Geometric Scale-Up of a Cell-Culture Bioreactor: 23-Chem-B4 Biochemical Engineering, Undated paper, Q4

Question 4: Geometric Scale-Up of a Cell-Culture Bioreactor (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	Small vessel	Large vessel
Working volume	V₁=0.01 m³	V₂=50 m³
Tank diameter	T₁=0.3 m	T₂=?
Impeller diameter	D₁=0.1 m	D₂=?
Agitation speed	N₁=100 rpm	N₂=?

Find. The large-vessel tank diameter T₂, impeller diameter D₂, and agitation speed N₂ under geometric similarity (constant H/T and constant D/T) with equal impeller tip speed.

Fig. 3 — geometric scale-up from a 0.01 m³ bench vessel to a 50 m³ production vessel (schematic, not to true relative scale); every linear dimension scales by the same factor, agitation speed scales inversely to preserve tip speed.

Approach. Geometric similarity means every linear dimension of the vessel (tank diameter, impeller diameter, liquid height) scales by the same factor S, found from the volume ratio; then equal impeller tip speed fixes the large-vessel agitation speed.

Linear scale factor from the volume ratio. Since $V\propto T^3$ under geometric similarity (constant H/T), $$S=\left(\frac{V_2}{V_1}\right)^{1/3}=\left(\frac{50}{0.01}\right)^{1/3}=(5000)^{1/3}$$ $$\boxed{S=17.10}$$
Large-vessel tank and impeller diameters. Every linear dimension scales by S, so the impeller-to-tank ratio D/T=0.1/0.3=1/3 is automatically preserved: $$T_2=S\,T_1=17.10\times0.3=\boxed{5.130\ \text{m}}$$ $$D_2=S\,D_1=17.10\times0.1=\boxed{1.710\ \text{m}}$$ (Check: $T_2/D_2=5.130/1.710=3.00=T_1/D_1$ — geometric similarity confirmed.)
Agitation speed from equal impeller tip speed. Impeller tip speed is $u_{tip}=\pi ND$; setting $u_{tip,1}=u_{tip,2}$, $$\pi N_1D_1=\pi N_2D_2\quad\Rightarrow\quad N_2=N_1\frac{D_1}{D_2}=\frac{N_1}{S}$$ $$N_2=\frac{100}{17.10}=\boxed{5.848\ \text{rpm}}$$ The tip speed held constant in both vessels is $u_{tip}=\pi\times(100/60)\times0.1=0.524$ m/s.

Quantity	Value
Linear scale factor, S	17.10
Large-vessel tank diameter, T₂	5.130 m
Large-vessel impeller diameter, D₂	1.710 m
Large-vessel agitation speed, N₂	5.848 rpm

Topic: Geometric scale-up of stirred bioreactors under an equal-tip-speed criterion.

Key relations: $V\propto T^3$ (geometric similarity); $S=(V_2/V_1)^{1/3}$; all lengths scale by S; $u_{tip}=\pi ND$; equal tip speed $\Rightarrow N_2=N_1/S$.

Why this works: Holding H/T (and D/T) fixed means the vessel's shape, not just its size, is preserved, so a single scale factor S applies to every length. Equal tip speed is the classical shear-based scale-up rule for shear-sensitive cell cultures (it keeps the maximum fluid shear near the impeller blade tip roughly constant), at the cost of large drops in both agitation speed and specific power input per unit volume at larger scale — a well-known scale-up trade-off (equal tip speed under-mixes/under-aerates the larger vessel relative to equal P/V scale-up).

Common pitfall: Assuming N scales the same way regardless of the chosen scale-up criterion — equal tip speed ($N\propto1/S$), equal P/V ($N\propto S^{-2/3}$ approx.) and equal Reynolds number ($N\propto1/S^2$) give three different (and quite different numerically) values of N₂ for the same geometry; the question must specify which criterion to use, as it does here (equal tip speed).

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

Question 4 of 5: Geometric Scale-Up of a Cell-Culture Bioreactor

Question 4: Geometric Scale-Up of a Cell-Culture Bioreactor (20 marks)