iv-4 - discontinuous-batch chemical reactors

April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor

1

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorA discontinuous or batch reactor has no fluid streams flowing through.

The total feed is introduced before start, and no withdrawal is made until the reaction has reached the degree of completion desired.

Perfect mixedmedium

X A(t)

V(t)

C A(t)

The mathematical model results from mass balances for all species

Input of mass to Output of mass from Accumulation of mass

the system during the system during within the system

the time period the time period during the time periodt t t

amount of mass amount of mass

in the system in the system

at time at timen nt t t


2

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorThe Characteristic Equation

n n

Mass within the system Mass within the systemInput of mass to the Output of mass from the

at time t +Δt at time tsystem per unit time system per unit time

n

n

t t

t

dt

Rewriting the dynamic mass balance

n

Input of reactant Output of reactant by mass of reactant within mass of reactant wiper unit time reaction per unit time the system at time t t

0n

n n

n

t t

A A A A t t A A t

t A A A A

M r t V t dt M N M N

n

thinthe system at time t

n

n n

n

t t

A A t t A t

t

r t V t dt N N


3

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorUsing the Mean Value Theorem of Integral Calculus A

A

dNV r

dt

j

j j

A

A A

dNsign V r

dt

0 / 0j j

j jA A A A A A AdN d N M X N dX

a a

0

A A

A

dX r V

dt N

0 XA

1

Ar V

0

AreaA

t

N

0 0

AfXA

AA

dXt N

r V

The Characteristic Equation

Reaction time

Initial conditions

Performance

Operating conditions

Single Reaction Chemical Process 1

0S

j jj

A


4

The Discontinuous/Batch ReactorThe Discontinuous/Batch Reactor

Multiple reactions chemical processes

Rewriting the dynamic mass balance

1 1

0R S

ij ji j

A

01 1

j

j j

R RA

A ij i A ij ii i

dN dN V r V r

dt dt

1 1

1 R Ri

ij ij ii i

dr

V dt

1, 1,2, ,i

i

dr i R

V dt

Reaction rate with respect to Aj

Equivalent reaction rate for “i” reaction

Taking the derivative of the parenthesis

Holds for every term of the sum

The extent of reaction “i”, kmol

The vector of extents of all reactions, kmol


5

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorThe Characteristic Equation and the Volume of Reaction

A) The volume of reaction is constant during the chemical process

1) Single reaction chemical process

, 1, 2, ,V ii V

dr i R

dt

0

0

0

A A

A

X C

A AA

A A A AC

dX dCt C

r X r C

0gas

1 A

A

p

A

A Ap

dpt

R r p T

2) Multiple reactions chemical process

The volumic extent of reaction “i”, kmol ∙m-3

The vector of volumic extents of all reactions, kmol∙m-3


6

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorThe Characteristic Equation and the Volume of Reaction

B) The volume of reaction is variable during the chemical process

1) Single reaction chemical process

2) Multiple reactions chemical process

0 0 1 ,

AXA

AA A A

dXt C

X r X

0 1 AV V X

1 1 1, 1,2, ,i i mi

i i i m

d d dr r r i R

V dt m dt dt

The mass extent of reaction “i”, kmol ∙kg-1

The vector of mass extents of all reactions, kmol∙kg-1


7


Irreversible chemical processes

(the equilibrium constant has very large values, thus the reverse

reaction is negligible)


8

The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation

Single reaction chemical process

Irreversible first order chemical processes


productskA

0 1A A A Ar k C k C X 0 0

A

A

C tA

CA

dCk dt

C

0 01

AX tA

A

dXk dt

X

1 ktAX e

0kt

A AC C e


0

1

1A

A A AA

Xr k C k C

X

1

1

A

A

A

dXX

X

1 AX 0 0

AX t

k dt 1 ktAX e

Important Conclusion!! The volume behavior does not affect the performance


9

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a first-order irreversible reaction

0

t

Slope = k

ln 1

ln0

XA

CACA

0

1 ktAX e

0kt

A AC C e

The initial concentration does not affect the performance of the first order processes


