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9-21
P9-10 (c)
The unsteady energy balance for a batch reactor is showed in equation 9-11.
( ) ( )
!
"#"$#++
=
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:
P9-10 (d)
Put α = 1, β = 1, ΘB = 3 in the equation from part (c)
111
$
'
"+
P9-10 (e)
Use the reaction constant expression from the great Swedish chemist Arrhenius and develop
expression from 9-10d.
' (
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:
534.17_2.8944_10 +!"=!+= inverseTinverseTaay
P9-10 (g)
Follow the procedure from example 9-3 gives heat of reaction.
#
&
P9-11 (a)
9-23
See Polymath program P9-11-a.pol
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
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
9-25
P9-11 (b)
P9-12
9-27
P9-12 (a)
See Polymath program P9-12-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 0.03789 0.0584442 0.053671
Cb 2.12 2.12 2.1423214 2.1376461
Cc 0.143 0.1227539 0.143 0.1275442
kc 10 10 10 10
k 24.990212 13.770535 24.990212 15.702753
Fb0 1000 1000 1000 1000
Fm0 100 100 100 100
mc0 1000 1000 1000 1000
ra -0.9468791 -0.9468791 -0.7913918 -0.8427832
Q 8.325E+05 6.594E+05 8.325E+05 6.963E+05
9-28
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ca)/d(t) = 1/tau*(Ca0-Ca)+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
P9-12 (b)
See Polymath program P9-12-b.pol
P9-12 (c)
9-30
P9-13
9-31
No control: See Polymath program P9-13-a.pol
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
ODE Report (RKF45)
Differential equations as entered by the user
Explicit equations as entered by the user
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
Cb 2.12 2.12 95.168593 95.168593
Cc 0.143 -1.7469378 0.143 -1.7469378
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
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
Explicit equations as entered by the user
[1] Fa0o = 80
[2] T0 = 70
[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
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
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ma)/d(t) = 310*.83+raV
Explicit equations as entered by the user
[1] mco = 310*.17
P9-15 (b)
P9-15 (c)
P9-15 (d)
9-40
P9-16 (a)
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 140 140
