Coagulation process to remove microplastics from water

Factors: Coagulant dosage, pH, stirring speed

Response: Removal efficiency
(% microplastics)

How do process parameters affect microplastic removal efficiency?

Experimental Design: A structured approach to planning experiments

Goal: Maximize information while minimizing resources

Focus: Identify key factors and their interactions

Key Benefits:

Improves efficiency and reliability of results

Provides insights into cause-and-effect relationships

1: Method optimization

Focus: Identify the best combination of factors (dynamically exploring design space)

Goal: Maximize performance (e.g., sensitivity, efficiency)

Example: Optimizing pH, temperature, and time for a chemical reaction

What is DoE?

A systematic approach to plan, conduct, and analyze experiments to understand how multiple factors (e.g., water pH, pollutant concentration) affect key outcomes (e.g., degradation efficiency).

Goals of DoE

Characterize relationships between factors and the response to understand their influence on system behavior.

Develop a mathematical model that enables interpolation across the chemical space for predictive analysis.

Capture interactions between factors to identify combined effects on the response.

Core Components of DoE

Factors: Controlled variables such as pollutant type, pH, or catalyst concentration.

Levels: Specific values, e.g., pollutant concentrations of 10, 50, and 100 mg/L.

Response: Measurable outcomes like pollutant degradation (%) or toxicity reduction.

Why Structure Matters

A structured design helps uncover relationships like how pH influences heavy metal precipitation.

Efficient Experimentation

DoE minimizes tests needed to determine the optimal dosage of coagulants in water treatment.

Identifies Key Insights

Reveals how interactions between pollutant types and catalysts affect degradation rates.

Improves Process Understanding

Optimizes processes like advanced oxidation or nutrient recovery from wastewater.

Full Factorial Design

Explores all combinations of factors, e.g., pH, temperature, and pollutant type, providing comprehensive insights.

Fractional Factorial Design

Tests a subset of combinations to reduce effort, e.g., identifying key factors in pesticide degradation.

Box-Behnken Design (BBD)

Efficiently explores factor interactions using midpoints of edges and the center of the design space, avoiding extreme conditions.

Objective

Investigate how temperature and pH influence pollutant removal efficiency in water samples.

Factors and Levels

Factor 1: Temperature → Levels: -1 (Low, 20°C), +1 (High, 40°C)

Factor 2: pH → Levels: -1 (Low, pH 5), +1 (High, pH 9)

Design Type

Full factorial design with 2 levels per factor (2 × 2 = 4 experiments).

Design Matrix

All combinations of factors and levels (-1, +1) form the matrix.

Fitting the Model

Using the data:

Run 1: Y = 70 Run 2: Y = 75 Run 3: Y = 85 Run 4: Y = 90

Estimated Coefficients

β₀: Average response (intercept)

β₁: Effect of temperature on removal efficiency

β₂: Effect of pH on removal efficiency

Example Result

Y = 80 + 7.5X₁ + 2.5X₂

Insights from Coefficients

β₁ (7.5): Higher temperature significantly improves removal efficiency.

β₂ (2.5): Higher pH has a smaller positive effect.

Predictions

Predict removal efficiency for any combination of temperature and pH using the regression equation.

Visualization

Create 2D or 3D plots to illustrate the effects of temperature and pH on removal efficiency.

Hypothesis Testing

For each coefficient (β₁, β₂):

- Null hypothesis (H₀): The coefficient is not significant (β = 0).

- Alternative hypothesis (H₁): The coefficient is significant (β ≠ 0).

t-Test for Significance

Use the formula for the t-statistic:

t = β / SE(β)

Where SE(β) is the standard error of the coefficient.

Understanding Coefficients

Regression coefficients in coded levels (\([-1, 1]\)) describe the change in response when the factor changes by the full range (\(+2\) units in coded space).

Converting to Real Units

The effect of a factor in real units is calculated by dividing the total response change by the real range of the factor.

Response Change per Real Unit = 2 × β (coded) / Real Range

Number of Experiments

In a full factorial design, the total number of experiments (\(N\)) depends on the number of factors (\(k\)) and the levels per factor (\(L\)).

Formula

\( N = L^k \)

Where: \(L\): Number of levels per factor (e.g., 2 for \([-1, +1]\)) \(k\): Number of factors

Example

For \(k = 3\) factors (\(X_1, X_2, X_3\)) and \(L = 2\) levels (\([-1, +1]\)):

\( N = 2^3 = 8 \) experiments

Degrees of Freedom (df) in Full Factorial Designs

The degrees of freedom (\(df\)) are calculated as:

df = N - p - 1

Where: \(-\) \(N\): Number of experiments \(-\) \(p\): Number of coefficients (factors or interactions) \(-\) \(1\): For the intercept

Model Scenarios

Main Effects Model: Only main factors are included (\(p = k\)).

Main Effects + Pairwise Interactions: Includes main factors and pairwise interactions (\(p = k + \frac{k(k-1)}{2}\)).

What is an Interaction?

An interaction occurs when the effect of one factor on the response depends on the level of another factor.

Interactions highlight synergies or conflicts between factors that cannot be explained by their individual effects alone.

Example

Consider two factors:
Temperature (\(X_1\)) and pH (\(X_2\)):

Without interaction: \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2\).

