Stirred Tank CFD
Stirred Tank CFD: Theoretical Foundations
Overview
Teacher! In what situations is CFD analysis of stirred tanks used?
It's a technology to predict the flow patterns, mixing time, and power consumption of stirred tanks used in chemical plants, pharmaceutical manufacturing, food processing, water treatment, etc., using CFD. It solves the complex three-dimensional flow created by the rotation of the impeller (stirring blade) using the Navier-Stokes equations.
Governing Equations
Please teach me the basic equations for stirred tanks.
First, dimensionless numbers are important. The impeller Reynolds number and the power number are fundamental.
$N$ is the rotational speed [rps], $D$ is the impeller diameter, and $P$ is the stirring power, right?
Correct. For $Re_{imp} > 10^4$, it's considered fully turbulent; for $Re_{imp} < 10$, it's laminar. The transition region (10 to 10,000) is the most difficult to analyze.
The mixing time $\theta_m$ is defined from the tracer response.
$T$ is the tank diameter, right? $\theta_m N$ is the dimensionless mixing time, and in fully turbulent flow, it becomes a constant (depending on impeller shape).
Exactly. For a 6-blade flat blade turbine (Rushton Turbine), $\theta_m N \approx 30$ to $50$ is a typical value.
Impeller Classification
| Impeller Shape | Np (Turbulent Region) | Flow Pattern | Application |
|---|---|---|---|
| Rushton Turbine (6 flat blades) | 5.0~5.5 | Radial | Gas-liquid mixing, general reactions |
| Pitched Blade Turbine (45°) | 1.2~1.7 | Axial-Radial | Solid-liquid suspension, mixing |
| Hydrofoil (A310, A320) | 0.3~0.4 | Axial | Low shear mixing |
| Anchor | 0.4~0.8 (Laminar Flow) | Tangential | High viscosity fluids |
| Helical Ribbon | 0.5~1.0 (Laminar Flow) | Axial+Tangential | Very high viscosity |
The power number is completely different depending on the impeller shape. Rushton is over 5, and Hydrofoil is around 0.3, right?
Rushton creates a strong shear field, making it suitable for gas-liquid dispersion, but power consumption is high. Hydrofoil efficiently circulates liquid with axial flow but has low gas-liquid dispersion capability. They are used according to the application.
Practical Considerations
- When deformation of the free surface (liquid surface) is large, the VOF method is necessary.
- The flow pattern changes significantly depending on the presence of baffles.
- CFD is effective for verifying scale-up rules (constant Np/V, constant tip speed).
- For non-Newtonian fluids (power-law, Herschel-Bulkley), the distribution of apparent viscosity is important.
The Father of Agitation Engineering, Rushton—Establishment of the Rushton Turbine and the Dimensionless Power Number (1950)
The foundation of stirred tank engineering was laid by the American J. H. Rushton. In his 1950 paper "Power Characteristics of Mixing Impellers," he defined the dimensionless power number for impellers, Np = P/(ρN³D⁵), and experimentally proved that Np converges to a constant value in the high Re region (approximately 5 for disc turbine types in fully turbulent flow). This "Rushton turbine" and power number correlation became the de facto standard for agitation design for the next 70 years. In modern CFD, his experiments are used as benchmarks for validating turbulence models, and numerous validation papers have confirmed that the prediction error of the standard k-ε model for Np is around 10-15%. The value of classical experimental data for understanding the limits of CFD accuracy remains unchanged to this day.
Computational Methods for Stirred Tank CFD
Details of Numerical Methods
How do you solve the flow where the impeller rotates in a stirred tank?
There are mainly three methods to model impeller rotation in CFD.
Rotation Model Selection
| Method | Overview | Computational Cost | Accuracy |
|---|---|---|---|
| MRF (Multiple Reference Frame) | Treats the rotating region in a steady-state manner | Low (Steady) | Medium |
| Sliding Mesh (SM) | Actually rotates the mesh of the rotating region | High (Unsteady) | High |
| Overset Mesh | Rotates using overlapping meshes | High (Unsteady) | High |
How do you decide between MRF and Sliding Mesh?
MRF is a method to obtain a steady-state solution, used for predicting time-averaged flow patterns or power numbers. Sliding Mesh provides an unsteady solution and is necessary for periodic force fluctuations (torque fluctuations) due to interference between the impeller and baffles, or for tracer tracking to determine mixing time.
