Centrifugation

CentrifugationTheory of centrifugationTypes of centrifugesApplicationsCentrifugal separationsSedimentation operation accelerated by centrifugal forcePre-requisite for the separation is a difference in density between the phasesThis applies to both solidliquid separationliquidliquid separationSedimentation by GravityA particle suspended in a liquid medium of lesser density tends to sediment downward due to the force of gravity (Fg)Two forces oppose the gravitational force the buoyancy force, Fbthe frictional force, Ff

Buoyancy force

mM = the mass of the fluid medium displaced Vp = the volume of the particle M = the density of the displaced fluid The net gravitational effect, taking into account the buoyancy force is

M = the density of the medium (g.cm-3) P = the particle density (g.cm-3)r = the particle radius (cm).Frictional forceThe movement of a particle through a fluid medium is hindered by the viscosity of the medium, as described for a spherical particle by Stokes equation

= viscosity of the medium in poise, P (g cm-1s-1); r = the radius of the particle (cm);(dx/dt) = the velocity of the moving particle (cm.s-1). At low velocities and pressures, the frictional force is negligible in a gas.

At higher velocities, even in gases, this force becomes substantial, combining with the buoyancy force eventually to exactly oppose the gravitational force, resulting in no further acceleration of the particle (the limiting or terminal velocity)

Mathematically, the conditions for attaining terminal velocity are met when:Frictional forceEffect of DiffusionRandom Brownian motion results in the net movement of solute or suspended particles from regions of higher concentration to regions of lower concentrationDiffusion works in opposition to centrifugal sedimentation, which tends to concentrate particles

D = the diffusion coefficient which varies for each solute and particle

A = the cross sectional area through which the particle diffuses

dP/dx = the particle concentration gradientFicks law:Sedimentation in a Centrifugal FieldA particle moving in a circular path continuously experiences a centrifugal force, Fc.

This force acts in the plane described by the circular path and is directed away from the axis of rotation.

The centrifugal force may be expressed as

m = the particle mass (g); a = the acceleration (cm.s-2) = the angular velocity (radians s-1 .rpm/60)x = the radial distance from the axis of rotation to the particle (cm)Relative centrifugal force (RCF)

Alternatively RCF is given by

Ratio of acceleration of the centrifugal field to that of acceleration owing to the earths gravitym = the particle mass (g); a = the acceleration (cm.s-2); = the angular velocity (radians s-1 2.rpm/60); x = the radial distance from the axis of rotation to the particle (cm).Forces acting on a particle in a centrifugal field

Fb = buoyancy forceFf = frictional forceFc = centrifugal forceFg = gravitational forceWhen centrifugal force is equaled by buoyancy and frictional forces

Assuming spherical particles the above equation becomes

Solving for (dx/dt)

Forces acting on a particle in a centrifugal field

In terms of particle diameter, d and particle velocity, v Upon integration the equation above yields the time required for a particle to traverse a radial distance from x0 to x1

x0 is the initial position of the particle x1 is the final position of the particle Parameters that govern settling velocityThe sedimentation rate of a particle in a centrifugal fieldincreases as the square of the particle diameter and rotor speeddoubling the speed or particle diameter will lessen the run time by a factor of fourincreases proportionally with distance from the axis or rotationinversely related to the viscosity of the carrier medium

Sedimentation Coefficient (Sr)In a homogeneous medium, the following parameters are constant for a given particleThe viscosityParticle size Particle densityDensity of the mediumThe sedimentation rate is proportional to 2x expressed in terms of the sedimentation coefficient, S.Measure of the sedimentation velocity per unit of centrifugal force

For a given set of run conditions

Sedimentation CoefficientThe sedimentation coefficient has the dimensions of secondsexpressed in Svedberg units equal to 10-13 sSedimentation coefficient is dependent on the particle being separatedthe centrifugal forcethe properties of the sedimentation medium.Useful to compare sedimentation coefficients obtained under differing conditions sedimentation media by reference to the behaviour of the particle in water at 20oC

