Mathematics Applied in Science and Technology.Volume 2, Number 1 (2010), pp. 71–80© Research India Publicationshttp://www.ripublication.com/mast.htm

A New Non-linear Parabolic Equation DescribingCentrifugal Segregation of Particles in a Fluid

D. O. Besong

Imperial College London, Department of Earth Science and Engineering

Abstract

In this work, a nonlinear parabolic equation is developed, describing the centrifu-gal segregation of particles in a fluid. The general form of this equation has beenlisted in a very useful work on nonlinear PDEs, but this particular equation is notmentioned under equations of that general form. This equation describing the cen-trifugal segregation of particles is new, to the best of our knowledge. We consideronly mono-disperse spherical particles in an isothermal fluid in a 1-D system in thederivation of our equation.

AMS subject classification:Keywords:

1. Introduction

Centrifugal systems have been used to study segregation in various areas of science. Inmedicine, Constandoulakis et al. [2] observed the separation of erythrocytes (i.e. particleseparation) in blood by centrifugation, while in petroleum engineering, Ratulowski et al.[11] studied the separation of miscible hydrocarbons by using a centrifugal force as asubstitute for gravitational force. In metal casting, Fu and Xing [3] as well as Panda et al.[9] have studied the effect of a centrifugal force on the extent that a steel roll segregatesduring casting. Fu and Xing [3], as well as Panda et al. [9] analyzed the effect of acentrifuge to the segregation of particles in molten metal during metal casting throughmathematical modelling. Moreover, Panda et al. [9] found an expression of the positionof a single particle with time, based on the centrifugal force.

Under zero acceleration, Fu and Xing [3] and Panda et al. [9] expressed the velocityv of each particle as a result of the centrifugal force as

v = Wr, (1.1)

35K55, 70–02, 70–08, 35–01.Centrifuge, Particle Segregation, First Principles, Derivation of

Nonlinear Equation.

72 D. O. Besong

where

W = 2a2�ρω2

9η, (1.2)

where �ρ is the density of pure solid minus that of pure liquid (kgm−3), a the radius ofthe suspended particles (m), r the distance (m) of the particle from the axis of rotation,ω the rotational speed (rad s−1), and η the dynamic viscosity of the fluid medium (Pa.s).

Assuming laminar flow, Panda et al. [9] performed force balance on a particle in afluid in centrifuge. This resulted in a second-order ordinary differential equation withtime t as the independent variable and radial position of particle r(t) as the dependentvariable. Solving for r(t), Panda et al. obtained the position of a particle at any time [9]

r(t) = R0eWt , (1.3)

where W is given in equation (1.2), R0 is the position of the particle at time t = 0. Wesee that at large time, r will be at infinity. It is our desire to derive an equation that canmodel real speed rolls where r is always finite that drove us to derive the new equation.

Gutierrez et al. [5] obtained the position r(t) of the particle by finding the generalsolution of the second-order differential equation mentioned in the above paragraph.This resulted in [5]

r(t) = A1eα1t + A2e

α2t , (1.4)

where α1 and α2 are the solutions of the characteristic equation of the second-orderdifferential equation [5]. Analysis shows that with the the particle density greater thanthat of pure fluid (i.e. �ρ > 0), α1 and α2 are real and of opposite signs. The constantsA1 and A2 depend on the initial position of the particle, and imposing that at t = 0,∂r

∂t= 0 [5].

Gutierrez et al. [5] found an analytical expression for r(t) for an initially uniformdistribution of particles in the fluid by considering the initial position of the differentparticles as random numbers [5]. So they generated random numbers representing theinitial positions of the uniformly distributed particles, and found the position of eachof these particles by equation (1.4). Then they repeated the process with many randomparticles and found the spatial distribution of particles at any particular time t . Gutierrezet al. [5] Panda et al. [9] offer an important way of solving for the position of eachparticle, which has been very beneficial to the centrifugal casting of matrix composites,but an equation has not yet been derived to fully describe the transient concentration ofthe particles in space. An advantage of an equation is that it is more efficient to solvenumerically since only the volume-fraction of the particles is solved for, rather than forthe position of many random particles at each time t . In this work, we derive such anequation so that any consistent boundary condition can be imposed.

