A mathematical model for the determination of a cyclone performance

Vol.27, No. 2, pp. 263-272, 2000 Copyright© 2000ElsevierScienceLtd Printed in the USA.All rightsreserved 0735-1933/00IS-seefrontmatter

Int. Comte. Heat Mass Transfer,

Pergamon

P I I S0735-1933(00)00107-X

A MATHEMATICAL

MODEL FOR THE DETERMINATION

OF A

CYCLONE PERFORMANCE

A. Avci and I. Karagoz Department of Mechanical Engineering University of Uludag TR-16059 Bursa, Turkey

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT A mathematical model of two phase flow in tangential cyclone separators is presented, by definitions of new parameters including the effects of cyclone geometry, surface roughness and concentration of particles. The critical diameters, fractional efficiencies and pressure losses are calculated under the assumptions that each phase has the same velocity and the same acceleration in the spiral motion of the flow, a relative velocity occurs in the radial direction, and drag coefficient remains constant under certain conditions. The results obtained under these assumptions are compared with their experimental and theoretical counterparts in literature, and very good agreement is observed with the experimental values. © 2000 Elsevier Science Ltd

Introduction

Cyclones have been widely used in different industrial processes. As a separating unit, the capabilities of cyclones extend to fluid mixtures which may consist of two or more immiscible liquids, liquid and gaseous components, or liquid and solid particles. Cyclones have also been used as combustors for the combustion of fuels having low calorific value or fuels requiting long residence time for a complete combustion

Having relatively simple construction and without any moving parts, cyclones can easily be manufactured and their running costs are low. But theoretical or numerical analysis of the flow in cyclones is difficult since it is a 3D multi-phase flow and effected by many geometrical parameters. 263

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A. Avci and I. Karagoz

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Therefore, a lot of work on the cyclones have been performed under some simplifying assumptions, and the models have been proposed in literature give acceptable results only under certain limitations or for certain types of cyclones.

One of the simple theoretical models for the calculation of efficiency is the one dimensional Rosin-Rammler-Intelman theory [1,2]. Similar equations by making different assumptions are reported by Batel [3], Stairmand [4], and Ter Linden [5]. These equations generally have similar structure, but they are simple or complex depending on the assumptions, and they predict the efficiency roughly. On the other hand, the equations in exponential form seem to be good comparing with experimental fractional efficiency curves [6,7]. None of these are successful in giving accurate results or show the effects of all parameters.

In order to obtain more general, useful and simple equations for cyclone performance, a mathematical model is developed with the definition of new parameters, accounts for the effects of more geometrical and physical properties such as geometry, friction, concentration, etc., and can be used for calculation of critical diameter and fractional efficiencies

Mathematical Modelin2 Geometrical dimensions of a typical cyclone separator are given in Fig. 1. Experiments show that tangential velocity of the flow during spiral motion in a cyclone, increases especially in the conical part and attain a maximum value at the bottom of the cone but the axial and radial velocities do not change significantly. Therefore, increase of the flow velocity in spiral motion leads to a decrease of the flow cross section. Additionally, secondary flows which occur from the core of the vortex to the outlet pipe, before attaining to the cone apex, also decrease the flow cross section. On the contrary, friction leads to decrease of acceleration and increase of the flow cross section.

The flow behavior and performances of the cyclone separator are effected by many geometrical and kinematic parameters. It is very difficult to analyze the effects of the all parameters separately. Therefore, it would be convenient to reduce the number of those parameters. This was done by the following definitions of parameters.

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DETERMINATION OF A CYCLONE PERFORMANCE

265

hi

I °I DI

Top wiew

FIG.1 Typical dimensions of a cyclone

Cyclone Nominal Diameter (D.) Although cyclone bodies are generally cylindrical, they can have ellipse shapes due to their positive effect on the efficiency. This effect was taken into account by the definition of nominal diameter given by 2

1 -

Dn

-

-

2r01

~

1 2r02

(1)

where rol and ro~ are the ellipse radii.