Rotor Efficiency or Pelleting Efficiency (k-factor)Pelleting efficiency or k-factor The time required for a particle to traverse a rotork-factor calculated at the maximum rated rotor speed, is a function of rotor design and is a constant for a given rotor.k-factors are useful for determining the minimum residence time required to pellet a particle in a given rotorcomparing sedimentation times for different rotors

The k-factor is derived from the equation

rmax and rmin are the maximum and minimum distances from the centrifugal axisIf the sedimentation coefficient of a particle is known, then the rotor k-factor can also be calculated from the relation:

T = time in hours required for pelleting S = the sedimentation coefficient in Svedberg unitsFor runs conducted at less than the maximum rated rotor speed, the k-factor may be adjusted according to

k-Factors are also useful when switching from a rotor with a known pelleting time, t1, to a second rotor of differing geometry by solving for t2 in the relationProblem: Separating cells growing on a supportAnimal cells can be cultivated on the external surface of dextran beads. These cell laden beads or microcarriers have a density of 1.02 g/ml and a diameter of 150 m.

A 50 litre stirred tank is used to cultivate cells grown on microcarriers to produce a viral vaccine. After growth, the stirring is stopped and are allowed to settle. The microcarrier-free fluid is then withdrawn to isolate the vaccine.

The tank has a liquid height to diameter ratio of 1:1.5; the carrier-free fluid has density of 1 g/ml and a viscosity of 1.1 cP.

Estimate the settling time to reach the velocityEstimate the time to reach this velocityUsing the equation for terminal velocity Substituting the values we get

(a) Stokes law is applicable if the condition is satisfied that

Hence the use of Stokes equation is justified

To calculate the liquid height from tank volume

The terminal velocity of the particle can be made use of to calculate the settling time

t = l/vg = 52.3 cm/0.022 cm/sec = 39.6 min

Approximately it will take 40 min for the microcarriers to completely settle

(b) Assuming the velocity of the microcarriers is originally zero we find the change of settling velocity by a force balance on the particle

Subject to initial condition t=0 and v=0Integrating the above equation we can find that

Hence assuming steady state condition where

Since our settling time is 40 minutes we easily meet this criterionProblem: Centrifugation of yeast cellsA laboratory bottle centrifuge consists of a number of cylinders rotated perpendicular to the axis of rotation.

During centrifugation the distance between the surface of liquid and the axis of rotation is 3 cm, and the distance from the bottom of the cylinder to that axis is 10 cm.

The yeast cells can be assumed to be spherical, with a diameter of 8.0 m and a density of 1.05 g/ml.

The fluid has physical properties close to those of pure water.

The centrifuge is to be operated at 500 rpm.

How long does it take to have complete separation?

From the equationIt was found that We are interested in the yeast cell which takes longest to settle, which is that starting near the liquid surface, t = 0; r = 3 cm.

Integrating the initial equation, we findSubstituting the values, we getTypes of Centrifugal SeparationAccording to the phase of the medium and the phase of the material to be purifiedGas-gasLiquid-liquid Liquid-solidAccording to the method by which purified fractions are recoveredBatch modeSemi-batch modecontinuous mode

Types of centrifugesTubular bowl centrifugesSimple yet can provide very high GCan be cooledDisadvantage: Requirement for intermittent dismantling for cleaning Disc type centrifuges: Three typesSolids-retainingIntermittent solids-ejectingContinuous solids-ejectingBasket type centrifugesUsed for centrifugal filtration

Tubular bowl centrifugeUtilize a vertically mounted, imperforate cylindrical-bowl design to process feed streams with a low solids contentLiquid is discharged continuously and solids are manually recovered after the rotor capacity is reachedIndustrial models are available with Diameters up to 1.8 mHolding capacities up to 12 kg Throughput rates of 250 m3 h-1 Centifugal forces ranging up to 20000 g.Laboratory models are available with Diameters of 4.5 cm Throughput rates of 150 L.h-1 Centrifugal forces ranging up to 62000g.Performance analysis of a tubular centrifugeAnalysis depends on finding the position of a particle as a function of time

Assumptions Particle located at a distance z from the bottom of the centrifugeIt is also located at position r from the axis of rotationThis position is between the liquid surface R1 and bowl radius R0 wFeed freely flows in the bottom and out the top Solids are thrown out by centrifugal force and trapped against the wall, located at R0The centrifugal force is so high that the liquid interface R1 is constantlR0zrR1Liquid InterfaceIdealization of the tubular bowl centrifugeThe particle is moving in both the z and r directions.

Its movement in the z direction comes from the convection of the feed pumped in the bottom of the centrifuge

Q is the feed flow rateThe particle movement in the r direction is related to its radial position by

Which can be rewritten in terms the velocity of a particle settling under the influence of gravity

To find the trajectory of the particle within the centrifuge

Performance analysis of a tubular centrifugeFor particles which are most difficult to capture, they enter the centrifuge at r=R1 and do not reach r=R0 until the end of the unit at z=l Integration and rearrangement of the equation for the particle trajectory gives the maximum flow possible flow rate in the centrifuge as a function of both particle properties and centrifuge characteristics

In most tubular centrifuges as R0 and R1 are approximately equal, we can simplify the above equation

The Generalized FormulaThe most-used quantity to characterize centrifuges, the concept

Where, Assumptions

Viscous drag is determining the particle movement.The flow in disk bowls between the disks is laminar and symmetrical.The liquid rotates at the same speed as the bowlThe particle concentration is low (no hindered settling).The particle always moves at its final settling velocity.This settling velocity (Vc) is proportional to the g force.The equation for critical diameter becomesProblem:

A laboratory tubular-bowl centrifuge has the following dimensions, with respect to Figure 19.28, and operating conditions: bowl speed 800 rps, R0 = 0.875 inch, R1= 0.65 in., and bowl length = L = 4.5 inches. When used to remove E. coli cells from the following fermentation broth, a satisfactory volumetric feed capacity of the centrifuge, Q, of 0.11 gpm is achieved.

Broth: f = 1.01 g/cm3 and = 1.02 x 10-3 kg/m-s

E. coli: smallest diameter, dp,min = 0.7 mm and p = 1.04 g/cm3

Assuming the applicability of Stokes law, estimate the feed capacity of the centrifuge.Compute the sigma factor for the laboratory centrifuge from using the given dimensions and operating conditions.

The rotation rate, v, in radians/s=2(3.14)(800)=5,030 s-1.

Solids-retaining Disc Centrifuge

Appropriate for liquid-solid or liquid-liquid separations where the solids content is less than about 1% by volumeFor liquid-solid separations, the solids that accumulate on the bowl wall are recovered when the rotor capacity is reached and the centrifuge is stoppedRemovable baskets are incorporated into some designs to facilitate solids removalRecovery of two liquid streams can be achieved by positioning exitPorts at different radial distances as dictated by the relative concentration of the liquidsIntermittent solids ejecting disc centrifuge

Suitable for processing samples with solids contents to about 15% by volumeSolids or sludge that accumulate on the bowl wall are intermittently discharged through a hydraulically activated peripheral openingLaboratory models to 18 cm diameter and industrial units to 60 cm

Industrial centrifuges capable of throughputs in excess of 100 m3 h-1

FeedDischargePhotocellDiscsSediment holding spaceSolids ejection portsOperating water valveDrain holeOpening chamberClosing chamberAnnular pistonTiming unitDischarge pumpBowl section of a self-cleaning disc stack centrifuge indicating direction of fluid flow and ejection of sedimented solids through passages controlled with hydraulically operated pistonsNozzle machines allow for continuousdischarge of solids through throttled nozzles

Solid bowl machines without solid discharge mechanisms require manual cleaning from time to time depending upon feedstock solidsDischarge is intermittent Continuous solids ejecting disc centrifuge

Solids contents ranging from 5 to 30% by volumeSolids are continuously discharged via backward-facing orificesNewer designs discharge to an internal chamber where the discharge is pumped out as a product streamIndustrial units are available to 200 m3 h-1 throughput capacity, elevated temperature (