What a recipe puts into the equilibrium

Before this page

Proving that an answer is the answer, where the budget and the conservation matrix enter the minimization, and the manual page Recipes, which shows the same objects at work.

A cement paste is described by the materials it was mixed from, their masses and the fraction of each that has reacted, whereas an equilibrium calculation expects the amounts of the components it must conserve. This page writes the conversion from the first description to the second as budget performs it, together with what is kept out of the equilibrium, and the identities that follow and that the tests check.

1. Masses and extents

A recipe holds grams of binder, made of materials in the mass fractions , and the water  . A material is a list of constituents , each a mass fraction of it, so that the mass of a constituent in the recipe is

An addition outside the binder (a salt, an admixture) enters with its own mass in place of . The degree of reaction of a constituent at the time is the product of that of its material and its own,

which lets a degree measured on a material as a whole, that of a slag by image analysis for instance, apply to the constituents that react, while the crystals of the same material are held at   (effective_extent).

2. The budget

A constituent is described either by a database species of molar mass , or by its oxides alone, the mass fraction of each oxide of molar mass . The reacted part of the first kind enters the equilibrium as that species,

the water as , and the reacted part of the second kind as the elements of its oxides, each oxide decomposed over the primaries of the system by the vector of primary_decomposition. The budget is then

where is the set of the oxides of whose element the system has a primary for. The decomposition of a basic oxide takes protons from the water,     , and that of an acidic one gives them back, so that holds negative entries; only as a whole has to be a budget the species can reach. The analysis is not renormalized: falls short of one by what the analysis does not report as an oxide, a loss on ignition for instance, and scaling it up would put into the paste matter that the analysis does not attest.

3. What is kept aside

The unreacted part of each constituent,   , stays in the paste as a solid that takes no part in the chemistry. Its volume is that of its species,   , or its mass over the density of the constituent, and its enthalpy is the enthalpy of formation of its species. A constituent known by its oxides has neither a molar volume nor an enthalpy in any database, and both are left unknown unless its source gives them. The reacted mass of an oxide outside , , is kept aside in the same way, with the reason :not_in_system: the titanium of a cement in a system without titanium has reacted, but nothing in the system can hold it.

4. The mass balance

Since decomposes one formula unit of the oxide without loss, the equilibrium holds the mass of every species put into it and of every oxide of . It follows that

with the mass of the equilibrium, that of what is kept aside and that of the additions. The last term, the part of the analyses not reported as oxides, vanishes for a constituent described by a species and for an analysis that sums to one.

5. Volume and heat

The initial volume is that of the reacted species, of the water and of what is kept aside; the porosity and the volume fractions of volume_fractions are referred to it, the void    being the chemical shrinkage that the products do not refill (Recipes). The enthalpy of the paste is that of the equilibrium plus that of what is kept aside,   , and the heat released at constant temperature and pressure between two states of one paste is   . A part kept aside whose mass is the same in both states contributes the same enthalpy to both and cancels, known or not; a part whose mass changes and whose enthalpy is unknown makes unknown, and heat_release returns NaN rather than a heat that leaves it out.

6. A constituent given a rate

When a constituent is given a rate law (KineticsProblem of a recipe), the rate decides its degree of reaction, and the constituent enters the problem whole and unreacted,  , whatever . A constituent known by its oxides has no formula to dissolve, so it is first given one: glass_species writes the elements of the oxides of that grams of it carry into one formula unit, being the mass of material the unit stands for, and with_species makes it a constituent of that species. Dissolving   formula units then adds to the budget

the same contribution as the reacted part of the oxide constituent in Section 2, so that the kinetic route and the route of imposed extents put the same elements into the equilibrium at the same degree of reaction. The two masses differ, however: the solid loses while the equilibrium gains , the rest being what Section 3 keeps aside on the route of imposed extents and what the formula unit does not model on the kinetic one (modeled_mass_fraction).