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L7-1
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Liquid Phase Reaction in PFRLIQUID PHASE: Ci ≠ f(P) → no pressure drop
Calculate volume required to get a conversion of XA in a PFR
2A → B -rA = kCA2 2nd order reaction rate
Mole balance
Rate law
Stoichiometry (put CA in terms of X)
AA
A0
d rX
dV F
2A Ar kC
A A0 AC C (1 X )
Combine
A0 A
A0
2A
2C 1X Xd
V F
k
d
X VAA0 A
220 0A0 A
F dXdV
k C 1 X
A0 A
2AA0
F XV
1 Xk C
Liquid-phase 2nd order reaction in PFR
Be
ab
le t
o d
o t
he
se
4 s
tep
s,
inte
gra
te &
so
lve
fo
r V
fo
r A
NY
O
RD
ER
RX
N
See Appendix A for integrals frequently used in reactor design
L7-2
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Liquid Phase Reaction in PBRLIQUID PHASE: Ci ≠ f(P) → no pressure drop
Calculate catalyst weight required to get a conversion of XA in a PBR
2A → B -r’A = kCA2 2nd order reaction rate
Mole balance
Rate law
Stoichiometry (put CA in terms of X)
AA
A0
rX 'd
dW F
A2
Ar ' kC
A A0 AC C (1 X )
Combine
A0 A
A0
2A
2C 1X Xd
W F
k
d
X WAA0 A22
0 0A0 A
F dXdW
k C 1 X
A0 A
2AA0
F XW
1 Xk C
Liquid-phase 2nd order reaction in PBRBe
ab
le t
o d
o t
he
se
4 s
tep
s, in
teg
rate
&
so
lve
fo
r V
fo
r A
NY
OR
DE
R R
XN
L7-3
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Isobaric, Isothermal, Ideal Gas-Phase Rxns in Tubular Reactors
GAS PHASE:j0 j A0 A 0 0
jA 0
C C X T ZPC
1 X P T Z
1 1 1
j0 j A0 Aj
A
C C XC
1 X
Gas-phase reactions are usually carried out in tubular reactors (PFRs & PBRs)
• Plug flow: no radial variations in concentration, temperature, & ∴ -rA
• No stirring element, so flow must be turbulent
FA0 FA
Stoichiometry for basis species A:
A0 AA0 A0 AA A
A A
C 1 XC C XC C
1 X 1 X
L7-4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Effect of e on u and XATf T0 A
T0
N N Change in total # moles at X 1
N total moles fed
: expansion factor, the fraction of change in V per mol A reactedu0: volumetric flow rate
00 A
0 0
PZ T1 X
Z T Pu u
u varies if gas phase & moles product ≠ moles reactant, or if a DP, DT, or DZ
occursNo DP, DT, or DZ occurs, but moles product ≠ moles reactant → 0 A1 Xu u
• = 0 (mol product = mol reactants): u u0: constant volumetric flow rate as XA ↑ < 0 (mol product < mol reactants): u < u0 volumetric flow rate ↓ as XA ↑
Q1: For an irreversible gas-phase reaction, how does the residence time and XA change when < 0?
a)They don’tb)The residence time is longer & XA increasesc)The residence time is longer & XA decreasesd)The residence time is shorter, & XA decreasese)The residence time is shorter & XA increases
L7-5
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Effect of e on u and XATf T0 A
T0
N N Change in total # moles at X 1
N total moles fed
: expansion factor, the fraction of change in V per mol A reactedu0: volumetric flow rate
00 A
0 0
PZ T1 X
Z T Pu u
u varies if gas phase & moles product ≠ moles reactant, or if a DP, DT, or DZ
occursNo DP, DT, or DZ occurs, but moles product ≠ moles reactant → 0 A1 Xu u
• = 0 (mol product = mol reactants): u u0: constant volumetric flow rate as XA increases
• < 0 (mol product < mol reactants): u < u0 volumetric flow rate decreases as XA increases
• Longer residence time than when u u0
• Higher conversion per volume of reactor (weight of catalyst) than if u u0
• > 0 (mol product > mol reactants): u > u0 with increasing XA
• Shorter residence time than when u u0
• Lower conversion per volume of reactor (weight of catalyst) than if u u0
L7-6
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Isobaric, Isothermal, Ideal Rxn in PFRGAS PHASE:
Ci = f( ) → no DP, DT, or DZ
Calculate PFR volume required to get a conversion of XA
2A → B -rA = kCA2 2nd order reaction rate
Mole balance
