Extraneous and missing solutions

Glass batch calculation or glass batching is used to determine the correct mix of raw materials (batch) for a glass melt.

Principle

The raw materials mixture for glass melting is termed "batch". The batch must be measured properly to achieve a given, desired glass formulation. This batch calculation is based on the common linear regression equation:

$N_B=(B^T\cdot B{)}^{-1}\cdot B^T\cdot N_G$

with N_B and N_G being the molarities 1-column matrices of the batch and glass components respectively, and B being the batching matrix. The symbol "^T" stands for the matrix transpose operation, "^-1" indicates matrix inversion, and the sign "·" means the scalar product. From the molarities matrices N, percentages by weight (wt%) can easily be derived using the appropriate molar masses.

Example calculation

An example batch calculation may be demonstrated here. The desired glass composition in wt% is: 67 SiO₂, 12 Na₂O, 10 CaO, 5 Al₂O₃, 1 K₂O, 2 MgO, 3 B₂O₃, and as raw materials are used sand, trona, lime, albite, orthoclase, dolomite, and borax. The formulas and molar masses of the glass and batch components are listed in the following table:

Formula of glass component	Desired concentration of glass component, wt%	Molar mass of glass component, g/mol	Batch component	Formula of batch component	Molar mass of batch component, g/mol
SiO2	67	60.0843	Sand	SiO2	60.0843
Na2O	12	61.9789	Trona	Na3H(CO3)2*2H2O	226.0262
CaO	10	56.0774	Lime	CaCO3	100.0872
Al2O3	5	101.9613	Albite	Na2O*Al2O3*6SiO2	524.4460
K2O	1	94.1960	Orthoclase	K2O*Al2O3*6SiO2	556.6631
MgO	2	40.3044	Dolomite	MgCa(CO3)2	184.4014
B2O3	3	69.6202	Borax	Na2B4O7*10H2O	381.3721

The batching matrix B indicates the relation of the molarity in the batch (columns) and in the glass (rows). For example, the batch component SiO₂ adds 1 mol SiO₂ to the glass, therefore, the intersection of the first column and row shows "1". Trona adds 1.5 mol Na₂O to the glass; albite adds 6 mol SiO₂, 1 mol Na₂O, and 1 mol Al₂O₃, and so on. For the example given above, the complete batching matrix is listed below. The molarity matrix N_G of the glass is simply determined by dividing the desired wt% concentrations by the appropriate molar masses, e.g., for SiO₂ 67/60.0843 = 1.1151.

$\mathbf{B}=\left[\begin{array}{ccccccc}1& 0& 0& 6& 6& 0& 0\\ 0& 1.5& 0& 1& 0& 0& 1\\ 0& 0& 1& 0& 0& 1& 0\\ 0& 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 2\end{array}\right]$            ${\mathbf{N}}_{\mathbf{G}}=\left[\begin{array}{c}1.1151\\ 0.1936\\ 0.1783\\ 0.0490\\ 0.0106\\ 0.0496\\ 0.0431\end{array}\right]$

The resulting molarity matrix of the batch, N_B, is given here. After multiplication with the appropriate molar masses of the batch ingredients one obtains the batch mass fraction matrix M_B:

${\mathbf{N}}_{\mathbf{B}}=\left[\begin{array}{c}0.82087\\ 0.08910\\ 0.12870\\ 0.03842\\ 0.01062\\ 0.04962\\ 0.02155\end{array}\right]$            ${\mathbf{M}}_{\mathbf{B}}=\left[\begin{array}{c}49.321\\ 20.138\\ 12.881\\ 20.150\\ 5.910\\ 9.150\\ 8.217\end{array}\right]$   or   ${\mathbf{M}}_{\mathbf{B}}\mathbf{(}\mathbf{1}\mathbf{0}\mathbf{0}\%\mathbf{n}\mathbf{o}\mathbf{r}\mathbf{m}\mathbf{a}\mathbf{l}\mathbf{i}\mathbf{z}\mathbf{e}\mathbf{d}\mathbf{)}=\left[\begin{array}{c}39.216\\ 16.012\\ 10.242\\ 16.022\\ 4.699\\ 7.276\\ 6.533\end{array}\right]$

The matrix M_B, normalized to sum up to 100% as seen above, contains the final batch composition in wt%: 39.216 sand, 16.012 trona, 10.242 lime, 16.022 albite, 4.699 orthoclase, 7.276 dolomite, 6.533 borax. If this batch is melted to a glass, the desired composition given above is obtained. During glass melting, carbon dioxide (from trona, lime, dolomite) and water (from trona, borax) evaporate.

Another simple glass batch calculation can be found at the website of the University of Washington.

Advanced batch calculation by optimization

If the number of glass and batch components is not equal, if it is impossible to exactly obtain the desired glass composition using the selected batch ingredients, or if the matrix equation is not soluble for other reasons (e.g., correlation), the batch composition must be determined by optimization techniques.

See also

• Glass ingredients
• Calculation of glass properties

References

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