Centrifugation Theory of centrifugation Types of centrifuges Applications Centrifugal separations • Centrifugal separation is a sedimentation operation accelerated by centrifugal force. • Prerequisite for the separation is a difference in density between the phases. • This applies to both – solid–liquid separation – liquid–liquid separation Sedimentation by Gravity • A particle suspended in a liquid medium of lesser density tends to sediment downward due to the force of gravity (Fg) Fg mg m 980cm.s 2 • There are two forces that oppose the gravitational force; – the buoyancy force, Fb – the frictional force, Ff Buoyancy force Fb mM g V p M g mM = mass of the fluid medium displaced, Vp = volume of the particle, ρM = density of the displaced fluid The net gravitational effect, taking into account the buoyancy force is Fg net 4 3 4 3 2 r P M g r P M 980cm.s 3 3 ρM is the density of the medium (g.cm-3); ρP is the particle density (g.cm-3) r is the particle radius (cm). Frictional force The movement of a particle through a fluid medium is hindered by the viscosity of the medium, η as described for a spherical particle by Stokes’ equation dx Ff 6 r dt η, is the viscosity of the medium in poise, P (g cm-1s-1); r is the radius of the particle (cm); (dx/dt) is the velocity of the moving particle (cm.s-1). Terminal velocity At low velocities and pressures, the frictional force is again 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. Mathematically, the conditions for attaining terminal velocity are met when: Fg Fb F f Effect of Diffusion • Random Brownian motion results in the net movement of solute or suspended particles from regions of higher concentration to regions of lower concentration • Diffusion works in opposition to centrifugal sedimentation, which tends to concentrate particles. • The rate of diffusion of a particle is given by Fick’s law: dP dP DA dt dx D is the diffusion coefficient which varies for each solute and particle; A is the cross sectional area through which the particle diffuses dP/dx is the particle concentration gradient. Sedimentation in a Centrifugal Field A 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 Fc ma m x 2 m is the particle mass (g); a is the acceleration (cm.s-2) ω is the angular velocity (radians s-1 π.rpm/60) x is the radial distance from the axis of rotation to the particle (cm) Relative centrifugal force (RCF) Ratio of acceleration of the centrifugal field to that of acceleration owing to the earth’s gravity Fc RCF Fg 2 m x mg 2 x g Alternatively RCF is given by RCF 1.119 10 5 rpm 2 x m = particle mass (g); a = acceleration (cm.s-2); ω = angular velocity (radians s-1 2π.rpm/60); x = radial distance from the axis of rotation to the particle (cm). Forces acting on a particle in a centrifugal field When centrifugal force is equaled by buoyancy and frictional forces, Fc Fb Ff dx m x VP M x 6 r dt 2 2 Assuming spherical particles the above equation becomes Fb = buoyancy force Ff = frictional force Fc = centrifugal force Fg = gravitational force 4 3 4 3 dx 2 2 r P x r M x 6 r 3 3 dt Solving for (dx/dt) 2 2 dx 2r P M x dt 9 Forces acting on a particle in a centrifugal field In terms of particle diameter, d and d 2 P M 2 x particle velocity, v v 18 Upon integration the equation above yields the time required for a particle to traverse a radial distance from x0 to x1 x 18 1 t 2 ln 2 d P M x0 x0 is the initial position of the particle x1 is the final position of the particle Problem: The effect of the microorganisms’s size on settling velocity Compare the settling velocities of E.coli and yeast at a cell concentration of 1.5% Parameter Yeast E. coli Size Density Apparent viscosity 7 micron 1.03 g/ml 1.8 cP 1 micron 1.03 g/ml 1.8 cP Assume isothermal conditions and the settling behaviour same as that of a sphere Sedimentation is governed by v 2 2 d P M x 18 Settling velocity for E. coli = 0.0033 cm/sec Settling velocity for yeast cells = 22.7 cm/sec Parameters that govern settling velocity • The sedimentation rate (i.e. limiting velocity) of a particle in a centrifugal field – increases as the square of the particle diameter and rotor speed, i.e. doubling the speed or particle diameter will lessen the run time by a factor of four – increases proportionally with distance from the axis or rotation – is inversely related to the viscosity of the carrier medium Sedimentation Coefficient • In a homogeneous medium, the