lec 2 path2 single reaction
TRANSCRIPT
Summary Summary -- Design Equations of Ideal ReactorsDesign Equations of Ideal Reactors
Differential
Equation
Algebraic
Equation
Integral
EquationRemarks
Vrdt
dnj
j )(= ∫=j
jO
n
n j
j
Vr
dnt
)(
Conc. changes with time
but is uniform within the
reactor. Reaction rate
varies with time.
Batch
Conc. inside reactor is
CSTR)( j
jjo
r
FFV
−
−=
Conc. inside reactor is
uniform. (rj) is constant.
Exit conc = conc inside
reactor.
PFRj
j rdV
dF= ∫=
j
jO
F
F j
j
r
dFV
)(
Concentration and
hence reaction rates
vary spatially.
IdealIdeal--Reactor Design EquationReactor Design Equation
Based on Mole Balance for Based on Mole Balance for
SingleSingle--Reaction SystemsReaction SystemsSingleSingle--Reaction SystemsReaction Systems
Consider the general reaction
dDcCbBaA +→+
We will choose A as our basis of calculation
Da
dC
a
cB
a
bA +→+ The basis of calculation is most always the
limiting reactant.
Basis of calculationBasis of calculation
What’s the limiting reactant?!!!
Definition of conversionDefinition of conversion
Da
dC
a
cB
a
bA +→+
How can we quantify how far the reaction proceeds to the right ???
How many moles of C are formed for every moles of A consumed ???How many moles of C are formed for every moles of A consumed ???
batch reactor batch reactor
0A
A0AA N
NNX
−=
flow reactor flow reactor
0A
A0AA F
FFX
−=
XA=
Moles of A reacted
Moles of A fed
Convenient way is given!!! “Conversion”
Further Discussions on ConversionFurther Discussions on Conversion
For irreversible reactions, the maximum value of
conversion, X, is that for complete conversion, i.e. X=1.0.
• Maximum conversion for irreversible reactions
dDcCbBaA +→+
conversion, X, is that for complete conversion, i.e. X=1.0.
• Maximum conversion for reversible reactions
For reversible reactions, the maximum value of
conversion, X, is the equilibrium conversion, i.e. X=Xe.
bBaA+ dDcC +
Consider to example rate equations
A A Br kC C− = 2 2A A Br kC C− = 1
2
AA
A
k Cr
k KC− =
+
Note that
( )Ar f Concentration=
Further Discussions on ConversionFurther Discussions on Conversion
We will discuss this issue in the later courses.
A
For only reaction occurring, Conc. can be expressed in term of X
What is the relationship between X and What is the relationship between X and rrAA ??
We need only -rA = f (X) and FA0 to design a variety of reactors !
The heart of the design of an ideal reactor:
(-rA) as a function of conversion (concentration, partial pressure etc.)
We’ll develop the reactor design equationWe’ll develop the reactor design equation in in
term of conversion term of conversion for single reaction systemfor single reaction system
(-rA) as a function of conversion (concentration, partial pressure etc.)
(We will discuss this issue in the later courses.)
( )A
A
dndt
r V= −
−
Consider to design equation of batch reactor
0 (1 )
( )A A
A
dn Xdt
r V
−= −
−
( )0 1A A An n X= −
Design eq. for batch reactor design
Where;
(-rA) = moles A reacting / (unit
volume) (time)
V = Reactor volume, but really
refers to the volume of fluid in
reactor.
0
( )A A
A
n dXdt
r V=
− 0 ( )AA A
dXn r V
dt= −Or
Design equation of batch reactor in conversion term
Integration form with t = 0 (XA=0) to t=t1 (XA=XA1)
1
0
0 ( )
AX
AA
A
dXt n
r V=
−∫The longer the reactants are
left in the reactor, the greater
will be the conversion.
•• Batch:Batch: Vrdt
dXn AA )(0 −=
? ?
Design eq. for batch reactor design
– The conversion is a function of the time
the reactants spend in the reactor.
– We are interested in determining how – We are interested in determining how
long to leave the reactants in the reactor
to achieve a certain conversion X.
-1/rA
X
Area =00 ( )
t
A At
dX Vt
r n=
= −
∫
Design eq. for batch reactor design (For example)
We are planning to operate a batch reactor to convert A into R.
