way but normally in the opposite direction, i.e., the reduced cut size is found through the use of Figure 1 or 2, and then the reduced total efficiency is obtained from Figure 4. When establishing this method, some operating conditions were fixed. If in a given application these operation conditions are different from those established here, eq. (4) can be used for correcting the reduced cut size, where StksoEu is given by eq. (11).
s t k , oSU - , Co, -
p) z c (a',0) 2 36,u,oa
(11)
Bradleyand Rietemahydroelones
189
For instance, Eqs. (12) and (13) give the corrections to be made when using a different flow ratio or feed concentration, respectively.
d'5o~
_[l-L~(I/R., n(1/Rw~l]~
d'501 d'
n,, _ ,o~ _ exP[2...(Cv,
(12)
]l
Cv, q
(13)
As Eqs. (4), (5), and (6) were established for feed concentrations up to 10% by volume, that is the maximum concentration that should be used in combination with this procedure. For higher concentrations a new set of equations would have to be established (Ortega-Rivas and Svarovsky, 1998). Validation of the method
Tables 5 and 6 present a comparison between experimental results obtained by Silva (1989) and Rietema (1961) and the data obtained by using Figures 1 and 2, respectively. Additionally, the combined use of Figures 1 and 4 allows a comparison with Silva's experimental data on efficiency. Using a Bradley hydrocyclone with De=I.5 cm, AP=2.76 bar, Cv=l% and powder with feed size distribution given by eq. (8) with k=3.33 ~tm and m=1.54, Silva (1989) obtained a flow rate of 51 cm3/s, a reduced cut size of 2.8 ~tm, a flow ratio of 27.4% and a reduced total efficiency of 54%. From Figure 1, for Dc=l.5 cm and AP=2.76 bar, the flow rate is 54 cm3/s (AQ=6.5%) and d'5o is 3.0 lxm. Correcting the latter by means of eq. (12), d'5o.corrmed=2.5 ~tm (Ad'5o=-10.7%). Using Figure 4, the reduced total efficiency is found to be 54% (AE'T=0.0%). TABLE 5 Comparison of experimental data of Silva (1989) with Bradley hydrocyclones and results obtained with Figure I
Exp. D,: AP Qexp Q AQ dso~:p d'50exp d'5o (%) No. (cm) (bar) (cm3/s) (cm3/s) (~tm) (p,m) 0.tm) 7 1.5 2.07 44 48 9.1 3.2 3.3 3.2 19 3.0 2.07 170 175 2.9 3.5 3.7 4.0" 7.1 7.4 29 6.0 0.69 356 380 6.7 7.9 8.4 8.3 8.8 9.458 6.0 1.38 480 520 From Fig. 1: d'5o=4.3 ~tm. After correction for Rw.ex =14.74% with eq. (12): d'5o = 4.0 ~tm. **From Fig. 1:d'50=6.3 ~tm. After correction for Cv=7.75% with eq. (13): d'5o = 9.4 ~tm.
Ad'5o (%) -3.0 8.1 6.8 6.8
TABLE 6 Comparison of experimental data of Rietema (1961) with Rietema hydrocyclones and results obtained with Figure 2
Exp. D¢ AP" Qexj Q No. (cm) (bar) (cm°/s) (cm3/s) 151 7.:5 0.63 1457 1500 152 7.5 1.60 2108 2250 153 7.5 2.11 2699 2600 •Calculated from data presented by Rietema (1961 ).
AQ (%) 3.0 6.7 -3.7
d5oexp
d'5o~xp"
(lain) 11.5 10.0 8.5
(~tm) 12.3 10.8 9.0
d' 50
Ad'5o
13.5 10.5 9.7
9.8 -2.8 7.8
(~m)
(%)
Therefore, Figures 1, 2 and 4 produce relatively low errors, when results are compared with experimental data from other authors.
