Optimization of the cyclone separator geometry for minimum pressure drop using mathematical models and CFD simulations

Chemical Engineering Science 65 (2010) 6048–6058

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

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

Optimization of the cyclone separator geometry for minimum pressure dropusing mathematical models and CFD simulations

Khairy Elsayed �, Chris Lacor

Vrije Universiteit Brussel, Department of Mechanical Engineering, Research Group Fluid Mechanics and Thermodynamics, Pleinlaan 2, B-1050 Brussels, Belgium

a r t i c l e i n f o

Article history:

Received 1 July 2010

Received in revised form

3 August 2010

Accepted 31 August 2010Available online 7 September 2010

Keywords:

Cyclone separator

Discrete phase modeling (DPM)

Mathematical models

Response surface methodology (RSM)

Design of experiment (DOE)

Optimization

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.08.042

esponding author. Tel.: +32 26 29 2368; fax:

ail address: [email protected] (K. Elsayed).

a b s t r a c t

The response surface methodology has been performed based on the Muschelknautz method of

modeling (MM) to optimize the cyclone geometrical ratios. Four geometrical factors have significant

effects on the cyclone performance viz., the vortex finder diameter, the inlet width and inlet height, and

the cyclone total height. There are strong interactions between the effect of inlet dimensions and vortex

finder diameter on the cyclone performance. CFD simulations based on Reynolds stress model are also

used in the investigation. A new set of geometrical ratios (design) has been obtained (optimized) to

achieve minimum pressure drop. A comparison of numerical simulation of the new design and the

Stairmand design confirms the superior performance of the new design compared to the Stairmand

design.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Cyclones are widely used in the air pollution control and gas–solid separation for aerosol sampling and industrial applications.With the advantages of relative simplicity to fabricate, low cost tooperate, and well adaptability to extremely harsh conditions andhigh pressure and temperature environments, the cycloneseparators have become one of the most important particleremoval devices which are preferably utilized in scientific andengineering fields. Cyclones are frequently used as final collectorswhere large particles are to be caught. Efficiency is generally goodfor dusts where particles are larger than about 5mm in diameter.They can also be used as pre-cleaners for a more efficient collectorsuch as an electrostatic precipitator, scrubber or fabric filter(Swamee and Aggarwal, 2009).

1.1. Cyclone performance

In addition to separation efficiency, pressure drop is consid-ered as a major criterion to design cyclone geometry and evaluatecyclone performance. Therefore, an accurate mathematical modelis needed to determine the complex relationship betweenpressure drop and cyclone characteristics. The pressure drop ina cyclone separator can also be decreased or increased by varying

ll rights reserved.

+32 26 29 2880.

the cyclone dimensions. For an accurate optimal design of acyclone, it is quite necessary to use a reliable pressure dropequation for it.

Currently, the pressure drop models for cyclone separators canbe classified into three categories (Zhao, 2009): (1) the theoreticaland semi-empirical models, (2) statistical models and (3)computational fluid dynamics (CFD) models.

The theoretical or semi-empirical models were developed bymany researchers, e.g. Shepherd and Lapple (1940), Alexander(1949), First (1949), Stairmand (1951), Barth (1956), Avci andKaragoz (2001), Zhao (2004), Karagoz and Avci (2005) and Chenand Shi (2007). These models were derived from physical descrip-tions and mathematical equations. They require a very detailedunderstanding of gas flow pattern and energy dissipation mechan-isms in cyclones. In addition, due to using different assumptions andsimplified conditions, different theoretical or semi-empirical modelscan lead to significant differences between predicted and measuredresults. Predictions by some models are twice more than experi-mental values and some models are even conflicted as to whichmodels work best (Swamee and Aggarwal, 2009).

In the 1980s, statistical models, as an alternative approach,were used to calculate cyclone pressure drop. For instance, themodels proposed by Casal and Martinez-Benet (1983) and Dirgo(1988) were developed through multiple regression analysisbased on larger data sets of pressure drop for different cycloneconfigurations. Although statistical models are more convenientto predict the cyclone pressure drop, it is significantly moredifficult to determine the most appropriate correlation function

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K. Elsayed, C. Lacor / Chemical Engineering Science 65 (2010) 6048–6058 6049

for fitting experimental data in this approach especially with thelimited computer statistical softwares and robust algorithmsavailable at that time.

Recently, the computational fluid dynamics (CFD) techniquehas presented a new way to model cyclone pressure drop. Forinstance, Gimbun et al. (2005) successfully applied CFD to predictand to evaluate the effects of temperature and inlet velocity onthe pressure drop of gas cyclones (Zhao, 2009). Undoubtedly, CFDis able to provide insight into the generation process of pressuredrop across cyclones but additional research is still needed tohave a good matching with experimental data. CFD is alsocomputationally expensive in comparison with the mathematicalmodels approach.