10

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a first-order irreversible reaction

0 20 40 60 80 1000

0.5

1

k(T)k(T-Ta) < k(T)k(T+Ta) > k(T)

k(T)k(T-Ta) < k(T)k(T+Ta) > k(T)

time, s

XA


11


Irreversible second order elementary chemical processes


0

0 01

AX tA

AA BA A

dXC k dt

X M X

productskA B

20 1A A B A A BA Ar k C C k C X M X

0 1 ln1

BA AA BA

BA A

M Xk C M t

M X

020 01

AX tA

A

A

dXC k dt

X

0 1A

AA

Xk C t

X

1 21

1 1A BA A A BA A

Z Z

X M X X M X

1 2 1 1BA A AZ M X Z X 1

2

11

1

1

1

ABA

A BABA

X ZM

X M ZM

1BAM

1BAM


12

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA ≠ 1

Constant volume case

lnCCB

A

0 0

Slope= 0 0( - )B Ak C C

0

0

0

ln1

ln

BA A

BA A

B A

A B

M XM X

C CC C

0

t

Slope= 0 0 ( - )B Ak C C

t

Intercept= ln( )BAM


13

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA ≠ 1Constant volume case

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

k(T)k(T- a) < k(T)k(T+ a) > k(T)

k(T)k(T- a) < k(T)k(T+ a) > k(T)

CA0 = 0.5

time, s

XA

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

k(T)k(T- a) < k(T)k(T+ a) > k(T)

k(T)k(T- a) < k(T)k(T+ a) > k(T)

CA0 = 1.5

time, s

XA

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

MBA = 2MBA = 4MBA = 2MBA = 4

CA0 = 0.5

time, s

XA

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

MBA = 2MBA = 4MBA = 2MBA = 4

CA0 = 1.5

time, s

XA


14

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA = 1


0 t

XAXA

1

0

Slope = kCA0

t 0

1CA

0 Slope=k

1/CA0


15

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA = 1


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

k(T)k(T- a) < k(T)k(T+ a) > k(T)

k(T)k(T- a) < k(T)k(T+ a) > k(T)

CA0 = 0.5

time, s

XA

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

k(T)k(T- a) < k(T)k(T+ a) > k(T)

k(T)k(T- a) < k(T)k(T+ a) > k(T)