It is useful to derive partial differential equations to describe physical phenomenaboth in and outside the laboratory. Polianin [10] presents an up-to-date collection ofnonlinear partial differential equations that arise in mathematics and physics. An equationdescribing the centrifugal segregation of particles would be desirable so that simulations

A New Non-linear Parabolic Equation Describing Centrifugal Segregation 73

Figure 1: Particle distribution after after 50 seconds of centrifuge.

representative of laboratory experiments can be carried out, especially when a numericalmethod must first be done in order to find the required rotational speed in the laboratoryor industry.

1.1. Analaytical Solution of Gutierrez et al.

Gutierrez et al. [5] generated random numbers representing the initial positions of theuniformly-distributed particles, and using equation (1.4) plotted the number of particlesat each position along a cylinder filled with aluminium melt. At a constant temperatureof T = 660◦C, the liquid (molten metal) density is ρL = 2.4g cm−3, the density of eachparticle is ρs = 3.19g cm−3 and the liquid viscosity η = 1.38 mPa ·s. The distance fromthe near end of the cylinder to the axis of rotation was R0 = 19.50 cm, and the far endwas R = 32.00 cm from the rotational axis. In the model of Gutierrez et al., the cylinderwas rotated at a constant rotational speed of ω = 250 RPM (rotations per minute). Theaverage radius of each particle was a = 3.75 × 10−4 cm. Figure 1 is the solution after50 seconds, from an initially uniform distribution [5].

2. Mathematical Formulation

The particles in a fluid column free from external forces are subjected to Brownianmotion which tends to even out the particle distribution, as well as gravity segregationwhich tends to separate the particles to the bottom of the column if they are denser thanthe liquid (and vice-versa).

74 D. O. Besong

Brownian motion is governed by the diffusion equation. The volume flux of particlesdue to Brownian motion is given by (see for instance [1], [8])

JD = −D∂n

∂zor JD = −Dnt

∂u

∂z, (2.1)

where z is the spatial coordinate (directed upward), D the diffusion coefficient, nt the totalnumber of particles per volume at the point z (both solid and supposed fluid 1 particles),n the number of solid particles per volume at point z and u the volume-fraction of thesolid at point z.

Einstein had derived the diffusion coefficient of suspended particles in a fluid basedon Stokes law and the Boltzmann law [7], given by

D = kT

6πηa, (2.2)

where T is the constant fluid temperature (K) in an isothermal container, k the Boltzmannconstant (in J.K−1), η the dynamic viscosity of the fluid medium (Pa.s), and a the radiusof the suspended particles (m).

Besong [1] derived a flux JG for the gravity segregation of particles. He derived anexpression for the gravity flux identical to the one earlier proposed (e.g. [12], [4]) andhence validated the previous claims from first principles. The gravity flux for particlesin terms of volume-fraction of the particles is (see for instance [1], [12] and [4])

JG = 4

3πa3 nD

kT· �ρ(1 − u)g = 4

3πa3 untD

kT· �ρ(1 − u)g (2.3)

since the number of solid particles per unit volume n = unt . Besong added the twofluxes and derived a conservation equation as follows:

nt

∂u

∂t= −∂JD

s

∂z− ∂JG

s

∂z. (2.4)

Since the system is assumed incompressible, and the fluid is considered as particles ofsame radius as the solid particles, the total number of particles (fluid and solid) per unitvolume is constant everywhere. Therefore the nt ’s cancel out leading to

∂u

∂t= D

∂2u

∂z2+ 4

3πa3 D

kT�ρg

∂

∂z[u (u − 1)] , (2.5)

which is the segregation-mixing equation of particles under gravity forces. Here, �ρ isthe density of pure solid minus that of pure liquid (kgm−3), a the radius of the suspendedparticles (m), g the acceleration due to gravity (ms−2) and t the time (s). The derivationof equation 2.5 is elaborated in this work for revision purposes only, but it can be foundin [1].

1To visualize nt , the fluid is imagined as made of fluid particles of the same size as the solid particles[1].

