general material balance of reacting system batch …bernauem/ark/lectures/chapter 4.pdfsummary •...
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Summary • General material balance of reacting system • Batch reactor • Continuous-flow reactors: CSTR (Continuous Stirred Tank Reactor) PFR (Plug Flow Reactor) • Steady state of CSTR and PFR • Design tasks : outlet (final conversion), given volume of reactor x volume of reactor, given outlet conversion
1
2
.
.
.
o
o
o
N
F
F
F
1
2
.
.
.
N
F
F
F
Inlet convective molar
flows (mol/s)
Outlet convective molar
flows (mol/s)
1
0 1,
( )
( )
N
ki i
i
i
A k NR
n t
V t
0
,
1
NR
i
i i ki V k iV
k V
nF F r dV c dV
t t
, ,
1 1 1 1
N N NR NR
o o
i i ki V k k V kV V
i i k k
nF F r dV F F r dV
t
T=const.
4. The material balances for isothermal ideal reactor models
Molar balance of species i
Overall molar balance
1
N
ki k
i
Arbitrary volume element
Batch reactors
0
,
1R
NR
i
i i ki V kV
k
nF F r dV
t
RkV
NR
k
ki
i Vrdt
dn.
,
1
Manometer
P=f(t)
Mixer
Q
Laboratory
Pharmaceutical industry
Special chemicals
Polymers
.
.
.
Heat flux
Vconst.
T=const.
Perfect mixing
Molar balance of species i
1
0 1,
( )
( ) V olum e of reaction m ixture
N
ki i
i
i
R
A k NR
n t
V t
Reactor volume constant (VR=const.)
, 1 2
1
1, , ...
i
NR
Ri i
ki V k N
kR
ndVdn dc
r c c cV dt dt dt
Reactor pressure constant (P = konst.)
RiiVcn .
kV
NR
k
ki
R
i
iRi
R
i
R
rdt
Vdc
dt
dc
dt
Vcd
Vdt
dn
V,
1
ln.11
dt
Vdcr
dt
dcR
ikV
NR
k
ki
iln
,
1
state equation f(T ,P,V,
To calculate ( ) o
composition)
r ( )
is needed = 0
RP t V t
Example Constant volume isothermal batch reactor, ideal gas mixture
1
N
kj k
j
1
, ,
1 1 1 1
,
1
ln( )
N
R j
j
N N NR NRjR R
R kj V k k V k
j j k k
NR
R
k V k
k
RTV n
P
dndV RTVRT RTV r r
dt P dt P P
d V RTr
dt P
1
, ,
1 1 1 1
N
j
jR
N N NR NRj
R kj V k k V k
j j k kR R
RTP n
V
dndP RT RTV r RT r
dt V dt V
Constant pressure isothermal batch reactor, ideal gas mixture
, ,
1 1
, ,
1 1
NR NR
i
ki V k i k V k
k k
NR NR
ki V k i k V k
k k
dc RTr c r
dt P
r y r
*i i
Pc y
RT
Example Constant volume batch reactor, liquid mixture constant pressure
Second order irreversible reaction
A1(l) + A2(l) A3(l)
2 1
1
1
1 2
1
1 2
2
1 2
3
1 2
1 1 2 2 2 1 1 2 3 1 1
1 11 2
1
2
1 1 1 2
01 1 1 2
1
1
,
,
o oo
i
i V i
o o o o o
c t
o o
o o
o o
o
c
oc c kt
dcr kc c
dt
dckc c
dt
dckc c
dt
dckc c
dt
c c c c c c c c c c c
dc dckc c c c
ck dt
dt c c
c
ec cc
c
c
1
1 2 1
1
1
o o
oc c c kt
c
1 2 1 1 2 2 3, 0, , , 0
o o
Vr kc c t c c c c c
0
0.5
1
1.5
2
2.5
0 2 4 6
ci mol/l
t/min
c1
c2
c3
1
2
3
2 m ole/l
1 mole/l
0
o
o
o
c
c
c
Continuous-flow reactors
Continuous Stirred Tank Reactor – CSTR
0
,
1
NR
i
i i ki V k R
k
dnF F r V
dt
We need suplementary information
• state behavior of reaction mixture
