Fluid-solid non-catalytic reactions: shrinking core model

Why it matters

Roasting of sulphide ores, reduction of iron ore, combustion and gasification of coal, calcination of limestone and regeneration of coked catalyst all involve a gas reacting with a solid that is itself consumed. The shrinking core model (SCM) predicts how long a particle takes to react, which step controls, and how conversion depends on particle size — the information needed to size roasters, kilns and fluidised-bed reactors.

Key ideas

Reaction type. A(fluid) + b B(solid) → fluid and/or solid products. Two particle behaviours:

• Particle of unchanging size: a porous ash (product) layer forms around a shrinking unreacted core (e.g. ZnS roasting, iron-ore reduction).
• Shrinking particle: no ash remains (e.g. carbon burning to CO₂); the particle itself shrinks.

Shrinking core model (unchanging size). Reaction occurs at a sharp front between the ash and the unreacted core, which moves inward. Three resistances act in series:

1. Diffusion of A through the gas film around the particle.
2. Diffusion of A through the ash layer.
3. Reaction at the surface of the unreacted core. The model assumes a pseudo-steady state: the core shrinks slowly compared with the rate at which the concentration profile adjusts (true for gas–solid systems, where the gas is far less dense than the solid).

Conversion and core size. For a sphere, 1 − XB = (rc/R)³, where rc is the core radius and R the particle radius.

Each controlling regime gives its own time–conversion relation, written with τ = time for complete conversion:

• Gas-film control: the surface concentration of A is zero and the flux through a film of fixed area is constant, so conversion is linear in time: t/τ = XB.
• Ash-diffusion control: the ash layer thickens, so the rate slows steeply: t/τ = 1 − 3(1 − XB)^(2/3) + 2(1 − XB).
• Chemical-reaction control: the rate is proportional to the core surface area: t/τ = 1 − (1 − XB)^(1/3) = 1 − rc/R.

Size dependence of τ — a key diagnostic. Film control: τ ∝ R (for particles of fixed size; roughly R^1.5–2 for shrinking particles). Ash diffusion: τ ∝ R². Reaction control: τ ∝ R. Temperature also helps: reaction control is strongly temperature-sensitive (Arrhenius), whereas diffusion controls are only weakly so. Experiments with different particle sizes and temperatures identify the controlling step.

Combined resistances. When no single step dominates, the times add approximately: t_total(XB) = t_film(XB) + t_ash(XB) + t_reaction(XB), each evaluated as if it alone controlled. As the ash layer grows, ash diffusion becomes progressively more important.

Shrinking particles. With reaction control, t/τ = 1 − (1 − XB)^(1/3), the same as for unchanging size. With gas-film control in the Stokes regime (small particles), t/τ = 1 − (1 − XB)^(2/3).

Reactor design. For a batch of uniform particles in uniform gas, use the relations directly. For solids in plug flow (rotary kiln, moving grate) with uniform gas composition, every particle has the same residence time. For solids in mixed flow (fluidised bed), average the single-particle conversion over the exponential residence-time distribution of the solids: 1 − X̄B = ∫(0→τ) (1 − XB(t))·E(t) dt.

Limitations. Real solids may react throughout their volume (progressive conversion model), sinter, crack or change porosity. The SCM works best for non-porous or low-porosity solids.

Formulas

1 − XB = (rc/R)³ (sphere)

• rc: radius of unreacted core (m); R: particle radius (m); XB: fractional conversion of solid.

t/τ = XB, τ = ρB·R/(3·b·kg·CAg) (gas-film control)

• ρB: molar density of B in the solid (mol/m³); b: moles of B per mole of A; kg: gas-film mass-transfer coefficient (m/s); CAg: bulk gas concentration of A (mol/m³).

t/τ = 1 − 3(1 − XB)^(2/3) + 2(1 − XB), τ = ρB·R²/(6·b·De·CAg) (ash-layer diffusion control)

• De: effective diffusivity of A in the ash (m²/s).

t/τ = 1 − (1 − XB)^(1/3), τ = ρB·R/(b·k''·CAg) (chemical-reaction control, first order in A)

• k'': first-order surface rate constant (m/s).

t/τ = 1 − (1 − XB)^(2/3) (shrinking particle, gas-film control, small particles)

t_total ≈ t_film + t_ash + t_reaction (resistances in series)

Worked examples

Example 1 (standard, same τ, different controls). Spherical particles need τ = 1 h for complete conversion. Find the time to reach XB = 0.875 if the controlling step is (a) gas film, (b) ash diffusion, (c) reaction.

1. 1 − XB = 0.125, so rc/R = 0.125^(1/3) = 0.5 and (1 − XB)^(2/3) = 0.25.
2. (a) Film: t = τ·XB = 0.875 h.
3. (b) Ash: t/τ = 1 − 3 × 0.25 + 2 × 0.125 = 1 − 0.75 + 0.25 = 0.50 → t = 0.50 h.
4. (c) Reaction: t/τ = 1 − 0.5 = 0.50 → t = 0.50 h.

0.875 h, 0.50 h and 0.50 h. The curves cross here, but at lower conversions they differ: at XB = 0.5, ash control needs only 0.110 h and reaction control 0.206 h.

Example 2 (GATE level, τ from properties and size effect). A solid (ρB = 20 000 mol/m³, b = 1) in spheres of radius 5 mm reacts with gas at CAg = 10 mol/m³; ash diffusion controls with De = 2 × 10⁻⁶ m²/s. Find τ, and τ for 2.5 mm particles.

1. τ = ρB·R²/(6·b·De·CAg) = 20 000 × (5 × 10⁻³)²/(6 × 1 × 2 × 10⁻⁶ × 10).
2. Numerator = 20 000 × 2.5 × 10⁻⁵ = 0.5; denominator = 1.2 × 10⁻⁴.
3. τ = 0.5/1.2 × 10⁻⁴ = 4167 s = 1.16 h.
4. τ ∝ R², so halving R gives τ = 4167/4 = 1042 s = 0.29 h.

τ ≈ 1.16 h for 5 mm particles and 0.29 h for 2.5 mm. If instead τ had halved, the data would point to reaction or film control.

Common mistakes

• Using 1 − XB = rc/R instead of (rc/R)³ for a sphere.
• Swapping the ash and reaction formulas — ash diffusion has the (2/3) and linear terms; reaction has the cube root.
• Using mass density of the solid instead of molar density ρB (mol/m³) in τ.
• Forgetting the stoichiometric factor b.
• Assuming the controlling step stays the same at all conversions; ash resistance grows as the layer thickens.

For GATE CH

Expect NAT questions on time for a given conversion under each control regime, τ from physical properties, core radius from conversion, and the effect of particle size or temperature on τ to identify the controlling step. Practise the three t/τ expressions until you can evaluate them quickly, and remember τ ∝ R² only for ash diffusion.

Quick check

1. Under gas-film control, how long does 60 % conversion take if τ = 2 h?
2. A particle is 87.5 % converted. What is rc/R?
3. Doubling particle size quadruples τ. Which step controls?
4. Which controlling step is most sensitive to temperature?

Answers: 1. 1.2 h; 2. 0.5; 3. ash-layer diffusion; 4. chemical reaction.