P9-10 (c)
The unsteady energy balance for a batch reactor is showed in equation 9-11.
( ) ( )
!
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=
ii
ARxs
CpN
VrHWQ
dt
dT &
&
Adiabatic operation (
Q
&
=0), neglecting
s
W
&
and assuming ΔCp=0 gives the following using the
expression for –rA:
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000
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1
TTTTTT
TT
Ck
Cp
H
TT
TTTTTTCk
CCp
H
Vr
CpN
H
dt
dT
BfBf
f
A
ii
Rx
f
BfBfA
Aii
Rx
A
iiA
Rx
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–
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P9-10 (d)
Put α = 1, β = 1, ΘB = 3 in the equation from part (c)
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0
0
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00
1
11
0
111
0
23
33
TTTTT
TT
Cp
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Ck
Cp
H
dt
dT
ff
f
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ii
Rx
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Rx
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P9-10 (e)
Use the reaction constant expression from the great Swedish chemist Arrhenius and develop
expression from 9-10d.
( )
( ) ( ) ( )
( )
0
0 0
2
0
3 3
Rx A
f f
i i f
Hk C
dT T T T T T T
dt Cp T T
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=# % % # +# % &
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0
1
2
1
0 0 0
exp
3 3
A
f f f
dT
CE E
dt kR T R T
T T T T T T T T
! “
=# # $ %
& ‘
# #
$ # # $ +$ # ( )
$
1 0
2
ln ln
A
dT
k C E E
dt
R T R T
! ” ! “
# $ %
# $
= + &
# $ # $ % %
‘ (
9-22
P9-10 (f)
First use a plot of T vs. t to get T0 (329 K) and Tf (439 K). Checking the concentration of species
B gives ΘB=3. Make a table showing t, T, dT/dt, left hand side of equation P9-10.7 and 1/T. Plot
left hand side of equation P9-10.7 vs. 1/T in Polymath and use linear regression to get E from the
slope and k1 from the intercept.
Regression equation as shown in Polymath:
Activation energy and Arrhenius constant from slope and intercept:
8944.2 8.314 74.36
E
slope E
R
=! ” =! # =!
kJ/mol
( ) {
17.534 11
1 0 1
1
2
0
0
6.7 12100
0 17.534 ln ln exp 1.298 10
12100 6.7
A
f
neglect
k C k
E
a k
R T
T T
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= = + = &=#=
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)
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P9-10 (g)
Follow the procedure from example 9-3 gives heat of reaction.
X=1 and T=Tf gives
Kmol
#
&
(
P9-11 (a)
9-23
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 1000 1000
Na 50 0.3324469 50 0.3324469
vb 1.5 1.5 1.5 1.5
k 5.0E-04 2.3E-04 5.0E-04 4.73E-04
V 10 10 1510 1510
Fbo 6 6 6 6
Ca 5 2.202E-04 5 2.202E-04
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Na)/d(t) = ra*V
[2] d(Nb)/d(t) = 2*ra*V+Fbo
[3] d(Nc)/d(t) = –ra*V
9-24
Explicit equations as entered by the user
[1] vb = 1.5
[2] k = .0005*exp((8000/1.987)*(1/300-1/T))
[3] V = 10+(vb*t)
[4] Fbo = 4*vb
[5] Ca = Na/V
9-25
P9-11 (b)
P9-12
9-26
9-27
P9-12 (a)
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 4 4
Ca 0.03789 0.03789 0.0584442 0.053671
Cb 2.12 2.12 2.1423214 2.1376461
Cm 0.2265 0.2265 0.226519 0.226519
T 138.53 125.70694 138.53 128.49299
I 0 -39.636625 0 -39.636625
Tsp 138.5 138.5 138.5 138.5
UA 1.6E+04 1.6E+04 1.6E+04 1.6E+04
Fb0 1000 1000 1000 1000
