07_08_09_ratb

7/24/2019 07_08_09_ratb

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REAKTOR ALIR TANGKI

BERPENGADUK (RATB)

Pertemuan 7, 8, dan 9

7/24/2019 07_08_09_ratb


Siat!"iat menda"ar #ada RATB

$% P&'a a'ir ada'a er*am#ur "em#urna (a*+ mied '&- atau B./)

2. .e"+i#un a'iran me'a'ui RATB ada'a +&ntin0u,ta#i +e* 1&'umetri" a'iran #ada #ema"u+an dan#en2e'uaran da#at ereda, di"ea+an &'eter3adin0a #eruaan den"iti

3. B./ me'i#uti #en2adu+an 0an2 "em#urnada'am 1&'ume rea+t&r, 02 erim#'i+a"i #ada

"emua "iat!"iat "i"tem men3adi "era2amdi"e'uru rea+t&r

4. Pen2adu+an 02 "em#urna 3u2a men2a+iat+an"emua +&m#&nen d'm rea+t&r mem#un0ai+e"em#atan 02 "ama ut+ menin22a'+an rea+t&r

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Siat!"iat menda"ar #ada RATB

(Lan3ut)5. Sea2ai a+iat #&in 4, terda#at di"triu"i

+&ntin0u dari -a+tu tin22a'

5% Sea2ai a+iat dari #&in 4, a'iran +e'uaranmem#un0ai "iat!"iat "ama den2an 'uidada'am rea+t&r

7% Sea2ai a+iat dari 5, terda#at "atu 'an2+a#eruaan 02 men3e'a"+an #eruaan "iat!

"iat dari in#ut dan &ut#ut8. .e"+i#un terda#at #eruaan di"triu"i -a+tu

tin22a', #en*am#uran "em#urna 'uida #adatin2+at mi+r&"+&#i+ dan ma+r&"+&i+memimin2 ut+ merata!rata "iat!"iat "e'urue'emen 'uida

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Keuntun2an dan Keru2ian

.en22una+an RATB

1 Keuntun2an1 Re'ati mura untu+ dian2un

2 .uda men2&ntr&' #ada tia# tin2+at, +arena tia#&#era"i #ada +eadaan teta#, #ermu+aan#er#indaan #ana" muda diada+an

3 Se*ara umum muda erada#ta"i d2 +&ntr&'&t&mati", memeri+an re"#&n *e#at #ada #eruaan

+&ndi"i &#era"i ( mi"a'6 +e* um#an dan +&n"entra"i)4 Pera-atan dan #emer"ian re'ati muda

5 Den2an #en2adu+an ei"ien dan 1i"+&"ita" tida+

ter'a'u tin22i, da'am #ra+te+ +e'a+uan m&de' da#atdide+ati 'ei te#at untu+ mem#redi+"i un3u+ +er3a%

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Keru2ian

Se*a

r a +&n"e#

da"ar"an2atmeru2i+an dari+en0ata

an+arenaa'iran+e'uar"amaden2ani"i 1e"e'

1 a' inimen0ea+an

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"em

ua r ea+"ier 'an

2"un2 #

ada+&n"entr a"i 0an2

'eirenda(+ata+anrea+tan

A, A)antara+e'uardanma"u+

2 Se*ara+ineti+an&rma'r A turuni'a A er+uran2, inierartidi#er'u+

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an

1&'um

e r ea+

t&r '

eie"ar

untu+mem#er&'e+&n1er"i 02diin2in+

an3 (

Unt

u++in

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et

i+a tida+

n&r

ma'

i"ater3adi+

ea'i+

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an

n0a, ta#i

ini

ti

da+ia"a,a

#a+a

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*&nt&n0a

dar

i"

atu"itua"id

emi+ia

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n

)

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Per"amaan #eran*an2an untu+

RATB

Pertiman2an "e*ara umum6

Nera*a ma"aNera*a Ener2i

Peran*an2an #r&"e" RATB "e*ara +a"

dian2un untu+ menentu+an 1&'ume1e"e' 0an2 di#er'u+an 2una men*a#ai+e*e#atan #r&du+"i 0an2 diin2in+an

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Parameter 0an2 di*ari me'i#uti6

1 :um'a "ta2e 02 di2una+an untu+ &#era"i

&#tima'

