Calculator Inputs
Formula Used
The ideal cone-and-plate geometry produces a nearly uniform shear rate across the radius when the cone angle is small. The primary relationship is:
γ̇ = Ω / tan(α)
- γ̇ is shear rate in s−1.
- Ω is angular velocity in rad/s.
- α is cone angle in radians.
If a Newtonian viscosity μ is provided, the calculator estimates shear stress τ = μ γ̇. A common torque approximation for uniform shear stress is:
M = (2π/3) R³ τ
Real instruments include edge effects, inertia, and temperature-dependent viscosity. Use these results as engineering estimates unless you have instrument calibration data.
How to Use This Calculator
- Enter the cone angle in degrees (often 1–4°).
- Enter the cone radius and choose its unit.
- Enter rotation speed and select the speed unit.
- Optionally enter viscosity to estimate stress and torque.
- Click Calculate to view results above the form.
- Use Download CSV or Download PDF for reporting.
For consistency, keep viscosity and temperature aligned with your lab conditions, and note whether the fluid is Newtonian or shear-thinning.
Example Data Table
| Cone angle (°) | Radius (mm) | Speed (rpm) | Shear rate (s⁻¹) | Viscosity (mPa·s) | Shear stress (Pa) |
|---|---|---|---|---|---|
| 2 | 20 | 60 | 1800 | 1000 | 1800 |
| 1 | 25 | 10 | 600 | 500 | 300 |
| 4 | 15 | 120 | 1800 | 50 | 90 |
Table values are illustrative for typical lab ranges. Shear stress assumes Newtonian viscosity.
1) Why Cone-and-Plate Testing Matters
Cone-and-plate fixtures are widely used in rotational rheometry because they deliver a near-uniform shear field for small cone angles. This makes it easier to compare materials, validate process changes, and document quality targets using one consistent shear rate definition.
2) Typical Geometry and Operating Ranges
Laboratory cones often use 10–30 mm radii and 1–4° angles. Rotation speeds can span 0.1–300 rpm, depending on sample stability. In these ranges, shear rates commonly fall between about 1 s−1 and 10,000 s−1, which covers many polymer, food, and coating workflows.
3) The Shear Rate Relationship
For an ideal cone-and-plate system, shear rate is computed from angular velocity and cone angle using γ̇ = Ω / tan(α). Small angles keep the gap nearly linear with radius, which reduces shear-rate variation and improves repeatability.
4) Unit Conversions You Must Track
Instrument speed may be reported in rpm, rad/s, or deg/s. Because the formula requires rad/s, conversions are essential: 60 rpm equals 2π rad/s, while 1 deg/s equals π/180 rad/s. Radius is used for torque estimates, so consistent length units matter too.
5) Using Viscosity to Estimate Shear Stress
If your fluid is Newtonian at the test conditions, shear stress follows τ = μ γ̇. For example, water at room temperature is near 1 mPa·s, while many syrups may exceed 1 Pa·s. Higher viscosity at the same shear rate yields proportionally higher stress.
6) Torque as an Instrument Load Indicator
Torque helps predict whether a test stays within transducer limits. A common approximation is M = (2π/3) R³ τ. Because torque scales with R³, a small increase in radius can significantly raise instrument load.
7) Data Quality: Temperature and Edge Effects
Temperature control is critical because viscosity can change by several percent per degree for many fluids. Edge effects, slip, and sample evaporation can also bias results, especially at high shear rates or long runs. Record temperature, trim time, and sample handling steps in your report.
8) Practical Reporting for Production Decisions
When using cone-and-plate results for batch acceptance or formulation tuning, report cone angle, radius, speed, calculated shear rate, and temperature. If you compute stress or torque, state the viscosity basis and whether it was measured or assumed. Clear reporting supports repeatable comparisons across labs and dates.
FAQs
1) Why is the cone angle limited to small values?
Small angles help keep shear rate nearly uniform across the radius. Large angles increase non-uniformity, raise geometric errors, and can reduce the validity of the ideal cone-and-plate assumptions.
2) What shear rate range is reasonable for typical tests?
Many labs operate from about 1 s⁻¹ up to 10,000 s⁻¹, depending on sample stability, viscosity, and instrument limits. Always confirm with your rheometer’s recommended operating window.
3) Does this calculator work for non-Newtonian fluids?
It computes geometric shear rate for the fixture. Stress and torque estimates assume Newtonian viscosity. For non-Newtonian fluids, use viscosity at the target shear rate, or rely on instrument-measured torque data.
4) How do I choose viscosity units correctly?
Use Pa·s for SI reporting. Many datasheets use cP or mPa·s. Note that 1 cP equals 1 mPa·s and both equal 0.001 Pa·s.
5) Why does radius affect torque but not shear rate?
Ideal shear rate depends on angular speed and cone angle. Torque depends on the stressed area and lever arm, which grow with radius, producing a strong R³ scaling in common cone-and-plate approximations.
6) What can cause unrealistic results?
Incorrect unit selection, entering degrees where radians are expected, and using viscosity at the wrong temperature are common causes. Also watch for slip, evaporation, or sample fracture at extreme speeds.
7) Can I use this for quick process checks?
Yes, for quick engineering estimates and consistent reporting. For formal QC decisions, pair these calculations with calibrated instrument readings, controlled temperature, and documented test protocols.