大部分工业生物分离的第一步往往是将不溶物质从发酵液

Chapter3Centrifugation

第三章离心Thefirststepinmostindustrialbioseparationsistheremovalofinsolublesolidsfromthefermentationbeer.Theconcentrationandsizeoftheseinsolublesvarieswidely.

大部分工业生物分离的第一步往往是将不溶物质从发酵液中除去。这些不溶性固体的浓度和颗粒大小的变化范围很宽。Theconcentrationcanbeashighas60%pervolumeoraslowas0.1%.

浓度可高达每单位体积含60%的不溶性固体，又可低至每单位体积仅含0.1%。Thesizerangesfrommicroorganismsofperhaps1umdiameteruptoinsolublenutrientscharacteristically1mmindiameter

粒径的变化可以从直径约为1um的微生物，到直径为1mm的不溶性物质。Whentheseinsolublesaredilute,large&rigid,theycanbeeasilyseparatedbyfiltration.Inthepreviouschapter,wedescribedfiltration,includingtheuseoffilteraids.Formanybeers,thesefilteraidsfacilitatefiltration.Forotherbeers,filteraidsareineffective.

对于这些浓度较小，粒径较大，硬度较强的不溶物，，我们可以采用过滤方法分离。在前一章中，我们讲述了过滤法，包括助滤剂的应用。有些发酵液中，使用助滤剂有利于过滤分离，而还有一些发酵液则不行。Whenthebeersarenoteasilyfiltered,theysometimescanbeseparatedbycentrifugation,whichisthesubjectofthischapter.

当发酵液不易被过滤纯化时，我们可以采用离心的方法来分离，这也是这章的主题。Centrifugationrequiresmoreexpensiveequipmentthanfiltration.However,itisofteneffectivewayevenwhenthesolidparticlesaresmall&hencehardtofilter.

与过滤设备相比，离心设备的价格昂贵。但当固体颗粒细小而难以过滤时，离心操作往显得十分有效。Centrifugationutilizesthedensitydifferencebetweenthesolidsandthesurroundingfluid.Whenasuspensionisallowedtostand,thedensersolidsslowlysettletothebottomofthecontainerundertheinfluenceofgravity,aprocesscalledsedimentation.Whenthissettlingisacceleratedwithacentrifugalfield,theprocessiscalledcentrifugation.Becausesedimentation¢rifugationaresimilar,botharediscussedhere.

离心分离是基于固体颗粒和周围液体密度存在差异，在离心场中使不同密度的固体颗粒加速沉降的分离过程，当静置悬浮液时，密度较大的固体颗粒在重力作用下逐渐下沉，这一过程称为沉降。由于沉降和离心相似，这儿就放在一块讨论。Thesolidconcentrateproducedbycentrifugationdiffersfromthatproducedbyfiltration。离心产生的固体浓缩物和过滤产生的浓缩不同。Abestcentrifugationproducesapaste,oftenityieldsonlyamoreconcentratedsuspension.

通常情况下离心只能得到一种较为浓缩的悬浮液或浆体。Filtrationproduces,relativelydrycake,whichisamajoradvantage.

而过滤可获得的水分含量较低的滤饼。However,manybiologicalfeeds,whichcanbecentrifugedcannotbeeffectivelyfiltered,sothatcentrifugationisoftenanattractivealternative.

但是，对大多数生物发酵液可以离心但不能有效地过滤分离，所以离心往往是很有效的方法。A.SettlingofSolids

颗粒的沉降

Whenasolidparticlemovesthroughaninfinitecontinuum,itsvelocityisaffectedbytwoforces.

当一固体微粒通过无限连续介质时，它的运动速度受两种力的影响：Firstistheparticleisacceleratedbythebuoyantforceresultingfromthedensitydifferencebetweentheparticle&thefluid.Thesurroundingbroth.

一是微粒受到因微粒和流体介质间密度不同而产生的浮力作用；Secondisarticleisretardedbythedragforcecasedbythismotion.

