Introduction & Context

The gassed-to-ungassed power ratio ($P_g/P_0$) is a key metric in aerobic stirred-tank bioreactors: it tells the Process Engineer how much mechanical power the impeller shaft sheds when gas is sparged into the liquid. Accurately estimating this drop is vital for designing adequate drive-motor sizing, matching oxygen-transfer demands, and ensuring stable fermenter operation—particularly in high-cell-density cultures or antibiotic fermentations where aeration is continuous yet agitator power must remain above a minimum threshold for bulk mixing and bubble dispersion.

Methodology & Formulas

Step 1: Compute the dimensionless aeration number

\[Na = \frac{Q}{N D^3}\]

Step 2: Apply the empirical power-drop correlation

\[\frac{P_g}{P_0} = 1 - 1.2\,Na\qquad \text{(valid for $Na\le 0.08$)}\]

Parameter Range	Correlation Validity
$0.02 \le Na \le 0.08$	Correlation is trustworthy; cavity flooding not excessive.
$Na \lt 0.02$	Too little aeration—negligible power loss; correlation still valid but underpredicts small gassing losses.
$Na \gt 0.08$	Cavity flooding onset; correlation is NOT reliable—switch to higher-order flooding models.

Step 3: Determine fractional power loss

\[\text{fractional loss} = 1 - \frac{P_g}{P_0}\]

Step 4: Calculation of actual gassed power

With the un-gassed shaft power $P_0$ known from motor measurements or from the un-gassed power number, obtain:

\[P_g = P_0\left(\frac{P_g}{P_0}\right)\]

All fluid properties, geometries, and operating conditions must be supplied in consistent SI units prior to evaluating these formulas.

The Gassed Power Number (N_p,g) is a dimensionless group that relates the power draw of an impeller under gassed conditions to the fluid density, rotational speed, and impeller diameter. It differs from the ungassed Power Number (N_p) because gas present inside the impeller cavity reduces the torque on the shaft. For conservative mechanical design, always compare both values and use the lower calculated power draw to size gearboxes and motors.

Rushton turbine: P_g/P₀ = 1 – 0.12·(Q/N·D³)^0.75 valid up to 0.2 vvm
Smith turbine: P_g/P₀ = 1 – 0.18·(Fl_g)^0.65 with Fl_g < 0.18
Pitched-blade down-pumping 4-bladed: multiply ungassed power by 0.75 for Q_s/ND³ < 0.03

If the calculated P_g/P₀ > 1, physically clamp it to 1.0 and increase safety factor on motor sizing.

Flooding occurs when the gas cavity behind the blades becomes so large that the impeller can no longer recirculate liquid through it; typically at Fl_g ≈ 0.18 for Rushtons. Beyond this point, power draw rises again as bubbles re-attach. Always plot P_g vs. Q curve; the minimum power point sets the mechanical load for the motor, not the flooded end. Failure to do so risks undersized motors that trip on low-torque overload when gas flow is later reduced.

Torque (Nm) from either load cell or motor current and efficiency curve
Rotational speed (rpm) verified with a stroboscope or encoder
Actual gas volumetric flow at reactor conditions (m³ s^-1), not just rotameter reading at 20 °C
Liquid density and viscosity at operating temperature
Impeller diameter (m)

Insert these into the definition N_p,g = 2πτ/(ρN³D⁵) and compare to correlation values; aim for ±15 % agreement before scaling-up.

Worked Example: Gassed Power Number Calculation

A process engineer needs to estimate the reduction in impeller power draw when sparging air into a yeast fermentation broth. The system operates at steady state with fully-developed cavities, and the broth's physical properties are approximated by water at 30°C and 1 bar absolute pressure.

Knowns (Input Parameters):

T = 303.0 K (process temperature)
P = 1.0 bar (absolute pressure)
D = 0.1 m (impeller diameter)
N = 5.0 rps (impeller rotational speed)
Q = 0.0003 m³ s⁻¹ (volumetric gas flow rate at process conditions)

Step-by-Step Calculation:

Compute the dimensionless aeration number, $ N_{aeration} = \frac{Q}{N D^3} $. Using the provided input values, the calculation yields $ N_{aeration} = 0.06 $.
Apply the empirical correlation for the gassed to ungassed power ratio: $ \frac{P_g}{P_0} = 1 - 1.2 N_{aeration} $, valid for $ N_{aeration} \leq 0.08 $. Substituting $ N_{aeration} = 0.06 $, we find $ \frac{P_g}{P_0} = 0.928 $.
Verify the correlation's validity range: $ 0.02 \leq N_{aeration} \leq 0.08 $. Here, $ N_{aeration} = 0.06 $, which satisfies the condition.

Final Answer:

The gassed to ungassed power ratio $ \frac{P_g}{P_0} = 0.928 $. This indicates that under the given aeration conditions, the impeller shaft power draw is reduced to 92.8% of the ungassed power, corresponding to a 7.2% reduction.

"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

Parameter Range	Correlation Validity
\(0.02 \le Na \le 0.08\)	Correlation is trustworthy; cavity flooding not excessive.
\(Na \lt 0.02\)	Too little aeration—negligible power loss; correlation still valid but underpredicts small gassing losses.
\(Na \gt 0.08\)	Cavity flooding onset; correlation is NOT reliable—switch to higher-order flooding models.