Topic C — Catalyst Pellet and Monolith Channel

reaction–diffusion with Arrhenius heat release · conjugate heat transfer in a monolith · thermal stress · tier: standard (pellet) → advanced (monolith)

Motivation

A catalyst is only as good as the fraction of it that the reactants reach. In a porous pellet, diffusion limits the rate (Thiele modulus, effectiveness factor) and the heat of reaction can make the inside much hotter than the surface. In a monolith, the reaction sits in a thin washcoat on channel walls; the wall conducts heat along the channel and couples the channels. You will compute effectiveness factors and hot spots, and the thermal stress they cause in the support.

Related fields: heterogeneous catalysis (metal-oxide and zeolite catalysts, reaction kinetics); nanostructured catalysts and adsorbents for clean energy. Course weeks: 3 (thermal stress), 12 (reactive flow, SUPG), 13 (conjugate heat transfer with submeshes).

Model

Stage 1 — pellet (standard). Sphere or cylinder (axisymmetric), effective diffusivity D_e and conductivity k_e: -\nabla\cdot(D_e\nabla c)=-r(c,T),\qquad -\nabla\cdot(k_e\nabla T)=(-\Delta H)\,r(c,T),\qquad r=k_0e^{-E/RT}c^n, surface conditions c=c_s, T=T_s (or external film resistances, Robin). Effectiveness factor \eta=\frac{\int r\,dV}{r(c_s,T_s)\,V}. Then the thermo-elastic stress in the pellet from the temperature field (Week 3, one-way coupled).

Stage 2 — monolith channel (advanced). One channel with its wall (square or circular, axisymmetric for circular), laminar flow in the channel (Navier–Stokes or the fully developed profile), species and energy in the gas, a washcoat with the reaction, heat conduction in the wall solved on a separate submesh and coupled at the interface (Week 13). Inlet temperature ramp for light-off.

Design outputs: \eta(\phi) and the maximum internal temperature against the Thiele modulus and the Prater number; for the monolith, the light-off curve (conversion against inlet temperature) and the peak wall temperature.

Suggested parameters

Pellet: radius 1–5 mm, Thiele modulus \phi=R\sqrt{k(T_s)/D_e} from 0.1 to 20, Prater number \beta=\frac{(-\Delta H)D_ec_s}{k_eT_s} = 0–0.3, Arrhenius number \gamma=E/RT_s = 10–30. Monolith: channel 1 mm, length 50 mm, Reynolds number 50–500, cordierite wall.

Required tasks

  1. Isothermal pellet, first order: \eta(\phi) against the exact curve; then n=2 (no closed form — mesh independence).
  2. Non-isothermal pellet: \eta and T_{\max} for several \beta, \gamma; Newton with continuation in \phi.
  3. Thermal stress in the pellet from the computed T (sphere: the closed-form stress for a given radial temperature, Week 3 analogy).
  4. (Advanced) the monolith channel with conjugate heat transfer: light-off curve.

Verification specific to this topic

  • Effectiveness factor: first-order isothermal sphere \eta=\frac{3}{\phi^2}(\phi\coth\phi-1), slab \eta=\tanh\phi/\phi — to discretisation error, with rates.
  • Prater relation: in the pellet, T-T_s=\frac{(-\Delta H)D_e}{k_e}(c_s-c) holds pointwise for any kinetics (combine the two equations) — so T_{\max}-T_s\le\beta T_s. Check it at every point.
  • Thermal stress: for a sphere with a radial temperature T(r) the stresses are closed-form integrals of T (Timoshenko & Goodier); compare your FE stress with the integrals of your FE temperature.
  • Monolith: fully developed laminar flow without reaction — the Graetz/Nusselt limit (\mathrm{Nu}=3.66 for a circular channel at constant wall temperature), and the energy balance between gas and wall.

Stretch goals

  1. Multiple steady states of the non-isothermal pellet (Weisz–Hicks): pseudo-arclength continuation in \phi.
  2. Langmuir–Hinshelwood kinetics with inhibition.
  3. Pellet shape optimisation (hollow cylinder, trilobe) at equal volume.

Milestones

M1 isothermal pellet and \eta(\phi) · M2 non-isothermal pellet with the Prater check · M3 thermal stress · M4 monolith flow and Graetz check · M5 coupled monolith and light-off.

Pitfalls

  • Large \phi: a boundary layer of thickness R/\phi at the surface — grade the mesh, otherwise \eta converges slowly from above.
  • Newton from c=c_s, T=T_s fails for large \beta\gamma: continuation (Week 7 load steps, Week 9 HW).
  • Axisymmetric sphere: the r-weight in every integral (Week 2), and no condition on the axis.

References

  • R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon 1975.
  • G. F. Froment, K. B. Bischoff, J. De Wilde, Chemical Reactor Analysis and Design, 3rd ed., Wiley 2011.
  • S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970 — thermal stresses in a sphere.