二是微粒所受到的流体阻力作用。FbFg

Fdd=2R球形颗粒沉降的受力情况FB=﹝πd3(ρs-ρ)/6﹞a.(3.1)disthesphere‘sdiameter微粒半径，mρs-ρrepresentthedensityofthesphere&fluid,ρs-ρ分别为微粒和液体介质密度，kg/m3aisaccelerationduetosomeappliedforce.

微粒加速度,m/s2

Thedragforce,FD,actingonasinglesphereindilutesolutionisgivenbyStoks‘Law.

在稀溶液中，作用于单个球形微粒上的阻力FD，可用Stoks（斯托克斯）定律表示。

FD=3πdμν(3.2)μistheviscosityofthecontinuum,

连续介质粘度νisthesphere‘svelocity.

微粒运动速度

Thisrelationisaccurateonlyifthesphereissmall.

这个等式仅当球形微粒较小时方能成立。Quantitatively,thismeansthat:

当Re<1时

Re=dρν/μ<1(3.3)Eq.(3.3)isalmostalwayssatisfiedforbiologicalsolutes,soEq.(3.3)willbebasictotheanalysisinthischapter.

等式3.3基本上满足所有的生物溶液，所以，本章主要分析这个等式。IfRe.1.greaterthan1.Thesereplacementshavetheform

如果Re>1时，阻力为

FD=f(ρν2/2)(πd2/4)(3.4)Wherefisfrictionfactor.f是摩擦因子

Whenthesphericalparticlebeginstomoreinsolution,thedragforceissmallbecausetheparticle‘svelocityissmall.Asaresult,theparticleacceleratesuntilthedrag&buoyancyforcesareequal.WecancombineEqs.(3.1)&(3.2),FD=FB,thenrewrite:

当球形粒子在介质中运动时速度较小，因此作用其上的阻力也较小，当阻力与浮力平衡时，微粒加速度为零。联立方程3.1和3.2，得到

ν=d2(ρs-ρ)a/(18μ)(3.5)Thisrelationgivessteadystateorterminalvelocityofthesphere.

此式给出了微粒稳定状态和最终速度

Forsettling,theaccelerationisobviouslyduetogravityg.

对于沉降，重力沉降加速度为重力加速度）

νg=d2(ρs-ρ)g/(18μ)(3.6)Second,forcentrifugation,theaccelerationisdifferent

离心沉降加速度则不同

a=rω2νω=d2(ρs-ρ)rω2/(18μ)(3.7)Whereωistheangularrotationinrad/sec,ω-转鼓回转角速度，r/s）risradialdistancefromthecenterofthecentrifugetothesphere.r为转鼓中心轴线与微粒间距离，mB.

Centrifuges离心机

Wefirstdescribethethreemostusefultypesofcentrifuges.

我们先来描述这三种最有用的离心机：

a、Tubularbowl:管式离心机simpleyet,canprovideaveryhighcentrifugalforce；

最简单，可提供较大离心力；Tubularcentrifugescanbecool,arealadvantageinproteinwork.

管状离心机可以冷却，在蛋白质生产中很有利；Suspensionisusuallyfedthroughbottom,andclarifiedliquidisremovedfromthetop.

悬浮液由管底进，澄清液由管口流出。TubularbowlSolidsdepositonthebowl‘swallasathickpaste.

管壁上沉积物为浓浆*Thesuspensioncanbefeduntilsolidlossintheeffluentbecomesprohibitive.

可连续加料至流出物固体损失使离心不能正常进行*Thebowlmustbedismantled&cleaned.Suchintermittentoperationmaybeasignificantdisadvantage.

须定时拆卸、清洗，这种间断性操作也是最大的缺点Theanalysisoftubularbowlcentrifugedependsonfindingthepositionofatypicalparticleasafunctionoftime.

取不同位置上的典型离子分析

R0R1zrLiquidinterfacelIdealizationofthetubularbowlcentrifugeTobeginthisanalysis,assumethatthistypicalparticleislocatedat

为了便于分析，假设典型粒子位于以下几种情况：(1)AdistanceZfromthebottomofthecentrifuge

位于离心机底部Z向上(2)Locatedatpositionrfromtheaxisofrotation.

