<ation <ation o& motion = >#"ocity #"ocity in a c#ntri& c#ntri&( (a" a" &i#"d &i#"d at st#a st#ady dy sta stat# t# or or c#ntri&(ation #ation ν: υ =
<ation <ation o& motion ?& th# 'artic"# mo)in( ot R is not constant at t + 0 R + R0. = h# #atio tion #com #com#: #: R 2a 2 ( ρ − ρ 0 )ω 2t = ln ......( Eq.4.4) Bh#r# t + tim# 9 µ R0
= C#ntri C#ntri& &(# (# #a #atio tion n A s#& s#&"" that that r#"a r#"at# t#ss tim# tim# to th# th# distanc# tra)#"#d y th# 'artic"#.
Cr##'in( &"o o
Cr##'in( &"o conditions ar# sa""y satis&i#d in s#dim#ntation hos# R#yno"ds nm#r is )#ry sma"" R# 1. Re =
2aυρ µ
......( Eq.4.5)
<i"irim s#dim#ntation iso'ycnic o
o
o
Bh#n 'artic"# d#nsity and so")#nt d#nsity is #a" th# )#"ocity is z#ro and th# 'roc#ss is ca""#d iso'ycnic s#'aration %#'aration as#d on d#nsity and oyancy C#ntri&(ation sita"# to d#t#rmin# 'artic"# or macromo"#c"# d#nsity
Bh#n conc#ntration o& s#dim#ntin( 'artic"#s incr#as# )#"ocity #com# d#cr#as#. ?n hind#r#d s#tt"in( 'artic"#s ar# so c"os# and th#y ha)# a maHor #&&#ct on #ach oth#r.
%#dim#ntation co#&&ici#nt o
o
Bh#n a ody &orc# is a''"i#d )#"ocity thro(h a )iscos m#dim is sa""y 'ro'ortiona" to th# acc#"#ratin( &i#"d ?n th# cas# o& s#dim#ntation th# r#s"tin( constant a 'ro'#rty o& oth 'artic"# and m#dim is $s#dim#ntation co#&&ici#nt* d#&in#d asJ
S ≡
υ 2
ω R
Where ν = dR/dt S = the Svedberg (1S = 10-13s)
rom <. .3J
o
S =
2a 2 ( ρ − ρ 0 ) 9 µ
......( Eq.4.6)
<i)a"#nt tim# o
irst # d#&in# G as th# dim#nsion"#ss acc#"#ration th# ratio o& c#ntri&(a" to (ra)itationa" acc#"#ration &or a 'artic"ar c#ntri&(#: 2
G ≡
o
ω
R
g
h# nit is (Ks m"ti'"#s o& #arthKs (ra)itationa" acc#"#ration.
<i)a"#nt tim# o
ro(h a''roFimation o& th# di&&ic"ty o& a (i)#n s#'aration y c#ntri&(ation is th# 'rodct o& th# dim#nsion"#ss acc#"#ration G and tim# r#ir#d &or s#'aration t ca""#d as L#i)a"#nt tim#K: 2
Equivalent time
o
≡
Gt
=
ω
R
g
t ......( Eq .4.7)
y'ica" )a"#s o& Gt: o
<!aryotic c#""s
+ 0.3 F 106 s
o
Drot#in 'r#ci'itat#s
+ 9 F 106 s
o
act#ria
+ 18 F 106 s
o
Riosom#
+ 1100 F 106 s
%i(ma ana"ysis o o
Common"y s#d in indstry Ns# th# o'#ration constant O to charact#riz# a c#ntri&(# into hich &##d &"os at )o"m#tric &"orat# P: P + Q ν(QO Bh#r#J vg + s#dim#ntation )#"ocity at 1F(: υ g =
2a 2 ( ρ − ρ 0 ) g 9 µ
O + th# (#om#try and s'##d o& c#ntri&(# or crosss#ctiona" ar#a o& c#ntri&(#
%i(ma ana"ysis o