1.2. Stairmand design

In 1951, Stairmand (1951) presented one of the most populardesign guidelines which suggested that the cylinder height andthe exit tube length should be, respectively, 1.5 and 0.5 times ofthe cyclone body diameter for the design of a high efficiencycyclone (Safikhani et al., 2010) (Fig. 1 and Table 1). In theStairmand model for pressure drop calculation (Stairmand, 1949),the velocity distribution has been obtained from a moment-of-momentum balance, estimating the pressure drop as entrance andexit losses combined with the loss of static pressure in the swirl.The main drawbacks of the Stairmand model are: (1) neglectingthe entrance loss by assuming no change of the inlet velocityoccurs at the inlet area; (2) assuming constant friction factor; (3)the effect of particle mass loading on the pressure drop is not

Fig. 1. Schematic diagram for Stairmand cyclone separator.

Table 1The geometrical parameters values for Stairmand design (barrel diameter

D¼0.205 m).

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/D Li/D Le/D

Stairmand design 0.5 0.2 0.5 4 1.5 0.5 0.36 1.0 0.618

included. All these drawbacks are overcome in the Muschelknautz

method of modeling (MM) (Hoffmann and Stein, 2008) introducedby Muschelknautz and Trefz (1990, 1991). The main benefit ofMM over other models is its ability to take the following effectsinto account: (a) wall roughness due to both the physicalroughness of the materials of construction and to the presenceof collected solids; (b) the effect of mass loading and Reynoldsnumber on cyclone performance; (c) the change of flow velocitythroughout the cyclone (Hoffmann and Stein, 2008).

The present paper is an attempt to obtain a new optimizedcyclone separator based on the MM model and to investigate theeffect of each cyclone geometrical parameter on the cycloneperformance using response surface methodology and CFDsimulation.

1.3. The Muschelknautz method of modeling (MM)

Hoffmann and Stein (2008) stated that the most practicalmethod for modeling cyclone separators at the present time is theMuschelknautz method (MM) (Muschelknautz and Kambrock,1970; Muschelknautz, 1972; Muschelknautz and Trefz, 1990;Trefz, 1992; Trefz and Muschelknautz, 1993; Cortes and Gil, 2007;Hoffmann and Stein, 2008). The roots of the Muschelknautzmethod (MM) extend back to an early work performed by Barth(1956) as it is based on the equilibrium orbit model (Hoffmannand Stein, 2008).

1.3.1. The pressure loss in cyclone

According to MM model, the pressure loss across a cycloneoccurs, primarily, as a result of friction with the walls andirreversible losses within the vortex core, the latter often dominatingthe overall pressure loss, Dp¼DpbodyþDpx. In dimensionless form,it is defined as the Euler number:

Eu ¼1

12rv2

in

½DpbodyþDpx� ð1Þ

The wall loss, or the loss in the cyclone body is given by

Dpbody ¼ fAR

0:9Q

r2ðvywvyCSÞ

1:5ð2Þ

where vin is the area average inlet velocity, r is the gas density, Q

is the gas volume flow rate, AR is the total inside area of thecyclone contributing to frictional drag. The wall velocity, vyw is thevelocity in the vicinity of the wall, and vycs is the tangentialvelocity of the gas at the inner core radius.

The second contribution to pressure drop is the loss in the coreand in the vortex finder is given by

Dpx ¼ 2þvycs

vx

� �2

þ3vycs

vx

� �4=3" #

1

2rv2

x ð3Þ

where vx is the average axial velocity through the vortex finder(for more details refer to Hoffmann and Stein, 2008).

1.3.2. Cut-off size

A very fundamental characteristic of any lightly loaded cycloneis its cut-point diameter or cut-off size x50 produced by the spin ofthe inner vortex. This is the practical diameter that has a 50%probability of capture. The cut size is analogous to the screenopenings of an ordinary sieve or screen (Hoffmann and Stein,2008). In lightly loading cyclone, x50 exercises a controllinginfluencing on the cyclone’s separation performance. It is theparameter that determines the horizontal position of the cyclonegrade-efficiency curve (fraction collected versus particle size). Forlow mass loading, the cut-off diameter can be estimated in MM

K. Elsayed, C. Lacor / Chemical Engineering Science 65 (2010) 6048–60586050

via the following equation (Hoffmann and Stein, 2008):

x50 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi18mð0:9Q Þ

2pðrp�rÞv2ycsðHt�SÞ

sð4Þ

where m is the gas dynamic viscosity, rp is the particle density, Ht

is the cyclone total height, and S is the vortex finder length.

