CHAPTER 9: AGITATION AGITATION AND AERATION 9.1 Derive the relationship between the overall ass trans!er "oe!!i"ient !or #as phase $ G an% the in%ivi%&al ass'trans!er "oe!!i"ients( ) * an% ) G. How "an this relationship be sipli!ie% !or sparin#l+ sol&ble #ases, Given: a. Overall Overall ass ass trans!er trans!er "oe!!i"ient "oe!!i"ient !or #as phase( $ G b. In%ivi%&al ass trans!er "oe!!i"ients( "oe!!i"ients( ) * an% ) G Re-&ire%: a. Derive Derive the relati relations onship hip betwee between n $ G( an% ) G an% ) * b. How "an the relationship be sipli!ie% !or sparin#l+ sol&ble sol&ble #ases, ol&tion: E/pressin# the olar !l&/ in ters o! overall %rivin# !or"e !or ass trans!er an% an overall ass trans!er "oe!!i"ient(
P∗¿ A P A , G− ¿ N A = K G ¿ c∗¿ A− c A , L
¿
N A = K L ¿ where $ G is tere% the 0overall ass trans!er "oe!!i"ient base% on #as phase %rivin# !or"es. In an e-&ivalent anner( we also "an write 2sin# Henr+3s *aw(
mc∗¿ A P A , G=¿ where is the Henr+3s law "onstant. 4e "an rewrite the !irst two e-&ations as(
N A P A , G− P A ,i = k G
N A c A , i−c A , L= k L L &bstit&tin# the e-&ilibri& relationships( P∗¿ A=
m N A k L L
P A , i− ¿ A%%in# the partial press&re %i!!eren"es we obtain
N A mN A 1 m P∗¿ A= + = N A ( + ) k G k L k G k L L L P∗¿ A= P A ,G −¿ P A , G− P A ,i + P A , i−¿ ipli!+in# !&rther( 1 1 m = + K G k G k L
5or sparin#l+ sol&ble #ases( 1
k G
=0
There!ore( 1
K G
=
m k L
N A c A , i−c A , L= k L L &bstit&tin# the e-&ilibri& relationships( P∗¿ A=
m N A k L L
P A , i− ¿ A%%in# the partial press&re %i!!eren"es we obtain
N A mN A 1 m P∗¿ A= + = N A ( + ) k G k L k G k L L L P∗¿ A= P A ,G −¿ P A , G− P A ,i + P A , i−¿ ipli!+in# !&rther( 1 1 m = + K G k G k L
5or sparin#l+ sol&ble #ases( 1
k G
=0
There!ore( 1
K G
=
m k L
CHAPTER 9: AGITATION AGITATION AND AERATION 9.6 Prove that E-. 79.68 is the sae with E-. 79.6( an% E-. 79.6; is the sae with E-. 79.6< 79.6< Given: a E-. E-. 79.6 79.68 8 an% an% E-. E-. 79. 79.6 6 b E-. 79.6; an% E-. 79.6< Re-&ire%: a Prove th that −2
E-. 79.68
k L =0.31 N Sc
3
(
1/ 3
∆ ρ μc g ρ
Sh =0.31 N Sc Sc N Gr Gr E-. 79.6 N Sh
2
1 /3
)
is the sae as
1/ 3
b Prove that E-. 79.6;
k L =
2 D AB
D32
−2
+ 0.31 N Sc ( 3
1 /3
∆ ρ μc g
E-. 79.6< N Sh=2.0 + 0.31 N Sc N Gr
ρ
2
1 /3
)
is the sae sae as
1/ 3
ol&tion: a . −2 1/ 3
N Sh=0.31 N Sc N Gr
1/ 3
k L =0.31 N Sc
3
(
∆ ρ μc g ρ
2
1 /3
)
&bstit&tin# E-. 79.61( E-. 79.66 an% E-. 79.6= into E-. 79.6: 1/ 3
1 /3
3
k L D 32 μ D ρ g ∆ ρ =0.31 ( c ) ( 32 c2 ) D AB ρc D AB μc μc
1 /3
k L =0.31 1/ 3 −2 / 3 ρc D AB
ρc
1 /3
g
1/ 3
Dividing both sides by
1 /3
∆ρ
2 /3
μc
Combining similar − 1/ 3
μc
k L =0.31 1/ 3 −2 / 3 ρc D AB − 1/ 3
μc
k L =0.31 1/ 3 −2 / 3 ρc D AB
ρc
1 /3
g
1/ 3
1 /3
∆ρ
1
ρc
1 /3
g
1/ 3
1 /3
∆ρ
Multiplying
1
−1/ 3
μ c
1/ 3
μ c
− −2 / 3
μc μc k L =0.31 −2/ 3 −2 / 3 ρc D AB
1/ 3
g
1 /3
ρ
1 /3
∆ρ
2/ 3
c
μ c ρc D AB
¿ ¿
k L =0.31 ¿ −2
k L =0.31 N Sc
3
(
∆ ρμ c g ρ
2
1 /3
−2
)
k L =0.31 N Sc
3
(
∆ ρ μc g
b.
N Sh=
2 D AB
D 32
+ 0.31 N Sc 1/ 3 N Gr1 /3 −2
k L =2.0 + 0.31 N Sc
3
(
∆ ρ μc g ρ
2
1/ 3
)
&bstit&tin# E-. 79.61( E-. 79.66 an% E-. 79.6= into E-. 79.6<: 1/ 3
3
1/3
k L D 32 μ D ρ g ∆ ρ =2.0 + 0.31 ( c ) ( 32 c 2 ) D AB ρc D AB μ c
Dividing both sides by
ρ
2
1 /3
)
1/ 3
D AB μ ρ k L =2 + 0.31 1/ 3 c −2 /3 c D32 ρc D AB
1/ 3
1 /3
g
1 /3
∆ρ
2 /3
μc
Combining similar −1 / 3
D AB μ ρ k L =2 + 0.31 1/ 3 c −2 /3 c D32 ρc D AB −1 / 3
D AB μ ρ k L =2 + 0.31 1/ 3 c −2 /3 c D32 ρc D AB
1/ 3
1 /3
g
1 /3
∆ρ
1 1/ 3
1 /3
g
1 /3
∆ρ
1
Multiplying −1/ 3
μ c
(
−2/ 3
D AB μ μ k L =2 + 0.31 −2/ 3c −2 /3 c D32 ρc D AB
1/ 3
μc ρ c D AB
¿ ¿
D AB k L =2 + 0.31 ¿ D32 −2
1 /3
−2
1 /3
D AB ∆ ρ μc g k L =2 + 0.31 N Sc 3 ( ) 2 D32 ρ
D AB ∆ ρμ c g k L =2 + 0.31 N Sc 3 ( ) 2 D32 ρ
1 /3
g
ρ
∆ρ
2/ 3
c
1 /3
−
1/ 3
μ c
)(
)
CHAPTER 9: AGITATION AND AERATION 9.> A "+lin%ri"al tan) 71.66 %iaeter is !ille% with water to an operatin# level e-&al to the tan) %iaeter. The tan) is e-&ippe% with !o&r e-&all+ spa"e% ba!!les( the wi%th o! whi"h is one tenth o! the tan) %iaeter. The tan) is a#itate% with a ?.= %iaeter( !lat'bla%e %is) t&rbine. The ipeller rotational spee% is >.>= rps. The air enters thro&#h an open en%e% t&be sit&ate% below the ipeller an% its voletri" !low rate is ?.?61; =@s at 1 .?< at an% 68 %e# C. %ensit+ 99;.?< )#@= Cal"&late: a. b. ". %. e.