2 /ra+"i +&n1er"i dan "uu da'am tia# "ta2e

3 Dimu'ai den2an mem#ertiman2+an nera*a

ma""a dan nera*a ener2i untu+ tia# "ta2e

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Nera*a ma""a, 1&'ume rea+t&r,

dan +e*e#atan #r&du+"i

Untu+ &#era"i +&ntin0u dari

RATB 1e"e' tertutu#, tin3au

rea+"i6

A ; <Æ ; <

den2an +&ntr&' 1&'ume

dideini"i+an "ea2ai

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1&'ume 'uida da'am rea+t&r

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($)

Se*ara &#era"i&na'6

(=)

Da'am term +e*e#atan 1&'umetri+6(>)

Da'am term +&n1er"i A, den2an an0a A 02 tida+

erea+"i da'am um#an (f A0 ? @)6

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(4)

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Untu+ &#er"a"i tuna+ ("tead0 "tate)Æ dn Adt ? @

()

(5)

Re"iden*e time6

S#a*e time6

Ke*e#at

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an #r&du+"i6

(7)

(8)

(9)

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Nera*a Ener2i1 Untu+ rea+t&r a'ir +&ntin0u "e#erti RATB, nera*a

ener2i ada'a nera*a enta'#i (), i'a +itamen2aai+an #eredaan ener2i +ineti+ danener2i #&ten"ia' da'am a'iran, dan +er3a "at

antara #ema"u+an dan #en2e'uaran2 A+an teta#i, da'am #erandin2ann0a den2anBR, +e"etiman2an aru" me'i#uti enta'#ima"u+ dan +e'uar &'e a'iran

3 Da'am a' era2ai tran"er #ana" dari ataumenu3u +&ntr&' 1&'ume, dan #ementu+an atau#e'e#a"an enta'#i &'e rea+"i da'am +&ntr&'1&'ume%Se'an3utn0a #er"amaan ener2i (enta'#i)din0ata+an "26

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($@)

Untu+

&#er

a"ituna+m

=

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m

0 ($$)

Su

"ti

tu"i

F A

0

f A

untu+(!

r A)

C

($=)

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uun2an f den2an "uu rea+"i (T) A

($>)

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Sistem densiti konstanUntu+ "i"tem den"iti +&n"tan, eera#a a"i'

#en0ederanaan antara 'ain6

Pertama , tan#a mem#erati+an ti#e rea+t&r,ra+"i +&n1er"i 'imitin2 rea*tant, A, da#atdin0ata+an da'am +&n"entra"i m&'ar

($4)

Kedua, untu+ a'iran rea+t&r "e#erti RATB, mean

residence time "ama den2an space time, +arena

q ? q0

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($)

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Ketiga, untu+ RATB, term a+umu'a"i da'am

#er"amaan nera*a ma""a men3adi6

($5)

Terakhir, untu+ RATB, #er"amaan nera*a

ma""a +eadaan tuna+ da#at di"ederana+an

men3adi6

($7)

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O#era"i +eadaan tuna+ #ada tem#eratur T

Untu+ &#era"i +eadaan tuna+, term a+umu'a"i

da'am #er" nera*a ma""a dii'an2+an

Atau, untu+ den"iti +&n"tan

Bi'a T tertentu, C da#at diitun2 dari #er" nera*a

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ma""a tan#a me'iat+an nera*a ener2i

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&nt& $%

/&r te 'iuid!#a"e rea*ti&n A ; B Æ #r&du*t"at =@ "u##&"e 4@F *&n1er"i&n & A i"de"ired in "tead0! "tate &#erati&n% Te rea*ti&ni" #"eud&!ir"t!&rder -it re"#e*t t& A, -it + A ? @%@=7

!$ at =@% Te t&ta' 1&'umetri* '&- rate i" $%8 m> !$, and te in'et m&'ar '&- rate"

& A and B are / AO and /BO m&' !$

,re"#e*ti1e'0% Determine te 1e""e' 1&'ume

reuired, i , &r "aet0, it *an &n'0 e i''ed t& 7F*a#a*it0%

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&nt& =%

A 'iuid!#a"e rea*ti&n AÆ B i" t& e

*&ndu*ted in a STR at "tead0!"tate at $5>%

Te tem#erature & te eed i" =@ and 9@F*&n1er"i&n & A i" reuired% Determine te

1&'ume & a STR t& #r&du*e $>@ +2 B !$

,and *a'*u'ate te eat '&ad () &r te #r&*e""%

D&e" ti" re#re"ent additi&n &r rem&1a' & eatr&m te "0"tem

Data6 . A ? .B ? =@@ 2 m&'!$

H *# ? =%@ : 2!$

K!$

H

? @%9 2 *m!>H RA ? !87 +: m&'!$H + A ? @%8@!$ at $5>

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&nt& >

&n"ider te "tartu# & a STR &r te 'iuid!#a"e rea*ti&n AÆ #r&du*t"% Te rea*t&r i"

initia''0 i''ed -it eed -en "tead0 '&- & eed() i" e2un% Determine te time (t) reuired t&a*ie1e 99F & te "tead0!"tate 1a'ue & fA%