A New Non-linear Parabolic Equation Describing Centrifugal Segregation 75

2.1. The Derivation of the Equation for Particle Segregation in a Centrifuge

Besong [1] has clearly presented a derivation for the volume flux JG of particles ina colloid due to gravity under the following assumptions (some of these assumptionsmentioned in [4], [7], [6], [13]):

i All particles behave as in a dilute particle system, so that they do not exert forceson each other.

ii The velocity of the particles depends only on the local particle density.

iii No wall effects.

iv All particles are of the same size and are spherical in shape.

v The particle size is small compared with the container.

vi There is no bulk motion. In the case of bulk motion, the gravity flux would beresolved only relative to the bulk velocity.

With the same assumptions applied for the gravity segregation of particles or centrifu-gal segregation, an analogical derivation as in [1] obtains a similar expression but withthe centripetal acceleration ω2r instead of g, and −r instead z for gravity. In centrifugalsegregation, −r is analogous to z because both the centrifugal force and r are oriented inthe same direction, whereas in the gravitational case g and z are in opposite directions.The derivation performed by substituting g in the flux due to gravity (equation (2.3)) byω2r , and the space variable z by −r results in the volume flux of particles due to thecentrifuge, given by

J� = −4

3πa3 untD

kT· �ρ(1 − u)ω2r. (2.6)

The group of parameters4

3πa3 untD

kT· �ρr = 2a2�ρω2r

9η= v (see equation 1.1).

Summing JD and J� and then applying mass conservation within a control volume(as in equation (2.4)) and simplifying, the nts cancel out leading to

∂u

∂t= D

∂2u

∂r2− 4

3πa3 D

kT�ρω2 ∂

∂r[ru (u − 1)] . (2.7)

Letting α = 4πa3

3kT�ρω2, the centrifugal segregation-mixing equation of particles in a

fluid medium is given by

∂u

∂t= D

∂2u

∂r2− αD

∂

∂r[ru (u − 1)] , (2.8)

76 D. O. Besong

hence∂u

∂t= D

∂2u

∂r2− αDr

∂

∂r[u (u − 1)] − αDu (u − 1) . (2.9)

In order to identify the general form of equation (2.9) on the list of nonlinear PDEspresented in [10] let us write equation (2.9) as

∂u

∂t= D

∂2u

∂r2+ f (r, u)

∂u

∂r+ g(u). (2.10)

where f (r, u) = −αDr∂

∂uu(u − 1) = −αDr(2u − 1) and g(u) = −αDu(u − 1).

Polianin [10] lists various parabolic PDEs with one space variable as equations ofthe form

∂u

∂t= D

∂2u

∂z2+ f (z, t, u)

∂u

∂z+ g(z, t, u) (2.11)

(i.e. with the general form of equations (2.10)) or (2.11). However, among the equationslisted under parabolic equations of the form of equations (2.10)) or (2.11), none of themhave f (z, u) = −αDz(2u − 1) and g(u) = −αDu(u − 1) as in the newly-derivedequation. Moreover, no connection has been made between centrifugal segregation andthe general form (equations (2.10)) and (2.11)).

3. Results and Discussion

In this section, we present the results of the finite difference simulation of equation (2.9)at various times. We compare the result at t = 50 seconds to that of Gutierrez et al. [5].

Since the molten metal is in a closed cylinder, the boundary condition we used is thatof zero-flux at the boundaries:

D∂u

∂r− αDru (u − 1) at r = R0 and r = R (3.1)

Gutierrez et al. [5] applied a uniform distribution, but they did not mention the initialconcentration. We prefer to use an initial condition of uniform equal volume fraction ofliquid and solid particles for our simulation of equation (2.9), i.e. u0 = 0.5. Figure 2shows the numerical solution of equation (2.9) at various times, using the same data asGutierrez et al. [5].

If we assume that the cross-sectional area A of the cylinder perpendicular to the radialdirection is 1cm2, then our initial number of particles is given by

N = A

43πa3

∫ 32

19.5udr (3.2)

For particles of radius a = 3.75 × 10−4 cm, we have N = 1012

43π × 3.753

× 6.25 =2.8 × 1010 particles. In our simulation, this number stayed constant at all time-steps,

A New Non-linear Parabolic Equation Describing Centrifugal Segregation 77

Figure 2: Numerical simulation of the derived PDE.

