3 matematika teknik kimia 1

7/16/2019 3 Matematika Teknik Kimia 1

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PEMODELAN MATEMATIKA

Ir. Heriyanto, M.T.

MATEMATIKAMATEMATIKA

TEKNIK KIMIATEKNIK KIMIA-1-1


2/62

DEFINISI MODEL (TERMINOLOGI)

Se!"a# o!$e% M (!en&a, 'i'te 'i%a ata"%iia, ata" *ro'e') a&a+a# o&e+ a*a!i+ater&a*at ana+oi antara o!$e% M &an o!$e%nyata R 'e#ina %e'i*"+an enenai R

&a*at &i!"at.


3/62


Mo&e+ M Re*re'enta'i o!$e% R

Ta%'iran o!$e% R yan &ii'o+a'i &ari 'e+"r"#rea+ita',

Mena!ar%an %enyataan ata" !aian &ari%enyataan.

Da*at &i'e&er#ana%an en$a&i !aian &ari%enyataan $i%a *er+" %e'i*"+an tertent"

'a$a.


4/62


Keter!ata'an ana+oi o&e+ M &an o!$e% RKeter!ata'an %e'e'"aian /"n'i,Keter!ata'an +e'e'"aian 'tr"%t"r &an

*eri+a%",

Keter!ata'an a%"ra'i.Mo&e+ M &an o!$e% R !o+e# !er!e&a '%a+a.Ha'i+ o&e+ !a"' a*a!i+a 0aria!e+ &an

/enoena *entinnya &ire*re'enta'i%an

'eara !enar &a+a %onte%' ata" in0e'taitertent".


5/62


Ana+oi antara o&e+ M &an o!$e% R &a*at&i!"at &a+a !ent"% *er'aaanateati'.

Mo&e+ ateati' ena!ar%an'e*eran%at *er'aaan a+$a!ar &an2ata"&i/eren'ia+ &an2ata" intera+ yan&i"na%an "nt"% en$e+a'%an *eri+a%"o!$e% R.


6/62

T3GAS 4HEMI4AL ENGINEERMeno*era'i%an &an eno*tia+%an

*ro'e' yan a&a

Meranan *a!ri% !ar" &an eo&i%a'i*a!ri% yan a&a.


7/62

APLIKASI MODEL MATEMATIS DIIND3STRI KIMIAPero!aanSi"+a'i

Ana+i'i' 'en'iti0ita'

Ken&a+i &an o*era'iO*tii'a'i

E%'*+ora'i


8/62

KETER5ATASAN MODELMATEMATIS1. 6eni', $"+a# 'erta %ea%"ratan &ata

7. Per%a%a' ateati'

8. Inter*reta'i #a'i+ o&e+.


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INTERPRETASI HASIL MODEL


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PEN93S3NAN DANPEN93S3NAN DAN

KLASIFIKASI MODELKLASIFIKASI MODEL

MATEMATIKA TEKNIK KIMIA-1


11/62

PEN93S3NAN MODELMATEMATIKAPeny"'"nan o&e+ ateati%a a&a+a#

*ene'etan 'e*eran%at *er'aaanateati%a.

Per'aaan ateati%a a&a+a# #"!"nanantara 0aria!e+ *ro'e'.


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TAHAP-TAHAP PEMODELAN1. For"+a'i *er'oa+an, *en"*"+an

o!$e%ti/ &an %riteria %e*"t"'an

7. Penaatan ter#a&a* *ro'e' &an%+a'i%a'inya "nt"% e!ai *ro'e'en$a&i !e!era*a '"!'i'te (e+een*ro'e')

8. Penent"an #"!"nan antara '"!'i'te

:. Ana+i'i' 0aria!e+ &an #"!"nan antar0aria!e+ *a&a 'etia* e+een *ro'e'


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TAHAP-TAHAP PEMODELAN;. Pe!ent"%an *er'aaan ateati%a

&enan en"na%an 0aria!e+ &an*araeter Pen"*"+an &ata


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TAHAP-TAHAP PEMODELAN

>. Ana+i'i' 'en'iti0ita' o&e+ "nt"%eni&enti%a'i *araeter yan!er*enar"# %"at &an +ea# ter#a&a*re'*on' o&e+

?. Penye&er#anaan o&e+.

1@. Ta#a* : ? &i"+an, 'a*ai inter*reta'i#a'i+ o&e+ 'e'"ai &enan %riteriao!$e%ti/ &an 'o+"'i yan &i#ara*%an.