Data6 C ? 8@@@ LH ? = L "!$H A& ? $% m&' L

!$H

+ A ? $% '@!4

"!$

%

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REAKTOR ALIR TANGKI

BERPENGADUK (RATB)

Pertemuan 8

7/24/2019 07_08_09_ratb


TIN:AU ULANG NERAA ENERGI SISTE. ALIR

/i

in

"0"tem

/i

&ut

iin

i&ut

Eiin J" Ei&ut

Nera*a Ener2iR

inR

&ut; R

2en? R

a**

n

n

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Q

KW

;

?

F i

E i

?

F i

E i

? dt i1

ini1

out System

Ei ? Ener20 & *&m#&nent i

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n n

(8!=)W ?W s

F i

PV i F i PV ii?1 ini?1 out

J&r+ d& t& '&- 1e'&*it0

/&r *emi*a' rea*t&r Ki, Pi, and M&ter ener20 arene2'e*ted "& tat6

E i ?U i (8!>)and

(8!4) H i ?U i ; PV i

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&mined te e% 8!4, 8!>, 8!=, and 8!$ e re"u't,

s ;n

n ?dE (8!)Q W F i H

i F i H

i

? ? dt i 1

ini 1

out System

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General Energy Balance:

For steady state operation:

Je need t& #ut te a&1e euati&n int& a &rm tat-e *an ea"i'0 u"e t& re'ate and T in &rder t& "iQerea* t&r"% T& a*ie1e ti" 2&a', -e -rite te m&'ar'&- rate" in term" & *&n1er"i&n and te enta'#ie"

a" a un*ti&n & tem#erature% Je n&- -i'' di""e*t

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&t /i and i%

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/'&- Rate", /i

/&r te 2enera'iQed rea*ti&n6

In 2enera',

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Enta'#ie" , i A""umin2 n& #a"e *an2e6

.ean eat *a#a*itie"6

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Self Test

a'*u'ate , , and

&r te rea*ti&n,

Tere are inert" I #re"ent in te "0"tem%

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dditional Information:

Solution

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Energy Balance !ith"dissected" enthalpies:

For constant or mean heat capacities:

dia#atic Energy Balance:

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dia#atic Energy Balance for$aria#le heat capacities:

%ST& lgorithm 'Section ()* Fogler+

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Self Test

/&r and adiaati* rea*ti&n -it and P?@,"+et* *&n1er"i&n a" a un*ti&n & tem#erature%S&'uti&n

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A

% /&r an e&termi* rea*ti&n

,R i" ne2ati1e

(!), EB in*rea"e" -it in*rea"in2 T%

e%2%, R? !$@@ +:m&'e A

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B% /&r an end&termi* rea*ti&n, R i" #&"iti1e (;),EB in*rea"e" -it de*rea"in2 T%

e%2%, R? ;$@@ +:m&'e A

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/&r a ir"t &rder rea*ti&n,

B&t te m&'e and ener20 a'an*e" are "ati"ied-en .B?EB% Te "tead0 "tate tem#eratureand *&n1er"i&n are TSS and SS, re"#e*ti1e'0, &r

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an enterin2 tem#erature TO%

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E$aluating the eat E-change Term, .