• start-up (shut-down) molar flow rates of individual species
1
0 1,
( )
V olum e of reactor
N
ki i
i
i
R
A k NR
n t
V const
Molar balance of species i
Overall molar balance
,
1
NR
o
k V k R
k
dnF F r V
dt
Constant pressure isothermal CSTR, ideal gas mixture
It arises from overall molar balance (volume of reactor, VR is constant)
1
, ,
1 1 1 1
,
1
0
N
j N N NR NR
o oi i
R ki V k R k V k
i i k k
NR
o
R k V k
k
d ndn
F F V r F F V rdt dt
F F V r
and molar balance of species i becomes
, ,
1 1
,
1
,
1
( ) ( )
= ( ) ( ) 1,
NR NR
o oi
i i R k V k R ki V k
k ki
NR
o oi iR
i i R ki i k V k
k
o NR
oi
i i ki i k V k
kR
dnF y F V r V r
dt
dn dyPVF y y V y r
dt RT dt
dy RTF RTy y y r i N
dt PV P
Inlet volumetric flow rate is
o
o
RTFV
P
R R
o o
o
V PV
V RTF
Mean residence time of reaction mixture (based on inlet flow rate) is given by
1
N
j R
i
RT n pV
o o o
i iF y F
and molar balance of species i becomes finally
,
1
1= ( ) ( ) 1,
NR
oi
i i ki i k V k
ko
dy RTy y y r i N
dt P
If only one reaction takes place:
1= ( ) ( ) 1,
oi
i i i i V
o
dy RTy y y r i N
dt P
Example
Start-up of an isothermal CSTR
2
2 g 2 g 2 g
4 3 o
5 1 1 -3
1 2 3 4 2 2 2
1 2 3 4
N O N 1 / 2O
0.1 l =10 m , T =320 C , 101 kPa
= , 2.028x10 m ole.Pa . m in .m
1, 0 (1=N O , 2=O , 3=N , 4=H e)
0, 0 0, 1
1
R
V N O
o o o o
o
V P
r kP k
y y y y
t y y y y
0.00
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4 5
yi (-)
t (s)
N2O
He
X1 =(F1o – F1)/F1
o
N2 O2
Numerical solution of balance
equations
(fractional) conversion of N2O*)
*) The equation is valid only at steady state
Steady state
,
1
, 1,
NR
o
i i R ki V k
k
F F V r i N
, 1,o
i i R i VF F V r i N
If only one reaction takes place:
( )
a) T he volum e of reactor for given outle t conversion
( )
b) T he outlet conversion for given volum e of reactor
( ) ( ) 0
o
j j R j V j
jo
R j
j V j
o
j
j j V j j
R
F X V r X
XV F
r X
FX r X f X
V
using fractional conversion of key species j
o
j j
j o
j
F FX
F
Graphical or numerical (iterative)
solution
Simple substitution
1
( )j V jr X
jX
( )
j
j V j
X
r X
Graphical assessment of the CSTR volume
Levenspiel
diagram
usually 1jv
( )j V jr X
jX
o
j
j
R
FX
V
Graphical assessment of the outlet conversion
Cascade of CSTR
( ) ( 1) ( ) ( )
( )
( )
( ) (
( )
)
( ) (
)
)
)
( 1) (
(
, 1, 1,
(1 )
( ), 1,
n n n n
i i R i V CSTR
o n
j jn
j o
j
n o n
j j j
n o o ni
o
j n n n
j j j V j CSTRn
R
i i j j
j
F F V r i N n N
F FX
F
F F X
F F F
FX X r X n N
V
X
……… ………
n-th member of
the cascade
( 1)n
jF
( )n
jF
Steady state
( )
( )
n
R
n
j
V
X
( )j V jr X
jX
(1)
o
j
j
R
FX
V
Graphical assessment of the outlet conversion in the cascade of CSTR
(1)
jX
(1)
( 2 )
o
j
j j
R
FX X
V
( 2 )
jX
Homework 6
Prove that for first order constant-volume reaction.