Power Re-&ireent Gas Hol%'&p a&ter'ean Diaeter Inter!a"ial area ol&etri" ass'trans!er "oe!!i"ient
Given: DT 1.66 4 1@1? DT DI ?.= N >.>= rps ?.?61; =@s
Re-&ire%: a. b. ". %. e.
P H D=6 a )la
vis"osit+ <.9?> / 1?B'> )#@'s
ol&tion: a. Nre 799;.?<7>.>=7.=B6 <.9?> / 1?B'> Nre >6(918.?=> 1?(???
%. Inter!a"ial Area: a H@ D=6 a 9<.8>>; @
Po 799;.?< 7>.>=B= 7.= B8 =1>>.<<6 4
e. ol&etri" ass'trans "oe!!.
2sin# e-&ation 9.8= o! Faes *ee
$l >.8 @s
P 1=>1.=61< 4
$la >.8 79<.8>>;
b. v 7@>71.6671.66B6 v 1.>= = s 7>/ .?61; @ 71.66B6 ?.?1< @s ?.?6 @s 2sin# e-&ation 9.>< o! Faes *ee H ?.?;9? ". 2sin# e-&ation 9.>6 D=6 >.<1=6 / 1?B'= D=6 >.<1=6
CHAPTER 9: AGITATION AND AERATION
$la ?.?>81 @s
9.8 Estiate the vol&etri" ass'trans!er "oe!!i"ient ) *a !or the #as'li-&i% "ontra"tor %es"ribe% in Proble 9.> b+ &sin# a "orrelation !or ) *a an% "opare the res< with the e/periental val&e. Given: Rea"tor vol&e( v 1.>== s ?.?1< @s P 1=>6 4atts Re-&ire%: ) *a 7&sin# e-&ation 9.;1 b+ Faes *ee J ) *a "opare% with e/periental val&e ol&tion: 1342 ) *a ?.?67 1.43 ?.> 7?.?1<?.8 ?.?8>< s'1
E/periental
Estiate% val&e ?.?>81 s'1
( 0.0548−0.0451 ) x 100 J !or vol&etri" ass'trans!er "oe!!i"ient 1;.;??;J error (0.0548 )
CHAPTER 9: AGITATION AND AERATION 9. The power "ons&ption b+ an a#itator in an &nba!!le% vessel "an be e/presse% as
( ) 2
P mo
ρND I = f 3 5 μ ρN D I
Can +o& %eterine the power "ons&ption an% ipeller spee% o! a 1(???'#allon !erenter base% on !in%in#s o! the opti& "on%ition !ro a one'#allon vessel b+ &sin# the sae !l&i% s+ste, Is +o&r "on"l&sion reasonable, 4h+ or wh+ not,
Given:
( ) 2
P mo
ρND I = f 3 5 μ ρN D I
P 1??? #allons 1 #allon Re-&ire%: Can the power "ons&ption an% ipeller spee% o! P be %eterine% on !in%in#s o! the opti& "on%ition !ro b+ &sin# the sae !l&i% s+ste, 4h+, ol&tion:
V P V m
=1000 The s"ale ratio is 1 D I , P = 1000 3 =10 D I , m
To a"hieve %+nai" siilarit+( the three n&bers !or the protot+pe an% o%el &st be e-&al
[
Pmo 3
5
] [
ρN D I P
=
P mo 3
5
ρN D I
[ ] [ ] 2
2
ρND I ρND I = μ P μ
m
]
m
2sin# the sae !l&i% !or o%el an% protot+pe( ρ P = ρm K μ P = μm
( Pmo ) p =10 [ P mo ] m 5
[ ] N P
3
N m
The e-&alit+ o! Re+nol%3s n&ber re-&ires
N P =0.01 N m while the e-&alit+ o! 5ro&%e n&ber re-&ires 1 N P = N m √ 10 whi"h shows two "on!li"tin# "on"epts. I! ρ P ≠ ρm K μ P ≠ μm K
[]
[]
1 μ μ = ρ m 31.6 ρ
P
There!ore( i! )ineati" vis"osit+ o! protot+pe is siilar to water( the )ineati" vis"osit+ o! the !l&i% whi"h nee%s to be eplo+e% !or the o%el sho&l% be 1@=1. o! the )ineati" vis"osit+ o! water. It is ipossible to !in% the !l&i% whose )ineati" vis"osit+ is that sall. As a "on"l&sion( i! all three %iensionless #ro&ps are iportant( it is ipossible to satis!+ the %+nai" siilarit+.
CHAPTER 9: AGITATION AND AERATION The spe"i!i" o/+#en %ean%s an% "riti"al o/+#en "on"entrations !or t+pi"al i"robial plants an% anial "ell "<&res are liste% below Cell "<&re
-o
Con"entration( ol@*
Es"heri"hia Coli
?.8 ol7#%w'1h'1
?.??<6
itisvini!era 7#rape
?. ol7#%w'1h'1
?.?88
Chinese Haster Ovar+
=.?/1?'1? ol7#%w'1h'1
?.?6?
a.Estiate the )*a re-&ireent to a"hieve "ell "on"entrations o! 68 #ra %r+ wei#ht@* !or E. Coli an% . ini!era an% =.?/1?9 !or CHO "ells( while aintainin# %issolve% o/+#en "on"entration above "riti"al. The o/+#en sol&bilit+ in the e%ia &se% !or the "<&res is ;.6/1?'= )#@= b. The relationship between )*a an% the power inp&t to a 1'= stirre% biorea"tor is )*a
Pt ¿ Vl
?.8
( "opare the biorea"tor power re-&ireents !or "<&re o! the three %i!!erent "ell t+pes
&n%er the "on%itions %es"ribe% in a. ol&tion: a.