Ener20 tran"erred et-een te rea*t&r and te

*&&'ant6

A""umin2 te tem#erature in"ide te STR, T, i"

"#atia''0 uni&rm6

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/anipulating the Energy E-change Term

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&minin26

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At i2 *&&'ant '&-rate" te e#&nentia' term -i''e "ma'', "& -e *ane#and te e#&nentia'term a" a Ta0'&r Serie",

-ere te term" & "e*&nd &rder &r 2reater are ne2'e*ted, ten6

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Sin*e te *&&'ant '&-rate i" i2, Ta$ ≅ Ta= ≅ Ta6

&e$ersi#le &eactions'%hp( Fogler,ppendi- %+

/&r Idea' 2a"e", K

and KP are re'ated 0

K P =

K C (RT)δ

δ?Σ

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νi

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/&r te "#e*ia' *a"e & 6

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lgorithm for dia#atic &eactions:

Le1en"#ie' P'&t &r

an e&terma',adiaati* rea*ti&n%

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(Te

P/R

te 1&'ume%)

/&r an eit *&n1er"i&n & /&r an eit *&ner"i&n & 4@F 7@F

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STR Saded area i" te rea*t&r 1&'ume%

/&r an eit *&n1er"i&n & /&r an eit *&ner"i&n & 4@F 7@F

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STR;P/R

/&r an intermediate *&n1er"i&n & 4@F and eit

*&n1er"i&n & 7@F

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E-ample: E-othermic, &e$ersi#le &eaction

J0 i" tere a maimum in te rate & rea*ti&n -it

re"#e*t t& *&n1er"i&n (and en*e, -it re"#e*t t&tem#erature and rea*t&r 1&'ume) &r an adiaati*

rea*t&r

&ate 0a!:

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&eactor Inlet Temperature and Interstage %ooling

1ptimum Inlet Temperature:

/ied C&'ume E&termi* Rea*t&r

%ur$e : Rea*ti&n rate "'&-, *&n1er"i&n di*tated 0

rate & rea*ti&n and rea*t&r 1&'ume% A" tem#eraturein*rea"e" rate in*rea"e" and tere&re *&n1er"i&nin*rea"e"%%ur$e B: Rea*ti&n rate 1er0 ra#id% Cirtua' eui'irium

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rea*ed in rea*ti&n *&n1er"i&n di*tated 0eui'irium *&n1er"i&n%

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Interstage %ooling:

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Self Test

An inert I i" in3e*ted at te #&int" "&-n e'&-6

S+et* te *&n1er"i&n!tem#erature tra3e*t&r0 &r an

end&termi* rea*ti&n%

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Solution

/&r an end&termi* rea*ti&n, te eui'irium*&n1er"i&n in*rea"e" -it in*rea"in2 T%

/&r and Ke?%$ and T=

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/r&m te ener20 a'an*e -e +n&- te

tem#erature de*rea"e" -it in*rea"in2 *&n1er"i&n%

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Ener20 Ba'an*e ar&und 3un*ti&n6

S&'1in2 T=

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E-ample %2(34

Se*&nd Order Rea*ti&n arried Out Adiaati*a''0in a STR

Te a*id!*ata'0Qed irre1er"i'e 'iuid!#a"e rea*ti&n

i" *arried &ut adiaati*a''0 in a STR%

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Te rea*ti&n i" "e*&nd &rder in A% Te eed, -i*

i" euim&'ar i n a "&'1ent (-i* *&ntain" te*ata'0"t) and A, enter" te rea*t&r at a t&ta'1&'umetri* '&-rate & $@ dm>min -it te*&n*entrati&n & A ein2 4.% Te enterin2

tem#erature i" >@@ K%1) Jat STR rea*t&r 1&'ume i" ne*e""ar0 t&

a*ie1e 8@F *&n1er"i&n2) Jat *&n1er"i&n *an e a*ie1ed in a $@@@

dm> STR Jat i" te ne- eit tem#erature3) &- -&u'd 0&ur an"-er" t& #art () *an2e, i

te enterin2 tem#erature & te eed -ere =8@K

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Additi&na' In&rmati&n6

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Eam#'e D8!= S&'uti&n, Part A


'a+ Je -i'' "&'1e #art (a) 0 u"in2 te n&ni"&terma'

rea*t&r de"i2n a'2&ritm di"*u""ed in a#ter 8%5) STR De"i2n Euati&n6

4) Rate La-6

3. St&i*i&metr06 'iuid,

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4. &mine6

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Gi1en *&n1er"i&n () , 0&u mu"t ir"t determine te

rea*ti&n tem#erature (T), and ten 0&u *an

*a'*u'ate te rea*t&r 1&'ume (C)%5. Determine T6 /&r

ti" #r&'em6

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-i* 'ea1e" u" -it6

Ater "&me rearran2in2 -e are 'et -it6

Su"titutin2 &r +n&-n 1a'ue" and "&'1in2 &r T6

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6) S&'1e &r te Rate &n"tant (+) at T ? >8@ K6

