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L3-1
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Rate of generation of reactant A in reactor due to rxn
Rate of accumulation ofreactant A in reactor =
Review: Batch Reactor Basic Molar Balance
• No material enters or leaves the reactor• In ideal reactor, composition and temperature are spatially
uniform (i.e. perfect mixing)• No flow in or out of reactor. Fj0 and Fj = 0.
dt
dNdVr jVj Batch Reactor
Design Equation
dt
dNVr jj
Ideal Batch Reactor Design Equation
Ideal (perfectly mixed) reactor: spatially uniform
temp, conc, & reaction rate
L3-2
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: CSTR Basic Molar Balance
Accumulation = In - Out + Generation by rxn
0 = Fj0 - Fj +
Vrj
V
jdVr
No spatial variation:
0 0 0
j Cj j A Aj
j A
F F C CV F V
r r
• Continuously add reactants and remove products• In an ideal reactor, composition and temperature
are spatially uniform (i.e. perfect mixing) • At steady state- no accumulation
Fj0 Fj
Ideal Steady State CSTR Design Equation:
in terms of concentration
in terms of flow
(upsilon)
L3-3
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
ΔV
FA0 FA
Review: Molar Balance – PFR
jVjVVj r
VFF
0Vlim
VrjFj0 Fj dtdNj+- =
0VrFF jVVjVj
jj r
dVdF
Ideal SS PFR Design Eq.
• Flow reactor operated at steady state (no accumulation per Δ)• Composition of fluid varies down length of reactor (material
balance for differential element of volume V
L3-4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
• Heterogeneous rxn: reaction occurs at catalyst particle surface • Concentration gradient of reactant and product change down
length of the reactor• Rxn rate based on the mass of catalyst W, not reactor volume V
Review: Molar Balance- Packed Bed Reactor (PBR)
jj r
dVdF
Similar to PFR, but expressed in terms of catalyst weight instead of reactor volume
Units for the rate of a homogeneous rxn (rj) :
Units for the rate of a catalytic rxn (rj’) : catalyst kgs
mol3ms
mol
So in terms of catalyst weight instead of reactor volume:
catalyst the of weightthe is W where'rdWdF
jj
L3-5
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L3: Conversion and Reactors in Series
FA0 X0
FA1, X1
V1 V2
FA2 X2
FA0 X0
FA1, X1
V1
V2
FA2, X2
VCSTR1 VPFR2
XA
FA0/
-rA
(m3)
L3-6
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Conversion, XAConversion is convenient for relating: rj, V, υ, Nj, Fj, and Cj
fed A moles reacted A moles Aon based conversionXA
D d Cc B b A a
Choose limiting reactant A as basis of calculation and normalize:
D ad C
ac B
ab A
BATCHSYSTEM: “Moles A fed” is the amount of A at the start of the reactor (t=0)
FLOWSYSTEM: “Moles A fed” is the amount of A entering the reactor
Usually pick the basis to be the limiting reagent
L3-7
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Conversion ExampleA + 2B → 2C
Start with 1 mole of A & 1 mole of B
If A is the basis and at the end we have:
1 mole A, 1 mole B ↔ XA = 0/1 = 0 (no reaction)½ mole A, 0 mole B ↔ XA = 0.5/1 = 1/20 mole A, -1 mole B ↔ XA = 1/1 = 1 (complete reaction)
Not possible!
The correct approach is to take B as the basis because B is the limiting reagentAt the end we have:
1 mole A, 1 mole B ↔ XB = 0/1 = 0 (no reaction)½ mole A, 0 mole B ↔ XB = 1/1 = 1 (complete reaction)
L3-8
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Expressing other Components in Terms of Conversion of A (XA)
D ad C
ac B
ab A
BATCHSYSTEM:
Longer reactant is in reactor, more reactant is converted to product (until reactant is consumed or the reaction reaches equilibrium)∴ Conversion (Xj) is a function of time (t) in the batch reactor
A0A0AA XN N N Moles A in
reactor at time t = Moles A fed - Moles A
consumed
A0AA X 1NN
A0A0BB X NabNN A0A0CC X N
acNN
A0A0DD X NadNN component inert NN 0II
reactant product
fed A moles reacted A moles XA
L3-9
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Expressing other Components in Terms of Conversion of A (XA)
jA0A0TjT XN1
ab
ac
adNNN
ad
ac 1
ab
dcAB
j≡ stoichiometric coefficient; positive for products, negative for reactants
A0Aj0jj XNNN
jA0A
jj0TjT XNNNN
Total moles in reactor at time t = Total
moles fed + total moles products formed minus reactants consumed
D ad C
ac B
ab A
fed A moles reacted A moles XA
L3-10
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Batch Reactor Design Equation with Xj
A0A0AA XN N N In terms of A:
Vr dt
dNA
A Ideal Batch Reactor Design Eq:
Want to determine how long to leave reactants in reactor to achieve a desired value for the conversion
A0A0AA XN N dtd N
dtd
dtdXN 0
dtdN A
0AA
dt
dXN dt
dN A0A
A ←Substitute into batch reactor design eq
Vr dt
dXN AA
0A Ideal Batch Reactor Design Eq with Xj:
AX
0 A
A0A Vr
dXNt
D ad C
ac B
ab A
fed A moles reacted A moles XA
→ take derivative of “NA” equation w/ respect to time
L3-11
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Flow and Conversion
A0A0AA XF F F
Molar flow rate that A leaves the reactor =
Molar flow rate A is fed to reactor
- Molar rate A is consumed in reactor
D ad C
ac B
ab A
fed A moles reacted A moles XA
FLOW SYSTEM:
For a given flow rate, the larger the reactor, the more time it takes the reactant to pass through the reactor, the more time to react∴ Conversion (Xj) is a function of reactor volume (V)
A0AA X 1FF
A0Aj0jj XFFF :general in
jA0A
jj0TjT XFFFF
L3-12
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
CSTR Design Equation & Xj
r
FFVj
A0A
Ideal SS CSTR:
A0A0AA XFFF Substitute for FA
r
XFFFVA
A0A0A0A
rXFVA
A0A
Ideal CSTR design eq in terms of XA
V ≡ CSTR volume required to achieve a specified conversionNote: XA and –rA are evaluated at the exit of the CSTR
L3-13
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
PFR Design Equation & Xj
Ideal SS PRF:
A0A0AA XFFF
AA r
dVdF
Want to determine the reactor volume required to achieve a desired amount of conversion
A0A0AA XF F dVd F
dVd
dVdXF 0
dVdF A
0AA
dV
dXF dVdF A
0AA ←Substitute into PFR design eq
AA
0A rdV
dXF Ideal SS PFR Design Eq with Xj:
AX
0 A
A0A r
dXFV
Applies for no pressure drop down PFR!
→ take derivative of “FA“ expression with respect to volume
L3-14
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
PBR Design Equation & Xj
Ideal SS PBF:
A0A0AA XFFF
'rdWdF
AA
Want to determine the weight of catalyst that is required to achieve a desired amount of conversion
A0A0AA XF F dWd F
dWd
dWdXF 0
dWdF A
0AA
dWdXF
dWdF A
0AA ←Substitute into PBR design eq
'rdWdXF A
A0A Ideal SS PBR
Design Eq with Xj:
AX
0 A
A0A 'r
dXFW
Applies for no pressure drop down PBR!
→ take derivative of FA expression with respect to W
L3-15
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Sizing CSTRsWe can determine the volume of the CSTR required to achieve a specific conversion if we know how the reaction rate rj depends on the conversion Xj
AA
0ACSTR
A
A0ACSTR X
rFV
rXFV
Ideal SS CSTR
design eq.
Volume is product of FA0/-rA and XA
• Plot FA0/-rA vs XA (Levenspiel plot)• VCSTR is the rectangle with a base of XA,exit and a height of FA0/-rA
FA 0 rA
X
Area = Volume of CSTR
X1
V FA 0 rA
X1
X1
L3-16
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Sizing a CSTR with a Levenspiel PlotXA 0 0.1 0.2 0.4 0.6 0.7 0.8
FA0/-rA 0.89 1.08 1.33 2.05 3.56 5.06 8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10123456789
XA
FA0/
-rA
(m3)
VCSTR for XA = 0.4?
AA
0ACSTR X
rFV
3
CSTR
m 82.0
4.005.2V
VCSTR for XA = 0.8?
3
CSTR
m 4.6
8.08V
Value of FA0/-rA for XA=0.4
L3-17
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Sizing PFRsWe can determine the required volume of a PFR to achieve a specific conversion if we know how the reaction rate rj depends on the conversion Xj
Aexit,AX
0 A
0APFR
exit,AX
0 A
A0APFR dX
rFV
rdXFV
Ideal PFR
design eq.
• Plot FA0/-rA vs XA (Experimentally determined numerical values) • VPFR is the area under the curve FA0/-rA vs XA,exit
FA 0 rA
Area = Volume of PFR
V 0
X1FA 0 rA
dX
X1
L3-18
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Sizing a PFR with a Levenspiel PlotXA 0 0.1 0.2 0.4 0.6 0.7 0.8
FA0/-rA 0.89 1.08 1.33 2.05 3.56 5.06 8
VPFR for XA = 0.4?
Aexit,AX
0 A
0APFR dX
rFV
We do not have an expression for –rA(XA)
L3-19
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Sizing a PFR with a Levenspiel PlotXA 0 0.1 0.2 0.4 0.6 0.7 0.8
FA0/-rA 0.89 1.08 1.33 2.05 3.56 5.06 8
VPFR for XA = 0.4?