h# s#dim#ntation )#"ocity at 1F( can dir#ct"y d#t#rmin# sin(: υ g =
o
2a 2 ( ρ − ρ 0 ) g 9 µ
?n "aoratory s#&" #ation &or d#t#rmin# vg : R g ln R 0 υ
g
=
2
t
C#ntri&(#: asis o& s#'aration = c#ntri&(# is s#d to s#'arat# 'artic"#s or #)#n macromo"#c"#s: c#""s s-c#"""ar com'on#nts 'rot#ins nc"#ic acids #tc n o i t a r a ' # s & o s i s a M
%iz# %ha'# 4#nsity
C#ntri&(#: M#thods o& s#'aration o
M#thodo"o(y: o
o
o
Nti"iz#s d#nsity di&&#r#nc# #t##n th# 'artic"#s/macromo"#c"#s and th# m#dim in hich th#s# ar# dis'#rs#d 4is'#rs#d syst#ms ar# sH#ct#d to arti&icia""y indc#d (ra)itationa" &i#"ds
h# most common ty'#s: o
"ar o" m"ticham#r dis!-nozz"# dis!-int#rmitt#n dischar(# scro"" and as!#t
Drinci'"# o& c#ntri&(ation: %to!#Ks Ea = h# conc#'t o& sin( c#ntri&(a" &orc# to #&&ici#nt"y s#'arat# com'on#nts o& di&&#rin( d#nsiti#s = h# &ndam#nta" 'rinci'"# that a''"i#s to c#ntri&(ation is %to!#Ks Ea orm"a &or d#t#rminin( th# rat# o& s#dim#ntation ?t stat#s that a 'artic"# mo)in( thro(h )iscos "iid attains a
constant )#"ocity or s#dim#ntation rat# h# rat# can # )#ry s"o &or 'artic"#s hos#: 1 d#nsity is c"os# to that o& th# "iid 2 &or 'artic"#s hos# diam#t#r is sma"" 3 h#r# th# )iscosity is hi(h
Bh#n # int#(rat# this #ation at t+0 and R+R 0 distanc# &rom c#nt#r o& rotation to th# 'artic"#s n#ar#st to c#nt#r o& rotation: 2
ω
st = ln
R R0
R ln R0 t = ω
2
s
ln (5 4 )
=
1h
3600 s = 8.1 h 2 rev 2π rad 1 min −13 x x 10,000 (70x10 s ) min rev 60 s
Gt =
Gtg t ⇒ ω = g Rt
ω
2
R
1 2
54 x 10 s x 9.81 = 0.05 m x 2(3600 )s 6
= 1,213
rad s
m s2
x
= 11,590 rpm
1 rev 2π rad
x
1 2
= 1,213
60 s min
rad s
⇒
13360 r'm
r + 100 mm RC + 8000 F (
⇒
852 r'm
a RDM + 0000 r + 105. mm
⇒
188877F(
RDM + 80000 r + 210.8 mm
⇒
151101F(
2 %o")# RC )a"# &or:
3 Ca"c"at# th# radis o& rotation in mm i&: a RC + 5000 F ( RDM + 10000
⇒
.6 cm
RC + 7000 F ( RDM + 5000
⇒
25 cm
Drodction c#ntri&(# = h# conc#'t o& c#ntri&(a" &orc# a.!.a c#ntri'#ta" &orc# can # a''"i#d in rotatin( d#)ic#s sch as c#ntri&(#s h#n th#y ar# ana"yz#d in a rotatin( coordinat# syst#m. = c#ntri'#ta" &orc# is a &orc# that ma!#s a ody &o""o a cr)#d 'ath.
= ody #F'#ri#ncin( ni&orm circ"ar motion r#ir#s a c#ntri'#ta" &orc# toards th# aFis as shon to maintain its circ"ar 'ath.
Drodction c#ntri&(#s
Drodction c#ntri&(#s: "ar o" c#ntri&(# Eiid #nt#rs th# o" thro(h an o'#nin( in th# c#nt#r o& th# "o#r o" h#ad.
