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Lecture (10)
Reactor Sizing and Design
1. General Mole Balance Equation
Mole balance on species j at any instance in time t ;
…………..4.1
Fj0 = Entering molar flow rate of species j (mol/time) Fj = Exiting molar flow rate of species j (mol/time)
Gj = Rate(total rate) of generation(formation) of species j (mol/time)=rj .V V = Volume (e.g. m3)
rj = rate of generation(formation) of species j (mole/time.vol) Nj = number of moles of species j inside the system Volume V (mole)
If rj varies with position in the system,
Then general mole balance:-
system within j of
onaccumulati of rate
system ofout j of
flow of rate
rxnby systemin j of
generation of rate
system into j of
flow of rate
dt
dN j
jjjo FGF
6V5V
4V
3V
2V
1V
3jr
1jr
4jr 5jr6jr
2jr
m
i
iij
m
i
ijj
jj
VrG
Vr
1
,
1
,
11,1,
G
G
0 ,m VLet
V
jj dVrG
VsystemVolumn
…………4.2
From this general mole balance equation we can develop the design equations for the various types of industrial reactors: batch, semi-batch. and continuous-flow reactors.
o Operate under unsteady state o Neither inflow nor outflow of reactants or products
If the reaction mixture is perfectly mixed so: o Constant rate of reaction throughout the reactor volume o Composition ≠ f (Position)
o Composition =f (time) ideal restrictions
o Temperature ≠ f (Position)
o Temperature ≠ f (time)
Mole Balance
REACTOR SIZING AND DESIGN
PART ONE
Batch Reactor
dt
dNdVr
j
j
V
jjo FF
0, joj FF
Isothermal Operation
...............................4.3
………………………..4.4
Let's consider the isomerization of species A in a batch reactor
As the reaction proceeds. the number of moles of A decreases and the number of moles of B increases, as shown in Figure below
The time t necessary to reduce the initial number of moles NAo to a final number of mole NA can be estimated as : from equation 4.4 ………………4.4
integrating with limits that at :
t = 0 NA = NA0 ← stat of reaction and at t = t NA = NA reaction time (end of reaction ) we obtain
…………..4.5
dt
dNdVr
j
j
V
jjo FF
dt
dNdVr
jV
j
fedmoles
reactedmolesmoles
A of
A of
0at t
fedinitially
A of
consumedor reacted
A of moles
BA
dt
dNVr
j
j
dt
dNVr A
A
Vr
dNdt
A
A
0A
A
N
N A
A
Vr
dNt
XN A
0consumedor reacted
A of moles
number of mole NA remain un-reacted after time t ,
Sub in equation 4.5 and 4.4
……….4.6 …4.7
Differential form Integral form Batch Reactor Design Equation Used in the Interpretation of m Lab Rate Data
XNNN AAA 00
consumedor reacted
A of moles
0at treactor the
tofedinitially
A of
ttime
at (remain)reacter in
A of moles moles
XNN AoA 1
Ao
AAo
N
NNX
Vrdt
dXN AAo
Vrdt
dNA
A 0A
A
N
N A
A
Vr
dNt
XNN AoA 1
tX
A
AoVr
dXNt
0
Space time or Mean Residence Time= is the time necessary to process one reactor mmmmmmmmmmmmmmmmmmm volume of fluid based on entrance conditions.
tB=t+tD
At constant volume batch reactor
i.e constant density reaction mixture.
NAo = CAo * V → then; equations 4.4 and 4.5 become ( ) :
….……….4.8….(Reaction Time)
Evaluation of Reaction Time Graphically:
From equation 4.7 plot vs. X and evaluate the area under the curve
to estimate reaction time
X1 X X
Or
From equation 4.7 plot vs. CA and evaluate the area under the curve
to estimate reaction time
dt
dCr A
A
A
Ao
C
CA
A
r
dCt
V
NC i
i
Ar
1
Ar
1
Ar
1
tX
A
AoVr
dXNt
0
AreaV
Nt Ao *
Area
Ar
1
A
Ao
C
CA
A
r
dCt
CA CA CAo
Example
Evaluation of Reaction Time Numerically:
Need to size reactors or calculate reaction time
o For the reactions in which the rate depends only on the concentration of
one species then
First order and Irreversible :-
,
Second order and Irreversible :-
,
,
BAAA kCr
A
Ao
A
Ao
C
CA
AC
CA
A
C
dC
kkC
dCt
1
Ao
A
C
C
kt ln.