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7) a'*u'ate te STR Rea*t&r C&'ume (C)6

Re*a'' tat6

Su"titutin2 &r +n&-n 1a'ue" and "&'1in2 &r C6

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E-ample %2(34 Solution, Part B


'#+ /&r #art () -e -i'' a2ain u"e te

n&ni"&terma' rea*t&r de"i2n a'2&ritmdi"*u""ed in a#ter 8% Te ir"t &ur "te#" & te a'2&ritm -e u"ed in #art (a) a##'0 t& &ur"&'uti&n t& #art ()% It i" at "te# numer ,

-ere te a'2&ritm *an2e"%

81TE: Je -i'' ind it m&re *&n1enient t& -&r+ -it

ti" euati&n in term" & "#a*e time, rater tan

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1&'ume6

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S#a*e time i" deined a"6

Ater "&me rearran2in26

Su"titutin26

Gi1en rea*t&r 1&'ume (C), 0&u mu"t "&'1e te

ener20 a'an*e and te m&'e a'an*e"imu'tane&u"'0 &r *&n1er"i&n (), "in*e it i" aun*ti&n & tem#erature (T)%

9) S&'1e te Ener20 Ba'an*e &r a" a un*ti&nEB

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& T6

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/r&m te adiaati* ener20 a'an*e (a" a##'ied t&

STR")6

6) S&'1e te .&'e Ba'an*e &r .B a" a un*ti&n &

T6

Je'' rearran2e &ur *&mined euati&n r&m

"te# 4 t& 2i1e u"6

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Rearran2in2 2i1e"6

S&'1in2 &r 2i1e" u"6

Ater "&me ina' rearran2in2 -e 2et6

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Let" "im#'i0 a 'itt'e m&re, 0 intr&du*in2 te

Dam+V'er Numer, Da6

Je ten a1e6

7) P'&t EB and .B6

W&u -ant t& #'&t EB and .B &n te "ame 2ra# (a"

un*ti&n" & T) t& "ee -ere te0 inter"e*t% Ti" -i'' te''0&u -ere 0&ur "tead0!"tate #&int i"% T& a**&m#'i"

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ti", -e -i'' u"e P&'0mat (ut 0&u *&u'd u"e a"#read"eet)%

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? @%87 and T ? >87 K

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Our *&rre"#&ndin2 P&'0mat #r&2ram '&&+" 'i+e

ti"6

NOTE6 Our u"e &d(T)d(t)?= in te a&1e

#r&2ram i" mere'0 a -a0

&r u" t& 2enerate a ran2e

& tem#erature" a" -e

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#'&t *&n1er"i&n a" a

un*ti&n & tem#erature%

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Eam#'e D8!= S&'uti&n, Part


'c+ /&r #art (*) -e -i'' "im#'0 m&di0 te P&'0mat

#r&2ram -e u"ed in #art (), "ettin2 &ur initia'

tem#erature t& =8@ K% A'' &ter euati&n" remainun*an2ed%

7) P'&t EB and .B6

Je "ee tat &ur *&n1er"i&n -&u'd e

a&ut @%7, at a tem#erature & > K%

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/ultiple Steady States

Pertemuan +e 9

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/ultiple Steady States

/a*t&r / A@ P@ and ten di1ide 0 / A@

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/&r a STR6 / ? !r C A@ A

-ere

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an tere e mu'ti#'e "tead0 "tate" (.SS) &r a

irre1er"i'e ir"t &rder end&termi* rea*ti&n

Solution

/&r an end&termi* rea*ti&n R i"

#&"iti1e, (e%2%, R?;$@@ +:m&'e)

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Tere are n& mu'ti#'e "tead0 "tate" &r an end&termi*, irre1er"i'e ir"t &rd er rea*t&r% Te "tead0

"tate rea*t&r tem#erature i" TS% Ji'' a reversible

end&termi* ir"t &rder rea*ti&n a1e

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.SS

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N&- -e need t& ind % Je d& ti" 0 *&minin2

te m&'e a'an*e, rate 'a-, Arreniu" Euati&n,and "t&i*i&metr0%

/&r te ir"t!&rder, irre1er"i'e rea*ti&n AÆ B, -ea1e6

-ere

At "tead0 "tate6

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Un"tead0 State STR

Ba'an*e &n a "0"tem 1&'ume tat i" -e''!mied6

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&TB Bertingkat '/ultistage+

1 RATB ertin2+at terdiri ata" = atau 'ei rea+t&r tan2+ier#en2adu+ 0an2 di"u"un "eri

2 Keuntun2an RATB ertin2+at dua atau 'ei, untu+men*a#ai a"i' 02 "ama u+uran 1&'ume rea+t&r 'ei+e*i' diandin2+an RATB tun22a'