"occ"ation and s#dim#ntation = %#dim#ntation )#"ocity o& &"ocs: υ =
2 a 2 (1 − ε )( ρ − ρ 0 )ω 2 R 9 µ Ω (ε , a )
X >oid )o"m# &raction Y r#dction o& d#nsity X 4ra( r#dction &actor T X "oc radis a
%#dim#ntation at "o acc#"#rations = t "o ( &orc#s s#dim#ntation rat# s"os don and som#tim#s # simi"ar ith th# rat# o& trans'ort y di&&sion = 4i&&sion ronian motion X cons##nc# o& a $random a"!* y 'artic"#s d# to th#rma" #n#r(y k k + o"tzmanKs constant hich r#'r#s#nt#d y: x 2
=
2 Dt
Bh#r#J FZ[ + m#an sar# distanc# D + di&&sion co#&&ici#nt t + tim#
%#dim#ntation at "o acc#"#rations •
or s'h#rica" 'artic"#s o& radis \ nd#r(oin( ronian motion in a &"id )iscosity ; r#"ationshi' o& di&&sion co#&&ici#nt to th#rma" #n#r(y !:
D
•
•
=
kT 6 πµ a
?n a conc#ntration (radi#nt th# nidir#ctiona" &"F o& 'artic"#s is 'ro'ortiona" to 4 and th# (radi#nt dc/dF o& th# 'artic"# conc#ntration c. 4i&&sion is not a&&#ct#d y (ra)ity.
%#dim#ntation at "o acc#"#rations = ?n ord#r to com'ar# s#dim#ntation and di&&sion rat#s:
%#dim#ntation at "o acc#"#rations: ?soth#rma" s#tt"in( •
•
•
?& t#m'#ratr# T) do#s not chan(# o)#r th# h#i(ht h o& an #ns#m"# o& 'artic"#s th#n th# m#an !in#tic #n#r(y hich is 'ro'ortiona" to kT o& a"" 'artic"#s is th# sam# at a"" h#i(hts. h# 'ot#ntia" #n#r(y o& a 'artic"# o& mass m is sa""y #F'r#ss#d as m(h. ?& 'artic"# ar# sH#ct to oyant &orc# 'ot#ntia" #n#r(y #com#s >-o(h.
%#dim#ntation at "o acc#"#rations: ?soth#rma" s#tt"in( •
rom o"tzmann distrition r"# th# conc#ntration o& 'artic"#s at h at #i"irim is: c(h)
•
=
V ( ρ − ρ 0 ) gh kT
c ( 0 ) exp −
h# o"tzmann distrition r"# #F'"ain that: o
o
o
conc#ntration c is an #F'on#ntia" &nction o& h#i(ht nd#r isoth#rma" condition. or "ar(# d#ns# 'artic"# ith 'ot#ntia" #n#r(y (r#at#r than ! mamma"ian c#""s i"" # conc#ntrat#d at h+0. %ma"" 'artic"#s i"" ha)# ch]constant
%#dim#ntation at "o acc#"#rations: ?soth#rma" s#tt"in( = Con)#cti)# motion and D^c"#t ana"ysis o
D# nm#r is th# ratio o& th# s#dim#ntation )#"ocity to th# charact#ristic rat# o& di&&si)# trans'ort o)#r distanc# E: Pe
=
v D/L
o
?& D# 0.1: di&&sion is dominant and c(h) is distrit#d
o
?& D# [ 10: s#dim#ntation dominat#s
%#dim#ntation at "o acc#"#rations: ?nc"in#d s#dim#ntation •
Ra'id r#mo)a" o& hi(h d#nsity so"ids can # achi#)#d at 1 ×( y sin( inc"in#d s#dim#ntation. ##d containin( ss'#nd#d 'artic"#s is 'm'#d into th# s#tt"#r at its "o#r #nd
Dartic"#-&r## o)#r&"o #Fits th# ''#r #nd
Dartic"#-rich ss'#nsion "#a)#s in th# nd#r&"o
%#dim#ntation at "o acc#"#rations: ?nc"in#d s#dim#ntation •
•
?nc"in#d s#tt"#rs ar# d#si(n#d so that th# 'ath to th# s#dim#ntation com'"#tion o& a 'artic"# is #Ftr#m#"y short on"y &# mm #&or# th# s#dim#nt#d 'artic"#s #(in to # con)#ct#d toard th# nd#r&"o. ?& 'artic"at# &raction d#sir#d it can # atch conc#ntrat#d y r#cyc"# o& nd#r&"o ac! to th# tan! hi"# th# o)#r&"o "##ds o&& th# s'#rnatant.
