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13-20$
P13-3)(a)$Continued$
13-21$
P13-3)(a)$Continued$
$
$
P13-3)(b))
This$is$the$same$as$part$(a)$except$the$energy$balance.$
Energy$balance:$
$
13-22$
P13-3)(b)$Continued$
$
$
$
P13-3)(c))
This$is$the$same$as$part$(b)$except$the$reaction$is$now$reversible.$
$
$
13-23$
P13-3)(c))
$
$
$
13-24$
P13-4)(a))
)
)
$
$
P13-4)(b))
)
13-25$
P13-4)(b)$Continued$
)
$
P13-5)
)
13-26$
P13-5)Continued$
$
$
POLYMATH)Results
Calculated)values)of)the)DEQ)variables
Variable initial value minimal value maximal value final value
t 0 0 10 10
X 0 0 0.2504829 0.2504829
T 373 373 562.91803 562.91802
ODE)Report)(RKF45)
Differential equations as entered by the user
[1] d(X)/d(t) = -ra/.1
[2] d(T)/d(t) = ((40000+(10*(T-298)))*(-ra)*(1/.1))/(56.25-(10*X))
13-27$
P13-5)Continued$
$
$
$
P13-6)(a))
A$+$2B$→$C$
$
$
13-28$
P13-6)(a))Continued$
$
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
Nb 0 0 5900.6649 5900.6649
Cc 0 0 0.0718765 0.0328924
ra 0 -0.0028611 0 -1.59E-06
ODE)Report)(RKF45)
Differential equations as entered by the user
[1] d(Na)/d(t) = ra*V
Explicit equations as entered by the user
[1] vb = 1.5
[2] k = .0005*exp((8000/1.987)*(1/300-1/T))
13-29$
P13-6)(a))Continued$
$
$
$
)
P13-6)(b))
$
$
13-30$
P13-6)(b))Continued$
$
$
P13-7)
A$+$B$→$C$
$
$
$
13-31$
P13-7)(a))
$
$
See$Polymath$program$P13-9-a.pol$
POLYMATH)Results
Calculated)values)of)the)DEQ)variables
Variable initial value minimal value maximal value final value
t 0 0 0.2 0.2
Ca 0.3 1.34E-65 0.3 1.34E-65
Cc 0 0 0.3 0.3
ODE)Report)(STIFF)
Differential equations as entered by the user
[1] d(Ca)/d(t) = ra1
Explicit equations as entered by the user
[1] UA = 0
$ $
13-32$
P13-7(b))
$
$
$
$
$
$
)
)
$
$
P13-7)(c))
$
$ $
$
$
P13-8)(a)$Yes.$As$more$of$C$is$produced,$the$reaction$rate$of$reaction$3$may$increase.$Since$reation$3$is$
high$exthothermic,$there$is$still$possibility$that$the$reactor$will$run$away$if$Qr$>$Qg$at$t$=$0.$
)
P13-8)(b)$
Qrt=0=UA(T0−Ta)=(100)(450 −400) =5000'cal/s
)
( )( )
( )( )
( )( )
2
2
10 10 1 0 0 3
3
20 20 2 0 0 3
3
30 30 3 0 0 3
10 1 20
0
2 2 2(0.001) 1.0 0.5 0.0005
33 31
( 10 ) 1.0 0.2 0.0001
22 23
11 1
(0.6 10 ) 0.5 0.2 0.00002
33 3
BA AAB
CA AAC
BC CBC
gBRxBC
t
mol
rr kCC s dm
mol
rr kCC s dm
mol
rr kCC s dm
QrVHrV
−
−
=
==−=−=−
==−=−× =−
==−=−× =−
=Δ+Δ
230 3
( 0.0005)(2000)( 5000) ( 0.0001)(2000)(10000) ( 0.00002)(2000)( 50000)
5000 2000 2000
5000 cal/s
Rx C B Rx B
HrVH+Δ
=− − +−+− −
=−+
=
$
P13-8)(d)$
0000
0
gr
tAP BP CP
QQ
dT
dt N C N C N C
=
−
==
++
)
13-34$
P13-8)(e))Continued$
2000[(1)(10) (0.5)(10) (0.2)(50)]
Ks
==−
++
)
P13-8)(f)$
See$Polymath$program$P13-8-f.pol$
P13-9))
Semi)batch)with)parallel)reactions)
( )
( )
dNa r1 r 2 V
dt
dNb Fbo r1 r2 V
dt
dNd r1 V
dt
dNu r2 V
dt
=+
=++
=− ⋅
=− ⋅
)
Rate$laws:$$
r1 k1 Ca
r2 k2.Cb
k1 10exp( 2000 / T)
k2 20exp( 3000 / T)
−=⋅
−=
=−
=−
)
Stoichiometry:$
0b
b
33
Na Nb
Ca Cb
VV
Nd Nu
Cd Cu
VV
VV vt
Fbo
v
Cbo
Cao 5mol / dm Cbo 1mol / dm
==
==
=+⋅
=
==
)
)
)
)
13-36$
P13-9)Continued$
See$polymath$program$P13-9$
Calculated values of DEQ variables
Variable
Initial value
Minimal value
Maximal value
Final value
1
Ca
5.
0.0402932
5.
0.0402932
2
Cao
5.
5.
5.
5.
3
Cb
1.
0.1879238
1.
0.3260075
4
Cbo
1.
1.
1.
1.
5
Cd
0
0
0.7888994
0.6235848
6
Cu
0
0
0.0504077
0.0504077
7
Fbo
10.
10.
10.
10.
8
k1
0.0204583
0.0204583
0.0464803
0.0460806
9
k2
0.0018507
0.0018507
0.0063377
0.0062561
10
Na
500.
28.20523
500.
28.20523
11
Nb
100.
68.89056
228.2052
228.2052
12
Nd
0
0
436.5094
436.5094
13
Nu
0
0
35.28541
35.28541
14
r1
-0.1022916
-0.1022916
-0.0018567
-0.0018567
15
r2
-0.0018507
-0.0020395
-0.001066
-0.0020395
16
T
323.
323.
372.3484
371.7507
17
t
0
0
60.
60.
18
Tao
373.
373.
373.
373.
19
Tbo
323.
323.
323.
323.
20
V
100.
100.
700.
700.
21
vb
10.
10.
10.
10.
Differential equations
1
d(Na)/d(t) = (r1+r2)*V
2
d(Nb)/d(t) = Fbo + (r1+r2)*V
3
d(Nd)/d(t) = -r1*V
4
d(Nu)/d(t) = -r2*V
5
d(T)/d(t) = (-30*Fbo*(T-Tbo) + 3000*(-r1*V) + (5000 + 10*(T-300))*(-r2*V))/(20*Na+30*Nb+50*Nd+40*Nu)
Explicit equations
1
Cbo = 1
2
Fbo = 10
3
k1 = 10*exp(-2000/T)
4
k2 = 20*exp(-3000/T)
5
vb = Fbo/Cbo
6
V = 100 + vb*t
7
Ca = Na/V
8
Cb = Nb/V
13-37$
P13-9$continued
9
r1 = -k1*Ca
10
r2 = -k2*Cb
11
Cao = 5
12
Cd = Nd/V
13
Tao = 373
14
Tbo = 323
15
Cu = Nu/V
)
$
)