α 7
!or e "oli: - <.8ol@h 768@*696.8 ol@*.h v.vini!era: - ?. ol@h 768@*18 ol@*.h CHO: - =/1?'1?ol@h 7=/1?9@*?.9 ol@*.h
O/+#en ol&bilt+: ;.=/1?'= )#@= 696.8 ol@*.h L;.6 /1?'= )#@= 71)ol@=6)#71/1? ol@)ol'?.???6)la )*a 9
18 ol@*.h L;.6 /1?'= )#@= 71)ol@=6)#71/1? ol@)ol'?.?88)la )*a <<.6=8=@hr ?.9 ol@*.h L;.6 /1?'= )#@= 71)ol@=6)#71/1? ol@)ol'?.?6)la )*a >.=9?6@hr
Pt
b. )*a ) ( Vl )
?.8
PT E.C. 9?;68.8<= =@hr6 $ 6 PT v.v. ;;8.><6 =@hr6 $ 6 PT CHO 19.6;=9 =@hr6 $ 6 PT E.C. PT v.v. PT CHO
CHAPTER 9: AGITATION AND AERATION ETIATING $ *a 2ING THE IP*E DMNAIC ETHOD A stirre% !erenter is &se% to haeatopoieti" "ells isolate% !ro &bili"al "or% bloo%. The li-&i% vol&e is 18 *iters. The siple %+nai" etho% is &se% to %eterine $ *a. The air !low is sh&t o!! !or a !ew in&tes an% the %issolve% o/+#en level %ropsK the air s&ppl+ is then re"onne"te% at a !low rate o! ?.68 *@s. The !ollowin# res<s are obtaine% at a stirrer spee% o! 8? rp. Tie( s 8 6? O/+#en tension 7Jair sat&ration 8? 4hen stea%+ state is establishe%( the %issolve% o/+#en tension is ;
? rp. The probe response tie &n%er these "on%itions was 6.< se"on%s. 4hen the $*a eas&reent was repeate% &sin# nitro#en to %eo/+#enate the i/t&re( the res<s !or o/+#en tension as a !&n"tion o! tie were siilar to those liste%. Estiate $ *a.
Solution: ¿
ln (
$*a
AL− AL 1 ¿
AL− AL 2
)
t 2− t 1 78 −50 ) 78− 66 ( 20 −5 ) !
ln ( $*a
0.056/s
CHAPTER 9: AGITATION AND AERATION TEADM'TATE $*a EA2REENT A 6?'* stirre% !erenter "ontainin# a"ill&s th&rin#iensis is &se% to pro%&"e a i"robial inse"ti"i%e. The o/+#en balan"e etho% is applie% to %eterine $*a. The !erenter operatin# press&re is 18? $Pa an% the "<&re teperat&re is =?oC. The o/+#en tension in the broth is eas&re% as <6J &sin# a probe "alibrate% to 1??J in sit& &sin# H 6O an% air at =?oC an% 18?$Pa. The sol&bilit+ o! o/+#en in the "<&re !l&i% is the sae as in H6?. Air is spar#e% into the vesselK the inlet #as !low rate eas&re% o&tsi%e the !erenter at 1 at press&re an% 66oC is ?.6=@s. The e/it #as !ro the !erenter "ontains 6?.1J o/+#en an% has a !lowrate o! <.91 in'1. a. Cal"&late the vol&etri" rate o! o/+#en &pta)e b+ the "<&re b. 4hat is the val&e o! $ *a. Solution: 1
)
NA " V L
[( ) ( ) ] # g P Ag # g P Ag i− o $ $
0.201 x 1.48
(
¿
1 60
8.91 x
(
(
0.23 L
!
)(¿ (
)(
30 + 273 ) K ¿)
¿
0.2099 %tm )
¿
1 0.08205 L& %tm
mol&K
¿
)
−¿
( 22 +273 ) K
1
¿ ( 20 L)
( 1.6357 x 10− −1.4572 x 10− ) =1.0878 x 10− 4
0.08205 ( 20 )
4
5
mol L & !
e"a&se o! stea%+ state the rate o! o/+#en trans!er is e-&al to the rate o! o/+#en &pta)e b+ the "ellsK the vol&etri" rate o! o/+#en &pta)e b+ the "<&re is 1.0878x10 -5 mol/ L.s b) Ass&e that the #as phase is well'i/e% so that the o/+#en "on"entration in the b&bbles "ontainin# the li-&i% is the sae as the o&tlet #as( that is 6?.1J. As the %i!!eren"e in the "oposition o! the #as phase to be "onstant thro&#ho&t the !erenter. The sol&bilit+ o! o/+#en in H6? =?oC an% 1at air press&re is <.?8/1?'= )#@=. Deterine the sol&bilit+ at the !erenter operatin# press&re o! 1.>< at an% #as phase o/+#en ole !ra"tion o! ?.6?1. 1.48 %tm ( 0.201 ) 8.05 x 10
−3
P$ 2 ' AG 1
CA*(6
¿
P$ 1 ' AG 2 CA*(1
1 %tm ( 0.2099 )
(
L
g
)
?.?11>#@*
CA* in the !erenter is <6J o! the o/+#en sol&bilit+ at =? oC an% 1.>
(
1.48 %tm 1 %tm −5
)
−3
(
8.05 x 10
1.1 x 1 0 mol / L & ! (
$ *a
L
g
) 9.;;/1?'= #@*
32 g ) mol −3
0.0114 g 9.77 x 10 g − L L
CHAPTER 9: AGITATION AND AERATION DMNAIC TECHNI2E
K La = 0.22 s-1
A strain o! AQoba"ter vinelan%ii is "<&re% in a 18 = stirre% !erenter !or al#inate pro%&"tion. 2n%er "&rrent operatin# "on%itions( $ *a is ?.1; s'1. The sol&bilit+ o! o/+#en in the broth is appro/iatel+
( )( ( )( 0.17
!
a) Sa/
mmol 1 hr 12.5 g&hr 3600 !