In the limit
( )
( )1 (1 ) ,
n
n n R
j o o
o
VX k
V
( )
( )lim 1 ,oR
n
kn R
j oRn
o
nVX e
V
Plug Flow reactor – PFR
z + z
VR + VR
z
VR
( )i RF V ( )
i R RF V V
v
R RV S z
,
1
( , ) ( , )1= ( , ) 1,
NR
i i
ki V k
kR
c t z F t zr t z i N
t S z
Molar balance of species i
Tubular reactors
High production capacity
Catalytic reactors (e.g. ammonia
synthesis)
.
.
.
Steady state
If only one reaction takes place:
using fractional conversion of key species j
o
j j
j o
j
F FX
F
,
1
1,
NR
i
ki V k
kR
dFr i N
dV
1,i
i V
R
dFr i N
dV
( )jo
j j V j
R
dXF r X
dV
( 2 )
(1)
( )
j
j
X
jo
j R
X j V j
dXF V
r X
1 1
3 1
,
m olar volum e (m m ole )
i i i i
i iN N
m
j m j
j j
m
F F y Fy c
V VF V F
V
useful relations:
BATCH, CSTR, PFR (PBCR), one reaction, fractional conversion of key component
BATCH 1
1
0 0
0 0
( )
( )
j R
j R
X to
j j
R
j R V j
X to
j j
R
V jj
n dXdt t
V r X
c dXdt t
r X
CSTR 1 1
1 1( ) ( )
o o
j j j j
R
j V j V jj
F X F XV
r X r X
PFR 1 1
0 0( ) ( )
j jX Xo o
j j j j
R
j V j V jj
F dX F dXV
r X r X
1 1
0 0( )
j jX Xo o
j j j j
j M M jj
F dX F dXW
r r X
Packed Bed Catalytic Reactor
PBCR
1
j Vr
jX
CSTR PFR
1
jX
Mean residence time
RV
V
1
0( )
jXo
j j
o
V jj
c dX
r X
1
1( )
o
j j
o
V jj
c X
r X
R
o
o
V
V
BATCH, PFR
CSTR
volumetric flow rate (m3/s) of reaction
mixture at inlet conditions oV
3
3
G as volum etric flow rate(m /hr)
V olum e of reactor (m )GHSV
3
3
Liquid volum etric flow rate (m /hr)
V olum e of reactor (m )LHSV
Gas Hourly Space Velocity
Liquid Hourly Space Velocity
Tasks
1. Calculate the volume of reactor (mass of catalyst) (CSTR,
PFR, PBCR) or time of reaction (BATCH) to obtain given
conversion.
2. Calculate the outlet conversion for given volume of reactor.
3. Calculate the reaction rate in laboratory reactor to obtain
kinetic law and estimate the kinetic parameters.
1 1
0 =
N N
i i i
i i
A
NR = 1 j – subscript of key component
BATCH
FLOW
.o o oi
i i i i j j
j
n n n n X
. .o o oi
i i i V R i j j
j
F F r dV F F X
1 1
. .
N N
o o o o
i i j j
i i j
n n n n n X
.o o
j j
j
F F F X
. .
. 1 .
o oi o oii j j i j j
ji k
io o o
j j j j
j j
n n X x x Xn
xn
n n X x X
. .
. 1 .
o o o oi i
i j j i j j
j ji
io o o
j j j j
j j
F F X x x XF
xF
F F X x X
3
.1
.1 .
- m olar volum e of reaction m ixture
(m /m ol)
o oi
i j j
ji i i
io
m m mj j
j
m
x x Xn n x
cV n V V V
x X
V
.1
.1 .
o oi
i j j
ji i i
io
m m mj j
j
x x XF F x
cV F V V V
x X
Gas phase - Ideal state behavior
.