With interaction: \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2\).

The interaction term (\(\beta_{12} X_1 X_2\)) modifies the response based on the combination of \(X_1\) and \(X_2\).

What is a Quadratic Term?

A quadratic term models the nonlinear effects of a factor on the response. It represents the curvature of the relationship between a factor and the response.

Quadratic terms allow for the identification of optimal factor levels, where the response is maximized or minimized.

Example

Consider two factors:
Temperature (\(X_1\)) and pH (\(X_2\)):

Without quadratic terms: \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2\).

With quadratic terms: \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2 + \beta_{11} X_1^2 + \beta_{22} X_2^2\).

The quadratic terms (\(\beta_{11} X_1^2, \beta_{22} X_2^2\)) allow for curvature in the response.

Why are Additional Points Needed?

To model quadratic terms, additional experiments are required to capture the curvature of the response.

These additional points ensure sufficient information to estimate quadratic effects accurately.

Types of Additional Points

1. Center Points (\(X_1 = 0, X_2 = 0\)): Measure the response at the geometric center of the design space.

2. Edge-Center Points (\(X_1 = 0\) or \(X_2 = 0\)): Measure the response at the midpoint of each factor's range, holding the other factor constant.

Key Insight

Without these points, the curvature of the response cannot be determined, and quadratic terms cannot be estimated reliably.

Why Use Fractional Designs?

Full factorial designs require exponentially more experiments as the number of factors increases (\(N = 2^k\)).

Fractional designs allow us to study key effects with fewer experiments, saving time and resources.

What is a Fractional Design Plan?

A fractional design tests only a subset (fraction) of the combinations in a full factorial design.

It uses mathematical rules to carefully select combinations that maximize the information gained.

Steps to Create a Fractional Design

1. Start with a full factorial design for a smaller set of factors (e.g., \(X_1, X_2\)).

2. Define additional factors (\(X_3, X_4, \dots\)) as interactions of existing factors.

3. Construct the design matrix by combining these relationships.

Example: Fractional \(2^{3-1}\) Design

\(k = 3\) factors (\(X_1, X_2, X_3\)).

Full factorial: \(2^3 = 8\) experiments.

Fractional plan: \(2^{3-1} = 4\) experiments.

Define \(X_3 = X_1 X_2\).

Assumptions of Fractional Designs

Fractional designs assume that higher-order interactions are negligible.

If higher-order interactions are significant, the results can be misleading.

Confounding

In fractional designs, some effects are confounded, meaning they cannot be distinguished from each other (same column pattern).

For example, if \(X_3 = X_1 X_2\), the effect of \(X_3\) is inseparable from the interaction \(X_1 X_2\).

Validation is Required

Fractional designs are suitable for initial screening, but full designs or additional experiments may be needed for confirmation.

What is a Box-Behnken Design?

A Box-Behnken design (BBD) is a response surface methodology (RSM) used for modeling nonlinear relationships between factors and responses.

Unlike full factorial designs, BBDs avoid extreme corner points, reducing the risk of running impractical or unsafe experiments.

Key Features

Includes center points (\(X = 0\)) and edge-center points.

Provides sufficient data to estimate quadratic effects.

Requires fewer experiments than a full factorial design.

How is a Box-Behnken Design Built?

Combines edge-center points and center points in the design space.

Requires a minimum of 3 factors (Box-Behnken is not defined for 2 factors).

For 3 factors (\(X_1, X_2, X_3\)), a Box-Behnken plan consists of 12 edge-center points and 1 center point.

Design Matrix Example: 3 Factors

Key Limitations

Not defined for 2 factors; requires at least 3 factors.

Inefficient for large numbers of factors compared to other RSM methods (e.g., Central Composite Designs).

Assumes quadratic effects dominate; may miss higher-order interactions.

Practical Challenges

Limited flexibility for factor ranges; scaling is often required.

Requires careful validation to ensure results are robust.

Key Considerations

The choice of experimental design depends on:

The number of factors (\(k\)) and their levels (\(L\)).

The type of effects to be studied (main effects, interactions, quadratic terms).

The available resources (time, budget, and experimental feasibility).

Full Factorial

Includes main effects and all interactions:

\( Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{12} X_1 X_2 + \beta_{13} X_1 X_3 + \beta_{23} X_2 X_3 + \beta_{123} X_1 X_2 X_3 \)

Fractional Factorial

Includes only main effects and selected interactions (e.g., confounded terms):

\( Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2 \)
where \(X_3 = X_1 X_2\)

Box-Behnken

Captures quadratic effects without corner points:

\( Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{12} X_1 X_2 + \beta_{13} X_1 X_3 + \beta_{23} X_2 X_3 + \beta_{11} X_1^2 + \beta_{22} X_2^2 + \beta_{33} X_3^2 \)

Backstory

A water treatment facility is seeking to optimize its process for removing pollutants. The response variable of interest is the removal efficiency (%) .

Two factors have been identified as critical to the process:

Temperature (\(X_1\)) : Range \(20^\circ\text{C}\) to \(60^\circ\text{C}\)

pH (\(X_2\)) : Range 6 to 8

Objective

To better understand the influence of these factors, you need to design and execute an experimental plan.

The goal is to identify the factor levels that maximize removal efficiency.