15/62

KEG3NAAN MODELFor"+a'i /enoena 'i%a &an 'i%o%iia,

yait" *er*in&a#an *ana', *er*in&a#ana''a &an *er*in&a#an oent", 'ertarea%'i %iia &i &a+a 'i'te #ooen &an

#eteroen.De'ain o*era'i *er*in&a#an a''a,

en#it"n *en"%ar *ana', ere%aya'area%'i %iia, &an enen&a+i%an *ro'e'.


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KLASIFIKASIMODEL MATEMATIKA


17/62

MODEL 5ERDASARKANPRINSIP FISIKOKIMIA

Di"na%an "nt"% e/or"+a'i/enoena *er*in&a#an.

Pro'e' &i!ai en$a&i 'e$"+a# e+een*ro'e' yan &i$e+a'%an &enan #"%"%e%e%a+an a''a, oent", &aneneri.


18/62


Mo&e+ &eterini'ti% ata" e+een o&e+BNi+ai ata" 'e*eran%at ni+ai 'etia* 0aria!e+ata" *araeter o&e+ *a&a %on&i'i tertent"te+a# &itent"%an.

Mo&e+ 'tati'ti% ata" e+een o&e+ 'tati'ti%Caria!e+ &an *araeter o&e+ er"*a%an!e'aran 'tati'ti%, !er"*a *ro!a!i+ita' ata"oen &ari /"n'i &en'ita' *ro!a!i+ita'.Mi'a+nya6i%a /"n'i &en'ita' *ro!a!i+ita' P(Y )

!er+a%" "nt"% 0aria!e+ 'tati'ti% Y, a%a P(Y)&Y a&a+a# *ro!a!i+ita' 0aria!e+ ter'e!"tyan !era&a &a+a rentan &Y &i 'e%itar Y.


19/62


K+a'i%a'i !er&a'ar%an $eni' *er'aaan

Tingkat kesulitan metode penyelesaianberkurang dari kanan ke kiri.


20/62

MODEL PDFMo&e+ !er!a'i' *er'aaan tran'*ort &a+a !ent"%

/"n'iona+ P(1, . . . , n).

Pro!a!i+ita' ene"%an 0aria!e+ teri%at (1, . . . , n)&a+a rentan &1, . . . , &n&i 'e%itar /"n'i 1(x,t), . . ., n(x,t) a&a+a# P(1, . . . , n)&1, . . . , &n.

Me!eri in/ora'i 'tati'ti% *ro'e' 'tati'ti%.Me!eri /"n'i &i'tri!"'i 0aria!e+ *ro'e'.

4onto#B

e%ani%a 'tati'ti%, teori %ineti% a', a*"ran a%ro&a+a &i'tri!"'i a%t" tina+, &i'tri!"'i "%"ran%ri'ta+, &i'tri!"'i a%ti0ita' *a&a *e+et %ata+i', &an&i'tri!"'i ""r &an "%"ran !ia%an i%ro!io+oi.


21/62

MODEL EMPIRISMODEL EMPIRIS

Kore+a'i re'*on' *ro'e' ter#a&a* *er"!a#an'at" ata" !e!era*a 0aria!e+ *ro'e'.

4onto#BFitting*o+inoia+ *a&a &ata e%'*erien, re'*on'*ro'e' *a&a *enen&a+ian *ro'e' &a+a !ent"%

/"n'i tran'/er *a&a &oain a%t" ata" /re%"en'i.Mer"*a%an o&e+ 'tati'ti% %arena &ata

&i*ero+e# 'eara e%'*erien &an !eri'i%e'a+a#an 'tati'ti%.

Mei+i%i a%na ter!ata' &a+aen$e+a'%an *ro'e' ata" e+een *ro'e'

Mi'a+B *re&i%'i !era&a &i +"ar rentan *ero!aan.


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MODEL 5ERDASARKANMODEL 5ERDASARKAN

PRINSIP FISIKOKIMIAPRINSIP FISIKOKIMIA

MATEMATIKA TEKNIK KIMIA-1


23/62

5a%ro"n&

Chemical

Process

Mathematical

Formulation

assumption

Result

Programming/

Analitical

Calculation


24/62

4#eia+ Enineerin Too+'

Ma'' 5a+ane (Neraa Ma''a)Enery 5a+ane (Neraa Pana')

E"i+i!ri" (Ke'eti!anan)

Rate *roe''e' (Pro'e'-*ro'e' Kee*atan)

H"anity (H"anita')H"anity (H"anita')

Eonoi (e%onoi)Eonoi (e%onoi)