Figure 3: Comparison between the derived PDE and that of Gutierrez after 50 seconds.

since equation (2.9) was derived from conservation laws and there was no flux at theboundaries (equation 3.1). The number of particles in the whole body of molten metal hasto de conserved. One great disadvantage of just making do with a differential equationof the spatial position of the particles as done by Gutierrez et al. is that the particles willnot be conserved within the molten metal, as we shall show. Another great advantageof the new nonlinear pde is is that at each time-step only the volume-fraction for say 21grid-points are solved for, whereas with the method used by Gutierrez et al., the positionof each of the 2.8 × 1010 particles have to be simulated as random numbers thereforenumerically inefficient.

3.1. Comparison with the Solution of Gutierrez et al.

Figure 3 compares the numerical simulation of equation (2.9) to the analytical solutionof Gutierrez et al. [5].

To allow for a consistent comparison, we assume that 150 particles in [5] (see figure1) is equivalent to complete segregation (i.e. a volume fraction of 1). This is because

78 D. O. Besong

the concentration of the uniform initial distribution was not revealed in [5], nor thecross-sectional area of the cylinder. The cross-sectional area of the cylinder would benecessary to transform number of particles to volume-fraction. As said above, let usassume the cross-sectional area was 1 cm2.

The large deviation between the simulation of the newly-derived equation and thatof Gutierrez et al. [5] can be attributed to the following reasons:

i In [9] and [5] no boundary condition was possible to implement at the boundariesof the cylinder, which means in their solution, particles closer to the far boundary(i.e. at r = R = 32.00 cm) would go out of the cylinder. In our equation, therewas a zero-flux boundary condition and so all the particles are conserved in thefluid. Consequently the solutions have to be different.

ii The analytical solutions in [9] and [5] (see equations (1.3) and (1.4)) mean that atlarge time all the particles would have position r = ∞. Therefore even at smalltime, the particles initially close to the far boundary are lost. However, if completeseparation occurs at the steady state of new equation, all the particles still in thecylinder (see figure 2).

iii All the above points imply that the particles were not totally conserved in thesolution represented in [5], figure 1, whereas the number of particles is conservedwith the new equation when no-flux boundaries are imposed. The centrifugally-induced flux of the particles in equation (2.9) has threshold solid volume fractions(i.e. J�(u) = 0 when u = 0 or when u = 1). This is a necessary property ofa segregating flux which does not allow for non-physically valid concentrations.In the solution of Gutierrez et al. [5], the motion of the particles were onlycharacterized by a constant velocity v. In that case, even if some sort of numericalbarrier were to be imposed at the outer boundary where r = R, particles wouldkeep piling at the boundary above physically possible numbers. Therefore theresults would still be different from our simulation

Since our simulation is conservative, any solution for the same problem which isnot conservative will differ from our simulation. However, there is a common trendbetween our solution and that of Gutierrez et al. [5], of the progressive segregation ofthe particles from the fluid. The solution of Gutierrez et al. [5] offered a qualitativestudy of the variation of the distribution of particles in a centrifuge at a time when anequation closely describing the phenomenon was not available. The new PDE derivedin this work from conservation principles offers an enhancement in the modelling andmathematical analysis of the centrifugal segregation of particles.

4. Conclusions

A new nonlinear parabolic equation describing particle segregation in a centrifugal sys-tem is derived. The derived equation (equation (2.9)) has not yet appeared in literature,

A New Non-linear Parabolic Equation Describing Centrifugal Segregation 79

to the best of my knowledge. To the best of my knowledge, an equation has not been doc-umented to fully describe the centrifugal segregation of particles in a fluid. The generalform (equation (2.10)) of the newly-derived equation has appeared in literature [10], butnone of the particular equations resembles the newly derived equation (equation (2.9)).Even the most recent contributions to the field of centrifugal segregation [5], [3] and [9]did not implement this equation, nor cited any previous work that did so. I think that thisconnection with such a useful physical system will add one more important equation tothe list listed presented by Polianin [10]. The new equation will be useful in modellingthe centrifugal segregation of particles.

5. Future Work

An analytical solution of equation (2.9) for the no-flux boundary conditions in this workwould be desirable. This work can also be extended to poly-disperse particles. Ananalytical solution for this problem with other boundary conditions in finite space wouldalso be desirable.

References

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80 D. O. Besong

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