CA0 = 1.5

time, s

XA


16




productskA B

20

1

1 1A BA A

A A B AA A

X M Xr k C C k C

X X

0 0 02 0

0

111 1

11 1

A AX X A AAA

A BA A A A BA AA A

A A

X dXdXt C

X M X k C X M XX k C

X X

1 21

1 1A

A BA A A BA A

X Z Z

X M X X M X

1 2 1 1BA A A AZ M X Z X X

1

2

11

1

1

1

ABA

BAA BA

BA

X ZM

MX M Z

M

1 1

0

11 ln ln

1

BAM

BABA A

A BA A

MM k C t

X M X


17

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA ≠ 1Variable volume case

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

eps = 0eps > 0eps < 0


CA0 = 0.5, MBA = 2

time, s

XA

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8



CA0 = 1.5, MBA = 2

time, s

XA

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8



CA0 = 0.5, MBA = 4

time, s

XA

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8



CA0 = 1.5, MBA = 2

time, s

XA


18


1

2 1

Z

Z



productskA B

2

20

1

1A

A A B AA

Xr k C C k C

X

20

0

11

1

AX A A

A A

X dXt

k C X

1 2

2 2

1

11 1A

AA A

X Z Z

XX X

1BAM

0

11 ln

1 1A

AA A

Xk C t

X X


19

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA = 1Variable volume case

0 5 10 15 20 250

0.2

0.4

0.6

0.8



CA0 = 0.5, k = 0.05

time, s

XA

0 5 10 15 20 250

0.2

0.4

0.6

0.8



CA0 = 1.5, k=0.05

time, s

XA

0 5 10 15 20 250

0.2

0.4

0.6

0.8



CA0 = 0.5, k=0.075

time, s

XA

0 5 10 15 20 250

0.2

0.4

0.6

0.8



CA0 = 1.5, k=0.075

time, s

XA


20


Irreversible second order chemical processes


0

0 01

AX tA

A

A BA A

dXC k dt

bX M X

a

productskaA bB

20 1A A B A A BA A

br k C C k C X M X

a

BA

bM

a

0 ln1

BA A

A BABA A

bM Xb akC M t

a M X

BA

bM

a 0 1

AA

A

XbkC t

a X


21


Irreversible second order chemical processes


0

0 0

1

1

AX tA A

A

A BA A

X dXC k dt

bX M X

a

productskaA bB

2

0 2

1

1

A BA A

A A B A

A

bX M X

ar k C C k C

X

BA

bM

a

1

1

0

1ln ln

1

BAa M

b

BAA BA

ABA A

Mbk C M t

ba X M Xa

BA

bM

a 0

11 ln

1 1A

AA A

Xbk C t

a X X


22


Irreversible trimolecular type third-order reactions


2

0

0 01

AX tA

AA BA A DA A

dXC k dt

X M X M X

productskA B D

30 1A A B D A A BA A DA Ar k C C C k C X M X M X

1

1BA

DA

M

M

20

1ln ln ln

1

1 1 1 1

BA DA

A BA A DA AA

BA DA BA DA BA DA BA DA

M M

X M X M XkC t

M M M M M M M M

1

1 1A BA A DA A A BA A DA A

Y Z W

X M X M X X M X M X

1 1 1

; ;1 1 1BA DA A BA DA BA DA BA DA

Y Z WM M X M M M M M M


23

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible third-order elementary reactions, MBA > 1 , MDA > 1Constant volume case

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

k(T0)k(T>T0)k(T<T0)

k(T0)k(T>T0)k(T<T0)

CA0 = 0.5, MBA=2, MDA=2.5

time, s

XA

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

k(T0)k(T>T0)k(T<T0)

k(T0)k(T>T0)k(T<T0)

CA0 = 0.5, MBA=4, MDA=5

time, s

XA

0 5 10 15 20 250

0.2

0.4

0.6

0.8

k(T0)k(T>T0)k(T<T0)

k(T0)k(T>T0)k(T<T0)

CA0 = 1.5, MBA=2, MDA=2.5

time, s

XA

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

k(T0)k(T>T0)k(T<T0)

k(T0)k(T>T0)k(T<T0)

CA0 = 1.5, MBA=4, MDA=5

time, s

XA

Drawback

Increasing MBA and/or MDA results in large quantities of B & D remained unreacted


24


Irreversible trimolecular type third-order reactions


2

20

0 0

1

1

AX tA A

AA BA A DA A

X dXC k dt

X M X M X

productskA B D

30 3

1

1

A BA A DA AA A B D A

A

X M X M Xr k C C C k C

X

1

1BA

DA

M

M

2 2 2

20

11 ln 1 ln 1 ln

1

1 1 1 1

BA DABA DA

A BA A DA AA

BA DA BA BA DA DA BA DA

M MM M

X M X M XkC t

M M M M M M M M

21

1 1A

A BA A DA A A BA A DA A

X Y Z W

X M X M X X M X M X


25

The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible third-order elementary reactions, MBA > 1 , MDA > 1Variable volume case

Drawback

Increasing CA0 results in faster reactions, but large reactants quantities remaine unreacted