This is a liquid reaction, the stoichiometry is A+R, and the rate of
reaction is given in below Table. How long must we react each
batch to achieve 90% conversion? The initial concentration of A is 2.0 M
Design eq. for batch reactor design (For example)
Known and unknown
Specify problem: Determine time to achieve 90% conversion of the bath reactor
0 2.0AC =
0.9X =?V =?t =
Relation between –rA and CA
Mol/l
Basis: (1) Using A as the basis species
(2) Basis calculation based in 1 liter of batch reactor volume
Assumption: (1) Unsteady state operation
(2) Well-mixed condition
(3) Constant volume (because of liquid reaction)
Relation between –rA and CA
Design eq. for batch reactor design (For example)
( )
0.9
0
0
AA
dXt C
r=
−∫
( )
0.9
0
?A
dXI
r= =
−∫
t C I= × 0.00
5.00
10.00
15.00
20.00
25.00
1/(-rA)
0At C I= ×
Using numerical technique for integration to solve IUsing numerical technique for integration to solve I
14.87I =
2.0 14.87 29.74t = × = min
0.00
0.00 0.20 0.40 0.60 0.80 1.00 xi
Numerical Evaluation of IntegralsNumerical Evaluation of Integrals
Integration with unequal segmentsIntegration with unequal segments
Suitable for experimental data, in-equally space data points
( )b
a
I f x dx= ∫
0 1 11 2( ) ( ) ( ) ( )( ) ( )... n nf x f x f x f xf x f x
I h h h −+ ++≅ + + +0 1 11 2
1 2
( ) ( ) ( ) ( )( ) ( )...
2 2 2n n
n
f x f x f x f xf x f xI h h h −+ ++≅ + + +
where,
( )1i i ih x x −= − or the width of segmentsf(x)
xi
The entering molar flow rate of species A, FA0 (mol/s), is just the product of the
entering concentration, CA0 (mol/dm3), and the entering volumetric flow rate, v0
(dm3/s):
00A0A vCF =
For liquid system, CFor liquid system, CAA00 is commonly given in terms of is commonly given in terms of molaritymolarity, for , for
example, Cexample, CAA00 = = 2 2 mol/dmmol/dm33
RelationshipRelationship ofof molarmolar//volumetric volumetric flowrateflowrate and and
concentrationconcentration
example, Cexample, CAA00 = = 2 2 mol/dmmol/dm
ForFor gasgas system,system, CCAA00 cancan bebe calculatedcalculated fromfrom thethe enteringentering temperaturetemperature andand
pressurepressure using the ideal gas law or some other gas law. For an ideal gas using the ideal gas law or some other gas law. For an ideal gas
lawlaw
0
00A
0
0A0A RT
Py
RT
PC ==
0
00A00A00A RT
PyvCvF ==
yA0 = entering mole fraction of A (-)
P0 = entering total pressure (kPa)
PA0 = yA0P0 = entering partial pressure of A (kPa)
T0 = entering temperature (K)
R = ideal gas constant (=8.314 kPa⋅dm3/mol⋅K)
A gas of pure A at 830 kPa enters a reactor with a volumetric flow
rate, v0, of 2 dm3/s at 500 K. Calculate the entering concentration of
A, CA0, and the entering molar flow rate, FA0.
Solution
300 /20.0)830)(1(
dmmolkPaPy
C A ===
RelationshipRelationship ofof molarmolar//volumetric volumetric flowrateflowrate and and
concentrationconcentration
33
0
000 /20.0
)500)(/314.8(
)830)(1(dmmol
KKmolkPadm
kPa
RT
PyC A
A =⋅⋅
==
smolsdmdmmolvCF AA /4.0)/2)(/2.0( 33000 ===
This feed rate (FA0 = 0.4 mol/s) is in the range
of that which is necessary to form several
mill ion pounds of product per year.
Design Equations for Design Equations for CSTRCSTR
Consider to design equation of CSTR reactor
A
A0A
r
FFV
−−
=
FA0
FAC
0A0A )X1(FFV
−−=
)X1(FF 0AA −=
FA
CACA
CA
A
0A0A
r
)X1(FFV
−−−
=
exitA
A
r
XFV
)(0
−=
Because the reactor is perfectly
mixed, the exit composition from the
reactor is identical to the composition
inside the reactor, and the rate of
reaction is evaluated at the exit
conditions.