位于旋转轴r轴向上的微粒(3)ThispositionisbetweentheliquidsurfaceR1&thebowlradiusR0.

位于液体界面半径r1和管心半径r0的微粒之间(4)Theparticleismovinginboththedirections.

粒子同时在Z和r两个方向上移动

ItsmovementintheZdirectioncomesfromconvectionofthefeedpumpedinthebottomofthecentrifuge:Z方向的动力来源于离心机底部泵入料液的对流）

dz/dt=Q/[π(R02-R12)](3.8)WhereQisthefeedflow.料液流速Theeq(3.8)impliesthatgravityhasonlyanegligibleeffectintheZdirection.

等式3.8表示Z方向重力可忽略

WewillassumethatthecentrifugalforceissohighthattheliquidinterfaceR1isconstant,independentofZ.

假定离心力很大，R1是常数，由Z决定Theparticlemovementintherdirectionisrelatedtoitsradialpositionrby(3.7).r方向上运动与半径r有关dr/dt=d2(ρs-ρ)rω2/(18μ)(3.9)Wewillrewritethiseq.Intermsofthevelocityofaparticlesettlingundertheinfluenceofgravity:dr/dt=νg(rω2/g)(3.10)WhereνgisvelocitygivenbyEq.(3.6).Wecombine(3.8)&(3.9)tofindthetrajectoryofaparticlewithinthecentrifuge:

结合3.8和3.9，得出离心机内部微粒的运动轨迹dr/dz=(dr/dt)/(dz/dt)=νg(rω2/g)π(R02-R12)/Q(3.11)Ifνgislarge,theparticlewillquicklyreachthewall,ifQ↑,theparticlewillbesweptfartherupthetube.

如果νg

很大，微粒将很快到达管壁；如果泵入流速Q增大，悬浮固体微粒将向上走得更远方能到达管壁。Wenowfocusonthoseparticles,whicharemostdifficulttocapture.Theseparticlesenterthecentrifugeatt=R1&donotreachr=R0untiltheendoftheunit:atZ=l,wecanintegrateEq.(3.11)forthesehardtocatchparticlestofind,aftersomerearrangement.对于那些难以到达管壁的微粒分析，在r=R1时进离心机，在r=R0时也不会碰到管壁这时Z=l，对（3.11）积分。

Q=νgω2πl(R02-R12)/[g㏑(R0/R1)](3.12)Inmosttubularcentrifuges,thiseq.canbesimplifiedbecauseR0&R1areapproximatelyequal.Asaresult:

对于大部分管式离心机，这个等式可以简化，因为R0和R1近似相等(R02-R12)/㏑(R0/R1)=(R0+R1)(R0-R1)/㏑[1+(R0-R1)/R1]=…=2R2(3.13)WhereRisanaverageradius,aboutequaltoR0orR1.ThusEq.(3.12)becomes:R是平均粒径

Q=νg(2πlR2ω2/g)=νg[∑](3.14)Velocityνgisafunctiononlyofparticlesthemselves&independentoftheparticularcentrifugebeingused.νg为微粒本身的函数，与离心机无关[∑]quantityinsquarebrackets,whichhasdimensionsof[length]2isnotafunctionofparticlesbutonlyoftheparticularcentrifuge.

[∑]的量纲是长度的平方，表达离心机函数与微粒性质无关，代表离心机的分离特性。b、Disctypecentrifuge碟片式离心机（1）thexdirection