Cyclone Dimensionless Diameter (DD) The surface area in comaet with the fluid flowing inside the cyclone is very important for the friction losses and efficiency. This area is effected by many geometrical parameters. But all these effects could be combined in one dimensionless parameter, DD, which defines the mean cyclone diameter having the same contact area with the same cyclone height. This parameter also takes into account of the effects of conical part and the geometry of the exit pipe as follows: DO -- 0.5(1 + D3)(1 - c ) + c ÷ O2h Dn L L D,L

Mean Curvature Radius (to) In cyclones, the main force for the efficiency of trapping the particles is centrifugal force which

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is inversely proportional to the curvature. Although it is variable, a constant mean curvature radius can be defined as geometrical mean of the diameters Dn and D3, to simplify the calculations;

ro =DD D~D~/2

(2)

Mean Acceleration in Spiral Motion (bd

Since separation of particles is mainly performed in the motion towards to the cone tip rather than in the flow from the cone tip to the exit, assuming that radial velocity in the cyclone is constant, flow is one-dimensional and there is no separation, mean acceleration can be calculated as bo

-

V32 - V 2 ' 2L e

(3)

where V3 and /4, are the flow velocities at the inlet and the cone apex, respectively, Le- L+b is the effective cyclone height, since fluid flows below the cone apex as about b.

Mean Flow W i d t h (a.)

Analyses of experimental results shows that (30..-40) percent of the cone tip diameter is not used by the flow and only (10+12) percent is used by the flow towards to exit [8]. Actually, the width and the height of the flow layer doing spiral motion, vary starting from the dimensions of the inlet section, a and b respectively, depending on many geometrical and flow parameters. Assuming that the height of the flow layer is constant and there is no geometrical constraint, a mean width for the flow layer will be defined first and then it will be modified to take into account the other parameters so as to simplify the calculations. Under these explanations, mean width of the flow layer which is constant can be defined as a geometrical mean of the heights at the inlet and cone apex as,

a. = ~

(4)

The Effects of Entrance Losses

In cyclones, the second phase is separated mainly during the flow from the inlet to the cone apex and is less effected by the reverse flow to the exit. Nevertheless, the dimensions of exit channel are important, and therefore, it would be appropriate to consider the effects of the reverse flow together with the dimensions of the exit channel. As can be seen from Fig2a, increasing the exit channel diameter (D2) may decrease the width and increase the height of the flow cross section. This may lead to an increase in efficiency and pressure losses. Smaller the exit channel diameter would cause reverse effects. Since the flow rate is a function of a, b and V,, each variable will be effected by the

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parameter of

DETERMINATION OF A CYCLONE PERFORMANCE

267

13=2a/(D1-192).In this study, it is assumed that the three variables are equally effected by

ratio 13.

The height of the channel (h) has also an effect on the performance of the cyclone. Experiments [9] show that if the height of the flow layer due to high b or high 132,is greater than the exit channel height, efficiency ~11 be smaller, as expected. Because some of the fluid will flow to the exit directly (Fig.2b). In the case ofb(h, the efficiency would increase for smaller values o f h and decrease for larger values of h, because of increase in the contact surface of the exit channel. According to experimental results, the optimum value o f h is thought to be equal tol.25b, approximately. In this study, the mean flow width is modified as follows, so as to incorporate all effects of the exit channel described above.

ano=a,[ l+0.5( 1"25 fl'/3b-1)] DD

(5)

I __ (a)

Co)

FIG.2. Effects of the exit channel on the flow

The Effects of Roughness Theoretical analysis in literature generally neglect surface roughness for simplicity. But it is known from experiments that roughness of the surfaces decreases the efficiency and pressures losses [9,10]. The reason is that the fluid flow is not able to accelerate sufficiently due to friction and therefore velocity of the flow is not able to reach to the ideal value at the cone apex. This causes the larger mean flow width and increase of separation from the surface, both decreasing the efficiency. Since it is proportional to the square of velocity, pressures losses will decrease with decreasing velocity. Therefore, the effects of roughness are taken into account with the modification of the mean width of the flow cross section. On the bases of considering energy losses, the final result for the flow width can be expressed by the following relation, since the variation in the flow width due to variation in the velocity is proportional square root of the kinetic energy;

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A. Avci and I. Karagoz

a.op = a.o (

1 l+ Kpf

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(6)

)o.s L~ a

where Kp can be taken bigger than 1 account for the other losses, andfis the friction factor.