Table 2The values of the independent variables.

Variables Minimum Center Maximum

Inlet height, a/D¼X 1 0.4 0.55 0.7

Inlet width, b/D¼X 2 0.14 0.27 0.4

Vortex finder diameter, Dx/D¼X 3 0.2 0.475 0.75

Total cyclone height, Ht/D¼X 4 3.0 5.0 7.0

Cylinder height, h/D¼X 5 1.0 1.5 2.0

Vortex finder length, S/D¼X 6 0.4 1.2 2.0

Cone tip diameter, Bc/D¼X 7 0.2 0.3 0.4

2. Response surface methodology (RSM)

The cyclone separator performance and the flow field areaffected mainly by the cyclone geometry where there are sevengeometrical parameters, viz. inlet section height a and width b,vortex finder diameter Dx and length S, barrel height h, cyclonetotal height Ht and cone tip diameter Bc. All of these parametersare always expressed as a ratio of cyclone diameter D, as shown inFig. 1 and Table 1.

In this study only the effect of geometry was taken into account.But what about the effect of flow rate on the performance.Overcamp and Scarlett (1993) studied the effect of changingReynolds number on the cut-off diameter (Stokes number) andfound that for Reynolds number values beyond 1E4, the effect ofincreasing Reynolds number on the cut-off diameter is very limited.Further more, Karagoz and Avci (2005) studied the effect ofincreasing the Reynolds number on the pressure drop and foundthat beyond Reynolds number of 2E4 any increase in the Reynoldsnumber have nearly no effect on pressure drop. As the Reynoldsnumber for all cases have Reynolds number higher than 2E4. (All thetested cyclones have the same flow rate.) So the effect of flow ratecan be safely neglected. As this optimization study does not includechanging the cyclone diameter or the number of cyclones to get theoptimum cyclone diameter. So no need to add the capital cost of thecyclone separator as a design parameter in this study (Swamee andAggarwal, 2009). Consequently, only the effect of changing thegeometrical parameters on the performance will be considered.

The usual method of optimizing any experimental set-up is toadjust one parameter at a time, keeping all others constant, untilthe optimum working conditions are found. Adjusting oneparameter at a time is necessarily time consuming, and may notreveal all interactions between the parameters. In order to fullydescribe the response and interactions of any complex system amultivariate parametric study must be conducted (Cowpe et al.,2007). As there are seven geometrical parameters to be investi-gated, the best technique is to perform this study via the responsesurface methodology (RSM).

RSM is a powerful statistical analysis technique which is wellsuited to modeling complex multivariate processes, in applica-tions where a response is influenced by several variables and theobjective is to optimize this response. Box and Wilson (1951) firstintroduced the theory of RSM in 1951, and RSM today is the mostcommonly used method of process optimization. Using RSM onemay model and predict the effect of individual experimentalparameters on a defined response output, as well as locating anyinteractions between the experimental parameters which other-wise may have been overlooked. RSM has been employedextensively in the field of engineering and manufacturing, wheremany parameters are involved in the process (Haaland, 1989;Myers and Montgomery, 2002; Liyana-Pathirana and Shahidi,2005).

In order to conduct a RSM analysis, one must first design theexperiment, identify the experimental parameters to adjust, anddefine the process response to be optimized. Once the experimenthas been conducted and the recorded data tabulated, RSM analysissoftware models the data and attempts to fit second-orderpolynomial to this data (Cowpe et al., 2007). The generalized

second-order polynomial model used in the response surfaceanalysis was as follows:

Y ¼ b0þX7

i ¼ 1

biXiþX7

i ¼ 1

biiX2i þ

XXio j

bijXiXj ð5Þ

where b0, bi, bii, and bij are the regression coefficients for intercept,linear, quadratic and interaction terms, respectively. While Xi and Xj

are the independent variables, and Y is the response variable (Eulernumber).

2.1. Design of experiment (DOE)

The statistical analysis is performed through three main steps.Firstly, construct a table of runs with combination of values of theindependent variables via the commercial statistical softwareSTATGRAPHICS centurion XV by giving the minimum andmaximum values of the seven geometrical factors under inves-tigation as input. Secondly, perform the runs by estimating thepressure drop (Euler number) using MM model. Thirdly, fill in thevalues of pressure drop in the STATGRAPHICS worksheet andobtain the response surface equation with main effect plot,interaction plots, Pareto chart and response surface plots besidethe optimum settings for the new cyclone design.

Table 2 depicts the parameters ranges selected for the sevengeometrical parameters. The study was planned using Box–Behnken design, with 64 combinations. A significant level ofPo0:05 (95% confidence) was used in all tests. Analysis ofvariance (ANOVA) was followed by an F-test of the individualfactors and interactions.

2.2. Fitting the model

Analysis of variance (ANOVA) showed that the resultantquadric polynomial models adequately represented the experi-mental data with the coefficient of multiple determination R2

being 0.92848. This indicates that the quadric polynomial modelobtained was adequate to describe the influence of the indepen-dent variables studied (Yuan et al., 2008). Analysis of variance(ANOVA) was used to evaluate the significance of the coefficientsof the quadric polynomial models (see Table 3). For any of theterms in the models, a large F-value (small P-value) wouldindicate a more significant effect on the respective responsevariables.

Based on the ANOVA results presented in Table 3, the variablewith the largest effect on the pressure drop (Euler number) wasthe linear term of vortex finder diameter, followed by the linearterm of inlet width and inlet height ðPo0:05Þ; the other fourlinear terms (barrel height, vortex finder length, cyclone totalheight and cone tip diameter) did not show a significant effectðP40:05Þ. The quadric term of vortex finder diameter also had asignificant effect ðPo0:05Þ on the pressure drop; however, theeffect of the other six quadric terms was insignificant ðP40:05Þ.

Table 3Analysis of variance of the regression coefficients of the fitted quadratic equation.a

Variable Regression coefficient F-ratio P-value

b0 �43.0742

Linear

b1 178.176 8.11 0.0075

b2 372.26 19.79 0.0001

b3 �161.452 232.04 0.0000

b4 �1.55344 0.6 0.446

b5 8.5875 0 0.9691

b6 �7.23112 0.1 0.757

b7 19.5663 0 0.9537

Quadric

b11 1.08238 0 0.9931

b22 �12.2111 0 0.9446

b33 403.419 107.8 0.0000

b44 �0.223597 0.09 0.7641

b55 �2.67108 0.05 0.8223

b66 1.81257 0.15 0.6994

b77 �62.1739 0.04 0.8364

Interaction

b12 91.0488 0.22 0.6427

b13 �355.892 14.75 0.0005

b14 0.459314 0 0.9726

b15 �3.27883 0 0.9514

b16 2.19997 0 0.9465

b17 26.2787 0.01 0.9191

b23 �720.685 42.42 0.0000

b24 1.03571 0 0.9467

b25 �2.53478 0 0.9675

b26 4.2616 0.01 0.9112

b27 �5.28466 0 0.9862

b34 5.2034 0.51 0.4799

b35 2.77536 0.01 0.9249

b36 0.985086 0 0.9568

b37 32.579 0.05 0.8221

b45 �0.0452174 0 0.9911

b46 0.345301 0.02 0.8902

b47 �1.5016 0.01 0.9404

b56 �0.422227 0 0.9667

b57 3.82354 0 0.9622

b67 �6.40134 0.02 0.8945

R2 0.92848

a Bold numbers indicate significant factors as identified by the analysis of

variance (ANOVA) at the 95% confidence level.


Furthermore, the interaction between the inlet dimensions andvortex finder diameters ðPo0:05Þ also had a significant effect onthe pressure drop, while the effect of the remaining terms wasinsignificant ðP40:05Þ.

2.3. Analysis of response surfaces

For visualization of the calculated factor, main effects plot,Pareto chart and response surface plots were drawn. The slope ofthe main effect curve is proportional to the size of the effect andthe direction of the curve specifies a positive or negative influenceof the effect (Gfrerer and Lankmayr, 2005) (Fig. 2(a)). Based on themain effect plot, the most significant factor on the Euler numberare (1) the vortex finder diameter, with a second-order curve witha wide range of inverse relation and a narrow range of directrelation, (2) direct relation with inlet dimensions, (3) inverserelation with cyclone total height and insignificant effects for theother factors.

Pareto charts were used to graphically summarize and displaythe relative importance of each parameter with respect to the Euler

number. The Pareto chart shows all the linear and second-ordereffects of the parameters within the model and estimate thesignificance of each with respect to maximizing the Euler numberresponse. A Pareto chart displays a frequency histogram with thelength of each bar proportional to each estimated standardizedeffect (Cowpe et al., 2007). The vertical line on the Pareto chartsjudges whether each effect is statistically significant within thegenerated response surface model; bars that extend beyond this linerepresent effects that are statistically significant at a 95% confidencelevel. Based on the Pareto chart (Fig. 2(b)) and ANOVA table (Table 3)there are four significant parameters (six terms in the ANOVA table)at a 95% confidence level: the negative linear vortex finder diameter;the linear inlet width; the linear total cyclone height; a second-ordervortex finder diameter; negative interaction between vortex finderdiameter and inlet dimensions. These are the major terms in apolynomial fit to the data. Therefore, the pareto chart is a perfectsupplementation to the main effects plot.