NCp 3372.5882 3372.5882 3385.3399 3383.2355
ThetaCp 284.375 284.375 284.375 284.375
v0 441.46403 441.46403 441.46403 441.46403
Ca0 0.1812152 0.1812152 0.1812152 0.1812152
Cb0 2.2651902 2.2651902 2.2651902 2.2651902
tau 0.1513355 0.1513355 0.1513355 0.1513355
mc 1000.3 872.17351 1000.3 899.92991
Ta2 106.23674 102.00022 106.23674 102.98479
9-28
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ca)/d(t) = 1/tau*(Ca0-Ca)+ra
[2] d(Cb)/d(t) = 1/tau*(Cb0-Cb)+ra
[3] d(Cc)/d(t) = 1/tau*(0-Cc)-ra
Explicit equations as entered by the user
[1] Fa0 = 80
[2] T0 = 70
[3] V = (1/7.484)*500
[4] Tsp = 138.5
[5] UA = 16000
[6] Ta1 = 60
[7] kc = 10
[8] k = 16.96e12*exp(-32400/1.987/(T+460))
[9] Fb0 = 1000
[10] Fm0 = 100
[11] mc0 = 1000
9-29
P9-12 (b)
See Polymath program P9-12-b.pol
P9-12 (c)
9-30
P9-13
9-31
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 400 400
Ca 0.001 1.027E-04 0.0732572 1.027E-04
Cc 0 0 0.099864 0.099864
T 300 300 2.574E+05 2.574E+05
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ca)/d(t) = ((.1-Ca)/50)+ra
[2] d(Cb)/d(t) = ((.1-Cb)/50)+ra
[3] d(Cc)/d(t) = (-Cc/50)-ra
Explicit equations as entered by the user
[1] V = 100
9-32
See Polymath program P9-13-b.pol
See Polymath program P9-13-c.pol
9-33
P9-14 (a)
See Polymath program P9-14-a.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 4 4
Ca 0.03789 -27.922973 0.0898852 -27.922973
Cc 0.143 -1.7469378 0.143 -1.7469378
T 138.53 17.999228 138.53 17.999228
I 0 -307.34656 0 -307.34656
Tsp 138.5 138.5 138.5 138.5
UA 1.6E+04 1.6E+04 1.6E+04 1.6E+04
k 24.990212 0.0260082 24.990212 0.0260082
9-34
Fm0 100 100 100 100
mc0 1000 1000 1000 1000
ra -0.9468791 -0.9468791 0.723285 0.723285
NCp 3372.5882 3372.5882 5.574E+04 5.574E+04
Cb0 2.2651902 -183.76896 364.28299 -4.45998
tau 0.1513355 -12.277456 24.337453 -0.2979677
mc 1000 938.55771 1000 938.55771
Ta2 106.24535 34.305432 106.24535 34.305432
ODE Report (STIFF)
Differential equations as entered by the user
[1] d(Ca)/d(t) = 1/tau*(Ca0-Ca)+ra
[2] d(Cb)/d(t) = 1/tau*(Cb0-Cb)+ra
[3] d(Cc)/d(t) = 1/tau*(0-Cc)-ra
[4] d(Cm)/d(t) = 1/tau*(Cm0-Cm)
Explicit equations as entered by the user
[1] Fa0o = 80
[7] kc = .2
[8] k = 16.96e12*exp(-32400/1.987/(T+460))
[9] Fb0 = 1000
[10] Fm0 = 100
[11] mc0 = 1000
[12] ra = -k*Ca
[13] NCp = Ca*V*35+Cb*V*18+Cc*V*46+Cm*V*19.5
[14] Fa0 = Fa0o+(kc/.1)*I
[15] ThetaCp = 35+Fb0/Fa0*18+Fm0/Fa0*19.5
[16] v0 = Fa0/0.923+Fb0/3.45+Fm0/1.54
[23] Q = mc*18*(Ta2-Ta1)
[24] X = (Ca0-Ca)/Ca0
9-35
P9-14 (b)
Ta1 = 55°F
See Polymath program P9-14-b.pol
P9-14 (c) No solution will be given
P9-15
9-36
9-37
P9-15 (a)
See Polymath program P9-15-a.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 10 10
Ma 500 399.06704 500 399.06704
T 516 514.15817 517.83877 514.15817
mco 52.7 52.7 52.7 52.7
k 0.7028594 0.6447579 0.7677849 0.6447579
mb 257.3 257.3 257.3 257.3
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ma)/d(t) = 310*.83+raV
[2] d(Mb)/d(t) = –mb-raV
Explicit equations as entered by the user
[1] mco = 310*.17
[2] mao = 310*.83
9-38
P9-15 (b)
See Polymath program P9-15-b.pol
P9-15 (c)
9-39
See Polymath program P9-15-c.pol
P9-15 (d)
9-40
See Polymath program P9-15-d.pol
P9-16 (a)
See Polymath program P9-16-a.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 140 140
T 520 520 660 660
vo 400 400 400 400
UA 3.75E+04 3.75E+04 3.75E+04 3.75E+04