This guide explains the principles behind our Mixing Scale-Up Calculator. It covers the core concepts, methodologies, and practical considerations for successfully translating a mixing process from a small-scale laboratory or pilot environment to a full-scale production vessel.

What This Calculator Does

The primary function of this tool is to predict the operating parameters and vessel geometry for a large-scale mixing system based on a proven, smaller-scale setup. It uses the principle of geometric similarity, where the large-scale vessel is a proportionally scaled version of the small one. By selecting a specific scale-up criterion, the calculator determines the required impeller speed (N₂) for the large vessel to achieve a similar process result, and from that, calculates key performance indicators like power draw, torque, and impeller tip speed.

When to Use It

This calculator is essential during the process development and engineering design phases in various industries, including:

• Chemical Manufacturing: Scaling up synthesis reactions, blending, or crystallization processes.
• Pharmaceuticals: Ensuring consistent API properties or formulation homogeneity from lab to production batches.
• Biotechnology: Scaling up bioreactors or fermenters while maintaining cell viability and product yield.
• Food & Beverage: Maintaining product texture, consistency, and stability when moving from pilot to industrial-scale production.

Use it when you have a well-characterized small-scale process and need to design or specify equipment for a larger production volume.

Inputs Explained

Scale 1: Lab/Pilot Scale (Known)

• Vessel Volume (V₁): The total volume of the liquid in your existing small-scale vessel.
• Tank Diameter (T₁): The internal diameter of your small-scale tank.
• Liquid Height (Z₁): The height of the liquid in the tank. The ratio Z₁/T₁ helps define the system's geometry.
• Impeller Type: The type of mixing impeller used. Each type has a unique Power Number (Np), which is a dimensionless constant used in power calculations for turbulent flow.
• Impeller Diameter (D₁): The diameter of your small-scale impeller. The ratio D₁/T₁ is a critical geometric parameter.
• Impeller Speed (N₁): The rotational speed (in RPM) of the impeller that provides the desired mixing result in the small-scale system.
• Liquid Density (ρ): The mass per unit volume of the fluid being mixed.
• Liquid Viscosity (μ): A measure of the fluid's resistance to flow. This is critical for determining the flow regime (laminar, transitional, or turbulent) via the Reynolds Number.

Scale 2: Production Scale (Target)

• Primary Target: You can define your scale-up target by either the desired production volume (V₂) or the diameter of the available production tank (T₂). The calculator will determine the other dimension based on geometric similarity.

Scale-Up Criteria

This is the core principle you choose to maintain between the two scales. Each has a different impact on the final design:

• Constant Power per Unit Volume (P/V): Aims to provide the same level of power intensity to the fluid. Often used for processes where mass transfer (like gas dispersion) is critical.
• Constant Impeller Tip Speed (v): Aims to maintain the same maximum shear rate at the impeller tip. Useful for shear-sensitive products or processes where tip speed relates to droplet size in emulsions.
• Constant Blend Time (θ) / Constant Reynolds Number (Re): Aims to achieve similar fluid flow patterns and blending performance. This criterion often results in very high power requirements at large scales.
• Geometric Similarity Only (Manual N₂): Allows you to input your own target speed (N₂) and see the resulting performance metrics.

Results Explained

After calculation, the tool provides a comprehensive overview of the scaled-up system:

• Operating Parameters: These are the key outputs for the large-scale system, including the required Impeller Speed (N₂), the estimated Power Draw (P₂) of the motor, the Torque (τ₂) on the shaft, and the resulting Impeller Tip Speed (v₂).
• Calculated Geometry: Provides the required dimensions (Tank Diameter, Liquid Height, Impeller Diameter) for the large-scale vessel to maintain geometric similarity with the small-scale system.
• Criteria Comparison: A powerful table that shows what the results would be for each of the main scale-up criteria. This allows you to quickly compare the trade-offs; for example, you can see how much more power is required to maintain a constant Reynolds number versus a constant P/V.
• Flow Regime Note: The calculator determines the Reynolds Number for both scales. If the flow regime changes (e.g., from turbulent to transitional), a warning is displayed, as the underlying assumptions (like a constant Np) may become less accurate.

Formula / Method

The calculator is built on fundamental chemical engineering principles for agitated vessels. Assuming geometric similarity and a turbulent flow regime (Re > 10,000), where the Power Number (Np) is constant:

Core Equations

• Reynolds Number (Re): Re = (D² * N * ρ) / μ
  This dimensionless number characterizes the flow regime. N is speed in rev/sec.
• Power (P): P = Np * ρ * N³ * D⁵
  This equation calculates the power consumed by the impeller.
• Impeller Tip Speed (v): v = π * N * D
  This is the linear speed at the outermost edge of the impeller.

Scale-Up Relationships

Based on these equations, we can derive the relationships used to calculate the target speed (N₂) from the known speed (N₁) and the ratio of impeller diameters (D₁/D₂):

• For Constant P/V: N₂ = N₁ * (D₁/D₂)⁵/³
• For Constant Tip Speed: N₂ = N₁ * (D₁/D₂)
• For Constant Reynolds Number: N₂ = N₁ * (D₁/D₂)²

Step-by-Step Example

Let's scale up a pilot process from a 10-liter vessel to a 10,000-liter (10 m³) production vessel, using the Constant Power per Unit Volume (P/V) criterion.