To visualize the effect of the independent variables on thedependent ones, surface response of the quadric polynomialmodels were generated by varying two of the independentvariables within the experimental range while holding the otherfactors at their central values (Yuan et al., 2008). Thus, Fig. 2(c)was generated by varying the inlet height and the inlet widthwhile holding the other five factor. The trend of the curve is linear,with more significant effect for inlet width, with no interactionbetween the inlet height and width. The response surface plotsgiven by Figs. 2(d)–(f) show that there are interactions betweenboth inlet width and inlet height with the vortex finder diameter.The effect of cyclone total height is less significant with respect tothe vortex finder diameter, but its effect is higher than that of thevortex finder length, the barrel height and the cone tip diameter.

2.4. Optimization (downhill simplex method)

The Nelder–Mead method, also known as downhill simplex

method is a commonly used nonlinear optimization technique. Thetechnique was proposed by Nelder and Mead (1965) and is atechnique for minimizing an objective function in a many-dimensional space (Wikipedia). It requires only function evalua-tions, and no calculation of derivatives (Press et al., 1992). Thedownhill simplex nonlinear optimization technique has been usedby many researchers (e.g. Bernon et al., 2001; Amoura et al.,2010). According to Bernon et al. (2001), Powell’s algorithm andthe downhill simplex one are ones of the most used minimizationalgorithms; the downhill-simplex algorithm became the mostperformant. Further more, the STATGRAPHICS XV package hasbeen used for design of experiment and optimization, with theonly available technique is downhill simplex. In this study, thetarget is to obtain the global optimum values. Consequently, nolinear constrains applied.

The idea of downhill simplex method is to employ a movingsimplex in the design space to surround the optimal point andthen shrink the simplex until its dimensions reach a specifiederror tolerance (Kiusalaas, 2010). In n-dimensional space asimplex is a figure of n+1 vertices connected by straight linesand bounded by polygonal faces. If n¼2, a simplex is a triangle; ifn¼3, it is a tetrahedron.

The allowed moves of the simplex are illustrated in Fig. 3 forn¼2 as an example. By applying these moves in a suitablesequence, the simplex can always hunt down the minimum point,enclose it and then shrink around it. The direction of a move isdetermined by the values of Y (the objective function to beminimized) at the vertices. The vertex with the highest value of Y

is labeled high and low denotes the vertex with the lowest value.The magnitude of a move is controlled by the distance d measured

Fig. 2. Analysis of design of experiment: (a) main effect plot, (b) Pareto chart, (c) response surface plot (X 1 versus X 2), (d) response surface plot (X 1 versus X 3), (e)

response surface plot (X 2 versus X 3) and (f) response surface plot (X 3 versus X 4).



from the high vertex to the centroid of the opposing face (in thecase of the triangle, the middle of the opposing side) (Kiusalaas,2010).

Kiusalaas (2010) stated the outline of the algorithm as:

�

hig

lo

F

TabOpt

Fa

X

X

X

X

X

X

X

TabThe

C

St

N

Choose a starting simplex.
� Cycle until dre (e being the error tolerance) � Try reflection.
3 If the new vertex is lower than previous high, acceptreflection.

3 If the new vertex is lower than previous low, try expansion.3 If the new vertex is lower than previous low, accept

expansion.3 If reflection is accepted, start next cycle.

h

w

h

l

ig. 3

le 4imiz

cto

1

2

3

4

5

6

7

le 5val

yclo

airm

ew

�

Try contraction.3 If the new vertex is lower than high, accept contraction and

start next cycle.
� Shrinkage. � End cycle
Table 4 gives the optimum values for cyclone geometricalparameters for minimum pressure drop estimated by MM using

high

low

high

low

Expansion

Re–ectio

n

igh

ow

Contraction

Shrink

age

. Basic operations in the downhill simplex method for two dimensions.

e response for minimum pressure drop.

r Low High Optimum

0.5 0.75 0.618025

0.14 0.4 0.23631

0.2 0.75 0.621881

3 7 4.23618

1.618 1.618 1.618

0.4 2 0.620421

0.2 0.4 0.381905

ues of geometrical parameters for the two designs (D¼0.205 m).

ne a/D b/D Dx/D Ht/D

and design 0.5 0.2 0.5 4

design 0.618 0.236 0.618 4.236

the downhill simplex optimization technique available in STAT-GRAPHICS XV software.

