Hydrocyclones for processing food liquids

Journal ofFood Engineering 14 (1991) 129-138

Hydrocyclones

for Processing Food Liquids Lev A. Rovinsky

19- 1, Apt. 284, Krasny Kazanets St, Moscow, 111395, USSR (Received 25 August 1990; accepted 16 October 1990)

ABSTRACT A model of the dynamics of the rotating flow of a fluid with suspended particles is presented. The research describes the conditions for particle separation and the dimensions of the separated particles. It further suggests a new design of hydrocyclone, calculation methods for the separation process and design parameters for the apparatus. Calculations are also presented for numerical parameters of hydrocyclones of both types and the latter were compared to evaluate the prospects of hydrocyclone usage for the separation of iarge fat particles in milk.

NOTATION Particle diameter Cross-sectional area :: c.f Centrifugal and resistance forces Gravitational constant g Interplate distance 0, H Pressure loss Stoke’s coefficient k Path length, total path length lL Mass m Empirical parameter ;f, Pressure Minimum pressure (must be > vapour pressure) Pmin PY Inlet pressure Flow rate Q r,R Radial position Rll Hydraulic radius t Time 129 Journal of Food Engineering 0260-8774/91/$03.50 Publishers Ltd. England. Printed in Great Britain

- 0

1991

Elsevier

Science

L. A. Rovinsky

130

V z

Time for movement to length I Flow width Radial velocity Tangential velocity Volume Number of interplate spaces

IT P 8 p,Ap 0 I;

Angle of plate Coefficient loss Dynamic viscosity Residence or transit time Density, density difference Angular velocity Resistance coefficient

t/ U V,

ut

Subscripts 0 1 2 i

m min

Initial values Values at inlet to hydrocyclone Middle of 2nd ring Middle of i-th ring Middle of m-th ring Minimum value

or middle of 1st ring

INTRODUCTION Hydrocyclones for processing food liquids are used on a limited scale in separation operations such as solid particles from wash waters during meat processing or large undissolved particles in starch production. The use of standard hydrocyclones (Fig. 1) for separation of highly dispersed suspensions and food emulsions (e.g. milk) is little known and does not give satisfactory results. Of practical interest is the use of hydrocyclones for separation of impurities and large milk fat particles for their subsequent dispersion (method of separate homogenizing). The analysis of the separation process of two-phase liquid systems in hydrocyclones was normally made on the basis of experimental data and described by empirical formulae. This paper suggests a calculation method for the separation process based on particle movement through a liquid medium when subjected to centrifugal forces.

Hydrocyclones for processing food liquids

Fig. 1.

131

A standard hydrocyclone.

THEORY In the rotating flow of a fluid (Fig. 2) a particle with mass m is influenced by centrifugal force F, = mu f/r, where U, = tangential (circumferential) speed and r = rotation radius, and by the resistance force Ff = kv,, where V, = radial velocity. For a spherical particle of diameter D < O-175 mm, according to Stokes’ law the coefficient is k = 3~r,uD, where p = dynamic viscosity. For a particle moving to the center of rotation v r = - dr/dt. The differential equation of particle movement is as follows: mdu,/dt=mv:/r-kv,

or d2r/dt2+kdr/mdt+vf/r=0

For the common relation is suggested

case of rotation

(1)

of viscous liquids the following

U,r”=const.=c

(2)

where n = empirical constant, and eqn ( 1) becomes d’r/dt”

+ k&-/m

dt +c2r-(2n+‘)=

0

(3)

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Q

Fig. 2.

Liquid path into hydrocyclone

and forces acting on a particle.