1 .
o oi
i j j
ji i
io
j j
j
x x Xx P P
cRT RT RT
x XP
.
. 1 .
o oi
i j j
ji i i
io
j j
j
x x XF F P P
cRTV RT RT
F x XP
(1 )
. .1
1 1
i i
i
oo
o j j
o j
o o o oi i
i j j i j j
j jo o
o oo o o
j j j j
j j
n nc
PTVV x XT P
n n X c c XT TP P
T P V T Px X x X
1
. .1
1 1
i i
i
oo
o j j
o j
o o o oi i
i j j i j j
j jo o
o oo o o
j j j j
j j
F Fc
V PTV x XT P
F F X c c XT TP P
T P V T Px X x X
m
RTV
P
Homework 7 Calculate volume of PFR to produce 150 kt ethylen/year. Reaction of the 1st order
C2H6(g) C2H4(g) + H2(g)
takes place at 1100 K and 0,6 Mpa. Final conversion of ethan is 80 %. Reaction rate is given by
rV = k. cA
k (1000 K) = 0,072 s-1 E = 343,6 kJ/mol
Pure ethan is fed into the reactor.
Assumptions:
Ideal gas, molar weight of ethylen is 28,054 kg/kmol.
K. J. Laidler and B. W. Wojciechowski: Kinetics and Mechanisms of the Thermal Decomposition of Ethane.
I. The Uninhibited Reaction, Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences, Vol. 260, No. 1300 pp. 91-102
Homework 4 (due after Chapter 4) Calculate volumes of CSTR a PFR working at 150 oC and 300 kPa to produce 1 t COCl2/day
with CO conversion equal to 95 %. A mixture of CO and Cl2 (molar ratio 1:1) is fed at 300
kPa and 150 oC.
Data
k(423 K) = 0.07 (m3mol-1)3/2.s-1
MCOCl2 = 98.92 kg/kmol.
Answer:
VCSTR = 0.053 m3 VPFR = 0.0021 m3
IN OUT
CO (1)
Cl2 (2)
COCl2 (3) 0
1
oF
1
oF
12
o oF F
1 1 11
oF F X
2 1 11
oF F X
3 1 1
oF F X
1 12
oF F X
CO(g) + Cl2(g) → COCl2(g) o
iF i
F
steady state; j - key com ponent
o oi
i i j j
j
F F F X
2
2
3
1 1 1 31 1 1
1 2
1 1 1
5 / 25 / 2
3 / 2 31
1
6
3
1 1
1
ideal gas [m /m ol]
1 1 [m ol/m ]
2 2
1[m ol/m /s]
2
1 10 / 98.92 / 24 3600
0.90.1231 m
563 o
m
o
CO Clo
m
V CO Cl
o
RTV
P
F X XF F FP P Pc c c c
V V F RT F RT F X RT X
XPr kc c k
RT X
FF
X
1 1
1 1 1 1
5 / 2 5 / 21 5 / 25 / 2 311 1
1
1
1
3
l/s
0.1231
( ) 1 300 10 1 0.9510.07
8.3145 423 2 0.952
0.0503
63 0.
m
95o o
CSTR
V
F X F XV
r X XPkRT X
1 5 / 25 / 20.95 0.95
1 1 1 1
15 / 25
3
/ 2
10 0 01
1
5 / 2
3
3
2
( ) 1 11
2
8.310.12316379.42692225 2.08 10
45 423
0.07 300 1m
0
jXo o o
j j
PFR
V jj
F dX F dX F XRTV dX
r X k P XXPkRT X
Summary • General material balance of reacting system • Batch reactor • Continuous-flow reactors: CSTR (Continuous Stirred Tank Reactor) PFR (Plug Flow Reactor) • Steady state of CSTR and PFR • Design tasks : outlet (final conversion), given volume of reactor x volume of reactor, given outlet conversion