Introduction & Context

In aerobic fermenters and bioreactors, mechanical agitation is used to disperse sparged gas into fine bubbles, increasing interfacial area and mass-transfer rates. The power drawn by the impeller under aerated conditions, P_g, is almost always lower than the un-gassed power, P₀, because the gas cavity behind the blades reduces the hydrodynamic torque. Accurate prediction of P_g is essential for:

Correct motor sizing and energy audits
Heat-removal calculations (power ≈ heat load)
Mass-transfer correlations that use P_g/V_L as the energy dissipation term
Scale-up/scale-down studies where constant P_g/V_L is often the design criterion

The calculation is routinely performed for Rushton turbines in standard baffled vessels operating at turbulent Reynolds numbers.

Methodology & Formulas

Dimensionless groups
Aeration number (gas flow characteristic):
\[ N_a = \frac{Q_g}{N d^3} \]
Reynolds number (flow regime):
\[ Re = \frac{\rho N d^2}{\mu} \]
Froude number (gravity effect):
\[ Fr = \frac{N^2 d}{g} \]
Power drawn without gas
\[ P_0 = P_o \rho N^3 d^5 \] where P_o is the impeller power number for the un-gassed system.

Regime-dependent power reduction
The ratio P_g/P₀ is obtained from the aeration number as follows:

Range of N_a	Correlation
\( N_a \le 0.02 \)	\( P_g/P_0 = 1 \)
\( 0.02 < N_a \le 0.08 \)	\( P_g/P_0 = 1 - 2 N_a \)
\( 0.08 < N_a \le 0.12 \)	\( P_g/P_0 = 0.4 - 3.75 (N_a - 0.08) \)
\( N_a > 0.12 \)	Impeller flooded; correlation invalid

Gassed power
Once P_g/P₀ is known, the gassed power is simply
\[ P_g = \left(\frac{P_g}{P_0}\right) P_0 \]
Validity limits
The empirical correlation is valid only for turbulent conditions; the Reynolds number should exceed 20 000. Below this value, the flow is insufficiently turbulent and the above relations do not apply.

The gassed power number (N_P,g) is a dimensionless constant that relates impeller power draw to gas dispersion in an agitated vessel. It is always lower than the ungassed power number (N_P) because the gas cavity that forms behind each blade reduces the effective pressure difference across the impeller, lowering torque. Typical reductions range 30–70% depending on impeller type, flow regime, and aeration rate.

Measure shaft torque (τ) and rotational speed (N) under gassed conditions.
Calculate gassed power: P_g = 2πNτ.
Obtain liquid density (ρ) and impeller diameter (D).
Compute Reynolds number to confirm turbulent regime (Re > 10,000).
N_P,g = P_g / (ρN³D⁵).

For Rushton turbines: N_P,g/N_P = 1 – 1.2(Q_g/ND³) for Q_g/ND³ < 0.03.
For 6-blade disk turbines in water, Michel & Miller (1962): P_g/P = 0.1(Q_g/N)^0.25.
For modern hollow-blade or concave impellers, use vendor-supplied curves; cavity size is smaller so N_P,g stays within 80% of N_P.
Always validate with small-scale experiments if scale-up accuracy is critical.

Upper impellers operate in lower gas hold-up regions, so their individual N_P,g is closer to N_P.
Lower impeller carries the bulk of the gas load; its N_P,g dominates the total power.
Overall gassed power for multiple impellers is approximately the sum of individual gassed powers, corrected for interaction factors (0.85–0.95) depending on spacing.
Spacing of at least one impeller diameter is recommended to minimize interaction and maintain predictable N_P,g.

Worked Example – Estimating Power Draw of a Gassed Agitator in a Fermenter

A biochemical engineer needs to verify that the 0.4 m diameter Rushton turbine installed in a 1 m³ pilot fermenter will still draw less than 4 kW after aeration is started. The broth is a water-like medium at 30 °C. Use the gassed power number approach to predict the gassed power and the percentage power reduction.

Knowns

Gravitational acceleration, \(g\) = 9.81 m s⁻²
Liquid density, \(\rho\) = 1050 kg m⁻³
Liquid viscosity, \(\mu\) = 0.003 Pa·s
Impeller diameter, \(d\) = 0.4 m
Rotational speed, \(N\) = 4.0 s⁻¹ (240 rpm)
Ungassed power number, \(P_o\) = 5.5 (dimensionless)
Gas volumetric flow-rate, \(Q_g\) = 0.01 m³ s⁻¹ (0.6 m³ min⁻¹)

Step-by-step calculation

Calculate the impeller Reynolds number to confirm turbulent regime: \[ Re = \frac{\rho N d^{2}}{\mu} = \frac{1050 \times 4.0 \times 0.4^{2}}{0.003} = 224\,000 \] \(Re \gg 10\,000\); flow is fully turbulent.
Evaluate the Froude number: \[ Fr = \frac{N^{2} d}{g} = \frac{4.0^{2} \times 0.4}{9.81} = 0.652 \]
Compute the aeration number \(N_a\) (dimensionless): \[ N_a = \frac{Q_g}{N d^{3}} = \frac{0.01}{4.0 \times 0.4^{3}} = 0.039 \]
Estimate the ratio of gassed to ungassed power with the commonly used correlation for Rushton turbines: \[ \frac{P_g}{P_0} = 1 - 12.5\, N_a = 1 - 12.5 \times 0.039 = 0.922 \]
Determine the ungassed power draw: \[ P_0 = P_o\, \rho\, N^{3}\, d^{5} = 5.5 \times 1050 \times 4.0^{3} \times 0.4^{5} = 3.785\ \text{kW} \]
Compute the gassed power: \[ P_g = \left(\frac{P_g}{P_0}\right) P_0 = 0.922 \times 3.785\ \text{kW} = 3.489\ \text{kW} \]
Find the percentage power reduction: \[ \text{Power reduction} = (1 - 0.922) \times 100\ \% = 7.8\ \% \]

Final Answer

Under the specified aeration conditions, the impeller draws 3.49 kW, representing a 7.8% drop compared with the ungassed case. The requirement to stay below 4 kW is therefore satisfied.

"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