Controlling Characteristics of Hydrocyclone via Additional Water ...

ISSN 00405795, Theoretical Foundations of Chemical Engineering, 2012, Vol. 46, No. 3, pp. 296–306. © Pleiades Publishing, Ltd., 2012. Original Russian Text © J. Dueck, A.V. Krokhina, L.L. Minkov, 2012, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2012, Vol. 46, No. 3, pp. 342–352.

Controlling Characteristics of Hydrocyclone via Additional Water Injection J. Duecka*, A. V. Krokhinab, and L. L. Minkovc a

b

Universitát ErlangenNürnberg, Schlossplatz 4, Erlangen, 91054 Germany Bauman Moscow State Technical University, ul. Vtoraya Baumanskaya 5, Moscow, 105005 Russia c Tomsk State University, pr. Lenina 36, Tomsk, 634050 Russia * email: [email protected]erlangen.de Received September 6, 2010; in final form, November 17, 2010

Abstract—The results of measuring the parameters of the classification of suspended solid particles by their sizes in a hydrocyclone equipped with a system for additional water injection are considered. The additional injection of water considerably changes the characteristics of the hydrocyclone. A mathematical model that predicts the trend in the change of hydrocyclone characteristics upon the injection of water is described. The mechanisms of the action of radial and tangential injection on classification have been explained. An efficient regime of injection is recommended to attain the required classification characteristics. DOI: 10.1134/S0040579512030037

INTRODUCTION Despite the widespread application of hydrocy clones in the technologies for the treatment of waste water and soils, for the separation of suspensions into phases, and for the fractional classification of solid particles and granular materials by their sizes in liquid flows [1–4], and the considerable progress in the opti mization of hydrocyclone design, attempts at improv ing the parameters of separation and classification are still made [5–10]. First of all, this concerns the attempts to decrease the concentration of fine parti cles entrained with the coarse product. One method of improving the operational charac teristics of a hydrocyclone is the additional injection of water into the apparatus. This method has been known for a rather long time [11–16], but there are no sys tematic studies of the effect of the injection parame ters on the characteristics of the classification of a sus pension, a problem that the present work seeks to alle viate. A laboratory hydrocyclone with an injector is sche matically shown in Fig. 1. It is supposed that injected water jets must impart an additional radial velocity to solid particles in the direction from the wall of the appa ratus to its axis. Fine particles caught by the ascending central vortex will be entrained to the overflow pipe and, thus, will not contaminate the condensed coarse grained product in the underflow pipe. In the present study, the location of the injector in the hydrocyclone was optimized in compliance with the recommendations given in [16]. For the practical application of the additional injection of water into the

apparatus in order to control the process of classifica tion, it is necessary to know the influence of injection parameters. To accomplish this, in this work, we per formed the integrated theoretical and experimental study, which allowed us to reveal the main regularities of the process. EXPERIMENTAL SETUP In experiments, we used a cylindroconical hydro cyclone of 50 × 10–3 m in diameter with a builtin water injection system at the bottom of its conical part. The diameter of the hydrocyclone’s overflow and underflow nozzles is 14 × 10–3 and 7.2 × 10–3 m, respectively. The experimental setup (Fig. 1a) includes hydrocy clone 1 with builtin injector 2 placed over tank 3 filled with a suspension. The suspension is fed into the hydrocyclone with a centrifugal pump. The hydrocy clone’s inlet pressure is adjusted with valve 4 and mea sured with pressure gauge 5 placed at the hydrocy clone’s inlet. The injected water flow rate is measured with digital flowrate gauge 6. Water injector 2 consists of an external ring with an internal ring inside of it and has two strictly antipodal holes 2 × 10–3 m in diameter. An adjusting valve sup plies the injector with water, which fills the latter and enters the hydrocyclone through the holes. The injec tor is installed at the bottom of the conical part of the cyclone. A set of rings allows water to be injected in the radial direction orthogonally to the main flow inside

