Geometric Bioreactor Scale-Up — Four Impeller-Speed Criteria: 23-Chem-B4 Biochemical Engineering, December 2016, Q2

Question 2: Geometric Bioreactor Scale-Up — Four Impeller-Speed Criteria (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Symbol	Value
Lab-scale working volume	V₁	50 L
Plant-scale working volume	V₂	10,000 L
Lab-scale impeller speed	N₁	300 RPM
Geometry (both scales)	H=3D, D=3D_i	geometric similarity

Find. The plant-scale impeller speed N₂ under each of the four scale-up criteria (a)–(d).

Approach. Under geometric similarity every linear dimension of the plant vessel is the same multiple R of the lab vessel's, and because H=3D and D=3D_i at both scales, that includes the tank volume itself (V∝D³), so R=(V₂/V₁)^1/3. Each scale-up rule holds a different turbulent-regime dimensionless (or physical) group constant across scales — impeller Reynolds number Re=ND_i²/ν, power per volume P/V∝N³D_i² (constant power number), pumping capacity per volume Q/V∝N (constant flow number, Q∝ND_i³), and impeller tip speed v_tip=πND_i — giving four different power-law relations N₂=N₁R^p.

Fig. 2 — lab- and plant-scale vessels under geometric similarity (H=3D, D=3D_i at both scales); the plant vessel's every linear dimension is R times the lab vessel's.

Scale ratio. Since V∝D³ under this fixed-proportion geometry, $$R=\frac{D_2}{D_1}=\left(\frac{V_2}{V_1}\right)^{1/3}=\left(\frac{10{,}000}{50}\right)^{1/3}=200^{1/3}=\boxed{5.848}$$
(a) Equal impeller Reynolds number. Re=ND_i²/ν constant (same fluid) ⇒ N₁D_i1²=N₂D_i2²: $$N_2=\frac{N_1}{R^2}=\frac{300}{5.848^2}=\boxed{8.77\ \text{RPM}}$$
(b) Equal power per unit volume. P∝N³D_i&sup5; (constant power number, turbulent regime) and V∝D_i³, so P/V∝N³D_i² constant: $$N_2=\frac{N_1}{R^{2/3}}=\frac{300}{5.848^{2/3}}=\boxed{92.4\ \text{RPM}}$$
(c) Equal liquid pumping rate per unit volume. Pumping capacity Q∝ND_i³ (constant flow number), and since V∝D_i³ too, Q/V∝N directly — the D_i³ cancels exactly, so this criterion is scale-independent: $$N_2=N_1\cdot R^{0}=\boxed{300\ \text{RPM}}$$ This is a genuinely counter-intuitive result worth stating explicitly: matching pumping rate per unit volume requires the SAME impeller speed at any scale under geometric similarity, not a reduced one.
(d) Equal impeller tip speed. v_tip=πND_i constant ⇒ N₁D_i1=N₂D_i2: $$N_2=\frac{N_1}{R}=\frac{300}{5.848}=\boxed{51.3\ \text{RPM}}$$

The four criteria span more than a 30-fold range (8.8–300 RPM) for the identical scale-up, which is the central lesson of this problem: there is no single "correct" scale-up rule — the choice must be driven by whichever physical phenomenon (shear damage, gas–liquid mass transfer, blend/circulation time) actually limits performance in the specific process being scaled.

Criterion	N₂ at plant scale
Scale ratio, R	5.848
(a) Equal impeller Reynolds number	8.77 RPM
(b) Equal power per unit volume	92.4 RPM
(c) Equal pumping rate per unit volume	300 RPM (=N₁)
(d) Equal impeller tip speed	51.3 RPM

Topic: Geometric similarity and impeller-speed scale-up rules for stirred-tank bioreactors.

Key relations: $R=(V_2/V_1)^{1/3}$; $Re=ND_i^2/\nu\Rightarrow N_2/N_1=R^{-2}$; $P/V\propto N^3D_i^2\Rightarrow N_2/N_1=R^{-2/3}$; $Q/V\propto N\Rightarrow N_2/N_1=R^{0}$; $v_{tip}=\pi ND_i\Rightarrow N_2/N_1=R^{-1}$.

Why this works: Each criterion freezes a different dimensionless or physical group that governs a different transport phenomenon in the vessel — Reynolds number governs the flow regime/shear field, power/volume governs bulk mixing intensity and heat removal, pumping rate/volume governs bulk circulation and blend time, and tip speed governs the maximum shear rate cells or particles experience at the impeller. Because all four groups cannot be held constant simultaneously across a scale change, the four rules necessarily diverge.

Common pitfall: Assuming impeller speed must always fall with increasing scale — the equal-pumping-rate-per-volume criterion is scale-invariant (N₂=N₁) precisely because Q/V's D_i³ dependence cancels the D_i³ growth of the working volume itself.

Source: Bailey & Ollis, Biochemical Engineering Fundamentals, 2nd ed., Ch. 8 (bioreactor design and scale-up); Shuler & Kargi, Bioprocess Engineering, 2nd ed., Ch. 9 (agitation and aeration).

Question 2 of 5: Geometric Bioreactor Scale-Up — Four Impeller-Speed Criteria

Question 2: Geometric Bioreactor Scale-Up — Four Impeller-Speed Criteria (20 marks)