3 Keru2ian utama RATB ertin2+at er&#era"i #ada+&n"entra"i 0an2 'ei renda diantara #ema"u+an dan#en2e'uaran

4 Untu+ RATB tun22a', erarti a-a er&#era"i #ada+&n"entra"i da'am "i"tem "erenda mun2+in, dan untu++ineti+a n&rma', di#er'u+an 1&'ume rea+t&r "ema+in e"ar

5 Bi'a = tan2+i (er&#era"i #d T "ama) di"u"un "eri, 0an2+edua er&#era"i #ada +&n"entra"i "ama "#t tan2+itun22a' diata", ta#i 02 #ertama er&#era"i #ada+&n"entra"i 'ei tin22i, 3adi 1&'ume t&ta' +edua tan2+i'ei +e*i' dari#ada tan2+i tun22a'

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&angkaian &TB #ertingkat 8

Per" nera*a ma""a #ada RATB +e i

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Grai+ i'u"tra"i &#era"i > RATB "eri

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Pen0e'e"aian #er" $4 %4!$ untu+ men*ari C (dieri

) atau men*ari (dieri C) da#at di'a+u+an"e*ara 2rai+ atau "e*ara ana'iti"% ara 2rai+da#at di2una+an untu+ men*ari , atau i'aentu+ ana'iti" (!r ) tida+ di+etaui

Pen0e'e"aian 2rai" untu+ N ? =

Untu+ "ta2e $6

Untu+ "ta2e =6

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Untu+ "ta2e =6

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E-ample 53;

A tree!"ta2e STR i" u"ed &r te rea*ti&n A Æ#r&du*t"% Te rea*ti&n &**ur" in aue&u" "&'uti&n,and i" "e*&nd!&rder -it re"#e*t t& A, -it + ?

A

@%@4@ L m&' ! $ min!$% Te in'et *&n*entrati&n & A andte in'et 1&'umetri* '&- rate are $% m&' L

!$ and =% L

min!$

, re"#e*ti1e'0% Determine te ra*ti&na'*&n1er"i&n ( A) &tained at te &ut'et, i C$

? $@ L, C= ? =@ L, and C> ? @ L, (a) ana'0ti*a''0, and () 2ra#i*a''0%

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S&'u"i

Untu+ "ta2e $ dari #er"amaan +e*e#atan

Karena den"ita" +&n"tan

La+u+an #en2aturan "ein22a di#er&'e #er" +-adrat

Atau den2anmema"u++an i'an2annumeri+

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Di#er&'e A$ ? @%$57 Simi'ar'0, &r "ta2e" = and >, -e &tain

A=

? @%>5=, and A> ? @%77, -i* i" te&ut'et ra*ti&na' *&n1er"i&n r&m te tree!"ta2e STR%

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Pen0e'e"aian *ara 2rai" "

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() Te 2ra#i*a' "&'uti&n i" "&-n in /i2ure

$4%$=% Te *ur1e &r (! r A) r&m te rate 'a- i"ir"t dra-n% Ten te &#eratin2 'ine AB i"*&n"tru*ted -it "'&#e / A@C$ ? * A@@C$ ?@%>7 m&' L

!$min

!$ t& inter"e*t te rate *ur1e at

A$ ? @%$57H "imi'ar'0, te 'ine" D and E/, -it*&rre"#&ndin2 "'&#e" @%$87 and @%@7,re"#e*ti1e'0, are *&n"tru*ted t& inter"e*t @%>5and A> ? @%8, re"#e*ti1e'0% Te"e are te"ame te rate *ur1e at te 1a'ue" A= ? 1a'ue"

a" &tained in #art (a)%

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O#tima' O#erati&n

Te &''&-in2 eam#'e i''u"trate" a "im#'e *a"e &&#tima' &#erati&n & a mu'ti"ta2e STR t& minimiQete t&ta' 1&'ume% Je *&ntinue t& a""ume a *&n"tant!

den"it0 "0"tem -it i"&terma' &#erati&nE#% $4!$@

&n"ider te 'iuid!#a"e rea*ti&n A ; % % % Æ #r&du*t" ta+in2 #'a*e in a t-&!"ta2e STR% I te

rea*ti&n i" ir"t!&rder, and &t "ta2e" are at te"ame T, &- are te "iQe" & te t-& "ta2e" re'ated t& minimiQe te t&t a' 1&'ume C &r a 2i1en

eed rate (/ A&) and &ut'et *&n1er"i&n ( A=)

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S&'u"i

/r&m te materia' a'an*e, euati&n $4%4!$, te

t&ta' 1&'ume i"

A

/r&m te rate 'a-,

B

Su"titutin2 (B) and () in (A), -e &tain

D

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D

?