296

CONTROLLING CHARACTERISTICS OF HYDROCYCLONE (а)

297 (b)

5 Tangential injector

Clarified product out Suspension in

1 Injected water in

4

6 2

Injector

Radial injector Condensed product out 3

Pump Fig. 1. Scheme of (a) an experimental setup and (b) injectors: (1) hydrocyclone, (2) injector; (3) tank filled with a suspension, (4) pressureadjusting valves, (5) pressure gauge, and (6) digital flow rate gauge.

the apparatus and in the tangential direction concur rently with the main flow of a suspension in the hydro cyclone. Feldspar (particle size distribution, d10 = 3.1 × 10–6 m, d50 = 12.5 × 10–6 m, d90 = 31.2 × 10–6 m; den sity, 2600 kg/m3) was used as a material for preparing the suspension, and the suspension concentration of 50 kg/m3 was chosen for tests. The size classification of a material is characterized by the socalled separation function T (di ), which determines the proportion of the particle fraction that is withdrawn through the underflow pipe and has the size d i . In the general form, it is expressed as [3]

T (d j ) =

m uf µuf , j , m f µ f , j

(1)

where m uf and m f are the mass flow rates of a solid sub stance through the underflow pipe and at the inlet of the hydrocyclone, respectively, and µ uf , j and µ f , j are the volumetric concentration of the jth fraction in the mass flows of the solid phase through the underflow pipe and at the inlet of the apparatus, respectively. The mass flows of the solid phase were determined from the measured density of the suspension, and the fractional composition of particles was measured opti cally on a Malvern MastersizerX diffractor. Using

these measurements as the base, we calculated the sep aration function by Eq. (1). MODEL The purpose of the mathematical model suggested below is to demonstrate the mechanism of the action of injection on the characteristics of classification and the trend in the change of these characteristics as func tions of the intensity of injection. The operation of hydrocyclones is based on the separation of solidphase particles in a swirling liquid flow. The velocity of the separation of a particle in the centrifugal field of a hydrocyclone may be several hun dred times higher than the velocity of the sedimenta tion of equivalent particles in a gravity field. The flow in a hydrocyclone is swirling and highly turbulent. Particles are separated mainly in the cylin drical part of a hydrocyclone at a certain distance from the inlet pipe and in its conical part. Here, the steady state concentration distribution of solid material frac tions settles predominantly towards the wall. By neglecting the distribution of hydrodynamic characteristics in the working zone of a hydrocyclone (Schubert–Neesse model [2, 3]), we can calculate the concentration distribution of particles of different fractions from the transport equation.

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The framework of this model is determined by the following assumptions. (1) The sedimentation of particles onto the wall occurs under the action of a centrifugal force in a swirling flow. The sedimentation velocity Vs, j for each jthsized fraction of particles with a diameter d j depends only on the size of the particle (according to the Stokes law). (2) The diffusion flow of particles in a turbulent field is characterized by a constant turbulent diffusion coefficient Dt . The transport equation, which determines the local concentration c j of the fraction of particles with a diameter d j in the apparatus, is written as ∂c ∂ ⎡ V +V (2) ( s, j inj ,r ) c j − Dt j ⎤⎥ = 0, ⎢ ∂r ⎣ ∂r ⎦ where Vinj,r is the radial component of the velocity of an injected jet. At the edge of an injector, this velocity may be expressed via the injection flow rate Qinj and the overall area of injected jets S inj as Vinj,e = Qinj S inj . The boundary conditions on the wall and axis of the apparatus are ∂c (V s, j + Vinj ,r ) c j − Dt j = 0 (3) ∂y for r = Dc 2 and r = 0, where Dc is the diameter of the hydrocyclone. The solution of Eq. (1) with conditions (2) can be written as r

ln

c j (r ) V s, j = r + 1 Vinj,r (s)ds. c j (0) Dt Dt

∫

(4)

0

The right side of Eq. (4) may change its sign depending on the intensity of the injection. On the inlet segment of an injected jet, its velocity is originally constant and then drops down to zero, as the jet approaches to the axis of the apparatus. Assum ing that n

⎛ ⎞ Vinj,r = −Vinj,e ⎜ 2r ⎟ , ⎝ Dc ⎠ from Eq. (4), we obtain

ln

c j (r ) V s, j V Dc ⎛ 2r ⎞ r − inj,e = c j (0) Dt Dt 2(n + 1) ⎜⎝ Dc ⎟⎠

(5)

n +1

.

