Introduction & Context

Motor sizing for mixing applications is a critical task in process engineering, ensuring that the mechanical drive system can overcome fluid resistance to achieve desired mass transfer, heat transfer, or homogenization. This calculation determines the required power rating for an electric motor driving a stirred tank impeller. It is essential for preventing motor burnout, ensuring operational longevity, and optimizing energy consumption in industrial reactors, fermenters, and blending vessels.

Methodology & Formulas

The sizing process relies on fluid dynamics principles to relate the impeller geometry and rotational speed to the power required to overcome fluid drag. The calculation follows these sequential steps:

Fluid and Kinematic Conversion: First, convert rotational speed from revolutions per minute to revolutions per second:

\[ N = \frac{N_{\text{rpm}}}{60} \]

Convert dynamic viscosity from centipoise to Pascal-seconds:

\[ \mu = \mu_{\text{cP}} \cdot 0.001 \]

Flow Regime Determination: Calculate the Reynolds number to characterize the flow regime, which dictates the selection of the power number:

\[ Re = \frac{\rho \cdot N \cdot d^2}{\mu} \]

Impeller Power Calculation: Determine the power absorbed by the impeller using the power number, which is a function of the impeller geometry and the flow regime:

\[ P_{\text{impeller}} = N_p \cdot \rho \cdot N^3 \cdot d^5 \]

Motor Power Sizing: Account for mechanical losses in the drive train (gearbox/coupling) and the motor efficiency, while applying a service factor to accommodate process fluctuations or startup torque requirements:

\[ P_{\text{motor}} = \frac{P_{\text{impeller}}}{\eta_{\text{drive}} \cdot \eta_{\text{motor}}} \cdot SF \]

Parameter	Regime / Condition	Criteria
Flow Regime	Turbulent	\( Re > 10^4 \)
Flow Regime	Laminar	\( Re < 10 \)
Service Factor	Standard Duty	\( 1.0 \leq SF \leq 2.0 \)
Efficiency	Typical Mechanical	\( 0.8 \leq \eta \leq 0.96 \)

To calculate the required torque for high-viscosity fluids, you must account for the impeller geometry and the fluid rheology. Follow these steps:

Determine the Reynolds number to identify the flow regime.
Calculate the Power Number (\(N_p\)) based on the specific impeller type.
Apply the torque formula: \(T = \frac{P \cdot 60}{2 \pi \cdot N_{\text{rpm}}}\), where \(P\) is power in watts and \(N_{\text{rpm}}\) is rotational speed in RPM.
Include a safety factor of at least 1.25 to account for potential viscosity spikes during batch processing.

Impeller speed directly dictates the gear reduction requirements and the resulting output torque. Higher speeds generally allow for smaller motor frames, but you must consider:

The mechanical limits of the shaft and seal assembly.
The potential for vortex formation at high RPMs.
The torque multiplication provided by the gearbox, which may necessitate a larger frame to handle the increased load.

A VFD is recommended for mixing applications when you need to manage process flexibility or startup conditions. You should specify a VFD if:

The process requires varying shear rates for different product grades.
You need to perform a soft start to prevent mechanical stress on the impeller shaft.
The fluid viscosity changes significantly during the reaction cycle, requiring speed adjustments to maintain constant power draw.

Worked Example: Motor Sizing for a Chemical Mixing Tank

A process engineer must select an electric motor for a stirred tank mixer used to blend a water-like Newtonian fluid in a pilot plant. The tank is equipped with a standard Rushton turbine impeller. The goal is to determine the minimum motor power rating required for steady-state operation.

Knowns (Input Parameters):

Fluid density, \(\rho = 1000.0 \, \text{kg/m}^3\)
Fluid viscosity, \(\mu = 1.0 \, \text{cP} = 0.001 \, \text{Pa·s}\)
Impeller diameter, \(d = 0.5 \, \text{m}\)
Impeller rotational speed, \(N_{\text{rpm}} = 100.0 \, \text{rpm}\)
Power number for Rushton turbine in turbulent flow, \(N_p = 5.0\) (dimensionless)
Mechanical drive efficiency, \(\eta_{\text{drive}} = 0.95\)
Motor efficiency, \(\eta_{\text{motor}} = 0.92\)
Service factor for continuous duty, \(SF = 1.2\) (dimensionless)

Step-by-Step Calculation:

Convert rotational speed to SI units (s⁻¹). The formula is \(N = \frac{N_{\text{rpm}}}{60}\). From the numerical results, \(N = 1.6667 \, \text{s}^{-1}\).
Calculate the Reynolds number to confirm the flow regime. The formula is \(Re = \frac{\rho \cdot N \cdot d^2}{\mu}\). Using the known values, the numerical result gives \(Re = 416,667\). Since \(Re > 10,000\), the flow is turbulent, validating the use of a constant \(N_p = 5.0\).
Compute the power absorbed by the impeller. The formula is \(P_{\text{impeller}} = N_p \cdot \rho \cdot N^3 \cdot d^5\). From the numerical results, \(P_{\text{impeller}} = 723.4 \, \text{W}\).
Determine the required motor power accounting for drive losses and a safety margin. The formula is \(P_{\text{motor}} = \frac{P_{\text{impeller}}}{\eta_{\text{drive}} \cdot \eta_{\text{motor}}} \cdot SF\). Using the efficiencies and service factor, the numerical result yields \(P_{\text{motor}} = 993.2 \, \text{W}\).

Final Answer:

The motor must have a rated power of at least 993.2 W (approximately 1 kW) to safely drive the impeller under the specified conditions.

"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