Freeman-Carroll Method

freeman_carroll_method(temperature: np.ndarray, alpha: np.ndarray, time: np.ndarray) → Tuple[float, float, float, np.ndarray, np.ndarray, np.ndarray, np.ndarray]

Perform Freeman-Carroll analysis to determine kinetic parameters for non-isothermal decomposition reactions.

Parameters:

  • temperature (np.ndarray) – Temperature data in Kelvin.

  • alpha (np.ndarray) – Conversion data.

  • time (np.ndarray) – Time data in minutes.

Returns:

Tuple containing activation energy (E_a in J/mol), reaction order (n), R-squared value, and arrays for plotting (x, y, x_filtered, y_filtered).

Return type:

Tuple[float, float, float, np.ndarray, np.ndarray, np.ndarray, np.ndarray]

Theory

The Freeman-Carroll method is used for determining the activation energy and reaction order from thermogravimetric data. It is based on the following equation:

\[\frac{\Delta \ln(d\alpha/dt)}{\Delta \ln(1-\alpha)} = n - \frac{E_a}{R} \cdot \frac{\Delta(1/T)}{\Delta \ln(1-\alpha)}\]
where:

  • dα/dt is the rate of conversion

  • α is the conversion

  • T is the temperature

  • E_a is the activation energy

  • n is the reaction order

  • R is the gas constant

Usage Example

import numpy as np
from pkynetics.model_fitting_methods import freeman_carroll_method
from pkynetics.result_visualization import plot_freeman_carroll

# Generate sample data
time = np.linspace(0, 100, 1000)
temperature = 300 + 5 * time
alpha = 1 - np.exp(-0.01 * time)

# Perform Freeman-Carroll analysis
e_a, n, r_squared, x, y, x_filtered, y_filtered = freeman_carroll_method(temperature, alpha, time)

print(f"Activation energy (E_a): {e_a/1000:.2f} kJ/mol")
print(f"Reaction order (n): {n:.2f}")
print(f"R-squared: {r_squared:.4f}")

# Visualize results
plot_freeman_carroll(x, y, x_filtered, y_filtered, e_a, n, r_squared)

Parameters

  • temperature (np.ndarray): Temperature data in Kelvin. Must be in ascending order.

  • alpha (np.ndarray): Conversion data. Must be between 0 and 1.

  • time (np.ndarray): Time data in minutes. Must be in ascending order.

Returns

A tuple containing:

  1. e_a (float): Activation energy in J/mol.

  2. n (float): Reaction order.

  3. r_squared (float): Coefficient of determination (R^2) of the fit.

  4. x (np.ndarray): x-values for plotting (Δ(1/T) / Δln(1-α)).

  5. y (np.ndarray): y-values for plotting (Δln(dα/dt) / Δln(1-α)).

  6. x_filtered (np.ndarray): Filtered x-values used for linear regression.

  7. y_filtered (np.ndarray): Filtered y-values used for linear regression.

Raises

  • ValueError: If input arrays have different lengths or contain invalid values.

Notes

  • The method uses smoothed and differentiated data to improve accuracy.

  • The analysis focuses on the most relevant part of the reaction, typically between 20% to 80% conversion.

  • Outliers are removed using the Interquartile Range (IQR) method to improve the fit.

  • Use the returned x, y, x_filtered, and y_filtered arrays with the plot_freeman_carroll function for visualization.

See Also

  • coats_redfern_method(): For solid-state reaction kinetics

  • plot_freeman_carroll(): For visualizing Freeman-Carroll analysis results