Forest Glasshouse

What is modelled

A spherical gather of glass loses heat from its surface by radiation and convection, and the heat inside reaches the surface only by conduction, so the skin cools much faster than the core. Glass has no melting point: its viscosity rises by many orders of magnitude over a few hundred degrees. Once the skin is too stiff to work, the gather goes back into the glory hole. At the end, the finished vessel must pass the annealing range slowly or temperature differences freeze in as stress. The 3D page shows the furnace, bench and vessels and has no controls, so every input here is the model's own, with the potash flux idea taken from its batch station.

Equations

• Viscosity (Vogel-Fulcher-Tammann) log10 η = A + B / (T - T0), A = -2.5, B = 4500 K, T₀ = 250 C, shifted lower by extra alkali; working 10³, softening 10^6.6, annealing 10¹², strain 10^13.5 Pa s.
• Radial conduction ρ c ∂T/∂t = (1/r2) ∂/∂r (k r2 ∂T/∂r), surface loss ε σ (T4 - Tenv4) + h (T - Tair).
• Biot number Bi = heff R / k, large here, so no single temperature describes the gather.
• Annealing: a wall of thickness d cooled at rate β has ΔT = β d2 / (8 α) between centre and surface and a stress σ = E αth ΔT / (3 (1 - ν)).

Checks run in the page

• Finite-volume sphere against the exact series for a fixed surface temperature.
• Energy lost by the glass against heat radiated and convected from the surface.
• Viscosity landmarks round-trip and bisection of the annealing point.
• Slab cooling at a constant rate against beta d2 / 8 alpha.

Honest simplifications

The viscosity constants are those of a typical soda-lime-like glass with a simple shift for potash, not data for any real forest glass. Conductivity, heat capacity and emissivity are constants, internal radiative transfer in the hot glass is hidden inside the conductivity, and the blown shape is a sphere of the same mass rather than a thin bubble. Stress is the elastic estimate below the annealing point.

• Choose Large gather: the core stays hot for a long time while the skin has already stiffened.
• Choose High potash: all the viscosity landmarks move to lower temperatures.
• Choose Thick wall, quick cooling and see the wall stress go far above the allowed value.