3. Comparison between the two designs using CFD

3.1. Numerical settings

For the turbulent flow in cyclones, the key to the success ofCFD lies with the accurate description of the turbulent behavior ofthe flow (Griffiths and Boysan, 1996). To model the swirlingturbulent flow in a cyclone separator, there are a number ofturbulence models available in FLUENT. These range from thestandard k�e model to the more complicated Reynolds stressmodel (RSM). Also Large eddy simulation (LES) is available as analternative to the Reynolds averaged Navier–Stokes approach. Thestandard k�e, RNG k�e and Realizable k�e models were notoptimized for strongly swirling flows found in cyclones (Chuahet al., 2006). The Reynolds stress turbulence model (RSM) requiresthe solution of transport equations for each of the Reynolds stresscomponents. It yields an accurate prediction on swirl flowpattern, axial velocity, tangential velocity and pressure drop incyclone simulations (Slack et al., 2000).

The air volume flow rate Qin¼0.08 m3/s for the two cyclones(inlet velocity for Stairmand design is 19 and 13.1 m/s for the newdesign), air density 1.0 kg/m3 and dynamic viscosity of 2.11E�5Pa s. The turbulent intensity equals 5% and characteristic lengthequals 0.07 times the inlet width (Hoekstra et al., 1999). Velocityinlet boundary condition is applied at inlet, outflow at gas outletand wall boundary condition at all other boundaries.

The finite volume method has been used to discretize thepartial differential equations of the model using the SIMPLEC(semi-implicit method for pressure-linked equations-consistent)method for pressure velocity coupling and QUICK scheme tointerpolate the variables on the surface of the control volume. Theimplicit coupled solution algorithm was selected. The unsteadyReynolds stress turbulence model (RSM) was used in this studywith a time step of 0.0001 s. The residence time (cyclone volume/gas volume flow rate) of the two cyclones are close ð � 0:25 sÞ.

Different grades of grid refinement were tested, with the firstgrid point located in various regions of the boundary layer. Thegrid refinement study shows that a total number of about 134 759hexahedral cells for Stairmand cyclone and 154 746 hexahedralcells for the new design are sufficient to obtain a grid-independent solution, and further mesh refinement yields onlysmall, insignificant changes in the numerical solution. Thesesimulations were performed on an 8 nodes CPU Opteron 64 Linuxcluster using FLUENT commercial software. The geometricalvalues are given in Table 5 for the two cyclones (cf. Fig. 1)

4. Results and discussion

4.1. Validation of results

In order to validate the obtained results, it is necessary tocompare the prediction with experimental data. The comparisonperformed with the measurements of Hoekstra (2000) of the

h/D S/D Bc/D Li/D Le/D

1.5 0.5 0.36 1.0 0.618

1.618 0.618 0.382 1.0 1.618

Fig. 4. Comparison of the time-averaged tangential and axial velocity between the LDA measurements, Hoekstra (2000) and the current Reynolds stress model (RSM)

results at section S6. From left to right: tangential velocity and axial velocity, Dx/D¼0.5. (The radial position is divided by the cyclone radius and the axial and tangential

velocity are divided by the inlet velocity.)


Stairmand cyclone using Laser Doppler Anomemetry (LDA). Thepresent simulation are compared with the measured axial andtangential velocity profiles at an axial station located at 94.25 cmfrom the cyclone bottom (Dx/D¼0.5), Fig. 4. The RSM simulationmatches the experimental velocity profile with underestimationof the maximum tangential velocity, and overestimation of theaxial velocity at the central region. Considering the complexity ofthe turbulent swirling flow in the cyclones, the agreementbetween the simulations and measurements is considered to bequite acceptable.

4.2. Flow field pattern

4.2.1. The pressure field

Fig. 5 shows the contour plot at Y¼0 and at section S7 (at themiddle of inlet section, Table 6). In the two cyclones the time-averaged static pressure decreases radially from wall to center. Anegative pressure zone appears in the forced vortex region(central region) due to high swirling velocity. The pressuregradient is largest along the radial direction, while the gradientin axial direction is very limited. The cyclonic flow is notsymmetrical as is clear from the shape of the low pressure zoneat the cyclone center (twisted cylinder). However, the twocyclones have almost the same flow pattern, but the highestpressure of the Stairmand design is nearly twice that of the newdesign, implying that the new design has a lower pressure drop.

The pressure distribution presented in Figs. 6 and 7 of the twocyclones at sections S1–S6 depict the two parts pressure profile(for Rankine vortex). Again the highest static pressure forStairmand design is more than twice that of the new design atall sections while the central value is almost the same for the twocyclones irrespective to the section location. This indicates that,the new design has a lower pressure drop with respect to theStairmand design.

