Two particles will fuse together, sinter, via the combination of surface tension and external forces according to the Rumpf Equation. Because the sintering depends upon viscosity η, it's important to know how quickly the viscosity decreases when the powder has absorbed water. This is worked out via a WLF-style dependence on Tg, which depends on water via the Tg-Moisture app, which in turn depends on the Moisture Sorption app.
These food science apps are produced in conjunction with expert Prof Emmanuel Purlis, CONICET, Argentina. This app is the first part of a chain of logic1 from a Nestlé & U Sheffield team.
Powders and Moisture
We are interested in the effect of water vapour on food powders. [The effects of liquid water are a different topic.] It follows a chain of logic:
- The powder absorbs water following an isotherm, as in the Moisture Sorption app
- The water lowers the glass transition temperature, Tg as in the Tg-Moisture app
- At a lower Tg the effective viscosity of the powder becomes lower so the particles can fuse or "sinter" together either on their own or aided by some extra force such as the weight of the powder in storage.
The theory used in the 3 apps comes from a Nestlé/U Sheffield team led by Christine Haider1. The simple calculation of particle interactions is, of course, approximate but seems to work well for the real foodstuffs studied by the team. Two particles of radius r, viscosity η, surface energy γ will sinter (perhaps with an extra applied force F from the weight of the powder) after time t to create a "bridge" of cross-sectional radius x given by the Rumpf Equation:
`(x/r)^2 = (0.8γ/r+0.4F/(πr^2))t/η`
The calculations are valid only to small values of x/r, so the app stops calculating at larger values.
Although surface energy is part of the equation, as is recognised by the Nestlé team the changes in surface energy are minimal (and uncertain) so here the value is fixed at 40mN/m.
If your ambient temperature, T, is greater than the (reduced) Tg then the viscosity (η0) is reduced via the WLF equation using C=17.4 and B=51.6 (or your own values if you've fitted data):
`η = η_0 10^(-(C(T-T_g))/(B+T-T_g))`
Now you have the viscosity at that humidity you get the x/r value and using a common rule of thumb if this is greater than 0.1 we can say that the powder will be caked. This will depend somewhat on porosity, ε, but this factor is only (1-ε)/ε which is not a huge shift across typical powders.
1Christine I. Haider, Gerhard Niederreiter, Stefan Palzer, Michael J. Hounslow, Agba D. Salman, Unwanted agglomeration of industrial amorphous food powder from a particle perspective, Chem. Eng. Res. & Design, 132, 2018, 1160–1169