25/62

PRINSIP

Ma'' 5a+ane

Ke. a''a in %e. a''a o"t %e.a''a a%""+a'i

2', o+2$a 2', o+2$a 2',o+2$a

(Sat"an #ar"' 'aaSat"an #ar"' 'aa)

Accumulation

Input Output


26/62

Heat 5a+ane

Ke *ana' in %e *ana' o"t %e *ana'a

a+2', 5T32$a a+2', 5T32$a a+2',

5T32$a

(Sat"an #ar"' 'aaSat"an #ar"' 'aa)

Accumulation

Input Output


27/62

Rate Proe''e' B F+"%' (*er*in&a#an tia*'at"an +"a' 'at"an a%t")

ProsesProses PersamaanPersamaan SatuanSatuan KonstantaKonstanta

MassaMassaMol/(cmMol/(cm22.dt).dt)

DDee di!usi"itas e!e#ti!$ di!usi"itas e!e#ti!$

cmcm22/dt/dt

CC

AA #onsentrasi$ mol/cm #onsentrasi$ mol/cm%%

& 'ara#$ cm& 'ara#$ cm

PanasPanasCal/(cmCal/(cm22.dt).dt)

# #onst.PP #ondu#si$# #onst.PP #ondu#si$

Cal/(cmCal/(cm22.dt./cm).dt./cm)

temperatur$ temperatur$

& 'ara#$ cm& 'ara#$ cm

MomentumMomentum

(gaya)(gaya)

d*ne/cmd*ne/cm22

oror

g/(cm.g/(cm.ss22))

+ "is#ositas$ gr/(cm.dt)+ "is#ositas$ gr/(cm.dt)$$poisepoise

"" #ecepatan$ cm/dt #ecepatan$ cm/dt

** 'ara# 'ara#$$ cmcm

dx

dcDn AeA =

dx

dTkqA=

dy

dvxyx =


28/62

E+een Co+"e

5i&an Datar

A

,&

-lemen "olume

A. ,&


29/62

,r

r

Si+in&er

-lemen "olume

uas selimut silinder. Arah

2 r . ,r


30/62

5o+a Pe$a+

,r

r -lemen "olume

uas selimut 0ola. Arah 1 r2. ,r


31/62

PROSED3R PEMODELANPROSED3R PEMODELAN

1.Ga!ar '%et'a 'i'te &an &eni'i%an !e'aran%iia, 'i%a &an eoetri.

7.Pi+i# 0aria!e+ teri%at (re'*on').8.Pi+i# 0aria!e+ !e!a' (i'a+ J, t).:.5"at &a/tar *araeter (%on'tanta 'i%, "%"ran

&an !ent"%) !"at *"+a &a/tar *araeter ta%%on'tan (i'a+ 0i'%o'ita' yan !er"!a#ter#a&a* te*erat"r).

;.Ga!ar '%et'a *eri+a%" 0aria!e+ teri%at, 'e*erti*ro+ te*erat"r yan &i#ara*%an.


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APLI4ATION

1. 6i%a a&a tan%i &enan 0o+"e 1@@@ L a%an &ii'i&enan air &enan %ee*atan 7@ L2enit. 5era*aa%t" y &i!"t"#%an aar tan%i *en"#


33/62

Neraa Ma''aKe a''a in %e a''a o"t %e a''a a

Fi @ (&C2&t)A%an &i*ero+e# B

&C2&t Fi

&C Fi &t

6i%a &iintera+%an &enan Kon&i'i !ata'

t @ C @

t C 1@@@

A%an &i*ero+e#

C 7@. t

1@@@ +t 7@ +t2nt . T

t 1@@@27@ nt ;@ nt.

=1000

0 0

20

t

dtdV


34/62

LATIHAN-1

6i%a a&a tan%i &enan 0o+"e 1@@@ Lteri'i >@ &enan air. Tan%i&i%o'on%an &enan %ee*atan 7;L2enit. 5era*a a%t" y &i!"t"#%anaar tan%i tina+ !eri'i 7@ air


35/62

7. Tan%i &enan %a*a'ita' 1@@@ L. A%an &ii'i&enan +ar"tan !enJene &enan %ee*atan 7@ L

*erenit, 'eentara !enJene &a+a tan%i&i%e+"ar%an &enan %ee*atan 1@ L *erenit.5era*a 0o+"e tan%i yan teri'i air 'ete+a# 8@enit

Fi = 20 L/menit

Fo = 10 L/menit


36/62

Neraa Ma''a

Ke a''a in %e a''a o"t %e a''a a

Fi Fo (&C2&t)A%an &i*ero+e# B

&C2&t Fi - Fo

&C (Fi Fo) &t


t @ C @

t 8@ C

A%an &i*ero+e#

C (Fi Fo) t (7@ 1@) +t2nt . 8@ nt

C 8@@ +t.