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

eps = 0eps >0eps < 0


CA0 = 0.5, MBA=2, MDA=2.5, k=0.05

time, s

XA

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8



CA0 = 0.5, MBA=2, MDA=2.5, k=0.075

time, s

XA

0 5 10 15 200

0.2

0.4

0.6

0.8



CA0 = 1.5, MBA=2, MDA=2.5, k=0.05

time, s

XA

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8



CA0 = 0.5, MBA=2, MDA=2.5, k=0.075

time, s

XA


26


Irreversible third-order reactions of particular type


2

0

0 01

AX tA

AA BA A

dXC k dt

X M b X

2 products

products

k

k

A B

A B

2302

230

1 2

1

A A BA AA A B

A A BA A

k C X M Xr k C C

k C X M X

2BAM

2

0

2 212 ln

2 1 2BA A A

A BABA A BA BA A

M X XkC M t

M X M M X

2BAM

20 2

22

1

A AA

A

X XkC t

X

2b

1BAM

2 0

0

11 lnBA BA B A B

BAB B BA A

M M C C Ck M t

C C M C

1BAM

2 20

1 12

A A

k tC C

1b


27


Irreversible third-order reactions of particular type


2 products

products

k

k

A B

A B

This case is homework


28


Empirical rate equations of nth order productskA

0

0

1 , 0

1, 0

1

nnA A

n nA A n A

AA

k C X V

r k C Xk C V

X

0V

1 10 1 , 1n n

A AC C n kt n 0V

•The reactions with order n >1 can never go to completion in finite time (but this is a characteristic of first order chemical processes).

•For orders n < 1, this rate form predicts that the reactant concentration will fall to zero and then become negative at some finite time

1

0max 1

nAC

tn k

1

10

0

1

1

AnX

A AnA n

A

X dXk C t

X


29


Empirical rate equations of nth order productskA

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

k(T)k(T<T0)k(T>T0)

k(T)k(T<T0)k(T>T0)

CA0 = 0.5, k=0.05, n=1.75

time, s

XA

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

k(T)k(T<T0)k(T>T0)

k(T)k(T<T0)k(T>T0)

CA0 = 0.5, k=0.05, n=0.05

time, s

XA

0 5 10 15 200

0.2

0.4

0.6

0.8



CA0 = 0.5, n=0.25, k=0.05

time, s

XA

0 10 20 30 40 500

0.2

0.4

0.6

0.8



CA0 = 1.5, n=0.25, k=0.05

time, s

XA


30


Reactions of Shifting Order productskA

1

21A

AA

k Cr

k C

A) Shift from low to high as the reactant concentration drops

•at high CA (or k2CA»1) - the reaction rate is of zero order with rate constant k1/k2.•at low CA (or k2CA«1) - the reaction is of first order with rate constant k1.

rAk2 / k1

CA

First order

Zero order

CA

t

Zero order

First order

12

0 0

1ln

1 A

A A A A

X k tk

C X C X


31


Reactions of Shifting Order productskA

1 2A Ar k k C

B) Shift from high to low as the reactant concentration drops(two competing reactions of different orders)

•at low CA - the reaction rate is of zero order with rate constant k1.•at high CA- the reaction is of first order with rate constant k2. rA

CA

First order

Zero order

CA

t

Zero order

First order

21 2 0

1 2

k tA

A

k k Ce

k k C


32


Autocatalytic Reactions kA R R R

20 1A A R A A RA Ar k C C k C X M X

One of the products of reaction acts as a catalyst

The total number of moles of A and R remains unchanged

max 1

2RA

A

MX

0 0 0A R A RC C C C C

There is a maximum, due to the antagonic effects

20 1 2 0A

A RA AA

rk C M X

X

0

0t

1

XA

Low rate

High rate

Low rate

Progress in time

Parabolic

Start with some R present

00

rA

1CA = CR CA/CA0

0ln 11

RA AA RA

RA A

M XC M kt

M X


33


Autocatalytic Reactions kA R R R

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

MRA = 0.01MRA = 0.05MRA = 0.1

MRA = 0.01MRA = 0.05MRA = 0.1

CA0 = 0.5, k=0.075

time, s

XA

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

MRA = 0.01MRA = 0.05MRA = 0.1

MRA = 0.01MRA = 0.05MRA = 0.1

CA0 = 1.5, k=0.075

time, s

XA

0 100 200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

k(T0)k(T<T0)k(T>T0)

k(T0)k(T<T0)k(T>T0)