(6)

In the general case, the radial distribution of con centrations will not be monotonic. If V s, j < Vinj,e the c j (r ) function will have a maxi 1/ n

⎡V ⎤ mum at the point 2r = ⎢ s, j ⎥ . The finer the parti Dc ⎣Vinj,e ⎦ cles, the longer the distance of this maximum from the wall of the apparatus.

Replacing the mass flows of a solid substance by the flows of the suspension in Eq. (1), e.g., m uf = W uf cuf , and the volumetric concentrations for the concentra tions of particles, e.g., c j,uf = cuf µ ufj , where cuf is the overall concentration of the solid phase, we determine the separation function T(dj) as follows:

T (d j ) =

Wuf cuf , j . Wuf cuf , j + Wof cof , j

(7)

The flow of particles fed into the apparatus is with drawn through the underflow (predominantly, for coarse particles) and overflow (for fine fractions) noz zles. Assuming that the particles from the nearaxis and nearwall zones are entrained through the over flow and underflow nozzles, respectively, we obtain −1

⎛ c ⎞ (8) T (d j ) = ⎜1 + S j,of ⎟ , c j,uf ⎠ ⎝ W where the notation S = of of the socalled split W uf parameter equal to the ratio of the flows of a suspen sion through the nozzles is used. For the separation curve, we obtain

T (d j ) =

1 . Vinj,e ⎞⎤ ⎡ Dc ⎛ 1 + S exp ⎢− ⎜V s (d j ) − ⎟ n + 1⎠⎥⎦ ⎣ 2Dt ⎝

(9)

If the sedimentation velocity V s (d j ) grows with an increase in the diameter of particles d j , as predicted by the Stokes formula, the separation function T (d j ) also grows. The measured separation function at a hydrocy clone inlet pressure of 0.1 MPa and different injection intensities for the cases of tangential and radial injec tion is shown in Fig. 2. The error in the measurement of the separation function at its maximum point does not exceed 5%, which allows us to reliably establish the influence of the injection. It is obvious that the interaction of the particles of different fractions with each other must be taken into account within a region of particle sizes of less than 5 µm, which causes the fine particles to be entrained with coarse particles and precipitated much more quickly [17, 18] than predicted by the Stokes formula. This explains the socalled fishhook effect, i.e., the increase of the separation function with a decrease in the size of particles [19]. Computeraided calculations of the separation function with consideration for this effect can be found in [20, 21]. The separation function and classification are characterized by the following parameters: (1) the diameter d50 of particles, 50% of which enter the underflow pipe (separation grain);

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CONTROLLING CHARACTERISTICS OF HYDROCYCLONE T(d) 1.0 0.8

(а)

T(d) 1.0

1 2 3 4

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

1

10

100 d, 10–6 m

0

299

(b) 1 2 3 4

1

10

100 d, 10–6 m

0

Fig. 2. Separation function T(d) at Q f = 9 × 10–4 m3/s and p = 0.1 MPa (1) without injection and at Qinj = (2) 3.3 × 10–5, (3) 6.6 × 10–5, and (4) 10–4 m3/s for (a) tangential and (b) radial injection.