dx/dt=ν0-νωsinθ(3.15)Ifθ=0,particlewouldmoveonlybyconvection&thisequationwouldbeequivalenttoeq.(3.8).ν0=[Q/(n2πrl)](3.16)Fromamassbalance,thevolumeofν0averagedoverymustequalthisconvectivevelocity:1/l∫0lν0dy=[Q/(n2πrl)](3.17)Performingthisintegrationgives:1/l∫0lf(y)dy=1(3.18)Finally,combiningEq.(3.15)&3.18)dx/dt=ν0-νωsinθ≒ν0=[Q/(n2πrl)]f(y)(3.19)（2）Theydirection:dy/dt=νωcosθ(3.20)dy/dt=νg(ω2r/g)cosθ(3.21)dy/dx=(dy/dt)/dx/dt)=(2πnlνgω2/[Qgf(y)])r2cosθ(3.22)r=R0-xsinθdy/dx=(2πnlνgω2/[Qgf(y)])(R0-xsinθR0-xsinθ)2

cosθ(3.23)Q=νg[2πnω2(R03-R13)cosθ/(3g)]=νg[∑](3.24)*Thesecentrifugesaremostcommontypeforbioseparations,offercontinuousoperationatthepriceofcomplexequipment.

这种离心机在生物分离中非常常见，可连续操作但结构复杂，价格较高。*Feedusuallyentersatthetop&clarifiedliquidflowsoutanannularslitnearthefeed.料液由管顶进，清液从加料口附近环行裂口流出Thedistinguishingcharacteristicbetweentubularbowltypesisthemethodofsolidsareeitherremovedintermittently,asinthetubularcentrifugesorcontinuously,outoforificesonthesideofthecentrifuge.

和管式离心机最显著的区别在固体即非间歇式的被移出也不通过离心机管壁上的孔连续的去除Thusthenatureofthepackedsolidsdictatesthetypeofdischargeused.

填充固体的性质决定离心机类型Weconsideraparticlelocatedatposition(x,y).

假定一固体微粒位于（x，y）的位置wherexisthedistancefromtheedgeoftheouterdiscsalongthegapbetweenthediscs,yisthedistancenormaltothelowerdiscs,x沿碟片间隙方向与碟片外沿距离

y为微粒与最下面碟片外缘的距离R1istheinteredgeofthediscs.R1内缘半径Liquidisfedintothecentrifugesothatitflowsupwardthroughthegapbetweenthediscs,enteringatR0&leavingatR1.

料液延碟片间隙向上运动，进入时在R0处，流出时R1处Theparticleismovingbothinthex&ydirections.Itsvelocityinthexdirectionisduetoconvection&tosedimentation:

微粒、y向运动，在对流作用和离心沉降作用下dx/dt=ν0-νωsinθ(3.15)Whereν0istheconvectiveliquidvelocity.

泵送作用下的流体速度νωisparticle‘svelocityundercentrifugation.

微粒在离心力作用下的运动速度θisangleatwhichthediscsaretiltedfromvertical.

碟片与垂直方向上的夹角

Ifθ=0,thenparticlewouldmoveonlybyconvection,&thisequationwouldbeequivalenttoEq.(3.8).θ=0微粒只在对流作用下运动，等式就与3.8相等

Thevelocityv0hasthreeimportantcharacteristics：(1)itismuchlargerthanthesedimentationvelocityVwsince比沉降速度Vw大很多(2)

itisafunctionofradius&getsbiggerasthefluidflowsinwardtowardtheaxisofrotation.thisisbecausethetotalflowQisaconstant,buttheareaforflowgetssmallerastheaxisisapproached.v0是半径的函数，流体流向轴心时v0变大，因为流量Q是常数，半径变小，流动空间也变小(3)

V0isafunctionofy,foritmustreachzeroatthesurfaces.V0是y的函数，即在碟片表面V0=0

Thesethreecharacteristicsleadustoassumethatν0=[Q/(2nπrl)]f(y)（3.16）

WhereQ---totalflow液体的流量

n---thenumberofdiscs碟片数

r---thedistancefromtheaxisofrotation

微粒与转鼓轴线间距离

l---thedistancebetweendiscs相邻碟片间隙宽度f(y)---somefunctiongivingthevelocityvariationacrossthedistancebetweendiscs.