−3
8 x 1 0
kg
3
m 1 mol 1000 mol
)
)( )( 32 g 1 mol
1 kg 1000 g
)
=
12240 g
m
3
b) Ass&e that a%%ition o! "opper s&lphate %oes not a!!e"t CA* o! $ *a
( )( ( )( 0.17 !
Sa/ 3
mmol 1 hr g&hr 3600 !
−3
8 x 1 0 kg 3
m 1 mol 1000 mol
)
)( )( 32 g 1 mol
1 kg 1000 g
)
=
51000 g 3 m
CHAPTER 9: AGITATION AND AERATION GA HAND*ING 4ITH R2HTON T2RINE A !erenter o! %iaeter an% li-&i% hei#ht 1.> is !itte% with a R&shton ipeller o! %iaeter ?.8 an% o!!'botto "learan"e ?.=8 operate% at ;8 rp. The !erentation broth is spar#e% with air at a vol&etri" !low rate o! ?.6<=@in. Hal!'wa+ thro&#h the "<&re soe bearin#s in the stirrer %rive be#in to !ail an% stirrer spee% &st be re%&"e% to a a/i& o! >8 rp !or the reain%er o! the pro"ess. a. 2n%er the noral operatin# "on%itions( is the #as "opletel+ %isperse%, b. A!ter the stirrer spee% is re%&"e%( is the ipeller !loo%e% or loa%e%, Solution: a) Ni
75 / mi(
( ) 1 mi( 60 !
1.68s'1 −1
1.25 !
2
5r
¿ Di g
¿2 (0.5 m ) ¿ ?.?;9 ¿ ¿
5or "oplete #as %ispersion 5l# ?.6
( )
5l#: 5loo%in#' loa%in# transition
0.5
D I D$
#r
( ) 0.5 m 1.4 m
0.5
?.6
0.5
( 0.0796 )0.5 ?.?==;
5# 5l#NiDi= 7?.?==;71.68s'1 7?.8= 5.27x10 -3 m3/s ?.6< =@in 5#( vol&etri" !lowrate o! #as #reater than the operatin# !low rate( we "an "on"l&%e that the air provi%e% is compltl! "isp#s" &n%er noral "on%itions. b)
Ni
45 / mi( (
1 mi( ) ?.;8s'1 60 ! −1
0.75 !
2
5r
¿ Di g
¿ 2( 0.5 m) ¿ ?.?6<; ¿ ¿
5loo%in#'loa%in# transition 5l# =?
( ) D I D$
3.5
#r =?
( ) 0.5 m 1.4 m
3.5
?.?6=>
5# 5l#NiDi= 7?.?6=>7?.;8s'1 7?.8= 0.0021$ m3/s
At re%&"e% stirrer spee%( a/i& air !low rate "an be han%le% witho&t ipeller !loo%in# as operatin# !low rate 7?.6<=@in is #reater than this. The ipeller is %L&&'('.
CHAPTER 9: AGITATION AND AERATION Clostri%i& a"etob&t+li"& "arries o&t anaerobi" !erentation an% "onverts #l&"ose into a"etone( b&tanol alon# with saller "on"entrations o! b&t+rate( a"etate( et". In !erentation the !ollowin# pro%&"ts were obtaine% !ro 1?? oles o! #l&"ose an% 11.6 oles o! NH =( as nitro#en so&r"e. Pro%&"ts !ore%: Cells 1=oles &tanol 8oles A"etone 66oles &t+ri" a"i% ?.> oles
a"eti" a"i% 1> oles CO6 661 oles H6 1=8oles Ethanol ?.; oles
+ per!orin# a "arbon( nitro#en( h+%ro#en( an% o/+#en balan"e( %eterine the "hei"al "oposition o! the "ells. Solution:
+ per!orin# a "arbon( h+%ro#en( nitro#en an% o/+#en balan"e( %eterine the eleent "oposition o! the "ells. 1??CH16O 11.6 NH=
1=CaH bO" N% 8C>H1?O 7b&tanol 66C=HO7a"etone
?.>C>HC6H1>O6 7a"eti" a"i% 661CO6 1=8H6 ?.; C6HO7ethanol where CaH bO" N% represents eleental "oposition o! "lostri%i& "ells Carbon alan"e: 1??7 11.67? 1=7a 87> 667= ?.>7> 1>76 66171 ?.;76 : a >.> H+%ro#en alan"e: 1??716 11.67= 1=7b 871? 667 ?.>7< 1>71> 1=876 ?.;7 : b 1.?6 O/+#en alan"e: 1??7 11.67? 1=7" 871 6671 ?.>76 1>76 66176 ?.;71 : " =.<< Nitro#en alan"e: 11.671 1=7% : % ?.<
Chei"al Coposition o! the "ell *.*6+16.02&3.88,0.86
CHAPTER 9: AGITATION AND AERATION
Aerobi" C<&re o! a""haro+"es "erevisae in a s+ntheti" e%i& pro%&"e% the !ollowin#: U ?.6@hr R 1.? M+@s 9? #ras %r+ "ell@#ol #l&"ose #l&taate ?.?>= #ras #l&taate@ # "ell'hr Ass&in# that "arbon "ontent o! the "ell is >8J( "he") "arbon an% o/+#en balan"e with respe"t to #iven "<&re. aCH16O bC8H9O> N "O6 #l&tai" a"i%
%CHVOWOX e CO6 !H6O
Solution: R 1.? ?6 CO6 " e U ?.6 hr '1 an% >8J C in bioass K 1ol "ell bioass 16#C
%
0.2 ( 0.45 ) hr ;.8/ 1?'= ol C@# %r+ "ell. Hr 12 g / mol
1?? # "ell bioass "ontainin# >8# C 7>8J C
g c)ll *iom%!! g%r*o( >8J (100 ) + c)ll *iom%!! 12 ( 100 )
4"ell
45
=26.67
4 6.; #@ol CHVOW NX 6.; 1716 V71 W71 X71> 1>.; V 1W 1>X 7E-&ation 1
a #l&"ose
= ' . /S
0.2 / hr 90
g /r0 c)ll g gl1co!)