(2) the separation function minimum Tmin; (3) the separation function of the finest fractions T0; and (4) the fishhook effect, estimated as F = T0 − Tmin. At d j → 0 , we obtain the separation function for the finest particles from Eq. (9) as

T (0) =

1 ⎡ ⎤ Dc Vinj,e ⎥ 1 + S exp ⎢ ⎣2 ( n + 1) Dt ⎦

,

(10)

which grows with increasing injection rate. This means that the model predicts the decrease in the portion of fine particles entrained into the coarse product. Setting the right side of Eq. (9) equal to 0.5, we can estimate the separation grain d 50 from the equation Vinj,e 2Dt + ln S. Denoting the separation V s (d50 ) = n + 1 Dc 0 grain size in the absence of injection as d 50 , we obtain d50 Dc = 1+ Vinj,e. 0 D n + 1) ln S 2 ( d50 t

(11)

Hence, the separation curve shifts towards coarser particles as the injection becomes more intense. Equa tions (10) and (11) reflect the influence of the injec tion rate on the characteristics of the separation curve, which are in qualitative agreement with experimen tally observed regularities and were derived under the assumption of a low suspension concentration and a rather large diameter of the inlet pipe for injected water. These approximate theoretical considerations must be complemented by certain measurements.

MEASUREMENT RESULTS The main separation characteristics of a hydrocy clone with a shutdown injector were experimentally predetermined as basic data. One of the main parameters that determine the separation capacity of hydrocyclones is the separation grain, the size of which depends on the parameters of the apparatus and the characteristics of a suspension [1–3, 22]. For each particular apparatus and suspen sion, the separation grain also depends on the feed pressure of the hydrocyclone. For the hydrocyclone used in our work, the change in the separation grain diameter depending on the inlet pressure of the 0 = 4.39 × 10–6 p–0.27, hydrocyclone is expressed as d 50 which corresponds to conventional dependences 0 d50 ∞ p −0.25 (pressure in MPa, particle size in m) given, e.g., in [1–3]. As shown by our measurements, the separation function for fine fractions grows in compliance with 0 0 the relationships Tmin = 0.6 p 0.37 and T (0) = 1.14 p 0.25 with increasing pressure. The numerical empirical coefficients in the given formulas incorporate all the geometrical characteristics of the hydrocyclone. The classification characteristics of the hydrocy clone in the presence of water injection depend on the design of an injector, which provides the tangential or radial injection of water jets. It can be seen from Fig. 2 that the effect of decreasing the amount of fine frac tions in the bottom product is more pronounced for the tangential injection of additional water than at the same flow rate of water injection, but in the radial direction. This is explained by the character of the influence of additional injection on the hydrodynamic conditions in the apparatus. As shown in [23, 24], in the case of the tangential injection of water, a specific toroidal blocking layer is


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DUECK et al. 0

0

d 50 /d 50 1.8

d 50 /d 50 1.4

(а)

(b)

1.6 1.2

1.4 1.2

1 2 3

1.0

1 2 3

1.0

0.8

0.8 0

0.05

0.10

0.15 0.20 Qinj/Qf, inj

0

0.05

0.10


Fig. 3. Separation grain diameter vs. flow rate of injected water at a hydrocyclone inlet pressure of (1) 0.06, (2) 0.1, and (3) 0.14 MPa for (a) tangential and (b) radial injection.

formed that turns the main flow, which contains fine fractions, up towards the overflow pipe for discharge. The higher the flow rate of injected water, the thicker the blocking layer. Coarse particles, which are located in the nearwall layer, are freely discharged through the underflow pipe. Due to the tangential jets of addi tional injection, the circumferential velocity of the main flow grows, which allows coarse particles to be separated more effectively in a centrifugal field. In the case of radial injection, injected jets are directed orthogonally to the main flow, which leads to its fragmentation and, consequently, the deceleration at the point of injection [25], i.e., a decrease in its tan gential velocity. The higher the flow rate of injected water, the more pronounced the effect of deceleration in the tangential direction. Hence, the velocity of the main flow is reduced at the point of injection, which allows the water of the main flow containing fine frac tions to be withdrawn through the underflow pipe. Note that only some injected water passes through the overflow pipe after it breaks through the main flow and reaches the center, thus partially withdrawing fine par ticles. This is why the relatively weak effect of improv ing the separation curve is observed for radial injection upon an increase in the flow rate of injected water. Let us represent the change in the classification characteristics d 50, T0, Tmin, and F as a function of the injection flow rate Qinj in more detail (Figs. 3–6). As was done in the processing of water flow rates through each inlet pipe of the hydrocyclone [23], we may attempt to describe all of the results that corre spond to different pressures with a unified curve using 0 universal variables as follows: Qinj Q f ,inj , d 50 d 50 , 0 0 0 T0 T0 , Tmin Tmin , and F F . These dimensionless variables, including the main characteristics of the separation curve at zero injection as basic parameters, will probably have a universal character.

The injection of water creates additional hydrody namic resistance for the flow of the suspension through the feed pipe, which leads to some decrease in the inlet flow rate of the hydrocyclone Q f ,inj at a con stant pressure in keeping with the experimentally established relationship

Q f ,inj 0 Qf

= 1 − 0.12

Qing 0

Qf

.

The separation grain diameter as a function of the flow rate of injected water at different inlet pressures of the apparatus is plotted in Fig. 3. These dependences may be described by the following expressions: tan d50

=

⎛

0 ⎜1 + d50

⎜ ⎝

⎛Q ⎞ ⎛Q ⎞ 0.9 ⎜ inj ⎟ + 27.1 ⎜ inj ⎟ ⎝ Q f ,inj ⎠ ⎝ Q f ,inj ⎠

⎞ ⎟, ⎟ ⎠

(12)

⎞ (13) ⎟. ⎜ ⎟ ⎝ ⎠ As can be seen from the plots, the diameter of the separation grain increases with increasing injection flow rate. This growth is more pronounced than that given by model equation (11). It seems likely that the weakness of the assumption that the parameters S and Dt are independent of the injection flow has some effect here. The observed growth is sharper for tangential injec tion than for radial injection. The dependence of T0 on the flow rate of injected water for the cases of tangen tial and radial injection at different inlet pressures of the hydrocyclone is plotted in Fig. 4. According to these plots, the changes in the separa tion function for the finest fractions can be described by the following expressions: rad d50

=

⎛

2

0 d50 ⎜1 +

tan T0

⎛Q ⎞ ⎛Q ⎞ 0.7 ⎜ inj ⎟ + 4.1 ⎜ inj ⎟ ⎝ Q f ,inj ⎠ ⎝ Q f ,inj ⎠

=

0 T0 e

⎛ Q ⎞ −4.26⎜⎜ inj ⎟⎟ ⎝ Q f ,inj ⎠

2

(14)

,


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CONTROLLING CHARACTERISTICS OF HYDROCYCLONE T0/T0(0) 1.0

T0/T0(0) 1.2

(а)

301

(b)

1.0

0.8

0.8

0.6

0.6 1

0.4 0.2 0

1

2

0.4

2

3

0.2

3

0.05

0.10


0

0.05

0.10


Fig. 4. T0 vs. flow rate of injected water at a feed pressure of (1) 0.06, (2) 0.1, and (3) 0.14 MPa for (a) tangential and (b) radial injector.

0

0

T min /T min

T min /T min

(а)

1.2

1

1.0

2

0.8

3

1.0 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.05

0.10

(b)

1.2


1 2 3

0

0.10

0.05


Fig. 5. Tmin vs. the floe rate of injected water at a feed pressure of (1) 0.06, (2) 0.1, and (3) 0.14 MPa for (a) tangential and (b) radial injector.

F/F0 1.2

F/F 0 1.05

(а)

1.0

(b)

1.00

0.8

0.95

0.6 1

0.4 0.2 0

1

0.90

2

2

0.85

3

0.05

0.10


0.80

3

0

0.05

0.10


Fig. 6. Fishhook effect vs. flow rate of injected water at a hydrocyclone inlet pressure of (1) 0.06, (2) 0.1, and (3) 0.14 MPa for (a) tangential and (b) radial injector. THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46

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C/C0 (а)

(b)

1.2

1.2 1.0 1.0 0.8 0.8

1

1

2

2

0.6

0.6 0

0.05

0.10

0.15 Qinj/Qf, inj

0

0.05

0.10

0.15 Qinj/Qf, inj

Fig. 7. Concentration of suspension in (1) underflow and (2) overflow nozzles vs. injection flow rate at a hydrocyclone inlet pres sure of 0.1 MPa for (a) tangential and (b) radial injection.

