SCALE 1 (PILOT BASELINE) SOURCE

V₁ Working Volume Pilot scale

D₁ Impeller Diameter Agitator

N₁ Agitation Speed Pilot RPM

RPM

N_p Impeller Power Number

ρ Broth Density Fluid mass

kg / m³

Scale 1 Operating Baseline

P₁/V₁: 1,180 W/m³
v_tip₁: 1.25 m/s
Ungassed Power P₁: 5.9 W

SCALE 2 (PRODUCTION TARGET) TARGET SPEC

V₂ Target Working Volume Commercial

H_L / T Aspect Ratio Height / Diameter

D / T Impeller / Tank Ratio Standard: 0.33

Computed Tank Geometry (Scale 2)

Tank Diameter T₂: 1.62 m
Liquid Height H₂: 2.43 m
Impeller Diam D₂: 0.540 m
Ungassed Power P₂: 5.9 kW

Scale-Up Recommendation

Constant P/V Preferred

Maintains oxygen mass transfer capability (kLa). Tip speed stays within acceptable shear thresholds.

SCALE-UP REGIME REGRESSION (MATPLOTLIB STYLE) FIGURE 2.1 • LOG-LOG TRANSLATION

REGIME TRANSLATION GUIDE:
• Constant P/V (Blue): Preserves kLa & turbulent dissipation.
• Constant Tip Speed (Orange): Preserves maximum shear stress (τ_max).
• Constant Mixing Time (Green): Constant blend time (θ95) leads to excessive power escalation.

📚 Stirred-Tank Bioreactor Scale-Up: Constant P/V, Tip Speed & Regime Mapping Upstream Bioprocess • Step 2 of 7

Theoretical Principles & Engineering Fundamentals

Scaling upstream bioprocesses from benchtop fermenters (1L–5L) to pilot (50L–500L) and industrial manufacturing plants (5,000L–25,000L) is a multi-objective optimization problem. Because geometric, kinematic, and dynamic similarity cannot be satisfied simultaneously, bioprocess engineers must select a primary scale-up invariant.

The standard industrial scale-up invariants include:

Constant Volumetric Power Input ($P/V$): Preserves average turbulence energy dissipation rate and volumetric mass transfer coefficient ($k_L a$), recommended for high-oxygen-demand bacterial cultures.

Constant Impeller Tip Speed ($v_{ ext{tip}}$): Restricts maximum hydrodynamic shear stress, critical for shear-sensitive CHO mammalian cell cultures.

Constant Mixing Time ($ heta_{95}$): Prevents glucose and pH concentration gradients, though requiring steep increases in agitator motor power at large volumes.

Governing Equations & Mathematical Formulations

Ungassed Agitation Power P = N_p \times \rho \times N^3 \times D^5

Calculates total mechanical power drawn by an impeller set where $N_p$ is impeller power number, $N$ is agitator speed in rev/s, and $D$ is impeller diameter in meters.

Impeller Tip Speed v_{\text{tip}} = \pi \times N \times D

Governs maximum localized fluid shear stress at the impeller tip, typically constrained to $< 1.8\,\text{m/s}$ for mammalian cells and $< 5.5\,\text{m/s}$ for fungal mycelia.

Scale-Up Speed Translation (Constant P/V) N_2 = N_1 \times \left(\frac{D_1}{D_2}\right)^{5/3} \times \left(\frac{V_2}{V_1}\right)^{1/3}

Relates large-scale agitator RPM ($N_2$) to laboratory scale ($N_1$) under invariant volumetric power dissipation.

Industrial Benchmark Data & Parameter Reference

Scale-Up Parameter	Constant P/V Scaling	Constant Tip Speed Scaling	Constant Mixing Time Scaling
Agitator Speed (N)	Decreases as (1/Scale)^(2/9)	Decreases as (1/Scale)^(1/3)	Remains constant (requires high power)
Volumetric Power (P/V)	1.0× (Invariant)	Decreases by (1/Scale)^(2/3)	Increases by (Scale)^(2/3)
Tip Speed (v_tip)	Increases by (Scale)^(1/9)	1.0× (Invariant)	Increases by (Scale)^(1/3)
Blend Time (θ95)	Increases by (Scale)^(2/9)	Increases by (Scale)^(1/3)	1.0× (Invariant)

Frequently Asked Questions (Bioprocess Engineering FAQ)

Why can't I keep both P/V and mixing time constant during scale-up?

As volume expands by a factor of 1,000, maintaining constant mixing time would require agitator power to scale by $1000^{5/3} = 100,000$, which is physically impossible and would boil the fermentation broth from metabolic and motor heat dissipation.

What is the typical P/V threshold for industrial E. coli fermenters?

Production-scale microbial fermenters typically operate at $1.5 ext{–}3.0 ext{ kW/m}^3$ ($W/L$) under gassed conditions, whereas mammalian cell bioreactors operate at $0.05 ext{–}0.15 ext{ kW/m}^3$.

How does gassing affect impeller power consumption?

Gas sparging creates gas cavities behind impeller blades, lowering the effective fluid density and drag coefficient, typically reducing power draw by 15% to 50% ($P_g/P_0 = 0.5 ext{–}0.85$).

Free Online Bioreactor Sizing & Scale-Up Calculator

📚 Stirred-Tank Bioreactor Scale-Up: Constant P/V, Tip Speed & Regime Mapping Upstream Bioprocess • Step 2 of 7

Theoretical Principles & Engineering Fundamentals

Governing Equations & Mathematical Formulations

Industrial Benchmark Data & Parameter Reference

Frequently Asked Questions (Bioprocess Engineering FAQ)

BioFlo Bioprocess Engineering Network