=V

dtFoFidV

0

30

0

)(


37/62

LA9IHAN-7

Tan%i &enan %a*a'ita' 1@@@ L teri'i>@ &enan !enJene. Tan%i en&a*ata+iran !enJene &enan %ee*atan 1@ L*erenit, 'eentara !enJene &a+atan%i &i%e+"ar%an &enan %ee*atan 7@L *erenit. 5era*a 0o+"e tan%i yanteri'i !enJene 'ete+a# 8@ enit


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8. Tan%i &enan %a*a'ita' 1@@@ L. M"+a-"+a !eri'iair &enan 0o+"e 1@@ L. Pa&a '"at" 'aat &ii'i

&enan +ar"tan ara &enan %on'entra'i 1@ 2L&enan %ee*atan *eni'ian 7@ L2enit. 5era*a0o+"e +ar"tan &a+a tan%i &an !era*a%on'entra'inya 'ete+a# 8@ enit.

Fi = 10 L/menit

Ci = 10 g/L

Vo = 100 lt


39/62

3nt"% enye+e'ai%an *er'aaan ini *er+"&i!"at &"a neraa a''a B

Neraa a''a tota+ 0o+"e

Neraa a''a %o*onen ara


40/62

Neraa Ma''a Tota+Ke a''a in %e a''a o"t %e a''a a

Fi @ (&C2&t)A%an &i*ero+e# B

&C2&t Fi &C2&t 7@

&C 7@ &t


t @ C @

t 8@ C

A%an &i*ero+e# *er'aaan

C 1@@ 7@t

Pa&a t 8@ enit a%an &i*ero+e# 0o+"e +ar"tan

C 1@@ [email protected]@ =@@ +t

=V

dtdV

100

30

0

20


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Neraa Ma''a Ko*onen araKe a''a in %e a''a o"t %e a''a a

Fi.4i @ &(C4) 2 &t

dt

dVC

dt

dCVFi.Ci +=

C(20)

dt

dC20t)(10020.10 ++=

dt

dC20t)(10020C200 +=

)20200()20100( CdC

t

dt

=

+


42/62

6i%a &iintera+%an &enan %on&i'i !ata'

t @ 4 @

t 8@ 4 A%an &i*ero+e# B

Pada waktu 30 menit :Ln (7) = ln (200) ln(200-20C)

1.94591 = 5.298317 ln (200 20C)

ln (200 20C) = 3.352407

(200 20C) =e!(3.352407)

200 20C = 28.57143

20C = 171.428"

C = 8.571429 #$% lt

&adi 'lume la$utan dalam tan#ki e*ea$ 700 lt danknent$ain+a adala, 8.571429 #$% lt

=

+

200

20200ln

20

1

100

20100ln

20

1 Ct


43/62

LATIHAN-8

Tan%i &enan %a*a'ita' 1@@@ L. M"+a-"+a !eri'i air ara &enan%on'entra'i @,1 % ara2L 'e!anya%7@@ L. Pa&a '"at" 'aat &ii'i &enan air"rni *a&a %ee*atan *eni'ian 7@L2enit. 5era*a 0o+"e +ar"tan &a+atan%i &an !era*a %on'entra'inya

'ete+a# 8@ enit.


44/62

A*+i%a'i yan e+i!at%an E+een0o+"e:. S"at" &in&in &atar yan !er"*a *e+at !a$a &enan

+"a' *er"%aan 1@@ /t7&enan %ete!a+an 1 in. *a&a!aian &a+a !er'ent"#an &enan airan &enante*erat"r %on'tan :@@ K 'e&anan !aian +"ar!er'ent"#an &enan "&ara +"ar &enan te*erat"r

%on'tan 8@@ K.a. 5"at+a# *er'aaan ateati% yanena!ar%an

*ro'e' !ai% *a&a %on&i'i "n'tea&y 'tate a"*"n

'tea&y 'tate !. Hit"n *ro+e te*erat"r tia* 'at"an $ara% @,1 in


45/62

A

xx !x

Input A.3A4&.A.(5# d/d&)

Output A. 3A4&6,7.A.(5# d/d&)

A#umulasi mc

Catatan

m 8. 9 8.A.,&

"00 #00


46/62

Neraa Ma''a Tota+

e maa in ke maa ut = ke maa a

/.(-k d%d) - /.(-k d%d)= d(m)%dt

/.(k d%d)- /.(k d%d) = ./...(d%dt)

&ika di*a#i den#an elemen lum = /.

