Calculate K and N From Shear Rate and Shear Stress

This calculator determines the consistency index (k) and flow behavior index (n) from shear rate and shear stress data, which are fundamental parameters in rheology for characterizing fluid behavior.

Introduction

In rheology, the relationship between shear stress (τ) and shear rate (γ) is described by the power-law model: τ = kγⁿ. The consistency index (k) represents the fluid's resistance to flow, while the flow behavior index (n) indicates how the fluid's viscosity changes with shear rate.

This calculator uses linear regression to determine k and n from experimental shear stress and shear rate data. The results help characterize fluid behavior, predict processing conditions, and optimize industrial applications.

Methodology

Power-Law Model

The power-law model is expressed as:

Formula

τ = kγⁿ

Where:

τ = shear stress (Pa)
γ = shear rate (s⁻¹)
k = consistency index (Pa·sⁿ)
n = flow behavior index (dimensionless)

To linearize the equation for regression analysis, take the natural logarithm of both sides:

Linearized Form

ln(τ) = ln(k) + n·ln(γ)

This linear form allows calculation of k and n using linear regression on the logged data.

Worked Example

Consider the following shear stress and shear rate data points:

Shear Rate (γ, s⁻¹)	Shear Stress (τ, Pa)
10	20
20	40
30	60
40	80
50	100

Using linear regression on the logged data:

Calculate ln(γ) and ln(τ) for each data point
Perform linear regression of ln(τ) on ln(γ)
The slope of the regression line equals n
The intercept equals ln(k)

For this example, the calculated values are:

n = 1.00 (Newtonian fluid)
k = 20 Pa·s (consistency index)

Interpreting Results

Flow Behavior Index (n)

n = 1: Newtonian fluid (linear relationship)
n < 1: Shear-thinning (pseudoplastic) fluid
n > 1: Shear-thickening (dilatant) fluid

Consistency Index (k)

Represents the fluid's resistance to flow at a reference shear rate. Higher k values indicate more viscous fluids.

Note

The power-law model is valid for shear rates within the fluid's linear viscoelastic range. Extrapolation beyond this range may not be accurate.

FAQ

What is the difference between k and n?: k (consistency index) measures the fluid's resistance to flow, while n (flow behavior index) describes how viscosity changes with shear rate.
How many data points are needed for accurate results?: At least 5-10 data points spanning at least one decade of shear rates are recommended for reliable regression results.
Can this calculator handle non-linear data?: Yes, the calculator uses logarithmic transformation to linearize the power-law relationship, making it suitable for both linear and non-linear fluid behavior.
What units should I use for input data?: Shear stress should be in Pascals (Pa) and shear rate in reciprocal seconds (s⁻¹) for consistent results.