Rate law
Stoichiometry (put CA in terms of X)
AA
A0
d rX
dV F
2A Ar kC
Combine
A0
2A
A0
2
2A
A
C 1 XdX
d 1 X
k
V F
2XA AA0A22
0A0 A
1 XFV dX
k C 1 X
22 AA0A A2
AA0
1 XFV 2 1 ln 1 X X
1 Xk C
Gas-phase 2nd order rxn in PFR no DP, DT, or DZ
A0 AA
A
C 1 XC
1 X
Integral A-7 in appendix
Be
ab
le t
o d
o t
he
se
5 s
tep
s, &
so
lve
for
V
for
AN
Y O
RD
ER
RX
N
L7-7
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Pressure Drop in PBRs
A0 AA
A 0
C 1 X PC
1 X P
AA
A0dX
Fd
r 'W
GAS PHASE: A → B -r’A = kCA2
Calculate dXA/dW for an isothermal ideal gas phase reaction with DP
2nd order reaction rate
Mole balance
Rate law A2
Ar ' kC
Stoichiometry (put CA in terms of X)
Combine
A0 A
A
2 22
2A
A 00
PP
k C 1 X
1 X
dX
dW F
Relate P/P0 to W (Ergun equation)
0A
0 0
PdP T1 X
dW 2 T P P
Ergun Equation can be simplified by using y=P/P0 and T=T0:
Ady
1 XdW 2y
Simultaneously solve dXA/dW and dP/dW (or dy/dW) using Polymath
L7-8
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Ergun Equation
A0
dy T1 X
dW 2y T
0
c c 0
2
A 1 P
Differential form of Ergun equation for pressure drop in PBR:
0
Py
P Tf T0
A0T0
N Ny
N
AC: cross-sectional area C: particle density
: constant for each reactor, calculated using a complex equation that depends on properties of bed (gas density, particle size, gas viscosity, void volume in bed, etc)
: constant dependant on the packing in the bed
volume of solid1 : fraction of solid in bed =
total bed volume
0A
0 0
PdP T1 X
dW 2 T P P
Calculates pressure drop in a packed bed.This equation can be simplified to:
L7-9
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L7: Unsteady-State Isothermal Reactor Operation: CSTR Start-Up
and Semi-Batch Reactors
V0 Vf
start
CBu0
V0 + u0t
time t end
Semi-batch
• Time required to reach steady-state after CSTR start-up
• Predicting concentration and conversion as a function of time
A+BA
L7-10
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Start-Up of a Fixed-Volume CSTRIsothermal (unusual, but simple case), well-mixed CSTR
Unsteady state: concentrations vary with time & accumulation is non-zero
Goal: Determine the time necessary to reach steady-state operation
moles A in CSTR changes with time until steady state is reached
In Out- +Generation = Accumulation
AA0 A A
dNF F r V
dt
CA0u0u0CA
Use concentration rather than conversion in the balance eqs
AA0 0 A 0 A
dNC C r V
dtu u
Divide by V to convert dNA to dCA
A AA0 A0 0
V
C C dN 1d
r
V
V
t
u u
0
V u
A0 AA
AC C
dt
Cr
d
A
A0 A AdC
C C rdt
Multiply
by t
L7-11
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
CSTR Start-Up: 1st Order Reaction A
AA0 A
dCC C
dtr AAr kC A
AA0 A
dCC C
dtCk
Integrate this eq to find CA (t) while 1st order rxn in CSTR is at unsteady-state:
Combine
Bring variables to one side & factor A
A0 AdC 1
C C 1 kdt
A0A
ACdC 1
1 k Cdt 1 k
C tAA
A00 0A
1 kdCdt
CC
1 k
A0
A
A0
CC
1 k 1 kln 0 t
C0
1 k
1 kt
A
A0
C1 e
C
1 k
t 1 kA0A
C1 e C
1 k
Put like variables with their integrals
L7-12
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
CSTR Start-Up: 1st Order Reaction
t 1 kA0A
C1 e C
1 k
AA0 A A
dCC C kC
dt We integrated this eq to find CA (t) while
CSTR of 1st order rxn is in unsteady-state:
At steady state, t is large and: 0
A0
ASC
C1 k
AA0 A A
dCC C kC
dt Is this consistent with steady
state balance eq for CSTR? No accumulation at steady state
0
A0
A0 A A ASC
C C kC 0 C1 k
In the unsteady state, when CA = 0.99CAS:
0t 1 kA0 s ACC1 e
1 k0.99
1 k
t 1 k t 1 k t 1 ks s s1 e 0.99 0.01 e ln 0.01 ln e
s1 k
4.6 t
s4.6 t1 k
time to reach 99% of steady-state concentration in terms of tk
Solve for ts to determine time to reach 99% of steady-state concentration
Goal: combine start-up and SS eqs to estimate time to reach SS (ts)
Yes, same!