碟片间流速变化的函数Notethatfromamassbalance,thevolumeofv0averagedoverymustequalthisconvectiveVelocity:

液体在y方向上v0的平均速度与其对流速度相等1/l∫0lν0dy=[Q/(n2πrl)](3.17)Performingthisintegrationgives1/l∫0lf(y)dy=1(3.18)

Finally,combiningEq.(3.15)&(3.16)wefinddx/dt=ν0-νωsinθ≒ν0=[Q/(n2πrl)]f(y)(3.19)Again,rememberthatwearerecognizingthattheconvectivevelocityismuchgreaterthanthatofsedimentation.

因为对流速度要远大于沉降速度

Wenowturntomotionintheydirection.

现在对y方向上的运动进行分析

dy/dt=νωcosθ(3.20)Fromeq.(3.6)&(3.7)rewrite

dy/dt=νg(ω2r/g)cosθ(3.21)Wecombinethisresultwitheq.(3.19)

dy/dx=(dy/dt)/dx/dt)=(2πnlνgω2/[Qgf(y)])r2cosθ(3.22)Finallyr=R0-xsinθSothatdy/dx=(2πnlνgω2/[Qgf(y)])(R0-xsinθR0-xsinθ)2cosθ(3.23)Thisgivesthetrajectoryofaparticlebetweenthediscsofthiscentrifuge.

上式给出了碟片式离心机中的微粒的运动轨迹方程Wenowfocusontheseparticleswhicharemostdifficulttocapture.

对难分离的微粒进行研究

Theseparticleenterattheouteredgeofthediscs,wherey=0&x=0.

这些微粒在碟片外缘进入，此时x=0，y=0theyarecapturedattheinneredgeofthediscs,aty=l&x=(R0-R)/sinθ.

如果在其离开隙道前刚好抵达上碟片底部,其坐标为x=(r0-r)/sinθ，y=lAftercapture,they&otherparticlesareforcedalongthediscssurfacetotheouteredge,Wheretheyaredischarges.

微粒在离心力场作用下,将沿碟片底部运动到碟片外边缘,汇集到滤渣中，再清除掉。Wecanintegrateeq.(3.23)forthesehardtocaptureparticlestoobtain,aftersomerearrangement

根据上述临界条件分析，由微分方程（3.3-18）可写出其定积分方程

Q=νg[2πnω2(R03-R13)cosθ/(3g)]=νg[∑](3.24)Vg---propertiesofparticleVg反映微粒特性

∑---reflectsthegeometryofthecentrifuge.∑为离心机特性Δp/l=(μαρ0)ν(3.25)-dp/dr=(μαρ0)ν(3.26)2πrlυ=Q(3.27)-dp/dr=μαρ0Q/(2πrl)(3.28)Δp=ρω2(R02-R12)/2(3.29)Q=(πω2ρl/μαρ0)×(R02-R12)/(R0/Rc)(3.30)C.centrifugalfiltration离心过滤Q=dν/dt(3.31)ρcπ(R02-Rc2)l=ρ0ν(3.32)t=μαρcRc2/[αρω2(R02-R12)][(R0/Rc)2-1-α㏑(R0/Rc)](3.33)t=(μαρ0/αΔp)(V/A)2(3.34)V/A=(ρc/ρ0)(ρ0V/ρcA)=(ρc/ρ0)(R0-Rc)(3.35)t=(μαρc2/αρ0Δp)(R0-Rc)2(3.36)Thethird,centrifugalfiltration,thebasketunitisreallyacombinationofacentrifuge&afilter.Itconsistsofaperforatedbaskedwhichrotatesrapidly.Suspensionisagainfedalongtheaxisofthebowl&solidaccumulateonthewallofthebasket.

第三种篮式过滤机是离心和过滤方法的结合。它具有一个多孔可高速旋转的圆筒。SRBA2直通篮式过滤器Liquidflowsundercentrifugalforcethroughthecakewhichaccumulatesonthebasketwall,andoutthroughtheperforationsinthatwall.

悬浮液延着圆筒的旋转轴连续加入，通过筒壁上的小孔流出ThesolutionisthrownupagainsttheinneratradiusR1,whichiskeptconstantbyaddingmorefeed.

