Reference ID: MET-CC9F | Process
Engineering Reference Sheets Calculation Guide
Introduction & Context
Geometric similarity is a fundamental principle in Process
Engineering used to ensure that the flow patterns and mixing
characteristics observed at a pilot scale are preserved when
scaling up to production volumes. By maintaining constant ratios
between all linear dimensions—such as impeller diameter, tank
diameter, and liquid height—engineers can reliably predict the
performance of larger vessels. This methodology is critical in
chemical and biochemical manufacturing, where consistent mixing
is required to maintain reaction kinetics, heat transfer, and
mass transfer rates across different scales.
Geometric similarity maintains dimensional ratios but does
NOT guarantee equivalent mixing performance. Additional
criteria (constant power/volume, tip speed, or Reynolds
number) must be selected based on process requirements.
Methodology & Formulas
The scale-up process relies on the linear scale factor derived
from the volumetric ratio of the two vessels. Once the scale
factor is established, it is applied to all geometric dimensions
to maintain similarity. The power requirements are then
calculated based on the assumption of a fully turbulent regime,
where the Power Number remains constant.
The following algebraic formulas are used to determine the
scaled parameters:
1. Geometric Scale Factor:
\[ S = \left( \frac{V_2}{V_1} \right)^{1/3} \]
2. New Impeller Diameter:
\[ D_2 = D_1 \cdot S \]
3. Reynolds Number (Flow Regime Verification):
\[ Re = \frac{\rho \cdot N \cdot D^2}{\mu} \]
4. Power Requirement:
\[ P = N_p \cdot \rho \cdot N^3 \cdot D^5 \]
Parameter
Condition / Regime
Criteria
Turbulent Regime
Validity of Constant Np
\( Re \geq 10,000 \)
Volume Input
Physical Constraint
\( V_1, V_2 > 0 \)
Viscosity Input
Mathematical Stability
\( \mu > 0 \)
Maintaining geometric similarity ensures that the flow patterns
and turbulence characteristics remain consistent across different
vessel sizes. If the ratios of the impeller diameter to the tank
diameter and the liquid height to the tank diameter are not
preserved, the following issues may occur:
Unpredictable changes in power consumption per unit volume.
Inconsistent shear rate distribution throughout the fluid.
Variations in blending time that deviate from pilot-scale
performance.
To achieve true geometric similarity, process engineers must
maintain constant ratios for all critical dimensions relative to
the tank diameter (T). Key ratios include:
Impeller diameter to tank diameter (D/T).
Impeller off-bottom clearance to tank diameter (C/T).
Liquid depth to tank diameter (H/T).
Baffle width to tank diameter (W/T).
While it is possible to achieve specific process objectives
without strict geometric similarity, it is generally discouraged
for complex mixing tasks. Deviating from geometric similarity
requires:
Extensive empirical testing to correlate performance.
Adjustments to impeller rotational speed to compensate for
geometry changes.
A higher risk of localized dead zones or stagnant regions in
the vessel.
Worked Example: Geometric Similarity for Mixing Scale-Up
A process engineer must scale up a laboratory mixing tank to
production size while maintaining identical flow patterns. The
pilot tank has a volume of 100 liters and uses a standard turbine
impeller. The target production volume is 10,000 liters, mixing
water at a constant rotational speed.
Knowns (Input Parameters):
Pilot tank volume, \( V_1 = 100.0 \, \text{L} \)
Production tank volume, \( V_2 = 10000.0 \, \text{L} \)
Pilot impeller diameter, \( D_1 = 0.2 \, \text{m} \)
Fluid density (water), \( \rho = 1000.0 \, \text{kg/m}^3 \)
Impeller rotational speed, \( N = 2.0 \, \text{rps} \)
Power number for turbulent regime, \( N_p = 5.0 \)
Step-by-Step Calculation:
Calculate the geometric scale factor, \( S \), using the cube
root of the volume ratio: \( S = (V_2 / V_1)^{1/3} \). From the
calculation, \( S = 4.642 \).
Determine the new impeller diameter for geometric similarity:
\( D_2 = D_1 \times S \). Using \( S = 4.642 \), \( D_2 = 0.2
\times 4.642 = 0.9284 \, \text{m} \), which rounds to \( 0.928
\, \text{m} \).
Verify the mixing regime is fully turbulent by calculating the
Reynolds number for both scales. The formula is \( Re = (\rho
\cdot N \cdot D^2) / \mu \).
For the pilot scale: \( Re_1 = 80000.0 \).
For the scaled system: \( Re_2 = \frac{1000 \times 2
\times (0.9284)^2}{0.001} = 1723853.12 \).
Both values exceed 10,000, confirming turbulent flow where the
power number \( N_p \) is constant.
Calculate the power requirement for the pilot scale using \( P
= N_p \cdot \rho \cdot N^3 \cdot D^5 \). From the calculation,
\( P_1 = 12.8 \, \text{W} \).
Calculate the power requirement for the scaled system using
the same formula: \( P_2 = 5 \times 1000 \times 2^3 \times
(0.9284)^5 = 27588.9248 \, \text{W} \).
Final Answer: To maintain geometric similarity and
turbulent mixing conditions, the production-scale tank requires an
impeller diameter of 0.928 m and a power input of
27589 W (approximately 27.6 kW).
Warning : Power per volume increased from 0.128 W/L to 2.76 W/L
(21.6×). This may significantly alter mixing performance despite
geometric similarity.
"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
