🧪 Non-Newtonian Yield-Pseudoplastic Hydraulics (Herschel-Bulkley)

Calculate non-Newtonian yield-pseudoplastic pipe flow using Herschel-Bulkley 3-parameter model: pipe pressure drop (kPa), wall shear stress (tau_w), solid unyielded plug core radius (rp), and generalized Reynolds number.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

Non-Newtonian Yield-Pseudoplastic Hydraulics (Herschel-Bulkley)

Fluid Mechanics

📊 Solver Telemetry ● ACTIVE

👁️ Views 31

⚡ Solves 28

💾 Downloads 377 📦 Fortran Code 4.5 KB

📅 Released Jun 2026

⏱️ Latency < 1 ms

🧪 Yield-Pseudoplastic Plug Flow Velocity Profile

Real-time visual simulation: Unyielded solid plug core ($r \le r_p$) surrounded by sheared boundary layer

📝 Configuration & Presets

🛢️ Bentonite Mud (τy = 12 Pa) ⛏️ Mining Tailings Slurry 🍅 Concentrated Tomato Paste 🧴 Carbopol Hydrogel

🔬 Herschel-Bulkley Rheology Parameters

Yield Stress τ_y [Pa]:

Consistency Index K [Pa·sⁿ]:

Flow Behavior Index n: <1: Shear-thinning, =1: Bingham, >1: Dilatant

Fluid Density ρ [kg/m³]:

📏 Pipe Dimensions & Target Flow

Pipe Inner Diameter D [mm]:

Pipe Length L [m]:

Volumetric Flow Rate Q [m³/h]:

Herschel-Bulkley Constitutive Equation:
• Rheology Law: τ = τ_y + K · (γ̇)ⁿ (for τ > τ_y)
• Wall Shear Stress: τ_w = ΔP · D / (4 L) [Pa]
• Unyielded Plug Core Radius: r_p = R · (τ_y / τ_w) [mm]
• Generalized Reynolds: Re_gen = ρ · V²⁻ⁿ · Dⁿ / [ K · ((3n+1)/4n)ⁿ · 8ⁿ⁻¹ ].

📊 Hydraulics Results

📊 Output Summary
💾 Fortran Source

Total Pipe Frictional Pressure Drop

383.74 kPa (3.837 bar)

Wall Shear τ_w: 119.92 Pa | Plug Radius r_p: 9.4 mm

LAMINAR (UNYIELDED PLUG CORE)

Solid Plug Core Radius (r_p) 9.4 mm Pipe Radius R = 25.0 mm

Unyielded Plug Area Fraction 14.1 % Central rigid moving core

Wall Shear Stress (τ_w) 119.9 Pa Yield Ratio τ_y/τ_w = 0.375

Generalized Reynolds (Re_gen) 94 Transition at ~2100

📈 Pipe Pressure Drop ΔP (kPa) vs Flow Rate Q (m³/h)

📉 Plug Radius rp (mm) vs Yield Stress τy (Pa)

=================================================================
 THERMOFLUIDCALC — HERSCHEL-BULKLEY HYDRAULICS REPORT
=================================================================
Case Title                 : Food Processing Concentrated Tomato Paste
Rheology Law Parameters    : tau_y = 45.00 Pa, K = 8.200 Pa.s^n, n = 0.400 (Density = 1080.0 kg/m3)
Piping Dimensions          : ID = 50.0 mm, Length = 40.0 m, Target Flow Q = 6.00 m3/h
-----------------------------------------------------------------
TOTAL PIPE PRESSURE DROP   : 383.74 kPa (3.8374 bar)
Wall Shear Stress (tau_w)  : 119.92 Pa
Unyielded Plug Radius (rp) : 9.38 mm (14.1 % of pipe cross-section)
Generalized Reynolds Number: 93.7
PREDICTED FLOW REGIME      : LAMINAR (UNYIELDED PLUG CORE)
=================================================================

📘 Calculation Methodology & Herschel-Bulkley Rheology

Unified 3-Parameter Rheology

Herschel-Bulkley is the most versatile non-Newtonian model. It reduces to Newtonian fluid ($\tau_y=0, n=1$), Power-Law fluid ($\tau_y=0$), and Bingham plastic ($n=1$).

Plug Flow Phenomenon

Near the pipe centerline, shear stress $\tau(r) \le \tau_y$, causing the fluid to move as a rigid unsheared solid cylinder (plug) at uniform maximum velocity.

Key Engineering Assumptions

Fully developed, steady isothermal laminar/transitional pipe flow.
Darby / Metzner-Reed generalized Reynolds formulations.
Applicable to drilling fluids, mining tailings, polymer pastes, and food purees.

ThermoFluidCalc Engineering Report