8-21
8-22
P8-5
2A B C+!
A
B
C
Fio
lb mole
hr
!
” #
$ %
& ‘
10
10
0.0
Tio(F)
80
80
–
,
Energy balance with work term included is:
Substituting into energy balance,
[ ]
0 0 0
( )
S S A R AF A pA pB
UA T T W F H X F C C T T
! “
# # # $ = + #
% &
P8-6
A B C+!
8-23
Adiabatic:
0
0
[ ( )]
i
R
i P P
X H T
T T C X C
!
“#
= + +#
$% %
P8-6 (a)
0
0
PFR A
A
A
CSTR
A
dX
V F
r
F X
V
r
=!
=!
“
For the PFR, FA0 = CA0v0 = (.1)(2) = .2 mols/dm3
See Polymath program P8-6-a.pol.
Calculated values of DEQ variables
Variable
Initial value
Minimal value
Maximal value
Final value
1
X
0
0
0.85
0.85
Differential equations
1
d(V)/d(X) = -Fa0 / ra
Explicit equations
1
Ca0 = .1
8-24
For the CSTR,
X = .85, T = 300+(200)(85) = 470 K.
P8-6 (b)
0
[ ]
i
R
i P
X H
T T
C
!
“#
= + $
For boiling temp of 550 k,
550 = T0 + 200
T0 = 350K
P8-6 (c)
P8-6 (d)
0
0
( )
A
CSTR
A
CSTR
A
A
F X
V
r
V
X r
F
=!
“=!
8-25
Calculated values of NLE variables
Variable
Value
f(x)
Initial Guess
1
T
484.4136
0
480.
2
ra
0.0003688
Nonlinear equations
1
f(T) = 300 + 200 * X – T = 0
2
f(X) = 500 – .2 * X / ra = 0
Explicit equations
1
k = .01 * exp(10000 / 1.98 * (1 / 300 – 1 / T))
Calculated values of NLE variables
Variable
Value
f(x)
Initial Guess
1
T
476.482
1.137E-13
480.
Nonlinear equations
1
f(T) = 300 + 200 * X – T = 0
T = 476.48 ad X = .8824
Hence, in the second reactor,
8-26
0 1
( )
A
CSTR
F X X
V
!
=!
See Polymath program P8-6-d-2.pol.
Calculated values of NLE variables
Variable
Value
f(x)
Initial Guess
1
T
493.8738
0
480.
Nonlinear equations
1
f(T) = 476.48 + 200 * (X – X1) – T = 0
2
f(X) = 250 – .2 * (X – X1) / ra = 0
Explicit equations
P8-6 (e) Individualized solution
P8-6 (f) Individualized solution
P8-7 (a)
C
K
& ‘
Stoichiometry:
0
C A
C C X
=
POLYMATH Results
No Title 03-21-2006, Rev5.1.233
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
V 0 0 10 10
X 0 0 0.0051176 0.0051176
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(X)/d(V) = -ra / Fa0
Explicit equations as entered by the user
[1] T = 300+200*X
8-28
301
301.2
0.005
0.006
0.829
0.8295
0.83
P8-7 (b)
When heat exchanger is added, the energy balance can be written as
[ ]
0
( ) ( ) ( )
ˆ
( )
a A Rxn
A i pi P
Ua T T r H T
dT
dV F C C
!
“+” “#
=+#
$
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
V 0 0 10 10
X 0 0 0.3634806 0.3634806
T 300 300 455.47973 450.35437
8-29
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(X)/d(V) = -ra / Fa0
[2] d(T)/d(V) = ((ra*DH)-Ua*(T-Ta))/(Fao*sumcp)
Explicit equations as entered by the user
[1] k = .01 * exp((10000 / 1.987) * (1 / 300 – 1 / T))
[11] sumcp = 30
350
400
450
500
T [K]
0.15
0.2
0.25
0.3
0.35
0.4
X
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5 6 7 8 9 10
V [m3]
Xe
P8-7 (c)
8-30
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
V 0 0 10 10
X 0 0 0.3611538 0.3611538
T 300 300 442.15965 442.15965
Ta 450 434.90618 450 441.60853
ODE Report (RKF45)
Differential equations as entered by the user
Explicit equations as entered by the user
[1] k = .01 * exp((10000 / 1.987) * (1 / 300 – 1 / T))
400
450
500
Ta
0.4
0.5
0.6
0.7
0.8
0.9
Xe
P8-7 (d)
For counter-current flow,
See Polymath program P8-7-d.pol.