Design Equations for Design Equations for PFRPFR
PFR Design EquationsPFR Design Equations
FA FA +dFdV
bA B
a+
l mL M
a a+
AA r
dV
dF=
)X1(FF 0AA −= dXFdF 0AA −=
dXFdF 0AA −=A0A r
dV
dXF −= differential form
∫ −=
X
0A
0A r
dXFV integral form
REACTOR DIFFERENETIAL ALGEBRAIC INTEGRAL
FORM FORM FORM
Vrdt
dXN AAO )(−= ∫ −=
X
AAO Vr
dXNt
0BATCHBATCH
Design Equations Design Equations in Terms of Conversionin Terms of Conversion
)( AAO rdV
dXF −= ∫ −=
X
AAO r
dXFV
0
CSTRCSTR
PFRPFR
ExitA
AO
r
XFV
)(
)(
−=
Levenspiel Plots:
Illustration of Reactor Sizing for
Single-Reaction SystemsSingle-Reaction Systems
Octave Levenspiel
(PhD 1952): Octave Levenspiel also obtained an MS from
Oregon State and served as a faculty member for 25 years
until he retired in 1991. He published over 100 papers and
proceedings, two of which have been listed as "Citation
Classics." He was awarded 1977 American Institute of
Chemical Engineers W.K. Lewis Award, the 1979 R.H.
Wilhelm Award, and the 2003 Founders Award and Gold
Medal, the highest honor given by the society. In 2000 he
Wilhelm Award, and the 2003 Founders Award and Gold
Medal, the highest honor given by the society. In 2000 he
was inducted into the National Academy of Engineering. He
also received two honorary doctorates , one from France.
He is considered to be one of the founders of He is considered to be one of the founders of He is considered to be one of the founders of He is considered to be one of the founders of
Chemical Reaction Engineering.Chemical Reaction Engineering.Chemical Reaction Engineering.Chemical Reaction Engineering.
SOURCE: http://engr.oregonstate.edu/oregonstater/fame/1998/che/octavelevenspiel.html
Levenspiel Plots: Sizing of CSTR
•• CSTR:CSTR:
– We are interested in determining the size of
the reactor to achieve a certain conversion X.
XFV
][×=
exitAACSTR r
XFV
)(
][0 −×=
-1/rA
X
Area =1
( )A
Xr
×−
0CSTR AV F Area= ×
Evaluate VCSTR by Levenspiel Plot
Levenspiel Plots: Sizing of PFR
•• PFR:PFR:
– We are interested in determining the size of
the reactor to achieve a certain conversion X.
1dX× ∫
=
=PFRxx
A dXF
V 0
X
)(
1
Ar−
XPFRdX
( )A
dXr
×− ∫
= −=
x A
APFR dX
rV
0
0
Area =1
( )A
dXr
×−∑
0
1
( )
PFRX
AX
dXr=
×−∫
0PFR AV F Area= ×
Evaluate VPFR by Levenspiel Plot
or
The isomerization reaction
A→B
was carried out adiabatically in the liquid phase and the data in below table
were obtained.
X (-rA) kmol/l.hr
0.95 0.10
0.90 0.30
Calculate the volume of each of the reactors
for an entering molar flow rate of species A of
50 kmol/hr.
For example:
0.90 0.30
0.85 0.50
0.80 0.60
0.75 0.50
0.70 0.25
0.65 0.10
0.60 0.06
0.50 0.05
0.35 0.05
0.00 0.04
50 kmol/hr.