The important particular cases of eqn (2) are: (a) y1= - 1: all points have one angular velocity w = U,/T = const., which corresponds to a solid body; (b) n = 1: law of areas - corresponds to a non-viscous liquid without friction losses; (c) - 1 < n < 1: intermediate case for a viscous liquid. Equation (3) has an analytical solution for integer values n( - 1, 0, + 1): in intermediate cases it has a numerical solution. In practical applications for highly dispersed particles, e.g. in centrifugal separators, the term d*r/dt* is disregarded. Then F, = Ff and eqn (3) may be written as b dr/dt + c2r-12rl+1)= 0 where b = 18p/ApD2, Ap = difference in densities of liquid and particle. The solution of the simplified equation for the boundary condition r( 0) = R, where R = initial rotation radius is as follows: 2n+Z= r r.2n+2- (2n + 2)&/b (4) The time necessary for the transition of a particle from initial radius R to some radius r > R is given by:

(5) Equation (5) shows that a decrease in approximation of their density to that considerable increase in the parameter transition of particles along the rotation

particle dimensions (D -0) and of the liquid (Ap-+O) lead to a b and the time necessary for the radius.

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Therefore, the separation of highly dispersed particles in standard hydrocyclones is unrealistic and forces the development of special forms of hydrocyclones. Different methods can be used to reduce the time taken to separate particles from a liquid. By analogy with separators, the path of particle movement in liquid may be reduced by the insertion of a set of conical plates. Figure 3 shows the diagram of a hydrocyclone with fixed plates (external radius R), having vertical openings (channels) at a radius r, with interplate spacing h and angle of their slope a. The flow of liquid is directed tangentially and is uniform all along the height of the plate set. At worst when entering the interplate space the particle will settle down on the inner surface of the upper plate (for a given spacing) and will travel along a path I = h/sin cc in the radial direction in order to be separated with the liquid passing from R to r. Simultaneously the particle is involved in a rotating movement with the flow of fluid. Suppose the flat flow of liquid is spirally spinning in the hydrocyclone as in Fig. 2. retaining a rectangular cross-section with height h, = h/cos a and input width uO (width of an input opening), then the length of the flow path from input to finite radius r, may be estimated in the following way. Let us consider the spiral path of a liquid as a set of concentric rings. Taking radius r, = R -u,/2 for the middle of the first ring; radius rz = R - 3~42 for that of the second ring, etc., the m-th radius of the (last) ring becomes r, = R - uO(2m - 1)/2, where the number of rings m = (R - r)/u,. It is evident that ri may also be written as rm = r + z4,,/2. Radii of intermediate rings ri are expressed in terms of an arithmetic series with sum m c Y;=(Q + r&42, i= 1 whence : i=

r;=(R’-

r;)/2u,,.

1

Since the length of i-th ring is Li= 2nr,, the total length of all rings, i.e. the path of liquid flow, is L=2n Equation stretched

: r;=n(R’-r’)/u,. i=l

(6)

(6) can be verified by imagining that the spiral flow of liquid is lengthwise. Then its volume is V=uu,h,L, but on the other

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Fig. 3.

Diagram of a hydrocyclone with a set of plates.

hand the volume of flow in the hydrocyclone can be determined as a cylinder of volume I/= nh( R2 - Y’), whence L = V/u,h = n( R2 - rz)/uo, which agrees with the value established by eqn (6). Equation (6) was obtained on the supposition that the flow width u is constant and equals u,, = const., i.e. rate of flow v is constant. For those cases when rate v varies with radius r according to eqn (2), eqn (6) needs a more precise definition. The flow section fat any point at radius Yis f= Q/v, where Q is flow rate and from eqn (2): f= Qr”/c. Since f= uh,, the flow width is variable and equals u = Qr"/ch, . In this case the value c can be found from the initial condition zl,,R” = c according to eqn (2). The flowpath stretched lengthwise is a prism with height h, , base width II,,, II = Qr”/ch, and base height L. By approximating the figure at the prism base to a trapezium we find that the prism volume is V= (14 + u,,)h, L/2 and is equal to hydrocyclone volume V= nh ,( R’ - r2), whence L = 2&z,(R7 - r’)/( zq,ch, + Qr’l)or after substituting c: L = 2nR”(K’-

YJ)/u(,(K”+ Y”)

(7)

It is evident that with constant rate of flow (i.e. with n = 0) expression (7) transforms to eqn (6). Let us determine the number of plates 2 + 1 to provide for the separation of particles from liquid. The radial movement of a particle to length I will take time t,, determined according to eqn (5): ~,=[R?!N+?_ (R -f)2n+J]b/(2n

