Cocoa Bean Sun Drying

What is modelled

A bean is treated as a slab of half-thickness L whose surface sits at the equilibrium moisture of the surrounding air. Inside, water moves by Fick diffusion with a diffusivity D that rises with temperature following an Arrhenius law. Sunlight warms the beans above the air temperature during the day, so D changes through the day, and drying nearly stops at night.

Equations

• ∂M/∂t = D(T) ∂²M/∂x², surface M = Me, centre symmetric.
• Moisture ratio MR = (M − Me)/(M0 − Me) = (8/π²) ∑n odd (1/n²) exp(−n²π² θ / 4), with θ = ∫ D dt / L².
• Arrhenius D = D0 exp(−Ea/(R T)).
• Equilibrium moisture from an Oswin form in relative humidity.

Checks run in the page

• Series solution against a finite-difference solver at constant D.
• Time-varying D handled with the effective time θ against the finite-difference solver.
• Arrhenius ratio between two temperatures against the formula.
• Water balance: moisture lost from the profile equals the integrated surface flux.

Honest simplifications

Slab geometry, concentration-independent D, shrinkage and bed layering ignored, solar heating as a fixed temperature rise, and surface equilibrium instantly. Real beans are turned by hand and rewet by dew or rain; values are illustrative.

• Choose Sunny dry day and read how many days it takes to reach 7 per cent moisture.
• Choose Humid overcast and watch the equilibrium moisture slow the last stage.
• Thicken the bean and see the drying time grow roughly with its square.