190
L.R. Castilho and R. A. Medronho
CONCLUSIONS For a given hydrocyclone diameter and at the same operational conditions, Bradley hydrocyclones always provide lower reduced cut sizes and, consequently, higher efficiencies than Rietema hydrocyclones. However, the latter always give greater flow rates tharLthe former. Therefore, it can be said that Bradley hydrocyclones are high efficiency separators andRietema hydrocyclones are high capacity separators. A simple procedure to design and to predict performance of Bradley and Rietema hydrocyclones was presented in this work. The use of this procedure allows any user to build a custom-made hydrocyclone especially designed for a specific purpose. NOMENCLATURE Cv
d ds0
d'5o D~ Di Do D~ ET E'T Eu G' e L Li AP
Q R~ Re Stkso V
Y 0 P P~
Feed volumetric concentration Particle diameter (L) Cut size (L) Reduced cut size (L) Hydrocyclone diameter (L) Feed inlet diameter (L) Overflow diameter (L) Underflow diameter (L) Total efficiency Reduced total efficiency Euler number Reduced grade efficiency Vortex finder length (L) Hydrocyclone length (L) Height of hydrocyclone cylindrical part (L) Pressure drop (ML-lj,-2) Feed volumetric flow rate (LaT- 1) Flow ratio Reynolds number Stokes number Feed velocity based on the hydrocyclone diameter (LT-l) Cumulative particle size distribution (undersize) of feed suspension Liquid viscosity (ML- iT- t) Angle of the hydrocyclone cone Liquid density (ML -3) Solid density (ML -a) REFERENCES
Antunes, M. and Medronho, R.A., Bradley hydrocyclones: design and performance analysis. In Hydrocyclones: Analysis and Applications, eds. L. Svarovsky and M.T. Thew, Kluwer Academic Publishers, Dordrecht, Nethedands, 1992, pp. 3-13. Bradley, D., The Hydrocyclone, 1965, Pergamon Press, London. Bremei, E., US Patent No. 453, 105 (1891). Capela Moraes, C.A., Hackenberg, C.M., Russo, C. and Medronho, R.A., Theoretical analysis of oily water hydrocyclones. In Hydrocyclones'96, eds. D. Claxton, L. Svarovsky and M. Thew, Mechanical Engineering Publications, London & Bury Saint Edmunds, 1996, pp. 383-398. Cilliers, J.J. and Harrison, S.T.L., The application of mini-hydroeyelones in the concentration of yeast suspensions, The Chemical Engineering Journal, 1997, 65, 21-26.
Bradleyand Rietemahydroclones
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Coelho, M.A.Z. and Medronho, R.A., EquafSes para o projeto e performance de hidrociclones. In Proc. of the XXII Brazilian Congress of Particulate Systems, Vol. I, UFSC, Flodan6polis/SC, Brazil, 1994, pp. 273-280. Concha, F., Barrientos, A., Munoz, L., Bnstamante, O. and Castro, O., A phenomenological model of a hydrocyclone. In Hydrocyclones'96, eds. D. Claxton, L. Svarovsky and M. Thew, Mechanical Engineering Publications, London & Bury Saint Edmunds, 1996, pp. 63-82. Devulapalli, B. and Rajamani, R.K., A comprehensive CFD model for particle-size classification in industrial hydrocyclones. In Hydrocyclones'96, eds. D. Claxton, L. Svarovsky and M. Thew, Mechanical Engineering Publications, London & Bury Saint Edmunds, 1996, pp. 83-104. Driessen, M.G., Theory of flow in a cyclone. Revue de L'Industrie Minerale/Numero Special, 1951, SaintEtienne, 449-461. He, P., Salcudean, M. and Gartshore, I.S., A numerical simulation of hydrocyclones. Chemical Engineering Research and Design, 1999, 77, 429-441. Klima, M.S. and Kim, B.H., Dense-medium separation of heavy-metal particles from soil using a wideangle hydrocyclone. Journal of Environmental Science and Health, 1998, Part A, 33, 1325-1340. Marti, S., Analysis of gas carry-under in gas-liquid cylindrical cyclones. In Hydrocyclones'96, eds. D. Claxton, L. Svarovsky and M. Thew, Mechanical Engineering Publications, London & Bury Saint Edmunds, 1996, pp. 399-421. Medronho, R.A. and Svarovsky, L., Tests to verify hydrocyclone scale-up procedure. In Proc. 2nd International Conference on Hydrocyclones, BHRA, Bath, United Kingdom, 1984, pp. 1-14. Medronho, R.A., Scale-Up of Hydrocyclones at Low Feed Concentrations, 1984, Ph.D. Thesis, University of Bradford, United Kingdom. Ortega-Rivas, E. and Svarovsky, L., Generalized Stokes number for modeling settling of non-newtonian slurries in dynamic separators, Advanced Powder Technology, 1998, 9 (1), 1-16. Plitt, L.R., A mathematical model of the hydrocyclone classifier, CIMBulletin, 1976, 69 (Dee.), 114-123. Rietema, K., Performance and design of hydrocyclones--Parts I to IV. Chemical Engineering Science, 1961, 15, 298-325. Rosin, P. and Rammler, E., The laws governing the fmeness of powdered coal. Journal of the Institute of Coal, 1933, 7, 29-36. Discussion, ibid., 109-112. Silva, M.A.P., Hidrociclones de Bradley: Dimensionamento e ,4nMise de Desempenho, 1989, M.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro/RJ, Brazil. Smyth, I.C. and Thew, M.T., A study of the effect of dissolved gas on the operation of liquid-liquid hydrocyclones. In Hydrocyclones'96, eds. D. Claxton, L. Svarovsky and M. Thew, Mechanical Engineering Publications, London & Bury Saint Edmunds, 1996, pp. 357-368. Svarovsky, L., Efficiency of Separation of Particles from Fluids. In Solid-Liquid Separation, 3rd Edn., ed. L. Svarovsky, Butterworths, London, 1990, pp. 43-73. Svarovsky, L., ttydrocyclones, 1984, Holt, Rinehart and Winston, London. Yuan, H., Rickwood, D., Smyth, I.C. and Thew, M.T., An investigation into the possible use of hydrocyclones for the removal of yeast from beer, Bioseparation, 1996, 6, 159-163.
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