The Effects of Concentration Experimental results also show that efficiency increases with increasing particle concentration. This effect is handled in a different way excluding the forces between the particles. Dragging of the fluid by the particles (dense phase) under the inertial effect prevents separation from the surface and decrease turbulence resulting an increase in effectiveness of holding dense phase. This effect was taken into account by defining the new flow width as follows: (7)

a o = a,o p/(1 + sc x[~,)

where sc is an experimental constant and C,, is the inlet concentration.

Cyclone Efficiency If the inlet and exit concentrations (Ci, Co) or mass flow rates of second phase (m,, mo) at the inlet and at the exit are known, fractional or total effciencies can be written as rl = l - rh° &i

or

(8/

rl = l - C° C,

Assumptions The following assumptions were considered for the calculation of efficiency. The mixture is homogenous at the inlet, constant acceleration is present in one dimensional spkal motion, the relative motion of the second phase is in the radial direction and relative velocity is very small, and constant mean temperature and mean roughness prevail through the flow. Drag coefficient of a particle is constant under certain conditions, additional forces are not present due to global motion, mean flow width and curvature radius of the cyclone defined above are constant.

Under these assumptions, the conservation equations can be written for the control volume in Fig3. The mass of second phase entering the control volume (Fig.3a.) can be written as m - a b V C and the distance travel during time dt is d s = V d t .

The mass flow rate reached the wall, of the second

phase which is flowing with radial velocity Vr, is d m - C ds b V~=b C V V~ dr, which is also equal to the change of the second phase in the flow according to conservation of mass. Multiplying and dividing by

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DETERMINATION OF A CYCLONE PERFORMANCE

t

/ \

P

~

~

269

~ _._....

J

/

Co)

(a)

FIG. 3 Control volume (a) and the forces applied on a particle CO). (a), leads to the equation

drn _ V r at rh

(9)

a

The forces appfied to a particle in the radial direction are the centrifugal force (Fc), drag force

(Fo) and hydrostatic force (Fh) (Fig.3b). Equilibrium of forces (Fc-FD+Fh) gives V2 V2 V: mv__=Co A p, r +m-r

2

2

(10)

where rap is the particle mass; m and p are the mass and density of fluid element which occupies the same volume with the particle, CD is the drag coefficient, and Vis the flow velocity in the spiral motion tangential to the trajectory. The radial velocity can be solved from this equation assuming that C/) is constant for a particle under certain working conditions. Using definition of acceleration that is

dt=dV/'o,, Eq.9 and Eq. 10 can be combine to give drh 4d th-[3-~

Pp 0.5 V d V (p --1)] ab,

(11)

where pp is the particle density. It is difficult to solve this equation because r and a are both variable. But these variables can be converted to the constants using mean curvature radius and mean flow width defined above (Eq.2 and Eq.7). Thus, taking r=r0 and a=a0, Eq. 11 can be integrated easily from the inlet to the cone apex, and using Eq.8, efficiency can be obtained as

-e- [

4 d

pp

Le

l°'~

(12)

In some cases, instead of efficiency, it is more important to calculate the critical diameter

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A. Avci and I. Karagoz

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defined as the diameter of particle which is hold with efficiency of 50%. Taking efficiency of 0.5, the critical particle diameter d~r can be calculated from Eq. 12, as /~

dcr=0.36 CD

2

P 0a0 (pp - p)L~

(13)

In Eq.12 and Eql3, the geometrical parameters seem to be dominant but the fluid and flow parameters effect the drag coefficient. Therefore, determination of this coefficient is an important problem. Selection of relation for CD and shape factors will effect the results. On the other hand, estimating of radial velocity will be difficult due to its dependency to the geometrical dimensions. To overcome these difficulties, analysis of experimental data for different cyclones gave an interesting result which shows that there would be a critical constant CD corresponding to the critical diameter for a cyclone under standard conditions. Using this result, after calculation of critical diameter, drag coefficient can be obtained from the following equations,

24 CD Re k

(14)

where Reynolds number (Re) is based on radial velocity and particle diameter, k - S f for Re
k-O. 646Sffor 500>Re>l. Then, fractional efficiencies for other diameters can be calculated. To take into account shape of particles, the shape factor (SJ) was included in the parameter k, and Sf=l for spherical particles

