Stirred Tank Shear Rate (γ) & Stress (τ)

Calculation Explanation

This calculator estimates the average shear rate (\(\gamma\)) and shear stress (\(\tau\)) in a stirred tank fermenter.

Average Shear Rate (γ):
The average shear rate is estimated using the following correlation (Eq. 7.9):
γ = k_i ⋅ B ⋅ N
Where:

• k_i: Dimensionless impeller constant (depends on geometry). Typical values are provided in the dropdown.
• N: Agitation speed (rev/time, e.g., s⁻¹).
• B: Term dependent on the flow behavior index (n):
  • If n ≠ 1:   B = ( 4n / (3n + 1) )^{n / (n - 1)}
  • If n = 1 (Newtonian):   B = e^1/4 ≈ 1.2840 (This is the limit as n → 1)

The flow behavior index \(n\) characterizes the fluid type (n=1: Newtonian, n<1: shear-thinning, n>1: shear-thickening).

Shear Stress (τ):
Shear stress is calculated based on the shear rate and fluid properties:

• For Newtonian fluids (n=1):   τ = γ ⋅ μ_L   (Eq. 7.10)
  where \(\mu_L\) is the dynamic viscosity (e.g., Pa·s).
• For non-Newtonian fluids (n≠1) (Power Law Model):   τ = K ⋅ γⁿ
  where \(K\) is the consistency index (e.g., Pa·sⁿ).

Ensure consistent units for N (e.g., s⁻¹), μ_L (e.g., Pa·s), and K (e.g., Pa·sⁿ) to obtain γ in s⁻¹ and τ in Pa.