Chapter 1 Environmental Engineering: Chemical Reactors

Contents

Chapter 1 Environmental Engineering: Chemical Reactors#

  1. Introduction: Chemical Reactors

  2. Simulation: Reactors

  3. Self-Assessment

1. Introduction#

⚗️ Chemical Reactors in Environmental Systems – Concepts & Applications#

Chemical reactors are idealized systems used to model the transformation of chemical constituents through reaction, mixing, inflow, and outflow. These models are widely applied in environmental engineering to simulate biogeochemical processes such as pollutant decay, nutrient cycling, and oxygen dynamics#

🧪 Types of Chemical Reactors#

1. Batch Reactor#

  • No inflow or outflow during the reaction period

  • Well-mixed and closed system

  • Reaction occurs over time with changing concentration

2. Continuously Stirred Tank Reactor (CSTR)#

  • Continuous inflow and outflow

  • Perfect mixing: output concentration equals reactor concentration

  • Steady-state or dynamic behavior

3. Plug Flow Reactor (PFR)#

  • Continuous flow with no mixing along the flow path

  • Concentration changes along the reactor length

  • Ideal for modeling transport-dominated systems

image.png#

🔬 Reaction Types#

  • Decay reactions: pollutant breakdown (e.g., BOD, ammonia)

  • Production reactions: generation of compounds (e.g., nitrate from nitrification)

⚗️ Chemical Reaction Orders – Equations & Parameter Estimation#

Chemical reactions are classified by their order, which reflects how the reaction rate depends on the concentration of reactants. Understanding reaction order is essential for modeling pollutant decay, nutrient cycling, and designing treatment systems.

📘 1. Zero-Order Reaction#

✅ Description#

  • Reaction rate is independent of reactant concentration.

  • Common in systems with saturated enzymes or surface-limited reactions.

📐 Rate Equation#

\[ \frac{dC}{dt} = -k \]

📊 Integrated Form#

\[ C(t) = C_0 - kt \]

📈 Estimation#

  • Plot \(( C \)) vs. \(( t \))

  • Slope = \(( -k \))

📘 2. First-Order Reaction#

✅ Description#

  • Reaction rate is proportional to the concentration.

  • Common in biodegradation, radioactive decay, and BOD modeling.

📐 Rate Equation#

\[ \frac{dC}{dt} = -kC \]

📊 Integrated Form#

\[ C(t) = C_0 e^{-kt} \]

📈 Estimation#

  • Plot \(( \ln(C) \)) vs. \(( t \))

  • Slope = \(( -k \))

📘 3. Second-Order Reaction#

✅ Description#

  • Reaction rate is proportional to the square of concentration or to the product of two reactants.

  • Occurs in chemical oxidation, precipitation, and complex formation.

📐 Rate Equation#

\[ \frac{dC}{dt} = -kC^2 \]

📊 Integrated Form#

\[ \frac{1}{C(t)} = \frac{1}{C_0} + kt \]

📈 Estimation#

  • Plot \(( 1/C \)) vs. \(( t \))

  • Slope = \(( k \))

📘 General n-th Order Reaction#

📐 Rate Equation#

\[ \frac{dC}{dt} = -kC^n \]

  • For \(( n \neq 1 \)), integration yields: $\( C(t) = \left[ C_0^{1-n} - (1-n)kt \right]^{\frac{1}{1-n}} \)$

🧪 Parameter Estimation Summary#

Reaction Order

Plot Type

Slope or Intercept

Estimated Parameter

Zero-order

\(( C \)) vs. \(( t \))

Slope = \(( -k \))

\(( k \))

First-order

\(( \ln(C) \)) vs. \(( t \))

Slope = \(( -k \))

\(( k \))

Second-order

\(( 1/C \)) vs. \(( t \))

Slope = \(( k \))

\(( k \))

🧠 Conceptual Insight#

  • Reaction order reflects sensitivity to concentration.

  • Higher-order reactions are nonlinear and more sensitive to changes in ( C ).

  • Accurate estimation of \(( k \)) is essential for predictive modeling.

❓ Quiz Questions#

  1. What does a zero-order reaction imply about concentration dependence?

  2. How do you estimate the rate constant for a first-order reaction?

  3. Why are second-order reactions more sensitive to concentration?

  4. What plot would you use to identify a second-order reaction?

  5. How does reaction order affect pollutant removal in reactors?

🌎 Environmental Analogs#

Natural System

Reactor Type

Lakes, reservoirs

CSTR

Rivers, floodplains

PFR

Wetlands, ponds

Hybrid

These systems mimic reactor behavior through flow, mixing, and reaction kinetics.

📐 Mathematical Modeling#

Understanding reaction kinetics allows us to construct models that simulate:

  • Pollutant removal efficiency

  • Sensitivity to inflow concentration

  • Residence time effects

  • Design of treatment systems

Example: First-Order Decay in CSTR#

\[ C_{\text{out}} = \frac{C_{\text{in}}}{1 + k \tau} \]

Where:

  • \(( C_{\text{in}} \)): inflow concentration

  • \(( C_{\text{out}} \)): outflow concentration

  • \(( k \)): reaction rate constant

  • \(( \tau \)): hydraulic residence time

🧠 Conceptual Insight#

  • CSTRs are good for modeling systems with strong mixing (e.g., lakes)

  • PFRs are ideal for transport-dominated systems (e.g., rivers)

  • Combining reactor models helps simulate complex natural systems like wetlands and floodplains

Foundational Literature#

[Davis and Cornwell, 2013, Masters and Ela, 2008, Mihelcic, 1999] collectively provide a robust foundation in environmental reaction kinetics and water quality modeling. Each text introduces core concepts, such as reaction rate laws, first-order decay, and reactor dynamics, and applies them to pollutant transformation, biodegradation, and contaminant fate in natural systems. These resources emphasize both theoretical derivation and practical application, making them essential for understanding the kinetics of environmental processes and for designing systems to manage water quality.

2. Simulation#

Plug Flow Reactor (PFR) vs Continuous Stirred Tank Reactor (CSTR)#

This guide outlines the key features, equations, and performance characteristics of two foundational reactor models in chemical engineering: Plug Flow Reactor (PFR) and Continuous Stirred Tank Reactor (CSTR). Both are used to model conversion of reactants into products under different flow and mixing assumptions.

Reaction Assumptions#

We assume a single irreversible reaction:

\[ A \rightarrow B \]

With rate law:

\[ r_A = k C_A^n \]

Where:

  • \(( r_A \)): reaction rate (mol/L·min)

  • \(( k \)): rate constant (1/min or L/mol·min)

  • \(( C_A \)): concentration of A (mol/L)

  • \(( n \)): reaction order

CSTR (Continuous Stirred Tank Reactor)#

At steady-state, molar balance:

\( F_{A0} - F_A = V \cdot r_A \)

In terms of conversion \(( X \)):

\[ X = \frac{k \tau}{1 + k \tau} \quad \text{(1st order)} \]

Where:

  • \(( \tau = \frac{V}{Q} \)): residence time (min)

  • \(( V \)): reactor volume (L)

  • \(( Q \)): volumetric flow rate (L/min)

Characteristics:#

  • Complete mixing → uniform concentration throughout

  • Lowest conversion for given ( \tau ) among ideal reactors

  • Easy to scale and operate continuously

PFR (Plug Flow Reactor)#

Differential form of molar balance:

\[ \frac{dF_A}{dV} = r_A \]

In terms of conversion \(( X \)) for 1st order:

\[ X = 1 - e^{-k \tau} \]

Characteristics:#

  • No axial mixing → concentration varies along length

  • Higher conversion than CSTR for same residence time

  • Mimics tubular reactor flow

Performance Comparison (1st Order Reaction)#

Reactor Type

Conversion Equation

Conversion vs \(( \tau \)) Behavior

CSTR

\(( X = \frac{k \tau}{1 + k \tau} \))

Slower rise, saturates at 1

PFR

\(( X = 1 - e^{-k \tau} \))

Faster rise, reaches 1 more quickly

Design Insight#

  • PFR is more volume-efficient for achieving high conversion.

  • CSTR is better for control and handling fluctuations.

  • For n > 1 reactions, difference in conversion widens even more.

❓ Quiz Questions#

  1. What distinguishes a CSTR from a PFR?

  2. Why are lakes often modeled as CSTRs?

  3. How does residence time affect pollutant removal?

  4. What type of reactor best represents a river reach?

  5. How can reactor models help design wastewater treatment systems?

🧪 Applications#

  • Designing wastewater treatment reactors

  • Modeling nutrient removal in wetlands

  • Simulating oxygen dynamics in lakes and rivers

  • Assessing pollutant fate in floodplains

import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import FloatSlider, Dropdown, interact
%matplotlib inline

def cstr_conversion(k, C0, tau, order):
    return k * tau / (1 + k * tau) if order == 'First' else 1 - 1 / (1 + k * C0 * tau)

def pfr_conversion(k, C0, tau, order):
    return 1 - np.exp(-k * tau) if order == 'First' else (k * C0 * tau) / (1 + k * C0 * tau)

def plot_reactors(k, C0, V, Q, order):
    tau = V / Q
    tau_vals = np.linspace(0.1, 2 * tau, 200)
    plt.figure(figsize=(9, 5))
    plt.plot(tau_vals, cstr_conversion(k, C0, tau_vals, order), label="CSTR", linestyle='-.')
    plt.plot(tau_vals, pfr_conversion(k, C0, tau_vals, order), label="PFR", linestyle='-')
    plt.axvline(tau, color='red', linestyle=':', label=f'Reactor τ = {tau:.2f}')
    plt.xlabel("Residence Time (min)")
    plt.ylabel("Conversion of A (X)")
    plt.title(f"{order}-Order Reaction | V={V:.0f} L, Q={Q:.0f} L/min, k={k:.2f}, C₀={C0:.2f}")
    plt.ylim(0, 1)
    plt.grid(True)
    plt.legend()
    plt.show()
    print(f"Conversion at τ = {tau:.2f} min:")
    print(f"  • CSTR = {cstr_conversion(k, C0, tau, order):.3f}")
    print(f"  • PFR  = {pfr_conversion(k, C0, tau, order):.3f}")

interact(plot_reactors,
         k=FloatSlider(value=0.2, min=0.01, max=1.0, step=0.01, description='Rate k'),
         C0=FloatSlider(value=1.0, min=0.1, max=2.0, step=0.1, description='C₀'),
         V=FloatSlider(value=100, min=10, max=500, step=10, description='Volume V'),
         Q=FloatSlider(value=10, min=1, max=100, step=1, description='Flow Q'),
         order=Dropdown(options=['First', 'Second'], value='First', description='Order'));

3. Self-Assessment#

Reactor Design Quiz: CSTR vs PFR#

This markdown format includes multiple-choice quiz questions, answers, and explanations to reinforce core concepts and applications of plug flow and stirred tank reactors.

Conceptual Questions#

1. What defines the flow behavior in a Plug Flow Reactor (PFR)?#

  • Complete mixing throughout the reactor

  • Unmixed flow with axial concentration gradient

  • Recirculation zones for better heat transfer

  • Turbulent flow that simulates stirring
    Explanation: In a PFR, there is no axial mixing. Reactants flow in a plug-like fashion with changing concentration and reaction rate along the reactor length.

2. How does a CSTR differ from a PFR in terms of conversion for a first-order reaction?#

  • CSTR achieves lower conversion than a PFR for the same residence time

  • CSTR achieves higher conversion due to continuous stirring

  • PFR and CSTR give identical conversion

  • PFR has more temperature control
    Explanation: PFRs outperform CSTRs in conversion for first-order reactions due to the gradient in reactant concentration, which maintains higher reaction rates along the reactor.

3. What is the expression for conversion in a CSTR (first-order reaction)?#

  • \(( X = \frac{k \tau}{1 + k \tau} \))

  • \(( X = 1 - e^{-k \tau} \))

  • \(( X = \frac{C_A}{C_{A0}} \))

  • \(( X = 1 - \frac{C_{A0}}{C_A} \))
    Explanation: The CSTR conversion expression for a first-order reaction is derived from the steady-state mass balance, assuming complete mixing.

4. In a PFR, what is the driving mechanism for conversion increase along the reactor length?#

  • Continuous decrease in reactant concentration

  • Uniform reactant concentration

  • Recirculation induced by paddle mixing

  • Feed recycling
    Explanation: In a PFR, reactant concentration decreases along the length, allowing the rate to remain relatively high, leading to improved conversion.

5. What operational advantage does a CSTR have over a PFR?#

  • Easier to control and maintain steady operation

  • Higher conversion per volume

  • Lower space requirement

  • Temperature remains constant throughout PFR
    Explanation: CSTRs are preferred for processes needing consistent conditions and easy control, especially when dealing with varying feed or reaction dynamics.

Reflective Questions#

  • 💭 How does reactor geometry affect residence time and mixing behavior?

  • 💭 What challenges might arise when scaling up PFRs in industrial setups?

  • 💭 How would the performance shift if reaction kinetics were second-order or autocatalytic?

  • 💭 In what conditions would combining a PFR and CSTR in series be more effective than either alone?

Reference Equations#

  • CSTR (1st-order): \(( X = \frac{k \tau}{1 + k \tau} \))

  • PFR (1st-order): \(( X = 1 - e^{-k \tau} \))

Where:

  • \(( X \)): conversion

  • \(( k \)): rate constant

  • \(( \tau = \frac{V}{Q} \)): residence time

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