Results

The mathematical model described above were applied to four different types of cyclone, namely A, B, C and D whose ratios of geometrical dimensions given in ref [1]. Among these, D is the typical cyclone described as a widely recommended type. The results obtained in this study were compared with experimental values and five semi-empirical models given in literature [2, 3, 4, 6, 10]. The value of CD corresponding to the critical diameter was obtained from the experimental data given by Stairmand for his high effective cyclone, and this value was also used for the other cyclones. Table. 1 shows the results of critical diameters calculated for different types of cyclones and comparison with the results given in literature. As can be seen that agreement with the experimental values is very good compared with the other models. The fractional efficiencies were also calculated in this study, and compared with the values

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DETERMINATION OF A CYCLONE PERFORMANCE

TABLE 1 Comparison of critical diameters for different types of cyclones Cyclone Types de, Oxn) A B C Experimental 8.1 0.75 Stairmand 3 4.3 1.3 Rosin-Rammler 3.7 5.1 2.2 Bmh 4 7 2.5 ~cht 4.4 5.9 2.5 !~ch 2.9 5 2.2 P~sentsm~ 2 8.1 0.76

271

D

0.9 4 4.4 7.9 5 3.4 1.1

given in literature. Figure 4a shows that the estimation of fractional efficiency is very satisfactory. Additionally, taking the shape factor, 0.9 and 0.8, the fractional efficiencies were calculated and obtained results are given in Fig.4b, for the typical cyclone. Discrepancies between experimental values and the present model is small for Sf=0.9.

efficiency

efficiency

lio,oi.!iO!!,iiiii!

1

q

0.2 ......... i ...... i i 7 1 i 1 1 i O

-Exp.

1.5

0.15

0.8

(1.6

0.4

0.2

15

0.15

1.5

tn(d/dcr)

ln(d/dc~) (a)

(b)

FIG. 4 Variation of fractional efl]ciencies with particle diameter. (a) Spherical particle, (b) particle with general shape

Discussion

Despite of its simple structure, many parameters such as particle characteristics and particle distributions, shape factors, concentration, surface roughness and geometrical characteristics effect interactively the flow behavior in a cyclone and therefore it would not he expected to attain accurate

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results. However, making appropriate assumptions, it could be possible to obtain results surprisingly very close to the experimental values and more successful comparing with other relations as given above. Therefore, the equations proposed seem to be very attractive for the determination of a cyclone performance.

One of the important parameters is the cyclone height. Experiments prove that efficiency increase with increase of height. But it should be a limit for the height due to fi'iction and separation from the surfaces and this should be analyzed experimentally.

The fractional efficiencies obtained in this study were also satisfactory in despite of, especially, estimation of the shape factor. Although fractional efficiency curves are similar to each other some discrepancies occur, especially, at the below and upper limits of particle diameter.

As a conclusion, it can be said that practical and acceptable results were obtained by the equations proposed with another advantage of including more parameters. However, drag coefficients and surface roughness should be analyzed in detail with the support of experiments to get more reliable and practical equations which serve to design optimum cyclone separators.

References

1.

A.K. Gupta, D.C. Lilley, N. Syred, SwirlFlows, p.295, Abacus Press, (1984)

2.

W.H. Walton, Cyclone Dust Separators,, Pergamon Press, (1974)

3.

W. Batel, Dust Extraction Technology, Technicopy Ltd., England, (1976)

4.

C.J. Stairmand, R.M. Kelsey, Soc. Chem. Industry, 42,15 (1955)

5.

A.J. Ter Linden, Proc. Inst. Mech. Eng. 130 (1960)

6.

D. Leith, W. Licht, A.1.Ch.E. Symposium Series, 68, 126 (1972)

7.

A.C. Stern, H.C. Wohlers, Fundamentals of Air Pollution, Academic Press, New York (1973)

8.

A. Avci, I. Karagoz, 4~h Int. Combustion Symposium, Turkey (1995)

9.

R.H. Perry, C.H. Chilton, Chemical Engineers Handbook, Mc Graw-Hill, Tokyo (1977).

10.

R.G. Dorman, Dust Control and Air Cleaning, Pergamon Press Ltd., Germany (1974)

Received October 1, 1999