Introduction & Context

The power number calculation for an agitated vessel is a fundamental step in the design and scale‑up of mixing equipment used in chemical, pharmaceutical, and food process engineering. It quantifies the relationship between the mechanical power supplied to an impeller and the fluid properties, impeller speed, and vessel geometry. Accurate estimation of the required power ensures that the mixing intensity meets process specifications (e.g., mass transfer, suspension, heat removal) while avoiding over‑specification of motor size, which can increase capital and operating costs.

This reference sheet presents the theoretical basis for computing the Reynolds number, selecting the appropriate power number, and determining the motor power rating for a standard axial‑ or radial‑flow impeller in a cylindrical vessel.

Methodology & Formulas

Step 1 – Reynolds number (Re)

The Reynolds number characterises the flow regime around the impeller and is defined as:

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

where

\(\rho\) = fluid density (kg·m\(^{-3}\))
\(\mu\) = dynamic viscosity (Pa·s)
\(N\) = impeller rotational speed (rev·s\(^{-1}\))
\(D\) = impeller diameter (m)

Step 2 – Selection of Power Number (Po)

The power number is a dimensionless coefficient that relates impeller power to fluid inertia. For a given impeller geometry, it is typically constant within a flow regime. The regime is identified by the Reynolds number:

Flow Regime	Condition	Power Number
Laminar	\(Re < Re_{\text{crit}}\)	\(Po = C_{\text{lam}}\)
Turbulent	\(Re \ge Re_{\text{crit}}\)	\(Po = C_{\text{turb}}\)

\(Re_{\text{crit}}\) denotes the critical Reynolds number that separates laminar and turbulent regimes; \(C_{\text{lam}}\) and \(C_{\text{turb}}\) are empirical constants for the selected impeller type (e.g., \(C_{\text{turb}}\) ≈ 5 for a standard Rushton turbine).

Step 3 – Power required by the impeller (P)

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

This expression follows from dimensional analysis of the torque balance on the impeller and assumes that the vessel walls do not significantly alter the flow field.

Step 4 – Motor power rating with safety factor (P_motor)

\[ P_{\text{motor}} = S \; P \]

\(S\) is a safety factor (> 1) applied to accommodate transient loads, bearing losses, and future process variations. The resulting motor rating is selected from standard motor sizes that meet or exceed \(P_{\text{motor}}\).

By following these steps, engineers can systematically estimate the mechanical power needed for a given mixing operation and size the drive motor appropriately, ensuring reliable and efficient process performance.

The power number (Np) is a dimensionless quantity used to characterize the power consumption of an agitator. It is crucial for scaling up agitated systems because it relates the power required to the impeller speed, diameter, and fluid properties. Key importance includes:

Enabling accurate scaling of agitator performance from lab to industrial scale.
Helping optimize energy efficiency by predicting power demand.
Guiding the selection of appropriate impeller types and operating conditions.

The power number is calculated using the formula: Np = P / (ρN³D⁵), where:

P = Power input (W).
ρ = Fluid density (kg/m³).
N = Impeller rotational speed (rpm converted to rev/s).
D = Impeller diameter (m).

The calculation assumes laminar or turbulent flow conditions, and the value of Np depends on the impeller geometry and Reynolds number (Re). Empirical correlations are often used for specific impeller types.

The power number is influenced by several factors:

Impeller design: Blade geometry, number of blades, and clearance affect turbulence and shear.
Fluid properties: Viscosity and density directly impact power consumption.
Reynolds number: Determines whether flow is laminar or turbulent, altering Np behavior.
Tank configuration: Baffles, tank diameter-to-impeller diameter ratio, and liquid level influence mixing efficiency.

These variables must be accounted for when selecting or designing agitators for specific applications.

Engineers can determine the power number through:

Measuring the actual power input (P) using a torque meter or power sensor.
Calculating Np using the formula Np = P / (ρN³D⁵).
Repeating measurements across varying speeds or fluid viscosities to observe Np trends.
Comparing results to empirical correlations for the specific impeller type.

Experimental validation ensures accuracy, especially for non-standard or custom agitator designs.

Worked Example – Power Draw for a Paddle Mixer in a Fermenter

A small-scale fermenter is being designed to grow a shear-tolerant yeast. The vessel is fitted with a single 200 mm diameter flat-blade paddle turning at 2 rps. We need to know the mechanical power dissipated in the broth so that the motor can be sized with a 30% over-capacity.

Knowns

Liquid density ρ = 1050 kg m⁻³
Liquid viscosity μ = 0.001 Pa·s
Impeller speed N = 2.0 rps
Impeller diameter D = 0.200 m

Step-by-step calculation

Calculate the impeller Reynolds number \[ Re = \frac{\rho N D^{2}}{\mu} = \frac{1050 \times 2.0 \times 0.200^{2}}{0.001} = 84\,000 \]
From the correlation chart for a flat-blade paddle in the turbulent regime (Re > 10 000) read the power number \[ Po = 5.0 \]
Compute the shaft power \[ P = Po \cdot \rho N^{3} D^{5} = 5.0 \times 1050 \times 2.0^{3} \times 0.200^{5} = 13.4\ \text{kW} \]
Add 30% margin for motor sizing \[ P_{\text{motor}} = 1.30 \times 13.4 = 17.5\ \text{kW} \]

Final Answer

The fermenter requires a motor rated at 17.5 kW to guarantee the desired agitation under full-load 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