Introduction & Context

In process engineering, the scale-up of a stirred tank batch crystallizer is a critical operation that requires maintaining consistent physical conditions to ensure product quality, crystal size distribution, and yield. The primary objective of this calculation is to transition from laboratory-scale (benchtop) to production-scale volumes while preserving the mixing environment.

The most common strategy for scale‑up in turbulent regimes is maintaining constant Power per Unit Volume (P/V). This approach ensures that the energy dissipation rate remains uniform throughout the fluid, which is essential for maintaining consistent turbulence intensity, mass transfer rates, and solid distribution uniformity to achieve uniform particle dispersion.

Methodology & Formulas

The scale-up process relies on geometric similarity, where the ratio of impeller diameter to tank diameter remains constant across scales. The following formulas define the physical relationships used to determine the required production-scale parameters.

First, the scale factor S is determined by the volumetric ratio:

\[ S = \left( \frac{V_{2}}{V_{1}} \right)^{1/3} \]

The production impeller diameter D₂ is derived from the scale factor and the laboratory impeller diameter D₁:

\[ D_{2} = D_{1} \cdot S \]

To maintain constant power per unit volume (P/V) in a turbulent regime, the impeller rotational speed N₂ is calculated as follows:

\[ N_{2} = N_{1} \cdot \left( \frac{D_{1}}{D_{2}} \right)^{2/3} \]

The power input P is calculated using the Power Number (Po), which characterizes the impeller efficiency:

\[ P = Po \cdot \rho \cdot N^{3} \cdot D^{5} \]

The hydrodynamic regime and shear intensity are validated using the Reynolds number (Re) and the impeller tip speed (v_tip):

\[ Re = \frac{\rho \cdot N \cdot D^{2}}{\mu} \] \[ v_{\text{tip}} = \pi \cdot N \cdot D \]

Parameter	Condition/Threshold	Engineering Significance
Reynolds Number (Re)	Re ≥ 10,000	Ensures fully turbulent flow regime for valid Power Number application.
Impeller Tip Speed (v_tip)	v_tip ≤ 6.0 m/s	Prevents excessive shear damage to sensitive crystalline structures.
Power per Unit Volume (P/V)	(P/V)₁ = (P/V)₂	Maintains consistent turbulence intensity and mixing efficiency.

Scaling up crystallization processes involves managing non-linear changes in fluid dynamics and heat transfer. Key challenges include:

Maintaining consistent supersaturation levels across larger volumes.
Managing the increased impact of secondary nucleation due to higher impeller tip speeds.
Ensuring uniform mixing to prevent localized zones of high supersaturation that lead to impurity entrapment.
Addressing the significant reduction in surface-area-to-volume ratio, which limits cooling capacity.

Selecting the correct criterion depends on the rate-limiting step of your specific process. Common approaches include:

Constant power input per unit volume (P/V) to maintain similar turbulence levels.
Constant impeller tip speed to control shear-sensitive crystal breakage.
Constant mixing time to ensure uniform distribution of reagents or cooling.
Constant mass transfer coefficient (kLa) if the process is diffusion-limited.

The cooling profile is critical because the cooling rate directly dictates the supersaturation generation rate. During scale-up, you must adjust the cooling profile to account for:

The decreased heat transfer area per unit volume, which necessitates a slower cooling rate to avoid uncontrolled nucleation.
The potential for temperature gradients within the vessel that can cause inconsistent crystal growth.
The need for a non-linear cooling curve to maintain a constant level of supersaturation throughout the batch.

Worked Example: Scale-Up of a Protein Crystallization Process

A laboratory stirred tank crystallizer (10 L) is scaled up to production scale (1000 L) while maintaining constant power per unit volume (P/V) to ensure equivalent mixing intensity. Geometric similarity is assumed.

Knowns:

Density: \(\rho = 1000.0 \; \text{kg/m}^3\)
Viscosity: \(\mu = 0.001 \; \text{Pa·s}\)
Power number (Rushton turbine): \(Po = 5.5\)
Lab volume: \(V_1 = 0.01 \; \text{m}^3\)
Lab impeller diameter: \(D_1 = 0.15 \; \text{m}\)
Lab impeller speed: \(N_1 = 5.0 \; \text{s}^{-1}\)
Production volume: \(V_2 = 1.0 \; \text{m}^3\)

Step-by-Step Calculation:

Determine the linear scale factor \(S\): \[ S = \left(\frac{V_2}{V_1}\right)^{1/3} = \left(\frac{1.0}{0.01}\right)^{1/3} = 4.642 \]
Calculate the production impeller diameter \(D_2\): \[ D_2 = D_1 \cdot S = 0.15 \cdot 4.642 = 0.696 \; \text{m} \]
Calculate the production impeller speed \(N_2\) for constant \(P/V\): \[ N_2 = N_1 \cdot \left(\frac{D_1}{D_2}\right)^{2/3} = 5.0 \cdot \left(\frac{0.15}{0.696}\right)^{0.667} = 1.797 \; \text{s}^{-1} \]
Compute the lab-scale power input \(P_1\): \[ P_1 = Po \cdot \rho \cdot N_1^3 \cdot D_1^5 = 5.5 \cdot 1000.0 \cdot (5.0)^3 \cdot (0.15)^5 = 52.207 \; \text{W} \]
Based on constant \(P/V\), the production-scale power input \(P_2\) scales directly with volume: \[ P_2 = P_1 \frac{V_2}{V_1} = 52.207 \times 100 = 5220.7 \; \text{W} \]
(Using \(P_2 = Po \cdot \rho \cdot N_2^3 \cdot D_2^5\) with the calculated \(N_2\) and \(D_2\) yields the same value within rounding.)
Verify constant power per unit volume: \[ \frac{P_1}{V_1} = \frac{52.207}{0.01} = 5220.7 \; \text{W/m}^3 \] \[ \frac{P_2}{V_2} = \frac{5220.7}{1.0} = 5220.7 \; \text{W/m}^3 \]
Confirmed: \(P/V\) is constant.
Check the production-scale Reynolds number \(Re_2\): \[ Re_2 = \frac{\rho \cdot N_2 \cdot D_2^2}{\mu} = \frac{1000.0 \cdot 1.797 \cdot (0.696)^2}{0.001} = 871047 \]
This value is well above 10,000, confirming turbulent flow.
Check the production-scale impeller tip speed \(v_{\text{tip},2}\): \[ v_{\text{tip},2} = \pi \cdot N_2 \cdot D_2 = \pi \cdot 1.797 \cdot 0.696 = 3.93 \; \text{m/s} \]
This tip speed lies within the typical range (2–4 m/s) for protein crystallization, avoiding excessive shear.

Final Answer:

The production crystallizer should operate with an impeller diameter \(D_2 = 0.696\) m, an impeller speed \(N_2 = 1.797\) s⁻¹ (approx. 108 rpm), and a power input \(P_2 = 5220.7\) W. The resulting Reynolds number \(Re_2 = 871047\) and tip speed \(v_{\text{tip},2} = 3.93\) m/s are within acceptable limits, and the constant \(P/V\) of \(5220.7 \; \text{W/m}^3\) ensures mixing intensity equivalent to the lab scale.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle