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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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
7
The Discontinuous/Batch ReactorThe Discontinuous/Batch Reactor
Irreversible chemical processes
(the equilibrium constant has very large values, thus the reverse
reaction is negligible)
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
8
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Single reaction chemical process
Irreversible first order chemical processes
A) The volume of reaction is constant during the chemical process
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
B) The volume of reaction is variable during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
11
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible second order elementary chemical processes
A) The volume of reaction is constant during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
14
The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA = 1
Constant volume case
0 t
XAXA
1
0
Slope = kCA0
t 0
1CA
0 Slope=k
1/CA0
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
15
The Discontinuous/Batch ReactorThe Discontinuous/Batch ReactorCommon plot for a irreversible second-order elementary reactions, MBA = 1
Constant 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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
16
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible second order elementary chemical processes
B) The volume of reaction is variable during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
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
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 1.5, MBA = 2
time, s
XA
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 0.5, MBA = 4
time, s
XA
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 1.5, MBA = 2
time, s
XA
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
18
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
1
2 1
Z
Z
Irreversible second order elementary chemical processes
B) The volume of reaction is variable during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 0.5, k = 0.05
time, s
XA
0 5 10 15 20 250
0.2
0.4
0.6
0.8
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 1.5, k=0.05
time, s
XA
0 5 10 15 20 250
0.2
0.4
0.6
0.8
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 0.5, k=0.075
time, s
XA
0 5 10 15 20 250
0.2
0.4
0.6
0.8
eps = 0eps > 0eps < 0
eps = 0eps > 0eps < 0
CA0 = 1.5, k=0.075
time, s
XA
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
20
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible second order chemical processes
A) The volume of reaction is constant during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
21
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible second order chemical processes
B) The volume of reaction is variable during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
22
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible trimolecular type third-order reactions
A) The volume of reaction is constant during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
24
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible trimolecular type third-order reactions
B) The volume of reaction is variable during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
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
eps = 0eps >0eps < 0
eps = 0eps >0eps < 0
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
eps = 0eps >0eps < 0
eps = 0eps >0eps < 0
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
eps = 0eps >0eps < 0
eps = 0eps >0eps < 0
CA0 = 0.5, MBA=2, MDA=2.5, k=0.075
time, s
XA
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
26
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible third-order reactions of particular type
A) The volume of reaction is constant during the chemical process
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
27
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Irreversible third-order reactions of particular type
B) The volume of reaction is variable during the chemical process
2 products
products
k
k
A B
A B
This case is homework
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
28
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
29
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
eps = 0eps >0eps < 0
eps = 0eps >0eps < 0
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
eps = 0eps >0eps < 0
eps = 0eps >0eps < 0
CA0 = 1.5, n=0.25, k=0.05
time, s
XA
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
30
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
31
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
32
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
33
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
34
The Discontinuous/Batch ReactorThe Discontinuous/Batch Reactor
Reversible chemical processes
(The complete conversion is never achieved, due to the thermodynamic barriers)
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
35
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
36
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
37
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
38
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Second-order Reversible Reactions (V=ct)
1
1
11 2ln
1
X XA Ae
Ae
XAe
X XA
t
1Slope=2 1 0dk X CAe A
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
39
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Second-order Reversible Reactions (V=ct)
A d A B d R Sr k C C k C C
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
The integral form of the characteristic equation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
40
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
41
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Second-order Reversible Reactions (V≠ct)
A d A B d R Sr k C C k C C
0 0
11
AX A ARA SAd A
C A A
X dXM Mk C t
K X p X q
The integral form of the characteristic equation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
42
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
Multiple reactions chemical processes
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
43
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
44
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
45
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
46
The DC Reactor – Solving the Characteristic The DC Reactor – Solving the Characteristic EquationEquation
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
54
The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
55
The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
56
The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
57
The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
58
The method of finite molecular The method of finite molecular weight momentsweight moments
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
59
The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
60
The method of finite molecular The method of finite molecular weight momentsweight moments
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%.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
61
The method of finite molecular The method of finite molecular weight momentsweight moments
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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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The method of finite molecular The method of finite molecular weight momentsweight moments
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.
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
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The method of finite molecular The method of finite molecular weight momentsweight moments
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
April 8, 2023 FILS – Chemical ReactorsThe Discontinuous/Batch Reactor
64
The method of finite molecular The method of finite molecular weight momentsweight moments
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.
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