4.2.2. The velocity field

Based on the contour plots of the time-averaged tangentialvelocity, Fig. 5, and the radial profiles at sections S1–S6 shown inFigs. 6 and 7, the following comments can be drawn. The tangentialvelocity profile at any section is composed of two regions, an innerand an outer one. In the inner region the flow rotates approximatelylike a solid body (forced vortex), where the tangential velocityincreases with radius. After reaching its peak the velocity decreases

with radius in the outer part of the profile (free vortex). This profileis a so-called Rankine type vortex as mentioned before, including aquasi-forced vortex in the central region and a quasi-free vortex inthe outer region. The maximum tangential velocity may reach twicethe average inlet velocity and occurs in the annular cylindrical part.The tangential velocity distribution for the two cyclones are nearlyidentical in pattern and values (dimensionless), with the highestvelocity occurring at 1/4 of the cyclone radius for both cyclones. Thisimplies a nearly equal collection efficiency for both cyclones, as thecentrifugal force is the main driving force for particle collection inthe cyclone separator. The axial velocity profiles for the two cyclonesare also very close, exhibiting a M letter shape (also known asinverted W axial velocity profile in some other literatures (cf.Horvath et al., 2008)). Part of the flow in the central region movesdownward in the two cyclones. This phenomena has been shown inthe axial velocity pattern in other published articles (e.g. Slack et al.,2000; Horvath et al., 2008).

Physical interpretation of the inverted W axial velocity profile:The swirling motion of the gas generates a strong radial pressuregradient, the pressure being low in the center of the vortex andhigh at the periphery (Figs. 6 and 7). As the strongly swirling gasenters the confines of the vortex finder on its way out of thecyclone, the swirl is attenuated through friction with the wall.This means that further up the vortex finder the pressure in thecenter is higher than at the exit of the separation space: a reversepressure gradient is present (Hoffmann et al., 1996). This drivesan axial flow with dip in the center of the vortex finder (invertedW profile); this core flow prevails throughout the entireseparation space of the cyclone in spite of the attenuation ofswirl in the conical part of the cyclone (Hoffmann et al., 1996).

4.3. Discrete phase modeling (DPM)

The Lagrangian discrete phase model in FLUENT follows theEuler–Lagrange approach. The fluid phase is treated as acontinuum by solving the time-averaged Navier–Stokes equa-tions, while the dispersed phase is solved by tracking a largenumber of particles through the calculated flow field. Thedispersed phase can exchange momentum, mass, and energywith the fluid phase.

A fundamental assumption made in this model is that thedispersed second phase occupies a low volume fraction (usuallyless than 10–12%, where the volume fraction is the ratio betweenthe total volume of particles and the volume of fluid domain),

Fig. 5. The contour plots for the time-averaged flow variables at sections Y¼0 and S7. From top to bottom: static pressure (N/m2), tangential velocity (m/s) and axial

velocity (m/s). From left to right: Stairmand design and new design, respectively.



even though high mass loading is acceptable. The particletrajectories are computed individually at specified intervalsduring the fluid phase calculation. This makes the modelappropriate for the modeling of particle-laden flows. The particleloading in a cyclone separator is small (3–5%), and therefore, itcan be safely assumed that the presence of the particles does notaffect the flow field (one-way coupling).

In terms of the Eulerian–Lagrangian approach (one-way coupling), the equation of particle motion is given by

Table 6The position of different sections.a

Section S1 S2 S3 S4 S5 S6 S7

zu=Db 2.75 2.5 2.25 2.0 1.75 1.5 0.25

a Sections S1–S5 are located in the conical section, section S6 at the cylindrical

part and S7 located through the inlet section.b z’ measured from the inlet section top.

Distance from center (m)

Stat

ic p

ress

ure

(mba

r)

-0.1 -0.05 0 0.05 0.1-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20

22.5New designStairmand design

Distance f

Tan

gent

ial v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05-0.5

0

0.5

1

1.5

2

2.5


Stat

ic p

ress

ure

(mba

r)

-0.1 -0.05 0 0.05 0.1-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20


Distance f

Tan

gent

ial v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05-0.5

0

0.5

1

1.5

2

2.5


Stat

ic p

ress

ure

(mba

r)

-0.1 -0.05 0 0.05 0.1-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20


Distance f

Tan

gent

ial v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05-0.5

0

0.5

1

1.5

2

2.5

Fig. 6. The radial profile for the time-averaged tangential and axial velocity at different

S1–S3. From left to right: time-averaged static pressure, tangential velocity and axial v