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
V 0 0 10 10
X 0 0 0.3647241 0.3647241
T 300 300 463.44558 450.37724
Ta 440.71 440.71 457.98124 450.00189
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(X)/d(V) = -ra / Fa0
Explicit equations as entered by the user
[1] k = .01 * exp((10000 / 1.987) * (1 / 300 – 1 / T))
[2] Kc = 10 * exp(-6000 / 1.987 * (1 / 450 – 1 / T))
400
450
500
Ta
T
0.6
0.7
0.8
0.9
Xe
8-32
P8-7 (e)
We see that it is better to use a counter-current coolant flow as in this case we achieve the
maximum equilibrium conversion using a lesser volume of the PFR.
P8-7 (f)
If the reaction is irreversible but endothermic, we have
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
V 0 0 10 10
X 0 0 0.4016888 0.4016888
T 300 300 428.84625 424.16715
Cpc 1 1 1 1
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(X)/d(V) = -ra / Fa0
Explicit equations as entered by the user
[1] k = .01 * exp((10000 / 2) * (1 / 300 – 1 / T))
8-33
For counter-current flow,
See Polymath program P8-7-f-counter.pol.
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
V 0 0 10 10
X 0 0 0.3458817 0.3458817
T 300 300 449.27319 449.27319
Ta 423.8 423.8 450.01394 450.01394
Explicit equations as entered by the user
[1] k = .01 * exp((10000 / 2) * (1 / 300 – 1 / T))
[2] Kc = 10 * exp(6000 / 2 * (1 / 450 – 1 / T))
[3] Fa0 = 0.2
P8-7 (g)
For a runaway reaction, the following must be true:
2
r
r C
RT
T T
E
!>
8-35
P8-8 (a)
A B C+!
Species Balance:
Stoichiometry:
0
‘ (
Rate Law is:
See Polymath program P8-8-a.pol.
Calculated values of DEQ variables
Variable
Initial value
Minimal value
Maximal value
Final value
1
X
0
0
0.8
0.8
2
W
0
0
43.13711
43.13711
Differential equations
1
d(W)/d(X) = v0 * (1 + X) * T / k / (1 – X) / T0
8-36
Explicit equations
1
T = 450 + 500 * X
P8-8 (b)
Species Balance for CSTR:
0
A
CSTR
F X
Wr
=!
“
P8-8 (c) Individualized solution
P8-8 (d)
For pressure drop, an extra equation is added
8-37
Calculated values of DEQ variables
Variable
Initial value
Minimal value
Maximal value
Final value
1
W
0
0
0.8
0.8
2
X
0
0
0.0544753
0.0544753
Differential equations
1
d(X)/d(W) = k / v0 * (1 – X) / (1 + X) * T0 / T * P / P0
Explicit equations
1
T = 450 + 500 * W
P8-9 (a)
We use the same equations as problem P8-8, except that the energy balance changes as:
Rxn
8-38
See Polymath program P8-9-a.pol.
Calculated values of DEQ variables
Variable
Initial value
Minimal value
Maximal value
Final value
1
W
0
0
50.
50.
2
X
0
0
0.1376181
0.1376181
3
T
450.
381.1888
450.
381.1888
Differential equations
1
d(X)/d(W) = k * (1 – X) / (1 + X) * T0 / T / v0
2
d(T)/d(W) = (Uarho * (Ta – T) + rA * 20000) / v0 / CA0 / 40
Explicit equations
1
T0 = 450
2
v0 = 20
8-39
If
b
UA
!
was increased by a factor of 3000, we use the same program with the new value. The
profiles are in the graphs below.
P8-9 (b)
For non-constant jacket temperature, the equation for incorporating the flow needs to be
introduced.
co-current: Ta0 = 50 °C
( )
a
a
C pC
Ua T T
dT
dW m C
!
“
=&
8-40
P8-9 (c)
For a fluidized CSTR with W = 80 kg, UA = 500 J/s/K,
W
P8-9 (d)
For a reversible reaction, we have all the previous equations, but the rate law is modified as:
A f A r B C
r k C k C C
!=!
See Polymath program P8-9-d.pol.
Calculated values of DEQ variables
Variable
Initial value
Minimal value
Maximal value
Final value
1
W
0
0
80.
80.
2
X
0
0
0.057593
0.057593
3
T
450.
420.7523
450.
420.7523