%#dim#ntation at "o acc#"#rations: i"#d-&"o &ractionation •
•
•
•
•
?s d#si(n#d to s#'arat# 'artic"#s o& di&&#r#nt siz#s on th# asis o& hydrodynamics o& a )#ry thin "ay#r &"at horizonta" chann#" %am'"# ss'#nsion is 'm'#d and sH#ct#d to th# "aminar &"o )#"ocity (radi#nt. 4ri)in( &orc# at "o#r chann#" is (ra)ity or c#ntri&(a" s#dim#ntation. %t##' )#"ocity (radi#nt occr at th# ''#r and "o#r chann#": o
s 'artic"# trans'ort 'roc##d th# 'artic"#s nch ' accordin( to th#ir )#"ocity ths th# "o#r a"" is ca""#d accm"ation a"".
C#ntri&(a" #"triation = %imi"ar to inc"in#d s#dim#ntation and &i#"d-&"o &ractionat# s#dim#ntation in 'r#s#nc# o& &"id &"o. = ?n a c#ntri&(a" #"triation a.!.a cont#rstr#amin( c#ntri&(ation &"id is continos"y 'm'#d in th# o''osit# dir#ction to that s#dim#ntation.
?ndstria" a''"ication ?ndstria" c#ntri&(#s can # c"assi&i#d into to main ty'#s: s#dim#ntation and &i"t#rin( c#ntri&(#s. 1$ Sedimentation centrifuges
Ns# c#ntri&(a" &orc# to s#'arat# so"ids &rom "iids as #"" as to "iids ith di&&#r#nt s'#ci&ic (ra)iti#s d#cant#r dis!-stac! so"id-o" as!#t and t"ar o" c#ntri&(#s '$ i%tering centrifuges
Ns# c#ntri&(a" &orc# to 'ass a "iid thro(h a &i"tration m#dia sch as a scr##n or c"oth hi"# so"ids ar# ca'tr#d y th# &i"t#rin( m#dia. i"t#rin( c#ntri&(#s 'rimari"y d#a" ith '#r&orat# as!#t 'sh#r and '##"#r c#ntri&(#s
?ndstria" a''"ication = astewater processing 4#a"s ith s#'aration o& mnici'a" &arm 4 disso")#d air &"otation tra' (r#as# dri""in( md and #n)ironm#nta" ast#at#r s"d(#s. = Chemica% processing Bhich 'rodc#s ra 'rodcts sch as acids sa"ts oi" r#&in#ry y-'rodcts 'o"ym#rs oi"-at#r-so"ids and so on. = *harmaceutica% and Biotechno%og& industries hat man&actr# dr(s )accin#s m#dicin#s '#nici""in myc#"ia <-co"i act#ria a"(a# #nzymatic ast# #tc.
?ndstria" a''"ication = ue% and Biofue% industr& ?nc"din( synth#tic &#"s iodi#s#" #thano" c#"""osic #thano" a"(a# iomass d#at#rin(J &#" and "# oi" 'ri&ication #tc. = ood *rocessing Bhich d#a"s ith r#&inin( o& )#(#ta"# oi"s dairy mi"! ch##s# #tc.J 'o"try sin# and ##& r#nd#rin(J y#""o hit# and ron (r#as# s#'arationJ &rit and )#(#ta"# Hic#J ##r in# and "ior c"ari&ication #tc. = +ining and minera% processing ?nc"din( coa" tar sands co''#r 'r#cios m#ta"s ca"cim caronat# !ao"in c"ay and many mor#.