Practically, it's efficient to first check the general flow field with MRF, then perform a detailed evaluation with Sliding Mesh.
MRF Settings
Please teach me the steps to set up MRF in Fluent.
1. Create a cylindrical rotating zone around the impeller in the mesh.
2. Cell Zone Conditions → Set Frame Motion → Rotational Velocity for the rotating zone.
3. Connect the top and bottom surfaces of the rotating zone to the external zone using Interface.
4. Do not include baffles in the rotating zone (baffles are on the stationary side).
Guidelines for rotating zone dimensions:
- Diameter: 1.1 to 1.3 times the impeller diameter
- Height: 1.5 to 2.0 times the impeller height
- Distance between impeller and zone boundary: 5 to 15% of impeller diameter
What happens if the rotating zone boundary is too close to the impeller?
The wake generated by the impeller is unnaturally cut off at the rotating zone boundary, reducing the prediction accuracy for power number and pumping capacity. Ensure sufficient margin.
Mesh Strategy
Important points for stirred tank meshing:
| Region | Mesh Size | Remarks |
|---|---|---|
| Impeller blade surface | D/100~D/50 | Resolve pressure difference on upper/lower blade surfaces |
| Impeller blade tip | D/100 | Vortex generation point |
| Around baffles | T/100 | Vortex behind baffles |
| Near tank wall | T/50~T/20 | Wall boundary layer |
| Near liquid surface | Refine if free surface analysis | When using VOF |
What's a guideline for the total cell count?
For a standard single-stage impeller + 4 baffle stirred tank, 1 to 5 million cells is a guideline. For long-duration mixing simulations with Sliding Mesh, computation time requires several tens of impeller revolutions (hundreds to thousands of time steps).
Turbulence Model
For fully turbulent flow ($Re_{imp} > 10^4$), Realizable k-epsilon + Standard Wall Function is the standard for stirred tanks. Its high prediction accuracy for Np has been verified in many papers.
However, SST k-omega sometimes captures the vortex structure in the impeller wake better, and it may yield better results for predicting mixing time. LES is for research purposes and is used for detailed visualization of vortex structures.
Coffee Break Yomoyama Talk
MRF Method for Stirred Tank CFD—Numerical Treatment of Impeller Rotation and Its Accuracy Limits
The "MRF method (Multiple Reference Frame method)" most commonly used in stirred tank CFD solves the region around the impeller in a rotating coordinate system and the tank body in a stationary coordinate system. While it allows steady-state calculation and is fast, it cannot capture the unsteady interference between the impeller and baffles (Impeller-Baffle Interaction), reducing prediction accuracy for local flow immediately behind baffles. The more accurate "Sliding Mesh (SM) method" connects the rotating and stationary regions in real-time and performs unsteady calculation, so it is more accurate than MRF but computational cost is 5 to 10 times higher. A practical guideline for decision-making is: "For detailed flow around baffles, mixing time, gas dispersion behavior → SM method"; "For flow rate, pressure, overall flow patterns → MRF method."
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For fully turbulent flow ($Re_{imp} > 10^4$), Realizable k-epsilon + Standard Wall Function is the standard for stirred tanks. Its high prediction accuracy for Np has been verified in many papers.
However, SST k-omega sometimes captures the vortex structure in the impeller wake better, and it may yield better results for predicting mixing time. LES is for research purposes and is used for detailed visualization of vortex structures.
MRF Method for Stirred Tank CFD—Numerical Treatment of Impeller Rotation and Its Accuracy Limits
The "MRF method (Multiple Reference Frame method)" most commonly used in stirred tank CFD solves the region around the impeller in a rotating coordinate system and the tank body in a stationary coordinate system. While it allows steady-state calculation and is fast, it cannot capture the unsteady interference between the impeller and baffles (Impeller-Baffle Interaction), reducing prediction accuracy for local flow immediately behind baffles. The more accurate "Sliding Mesh (SM) method" connects the rotating and stationary regions in real-time and performs unsteady calculation, so it is more accurate than MRF but computational cost is 5 to 10 times higher. A practical guideline for decision-making is: "For detailed flow around baffles, mixing time, gas dispersion behavior → SM method"; "For flow rate, pressure, overall flow patterns → MRF method."