+ 2)~’

(8)

while the residence time of the particle in the hydrocycione is 8 = nh, Z( R2 - r*)/Q. The condition t9> t, must be met to ensure separa-

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tion of the particle, i.e. the required number of interplate spacings is z>bQ[R2”+2-(R-Z)2”+2]

(911

’ nc2h,(2n + 2)(R2 - r2)

Let us also estimate loss of fluid pressure in the hydrocyclone. Three sections may be distinguished in the diagram of fluid movement in Fig. 2: prior to entry to the hydrocyclone (pressure P,, velocity v 1, cross-section f, ), at the start of rotation (P,, v,,fO), and at the outlet from the set of plates (P, v,f). For these sections the Bernoulli equation will read as follows: P,lp+v:/2=P,,lp+v~/2+gH,,

H,=C,v:l2g

PO/p + v $2 = P/p + v */2 + gH,

(10)

where H, = energy losses on entry; H,= losses on passage through the set of plates; and 5, = resistance coefficient. Taking into account that ~,=v,f,,/f,; v=vofo/f=vOuo/u=v,,(R/r)n, eqn (10) will read as follows: (4 - P&/P = &k

+(W’“-

( h/h

121/2 + gH,

(11)

To provide for the operation of the hydrocyclone, its pressure P should exceed Pmin(a value that exceeds vapour pressure at the processing temperature - to avoid boiling) and should not exceed the input pressure P*. Then, the operating conditions of the hydrocyclone may be obtained from eqn ( 11) Q’~~,+IR/~)““~/~~+~~H,~~(P,-P)/P+~~

W)

which limits the allowed energy losses along H,. The calculation of H, is hampered by the fact that the formulae of Shezi and Darcy-Weisbach normally applied for this purpose are valid at v = const., while here v = c/Y”,i.e. speed v is variable. If v is const.: H,= ALU */8gR,,

where the hydraulic radius R, for the rectangular mined as R,=uh,/2(u

(13) flow section is deter-

+h,)

or after substituting u: R,=h,u,t”/2(hR”+u,r”).

(14)

The approximate estimation of energy losses lengthwise may be obtained by replacing the variable section flow rate with its constant

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L. A. Rovinsky

section equivalent. Since the flow width is u = Q/U& taking into account eqn (2) yields u = Qr”/cZ, or after substituting c = u ,R” = QR”/fO yields u=

u()(r/R)”

(15)

Considering that the flow radius varies from R to r the average width for the entire flow becomes U= ( u. + u)/2 or using eqn (15) becomes U =Q[l +(r/R)“]2Zh,, while the average flow rate is equal to U = Q/f=

Q/h,Zi

EXPERIMENTS The laboratory hydrocyclone (Fig. 1) used in experiments had the following operating variables: r,, = 40 mm, r= 6 mm (inner tube), fluid consumption Q= 8 litres/min. At maximum milk input pressure P= 5 x lo5 Pa, the inlet velocity was U, =&(2P/,o)= 22 m/s (at consumption coefficient ,u r = O-7). Having assumed (for the most favourable case) that there are no friction losses (n = l), one obtains c = z,,,r. = 0.88 m2/s. If milk is processed at 50°C it is known that Ap/ ,u = 15 x lo4 s/m2. Suppose the problem is to separate particles with diameter D > 2 pm, i.e. Dmin= 2 X 10 -’ m, then b = 3 X 10’ l/s. By substituting these values into eqn (4), separation time for particles becomes t = 1.5 x lo4 s, which is unreal since the average residence time of a particle in the hydrocyclone 8 is considerably less: 19= V/Q = 3.7 s. To separate particles (condition c < 0) the capacity of the hydrocyclone should not be less than 3.3 m3, which is unrealistic. Consider the possibilities of using the modified hydrocyclone (with plates). For the hydrocyclone with capacity Q = 1.39 x 10V4 m3/s and pressure differential on input and output A P = 3 X 1O5 Pa, the fluid inlet velocity 2/O= 20 m/s is ensured at an inlet section value equal to f.= Q/ z1,,= 69.4 x 10e6 m*. If the dimensions of the hydrocyclone are R = O-15 m and Y= 0.05 m, interplate distance is h = 0.5 X 10 -3 m and their angle of slope is a = 50”, then value h, = 0.778 mm and 1= 0.65 3 mm. From experiments with rotating flow of milk of similar dimensions it was established that the empirical parameter II in expression (2) is n = - O-722, whence c =zI,, R” = 75.45 WZO’~‘~ l/s. To separate particles with marginal diameter Dmin= 2 pm and b = 3 x 10’ l/s, calculation of the separation time using eqn (8) wilI give t = 7.35 s. To provide for the condition 8 2 t, the required number of plates according to eqn (9) is: Z + 1 = 2 10 and when the thickness of the plate material is 0.5 mm the

