Introduction & Context

The Power Number calculation is a fundamental procedure in process engineering used to determine the energy requirements for mechanical agitation in vessels. By utilizing dimensionless analysis, engineers can predict the power draw of an impeller based on fluid properties, rotational speed, and geometric configuration. This calculation is critical for scaling up laboratory processes to industrial production, ensuring that mixing intensity remains consistent across different vessel sizes while maintaining geometric similarity.

Methodology & Formulas

The calculation relies on the relationship between the Power Number (N_p) and the Reynolds number (Re) to characterize the flow regime. The following algebraic framework defines the relationship between these variables:

The Reynolds number is calculated as:

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

The power consumption is determined by the following formula:

\[ P = N_p \cdot \rho \cdot N^3 \cdot D^5 \]

Where:

P = Power (W)
N_p = Power Number (dimensionless)
ρ = Density (kg/m³)
N = Rotational speed (rev/s)
D = Impeller diameter (m)
μ = Viscosity (Pa·s)

Parameter	Condition / Threshold
Flow Regime	Re > 10,000 (Turbulent)
Viscosity Conversion	μ_Pa·s = μ_cP × 0.001
Validity Check	If Re < 10,000, the constant N_p assumption is invalid.

To determine the Power Number (N_p) for non-standard geometries, you must rely on experimental correlation or computational fluid dynamics (CFD) modeling. Follow these steps:

Perform a torque measurement test using a calibrated dynamometer on a scaled-down prototype.
Calculate the power consumption (P) using the measured torque and rotational speed.
Apply the Power Number formula: N_p = P / (ρ · N³ · D⁵), where ρ is fluid density, N is impeller speed, and D is impeller diameter.
Validate your results against established literature for similar blade profiles if available.

The Power Number is generally constant only within the fully turbulent regime (typically Reynolds numbers above 10,000). In other regimes, the relationship changes as follows:

Laminar flow: N_p is inversely proportional to the Reynolds number (N_p = K / Re).
Transitional flow: N_p varies significantly as the flow pattern shifts from viscous-dominated to inertia-dominated.
Turbulent flow: N_p reaches a plateau where it becomes independent of the Reynolds number.

Baffles are critical for preventing vortex formation and ensuring efficient power transmission from the impeller to the fluid. Without baffles, the Power Number will be significantly lower and unpredictable. Consider these impacts:

Standard baffling (four baffles at 90 degrees) ensures the impeller operates at its rated N_p.
Removing baffles leads to surface aeration and a drop in power draw, rendering standard N_p correlations invalid.
Wall-mounted baffles increase the drag force, which directly increases the measured torque and the resulting Power Number.

Worked Example: Power Calculation for an Agitated Vessel

A process engineer is designing a baffled agitated vessel to mix water at 20°C. A standard Rushton turbine impeller is specified. The engineer must verify the operating regime and calculate the mechanical power required for agitation under the given conditions.

Knowns (Input Parameters):

Fluid density, ρ = 998.0 kg/m³
Fluid viscosity, μ = 1.0 cP (equivalent to 0.001 Pa·s)
Impeller diameter, D = 0.5 m
Rotational speed, N = 2.0 rev/s
Power number for a Rushton turbine in a baffled tank, N_p = 5.0 (constant in turbulent flow)
Turbulent flow threshold, Re > 10,000

Calculation Steps:

Verify the flow regime by calculating the Reynolds number (Re).
Formula: Re = ρ N D² / μ
Using the known values: Re = 499,000.0
Since Re = 499,000.0 > 10,000, the flow is confirmed to be turbulent. Therefore, the constant N_p assumption is valid.
The Power Number is defined as N_p = P / (ρ N³ D⁵). For the turbulent regime, rearrange to solve for power (P):
P = N_p · ρ · N³ · D⁵
Calculate the power using the constant N_p and the known parameters.
Substituting the values: P = 1,247.5 W (as derived from the numerical results).

Final Answer: The required mechanical power for agitation under these conditions is 1,247.5 watts (W).

"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