⎛ Q ⎞ −1.46⎜⎜ inj ⎟⎟ ⎝ Q f ,inj ⎠

(15) = . These dependences also are in qualitative agreement with model equation (10). The dependence of Tmin on the flow rate of injected water for the cases of tangential and radial injection at different inlet pressures of the hydrocyclone is plotted in Fig. 5. According to these plots, the changes in the mini mum of the separation function as a function of the injection flow rate can be described by the following exponential functions: rad T0

0 T0 e

Tmin = Tmine tan

0

Tmin = Tmine rad

0

overflow pipe, through which fine fractions are dis charged, grows for tangential injection and remains nearly constant for radial injection. This is explained by the mechanism of the influence of the injection on the redistribution of mass flows. In the case of tangen tial injections, the amount of fine fractions is decreased in the underflow pipe and, correspondingly, increased in the overflow pipe. In the case of radial injection, the decrease in the concentration in the underflow pipe is explained by the effect of dilution, i.e., by an increase in the amount of water fed via injection [23]. RESULTS AND DISCUSSION

⎛Q ⎞ −6.8⎜⎜ inj ⎟⎟ ⎝ Q f ,inj ⎠

,

⎛Q ⎞ −2.6⎜⎜ inj ⎟⎟ ⎝ Q f ,inj ⎠

,

(16) (17)

which correspond to the exponential character of ana lytical dependence (10). Hence, it follows from these plots that the concen tration of fine particles in the underflow pipe decreases most efficiently for the tangential injection of water jets into the hydrocyclone. As was demonstrated in Fig. 6, injection also favors the weakening of the fishhook effect. The weakening of the fishhook effect is clearly seen for the tangential injection of water and very weakly pronounced for radial injection. Finally, injection has some effect on the concentration of the solid phase in the overflow and underflow nozzles, as graphically shown in Fig. 7. It can be seen from Fig. 7 that the decrease in the concentration of a suspension in the underflow pipe through which the coarse fraction is discharged has the same character for both tangential and radial injec tions. Here, the concentration of a suspension in the

At the same flow rate of injected water, tangential injection proves to be much more efficient than radial injection. It seems to be due to that different methods of injection generate various types of flows. They were partially studied in [23–25]. It is shown in [23, 24] that tangentially injected water increases the tangential velocity of the main flow in the conical part of the apparatus and, correspond ingly, the velocity of the motion of coarse suspension particles towards the walls of a hydrocyclone. At the same time, a certain blocking vortex, which turns the main flow containing fine fractions towards the over flow pipe, is formed at the point of injection. To illustrate the solid phase fields in the injection zone, we performed some calculations using the Flu ent 6.3.26 software package. The modeling of hydro dynamics is detailed in [26, 27]. Here, we represent the results of calculating the concentration distribu tion of monodisperse particles using the model of interpenetrating continua and the Euler description of the interaction of phases in the twodimensional axi ally symmetric approximation [28].


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(b)

303

(c)

1.25e–02 1.12e–02 1.00e–02 8.75e–03 7.50e–03 6.25e–03 5.00e–03 3.75e–03 2.50e–03 1.25e–03 0.00e+00 Fig. 8. Concentration field of 1µm particles (a) without injection and for (b) radial and (c) tangential injection (linear scale).