2.22x10 -3 mol lucos/ "#! cll. +#
0.043 b #l&taate
ggl1t%m%t) gc)ll&hr
2.$25x10 -* mol lutamat/ "#! cll. +# g 147 gl1t%mic %ci/ mol
Eleental alan"e: C: a 8b % e H: 16a 9b %V 6! O: a >b 6" W% 6e ! : 6"6e : a >b W% ! N: b X%
* /
−4
2.925 x 10 −3 7.5 x 1 0
0.03$0
2sin# H an% O: 16a 9b %V 6! '6 7a >b W% ! b V% ' 6W%
sin"e b 6.968/1?'> : " = 7.5x10-3
6.968/1?'> V 7;.8/1?'= ' 6W 7;.8/1?'= −4 2.925 x 1 0 + 0.015 2 −3 V 7.5 x 1 0 !ro e-&ation 1: 1>.; V 1W 1>X : 1>.; V 1W 1>7?.?=9? 1>.16> V 1W &bstit&tin# V: 1>.16>
2.925 x 10
−4
+ 0.015 2
7.5 x 10
−3
1W : = 0.7825 = 1.60*
2sin# e-&ation o! O: a >b W% ! K ! a >b Y W% ! 76.66/1?'= >76.968/1?'> Y ?.;<687;.8/1?'= = 8.6213x10 -3 2sin# e-&ation o! C: a 8b % e K e a 8b Y % e 7 6.66/1?'= 87 6.968/1?'> Y ;.8/1?'= = c = 7.2825x10-3 aCH16O bC8H9O> N "O6
%CHVOWOX e CO6 !H6O
ns4#: 2.22 6+12&6 2.$255+$&*, 7.2825 &27.5 +1.60*&0.7825&0.03$0 7.2828 & 8.6213+2&
CHAPTER 9: AGITATION AND AERATION A 8? = biorea"tor 7H@DT6.8K wor)in# vol&e?J e-&ippe% with two sets o! a stan%ar% !lat bla%e t&rbine is &se% !or +east #rowth( the biorea"tor is operate% "ontin&o&sl+ at a %il&tion rate o! ?.= hr '1. The or#anis obe+s the ono%3s e-&ation 7Z?.> hr '1 an% $ s6 )#@=. The inlet s&#ar !ee% "on"entration is 8? )#@=. The biorea"tor is aerate% an% a#itate% at ?.8 vv at ? rp. The +iel% o! bioass base% on #l&"ose is ?.8 # "ell 7%r+ per #ra #l&"ose "ons&e%. The %ensit+ an% vis"osit+ o! the broth are 16?? )#@= an% ?.?6 Pa[s. Cell !or&la: CH1.
D?.= hr '1 Z?.> hr '1 $s6 )#@= CO8? )#@= 0.5 gc)ll ( /r0 ) MS@ ggl1co!)
N? rp \ 16?? )#@= Z?.?6 Pa[s Cell !or&la: CH1.
ol3n:
3 =2.5 D$
H li-&i% hei#ht DT tan) %iaeter
4 2 4 2 4 3 V $ = D $ 3 = D $ ( 2.5 D $ )= ( 2.5 ) ( D $ )
4
4
4
T 8? = 3
50 m
4
√
50 m
3
= ( 2.5 ) ( D$ ) K D$ = 3 4 4 ( 2.5 ) 3
4
DT 6.9>6?
μ 0.02 P%6 ! −5 3 1.67 10 5= = = 7 m /! ρ 1200 kg / m3
D$
5ro io"hei"al En#ineerin# b+ Faes *ee: D I =3 K DI ipeller %iaeter D$ 2.9420 m D I = = =0.9807 m 3
3
5or an air'ele"trol+te sol&tion: k L % =2.0 7 10
−3
[ ] P m V L
0.70
V !
0.2
) *a vol&etri" ass' trans!er "oe!!i"ient P #asse% power * vol&e o! the li-&i% or broth s s&per!i"ial velo"it+
5or the power n&ber( N p(
7e-. 9.;6( p. 6;( Faes *ee
60 rpm
2
N D I ρ
8
")0(ol/ ! N1m*)r , ") =
=
μ
(
1 mi( 60 !)c
)(
2
kg
0.9807 m) ( 1200
3
m
)
0.02 P%6 !
¿ 57706.3494 Re ] 1????K N p 7p. 68<( Faes *ee N 7?@? rps
5or a !lat'bla%e t&rbine at Re] 1????
(
)
3
kg 60 5 Pmo= N P ρN D I =6 1200 3 ( rp!) ( 0.9807 m) m 60 3
5
Pmo=6531.5066 ( 1(g%!!)/ po9)r )
5or the #asse% power
( )
[ ] [ ] [ ] [ ]
P D log 10 m =−192 I Pmo D$
4.38
2
D I N 5
2 1.96 D I D$
0.115
D I N g
:
7e-.
3
N D I
Faes *ee 3
:=
0.5 m ;2 3
m *roth6mi(
( 50 m3 7 0.60 ) =0.25 m3 / ! 60 2 ¿ 60
(
¿
0.9807 1.96 2.9420
( 0.9807 )¿
)
¿
[
3
0.25 m / ! 60 ( )( 0.9807 )3 60
[
Pm
]
4.38
0.9807 m =−192 2.9420 m 2 ( 6531.5066 )
[
]
60 (0.9807 ) ( rp! ) 60 2
−5
1.67 7 10
log10 ¿
P 191. 199= 4 5or "op&tation o! ) *a( s&per!i"ial velo"it+ 1 mi( 3 0.5 m %ir 60 !)c
(
: V != = S
3
m m)/i%6mi( 4 ( 2.9420 m)2 4
)
( 50 m
3
7 0.60 )=0.0368 m / !
]
0.115
¿
9.8=(
p.
68<(
−3
k L % =2.0 7 10
[
6191.1993 3
50 m ( 0.60 )
]
0.70
m ( 0.0368 ) !
0.20
=0.0431 !−1
Ass&e: ol&tion o! H6O> 6.? ols@*( !in% the "orrespon%in# o/+#en sol&bilit+ !ro Table 9.6 o! io"hei"al En#ineerin# b+ Faes *ee( p. 66 CO6 1.?6 ol@*
2se the sol&bilit+ o! O6 !ro Table 9.6 to %eterine the Henr+3s "onstant an% the e-&ilibri& "on"entration o! O6 at that "on%ition. 1 %tm %tm6 L 3 ; = =0.9804 1.02 mmol / L mmol 2
0.21 %tm
¿
L =
%tm6 L mmol
= 0.2142
0.9804
mmol L
5or ) *a(C*( the vol&etri" ass'trans!er "oe!!i"ient at the e-&ilibri&: k L % , ¿ = L
](
[
0.0431 3600 ! mmol 0.2142 1 hr ! L
)[
1 mol 1000 mmol
][
1 kmol 1000 mol
][
][
32 kg 1000 L 3 1 kmol 1m
]
=1.0635
io"hei"al rea"tion liite%: 2sin# ono% E-&ation D = μ = μm
(
!
