SEPTEMBER
1
2025
,
LUMPED MATHEMATICA MODELS
VARIABLE
VOLUME
CSTR/
Separable
PYNtON-SPYDER IDE
OPERATORS FORMATHED PRINTING
,
,
ARRAYS ,
ODE
JUPTER
MODULES
PLOTTING
By
Substitution
NOTEBOOK
(NumPy , MAT, MATPLOTLId)
Example: Dynamic CSTR during start-up
J FINISHING
ANALYSIS
FROM
THURSDAY)
Let’s now model the start-up of a CSTR. Namely, consider a CSTR vessel that is initially empty. At
time t=0, a reacting solution containing a chemical species A at a concentration of CAo is fed at a
constant volumetric rate, Fo, to the reactor. The reactor is filled to a liquid hold-up of Vf, at which time a
valve controlling reactor effluent flow is opened and reactor effluent is pumped from the reactor at a
volumetric rate of Fo. During the entire start-up process, an impeller, in combination with built-in baffles
along the reactor walls, is employed to ensure good mixing of the reaction solution within the CSTR.
The liquid phase reaction A→B is first order in A such that the rate of reaction is given by rA=-kCA
[mol/m3s], where k is the reaction rate constant [1/s] for species A under the conditions in the reactor.
G
Derive mathematical expression(s) describing how the concentration in the well-mixed reactor changes
as a function of time throughout the start-up process.
oXt1
Fo [mP/s]
·
Cao[neol/m3]
ra
=
-
k
.
G(t)
f
Fo(m/s]
Godmit
(m]
tiff
-
Forms
SaME
G(t)
=
Ca(t)
ii) iexp[-E ]
+
OXtItf
SYSTEM
DEFFUED
STEADY
By
WHERE
CSTR
Of
,
CORRESPONDS
Ef
WHERE
INTO
FLOW
TO
,
THE
BUT
TIME
The
NOT But
Occurs
-
-
↳
Cao[neol/m3]
.
VARS :
Calt)
VIt) ,
Two
mD
DEPENDEt
VAR
...
Two
MODELS
-
E
:
Indep Var
.
d
No
Spareal
dependence
-
& SCALtI VIIY
.
=
f (t)
Fo
=
.
Go
-
1 CeCt) UCt)
Ca (t) . V(t)
= Foco-l
1 =fat
-
.
B/C
Is
(No OUTLET)
Fo [mP/s]
DEP
REACTOR
Well-Meed
Full
(Vf)
(du
u=
=
FoCto-lef
-xd +
df
=
=
-
du
E
= Sat
Inlu)
-
(App(1
.
2
-ten / Fo Co-lf(t))
=
et
.
=
& - Mn/FG-lf)
In 150 Co
+10)
o
lf)
=
t -
=
=
t +A
Call = 0- IuIFGol
Edu/Foc)
In /FoCol
-
=
o
+A
4
let
# Acti
-
-
=
dv(t)
.
-
2
explant)
let
F
Ca(t)
=
V It)
F C
.
.
& t o , VI0)
=
=
Fot
=0
=
D
-
expl-4e]
clt)
=
21 -expl-rt)]
oct
+
+f =?
V(t +) = Vf
=
Fotf
E-/F
.
G(t)
↳
I C
,
.
ii) iexp[-E. ]
+
=
&
t=
Calty)
=
D
=
+
+
Ef
+=
+)
Calty) Caf
=
,
= exp-t]
(Co - Caf (1 ut)] exp( + ]
+
=
:
em -