Reference ID: MET-324F | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Constant mixing time scale-up is a critical methodology in Process Engineering used to ensure that chemical reactions or physical blending processes maintain consistent performance when transitioning from pilot-scale to production-scale vessels. By maintaining a constant blend time (tmix), engineers ensure that the time-dependent concentration gradients remain uniform across different scales.
This approach is typically employed in the design of stirred tank reactors where reaction kinetics are sensitive to mixing rates. However, it is constrained by the Power Law, which dictates that maintaining constant mixing time often leads to exponential increases in power consumption, frequently hitting the mechanical limits of industrial motors.
Methodology & Formulas
The calculation follows a structured approach based on geometric similarity and fluid dynamics. The following formulas are derived from the provided logic:
1. Impeller Speed Requirement: To maintain a constant mixing time, the impeller speed is determined by the mixing constant K:
\[ N = \frac{K}{t_{mix}} \]
2. Power Scaling: The power requirement for the production vessel is calculated using the geometric scale ratio, assuming constant impeller speed N:
3. Reynolds Number (Flow Regime Check): To validate the use of constant power numbers, the flow regime must be confirmed as turbulent:
\[ Re = \frac{\rho \cdot N \cdot D^2}{\mu} \]
Regime
Condition
Implication
Turbulent
Re ≥ 10000
Correlation valid; Np is constant.
Transition/Laminar
Re < 10000
Correlation invalid; scale-up model requires correction.
Geometry
Dsmall or Dlarge ≤ 0
Invalid geometry; calculation cannot proceed.
Time
tmix ≤ 0
Invalid mixing time; must be positive.
Maintaining a constant mixing time ensures that the macro-scale blending performance remains consistent across different vessel sizes. This approach is particularly critical for:
Processes where reaction kinetics are limited by mass transfer rates.
Systems requiring uniform distribution of reagents to prevent localized concentration gradients.
Applications where the time-dependent history of the fluid elements significantly impacts product quality.
While effective for blending, this strategy can lead to challenges in other physical parameters. Engineers should be aware of the following:
Power consumption per unit volume often increases significantly as the scale increases.
Local shear rates may become excessively high, potentially damaging shear-sensitive biological cells or delicate crystals.
Heat transfer surface area to volume ratios decrease, which may necessitate external cooling loops to maintain isothermal conditions.
To maintain a constant mixing time, the impeller speed \( N \) must be kept constant across scales. It is calculated directly from the mixing time correlation for the specific impeller geometry. The steps are:
Determine the mixing time constant \( K \) for your impeller geometry from pilot-scale data or correlations, where \( t_{mix} \cdot N = K \).
Since the mixing time \( t_{mix} \) is constant for scale-up, solve for \( N \) using \( N = K / t_{mix} \). This \( N \) will be the same for both pilot and production scales.
The power requirement for the production scale then scales with the fifth power of the impeller diameter ratio, as \( P \propto D^5 \) when \( N \) is constant.
Worked Example
A process engineer must scale up a batch mixing operation for an aqueous solution from a pilot-scale vessel to a production vessel. The key product quality constraint is that the blending time must remain unchanged to ensure consistent homogeneity. The objective is to determine the power requirement for the larger agitator and verify the flow regime remains turbulent.
Knowns (Input Parameters):
Pilot-scale impeller diameter, Dsmall: 0.5 m
Production-scale impeller diameter, Dlarge: 1.0 m
Target mixing time, tmix: 30.0 s
Pilot-scale power draw, Psmall: 500.0 W
Fluid density (water), ρ: 1000.0 kg/m³
Fluid dynamic viscosity (water at 20°C), μ: 0.001 Pa·s
Mixing time constant for the geometry, K: 5.0 (dimensionless)
Power number for the impeller, Np: 5.0 (dimensionless)
Calculation Steps:
The constant mixing time scale-up requires a constant impeller speed. Using the correlation \( t_{mix} \cdot N = K \), the required speed N is calculated as \( N = K / t_{mix} = 5.0 / 30.0 = 0.1667 \text{ rev/s} \) (rounded to four decimal places).
Check that the flow in the large vessel is fully turbulent to validate the use of a constant power number. The Reynolds number is calculated as \( Re = (\rho \cdot N \cdot D_{large}^2) / \mu \). Using \( N = 0.1667 \text{ rev/s} \), \( \rho = 1000.0 \text{ kg/m}^3 \), \( D_{large} = 1.0 \text{ m} \), and \( \mu = 0.001 \text{ Pa·s} \), we get \( Re = 166666.7 \), which is significantly greater than 10,000, confirming turbulent flow.
With geometric similarity (scale factor = 2.0) and constant impeller speed, the power scales with the fifth power of the impeller diameter ratio. The scaled power requirement is \( P_{large} = P_{small} \cdot (D_{large} / D_{small})^5 \). From the calculation, this yields \( P_{large} = 500.0 \cdot (1.0 / 0.5)^5 = 500.0 \cdot 32.0 = 16000.0 \text{ W} \).
Final Answer:
To maintain the constant mixing time of 30.0 seconds, the production-scale agitator must operate at 0.167 rev/s (approximately). This requires a motor capable of delivering 16000.0 W (16.0 kW) of power, assuming the turbulent flow regime is maintained (Re ≈ 166667).
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