Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exam 04-Chem-B4, Biochemical Engineering — May 2014. 3 hours, Closed-Book Exam
(any non-communicating calculator permitted). Six questions are printed; per the exam notes any five (5)
constitute a complete paper (100 marks) and only the first five as they appear in the answer book are
marked. All six are solved below for completeness.
Find. The large-vessel impeller speed N2 for each of four scale-up criteria:
(a) equal Re, (b) equal P/V, (c) equal pumping rate, (d) equal tip speed.
Approach. Geometric similarity means every linear dimension — tank diameter,
liquid height, impeller diameter — scales by the same factor R, fixed by the volume ratio
(V ∝ L³). Each scale-up rule then reduces to a power-law relation between N and Di
whose exponent comes from how Reynolds number, power, pumping (flow) number, and tip speed depend on N
and Di for a stirred-tank impeller.
Fig. 1 — geometric similarity: every linear dimension scales by
R = (V2/V1)1/3.
Linear scale factor from the volume ratio. For geometrically similar vessels every
length scales by the same ratio R, so volume (which scales as length cubed) fixes R directly:
$$R=\frac{D_{i2}}{D_{i1}}=\frac{D_{t2}}{D_{t1}}=\left(\frac{V_2}{V_1}\right)^{1/3}=100^{1/3}=\boxed{4.642}$$
(a) Equal impeller Reynolds number. $Re=\rho N D_i^2/\mu$ is held constant between
scales (same fluid, so ρ, μ cancel), giving $N_1D_{i1}^2=N_2D_{i2}^2$:
$$N_2=N_1\left(\frac{D_{i1}}{D_{i2}}\right)^2=N_1R^{-2}=60\times(4.642)^{-2}=\boxed{2.79\ \text{rpm}}$$
Matching Reynolds number forces the impeller speed to drop sharply with scale, since the larger impeller
diameter alone already increases Re by R².
(b) Equal power per unit volume. In the fully turbulent regime the power number
$N_P=P/(\rho N^3D_i^5)$ is constant, so $P\propto \rho N^3D_i^5$; dividing by volume ($V\propto D_i^3$
at fixed geometry) gives $P/V\propto N^3D_i^2$. Holding this constant between scales:
$$N_1^3D_{i1}^2=N_2^3D_{i2}^2\ \Rightarrow\ N_2=N_1\left(\frac{D_{i1}}{D_{i2}}\right)^{2/3}=N_1R^{-2/3}
=60\times(4.642)^{-2/3}=\boxed{21.6\ \text{rpm}}$$
This is the criterion most often used for shear-sensitive but mass-transfer-dependent cultures, since P/V
tracks the volumetric mixing/mass-transfer intensity (kLa).
(c) Equal impeller pumping (circulation) rate. The flow number $N_{Fl}=Q/(ND_i^3)$
is constant, so the volumetric pumping rate $Q\propto ND_i^3$. Equal Q between scales gives
$N_1D_{i1}^3=N_2D_{i2}^3$:
$$N_2=N_1\left(\frac{D_{i1}}{D_{i2}}\right)^3=N_1R^{-3}=60\times(4.642)^{-3}=60/100=\boxed{0.600\ \text{rpm}}$$
(Consistent with the volume ratio itself: $D_{i2}^3/D_{i1}^3=R^3=100$, the same 100× that defined
the scale-up.)
(d) Equal impeller tip speed. Tip speed $v_{tip}=\pi ND_i$ constant gives
$N_1D_{i1}=N_2D_{i2}$:
$$N_2=N_1\left(\frac{D_{i1}}{D_{i2}}\right)=N_1R^{-1}=60/4.642=\boxed{12.9\ \text{rpm}}$$
Tip-speed matching is the criterion typically chosen to protect shear-sensitive cells (mammalian/plant
cultures), since maximum shear near the blade tip scales directly with $v_{tip}$.
