Kinetics and Rates ofReactions

CEE 373

Roadmap

SANDBOX

Modeling concepts,scales and approaches

SANDBOXProgramming

languages, softwareengineering &

numerical methods

DESIGN

IMPLEMENTATION

Examination ofEquilibrium-based

Code

IMPLEMENTATION

Examination ofReaction Rate-based

Code

IMPLEMENTATION

Examination ofExisting Models forComplex Systems

Project Proposal

IMPLEMENTATION

Visualization, InterfaceDesign and Usability

READINESS

Internal Testing andCode Freeze

RELEASE

Final Presentations("Rollout")

KINETICS AND RATE LIMITED REACTIONS

OBJECTIVES1. Build a modeling framework for reaction rate-

limited chemistry.2. Examine and understand computer code.3. Produce model results and interpret critically.

KINETICS AND RATE LIMITED REACTIONS

1. Rate-Limited Reactions2. 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 ReactionsSIMPLE IRREVERSIBLE REACTION EXAMPLES

FirstA ➞ B

ZeroA ➞ B

€

−d[A]dt

= k0

€

−d[A]dt

= k1[A]

€

t1/2 =[A]0

2k0

€

[A] = [A]0 − k0t

€

t1/2 =1k1

ln2

€

ln[A][A]0

= k1t

Reaction MechanismsThe Added Complexity of Reality

A0

A1

A2

A0 A1 A2

A0 A1 A0 A1 A2

A0

A11

A21

A12 A13

A22 A23

A0

A11

A21

A12 A13

A22 A23

CONSECUTIVE IRREVERSIBLE PARALLEL IRREVERSIBLE

REVERSIBLE CONSECUTIVE REVERSIBLE

PARALLEL CONSECUTIVE PARALLEL CONSECUTIVE

Nitrification Kinetics

Nitrification in a Batch ReactorDERIVATION

€

NH4

+ k1,nitrosomonas → NO2

− k2 ,nitrobacter → NO3

−

Pair of irreversible, first order kinetic reactions

€

d[NH4

+]

dt= −k1[NH4

+]

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[NH4

+] = [NH4

+]0e−k1t

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d[NO2

−]

dt= k1[NH4

+]− k2[NO2

−]

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[NO2

−] =

k1[NH4

+]0

k2 − k1

e−k1t − e−k2t{ }

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[NO3

−] = [NH4

+]0 − [NH4

+]− [NO2

−]

First order rate law for step 1

Integrated form for step 1

First order rate lawexpression for consecutivefirst order steps

Integrated formfor consecutivesteps

Mass balance expression

Nitrification in a Batch ReactorRELATING TO COMPUTER CODE

TC = TC + (TB / 10)S = S + 1DA = Exp(-K1 * TC)DB = Exp(-K2 * TC)N1(S) = CA * DAJ = K1 * CA / (K2 - K1)N2(S) = J * (DA - DB)N3(S) = CA - N1(S) - N2(S)

K1 = LA * Exp(A * (TA - 20))K2 = LB * Exp(B * (TA - 20))

€

[NH4

+] = [NH4

+]0e−k1t

€

[NO2

−] =

k1[NH4

+]0

k2 − k1

e−k1t − e−k2t{ }

€

[NO3

−] = [NH4

+]0 − [NH4

+]− [NO2

−]

Temperature Effect Adjustments

€

′ k i = kiea(T−20)

20°C Reference StateConstant

€

where a =Ea

RT1T2

Nitrification in a ColumnNUMERICAL SOLUTIONS (STEADY STATE)

€

v =QθA

Velocity in porous media

€

x = vtSimple Transport

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[NH4

+] = [NH4

+]0e− ′ K 1x

€

[NO2

−] =

′ K 1[NH4

+]0

′ K 2 − ′ K 1e− ′ K 1x − e− ′ K 2x{ }

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[NO3

−] = [NH4

+]0 − [NH4

+]− [NO2

−]

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where Ki =ki

v

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′ K i = ′ K iea(T−20)Temperature Effect Adjustments

where Q = application rate, v = porewater velocity, θ = volumetric watercontent, A = cross-sectional area

where x = distance, v = velocity, t = time

Reaction MechanismsThe Added Complexity of Reality

A0

A1

A2

A0 A1 A2

A0 A1 A0 A1 A2

A0

A11

A21

A12 A13

A22 A23

A0

A11

A21

A12 A13

A22 A23

CONSECUTIVE IRREVERSIBLE PARALLEL IRREVERSIBLE

REVERSIBLE CONSECUTIVE REVERSIBLE

PARALLEL CONSECUTIVE PARALLEL CONSECUTIVE

Biologically Controlled ReactionsGrowth, Decay, and Biodegradation

Michaelis-Menten Kinetics

E + S ES P + Ek1

k-1

kp

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µ =µmax[S]Km + [S]

Examples• Biodegradation of pesticides• Algal growth

Numeric Types: Visual BASIC