CA0 = 0.5,MRA = 0.01

time, s

XA

0 50 100 150 2000

0.2

0.4

0.6

0.8

k(T0)k(T<T0)k(T>T0)

k(T0)k(T<T0)k(T>T0)

CA0 = 1.5,MRA = 0.1

time, s

XA


34


Reversible chemical processes

(The complete conversion is never achieved, due to the thermodynamic barriers)


35


First-order Reversible Reaction d

r

k

kA P

0

11

1 ,1

1

PA

C dA d A r P d A A C

C r

C

MK k

r k C k C k C X KK k

K

At equilibrium

11

1

11 1 1PA

dC

C PA AA

Ae

Ad A

CC

C

K M XX

XK

K

dXk X

dt K

Mk k

K

0AA

dCr

dt

1

01

1

PA

CA

C

MK

X

K

1Pe PA Ae

CAe Ae

C M XK

C X

The mass balance – applying the differential form of the characteristic equation


36


First-order Reversible Reaction d

r

k

kA P

ˆˆ ˆ1A

A

dXk X

dt

11 ˆlnˆ1

d PA

PA AeA

k Mkt t

M XX

t

1ln

ˆ1 AX

1Slope d PA

PA Ae

k M

M X


37


Second-order Reversible Reactions (V=ct)

2

2 2

2

d

r

d

r

d

r

d

r

k

k

k

k

k

k

k

k

A B R S

A R S

A R

A B R

A d A B d R Sr k C C k C C

At equilibrium

1 2 11

ln 2 11

AAe

Aed Ao

A Ae

Ae

XX

Xk C t

X XX

0AA

dCr

dt

2 2 10,

1 11 1

Ae dAe C

r

C C

X kX K

kK K

The integral form of the characteristic equation

0 0 0 0& 0A B R RC C C C

2 20

21 11

1 11 1

AA d A A

C

C C

Xr k C X

KK K


38



1

1

11 2ln

1

X XA Ae

Ae

XAe

X XA

t

1Slope=2 1 0dk X CAe A


39




At equilibrium

1

ln 1Ad Ao

A C

p q Xp q k C t

q p X K

0AA

dCr

dt

1 1

2

the roots are denoted andq p

1 11 1 1 0, d

A BA A BA CC C r

kX M X M K

K K k


0 0 0 0& 0A B R RC C C C

2 20

111

1 11 1

BA A BAA d A A

C

C C

M X Mr k C X

KK K


40


Second-order Reversible Reactions (V=ct) 2

2 2

2

d

r

d

r

d

r

d

r

k

k

k

k

k

k

k

k

A B R S

A R S

A R

A B R

A d A B d R Sr k C C k C C 0 0 0 0& 0A B R RC C C C

2 20

2 20

22 2

0

11

1 11 1

12

11 1

1 1

21

2 11

11

RA SABA

BA RA SA ARA SA CA d A A

C

C C

RA SA

RA SA ARA SA CA d A A

C

C C

R

A RARAA d A A

C

C

M MM

M M M XM M Kr k C X

KK K

M MM M XM M K

r k C XK

K K

MX MM

r k C XK

K

22 2

0

11

21 2

11 1

1 1

A

C

C

RABA

BA RA A CRAA d A A

C

C C

K

K

MM

M M X KMr k C X

KK K

The solving procedure remains the same; case should be taken when computing the roots of the equilibrium equation.


41


Second-order Reversible Reactions (V≠ct)


