Dimensionless Groups in Bioreactor Systems — Re, Sh, Sc, Thiele Modulus, Damköhler Number: 23-Chem-B4 Biochemical Engineering, May 2017, Q5

Question 5: Dimensionless Groups in Bioreactor Systems — Re, Sh, Sc, Thiele Modulus, Damköhler Number (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.

All five groups compare two competing rates or transport mechanisms; together they let an engineer judge, without solving the full transport equations, whether a bioreactor process is limited by bulk fluid mechanics, by mass transfer across a film, or by the intrinsic biochemical reaction itself.

Fig. 5 — the shared physical picture: bulk fluid (Re, Sc set by the flow and fluid alone) flows past a boundary film (Sh governs transfer across it) surrounding a reacting pellet or cell (φ compares reaction to diffusion inside it; Da compares reaction to external transfer overall).

Group	Definition	Physical meaning
(i) Reynolds number	$Re=\dfrac{\rho UL}{\mu}=\dfrac{UL}{\nu}$	Inertial forces / viscous forces. Sets the flow regime (laminar vs. turbulent) around an impeller, bubble, or particle; in bioreactors it governs shear intensity, blend time, and the correlations used for h and k_L.
(ii) Sherwood number	$Sh=\dfrac{k_LL}{D}$	Convective mass transfer / diffusive mass transfer. The mass-transfer analogue of the Nusselt number; a higher Sh means convection (bulk fluid motion) carries dissolved species across the film far more effectively than molecular diffusion alone could. Correlations such as Sh=2+0.6Re^1/2Sc^1/3 (Frössling/Ranz-Marshall, for a sphere) predict k_L from the flow conditions.
(iii) Schmidt number	$Sc=\dfrac{\mu}{\rho D}=\dfrac{\nu}{D}$	Momentum diffusivity / mass diffusivity. A pure fluid-property group (no geometry or velocity), analogous to the Prandtl number for heat transfer; it appears inside Sh correlations to translate a known flow (Re) into a mass-transfer coefficient.
(iv) Thiele modulus	$\phi=L\sqrt{k/D_e}$ (L = characteristic length, e.g. V_p/S_p for a pellet)	Intrinsic reaction rate / internal diffusion rate, inside a porous pellet or immobilized-cell bead. Large φ (≫1) means diffusion into the pellet interior is rate-limiting, so only a thin outer rind of the pellet reacts (low effectiveness factor η); small φ means the whole pellet volume is used efficiently.
(v) Damköhler number	$Da=\dfrac{k\,L}{k_L}$ (first-order form; generally reaction rate / mass-transfer rate)	Reaction rate / external mass-transfer rate across the film surrounding the pellet or cell. Da≫1 means the reaction consumes substrate faster than the film can resupply it, so the process is external-mass-transfer-limited (the reaction "starves" at the surface); Da≪1 means transfer is fast relative to reaction, so the reaction itself is rate-limiting and the bulk and surface concentrations are essentially equal.

Used together, these five groups form a diagnostic chain: Re and Sc (pure flow/fluid properties) feed into a correlation for Sh, which gives k_L (external film transfer); Da then compares that external transfer rate to the reaction rate to say whether film transfer or reaction controls at the particle surface, while φ performs the analogous comparison for diffusion versus reaction inside a porous pellet/immobilized-cell bead. A process engineer uses this chain to decide, before investing in experiments, whether raising agitation/flow (which changes Re, Sh, Da but not φ) or making the pellet smaller (which changes φ) is the correct lever to relieve a mass-transfer bottleneck.

Group	Symbol/formula	Compares
Reynolds	Re = ρUL/μ	Inertial vs. viscous forces (flow regime)
Sherwood	Sh = k_LL/D	Convective vs. diffusive mass transfer
Schmidt	Sc = ν/D	Momentum vs. mass diffusivity (fluid property)
Thiele modulus	φ = L√(k/D_e)	Reaction vs. internal (intraparticle) diffusion
Damköhler	Da = kL/k_L	Reaction vs. external (film) mass transfer

Topic: Dimensionless groups for flow, mass transfer, and reaction-diffusion competition in bioreactor design.

Key relations: Re=ρUL/μ; Sh=k_LL/D; Sc=ν/D; Sh=f(Re,Sc) (e.g. Frössling correlation); φ=L√(k/D_e) with effectiveness factor η=f(φ) (e.g. η=(3/φ²)(φcothφ−1) for a sphere); Da=kL/k_L.

Why this works: Each group is a ratio of two rates or two diffusivities, which is exactly what non-dimensionalizing the governing transport PDEs produces — collapsing many physical variables (velocity, viscosity, diffusivity, rate constant, geometry) into a handful of numbers whose relative size (≫1 or ≪1), not their absolute value, tells you which physical mechanism controls the overall process rate.

Common pitfall: Confusing the Thiele modulus (internal, intraparticle reaction-vs-diffusion) with the Damköhler number (external, film reaction-vs-transfer) — both compare "reaction" to "transport" but at different physical locations (inside vs. outside the particle), and a process can be limited by one without being limited by the other.

Source: Bailey & Ollis, Biochemical Engineering Fundamentals, 2nd ed., Ch. 4 & 8 (transport phenomena and mass-transfer dimensionless groups in bioreactors); Shuler & Kargi, Bioprocess Engineering, 2nd ed., Ch. 10 (mass transfer correlations).

Question 5 of 5: Dimensionless Groups in Bioreactor Systems — Re, Sh, Sc, Thiele Modulus, Damköhler Number

Question 5: Dimensionless Groups in Bioreactor Systems — Re, Sh, Sc, Thiele Modulus, Damköhler Number (20 marks)