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total height of the plate set will be 326.7 mm, i.e. quite an acceptable value. With a constant width of flow ii = Q[l +(r/R)“]2Zh, =0*682 x 10e3 m, the hydraulic radius is also constant: R, = &r/2( h, + U) = 0.182 x 10e3 m. With an initial width of flow u0 =fJZh, = O-425 X lob3 m its length according to eqn (7) is L = 9.21 m and the average loss of pressure according to eqn ( 13) is equal to Hi U) = ALU*/SgR,,( U)= 2.08 X 1O5Pa. This loss of pressure is quite acceptable when compared with the pressure on entry to the hydrocyclone of 3 X lo5 Pa.

RESULTS

AND DISCUSSION

From the theoretical analysis and experimental data, the separation possibilities of standard hydrocyclones are sufficient for the separation of solid particles from liquids. With a decrease of particle size and small differences between particle and liquid densities, the separation capabilities of hydrocyclones decrease dramatically. Increasing the hydrocyclone dimensions only partially solves this problem. Standard hydrocyclones cannot be used to separate highly dispersed particles or in cases where the difference in densities between particles and liquid is too small (i.e. in emulsions such as milk). The suggested design of hydrocyclone facilitates a radical increase in separation capability owing to the reduction in the path length for separated particles. According to preliminary calculations, the relative complexity of the design is modified compensated by the increase in separation efficiency. For industrial milk processing high separation efficiency is provided by an apparatus with quite acceptable dimensions.

CONSTRAINTS (1) The calculation formulae did not take into account other factors that effect hydrocyclones, such as liquid turbulence, gravitational and Coriolis forces, turbulent fluctuations, etc. Their detailed account will increase the accuracy of the suggested model and the calculation methods. (2) Only Newtonian liquids were considered. The theoretical analysis is therefore applicable to food liquids which are non-Newtonian, but should first take into account such properties.

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L. A. Rovinsky

COMMERCIAL

PROMISE

In practice, for preliminary flow purification of liquids, centrifugal separators can be replaced by simpler and cheaper equipment - hydrocyclones of the suggested type with considerable reduction in weight and energy consumption. The use of such hydrocyclones instead of cream separators for milk before homogenization is very promising since only large fat particles and not the entire milk require homogenization. The combination of the suggested hydrocyclone with a homogenizer may increase the efficiency of the process.

CONCLUSION The analysis demonstrated the limited capabilities of standard hydrocyclones for separation of suspensions and emulsions. A broader application of hydrocyclones, including that for food liquids, is possible if hydrocyclones of the described type are used. The suggested approach to the analysis of the separation process facilitates the development and design of hydrocyclones for specific products with given separation requirements and application conditions. *

BIBLIOGRAPHY Driessen, M. G. (195 1). Theorie de 1’Ecoulement dans un cyclone. Revue de L ‘Industrie Minerale, 3 l( 567). Rovinsky, L. A., Yashin, V. K., Kuzmin, Yu. N. ( 1986). In The Effective Methods and Equipments for Food Products Processing. VNIEKIProdmash, Moscow, pp. 1 1 - 18 (in Russian).