0 0

11

AX A ARA SAd A

C A A

X dXM Mk C t

K X p X q


0 0 0 0& 0A B R RC C C C

2 2

0 2

11

1 11 1 1

RA SA RA SABA

BA RA SA AC CA d A A

A

C C

M M M MM

M M M XK Kr k C X

XK K

d

r

k

kA B R S

0 1 ln ln lnARA SA

d AC A A A

p q XM M q pk C p q t q p

K q p X q X p X


42


Multiple reactions chemical processes


43


Irreversible Reactions in Parallel (V=ct)1

2

1 1

2 2

,

,

kA

kA

A R r k C

A S r k C

1 220 0 0

0 1 2

1 k k tSA R S A R S

A

C kC C C C C C e

C k k

1 2

1 2 00

ln k k tAA A

A

Ck k t C C e

C

1 2 1 211 1 0

0 1 2

1k k t k k tR RA A

A

dC C kk C k C e e

dt C k k

The integration of equations gives

2

1 2 1 21

AA ij i A A A

i

dCr r k C k C k k C

dt

1R

R A

dCr k C

dt


44


Irreversible Reactions in Parallel (V=ct)1

2

1 1

2 2

,

,

kA

kA

A R r k C

A S r k C

t 0

0

CA CR

CS

CA0


45


Irreversible Reactions in Series (V=ct) 1 2k kA R S

1 22 1

00 1 2

1k t k t

SA A R S

A

k e k eCC C C C

C k k

11

0 0

ln k tA A

A A

C Ck t e

C C

12 1 0

k tRR A

dCk C k C e

dt

The integration of equations gives

2

11

AA ij i A

i

dCr r k C

dt

2

1 21

RR ij i A R

i

dCr r k C k C

dt

2 11

0 1 2

k t k tR

A

C ke e

C k k


46


Irreversible Reactions in Series (V=ct) 1 2k kA R S

2 1max

2 1

ln /k kt

k k

2 2 1/

,max 1

0 2

k k k

R

A

C k

C k


47

Polymerization in Batch ReactorsPolymerization in Batch Reactors

Calculation of molecular Calculation of molecular weight distribution in a batch weight distribution in a batch

thermal polymerization of thermal polymerization of styrene styrene


48

Polymerization in Batch ReactorsPolymerization in Batch ReactorsPolymer molecular weight distribution (MWD) is one of the most important polymer properties that dictate the physical, mechanical, and rheological properties of industrial polymers. These are commonly used:• Molecular weight averages

•Average/mean molecular number•Average/mean molecular mass

• Polydispersity - ratio of mean molecular mass and mean molecular number

There are cases of two polymers of different chain length distribution having identical molecular weight averages.

Sometimes, a slight variation in high or low molecular weight fractions of polymer causes a significant difference in the polymer’s enduse properties.

In practice, it is desired to predict or control the entire chain length distribution instead of molecular weight averages.


49

Polymerization in Batch ReactorsPolymerization in Batch Reactors

In modern industrial polymerization processes, polymerization process conditions are designed and operated to produce the polymers of desired molecular weight properties with minimum variance from their target values.

To do so, it is required that the relations between the polymer’s molecular weight properties and reaction conditions be quantitatively established.

A detailed polymerization process model can be utilized for such purposes.

If the main goal is to predict and control polymer molecular weight distribution in a given polymerization process, it will be necessary to have available an appropriate computational method to predict the chain length distribution during the progress of reaction.


50

Polymerization in Batch ReactorsPolymerization in Batch ReactorsWhen a polymerization kinetic model is available, molecular weight averages

and MWD can be calculated by:• numerical integration of the polymer population balance equations• using a moment generating function• z-transform• continuous variable transformation• the method of molecular weight moments• Markov chain approaches• integrating an instantaneous chain length distribution to a desired

conversion• fitting a pre-specified chain length distribution function with molecular

weight averages• Monte Carlo simulation techniques, for free radical polymerization• discrete weighted Galerkin methods, with expansion of a chain length

distribution into certain orthogonal polynomials.


51

Polymerization in Batch ReactorsPolymerization in Batch ReactorsThe Method of Finite Molecular Weight Moments (MFMWM)• Calculate the chain length distribution in free radical polymerization where chain termination is via disproportionation and combination mechanisms.• Instead of calculating the concentration or weight fraction of polymer with a certain chain length, the weight fraction of polymers in a finite chain length interval is calculated.• The function that defines the weight fraction of polymer in a certain chain length interval is analytically represented from the reaction kinetic model.