(Zhao et al., 2006)

dupi

dt¼ FDðui�upiÞþ

giðrp�rÞrp

ð6Þ

dupi

dt¼ upi ð7Þ

where the term FD(ui�upi) is the drag force per unit particle mass(Zhao et al., 2006):

FD ¼18mrpd2

p

CDRep

24ð8Þ

Rep ¼rpdpju�upj

m ð9Þ

In FLUENT, the drag coefficient for spherical particles iscalculated by using the correlations developed by Morsi andAlexander (1972) as a function of the relative Reynolds numbers

rom center (m)

0 0.05 0.1

New designStairmand design

Distance from center(m)

Axi

al v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

-0.25

0

0.25

0.5


rom center (m)

0 0.05 0.1



Axi

al v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

-0.25

0

0.25

0.5


rom center (m)

0 0.05 0.1



Axi

al v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

-0.25

0

0.25

0.5


sections on the X–Z plane (Y¼0) at sections S1–S3 . From top to bottom: sections

elocity, respectively.


Stat

ic p

ress

ure

(mba

r)

-0.1 -0.05 0 0.05 0.1-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20



Tan

gent

ial v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5

1

1.5

2



Axi

al v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

-0.25

0

0.25

0.5



Stat

ic p

ress

ure

(mba

r)

-0.1 -0.05 0 0.05 0.1-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20



Tan

gent

ial v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5

1

1.5

2



Axi

al v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

-0.25

0

0.25

0.5



Stat

ic p

ress

ure

(mba

r)

-0.1 -0.05 0 0.05 0.1-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20



Tan

gent

ial v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5

1

1.5

2



Axi

al v

eloc

ity/I

nlet

vel

ocity

-0.1 -0.05 0 0.05 0.1-0.5

-0.25

0

0.25

0.5


Fig. 7. The radial profile for the time-averaged tangential and axial velocity at different sections on the X–Z plane (Y¼0) at sections S4–S6 . From top to bottom: sections

S4–S6. From left to right: time-averaged static pressure, tangential velocity and axial velocity, respectively.

Table 7

The Euler number Eu, pressure drop Dp and the cut-off diameter (x50) for the two

cyclones.

Method Eu (dimensionless) Dp (N/m2) X50 ðmmÞ

Stairmand design MM 5.79 1045 1.54

CFD 6.592 1190 1.0


Rep. The equation of motion for particles was integrated along thetrajectory of an individual particle. Collection efficiency statisticswere obtained by releasing a specified number of mono-dispersedparticles at the inlet of the cyclone and by monitoring the numberescaping through the outlet. Collisions between particles and thewalls of the cyclone were assumed to be perfectly elastic(coefficient of restitution is equal to 1).

New design MM 5.24 450 1.77

CFD 5.672 487 1.6

4.3.1. The DPM results

In order to calculate the cut-off diameters of the two cyclones,5880 particles were injected from the inlet surface with zerovelocity and a particles mass flow rate _mp of 0.001 kg/s(corresponding to inlet dust concentration Cinð _mp=Q Þ ¼

11:891 gm=m3). The particle density rp is 860 kg/m3 and themaximum number of time steps for each injection was 200 000steps. The DPM analysis results and the pressure drops for the twocyclones are depicted in Table 7. An acceptable agreement

between the CFD results and the MM mathematical model hasbeen obtained. While the difference between the two cyclone cut-off diameters is small, the saving in pressure drop is considerable(nearly half the value of Stairmand cyclone).

Based on the flow pattern analysis and the DPM results, onecan conclude that the cyclone collection efficiency for the two


cyclones should be very close, with the advantage of low pressuredrop in the new design. The authors want to emphasis that onlysmall changes in the geometrical dimensions of the two designslead to this improvement in the performance.

5. Conclusions

Mathematical modeling (the Muschelknautz method of mod-eling (MM)) and CFD investigation have been used to understandthe effect of the cyclone geometrical parameters on the cycloneperformance and a new optimized cyclone geometrical ratiosbased on MM model has been obtained.

The most significant geometrical parameters are: (1) thevortex finder diameter, (2) the inlet section width, (3) the inletsection height and (4) the cyclone total height. There are stronginteraction between the effect of inlet dimensions and the vortexfinder diameter on the cyclone performance. The new cyclonedesign are very close to the Stairmand high efficiency design inthe geometrical parameter ratio, and superior for low pressuredrop at nearly the same cut-off diameter. The new cyclone designresults in nearly one-half the pressure drop obtained by the oldStairmand design at the same volume flow rate.

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http://en.wikipedia.org/wiki/NelderMead_method

http://en.wikipedia.org/wiki/NelderMead_method

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Optimization of the cyclone separator geometry for minimum pressure drop using mathematical models and CFD simulations