1. Enter Scale 1 Data:
   • Vessel Volume (V₁): 0.01 m³
   • Tank Diameter (T₁): 0.2 m
   • Impeller Diameter (D₁): 0.07 m
   • Impeller Speed (N₁): 300 rpm
   • Liquid Properties: Density 1000 kg/m³, Viscosity 0.001 Pa·s
2. Enter Scale 2 Target:
   • Target Vessel Volume (V₂): 10 m³
3. Select Criterion:
   • Choose "Constant Power per Unit Volume (P/V)".
4. Analysis of Results:
   • The volume ratio is 10 / 0.01 = 1000. The length ratio is the cube root of this, which is 10.
   • The calculator first determines the new geometry: T₂ = 0.2 * 10 = 2.0 m and D₂ = 0.07 * 10 = 0.7 m.
   • It then calculates the required speed N₂ using the P/V formula. N₂ will be significantly lower than N₁.
   • Finally, it computes the power, torque, and other metrics for the production scale based on the calculated N₂ and D₂.

Tips + Common Errors

• Check Your Units: Always ensure you are using a consistent set of units (SI or Imperial) for all inputs. The tool handles conversions, but consistency at the input stage prevents errors.
• Physical Properties Matter: The accuracy of density (ρ) and especially viscosity (μ) is crucial. Viscosity can change with temperature, so use the value at your process temperature.
• Turbulent Flow Assumption: The constant Np values used are for fully turbulent flow (Re > 10,000). If your Reynolds number is in the laminar or transitional range, the results are less accurate. The calculator will warn you if the regime changes upon scale-up.
• Impeller Selection: The impeller type significantly affects power draw and flow pattern. Ensure the selected impeller in the tool matches what you are using in practice.
• P/V is Not a Silver Bullet: While common, constant P/V can lead to excessively high shear rates (tip speed) in very large tanks. Always check the other metrics in the comparison table to ensure they are within acceptable limits for your process.

Frequently Asked Questions (FAQs)

1. Which scale-up criterion is the best?

There is no single "best" criterion. The choice depends on the process goal. Use Constant P/V for gas dispersion or mass transfer limited processes. Use Constant Tip Speed for shear-sensitive applications or emulsifications. Use Constant Re/Blend Time for simple liquid blending, but be aware of the high power cost.

2. Why did my required power increase so dramatically?

Power scales with impeller speed cubed (N³) and diameter to the fifth power (D⁵). Even with a lower speed in the larger tank, the massive increase in impeller diameter leads to a significant overall power increase. This is a fundamental reality of mixing scale-up.

3. The calculator showed a "Flow regime has changed" warning. What should I do?

This means your process has moved from turbulent to transitional/laminar, or vice versa. The Power Number (Np) is no longer constant, and the calculator's power prediction will be less accurate. This often happens with viscous fluids. You may need more advanced modeling or lab experiments at the new Reynolds number.

4. Can I use this calculator for non-Newtonian fluids?

This calculator is designed for Newtonian fluids (where viscosity is constant). For non-Newtonian fluids (shear-thinning or shear-thickening), viscosity is a function of shear rate, making the calculation much more complex. The results from this tool should be considered a very rough first approximation only.

5. What does the Power Number (Np) mean?

The Power Number is a dimensionless parameter that relates the power draw of an impeller to fluid density, impeller speed, and impeller diameter. It is determined experimentally and is unique to each impeller geometry in a baffled tank under turbulent conditions.

6. Why is my calculated large-scale speed (N₂) so much lower than my lab speed (N₁)?

To maintain a parameter like tip speed or P/V, speed must decrease as the impeller diameter increases. Since v = πND, if D increases by 10x, N must decrease by 10x to keep v constant. The relationships for other criteria also lead to a decrease in speed.

7. Does this calculator account for baffles?

The Np values used are standard literature values for baffled tanks, which is the most common industrial configuration for achieving good mixing and preventing vortexing. If you have an unbaffled tank, the actual power draw will be lower, and these calculations will not be accurate.

8. How do I choose between targeting a final volume vs. a final tank diameter?

If you need to process a specific batch size, target the Vessel Volume (V₂). If you are trying to fit a process into an existing tank with a known diameter, target the Tank Diameter (T₂). The calculator ensures geometric similarity in both cases.

References

The methodologies used in this calculator are based on established principles in chemical engineering. For further reading, consult these authoritative sources:

• Paul, E. L., Atiemo-Obeng, V. A., & Kresta, S. M. (Eds.). (2004). Handbook of Industrial Mixing: Science and Practice. Wiley-Interscience. Link
• Green, D. W., & Southard, M. Z. (Eds.). (2019). Perry's Chemical Engineers' Handbook (9th ed.). McGraw-Hill Education. (Section 21: Solid-Solid Operations and Equipment)
• Treybal, R. E. (1980). Mass-Transfer Operations (3rd ed.). McGraw-Hill. (Chapter 6: Interphase Mass Transfer)
• Geankoplis, C. J. (2003). Transport Processes and Separation Process Principles (4th ed.). Prentice Hall. (Chapter 3: Principles of Momentum Transfer and Overall Balances)

Disclaimer

This Mixing Scale-Up Calculator is intended for educational and estimation purposes only. The results are based on theoretical models and standard assumptions that may not apply to all specific process conditions. The calculations should not be used for final engineering design, equipment purchasing, or safety-critical applications without verification by a qualified professional engineer and, where appropriate, experimental validation. The user assumes all risk and liability for the use of this tool.