In MFMWM, a quasi-steady state approximation is used for live polymer radicals and the resulting recursive relationship between the concentrations of live polymer chains is used to derive the expressions for polymer weight fractions.The major advantage of the method of finite molecular weight moments is that the differential equation for the weight fraction of any finite chain length interval is explicitly expressed in terms of process variables.This method is different from simply discretizing the polymer population balance equations to finite difference form for numerical calculations.


52

The method of finite molecular The method of finite molecular weight momentsweight moments

The kinetic scheme

1

1 2

1

1

3 2i

p

p

fin

t

k

k

k

i i

k

i i

ki j i j

M R

R M R

R M R

R M P R

R R P

Thermal initiation

Propagation

Termination thru

•Disproportionation

•Combination

Thermal polymerization of styrene


53


Applying the quasi-steady state approximation to live polymers (radicals) → The concentrations of radicals (live polymers)

The chain length i

Total live polymer concentration

1 1R R

11 iiR R

p

p fin t

k M

k M k M k M

1/ 232 i

t

k MR

k

The probability of propagation

The Schulz-Flory Most Probable Distribution


54


The mass balance equation for polymer of chain length i

21 2 21 1 12

i ii tfin

dP kk M R i R

dt

The molecular weight averages can be conveniently calculated using the method of moments

11 1

Ri

i

RiR

22 2

1

1

1R

ii

Ri R

0

1

Ri

i

R R

11 1

Pi

i

PiP

22 2

1

1

1P

ii

Pi P

0

1

Pi

i

P P


55


The moments equation for polymer of chain length i

20 1

2

P

t fin

dk R k MR

dt

21 12

1

P

t fin

dk R k MR

dt

2 22 12 3 4

1

P

t fin

dk R k MR

dt


56


The number-average and weight-average chain lengths

1 1 1

0 0 0

P R P

n P R PX

2 2 2

1 1 1

P R P

w P R PX

2

0

Pw

Pn

XPolydispesity

X


57


For the calculation of a complete polymer chain length distribution, the polymer population balance equations need to be solved.

The method of integrating the ODE system for “all i” to compute the polymer molecular weight distribution is impractical because of the very large value of i.

Simpler is the method of finite molecular weight moments.

2

Weight of polymer chain lenght from m to n,

Total weight of polymer

n

ii m

ii

iPf m n

iP

Definition


58



59


Having defined the function f (m, n), the following equation is derived

1

1

, ,1 Pni

Pi m

df m n f m ndP di

dt dt dt dt

11

2

1 11

2

1 2

2

1

11

11, 1

1 2 1 1 11

2 1 1 2 2 2 1

,2

1

m nm n

fin

P n n n

t

m m m

t finP

m nk M R

df m n

dt n n n n n nkR

m m m m m m

f m nk R k MR


60


Batch near-isothermal bulk polymerization of 150oC

The reaction temperature reaches 150oC after about 60 min, during which period, the monomer conversion is quite low.

This operation as a “near” isothermal operation.

As expected, the molecular weight averages reach their stationary values when the monomer conversion exceeds about 50%.


61


Weight chain length distribution for near-isothermal polymerization at 150oC

Notice that after 3 h of reaction, the chain length distribution is almost time invariant.


62


The reactor temperature set point is changed from 150oC to 180oC at t = 120 min

With an increase in the reactor temperature, the polymer molecular weight decreases slightly and the polydispersity increases slightly.


63


Weight chain length distribution for non-isothermal polymerization

Although the polymer chain length distribution curves look quite similar, there is a distinct diffe-rence in the final chain length distributions bet-ween the two cases


64


Differential and cumulative chain length distributions for near-isothermal and nonisothermal polymerizations

The temperature variations yield a subtle but clear difference in the chain length distributions of the two polymers → a difference in the enduse properties.

If the polymer chain length distribution can be computed with a kinetic model, it is possible to discern the difference in the polymer molecular weight properties, otherwise difficult to do with molecular weight averages only.

Documents

iv-4 - discontinuous-batch chemical reactors