(b) Compare CSTR and PFR volume at 65%
conversion
(a) Compare CSTR and PFR at 90%
conversion
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
0.00 0.20 0.40 0.60 0.80 1.00
For example:
CSTR: CSTR: 9090% conversion% conversion
exitA
A
r
XFV
)(0
−=
kmol
hrl
hr
kmolV
.2.229.050 ××=
1000=V liter
-1/rA
X
PFR: PFR: 9090% conversion% conversion
( )∫ −=X
AA r
dXFV
0
0
∫ −×=9.0
0
50Ar
dX
hr
kmolV
78.91634.1850 =×=V liter
X 1/(-rA)
0.95 25.00
0.90 22.22
0.85 20.00
0.80 16.67
0.75 10.00
0.70 4.35
0.65 6.25
0.60 10.00
0.50 16.67
0.35 25.00
0.00 33.33
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
0.00 0.20 0.40 0.60 0.80 1.00
For example:
CSTR: CSTR: 6565% conversion% conversion
liter
( )0AA exit
XV F
r=
−
.50 0.65 6.25
kmol l hrV
hr kmol= × ×
203V =
-1/rA
XX 1/(-rA)
0.95 25.00
0.90 22.22
0.85 20.00
0.80 16.67
0.75 10.00
0.70 4.35
0.65 6.25
0.60 10.00
0.50 16.67
0.35 25.00
0.00 33.33
PFR: PFR: 6565% conversion% conversion
( )∫ −=X
AA r
dXFV
0
0
∫ −×=65.0
0
50Ar
dX
hr
kmolV
75307.1550 =×=V liter
X
Ideal Reactors in SeriesIdeal Reactors in Series
- X1 at point i=1 is the conversion
achieved in the CSTR
- X2 at point i=2 is the total
conversion achieved in at this
point in the CSTR and the PFR
- X3 is the total conversion
achieved by all three reactors
For reactors in series, the conversion X is the total number of moles
of A that have reacted up to that point per mole of A fed to the first
reactor.
only be used when the feed stream only
enters the first reactor in the series and
t h e r e n o s i d e s t r e a m s e i t h e r
fed or withdrawn.
Xi=
Total moles of A reacted up to point i
Moles of A fed to the first reactor
achieved by all three reactors
Ideal Reactors in SeriesIdeal Reactors in Series
CSTRs in SeriesCSTRs in Series
Reactor 1:
0AF1AF
2AF
1AX2AX
1Ar−
2Ar−
0VrFF 11A1A0A =+−
)X1(FF 10A1A −=1
1A
0A1 X
r
FV
−=
Reactor 1:
Reactor 2:0VrFF 22A2A1A =+−
)X1(FF 20A2A −=)XX(
r
FV 12
2A
0A2 −
−=
Ideal Reactors in SeriesIdeal Reactors in Series
-1/rA
11
01 X
r
FV
A
ACSTR
−=−
)( 120
2 XXF
V ACSTR −
=−
21 −−− += CSTRCSTRtCSTR VVV
-1/rA1
-1/rA2
CSTRs in Series: CSTRs in Series: LevenspeilLevenspeil plotplot
X
)( 122
2 XXr
VA
CSTR −
−
=−
( )
−×
−+
×
−= 12
201
10
11XX
rFX
rFV
AA
AA
( ) ( )2010 −− ×+×= CSTRACSTRA AreaFAreaFV
X1 X2
***To achieve the same overall conversion, the total volume for two CSTRs in series is less than that required for one CSTR
Ideal Reactors in SeriesIdeal Reactors in Series
CSTRs in SeriesCSTRs in Series ((For Example)For Example)
For the two CSTRs in series, 60% conversion is achieved in the first reactor.
What is the volume of each of the two reactors necessary to achieve 90% overall conversion of entering species A
Calculate the volume of each of the reactors
for an entering molar flow rate of species A of
50 kmol/hr.
X (-rA) kmol/l.hr
0.95 0.10
0.90 0.30 50 kmol/hr.0.90 0.30
0.85 0.50
0.80 0.60
0.75 0.50
0.70 0.25
0.65 0.10
0.60 0.06
0.50 0.05
0.35 0.05
0.00 0.04
0 50 /AF kmol hr=1 0A AF F=
2 0A AF F=
1 0.6AX =2 0.9AX =
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
CSTRs in SeriesCSTRs in Series ((For Example)For Example)
21 −−− += CSTRCSTRtCSTR VVV
CSTRCSTR--11: : 6060% conversion% conversion
exitA
ACSTR r
XFV
)(10
1 −=−
kmol
hrl
hr
kmolVCSTR
.106.0501 ××=−
3.633=
-1/rA
X
1000sin =− gleCSTRV liter
0.00
0.00 0.20 0.40 0.60 0.80 1.00
kmolhr
3001 =−CSTRV
CSTRCSTR--22: : 9090% conversion% conversion
( )exitAACSTR r
XXFV
−−
=−12
02
( )kmol
hrl
hr
kmolVCSTR
.22.226.09.0502 ×−×=−
3.3332 =−CSTRV liter
X 1/(-rA)
0.95 25.00
0.90 22.22
0.85 20.00
0.80 16.67
0.75 10.00
0.70 4.35
0.65 6.25
0.60 10.00
0.50 16.67
0.35 25.00
0.00 33.33
literX
Ideal Reactors in SeriesIdeal Reactors in Series
PFRs in SeriesPFRs in Series
21 −− += PFRPFRtotal VVV
It is immaterial whether you place two plug-flow reactors in
series or have one continuous plug-flow reactor; the total reactor
volume required to achieve the same conversion is identical.
