LSF Formula

This is derived from the CSA ternary and the CSAF quaternary phase diagram. It is the line of best fit beyond which free CaO is the primary phase to crystallise from the melt in these systems.

Dastgir,

Remember the origin the Bogue calculation. This calculation is based on the hypothesis that the main oxides in clinker are four (s,a,f,c) and that they combine in four phases (c3s,c2s,c3a,c4af). (the intitial assumption)

If you have a certain mix of the four main oxides (s,a,f,c), you can calculate how much of the four clinker phases you can get from that: (c3s,c2s,c3a,c4af). This is obtained by solving a set of four linear equations with four unknowns.
Indeed, the mass balance gives you:

s  =  0.263 c3s + 0.349 c2s
a  =  0.377 c3s + 0.210 c4af
f   =  0.329 c4af
c  =  0.737 c3s + 0.651 c2s + 0.623 c3a + 0.462 c4af

Solving for (c3s,c2s,c3a,c4af) gives:

c3s  =  56.1 c - 7.6 s - 6.72 a - 1.43 f
c2s  =  8.6 s + 5.07 a + 1.08 f - 3.07 c
c3a  =  2.65 a -1.69 f
c4af =  3.04 f

These equations are the Bogue equations under the initial assumption. You can easily see that if you increase the quantity of lime (c), at a certain level the amount of c2s becomes negative. This is where the "initial hypothesis" cannot be satisfied. Actually, free lime will form, while you reached the maximum possible amount of c3s. This occurs when

c2s  =  8.6 s + 5.07 a + 1.08 f - 3.07 c = 0

or equivalently when:

3.07 c / ( 8.6 s + 5.07 a + 1.08 f ) = 1

or equivalently when:

c / ( 2.80 s + 1.65 a + 0.35 f ) = 1

Therefore it is tempting to conclude that the logical definition for the LSF is:

LSF =  c / ( 2.80 s + 1.65 a + 0.35 f )

and indeed we are close to the common formula, except that the factor for iron oxide (f) is 0.35 instead of 0.65.

Why is this factor different?
Actually I don't remember, but as soon as I can locate a copy of the Taylor's book, I will have a look. If I find the explanation, I will post it here.

Of course, one can already bet that the "initial hypothesis" must be partly wrong, at least as far as the iron-phase is concerned. Indeed, here is a list of possible phases that can form in the (s,a,f,c) system:

(s,a,f,c,c3s,c2s,c3s2,cs,c3a,c12a7,ca,ca2,ca6,a3s2,cas2,c2as,c4af,c2f)

Furthermore, I think that the "c4af" assumption is an over-simplification. This is already enough to make plausible the need for another factor for (f) in the LSF.

Finally, if you have fun with these things, I suggest that you repeat the Bogue calculations with more components and phases. You could easily include free lime, SO3, Na2O, K2O .

Postscriptum

Taylor is partly available on Google: http://books.google.com/books?id=1BOETtwi7mMC&printsec=frontcover

Dear lalbatros,

And what are the Bogue's formula having Alkali and SO3.

Kind regards,

Gulam Dastgir