Kinetics and Rates of
Reactions
CEE 373
Roadmap
SANDBOX
IMPLEMENTATION
IMPLEMENTATION
Modeling concepts,
scales and approaches
Examination of
Equilibrium-based
Code
Visualization, Interface
Design and Usability
SANDBOX
IMPLEMENTATION
READINESS
Programming
languages, software
engineering &
numerical methods
Examination of
Reaction Rate-based
Code
Internal Testing and
Code Freeze
DESIGN
IMPLEMENTATION
RELEASE
Project Proposal
Examination of
Existing Models for
Complex Systems
Final Presentations
("Rollout")
KINETICS AND RATE LIMITED REACTIONS
OBJECTIVES
1. Build a modeling framework for reaction ratelimited chemistry.
2. Examine and understand computer code.
3. Produce model results and interpret critically.
KINETICS AND RATE LIMITED REACTIONS
1. Rate-Limited Reactions
2. Kinetics of Nitrification in a Batch Reactor
• Derivation of expressions used in model
• Temperature effect on rate constant
• Implementation in computer code
3. Kinetics of Nitrification in a Column Reactor
• Expressions used in model
4. Michaelis-Menten Kinetics
• Substrate-limited reaction rates
Rate-Limited Reactions
SIMPLE IRREVERSIBLE REACTION EXAMPLES
A➞B
Zero
d[ A]
−
= k0
dt
[ A] = [ A]0 − k0 t
t1/ 2
A➞B
First €
€
d[ A] €
−
= k1[ A]
dt
€
€
€
[ A]0
=
2k0
[ A]
ln
= k1t
[ A]0
t1/ 2
1
= ln 2
k1
Reaction Mechanisms
The Added Complexity of Reality
CONSECUTIVE IRREVERSIBLE
A0
A1
PARALLEL IRREVERSIBLE
A2
A0
A1
A2
REVERSIBLE
A0
CONSECUTIVE REVERSIBLE
A1
A0
PARALLEL CONSECUTIVE
A0
A1
A2
PARALLEL CONSECUTIVE
A11
A12
A13
A21
A22
A23
A0
A11
A12
A13
A21
A22
A23
Nitrification Kinetics
Nitrification in a Batch Reactor
DERIVATION
Pair of irreversible, first order kinetic reactions
+
−
k1,nitrosomonas
k2 ,nitrobacter
NH4    
→ NO2    → NO3
−
+
+
d[NH4 ]
= −k1[NH4 ]
dt
First order rate law for step 1
€
+
−
First order rate law
expression for consecutive
first order steps
Integrated form
for consecutive
steps
+
[NH4 ] = [NH4 ]0 e −k1t
Integrated form for step 1
€
€d[NO2 ] = k [NH + ] − k [NO − ]
1
4
2
2
dt
€
+
−
k [NH4 ]0 −k1t −k2 t
[NO2 ] = 1
e −e }
{
k2 − k1
−
+
+
−
[NO3 ] = [NH4 ]0 − [NH4 ] − [NO2 ]
Mass balance expression
€
€
Nitrification in a Batch Reactor
RELATING TO COMPUTER CODE
Temperature Effect Adjustments
ki′ = ki e a(T −20)
Ea
where a =
RT1T2
20°C Reference State
Constant
€
+
+
[NH4 ] = [NH4 ]0 e −k1t
€
+
−
k [NH4 ]0 −k1t −k2 t
[NO2 ] = 1
e −e }
{
k2 − k1
€−
+
+
−
[NO3 ] = [NH4 ]0 − [NH4 ] − [NO2 ]
K1 = LA * Exp(A * (TA - 20))
K2 = LB * Exp(B * (TA - 20))
TC = TC + (TB / 10)
S = S + 1
DA = Exp(-K1 * TC)
DB = Exp(-K2 * TC)
N1(S) = CA * DA
J = K1 * CA / (K2 - K1)
N2(S) = J * (DA - DB)
N3(S) = CA - N1(S) - N2(S)
Nitrification in a Column
NUMERICAL SOLUTIONS (STEADY STATE)
Simple Transport
x = vt
Velocity in porous media
€
Q
v=
θA
where x = distance, v = velocity, t = time
where Q = application rate, v = pore
water velocity, θ = volumetric water
content, A = cross-sectional area
K i′ = K i′e a(T −20)
Temperature Effect Adjustments
€
€
€
+
+
−
+
ki
where Ki =
v
[NH4 ] = [NH4 ]0 e −K1′x
+
€
−
′
K [NH4 ]0 €
− K1′x
− K2′ x
[NO2 ] = 1
e
−
e
{
}
K 2′ − K1′
+
−
[NO3 ] = [NH4 ]0 − [NH4 ] − [NO2 ]
Reaction Mechanisms
The Added Complexity of Reality
CONSECUTIVE IRREVERSIBLE
A0
A1
PARALLEL IRREVERSIBLE
A2
A0
A1
A2
REVERSIBLE
A0
CONSECUTIVE REVERSIBLE
A1
A0
PARALLEL CONSECUTIVE
A0
A1
A2
PARALLEL CONSECUTIVE
A11
A12
A13
A21
A22
A23
A0
A11
A12
A13
A21
A22
A23
Biologically Controlled Reactions
Growth, Decay, and Biodegradation
E + S
k1
k-1
ES
kp
Michaelis-Menten Kinetics
µmax [S]
µ=
K m + [S]
Examples
• Biodegradation of pesticides
• Algal €
growth
P + E
Numeric Types: Visual BASIC