/r&m (E) and (D), -e &tain

E

int& te materia'

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r&m-i*

int& te materia'a'an*e &r "ta2e =

(*&ntained in te 'a"tterm in

(D)), -e a1e

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1 Tat i", &r a ir"t!&rder rea*ti&n, te t-& "ta2e"

mu"t e & eua' "iQe t& minimiQe C%2 Te #r&& *an e etended t& an N!"ta2e STR%

/&r &ter &rder" & rea*ti&n, ti" re"u't i"a##r&imate'0 *&rre*t% Te *&n*'u"i&n i" tat

tan+" in "erie" "&u'd a'' e te "ame "iQe,-i* a**&rd" -it ea"e & ari*ati&n%

3 A't&u2, &r &ter &rder" & rea*ti&n, eua'!"iQed 1e""e'" d& n&t *&rre"#&nd t& te minimum

1&'ume, te dieren*e in t&ta' 1&'ume i""ui*ient'0 "ma'' tat tere i" u"ua''0 n&e*&n&mi* eneit t& *&n"tru*tin2 dierent!"iQed1e""e'" &n*e ari*ati&n *&"t" are *&n"idered%

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Eam#'e $$

A rea*t&r "0"tem i" t& e de"i2ned &r 8F*&n1er"i&n & A (A) in a "e*&nd!&rder 'iuid#a"e rea*ti&n, AÆ #r&du*t"H + A ? @%@7 L m&'

!$

min!

$

, @ ? = L min!$

, and A@ ? @%@4@ m&'L!$% Te de"i2n &#ti&n" are6

(1) t-& eua'!"iQed "tirred tan+" in "erie"H(2) t-& "tirred tan+" in "erie" t& #r&1ide a

minimum t&ta' 1&'ume%

Te *&"t & a 1e""e' i" X=9@, ut a $@F di"*&unta##'ie" i &t 1e""e'" are te "ame "iQe and2e&metr0% Ji* &#ti&n 'ead" t& te '&-er *a#ita'*&"t

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S&'u"i

a"e (a)% /r&m te materia'!a'an*e euati&n $4%4!$ a##'ied t& ea* & te t-& 1e""e'" $ and =,

Euatin2 C$ and C= r&m (A) and (B), and

"im#'i0in2, -e &tain

Ti" i" a *ui* euati&n &r f A1 in term" & f A26

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Ti" euati&n a" &ne #&"iti1e rea' r&&t, A$ ?@%59,

-i* *an e &tained 0 tria'%Ti" *&rre"#&nd" t& C $ ? C= ? %9 $@

4 L

(r&m euati&n (A) &r (B)) and a t&ta' *a#ita' *&"t

& @%9(=9@)(%9 $@4)=$@@@ ? X>$,@@@ (-it te $@F

di"*&unt ta+en int& a**&unt)

%ase '#+) Te t&ta' 1&'ume i" &tained r&meuati&n" (A) and (B)6

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/&r minimum C,

Ti" a'"& re"u't" in a *ui* euati&n &r A$, -i*,-it te 1a'ue A= ? @%8 in"erted, e*&me"

S&'uti&n 0 tria' 0ie'd" &ne #&"iti1e rea' r&&t6 A$

? @%55% Ti" 'ead" t& C$ ? 4%9 '@4 L, C=? 5%84

'@4

L and a *a#ita' *&"t & X>4 =@@

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'@ L, and a *a#ita' *&"t & X>4,=@@%

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&n*'u"i&n6

Te '&-er *a#ita' *&"t i" &tained &r *a"e

(a) (t-& eua'!"iQed 1e""e'"), in "#ite &

te a*t tat te t&ta' 1&'ume i" "'i2t'0

'ar2er ($$%9 '@4 L 1er"u" $$%8 $@

4 L)%

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