The volumetric concentration of particles at the inlet of the hydrocyclone was set equal to 0.01. We considered the two compositions of suspensions with 10–6 and 10–5m particles. The flow rate of the injected liquid was 10–4 m3/s at an inlet pressure of 0.1 MPa. The concentrations of identically sized particles withdrawn through the underflow pipe with respect to their flow at the inlet of the apparatus are given in the table. It can be seen from the table that tangential injec tion reduces the portion of fine and coarse particles withdrawn through the underflow pipe by 1.18 and 1.1 times, respectively, compared to the case without injection. Radial injection decreases the separation function of fine and coarse particles by 1.08 and 1.01 times, respectively. Hence, fine particles are more efficiently forced from the underflow pipe into the overflow pipe in the case of tangential injection. The fields of the volumetric concentration of fine10–6m particles in the zone of an injector are shown in Fig. 8. In the case of flow without injection, all particles are uniformly spread over the entire zone with a volumetric concentration of 0.01 except for the nearwall region, where the volumetric concentration is more than 15% higher than its initial value, and the nearaxis region, where the concentration of particles is 15% lower than its initial value. Tangential injection considerably changes this situ ation. A zone free of particles is formed in the vortex region and the nearwall region in the direction from the injection point to the underflow pipe. In the zone over the vortex, the volumetric concentration of parti cles is 15% higher than its initial value. On the axis, the

volumetric concentration is still lower than its initial value of 0.01. It can be seen that there is a maximum of the volumetric concentration of particles in the radial direction (from the axis to the wall) in the conical part downstream from the injector, i.e., the profiles will not be smooth. This conclusion reflects model equation (6). Hence, the fine particles located in the axial zone of the flow are entrained with the ascending flow and withdrawn through the overflow pipe. This circum stance also explains the change of the concentrations in the underflow and overflow nozzles with increasing injectionflow rate. This is why classifications using tangential injection is efficient. Radial injection does not create a zone free of par ticles in the nearwall region, although the concentra tion of particles at the wall is noticeably reduced com pared to the concentration of the delivered suspen sion. In this case, the volumetric concentration of particles in the conical part drops monotonically from the axis to the wall. Portions of particles withdrawn through underflow pipe Param eter

Without injection

Tangential injection

Radial injection

d, m

10–6

10–5

10–6

10–5

10–6

10–5

T(d)

0.185

0.824

0.102

0.749

0.171

0.815


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(b)

(c)

6.20e–01 3.26e–01 1.71e–01 9.01e–02 4.74e–02 2.49e–02 1.31e–02 6.88e–03 3.62e–03 1.90e–03 1.00e–03 Fig. 9. Concentration field of 10µm particles (a) without injection and for (b) radial and (c) tangential injection (logarithmic scale).

The volumetric concentrations of coarse 10–5m particles are shown in Fig. 9. In the absence of injec tion, all of the coarse particles are located near the wall of the hydrocyclone. Radial injection slightly drives the particles towards the axis in the region localized somewhat lower than the point of injection. Tangential injection produces a blocking effect, not only for the liquid, but also for coarse particles, concen trating them over the vortex. Here, the volumetric con centration of particles attains a maximum. Particles are localized near the wall in the conical part and forced towards the axis in the cylindrical part (injector). When the method of radial injection is used, the classifying effect on suspension particles is improved due to the force acting across the main flow [3] in the form of injected water jets. Fine fractions are thus forced from the nearwall layer to the center of the apparatus [25], but, at the same time, a portion of fine particles flows out through the underflow pipe and contaminates the bottom product, as the tangential velocity of the main flow becomes lower near the injector. As a result, the change of the classifying effect in the case of radial injection owing to the overlap of the two above described factors is not as efficient as for tangential injection. The aboveproposed model is based on the idea of forcing fine particles from the nearwall region by an injection jet. As can be seen from calculations, this forcing is much more efficient for tangential injection than for radial injection. In the case of tangential injection, a vortex that additionally withdraws fine particles into the nearaxis region is formed in addi