following parameters are For a given set of run conditions, the constant for a given particle sedimentation coefficient, Sr, may be – – – – The viscosity Particle size Particle density Density of the medium • The sedimentation rate is proportional to ω2x, calculated as dx 2r 2 dt P M Sr 2 9 x – expressed in terms of the sedimentation coefficient, S. • Measure of the sedimentation velocity per unit of centrifugal force Sedimentation Coefficient • The sedimentation coefficient, S, has the dimensions of seconds and is expressed in Svedberg units equal to 10-13 s S20, w ST ,MT ,M P M 20,W P T ,M • Sedimentation coefficient is dependent on – the particle being separated – the centrifugal force – the 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 (k-factor) • The time required for a particle to traverse a rotor is known as the pelleting efficiency or k-factor • k-factor is calculated at the maximum rated rotor speed, is a function of rotor design and is a constant for a given rotor. • k-factors provide a convenient means of – determining the minimum residence time required to pellet a particle in a given rotor – useful for comparing sedimentation times for different rotors The k-factor is derived from the equation ln rmax rmin 1013 k 2 3600 rmax and rmin are the maximum and minimum distances from the centrifugal axis k 2.531011 ln rmax rmin / rpm2 If the sedimentation coefficient of a particle is known, then the rotor k-factor can also be calculated from the relation: k TS T is the time in hours required for pelleting S is the sedimentation coefficient in Svedberg units For runs conducted at less than the maximum rated rotor speed, the k-factor may be adjusted according to kadj rpmmax k rpmact 2 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 relation Stress in the bowl wall limits centrifuge speed • Self stress (SS) – Stress in the bowl wall is due to rotation S S 4.1110 10 n Db m 2 2 • Stress due to the liquid in the bowl (Sl) 2 2 2 n D D D b b i 10 S 1.03 10 • Total stress (ST) ST = SS + Sl 2 2 D D b i 2 ST 4.1110 10n Db m Db 4 Problem: Separating cells growing on a support Animal 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 o 150 μm. 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 1:5; the carrierfree fluid has density of 1 g/ml and a viscosity of 1.1 cP. a) Estimate the settling time to reach the velocity b) Estimate the time to reach this velocity Solution Using the equation for terminal velocity d v 2 M x 2 P 18 Substituting the values we get v 0.022cm.s 1 Problem: Centrifugation of yeast cells A 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 μm 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 equation It was found that d2 v ( s ) 2 r 18 dr d2 v ( s ) 2 r r 18 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 find 2 r d 2 ln ( s ) t 3cm 18 Substituting the values, we get (8 104 cm) 2 g 500 2 2 10cm ln 0.05 3 1 1 cm 60sec 3cm 18(0.01g.cm .sec t 2500sec Types of Centrifugal Separation • According to the phase of the medium and the phase of the material to be purified – Gas-gas – Liquid-liquid – Liquid-solid • According to the method by which purified fractions are recovered – Batch mode – Semi-batch mode – continuous mode Types of centrifuges • Tubular bowl centrifuges – Simple yet can provide very high G – Can be cooled – Disadvantage: Requirement for intermittent dismantling for cleaning • Disc type centrifuges: Three types – Solids-retaining – Intermittent solids-ejecting – Continuous solids-ejecting • Basket type centrifuges – Used for centrifugal filtration Tubular bowl centrifuge Utilize a vertically mounted, imperforate cylindrical-bowl design to process feed streams with a low solids content Liquid(s) is discharged continuously and solids are manually recovered after the rotor capacity is reached Industrial models are available with Diameters up to 1.8 m Holding 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 62000 g Performance analysis of a tubular centrifuge Analysis depends on finding the position of a particle as a function of time R0 R1 l Assumptions 1. Particle located at a distance z from the bottom of the centrifuge 2. It is also located at position r from the axis of z rotation r Liquid Interface 3. This position is between the liquid surface R1 Idealization of the and bowl radius R0 w 4. Feed freely flows in the bottom and out the top Tubular bowl 5. Solids are thrown out by centrifugal force and centrifuge trapped against the wall, located at R0 6. The centrifugal force is so high that the liquid interface R1 is constant Performance analysis of a tubular centrifuge The 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 dz Q dt ( R0 2 R12 ) Q is the feed flow rate The particle movement in the r direction is related to its radial position by 2 dr d 2 r s r dt 18 r 2 dr vg dt g 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 2 2 R R r 2 0 1 dr dr / dt vg dz dz / dt g Q For 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 l R0 R 2 Q 2 1 v 2 g g ln R0 / R1 In most tubular centrifuges as R0 and R1 are approximately equal, we can simplify the above equation R0 R1 R0 R1 ( R0 2 R12 ) ln( R0 / R1 ) ln 1 R0 R1 / R1 R0 R1 R0 R1 R1 ( R0 R1 ) R0 R1 / R1 .... 2R2 2 lR 2 2 Q vg vg [] g The Generalized Σ Formula The most-used quantity to characterize centrifuges, the Σ concept Qtheor vg . Assumptions • • • • • • Where, 2 lR 2 2 g 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 bowl The 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 becomes 18. .Q Se theor dc . p f V . 2 .r 2 18. .Q theor dc . p f .g 1 2 1 2 Solids-retaining Disc centrifuge Appropriate for liquid-solid or liquid-liquid separations where the solids content is less than about 1% by volume For liquid-solid separations, the solids that accumulate on the bowl wall are recovered when the rotor capacity is reached and the centrifuge is stopped Removable baskets are incorporated into some designs to facilitate solids removal Recovery of two liquid streams can be achieved by positioning exit Ports at different radial distances as dictated by the relative concentration of the liquids Intermittent solids ejecting disc centrifuge Suitable for processing samples with solids contents to about 15% by volume Solids or sludge that accumulate on the bowl wall are intermittently discharged through a hydraulically activated, peripheral opening Laboratory models to 18 cm diameter and industrial units to 60 cm Industrial centrifuges capable of throughputs in excess of 100 m3 h-1 Bowl section of a self-cleaning disc stack centrifuge indicating direction of fluid flow and ejection of sedimented solids through passages controlled with hydraulically operated pistons Discharge is intermittent. Feed Discharge pump Nozzle machines allow for continuous discharge of solids through throttled nozzles Timing unit Solid bowl machines without solid discharge mechanisms require manual cleaning from time to time depending upon feedstock solidsAnnular piston Photocell Discharge Discs Sediment holding space Solids ejection ports Opening chamber Closing chamber Drain hole Operating water valve Continuous solids ejecting disc centrifuge Solids contents ranging from 5 to 30% by volume Solids are continuously discharged via backwardfacing orifices Newer designs discharge to an internal chamber where the discharge is pumped out as a product stream Industrial units are available to 200 m3 h-1 throughput capacity, elevated temperature (<200oC) or pressure (7 bar) capability, and particle removal to 0.1 μm. Performance analysis of disc type centrifuge ω Objective: To find the location of a given particle Consider a particle at position (x, y) θ The velocity in the x direction is due to convection and sedimentation dx v0 v sin dt Average Q v0 f ( y) n(2 rl ) y x R1 convective velocity The volume of v0 averaged over y must equal the convective velocity R0 Characteristics of v0 Much larger than vωsinθ Function of radius Function of y Performance analysis of disc type centrifuge ω Q 1 v0 dy l0 n (2 rl ) l θ Performing integration we get l 1 f ( y )dy 1 l0 y x R1 Again considering that the convective velocity is much greater than that of sedimentation dx v0 v sin v0 dt Q f ( y) n(2 rl ) R0 Performance analysis of disc type centrifuge ω The velocity in the y direction is due to convection and sedimentation θ dy v cos dt dy dy / dt dx dx / dt 2 nlvg 2 r 2 r cos Qgf ( y ) In terms of R0 y x R1 R0 2 2 nlv r dy g 2 ( R x sin ) cos 0 dx Qgf ( y ) This describes the trajectory of the particle between the discs Performance analysis of disc type centrifuge For particles which are most difficult to capture ω These particles enter at the outer edge of the discs where x=0 and y=0 θ They are captured at the inner edge of the discs at y=l and x=(R0-R1)/sinθ After capture they and other particles are forced along the disc surface to the outer edge, where they are discharged. y x R1 2 n 2 3 3 Q vg ( R0 R1 ) cot vg [] 3g R0 In both the cases the quantity in square brackets Σ has dimensions of (length)2 Essentially the term Σ is dependent on the geometry of the centrifuge Horizontal continuous-conveyer centrifuge Integrate an active mechanical solids discharge mechanism in an imperforate bowl for the continuous processing of larger sample volumes The solids-discharge mechanism: A helical screw turning at a slightly slower rate than the rotor Capable of very high throughput, up to 300 000 L h-1 Basket type filtering centrifuge Combination of a centrifuge and a filter with a rapidly rotating perforated basket Suspension is fed along the axis of the bowl and solids accumulate on the wall basket Liquid flows under centrifugal force through the cake which accumulates on the basket wall and out through the perforations in the wall Used to wash accumulated cake solids in filtration Drainage number d (G )1/ 2 Higher drainage numbers correspond to more rapid drainage The theoretical sizing of a centrifugal separator • 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 bowl • The 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. Problem: Complete recovery of bacterial cells in a tubular bowl centrifuge For complete recovery of bacterial cells from a fermentation broth with a pilot plant scale tubular centrifuge. It has been already determined that the cells are approximately spherical with a radius of 0.5 μm and have density of 1.1 g.cm-3 The speed of the centrifuge is 5000 rpm The bowl diameter is 10 cm The bowl length is 100 cm The outlet opening of the bowl has a diameter of 4 cm Estimate the maximum flow rate of the fermentation broth that can be attained. Problem: Tubular centrifugation of E.coli A bowl centrifuge is used to concentrate a suspension of E.coli prior to cell disruption. The bowl of this unit has an inside radius of 12.7 cm and a length of 73 cm. the speed of the bowl is 16000 rpm. The volumetric capacity is 200 litres/hr. Under these conditons the centrifuge works well. a) Calculate the settling velocity vg for the cells b) After disruption the diameter of the debris is about one half of the original cell diameter and the viscosity is increased four times. Estimate the volumetric capacity of this centrifuge under these new conditions Problem: Disc centrifugation of chlorella Chlorella cells are being cultivated in open ponds. We plan to harvest the biomass by passing the dilute stream of cells through an available disc bowl centrifuge. The settling velocity vg for these cells has been measured as 1.04 x 10-4 cm.s-1. The centrifuge has 80 discs with an angle of 40o Outer radius of 15.7 cm. Inner radius of 6 cm. We plan to operate the centrifuge at 600 rpm. Estimate the volumetric capacity Q for this centrifuge. Equivalent time • To assess the approximate properties of a particle type to be separated • Define a dimensionless acceleration G • This dimensionless unit is measured in terms of ‘g’s (multiples of earth’s acceleration) • A rough approximation of the difficulty of a separation by centrifugation is the product of – the dimensionless acceleration and – the time required for separation • Determination by – Centrifuging samples for various times until a constant PCV is reached – The equivalent time is calculated as the product of G and the time required for reaching the constant PCV • For scale up of centrifugation we can assume constant equivalent time G Gt R 2 g R 2 g t (Gt )1 (GT )2 Problem: Scale-up based on equivalent time If bacterial cell debris has Gt = 54 x 106 s, how large must be the centrifuge bowl and what centrifuge will be needed to effect a full sedimentation in reasonable amount of time? Assume the reasonable amount of time as 2 hours and the bowl radius as 10 cm. Characteristics of Separator Types Coagulants and flocculants • Metal salts – especially of aluminium or ferric iron • Natural flocculants • Starch, Gums, Tannin, Alginic acid, Sugar/sugar acid polymers, Polyglucosamine (chitosan) • Synthetic flocculants – Polyacrylamides, Polyamines/imines, Cellulose derivatives (e.g. carboxymethyl cellulose), Polydiallydimethyl ammonium chloride • Chilling temperatures below 20oC, particularly yeast cells • pH adjustment in range 3-6 • Concentration – increases particle concentration, increasing collision frequency