∫∫ −+
−=
2
1
1
00 0
X
XA
A
X
AA r
dXF
r
dXF
Ideal Reactors in SeriesIdeal Reactors in Series
PFRs in Series: PFRs in Series: LevenspeilLevenspeil plotplot
-1/rA21 −−− += PFRPFRtPFR VVV
∫ −=−
1
001
X
AAPFR r
dXFV
∫X dX
1−PFRV2−PFRV
X
∫ −=−
2
101
X
XA
APFR r
dXFV
∫∫ −+
−=−
2
1
1
00 0
X
XA
A
X
AAtPFR r
dXF
r
dXFV
• The overall conversion of two PFRs in series is the same as
one PFR with the same total volumn.
( ) ( )2010 −−− ×+×= PFRAPFRAtPFR AreaFAreaFV
• To achieve the same overall conversion, the total
volume for two CSTRs in series is less than that
required for one CSTR. -1/rA
X
Ideal Reactors in SeriesIdeal Reactors in Series
• The overall conversion of two PFRs in series is the same
as one PFR with the same total volumn.
• CSTRs in series :A PFR can be modelled using a number
of CSTR in series
– useful in modelling catalyst decay in a packed-bed reactor
– modelling transit heat effects in PFRs.
X
-1/rA
X
Ideal Reactors in SeriesIdeal Reactors in Series
Combined CSTRs and Combined CSTRs and PFRs in SeriesPFRs in Series
CSTRCSTR--PFR in seriesPFR in series
PFRCSTRtotal VVV +=
∫ −=
2
10
X
XA
APFR r
dXFV
11
0 Xr
FV
A
ACSTR
−=
( ) ∫ −+
−=
2
1)(0
10
X
X AA
AAtotal r
dXF
r
XFV
Ideal Reactors in SeriesIdeal Reactors in Series
Combined CSTRs and Combined CSTRs and PFRs in SeriesPFRs in Series
PFRPFR--CSTR in seriesCSTR in series
CSTRPFRtotal VVV +=
∫ −=
1
00
X
AAPFR r
dXFV
( )122
0 XXr
FV
A
ACSTR −
−=
( )
−−
+−
= ∫2
2
120
0
0 )( AA
X
AAtotal r
XXF
r
dXFV
Ideal Reactors in SeriesIdeal Reactors in Series
Combined CSTRs and PFRs in Series (For Example)
The isomerization of butane
nC4H10→iC4H10
was carried out adiabatically in the liquid phase and the data in below table
were obtained.It is real data for a real reaction carried out
adiabatically, and the reactor scheme X -rA
[kmol/m3.hr]adiabatically, and the reactor scheme
shown as below:
Calculate the volume of each of the reactors for an entering molar flow
rate of n-butane of 50 kmol/hr.
A
[kmol/m3.hr]
0 39
0.2 53
0.4 59
0.6 38
0.65 25
Space Time & Residence Time
Time is of EssenceTime is of Essence
• Two types of time-parameters are commonly used in chemical reaction engineering
– space time
– residence time
• Space time is often used as a scaling parameter in reactor design
Mole balance for “flow reactor”Space time and space velocity of flow reactor
Space time
0
Vτ
ν≡
** Sometime called “mean resident time”
Reactor volume
Volumetric flow rate at entrance
Actual Residence
Time:
The time actually
spent by fluid
inside the reactor.
Space velocity
** Sometime called “mean resident time”
0 1SV
V
ντ
≡ =
** Measured at STP condition
•LHSV - Liquid Hourly Space Velocity
•GHSV - Gas Hourly Space Velocity
Illustration of difference between space time (Illustration of difference between space time (ττττττττ) and ) and
residence time (residence time (ttresres))
The Pop Corn Example
Under what practical conditions do we expect space
time = residence time ?
Class Assignment 2
(1)
(2)
(3)
Class Assignment 2
(4)
Class Assignment 2