tion to the mechanism of forcing at the expense of a radially oriented injection jet. This results in a stronger influence of tangential injection compared to radial injection. The mechanism of the tangential injection of addi tional water into the hydrocyclone is optimal for appli cations in industrial technologies, and the flow rate of injected water can be estimated from the abovederived relationships, which are written in terms of universal variables and allow these repaltionships to be used for other hydrocyclones with injectors of different sizes. CONCLUSIONS The additional injection of water into a hydrocy clone has a considerable effect on the characteristics of the separation function of the apparatus. The method of injection (tangential or radial) produces different effects on the quality of the classification. In particular, the increase in the flow rate of injected water in the case of tangential injection leads to growth in the separation grain and to the decrease in the sep aration function minimum and makes the separation curve more monotonical. In the case of radial injec tion, the characteristics of the separation function change less appreciably. Based on an analysis of measurements, we explained the mechanisms of the influence of tangen tial and radial injection on the characteristics of clas sification and proposed parameters for optimizing the injection technique to attain the desired classification parameters.


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NOTATION c—concentration of the fraction of particles with a diameter d, kg/m3; Dc —diameter of the hydrocyclone, m; Dt —turbulent diffusion coefficient, m2/s; d—diameter of a particle, m; d 50—diameter of the particles, 50% of which enter the underflow pipe, m; F = T0 − Tmin —fishhook effect depth; m f —mass flow rate of solid matter, kg/s; n —parameter; p —inlet pressure of the hydrocyclone, MPa; Qinj —flow rate of injected water, m3/s; Q f ,inj —flow rate of a suspension at the inlet of a hydrocyclone with injection, m3/s; r —radial coordinate, m; S —split parameter in the presence of injection; S inj —total area of the injection jets, m2; T (di )—separation function; T0 —separation function for the finest fractions; Tmin—separation function minimum; Vin,r —radial component of the velocity of an injec tion jet, m/s; Vs, j —precipitation velocity for the jthsized frac tion of particles, m/s; Vin,e —velocity of the jet at the edge of an injector, m/s; W —flow rate of the suspension, m3/s; µ j —volumetric concentration of the jth fraction in the mass flow of the solid phase. SUBSCRIPTS AND SUPERSCRIPTS 0—without injection (superscript); f—inlet; inj—injected, with injection; j—fraction number; of—overflow pipe; rad, tan—radial or tangential injection, respec tively; uf—underflow pipe. REFERENCES 1. Ternovskii, I.G. and Kutepov, A.M., Gidrotsikloniro vanie (Hydrocycloning), Moscow: Nauka, 1994. 2. Schubert, H., Heidenreich, E., Liepe, F., and Neesse, T., Mechanische Verfahrenstechnik Leipzig.: Deutscher Verlag für Grundstoffindustrie, 1990. 3. Heiskanen, K., Particle Classification London: Chap man and Hall, 1993.

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19. Dueck, J.G. and Neesse, Th., Zum Verlauf der Trenn kurve des Hydrozyklons im Feinstkornbereich, Auf bereitungstechnik, 2003, vol. 44, no. 7, p. 17. 20. Dueck, J.G., Min’kov, L.L., and Pikushchak, E.V., Modeling of the fishhook Effect in a Classifier, J. Eng. Phys. Thermophys., 2007, vol. 80, no. 1, p. 64. 21. Dueck, J.G., Pikushchak, E.V., and Min’kov, L.L., Modelling of Change of the Classifier Separation Car acteristics by Water Injection into the Apparatus, Ther mophys. Aeromech., 2009, vol. 16, no. 2, p. 247. 22. Bradley, D., The Hydrocyclone, London: Pergamon, 1965. 23. Dueck, J.G., Krokhina, A., Minkov, L.L., and Neesse, T., Hydrodynamics of a Cyclone with Wash Water Injec tion, Proc. 7th World Conf. on Experimental Heat Trans fer, Fluid Mechanics and Thermodynamics, Krakow, 2009, p. 1953. 24. Krokhina, A.V., Dueck, J.G., and Pavlikhin, G.P., Hydrodynamics of Hydrocyclone at Addition Injection

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