L7-13
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
CSTR Start-Up: 1st Order Reaction t 1 kA0
AC
1 e C1 k
99% of the steady-state concentration is achieved at: A AS4.6 C 0.99C
1 k
When k is very small (slow rxn), 1>>k: st 4.6
When k is very big (fast rxn), 1<<k s
4.6t
k
63% of the steady-state concentration is achieved at: 1 k
CA = 0.63CAS
k
t 4.61
In the unsteady state, the time to reach CA = 0.99CAS is:
L7-14
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Better Selectivity in a Semi-Batch Reactor
To enhance selectivity of desired product over side product
kPA B P 2
P p A Br k C C Desired product P
kSA B S 2
S S A Br k C C Undesired side product S
Instantaneous selectivity, SP/S, is the ratio of the relative rates*: 2
P P A BP/S 2
S S A B
r k C CS
r k C C
Higher concentrations of A favor formation of the desired product P
Higher concentrations of B favor formation of the undesired side product S
Slowly feed B into the reactor containing A
Commonly used in bioreactors, when the enzyme is inhibited by excess substrate
P A
S B
k C
k C
To maximize the formation of the desired product:
*We’ll look at this concept of instantaneous selectivity in more detail in L9
L7-15
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Semi-Batch Reactor Design EquationCBu0
V0 + u0t
Do a mole balance on A since it does not enter or leave the reactor (assume the reactor is well-mixed)
In Out- + Generation = Accumulation
AA0 A A
dNF F r V
dt
AAd
0 0 r V N
dt
Use whatever units are most convenient (NA, CA, XA, etc)
AAA A
NN
VC C V A
Ad
VC
rV
dt
AA A
dC
dr
tV V C
dVdt
2 parts: how CA changes with t and how V changes with t
Convert NA to CA using:
L7-16
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Semi-Batch Reactor Design EquationCBu0
V0 + u0t
Do a mole balance on A since it does not enter or leave the reactor (assume the reactor is well-mixed)
In Out- + Generation = Accumulation
AA
dN0 0 r V t
dt
AA A
dCr V V C
dtdVdt
2 parts: how CA changes with t and how V changes with t
Reactor volume at any time can be found with a mole balance
In Out- + Generation = Accumulation
0 0
d V 0 0
dt
u
u = u0
0 0
dVdt
u0 0V t Vu
Substitute: AA 0A
dCr V V C
dtu Rearrange to get in terms of dCA/dt
AA A 0
dCr V C V
dtu A 0 A
AC dC
rV dt
u Balance on A
L7-17
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Semi-Batch Reactor Design EquationCBu0
V0 + u0t
Mole Balance on B
BB B0
dNr V F
dt
In Out- + Generation = Accumulation
BB0 B
dNF 0 r V
dt
0dVdt
u
BB B0
dNr V F
dt
B B0B
BdVC d
C V rt
Vd
Fdt
BB 0 B B0 0
dCC V r V C
dtuu
B B0 0B
BdC
C Vd
r V Ct t
dVd
u
0 B0 BBB
C CdCr
dt V
u
Substitute
Rearrange to get in terms of dCB/dt
Balance on B
L7-18
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Semi-Batch Reactor Design Equation: in Terms of NACBu0
V0 + u0t
AA
dNr V
dt
AA 0 0
dNr V t
dtu
AA
dN0 0 r V
dt
In Out- + Generation = Accumulation
0 0Substitute V V tu
Reactor design eq. provided that rA is a function of NA
AAdN
Vdt
r
A B
0
A
0
NdN
dk
Vt
N
tu
NB comes from BMB: BA B0
dNr V F
dt B A B
B00 0
dN N Nk F
dt V tu
The design eq in terms of XA can be messy. Sometimes it gives a single
equation when using Nj or Cj gives multiple reactor design equations.
A B A BA A 2
0 0
N N N Nr k r k
V V V tu-rA = kACACB
L7-19
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
+ H2O
V0 Vf
FD
V0 - u0t
Semi-batch
To improve product yield in a reversible reaction: A l B l C l D g • Start with A(l) and B(l) in the reactor• D(g) bubbles out of the liquid phase, pushing the equilibrium to the right
and forcing the reaction to go to completion
Common industrial reaction:
n
+ n
Boil off water to produce high MW polymer
nylon
A+BC+DA+B
Improving Yields of Reversible Rxns with Semi-Batch Reactors
L7-20
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
How do we account for the loss of product D in the material balance?
V0 Vf
FD
V0 - u0t
Semi-batch
To improve product yield in a reversible reaction: A l B l C l D g • Start with A(l) and B(l) in the reactor• D(g) bubbles out of the liquid phase, pushing the equilibrium to the right
and forcing the reaction to go to completion
A+BC+DA+B
Improving Yields of Reversible Rxns with Semi-Batch Reactors
L7-21
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Loss of Mass in Semi-Batch Reactor
In Out- + Generation = AccumulationOverall Mass balance:
0dm
0 dt
mu
↑want in terms of dV/dt
V0 + u0t
D(g)u = u0 0 A l B l C l D g elementary rxn
ggas leaving reactor
timem
mV
mV
Divide mass balance by
00
d 1 ddt dt
Vmm m uu
Relate ṁ to a rate: From stoichiometry, rD = -rA
Amoles
rvolume time
Next, convert units to:
masstime
Amoles
r Vtime
DD D D D
D
massmoles moles MW mass
MW A
massti
V MWme
r m
Substitute for ṁ
0A Dr VdV
dt
MWu
One of the diff. eq. that are simultaneously solved (by Polymath)
Conversion 1:
Conversion 2: