Immobilized-Enzyme Packed-Bed Reactor

Reactor Controls

Flow rate (superficial velocity) 6.0 mm/s

Bead radius 1.20 mm

Enzyme loading (V_max) 2.0 mM/s

Inlet substrate S₀ 10.0 mM

Column Profile

Inlet (fresh substrate) → outlet (depleted)

Outlet conversion X

0.0%

Residence time τ

0.0 s

Thiele modulus φ

0.00

Effectiveness η

1.00

Transport regime

Reaction-limited (kinetic)

How it works

Enzyme is immobilized inside porous spherical beads packed into a column. Substrate must first diffuse into a bead before the enzyme can convert it, so at high enzyme loading or large beads the reaction outruns diffusion and the bead's core sits at a lower concentration than its surface — the reaction becomes diffusion-limited.

Intrinsic rate (Michaelis–Menten):
  r = Vmax·S / (Km + S)
Linearized rate constant:  k1 = Vmax / Km

Thiele modulus (sphere, radius R, diffusivity Deff):
  φ = R·√(k1 / Deff)

Effectiveness factor (fraction of intrinsic
rate actually achieved, averaged over the bead):
  η = (3/φ²)·(φ·coth φ − 1)      [η → 1 as φ → 0]

Intraparticle profile (first-order limit):
  C(r)/Cs = (R/r)·sinh(φ r/R) / sinh(φ)

Axial mass balance down the packed bed
(plug flow, superficial velocity u):
  dS/dz = −η · Vmax·S/(Km+S) / u

Flow rate — sets the superficial velocity u and hence residence time τ; faster flow means less contact time and lower conversion.
Bead radius — larger beads increase the diffusion path length, raising φ and pulling the bed into the diffusion-limited regime (η falls).
Enzyme loading (Vmax) — more active enzyme per bead raises the intrinsic rate but also raises φ, so effectiveness can fall even as the bed converts more overall.
Inlet substrate S₀ — sets the feed concentration entering the top of the column.

The inset sphere (bottom-right of the scene) shows the steady-state concentration profile inside a single bead at the column inlet — bright at the surface, dimmer toward the core when diffusion limits the reaction. This is the standard reactor-engineering picture behind real industrial immobilized-enzyme processes: glucose isomerase columns making high-fructose corn syrup, immobilized lipases for biodiesel transesterification, and packed-bed penicillin-acylase reactors for semi-synthetic antibiotics all live or die by this same Thiele-modulus trade-off.

High substrate Partial conversion Depleted

InletOutlet

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