1
kt
AoA eCC
2
AA kCr BA
A
Ao
A
Ao
C
CA
AC
CA
A
C
dC
kkC
dCt
22
1
AoA CCkt
111
ktC
CC
Ao
AoA
1
)( AA Cfr
)(CAfrA
AreaAreat
nth order and Irreversible :-
,
Example
Bimolecular Reactions
o when the rate law depends on more than one species , we must relate the
concentrations of the different species to eac2h other "as a function of
conversion ". This relationship is most easily established with
the aid of a Stoichiometric table.
In formulating our stoichiornetsic table, we shall take species A component as our basis of calculation (i.e.. limiting reactant) and then divide through by the stoichiometric coefficient of A , in order to put everything on a basis of "pet mole of A ".
Stoichiornetsic table presents the following information
o Column I: the particular species o Column 2: the number of moles of each species initially present o Column 3: the change in the number of moles brought about by reaction o Column 4: the number of moles remaining in the system at time t o Column 5: concentrations as a function of conversion of each species
• Consider the general reaction;
Stoichiometry set up of equations with A as basis
The rate law is :
n
AA kCr BA
11
1
1
n
Ao
n
A CCn
kt
nn
AoAoA tkCnCC 1
1111
Da
dC
a
cB
a
bA
C
d
D
c
Cb
B
a
AAAK
CCCCkr
)(XfrA
Constant Volume (Constant Density)
liquid-phase and some of gas phase reaction system fall into this category.
Stoichiometric Table Batch System
Specie Initial Change Remaining Concentration A NAo
-NAo X
NA = NAo(1 – X)
AC XCA 10
B NBo = NAo B
-(b/a)NAo X
NB = NAo[B –(b/a)X] BC
X
a
bC BA0
C NCo = NAo C
+(c/a)NAo X
NC = NAo[C +(c/a)X] CC
X
a
cC CA0
D NDo = NAo D
+(d/a)NAo X
ND = NAo[D +(d/a)X] DC
X
a
dC DA0
I NI = NAo
NI = NAo I
IoC
NTo = NAo i NT = NTo +NAoX
Where
i = Nio/NAo = Cio/CAo= yio/yAo
= (d/a) + (c/a) – (b/a) - 1
• Express table in terms of concentrations
– Concentration (batch):
V
NC i
i
0VV
Xa
bCX
a
b
V
N
V
NC
XCV
XN
V
NC
BABAB
B
AAA
A
0
0
0
0
0
0 11
Mole balance equation and the rate law are coupled and then solved
Example
Variable Volume (Variable Density, but with Constant T and P )
Individual concentration can be determined by expressing the volume for
batch system as a function of conversion using the equation of state:
PV=ZNTRT………..at any time in the reaction
PoVo=ZoNToRTo……at any time =0;when reaction is initiated
Then,
0
0
00
0Z
Z
P
P
T
T
N
NVV
T
T………………….4.9
Change in the total number of moles during reaction in gas phase reaction system,
but with constant temperature and pressure, and the compressibility factor will not
change significantly during the course of the reaction ,
0
0
T
T
N
NVV
Where NT = NTo +NAoX
= (d/a) + (c/a) – (b/a) – 1
= (change in total number of mole) / (mole of A reacted)
XN
N
N
N
T
Ao
T
T 00
1
0T
Ao
AoN
Ny
Xa
dCX
a
d
V
N
V
NC
Xa
cCX
a
c
V
N
V
NC
DADAD
D
CACAD
C
0
0
0
0
0
0
Ao
T
Ao yN
N
0
…………………………4.10a.
Then
XN
N
T
T 10
XN
NN
T
ToT
0
………………….…………….4.10b
At complete conversion i.e X=1 , NT= NTf ; therefore ,
0T
ToTf
N
NN ………………………………….4.11
= (change in total number of mole for complete conversion ) / (total moles fed)
Then the volume as a function of conversion :
XVV 10 …………………………………….4.12
Concentration at variable volume or density
Specie
V
NC A
A V
XN A
10
)1(
10
XV
XN
o
A
)1(
10
X
XCA
V
NC B
B
V
XN B (b/a)- B0
)1(
(b/a)- B0
XV
XN
o
B
)1(
(b/a)- B0
X
XCB
V
NC C
C
V
XNCo (c/a) C
)1(
(c/a) C
XV
XN
o
Co
)1(
(c/a) C
X
XCCo
V
NC D
D
V
XND (d/a)- D0
)1(
(d/a)- D0
XV
XN
o
D
)1(
(d/a)- D0
X
XCD
V
NC I
I V
N IAo )1( XV
N
o
IAo
)1( X
C IAo
Example
Chemical reactors can liberate or absorb very large amounts of energy , and the handling of
this energy is a major concern in reaction engineering. It is important to estimate the
temperature increase or decrease in an adiabatic reactor in which no heat is add or
removed, and exothermic reactor and also the composition of the reaction mixture at any
time.
