Wax Seal Press

What is modelled

A viscous disc squeezed between two flat plates thins slowly: the thinner it is, the harder it is for the liquid to escape sideways, so the squeeze rate falls as the cube of the thickness (Stefan's law). At the same time the wax loses heat into the cold matrix and paper, and its viscosity rises steeply as it cools. The seal is good only if the matrix has pressed the wax thin enough before it sets.

Equations

• Stefan squeeze dh/dt = −2 F h3 / (3π η R4), with the radius from volume conservation R = R0 √(h0/h).
• Arrhenius viscosity η(T) = η80 exp[(Ea/Rg)(1/T − 1/353 K)].
• Cooling of the layer by its slowest conduction mode with time constant τ = 4 h2 / (π2 α).

Checks run in the page

• The numerical squeeze against the closed form for constant viscosity.
• Conservation of the wax volume.
• A finite-difference heat equation against the slowest-mode time constant.
• Halving of the squeeze time when the force is doubled.

Honest simplifications

The wax is treated as a Newtonian liquid with one temperature (the lumped slowest mode), no yield stress, no engraved relief, no slip at the plates and no latent heat of solidification. Real sealing wax is a blend of resins and fillers with a softening range, not a sharp setting temperature, and the numbers here are plausible rather than measured.

• Press Light press: the wax sets while still thick.
• Press Hot wax and see how a few degrees of extra temperature buy a thinner seal.
• Press Cold matrix to see the wax freeze at the matrix and stall the flow.