K ! + !
)
Deterine the E/it s&bstrate "on"entration( Cs 0.3
hr
=
0.4
hr
[ ] !
2
kg + ! 3 m
< !=6.0 kg / m
3
Then( %eterine the "ell "on"entration that was pro%&"e% x − xo = x == x ( !o− ! )= 0.5 !
g /r0 c)ll kg kg ( 50 −6 ) , 3 =22 3 g gl1co!)co(!1m)/ m m
5or ^O6( #ras o! o/+#en re-&ire% per #ra %r+ "ell pro%&"e% 7aleles( 19;1 32 + 8 3 −16 > ; = + 0.01 ;8 −0.0276 8 +0.01714 N 8 −0.08 3 8 2
= x 6 + !
4here: C( H( O Y is the n&ber o! atos present in the "arbon so&r"e C3( H3( O3( N3( ' is the per"ent "arbon( h+%ro#en( o/+#en an% nitro#en respe"tivel+ in the "ell
kg 3
m 6 hr
5ro CH16O 7"arbon so&r"e C H16 O 4 716 1671 71
> ; = 2
32 ( 6 ) + 8 ( 12)− 16 ( 16 ) 0.5 ( 180 )
> ; =0.7656 2
+ 0.01 ( 32.52)− 0.0276 (48.78 )+ 0.01714 ( 11.38 )− 0.08 (7.32)
g; 2 g /r0 c)ll
> ; 6 x 6 μ m=0.7658 2
5ro CH1.. /1?? ><.;< H 1.<71 H3 71.<71@ 6>. /1?? ;.=6 O ?.871 O3 7?.871@ 6>. /1?? =6.86 N ?.671> N3 7?.671>@ 6>. /1?? 11.=< _ 6>.
g ;2 g /r0 c)ll
( ) ( )= kg 3 m
22
0.4
hr
6.73904
kg m 6 hr 3
^O67/7Z ) *a( Cl io"hei"al rea"tion ass Trans!er Th&s( it is ass trans!er liite% be"a&se there3s ver+ little o/+#en that is bein# trans!erre% or %issolve% i broth.
CHAPTER 9: AGITATION AND AERATION
In an o/+#en absorption st&%+( the !ollowin# %ata were obtaine% on the o/+#en trans!er "apa"it+ o! an air %i!!&sion &nit. 9.>=9 =@s T 16`C H* >.6;
Air b&bble %iaeter: 6.8 / 1?'= Air b&bble velo"it+: ?.= @s DT 99.8
The %issolve% o/+#en "on"entration eas&reent was tab&late% as: Tie( in = 9 16 18 1< 61
Cs( )#@= ?. /1?'= 1. /1?'= =.1 /1?'= >.= /1?'= 8.> /1?'= .? /1?'= ;.? /1?'=
5ro the in!oration( "op&te the: a. $ *a an% ) * b. ass o! o/+#en per ho&r trans!erre% per 6<.=1; = an% Qero %issolve% o/+#en "on"entration an% the o/+#en trans!er e!!i"ien"+ ". How &"h o/+#en will be trans!erre% to waste with V?.
!m= !
[
P* 29.4
+
; t 42
]
P b ' absol&te press&re at the %epth o! air release( psi Ot Y "on"entration o! O6 in air leavin# the tan)( J Cs ' sat&ration "on"entration o! O6 e/periental teperat&re Cs Y ean o/+#en sat&ration "on"entration
1?J absorption ;t =
mol)! of 1(%*!or*)/ ; 2 tot%l mol)!of l)%?i(g %ir
7 100 =
21 ( 1 −0.1) 21 ( 1−0.1 ) + 79
7 100=19.3
5or Cs( −2 kg
!m=1.08 x 10
m
Tie( in = 9 16 18 1< 61
3
[
]
20.2 p!i 19.3 + 29.4 42
=0.01238
kg 3 m
Cs Y Cs( )#@= ?.?16=< ' ?. /1?'= ?.?16=< ' 1. /1?'= ?.?16=< ' =.1 /1?'= ?.?16=< ' >.= /1?'= ?.?16=< ' 8.> /1?'= ?.?16=< ' .? /1?'= ?.?16=< ' ;.? /1?'=
Plot Cs Y Cs vs. Tie −4
m = !lop)=−3.6071 7 10
[
?.?11;< ?.?1?;< 9.6< /1?'= <.?< /1?'= .9< /1?'= .=< /1?'= 8.=< /1?'=
]
60 mi( kg kg =−0.0216 3 m 6mi( 1 hr m 6 hr 3
Deterine the ) *a k L % =
!lop) = m)%(∆ co(c)(tr%tio( of ; 2
−0.0216
kg 3
m 6 hr =2.7 hr −1 ( 0 −8 7 10−3) kg3 m
Deterine the inter!a"ial area !or the "al"&lation o! the ass trans!er "oe!!i"ient( ) * 6 : 3 L A 4 2 =
4here : A@ Y is the inter!a"ial area % Y is the air b&bble %iaeter ' is the air b&bble velo"it+
V $ = A = V
4 ( 99.5 m )2 ( 4.27 m )=33,201.97 m3 4 6 ( 9.439 m
( 2.5 7 10− m ) 3
3
) (4.27 m)
( ) 0.3
m (33,201.97 m3) !
5or ) *( k L % 2.7 / hr m = =0.2677 k L = A /V 9.7114 / m hr
= 9.7114 m−1
.
$ −20
k L %( $ ) =k L %( 20℃ )( 1.02)
k L %( 12 ℃ ) 2.6 / hr 3.046 k L %( 20 ℃ )= = = 12 − 20 12 − 20 hr 1.02 1.02
() () @ ?
( 20 ℃ )
@ ?
( 32 ℃ )
=k L %(20 ℃) ( ! , 20 ℃− L, 20 ℃ )
7e-.1
=k L %(32℃ ) ( ! , 32℃ − L, 32℃ )
7e-.6
$ −20
k L %( 32℃) =k L %( 20℃ ) ( 1.02)
Deterine the ao&nt o! o/+#en trans!erre% per ho&r at 6?`C.
() @ ?
