Coats-Redfern Method

coats_redfern_method(temperature: np.ndarray, alpha: np.ndarray, heating_rate: float, n: float = 1) → Tuple[float, float, float, np.ndarray, np.ndarray, np.ndarray, np.ndarray]

Perform Coats-Redfern analysis to determine kinetic parameters for solid-state reactions.

Parameters:

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

  • alpha (np.ndarray) – Conversion data.

  • heating_rate (float) – Heating rate in K/min.

  • n (float) – Reaction order. Default is 1.

Returns:

Tuple containing activation energy (E_a in J/mol), pre-exponential factor (A in min^-1), 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 Coats-Redfern method is an integral method used for the determination of kinetic parameters from thermogravimetric data. It is based on the following equation:

\[\ln\left(\frac{g(\alpha)}{T^2}\right) = \ln\left(\frac{AR}{\beta E_a}\right) - \frac{E_a}{RT}\]
where:

  • g(α) is the integral form of the reaction model

  • T is the temperature

  • β is the heating rate

  • E_a is the activation energy

  • A is the pre-exponential factor

  • R is the gas constant

Usage Example

import numpy as np
from pkynetics.model_fitting_methods import coats_redfern_method
from pkynetics.result_visualization import plot_coats_redfern

# Generate sample data
temperature = np.linspace(300, 800, 500)
alpha = 1 - np.exp(-0.01 * (temperature - 300))
heating_rate = 10  # K/min

# Perform Coats-Redfern analysis
e_a, a, r_squared, x, y, x_filtered, y_filtered = coats_redfern_method(temperature, alpha, heating_rate)

print(f"Activation energy (E_a): {e_a/1000:.2f} kJ/mol")
print(f"Pre-exponential factor (A): {a:.2e} min^-1")
print(f"R-squared: {r_squared:.4f}")

# Visualize results
plot_coats_redfern(x, y, x_filtered, y_filtered, e_a, a, 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.

  • heating_rate (float): Heating rate in K/min. Must be positive.

  • n (float, optional): Reaction order. Default is 1.

Returns

A tuple containing:

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

  2. a (float): Pre-exponential factor in min^-1.

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

  4. x (np.ndarray): x-values for plotting (1000/T).

  5. y (np.ndarray): y-values for plotting (ln(g(α)/T^2)).

  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 focuses on the most linear part of the data, typically between 20% to 80% conversion.

  • The function automatically removes any invalid points (NaN or inf) before performing the linear regression.

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

See Also

  • freeman_carroll_method(): For non-isothermal decomposition kinetics

  • plot_coats_redfern(): For visualizing Coats-Redfern analysis results