Energy Balance
+ =
)( VrTH Ar )( TTUAQ a
dt
dTCCV iip,
T = reaction temperature K
Ta= wall temperature K
TR= reference temperature K
A = heat transfer area m2
Cpi = specific heat KJ/Kmol
U = overall heat transfer KJ/s.m2.K
Non-Isothermal Operation
Heat Generated by
Reaction
tion
Heat Addition and
Removal by wall
Heat Accumulated by
Reaction
rH =enthalpy change in the reaction per mole of Areacting
The number of moles of species i at any X is = XNN iiAi 0
Then energy balance is :
dt
dTNCTTUAVrTH iipaAr ,)()(
………………….4.13
Energy and mole balance equations with the rate law are coupled and then solved
Mole balance equation
rH is calculated as
T
TpR
o
rrR
dTCTHTH
The rate law is required as a function of temperature and composition
Variable Volume (Variable Density ,T and/or P)
"Variable T in non-isothermal"
The volume for batch system as a function of conversion as :-
0
0
00
0Z
Z
P
P
T
T
N
NVV
T
T
0
0
0
0 1Z
Z
P
P
T
TXVV
If the compressibility factor will not change significantly during the course of the
reaction Zo=Z
P
P
T
TXVV 0
0
0 1
Concentration at variable volume (density , T and/or P )
Specie
dt
dTCpXCNTTUAVrTH ipiAaAr ,0)()(
Vrdt
dXN AAo
V
NC A
A V
XN A 10
o
o
o
A
P
P
T
T
XV
XN
)1(
10
o
oA
P
P
T
T
X
XC
)1(
10
V
NC B
B
V
XN B (b/a)- B0
o
o
o
B
P
P
T
T
XV
XN
)1(
(b/a)- B0
o
oB
P
P
T
T
X
XC
)1(
(b/a)- B0
V
NC C
C
V
XNCo (c/a) C
o
o
o
Co
P
P
T
T
XV
XN
)1(
(c/a) C
o
oCo
P
P
T
T
X
XC
)1(
(c/a) C
V
NC D
D
V
XND (d/a)- D0
o
o
o
D
P
P
T
T
XV
XN
)1(
(d/a)- D0
o
oD
P
P
T
T
X
XC
)1(
(d/a)- D0
V
NC I
I V
N IAo
o
o
o
IAo
P
P
T
T
XV
N
)1(
o
oIAo
P
P
T
T
X
C
)1(
Example
A batch reactor is usually well mixed, so that may neglect the special variation in
temperature and species concentration .
Batch reactors operated adiabatically are often used to determine the reaction orders, activation energies, and specific reaction rates of exothermic reactions by monitoring the temperature-time trajectories for different initial conditions.
In adiabatic operation of a batch reactor
0Q
dt
dTNCVrTH iipAr ,)(
………………………….4.14
Energy and mole balance equations with the rate law are coupled and then
solved:
Adiabatic Operation of a Batch Reactor
dt
dTCpXCNVrTH ipiAAr ,0)(
;Where To = initial temperature
Example
The highest conversion that can be achieved in reversible reactions is the equilibrium conversion XEB. For endothermic reactions, the equilibrium conversion increases with increasing temperature up to a maximum of 1.0. For exothermic reactions, the equilibrium conversion decreases with increasing temperature Figure ( ) show the variation of the concentration equilibrium constant as a function of temperature for an exothermic reaction the corresponding equilibrium conversion XEB as a function of temperature.
Figure ( ) show the variation of the concentration equilibrium constant and equilibrium conversion as a function of temperature for an exothermic reaction.
TH
TTCX
r
oipi
)(,
CpXC
XTHTT
ipi
ro
,
Equilibrium Conversion
To determine the maximum conversion that can be achieved in an exothermic reaction carried out adiabatically, we find the intersection of the equilibrium conversion as a function of temperature ,with temperature –conversion relationships from the energy balance
……………..4.15
Graphical solution of equilibriurn and energy balance equations to obtain the adiabatic temperature
and the adiabatic equilibriurn
conversion XEB.
Example
TH
TTCX
r
oipi
EB
)(,