=k L %( 20 ℃) ( !¿− L, 20 ℃ ) =
( 20℃ )
@( 20℃ )=
3.046
hr
3.046 ( 1.07 7 10−2− 0 ) kg3 hr m
( 1.07 7 10− −0 ) kg ( 28.317 m )=0.923 kg 2
3
m
3
hr
C. 5ro e-.1 an% 6( in ters o! e-&al wor)in# vol&e( k L %( 32℃) ( ! , 32℃ − L, 32℃ ) @( 32℃ )= @ k L %(20 ℃) ( ! , 20℃ − L, 20℃ ) (20 ℃)
@ ( 32℃ ) 80 )ff =0.80
[
32 − 20
k L %( 20 ℃ ) ( 1.02)
32 −20
1.02 ¿
(
)
¿
kg ( 8.673 7 10−2−1.5 7 10−2 ) kg3 ¿=6.2778 kg h hr m
0.923
]
( ! , 32 ℃− L , 32℃ ) @(20 ℃) k L %( 20℃ ) ( ! , 20 ℃− L , 20 ℃)
¿
@( 32℃ ) 80 )ff =0.80 ¿
CHAPTER 9: AGITATION AND AERATION The &s&al pro"e%&re o! s"ale &p o! !erenter is to !i/ one o! several "riteria involvin# Re+nol%s N&ber( power "ons&ption per &nit vol&e o! li-&i%( tip velo"it+ o! an ipeller( the li-&i% "ir"&lation tie an% the vol&etri" o/+#en trans!er "oe!!i"ient. The "hoi"e o! "riterion will %epen% on the !erentation bein# st&%ie%. Estiate( &sin# two etho%s( the re-&ire% spee% o! an ipeller an% the power re-&ireents o! a pro%&"tion s"ale !erenter o! ? =( to at"h the vol&etri" ass trans!er "oe!!i"ient. 5ollowin# opti& "on%itions were obtaine% with a ?.?= = !erenter: Densit+ o! broth: 16?? )#@= Aeration rate: 1vv *i-&i% Hei#ht: 1.6 DT
*i-&i% vol&e: ?.?1< = O/+#en trans!er rate: ?.68 )ol@=[hr
Two sets o! stan%ar% !lat bla%e t&rbine ipellers were installe%. Given: \* 16?? )#@= 1.? vv 1 ?.?= = ol&tion:
H* 1.6 DT OTR ?.68 )ol@=[hr 6 ? =
* ?.?1< =
Deterine the tan) %iaeter( DT( an% ipeller %iaeter( DI( !or ea"h "on%itions. 4 4 2 2 3 V 1= D $ 3 L = ( D$ ) ( 1.2 D$ )=0.3 4 D$ 4 4 1
D $ = 1
1
( ) =[ 1 3
V 1
0.3 4
1 3
D I = D $ = 1
1
0.018 m 0.3 4
1
]=
3 1 3
1
0.267 m
1 ( 0.267 m) =0.089 m 3
7io"hei"al En#ineerin# b+ Faes *ee(
p#.6;>
3 L =1.2 D$ =1.2 ( 0.267 m )=0.32 m 1
1
]
1 3
1 ( 3.36 m )=1.12 m 3
D I = D $ = 2
[
( )
1
V 2 3 D $ = = 0.3 4 2
0.018 60 m 7 0.03 0.3 4 3
2
1 3
=3.36 m
3 L =1.2 D$ =1.2 ( 3.36 m )=4.03 m 2
2
Aeration !low rate: 3
:=
1 m ;2 3
m *roth6mi(
( 0.018 m3 )
[ ] 1 mi( 60 !)c
=3 7 10 −4 m3 / !)c
5or s&per!i"ial velo"it+ 2 0.267 m ¿ ¿ 4 ¿ 4
−4
: V != = A
3 71 0
3
[
m 3600 !)c 1 hr !)c
¿
]
Deterine the partial press&re o! o/+#en in the s+ste
(
1 %tm + 1 %tm +
P; =
3 L 10.3 m 3 2 ;
2 %tm
2
)
[
1 %tm + 1 %tm +
( 0.21 )=
0.32 m 3 2 ; 10.3 m 3 2 ;
2 %tm
]
( 0.21 ) =0.213 %tm
Then %eterine the ) *a 0.25
kmol 3
;$" m 6 hr kmol k L % = = =1.17 3 0.213 %tm P; m 6hr6%tm 2
5or the %eterination o! the power re-&ireent Cooper et. al. 719>> "orrelate% ) *a( #asse% power per &nit vol&e 7P@an% s&per!i"ial velo"it+ 7s !or arie% Dis) Ipellers as: kmol 0.67 V ! ¿ , 3 m 6hr6%tm 0.95 Pm k L % =0.0635 ¿ V
( )
5or tan%ar%'5lat la%e T&rbine( Aiba et.al. 7198 0.67
V ! ¿
,
kmol 3
m 6hr6%tm 0.95 Pm k L % =0.0318 ¿ V
( )
5or T&rbine T+pe Ipellers( an3t Riet 719;9 0.5
V ! ¿ ,
kg 3
m 6 hr 0.4 Pm k L % =0.026 ¿ V
( )
5or the #asse% power(P m 0.67 ¿ =0.1 hP hr Pm kmol 19.3
1.17
3
m 6hr 6 %tm
=0.0318
(
0.018 m
)
0.95
3
¿
Then %eterine the &n#asse% power( Po( !or two sets o! ipeller( N p 6/16 Pmo g c
3
N p ρ N i Di N p= < Pmo = 3 5 gc ρ N i Di
5
5
0.089 m ¿
¿ 3 N i ¿ ¿
(
)
kg ¿ 3 m
12 1200
3
5
N ρ N i Di Pmo= p =¿ gc
2sin# i"haelis "orrelation: Pm =0.5 1
[
2
( P ) D mo1
:1
0.1 hP=0.5
N i=18 rp!
