Chemical Kinetics and Reactor Design Course Review James B. Rawlings Department of Chemical Engineering University of Wisconsin 15th June 2004
1
Stoi Stoich chio iome metr try y
Chemical Equilibrium ns
dG
+ CO CO2 + H2 H2 O + H H2 + OH CO2 + H OH + CO A1 = H, A2 = H2 , A3 = OH, A4 = H2 O, A5 = CO, A6 = H2 O
CO2 .
νij Aj
j 1
=
= 0,
i
= ∂G ∂ξ i
µj
= 1, . . . , nr
νA
= 0,
ν
0 1 1
1 1 0
0 1 1
−1 −1 −1 0 − 0 −1
r i : reaction rate for reaction i. Rj : production rate for species j . Rj R
= =
i 1 T
=
=
= 0,
=
i
= Gj◦ + RT ln aj ,
aj
=
= 1, . . . , nr
= f j /f j◦
νij
aj
1 0 1
= ◦ ∆G = −RT ln K i
i
Standard Standard state: state: pure species species j at 1 atm atm and and syst system em temtemperature. Phase Equilibrium
f jα j
νij µj
j 1
j 1
= f jβ,
j
= 1, . . . , ns ,
all phases α and β
Know how to evaluate fugacities for ideal gases, ideal solutions, and Raoult and Henry law approximations.
= 1, . . . , ns
ν r
Extent of reaction: dnj
2
νij r i ,
T ,P
K i
nr
=
ns
ns
νij > 0, for Aj product; νij < 0, for Aj reactant.
= −
µj dnj
j 1
µj : chemical potential for species j .
ns
= −SdT + V dP
+
Temperature Dependence of K
nr i 1
= νij dεi ,
j
= 1, . . . , ns .
∂ ln K i ∂T K i2 ln K i1
=
◦
∆H i
RT 2 ◦ ∆H i R
=−
Thermo Thermodyn dynami amics cs
At equilibrium, for a given T and P , the Gibbs free energy is a minimum.
(Van’t Hoff Equation) 1 T 2
−
1 T 1
(assumption?)
How does raising temperature affect the equilibrium extent for exothermic and endothermic reactions? 1
3
Rate Expressions r
= k(T)f(cj ’s) Arrhenius expression for k: k(T) = k0 e−E /RT , E a is the aca
BATCH
tivation energy, always positive. How does raising temperature affect reaction rate? What is f (cj ’s) for a sequence of elementary steps. What is the difference between a mechanism and an overall stoichiometry. Quasi-steady-state assumption, equilibrium assumption and rate limiting step.
constant volume SEMI-BATCH CSTR
Langmuir Isotherms (chemisorption)
constant volume A
+ X A · X cA
steady state
= 1K +AcK Accm
PFR
4
= Rj V R
(1)
= Rj
(2)
= Qf cjf
+ Rj V R
(3)
= Qf cjf
(4)
=
− Qcj + Rj V R 1 (cjf − cj ) + Rj τ cjf + Rj τ
(5)
=
∂c j ∂(c j Q) = ∂t ∂V d(cj Q) = Rj dV dcj = Rj dτ
−
A A
steady-state
Hougen-Watson rate expressions Deciding which mechanism best explains rate data.
d(cj V R ) dt dcj dt d(cj V R ) dt d(cj V R ) dt dcj dt cj
constant density,
+ Rj
(6) (7) (8) (9)
Material and Energy Balances d dt
V
cj dV dU dt
= Q0 cj0 − Q1 cj1
+ V
Rj dV ,
j
= 1, . . . , ns
= m0 H ˆ0 − m1H ˆ1 + Q˙ − W ˙s BATCH CSTR
dT = dt ˆP dT = V R ρ C dt ˆP V R ρ C
− + − −
i
i
j
PFR (steady state)
ˆP Qρ C
dT = dV
∆H Ri r i V R
i
+ Q˙
+ Q˙ cjf Qf (H jf − H j )
(10)
∆H Ri r i V R
∆H Ri r i
+ q˙
(11)
(12)
Table 1: Summary of Mole and Energy Balances for Several Ideal Reactors.
2
˙ Q
˙ W
cj cjs
A V m1
m0
D
E 1 cj1
E 0 cj0
T cj
Figure 1: Reactor volume element.
R
460
1 0.9
440 r
0.8
extinction point
420
0.7
r
extinction point
400
0.6 x 0.5
T (K)
B
380
T cj
C
360
0.4
340
0.3
320
0.2
r
0.1
r
Figure 4: Expanded views of a fixed-bed reactor.
ignition point
300
ignition point
0
280 0
5
10
15
20 25 τ (min)
30
35
40
45
0
5
10
15
20 25 τ (min)
30
35
40
45
5. reaction of adsorbed reactants to adsorbed products (surface reaction)
Figure 2: CSTR steady-state multiplicity, stable and unsta ble steady states, ignition, extinction, hysteresis. 1
6. desorption of adsorbed products
400
7. diffusion out of pores
380
8. mass transfer from catalyst to bulk fluid
x(t) 360
x 0.5
9. convection of product in the bulk fluid
340 T( K) 320 300
T(t) 0 0
100
Simultaneous Reaction and Diffusion
200
300
400
500
2 cA + RA = 0
280 600
1st order reaction, spherical pellet:
time (min)
−
1 ∂ 2 ∂c r r 2 ∂r ∂r
Figure 3: CSTR oscillations.
5
Packed Bed Reactors
r r
Steps in a Catalytic Reactor 1. convection of reactant in the bulk fluid
1,
= =
(3Φ)2 c
=0
c 1 ∂c 0 ∂r
=
0,
=
(3Φr ) = r 1 sinh sinh(3Φ) V p n + 1 kρp csn−1 Thiele Modulus: Φ = Solution: c(r)
2. mass transfer from bulk fluid to catalyst 3. diffusion of reactant in the pores (molecular and Knudsen diffusion)
S p
2
De
Effectiveness Factor: actual rate in pellet η rate without diffusional limitations
=
4. adsorption of reactant on the active sites 3
1/2
Φ
1 0.8
c
dN j dV ˆp dT Qρ C dV
= 0.1 = 0.5
Φ
= Rj =−
0.4
Φ
= 1.0
0.2 Φ
= 2.0
i
0 0
0.5
1
1.5
2
2.5
ε
3
∆H Ri r i
i
(1 B ) Q Dp B3 A2c
(17)
=
(18) (19) (20)
i
= 1 − ρb /ρp ,
Q
r
6
+ R2 U o(T a − T )
(1 − B )µf 7 ρQ =− − +4A 150 Dp c Rj = (1 − B )R jp ∆H Ri r i = (1 − B ) ∆H Ri r ip dP dV
0.6
(16)
j
N j /c(P,T)
Mixing in Chemical Reactors
Residence-time distribution n n n n
1
=1 =2 =5 = 10
sphere(13) cylinder(14) slab(15)
1
p(θ)dθ,
probability that a feed molecule spends time θ to θ dθ in the reactor probability that a feed molecule spends time zero to θ in the reactor
+
0.1
P(θ),
η η 0.01
= t
0.1 0.1
1
10
0.001 0.01
0.1
1
Φ
10
ce (t)
100
Φ
Sphere
Cylinder
η =
Slab
η =
1 Φ
1 tanh 3Φ
1 I 1 (2Φ) Φ I 0 (2Φ) tanh Φ Φ
− 31Φ
cf (t )p(t
− t )dt,
cf (t)
= 0, t ≤ 0
(21)
Step response, impulse response for CSTR, PFR, n CSTRs, PFR with dispersion. Segregated reactor, maximally mixed reactor.
Figure 5: Effectiveness Factor Versus Thiele Modulus: Effect of Geometry and Reaction Order.
η =
0
Given a single reaction with convex (concave) reaction rate expression, the highest (lowest) conversion for a given RTD is achieved by the segregated reactor and the lowest (highest) conversion is achieved by the maximally mixed reactor.
(13)
(14)
7
Parameter Estimation
probability review.
(15)
p(x) Table 2: Effectiveness factor versus Thiele modulus for the sphere, semi-infinite cylinder, and semi-infinite slab.
p(x)
4
=√ 1
Normal distribution, mean, variance. 1 exp 2πσ 2
= (2π)n/2 |P |1/2 exp
−
−
1 (x m)2 σ 2 2
1 (x 2
−
T
(22)
− m) P −1 (x − m)
10 c0
8
c1
c2
τ1
1 r(c)
c3
τ3
τ2
6 r
τ3
= 1.07
4
r
r r
2
τ2 c3
= 13.9
τ1
c2
= 3.95
c1
c0
0 0
1
2
3
4
5
6
c
Figure 6: Inverse of reaction rate versus concentration. Optimal sequence to achieve 95% conversion is PFR–CSTR–PFR.
x T Ax Av i
Let y be measured as a function of x
Least squares.
=b
= λivi x2
y i
= mxi + b, i = 1, . . . n y = Xθ + e, e ∼ N( 0, σ 2 I ) The best estimate of θ in a least squares sense is given by
= θ
θ
P
(θ
= σ 2(X T X) −1
− θ)T X T X (θ − θ) ≤ χ2 (np , α) σ 2
− θ)T X T X (θ − θ) ≤ np F(np , nd − np , α) s2
s2
= nd −1 np (y − X θ)T (y − X θ)
b λ1 v 1
x1
˜11 bA
Figure 7: The geometry of quadratic form x T Ax
Confidence interval. α-level confidence region for estimating np parameters is given
(θ
˜22 bA
(X T X )−1 X T y
∼ N( θ, P ),
b λ2 v 2
(23)
(24) (25)
5
= b.