中空柱状料液的内径基本为一个常数，不随料液的加入而改变Cakeaccumulatesonthewall,thecakesinterfaceislocatedbyRc.

滤饼在筒壁上沉积，滤饼的内径为Rc

Forsimplicity,wewillassumethatthiscakeisincompressible.

为了简便起见，假定滤饼为不可压缩滤饼Theanalysisofcentrifugalfiltrationconsidersflowthroughacakeofcollectedsolids.

分析离心过滤应考虑到固体是通过液体流过滤饼进行分离Assuch,itismorelikethefiltrationanalysisinsection2.3&lesslikethedescriptionofothercentrifugesjustoutline.

正因如此，该过程相对来说,更象是2.3章提到的过滤，而不象离心过程WebegintheanalysisbyrecallingthatthepressuredropΔpinaflatcakeisproportionaltothefluidvelocityvthroughthatcake:

我们首先根据过滤压差Δp和流体通过滤饼的速度成正比进行分析Δp/l=(μαρ0)ν(3.25)Wherel---thethicknessofthecakel-滤饼的厚度μ---thefluidvelocityu-料液的粘度α---thespecificresistanceofthecakea-滤饼的比阻力ρ0---themassofsolidpervolumeofliquidfeedp0-单位体积料液所含的滤渣量Becauseourcakeisnotflat,thepressuredropvarieswiththeradius,andthisequationmustbereplaceby

由于转鼓壁上的滤饼并非平面状的，压差沿着半径方向而改变，故将式改写成为微分方程式

-dp/dr=(μαρ0)ν(3.26)Inaddition,thevelocitythroughthecakeisnotconstant,butishighertowardthecenter.thetotalvolumetricflowQisindependentofposition.

另外，通过滤饼的流速不是常数，轴心处流速较高。与过滤生产能力Q有关

Asaresult:2πrlυ=Q(3.27)Wherelisnowtheheightofthecentrifugel-离心机的高度

combining(3.27)&(3.26);wefind,

结合（3.27）、（3.26）两式可得

-dp/dr=μαρ0Q/(2πrl)(3.28)Wecaneasilyintegratethispressuredropisalmostallduetothecentrifugalforceonthefluid

对压降进行积分为

Δp=ρω2(R02-R12)/2(3.29)

Combingthispressuredroopwithintegratedformofeq(3.28)&(3.29),wefind,aftersomerearrangement,that

而在离心力场作用下转鼓壁上的料液层延径向的压力为：结合式（3.28）和（3.29）重新整理后得

Q=(πω2ρl/μαρ0)×(R02-R12)/㏑(R0/Rc)(3.30)p----thedensityofthesuspensionρ为悬浮液的密度

Remember,thatthisflowisnotconstant,butdropsasthecakethicknessincreases&henceRcdecreases.ThusbothQ&Rcarefunctionoftime.

流速不是常数随着滤饼厚度的增加而降低，因此Rc也是减小的。这样Q和Rc都是时间的函数。

Wearemostinterestedinthetimetoproduceagivenvolumeoffiltrate.Tofindthistime,werecognizethat对于产生一定体积的滤液所需的时间有下式成立

Q=dν/dt(3.31)

andthatρcπ(R02-Rc2)l=ρ0ν(3.32)wherePc---themassofsolidspervolumeofcake.Pc——单位体积滤饼中固体的质量

Whenwecombinetheseequationswitheq(3.30)wecanintegratetofindarelationgivingthecakeradiusRcasafunctionoftime.

把这些方程和式（3.30）结合，积分可得滤饼半径Rc是时间的函数

t=μαρcRc2/[αρω2(R02-R12)][(R0/Rc)2-1-α㏑(R0/Rc)](3.33)Thiseq(3.33)givesthetimetoobtainacakeofthickness(R0-Rc)&soisthedesiredresult.

式（3.33）给出了得到厚度为(R0-Rc)的滤饼所需要的时间

Wecanmakeaninterestingcomparisonbetweenthisresult&thatforaflatcake.

我们可将这个结果和平面滤饼进行比