About the Flow Behavior Index Calculator

This guide provides detailed information on the principles behind our Flow Behavior Index calculator. It covers the what, why, and how of analyzing fluid rheology using the Power Law model, helping users interpret their data and results accurately.

What This Calculator Does

The calculator analyzes experimental data of shear stress (τ) versus shear rate (γ̇) to characterize the flow properties of non-Newtonian fluids. Its primary functions are:

• Calculate Key Parameters: It determines the Flow Behavior Index (n) and the Consistency Index (K) by applying a linear regression to the logarithmic form of the Power Law model.
• Classify Fluid Type: Based on the calculated Flow Behavior Index (n), it classifies the fluid as Pseudoplastic (shear-thinning), Newtonian, or Dilatant (shear-thickening).
• Assess Model Fit: It calculates the Coefficient of Determination (R²), which indicates how well the Power Law model represents the provided experimental data. An R² value close to 1.0 suggests a strong fit.
• Visualize Data: It generates both arithmetic and log-log plots of the data, visually comparing the experimental points against the fitted Power Law model curve.

When to Use It

This tool is essential for scientists, engineers, and technicians in various fields who need to understand and quantify fluid behavior. Key applications include:

• Pharmaceuticals: Formulating creams, ointments, and suspensions with desired consistency and spreadability.
• Food Science: Quality control of products like ketchup, yogurt, and sauces, ensuring consistent texture and flow.
• Chemical Engineering: Designing pipelines and mixing processes for polymers, slurries, and paints.
• Cosmetics: Developing lotions, gels, and foundations with specific application properties.
• Materials Science: Characterizing novel polymers and colloidal suspensions.

Inputs Explained

To use the calculator, you need a set of corresponding measurements from a rheometer or viscometer.

• Shear Rate (γ̇): This is the rate at which a fluid is sheared or “worked.” It represents the velocity gradient within the fluid flow and is typically measured in reciprocal seconds (s⁻¹). You must enter at least two data points.
• Shear Stress (τ): This is the force per unit area required to move one layer of fluid relative to another. It’s the fluid’s internal resistance to flow. The calculator accepts common units like Pascals (Pa), dynes per square centimeter (dyn/cm²), and pounds per square foot (psf). Ensure you select the correct unit to match your data for accurate conversion.

Results Explained

After calculation, the tool provides a comprehensive analysis of your fluid’s properties:

• Flow Behavior Index (n): A dimensionless number indicating the degree of non-Newtonian behavior.
  • n < 1: Pseudoplastic (shear-thinning) fluid. Its viscosity decreases as the shear rate increases (e.g., paint, ketchup).
  • n = 1: Newtonian fluid. Its viscosity is constant regardless of the shear rate (e.g., water, simple oils).
  • n > 1: Dilatant (shear-thickening) fluid. Its viscosity increases as the shear rate increases (e.g., cornstarch-water mixture).
• Consistency Index (K): A measure of the fluid’s “thickness” or consistency. It is analogous to viscosity for a Newtonian fluid. The units are typically Pa·sⁿ, and they depend on the value of ‘n’. A higher K value indicates a more viscous fluid at a shear rate of 1 s⁻¹.
• Coefficient of Determination (R²): A statistical measure from 0 to 1 that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). In this context, it shows how well the Power Law model fits your data. An R² value above 0.95 is generally considered a good fit.

Formula / Method

The calculator is based on the Ostwald-de Waele relationship, commonly known as the Power Law model:

τ = K * (γ̇)ⁿ

Where:

• τ = Shear Stress
• K = Consistency Index
• γ̇ = Shear Rate
• n = Flow Behavior Index

To solve for n and K, the equation is linearized by taking the natural logarithm (or any base logarithm) of both sides:

log(τ) = log(K) + n * log(γ̇)

This equation is in the form of a straight line, y = c + mx, where:

• y = log(τ)
• c = log(K) (the y-intercept)
• m = n (the slope)
• x = log(γ̇)

The calculator performs a linear regression (method of least squares) on the log-transformed data points to find the best-fit line. The slope of this line gives n, and the antilog of the y-intercept gives K.

Step-by-Step Example

Let’s analyze a shear-thinning fluid with the following data (Stress in Pa, Rate in s⁻¹):

1. Data Transformation: The natural logarithm of each shear rate and shear stress value is calculated.
2. Linear Regression: A linear regression is performed on the log-log data pairs (e.g., (2.303, 1.609)).
3. Determine Slope (n): The regression analysis yields a slope for the best-fit line. For this data, the slope is approximately n = 0.70.
4. Determine Intercept (log K): The regression also provides the y-intercept, which is log(K). For this data, the intercept is approximately log(K) = 0.005.
5. Calculate K: The Consistency Index K is found by taking the antilog (eⁿ) of the intercept. K = e^0.005 ≈ 1.005 Pa·sⁿ.
6. Conclusion: Since n (0.70) is less than 1, the fluid is classified as Pseudoplastic (shear-thinning).

Tips + Common Errors

Frequently Asked Questions

References

1. Mezger, T. G. (2020). The Rheology Handbook: For Users of Rotational and Oscillatory Rheometers (5th ed.). Vincentz Network.
2. Barnes, H. A., Hutton, J. F., & Walters, K. (1989). An Introduction to Rheology. Elsevier.
3. Schramm, G. (2000). A Practical Approach to Rheology and Rheometry. Gebrueder HAAKE GmbH.
4. ASTM D2196-20, “Standard Test Methods for Rheological Properties of Non-Newtonian Materials by Rotational (Brookfield type) Viscometer,” ASTM International, West Conshohocken, PA, 2020, www.astm.org.