Aexit,AX
0 A
0APFR dX
rFV
We do not have an expression for –rA(XA)
Numerically evaluate (Appendix A.4) to estimate the area under the curve
Volume of PFR
L3-20
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Numerical Evaluation of Integrals (A.4)Simpson’s one-third rule (3-point):
2102X
0XfXf4Xf
3hdxxf
hXX 2
XXh 0102
Trapezoidal rule (2-point):
101X
0XfXf
2hdxxf
01 XXh
Simpson’s three-eights rule (4-point):
32103X
0XfXf3Xf3Xfh
83dxxf
3XXh 03
h2XX hXX 0201
Simpson’s five-point quadrature :
432104X
0XfXf4Xf2Xf4Xf
3hdxxf
4XXh 04
L3-21
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
XA 0 0.1 0.2 0.4 0.6 0.7 0.8FA0/-rA 0.89 1.08 1.33 2.05 3.56 5.06 8
Sizing a PFR with a Levenspiel PlotXA 0 0.1 0.2 0.4 0.6 0.7 0.8
FA0/-rA 0.89 1.08 1.33 2.05 3.56 5.06 8
VPFR for XA = 0.4?A
exit,AX
0 A
0APFR dX
rFV
Use Simpson’s one-third rule (3-point):
31 33 20 890 2 4 0 553
05 PFR .V ... m.
2102X
0XfXf4Xf
3hdxxf hXX
2XXh 01
02
0 00 4
3 0 2 00 4
AA
AF
AA
P RA
Fr X
hV Fr. .
Fr X X
2.02.00X 0.22
04.0h 1
= area under the curve
XA increments must be equal
0.89 1.33 2.05
L3-22
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Reactors in SeriesIn practice, reactors are usually connected so the exit stream of one reactor is the feed stream for the next reactor
Conversion up to point i (no side streams): reactor 1st into fed A Moles
i point to up reacted A of moles totalXi
FA0
FA1
i=1X1
V1 V3
FA3 i=3 X3
V2
FA2
i=2X2
i0A0AAi XFFF
L3-23
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
2 CSTRs in Series
FA0 X0
FA1, X1
V1 V2
FA2 X2
Materials balance reactor 1:In Out- + Gen. = Accum.
A1A0 A1 1F r VF 0
A1 A0 A0 1F F F X
A0 AA 10 A1 10FF r V- F X 0
0VrXF 11A10A
VXr
F1CSTR1
1A
0A
Need to express FA1 in terms of X1
L3-24
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
2 CSTRs in Series
FA0 X0
FA1, X1
V1 V2
FA2 X2
122A
0A2CSTR XX
rFV
Materials balance reactor 2:In Out- + Gen. = Accum.
0Vr F F 22A2A1A
20A0A2A XFFF
2A
2A1A2CSTR r
FFV
2A
20A0A10A0A2CSTR r
XFFXFFV
Value of FA0/-rA at X2
VXr
F1CSTR1
1A
0A
10A0A1A XFFF Materials balance reactor 1:
Need to express FA2 in terms of X2
L3-25
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
2 CSTRs in Series
Usually for the same overall conversion, VTOTAL, 2 CSTRs IN SERIES < VSINGLE CSTR
FA0 X0
FA1, X1=0.4
V1 V2
FA2 X2=0.8
XA 0 0.4 0.8FA0/-rA 0.89 2.05 8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
2
4
6
8
XAFA
0/-r
A (m
3)
VCSTR1 for XA1 = 0.4?
31CSTR m 82.04.005.2V
VCSTR2 for XA2 = 0.4 to 0.8?
122A
0A2CSTR XX
rFV
32CSTR m 2.34.08.08V
33321CSTR m 02.4m 2.3m 82.0V
VCSTR of single CSTR with XA = 0.8?
3CSTR m 4.68.08V <
L3-26
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
2 PFRs in Series
A2X
1X A
0AA
1X
0 A
0AA
2X
0 A
0APFR dX
rFdX
rFdX
rFV
FA0, X0
FA1 X1 FA2, X2
XA 0 0.2 0.4 0.6 0.8FA0/-rA 0.89 1.33 2.05 3.56 8
VPFR2 for XA2 = 0.4 to 0.8?
32PFR m 61.3856.3405.2
32.0V
33321PFR m 17.2m 61.1m 55.0V
8.0Xr
F6.0Xr
F44.0Xr
F3hV
A
0A
A
0A
A
0A
Same volume as 1 PFR with XA=0.8
2 PFRs in series, X1=0.4 and X2=0.8
When XA1= 0.4, VPFR1 =0.55 m3 (slide L3-20)
L3-27
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Combinations of CSTRs & PFRs in Series
FA0 X0
FA1, X1
V1
V2
FA2, X2
VCSTR1 VPFR2 VPFR1 VCSTR2
FA0, X0
FA1 X1
V1
V2
FA2, X2
( )VCSTR1 + VPFR2 ≠ VPFR1 + CCSTR2
L3-28
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Reactors in Series
for any combination of PFRs & CSTRs in series
then ,increasinglly monotonica is r-
F IfA
A0
CSTR onei j
)j(CSTR)i(PFR PFR one VVVV
In general, 1 PFR = any number of PFRs in series 1 PFR = ∞ number of CSTRs in series
Definitions:Space time (t): time necessary to process one reactor volume, also called mean residence time or holding time
Space velocity (SV): inverse of space time, but vo may be measured under different conditions than the space time
0
V
t
t 1V
SV 0
0 liquid @ 60 F or 75 FLHSVV
0 STPGHSVV
Liquid-hourly space
velocity
Gas-hourly space
velocity0| is the volumetric flow rate measured at specified condition