[
[
3 i1
0.56
] 3 2
( 0.108 7 10− N ) ( 0.089 m ) 3
( 3 7 10−4 m3 / !)c)0.56
]
60 !)c 1 mi( −3
−3
3
1
[ ] Pmo
=
1
]
=1080 rpm
Pmo =0.108 7 10 N i , hP=0.108 7 10 P m
3
i
( 1080 )3= 0.630 hP
0.10 =0.1587 0.630
5or Constant Power Inp&t in a a!!le% essel Pmo 3
ρ6 N 1 6 D I
3
ρ 6 N 2 6 D I
1
[
Pmo
= 5
5
< 9h)r) P mo=co(!t%(t ∧ ρ= + / V
2
1
+ 3 5 6 N 1 6 D I V 1 1
] [ =
mo/)l
1
+ 3 5 6 N 2 6 D I V 2 2
]
protot0p)
5or "onstant power in #eoetri"all+ siilar vessels 3
ρ6 N 1 6 D I
1
V 1
5
3
ρ 6 N 2 6 D I
=
A""%#. To Ghose
5
2
V 2
A""%# to Faes *eeK P@ "onstant
3
3
N 1 6 D I
N 2 =
5
1
V 1 V 2
D I
5
[ ][ ]
V 2 < N 2= N 1 V 1
1 3
D I
1
D I
2
[ ][ ] 3
D I N 1 = N 2 D I
2
2
< N 2= N 1
1
[ ] D I
2/ 3
2
D I
1
2
[ ][ ] 1 3
60 N 2=1080 0.03
0.089 1.12
5 3
,rpm
!or "onstant ipeller tip velo"it+ N 1 D I = N 2 D I 1
2
[ ] [ ] D I
1
D I
2
N 2= N 1
− 2/ 3
[ ] D I
2
D I
1
[ ]
0.089 =720 1.12
− 2/ 3
N 2=3895.39 rpm
N 2=199.8575 rpm
N 2= N 1
5 3
0.089 =720 1.12
N 2=57.2143 rpm
!or "onstant ipeller spee% 2/ 3 D I D I N 1= N 2 < N p = N m D I D I
[ ] 2
1
2 /3
[ ] m
p
N 2=3895.39 rpm Or ( N D I ) p=1.7 ( N D I )m
CHAPTER 9: AGITATION AND AERATION K ADDITIONA* PRO*E A strain o! aQoba"ter vinelan%ii is "<&re% in a 18 = stirre% !erenter !or pro%&"tion o! al#inate. 2n%er "on%itions the ass trans!er "oe!!i"ient( $ *a( is ?.68 s'1. O/+#en sol&bilit+ in a !erenter −3
7
broth is appro/iatel+ <.8
a/ "ell %ensit+ in the broth, RE3D: Sa/ ol3n: ¿
K L% 7 %l . m%x = @o 2
10
m
kg
3
. The spe"i!i" o/+#en &pta)e is 18 ol #@h. 4hat is the
( )( )( )(
0.25 −3 kg 8.5∗10 3 ! m
(
mmol 15 g∗h
1 3600 !
g
Sa/
¿
)
1 32 g 1 kg )( )( ) 1000 mmol 1 mol 1000 g
¿¿
g
=15.94 ¿ 189=;.8 3 L m
CHAPTER 9: AGITATION AND AERATION K ADDITIONA* PRO*E Biochemical Engineering and Biotechnology by Ghasem D. Najafpour Cal"&late the #as hol%'&p !or an a#itate% an% aerate% s+ste with power inp&t o! 1< hp in an
7
P V V!
¿ ¿ ¿ ¿0.4 ¿
4hereK P power in hp &n#asse% li-&i% vol&e in = s #as s&per!i"ial velo"it+ in @h
olvin# !or H: 18 80
¿ ¿
2.6 x 60 mi( / h
¿ ¿ ¿
H ?. The #as hol% &p "an be %e!ine% b+ the above %e!inition &sin# the #as hei#ht per vol&e( where H ?. !or aeration Ho
Vg V g + V L
Ho <.8J
0.6 0.6 + 6.5 ?.?<8
CHAPTER 9: AGITATION AND AERATION K ADDITIONA* PRO*E Biochemical Engineering and Biotechnology by Ghasem D. Najafpour Cal"&late the spee% o! an ipeller an% the power re-&ireents o! a pro%&"tion's"ale biorea"tor with ?= &sin# two %i!!erent etho%s. Also( at"h the vol&etri" ass trans!er "oe!!i"ient. The !ollowin# opti& "on%itions were obtaine% with a sall s"ale !erenter o! vol&e ?.== an% ?J o! the vessel wor)in#. The %ensit+ o! the broth( \ broth( 16?? )#.'=( wor)in# vol&e ?.1<=( aeration rate o! one vol&e o! #as per vol&e o! li-&i% 7vv( o/+#en trans!er rate ?.68 )ol.'=.h'1( li-&i% hei#ht insi%e the vessel( H*( 1.6Dt. two sets o! stan%ar%( !lat'bla%e t&rbine ipellers were installe%. Given: pro%&"tion's"ale( 6 ?= sall's"ale( ?.== ?J o! vessel wor)in# \ broth( 16?? )#.'= wor)in# vol&e( 1 ?.1<= 1vv OTR ?.68 )ol.'=.h'1 H* 1.6Dt Two sets o! stan%ar% !lat'bla%e t&rbine ipellers Re-&ire%: ipeller spee%( power re-&ireent( )la ol&tion: 1 7@>7Dt671.6Dt ?.=Dt= Diaeter o! the vessel: Dt 71@?.=1@= 7?.1<@=1@= ?.8; Diaeter o! the ipeller: Di(1 71@=7Dt ?.8;@= ?.196 Hei#ht o! li-&i% e%ia was ass&e% to be 1.6 ties the %iaeter o! the !erenter vessel. H*(1 1.6 / ?.8; ?.91
Diaeter o! the lar#er vessel: Dt6 71@?.=1@= 7?@?.=1@= =.= The ipeller siQe !or the lar#er vessel is: Di6 =.=@= 1.16 An% the li-&i% e%ia hei#ht in the se"on% !erenter is: H*6 1.6 / =.= >.?= Ass&e the !erentation broth has the sae vis"osit+ as water: U1 1"p Aeration rate !or 1vv is:
4 2s 7?.1< / ?@7
4
/ ?.8;6 >1.>8 @h
*et &s ta)e avera#e val&es !or the partial press&re o! o/+#en PO6 71 at L1 7H * @1?.= / atc@6 / ?.61 ?.61= at The o/+#en trans!er rate is OTR ?.68 )ol@=.h The ass trans!er "oe!!i"ient is )* ?.68@?.61= 1.1;> )ol@=.h.at 2se iperial "orrelation base% on the !ollowin# e-&ation !or ass trans!er in the biorea"tors. The #eneral e-&ation !or the eval&ation o! $la is $la /7P#@M 72s^ where /(+( an% Q are epiri"al "onstants. 5or Newtonian !l&i%s( non'"oales"in# broth an% #as b&bbles( the !ollowin# "orrelation is vali% !or a wor)in# vol&eo! less than > = an% a power per &nit vol&e o! 8?? Y 1?(??? 4@=