( )

dt

dT

k

c

dx

Td

dt

dTc

dx

Tdk

dt

dTc

dx

kd

dt

dTc

x

dx

dTk

dx

dTk

dx

dT

xxx

=

=

=

=

+

2

2

2

2

..


47/62

Per'aaan &a*at &i'e&er#ana%an en$a&i

Di*ero+e# *er'aaan &ieren'ia+ *ar'ia+(PDP)

c

k

t

T

x

T

=

=

12

2


48/62

Pa&a %on&i'i 'tea&y 'tate (a$e) B ti&a%a&a *er"!a#an te*erat"r ter#a&a*a%t", a%a *er'aaan a%an en$a&i

Di*ero+e# *er'aaan &ieren'ia+ or&iner

(PDO) &enan %on&i'i !ata' BQ @ T :@@

Q 1 T 8@@

02

2

=dx

Td


49/62

Denan ara yan 'aa a%an &i*ero+e# #a'i+yan ana+o "nt"% *er*in&a#an a''a &enan*erantian B

nA A a%""+a'i &(C.4)2&tDe % A. &42&t

4 T

0

1

2

2

2

2

=

=

dx

Cd

dt

dC

Dedx

Cd

TAM5AHAN


50/62

;. S"at" *i*a 'i+in&er &ari ar!on 'tee+ &enan&iaeter &a+a ri ; in &an &iaeter +"ar ro


51/62

$

r

Si+in&er

-lemen "olume

uas selimut silinder. Arah

2 r . ,r


52/62

Neraa Ma''a


26$L.(-k d%d$)$ - 26$L.(-k d%d$)$$= d(m)%dt26$L.(k d%d$)$$- 26$L.(k d%d$)$ = . 26$L.$..

(d%dt)

&ika di*a#i den#an elemen lum = 26L. $

( )

dt

dTcr

dr

Tdrk

dr

dTk

dtdTcr

drTdrk

drdk

drdTr

drdr

drdTk

dt

dTcr

dr

rkd

dt

dTcr

r

dr

dTrk

dr

dTrk

dr

dT

rrr

=+

=++

=

=

+

2

2

2

2

..

0


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dt

dT

dr

dT

rdr

Td

dt

dT

k

c

dr

dT

rdr

Td

dt

dT

cdr

dT

krdr

Td

k

11

1

1

2

2

2

2

2

2

=+

=+

=+


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(PDO) &enan %on&i'i !ata' Br ; T 1@@@

r


55/62

TAM5AHAN


nA

A a%""+a'i &(C.4)2&tDe % 7UrL.r&42&t

4 T

01

11

2

2

2

2

=+

=+

dr

dC

rdr

Cd

dt

dC

dr

dC

rdr

Cd


56/62


57/62

,r

r

%lemen &olume

= Luas selimut 'ola A$a

= "* $2 $

N M


58/62

Neraa Ma''a


46$2

.(-k d%d$)$ - 46$2

.(-k d%d$)$3$= d(m)%dt46$2.(k d%d$)$3$- 46$2.(k d%d$)$ = . 46$2.$..(d%dt)

&ika di*a#i den#an elemen lum = 46. $

( )

dt

dTcr

dr

Tdkr

dr

dTrk

dt

dT

crdr

Td

krdr

dk

dr

dT

rdr

dr

dr

dT

rk

dt

dTcr

dr

krd

dtdTcr

rdr

dTkr

dr

dTkr

drdT

rrr

2

2

2

2

2

2

2

22

2

2

2

22

2

2

..

=+

=++

=

=

+

0


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dt

dT

dr

dT

rdr

Td

dt

dT

k

c

dr

dT

rdr

Td

dt

dT

cdr

dT

krdr

Td

k

12

2

2

2

2

2

2

2

2

=+

=+

=+


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(PDO) &enan %on&i'i !ata' Br ; T 1@@@

r


61/62

TAM5AHAN


nA

A a%""+a'i &(C.4)2&tDe % 7Ur7.r &42&t

4 T

02

12

2

2

2

2

=+

=+

dr

dC

rdr

Cd

dt

dC

dr

dC

rdr

Cd


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3 matematika teknik kimia 1

Documents

ti an o i yan i

eraaan ateati nan

ana ii eniti0ita oe

erta ratan ata7

ataan aian ni

iia i a a ite

ai oe eai enan

an enan e